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+{"text":"\\section{Introduction} \\label{sec:intro}\n\nSmall bodies are the final frontier in the study of flux-limited populations in the Solar System. While these objects are primarily very small and often very distant, they are nevertheless critical to our understanding of the formation of the Solar System. For example, Trans-Neptunian Objects (TNOs) contain dynamically-unperturbed relics from the formation of the Solar System \\citep{Luu+Jewitt2002}. They provide a window into the early history of the Solar System and enable tests of planetary formation and migration hypotheses. The Nice model \\citep{NiceModel} suggests that all the giant planets formed well-interior to 20 au and migrated outwards due to interactions with planetesimals. Better knowledge of the dynamical populations of TNOs would enable tests of additional and alternative hypotheses regarding the dynamical history of the Solar System, such as the smooth migration of Neptune \\citep{Hahn+2005, NesvornyDavid2015JNCE, MorbidelliAlessandro2019Kbfa}, a stellar flyby \\citep{Kenyon+2004}, or rogue planetary embryos \\citep{Gladman+2006}. Improving our understanding of TNO size and orbital distributions, especially at the low-mass end where they are more poorly constrained, will be critical for our understanding of these and other hypotheses.\n\nWell beyond the edge of the Classical Kuiper Belt lies the Oort Cloud. Most famously, the Inner Oort Cloud includes the object Sedna, which is thought to be a member of a larger population of sednoids \\citep{Brown_2004}. Sedna is an Inner Oort Cloud object with a perihelion of $76\\pm4$ au, but a semimajor axis of $480\\pm40$ au. Therefore, it spends well over $90\\%$ of the time on its orbit beyond the detection limit of the survey that discovered it. According to the Minor Planet Center (MPC) database for TNOs, centaurs, and scattered disk objects (SDO), Sedna has the second largest perihelion (after 2012 VP113) of any detected Solar System object. It represents one of the only currently-observable links to the Oort Cloud, a region that contains a wealth of information about the history of the Solar System. If we could increase the number of known sednoids and other Oort Cloud objects, they would provide observational constraints on the formation environment of the Sun \\citep{BRASSER200659} and the Sun's dynamic history in the Milky Way after leaving its formation environment \\citep{KAIB2011491}.\n\nNew and upcoming approaches to survey astronomy provide exciting opportunities for the study of these populations. For example, the upcoming Legacy Survey of Space and Time \\citep[LSST; lsst.org;][]{Ivezic_2019} expects to survey over 18,000 square degrees of the sky 825 times over a period of 10 years, generating about 20 TB of data every 24 hours. \n\nLSST plans to detect Solar System objects from individual images, with a single-visit limiting magnitude in the r band of 24.7, and link these detections to measure orbits. Current projections \\citep{lsstsciencebook} show that LSST is expected to detect about 40,000 TNOs, which is by itself a large increase over the currently-known 4077 Centaurs, KBOs, and SDOs (MPC). However, if we could coadd the images to increase the signal-to-noise ratio (SNR) of the Solar System detections, then we could recover significantly more objects. Following the formula $\\Delta m = (5\/2) \\log{\\sqrt{N}}$, coadding just three months of LSST data would increase the limiting magnitude in the $r$ band from 24.7 to 26.1. This increase in depth means LSST would detect $\\sim 8.0$ times more TNOs compared to a single image, assuming the single power-law $r$ band KBO distribution of \\citet{FRASER2008827}. Instead of 40,000 TNOs, we could detect $\\sim 320,000$ TNOs. If we could coadd a year of LSST data, this increases to over 520,000 new TNOs detected (given our simplified assumptions). None of this requires any more data than LSST will already acquire.\n\nCoadding moving objects poses unique challenges compared to coadding stars. Because stars move very slowly compared to most survey cadences, coaddition of a stack of aligned images usually increases the limiting magnitude for stars compared to single images. Solar System objects, however, generally move at on-sky velocities of $>1''\\ \\mathrm{hr^{-1}}$, due to both the proper motion of the objects and the reflex motion caused by the Earth's orbit. This means that traditional image coaddition typically does not increase the number of detectable moving objects. Known moving objects may be tracked and aligned along their orbits to improve the quality of the detection, but to use image coaddition to detect new objects with unknown orbits, another approach is required.\n\nThe Kernel-Based Moving Object Detection \\citep[KBMOD;][]{kbmod} algorithm takes a time series of images of the same RA and Dec, uses a ``track before detect\" (TBD) approach to account for the potential motion of objects on an image,  and then coadds the shifted images (increasing the SNR of objects with the candidate trajectory). To sample all possible orbital parameters requires searching billions of candidate trajectories even within the footprint of a single charge-coupled device (CCD). Consequently, current implementations of TBD have generally been restricted  to narrow-field surveys \\citep{Bernstein_2004}. KBMOD addresses this by using GPU-accelerated computing to search over a wide range of trajectories for a stack of CCDs in of order 10 minutes.\n\nIn this paper, we present a number of algorithmic improvements to KBMOD that allow us to search for moving objects in difference images. We use the Dark Energy Camera (DECam) NEO Data Survey to validate our improvements. This is a larger survey with a longer and more irregular cadence than KBMOD has been applied to in the past. Successfully running on difference images and a more complicated survey shows that KBMOD is beginning to be applicable at the scale needed for upcoming big data surveys like LSST. In Section \\ref{sec:data}, we discuss the DECam NEO Data Survey and the processing we applied to it using the LSST Software Stack. In Section \\ref{sec:tech}, we discuss the KBMOD algorithm and present recent improvements. In Section \\ref{sec:results}, we discuss the results from our analysis, including the detection of unidentified outer Solar System objects. We discuss current limitations and future improvements in Section \\ref{sec:discuss}.\n\\section{Data} \\label{sec:data}\n\n\\subsection{The DECam NEO Survey Data} \\label{sec:neodata}\n\nThe DECam NEO Data Survey covered an area on the sky of greater than 2000 square degrees. The $\\sim$6.7 TB data set from the DECam NEO Data Survey (PI Lori Allen) uses the Dark Energy Camera on the 4m Blanco telescope at the Cerro Tololo Inter-American Observatory (CTIO) \\citep{Flaugher_DECam_Instrument}. The DECam NEO Data Survey consists of 32 nights of data.  In the first 10-night observing run in 2014, \\citet{Trilling2017} found 235 unique NEOs.\n\nEach individual image taken by DECam is a composite of 62 2K x 4K science CCDs, with a fill factor of 0.8 \\citep{HERNER2020100425}. Each CCD image covers an area of $\\sim$0.04 square degrees with a pixel scale of 0.27 arcseconds. This results in a total field of view for DECam of about 3 square degrees. The CCDs are $250\\ \\mathrm{\\mu m}$ thick fully depleted devices, with a peak quantum efficiency above $85\\%$ at $\\sim6500$\\AA \\citep{Flaugher_DECam_Instrument}. Gaps between CCDs are between 153 pixels (columns) and 201 pixels (rows). Observations for this data set were taken in the \\textit{VR} filter, a broad optical filter extending from 500 to 760 nm.\n\nWe separate this data set into 782 pointing groups based on RA and Dec. CCD 01 and 61 had no data in our images, leading to a set of 60 CCDs per pointing group. We define a pointing group as a set of DECam exposures within $25''$ of a common RA and Dec and define a pointing as an individual DECam exposure (i.e. a set of 60 CCDs) in a pointing group. Most pointing groups contain between 5 and 25 pointings. Pointing groups characteristically have 5 pointings per night, with all data taken over nearly-consecutive nights. The intra-night pointings are taken about five minutes apart for a total intra-night timespan of approximately 25 minutes.\n\n24 pointing groups had a high stellar number density, with more than 10000 sources detected in a CCD. When astrometrically calibrating these images (see Section \\ref{sec:dataprocess}), these pointing groups exceeded the memory limits of the available computational resources and were therefore excluded. The current limitations regarding the processing of dense fields with LSST Science Pipelines are described in \\citet{sullivan_ian_2021_5172677}. Detectability of moving objects with KBMOD, however, is driven strongly by the quality of the difference images.\n\n372 pointing groups contained data from at least four unique survey nights. Because of the short intra-night image cadence, which can cause slow-moving objects to exhibit minimal motion within a night, we only search over pointing groups with at least four unique survey nights. This ensures that any given KBMOD trajectory will search a sufficiently-large number of unique on-sky positions, thereby reducing the probability of linking of static objects.\n\nIn order to comply with computational limitations, we selected 43 pointing groups from the set of 372 pointing groups, focusing our research on higher-quality data. These 43 pointing groups have a total effective search area of approximately 132 square degrees. We refer to these 43 pointing groups as the ``search sample''. This down select from 372 pointing groups was as follows. 40 pointing groups existed where all pointings in the pointing group had a maximum seeing full width at half maximum (FWHM) of $1.25''$. These 40 pointing groups make up the bulk of the search sample. There were an additional 12 pointing groups that had over 20 total pointings, but with only 20 pointings with seeing $< 1.25''$. These pointing groups returned a greater number of erroneous candidate trajectories that required by-eye rejection. This is possibly due to the inclusion of poor-seeing images in the image differencing template (see Section \\ref{sec:dataprocess}). Due to computational limitations, we elected to run KBMOD on only 3 of these pointing groups, focusing our GPU resources on the 40 pointing groups where all 20 pointings had the required seeing limits. These 3 pointing groups make up the remainder of the search sample.\n\n\\subsection{Processing the DECam Data} \\label{sec:dataprocess}\n\nThe raw DECam images were processed by the DECam Community Pipeline \\citep{Valdes2014} resulting in a set of InstCal PROCTYPE images, as defined in the NOAO Data Handbook \\citep{NOAO+handbook}. These images are bias and linearity corrected, flat-fielded, and sky-subtracted by the community pipeline. Data quality masks and inverse variance arrays were provided. We downloaded the compressed InstCal data from the NOAO Data Archive between July and November of 2017.\n\nPrior to running the KBMOD pipeline, we first astrometrically calibrate the images in all 782 pointing groups. This was undertaken using the LSST Science Pipelines Software \\citep{LSST_DM}. Sources were detected in the individual images. Sources with a $\\mathrm{SNR}>=40$ were matched to the data from the GAIA Data Release 1 (DR1). The median astrometric scatter for the sources used to fit the CCD world coordinate systems (WCS) was 25 mas; 373 CCDs had an astrometric scatter worse than 100 mas. The median number of sources detected per CCD was 3575.\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.9\\textwidth,keepaspectratio]{ImDiff_Example.pdf}\n    \\caption{Single pointing  (pointing group 011, CCD 29, visit 303605) before (left) and after (right) image differencing. Similar to DS9, we applied an arcsinh filter to the pixels in this example in order to better show objects in each image.} \\label{fig:diffexp}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.9\\textwidth,keepaspectratio]{Source_comparison.pdf}\n    \\caption{Number of sources per CCD image for each visit in 10 pointing groups (pointing group 091 to 100). The median number of sources per science image (orange) is 3396 per CCD image. The median number of sources per difference image (blue) is 180 per CCD image. Differencing the science images therefore reduces the number of static sources in the image by a factor of about 18.} \\label{fig:sourcecomp}\n\\end{figure}\n\nAs a followup to \\cite{kbmod}, we use image differencing to remove non-variable and non-moving sources within an image (as opposed to just masking the sources). We used the LSST Stack to difference the images in the pointing groups using a method based on \\citet{Alard1998}. For the DECam NEO Survey data, we difference each pointing against a coadded template.  Given the short intra-night time separation between images of a given pointing group, objects moving slower than of order $1\"\\ \\mathrm{hr}^{-1}$ will not move a full psf width over a single night. We therefore separate a pointing group into two approximately equal groups such that each image in the first group will be separated in time from each image in the second group by at least twelve hours. A coadded template was independently generated from each group and used to difference the opposite group. Because our minimum search velocity is $\\geq 92$ pixels per day ($\\geq 24\"$ per day), this guarantees that objects of interest will be much greater than one PSF away from where they were in the coadded template. This means that pointings in the middle of a pointing group---with respect to time --- will have the shortest image differencing baseline, and will therefore set a theoretical limit on the slowest-moving objects we can detect.\n\nIn order to difference the science images against the coadded template \\citep{Alard1998,2017ApJ...836..188Z,2017ApJ...836..187Z,2016ApJ...830...27Z}, we need to find a convolution kernel $K$ such that for a science image $I(x)$ and a coadded template $\\Phi(x)$, $I(x) = K \\otimes \\Phi(x)$. Following the approach of \\citet{Alard1998}, we separated the template into local spatial cells of 128x128 pixels. We detected sources in both images, and grouped them into the spatial cells. Stamps of these sources were created with sizes between 21x21 pixels and 35x35 pixels, depending on the FWHM of the source. Stamps in each cell were used to find the local spatially-invariant convolution kernel solutions of each stamp. The local convolutional kernel was modelled as a set of Gaussian functions multiplied with a polynomial. The coefficients of the kernel were then found by solving a least-squares problem. One source (and thus one stamp) was selected for each grid cell based on the clipped mean of all the kernel solutions in the cell. This gave the local convolution kernel for that cell. Chebyshev polynomials of the first kind were fit to the local kernel coefficients in order to determine a model for spatially-variant global convolution kernel coefficients. This global kernel was then used to match the PSF of the coadded template to that of the science image. We matched the template to the science image, rather than the other way around, because the template has less noise and the convolution correlates noise. The two images were then subtracted. Finally, a decorrelation algorithm was run to remove the correlation in the noise of the difference image. After differencing the image, we warp all images in a pointing group to the sky plane of the first pointing in the pointing group. This ensures that a pixel in one pointing will correspond to the same RA and Dec as that of the same pixel in another pointing.\n\nAs an example, Figure \\ref{fig:diffexp} shows pointing group 023, CCD 35, visit 303665 before and after image differencing and warping. The final image size for the KBMOD image is set by the intersection of the image and the template that is subtracted. Slight misalignments of the pointings in a pointing group may reduce the final image sizes. All pointing groups were, however, aligned to within $50$ arcsec in RA and Dec, with all but 28 pointing groups aligned to better than $25$ arcsec in both RA and Dec. Therefore the reduction in image area was minimal.\n\nThe asteroid search was run for each aligned stack of DECam CCDs independently; we did not search trajectories across CCD boundaries. The effective area on which we are able to search for moving objects is, therefore, about 0.04 square degrees. In other words, a necessary requirement for the detection of a moving object with the KBMOD algorithm is that the object stays within the field of view of an individual CCD for at least two pointings. In practice, we require that an object stay in the field for at least 3 nights (typically 15 pointings). This means that an object must move slower than about $15\"\\ \\mathrm{hr}^{-1}$ to be detected by KBMOD.\n\n\\section{Techniques} \\label{sec:tech}\nKBMOD generates images of likelihood ($\\Psi_i$) and variance ($\\Phi_i$) from a series of CCD images as described in \\citet{kbmod}. Assuming a Gaussian likelihood function, a stack of $\\Psi_i$ and $\\Phi_i$ images can then be shifted along a potential asteroid trajectory and summed in order to get the coadded likelihood of a detection ($\\Psi_{coadd} = \\sum_i \\Psi_i$ and $\\Phi_{coadd} = \\sum_i \\Phi_i$). See \\citet{2018AJ....155..169O} for the optimal approach for source detection with Poisson noise. We define a SNR $\\nu$ for a detection such that $\\nu_{coadd} = \\Psi_{coadd}\/\\sqrt{\\Phi_{coadd}}$. In this $\\nu$ image, generated for each given angle and velocity vector, any points above some threshold $m$ can be considered to be $m$-sigma detections of a moving source. For a single trajectory, we can define the summed likelihood as $\\sum LH = \\nu^{\\mathrm{trajectory}}_{coadd}$. The interested reader is directed to \\citet{kbmod} for more detail.\n\nThe large number ($\\gg 10^9$) of potential asteroid trajectories means that these $\\Psi_i$ and $\\Phi_i$ images must be searched many times over. For this reason, KBMOD uses massively-parallel GPU computing for the core computations. The current software allows a user to search over $10^{10}$ potential moving object trajectories in a stack of 10-15 4K x 4K images in under a minute using a consumer-grade GPU (e.g., Nvidia 1080 Ti) \\citep{kbmod}. Our pointer-arithmetic approach means that we never actually shift and stack images. Rather, we merely sum the previously-calculated likelihoods, utilizing thousands of concurrent GPU threads to keep the computation feasible on consumer-grade hardware.\n\nThe DECam NEO data set presents unique filtering challenges compared to \\citet{kbmod} due to the increased number of potentially-valid trajectories, the short intra-night cadence, and image differencing artifacts. In \\citet{kbmod}, detected sources appearing in the same position in 2 or more images, pixels with counts above 120 counts, and other mask flags set by the DECam community pipeline or the LSST software stack were all masked. In the current data set, we use difference imaging to subtract static sources. This enables us to decrease the masked area of the image, only masking sources flagged as detected if they appear in 10 or more images. However, despite reducing the number of detected individual sources on the image by a factor of about 18 (see Figure \\ref{fig:sourcecomp}), leaving most of the image unmasked, coupled with difference imaging artifacts, increases the number of trajectories with $\\sum LH > 10$ by a factor of 10. This problem is worsened by the intranight cadence. The average time between images within a single night is about 5 minutes. This means that for a characteristic trajectory with a velocity of 100 pixels per day, objects will move by less than 1 pixel between images. Conversely, this also means that if a static source appears along the potential trajectory, flux from this object will most likely be present in at least five trajectory data points, introducing repeated outliers into the trajectory.\n\n\\subsection{$\\sigma_G$ Filtering}\n\nIn order to deal with the increased number of high likelihood trajectories (i.e. $10^7$ with $\\sum LH > 10$), we developed faster, more-effective filtering. First, we altered how the GPU and C++ code handed off data to the Python-based filtering, leading to a speed increase of up to 300\\%. Second, we replaced the Kalman filter used in \\citet{kbmod} with a more statistically-robust quantile-based filtering method. We describe this new filtering method below.\n\nWith a traditional quantile-based filter, the filter rejects data points that are greater than $n\\sigma$ from the central value of the distribution, where $\\sigma$ is a measure of the spread of the distribution. In the case of a Gaussian distribution, $\\sigma$ might be estimated by computing the standard deviation of the data and the central value estimated by computing the mean of the data. If we take $n=1$, then this simple filter would reject any data points that are greater than $1\\sigma$ from the mean.\n\nIn the presence of significant outliers, the mean and standard deviation become biased estimators for the central value and the spread of the underlying Gaussian distribution. Following the approach of \\citet{astroMLText}, we adopt a robust estimator for the central value and the true standard deviation of a Gaussian distribution with outliers. Consider the cumulative distribution function (CDF) of a Gaussian distribution\n\n\\begin{equation}\nf(x) = \\frac{1}{2} \\left[ 1+ \\mathrm{erf}\\left(\\frac{x-\\mu}{\\sigma_G \\sqrt{2}}\\right)\\right]\n\\end{equation}\n\n\\noindent where $\\mu$ is the mean, $\\sigma_G$ is the standard deviation of the Gaussian, and $\\mathrm{erf}$ is the error function. The inverse, then, is given by\n\n\\begin{equation}\nx = \\mu + \\sigma_G \\sqrt{2}\\ \\mathrm{erf}^{-1} \\left[2 f(x) -1 \\right]\n\\end{equation}\n\n\\noindent By sampling the Gaussian distribution at two quantiles $f(x_i)$ and $f(x_j)$, we can estimate $\\sigma_G$. To do this, we take the difference of the inverted CDF\n\n\\begin{align}\n    x_j - x_i &= \\sigma_G \\sqrt{2} \\left( \\mathrm{erf}^{-1} \\left[2 f(x_j) -1 \\right] -  \\mathrm{erf}^{-1} \\left[2 f(x_i) -1 \\right] \\right) \\\\\n    \\implies \\sigma_G &= \\frac{1}{ \\mathrm{erf}^{-1} \\left[2 f(x_j) -1 \\right] -  \\mathrm{erf}^{-1} \\left[2 f(x_i) -1 \\right]} \\left(x_j - x_i\\right) \\\\\n    \\implies \\sigma_G &= C \\left[ x_j - x_i \\right]\n\\end{align}\n\n\\begin{figure}[tbh]\n    \\centering\n    \\includegraphics[width=1.0\\textwidth,keepaspectratio]{Num_Results_Filt_Nofilt.pdf}\n    \\caption{Number of candidate trajectories at various stages of processing for variable numbers of results per pixel. The solid line shows the number of candidate trajectories with $\\sum LH > 10$ returned from the GPU for subsequent filtering. The dashed and dotted lines show the number of candidate trajectories passing CPU $\\sigma_G$ filtering (dashed) and central moment stamp filtering and clustering (dotted). GPU filtering decreases the total number of candidate trajectories with $\\sum{LH} > 10$, but increases the number of candidate trajectories that pass subsequent lightcurve filtering and stamp filtering and clustering. Because GPU memory constraints limit the number of candidate trajectories per starting pixel that can be saved for subsequent analysis, using a GPU filter means that the results that are passed out of the GPU are more likely to be potentially-valid. These results are then processed with the CNN filter and subject to human review. These data come from repeated reprocessings of pointing group 023, CCD 35.}\n    \\label{fig:res_per_pixel}\n\\end{figure}\n\n\\begin{figure}[tbh]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth,keepaspectratio]{filter_effectiveness.pdf}\n    \\caption{Number of results per CCD image stack from pointing group 190 requiring by-eye confirmation or rejection (hereafter ``candidate trajectories'') for likelihood limits of 10 (dashed line) and 15 (solid line), with (orange) and without (blue) RESNET 50 CNN filtering. These results are from pointing group 190, one of the search sample pointing groups. Here, the CNN was set to filter out any candidate trajectories with a probability of true that was less than 75\\%. When using a LH limit of 15 and the CNN, the number of candidate trajectories per CCD was reduced to eleven or less, an acceptable number of trajectories for a human to review.}\n    \\label{fig:CNNStats}\n\\end{figure}\n\n\\begin{figure}[tbh]\n    \\centering\n    \\includegraphics[width=.97\\textwidth,keepaspectratio]{2015_GQ56.pdf}\n    \\caption{Sample output for object 2015 GQ56 (pointing group 300, CCD 30) when using trajectory estimates from the JPL Horizons service. The first row shows the coadded stamp (left) and the flux lightcurve (right). Orange points in the flux lightcurve are points that pass $\\sigma_G$ lightcurve filtering. The remaining rows show the postage stamps for 2015 GQ56 in each individual image. The coadded stamp was generated by taking the median value at each pixel; this effectively removes image differencing artifacts. This trajectory was generated using orbital values from JPL Horizons. These figures were generated for all known KBOs in search sample in order to determine the unfiltered $\\sum LH$, as well as for debugging purposes. Each stamp shows the estimated SNR $\\nu$ of that stamp.}\n    \\label{fig:knownObject}\n\\end{figure}\n\nHere, C is a coefficient dependent only on the choice of quantiles. $x_j$ and $x_i$ are estimated by selecting values from the lightcurve. The choice of upper and lower quantiles is user-determinable. Here, we estimate $\\sigma_G$ using data from the 25th to 75th percentiles, for a coefficient of $C_{25,75} \\approx 0.7413$. Then, we can estimate $x_{25}$ and $x_{75}$ from the data by selecting the 25th and 75th percentile values respectively from the data. We can then estimate the standard deviation of the underlying Gaussian distribution with $\\sigma \\approx 0.7413 \\left( x_{75}-x_{25} \\right)$. Given a robust estimator of the spread of the distribution (i.e. $\\sigma_G$), we apply a filter that rejects any points that are not within $\\pm n \\sigma_G$ (e.g. $2 \\sigma_G$) of the median of the data.\n\nWe apply this method to the likelihood and\/or flux values of each trajectory. We then recompute $\\sum LH$ for the trajectory values that pass the filter and reject the trajectory if the recomputed likelihood ($\\sum LH'$) is less than 10. In practice, this filtering method successfully rejects of order $10^6$ erroneous candidate trajectories in approximately $60$s using 30 central processing unit (CPU) cores.\n\n\\subsection{In-line GPU Filtering}\n\nApplying a variant of the $\\sigma_G$ filter in the GPU while the search is running, instead of in post-processing, increases the number of potentially-valid trajectories returned to the CPU by KBMOD. In \\citet{kbmod}, KBMOD passed the four trajectories per pixel with the highest $\\sum LH$ from the GPU to the CPU. Other trajectories with the same starting pixel were discarded. Because KBMOD searches of order $10^{12}$ trajectories for a 2K x 4K image, it is computationally infeasible to keep the results of all evaluated trajectories in GPU RAM. The disadvantage of this approach is that lower-likelihood trajectories may get removed from the search even if they are valid trajectories of true objects. With reduced masking, there are many erroneous candidate trajectories with high likelihood. This means that removing the masks may have increased the probability of discarding valid trajectories.\n\nIn-line GPU filtering solves this problem by applying the filtering method to compute $\\sum LH'$ before the trajectory is passed back to the CPU. This in-line GPU filter means that if a trajectory has a high $\\sum LH$ only due to an outlier in the data, that trajectory is unlikely to supplant another valid trajectory when GPU results are passed back to the CPU. We also increased the number of returned results per pixel from four to eight. This means that we were able to process about four times as many results per pixel compared to \\citet{kbmod}. The in-line GPU filtering uses a single GPU and is about 10\\% faster than comparable CPU filtering using 30 CPU cores. Figure \\ref{fig:res_per_pixel} demonstrates how the in-line GPU filter returns more potentially-valid trajectories for a given number of trajectories per pixel.\n\n\\subsection{Median Stamp Coadd Generation}\n\nAs shown in Figure \\ref{fig:diffexp}, saturated cores and small image misalignments leave a number of artifacts in the difference image that also have to be accounted for in the filtering process. As in \\citet{kbmod}, we computed the central moments of postage stamps for candidate trajectories. Stamps were rejected if they did not have central moments that were consistent with a Gaussian. In this data, we required that the x, y, xy, xx, and yy moments be strictly less than 0.5, 0.5, 1.5, 36.5, and 36.5 respectively. These values were chosen empirically based on the central moments of known KBOs. We generated coadded stamps by computing the median pixel value for each pixel along the trajectory. This mitigates the effect of image differencing artifacts, improving the performance of the central moment filter.\n\n\\begin{figure}[b!]\n    \\centering\n    \\includegraphics[width=1.0\\textwidth,keepaspectratio]{object_recovery.pdf}\n    \\caption{Recovered known objects as a function of reported magnitude. We ran an untargeted KBMOD search on all CCDs in the search sample that had known KBOs on them.  Figure \\ref{fig:recoveryAnalysis} shows the recovery statistics for the recovered objects. 18 of the recovered objects were below the approximate upper-limit single-image $10\\sigma$ limiting magnitude.}\n    \\label{fig:objectRecovery}\n\\end{figure}\n\n\\begin{figure}[tbh]\n    \\centering\n    \\includegraphics[width=1.0\\textwidth,keepaspectratio]{ResultsAnalysis.pdf}\n    \\caption{Statistics for the known objects that were recovered with a untargeted KBMOD search on CCDs with known objects in the search sample. For object recovery, we discarded any results that had a starting position more than 5 pixels (approximately $1.35\"$, or one PSF FWHM) from the predicted location or had a velocity difference of more than 5 pixels per day (approximately $0.056\"\\ \\mathrm{hr}^{-1}$). The velocity cutoff was chosen based on the recovery distribution. As shown in bottom left and bottom right respectively, the median difference between predicted and recovered position and speed was significantly lower than these cutoff values. The upper left plot shows each trajectory's initial predicted and recovered position on the CCD image for each object. The upper right plot shows each trajectory's predicted and recovered x and y velocity on the CCD image for each object.}\n    \\label{fig:recoveryAnalysis}\n\\end{figure}\n\n\\subsection{CNN Filtering}\n\nTo further reduce the number of false positives, we filter using a convolutional neural network (CNN). We built a Residual Network with 50 layers (ResNet50\\footnote{\\url{https:\/\/github.com\/priya-dwivedi\/Deep-Learning\/blob\/master\/resnet_keras\/Residual_Networks_yourself.ipynb}}) \\citep{he2015deep}. Residual networks are a type of CNN that add ``shortcut connections'' into the network architecture, which help to train deeper networks. Training a CNN requires a large amount of representative data. In this case, we needed a large ($>10^4$) labeled set of 21x21 stamps containing approximately equal numbers of false positives and true positives. To generate false positives, we ran an untargeted search (with similar grid spacing as described in \\ref{sec:search-detect-recovery}) with a coadded likelihood limit of $\\sum LH > 10$ along trajectories unlikely to correspond to real objects (approximately $90^\\circ$ from the direction of the ecliptic). We ran a total of 53 searches with data from 34 unique pointing groups. These pointing groups were not constrained to the search sample. These searches yielded 113,549 21x21 false positive postage stamps. Because KBOs are relatively rare, we could not use real recovered objects to generate the thousands of true positives needed to train the CNN. To circumvent this limitation, we generated 44,950 simulated true positives. To make these stamps, we retrieved 25 21x21 postage stamps from a CCD along a semi-random trajectory. Next, we drew a random brightness from an exponential distribution (with dimmer objects being the most likely). Using this brightness, we added a Gaussian to each background stamp with a random standard deviation ($1-2.1$ pixels), a random central offset ($<2$ pixels), and a random linear offset ($<2$ pixels over the image time baseline). To train the CNN, we cut the false positive stamps and simulated true stamps down to 40,000 randomly-selected coadded stamps each. We used 70\\% of the data for training, 20\\% for validation, and the remaining 10\\% for testing. After 20 epochs, the training set accuracy was about 99\\%, while the validation set accuracy was about 96\\%. After training, the test set accuracy was also about 96\\%.\n\nThis CNN returns a predicted probability that a coadded postage stamp contains a simulated object. Because the stamps of simulated objects differ from the stamps of real objects, this probability is not a perfect representation of the likelihood that a coadded stamp contains a real object. However, it creates a user-programmable threshold that can be used to reduce false positives enough that the remaining candidate trajectories can be analyzed by-eye. We reject any stamps with a CNN probability of true less than 75\\%. As shown in Figure \\ref{fig:CNNStats}, when reviewing only objects with a $\\sum LH>15$ and using this CNN filter, there are generally fewer than 10 candidate trajectories per CCD that require human by-eye confirmation or rejection.\n\n\\section{Results} \\label{sec:results}\n\\subsection{Search, Detection, and Recovery} \\label{sec:search-detect-recovery}\n\\begin{figure}[tbh]\n    \\centering\n    \\includegraphics[width=0.93\\textwidth,keepaspectratio]{OrbFit.pdf}\n    \\caption{Best-fit barycentric distance $r_0$, inclination $i$, and longitude of ascending node $\\Omega$ (dots) with respective standard deviations (lines) of the detected known objects (left) and unidentified objects (right) using the method of \\citet{Bernstein2000}. $r_0$, $i$ and $\\Omega$ were also fit with Find\\_Orb. When the value from Find\\_Orb is inconsistent with \\citet{Bernstein2000} within 1$\\sigma$, we show the best-fit value from \\citet{Bernstein2000} with a square instead of a dot. For the known objects, the JPL Horizons value of the corresponding parameter is overplotted with an x marker. The short time baseline of the observations allows us only to constrain initial barycentric distance, inclination, and longitude of ascending node. The medians of the absolute value of the residuals between the best-fit values and the JPL Horizons values are 0.36 au, 0.32 degrees, and 0.92 degrees for $r_0$, $i$, and $\\Omega$ respectively. As reported by JPL Horizons, the median values of the known objects for $r_0$ and $i$ are $\\widetilde{r_0} = 41.55$ au and $\\widetilde{i} = 5.46^\\circ$ respectively. The median values of the unidentified objects for the best-fit $r_0$ and $i$ are $\\widetilde{r_0} = 41.28$ au and $\\widetilde{i}=7.67^\\circ$. }\n    \\label{fig:orbfit}\n\\end{figure}\n\nWe ran an untargeted KBMOD search on each stack of CCDs in the search sample for a total of 2580 searches. Similar to \\citet{kbmod}, an untargeted search looks for linear trajectories with velocities between 92 and 550 pixels per day ($1.04\"\\ \\mathrm{hr}^{-1}$ to $6.19\"\\ \\mathrm{hr}^{-1}$) with angles of $\\pm \\pi\/10$ from the ecliptic angle. Compared to \\citet{kbmod}, we doubled the resolution of the grid spacing from 256 velocity steps and 128 angle steps to 512 velocity steps and 256 angle steps. This ensured that trajectories would end up separated by no more than about two PSF FWHM from neighboring trajectories.\n\nIn order to test the efficiency of these new filtering methods, we generated a list of known objects in the search sample. We used Skybot \\citep{Skybot} and JPL Horizons \\citep{JPL_Horizons} to find all KBOs that were present in the search sample, with the additional requirement that they be present in the first image of the pointing group. We generated 21x21 pixel postage stamps of the object in each image in which it is present. We developed a variant of KBMOD that computes the likelihoods along a single trajectory then runs the aforementioned quantile-based filtering, and computed the central moments of the postage stamps.  Figure \\ref{fig:knownObject} shows these results for pointing group 300, CCD 30, object 2015 GQ56. We removed KBOs with an unfiltered $\\sum LH < 15$. This left us with a ``recovery sample'' of 26 KBOs.\n\nIn the untargeted search of the search sample, we recovered 22 out of 26 (or 84.6\\%) of the known objects in the recovery sample after all filtering was applied (see Figure \\ref{fig:objectRecovery}). The CNN probability threshold was kept at 75\\%. Recovery statistics for these objects are shown in Figure \\ref{fig:recoveryAnalysis}. For object recovery, we discarded any trajectories that had a starting position more than 5 pixels (approximately $1.35\"$, or one PSF FWHM) from the predicted location or had a velocity difference from the known velocity of more than 5 pixels per day (approximately $0.056\"\\ \\mathrm{hr}^{-1}$). The median position and speed residuals were $0.427\"$ and $0.0036\"\\ \\mathrm{hr}^{-1}$ respectively, significantly below the chosen cutoff values. This velocity error corresponds to approximately a 1.27 pixel position error over four days. Using the NOAO DECam Exposure Time Calculator (ETC), we estimate the single-image $10\\sigma$ depth to be at most $22.75$V. Because the pointing groups contain data from different nights, we computed this limit assuming a new Moon. It is therefore an upper limit. 18 of the recovered objects were fainter than the upper-limit single-image $10\\sigma$ depth. This confirms that KBMOD is able to use difference images to find moving KBOs that are too dim to detect in a single image at the $10\\sigma$ level, extending the result of \\citet{kbmod} to difference images.\n\nWe investigated each of the missed known objects individually. 2013 GY136 (pointing group 204, CCD 57) failed to process due to a CCD that failed image differencing. This reduced the total number of images in CCD 57 to fewer than 20, and CCD 57 was therefore not reprocessed. 2013 GZ137 (pointing group 202, CCD 52) failed CNN filtering with a threshold of 75\\%, but passes with a threshold of 50\\%. 2015 GY55 (pointing group 306, CCD 26) starts within 4 pixels of the chip edge, causing this trajectory not to be searched by KBMOD. 2013 GH137 (pointing group 192, CCD 41) has two fully-masked stamps, and two more with partial masking, which may have caused it to be filtered out.\n\nIn addition to the detected 22 known objects in the recovery sample, we detected 2 additional known KBOs. These KBOs had an unfiltered $\\sum LH < 15$ along the JPL Horizons trajectories, and were therefore not included in the recovery sample. The best KBMOD trajectories for these objects had a filtered $\\sum LH' > 15$. We then linked these objects back with known KBOs.\n\n\\begin{figure}[tbh]\n    \\centering\n    \\includegraphics[width=1\\textwidth,keepaspectratio]{Brown_Inc_KS.pdf}\n    \\caption{One-sided Kuiper variant of the Kolmogorov-Smirnov (K-S) test comparing our recovered inclinations with the inclination distribution predicted by \\citet{brown_inclinations}. We reject the null hypothesis that our inclinations came from the distribution of \\citet{brown_inclinations} with only 76.6\\% confidence (less than 1$\\sigma$). We therefore consider our observed inclinations to be consistent with the distribution predicted by \\citet{brown_inclinations}.}\n    \\label{fig:KS_inc}\n\\end{figure}\n\n\\subsection{Orbit Fitting and Analysis}\nWe detected 75 moving objects that we were unable to link to existing objects. Trajectories with $\\sum LH > 15$ that passed all filtering were accepted or rejected with a by-eye examination of the individual stamps, the coadded stamp, and the flux lightcurve.\n\nAs shown in Figure \\ref{fig:orbfit}, we used the method described in \\citet{Bernstein2000} to fit barycentric distance $r_0$, inclination $i$, and longitude of ascending node $\\Omega$ of both the recovered known objects and the unidentified objects. For the known objects, we compared the orbital parameters fit to the KBMOD trajectory with their respective parameters as reported by JPL Horizons. The medians of the absolute value of the residuals between the best-fit values and the JPL Horizons values are 0.36 au, 0.32 degrees, and 0.92 degrees for $r_0$, $i$, and $\\Omega$ respectively. The median values for $r_0$ and $i$ of the known objects reported by JPL Horizons are $\\widetilde{r_0} = 41.55$ au and $\\widetilde{i} = 5.46^\\circ$ respectively. The median values of the unidentified objects for the best-fit $r_0$ and $i$ are $\\widetilde{r_0} = 41.28$ au and $\\widetilde{i}=7.67^\\circ$. The three parameters (shown in Figure \\ref{fig:orbfit}) that are well-fit with our data constrain the plane of the orbit and the initial distance of the object from the Solar System barycenter. Individual values are shown in Table \\ref{sec:param_table}.\n\nIn addition to the method of \\citet{Bernstein2000}, we used Find\\_Orb\\footnote{\\url{https:\/\/github.com\/Bill-Gray\/find_orb}} to fit $r_0$, $i$, and $\\Omega$. This allowed us to compare the best-fit values between the two orbit fitting codes. When best-fit values from Find\\_Orb were not within 1$\\sigma$ of the best-fit value from the \\citet{Bernstein2000} code, we show the value as a square in Figure \\ref{fig:orbfit}. We discarded values with inconsistent inclinations from the remainder of the orbit analysis.\n\nThere were a few noteworthy limitations to our dataset and apparent outliers in our best-fit values. Because of the relatively short time baseline of about four days, we were unable to place any meaningful constraints on the other Keplerian elements individually. For three unidentified objects (unidentified object numbers 58, 69, and 74), the orbit fitting code did not return uncertainties. We therefore consider them inconsistent between \\citet{Bernstein2000} and Find\\_Orb. Unidentified object numbers 4, 6, and 8 have a best-fit inclination of $i_{\\mathrm{fit}}> 90^\\circ$. Similarly, known object number 20 (2000 EE173) has a best-fit inclination of $i_{\\mathrm{fit}} = 173.36 ^ \\circ \\pm 0.54 ^ \\circ$, but a JPL Horizons inclination of $i_{\\mathrm{Horizons}}=5.95^ \\circ$. However, these 4 objects are all marked as inconsistent between Find\\_Orb and \\citet{Bernstein2000}. As such, their best-fit values are removed from further orbital analysis.\n\n\\begin{figure}[tbh]\n    \\centering\n    \\includegraphics[width=1\\textwidth,keepaspectratio]{Expected_Inc_Histogram.pdf}\n    \\caption{Inclination distributions of our detected objects (orange) and the distribution predicted based on \\citet{brown_inclinations} (blue), after accounting for the search sample ecliptic latitudes. The \\citet{brown_inclinations} is weighted to the number of objects recovered by KBMOD. Uncertainties in the orange histogram are calculated as 1$\\sigma$ Poisson intervals of $\\sqrt{N}$.}\n    \\label{fig:expected_inc}\n\\end{figure}\n\nTo evaluate the consistency of the properties of our detected asteroids with published distributions, we apply the analysis of \\citet{kbmod} to the detected objects with consistent inclinations reported in this paper. We compared our observed inclination distribution with that of \\citet{brown_inclinations} by using a one-sided Kuiper variant of the Kolmogorov-Smirnov (K-S) test. We use a test statistic of $D\\sqrt{N}$ where $N$ is the number of objects, and $D$ is given by Equation 30 in \\citet{kbmod}.\n\n\\begin{equation}\n    D=\\mathrm{max}\\left(P_j-j\/N\\right)\n\\end{equation}\n\n\\noindent$P_j$ is the probability for a given inclination distribution that an object $j$ has an inclination equal to or below the actual inclination $i_j$. Some TNO sub-populations have non-uniform inclination distributions around the ecliptic. This is an unmodeled systematic in our test statistic. We compute $P_j$ using Monte Carlo methods. We take $10^5$ inclinations from the \\citet{brown_inclinations} distribution, place them randomly along circular orbits and take all objects within $\\pm0.5^\\circ$ of the ecliptic latitude $\\beta_j$ of discovery. These values allow us to find $P_j$ by calculating the probability that an object with a given $\\beta_j$ has an inclination at or below $i_j$. We run 1000 Monte Carlo simulations, using the mean $D\\sqrt{N}$ as our test statistic. See Section 4.2.1 of \\citet{kbmod} and Section 3 of \\citet{brown_inclinations} for more detail.\n\nOur mean value for $D\\sqrt{N}$ was 1.40. As shown in Figure \\ref{fig:KS_inc}, we reject the null hypothesis that our observed inclinations come from the distribution of \\citet{brown_inclinations} with only 76.6\\% confidence, which is less than the 1$\\sigma$ confidence level of 84.1\\% ($D\\sqrt{N} = 1.47$). This is to say that we cannot confidently reject the null hypothesis. We can therefore say that our observed inclinations are consistent with \\citet{brown_inclinations}.\n\nWe repeated the further comparison of \\citet{kbmod}, using an approximate survey simulation to identify the distribution of objects with a given inclination that we would expect to find given the central RA and Dec of our search sample. We modeled the DECam field of view as a circle with a diameter of $2.2^\\circ$. We used the inclinations and orbits from the Monte Carlo simulations used to generate Figure \\ref{fig:KS_inc} and recorded the objects visible within the simulated camera footprint. We then normalized this simulated object distribution to the number of detected objects in the search sample. Figure \\ref{fig:expected_inc} shows the simulated distribution (blue) and the observed distribution (orange). The $\\chi^2$ value between the simulated and expected distributions was 8.58, corresponding to a $p$-value of 0.48. We therefore again say that our observed inclinations are consistent with \\citet{brown_inclinations}.\n\n\\begin{figure}[tbh]\n    \\centering\n    \\includegraphics[width=1\\textwidth,keepaspectratio]{VR_Mag.pdf}\n    \\caption{Best-fit \\textit{VR} magnitudes for the previously-known objects (left) and unidentified objects (right).}\n    \\label{fig:VRmag}\n\\end{figure}\n\n\\begin{figure}[tbh]\n    \\centering\n    \\includegraphics[width=1\\textwidth,keepaspectratio]{Mag_Comparison.pdf}\n    \\caption{\\textit{VR} magnitude distribution (orange) of recovered objects (known and unidentified), along with the number of objects predicted by \\citet{FRASER2008827} assuming a circular camera footprint of 3 square degrees with no fill factor (blue) and a fill factor of 0.55 (green). Uncertainties in the orange histogram are calculated as 1$\\sigma$ Poisson intervals of $\\sqrt{N}$.}\n    \\label{fig:mag_compare}\n\\end{figure}\n\n\\subsection{Magnitude Estimation and Analysis}\\label{sec:mag}\nFigure \\ref{fig:VRmag} shows our estimates of the \\textit{VR} magnitude of the known and unidentified objects detected with KBMOD. To fit the \\textit{VR} magnitudes, we generated 25x25 pixel postage stamps in the undifferenced science images following the KBMOD linear trajectory. In each stamp, we fit for the location of the object by maximizing the value of the flux minus the stamp background. The flux was calculated by summing the counts within a circular top-hat psf with a radius of twice the FWHM of the stamp. The local stamp background was estimated from the region outside of this psf. The magnitude zero point was obtained from the InstCal images. We then took the median magnitude value from each set of 15 to 20 magnitude estimates.\n\nAs we did in \\citet{kbmod}, we compared our joint magnitude distribution with the apparent magnitude luminosity function presented in \\citet{FRASER2008827}, adjusting for ecliptic latitude by using the inclination distribution of \\citet{brown_inclinations}. We use the $<VR-R>$ KBO color reported by \\citet{FRASER2008827}. We note, however, that the DECam \\textit{VR} filter of our observations differs somewhat from the Mosaic2 \\textit{VR} filter used in \\citet{FRASER2008827}. They have similar central wavelengths, but different filter response curves. Individual magnitudes are shown in Table \\ref{sec:param_table}. The magnitude uncertainties listed in Table \\ref{sec:param_table} are reported as $\\sigma_G$ uncertainties estimated from each set of magnitude estimates.\n\nWe approximate the camera footprint as a 3 square degree circle. In practice, our trajectories do not cover the entire camera footprint. Each individual KBMOD search only uses data a single CCD, requiring 60 individual searches to cover a full camera footprint. We further require that each candidate trajectory have at least 15 observations, corresponding to a time baseline of about 3 days. Depending on the search velocity and angle, this means that any objects that start near a CCD edge will not be searched, as the trajectory will go off the CCD edge before the trajectory has the requisite 15 observations. We define an effective search fill factor as the fraction of the CCD that is actually searched with KBMOD. For our search parameters, the search fill factor varies from about 0.5 to about 0.9. Assuming a typical KBO speed and angle of 275 pixels per day with an in-image angle of 4.4 radians gives a typical search fill factor of around 0.7. Multiplying this by the camera active-pixel fill factor of about 0.8 gives a typical net fill factor of approximately 0.55.\n\nFigure \\ref{fig:mag_compare} shows a histogram of our observed \\textit{VR} magnitudes along with the number expected from \\citet{FRASER2008827} assuming a fill factor of 1.0 and 0.55. Our joint magnitude distribution is largely inconsistent with \\citet{FRASER2008827} assuming a fill factor of 1.0, but is consistent to within uncertainties up to about $VR=23.25$ assuming a fill factor of 0.55.\n\n\\section{Discussion} \\label{sec:discuss}\n\nThe improvements already presented in this paper helped enable KBMOD to detect 22 out of 26 known objects in the recovery sample. The trajectories of these known objects were recovered with a median error in starting position of less than two pixels. Furthermore, KBMOD was able to detect 75 objects that we were unable to link with any previously-known objects. Although the time baseline of the data was short, we were able to fit the barycentric distance, inclination, and longitude of ascending node of both the known objects and the unidentified objects. The inclination distribution of the recovered objects is consistent with the distribution from \\citet{brown_inclinations}. The number of objects detected as a function of magnitude is consistent with the distribution from \\citet{FRASER2008827} assuming a net fill factor of 0.55.\n\n\\citet{kbmod} validated KBMOD on the High Cadence Transient Survey (HiTS) \\citep{HITS}. This work validates algorithmic improvements to KBMOD filtering with a survey that has a time baseline of up to four nights, compared to the three nights used in \\citet{kbmod}. Furthermore, this work validates KBMOD as applied to images that have been differenced with a coadded template.\n\nIn so doing, we demonstrated that KBMOD can recover KBOs in difference images from a survey with a longer time baseline and an irregular cadence. However, this required more robust filtering methods. By adding GPU filtering, we have increased the effective number of potentially-valid candidate trajectories that can be passed out of the GPU for further filtering and analysis. With the $\\sigma_G$-based filtering, we have also implemented more robust lightcurve filtering that improves filtering with an irregular image cadence. The CNN ResNet50 stamp filter shows great promise for future stamp filtering methods.\n\nNext-generation astronomy surveys will soon be current-generation. This imminent wealth of data will require new computational tools in order to access its full potential. KBMOD has the potential to increase the number of TNOs detected with LSST from $\\sim 40,000$ to $\\sim 320,000$ as well as investigate the faint and mysterious class of objects at the very edge of our Solar System. In terms of probing the sednoids, with three months of coadded data we could detect a Sedna-like object at opposition at over 290 au, as opposed to $\\sim 210$ au for a single image. With a year of coadded data, 290 au increases to 310 au. If we could coadd the entire LSST survey, 310 au increases to over 415 au. Note that objects on elliptical orbits spend much more of their time further from the Sun. If Sedna, which was detected near its perihelion around 90 au \\citep{Brown_2004}, is representative of a larger population of sedoids, then most of these objects should be closer to apocenter than pericenter. Therefore, a linear increase in detection distance should yield a super-linear increase in the number of detected objects on a similar orbit. With this coaddition approach, it might even be possible to detect inner Oort Cloud objects with perihelion near 400 au, and aphelion well beyond.\n\nFurther work is needed before KBMOD will be able to run on LSST. We do not currently address the ``look-elsewhere'' effect in our search algorithm \\citep[e.g.][]{look_elsewhere}. However, our false positives are already dominated by image artifacts and real sources. Even after filtering, trajectories require human by-eye confirmation or rejection. Because of this requirement of human review, we consider this an acceptable limitation. Future work will further investigate necessary algorithmic improvements to enable machine-only object confirmation, including addressing the ``look-elsewhere'' effect.\n\nEnabling KBMOD to search across multiple CCDs will increase the effective fill factor, enabling greater completeness and longer time baselines. CCD chip gaps and camera edges will always keep the fill factor below 1.0 (relative to a circular footprint). However, with a CCD chip gap between 153 (columns) and 201 (rows) pixels, a KBO would move past the chip gap and onto the next CCD in about one night, assuming a typical KBO velocity of 275 pixels per day.\n\nImproving image astrometry and image differencing is likely to reduce the number of image differencing artifacts, thereby reducing the number of candidate trajectories requiring by-eye detection. The non-uniformity of the image time baseline in this survey means that artifacts appeared in approximately the same location in up to five images. This posed a unique challenge to filtering out artifacts from candidate trajectories. Because of these factors, and because we ultimately validate each detected object by-eye, we save a full efficiency analysis of KBMOD for a future survey.\n\nGiven the relatively low inclination (median value of $\\widetilde{i} = 7.67^\\circ$) and barycentric distances between 30 au and 50 (median value of $\\widetilde{r_0} = 41.28$ au), we find it likely that the majority of the unidentified objects presented in Figure \\ref{fig:orbfit} are Kuiper belt objects. However, because the short arcs prevent us from placing accurate constraints on semi-major axis and eccentricity, we are unable to confirm this prediction with the current data. Future follow-up or precovery attempts for these objects may be able to extend the observational arcs enough to accurately constrain them to the Kuiper belt, and perhaps place them within a Kuiper belt subpopulation (e.g. the cold classical Kuiper belt).\n\nThe authors acknowledge support from NASA awards NNG16PJ23C and 80NSSC21K1528, and NSF awards AST-1715122, AST-1409547, and OAC-1739419. This work used the Extreme Science and Engineering Discovery Environment \\citep[XSEDE; ][]{XSEDE}, which is supported by National Science Foundation grant number ACI-1548562. This work used the XSEDE Bridges GPU and Bridges-2 GPU-AI at the  Pittsburgh Supercomputing Center through allocation TG-AST200009. The authors acknowledge support from the DIRAC Institute in the Department of Astronomy at the University of Washington. The DIRAC Institute is supported through generous gifts from the Charles and Lisa Simonyi Fund for Arts and Sciences, and the Washington Research Foundation.\n\n\\software{KBMOD \\citep{kbmod}, LSST Science Pipelines \\citep{LSST_DM}, astropy \\citep{astropy}, scikit-image \\citep{skimage}, numpy \\citep{numpy}, CUDA \\citep{cuda}, scikit-learn \\citep{scikit}, pandas \\citep{pandas}, matplotlib \\citep{matplotlib}, tensorflow \\citep{tensorflow}}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nRecent work \\cite{Lanzetta95} has shown that at least some fraction of\nthe \\mbox{Ly$\\alpha$}\\ absorption lines seen in the spectra of low redshift QSOs\narises in the extended haloes of galaxies. At high redshift \\mbox{Ly$\\alpha$}\\\nabsorbers are found to be strongly clustered \\cite{Fernandez96} and\nmany contain ionised carbon \\cite{Cowie95}. This suggests that some\nfraction of high redshift \\mbox{Ly$\\alpha$}\\ absorbers may also be identified with\nthe haloes of galaxies. Thus it seems likely that the \\mbox{Ly$\\alpha$}\\ forest may\nexhibit large scale structure.\n \nHowever, it has been shown \\cite{Fernandez96} that significant\nclustering may be missed when using the classical tool of cluster\nanalysis, the two-point correlation function\n\\cite{Peebles,SYBT}. Here, we present a new technique to search for\nnon-randomness in the spatial distribution of the \\mbox{Ly$\\alpha$}\\ forest based\non the first and second moments of the transmission probability\ndensity function. This method is able to identify the strength, position\nand scale of individual structures since it retains spatial\ninformation.  It is fairly insensitive to noise and resolution\ncharacteristics and is easy to apply in practice. The new technique\nhas been tested with the help of synthetic spectra and it was found to\nbe substantially more sensitive than a two-point correlation function\nanalysis.\n\n\\section{Results}\n\nThe method was applied to the spectra of a close group of eight QSOs\nwith a mean redshift of 2.97. The data \\cite{Williger96} was kindly\nmade available to us by Gerry Williger (see also these proceedings).\n\nFigure \\ref{liskeF1} shows the result of the analysis. The most\nprominent feature is a $5.3\\sigma$ overdensity of absorption at\n3978~\\AA\\ ($z = 2.272$). It is due to the spectra of Q0041-2707 and\nQ0041-2658. The two lines of sight are separated by $2.4~h_{100}^{-1}$\nproper Mpc ($q_0 = 0.5$) and the feature covers $\\sim 2600$~km\/s in\nvelocity space. Williger et~al.~\\cite{Williger96} find metal absorption\nat redshift 2.2722 in the spectrum of Q0041-2658, which is remarkably\nconsistent with the redshift of the overdense structure.\n\nThere are also two noticeable voids at $\\sim 4490$~\\AA\\ and at\n4842~\\AA. The second void is possibly due to a foreground QSO which\nlies within 500~km\/s of the void.\n\n\n\\begin{figure}\n\\centerline{\\vbox{\n\\psfig{figure=figure1.ps,angle=-90,height=6.9cm}\n}}\n\\caption[]{Transmission in the spectra of a group of eight, closely\nspaced QSOs as a function of wavelength and size of window (smoothing)\nfunction. The transmission is measured relative to a theoretical\nexpectation value and in units of the theoretical standard deviation\nby convolving the spectra with a Gaussian of varying size. Each\n``triangle'' corresponds to one spectrum, where the base is the\noriginal spectrum itself and the tip is a value comparable to $1-D_A$,\nwhere $D_A$ is the flux deficit parameter \\cite{Oke82}.}\n\\label{liskeF1}\n\\end{figure}\n\n\n\n\n\n\\begin{iapbib}{99}{\n\\bibitem{Cowie95} Cowie L.L., Songaila A., Kim T., Hu E.M., 1995, \\aj 109, 1522\n\\bibitem{Fernandez96} Fern\\'andez-Soto A. \\et, 1996, \\apj 460, L85\n\\bibitem{Lanzetta95} Lanzetta K.M., Bowen D.B., Tytler D., \\& Webb J.K., 1995,\n\t\t\t\\apj 442, 538\n\\bibitem{Oke82} Oke J.B., Korycansky D.G., 1982, \\apj 255, 11\n\\bibitem{Peebles} Peebles P.J.E., 1993, {\\it Principles of Physical Cosmology},\n\t\t\tPrinceton Univ. Press\n\\bibitem{SYBT} Sargent W.L.W., Young P. J., Boksenberg A., \\& Tytler D., 1980,\n\t\t\tApJS 42, 41\n\\bibitem{Williger96} Williger G.M. \\et, 1996, ApJS 104, 145\n\n}\n\\end{iapbib}\n\\vfill\n\\end{document}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{1. Introduction}\n\nIn recent years there have been growing interest both in brane universes \\cite{RS,brane1}\n and in the higher-order gravity theories of which the simplest is $f(R)$ gravity \\cite{f(R)}. In this talk we are going to combine both ideas and formulate the higher-order gravities on the brane. It emerges that the formulation is a bit non-trivial, since one faces ambiguities of the quadratic delta function contributions to the field equations. We will say how to avoid these problems and show how the Israel junction conditions for such higher-order brane gravity models can be formulated.\n\n\\section{2. Fourth-order gravities.}\n\nWhen one considers the general gravity theories (e.g. \\cite{clifton,braneR2}):\n\\begin{equation}\n\\label{XYZ}\nS = \\chi^{-1} \\int d^D \\sqrt{-g} f(X,Y,Z)\n\\end{equation}\nin a D-dimensional spacetime ($\\chi =$ const.), where $X,Y,Z$ are curvature invariants\n\\begin{equation}\nX = R,\\hspace{0.4cm} Y= R_{ab}R^{ab},\\hspace{0.4cm} Z= R_{abcd}R^{abcd},\n\\end{equation}\nthen one immediately faces the 4th order field equations, except\nwhen they reduce to the theories with Euler densities of the n-th order $I^{(n)}$\n\\cite{lovelock}\n\\begin{eqnarray}\n\\label{euler}\nS = \\int_M d^D x \\sqrt{-g} \\sum_n \\kappa_n I^{(n)}~,\n\\end{eqnarray}\nthe lowest of them being the cosmological constant $I^{(0)} = 1$ ($\\kappa_0 = -2\\Lambda(2\\kappa^2)^{-1} = -2\\Lambda\/16 \\pi G$), the Ricci scalar $I^{(1)} =R$ ($\\kappa_1 = (2\\kappa^2)^{-1}$), and the Gauss-Bonnet density $I^{(2)} = R{GB} = R^2 - 4 R_{ab}R^{ab} + R_{abcd}R^{abcd}$ ($\\kappa_2=\\alpha(2\\kappa^2)^{-1}$, $\\alpha=$ const.).\n\nHowever, the theories based on the Lagrangians which are the functions of the Euler densities\nsuch as\n\\begin{eqnarray}\nf(R) = f(X), \\hspace{1.5cm} f(R_{GB}) = f(Z-4Y+X^2)~,\\hspace{1.5cm} f = f(I^{(n)})\n\\end{eqnarray}\nare again fourth-order.\n\n\\section{3. Formulation of the 4th order gravities on the brane - Israel formalism.}\n\nIn the context of the recent interest in string\/M-theory, it is interesting to formulate the general gravity theories (\\ref{XYZ}) within the framework of the brane models \\cite{PRD08}. The full brane action for such a theory reads as\n\\begin{eqnarray}\n\\label{XYZB}\nS &=& \\chi^{-1} \\int_{M} d^{D}x \\sqrt{-g} f(X,Y,Z) + S_{brane} + S_{m}~,\n\\end{eqnarray}\nwith the total energy-momentum tensor\n\\begin{eqnarray}\n\\label{Tab}\nT_a^{~b}=T_{a}^{~b~-}\\theta(-w) + T_{a}^{~b~+}\\theta(w) +\n\\delta(w)S_a^{~b},\n\\end{eqnarray}\nwhere $S_a^{~b}$ is the energy-momentum tensor on the brane, and\n$T_{a}^{~b~\\pm}$ are the energy-momentum tensors on the both sides of the brane, \n$\\theta(w)$ is the Heaviside step function, and $\\delta(w)$ is the Dirac delta function.\n\nWe assume Gaussian normal coordinates, i.e.,\n$(\\mu,\\nu = 0, 1, 2,\\ldots,D-2;w=D)$\n\\begin{eqnarray}\n\\label{bm}\nds^2=g_{ab} dx^a dx^b = \\epsilon dw^2+h_{\\mu\\nu}dx^{\\mu}dx^{\\nu}~,\n\\end{eqnarray}\nwhere $\\epsilon = \\vec{n} \\cdot \\vec{n} = +1$ for a spacelike hypersurface,\n$\\epsilon= -1$ for a timelike hypersurface, and $h_{ab} = g_{ab} - \\epsilon n_a n_b$\nis a projection tensor onto a $(D-1)$-dimensional hypersurface, $\\vec{n}$ is the normal vector to the hypersurface. In these coordinates the extrinsic\ncurvature is\n\\begin{eqnarray}\nK_{\\mu\\nu}=-{1\\over 2}{\\partial h_{\\mu\\nu}\\over\\partial w}~,\n\\end{eqnarray}\nand the Gauss-Codazzi equations read \\cite{brane2}\n\\begin{eqnarray}\n\\label{GC}\nR_{w\\mu w\\nu}&=& {\\partial K_{\\mu\\nu}\\over \\partial w}+K_{\\rho\\nu}K^{\\rho}_{\\,\\,\\mu},  \\\\\nR_{w\\mu\\nu\\rho}&=&\\nabla_{\\nu}K_{\\mu\\rho}-\\nabla_{\\rho}K_{\\mu\\nu}, \\\\\nR_{\\lambda\\mu\\nu\\rho}&=&~^{(D-1)} R_{\\lambda\\mu\\nu\\rho}+\n\\epsilon\\left[K_{\\mu\\nu}K_{\\lambda\\rho}\n-K_{\\mu\\rho}K_{\\lambda\\nu}\\right]~.\n\\end{eqnarray}\nIn the standard Israel approach \\cite{israel66} one assumes that at the brane position $w=0$:\n\\begin{eqnarray}\n\\label{cont1}\nh^{-}_{\\mu\\nu} &=& h^{+}_{\\mu\\nu}~,\\\\\n\\label{cont2}\nh^{-}_{\\mu\\nu,w} & \\neq & h^{+}_{\\mu\\nu,w}~, \\hspace{0.5cm} K^{-}_{\\mu\\nu}\n\\neq K^{+}_{\\mu\\nu}~,\n\\end{eqnarray}\ni.e., the {\\it metric is continuous} but it has a kink, its first derivative has {\\it a step function} discontinuity, and its second derivative gives the {\\it delta function} contribution.\n\nIn terms of $\\theta(w)$ and $\\delta(w)$ functions this is equivalent to\n\\begin{eqnarray}\nh_\\mu{_\\nu}(w) &=& h^{-}_{\\mu\\nu}(w) \\theta(-w) + h^{+}_{\\mu\\nu}(w)\n\\theta(w) ~,\\\\\n{\\partial h_{\\mu\\nu}} \\over \\partial w &=&  {\\partial h^{+}_{\\mu\\nu} \\over\n\\partial w} \\theta(-w) +  {\\partial h^{-}_{\\mu\\nu} \\over \\partial w} \\theta(w)~, \\\\\n{\\partial{^2} {h_\\mu{_\\nu}} \\over \\partial w{^2}}&=&  {\\partial{^2} h^{-}_{\\mu\\nu}\n\\over \\partial w{^2}} \\theta(-w) +  {\\partial{^2} h^{+}_{\\mu\\nu} \\over \\partial w{^2}}\n\\theta(w) \\nonumber \\\\ &+& \\left( {\\partial h^{-}_{\\mu\\nu} \\over \\partial w} -\n{\\partial h^{+}_{\\mu\\nu} \\over \\partial w} \\right)\\delta(w)~.\n\\end{eqnarray}\nFor the standard brane models with the Einstein-Hilbert action in the bulk\n\\begin{eqnarray}\nS = \\frac{1}{2\\kappa^2}\\int_{M} d^{D}x \\sqrt{-g} R + S_{brane} +\nS_{m}\n\\end{eqnarray}\nthe field equations read as \\cite{brane2}\n\\begin{eqnarray}\n\\label{Gww}\nG^w_{~w}&=&-{1\\over 2}~^{(D-1)}R+{1\\over 2}\\epsilon\\left[K^2-Tr(K^2)\\right]=\\kappa^2 T^w_{~w}, \\\\\n\\label{Gwm}\nG^w_{~\\mu}&=&\\epsilon\\left[\\nabla_{\\mu}K-\\nabla_{\\nu}K^{\\nu}_{\\,\\,\\mu}\\right]=\\kappa^2\nT^w_{~\\mu}, \\\\\n\\label{Gmm}\nG^{\\mu}_{~\\nu}&=&~^{(D-1)}G^{\\mu}_{~\\nu}\n             +\\epsilon\\left[{\\partial K^{\\mu}_{~\\nu}\\over\\partial w}-\\delta^{\\mu}_{~\\nu}\n{\\partial K\\over\\partial w}\\right]\\\\\n&+& \\epsilon\\left[-\nK K^{\\mu}_{~\\nu}+{1\\over 2}\\delta^{\\mu}_{~\\nu}Tr(K^2)+{1\\over 2}\\delta^{\\mu}_{~\\nu}\nK^2\\right]=\\kappa^2 T^{\\mu}_{~\\nu}~.\\nonumber\n\\end{eqnarray}\nand in the limit $ \\lim_{w \\to 0} \\int_{-w}^{w}$, which ``fishes out'' the delta function contributions, one gets the {\\it standard Israel junction conditions} as \\cite{brane2}:\n\\begin{eqnarray}\n\\label{jcE}\n\\epsilon \\{ [K^{\\mu}_{~\\nu}]-\\delta^{\\mu}_{~\\nu}[K]\\} &=& \\kappa^2 {S}^{\\mu}_{~\\nu},\n\\hspace{0.5cm} [K^{\\mu}_{~\\nu}] \\equiv K^{\\mu~+}_{~\\nu}-K^{\\mu~-}_{~\\nu}.\n\\end{eqnarray}\nBy $[X] = X^+ + X^-$ we define a jump of an appropriate quantity $X$ at the brane. \n\nHowever, for the general $f(X,Y,Z)$ theory on the brane, the standard continuity relations (\\ref{cont1})-(\\ref{cont2}) do not work. This can be seen from the field equations\nof the action (\\ref{XYZ})\n\\begin{eqnarray}\n\\label{XYZ1}\nP_{a b}&=&\\frac{\\chi}{2} T_{a b}, \\\\\n\\label{XYZ2}\nP^{a b} &=& -\\frac{1}{2} f g^{a b} + f_X R^{a b}+2 f_Y R^{c (a} {R^{b)}}_{c}+2\nf_Z R^{e d c (a} {R^{b)}}_{c d e} \\nonumber \\\\ &+& f_{X; c d}(g^{a\nb} g^{c d}-g^{a c} g^{b d}) + \\square (f_Y R^{a b}) + g^{a b} (f_Y\nR^{c d})_{;c d} \\nonumber \\\\ &-& 2 (f_Y R^{c (a})_{;\\; \\; c}^{\\;\nb)}-4 (f_Z R^{d (a b) c})_{;c d},\n\\end{eqnarray}\nwhere $f_X = {\\partial f \/ \\partial X}$ etc.\n\nTake, for example, the square of the Ricci scalar\n\\begin{eqnarray}\nR&=&~^{(D-1)}R+\\epsilon\\left[2h^{\\mu\\nu}{\\partial K_{\\mu\\nu}\\over\\partial w}\n+3Tr(K^2)-K^2\\right]~,\\nonumber\n\\end{eqnarray}\nwhere $K\\equiv K^{\\mu}_{\\,\\,\\mu}$ , $Tr(K^2)\\equiv K^{\\mu\\nu}K_{\\mu\\nu}$,\nand appropriately, of the Ricci tensor, and of the Riemann tensor. These squares produce the terms of the type\n\\begin{eqnarray}\n\\label{terms}\n{\\partial^2 h^{\\mu \\nu} \\over \\partial^2 w}{\\partial K_{\\mu \\nu} \\over \\partial w},\n   {\\partial K_{\\mu \\nu} \\over \\partial w}{\\partial K^{\\mu \\nu} \\over \\partial w},\n   \\left({\\partial K \\over \\partial w}\\right)^2~,\n\\end{eqnarray}\nwhich are proportional to $\\delta^2(w)$, and so they are {\\it ambiguous}.\n\nAmazingly, all these ambiguous terms cancel each other exactly in the case of the Euler densities \\cite{meissner01}. In fact, the junction conditions for one of the Euler densities -- the Gauss-Bonnet density, were already obtained as \\cite{deruelle00,davis}\n\\begin{eqnarray}\n2 \\alpha \\left( 3 [J_{\\mu\\nu}] - [J] h_{\\mu\\nu}\n- 2 [P]_{\\mu\\rho\\nu\\sigma} [K]^{\\rho\\sigma} \\right)\n+ [K_{\\mu\\nu}] - [K] h_{\\mu\\nu} = - \\kappa^2 S_{\\mu\\nu}~,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nP_{\\mu\\rho\\nu\\sigma} &=& R_{\\mu\\rho\\nu\\sigma} + 2 h_{\\mu[\\sigma}R_{\\nu]\\rho}\n+ 2 h_{\\rho[\\nu}R_{\\sigma]\\mu}\n+ R h_{\\mu[\\nu}h_{\\sigma]\\rho}~, \\\\\nJ_{\\mu\\nu} &=& \\frac{1}{3} \\left( 2KK_{\\mu\\sigma}K^{\\sigma}_{\\nu} +\nK_{\\sigma\\rho}K^{\\sigma\\rho}K_{\\mu\\nu} -\n2K_{\\mu\\rho}K^{\\rho\\sigma}K_{\\sigma\\nu}\n- K^2 K_{\\mu\\nu} \\right)~.\n\\end{eqnarray}\nIn the limit $\\alpha \\to 0$, they just give Einstein-Hilbert action junction conditions (\\ref{jcE}).\n\nIn view of the ambiguities of the terms in (\\ref{terms}), we find two ways to formulate the junction conditions for general $f(X,Y,Z)$ theories on the brane.\n\n\\subsection{A. Smoothing out the continuity conditions for the metric tensor at the brane}\n\nIn order to do that we impose more regularity onto the metric tensor at the brane position, i.e., we consider {\\it a singular hypersurface of the order three} \\cite{israel66} which fulfills the conditions (compare (\\ref{cont1})-(\\ref{cont2}))\n\\begin{eqnarray}\n\\label{hh1}\nh^{-}_{\\mu\\nu} &=& h^{+}_{\\mu\\nu}~,\\\\\n\\label{hh2}\nh^{-}_{\\mu\\nu,w} & = & h^{+}_{\\mu\\nu,w}~, \\hspace{0.5cm} K^{-}_{\\mu\\nu}\n= K^{+}_{\\mu\\nu}~,\\\\\n\\label{hh3}\nh^{-}_{\\mu\\nu,ww} &=& h^{-}_{\\mu\\nu,ww}~, \\hspace{0.5cm} K^{-}_{\\mu\\nu,w} =\nK^{+}_{\\mu\\nu,w}~,\\\\\n\\label{hh4}\nh^{-}_{\\mu\\nu,www} & \\neq & h^{+}_{\\mu\\nu,www}~, \\hspace{0.5cm} K^{-}_{\\mu\\nu,ww}\n\\neq K^{+}_{\\mu\\nu,ww}~,\n\\end{eqnarray}\ni.e., the metric and its first derivative are regular, the {\\it second derivative of the\nmetric is continuous}, but possesses a kink, the third derivative of the metric\nhas {\\it a step function} discontinuity, and no sooner than the fourth derivative of the\nmetric on the brane produces the {\\it delta function} contribution.\n\nThe physical interpretation as put in terms of the second-order theory can be that there is a jump of the first derivative of the energy-momentum tensor (e.g. jump of a pressure gradient) at the brane. \n\nIn his seminal work, Israel \\cite{israel66} proposed {\\it a singular hypersurface of order two}, which physically corresponded to a boundary surface characterized by a jump of the energy-momentum tensor (e.g. a boundary surface separating a star from the surrounding vacuum) which was characterized by\n\\begin{eqnarray}\n\\label{hhb1}\nh^{-}_{\\mu\\nu} &=& h^{+}_{\\mu\\nu}~,\\\\\n\\label{hhb2}\nh^{-}_{\\mu\\nu,w} & = & h^{+}_{\\mu\\nu,w}~, \\hspace{0.5cm} K^{-}_{\\mu\\nu}\n= K^{+}_{\\mu\\nu}~,\\\\\n\\label{hhb3}\nh^{-}_{\\mu\\nu,ww} &\\neq& h^{-}_{\\mu\\nu,ww}~, \\hspace{0.5cm} K^{-}_{\\mu\\nu,w} = K^{+}_{\\mu\\nu,w}~,\n\\end{eqnarray}\ni.e., the metric is regular, the {\\it first derivative of the\nmetric is continuous}, but possesses a kink, the second derivative of the metric has {\\it a step function} discontinuity, and the third derivative of the metric on the brane produces the {\\it delta function} contribution.\n\nThe appropriate junction conditions can be obtained as follows.\nWe rewrite the field equations (\\ref{XYZ1})-(\\ref{XYZ2}) as\n\\begin{eqnarray}\n\\label{Wabd}\n\\sqrt{-g}C_{ab}{W^{abd}}_{;d} + \\sqrt{-g}C_{ab}V^{ab} =\n{\\chi \\over 2} T^{ab}C_{ab}\\sqrt{-g}~,\n\\end{eqnarray}\nwhere we have introduced is an arbitrary tensor field $C_{ab}$, and\n\\begin{eqnarray}\n    W^{abd}&=&f_{X; c }(g^{a b} g^{c d}-g^{(a c} g^{b) d}) + (f_Y\n    R^{ab})^{;d} \\\\ \\nonumber\n    &+& g^{ab}(f_Y R^{cd})_{;c}  -2 (f_Y R^{d(a})^{;b)}\n    - 4(f_Z R^{d(ab)c})_{;c}~,\\\\\n    V^{ab} &=& -\\frac{1}{2} f g^{a b} + f_X R^{a b}+2 f_Y R^{c (a}\n    {R^{b)}}_{c} \\nonumber \\\\\n    &+& 2 f_Z R^{e d c (a} {R^{b)}}_{c d e}~,\n\\end{eqnarray}\ncontain third derivatives of the metric giving a step function discontinuity, so that ${W^{abd}}_{;d}$ is proportional to $\\delta(w)$. Then, we integrate both sides of the formula (\\ref{Wabd}) over the volume $V$ which contains the\nfollowing parts (cf. Fig. \\ref{fig1}): $G1$, $G2$ - are the\nleft-hand-side and the right-hand-side bulk volumes which are\nseparated by the brane, $A1=\n\\partial G1 + A0$, $A2= \\partial G2 - A0$ are the boundaries of\nthese volumes, and $A0$ is the brane which orientation is given by\nthe direction of the normal vector $\\vec{n}$. \n\n\\begin{figure}[h]\n\\includegraphics[width=8cm]{brane.png}\n\\caption{A schematic picture illustrating the domains of integration\n        used in derivation of the junction conditions. Here $V= G1+G2$ is the \n        total volume, $G1$, $G2$ - are the\nleft-hand-side and the right-hand-side bulk volumes which are\nseparated by the brane, $A1=\n\\partial G1 + A0$, $A2= \\partial G2 - A0$ are the boundaries of\nthese volumes, and $A0$ is the brane which orientation is given by\nthe direction of the normal vector $\\vec{n}$.}\n        \\label{fig1}\n\\end{figure}\n\nWe have\n\\begin{eqnarray}\n\\int_{G1+G2}{\\sqrt{-g}C_{ab}{W^{abd}}_{;d} d\\Omega}  \n+ \\int_{G1+G2} {\\sqrt{-g}C_{ab}V^{ab}d\\Omega}\n=\\int_{G1+G2}{{\\chi \\over 2}T^{ab}C_{ab}\\sqrt{-g}d\\Omega}~, \n\\end{eqnarray}\nand so\n\\begin{eqnarray}\n&& \\int_{G1+G2}\\sqrt{-g}(C_{ab}W^{abd})_{;d} d\\Omega\n- \\int_{G1+G2}\\sqrt{-g}C_{ab;d}W^{abd}\nd\\Omega  \n+ \\int_{G1+G2}\\sqrt{-g}C_{ab}V^{ab}d\\Omega  \\nonumber \\\\ \n&=& \\int_{G1}{\\chi \\over 2} T^{ab}C_{ab}\\sqrt{-g}d\\Omega + \\int_{G2}{\\chi \\over 2}\nT^{ab}C_{ab}\\sqrt{-g}d\\Omega \n+ \\int_{A0}{\\chi \\over 2} S^{ab}C_{ab}\\sqrt{-\\gamma}d\\sigma~,\n\\end{eqnarray}\nof which the first term can be integrated out to a boundary A1+A2 and then the limit $V \\to A0$ (or $ \\lim_{w \\to 0} \\int_{-w}^{w}$ in Gaussian coordinates) is taken.\n\nThe final form of the junction conditions which generalize (\\ref{jcE}) onto the fourth-order gravity are\n\\begin{eqnarray}\n\\label{ws}\n    [W]^{abd}n_d - {\\chi \\over 2} S^{ab} &=& 0~, \\hspace{.3cm} [W]^{abd} = W^{abd+} - W^{abd-}.\n\\end{eqnarray}\n\nIt is remarkable that these junction conditions involve the higher derivatives of the scale factor. To see this take for example $f(X,Y,Z)=f(R)$ theory in $D=5$ dimensions with metric\n\\begin{eqnarray}\n\\label{bw1}\nds^2=-dt^2\n+ a^2(t,w)[dr^2 +r^2(d\\Theta^2 +sin^2\\Theta d\\phi^2)]+dw^2~~.\n\\nonumber\n\\end{eqnarray}\nThe junction conditions (\\ref{ws}) give a jump of the third derivative of $a(t,w)$, as expected\n\\begin{eqnarray}\n    [a'''] &=&  {\\chi \\over 2}{a_0} {p_0}~, \\\\\n    p_0&=& \\rho_0~,\n\\end{eqnarray}\nwhere $(\\ldots)' = \\partial \/ \\partial w$, $a_0=a(w=0)$, and the\nbrane energy-momentum tensor is $S_{\\mu}^{\\nu} =\n(-\\rho_0,p_0,p_0,p_0)$.\n\n\\subsection{B. Reduction to an equivalent 2nd order theory}\n\nYet another way to obtain the junction conditions is the reduction of the action (\\ref{XYZ}) to a second-order action. This gives equivalent junction conditions, though at the expense of introducing a new tensor field $H^{abcd}$ (tensoron). In fact, starting from the action \\cite{kijowski}\n\\begin{eqnarray} \\label{r} S_{G} &=& \\chi^{-1} \\int_{M} d^{D}x \\sqrt{-g}\nf(g_{ab},R_{abcd}).\n\\end{eqnarray}\nwe may transform to an equivalent 2nd order action in the form\n\\begin{eqnarray} \\label{equiv}\n S_{I} = \\chi^{-1} \\int_{M} d^{D}x \\sqrt{-g} \\{ H^{ghij}(R_{ghij}-\n \\phi_{ghij}) + f(g_{ab},\\phi_{cdef}) \\}~,\n \\end{eqnarray}\nwhere\n\\begin{eqnarray} \\label{H} H^{ghij} \\equiv {\\partial f(g_{ab},\\phi_{abcd}) \\over\n\\partial \\phi_{ghij}}~, \\hspace{1.cm} det \\left[{\\partial^2 f(g_{ab},\\phi_{abcd}) \\over\n\\partial \\phi_{ghij} \\partial \\phi_{klmn}} \\right] \\neq 0.\n\\end{eqnarray}\nThis transition for $f(R)$ theory requires a new scalar $H=f'(Q)$ (a scalaron) with the condition that $f''(Q) \\neq 0$, and the equation of motion $Q=R$. Similarly, for $f(R_{GB})$ theory, one defines a scalar $H=f'(A)$, with the equation of motion  $A=R_{GB}$).\nIn order to get junction conditions, we have to slightly redefine the tensoron\n\\begin{eqnarray}\nA^{abcd}={1 \\over 2} \\{H^{acdb} &+& H^{abdc}-H^{cbda} - H^{acbd} - H^{abcd}+H^{cbad}\\}\n\\end{eqnarray}\nwhich in a particular case of $f(X,Y,Z)$ theory takes the form\n\\begin{eqnarray}\nA^{abcd}= f_{X}(g^{ad} g^{cb}-g^{cd} g^{ba}) + f_{Y}(2R^{ad}g^{bc} - R^{cd}g^{ba} -R^{ba}g^{cd}) + 4f_{Z}R^{acbd}~.\n\\end{eqnarray}\n\nThe field equations for an equivalent action (\\ref{r}) read as\n\\begin{eqnarray} \\label{em1}\nR_{ghij} &=& - {\\partial V(g_{ab},H^{cdef}) \\over \\partial H^{ghij}}~, \\\\\n\\label{em2} {1\\over 2} g^{ab}f &+& {\\partial f \\over \\partial\ng_{ab}} + H^{becd} \\phi ^{a}_{~ecd}(g_{ab},H^{klmn})\n+\\{A^{(ab)cd}\\}_{;dc}\n = - {\\chi \\over 2} T^{ab}~,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nV(g_{ab},H^{cdef}) = - H^{hgij} \\phi_{ghij}(g_{ab},H^{cdef}) +\nf(g_{ab},\\phi_{klmn}(g_{ab},H^{cdef})) \n\\end{eqnarray}\nIn fact, the possibility to express the fields $\\phi_{abcd}$ as a function of\n$g_{ab}$ and $H^{cdef}$ is guaranteed by the condition\n(\\ref{H}) (which is an analogue of the condition $f''(Q) \\neq 0$).\n\nOne can show that junction conditions of the second-order theory are equivalent to junction \nconditions of the fourth-order theory \\cite{PRD08}.\n\n Applying the same method as in the previous case (i.e. taking the limit of $V \\to A0$) we notice that the first three terms of (\\ref{em2}) do not give any contribution to the junction conditions (since they do not contain delta functions at all) which now have the form:\n\\begin{eqnarray}\n\\label{jc1}\n[{A^{(ab)cd}}_{;d}]n_{c} = - {\\chi \\over 2} S^{ab}~.\n\\end{eqnarray}\nAssuming that\n\\begin{eqnarray} \\label{f}\nf(g_{ab},\\phi_{abcd})=f({\\phi_{ab}}^{ab},{\\phi_{acb}}^{c}{\\phi^{acb}}_{c},\\phi_{abcd}\n\\phi^{abcd}),\n\\end{eqnarray}\nwe can get the same result as in the 4th theory\n\\begin{eqnarray} \\label{equivH}\n{[A^{(ab)cd}}_{;d}]n_{c}&=&{[A^{(ab)cd}}_{;c}]n_{d} = [- \\{f_{X; c\n}(g^{ab} g^{c d}-g^{c(a} g^{b)d}) \\nonumber \\\\  &+& (f_Y  R^{ab})^{;d} + g^{ab}(f_Y R^{cd})_{;c} \\\\\n\\nonumber -2(f_YR^{d(a})^{;b)} &-& 4(f_Z R^{d(ab)c})_{;c}\\}]n_{d}=\n-[W^{abd}]n_{d}. \\end{eqnarray}\n\nSimilar approach was used for less-general $f(R)$ theories of gravity on the brane by \nBorzeszkowski and Frolov \\cite{borzeszkowski}; Parry at al. \\cite{branef(R)}, Deruelle et al. \\cite{deruelle07}, and for $f(X,Y,Z) = aX^2 + bY + cZ$ ($a,b,c =$ const.) theories by Nojiri and Odintsov \\cite{braneR2}.\n\n\n\\section{4. Formulation of the 4th order gravities on the brane - Gibbons-Hawking Boundary Terms}\n\nIn this approach, following the idea of Gibbons and Hawking \\cite{GH}, we do not assume any vanishing of the first derivative of the variation of the metric tensor $\\delta g_{ab;c}$ on the boundary of the integration volume while using the variational principle. Strictly speaking, only the assumption of the vanishing of the normal derivative of the variation of the metric tensor $\\delta g_{ab,w}$ is required. Instead, we postulate that some extra terms to the action are added and that these terms ``kill'' the first derivatives of the metric variation. These terms are called Gibbons-Hawking boundary terms now. In fact, the Gibbons-Hawking boundary term for the Einstein-Hilbert action is composed of the trace of the extrinsic curvature and it was found by Gibbons and Hawking themselves \\cite{GH}. Then, for the action being the combination of the square of the Weyl tensor and an arbitrary function of the scalar curvature they were found by Hawking and Lutrell \\cite{lutrell} and Barrow and Madsen \\cite{madsen}.\nFor the Gauss-Bonnet and other Lovelock densities they were found by Bunch \\cite{bunch81}, Mueller-Hoissen and Myers \\cite{surface}, Davis \\cite{davis} and Gravanis and Willinson \\cite{gravanis}. The boundary terms for the action being an arbitrary function of the curvature invariants were found by Barvinsky and Solodukhin \\cite{barvinsky}.\n\nFor the theories which are of interest for this talk, the Gibbons-Hawking boundary terms have\nthe following form \\cite{JCAP09}: for the $f(R)$ theory the term reads as \n\\begin{eqnarray}\n\\label{gib}\nS_{GH,p}= -2(-1)^{p}\\epsilon \\int_{\\partial M_p} \\sqrt{-h} H K\nd^{D-1}x~,\n\\end{eqnarray}\nwhere $H=f'(Q)$ is the scalaron,\nwhile for the $f(X,Y,Z)$ theory it reads as\n\\begin{eqnarray}\n\\label{gib2}\nS_{GH,p} =\n - (-1)^{p} \\int_{\\partial M_p}\nd^{D-1}x \\sqrt{-h} A^{(ab)cd}n_{c} n_{d}\\mathcal{L}_{\\vec{n}}g_{ab}~,\n\\end{eqnarray}\nwhere $A^{(ab)cd}$ is the tensoron.\n\nUsing the method of the boundary terms we derived the most general Israel junction conditions for $f(R)$ theory as \\cite{JCAP09}:\n\\begin{eqnarray}\n\\label{jc2}\n[K]&=& 0~, \\\\\n\\label{jc21}\nS^{ab}n_{a}n_{b}&=& 0~, \\\\\n\\label{jc22}\nS^{ab}h_{ac}n_{b}&=& 0~, \\\\\n\\label{jc23}\n-(D-1)[H_{;c}n^{c}]-D[H]K &=& \\epsilon {\\chi \\over 2} S^{ab}h_{ab}~,\\\\\n\\label{jc24}\n-h_{ab}[H_{;c}n^{c}]-[H]Kh_{ab} &+& [HK_{ab}] \\\\\n&=&\n\\epsilon {\\chi \\over 2}S^{cd}h_{ca}h_{db}. \\nonumber\n\\end{eqnarray}\nA generality of these conditions refers to the fact that no assumption about the \ncontinuity of the scalaron on the brane has been made. They reduce to the conditions already obtained in the literature, if one assumes $[H] =0$ \\cite{deruelle07}.\n\nOn the other hand, the most general Israel junction conditions for the $f(X,Y,Z)$ theory,\nwith no assumption about the continuity of the tensoron on the brane, are \\cite{JCAP09}:\n\\begin{eqnarray}\n\\label{JCXYZ1}\n&&[KA^{(ab)cd}] n_{c} n_{d} + [\\mathcal{L}_{\\vec{n}}A^{(ab)cd}] n_{c} n_{d}\n\\\\ \\nonumber &-& \\epsilon[A^{(ab)cd}K_{cd}] - g^{ab}[A^{(ef)cd} K_{ef}]n_{c} n_{d}  \\\\\n\\nonumber &+& 2 \\epsilon [D_{s}A^{(ef)cd}n_{c} n_{d}]h^{s}_{e}h^{(a}_{f}n^{b)}\n-  2\\epsilon[{A^{(ab)cd}}_{;(c}]n_{d)} = {\\chi \\over 2} S^{ab}~,\n \\\\\n\\label{JCXYZ2}\n&& \n n_{b} n_{c}[\\mathcal{L}_{\\vec{n}}g_{ad}]-\n n_{a} n_{c}[\\mathcal{L}_{\\vec{n}}g_{db}]-n_{b} n_{d}[\\mathcal{L}_{\\vec{n}}g_{ac}]\n +n_{a} n_{d}[\\mathcal{L}_{\\vec{n}}g_{cb}]=0~.\n\\end{eqnarray}\nThey reduce to the conditions (\\ref{jc1}), if one assumes continuity of the tensoron on the brane \n\\begin{equation}\n[A^{(ab)cd}] = 0~.\n\\end{equation}\n\n\\section{5. Fourth-order gravities and statefinders}\n\nWe claim the fact that general $f(R,R_{ab}R^{ab},R_{abcd}R^{abcd})$ theories are fourth-order may have some advantageous consequences onto their observational verification\nby the application of statefinder diagnosis of the universe.\n\nIn fact, statefinders are the higher-order characteristics of the universe expansion which go\nbeyond the Hubble parameter $H$ and the deceleration parameter $q$:\n\\begin{eqnarray}\n\\label{hubb}\nH &=& \\frac{\\dot{a}}{a}~,\\hspace{0.5cm} q  =  - \\frac{1}{H^2} \\frac{\\ddot{a}}{a} = - \\frac{\\ddot{a}a}{\\dot{a}^2}~.\n\\end{eqnarray}\nThey can generally be expressed as ($i \\geq 2$)\n\\begin{eqnarray}\n\\label{dergen}\nx^{(i)} &=& (-1)^{i+1}\\frac{1}{H^{i}} \\frac{a^{(i)}}{a} = (-1)^{i+1}\n\\frac{a^{(i)} a^{i-1}}{\\dot{a}^{i}}~,\n\\end{eqnarray}\nand the lowest order of them are known as:\njerk, snap (\"kerk\"), crack (\"lerk\")\n\\begin{eqnarray}\n\\label{jerk}\nj &=& \\frac{1}{H^3} \\frac{\\dot{\\ddot{a}}}{a} =\n\\frac{\\dot{\\ddot{a}}a^2}{\\dot{a}^3}~, \\hspace{0.5cm} k = -\\frac{1}{H^4} \\frac{\\ddot{\\ddot{a}}}{a} = -\\frac{\\ddot{\\ddot{a}}a^3}{\\dot{a}^4}~, l = \\frac{1}{H^5} \\frac{a^{(5)}}{a} =\n\\frac{a^{(5)} a^4}{\\dot{a}^5}~,\n\\end{eqnarray}\nand pop (\"merk\"), \"nerk\", \"oerk\", \"perk\" etc. \\cite{statefind}.\n\nIn the case of the 4th order gravities, statefinders may become  powerful tools to constrain such theories observationally, since they enter observational relations\nin the higher orders of redshift $z$ (see \\cite{statefR} for non-brane case diagnosis). \n\nApparently, a blow-up of statefinders may also be linked to an\nemergence of exotic singularities in the universe \\cite{blowup}.\n\n\n\\section{6. Conclusions}\n\nWe conclude the following:\n\n\\begin{itemize}\n\\item The formulation of the fourth-order gravity theories on the brane is non-trivial because of the powers of {\\it delta function ambiguities}.\n\n\\item Two methods were applied: \\\\\nA. {\\it Smoothing out} the continuity conditions for the metric tensor at the brane; \\\\  \nB. Reduction to an {\\it equivalent} 2nd order theory.\n\\\\In both cases the Israel {\\it junction conditions} have been obtained and they {\\it are} also mutually {\\it equivalent}.\n\n\\item The method of the {\\it GH boundary terms} was also applied and the most general junction conditions (with no continuity of the scalaron and tensoron on the brane assumed) were obtained that way, too.\n\n\\item Higher-order brane gravities contain {\\it higher-order derivatives} of the geometric quantities (in a Friedmann model it is just the scale factor) which may manifest themselves in the {\\it higher-order characteristics of expansion} such as statefinders (jerk, snap, lerk\/crack, merk\/pop). \n    \n\\item A blow-up of statefinders may be linked to an {\\it emergence of exotic singularities} in the universe.\n\\end{itemize}\n\n\n\n\n\n\n\n\n\n\\begin{theacknowledgments}\nWe acknowledge the support of the Polish Ministry of Science\nand Higher Education grant No N N202 1912 34 (years 2008-10).\n\\end{theacknowledgments}\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nTaking cues from the concepts of location~\\cite{slater1975leaves}, neighbor-location~\\cite{slater1988locationdom} and locally identifying colorings~\\cite{LID-EGMOP12}, recently, two variants of graph coloring were introduced, namely, \\textit{locating coloring}~\\cite{Chartrand200289} and\n \\textit{neighbor-locating coloring}~\\cite{alcon2020neighbor,BA2014}. While the former concept has been \nwell-studied since 2002 \\cite{BA2014,BA2015,behtoei2011locating,BO2016,Chartrand200289,chartrand2003graphs,purwasih2017locating,purwasih2015bounds,syofyan2015locating,welyyanti2017locating,welyyanti2015locating}), our focus of study is the latter, which was introduced in 2014 in~\\cite{BA2014} under the name of \\emph{adjacency locating coloring}, renamed in 2020 in~\\cite{alcon2020neighbor} and studied in a few papers since then~\\cite{alcon2019neighbor,alcon2021neighbor,HERNANDO2018131,mojdeh2022conjectures}.\n\nThroughout this article, we will use the standard terminologies and notations used in ``Introduction to Graph Theory'' by West~\\cite{west2001introduction}.  \n\n\nGiven a graph $G$, a \\textit{(proper) $k$-coloring} is a function\n$f: V(G) \\to \\{1, 2, \\cdots, k\\}$ such that $f(u) \\neq f(v)$ whenever $u$ is adjacent to $v$. The value $f(v)$ is called the \\emph{color}\u00a0of $v$.\nThe \\textit{chromatic number} of $G$, denoted by $\\chi(G)$, is the minimum $k$ for which $G$ admits a $k$-coloring.\n\n\nGiven a (proper) $k$-coloring $f$ of $G$, its $i^{th}$ color class is the collection $S_i$ of vertices that have received the color $i$. The distance between a vertex $x$ and a set $S$ of vertices is given by \n$d(x, S) = \\min\\{d(x, y) : y \\in S\\}$,\nwhere $d(x,y)$ is the number of edges in a shortest path connecting $x$ and $y$. Two vertices $x$ and $y$ are \\textit{metric-distinguished} with respect to $f$ if either \n$f(x)\\neq f(y)$ or $d(x,S_i)\\neq d(y,S_i)$ for some color class $S_i$. A \n(proper) $k$-coloring $f$ of $G$ is a \\textit{locating $k$-coloring} if any two distinct vertices are metric-distinguished with respect to $f$.\nThe \\textit{locating chromatic number} of $G$, denoted by $\\chi_L(G)$, is the minimum $k$ for which $G$ admits a locating $k$-coloring. \n\n\n\nGiven a (proper) $k$-coloring $f$ of $G$, suppose that a neighbor $y$ of a vertex $x$ belongs to the color class $S_i$. In such a scenario, we say that \n$i$ is a \\textit{color-neighbor} of $x$ (with respect to $f$). \nThe set of all color-neighbors of $x$ is denoted by $N_f(x)$.\nTwo vertices $x$ and $y$ are \\textit{neighbor-distinguished} with respect to $f$ if \neither $f(x) \\neq f(y)$ or \n$N_f(x) \\neq N_f(y)$. \nA (proper) $k$-coloring $f$ is \\textit{neighbor-locating $k$-coloring} if each pair of distinct vertices are neighbor-distinguished. \nThe \\textit{neighbor-locating chromatic number} of $G$, denoted by $\\chi_{NL}(G)$, is the minimum $k$ for which $G$ admits a neighbor-locating $k$-coloring. \n\n\nObserve that a neighbor-locating coloring is, in particular, a locating coloring as well. Therefore, we have the following obvious relation among the three parameters~\\cite{alcon2020neighbor}:\n$$\\chi(G) \\leq \\chi_{L}(G) \\leq \\chi_{NL}(G).$$\nNote that for complete graphs, all three parameters have the same value, that is, \nequality holds in the above relation. Nevertheless, the difference between the pairs of values of parameters $\\chi(\\cdot), \\chi_{NL}(\\cdot)$ and $\\chi_L(\\cdot), \\chi_{NL}(\\cdot)$, respectively, can be arbitrarily large. Moreover, it was proved that for any pair $p,q$ of integers with $3\\leq p\\leq q$, there exists a connected graph $G_1$ with $\\chi(G_1)=p$ and $\\chi_{NL}(G_1)=q$~\\cite{alcon2020neighbor} and a connected graph $G_2$ with $\\chi_{L}(G_2)=p$ and $\\chi_{NL}(G_2)=q$~\\cite{mojdeh2022conjectures}. \nThe latter of the two results positively settled a conjecture posed in~\\cite{alcon2020neighbor}. We strengthen these results by showing that for any three integers $p,q,r$ with $2\\leq p\\leq q\\leq r$, there exists a connected graph $G_{p,q,r}$ with $\\chi(G_{p,q,r})=p$, $\\chi_L(G_{p,q,r})=q$ and $\\chi_{NL}(G_{p,q,r})=r$, except when $2=p=q<r$.\n\n\nOne fundamental difference between coloring and locating coloring (resp., neighbor-locating coloring) is that the restriction of a coloring of $G$ to an (induced) subgraph $H$ \nis necessarily a coloring, whereas the analogous property is not true for \nlocating coloring (resp., neighbor-locating coloring). Interestingly, we show that the \nlocating chromatic number (resp., neighbor-locating chromatic number) of an induced subgraph $H$ of $G$ can be arbitrarily larger than that of $G$.\n\n\n\nAlcon \\textit{et al.}~\\cite{alcon2020neighbor} showed that the number $n$ of vertices of $G$ is bounded above by $k(2^{k-1}-1)$, where $\\chi_{NL}(G)=k$ and $G$ has no isolated vertices, and this bound is tight. This exponential bound is reduced to a polynomial one when $G$ has maximum degree $\\Delta$, indeed it was further shown in~\\cite{alcon2020neighbor} that the upper-bound $n\\leq k\\sum_{j=1}^{\\Delta}{k-1\\choose j}$ holds (for graphs with no isolated vertices and when $\\Delta\\leq k-1$). It was left open whether this bound is tight. The \\textit{cycle rank} $c$ of a graph $G$, denoted by $c(G)$, is defined as $c(G) = |E(G)| - n(G) + 1$. Alcon \\textit{et al.}~\\cite{alcon2021neighbor} gave the upper bound $n\\leq \\frac{1}{2}(k^3+k^2-2k)+2(c-1)$ for graphs of order $n$, neighbor-locating chromatic number $k$ and cycle rank $c$. Further, they also obtained tight upper bounds on the order of trees and unicyclic graphs in terms of the neighbor-locating chromatic number~\\cite{alcon2021neighbor}, where a unicyclic graph is a connected graph having exactly one cycle.\n\nAs a connected graph with cycle rank $c$ and order $n$ has $n+c-1$ edges and a graph of order $n$ and maximum degree $\\Delta$ has at most $\\frac{\\Delta}{2}n$ edges, the two latter bounds can be seen as two approaches for studying the neighbor-locating coloring for sparse graphs. We generalize this approach by studying graphs with given average degree, or in other words, graphs of order $n$ having at most $an+b$ edges for some constants $a,b$ (such graphs have average degree $2a+2b\/n$).\nFor such graphs, we prove the upper bound $n\\leq 2b+k\\sum\\limits_{i=1}^{2a}(2a+1-i){k-1\\choose i}$. \nFurthermore, we show that this bound is asymptotically tight, by a construction of graphs with $an+b$ edges (where $2a$ is any positive integer and $2b$ any integer) and neighbor-locating chromatic number $\\Theta(k)$, whose order is $\\Theta(k^{2a+1})$. Moreover, when $b=0$, the graphs can be taken to have maximum degree $2a$. This implies that our bound and the one from~\\cite{alcon2020neighbor} are roughly tight.\n\n\nIn Section~\\ref{Basic results}, we study the connected graphs with prescribed values of chromatic number, locating chromatic number and neighbor-locating chromatic number. We also study the relation between the locating chromatic number (resp., neighbor-locating chromatic number) of a graph and its induced subgraphs. Finally, in Section~\\ref{Density} we study the density of graphs having bounded neighbor-locating chromatic number.\n\n\n\n\\section{Gap between $\\chi(G)$, $\\chi_L(G)$ and $\\chi_{NL}(G)$}~\\label{Basic results}\nThe first result we would like to prove involves three different parameters, namely, the chromatic number, the locating chromatic number, and the neighbor-locating chromatic number. \n\n\n\n\n\n\n\\begin{theorem}\nFor all $2 \\leq p \\leq q \\leq r$, except when $p=q=2$ and $r >2$, there exists a connected graph $G_{p,q,r}$ satisfying $\\chi(G_{p,q,r}) = p$, $\\chi_{L}(G_{p,q,r}) = q$, and $\\chi_{NL}(G_{p,q,r}) = r$. \n\\end{theorem}\n\n\n\n\n\\begin{proof}\nFirst of all, let us assume that $p=q=r$. In this case, \nfor $G_{p,q,r} = K_p$, it is trivial to note that \n$\\chi(G_{p,q,r})=\\chi_L(G_{p,q,r})=\\chi_{NL}(G_{p,q,r})=p$.\nThis completes the case when $p=q=r$. \n\n\n\\medskip\n\nSecond of all, let us handle the case when $p < q=r$. \nIf $2 = p < q = r$, then take $G_{p,q,r}= K_{1,q-1}$. Therefore, we have\n $\\chi(G_{p,q,r})=2$ as it is a bipartite graph, and it is known that $\\chi_L(G_{p,q,r})=\\chi_{NL}(G_{p,q,r})=q$~\\cite{alcon2020neighbor,Chartrand200289}.\n\n\nIf $3 \\leq p < q = r$, then we construct $G_{p,q,r}$ as follows: start with a complete graph $K_p$, on vertices $v_0, v_1, \\cdots, v_{p-1}$, take $(q-1)$ new vertices $u_1, u_2, \\cdots, u_{q-1}$, and make them adjacent to $v_0$.  It is trivial to note that $\\chi(G_{p,q,r})=p$ in this case. Moreover, note that we need to assign $q$ distinct colors to $v_0, u_1, u_2, \\cdots, u_{q-1}$ under any locating or neighbor-locating coloring. On the other hand, $f(v_i) = i+1$ and $f(u_j) = j+1$ is a valid locating $q$-coloring as well as neighbor locating $q$-coloring of $G_{p,q,r}$. Thus we are done with the cases when $p < q=r$. \n\n\\medskip\n\nThirdly, we are going to consider the case when $p=q < r$. \nIf $3 = p = q < r$, then let $G_{p,q,r} = C_n$ where $C_n$ is an odd cycle of  suitable length, that is, a length which will imply \n$\\chi_{NL}(C_n)=r$. It is known that such a cycle exists~\\cite{alcon2019neighbor,BA2014}. \nAs we know that $\\chi(G_{p,q,r})=3$, \n$\\chi_L(G_{p,q,r})=3$~\\cite{Chartrand200289}, and \n$\\chi_{NL}(G_{p,q,r})=r$~\\cite{alcon2019neighbor,BA2014}, we are done. \n\nIf $4 \\leq p = q < r$,  \nthen we construct $G_{p,q,r}$ as follows: start with a complete graph $K_p$ on vertices $v_0, v_1, \\cdots, v_{p-1}$, and an odd cycle $C_n$ on vertices $u_0, u_1, \\cdots, u_{n-1}$, and identify the vertices $v_0$ and $u_0$. \nMoreover, we say that the length of the odd cycle $C_n$ is a suitable length, that is, it is of a length which ensures \n$\\chi_{NL}(C_n)=r$.\nIt is known that such a cycle exists~\\cite{alcon2019neighbor,BA2014}. \nNotice that $\\chi(G_{p,q,r})=p$. \nA locating coloring $f$ can be assigned to $G_{p,q,r}$ as follows: $f(v_i)=i+1$, $f(u_j)=a$ for odd integers $1\\le j\\le n-1$ and $f(u_l)=b$ for even integers $2\\le l\\le n-1$, where $a,b\\in \\{2, 3, \\dots,p\\}$. A vertex $v_i\\in K_p$ (other than $v_0$) and a vertex $u_j \\in C_n$ such that $f(v_i)=f(u_j)$ are metric-distinguished with respect to $f$ since $d(v_i,S_l)=1\\neq d(u_j,S_l)$ for at least one $l\\in \\{2, 3, \\dots,p\\}\\setminus \\{a,b\\}$. Thus, $\\chi_L(G_{p,q,r})=p$. \nOn the other hand, as the neighborhood of the vertices of the cycle $C_n$ (subgraph of $G_{p,q,r}$) \ndoes not change if we consider it as an induced subgraph except for the vertex $v_0 = u_0$. Thus, we will need at least $r$ colors to color $C_n$ while it is contained inside $G_{p,q,r}$ as a subgraph. Assign a neighbor-locating coloring $c$ to $G_{p,q,r}$ as follows: assign $p$ distinct colors to the complete graph $K_p$. Use $p$ colors from $K_p$ and $r-p$ new colors to color the odd cycle $C_n$. A vertex $v_i\\in K_p$ (other than $v_0$) and a vertex $u_j\\in C_n$ such that $c(v_i)=c(u_j)$ are neighbor-distinguished with respect to $c$ since $v_i$ has $p-1$ ditinct color neighbors whereas $u_j$ can have at most two distinguished color neighbors. Hence $\\chi_{NL}(G_{p,q,r})=r$. Thus, we are done in this case also. \n\n\\medskip\n\n\nFinally, we are into the case when $p < q < r$.\nIf $2=p<q<r$, then refer~\\cite{mojdeh2022conjectures} for this case.\n\n\n\nIf $3 = p < q <r$, then we start with an odd cycle $C_n$ on vertices\n$v_1, v_2, \\cdots, v_{n}$, where\n$n=k=\\frac{r(r-1)(r-2)}{2}$ if $k$ is odd and $n=k-1$ if $k$ is even.\nIt is known that $\\chi_{NL}(C_n)=r$ from~\\cite{alcon2019neighbor,BA2014}. Take $q-1$ new vertices $u_1, u_2, \\cdots, u_{q-1}$ and make all of them adjacent to $v_0$. This so obtained graph is $G_{p,q,r}$. It is trivial to note that $\\chi(G_{p,q,r})=3$ in this case. Note that we need to assign $q$ distinct colors to $v_1,u_1, u_2, \\cdots, u_{q-1}$ under any locating or neighbor-locating coloring. \nNow, we assign a locating coloring $f$ to $G_{p,q,r}$ as follows: $f(v_1)=1$, $f(v_i)=2$ for all even integers $2\\le i\\le n$, $f(v_j)=3$ for all odd integers $3\\le j\\le n$, $f(u_s)=s+1$ for all $1\\le s\\le q-1$. This gives us $\\chi_L(G_{p,q,r})=q$. \n\nOn the other hand, as the neighborhood of the vertices of the cycle $C_n$ (subgraph of $G_{p,q,r}$) \ndoes not change if we consider it as an induced subgraph except for the vertex $v_1$. Thus, we will need at least $r$ colors to color $C_n$ while it is contained inside $G_{p,q,r}$ as a subgraph. Assign a neighbor-locating coloring $c$ to $G_{p,q,r}$ as follows:\nassign a neighbor-locating $r$-coloring to the odd cycle $C_n$ such that each vertex has two distinct color neighbors in case of $n=k$ and all vertices except the two vertices say $v_i$ and $v_j$, have two distinct color neighbors in case of $n=k-1$ (refer~\\cite{alcon2019neighbor} for such a neighbor-locating $r$-coloring). Assign distinct colors to the $q-1$ leaf vertices by choosing any $q-1$ colors from $r$ colors (except $c(v_i)$ and $c(v_j)$ in case of $n=k-1$) given to the cycle $C_n$. A vertex $v_i$ ($i\\neq 1$) in the cycle and a leaf vertex $u_j$ such that $f(v_i)=f(u_j)$ are neighbor distinguished since $v_i$ has two distinct color neighbors whereas $u_j$ has only one color neighbor. Hence we have $\\chi_{NL}(G_{p,q,r})=r$.\n\n\n\n\n\n\nIf $4 \\leq p < q <r$, then we start with \na path $P_n$ on $n$ vertices, where $n=\\frac{r(r-1)(r-2)}{2}$. \n It is known that $\\chi_{NL}(P_n)=r$ from~\\cite{alcon2019neighbor,BA2014}. \nLet $P_n = u_0 u_1 \\cdots u_{n-1}$. Now let us take a complete graph on $p$ vertices $v_0, v_1, \\cdots, v_{p-1}$. Identify the two graphs at $u_0$ and $v_0$ to obtain a new graph. Furthermore, take $(q-2)$ independent vertices \n$w_1, w_2, \\cdots, w_{q-2}$ and make them adjacent to $u_{n-2}$. This so obtained graph is $G_{p,q,r}$. It is trivial that $\\chi(G_{p,q,r})=p$. Note that under any locating or neighbor-locating coloring, $q$ distinct colors have to be given to the vertices $u_{n-2}, w_1, w_2, \\cdots, w_{q-2}, u_{n-1}$. Now, define a locating coloring $f$ of $G_{p,q,r}$ as follows: $f(v_0)=f(v_{n-3})=1$, $f(v_i)=2$ for all odd integers $1\\le i\\le n-2$, $f(v_j)=3$ for all even integers $2\\le j\\le n-2$, $f(v_{n-2})=2, f(v_{n-1})=3$ if $n$ is even ($f(v_{n-2})=3, f(v_{n-1})=2$ if $n$ is odd), colors $1,4,5,6,\\dots,q$ to the leaf vertices and $f(k_s)=s+1$ for all $1\\le s \\le p-1$. Thus, $\\chi_L(G_{p,q,r})=q$. \n\n\nMoreover, as the neighborhood of the vertices of the path $P_n$ (subgraph of $G_{p,q,r}$) \ndoes not change if we consider it as an induced subgraph except for the vertices $u_0$ and $u_{n-2}$. Thus, we will need at least $r$ colors to color $P_n$ while it is contained inside $G_{p,q,r}$ as a subgraph. Assign a neighbor-locating coloring $c$ to $G_{p,q,r}$ as follows: \nassign a neighbor-locating $r$-coloring to the path $P_n$ such that each vertex (except the end vertices $u_0$ and $u_{n-1}$) has two distinct color-neighbors (refer~\\cite{alcon2019neighbor} for such a neighbor-locating $r$-coloring). Choose any $p-1$ distinct colors from $r$ colors (except $c(u_0)$) used in neighbor-locating $r$-coloring of $P_n$ and assign them to the remaining $p-1$ vertices of the complete graph.\nAssign distinct colors to the $q-2$ leaf vertices by choosing any $q-2$ colors from $r$ colors of $P_n$ except the colors $c(u_0)$, $c(u_{n-2})$ and $c(u_{n-1})$. A vertex $u_i$ ($i\\neq 0, n-2,n-1$) in the path and a leaf vertex $w_j$ such that $c(u_i)=c(w_j)$ are neighbor distinguished since $u_i$ has two distinct color neighbors whereas $w_j$ has only one color neighbor. Hence, we have $\\chi_{NL}(G_{p,q,r})=r$. \n\\end{proof}\n\n\n\n\n\\bigskip\n\nFurthermore, we show that, unlike the case of chromatic number,  an induced subgraph can have an arbitrarily  higher locating chromatic number (resp., neighbor-locating chromatic number) than that of the graph. \n\n\n\n\n\\begin{theorem}\nFor every $k \\geq 0$, there exists a graph $G_k$ having an induced subgraph $H_k$ such that \n$\\chi_L(H_k) - \\chi_L(G_k) = k$  and \n$\\chi_{NL}(H_k) - \\chi_{NL}(G_k) = k$. \n\\end{theorem}\n\n\n\n\n\\begin{proof}\nThe graph $G_k$ is constructed as follows. \nWe start with $2k$ independent vertices $a_1, a_2, \\cdots, a_{2k}$ and $k$ disjoint edges $b_1b'_1, b_2b'_2, \\cdots, b_kb'_k$. After that we make all the above mentioned vertices adjacent to a special vertex $v$ to obtain our graph $G_k$. \nNotice that $v$ and the $a_i$s must all receive distinct colors under any locating coloring or neighbor-locating coloring.  On the other hand, \nthe coloring $f$ given by $f(v) = 1$, $f(a_i) = i+1$, $f(b_i) = 2i+1$, and \n$f(b'_i) = 2i$ is indeed a locating coloring as well as a \nneighbor-locating coloring of $G_k$.   \nHence we have \n$\\chi_L(G_k) = \\chi_{NL}(G_k) = (2k+1)$.\n\nNow take $H_k$ as the subgraph induced by $v$, $a_i$s and $b_i$s. It is \nthe graph $K_{1,3k}$.\nHence we have \n$\\chi_L(H_k) = \\chi_{NL}(H_k) = (3k+1)$~\\cite{alcon2020neighbor,Chartrand200289}.\n\nThis completes the proof. \n\\end{proof}\n\n\n\n\n\n\\section{Bounds and constructions for sparse graphs}~\\label{Density}\nIn this section, we study the density of graphs having bounded neighbor-locating chromatic number. \n\n\n\n\\subsection{Bounds}\nThe first among those results provides an upper bound on the number of vertices of a graph in terms of its neighbor-locating chromatic number. This, in particular shows that the number of vertices of a graph $G$ is bounded above by \na polynomial function of $\\chi_{NL}(G)$. \n\n\\begin{theorem}\\label{th_lower bound}\nLet $G$ be a connected graph on $n$ vertices and $m$ edges such that $m\\le an+b$, where $2a$ is a positive integer and $2b$ is an integer. If $\\chi_{NL}(G)=k$, then\n$$n  \\le 2b+k \\sum_{i=1}^{2a}(2a+1-i){k-1 \\choose i}.$$\nIn particular, any graph whose order attains the upper bound must be of maximum degree $2a+1$ and with exactly $k{k-1 \\choose i}$ number of vertices of degree $i$.\n\\end{theorem}\n\n\n\n\n\\begin{proof}\nLet $D_i$ and $d_i$ denote the set and the number of vertices in $G$ having degree equal to $i$, respectively, and let $D_i^+$ and $d_i^{+}$ denote the set and the number of vertices in $G$ having degree at least $i$, for all $i \\geq 1$. Using the handshaking lemma, we know that \n$$\\sum_{v \\in V(G)} deg(v) = 2|E(G)| = 2m \\leq 2(an+b).$$\nNotice that, as $G$ is connected, and hence does not have any vertex of degree $0$, it is possible to write \n$$\\sum_{v \\in V(G)} deg(v) = \\sum_{i=1}^{2a} i \\cdot d_i + \\sum_{v \\in D_{2a+1}^+} deg(v).$$\nMoreover, the number of vertices of $G$ can be expressed as \n$$n = (d_1 + d_2 + \\cdots + d_{2a}) + d_{2a+1}^+ = d_{2a+1}^+ + \\sum_{i=1}^{2a} d_{i}.$$\nTherefore, combining the above equations and inequalities, we have \n\\begin{equation}\n    \\sum_{i=1}^{2a} i \\cdot d_i + \\sum_{v \\in D_{2a+1}^+} deg(v)\n                \\leq 2b + 2a \\left( d_{2a+1}^+ + \\sum_{i=1}^{2a} d_{i}\\right) \\nonumber\n\\end{equation}\nwhich implies\n\n\n\\begin{equation}\n  d_{2a+1}^+ \\leq \\sum_{v \\in D_{2a+1}^+} \\left( deg(v) - 2a \\right)  \\leq  \\left(\\sum_{v \\in D_{2a+1}^+} deg(v)\\right) - 2a d_{2a+1}^+ \\leq 2b +  \\sum_{i=1}^{2a} (2a-i)d_{i}     \\nonumber\n\\end{equation}\nsince there are exactly $d_{2a+1}^+$ terms in the summation $\\sum_{v \\in D_{2a+1}^+} \\left( deg(v) - 2a \\right)$ where each term is greater than or equal to $1$, as $deg(v) \\geq 2a +1 $ for all $v \\in D_{2a+1}^+$.  \n\nLet $f$ be any neighbor-locating $k$-coloring of $G$. Consider an ordered pair $(f(u), N_f(u))$, where $u$ is a vertex having degree at most $s$. Thus, $u$ may receive one of the $k$ available colors, while its color neighborhood may consist of \nat most $s$ of the remaining $(k-1)$ colors. Thus, there are at most $k \\sum\\limits_{i=1}^{s} {k-1 \\choose i}$ choices for the ordered pair $(f(u), N_f(u))$. \nAs for any two vertices $u,v$ of degree at most $s$, the following ordered pairs $(f(u), N_f(u))$ and $(f(v), N_f(v))$  must be distinct, we have\n$$\\sum_{i=1}^s d_i \\leq k \\sum_{i=1}^{s} {k-1 \\choose i}.$$\nUsing the above relation, we can show that \n$$\\sum_{i=1}^{2a} (2a+1-i)d_i = \\sum_{s=1}^{2a} \\left( \\sum_{i=1}^s d_i \\right) \\leq  \n\\sum_{s=1}^{2a} \\left( k \\sum_{i=1}^{s} {k-1 \\choose i} \\right) = k \\sum_{i=1}^{2a} (2a+1-i) {k-1 \\choose i}.$$\nAs $$\\sum_{i=1}^{2a} (2a+1-i)d_i = \\sum_{i=1}^{2a}d_i + \\sum_{i=1}^{2a} (2a-i)d_i \\text{ and } \nd_{2a+1}^+ \\leq 2b +  \\sum_{i=1}^{2a} (2a-i)d_{i},$$\nwe have \n$$n = d_{2a+1}^+ + \\sum_{i=1}^{2a} d_{i} \\leq 2b + k \\sum_{i=1}^{2a} (2a+1-i) {k-1 \\choose i}.$$\nThis completes the first part of the proof. \n\n\\medskip\n\n\nFor the proof of the second part of the Theorem, we notice that if the order of a graph $G^*$ attains the upper bound, then equality holds in all of the above inequations. In particular, we must have $d_{2a+1}^+ = \\sum_{v \\in D_{2a+1}^+} (deg(v)-2a)$ which implies that $G^*$ cannot have a vertex of degree more than $2a+1$. Moreover, we also have the following equality.\n$$\\sum_{i=1}^s d_i = k \\sum_{i=1}^{s} {k-1 \\choose i} \\text{ for $s=1, 2, \\ldots , 2a+1$}.$$\nThis proves that $G^*$ has exactly $k{k-1 \\choose i}$ vertices of degree $i$.\n\\end{proof}\n\n\nNext we are going to present some immediate corollaries of  Theorem~\\ref{th_lower bound}. \nA \\textit{cactus} is a connected graph in which no two cycles share a common edge. \n\n\n\\begin{corollary}\\label{cactus}\nLet $G$ be a cactus on $n$ vertices and $m$ edges. If $\\chi_{NL}(G)=k$, then \n$$n \\le \\frac{k^4+11k^2-12k-6}{6}.$$\nMoreover, if the cactus has exactly $t$ cycles, then we have\n$$n \\le 2(t-1)+\\frac{k^3+k^2-2k}{2}.$$\n\\end{corollary}\n\n\\begin{proof}\nObserve that $G$ has at most $\\frac{3(n-1)}{2}$ edges. So, by substituting $a=\\frac{3}{2}$ and $b=-\\frac{3}{2}$ in the bound for $n$ established in Theorem~\\ref{th_lower bound}, we have \n\\begin{align*}\n    n  \\leq 2b + k \\sum_{i=1}^{2a} (2a+1-i) {k-1 \\choose i} &= -3 + k\\sum_{i=1}^{3} (4-i) {k-1 \\choose i} \\\\\n    &= -3 + 3k {k-1 \\choose 1} + 2k {k-1 \\choose 2} + k {k-1 \\choose 3}\\\\\n    &= \\frac{k^4+11k^2-12k-6}{6}. \n\\end{align*}\nNote that, if the cactus $G$ has has exactly $t$ cycles, then $G$ has exactly $(n+t-1)$ edges.\nHence, replacing $a=1$ and $b = (t-1)$ in the bound for $n$ established in Theorem~\\ref{th_lower bound}, we have\n\\begin{align*}\n    n  \\leq 2b + k \\sum_{i=1}^{2a} (2a+1-i) {k-1 \\choose i} &= 2(t-1) + k \\sum_{i=1}^{2} (3-i) {k-1 \\choose i} \\\\&= 2(t-1) + 2k{k-1 \\choose 1} + k{k-1 \\choose 2} \\\\&= 2(t-1)+\\frac{k^3+k^2-2k}{2}. \n\\end{align*}\n\\end{proof}\n\n\n\nA graph is \\textit{$t$-degenerate} if its every subgraph has a vertex of degree at most $t$. \n\n\\begin{corollary}\\label{t-degenerate}\nLet $G$ be a $t$-degenerate graph on $n$ vertices and $m$ edges. If $\\chi_{NL}(G)=k$, then\n$$n\\le k\\sum_{i=1}^{2t}(2t+1-i){k-1\\choose i}-t(t+1).$$\n\\end{corollary}\n\n\\begin{proof}\nObserve that the number of edges in a $t$-degenerate graph is \n$m\\le tn-\\frac{t(t+1)}{2}$. Substituting $a=t$ and $b=-\\frac{t(t+1)}{2}$ in the bound for $n$ established in Theorem~\\ref{th_lower bound}, we have\n\\begin{equation*}\n   n  \\leq 2b + k \\sum_{i=1}^{2a} (2a+1-i) {k-1 \\choose i} = -t(t+1) + k \\sum_{i=1}^{2t} (2t+1-i) {k-1 \\choose i}.\n\\end{equation*}\n\\end{proof}\n\n\nA planar graph is $5$-degenerate, thus using the above corollary, we know that for a planar graph $G$ one can obtain an upper bound of $|V(G)|$. However, since $|E(G)| \\le 3|V(G)|-6$, we are able to obtain a better bound. \n\n\n\\begin{corollary}\\label{planar}\nLet $G$ be a planar graph on $n$ vertices and $m$ edges. If $\\chi_{NL}(G)=k$, then \n$$n\\le k\\sum_{i=1}^{6}(7-i){k-1\\choose i}-12.$$\n\\end{corollary}\n\n\\begin{proof}\nNote that the number of edges in a planar graph is at most $3n-6$. Substituting $a=3$ and $b=-6$ in the bound for $n$ established in Theorem~\\ref{th_lower bound}, we have \n\\begin{equation*}\n    n  \\leq 2b + k \\sum_{i=1}^{2a} (2a+1-i) {k-1 \\choose i} = -12 + k \\sum_{i=1}^{6} (7-i) {k-1 \\choose i}.\n\\end{equation*}\n\\end{proof}\n\n\n\\subsection{Tightness}\nNext we show the asymptotic tightness of Theorem~\\ref{th_lower bound}. To that end, we will prove the following result. \n\n\\begin{theorem}\\label{graph construction}\nLet $2a$ be a positive integer and let $2b$ be an integer. Then, there exists a graph $G$ on $n$ vertices and $m$ edges satisfying $m \\leq an+b$ such that $n = \\Theta(k^{2a+1})$ and $\\chi_{NL}(G)=\\Theta(k)$. Moreover, when $b=0$, $G$ can be taken to be of maximum degree $2a$.\n\\end{theorem}\n\n\nThe proof of this theorem is contained within a number of observations and lemmas. Also, the proof is constructive, and the constructions depend on particular partial colorings. Therefore, we are going to present a series of graph constructions, their particular colorings, and their structural properties. We are also going to present the supporting observations and lemmas in the following. \n\n\n\\medskip\n\n\n\\begin{lemma}\\label{lem matrix matching}\nLet us consider a $(p \\times q)$ matrix whose $ij^{th}$ entry is $m_{i,j}$, where $p < q$.  Let $M$ be a complete graph whose  vertices are the entries of the matrix. Then there exists a \nmatching of $M$ satisfying the following conditions:\n\\begin{enumerate}[(i)]\n    \\item The endpoints of an edge of the matching are from different columns. \n    \\item Let $e_1$ and $e_2$ be two edges of the matching. If one endpoint of $e_1$ and $e_2$ are from the same column, then the other endpoints of them must belong to distinct columns. \n    \\item The matching saturates all but at most one vertex of $M$ per column. \n\\end{enumerate}\n\\end{lemma}\n\n\n\n\\begin{proof}\nConsider the permutation $\\sigma = (1\\ 2\\  \\cdots q)$. \nThe matching consists of edges of the type \n$m_{(2i-1),j}m_{2i,\\sigma^i(j)}$ \nfor all $i \\in \\{1, 2, \\cdots, \\lfloor \\frac{p}{2} \\rfloor \\}$ and \n$j \\in \\{1, 2, \\cdots, q\\}$. We will show that this matching satisfies all \nlisted conditions. \n\n\n\nObserve that, a typical edge of the matching is of the form \n$m_{(2i-1),j}m_{2i,\\sigma^i(j)}$. As the second co-ordinates of the subscript \n of the endpoints of the said edge is different, condition~$(i)$ \n from the statement is verified. \n \n \n Suppose that there are two edges of the type \n $m_{(2i-1),j}m_{2i,\\sigma^i(j)}$ and \n $m_{(2i'-1),j'}\\\\m_{2i',\\sigma^{i'}(j')}$. \n If $m_{(2i-1),j}$ and $m_{(2i'-1),j'}$ are from the same column, \n that is, $j = j'$, then we must have $i \\neq i'$ as they are different vertices. Thus\n $\\sigma^i(j) \\neq \\sigma^{i'}(j) = \\sigma^{i'}(j')$ as $i \\neq i'$. \n If $m_{(2i-1),j}$ and $m_{2i',\\sigma^{i'}(j')}$ are from the same column, \n then we have $j = \\sigma^{i'}(j')$. Moreover, if we have \n $j' = \\sigma^{i}(j)$, then it will imply that \n $$j = \\sigma^{i'}(\\sigma^{i}(j)) = \\sigma^{i+i'}(j).$$\n This is only possible if $q | (i+i')$, which is not possible as \n $i, i' \\in  \\{1, 2, \\cdots, \\lfloor \\frac{p}{2} \\rfloor \\}$. \n Therefore, we have verified condition $(ii)$ of the statement. \n\nNotice that, the matching saturates all the vertices of $M$ when $p$ is even,\n whereas it saturates all except the vertices in the $p^{th}$ row of the matrix when $p$ is odd. This verifies condition $(iii)$ of the statement. \n\\end{proof}\n\n\n\n\n\\begin{corollary}~\\label{cor_spanning_supergh}\nLet $G$ be a graph with an independent set $M$ of size $(p \\times q)$, where $M = \\{m_{ij}: 1 \\leq i \\leq p, 1 \\leq j \\leq q\\}$ and $p < q$. Moreover, let $\\phi$ be a $(k'+q)$-coloring of $G$ satisfying the following conditions:\n\\begin{enumerate\n    \\item $k'+1 \\leq \\phi(x) \\leq k'+ q$ if and only if $x \\in M$,\n    \n    \\item $x$ and $y$ are neighbor-distinguished unless \n    both belong to $M$,\n    \n    \\item $\\phi(m_{ij}) = k'+j$.\n\\end{enumerate}\nThen it is possible to find spanning supergraph $G'$ of $G$ by adding a matching between the vertices of $M$ which will make $\\phi$ a neighbor-locating $(k'+q)$-coloring of $G'$. \n\\end{corollary}\n\n\n\\begin{proof}\nFirst of all build a matrix whose $ij^{th}$ entry is the vertex $m_{ij}$. \nAfter that, build a complete graph whose vertices are entries of this matrix. Now using Lemma~\\ref{lem matrix matching}, we can find a matching of this complete graph that satisfies the three conditions mentioned in the statement of Lemma~\\ref{lem matrix matching}. We construct $G'$ by including exactly the edges corresponding to the edges of the matching, between the vertices of $M$. We want to show that after adding these edges and obtaining $G'$, indeed $\\phi$ is a neighbor-locating $(k'+q)$-coloring of $G'$. \n\nNotice that by the definition of $\\phi$, $(k'+q)$ colors are used. So it is enough to show that the vertices of $G'$ are neighbor-distinguished with respect to $\\phi$. To be precise, it is enough to show that two vertices $x,y$ from $M$ are neighbor-distinguished with respect to $\\phi$ in $G'$. \nIf $\\phi(x) = \\phi(y)$, then they must have different color-neighborhood inside $M$ according to the conditions of the matching. This is enough to make $x,y$ neighbor-distinguished. \n\\end{proof}\n\n\nNow we are ready to present our iterative construction. However, given the involved nature of it, we need some specific nomenclatures to describe it. For convenience, we will list down some points to describe the whole construction.  \n\n\n\\begin{enumerate}[(i)]\n\\item  An \\textit{$i$-triplet} is a $3$-tuple of the type $(G_i, \\phi_i, X_i)$ where $G_i$ is a graph, $\\phi_i$ is a neighbor-locating $(ik)$-coloring of $G_i$, $X_i$ is a set of $(i+1)$-tuples of vertices of $G_i$, each having non-repeating elements. Also, two $(i+1)$-tuples from $X_i$ do not have any entries in common. \n\n\n\n\n\n\\item Let us describe the $1$-triplet \n$(G_1, \\phi_1, X_1)$ explicitly. Here $G_1$ is the path \n    $P_t = v_1v_2\\cdots v_t$ on $t$ vertices where \n$t = 4\\left\\lfloor \\frac{k^2(k-1)}{8}\\right\\rfloor$. As \n$$\\frac{(k-1)^2(k-2)}{2} < 4\\left\\lfloor \\frac{k^2(k-1)}{8}\\right\\rfloor \\leq \\frac{k^2(k-1)}{2},$$\nwe must have $\\chi_{NL}(P_t) = k$ (see \\cite{alcon2019neighbor}).\nLet $\\phi_1$ be any neighbor-locating $k$-coloring of $G_1$ and \n$$X_1 = \\{(v_{i-1}, v_{i+1}) : i\\equiv 2,3 \\pmod 4 \\}.$$ \n\n\n\n\\item  Suppose an $i$-triplet $(G_i, \\phi_i, X_i)$  is given. We will (partially) describe a way to construct an $(i+1)$-triplet from it. \nTo do so, \nfirst we will  construct  an intermediate graph $G'_{i+1}$ as follows: \nfor each $(i+1)$-tuple $(x_1, x_2, \\cdots, x_{i+1}) \\in X_i$  we will add a \\textit{new vertex} $x_{i+2}$ adjacent to each \n    vertex from the $(i+1)$-tuple. \n    Moreover, $(x_1, x_2, \\cdots, x_{i+1}, x_{i+2})$ is designated as an $(i+2)$-tuple in $G'_{i+1}$. \n After that, we will take $k$ copies of $G'_{i+1}$ and call this so-obtained graph as $G''_{i+1}$. Furthermore, we will extend $\\phi_i$ to a function $\\phi_{i+1}$ by assigning the color $(ik+j)$ to the new vertices from the $j^{th}$ copy of  $G'_{i+1}$. The copies of the $(i+2)$-tuples are the $(i+2)$-tuples of $G''_{i+1}$. \n\n\n\n\\item Consider the $(i+1)$-triplet $(G''_{i+1}, \\phi_{i+1}, X''_{i+1})$ where $X''_{i+1}$ denotes the set of all $(i+2)$-tuples of $G''_{i+1}$.  \nThe \\textit{color of an $(i+2)$-tuple}\n$(x_1, x_2, \\cdots, x_{i+2})$ is the set \n$$C((x_1, x_2, \\cdots, x_{i+2}))= \\{\\phi_i(x_1), \\phi_i(x_2), \\cdots, \\phi_i(x_{i+2})\\}.$$ \nLet us partition the set of new vertices based on the colors used on the elements (all but the last one) of the $(i+2)$-tuple of which it is (uniquely) part of. To be explicit, the last elements of two $(i+2)$-tuples are in the same partition if and only if they have the same color neighbors. Let this partition be denoted by $X_{i1}, X_{i2}, \\cdots, X_{is_i}$, for some integer $s_i$.\n\n\n\\item First fix a partition $X_{ir}$ of $X_i$. Next construct a matrix with its $\\ell^{th}$ column having vertices from  $X_{ir}$ as its entries if they are also from the $\\ell^{th}$ copy of $G'_{i+1}$ in $G''_{i+1}$. Thus the matrix is a $(p \\times q)$ matrix where \n$p = |X_{ir}|$ and $q = k$. We are going to show that, $p < q$. However, for convenience, we will defer it \nto a later part (Lemma~\\ref{lem_X_ir}). \n\n\n\n\n\n\\item Let us delete all the new vertices from $G''_{i+1}$ except for the ones in $X_{ir}$. This graph has the exact same properties of the graph $G$ from Corollary~\\ref{cor_spanning_supergh} where $X_{ir}$ plays the role of the independent set $M$. Thus it is possible to add a matching and extend the coloring (like in Corollary~\\ref{cor_spanning_supergh}). We do that for each value of $r$ and add the corresponding matching to our graph $G''_{i+1}$. After adding all such matchings, the graph we obtain is $G_{i+1}$. \n\\end{enumerate}\n\n\\begin{figure}[h]\n    \\centering\n\t\\begin{tikzpicture}[scale=.42]\n\t\n\t\\foreach \\x in {-12,...,11} \\foreach \\y in {0}{\n\t\t\t\t\\node[circle, draw=black, scale=.3, fill=black] (\\x) at (\\x,\\y){};}\n\t\n\t\n\t\\foreach \\x\/\\y in {-12\/-11,-11\/-10,-10\/-9,-9\/-8,-8\/-7,-7\/-6,-6\/-5,-5\/-4,-4\/-3,-3\/-2,-2\/-1,-1\/0,0\/1,1\/2,2\/3,3\/4,4\/5,5\/6,6\/7,7\/8,8\/9,9\/10,10\/11} \n\t\\draw[-,draw=black, opacity=.5] (\\x) -- (\\y);\n\t\n\t\n\t\\foreach \\x in {-11,-10,-7,-6,-3,-2,1,2,5,9} \\foreach \\y in {2}{\n\t\t\t\t\\node[circle, draw=black, scale=.3, fill=black] (\\x) at (\\x,\\y){};}\n\t\n\t\n\t\\foreach \\x in {-11,9} \\foreach \\y in {2,6,10,14}{\n\t\t\t\t\\node[circle, draw=red, scale=.3, fill=red] (\\x) at (\\x,\\y){};}\n\t\n\t\n\t\n\t\\foreach \\x in {6,10} \\foreach \\y in {2,6,10,14}{\n\t\t\t\t\\node[circle, draw=blue, scale=.3, fill=blue] (\\x) at (\\x,\\y){};}\n\t\n\t\n\t\n\t\n\t\\foreach \\x\/\\y in {-11\/-12,-11\/-10,-10\/-11,-10\/-9,-7\/-8,-7\/-6,-6\/-7,-6\/-5,-3\/-4,-3\/-2,-2\/-3,-2\/-1,1\/0,1\/2,2\/1,2\/3,5\/4,5\/6,6\/5,6\/7,9\/8,9\/10,10\/9,10\/11}\n\t\\draw[-,draw=black, opacity=.5] (\\x,2) -- (\\y,0);\n\t\n\t\n\t\n\t\n\t\n\t\\foreach \\x in {-12,...,11} \\foreach \\y in {4}{\n\t\t\t\t\\node[circle, draw=black, scale=.3, fill=black] (\\x) at (\\x,\\y){};}\n\t\n\t\n\t\\foreach \\x\/\\y in {-12\/-11,-11\/-10,-10\/-9,-9\/-8,-8\/-7,-7\/-6,-6\/-5,-5\/-4,-4\/-3,-3\/-2,-2\/-1,-1\/0,0\/1,1\/2,2\/3,3\/4,4\/5,5\/6,6\/7,7\/8,8\/9,9\/10,10\/11} \n\t\\draw[-,draw=black, opacity=.5] (\\x) -- (\\y);\n\t\n\t\n\t\\foreach \\x in {-10,-7,-6,-3,-2,1,2,5} \\foreach \\y in {6}{\n\t\t\t\t\\node[circle, draw=black, scale=.3, fill=black] (\\x) at (\\x,\\y){};}\n\t\n\t\n\t\\foreach \\x\/\\y in {-11\/-12,-11\/-10,-10\/-11,-10\/-9,-7\/-8,-7\/-6,-6\/-7,-6\/-5,-3\/-4,-3\/-2,-2\/-3,-2\/-1,1\/0,1\/2,2\/1,2\/3,5\/4,5\/6,6\/5,6\/7,9\/8,9\/10,10\/9,10\/11}\n\t\\draw[-,draw=black, opacity=.5] (\\x,6) -- (\\y,4);\n\t\n\t\n\t\n\t\n\t\n\t\\foreach \\x in {-12,...,11} \\foreach \\y in {8}{\n\t\t\t\t\\node[circle, draw=black, scale=.3, fill=black] (\\x) at (\\x,\\y){};}\n\t\n\t\n\t\\foreach \\x\/\\y in {-12\/-11,-11\/-10,-10\/-9,-9\/-8,-8\/-7,-7\/-6,-6\/-5,-5\/-4,-4\/-3,-3\/-2,-2\/-1,-1\/0,0\/1,1\/2,2\/3,3\/4,4\/5,5\/6,6\/7,7\/8,8\/9,9\/10,10\/11} \n\t\\draw[-,draw=black, opacity=.5] (\\x) -- (\\y);\n\t\n\t\n\t\\foreach \\x in {-10,-7,-6,-3,-2,1,2,5} \\foreach \\y in {10}{\n\t\t\t\t\\node[circle, draw=black, scale=.3, fill=black] (\\x) at (\\x,\\y){};}\n\t\n\t\n\t\\foreach \\x\/\\y in {-11\/-12,-11\/-10,-10\/-11,-10\/-9,-7\/-8,-7\/-6,-6\/-7,-6\/-5,-3\/-4,-3\/-2,-2\/-3,-2\/-1,1\/0,1\/2,2\/1,2\/3,5\/4,5\/6,6\/5,6\/7,9\/8,9\/10,10\/9,10\/11}\n\t\\draw[-,draw=black, opacity=.5] (\\x,10) -- (\\y,8);\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\foreach \\x in {-12,...,11} \\foreach \\y in {12}{\n\t\t\t\t\\node[circle, draw=black, scale=.3, fill=black] (\\x) at (\\x,\\y){};}\n\t\n\t\n\t\\foreach \\x\/\\y in {-12\/-11,-11\/-10,-10\/-9,-9\/-8,-8\/-7,-7\/-6,-6\/-5,-5\/-4,-4\/-3,-3\/-2,-2\/-1,-1\/0,0\/1,1\/2,2\/3,3\/4,4\/5,5\/6,6\/7,7\/8,8\/9,9\/10,10\/11} \n\t\\draw[-,draw=black, opacity=.5] (\\x) -- (\\y);\n\t\n\t\n\t\\foreach \\x in {-10,-7,-6,-3,-2,1,2,5} \\foreach \\y in {14}{\n\t\t\t\t\\node[circle, draw=black, scale=.3, fill=black] (\\x) at (\\x,\\y){};}\n\t\n\t\n\t\\foreach \\x\/\\y in {-11\/-12,-11\/-10,-10\/-11,-10\/-9,-7\/-8,-7\/-6,-6\/-7,-6\/-5,-3\/-4,-3\/-2,-2\/-3,-2\/-1,1\/0,1\/2,2\/1,2\/3,5\/4,5\/6,6\/5,6\/7,9\/8,9\/10,10\/9,10\/11}\n\t\\draw[-,draw=black, opacity=.5] (\\x,14) -- (\\y,12);\n\t\n\t\n\t\\foreach \\x\/\\y in \n\t{-12\/-.4, -12\/3.6, -12\/7.6, -12\/11.6, -7\/-.4, -7\/3.6, -7\/7.6, -7\/11.6, \n\t-5\/-.4, -5\/3.6, -5\/7.6, -5\/11.6,\n\t-2\/-.4, -2\/3.6, -2\/7.6, -2\/11.6,\n\t0\/-.4, 0\/3.6, 0\/7.6, 0\/11.6,\n\t10\/-.4, 10\/3.6, 10\/7.6, 10\/11.6}\n\t\\node  at (\\x,\\y) {\\scriptsize $1$};\n\t\n\t\n\t\n\t\\foreach \\x\/\\y in \n\t{-10\/-.4, -10\/3.6, -10\/7.6, -10\/11.6, -8\/-.4, -8\/3.6, -8\/7.6, -8\/11.6, \n\t-6\/-.4, -6\/3.6, -6\/7.6, -6\/11.6,\n\t1\/-.4, 1\/3.6, 1\/7.6, 1\/11.6,\n\t6\/-.4, 6\/3.6, 6\/7.6, 6\/11.6,\n\t8\/-.4, 8\/3.6, 8\/7.6, 8\/11.6}\n\t\\node  at (\\x,\\y) {\\scriptsize $2$};\n\t\n\t\n\t\n\t\\foreach \\x\/\\y in \n\t{-11\/-.4, -11\/3.6, -11\/7.6, -11\/11.6, -9\/-.4, -9\/3.6, -9\/7.6, -9\/11.6, \n\t-4\/-.4, -4\/3.6, -4\/7.6, -4\/11.6,\n\t3\/-.4, 3\/3.6, 3\/7.6, 3\/11.6,\n\t5\/-.4, 5\/3.6, 5\/7.6, 5\/11.6,\n\t11\/-.4, 11\/3.6, 11\/7.6, 11\/11.6}\n\t\\node  at (\\x,\\y) {\\scriptsize $3$};\n\t\n\t\n\t\n\t\\foreach \\x\/\\y in \n\t{-3\/-.4, -3\/3.6, -3\/7.6, -3\/11.6, \n\t-1\/-.4, -1\/3.6, -1\/7.6, -1\/11.6, \n\t2\/-.4, 2\/3.6, 2\/7.6, 2\/11.6,\n\t4\/-.4, 4\/3.6, 4\/7.6, 4\/11.6,\n\t7\/-.4, 7\/3.6, 7\/7.6, 7\/11.6,\n\t9\/-.4, 9\/3.6, 9\/7.6, 9\/11.6}\n\t\\node  at (\\x,\\y) {\\scriptsize $4$};\n\t\n\t\n\t\n\t\n\t\\draw[-,dashed,draw=red, opacity=.5] (-11,2) -- (9,6);\n\t\\draw[-,dashed,draw=red, opacity=.5] (-11,6) -- (9,10);\n\t\\draw[-,dashed,draw=red, opacity=.5] (-11,10) -- (9,14);\n\t\\draw[-,dashed,draw=red, opacity=.5] (-11,14) -- (9,2);\n\t\n\t\n\t\n\t\\draw[-,dashed,draw=blue, opacity=.8] (6,2) -- (10,6);\n\t\\draw[-,dashed,draw=blue, opacity=.8] (6,6) -- (10,10);\n\t\\draw[-,dashed,draw=blue, opacity=.8] (6,10) -- (10,14);\n\t\\draw[-,dashed,draw=blue, opacity=.8] (6,14) -- (10,2);\n\t\n\t\n\t\\draw [decorate,\n    decoration = {calligraphic brace,mirror,raise=5pt,\n        amplitude=5pt},thick] (11,-.1) --  (11,2.1);\n    \n    \\node at (12.4,1) {$G'_2$};\n\t\n\t\n\n    \n   \n\t\n\t\\node at (-12.5,2) {\\textcolor{darkmagenta}{\\textbf{$5$}}};\n\t\\node at (-12.5,6) {\\textcolor{darkmagenta}{\\textbf{$6$}}};\n\t\\node at (-12.5,10) {\\textcolor{darkmagenta}{\\textbf{$7$}}};\n\t\\node at (-12.5,14) {\\textcolor{darkmagenta}{\\textbf{$8$}}};\n\t\n\n\n\n\n\t\n\t\n\t\n\t\\node at (-14.1,1.3) {\\textcolor{darkmagenta}{\\scriptsize{Colors of new}}};\n\t\\node at (-14,.7) {\\textcolor{darkmagenta}{\\scriptsize{vertices}}};\n\t\t\t\t\n\t\\end{tikzpicture}\n\t\\centering\n    \\caption{Construction of $G_2$ from $G_1=P_{24}$. Here $\\chi_{NL}(P_{24})=4$, the red and blue edges are the two sets of newly added matchings.}\n    \\label{fig:Construction of $G_i+1$ from $G_i$}\n\\end{figure}\n\n\n\\begin{lemma}~\\label{lem_X_ir}\nWe have $|X_{ir}| < k$, where $X_{ir}$ is as in Item(v) of the above list. \n\\end{lemma}\n\n\\begin{proof}\nWe show that the set of new vertices having the same color neighbors in  $G_1$ is strictly less than $k$ as follows:\n\nAny vertex (other than the end vertices) in $G_1$ has two color neighbors say $i$ and $j$ ($i$ may or may not be equal to $j$). Having fixed the two color neighbors, this vertex will have at most $k-1$ choices of colors. This implies that there can be at most $k-1$ new vertices with given color neighbors, which would also ensure that the set of new vertices having the same color neighbors in  $G_1$ (that is the set of $2$-tuples \nhaving the same color in $G_1$) is strictly less than $k$.\n\nAfter that we are done by induction. \n\\end{proof}\n\n\n\\begin{lemma}\\label{lem phii nl-col}\nThe function $\\phi_{i+1}$ is a neighbor-locating coloring of $G_{i+1}$. \n\\end{lemma}\n\n\\begin{proof}\nThe function $\\phi_{i+1}$ is constructed from $\\phi_i$, alongside constructing the triplet $G_{i+1}$ from $G_i$. While constructing, we use the same steps from that of Corollary~\\ref{cor_spanning_supergh}. \nThus, the newly colored vertices become neighbor-distinguished in $G_{i+1}$ under $\\phi_{i+1}$. \n\\end{proof}\n\n\nThe above two lemmas validate the correctness of the iterative construction of $G_i$s. However, it remains showing how $G_i$s help us prove our result. To do so, let us prove certain properties of $G_i$s. \n\n\\begin{lemma}\\label{lem max degree of Gi}\nThe graph $G_i$ is not a regular graph and has \nmaximum degree $(i+1)$. \n\\end{lemma}\n\n\\begin{proof}\nAs we have started with a path, our $G_1$ has maximum degree $2$ and is not regular. In the iteration step for constructing the graph \n$G_{i+1}$ from $G_i$, \nthe degree of an old vertex (or its copy) can increase at most by $1$, while a new vertex of $G_{i+1}$ is adjacent to exactly $(i+1)$ old vertices and at most one new vertex. Hence, a new vertex in $G_{i+1}$ can have degree at most $(i+2)$. Therefore, the proof is done by induction. \n\\end{proof}\n\n\n\n\n\\medskip\n\n\n\n\nFinally, we are ready to prove Theorem~\\ref{graph construction}.\n\n\n\\medskip\n\n\n\\noindent \\textit{Proof of Theorem~\\ref{graph construction}.}\nGiven $a$ and $b$, to build the example that will prove the theorem, one can consider $G = G_{2a+1}$. \\qed\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bigskip\n\n\\noindent \\textbf{Acknowledgements:}\nThis work is partially supported  by the following projects: ``MA\/\\\\IFCAM\/18\/39'', ``SRG\/2020\/001575'', ``MTR\/2021\/000858'' and ``NBHM\/RP-8 (2020)\/\\\\Fresh''. Research by the first and second authors is partially sponsored by a public grant overseen by the French National Research Agency as part of the ``Investissements d'Avenir'' through the IMobS3 Laboratory of Excellence (ANR-10-LABX-0016), the IDEX-ISITE initiative CAP 20-25 (ANR-16-IDEX-0001) and the ANR project \\\\ GRALMECO (ANR-21-CE48-0004-01).\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nInflationary expansion of the Universe is now believed to have existed \nbefore the radiation dominated era in the early Universe \\cite{uobs,planck15}.\nAlthough a lot of inflation models have been proposed by now \\cite{slowroll}, \nwe do not know which model can describe this phenomenon correctly. \nThe relation between inflation and particle physics is also unclear. \nHowever, recent results of the CMB observations seem to have ruled out \nmany of them already \\cite{planck15}. This has been done by comparing\nboth values of the scalar spectral index and the tensor-to-scalar \nratio obtained from the CMB observations and predicted values for them \nby each model.\n\nIf we take a large inflaton scenario in the slow-roll inflation \nframework, trans-Planckian values are required for realization of\nsufficient $e$-foldings. In that case, we cannot answer the question \nwhy higher non-renormalizable terms do not contribute\nto the inflaton potential. If we include such terms in \ninflaton potential, slow-roll conditions cannot be satisfied.\nHowever, such kind of difficulty can be reconciled by \ncompactifying the inflaton trajectory into a winding trajectory \nin the higher dimensional fields space. In that case, sufficient \n$e$-foldings along the trajectory can be obtained even if each field is \nkept in sub-Planckian regions. That possibility was studied in \\cite{Yang} \nby introducing an idea about a \\textit{spiralized inflation} in \ntwo-dimensional field space as a possible solution of the \nmultifield slow-roll inflation. Its realizations have been proposed \nin the several frameworks, such as in string models \\cite{Spiral}, \nSUSY models \\cite{susy}, axions-based models \n\\cite{axion, Natural, AlignNI, real, Dante} \nand a complex scalar model which has similar features with axions-based \nmodels \\cite{complex, mac}. The higher order corrections \nto the potential realizing the spiralized inflation are found \nto affect significantly the predicted tensor-to-scalar ratio \nwithout changing the spectral index substantially \\cite{corr}.    \n\nIn this paper we study an inflation model based on large but \nsub-Planckian inflaton. The model considered here has been \nproposed in \\cite{bks}.\nIt could have an intimate connection to neutrino mass generation.\nAlthough the required number for $e$-foldings is found to be realized \nin this model, the predicted tensor-to-scalar ratio by the model \nis too large compared with a central value of the up-dated observational \nresults \\cite{planck15,bkp}. Our main purpose is to study whether \nthe favorable values for them can be obtained in this model. We also discuss\na possible connection to a certain particle physics model.        \n\n\\section{A sub-Planckian inflaton model} \nThe model studied here is defined by a complex scalar $S$ \nwhich has $Z_2$ odd parity. Its $Z_2$ invariant potential is assumed to \nbe given such as \\cite{bks}   \n\\begin{equation}\nV_S = c_1\\frac{(S^\\dagger S)^n}{M_{\\rm pl}^{2n-4}}\n\\left[1+ c_2\\left\\{ \\left(\\frac{S}{M_{\\rm pl}}\\right)^{2m} \n\\exp\\left(i\\frac{S^\\dagger S}{\\Lambda^2}\\right)\n+ \\left(\\frac{S^\\dagger}{M_{\\rm pl}}\\right)^{2m}\n\\exp\\left(-i\\frac{S^\\dagger S}{\\Lambda^2}\\right)\n\\right\\} \\right]. \\nonumber \\\\\n\\label{model1}\n\\end{equation}\nIf we use the polar coordinate $S=\\frac{\\varphi}{\\sqrt 2}e^{i\\theta}$,\nthis potential can be written as\n\\begin{equation}\nV_{S}=c_1\\frac{\\varphi^{2n}}{2^nM_{\\rm pl}^{2n-4}}\\left[\n1+ 2c_2\\left(\\frac{\\varphi}{\\sqrt 2 M_{\\rm pl}}\\right)^{2m}\n\\cos\\left(\\frac{\\varphi^2}{2\\Lambda^2}+2m\\theta\\right)\\right],  \n\\label{infpot}\n\\end{equation}\nwhere $c_2$ is assumed to satisfy \n$c_2< 0.5 \\left(\\frac{\\varphi}{\\sqrt 2M_{\\rm pl}}\\right)^{-2m}$.\nAs shown in Fig.~\\ref{figpot}, \n$V_{S}$ has local minima with a potential barrier\n$V_b\\simeq \\frac{c_1c_2\\varphi^{2(n+m)}}{2^{n+m-2}M_{\\rm pl}^{2(n+m-2)}}$\nin the radial direction. These minima form a spiral-like valley\nwhose slope in the angular direction could be extremely small.\nAs a result, if we use the field evolution along this valley, \nslow-roll inflation is expected to be caused even for sub-Planckian \nvalues of $\\varphi$ \\cite{bks,mac}.  \n\n\\begin{figure}[t]\n\\begin{center}\n       \\includegraphics[width=9cm]{figpot2_black.eps}\n\\end{center}\n\\vspace*{5mm}\n\n\\caption{ Potential $V_S$ at a fixed $\\theta$ \nfor $n=2, m=1$ case. Other parameters are fixed \nat $c_1=1.1791 \\times 10^{-7}$, \n$c_2 = 1.4$ and $\\Lambda=0.05 M_{\\rm pl}$.}\n\\label{figpot}\n\\end{figure}\n\n\nThe evolution of the scalar field \n$S=\\frac{1}{\\sqrt{2}} (\\varphi_1 + i \\varphi_2)$ in this potential \nis dictated by the following equation of motion:\n\\begin{align}\n\\ddot{\\varphi_i} + 3 H \\dot{\\varphi_i} = \n    -\\frac{\\partial V_S}{\\partial \\varphi_i} \\qquad (i=1,2),\n\\label{evol}\n\\end{align}\nwhere the Hubble parameter $H$ of the system is now written as \n$H^2 =\\frac{1}{3M_{\\rm pl}^2} \\left(\\sum_i \\frac{1}{2} \\dot{\\varphi_i}^2 \n+V_S \\right)$ and $\\frac{\\partial V_S}{\\partial \\varphi_i}$ \ndenotes partial derivative of the potential $V_S$ in the direction of \nfield component $\\varphi_i$. \nTaking $m=1$, the terms $\\frac{\\partial V_S }{\\partial \\varphi_i}$  \ncould be simply written for any number of $n$ as follows,\n\\begin{align}\n\\frac{\\partial V_S}{\\partial \\varphi_1} = \n\\frac{c_1 \\left(S^\\dagger S\\right)^n}{M_{\\rm pl}^{2n-4}} \n\\left[ \\frac{n\\varphi_1}{\\left(S^\\dagger S\\right)} \\right.\n&+ \\left. \\frac{c_2 \\varphi_1}{M_{\\rm pl}^{2}} \n\\left\\{ \\frac{ n \\left(\\varphi_1^2 -\\varphi_2^2\\right)}\n{\\left(S^\\dagger S\\right)} + 2 \n- \\frac{2\\varphi_1 \\varphi_2}{\\Lambda^2} \\right\\}  \n\\cos\\left(\\frac{ S^\\dagger S}{\\Lambda^2}\\right) \\right. \\nonumber \\\\\n&- \\left. \\frac{c_2 \\varphi_1}{M_{\\rm pl}^{2}} \n\\left\\{ \\frac{ 2n\\varphi_1 \\varphi_2 }{\\left(S^\\dagger S\\right)} \n+ 2 \\frac{\\varphi_2}{\\varphi_1} \n+ \\frac{ \\left(\\varphi_1^2 -\\varphi_2^2\\right)}{\\Lambda^2} \\right\\} \n \\sin\\left(\\frac{ S^\\dagger S}{\\Lambda^2}\\right) \\right], \t\n\\end{align}\n\\begin{align}\n\\frac{\\partial V_S}{\\partial \\varphi_2} = \n\\frac{c_1 \\left(S^\\dagger S\\right)^n}{M_{\\rm pl}^{2n-4}} \n\\left[ \\frac{n\\varphi_2}{\\left(S^\\dagger S\\right)} \\right.\n&+ \\left. \\frac{c_2 \\varphi_2}{M_{\\rm pl}^{2}} \n\\left\\{ \\frac{ n \\left(\\varphi_1^2 -\\varphi_2^2\\right)}\n{\\left(S^\\dagger S\\right)} - 2 \n- \\frac{2\\varphi_1 \\varphi_2}{\\Lambda^2} \\right\\}  \n\\cos\\left(\\frac{ S^\\dagger S}{\\Lambda^2}\\right) \\right. \\nonumber \\\\\n&- \\left. \\frac{c_2 \\varphi_2}{M_{\\rm pl}^{2}} \n\\left\\{ \\frac{ 2n\\varphi_1 \\varphi_2 }{\\left(S^\\dagger S\\right)} \n+ 2 \\frac{\\varphi_1}{\\varphi_2} \n+ \\frac{ \\left(\\varphi_1^2 -\\varphi_2^2\\right)}{\\Lambda^2} \\right\\}  \n\\sin\\left(\\frac{ S^\\dagger S}{\\Lambda^2}\\right) \\right]. \t\n\\end{align}\nWe may solve eq.~(\\ref{evol}) numerically to see \nthe evolution of $\\varphi_i$.  \nThe initial value of each component $\\varphi_i$ cannot be selected \narbitrarily since the slow-roll behavior could be ruined depending on it.\nIf the initial position of the inflaton is located at a point higher \nthan its next potential barrier, the inflaton could cross over \nit without realizing the slow-roll motion along the angular direction. \nThe most simple setting for the initial value to realize the slow-roll \nis to take it at a potential minimum. \nAn example of the evolution of the scalar field components \nis illustrated in Fig. \\ref{fig:1}. \n\n\n\\begin{figure}[t]\n        \\begin{center}\n        \\begin{subfigure}[b]{0.46\\textwidth}\n                \\includegraphics[width=\\textwidth]{dynamics.eps}\n                \\caption{}\n                \\label{fig:dyn}\n        \\end{subfigure\n    ~\n     \n        \\begin{subfigure}[b]{0.55\\textwidth}\n                \\includegraphics[width=\\textwidth]{evolution.eps}\n                \\caption{}\n                \\label{fig:evo}\n        \\end{subfigure}\n\t\t\t\t\\end{center}\n    ~\n     \n     \n\\caption{  Inflaton evolution for $n=2, m=1$ case. \nParameters in the potential are fixed at the same values as\nthe ones used in Fig.~\\ref{figpot}. Inflaton is assumed to be \nat a potential minimum initially.  In this case, \nit is numerically proven that the end of \nsingle field inflation\nsigned as turning point in the panel (a) is mostly realized much before \n$\\varepsilon \\simeq 1$, such that it is illustrated in the panel (b).  }\n\\label{fig:1}\n\\end{figure}\n\nNow we describe features of the inflation induced by this field \nevolution in detail.  \nThe radial component $\\varphi$ is assumed to take a large initial \nvalue on a local minimum in the radial direction. \nEven if $\\varphi$ is not just on this minimum initially, \nit converges to a minimum point within a certain period of time \nas long as it starts to roll from the neighborhood of \na minimum point of the valley initially. \nIn that case, as shown in \\cite{bks},\nthe model could cause sufficient $e$-foldings through the inflaton \nevolution along this spiral-like valley even for sub-Planckian \nvalues of $\\varphi$. \nAn inflaton field $\\chi$ could be identified with\n\\begin{equation}\n\\chi\\equiv a_e+\\frac{\\varphi_{e}^3}{6m\\Lambda^2}-a=\n\\frac{\\varphi^3}{6m\\Lambda^2},\n\\end{equation}\nwhere the subscript $e$ of the fields stands for the field value \nat the end of single field inflation.\nThe field $a$ is defined by using $\\varphi$ as\n\\begin{equation}\nda=\\left[\\varphi^2+\\left(\\frac{d\\varphi}{d\\theta}\n\\right)^2\\right]^{1\/2}d\\theta\n=\\left[1 + 4m^2\\left(\\frac{\\Lambda}{\\varphi}\\right)^4\\right]^{1\/2}\n\\varphi d\\theta.\n\\label{infl}\n\\end{equation}\nThe number of e-foldings caused by $\\chi$ during its slow-roll \nis given as\n\\begin{equation}\nN=-\\frac{1}{M_{\\rm pl}^2}\\int_\\chi^{\\chi_e} d\\chi ~\\frac{V_{S}}{V_{S}^\\prime}\n\\equiv N(\\chi)-N(\\chi_e), \n\\label{efold0}\n\\end{equation}\nwhere $V_{S}^\\prime=\\frac{dV_{S}}{d\\chi}$ and $N(\\chi)$ is represented \nby using the hypergeometric function $F$ as\n\\begin{eqnarray}\nN(\\chi)&=&\\frac{1}{6m^2n}\n\\left(\\frac{M_{\\rm pl}}{\\Lambda}\\right)^4\n\\left(\\frac{\\varphi}{\\sqrt 2M_{\\rm pl}}\\right)^6\\left[~1 \n+\\frac{6c_2m}{n(3+m)}\\left(\\frac{\\varphi}{\\sqrt 2M_{\\rm pl}}\\right)^{2m}\n\\right. \\nonumber \\\\\n&&\\hspace*{2cm}\\left. \n\\times F\\left(1,~\\frac{3}{m}+1,~\\frac{3}{m}+2,~2c_2\\left(1+\\frac{m}{n}\\right)\n\\left(\\frac{\\varphi}{\\sqrt 2M_{\\rm pl}}\\right)^{2m}\\right)\\right].\n\\label{efold}\n\\end{eqnarray} \n\nSlow-roll parameters \n$\\varepsilon\\equiv \\frac{M_{\\rm pl}^2}{2}\n\\left(\\frac{V_{S}^{\\prime}}{V_{S}}\\right)^2$ and \n$\\eta\\equiv M_{\\rm pl}^2\\left(\\frac{V_{S}^{\\prime\\prime}}{V_{S}}\\right)$ \nfor single field inflation can be represented by \nusing the model parameters as\n\\begin{eqnarray}\n&&\\varepsilon=m^2\\left(\\frac{\\sqrt 2M_{\\rm pl}}{\\varphi}\\right)^6\n\\left(\\frac{\\Lambda}{M_{\\rm pl}}\\right)^4\\left[\n\\frac{n-2c_2(m+n)\\left(\\frac{\\varphi}{\\sqrt 2M_{\\rm pl}}\\right)^{2m}}\n{1-2c_2\\left(\\frac{\\varphi}{\\sqrt 2M_{\\rm pl}}\\right)^{2m}}\\right]^2, \\nonumber\\\\\n&&\\eta=m^2\\left(\\frac{\\sqrt 2M_{\\rm pl}}{\\varphi}\\right)^6\n\\left(\\frac{\\Lambda}{M_{\\rm pl}}\\right)^4\n\\frac{n(2n-3)-2c_2(m+n)(2m+2n-3)\\left(\\frac{\\varphi}{\\sqrt 2M_{\\rm pl}}\n\\right)^{2m}}\n{1-2c_2\\left(\\frac{\\varphi_1}{\\sqrt 2M_{\\rm pl}}\\right)^{2m}}.  \n\\label{slow}\n\\end{eqnarray}\nIf the $c_2$ term is neglected in these formulas, we can find very \nsimple formulas for these slow-roll parameters at the time \ncharacterized by the inflaton value $\\chi_\\ast$.\nThey can be represented by using the $e$-foldings $N_\\ast$, \nwhich is defined for $N(\\chi_\\ast)$ in eq.~(\\ref{efold0}), as \n\\begin{equation}\n\\varepsilon\\simeq\\frac{n}{6(N_\\ast+N(\\chi_e))}, \\qquad \n\\eta\\simeq\\frac{2n-3}{6(N_\\ast+N(\\chi_e))}.\n\\end{equation}\nThus, the scalar index $n_s$ and the tensor-to-scalar ratio\n$r$ can be derived as \\cite{bks}\n\\begin{equation}\nn_s=1-6\\varepsilon+2\\eta\\simeq 1-\\frac{n+3}{3(N_\\ast+N(\\chi_e))}, \\qquad \nr=16\\varepsilon\\simeq \\frac{8n}{3(N_\\ast+N(\\chi_e))}.\n\\end{equation}\n\nIn order to see the features of this model, it may be useful to \ncompare the model with single field inflation with monomial \npotential $\\varphi^{\\bar n}$. Since the $e$-foldings in this model\nis written as $N\\simeq \\frac{1}{2\\bar n}\\frac{\\varphi^2}{M_{\\rm pl}^2}$,\nthe sufficient $e$-foldings require a trans-Planckian value for the \ninflaton $\\varphi$. We note that the contribution of $\\varphi_e$ \nis negligible in this case.\nIf we use this number of $e$-foldings $N_\\ast\\simeq N(\\varphi_\\ast)$, \nthe slow roll parameters are expressed as \n\\begin{equation}\n\\varepsilon=\\frac{\\bar n}{4N_\\ast}, \\qquad \\eta=\\frac{\\bar n-1}{2N_\\ast},\n\\end{equation}\nand then $n_s$ and $r$ can be written as\n\\begin{equation}\nn_s=1-\\frac{\\bar n+2}{2N_\\ast}, \\qquad r=\\frac{4\\bar n}{N_\\ast}.\n\\end{equation}\n\nIt is easily found that $n_s$ and $r$ in both models \nhave the same expression for $\\bar n=\\frac{2}{3}n$ in the limit $c_2=0$. \nHowever, we should remind that the present model works well \nonly for the non-negligible $c_2$ since this term causes \nthe potential barrier $V_b$ in the radial direction. \n$V_b$ makes the inflaton $a$ evolve along the \nspiral-like trajectory formed by the potential minima  \nlike a single field inflation.\nThis brings about a different feature for the model \nfrom the $\\varphi^{\\bar n}$ inflation scenario.\n\nIn this model,the single field inflation \nis expected to end at the time when \n$\\frac{1}{2}\\dot{\\chi}^2\\simeq V_b$ is realized.\nIf we apply the slow-roll approximation $3H\\dot\\chi=-V_{S}^\\prime$\nto the slow-roll parameter $\\varepsilon$,\nthe inflation is found to end \nat $\\varepsilon=\\frac{3V_b\\left(V_b +V_{S}\\right)}{V_{S}^2}$.\nSince $V_{S}>V_b$ is satisfied, the end of inflation could happen \nmuch before the time when  $\\varepsilon\\simeq 1$ is realized. \nIn such a case, $N(\\chi)\\gg N(\\chi_e)$ is not satisfied \nand then $N(\\chi_e)$ has a substantial contribution to determine \nthe number of $e$-foldings $N_\\ast$ in eq.~(\\ref{efold0}).\nThus, the smaller $N_\\ast$ could be enough to\nrealize the same values for $n_s$ and $r$ in comparison with the \n$\\varphi^{\\bar n}$ inflation.\\footnote{ \nAlthough this becomes clear especially \nin the small $c_2$ case, $\\varphi_e$ could be well approximated \nas the $\\varphi$ value at $\\varepsilon=1$ in other cases.}.\nWe should also note that the values of $n_s$ and $r$ in this\nmodel could deviate largely from ones predicted in the $\\varphi^{\\bar n}$ \ninflation model due to the non-negligible contribution from the \n$c_2$ term. \nIllustration given in Fig. \\ref{fig:evo} justifies this argument \nnumerically when whole contributions, including non-negligible $c_2$ term, \nare taken in to account. As it is expected, the end of inflation \nat which inflaton starts to oscillate around global minimum of the \npotential takes place much before $\\varepsilon \\simeq 1$.   \nAfter this time, the inflaton falls in the reheating  process and \nproduces lighter particles.\n\n\\section{Spectral index}\nWe estimate the scalar spectral index $n_s$ and the \ntensor-to-scalar ratio $r$ by taking account of the $c_2\\not=0$ effect. \nBefore it, we constrain parameters in the potential by using \nthe normalization for the scalar perturbation found in the CMB.\nThe normalization for the scalar perturbation found in the CMB \nobservations gives the constraint on the inflaton potential\n$V_{S}$ at the time when the scale characterized \nby a certain wave number $k_\\ast$ exits the horizon. \nThe observation of CMB requires the spectrum of scalar perturbation\n${\\cal P}_{\\mathcal{R}}(k)=A_s\\left(\\frac{k}{k_\\ast}\\right)^{n_s-1}$ to\ntake $A_s\\simeq 2.43 \\times 10^{-9}$ \nat $k_\\ast=0.002~{\\rm Mpc}^{-1}$ \\cite{uobs}, \n$V_S$ should satisfy\n\\begin{equation}\n\\frac{V_{S}}{\\varepsilon}=(0.0275 M_{\\rm pl})^4,\n\\label{norm}\n\\end{equation}\nwhere we use $A_s=\\frac{V_S}{24\\pi^2M_{\\rm pl}^4\\varepsilon}$.\nIf the $c_2$ term does not dominate the potential, we can represent \nthe condition from this normalization constraint as\n$c_1\\simeq 9.5 \\times 10^{-8}\\frac{n}{N_\\ast}\n\\left(\\frac{\\sqrt 2M_{\\rm pl}}{\\varphi_\\ast}\\right)^{2n}$\nwhere $\\varphi_\\ast$ is a value of $\\varphi$ at the time when \nthe $e$-foldings is $N_\\ast$. \n  \nOn the other hand, the $e$-foldings $N_\\ast$ expected \nafter the scale $k_\\ast$ exits the horizon is dependent on the reheating \nphenomena and others in such a way as \\cite{slowroll}\n\\begin{equation}\nN_\\ast\\simeq 61.4-\\ln\\frac{k_\\ast}{a_0H_0}\n-\\ln\\frac{10^{16}~{\\rm GeV}}{V_{k_\\ast}^{1\/4}}\n+\\ln\\frac{V_{k_\\ast}^{1\/4}}{V_{\\rm end}^{1\/4}}-\n\\frac{1}{3}\\ln\\frac{V_{\\rm end}^{1\/4}}{\\rho_{\\rm reh}^{1\/4}}.\n\\end{equation} \nThis suggests that $N_\\ast$ should be considered to have a value \nin the range 50 - 60.  \nTaking these constraints into account, we estimate both $n_s$ and \n$r$ for the case where $N_\\ast$ is in this range.\n\n\\begin{figure}[t]\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\\hline\n$n$ &$c_1$ & $c_2$ & $\\frac{\\Lambda}{M_{\\rm pl}}$ & \n$\\frac{\\varphi_1^\\ast}{\\sqrt 2 M_{\\rm pl}}$ & $H_\\ast$\n& $N_\\ast$ & $n_s$ & $r$ & $n_s^\\prime$ \\\\ \n&   &&& &$(\\times 10^{13}{\\rm GeV})$&&&& \\\\ \\hline \n\\hline\n3 & 1.00 $\\times 10^{-6}$ &1.5 & 0.05& 0.417& 6.528& 60.0&  0.967& 0.070& -0.00047 \\\\\n  & 9.84 $\\times 10^{-7}$ &1.7 & 0.05& 0.411& 5.914& 60.0&  0.964& 0.056& -0.00043 \\\\\n\t& 8.62 $\\times 10^{-7}$ &1.9 & 0.05& 0.406& 5.399& 60.0&  0.959& 0.040& \n-0.00032 \\\\\n\\hline\n2 &1.32 $\\times 10^{-7}$& 1.1& 0.05& 0.394& 7.019& 60.0& 0.973& 0.058& -0.00043   \\\\\n\t&1.76 $\\times 10^{-7}$& 1.1& 0.05& 0.384& 6.725& 50.0& 0.968& 0.072& -0.00061   \\\\\n\t&1.22 $\\times 10^{-7}$& 1.6& 0.05& 0.383& 5.931& 60.0& 0.969& 0.039& -0.00040   \\\\\n\t&1.71 $\\times 10^{-7}$& 1.6& 0.05& 0.374& 5.767& 50.0& 0.964& 0.052& -0.00059   \\\\\n\t&1.03 $\\times 10^{-7}$& 1.9& 0.05& 0.374& 5.318& 60.0& 0.963& 0.026& -0.00035   \\\\\n\\hline\n1  & 1.36 $\\times 10^{-8}$ & 0.5&0.05 & 0.349& 5.079& 50.0& 0.975& 0.041& -0.00046\\\\ \n  &7.45 $\\times 10^{-9}$ & 1.6&0.05 & 0.333& 4.146& 60.0& 0.970& 0.015& -0.00036\\\\ \n  &1.02 $\\times 10^{-9}$ & 1.6&0.05 & 0.326& 4.102& 50.0& 0.966& 0.019& -0.00052\\\\ \n  &6.15 $\\times 10^{-9}$ & 1.8&0.05 & 0.327& 3.976& 60.0& 0.966& 0.011& -0.00035\\\\ \n  &8.77 $\\times 10^{-9}$ & 1.8&0.05 & 0.320& 3.944& 50.0& 0.962&  0.016& -0.00052\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\vspace*{2mm}\n\n{\\footnotesize Table~1\\ Examples of the predicted values for the \nspectral index $n_s$ and the tensor-to-scalar ratio $r$ \nin this scenario with $m=1$.  }\n\\end{figure}\n \nNumerical examples are shown in Table~1 for the cases $n=1,2,3$ with\na fixed $\\Lambda$.\\footnote{ We note that the first term of $V_S$ \nbecomes $c_1M_{\\rm pl}^2S^\\dagger S$ and $c_1(S^\\dagger S)^2$ for $n=1$ and 2, \nrespectively.}\nFor given values of $c_2$, the values of $c_1$ and $\\varphi_\\ast$ are \nfixed so that the normalization condition given in eq.~(\\ref{norm}) \nis satisfied and also $N_\\ast$ takes its value in the imposed range 50 - 60.\nBoth $n_s$ and $r$ are estimated for them.\\footnote{\nIf we apply the value of $A_s$ at \n$k_\\ast=0.05$~Mpc$^{-1}$ \\cite{planck15} \nto the present analysis using the same values of $c_{1,2}$ and \n$\\Lambda$, $\\varphi_\\ast$ and $N_\\ast$ are changed. \nThis effect on $r$ is found to be $r_{0.05}\\simeq 1.07r_{0.002}$ \nfor the fixed values of $c_1$ and $c_2$ which give $n_s \\simeq 0.971$ and\n$r_{0.002} \\simeq 0.1$ at $k_* = 0.002$ Mpc$^{-1}$. }\nIn  Fig.~\\ref{fig:3}, \nwe plot the predicted points in the $(n_s,r)$ plane by red and green \ncircles, which correspond to $N_\\ast=50$ and 60 respectively \nfor every 0.1 of $c_2$ starting from $c_2=0.1$ on the right-hand side \nwhile $\\Lambda$ is fixed as $\\Lambda=0.05M_{\\rm pl}$. \nWe show the boundary values of $c_2$ by the red and black stars, \nfor which either red or green circles are inside of the region of \n$2\\sigma$ CL and $1\\sigma$ CL of the latest Planck TT+lowP+ BKP+lensing+ext \ncombined data for the $n=3$ and $n=1,2$ panels, respectively. \nThey show that the present model with $c_2$ included in this interval \nare favored by the latest Planck data combined with others. \nThe best fit result is obtained for the $n=1$ case.\n\nAs discussed above, the present model shows the similar behavior to\nthe monomial inflation models at least for the spectral index \nand the tensor-to-scalar ratio in the limiting case with the negligible \n$c_2$. However, if $c_2$ is not negligible, this feature could be changed\nand these values largely deviate from the monomial inflation models.\nSince the predicted region in the $(n_s,r)$ plane could be distinctive\nfrom other inflation models, the model might be tested through \nfuture CMB observations. One of  the promising CMB \nobservations would be LiteBIRD which is expected to detect the signal \nof the gravitational wave with $r > 0.01$ at more \nthan $10 \\sigma$ \\cite{litebird}. \nThus the whole of the predicted region could be verified in near future. \n\nRecent CMB results suggest that the running of the spectral \nindex is consistent with zero at $1 \\sigma$ level.\nThus, this can be an another useful test of the model.  \nThe running of the spectral index is known to be expressed \nby using the slow-roll parameters as\n\\begin{equation}\nn_s^\\prime\\equiv \\frac{dn_s}{d\\ln k}\n\\simeq -24\\varepsilon^2+16\\varepsilon\\eta-2\\xi^2,\n\\end{equation}\nwhere $\\xi$ is defined as \n$\\xi^2\\equiv M_{\\rm pl}^4\\frac{V_S^\\prime V_S^{\\prime\\prime\\prime}}{V_S^2}$.\nIn the present model, it is written by using the model parameters as  \n\\begin{eqnarray}\n\\xi^2&=&2m^4\\left(\\frac{\\sqrt 2M_{\\rm pl}}{\\varphi}\\right)^{12}\n\\left(\\frac{\\Lambda}{M_{\\rm pl}}\\right)^8\n\\left[\\frac{n-2c_2(n+m)\\left(\\frac{\\varphi}{\\sqrt 2M_{\\rm pl}}\\right)^{2m}}\n{1-2c_2\\left(\\frac{\\varphi}{\\sqrt 2M_{\\rm pl}}\\right)^{2m}}\\right. \\nonumber \\\\\n&&\\left.\\times \\frac{n(n-3)(2n-3)-2c_2(n+m)(n+m-3)(2n+2m-3)\n\\left(\\frac{\\varphi}{\\sqrt 2M_{\\rm pl}}\\right)^{2m}}\n{1-2c_2\\left(\\frac{\\varphi}{\\sqrt 2M_{\\rm pl}}\\right)^{2m}}\\right].\n\\end{eqnarray}\nIf we use the parameters given in Table~1, the running of the \nspectral index can be estimated in each case by using these formulas.\nThe results are shown in the last column of Table~1.\nAlthough they are consistent with the latest Planck data,\nthey take very small negative values.\nWe might be able to use it for the verification of the model in future.\n\n\\begin{figure}[ht]\n        \\begin{center}\n\t\t\t\t      \\begin{subfigure}[b]{0.33\\textwidth}\n                \\includegraphics[width=\\textwidth]{n3-last.eps}\n                \\caption{}\n                \\label{fig:n1a}\n        \\end{subfigure\n\t\t\t\n        \\begin{subfigure}[b]{0.33\\textwidth}\n                \\includegraphics[width=\\textwidth]{n2-last.eps}\n                \\caption{}\n                \\label{fig:n2}\n        \\end{subfigure\n  ~\n   \n        \\begin{subfigure}[b]{0.33\\textwidth}\n                \\includegraphics[width=\\textwidth]{n1-last.eps}\n                \\caption{}\n                \\label{fig:n1}\n        \\end{subfigure}\n\t\t\t\t\\end{center}\n   ~\n    \n    \n\\caption{~ Predicted regions in the $(n_s, r)$ plane \nare presented in panel (a) for $n=3$, \nin panel (b) for $n=2$, and in panel (c) for $n=1$. \n$\\Lambda$ is fixed as $\\Lambda = 0.05 M_{\\text{pl}}$  \nin all cases. The values of $c_1$ and $\\varphi_\\ast$ are given \nin Table 1 for representative values of $c_2$. \nContours given in the right panel of Fig.~21 in Planck 2015 \nresults.XIII.\\cite{planck15} are used here. Horizontal \nblack lines $r=0.01$ represent a possible limit detected by LiteBIRD \nin near future. }\n\\label{fig:3}\n\\end{figure}\n\n\\section{Relation with particle physics}\nFinally, we discuss the relation of the model with particle \nphysics. Although we cannot clarify the origin of potential (\\ref{model1})\nat the present stage, we expect it might be produced through \nsome non-perturbative effects of Planck scale physics.  \nThe complex scalar $S$ can play an important role in particle physics \nif we embed it in an extended standard model. \nAs such an interesting example, we consider the radiative neutrino \nmass model proposed by Ma \\cite{ma}. \nThis model is given by the following Lagrangian for the neutrino sector: \n\\begin{eqnarray}\n-{\\cal L}&=&\\sum_{\\alpha,k=1}^3\\left(h_{\\alpha k} \\bar N_k\\eta^\\dagger\\ell_\\alpha\n+h_{\\alpha k}^\\ast\\bar\\ell_\\alpha\\eta N_k+\n\\frac{M_k}{2}\\bar N_kN_k^c \n+\\frac{M_k}{2}\\bar N_k^cN_k\\right) \\nonumber \\\\\n&+&m_\\phi^2\\phi^\\dagger\\phi+m_\\eta^2\\eta^\\dagger\\eta+\n\\lambda_1(\\phi^\\dagger\\phi)^2+\\lambda_2(\\eta^\\dagger\\eta)^2\n+\\lambda_3(\\phi^\\dagger\\phi)(\\eta^\\dagger\\eta)  \n+\\lambda_4(\\eta^\\dagger\\phi)(\\phi^\\dagger\\eta) \\nonumber \\\\ \n&+&\\frac{\\lambda_5}{2}\\left[ (\\eta^\\dagger\\phi)^2+\n(\\phi^\\dagger\\eta)^2\\right], \n\\label{ma-model}\n\\end{eqnarray}\nwhere $\\ell_\\alpha$ and $\\phi$ are the doublet leptons and \nthe ordinary doublet Higgs scalar in the standard model. \nTwo types new fields are introduced in this model, that is, \nan inert double scalar $\\eta$ and singlet fermions $N_k$. \nAll their masses are assumed to be of $O(1)$~TeV.\nNew fields $\\eta$ and $N_k$ are assigned odd parity of imposed $Z_2$ \nsymmetry, although all the standard model contents have its even parity.\nSince $\\eta$ is assumed to have no vacuum expectation value, this $Z_2$\nsymmetry is exact and then neutrino masses cannot be generated at tree \nlevel. Neutrinos get masses through a one-loop diagram which has $\\eta$ \nand $N_k$ in the internal lines as shown in the left-hand diagram of Fig.4. \nMoreover, the lightest neutral $Z_2$ odd field is stable\nto be a good dark matter (DM) candidate. Thus, DM is an inevitable ingredient\nfor the neutrino mass generation in this model.  \nThe model has been clarified quantitatively to have interesting features \nthrough a lot of studies \\cite{raddm,radlept}.\n\n\n\nWe can relate the present model to the Ma model by identifying \nthe $Z_2$ symmetry in the present model with that in the Ma model. \nWe assign its odd parity to the complex scalar $S$.    \nIf we take account of these symmetry, new terms which are subdominant\nduring the inflation period are introduced as invariant ones,\n\\begin{eqnarray}\n-{\\cal L}_S&=& \\tilde m_{S}^2S^\\dagger S+\\frac{1}{2} m_{S}^2S^2+\n\\frac{1}{2} m_{S}^2S^{\\dagger 2}+\\kappa_1(S^\\dagger S)^2 \n+\\kappa_2(S^\\dagger S)(\\phi^\\dagger\\phi)\n+ \\kappa_3(S^\\dagger S)(\\eta^\\dagger\\eta)\n\\nonumber \\\\\n&-& \\mu S\\eta^\\dagger\\phi - \\mu S^\\dagger\\phi^\\dagger\\eta.\n\\label{ext-model}\n\\end{eqnarray}\nHere we note that the $\\lambda_5$ term in eq.~(\\ref{ma-model}) \nis also allowed under the imposed symmetry. \nHowever, since its $\\beta$-function is proportional to itself\nif an interaction $\\mu S\\eta^\\dagger\\phi$ in the last line of \neq.~(\\ref{ext-model}) is neglected, \n$\\lambda_5=0$ is stable for radiative corrections.\nOn the other hand, if it is included in the Lagrangian, \nthe $\\lambda_5$ term can be induced through this interaction \nas the effective one at low energy regions after integrating out \nthe heavy $S$ field. \n\nThis can be easily seen through the neutrino mass generation. \nIn the present extended model, the neutrino masses can be generated \nthrough the right-hand diagram of Fig.~4.\nThe neutrino masses obtained through this diagram \ncan be described by the formula \n\\begin{equation}\n({\\cal M}_\\nu)_{\\alpha\\beta}=\\sum_{k=1}^3\\sum_{a=1,2}\n\\frac{h_{\\alpha k}h_{\\beta k}M_k\\mu_a^2\\langle\\phi\\rangle^2}{8\\pi^2}\nI(M_{\\eta}, M_k, m_{\\varphi_a}), \n\\label{nmtr2}\n\\end{equation}\nwhere $M_\\eta^2=m_\\eta^2+(\\lambda_3+\\lambda_4)\\langle\\phi\\rangle^2$ and\n$m_{\\varphi_a}$ represents the mass of the real and imaginary component \nof $S$ which can be expressed as $m_{\\varphi_1}^2=\\tilde m_S^2+m_S^2$ and\n$m_{\\varphi_2}^2=\\tilde m_S^2-m_S^2$. \n$\\mu_a$ stands for $\\mu_1=\\frac{\\mu}{\\sqrt 2}$ and \n$\\mu_2=\\frac{i\\mu}{\\sqrt 2}$, respectively.\nThe function $I(m_a,m_b,m_c)$ is defined as\n\\begin{eqnarray}\nI(m_a,m_b,m_c)&=&\\frac{(m_a^4-m_b^2m_c^2)~\\ln m_a^2}\n{(m_b^2-m_a^2)^2(m_c^2-m_a^2)^2}+\n\\frac{m_b^2~\\ln m_b^2}\n{(m_c^2-m_b^2)(m_a^2-m_b^2)^2} \\nonumber\\\\\n&+&\\frac{m_c^2~\\ln m_c^2}\n{(m_b^2-m_c^2)(m_a^2-m_c^2)^2}-\n\\frac{1}{(m_b^2-m_a^2)(m_c^2-m_a^2)}.\n \\label{mnu2}\n\\end{eqnarray} \nIf $m_{\\varphi_a}^2 \\gg M_k^2, M_\\eta^2$ is satisfied and it corresponds to \nthe present case, this formula is found to be reduced to\n\\begin{equation}\n{\\cal M}^\\nu_{\\alpha\\beta}\\simeq \n\\left(\\sum_{a=1,2}\\frac{\\mu_a^2}{m_{\\varphi_a}^2}\\right)\n\\sum_{k=1}^3\\frac{h_{\\alpha k}h_{\\beta k}\\langle\\phi\\rangle^2}{8\\pi^2}\n\\frac{M_k}{M_\\eta^2-M_k^2}\\left[1 +\\frac{M_k^2}{M_\\eta^2-M_k^2}\n\\ln\\frac{M_k^2}{M_\\eta^2}\\right],  \n\\end{equation}\nwhich is equivalent to the neutrino mass formula obtained through\nthe left-hand diagram of Fig.~4 for the Ma model. This shows that \n$\\lambda_5$ can be identified with $\\sum_a\\frac{\\mu_a^2}{m_{\\varphi_a}^2}$\nas the effective coupling obtained at the low energy regions much \nsmaller than $m_{\\varphi_a}$.  \nThe key coupling for the neutrino mass generation in the Ma model\ncould be closely related to the inflaton interaction term in this \nextension. \n\n\\begin{figure}[t]\n        \\begin{center}\n                \\includegraphics[width=15cm]{diag.eps}\n         \\end{center}\n   \\vspace*{5mm}\n\\caption{~One-loop diagrams contributing to the neutrino mass generation. \nThe left-hand diagram is the one in the Ma model.\nLepton number is violated through the Majorama mass of $N_K$. \nThe right-hand diagram is the one in the present extended model.\n$\\varphi_a$ represents the real and imaginary part of the singlet \nscalar $S$ defined by $S=\\frac{1}{\\sqrt 2}(\\varphi_1+i\\varphi_2)$.\n$\\mu_a$ is a dimensional coupling for $\\varphi_a$ which is expressed \nas $\\mu_1=\\frac{\\mu}{\\sqrt 2}$ and $\\mu_2=\\frac{i\\mu}{\\sqrt 2}$. }\n\\label{fig:4}\n\\end{figure}\n\n\nWe should also note that the interesting feature for DM in the Ma model \nis completely kept in this extended model. \nWe suppose that the $Z_2$ odd lightest field is the neutral real \ncomponent of the inert doublet $\\eta_R$. \nIts stability is guaranteed by the imposed $Z_2$ symmetry.\nSince its relic abundance is determined by the coannihilations among the\ncomponents of $\\eta$ which are controlled by the coupling constants \n$\\lambda_{3,4}$ in eq.~(\\ref{ma-model}), the results obtained \nin \\cite{ham,ks} can be applied to the present model without affecting\nthe analysis in this paper.\nThey shows the required relic abundance $\\Omega h^2=0.12$ could be\neasily realized if either $\\lambda_3$ or $\\lambda_4$ takes \na value of $O(1)$ for the $\\eta_R$ with the mass of $O(1)$~TeV.\nThus, this extended model could give a simple explanation not only for \nthe inflation but also for the neutrino masses and the DM abundance, \nsimultaneously.   \n \n\\section{Summary}\nWe have considered an inflation scenario based on a complex \nsinglet scalar. Special potential of this scalar constrains\nthe inflaton evolution along a spiral-like trajectory in the space \nof two degrees of freedom.\nThis makes the model behave like a single field inflation scenario. \nHowever, since the slop along this constrained direction is flat \nenough, inflaton can travel through trans-Planckian path.\nAs a result, the sufficient $e$-foldings can be realized even \nfor sub-Planckian inflaton values. \nSerious potential problem in the large field inflaton could \nbe solved in this model. \nBoth the spectral index and the tensor-to-scalar ratio \npredicted in this model can be consistent with recently \nup-dated CMB observational results. \nSince these could take values in distinctive regions from other\ninflation scenario, the model might be tested through future CMB \nobservations. \n\nThe inflaton in this model might be embedded into the extended \nstandard model as an important ingredient. \nAs such an example, we have discussed a possibility that the inflaton \nis an indispensable element in the radiative neutrino mass model,\nwhere a certain quartic scalar coupling plays a crucial role in the neutrino \nmass generation. Since the inflaton causes this coupling \nas an effective one at low energy regions, it could have a close \nrelation with particle physics in this extension.\nThe model might have another interesting feature. \nReheating through the inflaton decay might give the origin of \nbaryon number asymmetry through the generation of the lepton number \nasymmetry in a non-thermal way.  \nDetailed study of this subject will be presented in future \npublication \\cite{ks2}.\nIf it could be shown through explicit analysis, the problems in the standard \nmodel might be solved in a compact way in this extended model.\n \n\\section*{Acknowledgement}\nR.~ H.~ S.~ Budhi is supported by the Directorate General of Higher \nEducation (DGHE) of Indonesia (Grant Number 1245\/E4.4\/K\/2012). \nS.~K. is supported by Grant-in-Aid for JSPS fellows (26$\\cdot$5862).\nD.~S. is supported by JSPS Grant-in-Aid for Scientific\nResearch (C) (Grant Number 24540263) and MEXT Grant-in-Aid \nfor Scientific Research on Innovative Areas (Grant Number 26104009).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}