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+{"text":"\\section{Introduction}\n\nElectron pumps are devices that aim to generate a reference DC electric current by moving electrons one at a time in response to a periodic control signal at frequency $f$. They potentially offer a simple and elegant traceability route for small currents, direct to the SI definition of the ampere\\cite{kaneko2016review,scherer2019single}. A class of pumps fabricated from semiconductor materials\\cite{kaestner2015non} has demonstrated accurate and robust current generation at roughly the part-per-million (ppm) accuracy level, for currents $I_{\\text{P}}$ up to $160$~pA \\cite{giblin2019evidence}. However, important questions must be answered before electron pumps can confidently be adopted as reference current standards at the uncertainty levels of primary electrical metrology. Most significantly, the robustness and device independence of the current needs to be demonstrated at at least the $0.1$~ppm level, over a range of device designs and operating parameters. To date, two studies have focused on the robustness of the current from GaAs pumps, at current levels of $\\sim 100$~pA, at uncertainty levels for each data point of $\\sim 2$~ppm\\cite{giblin2017robust} and $\\sim 0.5$~ppm\\cite{stein2016robustness}. \n\nHowever, blind measurement techniques which have been implemented in other metrology areas to remove bias \\cite{schlamminger2015summary} have not yet been applied to the study of electron pumps where the pump current is treated as an unknown and compared to a known reference current. Addressing unconscious experimenter bias is particularly important in experiments where the expectation of the result is strongly constrained; in this case, we expect $I_{\\text{P}} = ef$, and there is a possibility that in a non-blind measurement, the experimenter may unconsciously favour pump control parameters that yield this result. Particularly important in the electron pump context is the lack of reproducibility in attempts to realise a capacitance standard based on pumping a known number of electrons onto a cryogenic capacitor \\cite{keller1999capacitance,scherer2017electron}. The authors of Ref. \\onlinecite{scherer2017electron} were unable to reproduce the results of Ref. \\onlinecite{keller1999capacitance}, and identified components in the capacitance measurement uncertainty which had previously been under-estimated.\n\nEvaluating the robustness of the pump current presents a challenge due to the time-scales involved: the small currents require many hours of averaging to resolve $0.1$~ppm for a single data point, and the time-scale of the whole measurement campaign challenges the stability of the measurement system and the electron pump itself\\cite{giblin2019evidence}. To reduce the measurement time, or equivalently, to allow more data points to be measured within the timescale of a measurement campaign, the pump current should be increased as much as possible. Custom gate drive waveforms which slow down the electron capture process have been used to operate GaAs pumps accurately at much higher frequencies than were possible with sine wave drive \\cite{giblin2012towards,stein2015validation,stein2016robustness}. With these pumps the upper frequency limit for accurate pumping was $f \\sim 1$~GHz even with the custom waveforms. Silicon pumps, on the other hand, have demonstrated accurate pumping at $f=1$~GHz with sine wave drive \\cite{zhao2017thermal,giblin2020realisation}, and the possibility of increasing the frequency further while maintaining sub-ppm pumping accuracy using custom waveforms has not yet been explored. \n\n\\begin{figure}\n\\includegraphics[width=9cm]{AWGGraph}\n\\caption{\\label{AWGGraphFig}\\textsf{(a): Grey-scale derivative pump map using sine wave drive at 2.062 GHz, $P_{\\text{RF}}=13.2$~dBm. (b): Pump map using a waveform from an AWG at repetition rate $f=2$~GHz. One cycle of the AWG waveform is shown in the inset. Note that this waveform is subsequently amplified by an inverting amplifier to yield the correct polarity of gate voltage, whereby the negative voltage pulse on the entrance gate raises the entrance barrier to pump an electron. (c): Log-scale plots of the pump current along the horizontal dashed lines in plots (a) and (b).}}\n\\end{figure}\n\n\\section{Experimental method and blind protocol}\n\nWe investigate a single well-characterised sample of silicon pump which has previously been the subject of two precision measurement campaigns\\cite{yamahata2016gigahertz,giblin2020realisation}. The pump is a silicon nanowire-MOSFET, in which charge carriers are induced by a positive voltage applied to a global top gate \\cite{fujiwara2008nanoampere,yamahata2016gigahertz} which was set to $4$~V for all the measurements. Two finger gates, denoted the entrance gate and exit gate, define the region of the nanowire where a single electron can be trapped. Negative DC voltages $V_{\\text{ENT}}$ and $V_{\\text{EXIT}}$ applied to these gates define the pump operating point, and the periodic pump drive signal is added to $V_{\\text{ENT}}$ using a room-temperature bias-tee. A $50$~Giga samples\/s arbitrary waveform generator (AWG, Tektronix 70001A) was used to generate a custom waveform for the pump drive. Because the AWG output had a maximum peak-peak amplitude of $V_{\\text{AC}} = 0.5$~V, the output was amplified by a wide-band inverting RF amplifier with $+15$~dB gain before the bias-tee. The AWG is referenced to a $10$~MHz frequency reference derived from a hydrogen maser.\n\nFigure \\ref{AWGGraphFig} shows characterisation data using both sine wave drive and the custom AWG waveform at a repetition frequency of $2$~GHz. It is clear from the log-scale plots of figure 1 (c) that there is a substantial plateau along the exit gate axis when using the AWG drive waveform, but not when using sine wave drive. The inset to figure 1(b) shows the AWG waveform used for all the measurements reported in this paper. Characterisation data at other frequencies is inlcuded in supplementary sections A and B.\n\nThe experimental apparatus and methods used for this study are in many respects identical to that used in Ref. \\onlinecite{giblin2020realisation}. As in those experiments, the pump is cooled to a temperature close to 4 K by suspending it above a liquid helium surface. The pump current $I_{\\text{P}}$ is measured using a noise-optimised ultrastable low-noise current amplifier (ULCA) \\cite{krause2019noise}, with a precision digital voltmeter (DVM) recording the ULCA output. As in Ref. \\onlinecite{giblin2020realisation}, the DVM was calibrated roughly once every hour by switching its input to a Josephson voltage standard (JVS). A single precision measurement typically lasted between 8 and 10 hours and included between 7 and 11 voltmeter calibrations. To remove offset drifts in the measurement system during precision measurements, the pump drive signal was toggled on and off with a cycle time of $228$~s. Roughly the first $34$ seconds of each data segment (300 out of 1000 data points) following each on or off switch was rejected from the analysis to remove transient effects. More details of the measurement protocol are given in supplementary section C. \n\nThe pump current $I_{\\text{P}}$ is calculated from the on-off difference in the DVM voltages $\\Delta V$ using the equation $I_{\\text{P}} = \\Delta V \/ A_{\\text{TR}}$. Here, $A_{\\text{TR}}$ is the trans-resistance gain of the ULCA, nominally equal to $10^9$~V\/A. This gain is calibrated against the quantum Hall resistance (QHR) in 2 stages\\cite{drung2015ultrastable} and via some intermediate transfer standards, using a cryogenic current comparator (CCC) with relative uncertainty less than $0.1$~ppm \\cite{giblin2019interlaboratory}. Detailed calibration results are reported in supplementary section E. The measurement of the pump current was therefore traceable to the SI unit ampere via the JVS, the QHR, and the relationship $I=V\/R$. For characterisation measurements such as those reported in figures 1, 2a, and small filled points in figures 2b and 2c, no offset subtraction was performed: the pump drive signal was left on, and each data point is a single $20$ power line cycle DVM measurement. \n\nA blind protocol was implemented so that the lead experimenter could not see the true value of $I_{\\text{P}}$ while the measurements and data analysis were in progress. This is achieved by multiplying all the DVM readings by a hidden scaling factor $\\beta=1.00000387$, at a low level in the measurement software. An exception occurs when when the DVM is connected to the JVS for calibration, in which case $\\beta =1$. While tuning the pump and performing measurements, the experimenter does not know the scaling factor and can only access the scaled pump current $I_{\\text{P,B}} = \\beta \\Delta V \/ A_{\\text{TR}}$. Therefore, the tuning of the pump operating parameters and the choice of parameters for the precision measurements can only be made with reference to the flatness of the current plateau, not the deviation of the current from $ef$. The scaling factor was programmed and password-protected by a member of the team who was not otherwise involved in the experiments. The experimenter knew that it was constrained such that $|1 - \\beta| < 5 \\times 10^{-6}$ so that gross failures of the pump or apparatus would be apparent during characterization measurements. The scaling factor was revealed after the experiments were finished and data analysis, including analysis of the ULCA calibrations, completed.\n\n\\section{precision measurement campaign}\n\nThe aim of the measurements was to study the pump current as a function of control parameters $V_{\\text{ENT}}$, $V_{\\text{EXIT}}$ and $V_{\\text{AC}}$. To this end, a total of 67 precision measurements were made during a 7-week campaign, employing the apparatus and blind protocol described in section II. The measurements were divided into 17 `runs'. For most of the runs, several measurements were made while varying one control parameter. Runs 11-13 consisted of single measurements without varying a parameter. Further detail of the measurement chronology is given in supplementary section D. To monitor the stability of the pump, a `fingerprint' pump map was obtained before and after each run, apart from a few occasions when it was prevented by an experimental difficulty. For completeness, all of these pump maps are shown in supplementary section H. Additional line scans of current as a function of one or more control parameters were also measured to assess the optimal values of fixed control parameters for the next precision run. Typically, these scans were used to find the value of the control parameter that maximised the plateau width. They used a single $20$~PLC measurement for each data point, with a relative uncertainty of approximately $10$~ppm per data point. A pass \/ fail stationary mean statistical test, described in supplementary section G, was applied at the data analysis stage to each precision measurement to evaluate whether the current was stable during the measurement time. \n\n\\begin{figure}\n\\includegraphics[width=9cm]{PumpMapandLogPlots}\n\\caption{\\label{LogPlotFig}\\textsf{(a): Derivative pump map measured after precision run 4 and before precision run 5. (b) and (c): Line-scans of the pump current measured along the gate voltage axes indicated by solid colored lines in (a), and plotted on a logarithmic scale. The vertical dashed lines indicate the fixed value of entrance (exit) gate voltage used for the exit (entrance) gate scan. The diagonal dashed lines are guides to the eye extrapolating the exponential edges of the plateau. Larger filled points are the precision measurement data for runs 1-5, with the run number indicated in the plot legend. Data points indicated with a star ($*$) failed the stationary-mean test.}}\n\\end{figure}\n\n\\section{Results of precision measurements}\n\n\\subsection{Precision results}\n\nAfter run 5, the pump became less stable, (supplementary section H), making it difficult to establish the flatness of plateaus along $V_{\\text{ENT}}$ and $V_{\\text{EXIT}}$ axes. For this reason, we concentrate here on the data from runs 1-5, although the full precision data set is presented in supplementary figure S9. In figure \\ref{LogPlotFig}, we present the data from the first 5 precision runs. Panel (a) shows a pump map recorded between runs 4 and 5, and panels (b) and (c) show line-scans on a log scale which highlight the deviation of the current from the ideal value on the $1ef$ plateau. The fixed value of $V_{\\text{ENT}}$ ($V_{\\text{EXIT}}$) for the $V_{\\text{EXIT}}$ ($V_{\\text{ENT}}$) line-scan was adjusted in order to maximise the width of the plateau in the log-scale plot. The results of precision runs 1-5 are plotted as solid points in figures \\ref{LogPlotFig} (b) and (c). Runs 1-4 were $V_{\\text{EXIT}}$ scans, plotted in figure \\ref{LogPlotFig} (c), and run 5 was a $V_{\\text{ENT}}$ scan, plotted in figure \\ref{LogPlotFig} (b). The 18 data points along the $V_{\\text{EXIT}}$ axis (figure \\ref{LogPlotFig} (c)) define a plateau in agreement with an extrapolation of the standard-accuracy measurement. The precision data point marked with a star ($*$), failed the stationary-mean test, presumably because it was close to the edge of the plateau, and small fluctuations in offset charge, equivalent to shifts in $V_{\\text{EXIT}}$, caused fluctuations in the pumped current to be resolved on the time-scale of the precision measurement.\n\nThe precision $I_{\\text{P}}(V_{\\text{EXIT}})$ data for runs 1-4 (apart from the point that failed the stationary mean test) are re-plotted on a linear y-axis in figure 3a as $\\Delta I_{\\text{P}} = (I_{\\text{P}} - ef)\/ef$. The mean of these 17 points is $\\Delta I_{\\text{P}} = 0.22$~ppm, with a standard deviation $\\sigma$ of $0.14$~ppm. The individual data points have a mean combined uncertainty $\\langle U_{\\text~{T}}\\rangle$ of $0.102$~ppm, although the uncorrelated random uncertainty, $U_{\\text{A}}$, for each data point is smaller, in the range $0.08-0.09$~ppm. The scatter of the points is therefore slightly larger than what would be expected from the type A uncertainty of each point ($\\sigma > \\langle U_{\\text~{A}}\\rangle$), although not statistically incompatible with the assumption that the data is sampling a stationary mean along the plateau. We can therefore conclude that this data is consistent with a plateau along the $V_{\\text{EXIT}}$ axis, flat at the $0.1$~ppm level, but significantly offset from $ef$ by $0.22$~ppm. Figure 4 (b) shows the same data re-analysed with the first 700 data points rejected from the beginning of each 1000-point data segment instead of the standard 300. This was to test for the presence of a time constant in the current, as discussed in section V.\n\n\\begin{figure}\n\\includegraphics[width=9cm]{LinearHighAccGraph}\n\\caption{\\label{HighAccFig}\\textsf{Results of precision measurements for runs 1-4 as a function of $V_{\\text{EXIT}}$, expressed as $\\Delta I_{\\text{P}} = (I_{\\text{P}} - ef)\/ef$. The data have been analysed with (a): 300 and (b): 700 data points rejected from the start of each 1000-point data segment. Error bars show the combined standard uncertainty $U_{\\text{T}}$ The horizontal dashed lines show the weighted means of each data set. Arrows highlight run 1, measurement 4 and run 4, measurement 4. A breakdown of the uncertainty for these measurements is given in table I.}}\n\\end{figure}\n\nOnly one precision run was performed along the $V_{\\text{ENT}}$ axis before the interruption, illustrated by the heavy filled points in figure \\ref{LogPlotFig} (b). One data point, marked with a $*$, failed the stationary-mean test. It is not clear from this single run whether this data point indicates real structure to the plateau at level of $\\sim 5$~ppm, or if it is the result of a drift in the device state. The remaining 4 data points mark a plateau region which, combined with the stability of the pump map from runs 1-5, gives confidence that the fixed value of of $V_{\\text{ENT}}$ selected for runs 1-4 is in the middle of an experimentally-determined plateau. The mean of the 4 measurements from run 5 is $\\Delta I_{\\text{P}} = 0.15$~ppm, with a standard deviation of $0.09$~ppm. This is consistent with the deviation measured in runs 1-4 given the much smaller sample size.\n\n\\subsection{Uncertainty}\n\nIn table I, the breakdown of the uncertainty is given for two measurements indicated by arrows in figure 3a. The uncertainties due to the two stages of the ULCA calibration are presented as separate components, with the uncertainty due to the drift of the ULCA gains in between calibrations included in these two terms. This was significantly reduced by performing frequent ULCA calibrations, with more detail given in supplementary sections D and E. Run 1, measurement 4 is a typical representative measurement, and run 4, measurement 4 had the lowest combined uncertainty of the campaign. As in previous measurement campaigns, the type A uncertainty of the pump measurement is the largest single contribution, but the larger pump current achieved in this study has reduced $U_{\\text{A}}$ to below $0.1$~ppm and the uncertainty in the ULCA calibration is now a significant contribution. Specifically, the uncertainty in the output stage gain $R_{\\text{IV}}$ (nominal value $1$~M$\\Omega$) is limited by the $0.04$~ppm uncertainty in the $100$~k$\\Omega$ reference resistor traceable to the QHR via a chain of 3 intermediate measurements \\cite{giblin2018reevaluation,giblin2019interlaboratory}.\n\n\\begin{table}\n\\caption{\\label{UncertTable} Uncertainty breakdown for run 1, measurement 4, and run 4, measurement 4. All entries in the table are dimensionless relative uncertainties ($k=1$) in parts per million.}\n\\centering\n\\setlength{\\tabcolsep}{8pt}\n\\begin{tabular}{c c c}\nContribution & Meas. 1.4 & Meas. 4.4 \\\\\n\\hline\\hline\nULCA $G_{\\text{I}}$ Cal. & 0.024 & 0.024 \\\\ \nULCA $R_{\\text{IV}}$ Cal. & 0.062 & 0.043 \\\\ \nULCA Temp. corr. & 0.023 & 0.023 \\\\ \nDVM Cal. & 0.014 & 0.016 \\\\\nPump $U_{\\text{A}}$  & 0.088 & 0.061 \\\\\n\\hline\nTotal $U_{\\text{T}}$& 0.111 & 0.084 \\\\\n\\end{tabular}\n\\end{table}\n\n\\subsection{stability of the pump}\n\nThe measurement campaign was divided into two parts by an instrument issue which forced a period of 8 days' down-time between runs 5 and 6. During this time, the pump was thermally cycled to room temperature and back to $4$~K twice. From examination of the pump maps in supplementary section H, it is clear that the pump became less stable after this interruption, although even before the interruption, small changes in the `nose' (the onset of pumped current as $V_{\\text{EXIT}}$ is made less negative) of the pump map are visible. This contrasts with the data of Ref. \\onlinecite{giblin2020realisation} showing this sample of pump to be extremely stable over multiple cool-downs in different laboratories, when driven with a sine wave at $\\sim 1$~GHz. We conjecture that at least some of the changes visible in the pump maps during the present campaign may be due to changes in the transmission of the cryogenic microwave line at frequencies $\\gg 1$~GHz. This could plausibly arise due to changes in the temperature gradient along the line, and would affect the high frequency components of the AWG waveform, causing distortion of the waveform at the pump entrance gate. \n\n\n\\section{discussion}\n\nThe study was complicated by instability in the pump map, which made it difficult in the later parts of the measurement campaign to interpret the results as sampling a stable state of the device. However, enough results were obtained from runs 1-5 to establish that the pump current is invariant in the exit gate voltage at the level of $1$ part in $10^{7}$. Averages over these data points presented in the previous section give $\\Delta I_{\\text{P}} \\sim 2 \\times 10^{-7}$, a significant offset from the ideal current $I_{\\text{P}} = ef$. The flatness of the plateau suggests that the offset is due to an error in the measurement system which applies a constant offset to all the measurements, rather than an error due to the physics of the pump itself.\n\nOne possible cause of error is a time constant in the pump current. This was discussed in Ref. \\onlinecite{giblin2020realisation}, and could plausibly arise from heating due to the relatively large RF powers applied to the device gate. Repeating the data analysis of runs 1-4 with 700 data points rejected from the start of each segment instead of 300 did indeed yield an average pump current closer to $ef$, as shown in figure 3b. However, the larger type A uncertainties in this analysis make it difficult to draw a firm conclusion regarding possible time constants. Measurements with much longer on-off cycle times could potentially resolve this question, but require the $1\/f$ noise corner of the ULCA current measurement to be at frequencies well below $1$~mHz. ULCA units have demonstrated this performance in bench tests \\cite{krause2019noise}, but the cryogenic wiring involved in a pump measurement introduces additional sources of noise and drift. Another possible cause of error is a non-linearity in the gain of the ULCA. The ULCA input stage gain $G_{\\text{I}}$ is calibrated at a current of $6$~nA, and the pump current is $320$~pA. Comparisons of the gains of two ULCA units with different input stage designs, detailed in supplementary section F, set an upper limit to possible non-linearity of a few parts in $10^8$, so this is unlikely to cause errors of a part in $10^{7}$. \n\nPossibly the most important cause of error could arise from the CCC calibration of the ULCA $G_{\\text{I}}$. This could result from rectification of noise by the CCC's SQUID detector leading to different SQUID offsets for the two polarities of current used in the ULCA calibration \\cite{drung2014ultrastable}. One study on CCC performance in the low-flux regime\\cite{drung2015improving} concluded that noise pickup might lead to this type of error at SQUID flux levels below $1$~$\\mu \\Phi_{0}$, although this number was based on a limited number of measurements and is specific to a particular CCC design\\cite{goetz2014compact}, different in detail to the CCC used to calibrate the ULCA in our experiments. We calibrated the ULCA $G_{\\text{I}}$ using a CCC\\cite{giblin2019interlaboratory} with a $10000:10$ turns ratio, and a sensitivity of $6$~$\\mu$A~turns$\/\\Phi_{0}$. The current in the large winding was approximately $\\pm 5$~nA, giving a full-signal ampere-turns product of $100$~$\\mu$A~turns, corresponding to a flux of $16.7$~$\\Phi_{0}$. A flux of $1$~$\\mu \\Phi_{0}$ therefore corresponds to $0.06$~ppm of the full signal in the ULCA $G_{\\text{I}}$ calibration, three times smaller than the observed discrepancy in the electron pump current. However, no investigations have yet been carried out on the performance of our CCC in the low-flux regime, so the size of possible noise-rectification errors is not known. Low flux ratio accuracy tests such as those presented in Ref. \\onlinecite{drung2015improving} should provide useful information on the scale of possible errors. We note that if these errors are affecting the ULCA calibrations in our experiment, they are remarkably constant in time, as shown by the $\\sim 5 \\times 10^{-8}$ relative stability of the ULCA input gain over the duration of the measurement campaign illustrated in supplementary section E. This indicates that if noise is affecting the SQUID, its most likely source is the CCC bridge electronics, rather than external sources.\n\nThe upper frequency limit for accurate pumping with tunable-barrier pumps has previously been empirically established at around $1$~GHz \\cite{giblin2019evidence}. We have shown that this can be increased, albeit in a rather exceptional sample of pump. In this study, the practical upper frequency limit was determined by a combination of plateau rounding, and increased incidence of switching events which shifted the pump operating point in the $V_{\\text{ENT}}-V_{\\text{EXIT}}$ plane. This hints at device-physics factors which may limit the practical upper operation frequency, possibly charge traps which are activated by high frequency components present in the drive signal. Further investigation of more samples of pump could shed fruitful light on this question.\n\n\\section{conclusions}\n\nPrecision measurements have been made of the current from a silicon electron pump driven at a frequency of $2$~GHz using a custom drive waveform applied to the entrance gate. The pump current is invariant in exit gate voltage with a precision of $0.1$~ppm (32 aA), but offset by roughly $0.2$~ppm from the expected current corresponding to one electron for each pump cycle. The application of a blind measurement protocol provides added confidence that this result is not affected by experimenter bias. At this accuracy level, the measurement of the pump current challenges the state of the art in existing electrical metrology methods, with scaling of small currents using CCCs at low flux levels posing a particularly interesting problem. The recent demonstration of current plateaus due to the dual Josephson effect \\cite{shaikhaidarov2022quantized} raises the possibility of a metrological investigation of the dual Josephson effect in the near future, providing added motivation for a better understanding of low current scaling.\n\n\\begin{acknowledgments}\nThe authors would like to thank Colin Porter and Scott Wilkins for making the NPL primary Josephson voltage standard available, and for assistance with setting up the voltmeter calibration. This research was supported by the UK department for Business, Energy and Industrial Strategy. A.F. and G.Y. are supported by JSPS KAKENHI Grant Number JP18H05258.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{Motivation}\n\n\\subsubsection{Centralized User\/Distributed (Hybrid) Model}\t\t\\label{motivation:hybrid:sec}\nIEEE 802.1 Time Sensitive Networking (TSN) provides a standardized\nframework of tools for providing deterministic ultra-low latency\n(ULL), e.g., for industrial control applications, automotive\nnetworking, and avionics communication\nsystems~\\cite{finn2018introduction,nas2019ult}. In particular, the\nIEEE 802.1Qbv Time Aware Shaper (TAS) has received extensive attention\nas a key tool for achieving deterministic ULL network service. The TAS\noperation requires careful planning of the synchronized time cycles\nand the gate times that are allocated to the scheduled traffic (ST)\nand the unscheduled best effort traffic (BE). The TAS parameter\nsettings specifying the timing characteristics (cycle time, gate slot\nallocations) are also commonly referred to as the Qbv schedule or the\nTAS schedule. For a given static networking scenario, the TAS\noperation with a properly configured Qbv schedule can ensure the\ndeterministic ULL required by demanding industrial and automotive\napplications~\\cite{nas2019per}.\n\nModern network scenarios often involve dynamic changes with varied use\ncases, such as changes in the network nodes and network topology, or\nthe traffic pattern. For instance, nodes or links may be dynamically\nadded or removed. Or, nodes may inject additional traffic flows or\ntraffic flows may terminate, or the latency requirements of flows may\nchange dynamically. Such dynamic changes have been included in the use\ncases defined by IEC\/IEEE 802.1 TSN TG~\\cite{usecas2018}.\n\nIn a typical industrial environment, sensors that periodically or\nsometimes sporadically send ambient measurements to a local gateway\nrequire certain Quality of Service (QoS) guarantees. In such a\nvolatile and dynamic environment, new machinery that requires\nprioritized execution (e.g., emergency cooling procedures or\nmaintenance tasks for network traffic tests) may be brought onto the\nfactory floor. To deal with such scenarios, the Time-Aware Shaper\n(TAS) Gate Control Lists (GCLs) in coordination with the Network\nManagement Entities (NMEs), e.g., Centralized Network Configuration\n(CNC), has to adapt to changing environment conditions by judiciously\napplying reconfiguration such that stream deadlines, QoS, and total\nstream utilization times (reported by a stream registration procedure)\nare satisfied.\n\nSimilarly, in a disaster management networking scenario, a network may\nhave physically lost large amounts of network resources (links and\nroutes) but needs to manage the flows that are currently present. A\ncentral orchestrator can prudently time share the reserved resources such\nthat a large number of streams can be completely serviced, i.e., the total\nexecution time for each stream is completed by the network.\n\nGenerally, in such dynamic networking scenarios, applying only\nadmission control will clearly guarantee (in accordance with a traffic\nshaper) the QoS metrics (of the admitted flows).  However, for a given\nstatic network configuration, the total number of admissible streams\nmay be well below the number of streams that seek network\nservice. Therefore, adding a dynamic reconfiguration strategy to\nmanage and configure the network appears to be a plausible and\nattractive solution that intuitively should lower capital and\noperational expenditures as it mitigates the over-provisioning of\nnetwork resources. The general idea for such an allocation scheme is\nto control network access in a timely and orderly fashion such that a\nmaximum number of streams can be effectively serviced.\n\nOur objective therefore is to maximize the number of admitted flows\n(i.e., tasks or streams) in such a dynamically changing and volatile\nenvironment whilst keeping the TSN QoS metric guarantees.  To the\nbest of our knowledge there are no prior detailed studies on a\ndynamic stream resource allocation and admission control policy in\nconjunction with a network reconfiguration policy being executed while\nflows are carried in a TAS time scheduled network. In this paper, we\nfocus on the IEEE 802.1Qbv~\\cite{IEEE8021Qbv} enhancements and design\na reconfiguration framework taking inspiration from the IEEE\n802.1Qcc~\\cite{IEEE8021Qcc} standards for managing, configuring, and\nreconfiguring a TSN network.\n\n\\subsubsection{Decentralized Model}\nThe IEEE 802.1Qcc standard specifies three models for configuring the\nTime-Aware Shaper (TAS) gating schedules (GCL\/GCE timing computation):\na fully-centralized model, a centralized network\/distributed user\nmodel (hybrid model, see Section~\\ref{motivation:hybrid:sec}), and a\nfully-distributed configuration model. The centralized model greatly\neases control and configuration messages sent across the network and\ncan precisely configure TAS schedules due to having complete knowledge\nof the network and full capabilities of each bridge. However the\ncentralized model suffers from common disadvantages, such as a\nsingle-point of failure, relatively large capital\/operational\n(CapEx\/OpEx) expenditures (as the centralized control may be\nsuperfluous in a small-scale network~\\cite{chen2017rap}), and adding\nunnecessary complexity to a small-scale network. Thus, a\nfully-distributed configuration model (e.g., SRP over MRP or RAP over\nLRP) may be attractive for some networks. The fully-distributed\nconfiguration model avoids the added complexity and single point of\nfailure of a centralized management entity.  Moreover, Chen et\nal.~\\cite{chen2017rap} have argued that the centralized configuration\nmodels can be an over-design for real-time applications with relaxed\nlatency requirement (order of magnitude of milliseconds). Chen et\nal. have also argued that the distributed model is more\nscalable. (However, studies of the fully distributed model with RAP\nover LRP targeted typically applications with relatively relaxed\nlatency requirements.)\n\nIn the absence of a Centralized Network Configuration (CNC) node, the\nTSN Task Group (TG) specifies the IEEE 802.1CS (Link-Local\nRegistration Protocol, LRP)~\\cite{IEEE8021CS} standard for\nregistration and distribution of application configuration parameters\nbetween point-to-point links targeting newly published TSN features. A\nlegacy protocol, such as the Stream Reservation Protocol\n(SRP)~\\cite{IEEE8021Qat} which is primarily used for AVB application,\nis intended to serve as the main resource reservation and admission\ncontrol protocol. However, extending and porting the SRP to be\nutilized for bridges that support TAS will not suffice since bandwidth\nreservation cannot directly apply TAS's time slot reservation\nnatively. Therefore, the Resource Allocation Protocol, IEEE 802.1Qdd\n(RAP)~\\cite{chen2017rap}, has been proposed to apply a distributed\nresource reservation that can exchange TSN features.\n\nIn our model, the switch computes the TAS time slot for all admitted\nstreams as follows. In the absence of admission control, we predefine\nthe TAS slot to be a minimum of 10\\% and a maximum of 90\\% of the\nCycle Time (CT), even when no streams are registered, so as to avoid\nstarvation of Best Effort (BE) traffic. With admission control, the\nstatic predefinition, which can potentially waste resources if unused,\ncan be eliminated. Essentially, as streams get registered, we keep\ntrack of the remaining load on each egress port until the load (which\ndepends on the slot size and CT) is negative (oversubscribed link).\nSuch a link over-subscription invokes a procedure call that increases\nthe slot time (by a step size of 1\\%, or more fine-grained increments)\nuntil the remaining load is positive. This procedure is iteratively\ncalled until all registered streams and the new stream are\nappropriately registered with sufficient Scheduled Traffic (ST) slot\ntime to transmit all frames during a single appropriately sized\nCT. Note that the TAS time slot is defined as the portion of the CT\nthat is allocated to high-priority ST traffic.\n\nOur proposed TAS configuration\/reconfiguration, is designed for the\nfully-distributed configuration model. In the distributed approach,\nthe GCE slot parameters are configured in a distributed manner by the\nswitches as per the distributed algorithm\/procedure explained in\nSec.~\\ref{tas:reconfig:prot}. In the centralized approach, the GCE\nslot parameters are configured centrally by the CNC with the\nCentralized User Configuration (CUC) node assisting in passing\nend-station (source\/sink, devices, etc.) capabilities and\nparameters. Similarly, the ``hybrid'' model also utilizes the CNC for\nconfiguration exchange and network side management (see\nSection~\\ref{motivation:hybrid:sec}).\n\nRegarding the differences between the hybrid model and the fully\ncentralized model, the main network-side difference is the way the\nUser\/Network configuration Information (UNI) is propagated. In the\nfully centralized model, the sources communicate Control Data Traffic\n(CDT) messages to\/from the CUC node, instead of having each source\ninteract directly with the CNC. According to the 802.1Qcc standard,\nthe general advantage is that the computation complexity (especially\nfor industrial\/automotive applications with computationally complex\nI\/O timing requiring detailed knowledge of the application's\nsoftware\/hardware within each end station) can be tolerated by having\nthe CUC handle end station discovery, retrieving end station\ncapabilities and user requirements, and configuring TSN features in\nend stations. The CNC is used to only manage and configure network\nside components. In terms of benefits, the configuration delay is\npotentially reduced, scheduling optimality increased, and hardware\nperformance overhead\/complexity reduced with a shorter network\nresponse time for networks involving command and control systems. The\ndevelopment and investigation of a reconfiguration approach for the\nfully centralized configuration model is left as future work.\n\nOur study focuses on the centralized network\/distributed user model\n(hybrid model) and the fully-distributed (decentralized) configuration\nmodel.  For brevity we refer to the centralized network\/distributed\nuser model (hybrid model) also as the centralized model or the\ncentralized topology.  We refer to the fully-distributed\n(decentralized) model also as the decentralized model or the\ndecentralized topology.\n\nOur proposed TAS reconfiguration architecture maps and propagates\nstream information and conducts dynamic TAS time slot reservations\n(including GCL\/GCE scheduling and provisioning) on local shared stream\ndatabase records to guarantee TSN QoS for all admitted streams and to\nmaximize stream admission using a reconfiguration strategy of TAS\ngating schedules within each bridge in a fully decentralized IEEE\n802.1Qcc model.\n\n\\subsection{Related Work} \\label{tsn:rel:sec}\nRaagaard et al.~\\cite{raagaard2017fog} presents a heuristic algorithm\nthat reconfigures TAS switches according to runtime network\nconditions. Feasible schedules are produced and forwarded using a\nconfiguration agent (composed of a Centralized User Configuration\n(CUC) and Centralized Network Configuration (CNC)). Raagaard et al's\nmodel places emphasis on appearing and disappearing synthetic flows in\na fog computing platform that takes into account the flow's properties\nand possible routes. Contrary to this approach our framework performs\nflow maximization with optimal reconfiguration based on firm bandwidth\ncomputation strategies at run-time. Further, we show the equilibrium\npoint for this algorithm.\n\nFurther related work that is complementary to\nour study has been conducted by\nPop et al.~\\cite{pop2018enabling}, Hackel et al.~\\cite{hackel2019software},\nHerlich et al.~\\cite{herlich2016proof},\nNayak et al.~\\cite{nayak2016time,nayak2017incremental,nayak2017routing},\nand Kobzan et al.~\\cite{kobzan2018secure}.\n\n\\subsection{Contributions}\nWe comprehensively evaluate the performance of TAS\nfor reconfigurations in the hybrid and fully distributed models\nwith respect to\nnetwork deployment parameters, such as, maximum window size for the\nGate Control List (GCL) repetition, gating ratio proportion, i.e.,\nGate Control Entry (GCE) proportion, to control delay perceived at the\nreceiving end, signaling impact on Scheduled Traffic (ST) and\nBest-Effort traffic (BE) classes, and packet loss rate experienced at\nthe receiving end. In particular, we make the following contributions:\n\\begin{itemize}\n\\item[i)] We design a CNC interface for a TSN network to globally\n  manage and configure TSN streams, including admission control and\n  resource reservation.\n\\item[ii)] We integrate the CNC in the control plane with TAS in the\n  data plane to centrally manage and shape traffic using the CNC as\n  the central processing entity for flow schedules as more flows are\n  added.\n\\item[iii)] We modify and test the model to operate in a distributed\n  fashion, i.e., the control and data planes are combined.\n\\item[iv)] We evaluate each design approach and for a range of numbers of\n  streams and sources with different TAS parameters. We show results\n  based on admission ratios, network signaling overhead, and QoS\n  metrics.\n\\end{itemize}\n\n\\subsection{Organization}\nThis article is organized as follows. Section~\\ref{tsn:back:sec}\nprovides background information and an overview of related work on the\n802.1 TSN standardization, focusing on the enhancements to scheduled\ntraffic and centralized management and\nconfiguration. Section~\\ref{tsn:design:sec} shows the complete\ntop-down design of the CNC (hybrid model) and main components that\nachieve ultra-low latencies and guaranteed QoS for a multitude of ST\nstreams. Similarly, Section~\\ref{tas:reconfig:prot} shows the approach\nused in implementing the decentralized (fully distributed) TAS\nreconfiguration model. The simulation setup as well as main\nparameters and assumptions are given in Section~\\ref{tsn:sim:setup} and\nresults are presented in Sections~\\ref{eval:cent:sec} and~\\ref{eval:dec:sec}. Finally conclusions and future work are outlined\nin Section~\\ref{concl:sec}.\n\n\\section{Background: IEEE 802.1 Time Sensitive Networking} \\label{tsn:back:sec}\n\n\\subsection{IEEE 802.1Qbv: Time Aware Shaper (TAS)}\nTAS's main operation is to schedule critical traffic streams in\nreserved time-triggered windows. In order to prevent lower priority\ntraffic, e.g., best effort (BE) traffic, from interfering with the\nscheduled traffic (ST) transmissions, ST windows are preceded by a\nso-called guard band. TAS is applicable for Ultra Low Latency (ULL)\nrequirements but needs to have all time-triggered windows\nsynchronized, i.e., all bridges from sender to receiver must be\nsynchronized in time. TAS utilizes a gate driver mechanism that\nopens\/closes according to a known and agreed upon time schedule for\neach port in a bridge. In particular, the Gate Control List (GCL)\nrepresents Gate Control Entries (GCEs), i.e., a sequence of on and off\ntime periods that represent whether a queue is eligible to transmit or\nnot.\n\nThe frames of\na given egress queue are eligible for transmission according to the\nGCL, which is synchronized in time through the 802.1AS time\nreference. Frames are transmitted according to the GCL\/GCE and\ntransmission selection decisions. Each individual software queue has\nits own transmission selection algorithm, e.g., strict priority\nqueuing. Overall, the IEEE 802.1Qbv transmission selection transmits a\nframe from a given queue with an open gate if: $(i)$ The queue\ncontains a frame ready for transmission, $(ii)$ higher priority\ntraffic class queues with an open gate do \\textit{not} have a frame to\ntransmit, and $(iii)$ the frame transmission can be completed before\nthe gate closes for the given queue. Note that these transmission\nselection conditions ensure that low-priority traffic is allowed to\n\\textit{start} transmission only if the transmission will \\textit{be\n  completed} by the start of the scheduled traffic window for\nhigh-priority traffic. Thus, this transmission selection effectively\nenforces a ``guard band'' to prevent low-priority traffic from\ninterfering with high-priority traffic~\\cite{finn2018introduction}.\n\n\\subsection{IEEE 802.1Qcc: Centralized Management and Configuration}\n\n\\begin{figure}[t!]  \\centering\n\t\\includegraphics[width=3.5in]{fig_cnc_topo.pdf} \\vspace{-0.5cm}\n\t\\caption{Illustration of Centralized Network Configuration (CNC): End stations interact with the network entities via the User-Network Interface (UNI). The CNC receives the requests, e.g., flow reservation requests, and provides corresponding management functions.  An optional CUC provides delay-optimized configuration, e.g., for closed-loop IACS applications.  The solid arrows represent the protocol, e.g., YANG or TLV, that is used as the UNI for exchanging configuration information between Talkers\/Listeners (users) and Bridges (network).  The dashed arrows represent the protocol, e.g., YANG or TLV, that transfers configuration information between edge bridges and the CNC.} \\label{fig_tsn_Qcc}\n\\end{figure}\n\nIEEE 802.1Qcc~\\cite{IEEE8021Qcc} provides a set of tools to globally\nmanage and control the network. In particular, IEEE~802.1Qcc enhances\nthe existing Stream Reservation Protocol (SRP) with a User Network\nInterface (UNI) which is supplemented by a Centralized Network\nConfiguration (CNC) node, as shown in Fig.~\\ref{fig_tsn_Qcc}. The UNI\nprovides a common method of requesting layer 2 services. Furthermore,\nthe CNC interacts with the UNI to provide a centralized means for\nperforming resource reservation, scheduling, and other types of\nconfiguration via a remote management protocol, such as\nNETCONF~\\cite{enns2006netconf} or RESTCONF~\\cite{bierman2017restconf};\nhence, 802.1Qcc is compatible with the IETF YANG\/NETCONF data modeling\nlanguage.\n\n\\section{Centralized Model Design and Framework Considerations} \\label{tsn:design:sec}\nThis section presents our design methodology and main signaling\nframework for the centralized network\/distributed user\nmodel (hybrid model). Our main goals behind designing the CNC is given by the\nfollowing constraints. Additionally, the CNC can be logically or\nphysically connected to the data-plane with in-band or out-of-band\nmanagement links.\n\\begin{enumerate}\n\\item Our focus is mainly on stream based network adaptation. By this\n  technique, fluctuating streams (already registered streams and new\n  incoming streams) and their requirements can be accommodated by the\n  network dynamically based on a single remote procedure call to the\n  CNC.\n\\item Identify and execute flow requirements by populating the\n  registration table. The control plane resource orchestration is\n  purely carried out by the monitoring of existing flows which have\n  been satisfied.\n\\item Optimizing resource allocation (maximize admitted streams\/flows)\n  based on a bounded latency and network utilization, i.e., prioritize\n  flows that request low network resources.\n\\end{enumerate}\nOur main assumption to accurately apply admission control and,\nconsequently, reconfiguration, is that each source must define a flow\nin terms of total resources needed (governed by the bandwidth\nrequirements) and the total time needed for the resource to be used\n(which in our traffic model is termed as the resource utilization\ntime). Essentially, the CNC uses this information (which is tagged in\nthe Ethernet frame header) to determine whether a frame is admitted or\nrejected.\n\n\\begin{figure}[t!] \\centering\n\\includegraphics[width=3.3in]{fig_cnc_framework.pdf} \\vspace{0.2cm}\n\\caption{Network Management Entity Framework for TSN Switches:\n  Centralized Network Configuration (CNC) is used to send and receive\n  Control Data Traffic (CDT) to configure routing segments and network\n  resources with the goal of maximizing network flows\/streams.}\n\t\\label{fig_cnc_reconfig}\n\\end{figure}\n\n\\subsection{Core Components}\nOur design is split into two layers, Control Plane and Data Plane,\nfollowing the decoupling SDN paradigm, thereby inheriting the benefits\nof the orthogonality of the two planes, as shown in\nFig.~\\ref{fig_cnc_reconfig}.\n\n\\subsubsection{Configuration Module}\nThe configuration module is the main component that interacts with the\nregistered flows and network components. It includes the global stream\nregistration table which records all approved stream transmitting in\nthe network, and the admission control element that encapsulates and\ndecapsulates CDT headers and forwards the information to the necessary\nmodule\/element.\n\n\\paragraph{Global Stream Registration Table}\nThe source streams (devices\/users) make a Remote Procedural Call (RPC)\nvia the stream registration interface for providing information that\ncan be mapped as a unique tuple structure identification $<Flow ID,\nBridge Gateway>$. Upon receiving the registration packet, i.e.,\nControl Data Traffic (CDT), the CNC determines whether the new stream\ncan be accepted in its stream table. To guarantee the QoS for all\nregistered streams, admission control principles are applied to all\nstreams according to the stream's path, required network resources,\nand available resources.\n\n\\paragraph{Admission Control}\nThe admission control element is the first element that the new\nstreams interacts with. The admission control element under the\nconfiguration module globally manages all streams transmitting in the\nTSN domain governed by the CNC. The admission control element extracts\nthe necessary information from the CDT packet and forwards the\ninformation according to the CDT type. The CNC apples several steps to\ndecide whether to accept or reject the stream transmission request.\n\\begin{enumerate}\n\\item The CNC checks the destination address(es) of the stream and\n  consults its resource manager module for network resources available\n  on the new stream's path, which is computed based on the path\n  computation element within the CNC.\n\\item According to the bandwidth required for the new stream\n  (calculated at the bridge gateway for the new stream), all links on\n  the path are checked to see if enough bandwidth is available for the\n  new stream.\n\\item In the event that not enough resources are available, the CNC\n  applies the TAS reconfiguration module to identify the bottleneck\n  link(s) and to check whether the gating ratio can be increased for that\n  specific traffic class whose current resource utilization and\n  deadline will not cause a late deadline by being added to the TAS\n  slot reservation.\n\\end{enumerate}\n\n\\subsubsection{Reconfiguration Module}\nThe reconfiguration module includes the flow scheduling element (for\nour network model, the Time-Aware Shaper (TAS) is used in the data\nplane), the reconfiguration calculus element which optimizes flow\nregistration according to each stream's total resource utilization and\nflow deadlines, and finally the path computation element which defines\nthe path for all stream according to the QoS constraint.\n\n\\paragraph{Flow Scheduling}\nThe flow scheduling element currently takes the Time-Aware Shaper into\nconsideration. Due to the TAS's requirements on time synchronization\nbetween network components (switches, hosts, etc.), the CNC follows\nthe same principle of scheduling flows according to a known timescale\n(initially set to be 50~$\\mu$s in our network model). The CNC then\npasses on this time synchronization information to the TSN enabled\nswitches within its domain. Any approved streams will transmit frames\naccording to the time scale specified by the flow scheduler in the\nCNC.\n\n\\paragraph{Reconfiguration Calculus}\nIn addition to centrally managing resources and providing admission\ncontrol policies to the network, the CNC can invoke the TAS\nreconfiguration strategy with the goal of borrowing BE time\nslots for pending ST traffic streams. This element consults the\nresource manager module on the bottleneck link and checks whether the\nadded stream will oversubscribe the link. The TAS reconfiguration\nincrementally (1\\% of total CT) increases the traffic class slot time\nand reserves it for the new stream.\n\n\\paragraph{Path Computation}\nFor large scale and complex LAN\/MAN topologies, it is often required\nto supplement streams with equal cost paths in the event of a path\ndisruption (e.g., link failure, stream saturation, and explicit\ncongestion). The CNC's path computation element is tasked with finding\nsuch paths as a fail-over approach to avoid any violations to any\nstream's QoS. Presently, our model has a rudimentary application of\npath computation, i.e., it is defined statically for all core network\ncomponents (shortest path), since the main emphasis was on\nreconfiguration based on stream characteristics as defined by the\nsource.\n\n\\subsubsection{Resource Manager Module}\nThe resource manager module centrally manages all network resources\nwithin the CNC's domain. It includes the network resource table that\nrecords all streams' usage of resources, and the resource allocation\nscheme element to which we delegate the task of calculating the required\nnetwork resources for a given stream according to an allocation\nscheme.\n\n\\paragraph{Network Resource Table}\nTo remove certain overheads on the configuration module, the network\nresource table operates in tandem with the global stream registration\ntable to accurately determine the required network resources (mainly\nbandwidth for our traffic model). It classifies streams based on\nperiodic stream properties. Any stream that has been approved by the\nCNC has a record attached to it in the network resource table.\n\n\\paragraph{Resource Allocation Scheme}\nSeveral allocation schemes can be implemented for all traffic classes\ndefined in the network. For periodic streams, the time slot given by\nthe flow scheduler (according to the TAS Cycle Time and number of\ntraffic classes) and the data rate defined by the source is used to\ncalculate the required bandwidth for each link on the path to the\ndestination (i.e., sink).\n\n\n\\subsubsection{Data Plane}\nThe data plane contains all core switches. Any TSN switch interfaced\nby the CNC is given a switch ID and has a local stream registration\ntable. The remaining switch elements compose the forwarding and\nqueuing operation with several traffic shapers (802.1Qbv TAS in our\nnetwork model).\n\n\\paragraph{Local Stream Registration Table}\nThis data plane registry contains the subset of source streams that\nare established for the corresponding bridge gateway and attached\nsources to each port. The CNC delegates some control to the bridge\ngateway to instruct and alert sources of any new network conditions\nand explicit changes.\n\n\\paragraph{Traffic Shaper --- Time-Aware Shaper}\nThe traffic shaper is the main shaping and scheduling mechanism that\ncontrols the gating schedules for all the traffic classes within the\nTSN domain. All bridges are synchronized to the same gating schedule\nGCL Cycle Time (CT) given by the CNC's flow schedule element (CT\nindicates the time period for the GCL to repeat).\n\n\n\\section{Decentralized Model Design and Framework Considerations} \\label{tas:reconfig:prot}\n\n\\begin{figure}[t!] \\centering\n\\includegraphics[width=3.5in]{fig_distributed_reconfig.pdf} \\vspace{-0.2cm}\n\\caption{A TSN fully distributed configuration model example\n  illustrating the general strategy and logic of each TSN switch with\n  TAS support. In the absence of a CNC to centrally manage network\n  parameters, each switch performs admission control and resource\n  reservation (according to the TAS time slot load) and propagates the\n  information to the next hop on the stream path. A single rejection on\n  one hop terminates the forwarding of the CDT, and sends another CDT\n  on the reverse path indicating the stream rejection outcome. If all\n  switches on the path accept the stream, then the source is notified\n  of the stream acceptance outcome and can begin forwarding in the\n  next TAS cycle. In our model, CDT traffic has higher priority than\n  non-CDT traffic (including ST). The formal definition of the CDT\n  traffic is left for future work. }\n\t\\label{fig_distributed_reconfig}\n\\end{figure}\n\nThis section presents our design methodology and framework for the TAS\nreconfiguration in the decentralized (fully distributed) model. Our\ncurrent proposed architecture generally follows the steps enumerated\nbelow and illustrated in Fig.~\\ref{fig_distributed_reconfig}. Our\ndescription focuses on the additions to the design of RAP over LRP,\ne.g., TAS slot computation\/reservations.\n\n\\begin{enumerate}\n\t\\item At each egress port (Port Identifier, PID), the TSN switch maintains a local stream registration table that includes information, such as flow ID, gateway (i.e., the first bridge that a talker is connected to), destination address(es), the traffic injection rate per GCL cycle time, and the calculated port bandwidth requirement. The traffic injection rate is not computed, rather the traffic injection rate is reported by the source (talker) to the network devices. It mainly indicates the bandwidth requirements of a stream. Bandwidth for a bridge egress port needed for a stream is computed using the ST injection rate (or ST rate), the average packet size, and the bridge TAS timing configuration (e.g., the CT and current traffic\tclass slot time). This information is carried and communicated between bridges using the CDT packet type identifier (or message type).\n\t\\item A source (talker) can send a stream transmission request, i.e., a CDT message of type Stream Transmission Request to\tregister its stream and use the TSN service for scheduled traffic.\n\t\\item Each switch maintains a resource manager module for each port. If the newly incoming stream is accepted (due to available resources and TAS slot space). The TAS slot size for a specific traffic class is governed by the CT and traffic class gating ratio (in time). The TAS ST slot can be configured\/reconfigured according to stream requests and terminations. The stream registration message is then propagated towards the next switch, and a map is maintained for the stream (and any other streams) pending approval.\n\t\\item If accepted by the last switch, then the stream registration record is added to the local stream registration table, and bandwidth resources are allocated for the stream and TAS slot space is modified (if necessary) on the reverse path. The main reason for allocating the resources in the reverse path is as follows. If we allocate the resources in the forward direction but a switch in the next hop rejects the stream (due to lack of resources),\tthen we have to release the resources reserved earlier for the stream. Therefore, we avoid the allocation until all hops provide\tassurance that the stream will be accommodated.\n\t\\item Each switch receiving the pending registration message adds the stream record to its local table, allocates the necessary resources and TAS slot reservation, and propagates the registration message towards the source gateway.\n\t\\item The source gateway receives the pending stream registration message and similarly allocates the resources and finally sends an approval granted message towards the source which prompts the source to start sending data in the next available TAS cycle.\n\\end{enumerate}\n\nWhile the previous enumerated points provide an overview of the procedural approach used, a key question remains, namely what happens if frames belonging to a given stream arrive after the gate for it's traffic class queue has been closed? Generally, the way the stream traffic end station (source\/talker) operates is by synchronizing to the switch's TAS ST slot time for the class the stream belongs to (in\tour case, it is ST) after the CDT stream registration is complete. Therefore, it will always send data traffic at the beginning of the ST slot. After the data traffic for the ST stream arrives at the first switch (where the gate for the ST queue will definitely be open due to source-network synchronization), the new ST traffic is queued at the back of the queue awaiting transmission. Since registration and resource allocation for all ST streams is enforced, the computed ST slot size will be large enough to transmit all ST packets (from all streams being serviced through that specific switch's egress port) queued that arrived from different sources during the cycle time. The delay bounds (a data traffic stream arriving before a specific time and not necessarily at the right time) is guaranteed since the cycle time for each switch along the path is configured to be large enough to transmit all ST frames registered on the multi hop path (up to 5 hops). Some potential ST traffic sent on cycle time $(t)$ can be transmitted at cycle time $(t+1)$, but would be blocked by some subsequent slots in the first cycle time belonging to different traffic classes (in our case, it is BE) leading to higher delays but still within tolerated delays since the cycle time is usually much smaller than the max delay bounds or threshold defined (e.g., 50 microseconds cycle time and 100 microseconds delay bounds).\n\nIn our simplified example in Fig.~\\ref{fig_distributed_reconfig}, each switch, upon initialization and with admission control, provisions all the CT to BE traffic. When the first stream transmission request occurs, the switches exchange the CDT message and start building the local stream registration table. A switch builds the GCE slot time for ST by mapping each stream request to its internal stream registration table upon granting the stream\tapproval. The switch builds the ST slot time iteratively by\tfollowing each stream admission\/termination. The switch passes each set of stream configuration parameters (UNI) through the 802.1Qcc registration and reservation protocol.\n\nST traffic stream load is viewed in two main ways in our implementation. $i)$ ST injection rate, and $ii)$ resource utilization time. The ST injection rate corresponds to the number of packets sent from an ST stream in one TAS cycle time, while the resource utilization time is the time that the stream reports to the network on how long the resources reserved will be used. The bandwidth requirement for a new stream request is calculated by taking the reported ST injection rate of the stream and multiplying that with the packet size for the stream. That quantity is then divided by the TAS slot time (at a specific switch). Essentially, an increase in the slot time corresponds to a decrease in the bandwidth requirement (since more time is given) and vice versa. Note that this only works for streams with periodic data transmissions and not for sporadic data streams since we cannot calculate how much time is needed for a sporadic data transmission. This procedure is only for ST traffic; BE traffic is transmitted with the leftover slot time after the ST traffic has been serviced.\n\n\\subsection{Core Components}\n\n\\begin{figure}[t!] \\centering\n\t\\includegraphics[width=3.5in]{fig_stream_registration.pdf} \\vspace{-0.2cm}\n\t\\caption{The main logical steps performed by each switch along the stream's path are shown to apply stream registration and reservation. Each switch generally waits for an event (addition, removal, or pending) for each stream. For instance, a stream removal is usually based on the resource utilization time (stream life time) that was specified at stream establishment. The bridges that allocated resources for the stream can remove the stream after the stream life time has expired. For the cases of stream addition or pending, the event is the CDT message received (whether in the forward or reverse direction). Towards completing (finalizing, confirming) a stream reservation (registration), the pending event is the event for a CDT message in the reverse direction where each switch (not the last switch) waits for the approval (confirmation of reservation) of the next-hop switch.}\n\t\\label{fig_stream_registration}\n\\end{figure}\n\nThis section outlines the main components required to successfully implement stream admission control and resource reservation within switches that support the TAS traffic shaper in a distributed fashion. Fig.~\\ref{fig_stream_registration} illustrates the typical registration\/reservation procedure for all streams within the TSN domain.\n\n\\subsubsection{Admission Control}\nThe admission control element extracts the necessary information from the CDT packet and forwards the information according to the CDT type. The switch forwarding mechanism applies several steps to accurately decide whether to accept or reject the stream transmission request Note that the stream transmission request corresponds to a CDT message. In particular, the switch consults the\nresource manager module to check if enough resources (bandwidth) is available for the new stream that is calculated by the reported traffic injection rate, the maximum cycle time, and the traffic class's TAS slot time. A given stream's bandwidth requirement is calculated by multiplying the ST injection rate with the average packet size and dividing by the current ST slot size. Note that the traffic class TAS slot time is the time during which the TAS gate is open to transmit frames belonging to the considered class.  Also note that all GCEs are executed during each CT. If the CT is smaller than the aggregate of the GCEs, then we need to either increase the CT or reject streams that cause the exceedance of the CT.\n\n\\subsubsection{Flow Scheduling}\nThis element currently takes the Time-Aware Shaper into consideration. Due to the TAS's requirements on time synchronization between network components (switches, hosts, etc.), all switches\/hosts follow the TAS GCL timescale cycle time (e.g., 50~$\\mu$s). Depending on the number of traffic classes supported, the TAS cycle time can be divided into appropriate slots for each traffic class load. The TAS CT is divided among all the traffic classes (in our evaluation model, we consider two traffic classes, BE and ST). Currently, in our evaluations, the CT is initially predefined to 50 microseconds. Note that the CT could be changed\/configured dynamically. The dynamic adaptation of the CT with respect to new stream additions, application specifications, or other events is a topic for future work.\n\n\\subsubsection{Stream Registration Table}\nStream creation follows a Poisson process with a mean duration. Different scenarios with varying mean duration enables analysis of how reconfiguration works in multiple settings. The stream life-time is defined by the duration. For example, how the switches in the path are aware of the Poisson parameter from the edge switch. All GCE entries and Queues are based purely on this slots allocated based on the Poisson parameter. Although, our approach does not intend to minimize delay, we follow a delay bound strategy that leads to finding the limiting value. Implicitly, the limiting value can be minimized that can provide a minimized delay. Additionally, the stream registration table contains the characteristics of the source streams that are established for the corresponding bridge egress port. Each record gets populated (if accepted) on the reverse path taken by the stream's registration message (after reaching the destination switch).\n\n\\subsubsection{Traffic Shaper - Time-Aware Shaper}\nThe main shaping and scheduling mechanism that controls the gating schedules for all the traffic classes within the TSN domain. All bridges are synchronized to the same gating schedule GCL CT that is initially predefined by network administrator. Ideally, we want the CT to be large enough for all streams from all\ntraffic classes to be accommodated and small enough that the all streams QoS fits the delay requirements.\nIn our current evaluations, the CT is predefined at the widely used 50 microseconds.\n\n\\subsubsection{Reconfiguration Calculus}\nIn essence, the reconfiguration (dynamic configuration) of the TAS schedules (switch GCL\/GCE) for each egress port is dynamically invoked according to two principle events, $i)$ adding a new stream, and $ii)$\nremoving an existing stream. The switch's gating ratio (for a particular stream belonging to a defined traffic class) reports certain parameters (e.g., packet injection rate, maximum packet size, latency requirement, deadline, application response time, etc.) which are then used to check if enough slot time is available (which corresponds to attempting bandwidth reservation). In the event that no slots are  available, the GCE slot size is recomputed (according to the registered stream properties within the registration table) generally by allocating more resources from Best Effort Traffic.\nThe stream life time is reported by the source to the network as user\/network information (UNI). Each UNI is propagated by each switch along the path which allows the switch to register the stream and store the stream's resource utilization time, (stream life-time), among other critical information. Any information\tpertaining to the UNI of a stream is recorded in the stream\tregistration table. In terms of GCEs for TAS with support of ST and BE traffic classes, only two GCEs within a GCL (1\/0 (ST\/BE) for the\tfirst entry and 0\/1 (BE\/ST) for the second) are necessary with a total of three outbound queues for each egress channel port in a TSN switch. Two queues for each traffic class, and another queue for CDT traffic (signaling traffic). Upon initialization, each switch allocates $0$ resources to ST, and BE therefore gets all the leftover CT slot. As streams get registered, the ST slot time is recomputed\n(according to the stream packet size, ST injection rate, and current slot time, if the slot exists). If the stream is the first stream to the switch, i.e., ST slot is $0$, then the ST slot size is defined\nto be 1\\% of the CT at a minimum by borrowing necessary time slot from BE.\n\n\\subsubsection{Path Computation}\nWhile this module is fundamentally necessary in any switch (in a decentralized\/distributed network), we manually define static routing tables for destination addresses and associated ports on each switch.\nEssentially, we assume a manual procedure to compute paths, i.e., we assume that there is a path computation module that is used in both centralized and distributed configuration models. We make this assumption to simplify operations and place emphasis on the TAS reconfiguration technique.\n\n\\subsubsection{Network Resource Table}\nTo remove certain overheads on the configuration procedure, the network resource table operates in tandem with the stream registration table to accurately determine the required network resources (mainly bandwidth for our traffic model) per switch egress port. It classifies streams based on periodic and sporadic streams properties, though currently the focus is on periodic ST streams. Any approved stream by the switch has a record attached to it in the network resource table, located within each switch, which can be called to compute and store current and remaining link\/port loads for each switch. Each egress port has a network resource table. More details on the network resource table will be provided in the next iteration of this document.\n\n\\section{Performance Evaluation}  \\label{tsn:sim:setup}\n\n\\begin{figure}[t!]  \\centering\n\\includegraphics[width=3.3in]{fig_tsn_indctrl} \\vspace{-0.2cm}\n\\caption{Industrial control loop\n  topology~\\cite{guck2016function}. Each source generates stream data\n  with varying hop counts and packets rates unidirectionally or\n  bidirectionally across the six switches ultimately destined to a\n  sink} \\label{fig_tsn_indLoop}\n\\end{figure}\n\n\\subsection{System Overview and Simulation Setup}  \\label{tsn:eval:sec}\nThis section explains the simulation setup and model. Furthermore, the\ntopology and simulation scenarios will be presented. Throughout, we\nemploy the OMNet++~\\cite{varga2008overview} simulation environment.\n\n\\begin{table}\n\t\\caption{Simulation Parameters}\n\t\\begin{tabularx}{\\columnwidth}{|X|X|X|}\n\t\t\\hline\n\t\t\\multicolumn{1}{|c|}{\\textbf{Key}} & \\multicolumn{1}{|c|}{\\textbf{Symbol}} & \\multicolumn{1}{c|}{\\textbf{Value}} \\\\\n\t\t\\hline\n\t\tSimulation Duration & $Sim_{limit}$ &  $100$ seconds \\\\\n\t\t\\hline\n\t\tInitialized Cycle Time & $GCL_{CT}$ & $50~\\mu$s \\\\\n\t\t\\hline\n\t\tInitialized Gating Ratio & $ST^{R}_{init}$ & $20$\\% (i.e., 10~$\\mu$s) \\\\\n\t\t\\hline\n\t\tAverage Streams per Second & $\\pi$ & $1 - 20$ \\\\\n\t\t\\hline\n\t\tAverage stream duration & $\\tau$ & $2 - 5$ seconds \\\\\n\t\t\\hline\n\t\tNumber of Frames per Cycle & $\\gamma$ & $1$ \\\\\n\t\t\\hline\n\t\tBE Traffic Intensity & $\\rho_{L}$ & $0.1, 1.0, 2.0$ (low, mid, and high) \\\\\n\t\t\\hline\n\t\tST sources & $S$ & $6$ \\\\\n\t\t\\hline\n\t\tHurst Parameter & $H$ & $0.5$ \\\\\n\t\t\\hline\n\t\tQueue Size & $Q_{size}$ & $512$~Kb\n\t\n\t\n\t\n\t\t\\\\\n\t\t\\hline\n\t\\end{tabularx}\n\t\\label{table: simulation parameters}\n\\end{table}\n\n\\begin{table}[t!]\n\t\\centering\n\t\\footnotesize\n\t\\caption{Traffic proportions relative to number of hops for the industrial control loop topology}\n\t\\label{hops}\n\t\\begin{tabular}{cccccc}\n\t\t\\textbf{Hops \\#} & 1 & 2 & 3 & 4 & 5 \\\\\n\t\t\\textbf{Range Distribution} & 20 \\% & 20 \\% & 20 \\% & 20 \\% & 20 \\% \\\\\n\t\\end{tabular}\n\\end{table}\n\n\\subsubsection{Network Model}\t\\label{tas:sec:net}\nThe network topology is modeled around an industrial control loop\ntopology that consists of six core switches in a ring topology\nconnected to the CNC as shown in Fig.~\\ref{fig_tsn_indLoop}. Each\nswitch-to-switch link operates as a full-duplex Ethernet link with a\ncapacity (transmission bitrate) $R = 1$~Gbps. Each switch can act as a\ngateway for a number of traffic sources and one sink. The distance\nbetween two successive switches along the ring is fixed to 100~m and\nthe switch-to-switch propagation delay is set accordingly to\n0.5~$\\mu$s. All switches are configured to use 802.1Qbv TAS as the\ntraffic shaper for each switch-to-switch egress port whose flow\nschedule (ST gating ratio and cycle time) is configured by the CNC in\nthe centralized (hybrid) model and independently in the decentralized\n(fully distributed) model.\n\n\\subsubsection{Traffic Model}\t\t\\label{tas:sec:traffic}\nWe consider periodic (pre-planned) traffic and sporadic self-similar\nPoisson traffic for ST traffic and for BE traffic, respectively. To\nemulate dynamic conditions in the network, we employ several\ndistributed ST sources that generate $\\pi$ ST streams according to the\nnetwork and traffic parameters shown in Table~\\ref{table: simulation\n  parameters}. The stream generation follows a Poisson distribution\nwhere $\\pi$ represents the average number of generated streams per\nsecond. Each stream within a source injects packet traffic with packet\nsize $64$~bytes and has an ST traffic injection rate that is uniformly\ndistributed with a value ($\\gamma$) statically defined at stream\ncreation time and a destination address assigned by the number of\nswitch-to-switch hops as shown in Table.~\\ref{hops}. Furthermore, at\nthe stream creation time, each stream is given a start time (usually\nthe current runtime), and a finish time based on $\\tau$. We consider\nadmission as the completion of the flow from start to the finish time\nreported by the source. Each source is attached to a core TSN switch\ngateway (first hop switch). While the TSN switches operate with time\nsynchronization, the ST sources (outside the TSN domain) do not need\nto be synchronized. Therefore, the gateways can in the worst case\ndelay ST traffic by a maximum of 1 cycle times. However, note that the\nST traffic follows an isochronous traffic class as specified by\nIEC\/IEEE 60802 where the sources are synchronized with the network\nafter stream registration is initiated.\n\n\\subsection{Centralized Model Evaluation}\t\\label{eval:cent:sec}\nIn evaluating the proposed solution described in\nSection~\\ref{tsn:design:sec}, we consider both periodic and sporadic\nsources for ST and BE traffic, as described in\nSection~\\ref{tas:sec:traffic}. We evaluated the CNC with TAS shaper on\nthe industrial control loop for the unidirectional and bi-directional\ntopologies and results are collected for tests following the simulation\nparameters shown in Table~\\ref{table: simulation parameters}.\n\n\\subsubsection{Unidirectional Ring Topology}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_2_cenUni.pdf}\n\t\\caption{Centralized Unidirectional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 2$ under different BE loads~$\\rho_{L}$, ST stream rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_2}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_3_cenUni.pdf}\n\t\\caption{Centralized Unidirectional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 3$ under different BE loads~$\\rho_{L}$, ST stream rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_3}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_4_cenUni.pdf}\n\t\\caption{Centralized Unidirectional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 4$ under different BE loads~$\\rho_{L}$, ST stream rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_4}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_5_cenUni.pdf}\n\t\\caption{Centralized Unidirectional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 5$ under different BE loads~$\\rho_{L}$, ST stream rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_5}\n\\end{figure}\n\nFigs.~\\ref{fig_delay_2}--\\ref{fig_delay_5} show the average mean\ndelay for ST traffic and for BE traffic for the centralized\nunidirectional ring topologies. The average delays in general are low\nand stable for both BE and ST traffic. Since the CNC manages the ST\ntraffic streams and therefore guarantees the bandwidth rates needed to\ntransmit across a single switch hop in one CT, the ST delays are less\nthan $100~\\mu$s for all $\\tau$ values. The ST delays with\nreconfiguration active at the CNC experience higher delays than ``No\nReconfiguration'' since we essentially push more frames into the\nnetwork that increase the queuing delay.\nBE traffic\nexperiences much higher delays than ST. With the ``No Reconfiguration''\napproach, the BE traffic delay is near constant since the gating ratio is\nleft unchanged throughout the simulation. For the test with\nreconfiguration, the BE mean delay increases dramatically since we\ntend to accept and exhaust the CT with ST streams over the course of\nthe simulation run.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_MaxDelay_50us_RhoI01_cenUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_MaxDelay_50us_RhoI10_cenUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_MaxDelay_50us_RhoI20_cenUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Unidirectional Topology: Max delay as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_maxDelay}\n\\end{figure}\n\nAs mentioned in the introduction section, TSN needs to guarantee and\nbound the maximum delay in order to deterministically forward traffic\nacross a TSN domain. Fig.~\\ref{fig_maxDelay} shows the maximum delay\nevaluation for ST traffic. For the unidirectional ring topology with a\nmaximum of five hop streams, the reconfiguration approach maximum\ndelay is bounded at $0.105$~ms, while for the ``No reconfiguration''\napproach, the max delay is bounded at nearly $60\\mu$~s. For TAS and\nthe CNC's registration and reservation procedure, the guarantee is\napplied to bandwidth as a share of the egress port using time division\nmultiplexing.\nWith the parameters chosen empirically, the\nmaximum delays is capped to approximately $100\\mu$~s which is ideal\nfor the topology chosen and critical ST traffic.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_admit_ratio_50us_RhoI01_cenUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_admit_ratio_50us_RhoI10_cenUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_admit_ratio_50us_RhoI20_cenUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Unidirectional Topology: Stream Admission as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_admin}\n\\end{figure}\n\nWhile QoS metrics are important, another factor that determines the\nperformance gains is the admission ratio for the\nsystem. Fig.~\\ref{fig_admin} shows the stream admission ratio that\nresults for both reconfiguration and no reconfiguration. In general,\neach generated stream needs a data rate of about $10.24$~Mbps for a\n50~$\\mu$s CT for each egress port on the stream's path with one ST\npacket rate per CT and fixed packet size of $64$~B. With an egress\nport channel capacity of $R=10^{9}$, approximately $100$ streams can\nbe guaranteed. Compared to the ``no reconfiguration'' approach, the\nreconfiguration significantly improves the admission rates at the cost\nof higher BE traffic delays since the ST slot borrows BE time slots to\naccommodate the ST streams.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_mean_signal_delay_ring_topo_totalDelay_50us_RhoI01_cenUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_mean_signal_delay_ring_topo_totalDelay_50us_RhoI10_cenUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_mean_signal_delay_ring_topo_totalDelay_50us_RhoI20_cenUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Unidirectional Topology: Stream Signaling delay as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_signalDelay}\n\\end{figure}\n\nCDT traffic that requests transmission guarantees from the CNC\nexperiences some delay before being either admitted or\nrejected. Fig.~\\ref{fig_signalDelay} shows the average signaling\nlatency for ST stream registration. Since the control plane is out-of\nband from the data plane within the TSN domain, the delay is constant\nthroughout the simulation run.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_sig_overhead_50us_RhoI01_cenUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_sig_overhead_50us_RhoI10_cenUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_sig_overhead_50us_RhoI20_cenUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Unidirectional Topology: Stream average signaling Overhead as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_signalOverhead}\n\\end{figure}\n\nStream registration and reservation introduces some control plane\noverhead. Fig.~\\ref{fig_signalOverhead} shows the signaling\nperformance overhead. More specifically, the overhead is measured at\nthe CNC or both incoming and outgoing control (CDT)\ntraffic. Generally, the reconfiguration approach introduces more\nsignaling overhead; however, Ethernet generally has large bandwidths,\nthus the CDT traffic rates are minimal compared to the link\ncapacities. Furthermore, when $\\tau = 2$, we observe higher signaling\noverhead due to accepting larger numbers of streams (rejections are\ninexpensive compared to acceptance) both with and without\nreconfiguration.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_ST_RhoI01_cenUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_ST_RhoI10_cenUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_ST_RhoI20_cenUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Unidirectional Topology: ST Total average throughput measured at the sink as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_avgTput_ST}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_BE_RhoI01_cenUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_BE_RhoI10_cenUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_BE_RhoI20_cenUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Unidirectional Topology: BE Total average throughput measured at the sink as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_avgTput_BE}\n\\end{figure}\n\nObserving the network's throughput gain, Fig.~\\ref{fig_avgTput_ST} and\nFig.~\\ref{fig_avgTput_BE} show the average throughput measured at the\nsink for both ST and BE traffic. While the total throughput that can\nbe achieved is around $6$~Gbps, the maximum throughput allowed is\naround $2$~Gbps since some switches get bottlenecked faster than other\nswitches which restricts the addition of more flows. For the ``no\nreconfiguration'' approach, BE traffic is varied between $0.1$ traffic\nintensity to $2.0$. Fig.~\\ref{fig_avgTput_BE}[a]--[b] shows an average\nthroughput of $0.1$~Gbps and $1.0$~Gbps, respectively, which is expected\nand no frames are dropped. However, as the load reaches $2.0$~Gbps and\nmore ST streams are accepted, i.e., the BE traffic time slot is shortened,\nBE traffic starts to suffer and caps at around $1.2$~Gbps, as shown in\nFig.~\\ref{fig_avgTput_BE}[c]. With reconfiguration, BE tends to suffer\nmore since we shift the time slot of BE to ST (maximum of $90\\%$) and\nthe throughput drops to $0.1$~Gbps as shown in\nFig.~\\ref{fig_avgTput_BE}[b][c].\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_ST_RhoI01_cenUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_ST_RhoI10_cenUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_ST_RhoI20_cenUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Unidirectional Topology: ST Frame loss ratio as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_lossProb_ST}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_BE_RhoI01_cenUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_BE_RhoI10_cenUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_BE_RhoI20_cenUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Unidirectional Topology: BE Frame loss ratio as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_lossProb_BE}\n\\end{figure}\n\nTo show the performance of the CNC management of streams,\nFig.~\\ref{fig_lossProb_ST} and Fig.~\\ref{fig_lossProb_BE} show the\npacket loss ratio for ST and BE traffic in the network. Since the CNC\nmanages only ST streams, the TSN guarantees (which include zero packet\nloss since retransmissions are in general too expansive for ST\ntraffic) are only valid for ST streams. For BE under the\nreconfiguration approach, as the load for ST traffic increases, the\npacket loss increases as well. For the ``no reconfiguration''\napproach, the loss typically is constant even for high loads of BE\ntraffic.\n\nOverall, the reconfiguration approach certainly provides a means to\nmanage and to ensure that the number of ST streams is maximized\naccording to the link capacity. However, we observe from the results\nand the topology used that any bottleneck switch can generally reduce\nthe link utilization which can significantly drop the throughput, even\nif the delay and loss are guaranteed. Selecting different paths (if\none exists) and modeling the queue to ensure maximum ST stream delays\ncan help potential bottleneck links and increase throughput throughout\nthe network.\n\n\\subsubsection{Bi-Directional Ring Topology}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_2_cenBi.pdf}\n\t\\caption{Centralized Bi-directional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 2$ under different BE loads~$\\rho_{L}$, ST stream rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_2_bi}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_3_cenBi.pdf}\n\t\\caption{Centralized Bi-directional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 3$ under different BE loads~$\\rho_{L}$, ST stream rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_3_bi}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_4_cenBi.pdf}\n\t\\caption{Centralized Bi-directional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 4$ under different BE loads~$\\rho_{L}$, ST stream rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_4_bi}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_5_cenBi.pdf}\n\t\\caption{Centralized Bi-directional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 5$ under different BE loads~$\\rho_{L}$, ST stream rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_5_bi}\n\\end{figure}\n\nThe unidirectional ring topology certainly simplifies the complexities of calculating the ST slot window. However, to show the a more pronounced gain in stream utilization and admission, a bi-directional ring topology is used with static shortest path routes. The two port switch now has two paths to the destination according to the hop count specified in Table.~\\ref{hops}, where about $60\\%$ of streams take one port and the rest ($40\\%$) take the other port. Note that the edge links (switch to sink and source to switch) are given higher link capacities to avoid congestion at the edges where the CNC currently does not control (at least $2$~Gbps for the bi-directional ring). Fig.~\\ref{fig_delay_2_bi} - \\ref{fig_delay_5_bi} shows the average mean delay evaluation for both ST and BE traffic under different stream lifetime values, $\\tau$. Compared to the unidirectional topology, the bi-directional provides significantly better delay results since an extra port with full-duplex link support now provides extra capacity to service streams giving more slot reservations to BE even at high ST stream load.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_MaxDelay_50us_RhoI01_cenBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_MaxDelay_50us_RhoI10_cenBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_MaxDelay_50us_RhoI20_cenBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Bi-directional Topology: Max delay as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_maxDelay_bi}\n\\end{figure}\n\nIn terms of maximum delays, the bi-directional topology configuration produces higher than expected max delays due to the increasing ST stream acceptance without taking into account queue delays. For the bi-directional topology tests, the queue sizes were left the same (see Table.~\\ref{table: simulation parameters}) and the CNC typically guarantees an upper bound delay per cycle per hop, i.e., each switch hop constitutes a max delay of one CT (or $50\\mu$s in our tests). Fig.~\\ref{fig_maxDelay_bi} shows the maximum delay evaluation for ST traffic for the bi-directional ring topology with CNC present. With the ``no reconfiguration'' approach, and since the ST slot size is kept at the initialized value ($20\\%$ of CT or $10~\\mu$s), the maximum delay is constant at $50~\\mu$s. However, if reconfiguration is active at the CNC, then the maximum delay is bounded at $300~\\mu$s due to the high admission rate and larger queuing delay per cycle. Note that the maximum hop traversal for any ST stream is kept at $3$~hops due to the bi-directional topology.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_admit_ratio_50us_RhoI01_cenBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_admit_ratio_50us_RhoI10_cenBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_admit_ratio_50us_RhoI20_cenBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Bi-directional Topology: Stream Admission as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_admin_bi}\n\\end{figure}\n\nWhile the max delay suffers due to not taking into account queuing delay in general, the admission rate is much higher (by about $40\\%$) at high ST loads. Fig.~\\ref{fig_admin_bi} shows the stream admission ratio results. With $\\pi = 20$ and $\\tau = 5$, the admission rate is close to $90\\%$ for the bi-directional topology with reconfiguration active at the CNC. In contrast, the ``no reconfiguration'' approach improves slightly (by about $20\\%$) compared to the unidirectional ring since the initialized gating ratio is too restrictive and can largely underutilize the link. Note that while we show the admission rates for different BE loads, $\\rho_{L}$, the admission ratio does not change since TAS effectively segments the traffic at the egress switch\/port, i.e., BE traffic does not block at ST traffic.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_mean_signal_delay_ring_topo_totalDelay_50us_RhoI01_cenBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_mean_signal_delay_ring_topo_totalDelay_50us_RhoI10_cenBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_mean_signal_delay_ring_topo_totalDelay_50us_RhoI20_cenBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Bi-directional Topology: Stream Signaling delay as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_signalDelay_bi}\n\\end{figure}\n\nSimilar to the unidirectional ring, the bidirectional ring topology provides constant signaling delay due to the CNC out-of band signaling channels. Fig.~\\ref{fig_signalDelay_bi} shows the signaling delay for ST stream registration. Note that the average signaling delay is lower than in the unidirectional ring since the edge links, specifically the source to switch link, is larger than in the unidirectional ring hence the transmission delay is shorter.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_sig_overhead_50us_RhoI01_cenBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_sig_overhead_50us_RhoI10_cenBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_sig_overhead_50us_RhoI20_cenBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Bi-directional Topology: Stream average signaling Overhead as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_signalOverhead_bi}\n\\end{figure}\n\nSince the topology is effectively the same (albeit having another port to the switch), the signaling overhead in general is very similar to the unidirectional topology. Fig.~\\ref{fig_signalOverhead_bi} shows the signaling performance overhead. Note that while the hop traversal is reduced (since the stream can take one of two paths to the destination governed by hop traversal), the number of sent and received CDT frames are the same in general. Clearly, similar to the unidirectional topology, the reconfiguration approach generates more CDT traffic. Note that rejections in general are less costly in terms of sent and received frames in the network. Therefore, the higher the admission rate, the more overhead is observed in the control plane, though based on Fig.~\\ref{fig_signalOverhead_bi}, the overall overhead is not even close to $1$~Mbps and therefore is much lower compared to the channel capacity.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_ST_RhoI01_cenBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_ST_RhoI10_cenBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_ST_RhoI20_cenBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Bi-directional Topology: ST Total average throughput measured at the sink as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_avgTput_ST_bi}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_BE_RhoI01_cenBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_BE_RhoI10_cenBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_BE_RhoI20_cenBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Bi-directional Topology: BE Total average throughput measured at the sink as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_avgTput_BE_bi}\n\\end{figure}\n\nIn terms of overall throughput, Fig.~\\ref{fig_avgTput_ST_bi} and Fig.~\\ref{fig_avgTput_BE_bi} shows the average throughput measured at the sink for both ST and BE traffic for the bi-directional ring topology. Compared to the unidirectional ring, the throughput for the bi-directional ring is much higher (sink maximum capacity is around $12$~Gbps from all switch to sink channels).\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_ST_RhoI01_cenBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_ST_RhoI10_cenBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_ST_RhoI20_cenBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Bi-directional Topology: ST Frame loss ratio as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_lossProb_ST_bi}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_BE_RhoI01_cenBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_BE_RhoI10_cenBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_BE_RhoI20_cenBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Centralized Bi-directional Topology: BE Frame loss ratio as a results of TAS with centralized configuration (CNC) management entity.}\n\t\\label{fig_lossProb_BE_bi}\n\\end{figure}\n\nSimilar to the unidirectional ring topology, bi-directional topology achieves zero loss to ST streams while significantly improving the loss rate for BE traffic. Fig.~\\ref{fig_lossProb_ST_bi} and Fig.~\\ref{fig_lossProb_BE_bi} shows the packet loss ratio for ST and BE traffic in the network for the bi-directional ring topology. Maximum BE loss as high BE traffic intensity, $\\rho_{L} = 2.0$, is around $30\\%$ which is a significant reduction from the unidirectional topology (of around $90\\%$).\n\nIn contrast to the unidirectional topology, the bi-directional topology with reconfiguration active at the CNC achieves improved QoS metrics and admission rates. However, without modeling the queue characteristics and guaranteeing queuing delay, it is difficult to guarantee maximum delays. Modeling the ST queue at each egress port in conjunction with TAS configuration properties grants the possibility to produce deterministic forwarding plane at the data plane for ST streams. This part is left for the next iteration of this report.\n\n\\subsection{Decentralized Model Evaluation}\t\\label{eval:dec:sec}\nIn evaluating the proposed solution describes in more detail in section~\\ref{tas:reconfig:prot}, we consider both periodic and sporadic sources for ST and BE traffic as discussed in section~\\ref{tas:sec:traffic} respectively. We evaluated network with TAS shaper on the industrial control loop unidirectional and bi-directional topology and results are collected for tests following the simulation parameters shown in Table.\\ref{table: simulation parameters}.\n\n\\subsubsection{Unidirectional Ring Topology}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_2_distUni.pdf}\n\t\\caption{Decentralized Unidirectional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 2$ under different loads~$\\rho_{L}$, mean traffic rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_2_dec}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_3_distUni.pdf}\n\t\\caption{Decentralized Unidirectional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 3$ under different loads~$\\rho_{L}$, mean traffic rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_3_dec}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_4_distUni.pdf}\n\t\\caption{Decentralized Unidirectional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 4$ under different loads~$\\rho_{L}$, mean traffic rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_4_dec}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_5_distUni.pdf}\n\t\\caption{Decentralized Unidirectional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 5$ under different loads~$\\rho_{L}$, mean traffic rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_5_dec}\n\\end{figure}\n\nDecentralized model essentially transfers some of the CNC functions (e.g., TAS reconfiguration and resource reservation modules) from the centralized model down to the data plane switch's egress TAS enabled ports. The main difference between the centralized and decentralized models is the signaling performance which is now in-band and can affect data traffic. Fig.~\\ref{fig_delay_2_dec} - \\ref{fig_delay_5_dec} shows the average mean delay evaluation for both ST and BE traffic. While the CDT traffic is in-band, the average delay is about the same as the centralized topology average delay in Fig.~\\ref{fig_delay_2} - \\ref{fig_delay_5}. Typically, the ST stream's average delay is minimal to near constant for both the reconfiguration and ``no reconfiguration'' approaches. For BE, the ``no reconfiguration'' approach produces constant average delay for each BE $\\rho_{L}$ traffic intensity. When used with reconfiguration, the mean delay start to democratically increase since BE traffic time slots are begin reserved for ST streams.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_MaxDelay_50us_RhoI01_distUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_MaxDelay_50us_RhoI10_distUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_MaxDelay_50us_RhoI20_distUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Unidirectional Topology: Max delay as a results of TAS.}\n\t\\label{fig_maxDelay_dec}\n\\end{figure}\n\nIn terms of maximum delays under the unidirectional topology using the decentralized model, Fig.~\\ref{fig_maxDelay_dec} shows the maximum delay evaluation for ST traffic. In contrast to the average delay, the maximum delay does get affected by in-band CDT traffic. In the decentralized model, the CDT traffic is given the highest priority above both ST and BE traffic. Therefore, the maximum delays can reach about $50~\\mu$s more than the ST threshold of $100~\\mu$s.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_admit_ratio_50us_RhoI01_distUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_admit_ratio_50us_RhoI10_distUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_admit_ratio_50us_RhoI20_distUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Unidirectional Topology: Stream admission as a results of TAS.}\n\t\\label{fig_admin_dec}\n\\end{figure}\n\nThe admission rate is very similar to the centralized model since the network parameters used are identical. Fig.~\\ref{fig_admin_dec} shows the stream admission ratio results.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_mean_signal_delay_ring_topo_totalDelay_50us_RhoI01_distUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_mean_signal_delay_ring_topo_totalDelay_50us_RhoI10_distUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_mean_signal_delay_ring_topo_totalDelay_50us_RhoI20_distUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Unidirectional Topology: Stream average signaling delay as a results of TAS.}\n\t\\label{fig_signalDelay_dec}\n\\end{figure}\n\nIn contrast to the centralized model, the decentralized model's in-band CDT traffic implies varied stream signaling delays as shown in Fig.~\\ref{fig_signalDelay_dec} which shows the signaling delay for ST stream registration. As the number of streams generated ($\\pi$) increase, the overall average signaling delay decreases which is due to the increased rejections as more streams attempt to request network resources. In the decentralized model, a rejection by an intermediate bottlenecked switch implies a termination of the CDT traffic and a notification to any previous pending stream records to cancel the potential reservation and eventually notify the source of the rejection. If this rejection happens closer to the source in the CDT registration procedure, then the average delay will be much shorter when compared to an stream acceptance. In general, the average stream signaling delay is in the order of microseconds which is reasonable for most industrial control systems applications.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_sig_overhead_50us_RhoI01_distUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_sig_overhead_50us_RhoI10_distUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_sig_overhead_50us_RhoI20_distUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Unidirectional Topology: Stream Signaling Overhead as a results of TAS.}\n\t\\label{fig_signalOverhead_dec}\n\\end{figure}\n\nGenerally, the decentralized model produced greater signaling overhead than the centralized model since CDT traffic is measured at each data traffic port for incoming and outgoing as shown in Fig.~\\ref{fig_signalOverhead_dec}. Analogous to the signaling delay, the more ST streams accepted, the more overhead is observed. Therefore, as $\\tau$ increases and consequently, the more rejections occur, the lower the overhead. As shown in Fig.~\\ref{fig_signalOverhead_dec}, the results for the signaling overhead with reconfiguration shows a more pronounced difference with varied $\\tau$ values.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_ST_RhoI01_distUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_ST_RhoI10_distUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_ST_RhoI20_distUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Unidirectional Topology: ST Total average throughput measured at the sink as a results of TAS.}\n\t\\label{fig_avgTput_ST_dec}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_BE_RhoI01_distUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_BE_RhoI10_distUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_BE_RhoI20_distUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Unidirectional Topology: BE Total average throughput measured at the sink as a results of TAS.}\n\t\\label{fig_avgTput_BE_dec}\n\\end{figure}\n\nThroughput results are generally the same when compared to the unidirectional centralized model. Fig.~\\ref{fig_avgTput_ST_dec} and Fig.~\\ref{fig_avgTput_BE_dec} shows the average throughput measured at the sink for both ST and BE traffic.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_ST_RhoI01_distUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_ST_RhoI10_distUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_ST_RhoI20_distUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Unidirectional Topology: ST Frame loss ratio as a results of TAS.}\n\t\\label{fig_lossProb_ST_dec}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_BE_RhoI01_distUni.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_BE_RhoI10_distUni.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_BE_RhoI20_distUni.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Unidirectional Topology: BE Frame loss ratio as a results of TAS.}\n\t\\label{fig_lossProb_BE_dec}\n\\end{figure}\n\nSimilarly, the packet loss rate is nearly similar to the unidirectional centralized model as shown  in Fig.~\\ref{fig_lossProb_ST_dec} and Fig.~\\ref{fig_lossProb_BE_dec} for ST and BE traffic in the network respectively. The unidirectional topology with either the centralized or decentralized approach generally get bottlenecked much faster compared to the bi-directional results. Therefore, BE traffic quickly suffers as more ST streams request TAS slot reservation. In terms of added improvements, changing the cycle time (GCL time) to different values depending on both ST and BE traffic proportions can generally improve BE traffic whilst still guaranteeing ST streams. In typical industrial environments, ST streams have generally low data rates, are less frequent, and smaller in size than BE traffic, and therefore can be admitted quite easily without much effect on BE traffic.\n\n\\subsubsection{Bi-Directional Ring Topology}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_2_distBi.pdf}\n\t\\caption{Decentralized Bi-directional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 2$ under different loads~$\\rho_{L}$, mean traffic rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_2_dec_bi}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_3_distBi.pdf}\n\t\\caption{Decentralized Bi-directional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 3$ under different loads~$\\rho_{L}$, mean traffic rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_3_dec_bi}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_4_distBi.pdf}\n\t\\caption{Decentralized Bi-directional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 4$ under different loads~$\\rho_{L}$, mean traffic rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_4_dec_bi}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\includegraphics[width=3.3in]{fig_tas_mean_delay_ring_topo_totalDelay_50us_5_distBi.pdf}\n\t\\caption{Decentralized Bi-directional Topology: Mean end-to-end delay for ST and BE traffic for $\\tau = 5$ under different loads~$\\rho_{L}$, mean traffic rates~$\\pi$, and initialized gating ratio of $20\\%$.}\n\t\\label{fig_delay_5_dec_bi}\n\\end{figure}\n\nIn the bi-directional topology using the decentralized model, in-band CDT traffic affects the data traffic similar to the unidirectional model. Fig.~\\ref{fig_delay_2_dec_bi} - \\ref{fig_delay_5_dec_bi} shows the average mean delay evaluation for both ST and BE traffic. As $\\tau$ is increased, i.e., the number of streams at any time increase, the BE slot reservations are reserved for ST streams which affects BE QoS mean delay.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_MaxDelay_50us_RhoI01_distBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_MaxDelay_50us_RhoI10_distBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_MaxDelay_50us_RhoI20_distBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Bi-directional Topology: Max delay as a results of TAS.}\n\t\\label{fig_maxDelay_dec_bi}\n\\end{figure}\n\nIn terms of maximum delay, Fig.~\\ref{fig_maxDelay_dec_bi} shows the maximum delay evaluation for ST traffic. While the reconfiguration approach looks very similar to the centralized model, the ``no reconfiguration'' approach gets affected by in-band CDT traffic that raises the max delay in some cases to $100~\\mu$s.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_admit_ratio_50us_RhoI01_distBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_admit_ratio_50us_RhoI10_distBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_admit_ratio_50us_RhoI20_distBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Bi-directional Topology: Stream admission as a results of TAS.}\n\t\\label{fig_admin_dec_bi}\n\\end{figure}\n\nAdmission rate is exactly the same as the centralized model as shown in Fig.~\\ref{fig_admin_dec_bi} given the same networking parameters.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_mean_signal_delay_ring_topo_totalDelay_50us_RhoI01_distBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_mean_signal_delay_ring_topo_totalDelay_50us_RhoI10_distBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_mean_signal_delay_ring_topo_totalDelay_50us_RhoI20_distBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Bi-directional Topology: Stream average signaling delay as a results of TAS.}\n\t\\label{fig_signalDelay_dec_bi}\n\\end{figure}\n\nFig.~\\ref{fig_signalDelay_dec_bi} shows the signaling delay for ST stream registration.It is observed that as the ST stream generation increases, so does the mean signaling delay up to a stream mean rate, $\\pi$. The mean delay starts to decrease as the load keeps increasing.\n\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_sig_overhead_50us_RhoI01_distBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_sig_overhead_50us_RhoI10_distBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_sig_overhead_50us_RhoI20_distBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Bi-directional Topology: Stream Signaling Overhead as a results of TAS.}\n\t\\label{fig_signalOverhead_dec_bi}\n\\end{figure}\n\nThe stream signaling overhead produces results that are the same as the decentralized model as shown Fig.~\\ref{fig_signalOverhead_dec_bi}. In general, the results between different $\\tau$ values are close since for any $\\tau$ value, almost all the streams are getting accepted generating the same overhead in total.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_ST_RhoI01_distBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_ST_RhoI10_distBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_ST_RhoI20_distBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Bi-directional Topology: ST Total average throughput measured at the sink as a results of TAS.}\n\t\\label{fig_avgTput_ST_dec_bi}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_BE_RhoI01_distBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_BE_RhoI10_distBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_ring_topo_AvgTput_50us_BE_RhoI20_distBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Bi-directional Topology: BE Total average throughput measured at the sink as a results of TAS.}\n\t\\label{fig_avgTput_BE_dec_bi}\n\\end{figure}\n\nFig.~\\ref{fig_avgTput_ST_dec_bi} and Fig.~\\ref{fig_avgTput_BE_dec_bi} shows the average throughput measured at the sink for both ST and BE traffic. The average throughput results are nearly identical to the centralized model. While it is difficult to see, the throughput is slightly less than the centralized since we now incorporate in-band control traffic while reduces the link utilization for data traffic.\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_ST_RhoI01_distBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_ST_RhoI10_distBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_ST_RhoI20_distBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Bi-directional Topology: ST Frame loss ratio as a results of TAS.}\n\t\\label{fig_lossProb_ST_dec_bi}\n\\end{figure}\n\n\\begin{figure} [t!] \\centering\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_BE_RhoI01_distBi.pdf}\n\t\t\\caption{Low $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_BE_RhoI10_distBi.pdf}\n\t\t\\caption{Mid $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\columnwidth} \\centering\n\t\t\\includegraphics[width=3.3in]{fig_tas_loss_pkt_prob_ring_topo_50us_BE_RhoI20_distBi.pdf}\n\t\t\\caption{High $\\rho_{L}$}\n\t\\end{subfigure}\n\t\\caption{Decentralized Bi-directional Topology: BE Frame loss ratio as a results of TAS.}\n\t\\label{fig_lossProb_BE_dec_bi}\n\\end{figure}\n\nFig.~\\ref{fig_lossProb_ST_dec_bi} and Fig.~\\ref{fig_lossProb_BE_dec_bi} shows the packet loss ratio for ST and BE traffic in the network. Similar to all the different models and topologies, ST streams have zero frame drops as required by TSN. BE traffic results are nearly identical to the centralized model under bi-directional topology. Similarly, the overall performance is largely improved under the bi-directional topology due to the additional port.\n\nWhile the decentralized model certainly proves to operate nearly identical to the centralized one in terms of QoS metrics and overall admission rate, the main disadvantage is in-band CDT traffic which can lead to delayed ST streams particularly affecting guaranteed maximum delays violations. A work around is to service all the ST streams first, and then service CDT frames before servicing the BE traffic, though this might lead to additional signaling delays depending on the ST load. However, some applications have a more relaxed constraint in guaranteeing maximum delay but require complete segregation of traffic based on the class of service which can be handled using the decentralized model without the overhead complexities of a CNC device. Furthermore, the decentralized model can struggle to find alternate paths without full network visibility, as opposed to the centralized model which can easily reroute streams in the event of failures due to having full network visibility, though this is out of scope for this phase of the project. Moreover, while not tested in our model, adding new devices and removing devices in the decentralized model leads to information flooding across the network that adds complexity and can skew TAS schedules if not handled appropriately. However, in the centralized model, this ``plug and play'' feature can be easily extended to our TSN domain since the CNC has full control and management of the data plane and can adjust and coordinate any scheduling issues in a timely and controllable manner.\n\n\\section{Conclusions and Future Work}  \\label{concl:sec}\nThe IEEE 802.1Qcc framework and the 802.1Qbv traffic shaper enable the\nimplementation of a deterministic forwarding plane that provides\nstrict guarantees to any scheduled traffic service without any flow or\ncongestion control mechanism at the source.  Using an automated\nnetwork configuration is an imperative tool set to provide a unified\ncommunication platform based on commercial of the shelf (COTS) full-duplex\nEthernet with high bandwidth\/low complexity compared to Controller\nArea Networks (CANs), Local Interconnect Networks (LINs), and\nspecialized field-buses in industrial control system applications\n(e.g., industrial control, automotive, avionics).  Such network\ndesigns can form a contract with the source to forward mission\ncritical traffic and to automate the network configuration process using\n802.1Qcc for the full lifetime of the stream. Additionally, depending\non the forwarding plane port traffic shaper (e.g., TAS), the required\nschedules can be passed to the switch servers using general\nuser\/network information protocols (e.g., TLV, Yang, SNMP).\n\nIn this paper, we have investigated the impact of TAS reconfigurations\nin response on dynamic network conditions. We have demonstrated the\neffectiveness of TAS with and without the CNC, i.e., for centralized\n(hybrid) vs. decentralized (fully distributed) models.  We have\nexamined network QoS traffic characteristics when maximizing stream\nadmission to the network whilst reserving some BE time slots in the\nevent of high ST transmission requests.\n\nBased on the insights from the present study we outline the following\nfuture research directions. First, it would be interesting to\njudiciously change the GCL time for switches during reconfiguration\nwhilst satisfying QoS requirements.\nThe studied reconfiguration techniques should also be examined\nin alternate approaches for providing deterministic Qos,\ne.g.,~\\cite{nas2019tsn,seaman2019pat} as well as in the context of related\nQoS oriented routing approaches, e.g.~\\cite{chu2018pre,guc2017uni}.\n\nIn the wider context of QoS networking and related applications,\ndeterministic networking should be examined in the context of\nemerging multiple-access edge computing\n(MEC)~\\cite{doa2019pro,gao2019dyn,mar2019mod,sha2018lay},\nin particular MEC settings for low-latency\napplications~\\cite{elb2018tow,xia2019red,zha2018mob}.\nAs an alternative approach to coordinating the\nreconfigurations, emerging softwarized control paradigms, such as\nsoftware defined networking can be explored~\\cite{ami2018sdn,der2019cou,des2018min,kel2019ada,san2018inf}.\nRegarding the reliability aspects, a potential future research direction is to\nexplore low-latency network coding mechanisms, e.g.,~\\cite{ace2018har,coh2019ada,eng2018exp,gab2018cat,luc2018ful,ma2019high,wun2017cat}, to enhance networking protocols targeting reliable low-latency communication.\n\n\\bibliographystyle{IEEEtran}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection*{Preliminaries}\n\\subsubsection*{Introduction and preliminaries.}\nThis paper gives a characterization of the class of languages recognized by a model of quantum automata, by using tools from algebraic theory, in particular, varieties of languages.\nMany models of one-way quantum finite automata are present in the literature: the oldest is the Measure-Once model \\cite{BC01,BP02}, characterized by unitary evolution operators and a single measurement performed at the end of the computation. On the contrary, in other models, evolutions and measurements alternate along the computation \\cite{Aal06,KW97}. The model we study is the Measure-Only Quantum Automaton (\\textsc{MOn-1qfa}), introduced in \\cite{BMP10}, in which we allow only measurement operations, not evolution.\nAll these quantum models are generalized by Quantum Automata with Control Language \\cite{BMP03}.\n\nA \\textsc{MOn-1qfa}\\ over the alphabet $\\Sigma$ is a tuple of the form \n$\nA = \\langle \\Sigma \\cup \\{\\#\\},$ $(O_c)_{c\\in\n  \\Sigma\\cup\\{\\#\\}}, \\pi_0, F \\rangle\n$. \nThe complex $m$-dimensional vector $\\pi_0\\in\\mathbb{C}^{1\\times m}$, with unitary norm $||\\pi_0||=1$, \nis called the quantum initial state of $A$.\nFor every $c\\in\\Sigma$, $O_c\\in\\mathbb{C}^{m\\times m}$\nis (the representative matrix of) an idempotent Hermitian operator and denotes an observable.\nThe subset $F\\subseteq V(O_{\\#})$ of the eigenvalues of $O_{\\#}$ is called the spectrum of the quantum final accepting states of $A$.\\mbox{}\\\\\nThe computation dynamics of automaton $A$ is carried out in the following way: \nlet $x=x_1\\ldots x_n\\in\\Sigma^*$, suppose we start from $\\pi_0$, then $A$ measures the system with cascade observables \n$O_{x_1}, \\ldots, O_{x_n}$ (by applying the associated orthogonal projectors) \nand then performs the final measure with the end-word observable $O_{\\#}$, that is the observable of the final accepting states $F$ of $A$.\\\\\nThis last measure returns, as a result, an eigenvalue\n$r\\in V(O_{\\#})$, if $r\\in F$ then we say that the automaton $A$ accepts the word $x\\in\\Sigma^*$, otherwise that $A$ does not accept it. \nWhat is remarkable in this computation dynamics is the probability $p_A(x)$ that $A$ accepts $x=x_1\\cdots x_n$.\nIn the specific case of \\textsc{MOn-1qfa}s\\, it turns out to be of some interest to express $p_A(x)$ using the well-known formalism of quantum density matrices.\nWe say that a language $L$ is recognized by $A$ with isolated cut point $\\lambda$ iff for all $x\\in\\Sigma^*$\\, $p_A(x)>\\lambda \\Leftrightarrow x\\in L$ and there exists a constant value $\\delta>0$ such that $|p_A(x)-\\lambda|\\geq\\delta$.\n\nWe now recall some general definitions and results from the algebraic theory of automata and formal languages.\nFor more details, we refer the reader to, e.g. \\cite{Ei76,Pi86}.\nLet $L$ be a regular language and let $\\langle \\Sigma, Q, \\delta, q_0,\nF \\rangle$ be the minimal deterministic automaton recognizing $L$. For a word $w=\\sigma_1\\cdots \\sigma_n\\in\\Sigma^*$, we define its variation as \n$\\text{Var}_L(w) =  \\#\\{0\\leq k < n \\mid \\delta(\\sigma_1\\cdots \\sigma_k)\\neq \\delta(\\sigma_1\\cdots \\sigma_{k+1})\\}$.\nWe say that $L$ has finite variation iff $ \\text{sup}_{x\\in\\Sigma^*} \\text{Var}_L(x) < \\infty $.\nUsing results and similar techniques as in \\cite{BMP10}, it is not difficult to show  \nthat the class $\\textbf{LMO}(\\Sigma)$ of languages recognized by a \\textsc{MOn-1qfa}\\ with isolated cut point is a boolean algebra of regular languages in $\\Sigma^*$ with finite variation.\nWe say that a language $L\\in\\Sigma^*$ is literally idempotent iff for all $x,y\\in\\Sigma^*$ and $a\\in\\Sigma$, $xa^2y\\in L\\Leftrightarrow xay\\in L$; we say that $L$ is literally idempotent piecewise testable if and only if it lies\nin the boolean closure\nof the following class of languages: \n$\\Sigma^* a_1 \\Sigma^* a_2 \\Sigma^*\\cdots \\Sigma^* a_k \\Sigma^*$, for $a_1, a_2, \\ldots, a_k\\in\\Sigma$ and $a_1\\neq \\cdots \\neq a_k$. \nWe denote by $\\text{liId}$ the class of literally idempotent languages and by $\\text{liId}\\textbf{PT}$ the class of literally idempotent piecewise testable languages.\nFor any language $L$, we call $M(L)$ its syntactic monoid.\nWe say that a class of finite monoids $\\mathbf{A}$ is a (literal) pseudovariety if and only if it is closed under (literal) substructures, homomorphic images and finite direct products, \\cite{KP08}.\nLet $\\mathbf{A}$ be a class of monoids and let $\\Sigma$ be an alphabet. We denote by $V_{\\Sigma}(\\mathbf{A})$ \nthe class of regular languages on $\\Sigma$ having syntactic monoid in $\\mathbf{A}$. \nLet $L, R$ and $J$ be the Green's relations determined by left, right and two-sided ideals, respectively.\nIn this paper we denote by $\\mathbf{R}$ the pseudovariety of $R$-trivial finite monoids and by $\\mathbf{J}$ the pseudovariety of $J$-trivial finite monoids.\nWe also define $\\overline\\mathbf{J}$ as the literal pseudovariety of $J$-trivial syntactic monoids $M(L)$ such that the associated morphism $\\phi_L:\\Sigma^*\\rightarrow M(L)$ satisfies the literal idempotent condition $\\phi_L(\\sigma)\\phi_L(\\sigma)=\\phi_L(\\sigma)$, for every $\\sigma\\in\\Sigma$.\nWe say that a class of regular languages $V:\\Sigma\\rightarrow 2^{\\Sigma^*}$ is a $*$-variety of \\emph{Eilenberg} if $V(\\Sigma)$ is closed under boolean operations, right and left quotient, and inverse homomorphism. Replacing closure under inverse homomorphism by closure under inverse literal homomorphism, we get the notion of literal variety of languages.\nA fundamental result is due to Eilenberg, who showed that there exists a bijection $V_\\Sigma$ from the psuedovarieties of monoids and the $*$-varieties of Eilenberg of formal languages \\cite{Pi86}.\nIn \\cite{KP08}, Kl\\'ima and Pol\\'ak showed the following\n\\begin{theorem}\\label{thm:klimapolak} Let $L\\subseteq\\Sigma^*$. \nIt holds that  \n$L\\in\\text{liId}\\textbf{PT}$ if and only if \n$L\\in V_{\\Sigma}(\\mathbf{J})\\cap \\text{liId}(\\Sigma)$ if and only if  $L\\in V_{\\Sigma}\\left(\\overline{\\mathbf{J}}\\right)$.\n\\end{theorem}\n\\subsubsection*{Results.}\nWe give a direct proof that the class of finite variation regular languages is a $*$-variety of Eilenberg.\nMoreover, we observe that a regular language $L$ has finite variation if and only if its syntactic monoid is $R$-trivial.\nWe proceed further on with our analysis by showing that the class of \\textsc{MOn-1qfa}\\ over $\\Sigma$ is in fact a sub-class of Latvian Automata. This class of \nautomata has been fully characterized algebraically in \\cite{Aal06} as the class of automata recognizing exactly regular languages having syntactic monoids in the class $\\mathbf{BG}$ of block groups. \n\\begin{theorem}\nLet $A$ be a \\textsc{MOn-1qfa}\\ on $\\Sigma$ and let $L_A$ be a language recognized by $A$ with cut-point $\\lambda$ isolated by $\\delta$. \nThen there exists a Latvian automaton $A'$ recognizing \n$L_{A'}=L_{A}$ with cut-point $\\lambda' = \\frac{1}{2}$ isolated by $\\delta' = \\frac{\\delta}{2\\cdot \\text{max}(\\lambda, 1-\\lambda)}$.\n\\end{theorem}\nThis directly implies that $\\textbf{LMO}(\\Sigma)\\subseteq V_{\\Sigma}(\\mathbf{BG})$.\n\nCombining our analysis with the results of \\cite{Aal06} on block groups syntactic monoids and the results of \\cite{BMP10} \non finite variation languages, we prove the following\n\\begin{theorem}\nLet $L\\in\\textbf{LMO}(\\Sigma)$ be a language recognized by some \\textsc{MOn-1qfa}\\ with isolated cutpoint. Then its syntactic monoid $M(L)$ \nis an $R$-trivial block group, formally speaking $ M(L) \\in \\mathbf{BG}\\cap \\mathbf{R} $.\n\\end{theorem}\nSince an $R$-trivial block group is also $J$-trivial, and since $\\textbf{LMO}(\\Sigma)$ is a boolean algebra, we have $\\textbf{LMO}(\\Sigma)\\subseteq V_\\Sigma(\\mathbf{J})$. This, together with Theorem \\ref{thm:klimapolak} and the fact that languages in $\\textbf{LMO}(\\Sigma)$ are literally idempotent, leads to the following\n\\begin{theorem}\n$\\textbf{LMO}(\\Sigma)\\subseteq V_\\Sigma(\\overline\\mathbf{J})$.\n\\end{theorem}\n\nWe now show how languages in \\text{liId}\\textbf{PT}\\ can be recognized by \\textsc{MOn-1qfa}s.\nConsider the language \n$L[a_1, \\ldots, a_k] = \\Sigma^*a_1\\Sigma^*\\cdots\\Sigma^*a_k\\Sigma^*$,\nwhere $a_1,\\ldots, a_k \\in\\Sigma$, $a_i \\neq a_{i+1}$ for $1\\leq i< k$, and let $S = \\{a_1, \\ldots, a_k\\}$.\nFor every $\\alpha \\in S$, let $\\#\\alpha$ be the number of times that $\\alpha$ appears as a letter in the word $a_1a_2\\cdots a_k$.\nLet $j^{(\\alpha)}_1 < j^{(\\alpha)}_2 < \\cdots < j^{(\\alpha)}_{\\#\\alpha}$ be all the indexes such that $\\alpha =\na_{j^{(\\alpha)}_1} = \\ldots = a_{j^{(\\alpha)}_{\\#\\alpha}}$. We define, for every $\\alpha\\in S$, two orthogonal projectors of dimension $(k+1)\\times (k+1)$: the up operator $P^{(k)}_{\\nearrow}(\\alpha)$ and the down operator $P^{(k)}_{\\searrow}(\\alpha)$, such that\n\\[\n\\left(P^{(k)}_{\\nearrow}(\\alpha)\\right)_{rs} =\n\\left\\{\n\t\\begin{array}{cl}\n\t1 & \\mbox{ if } r=s \\mbox{ and } \\forall\\, 1\\leq i\\leq \\#\\alpha \\mbox{ it holds } r,s\\notin\\{j_{i}^{\\alpha}, j_{i}^{\\alpha}+1\\},\\\\\n\t\\frac{1}{2} & \\mbox{ if } \\exists\\, 1\\leq i\\leq \\#\\alpha \\mbox{ such that } r,s\\in\\{j_{i}^{\\alpha}, j_{i}^{\\alpha}+1\\},\\\\\n\t0 & \\mbox{ otherwise,}\n\t\\end{array}\n\\right.\n\\]\n\\[\n\\left(P^{(k)}_{\\searrow}(\\alpha)\\right)_{rs} =\n\\left\\{\n\t\\begin{array}{cl}\n\t\\frac{1}{2} & \\mbox{ if } r=s \\mbox{ and } \\exists\\, 1\\leq i\\leq \\#\\alpha \\mbox{ such that } r,s\\in\\{j_{i}^{\\alpha}, j_{i}^{\\alpha}+1\\},\\\\\n\t-\\frac{1}{2} & \\mbox{ if } r\\neq s \\mbox{ and } \\exists\\, 1\\leq i\\leq \\#\\alpha \\mbox{ such that } r,s\\in\\{j_{i}^{\\alpha}, j_{i}^{\\alpha}+1\\},\\\\\n\t0 & \\mbox{ otherwise.}\n\t\\end{array}\n\\right.\n\\]\nBy calling $e_j$ the boolean row vector such that $(e_j)_i=1\\Leftrightarrow i=j$, we define $A[a_1,\\ldots,a_k]=\\langle \\Sigma\\cup\\{\\#\\}, \\pi_0^{(k)}, \\{O_\\sigma^{(k)}\\}_{\\sigma\\in\\Sigma\\cup\\{\\#\\}}, F^{(k)} \\rangle$ as the \\textsc{MOn-1qfa}\\ where\n\\begin{itemize}\n\t\\item $\\pi_0^{(k)}=e_1\\in\\mathbb{C}^{1\\times(k+1)}$,\n\t\\item for $\\alpha\\in S$, the associated projectors of $O_\\alpha^{(k)}$ are $P^{(k)}_{\\nearrow}(\\alpha)$ and $P^{(k)}_{\\searrow}(\\alpha)$,\n\t\\item with each $O_\\sigma^{(k)}$ such that $\\sigma\\in\\Sigma\\setminus S$, we associate the projector $I_{(k+1)\\times(k+1)}$,\n\t\\item the projector of the accepting result of $O_\\#^{(k)}$ is $(e_{k+1})^Te_{k+1}$, i.e. the $(k+1)\\times(k+1)$ boolean matrix having a 1 only in the bottom right entry.\n\\end{itemize}\nA careful analysis of the behavior of $A[a_1, \\ldots, a_k]$ leads to the following\n\\begin{theorem}\\label{th:PTtoMON}\nThe automaton $A[a_1, \\ldots, a_k]$ recognizes $L[a_1, \\ldots,\na_k]$ with cutpoint $\\lambda = \\frac{1}{2^{2k+1}}$ isolated by \n$\\delta = \\frac{1}{2^{2(k+1)}}$.\n\\end{theorem}\nSince the class \\text{liId}\\textbf{PT}\\ is the boolean closure of languages of the form $L[a_1, \\ldots, a_k]$, and $\\textbf{LMO}(\\Sigma)$ is a boolean algebra, Theorem \\ref{th:PTtoMON} implies that all literally idempotent piecewise testable languages can be recognized by \\textsc{MOn-1qfa}s.\nThe observations made up to this point imply our main result:\n\\begin{theorem}\\label{CharacterizationResult}\n$\n\\textbf{LMO}(\\Sigma) = V_{\\Sigma}(\\overline{\\mathbf{J}}) = \\text{liId}\\textbf{PT}(\\Sigma).\n$\n\\end{theorem}\nTheorem \\ref{CharacterizationResult} allows us to prove the existence of a polynomial time algorithm for deciding $\\textbf{LMO}(\\Sigma)$ membership:\n\\begin{theorem}\\label{th:LMOalgo}\nGiven a regular language $L\\in\\Sigma^*$, the problem of determining whether $L\\in\\textbf{LMO}(\\Sigma)$ is decidable in time $O((|Q|+|\\Sigma|)^2)$, where $|Q|$ is the size of the minimal deterministic automaton for $L$.\n\\end{theorem}\nThis algorithm first constructs the minimal deterministic automaton $A_L$ for $L$ in time $O(|Q|\\log(|Q|))$ as shown in \\cite{H71}. Then, in time $O(|Q|)$, it checks whether $L$ is literally idempotent by visiting all the vertices in the graph of $A_L$. Finally, it verifies whether $L$ is piecewise testable in time $O((|Q|+|\\Sigma|)^2)$ with the technique shown in \\cite{T01}. The fact that $\\textbf{LMO}(\\Sigma)=\\text{liId}\\textbf{PT}(\\Sigma)$ completes the proof.\n\n\\subsubsection*{Acknowledgements:} The authors wish to thank Alberto Bertoni for the stimulating discussions which lead to the results of this paper.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction\\label{introduction}}\n\nThe theory of cosmic inflation\n\\cite{starobinsky-1980,guth-1981,linde-1982,albrecht-1982,linde-1983}\nstands now on solid observational foundations, as several of its key predictions\nhave been confirmed by precise measurements of the\ncosmic microwave background radiation \\cite{Planck-inflation-2015}.\n\nAccording to the inflationary paradigm, the universe undergoes a period of\naccelerated expansion in the early stages of its evolution,\nwhich is driven by a scalar field --- the inflaton --- slowly rolling down its potential.\nSuch an early inflationary period not only solves the flatness, horizon, homogeneity,\nisotropy, and primordial monopole problems,\nbut also provides the seeds for the formation of the observed large-scale structures\nof the universe.\n\nA period of accelerated expansion is not exclusive to the early stages of\nevolution of the universe. In fact, cosmological observations have shown, not\nwithout surprise, that accelerated expansion is also taking place at the\npresent time \\cite{riess-1998,perlmutter-1999}, implying the existence of an\nunknown form of energy\n--- dubbed as dark energy --- which accounts for a substantial part of the total\nenergy density of the universe \\cite{Planck-parameters-2015}.\nWithin the $\\Lambda$CDM concordance model, this dark energy is assumed to be\na cosmological constant.\nHowever, this simple explanation raises problems of its own \\cite{weinberg-1989},\na circumstance that led to the hypothesis that dark energy could be identified\nwith a scalar field \\cite{caldwell-1998}, as in the inflationary paradigm.\n\nIn addition to dark energy, the concordance cosmological model also includes\ncold dark matter, which accounts for about one quarter of the total energy density\nof the universe \\cite{Planck-parameters-2015}.\nAlthough the existence of dark matter has been inferred by its gravitational\neffects on a multiplicity of astrophysical and cosmological phenomena,\nit has so far eluded a direct detection and, after decades of intense experimental\nefforts, its physical nature remains a mystery \\cite{bertone-2018}.\nSuch circumstances led to the consideration of a wider range of dark-matter candidates,\nincluding the possibility that a scalar field, similar to those appearing in the models\nof inflation and dark energy, could play the role of dark matter.\n\nSince inflation, dark energy, and dark matter can all be identified with\nscalar fields, it is natural to try to unify these seemingly disparate phenomena\nunder the same theoretical roof using these fields.\n\nSuch a unified description was proposed in Refs.~\\cite{liddle-2006,liddle-2008}.\nThere, inflation was assumed to be of the usual (cold) type,\nfollowed by a post-inflationary reheating period,\nin which the decay of the inflaton field was required to be incomplete,\nleaving a remnant that behaved like cold dark matter.\nIn order to reduce the energy density of the remnant to the level required by\ncosmological observations, two possibilities were considered:\nmodification of the decay rate during the reheating process \\cite{liddle-2006} or\nintroduction of an additional period of thermal inflation,\ndriven by a separate field, at lower energy densities \\cite{liddle-2008}.\nIn what concerns dark energy, this scenario assumed a non-zero vacuum energy\nfor the inflaton\/dark-matter field, motivated by a combination of the string\nlandscape picture and the anthropic principle.\n\nA new scenario for a triple unification, in which dark energy is described not\nby a cosmological constant, but rather by a dynamical scalar field, was soon\nafterwards proposed \\cite{henriques-2009}. Within a two-scalar-field\ncosmological model inspired by supergravity, one of the fields played the role\nof dark energy, inducing the present accelerated expansion of the universe,\nwhile the second field played the roles of both inflaton and dark matter.\nBecause inflation was assumed to be of the warm type \\cite{berera-1995}, no\ndistinctive post-inflationary reheating phase was required; soon after the\nsmooth transition to the radiation-dominated era, the energy transfer from the\ninflaton field to the radiation bath ceased and the former began to oscillate\naround the minimum of the potential, thus mimicking the behavior of a\ncold-dark-matter fluid. However, despite its success in unifying inflation,\ndark energy, and dark matter within a single framework, this two-scalar-field\ncosmological model was not entirely satisfactory, since it accounted for just\na fraction of the dark matter content of the universe.\n\nThe purpose of the present article is to provide a unified description\nof inflation, dark energy, and dark matter in a more general setting, namely,\nwithin a two-scalar-field cosmological model\ngiven by the action\\footnote{Throughout this article we will adopt the natural\nsystem of units and use the notation\n$\\kappa\\equiv \\sqrt{8\\pi G}= \\sqrt{8\\pi}\/m_\\texttt{P}$,\nwhere $G$ is the gravitational constant and\n$m_\\texttt{P} = 1.22 \\times 10^{19}\\, {\\rm GeV}$ is the Planck mass.}\n\\begin{align}\n S = {} & \\int d^4x \\sqrt{-g} \\bigg[\n  \\frac{R}{2\\kappa^2} - \\frac12 (\\nabla \\phi)^2\n    \\nonumber\n  \\\\\n  & - \\frac12 e^{-\\alpha\\kappa\\phi} (\\nabla \\xi)^2\n  - e^{-\\beta\\kappa\\phi} V(\\xi) \\bigg],\n \\label{action 2SF}\n\\end{align}\nwhere $g$ is the determinant of the metric $g_{\\mu\\nu}$, $R$ is the Ricci\nscalar, $\\phi$ and $\\xi$ are scalar fields, and $\\alpha$ and $\\beta$ are\nindependent dimensionless parameters\\footnote{The triple unifications of\ninflation, dark energy, and dark matter proposed in\nRefs.~\\cite{liddle-2006,liddle-2008} and Ref.~\\cite{henriques-2009} correspond\nto models given by action~(\\ref{action 2SF}) with, respectively,\n$\\alpha=\\beta=0$, $\\phi=0$, and $V(\\xi)=V_0+\\frac12 M^2 \\xi^2$, where $V_0$\nand $M$ are arbitrary constants, and $\\alpha=0$, $\\beta=-\\sqrt2$, and\n$V(\\xi)=A_1[1-2A_2 \\exp(-\\sqrt2\\kappa \\xi)+A_3 \\exp(-2\\sqrt2\\kappa \\xi)]$,\nwhere the constants $A_i$ are related to fundamental quantities of the\nSalam-Sezgin six-dimensional supergravity theory.}. Such an action, with a\nnon-standard kinetic term and an exponential potential, arises in a great\nvariety of gravity theories, such as the Jordan-Brans-Dicke theory,\nKaluza-Klein theories, $f(R)$-gravity, and string theories (see\nRefs.~\\cite{berkin-1991,starobinsky-2001} for a derivation of the above action\nin the context of these theories). More recently, it has been shown that this\naction also arises in the context of hybrid metric-Palatini theories of\ngravity \\cite{harko-2012,tamanini-2013}.\n\nFor an appropriate choice of the potential $V(\\xi)$, the two-scalar-field\ncosmological model given by action~(\\ref{action 2SF}) allows for a unified\ndescription of inflation, dark energy, and dark matter, in which the scalar\nfield $\\xi$ plays the roles of both inflaton and dark matter, while the scalar\nfield $\\phi$ plays the role of dark energy.\n\nThe simplest potential providing such a triple unification has the form\n\\begin{align}\n V(\\xi) & = V_a + \\frac12 m^2 \\xi^2,\n  \\label{potential xi}\n\\end{align}\nwhere $V_a$ and $m$ are constants, related, respectively, to the energy density\nof dark energy and to the $\\phi$-dependent mass of the scalar field\n$\\xi$, defined as\n\\begin{align}\n M_\\xi^2 (\\phi) & = m^2 e^{-\\beta\\kappa\\phi}.\n   \\label{mass xi}\n\\end{align}\nWhile emphasizing that a triple unification such as the one proposed in this\narticle could be achieved by any potential whose expansion around its minimum\nhas the form $A+B\\xi^2+\\dots$, for definiteness we will use the potential\ngiven by Eq.~(\\ref{potential xi}).\n\nIn what follows, let us briefly outline the key aspects of the triple unification\nproposed in this article.\n\nInflation is assumed to be of the warm type.\nEnergy is continuously transferred from the inflaton field $\\xi$\n(and also from the dark-energy field $\\phi$) to a radiation bath,\nthereby ensuring that the energy density of the latter is substantial\n--- albeit sub-dominant --- throughout the inflationary expansion\nand that a smooth transition to a radiation-dominated era takes place\nwithout the need for a distinctive post-inflationary reheating phase.\n\nThe dissipation coefficients, mediating the energy trans\\-fer from the scalar\nfields to the radiation bath, have a generic dependence on the temperature,\nnamely, $\\Gamma \\propto T^p$ ($p$ constant), and, immediately after the end of\nthe inflationary period, are exponentially suppressed, becoming negligible\nshortly afterwards.\n\nShortly after the end of the inflationary period, the inflaton $\\xi$ decouples from\nradiation and begins to oscillate rapidly around the minimum of its potential,\nthus behaving on average like a pressureless nonrelativistic fluid, i.e.,\nlike cold dark matter.\n\nDue to the non-standard kinetic term and the exponential factor in the potential,\nthe energy density of cold dark matter depends explicitly on the scalar field $\\phi$,\nimplying that, in general, this quantity does not evolve exactly\nas ordinary baryonic matter.\n\nAfter a radiation-dominated era, encompassing the primordial nucleosynthesis period,\ncold dark matter, together with ordinary baryonic matter,\ndominates the dynamics of the universe, giving rise to a matter-dominated era,\nlong enough to allow for structure formation.\n\nAt recent times, the scalar field $\\phi$ finally emerges as the dominant component\nof the universe, giving rise to a second era of accelerated expansion,\nthus behaving like dark energy.\n\nThese key aspects of the proposed triple unification will be detailed\nin the body of the article.\n\nTo conclude this introductory section, let us point out that unified\ndescriptions of inflation, dark energy, and dark matter have been proposed in\nseveral other\ncontexts~\\cite{capozziello-2006,bose-2009,santiago-2011,odintsov-2019,lima-2019,ketov-2020,odintsov-2020}.\n\nThis article is organized as follows. The evolution equations for the\ntwo-scalar-field cosmological model are presented in the next section. For\nclarity, the cosmic evolution is divided into two stages, the first\ncorresponding to the inflationary period and the transition to the\nradiation-dominated era (Sect.~\\ref{1st stage}) and the second encompassing\nthe radiation-, matter-, and dark-energy-dominated eras (Sect.~\\ref{2nd\nstage}). The continuity of the different physical quantities at the transition\nbetween the first and second stages of evolution is analyzed in\nSect.~\\ref{transition}. Numerical solutions are presented in\nSect.~\\ref{numerical}, which is divided into three subsections. In the first,\nwe analyze the case $\\alpha=\\beta$, while the second is devoted to the case\n$\\alpha\\neq\\beta$. Dissipative effects during inflation are analyzed in the\nlast subsection. Finally, in Sect.~\\ref{conclusions}, we present our\nconclusions.\n\n\n\\section{Two-scalar-field cosmological model\\label{2SF model}}\n\nOur analysis of the cosmic evolution is divided into two stages: the first\ncorresponds to the inflationary period and the transition to the\nradiation-dominated era, while the second encompasses the radiation-, matter-,\nand dark-energy-dominated eras.\n\n\n\n\\subsection{First stage of evolution: the inflationary era\\label{1st stage}}\n\nWe assume inflation to be of the warm type (for reviews, see\nRefs.~\\cite{berera-2009,bastero-gil-2009}).\nIn this inflationary paradigm, a continuous transfer of energy from the\ninflaton field to radiation ensures that the energy density of the latter\nremains substantial --- albeit sub-dominant --- throughout the inflationary era.\nThis energy transfer also guarantees that the\ntransition to a radiation-dominated era takes place in a smooth manner.\nIt contrasts with the usual (cold) inflationary paradigm,\nin which radiation is severely diluted during inflation, a circumstance that\nmakes a distinctive post-inflationary reheating process necessary in order\nto recover the standard cosmic evolution.\n\nIn our model, radiation is described by a perfect fluid with an equation-of-state\nparameter $w_\\texttt{R}=p_\\texttt{R}\/\\rho_\\texttt{R}=1\/3$, where $p_\\texttt{R}$\nand $\\rho_\\texttt{R}$ are the pressure and the energy density of the fluid, respectively.\n\nThe energy density of radiation is sustained, during the inflationary period,\nby a continuous transfer of energy from the scalar fields $\\xi$ and $\\phi$,\nwhich is  accomplished by the introduction of dissipative terms with\ncoefficients $\\Gamma_\\xi$ and $\\Gamma_\\phi$ into the equations of motion. This\nenergy transfer prevents the radiation bath from being diluted, keeping the\nexpanding universe ``warm''.\n\nTo implement warm inflation, it would suffice to have a continuous and\nsignificative transfer of energy from the inflaton field $\\xi$ to the\nradiation bath, in which case one could simply set the dissipation coefficient\n$\\Gamma_\\phi$ to zero and ignore any energy exchange between the dark-energy\nfield $\\phi$ and radiation. However, as it follows from action (\\ref{action\n2SF}), a direct transfer of energy between the two scalar fields also takes\nplace (for nonvanishing $\\alpha$ and\/or $\\beta$). Therefore, it seems natural\nto also allow for a direct energy transfer from the scalar field $\\phi$ to the\nradiation bath, mediated by a non-zero $\\Gamma_\\phi$, although we shall\nemphasize that this is not essential for the implementation of the\nwarm-inflation scenario.\n\nInflation comes to an end when, due to an increase of dissipative effects,\nthe energy density of the radiation bath smoothly takes over and begins to\ndominate the evolution of the universe.\nAt this point, the dissipation coefficients $\\Gamma_\\xi$ and $\\Gamma_\\phi$ are\nexponentially suppressed and, consequently, the radiation bath decouples from\nthe scalar fields $\\xi$ and $\\phi$ and begins to evolve in the usual manner.\n\nLet us now present the equations governing the cosmic evolution during the\ninflationary period and the transition to a radiation-dominated era.\n\nWe assume a flat Friedman-Robertson-Walker universe\\footnote{Since the\ncurrent cosmological measurements constrain the present-time value of the\ncurvature density parameter $\\Omega_k$ to be very small\n\\cite{Planck-parameters-2015}, a spatially flat universe can be assumed without\nmuch loss of generality.}, given by the metric\n\\begin{align}\nds^2 = -dt^2 + a^2(t) d\\Sigma^2,\n  \\label{FRW metric}\n\\end{align}\nwhere $a(t)$ is the scale factor and $d\\Sigma^2$ is the metric of the\nthree-dimensional Euclidean space.\n\nThe equations of motion for the scalar fields $\\xi(t)$ and $\\phi(t)$ and for\nthe energy density of radiation $\\rho_\\texttt{R}(t)$ are then\n\\begin{align}\n & \\ddot{\\xi} + 3 \\frac{\\dot{a}}{a} \\dot{\\xi} - \\alpha\\kappa \\dot{\\phi}\\dot{\\xi}\n + \\frac{\\partial V}{\\partial \\xi} e^{(\\alpha-\\beta)\\kappa\\phi}\n = - \\Gamma_\\xi \\dot{\\xi} e^{\\alpha\\kappa\\phi}, \\label{ddot-xi-1}\n  \\\\\n & \\ddot{\\phi} + 3 \\frac{\\dot{a}}{a} \\dot{\\phi} + \\frac{\\alpha\\kappa}{2}\n \\dot{\\xi}^2 e^{-\\alpha\\kappa\\phi}\n -\\beta \\kappa V e^{-\\beta \\kappa \\phi}\n = - \\Gamma_\\phi \\dot{\\phi}, \\label{ddot-phi-1}\n  \\\\\n & \\dot{\\rho_\\texttt{R}} + 4 \\frac{\\dot{a}}{a} \\rho_\\texttt{R} = \\Gamma_\\xi\n \\dot{\\xi}^2 + \\Gamma_\\phi \\dot{\\phi}^2, \\label{dot-rho-1}\n\\end{align}\nwhile the Einstein equations for the scale factor $a(t)$ are given by\n\\begin{align}\n& \\bigg( \\frac{\\dot{a}}{a} \\bigg)^2 = \\frac{\\kappa^2}{3} \\bigg(\n \\frac{\\dot{\\phi}^2}{2} + \\frac{\\dot{\\xi}^2}{2} e^{-\\alpha\\kappa\\phi}\n + V e^{-\\beta\\kappa\\phi} + \\rho_\\texttt{R} \\bigg), \\label{dot-a-1}\n   \\\\\n& \\hspace{2.7mm}\\frac{\\ddot{a}}{a} =  - \\frac{\\kappa^2}{3} \\bigg( \\dot{\\phi}^2 +\n \\dot{\\xi}^2 e^{-\\alpha\\kappa\\phi} - V e^{-\\beta\\kappa\\phi} + \\rho_\\texttt{R} \\bigg),\n \\label{ddot-a-1}\n\\end{align}\nwhere an overdot denotes a derivative with respect to time $t$ and the\npotential $V$ is given by Eq.~(\\ref{potential xi}).\n\nNote that the above evolution equations differ from the usual ones in warm\ninflationary models in that they contain extra terms arising due to the presence,\nin action (\\ref{action 2SF}), of a non-standard kinetic term for the field $\\xi$.\n\nInstead of the comoving time $t$, let us use a new variable $u$, related to the\nredshift $z$,\n\\begin{align}\n  u = -\\ln \\left( \\frac{a_0}{a} \\right) = - \\ln (1+z),\n  \\label{variable u}\n\\end{align}\nwhere $a_0\\equiv a(u_0)$ denotes the value of the scale factor at the\npresent time $u_0=0$.\n\nWith this change of variables, the above equations for $\\xi$, $\\phi$, and\n$\\rho_\\texttt{R}$ become\n\\begin{align}\n  &  \\hspace{-0.5mm}  \\xi_{uu} = - \\bigg\\{\n     \\bigg[ \\frac{\\ddot{a}}{a} + 2 \\bigg( \\frac{\\dot{a}}{a} \\bigg)^2\n           +\\frac{\\dot{a}}{a} \\Gamma_\\xi e^{\\alpha\\kappa \\phi} \\bigg]\\xi_u\n    \\nonumber\n  \\\\\n  & \\hspace{8.5mm}\n    - \\alpha\\kappa  \\bigg( \\frac{\\dot{a}}{a} \\bigg)^2 \\phi_u \\xi_u\n    + m^2 \\xi e^{(\\alpha-\\beta)\\kappa \\phi}\n  \\bigg\\} \\bigg( \\frac{\\dot{a}}{a} \\bigg)^{-2},\n      \\label{Eq xi E1}\n  \\\\\n  & \\hspace{-1.0mm}\\phi_{uu} =  - \\bigg\\{\n     \\bigg[ \\frac{\\ddot{a}}{a} + 2 \\bigg( \\frac{\\dot{a}}{a} \\bigg)^2\n           +\\frac{\\dot{a}}{a} \\Gamma_\\phi\n     \\bigg]\\phi_u\n  + \\frac{\\alpha\\kappa}{2} \\bigg( \\frac{\\dot{a}}{a} \\bigg)^2 \\xi_u^2\n  e^{-\\alpha\\kappa\\phi} \\nonumber\n  \\\\\n  & \\hspace{8.5mm}\n   -\\beta\\kappa \\left( V_a +\\frac12 m^2 \\xi^2 \\right)\n   e^{-\\beta\\kappa\\phi}\n  \\bigg\\} \\bigg( \\frac{\\dot{a}}{a} \\bigg)^{-2},\n      \\label{Eq phi E1}\n  \\\\\n  & \\hspace{0mm} \\rho_{\\texttt{R}u} = - 4 \\rho_\\texttt{R}\n  + \\frac{\\dot{a}}{a} \\left( \\Gamma_\\xi \\xi_u^2 + \\Gamma_\\phi \\phi_u^2 \\right),\n      \\label{Eq rho E1}\n\\end{align}\nwhere the subscript $u$ denotes a derivative with respect to $u$; $\\dot{a}\/a$\nand $\\ddot{a}\/a$ are functions of $u$, $\\xi$, $\\xi_u$, $\\phi$, and $\\phi_u$,\ngiven by\n\\begin{align}\n \\left( \\frac{\\dot{a}}{a} \\right)^2 = 2\\kappa^2\n \\frac{ \\left( V_a + \\frac12 m^2 \\xi^2 \\right) e^{-\\beta\\kappa\\phi} +\n \\rho_\\texttt{R} }{6 - \\kappa^2 \\phi_u^2\n - \\kappa^2 \\xi_u^2 e^{-\\alpha\\kappa\\phi}}\n \\label{Eq friedman E1}\n\\end{align}\nand\n\\begin{align}\n& \\frac{\\ddot{a}}{a} = \\frac{\\kappa^2}{3}\n \\Bigg\\{ 2\\kappa^2 \\left[ \\left( V_a+\\frac12 m^2 \\xi^2 \\right)\n e^{-\\beta\\kappa\\phi} + \\rho_\\texttt{R} \\right] \\nonumber\n \\\\\n& \\hspace{6.4mm} \\times \\frac{\\phi_u^2 + \\xi_u^2 e^{-\\alpha\\kappa\\phi}}\n {\\kappa^2 \\phi_u^2 + \\kappa^2 \\xi_u^2 e^{-\\alpha\\kappa\\phi} - 6}\n \\nonumber\n \\\\\n& \\hspace{6.4mm} + \\left( V_a + \\frac12 m^2 \\xi^2 \\right)\n  e^{-\\beta\\kappa\\phi} - \\rho_\\texttt{R}\n    \\Bigg\\}.\n   \\label{Eq Dotdota E1}\n\\end{align}\n\nIn order to solve the above system of equations, one has to specify the\ndissipation coefficients $\\Gamma_\\xi$ and $\\Gamma_\\phi$.\n\nOver the years, a variety of forms has been adopted for these coefficients,\nfrom the simplest, based on general phenomenological considerations, to the\nmore elaborate ones, derived from microscopic quantum field theory. In\ngeneral, the dissipation coefficients $\\Gamma$ appearing in the literature are\nfunctions of the temperature $T$ and\/or the inflaton field $\\xi$, as, for\ninstance, $\\Gamma \\propto T^3\/\\xi^2$\n\\cite{bastero-gil-2013,lima-2019}, $\\Gamma \\propto T$\n\\cite{bastero-gil-2016,rosa-2019,rosa-2019b}, or $\\Gamma \\propto T^{-1}$\n\\cite{bastero-gil-2019}.\n\nIn this article, we will not be concerned with the specific microscopic models\nused to derive the dissipation coefficients. We will adopt instead a\nmodel-independent approach, assuming that, during inflation, these\ncoefficients have a generic dependence on the temperature of the radiation\nbath, namely, $\\Gamma \\propto T^p$. We also assume that, immediately after the\nend of the inflationary period, the dissipation coefficients are exponentially\nsuppressed, becoming negligible soon afterwards. In short, we assume\nthe dissipation coefficients $\\Gamma_\\xi$ and $\\Gamma_\\phi$ to be given by\n\\begin{align}\n \\Gamma_{\\xi,\\phi} = f_{\\xi,\\phi} \\times \\left\\{\n    \\begin{aligned}\n     & T^p,\n     & T\\geq T_\\texttt{E},\n    \\\\\n     & T^p  \\exp \\left[ 1- \\left( \\frac{T_\\texttt{E}}{T} \\right)^q \\right],\n     & T\\leq T_\\texttt{E},\n    \\end{aligned}\n \\right.\n \\label{gammas}\n\\end{align}\nwhere $T_\\texttt{E}$ is the temperature of the radiation bath at the end of\nthe inflationary period, $f_\\xi$ and $f_\\phi$ are positive constants with\ndimension $(\\textrm{mass})^{1-p}$ encoding the details of the microscopic\nmodels used to derive the dissipation coefficients, and $q>0$ and $p$ are\nparameters determining the temperature dependence of these coefficients.\n\nThe suppression of the dissipation coefficients immediately after the\ninflationary period is of paramount importance in our unification proposal in\norder to guarantee that the inflaton field $\\xi$ survives and that it has\nenough energy to mimic the behavior of cold dark matter in a way consistent\nwith cosmological observations (see Sect.~\\ref{2nd stage} below for details).\nA microscopic model in which this suppression is achieved naturally has been\nproposed recently \\cite{rosa-2019}, in the context of the warm little inflaton\nscenario \\cite{bastero-gil-2016}. More specifically, the dissipation\ncoefficient mediating the energy transfer from the inflaton field to\nradiation, which initially is proportional to $T$, becomes exponentially\nsuppressed when the temperature drops below a certain threshold value (roughly\ncoinciding with the end of the inflationary period), thus leading to a stable\ninflaton remnant. Such suppression of the dissipative effects below a\nthreshold temperature is also present in another warm-inflation model\n\\cite{bastero-gil-2019} in which the dissipation coefficient during inflation\nis proportional to $T^{-1}$.\n\nAs will be shown in Sect.~\\ref{num-dissipative},\nfor dissipation coefficients proportional to $T^p$, with $p>2$,\nsuppression of the dissipation coefficients below a threshold\ntemperature occurs naturally, as a result of the background dynamics,\nmaking it unnecessary, for such values of p, to introduce explicitly the\nexponential factor in Eq.~(\\ref{gammas}).\n\nFor our base scenario we will choose $p=1$,\ncorresponding to dissipation coefficients linearly dependent on the temperature,\nand $q=2$ (see Sect.~\\ref{num-base}); other values of the parameters $p$ and $q$\nwill be considered in Sect.~\\ref{num-dissipative}.\n\nFinally, let us recall that the temperature $T$ of the radiation bath is\nrelated to its energy density by\n\\begin{align}\n \\rho_\\texttt{R} = \\frac{\\pi^2}{30}g_* T^4,\n   \\label{temp-rho}\n\\end{align}\nwhere $g_*$ denotes the effective number of relativistic degrees of freedom at\ntemperature $T$. Assuming the standard model of particle physics and taking\ninto account that, at the relevant temperatures, all the degrees of freedom of\nthis model are relativistic and in thermal equilibrium, $g_*$ takes the value\n$106.75$.\n\nSolving Eqs.~(\\ref{Eq xi E1})--(\\ref{Eq Dotdota E1}) allows us to determine\nthe density parameters for radiation and for the scalar fields $\\xi$ and\n$\\phi$,\n\\begin{align}\n& \\Omega_\\texttt{R} = \\frac{\\rho_\\texttt{R}}{\\rho_c}=\n  \\frac{\\kappa^2}{3}\\rho_\\texttt{R} \\left( \\frac{\\dot{a}}{a} \\right)^{-2},\n\\\\\n& \\Omega_\\xi = \\frac{\\rho_\\xi}{\\rho_c}= \\frac{\\kappa^2}{6} \\bigg[\n  \\xi_u^2 \\, e^{-\\alpha\\kappa\\phi}\n  + m^2 e^{-\\beta\\kappa\\phi} \\xi^2\n  \\left( \\frac{\\dot{a}}{a} \\right)^{-2} \\bigg],\n\\\\\n& \\Omega_\\phi = \\frac{\\rho_\\phi}{\\rho_c}= \\frac{\\kappa^2}{3} \\bigg[\n  \\frac{\\phi_u^2}{2} + V_a e^{-\\beta\\kappa\\phi}\n  \\left( \\frac{\\dot{a}}{a} \\right)^{-2} \\bigg], \\label{Omega-phi-E1}\n\\end{align}\nas well as the effective equation-of-state parameter,\n\\begin{align}\n w_{\\rm eff} = \\frac13 \\bigg( \\Omega_\\texttt{R}\n + 3 \\Omega_\\phi \\frac{p_\\phi}{\\rho_\\phi}\n + 3 \\Omega_\\xi \\frac{p_\\xi}{\\rho_\\xi} \\bigg), \\label{eos parameter E1}\n\\end{align}\nwhere $\\rho_c=(3\/\\kappa^2)(\\dot{a}\/a)^2$ is the critical density and the\nenergy density and pressure of the scalar fields $\\xi$ and $\\phi$ are given by,\nrespectively,\n\\begin{align}\n \\rho_\\xi = \\frac12 \\left( \\frac{\\dot{a}}{a} \\right)^2 \\xi_u^2\n e^{-\\alpha\\kappa\\phi}\n + \\frac12 m^2 e^{-\\beta\\kappa\\phi} \\xi^2,\n  \\label{rho xi E1}\n\\\\\n p_\\xi = \\frac12 \\left( \\frac{\\dot{a}}{a} \\right)^2 \\xi_u^2\n e^{-\\alpha\\kappa\\phi}\n - \\frac12 m^2 e^{-\\beta\\kappa\\phi} \\xi^2,\n    \\label{p xi E1}\n\\end{align}\nand\n\\begin{align}\n \\rho_\\phi = \\frac12 \\left( \\frac{\\dot{a}}{a} \\right)^2 \\phi_u^2\n + V_a e^{-\\beta\\kappa\\phi},\n  \\label{rho phi E1}\n \\\\\n  p_\\phi = \\frac12 \\left( \\frac{\\dot{a}}{a} \\right)^2 \\phi_u^2\n - V_a e^{-\\beta\\kappa\\phi}.\n  \\label{p phi E1}\n\\end{align}\n\nIn this article we will be focused on the background dynamics of the\ntwo-scalar-field cosmological model and its capability to reproduce the\nmain stages of evolution of the universe, leaving the analysis of the\nprimordial spectrum of density perturbations and its agreement with\ncosmological observations to future work.\nNevertheless, we would like to emphasize here that warm-inflation models\nhave been shown to be consistent with cosmic microwave background (CMB) data\nfor a large variety of potentials and dissipation coefficients.\nIn particular, predictions of the tensor-to-scalar ratio and the spectral tilt\nof the primordial spectrum were shown to agree with cosmological data for models\nwith a quartic potential and dissipation coefficients proportional to $T$ and $T^3$ \\cite{benetti-2017,arya-2018,bastero-gil-2018,bastero-gil-2018a,motaharfar-2019},\nand also in the case of a quadratic potential and dissipation\ncoefficient proportional to $T^{-1}$ \\cite{bastero-gil-2019}.\nThe former models favor mostly the weak dissipative regime, while the latter is\nconsistent with strong dissipation.\n\nWe now turn to the description of the second stage of evolution, which\nencompasses the radiation-, matter-, and dark-energy-dominated eras.\n\n\n\n\n\\subsection{Second stage of evolution: the radiation-, matter-, and\ndark-energy-dominated eras\\label{2nd stage}}\n\nAs mentioned above, at the end of the inflationary period, the dissipation\ncoefficients $\\Gamma_\\xi$ and $\\Gamma_\\phi$ are exponentially suppressed and,\nsoon afterwards, become negligible, allowing us to set them exactly to zero.\nThis marks the end of the first stage of evolution.\n\nDuring the second stage of evolution, in the absence of dissipation, radiation\ndecouples from the scalar fields $\\xi$ and $\\phi$ and Eq.~(\\ref{Eq rho E1})\nyields the solution\n\\begin{align}\n \\rho_\\texttt{R}=\\rho_{\\texttt{R}0}\\,e^{-4u},\n   \\label{Eq rho E2}\n\\end{align}\nwhere $\\rho_{\\texttt{R}0}\\equiv \\rho_{\\texttt{R}}(u_0)$ denotes the energy\ndensity of radiation at the present time $u_0=0$.\n\nFor its part, the scalar field $\\xi$ begins to oscillate rapidly around its\nminimum, behaving like a nonrelativistic dark-matter fluid with equation of state\n$\\langle p_\\xi \\rangle=0$ \\cite{turner-1983},\nwhere the brackets $\\langle ... \\rangle$\ndenote the average over an oscillation\\footnote{These oscillations take place\nif the mass of the scalar field $\\xi$, given by Eq.~(\\ref{mass xi}),\nis much bigger than the Hubble parameter, $M_\\xi \\gg H \\equiv \\dot{a}\/a$,\na condition that can be easily satisfied by choosing a large\nenough value for the constant $m$.}.\n\nLet us derive an expression for the energy density of the dark-matter fluid\nin terms of $u$ and $\\phi(u)$.\nTo that end, we multiply Eq.~(\\ref{Eq xi E1}) by $\\xi_u$ and use the\ndefinition of $\\rho_\\xi$ given by Eq.~(\\ref{rho xi E1}) to obtain\n\\begin{align}\n& \\rho_{\\xi u}\n +3 \\left( \\frac{\\dot{a}}{a} \\right)^2 \\xi_u^2 e^{-\\alpha\\kappa\\phi}\n -\\frac{\\alpha\\kappa}{2} \\left( \\frac{\\dot{a}}{a} \\right)^2\n   \\xi_u^2 \\phi_u e^{-\\alpha\\kappa\\phi} \\nonumber\n \\\\\n& \\hspace{15mm} + \\frac{\\beta\\kappa}{2} m^2 \\xi^2 \\phi_u\n  e^{-\\beta\\kappa\\phi}\n = 0.\n\\end{align}\n\nAveraging over an oscillation period and taking into account that\n$\\big<p_\\xi\\big>=0$ implies $\\big< \\xi^2 \\big>\n=\\rho_\\xi m^{-2} e^{\\beta\\kappa\\phi}$ and\n$\\big< \\xi_u^2 \\big> = \\rho_\\xi (\\dot{a}\/a)^{-2} e^{\\alpha\\kappa\\phi}$,\nthe above equation can be written as\n\\begin{align}\n \\rho_{\\xi u} + 3 \\rho_\\xi\n - \\frac{(\\alpha-\\beta)\\kappa}{2} \\rho_\\xi \\phi_u = 0,\n\\end{align}\nyielding the solution\n\\begin{align}\n \\rho_\\xi = C e^{-3u} e^{\\frac{(\\alpha-\\beta)\\kappa}{2}\\phi},\n   \\label{rho dark matter}\n\\end{align}\nwhere $C$ is a constant whose value is fixed by current cosmological\nmeasurements [see Eq.~(\\ref{condition 2}) below].\n\nAs expected, the energy density of dark matter is proportional to $e^{-3u}$\n(or, in terms of the scale factor, proportional to $a^{-3}$), due to the fact that\nthe potential $V(\\xi)$ was chosen to be quadratic.\nBut it also depends directly on the scalar field $\\phi$,\nthrough an exponential factor, as a consequence of both the non-standard\nkinetic term of the scalar field $\\xi$ and the exponential potential\n[see action~(\\ref{action 2SF})].\nAs will be seen in Sect.~\\ref{numerical}, such dependence of $\\rho_\\xi$ on the\ndark-energy field $\\phi$ has implications on the cosmic evolution, leading\nto a non-simultaneous peaking of the energy densities\nof dark matter and ordinary baryonic matter.\n\nNow, taking into account the above expressions for the energy densities of\nradiation and dark matter,\nthe evolution Eqs.~(\\ref{Eq xi E1})--(\\ref{Eq Dotdota E1})\ncan be considerably simplified, yielding\n\\begin{align}\n \\phi_{uu} = {}& - \\bigg\\{\n  \\bigg[ \\frac{\\ddot{a}}{a} + 2 \\bigg( \\frac{\\dot{a}}{a} \\bigg)^2 \\bigg] \\phi_u\n  -\\beta\\kappa V_a e^{-\\beta\\kappa\\phi} \\nonumber\n\\\\\n  & +\\frac{(\\alpha-\\beta)\\kappa C}{2}\n  e^{\\frac{(\\alpha-\\beta)\\kappa}{2}\\phi} e^{-3u} \\bigg\\} \\left(\n\\frac{\\dot{a}}{a} \\right)^{-2}, \\label{Eq phi E2}\n\\end{align}\nwith\n\\begin{align}\n \\left( \\frac{\\dot{a}}{a} \\right)^2= {}&\n 2\\kappa^2 \\bigg[ V_a e^{-\\beta\\kappa\\phi}\n + \\left( \\rho_{\\texttt{BM}0}\n + C e^{\\frac{(\\alpha-\\beta)\\kappa}{2}\\phi} \\right) e^{-3u} \\nonumber\n\\\\\n & + \\rho_{\\texttt{R}0}e^{-4u} \\bigg]\n \\left( 6-\\kappa^2 \\phi_u^2 \\right)^{-1},\n\\label{Eq friedman E2}\n\\end{align}\nand\n\\begin{align}\n \\frac{\\ddot{a}}{a} = {}& \\frac{\\kappa^2}{6} \\bigg\\{\n 4\\kappa^2 \\bigg[ V_a e^{-\\beta\\kappa\\phi}\n     + \\left( \\rho_{\\texttt{BM}0}\n     + C e^{\\frac{(\\alpha-\\beta)\\kappa}{2}\\phi} \\right) e^{-3u} \\nonumber\n\\\\\n & + \\rho_{\\texttt{R}0} e^{-4u} \\bigg]\n \\phi_{u}^2 \\left( \\kappa^2 \\phi_u^2 - 6 \\right)^{-1}\n + 2 V_a e^{-\\beta\\kappa\\phi} \\nonumber\n\\\\\n & - \\left( \\rho_{\\texttt{BM}0}\n   + C e^{\\frac{(\\alpha-\\beta)\\kappa}{2}\\phi} \\right) e^{-3u}\n   - 2\\rho_{\\texttt{R}0} e^{-4u} \\bigg\\}.\n \\label{Eq Dotdota E2}\n\\end{align}\n\nIn the above equations, we have introduced ordinary baryonic matter, described\nas a perfect fluid with pressure $p_\\texttt{BM}=0$ and energy density\n\\begin{align}\n \\rho_\\texttt{BM} = \\rho_{\\texttt{BM}0}\\,e^{-3u},\n  \\label{rho baryonic matter}\n\\end{align}\nwhere, as usual, the subscript $0$ indicates present-time values.\n\nAgreement with current cosmological measurements \\cite{Planck-parameters-2015}\nrequires $\\rho_{\\texttt{R}0}=9.02\\times10^{-128}\\, m_\\texttt{P}^4$ and\n$\\rho_{\\texttt{BM}0}=8.19\\times10^{-125}\\, m_\\texttt{P}^4$, as well as\n\\begin{align}\n & \\frac12 \\bigg[ \\left( \\frac{\\dot{a}}{a} \\right)^2 \\phi_{u}^2 \\bigg]_{u=u_0}\n + V_a e^{-\\beta\\kappa\\phi_0} = \\rho_{\\texttt{DE}0},\n\\label{condition 1}\n\\\\\n& C e^{\\frac{(\\alpha-\\beta)\\kappa}{2}\\phi_0} = \\rho_{\\texttt{DM}0},\n\\label{condition 2}\n\\end{align}\nwhere the present-time energy densities of dark energy and dark matter are\n$\\rho_{\\texttt{DE}0}=1.13\\times10^{-123}\\, m_\\texttt{P}^4$ and\n$\\rho_{\\texttt{DM}0} = 4.25\\times10^{-124}\\, m_\\texttt{P}^4$,\nrespectively\\footnote{Note that these values for $\\rho_{\\texttt{R}0}$,\n$\\rho_{\\texttt{BM}0}$, $\\rho_{\\texttt{DM}0}$, and $\\rho_{\\texttt{DE}0}$\ncorrespond to a Hubble constant $H_0\\equiv(\\dot{a}\/a)_0= 1.17 \\times 10^{-61}\n\\, m_\\texttt{P}$ or, in more familiar units, $H_0=67\\,{\\rm km}\\,{\\rm\ns}^{-1}\\,{\\rm Mpc}^{-1}$.}.\n\nAs will be shown in Sect.~\\ref{numerical}, for a choice of $V_a$ and $C$\nsatisfying the above conditions, Eqs.~(\\ref{Eq phi E2})--(\\ref{Eq Dotdota E2})\ndescribe a radiation-dominated era, encompassing the primordial\nnucleosynthesis period, followed by an era dominated by the scalar field $\\xi$\n(dark matter) and ordinary baryonic matter, lasting long enough for structure\nformation to occur, and, finally, a dark-energy-dominated era, induced by the\nscalar field $\\phi$, during which the universe undergoes accelerated\nexpansion. The requirement that the transition from the radiation- to the\nmatter-dominated era does not occur too early in the cosmic history and,\nconsequently, does not conflict with primordial nucleosynthesis, as well as\nthe requirement that the expansion of the universe is accelerating at the\npresent time, imposes constraints on the parameters $\\alpha$ and $\\beta$,\nnamely, $|\\alpha-\\beta| \\lesssim 1$ and $|\\beta| \\lesssim 3\/2$.\n\nDuring the second stage of evolution, the density parameter for the scalar\nfield $\\phi$ is given by Eq.~(\\ref{Omega-phi-E1}), while the density\nparameters for radiation, baryonic matter, and the scalar field $\\xi$, are\ngiven by, respectively,\n\\begin{align}\n & \\Omega_\\texttt{R} = \\frac{\\rho_\\texttt{R}}{\\rho_c} =\n \\frac{\\kappa^2}{3}\\rho_{\\texttt{R}0} e^{-4u}\n \\left( \\frac{\\dot{a}}{a} \\right)^{-2},\n\\\\\n & \\Omega_\\texttt{BM} = \\frac{\\rho_\\texttt{BM}}{\\rho_c} =\n \\frac{\\kappa^2}{3}\\rho_{\\texttt{BM}0} e^{-3u}\n \\left( \\frac{\\dot{a}}{a} \\right)^{-2},\n\\\\\n & \\Omega_\\xi = \\frac{\\rho_\\xi}{\\rho_c} =\n \\frac{\\kappa^2}{3} C e^{\\frac{(\\alpha-\\beta)\\kappa}{2}\\phi} e^{-3u}\n \\left( \\frac{\\dot{a}}{a} \\right)^{-2}.\n\\end{align}\n\nThe effective equation-of-state parameter, during this stage of evolution, is\n\\begin{align}\nw_{\\rm eff} = \\frac13 \\bigg[ 1 - \\Omega_\\texttt{BM} - \\Omega_\\xi - \\Omega_\\phi\n\\bigg( 1- 3 \\frac{p_\\phi}{\\rho_\\phi} \\bigg) \\bigg], \\label{eos parameter E2}\n\\end{align}\nwhere $\\rho_\\phi$ and $p_\\phi$ are given by Eqs.~(\\ref{rho phi E1}) and\n(\\ref{p phi E1}), respectively.\n\nA detailed analysis of this stage of evolution for the case $\\alpha=2\/\\sqrt6$\nand arbitrary $\\beta$ can be found in Ref.~\\cite{sa-2020},\nwhere a unified description of dark matter and dark energy was proposed\nwithin the generalized hybrid metric-Palatini theory of gravity.\n\n\n\\subsection{Transition between the first and second stages of evolution\n\\label{transition}}\n\nAs we have just seen, the first stage of evolution, corresponding to the\ninflationary era, is described by Eqs.~(\\ref{Eq xi E1})--(\\ref{Eq Dotdota\nE1}), while the second stage of evolution, encompassing the radiation-, matter-,\nand dark-energy-dominated eras, is described by Eqs.~(\\ref{Eq phi E2})--(\\ref{Eq\nDotdota E2}).\n\nThe transition from the first to the second stage of evolution occurs shortly\nafter the end of inflation, at the beginning of the radiation-dominated era,\nwhen the dissipation coefficients $\\Gamma_\\xi$ and $\\Gamma_\\phi$, given by\nEq.~(\\ref{gammas}), are exponentially suppressed and become negligible. The\nmoment at which the dissipation coefficients are set exactly to zero marks the\nend of the first stage of evolution and the beginning of the second. In what\nfollows, this transition moment will be denoted by $u=u_*$.\n\nAt the transition between the first and second stages of evolution the\ndifferent physical quantities should be continuous.\n\nFor the scalar field $\\phi$ this is achieved by\nsimply requiring the initial value of the second stage to be equal to the\nfinal value of the first stage.\n\nFor the energy density of the scalar field $\\xi$, continuity at the\ntransition requires\n\\begin{align}\n \\rho_\\xi(u_*) =\n C e^{-3u_*}\n e^{\\frac{(\\alpha-\\beta)\\kappa}{2}\\phi_*},\n      \\label{continuity rho-xi}\n\\end{align}\nwhere $\\phi_*\\equiv\\phi(u_*)$ and $\\rho_\\xi(u_*)$ denotes the energy density\nof the $\\xi$ field during the first stage of evolution, given by Eq.~(\\ref{rho\nxi E1}), evaluated at $u=u_*$. Since the constant $C$ is fixed by\nEq.~(\\ref{condition 2}), satisfying the above continuity condition amounts to\nfix the value of $u_*$ or, equivalently, the duration of the first stage of\nevolution,\n\\begin{align}\n \\Delta^{[\\texttt{I}]}u \\equiv {} & u_*-u_i \\nonumber\n \\\\\n = {} & - \\frac13 \\ln \\frac{\\rho_\\xi(u_*)}{\\rho_{\\texttt{DM}0}}\n -\\frac{(\\alpha-\\beta)\\kappa}{6}[\\phi_0-\\phi_*] - u_i,\n  \\label{duration I}\n\\end{align}\nwhere $u_i$ denotes the value of $u$ at the beginning of the first stage of\nevolution.\n\nFinally, for the energy density of radiation, continuity at the transition\nrequires that\n\\begin{align}\n \\rho_\\texttt{R}(u_*) = \\rho_{\\texttt{R}0} e^{-4u_*},\n   \\label{continuity rho-R}\n\\end{align}\nwhere $\\rho_\\texttt{R} (u_*)$ denotes the energy density of radiation during\nthe first stage of evolution [i.e., the solution of Eq.~(\\ref{Eq rho E1})]\nevaluated at $u=u_*$. Because $\\rho_{\\texttt{R}0}$ is fixed by current\ncosmological measurements (see the discussion in Sect.~\\ref{2nd stage}), the\nabove continuity condition leads, in general, to a value of $u_*$ different\nfrom the one determined from Eq.~(\\ref{continuity rho-xi}). To avoid this and,\nat the same time, to maintain adherence to the convention $u_0=0$, the value\nof the variable $u$ at the beginning of the first stage of evolution, $u_i$,\nhas to be shifted by an appropriate amount, i.e., by an amount ensuring that\n$u_*$, determined from Eq.~(\\ref{continuity rho-xi}), also satisfies\nEq.~(\\ref{continuity rho-R}). This procedure fixes the duration of the second\nstage of evolution to be\n\\begin{equation}\n \\Delta^{[\\texttt{II}]}u \\equiv u_0-u_*\n = \\frac14 \\ln \\frac{\\rho_\\texttt{R}(u_*)}{\\rho_{\\texttt{R}0}}.\n  \\label{duration II}\n\\end{equation}\n\nNow, we can proceed to the numerical analysis of the equations of our\ntwo-scalar-field cosmological model.\n\n\n\\section{Numerical solutions\\label{numerical}}\n\nLet us solve numerically Eqs.~(\\ref{Eq xi E1})--(\\ref{Eq Dotdota E1}) and\nEqs.~(\\ref{Eq phi E2})--(\\ref{Eq Dotdota E2}), corresponding to the first and\nsecond stages of evolution, respectively, and present a unified description of\ninflation, dark energy, and dark matter within the two-scalar-field\ncosmological model given by action~(\\ref{action 2SF}).\n\nFor appropriate choices of the initial values for the variables $\\xi$,\n$\\xi_u$, $\\phi$, $\\phi_u$, and $\\rho_\\texttt{R}$ and the values of the\nconstants $\\alpha$, $\\beta$, $V_a$, $m$, $C$, $f_\\xi$, $f_\\phi$, $p$, and $q$ our\nmodel allows for a cosmic evolution consistent with current cosmological\nobservations.\n\nNote that, because of the symmetries of action~(\\ref{action 2SF}), we can assume\n$\\alpha\\geq0$ without loss of generality. The solutions corresponding to\nnegative values of $\\alpha$ can be obtained from the solutions with positive values\nusing the transformation $\\alpha\\rightarrow-\\alpha$, $\\beta\\rightarrow-\\beta$,\nand $\\phi\\rightarrow-\\phi$.\n\nFor clarity of presentation, this section is divided into three subsections,\nwhere we analyze the case $\\alpha=\\beta$, the case $\\alpha\\neq\\beta$, and the\ndependence of the dissipative effects on the parameters $p$ and $q$.\n\n\n\\subsection{Case $\\alpha=\\beta$\\label{num-base}}\n\nWe first analyze the case $\\alpha=\\beta=1$, choosing representative values for\nthe initial conditions, namely $\\xi(u_i)=0.52\\, m_\\texttt{P}$,\n$\\phi(u_i)=10^{-3}\\, m_\\texttt{P}$, $\\xi_u(u_i)=10^{-2}\\, m_\\texttt{P}$,\n$\\phi_u(u_i)=10^{-5}\\, m_\\texttt{P}$, $\\rho_\\texttt{R}(u_i)=0.25\\times\n10^{-12} \\,m_\\texttt{P}^4$, and for the parameters\\footnote{The parameters\n$f_\\xi$ and $f_\\phi$, which determine the amplitude of the energy transfer\nbetween the scalar fields and the radiation bath, are, in general, different.\nHere, for simplicity, they are assumed to be equal.}, namely,\n$V_a=7.64\\times10^{-123} \\, m_\\texttt{P}^4$, $m=10^{-5} \\, m_\\texttt{P}$,\n$f_\\xi=f_\\phi=2$, $p=1$, $q=2$. We call this case the base scenario.\n\n\nInitially, the scalar field $\\xi$ (the inflaton) slowly rolls down its\nquadratic potential, leading to an accelerated expansion of the universe ($60$\n$e$-folds). The energy scale of inflation, defined as\n\\begin{align}\nE_{\\rm inf}=\\left[ e^{-\\beta\\kappa\\phi(u_i)} V[\\xi(u_i)] \\right]^{1\/4},\n\\end{align}\nis $2.3\\times10^{16} \\, {\\rm GeV}$. Because of the dependence on the scalar\nfield $\\phi$, the inflaton mass $M_\\xi$ decreases from $10^{-5} \\,\nm_\\texttt{P}$ to $4.3\\times 10^{-6} \\, m_\\texttt{P}$ during this stage of\ncosmic evolution. Dissipative effects guarantee that energy is continuously\ntransferred from both fields $\\xi$ and $\\phi$ to the radiation bath,\npreventing it from being diluted away by accelerated expansion. Throughout the\ninflationary period, the dissipation ratios, defined as\\footnote{Since we are\nconsidering $\\Gamma_\\phi=\\Gamma_\\xi$, in what follows, for\nsimplicity, both dissipation ratios $Q_\\phi$ and $Q_\\xi$ will\nbe denoted by $Q$.}\n\\begin{align}\n Q_{\\xi,\\phi} = \\frac{\\Gamma_{\\xi,\\phi}}{3H},\n   \\label{q}\n\\end{align}\nremain larger than unity (strong dissipative regime) and the temperature of\nthe radiation bath decreases only slightly (from $3.5\\times10^{15}\\, {\\rm\nGeV}$ to $7.9\\times10^{14}\\, {\\rm GeV}$). The evolution of the relevant\nquantities during the inflationary period is shown in\nFig.~\\ref{Fig_inflation}.\n\n\\begin{figure}[t]\n\\includegraphics[width=7.9cm]{Fig1x.pdf}\n\\caption{Evolution of the inflaton field $\\xi$, its mass $M_\\xi$, the\ndissipation ratio $Q$, the temperature $T$ of the radiation bath, the Hubble\nparameter $H$, and the ratio $\\rho_\\texttt{R}\/\\rho_\\xi$ during the\ninflationary period, which extends from $u\\approx-124.7$ to $u\\approx-64.7$\n($60$ $e$-folds of expansion). The energy scale of inflation is $E_{\\rm\ninf}\\approx 2.3\\times10^{16} \\, {\\rm GeV}$.}\n \\label{Fig_inflation}\n\\end{figure}\n\nAt a certain point of the evolution, radiation emerges as the dominant\ncomponent of the universe and the inflationary period comes to an end (see\nFig.~\\ref{Fig_rhoxi-rhoR-E1}). At that moment, when the temperature of the\nradiation bath is $T_\\texttt{E} \\approx 7.9 \\times 10^{14} \\, {\\rm GeV}$, the\ndissipation coefficients $\\Gamma_\\xi$ and $\\Gamma_\\phi$ begin to decrease\nexponentially, implying that soon afterwards the dissipation ratio $Q$ becomes\nnegligible (see Fig.~\\ref{Fig-Gamma}). At $u=u_*\\approx-63.7$, the dissipation\ncoefficients $\\Gamma_\\xi$ and $\\Gamma_\\phi$ are set exactly to zero and the\ndynamics of the cosmic evolution becomes governed by Eqs.~(\\ref{Eq phi\nE2})--(\\ref{Eq Dotdota E2}). This marks the beginning of the second stage of\nevolution.\n\n\\begin{figure}[t]\n\\includegraphics[width=7.9cm]{Fig2x.pdf}\n\\caption{Evolution of the energy densities of the scalar field $\\xi$ and of\nradiation. During the inflationary period, due to dissipative effects, the\nlatter remains almost constant. At $u\\approx-64.7$, radiation emerges as the\ndominant component of the universe and the inflationary period comes to an end\n(inset plot). During the radiation-dominated era, the scalar field $\\xi$\nbehaves like a pressureless nonrelativistic fluid (dark matter), becoming\ndominant, together with ordinary baryonic matter, at $u\\approx-8.6$.}\n \\label{Fig_rhoxi-rhoR-E1}\n\\end{figure}\n\n\\begin{figure}[t]\n\\includegraphics[width=7.9cm]{Fig3x.pdf}\n\\caption{Evolution of the dissipation ratio $Q$. Strong dissipation, $Q\\gg1$,\nis maintained throughout inflation. At the end of the inflationary period\n($u\\approx-64.7$, vertical grey line), the dissipation coefficients\n$\\Gamma_\\xi$ and $\\Gamma_\\phi$ are exponentially suppressed, implying that\nsoon afterwards $Q$ becomes negligible. At $u=u_*\\approx-63.7$, the\ndissipation coefficients are set exactly to zero, marking the end of the first\nstage of evolution.}\n \\label{Fig-Gamma}\n\\end{figure}\n\nIn the scenario we have been considering, the initial conditions\nand the values of the parameters are such that a strong dissipative regime ($Q>1$)\nis maintained throughout inflation. However, as will be shown below in\nSect.~\\ref{num-dissipative},\na weak dissipative regime can also be obtained for other choices of initial\nconditions and values of the parameters.\n\nThe value of the energy density of the field $\\xi$ at the transition between\nthe first and second stages of evolution, $\\rho_\\xi(u_*)$, is of crucial\nimportance (see Fig.~\\ref{Fig_rhoxi-rhoR-E1}). It cannot be too large,\notherwise the radiation-dominated era will be too short, conflicting with\nprimordial nucleosynthesis, but it cannot be too small, otherwise the\nmatter-dominated era will not be long enough for structure formation to take\nplace or, worse, such an era may not even occur. As discussed above [see\nEq.~(\\ref{duration I})], in order to guarantee an adequate value of\n$\\rho_\\xi(u_*)$, the duration of the first stage of evolution should be chosen\ncarefully; in the example we have been considering (base scenario),\n$\\Delta^{[\\texttt{I}]}u \\approx 61.0$, implying $\\rho_\\xi(u_*) \\approx 4.2\n\\times 10^{-41} \\, m_\\texttt{P}^4$, which allows for radiation- and\nmatter-dominated eras with durations consistent with current cosmological\nobservations.\n\nLet us open here a parenthesis to briefly comment on a recent proposal of\nunification of inflation and dark matter \\cite{rosa-2019}, in which inflation\nis also assumed to be of the warm type. There, the potential of the inflaton\nfield, in addition to a quadratic term, also has a quartic one, which\ndominates for large field values, implying that this scalar field behaves,\nduring most of the radiation-dominated era, including the primordial nucleosynthesis\nperiod, as dark radiation. It begins to behave like dark matter\njust before the transition between the radiation- and the matter-dominated eras.\nThis contrasts with the situation in our model,\nin which the scalar field $\\xi$ behaves like dark matter from the very beginning\nof the radiation-dominated era [see Eq.~(\\ref{rho dark matter}) and\nFig.~\\ref{Fig_rhoxi-rhoR-E1}].\n\nReturning to our base scenario, we point out that, as discussed in\nSect.~\\ref{transition}, continuity of the energy density of radiation at the\ntransition between the first and second stages of evolution fixes the duration\nof the second stage to be $\\Delta^{[\\texttt{II}]}u \\approx 63.7$; at the\ntransition, $\\rho_\\texttt{R}(u_*) \\approx 4.2 \\times 10^{-17} \\,\nm_\\texttt{P}^4$.\n\nDuring the second stage of evolution, the energy densities $\\rho_\\texttt{R}$ and\n$\\rho_\\xi$ evolve according to Eqs.~(\\ref{Eq rho E2}) and (\\ref{rho dark matter}),\nleading successively to radiation- and dark-matter-dominated eras.\nMeanwhile, the scalar field $\\phi$ plays no\nsignificant role in the dynamics of the universe, only becoming dominant at\nrecent times ($u\\approx-0.3$) when it  induces a period of accelerated expansion\n(see Fig.~\\ref{Fig-Omega}).\n\n\\begin{figure}[t]\n\\includegraphics[width=7.9cm]{Fig4x.pdf}\n\\caption{Evolution of the density parameters for radiation, baryonic matter,\nand the scalar fields $\\xi$ (inflaton and dark matter) and $\\phi$ (dark\nenergy). The successive inflationary, radiation, matter, and dark energy eras\nare clearly delimited. At the present time, $u_0=0$, the density parameters\nare $\\Omega_{\\phi}(u_0)\\approx 0.69$, $\\Omega_{\\xi}(u_0)\\approx 0.26$,\n$\\Omega_{\\texttt{BM}}(u_0)\\approx0.05$, and\n$\\Omega_{\\texttt{R}}(u_0)\\approx5.5\\times10^{-5}$, in agreement with\ncosmological measurements.}\n \\label{Fig-Omega}\n\\end{figure}\n\nFor our choice of the initial conditions and parameters (and $C \\approx\n4.25\\times10^{-124} \\, m_\\texttt{P}^4$) the conditions given by\nEqs.~(\\ref{condition 1}) and (\\ref{condition 2}) are satisfied, implying that\nthe density parameters at the present time $u_0=0$ become\n$\\Omega_{\\phi}(u_0)\\approx 0.69$, $\\Omega_{\\xi}(u_0)\\approx 0.26$,\n$\\Omega_{\\texttt{BM}}(u_0)\\approx0.05$, and\n$\\Omega_{\\texttt{R}}(u_0)\\approx5.5\\times10^{-5}$, in agreement with\ncosmological measurements \\cite{Planck-parameters-2015}.\n\nIn the future (i.e., for $u>0$), the density parameters for radiation\n$\\Omega_{\\texttt{R}}$, baryonic matter, $\\Omega_{\\texttt{BM}}$,\nand dark matter $\\Omega_{\\xi}$ become negligible in comparison\nwith the density parameter for dark energy $\\Omega_{\\phi}$,\nimplying that the effective equation-of-state parameter $w_{\\rm eff}$\ntends to the value $-1+\\beta^2\/3$ (see Fig.~\\ref{Fig-weff}).\n\n\\begin{figure}[t]\n\\includegraphics[width=7.9cm]{Fig5x.pdf}\n\\caption{Evolution of the effective equation-of-state parameter $w_{\\rm eff}$,\nclearly showing the inflationary, radiation, matter, and dark-energy eras\n(the solid blue line corresponds to the base scenario, $\\alpha=\\beta=1$).\nThe value of $w_{\\rm eff}$ for $u\\rightarrow+\\infty$ depends on the parameter\n$\\beta$ (but not on $\\alpha$), such that, for $|\\beta|<\\sqrt2$, the universe\nenters a period of everlasting accelerated expansion. For $\\beta=3\/2$ this\naccelerated expansion is only temporary, since asymptotically $w_{\\rm eff}$\napproaches the value $-1\/4$.}\n \\label{Fig-weff}\n\\end{figure}\n\nIndeed, for negligible $\\Omega_{\\texttt{R}}$, $\\Omega_{\\texttt{BM}}$,\nand $\\Omega_{\\xi}$, Eq.~(\\ref{Eq phi E2}) simplifies considerably,\nbecoming\n\\begin{align}\n \\phi_{uu} =  {} &- \\frac{1}{2\\kappa}(\\kappa\\phi_u -\\beta) (6-\\kappa^2\\phi_u^2).\n\\end{align}\nThis differential equation admits an analytical solution, which, in\nthe limit $u\\rightarrow+\\infty$ and for $\\beta^2<6$, becomes\n$\\phi_u=\\beta\/\\kappa$.\nNow, inserting this asymptotic solution, in Eq.~(\\ref{Eq friedman E2}), one obtains\n$V_a e^{-\\beta\\kappa\\phi}=(6-\\beta^2)H^2\/(2\\kappa^2)$.\nFinally, substituting the above expressions for $\\phi$ and $\\phi_u$\ninto Eq.~(\\ref{eos parameter E2}), one obtains\n$w_{\\rm eff}=-1+\\beta^2\/3$ for $u\\rightarrow+\\infty$.\n\nThe asymptotic behavior of the effective equation-of-state parameter implies that,\nfor $|\\beta|<\\sqrt2$, the universe enters a period of everlasting\naccelerated expansion.\nFor values of $|\\beta|$ slightly above $\\sqrt2$, this accelerated expansion\nstill takes place, but does not last forever.\nFor instance, in the case $\\beta=3\/2$, shown in Fig.~\\ref{Fig-weff},\naccelerated expansion occurs at the present time ($w_{\\rm eff}<-1\/3$)\nand then ceases as $w_{\\rm eff}$ tends to $-1\/4$.\n\nIn summary, in the example we have been considering (base scenario),\ninflation, driven by the scalar field $\\xi$, begins at $u\\approx-124.7$ and\nextends for $60$ $e$-folds, till $u\\approx-64.7$. During this period, a\nradiation bath with temperature of about $10^{15} \\, {\\rm GeV}$ is sustained\nby a continuous and copious energy transfer from the scalar fields $\\xi$ and\n$\\phi$. At the end of inflation, the dissipation coefficients are\nexponentially suppressed; as a consequence, radiation decouples from the\nscalar fields $\\xi$ and $\\phi$ and begins to evolve in the usual manner,\ndominating the dynamics of the universe till $u\\approx-8.6$. In the absence of\ndissipative effects, the scalar field $\\xi$ oscillates around its minimum,\nbehaving like a pressureless nonrelativistic fluid (dark matter), and,\ntogether with ordinary baryonic matter, becomes dominant at $u\\approx-8.6$.\nThe scalar field $\\phi$ (dark energy), having played no significant role in\nthe dynamics of the universe during the preceding eras, finally becomes\ndominant at $u\\approx-0.3$, giving rise to an everlasting period of\naccelerated expansion of the universe.\n\nBecause we are only interested in models that support accelerated expansion at\nthe present time, we restrict our analysis to the cases $\\beta\\lesssim3\/2$.\nFurthermore, as pointed out above, we can assume $\\alpha\\geq0$ without loss of\ngenerality. Our numerical simulations show that, for values of $\\alpha=\\beta$\nlying in this interval, the cosmic evolution is quite similar to the base\nscenario, making it unnecessary to present here a detailed analysis. We just\nrefer the reader to the base scenario and also to Fig.~\\ref{Fig-weff}, where,\nthe cosmic evolution is outlined for three more cases, namely\n$\\alpha=\\beta=0,1\/2$ and $3\/2$.\n\n\n\n\\subsection{Case $\\alpha\\neq\\beta$\n\\label{num-deviations}}\n\nLet us start by considering a varying $\\beta$ for fixed $\\alpha$ (say $\\alpha=1$,\nas in the base scenario considered in the previous subsection).\nThe evolution of the density parameters for the scalar fields $\\xi$ and $\\phi$,\nas well as for radiation and baryonic matter, are shown in\nFigs.~\\ref{Fig_Omegas-beta05} and \\ref{Fig_Omegas-beta0} for the cases\n$\\beta=1\/2$ and $\\beta=0$, respectively,\nwhile the evolution of the effective equation-of-state parameter\nis shown in Fig.~\\ref{Fig-weff-beta} for $\\beta=0,1\/2,1$ and $3\/2$.\n\n\\begin{figure}[t]\n\\includegraphics[width=7.9cm]{Fig6x.pdf}\n\\caption{Evolution of the density parameters for the case $\\alpha=1$ and\n$\\beta=1\/2$. The transition from the radiation- to the matter-dominated era\ntakes place at $u\\approx-9.4$, a little earlier than in the base scenario.\nDuring the matter-dominated era, the density parameter of dark energy is a\nnon-negligible fraction (about $3\\%$)\nof the density parameter of matter (dark plus baryonic).\nThe density parameter of dark matter reaches its maximum value at $u\\approx -4.8$,\nwhile for baryonic matter this peaking occurs later, at $u\\approx -1.4$.}\n \\label{Fig_Omegas-beta05}\n\\end{figure}\n\n\\begin{figure}[t]\n\\includegraphics[width=7.9cm]{Fig7x.pdf}\n\\caption{Evolution of the density parameters for the case $\\alpha=1$ and\n$\\beta=0$. The transition from the radiation- to the matter-dominated era\ntakes place at $u\\approx-15.2$, much earlier than in the base\nscenario.\nDuring the matter-dominated era, the density parameter of dark energy is a\nsignificant fraction (about $20\\%$)\nof the density parameter of matter (dark plus baryonic).\nThe density parameter of dark matter reaches its maximum value at $u\\approx -7$,\nwhile for baryonic matter this peaking occurs much later, at $u\\approx -0.8$.}\n \\label{Fig_Omegas-beta0}\n\\end{figure}\n\n\\begin{figure}[t]\n\\includegraphics[width=7.9cm]{Fig8x.pdf}\n\\caption{Evolution of the effective equation-of-state parameter $w_{\\rm eff}$\nfor $\\alpha=1$ and different values of $\\beta$ (the solid blue line corresponds\nto the base scenario, $\\beta=1$).\nDuring the matter-dominated era, the value of $w_{\\rm eff}$ is approximately zero\nfor $\\beta=1\/2$ and $3\/2$, similarly to the base scenario (inset plot).\nHowever, for $\\beta=0$, the situation changes drastically, with\nthe value of $w_{\\rm eff}$ departing significantly from zero, due to a\ngreater influence of dark energy during this era.\nFor $\\beta=0,1\/2$, and~$1$, the universe initiates, at recent times, a\nperiod of everlasting accelerated expansion.\nFor $\\beta=3\/2$ accelerated expansion takes place at the present time $u_0=0$,\nbut does not last forever, as $w_{\\rm eff}$ asymptotically approaches the value~$-1\/4$.}\n \\label{Fig-weff-beta}\n\\end{figure}\n\nA first change in the cosmic evolution, as compared with the case $\\alpha=\\beta$,\nis related to the duration of~the matter-dominated era.\nThe more $\\beta$ differs from $\\alpha$,\nthe earlier the transition from a radiation to a matter-dominated universe\ntakes place and the longer the duration of the latter.\nThis effect is quite mild for $|\\alpha-\\beta|\\lesssim1$,\nhaving no implications on the viability of the cosmological solutions\n(see Fig.~\\ref{Fig_Omegas-beta05} and the inset of Fig.~\\ref{Fig-weff-beta}).\nHowever, for $|\\alpha-\\beta|\\gtrsim 1$, the effect becomes so strong that it\nbegins to conflict with primordial nucleosynthesis.\nFor instance, in the case $\\alpha=1$ and $\\beta=0$, the transition from the\nradiation- to the matter-dominated era takes place already at $u\\approx-15.2$\n(see Fig.~\\ref{Fig_Omegas-beta0}),\nquite near to the  primordial nucleosynthesis value $u\\approx -18$.\nIf one further increases the value of $|\\alpha-\\beta|$,\nconflict with primordial nucleosynthesis can only be avoided by dropping the\nrequirement that the present-time density parameter $\\Omega_{\\xi}(u_0)$ must be\nequal to the observational value $\\Omega_\\texttt{DM0}$.\nIn other words, one must accept that the\nscalar field $\\xi$ accounts only for part of the dark-matter content of the\nuniverse; the rest must be introduced by hand, together with ordinary baryonic\nmatter. In fact, this was the situation reported in Ref.~\\cite{henriques-2009},\nwhere, for $\\alpha=0$ and $\\beta=-\\sqrt{2}$, the scalar field contributed only\nabout 6\\% to the total matter content of the universe.\n\nAnother change in the cosmic evolution concerns the behavior of the scalar field\n$\\phi$ (dark energy).\nFor $\\alpha\\neq\\beta$, it starts to influence the dynamics of the universe much earlier,\nat the beginning of the matter-dominated era, and its energy density is\na non-negligible fraction of the total energy density throughout\nthe matter-dominated era.\nThe higher $|\\alpha-\\beta|$, the greater the fraction of dark energy during this era.\nFor instance, in the case $\\alpha=1$ and $\\beta=1\/2$,\nshown in Fig.~\\ref{Fig_Omegas-beta05}, the density parameter of dark energy is about\n$3\\%$ of the density parameter of matter (dark plus baryonic),\nwhile in the case $\\alpha=1$ and $\\beta=0$, shown in Fig.~\\ref{Fig_Omegas-beta0},\nthis percentage increases to $20\\%$.\nThis behavior of dark energy implies that, during the matter-dominated era,\nthe value of the effective equation-of-state parameter $w_{\\rm eff}$ differs from zero\n(see Fig.~\\ref{Fig-weff-beta}).\nNote, however, that for the cases $\\beta=1\/2$ and $3\/2$ the dark-energy fraction\nof the total energy density during the matter-dominated era is much smaller than\nin the case $\\beta=0$, meaning that the value of $w_{\\rm eff}$\nin these cases remains close to zero during the matter-dominated era.\n\nA third change occurring in the cosmic evolution is related to the peaking of\nthe energy densities of dark and baryonic matter. In the case $\\alpha=\\beta$,\nthe energy density of dark energy, given by Eq.~(\\ref{rho dark matter}),\nevolves exactly as the energy density of ordinary baryonic matter (i.e., as\n$e^{-3u}$ or, in terms of the scale factor, as $a^{-3}$), meaning that the\nratio between $\\rho_\\xi$ and $\\rho_\\texttt{BM}$ is constant throughout time\nand, consequently, the peaking of these two quantities occurs simultaneously.\nFor $\\alpha\\neq\\beta$ the situation is quite different. The ratio\n$\\rho_\\xi\/\\rho_\\texttt{BM}$ depends directly on the behavior of the scalar\nfield $\\phi$, namely, $\\rho_\\xi\/\\rho_\\texttt{BM} \\propto \\exp\n[(\\alpha-\\beta)\\kappa\\phi\/2]$. This implies that the peaking of the energy\ndensities of dark matter and ordinary baryonic matter does not occur at the\nsame time. For instance, in the case $\\alpha=1$ and $\\beta=0$, shown in\nFig.~\\ref{Fig_Omegas-beta0}, the density parameter of dark matter reaches its\nmaximum value at $u\\approx -7$, while for baryonic matter this peaking occurs\nmuch later, at $u\\approx -0.8$.\n\nAs seen above, agreement with current cosmological data requires\n$|\\alpha-\\beta| \\lesssim 1$, in order to guarantee that the radiation-dominated era\nlasts long enough to encompass primordial nucleosynthesis,\nand $|\\beta|\\lesssim 3\/2$, in order to guarantee accelerated expansion at the\npresent time ($|\\beta|<\\sqrt{2}$ to guarantee that this accelerated expansion\nlasts forever).\nFor such values of the parameters $\\alpha$ and $\\beta$,\nthe two-scalar-field cosmological model given by action~(\\ref{action 2SF})\nallows for a triple unification of inflation, dark energy, and dark matter\nwhich is, at least qualitatively, consistent with observations.\n\nThe above conclusions are confirmed by the numerical solutions obtained for\nother values of the parameters $\\alpha$ and $\\beta$.\nFor completeness, the evolution of the effective equation-of-state parameter for the\ncase $\\beta=1$ and varying $\\alpha$ is presented in Fig.~\\ref{Fig-weff-alpha}.\n\n\\begin{figure}[t]\n\\includegraphics[width=7.9cm]{Fig9x.pdf}\n\\caption{Evolution of the effective equation-of-state parameter $w_{\\rm eff}$\nfor $\\beta=1$ and different values of $\\alpha$ (the solid blue line corresponds\nto the base scenario, $\\alpha=1$).\nAt recent times, the universe experiences a second period of accelerated expansion;\nas clearly shown in the bottom right inset plot, independently of the value of\nparameter $\\alpha$, $w_{\\rm eff}$ asymptotically approaches $-2\/3$.\nDue to the earlier influence of dark energy, during the matter-dominated era,\n$w_{\\rm eff}$ differs significantly from zero in the\ncases $\\alpha=0$ and $\\alpha=2$ and mildly in the cases $\\alpha=1\/2$ and\n$\\alpha=3\/2$ (see upper left inset plot).}\n \\label{Fig-weff-alpha}\n\\end{figure}\n\n\n\\subsection{Dissipative effects \\label{num-dissipative}}\n\nWe conclude the present section with an analysis of the\ndissipative effects during both the inflationary period and the transition\nto the radiation-dominated era.\n\nSo far we have considered the dissipation coefficients $\\Gamma_\\xi$ and\n$\\Gamma_\\phi$ to depend linearly on the temperature, i.e., we have chosen\n$p=1$ in Eq.~(\\ref{gammas}). However, as already referred to in Sect.~\\ref{1st\nstage}, in the context of warm inflation several other possibilities have been\nconsidered, from the simplest, based on general phenomenological\nconsiderations, to the more elaborate ones, derived from microscopic quantum\nfield theory. Accordingly, we can extend our analysis to include other values\nof $p$, corresponding, in particular, to dissipation coefficients constant\nthroughout the inflationary period ($p=0$) and dissipation coefficients\ninversely proportional to the temperature ($p<0$).\n\nOur numerical simulations show that, for these values of $p$, the cosmic\nevolution proceeds in a similar way to the case $p=1$. In\nFig.~\\ref{Fig_Gammas_p} the evolution of the dissipation ratio $Q$ is shown\nfor $p=-1,0,1$. For comparison purposes, we choose the duration of the\ninflationary period to be the same in all three cases, which in turns requires\nthe choice $f_\\xi=f_\\phi=2$ for $p=1$, $f_\\xi=f_\\phi=3.1\\times10^{-4}\\,\nm_\\texttt{P}$ for $p=0$, and $f_\\xi=f_\\phi=3.8\\times10^{-8}\\, m_\\texttt{P}^2$\nfor $p=-1$ (the initial conditions and the values of the other parameters are\nthe same as those of the base scenario).\n\n\\begin{figure}[t]\n\\includegraphics[width=7.9cm]{Fig10x.pdf}\n\\caption{Evolution of the dissipation ratio $Q$ for $q=2$ and $p=-1,0,1$,\ncorresponding to dissipation coefficients inversely proportional to the\ntemperature, constant, and  proportional to the temperature, respectively.}\n \\label{Fig_Gammas_p}\n\\end{figure}\n\nIn warm inflation, the dissipative effects --- and the consequent energy\ntransfer from the inflaton to the radiation bath --- play an essential role,\nbut they should vanish soon after the end of the inflationary period, allowing\ncosmic evolution to further proceed in the usual way. In our cosmological\nmodel, this is achieved by assuming that, immediately after the end of\ninflation, the dissipation coefficients are suppressed by an exponential term\nparameterized by $q$ [see Eq.~(\\ref{gammas})]. In all the cases considered so\nfar, we have choose $q=2$. However, the suppression of dissipative effects\nafter the inflationary period can be chosen to proceed slower or faster. This\nis illustrated in Fig.~\\ref{Fig_Gammas_q}, where the dissipation ratio $Q$ is\nshown for different values of $q$, namely $q=1,2,3$, with $p=1$ and\n$f_\\xi=f_\\phi=2$.\n\n\\begin{figure}[t]\n\\includegraphics[width=7.9cm]{Fig11x.pdf}\n\\caption{Evolution of the dissipation ratio $Q$ for $p=1$ and $q=1,2,3$. The\nhigher the value of $q$, the faster the dissipation ratio $Q$ is suppressed\nafter the inflationary period.}\n \\label{Fig_Gammas_q}\n\\end{figure}\n\nLet us note that for $p>2$, the dissipation coefficients $\\Gamma_\\xi$ and\n$\\Gamma_\\phi$ are suppressed naturally after inflation, with no need for an\nexplicit exponential suppression term in Eq.~(\\ref{gammas}). This can be seen\nas follows.\n\nAfter the end of the inflationary period ($u>u_\\texttt{E}$), cosmic evolution is\ndominated by radiation.\nLet $\\tilde{u}>u_\\texttt{E}$ be the value of $u$ above which the dissipative terms in\nEq.~(\\ref{Eq rho E1}) are much smaller than the energy density of radiation,\ni.e., $(\\dot{a}\/a)(\\Gamma_\\xi \\xi_u^2 + \\Gamma_\\phi \\phi_u^2) \\ll\n\\rho_\\texttt{R}$. Then, for $u>\\tilde{u}$, the energy density of radiation\nevolves approximately as\n$\\rho_\\texttt{R}(u)\\simeq\\tilde{\\rho}_\\texttt{R}e^{-4(u-\\tilde{u})}$, where\n$\\tilde{\\rho}_\\texttt{R} \\equiv \\rho_\\texttt{R}(\\tilde{u})$. Inspection of the\nnumerical solutions of Eq.~(\\ref{Eq friedman E1}) reveals that, for\n$u>\\tilde{u}$, the Hubble parameter is given, in a good approximation, by $H\n\\simeq \\kappa (\\rho_\\texttt{R}\/3)^{1\/2}$. Then, from Eqs.~(\\ref{gammas})\nand (\\ref{q}) one obtains an approximate analytical\nexpression for the dissipation ratio as a function of $u$, valid for\n$u>\\tilde{u}$,\n\\begin{align}\n Q \\simeq\n A_1 \\exp \\left[ 1-(p-2)(u-\\tilde{u})- A_2 e^{q(u-\\tilde{u})} \\right],\n   \\label{Q-approx}\n\\end{align}\nwhere $A_1$ and $A_2$ are given by\n\\begin{align}\n A_1 = {}& \\frac{f_{\\xi,\\phi}}{\\sqrt3\\kappa}\n  \\left( \\frac{30}{\\pi^2 g_*} \\right)^{\\frac{p}{4}}\n  \\tilde{\\rho}_\\texttt{R}^{\\frac{p-2}{4}}, \\\\\n A_2 = {}& \\left( \\frac{\\rho_\\texttt{RE}}{\\tilde{\\rho}_\\texttt{R}}\n           \\right)^{\\frac{q}{4}},\n\\end{align}\nand $\\rho_\\texttt{RE} \\equiv \\rho_\\texttt{R}(u_\\texttt{E})$ denotes the energy\ndensity of radiation at the end of the inflationary period. For $p\\leq2$, the\ndissipation ratio $Q$ does not decrease if $q=0$, but for $p>2$ the parameter\n$q$ can be set to zero and suppression nevertheless takes place (see\nFig.~\\ref{Fig_Gammas_q0}). This result is in agreement with\nRef.~\\cite{lima-2019}, where, in the context of a quintessential inflationary\nmodel, the dissipation coefficient, assumed to depend both on the temperature\nand the inflaton field as $\\Gamma\\propto T^p \\xi^c$, is shown to be a\ndecreasing function of the number of $e$-folds for $p>2$.\n\n\\begin{figure}[t]\n\\includegraphics[width=7.9cm]{Fig12x.pdf}\n\\caption{Evolution of the dissipation ratio $Q$ for $q=0$ and $p=1,2,3$. For\n$p>2$, as a result of the background dynamics, suppression takes place even in\nthe absence of the exponential factor in Eq.~(\\ref{gammas}).}\n \\label{Fig_Gammas_q0}\n\\end{figure}\n\nSo far, the cases considered correspond to a strong dissipative regime, for which\n$Q>1$. However, it is of relevance to show that a weak dissipative regime can also\nbe obtained in our two-scalar-field cosmological model for other choices of the\ninitial conditions and the values of the parameters.\nIndeed, as already mentioned above, the confrontation of warm-inflation predictions\nwith the latest CMB data\n\\cite{bastero-gil-2019, benetti-2017,arya-2018,bastero-gil-2018,bastero-gil-2018a,motaharfar-2019}\nhas shown a preference for the weak dissipation regime for certain combinations\nof the inflaton's potential and the dissipation coefficients (for instance,\nfor a quartic potential and a dissipation coefficient proportional to $T$ or $T^3$).\n\nFrom Eq.~(\\ref{q}), it follows that a smaller value of the dissipation ratio\nat the onset of the first stage of evolution, $Q_\\xi(u=u_i)$, can be obtained\nby decreasing $f_\\xi$ and\/or increasing $\\xi(u_i)$ and $\\rho_\\texttt{R}(u_i)$.\nStarting from the base scenario, considered in detail in Sect.~\\ref{num-base},\nit is then straightforward to find a set of initial conditions and values of\nthe parameters corresponding to weak dissipation, as, for instance,\n$\\xi(u_i)=3.19\\, m_\\texttt{P}$, $\\phi(u_i)=10^{-3}\\, m_\\texttt{P}$,\n$\\xi_u(u_i)=10^{-2}\\, m_\\texttt{P}$, $\\phi_u(u_i)=10^{-5}\\, m_\\texttt{P}$,\n$\\rho_\\texttt{R}(u_i)=2.2 \\times 10^{-12}\\, m_\\texttt{P}^4$, $V_a=2.69 \\times\n10^{-123}\\, m_\\texttt{P}^4$, $m=10^{-5}\\, m_\\texttt{P}$, $f_\\xi=0.05$,\n$f_\\phi=25$, $p=1$, and $q=2$. The evolution of the dissipation ratio $Q_\\xi$\nfor this case is shown in Fig.~\\ref{Fig_Q_xi} (the effective equation-of-state\nparameter and the density parameters evolve similarly as in the base scenario,\nmaking it unnecessary to present them here).\n\n\\begin{figure}[t]\n\\includegraphics[width=7.9cm]{Fig13x.pdf}\n\\caption{Evolution of the dissipation ratio $Q_\\xi$. Weak dissipation is maintained\nthroughout most of the inflationary period. Towards its end, the dissipation ratio\nsharply increases, reaching values above unity, which facilitates the transition to\nthe radiation-dominated era.}\n \\label{Fig_Q_xi}\n\\end{figure}\n\nThe primordial spectrum of density perturbations of our unification model and\nits agreement with observational data, for the quadratic potential, given by\nEq.~(\\ref{potential xi}), and the dissipation coefficients, given by\nEq.~(\\ref{gammas}), both in the strong and weak dissipative regimes,\nwill be explored in future work.\n\n\n\\section{Conclusions\\label{conclusions}}\n\nIn this article we have presented a unified description of inflation, dark energy,\nand dark matter in a two-scalar-field cosmological model with a non-standard\nkinetic term and an exponential potential.\nSuch models arise in a great variety of gravity theories, such as the\nJordan-Brans-Dicke theory, Kaluza-Klein theories, $f(R)$-gravity, string theories,\nand hybrid metric-Palatini theories of gravity.\n\nIn the proposed triple unification, one of the scalar fields plays the role of\ninflaton and dark matter and the other plays the role of dark energy.\nMore specifically, inflation, assumed to be of the warm type, is driven by the\nscalar field $\\xi$, which, shortly after the end of the inflationary period,\ndecouples from radiation and begins to oscillate rapidly around the minimum\nof its potential, thus behaving like a cold-dark-matter fluid; the second scalar\nfield $\\phi$ emerges, at recent times, as the dominant component of the universe\nand gives rise to an era of accelerated expansion. Thus, seemingly disparate\nphenomena like inflation, dark energy, and dark matter are unified under the\nsame theoretical roof using scalar fields.\n\nThe two-scalar-field cosmological model given by action~(\\ref{action 2SF})\ncontains two dimensionless parameters, $\\alpha$ and $\\beta$ (for $\\alpha=0$,\nthe kinetic term for the scalar field $\\xi$ becomes canonical; for $\\beta=0$,\nthe direct coupling in the potential between the two scalar fields $\\xi$\nand $\\phi$ disappears). These parameters could, in principle, be chosen freely;\nhowever, as detailed in Sect.~\\ref{num-deviations}, the requirement that the\ntransition from the radiation- to the matter-dominated era does not occur too\nearly in the cosmic history and, consequently, does not conflict with primordial\nnucleosynthesis, as well as the requirement that the expansion of the universe is\naccelerating at the present time, imposes constraints on the parameters $\\alpha$\nand $\\beta$, namely, $|\\alpha-\\beta| \\lesssim 1$\nand $|\\beta| \\lesssim 3\/2$.\n\nFor such values of $\\alpha$ and $\\beta$, the picture emerging in our unified\ndescription of inflation, dark energy, and dark matter is consistent with the\nstandard cosmological model.\nIndeed, the inflationary period is followed by a radiation-dominated era that\nencompasses the primordial nucleosynthesis epoch; the matter-dominated era lasts\nlong enough for structure formation to occur; the transition to a\ndark-energy-dominated universe takes place in a recent past;\nand the density parameters for dark matter and dark energy, as well as for radiation\nand ordinary baryonic matter, evaluated at the present time, are in agreement with\ncurrent cosmological observations (see Fig.~\\ref{Fig-Omega} for the case $\\alpha=\\beta=1$,\nFig.~\\ref{Fig_Omegas-beta05} for the case $\\alpha=1$, $\\beta=1\/2$,\nand Fig.~\\ref{Fig_Omegas-beta0} for the case $\\alpha=1$, $\\beta=0$).\n\nOf crucial importance for consistency with the standard cosmological model is\nthe value of the energy density of the scalar field $\\xi$ at the moment when\nthis field begins to oscillate rapidly around its minimum, changing its\nbehavior from an inflaton field to a nonrelativistic dark-matter fluid (i.e.,\nat the transition from the first to second stage of cosmic evolution). If this\nenergy density is too large, the radiation-dominated era is too short,\nconflicting with primordial nucleosynthesis; if it is too small, the\nmatter-dominated era is not long enough for structure formation to take place\n(see Fig.~\\ref{Fig_rhoxi-rhoR-E1}). In our model, an appropriate value of the\nenergy density of the inflaton\/dark-matter field at the transition between the\nfirst and second stages of evolution is achieved by ensuring that, immediately\nafter the end of the inflationary period, the dissipation coefficients,\nresponsible for the energy transfer from the scalar fields $\\xi$ and $\\phi$ to\nthe radiation bath, are rapidly suppressed, becoming negligible soon\nafterwards. When this happens, the scalar field $\\xi$ decouples from radiation\nand, because its effective mass is larger than the Hubble parameter, begins to\noscillate rapidly around its minimum, behaving like dark matter.\n\nIn our cosmological model, the behavior of the dissipation coefficients is\ncontrolled by two parameters, $p$ and $q$. The first sets the temperature\ndependence of the coefficients and the second determines how fast these\ncoefficients are suppressed after the end of the inflationary period [see\nEq.~(\\ref{gammas})]. As we have shown, for $p>2$, the dissipation coefficients\nare suppressed naturally after inflation, with no need for an explicit\nsuppression term, allowing us to set $q=0$. This does not mean, however, that\nthe models with $p\\leq2$, requiring $q\\neq0$, are less admissible. Actually,\nsuch models, have been proposed in recent years\n\\cite{bastero-gil-2016,rosa-2019,rosa-2019b,bastero-gil-2019} and are quite\nsound, both theoretically and observationally.\n\nAs already mentioned above, shortly after the end of the inflationary period,\nthe scalar field $\\xi$ begins to oscillate rapidly around the minimum of its\npotential. We have derived an expression for its energy density during this\noscillating phase [see Eq.~(\\ref{rho dark matter})],\nshowing that it is proportional to\n$\\exp[(\\alpha-\\beta)\\kappa\\phi\/2]$.\nIn the case $\\alpha=\\beta$, this exponential equals unity and dark matter behaves\nexactly as ordinary baryonic matter, i.e., evolves as $a^{-3}$, where $a$ is\nthe scale factor. But for $\\alpha\\neq\\beta$ the situation is quite different;\nthe energy density of dark matter depends directly on the dark-energy field $\\phi$,\nleading to a non-simultaneous peaking of the energy densities of dark matter and\nordinary baryonic matter\n(see Figs.~\\ref{Fig_Omegas-beta05} and \\ref{Fig_Omegas-beta0}).\nFurthermore, in the case $\\alpha\\neq\\beta$, the energy density of the dark energy\nis a non-negligible fraction of the critical energy density throughout the\nmatter-dominated era.\n\nFinally, we have shown that the effective equation-of-state parameter $w_{\\rm\neff}$ tends, asymptotically, to $-1+\\beta^2\/3$, implying that, for\n$|\\beta|<\\sqrt2$, the universe enters a period of everlasting accelerated\nexpansion (for values of $|\\beta|$ slightly above $\\sqrt2$, this accelerated\nexpansion still takes place, but does not last forever).\n\nIn this article, we have proposed a triple unification of inflation, dark\nenergy, and dark matter in a two-scalar-field cosmological model. This is a\nfirst approach, intended to show that such a unification is, in principle,\npossible and that it reproduces, at least qualitatively, the main features of\nthe observed universe. We expect to explore and deepen this model in future\npublications.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\@startsection {section}{1}{\\z@}%\n\n\n\n\\newcommand{\\mc}[1]{\\textcolor{red}{#1}}\n\n\\begin{document}\n\n\\title{Rotating Black Holes in Higher Order Gravity}\n\n\\author{Christos Charmousis}\n\\affiliation{Laboratoire de Physique Th\\'eorique, CNRS, Univ.\\ Paris-Sud, \nUniversit\\'e Paris-Saclay, 91405 Orsay, France}\n\\author{Marco Crisostomi} \n\\affiliation{Laboratoire de Physique Th\\'eorique, CNRS, Univ.\\ Paris-Sud, \nUniversit\\'e Paris-Saclay, 91405 Orsay, France}\n\\affiliation{Institut de physique th\\'eorique, Univ.\\ Paris Saclay, CEA, \nCNRS, 91191 Gif-sur-Yvette, France}\n\\affiliation{AIM, CEA, CNRS, Univ.\\ Paris-Saclay, Univ.\\ Paris Diderot,\nSorbonne Paris Cit\\'e, F-91191 Gif-sur-Yvette, France}\n\\author{Ruth Gregory}\n\\affiliation{Centre for Particle Theory, Durham University, South Road,\nDurham, DH1 3LE, UK}\n\\affiliation{Perimeter Institute, 31 Caroline St., Waterloo, Ontario,\nN2L 2Y5, Canada}\n\\author{Nikolaos Stergioulas}\n\\affiliation{Department of Physics, Aristotle University of Thessaloniki, \n54124 Thessaloniki, Greece}\n\\date{\\small \\today}\n\n\\begin{abstract}\nWe develop a new technique for finding black hole solutions in modified \ngravity that have ``stealth'' hair, i.e., hair whose only gravitational effect\nis to tune the cosmological constant. We consider scalar-tensor theories \nin which gravitational waves propagate at the speed of light, and show that \nEinstein metrics can be painted with stealth hair provided there exists a\nfamily of geodesics always normal to spacelike surfaces. We also present\na novel scalar-dressed rotating black hole that has finite scalar field\nat both the black hole and cosmological event horizons.\n\\end{abstract}\n\n\n\\maketitle\n\n\n\nBlack Holes are a cornerstone in the study of General Relativity (GR), \nbe it theoretical or practical.\nFrom the theoretical perspective, we have several exact solutions \ncorresponding to black holes in GR, with charges such as mass, angular \nmomentum, and even acceleration or Taub NUT parameters. \nOf these, the most important is the Kerr solution\nthat describes the rotating black hole; most astrophysical \nblack holes are believed to be rotating, indeed, the first\ndetection of gravitational waves was from the merger of two \nspinning Kerr black holes \\cite{Abbott:2016blz}. \n\nAnother key objective in gravity is to explain our universe. One of\nthe main challenges in cosmology is to explain late time acceleration:\nIs it due to dark energy, modified gravity, or a very \nsmall cosmological constant~$\\Lambda$, finely tuned by some as \nyet undiscovered mechanism? One fruitful means of explaining \nthe small late time acceleration is to modify gravity in such a way as\nto induce (or tune) a cosmological constant, but if gravity is modified,\nthen it is vital to understand how these modifications affect black holes. \nIf it is not possible to construct astrophysically realistic black hole \nsolutions, then the theory cannot be considered viable. \n\nIn GR, the Kerr solution (see \\cite{Carter:1968ks,CARTER1968399}\nfor inclusion of a cosmological constant) describes the most general \naxisymmetric stationary rotating black hole, moreover, the event horizon \ntelescope \\cite{Doeleman:2009te} hopes soon to be able to directly \nimage the shadow of the black hole at the centre of our galaxy, possibly \nsetting constraints on deviations\nfrom Kerr \\cite{Johannsen:2016vqy}. It is important therefore to investigate \nwhether spinning black hole solutions exist in modified gravity, and if so, \ndo they carry non-trivial extra degrees of freedom?  \n\nIn this letter, we focus on a particular family of modified scalar-tensor \ntheories of gravity, developing a new technique to find, for the first time, \nastrophysically realistic rotating black hole solutions \nwith a nontrivial scalar field, or \\emph{stealth hair}.\nOur method is based on the Hamilton-Jacobi approach\nto finding geodesics. In brief, we prove that a GR solution can also\nbe a solution to modified gravity if the scalar field is the Hamilton-Jacobi\npotential for a geodesic congruence in the spacetime in question.\nThis allows for a wide range of physically interesting stealth solutions,\nin particular we present examples of rotating black holes with stealth hair.\n\nWe focus for definiteness on shift-symmetric scalar-tensor theories \nof gravity in the family of Degenerate Higher Order Scalar-Tensor (DHOST) \ntheories \\cite{Langlois:2015cwa,Crisostomi:2016czh,BenAchour:2016fzp}, \nalthough the technique can be generalised to \nother modifications of gravity. In particular we will focus on the subset of \ntheories where gravitational waves propagate at the speed of \nlight,  $c_T=1$, in accord with the recent multi-messenger neutron \nstar binary merger observation~\\cite{GBM:2017lvd}. \n\nThe most general shift-symmetric scalar-tensor theory of gravity in\nwhich gravitational waves propagate with the speed of light is\n\\begin{equation}\n\\begin{aligned}\n{\\cal L}\\, &= \\, K(X) + G(X) R \n+ A_3 \\phi^{\\mu} \\phi_{\\mu \\nu} \\phi^{\\nu} \\Box \\phi \\\\\n&+ A_4  \\phi^{\\mu} \\phi_{\\mu \\rho} \\phi^{\\rho \\nu} \\phi_{\\nu} \n+ A_5  (\\phi^{\\mu} \\phi_{\\mu \\nu} \\phi^{\\nu})^2 \\,, \n\\end{aligned}\n\\label{ESTlag}\n\\end{equation}\nwhere $K$, $G$ and $A_i$ are all functions of $X = (\\partial\\phi)^2$, \nand we abbreviate $\\partial_\\m \\phi$ as $\\phi_\\mu$, and  $\\nabla_\\n \\nabla_\\m \\phi$\nas $\\phi_{\\m\\n}$. In order to propagate a single scalar degree of freedom \nand avoid Ostrogradski instabilities, $A_{4,5}$ are constrained by\n\\begin{equation}\n\\begin{aligned}\nA_4&= -A_3 +\\frac{1}{8 G}(4 G_X + A_3 X)(12 G_X + A_3 X) \\,, \\\\\nA_5 &= \\frac{A_3}{2 G}(4 G_X + A_3 X)\\,,\n\\end{aligned}\n\\end{equation}\nwhere $G_{X} = \\partial G \/ \\partial X$. These constraints reduce the number of free \nfunctions to three ($K, G$ and $A_3$) and for simplicity we do not consider \nthe cubic Horndeski term in our Lagrangian~(\\ref{ESTlag}), though this can \neasily be reinstated if required.\n\nA family of exact static spherically symmetric solutions, with the scalar field \nplaying the role of dark energy, was initially found in a class of shift-symmetric \nHorndeski theories \\cite{Babichev:2013cya}. It has the nice feature of locally \ndescribing a Schwarzschild geometry while asymptotically approaching a \nself-tuned accelerating cosmology. Since these solutions acquire a metric \nsimilar to that of GR while having a non trivial scalar field, they have been \nwidely called {\\it stealth} solutions\\footnote{These solutions were extended \nand generalised in different modified gravity theories with similar properties, \nsee for example \\cite{Rinaldi:2012vy,Kobayashi:2014eva,Babichev:2017guv,\nChagoya:2016aar,Babichev:2017rti}.}. \nThey can be mapped via disformal transformations to stealth solutions \nof DHOST theories with unitary speed of gravitational waves \n\\cite{Babichev:2017lmw}, and are free of ghost and gradient instabilities \n\\cite{Babichev:2018uiw}. For spherically symmetric stealth solutions in \nDHOST theories see \n\\cite{Chagoya:2018lmv,BenAchour:2018dap,Motohashi:2019sen}.\n\nKnown stealth solutions are spherically symmetric and all feature \nthe same characteristic: a constant kinetic term, $X$, for the scalar field\nthat does not deform an underlying Einstein geometry.\nWe suspect that it is this feature of a constant magnitude of $\\partial \\phi$\nthat allows stealth hair, thus we look for an Einstein manifold,\n$R_{\\m\\n}= \\L g_{\\m\\n}$, admitting such a solution for $\\phi$.\nFirst, the equations of motion under these assumptions \nbecome \n\\begin{equation}\n\\begin{aligned}\n&\\left[ A_3(X_0) \\left({\\cal E}_3 - \\L X_0 \\right) \n- 2 \\left(K_X +4 \\L G_X\\right)|_{_{X_0}} \\right] \\phi_\\m \\phi_\\n\\\\\n& +\\left(K + 2\\L G \\right)|_{_{X_0}}g_{\\m\\n} = 0\n\\end{aligned}\n\\label{EOMg}\n\\end{equation}\nfor the metric and\n\\begin{equation}\n\\begin{aligned}\n&A_3(X_0) \\left( {\\cal E}_4 +2R_{\\m\\n\\rho\\s} \\phi^{\\n\\s}\\phi^\\m \\phi^\\rho \n-3 \\L X_0 \\Box \\phi \\right) \\\\\n& - 2 \\left( K_X +4 \\L G_X \\right)|_{_{X_0}} \\Box \\phi =0 \n\\end{aligned}\n\\label{EOMphi}\n\\end{equation}\nfor the scalar, where for compactness we have defined \n\\begin{equation}\n\\begin{aligned}\n{\\cal E}_3 &\\equiv \\left( \\Box \\phi \\right)^2 - \\left( \\phi_{\\m\\n} \\right)^2 \\,, \\\\\n{\\cal E}_4 &\\equiv \\left( \\Box \\phi \\right)^3 -3 \\Box \\phi \\left( \\phi_{\\m\\n} \\right)^2 \n+ 2 \\left( \\phi_{\\m\\n} \\right)^3 \\,.\n\\end{aligned}\n\\end{equation}\nSetting aside the problem of finding a solution for $\\phi$ momentarily,\nnote that in the above, $A_3$, $G$ etc.\\ are all constants,\nevaluated at some $X=X_0=(\\nabla\\phi)^2$. Since $\\phi_\\mu$ itself\nis not necessarily a constant vector, to satisfy the above in general \nwe must choose subspaces of the general parameter space for\n$\\{A_3,G,K\\}$. Starting with the top line of \\eqref{EOMphi}, we\ndeduce $A_3(X_0)=0$ (unless spacetime has very special\nsymmetries as we will see later), hence \n\\begin{equation}\n(K_X+4\\Lambda G_X)|_{X_0}=0 \\,, \\label{Gconstr}\n\\end{equation}\nwhere we emphasise that this is \\emph{at the specific value} of $X$,\n$X_0$. These two constraints now imply that \\eqref{EOMg} is\nsatisfied, provided we set $\\Lambda = -K\/(2G) |_{X_0}$. In other words,\nthe cosmological constant appearing in the Einstein manifold is no\nlonger the bare cosmological constant included in \nthe constant part of the $K$ function. This is the \\emph{self-tuning}\nproperty of these gravity theories. \n\nTo sum up: Given a general Lagrangian \\eqref{ESTlag}, we first look for \nzeros of $A_3$ that determine the value(s) of $X_0$, then ask that the \nderivatives of $G$ and $K$ are related at $X=X_0$ as required above. The \neffective cosmological constant is then fixed by the ratio of~$K$ to~$G$.\n\nHaving established the conditions under which Einstein-like metrics can be\nsolutions to modified gravity, we now make a key observation that allows \nus to construct a stealth solution on the Einstein manifold: \nGiven sufficient symmetry in a spacetime, the geodesic equation\n\\begin{equation}\n\\frac{d^2 x^\\m}{d \\l^2} + \\Gamma^\\m_{\\rho\\s} \\frac{d x^\\rho}{d \\l} \n\\frac{d x^\\s}{d \\l} = 0 \\label{geo}\n\\end{equation}\ncan be solved using a Hamilton-Jacobi potential $S$, such that the gradient\nof the potential gives the tangent vector of the geodesic\n\\begin{equation}\n\\frac{\\partial S}{\\partial x^\\m} = p_\\m =  g_{\\mu\\nu}\n\\frac{d x^\\n}{d \\l}  \\,.\n\\label{HJp}\n\\end{equation}\nTypically, this method is used to simplify the solution of a particular\ngeodesic (such as the orbit of a planet), however, the form of the potential\ncan be used over a wider range of co-ordinate values that in the case of\na hypersurface orthogonal geodesic congruence becomes effectively\nthe whole of the spacetime. Thus, given that $\\phi_\\mu$ has constant \nmagnitude, as does the tangent vector of an affinely parametrised geodesic, \nit is natural to make the identification\n\\begin{equation}\n\\phi \\leftrightarrow S \\,, \n\\label{parallel}\n\\end{equation}\nthe properties of the geodesic congruence then will ensure that $\\phi$\nhas the requisite properties to be a stealth solution to the extended\ngravity equations of motion. Moreover, this provides a nice physical \ninterpretation of the constants appearing in the solution. \n\nWe will now illustrate this technique and find a rotating black hole\nwith stealth hair. Consider the Kerr-(A)dS geometry \\cite{CARTER1968399}\n\\begin{equation}\n\\begin{aligned}\nds^2 &= - \\frac{\\Delta_r}{\\Xi^2 \\rho^2} \\left[ dt - a\\, \\sin^2\\theta d \\varphi \\right]^2 \n+ \\rho^2 \\left( \\frac{d r^2}{\\Delta_r} + \\frac{d \\theta^2}{\\Delta_\\theta} \\right) \\\\\n&+ \\frac{\\Delta_\\theta \\sin^2\\theta}{\\Xi^2 \\rho^2} \n\\left[ a\\, dt - \\left( r^2 + a^2 \\right) d \\varphi \\right]^2 \\,, \n\\end{aligned}\n\\label{Kerr}\n\\end{equation}\nwhere\n\\begin{equation}\n\\begin{aligned}\n\\Delta_r &= \\left( 1 - \\frac{r^2}{\\ell^2}\\right) \\left( r^2 + a^2 \\right) -2Mr \\,, \\;\\;\n\\Xi = 1 + \\frac{a^2}{\\ell^2} \\,, \\\\\n\\Delta_\\theta &= 1 + \\frac{a^2}{\\ell^2}\\cos^2\\theta \\,,\\qquad\n\\rho^2 = r^2 + a^2 \\cos^2\\theta \\,, \n\\end{aligned}\n\\end{equation}\n$M$ is the black hole mass, $a$ the angular momentum parameter \nand $\\ell=\\sqrt{3\/\\Lambda}$ is the de Sitter radius, related to the \neffective cosmological constant (for AdS reverse the sign of $\\ell^2$).\n\nApplying the Hamilton-Jacobi technique, we note that the components of \nthe metric \\eqref{Kerr} are independent of $t$ and $\\varphi$, thus $E=-p_t$ \nand $L_z=p_\\varphi$ are two constants of the motion, identified with \nthe energy and the azimuthal angular momentum respectively.\nA third constant of motion is the magnitude of the tangent vector \n$g^{\\m\\n}p_\\m p_\\n=X_0=-m^2$, associated with the rest mass of the\ntest particle\\footnote{Note, for illustration we take timelike geodesics,\nshould a spacelike congruence be required, substitute $m^2 \\to -m^2$\nin the derivation.}. Most importantly however, a fourth\nconstant was discovered by Carter \\cite{Carter:1968rr}\n(here generalised to include $\\Lambda$):\n\\begin{equation}\n\\begin{aligned}\n{\\cal Q} &= \\Delta_\\theta \\, p_\\theta^2 + m^2 a^2 \\cos^2\\theta \\\\\n&- \\Xi^2 \\left[ \\left( a \\, E - L_z\\right)^2 - \\frac{\\sin^2\\theta}{\\Delta_\\theta} \n\\left( a \\, E - \\frac{L_z}{\\sin^2\\theta}  \\right)^2 \\right] \\,,\n\\end{aligned}\n\\end{equation}\nwho demonstrated that the geodesic equation was separable. We can\ntherefore write\n\\begin{equation}\nS = - E\\, t + L_z \\varphi + S_r (r) + S_\\theta (\\theta) \\,, \\label{S}\n\\end{equation}\nwhere\n\\begin{equation}\nS_r = \\pm \\int \\frac{\\sqrt{R}}{\\Delta_r} dr \\,, \\qquad \nS_\\theta = \\pm \\int \\frac{\\sqrt{\\Theta}}{\\Delta_\\theta} d\\theta \\,,\n\\label{sdefns}\n\\end{equation}\nwith\n\\begin{figure}\n\\includegraphics[width=0.4\\textwidth]{fullpenrose.pdf} \n\\caption{\\label{fig:penrose} Contours of constant $\\phi$ for \n$a=GM=0.1\\ell$, $\\eta_c = 0.612$ in local Kruskal coordinates \nfor the future event horizons, $\\kappa_b U = -e^{-\\frac{\\kappa_b (t-r^\\star)}{2}} ,\n \\kappa_cV = - e^{-\\frac{\\kappa_c  (t+r^\\star)}{2}}$, ($\\kappa_i$ being\nthe absolute values of the surface gravities of each horizon).}\n\\end{figure}\n\\begin{eqnarray}\nR &=& \\Xi^2 \\left[ E \\left( r^2 + a^2 \\right) -a\\,L_z \\right]^2 \\nonumber \\\\ \n&-& \\Delta_r \\left[ {\\cal Q} +  \\Xi^2 \\left( a \\, E - L_z \\right)^2 + m^2 r^2  \\right] \\,, \\\\\n\\Theta &=& - \\Xi^2 \\sin^2\\theta \\left( a\\,E - \\frac{L_z}{\\sin^2\\theta} \\right)^2 \\nonumber \\\\\n&+& \\Delta_\\theta \\left[ {\\cal Q} +  \\Xi^2 \\left( a \\, E - L_z \\right)^2 \n- m^2 a^2 \\cos^2\\theta \\right] \\,.\n\\end{eqnarray}\n\nNow let us look for explicit solutions for the scalar field $\\phi=S$. This places\nfurther constraints on the potential, as we require $\\phi_\\mu$ to\nbe regular throughout the spacetime. Checking regularity on the axes\nrequires $\\partial S\/\\partial \\theta\\to0$ as $\\theta\\to 0,\\pi$, i.e.\\ \n$\\Theta \\propto \\sin^2\\theta$. This in turn requires $L_z=0$ and \n${\\cal Q} + \\Xi^2 a^2 E^2 = m^2 a^2$, and writing $\\Xi E=\\eta m$,\nwe get:\n\\begin{equation}\n\\begin{aligned}\n\\Theta &= a^2 m^2 \\sin^2\\theta \\left ( \\Delta_\\theta - \\eta^2 \\right) \\,, \\\\\nR &= m^2 (r^2+a^2)\\left (\\eta^2 (r^2+a^2) - \\Delta_r\\right ) \\,.\n\\end{aligned}\n\\label{RTheta}\n\\end{equation}\nThis has now reduced the parameter space to an overall scaling, $m$,\nand a ``relative energy'' $\\eta$, constrained to lie in $\\eta\\in[\\eta_c,1]$;\nthe upper limit coming from $\\Theta\\geq0$, and the lower limit from \n$R\\geq0$ in \\eqref{RTheta}.\n\\begin{figure}\n\\includegraphics[width=0.35\\textwidth]{nearhorizoncontours.pdf} \n\\caption{\\label{fig:phicont} \nContours of $\\phi$ at constant $v=t+r^\\star$ in the $\\{r,\\theta\\}$ plane \nnear the black hole horizon with the same parameter values\nas figure \\ref{fig:penrose}, taking $m=100$.\n}\n\\end{figure}\n\nAt first sight, it appears we have four distinct solutions coming from\nthe choice of signs in \\eqref{sdefns},\nhowever, an interesting restriction occurs when $\\eta=1$ or $\\eta_c$.\nIn this case $\\Theta$ (or $R$) vanishes for some value of $\\theta$\n(or $r$), and the branch choice changes. This is most easily seen for\n$\\eta=1$, here $\\Theta_1 = \\frac{m^2 a^4}{\\ell^2} \\sin^2\\theta \\cos^2\\theta$,\nand the natural root is $\\cos\\theta$ which changes sign across the \nhemisphere. The same phenomenon occurs for $R$, but this\nleads to an important consequence as we now discuss.\n\nInspection of \\eqref{sdefns} shows that \n$S_r \\sim \\pm m\\eta r^\\star$ near the event horizons, where\n$r^\\star = \\int dr(r^2+a^2)\/\\Delta_r$ is the tortoise coordinate,\ntherefore, if we interpret $\\sqrt{R}$ as being the positive root,\nour scalar field will be divergent at one or the other horizon\n(dependent on the branch choice). \nNote however, that for $\\eta_c$, $R$ has a quadratic zero at\nsome $r_0$: $R\\sim R''(r_0) (r-r_0)^2\/2$, thus the true root,\n$\\sqrt{R}\\sim(r-r_0)$, changes sign at $r_0$. This means\nthat for $\\phi$ to be differentiable, we must change the sign \nof $\\sqrt{R}$ across $r_0$ and set\n\\begin{equation}\nS_r = (H[r-r_0] - H[r_0-r]) \\int_{r_0}^r \\frac{|\\sqrt{R}|\\,dr}{\\Delta_r}\n\\end{equation}\nwhere $H$ is the Heaviside step function.\nThis now renders $\\phi$ finite at both future event horizons,\nand infinitely differentiable between the horizons as shown\nin figure \\ref{fig:penrose}.\n\nIt is worth emphasising this last point: All black hole solutions in\nthe literature for higher order scalar-tensor gravity\nare spherically symmetric, and\nhave scalar fields that diverge either on the black hole or\ncosmological event horizon. While this is not a physical\nproblem when $\\phi$ interacts with gravity only through its gradient, \nit is nonetheless a less appealing feature of these\nsolutions. Here, we have constructed a \\emph{rotating}\nblack hole with \\emph{finite} stealth scalar hair. This scalar\nwill be manifestly continuous across each horizon, and be\nstraightforward to analyse in perturbation theory.\nFinally, the integral for the $\\theta-$potential $S_\\theta$ gives\n\\begin{equation}\n\\begin{aligned}\n\\pm S_\\theta = &\\eta \\,\\log \\left [\n\\frac{ \\sqrt{1-\\eta^2+\\frac{a^2}{\\ell^2}\\cos^2\\theta}\n+ \\frac{a}{\\ell} \\cos\\theta}{\\sqrt{(1-\\eta^2)\\Delta_\\theta}} \\right]\\\\\n&\\;\\;\\;-\\log \\left [\n\\frac{ \\sqrt{1-\\eta^2+\\frac{a^2}{\\ell^2}\\cos^2\\theta}\n+ \\frac{a}{\\ell} \\cos\\theta}{\\sqrt{1-\\eta^2}} \\right]\\;,\n\\end{aligned}\n\\end{equation}\nleading to an ``off-centre'' behaviour in the scalar as shown in \nfigure \\ref{fig:phicont}.\n\nFor $\\eta>\\eta_c$, the radial function $R$ has no zero, and \nthe scalar field diverges on one horizon, in common with the \nknown solutions in the literature. The field also demonstrates a \nsimilar asymmetry in $\\theta$, except for $\\eta=1$, when \n\\begin{equation}\nS_\\theta =  \\pm \\frac{m\\ell}{2} \\log \\Delta_\\theta \\,,\n\\end{equation}\nrendering the angular variation symmetric about the equator\nas shown in figure \\ref{fig:nari}.\n\\begin{figure}\n\\includegraphics[width=0.35\\textwidth]{phicontoursnari.pdf} \n\\caption{\\label{fig:nari} \nContours of $\\phi$ for $\\eta=1$ at constant $v=t+r^\\star$ in the \n$\\{r,\\theta\\}$ plane. ($a=GM=0.22\\ell$, $m=100$).\n}\n\\end{figure}\n\nOur solution for $\\phi$ shows a clear dependence on both $\\theta$ and $r$,\nas well as the time dependence in common with known stealth solutions \nhaving spherical symmetry \\cite{Babichev:2013cya}. The key \ndifference is that we can construct a scalar that is finite on\nboth the black hole and cosmological horizon. To compare to\nsolutions in the literature, we take spherical symmetry ($a\\to0$)\nand find general solutions of the form $\\phi = -m \\eta t \\pm S_r$.\nOnce again, we have a finite-$\\phi$ solution for $\\eta_c^2 \n= 1-3(M\/\\ell)^{2\/3}$, \n\\begin{equation}\n\\phi = - m \\left [ \\eta_c t + \\int \\frac{r(r-r_0) \\sqrt{r(r+2r_0)}}{\\ell\\Delta_r} dr\\right]\\,,\n\\label{regso3phi}\n\\end{equation}\nwhere $r_0 = M^{1\/3} \\ell^{2\/3}$. It is interesting to compare this to\nthe time-dependent solution of a black hole in slow-roll inflation\n\\cite{Chadburn:2013mta,Gregory:2018ghc}. \nThere, the scalar profile $\\phi_{SR}\\propto T$ (for a suitable time\ncoordinate) is also finite at both horizons, but \\eqref{regso3phi}\nhas constant gradient, whereas $\\phi_{SR}$ \nsolves a wave equation, resulting in a slightly different\nradial profile between the horizons.\n\nAlso, note that in the case of spherical symmetry, we can relax our \nconstraint $A_3(X_0)=0$. In this case ${\\cal E}_3, \\,{\\cal E}_4$ and the \nRiemann tensor term appearing in (\\ref{EOMg},\\ref{EOMphi}) have a \nsimple form, and combine to require\n\\begin{equation}\n(K_X+4\\Lambda G_X+\\frac{3}{2}\\Lambda X A_3)|_{X_0}=0 \\,,\n\\end{equation}\nwith the same self-tuning condition for the cosmological \nconstant\\footnote{See \\cite{Babichev:2016kdt} for a precise analysis \nof self tuning conditions in spherical symmetry.}, however, note\nthat $A_3(X_0)\\neq0$ \\emph{requires} $\\eta\\equiv1$, thus we\nno longer have the $\\eta-$degree of freedom.\nWhat now happens is that spacetime\nbecomes foliated by surfaces of constant $\\phi$ that are\nflat (for $\\L=0$).\n\nAnother interesting possibility for spherical symmetry is a static \nsolution, found by setting $E=0$, hence $X_0>0$ and $\\phi$\ncorresponds to a congruence of spacelike geodesics:\n\\begin{equation}\n\\phi_s = S_r(r) = \\sqrt{X_0} \\int \\frac{r}{\\sqrt{\\Delta_r}}\\,dr \\,,\n\\end{equation}\nagreeing with a solution reported in \\cite{Motohashi:2019sen}.\n\nIt is also worth noting a side result of our analysis: A search for \nsolutions with $X=X_0$ and spherical symmetry in $c_T=1$ \ntheories (\\ref{ESTlag}) allows \\emph{only} for Einstein geometries. \nThis is unlike Horndeski theories (with $c_T\\neq 1$) where solutions \nof black holes and solitons were found that have $X=X_0$ but are \n\\emph{not} Einstein spaces (for a concise up to date review see \n\\cite{Lehebel:2018zga}). This is an interesting feature, hinting that \nsolutions of $c_T=1$ theories, where $X$ is not constant belong to \nbranches that eventually flow towards $X$ constant solutions with \nan Einstein space metric.\n\nTo sum up: we have presented the first exact solutions for a rotating black\nhole with scalar hair in shift-symmetric \nscalar-tensor theories with unitary speed for gravitational waves.\nOur method was based on an interesting correspondence between \nfamilies of black hole geodesics and stealth solutions in modified gravity. \nThe geodesic correspondence can be used to find other stealth solutions, \nfor instance for more general type D spacetimes,\nor even non-stealth solutions that do not exist in GR --\none only needs to compute the Hamilton-Jacobi potential for a \ngeodesic in the relevant spacetime. \nIt is also plausible that this technique can be extended to other modifications of \ngravity, such as vector-tensor theories where we know that certain stealth \nsolutions exist. Although we did not consider charged black holes, clearly one\ncan also use this method to find stealth Kerr-Newman solutions.\nPerhaps most importantly, we have\npresented a scalar solution that is finite at both the cosmological and \nevent horizons, thus manifestly extendible beyond the cosmological horizon.\nThe angular asymmetry of this solution could provide a distinctive\nsignature of this hair, although this would require a full perturbation \nanalysis beyond the scope of this investigation.\n\n\n\\SEC{Acknowledgments.} \\;\\;\nIt is a pleasure to thank Eugeny Babichev, Gilles Esposito-Far\\`ese, \nAntoine Leh\\'ebel, Karim Noui, Eric Gourgoulhon, \nKarim Van Aelst for many interesting discussions. CC and NS would like \nto acknowledge networking support by the GWverse COST Action CA16104, \n\\emph{Black holes, gravitational waves and fundamental physics}. \nCC thanks the Laboratory of Astronomy of AUTh in Thessaloniki for \nhospitality during the course of this work.\nMC is supported by the Labex P2IO and the Enhanced Eurotalents Fellowship.\nRG is supported in part by the STFC [consolidated grant ST\/P000371\/1], \nand in part by the Perimeter Institute. Research at Perimeter Institute is \nsupported by the Government of Canada through the Department of Innovation, \nScience and Economic Development and by the Province of Ontario \nthrough the Ministry of Research and Innovation.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}