diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzlgvt" "b/data_all_eng_slimpj/shuffled/split2/finalzzlgvt" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzlgvt" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n {As the new technologies in 5G mobile communication systems,small cells and massive multiple-input multiple-output (MIMO)\\cite{Andrews} can greatly improve the spectral efficiency and energy efficiency (EE), which have received much attention in recent years.}\n Although small cells and massive MIMO have been compared in\\cite{Liu1}, they are not necessarily rivals.\n Actually, a two-tier network architecture that incorporates the small cells and massive MIMO can attain the benefits of both technologies\\cite{Hoydis,Li}.\n In the two-tier network architecture, massive MIMO ensures full-area coverage, while small cells mainly serve indoor and outdoor hotspots.\n {The power consumption of the two-tier network was characterized and analyzed in \\cite{Sanguinetti}.\n A spatial interference coordination scheme was proposed to protect small cell users from macro-cell interference in \\cite{Dommel}.\n Three low-complexity strategies for explicit inter-tier interference coordination through spatial blanking were developed in \\cite{Adhikary}.}\n\n Although the two-tier network may provide capacity enhancements, it poses many challenges to the power control of the two-tier network.\n Firstly, due to the large number of small cells, centralized power control algorithm may be infeasible for the self-organized small cells and low-complexity power control algorithm is needed.\n Secondly, there exists not only fairness problem between the macrocell and the small cells, but also fairness problem among the small cells in the two-tier network.\n Game theory is suitable to address the problem of power control in self-organizing small cells since it allows the players to learn from the environment and take individual decisions for attaining the equilibrium with minimum information exchange.\n\n Non-cooperative game theory (NGT) was applied to describe the system model and implement the distributed resource allocation in \\cite{Eun,Liang,Hanjun1}.\n A distributed power control scheme for closed access femtocell networks was proposed in \\cite{Eun} to minimize interference to macro users.\n A distributed coverage optimization algorithm using game theory was proposed in \\cite{Liang} for the self optimization network architecture of small cell cluster.\n The joint uplink subchannel and power allocation problem in cognitive small cells using cooperative Nash bargaining game theory was investigated in \\cite{Hanjun1}.\n\n {\n However, methods to guarantee the effectiveness of Nash equilibrium influencing the fairness among the users still need to be investigated.\n Thus, the power control algorithm to improve the fairness among the users is also needed in the two-tier networks.}\n In \\cite{Hanjun2}, the authors proposed a resource allocation scheme to maximize the total capacity and improve the fairness.\n However, the power control algorithms in \\cite{Eun,Liang,Hanjun1,Hanjun2} were proposed for small cells underlaying traditional macro cellular networks.\n {There is little work about power control on the two-tier networks where a macrocell tier with a massive MIMO base station is overlaid with a small cell tier.}\n\n {Motivated by the aforementioned results, we propose a distributed energy efficient power control algorithm for the uplink two-tier networks with small cells and massive MIMO.\n The distributed power control algorithm is implemented by decoupling the EE optimization problem into two steps for the multi-user and multi-cell scenario.\n In the first step, we propose to assign the users into different groups.\n In the second step, multiple power control games based on evolutionary game theory (EGT) are formulated for each group.\n Then, each user in the two-tier networks can take individual decisions to optimize its own energy efficiency.\n Thus, the computational complexity can be greatly reduced in comparison with the brute force approach and the fairness among the users can be improved significantly.\n}\n\n The rest of this paper is organized as follows.\n In Section II, the system model of the two-tier { networks with} small cells and massive MIMO is presented.\n In Section III, the distributed energy efficient power control algorithm is proposed. Simulation results are shown in Section IV, and final conclusions are drawn in Section V.\n\n\\section{System Model}\n{In this paper, the two-tier networks, where a macrocell tier with a massive MIMO base station is overlaid with a tier of small cells, are considered.\nWe investigate the uplink transmission based on the orthogonal frequency division multiple access (OFDMA) operation.\nThe two-tier networks }are illustrated in Fig. \\ref{smallcell_massiveMIMO}, which includes a macro base station (MBS) with \\({N_\\text{MBS}}\\) antennas and $K$ SBSs with \\({N_\\text{SBS}}\\) antennas.\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{{twotier}}\n\\captionstyle{mystyle3}\n\\caption{System model of the two-tier network.}\n\\label{smallcell_massiveMIMO}\n\\end{figure}\n{We assume} that \\({N_\\text{MBS}} \\gg {N_\\text{SBS}}\\).\nThe small cells share the same set of orthogonal subcarriers with the macrocell and the number of the subcarriers is assumed to be $N$.\n{We suppose }that the number of users served by the MBS is \\({N_0}\\), and the number of users served by the $k$-th SBS is \\({N_k} (k = 1,2,...,K)\\).\nEach user selects a transmit power level from a finite set of values denoted by \\(\\mathcal{L}=\\{ 1,2,...,L\\} \\).\n\nThe impulse responses of all the channels are assumed to be flat fading.\nThe received signal at the MBS can be expressed as:\n\\begin{equation}\\begin{split}\\label{Macro SIN}\n{\\boldsymbol{y}_0}{\\bf{ = }}{{\\boldsymbol{G}}_{00}}{{\\boldsymbol{p}}_0}^{1\/2}{{\\boldsymbol{x}}_0} + \\sum\\limits_{k = 1}^K {{{\\boldsymbol{G}}_{k0}}{{\\boldsymbol{p}}_k}^{1\/2}{{\\boldsymbol{x}}_k}} { +\\boldsymbol{ n}}\n\\end{split}\\end{equation}\nwhere\n${{\\boldsymbol{G}}_{00}} = {{\\boldsymbol{H}}_{00}}{{\\boldsymbol{D}}_{00}}^{1\/2} \\in {\\mathcal{C}^{{N_\\text{MBS}} \\times {N_0}}}$ is the channel coefficient matrix between the MBS and its users,\n${{\\boldsymbol{G}}_{k0}} = {{\\boldsymbol{H}}_{k0}}{{\\boldsymbol{D}}_{k0}}^{1\/2} \\in {\\mathcal{C}^{{N_\\text{MBS}} \\times {N_k}}}$ is the channel coefficient matrix between the MBS and the users served by $k$-th SBS,\n${{\\boldsymbol{H}}_{00}} $ and ${{\\boldsymbol{H}}_{k0}} \\in {\\mathcal{C}^{{N_\\text{MBS}} \\times {N_0}}}$ are the realizations of fast fading channels and their components are always assumed to be i.i.d. Rayleigh flat-fading random variables $\\mathcal{N}(0,1)$,\n ${{\\boldsymbol{D}}_{00}}$ and ${{\\boldsymbol{D}}_{k0}} \\in {\\mathcal{C}^{{N_0} \\times {N_0}}}$ are the diagonal matrices whose elements represent large scale fading factors between the MBS and the users,\n${{\\boldsymbol{p}}_0}$ and ${\\boldsymbol{x}}_0^{}$ are the transmit power and the transmit data symbol of the users served by the MBS, respectively,\n ${{\\boldsymbol{p}}_k}$ and ${\\boldsymbol{x}}_k^{}$ are the transmit power and the transmit data symbol of the users served by the $k$-th SBS, respectively,\n ${\\boldsymbol{n}}$ is the additive white Gaussian noise.\n\nThe received signal at the $k$-th SBS can be expressed as:\n\\begin{multline}\\label{Smallcell SIN}\n{{\\boldsymbol{y}}_k}{\\boldsymbol{ = }} \\\\\n{{\\boldsymbol{G}}_{kk}}{{\\boldsymbol{p}}_k}^{1\/2}{{\\boldsymbol{x}}_k}{ +\\boldsymbol{ G}}_{0k}^{}{{\\boldsymbol{p}}_0}^{1\/2}{{\\boldsymbol{x}}_0}{ +\\boldsymbol{ }}{\\sum\\limits_{l = 1, l \\ne k}^{K}} {{{\\boldsymbol{G}}_{lk}}{{\\boldsymbol{p}}_l}^{1\/2}{{\\boldsymbol{x}}_l}} { +\\boldsymbol{ n}}\n\\end{multline}\nwhere ${{\\boldsymbol{G}}_{kk}} = {{\\boldsymbol{H}}_{kk}}{{\\boldsymbol{D}}_{kk}}^{1\/2} \\in {\\mathcal{C}^{{N_\\text{SBS}} \\times {N_k}}}$ is the channel coefficient matrix between the $k$-th SBS and its users,\n ${{\\boldsymbol{G}}_{0k}} = {{\\boldsymbol{H}}_{0k}}{{\\boldsymbol{D}}_{0k}}^{1\/2} \\in {\\mathcal{C}^{{N_\\text{SBS}} \\times {N_0}}}$ is the channel coefficient matrices between the $k$-th SBS and the users served by the MBS,\n ${{\\boldsymbol{G}}_{lk}} = {{\\boldsymbol{H}}_{lk}}{{\\boldsymbol{D}}_{lk}}^{1\/2} \\in {\\mathcal{C}^{{N_\\text{SBS}} \\times {N_l}}}$ is the channel coefficient matrix between the $k$-th SBS and the users served by the $l$-th SBS,\n ${{\\boldsymbol{H}}_{kk}} $, ${{\\boldsymbol{H}}_{0k}}$ and ${{\\boldsymbol{H}}_{lk}}\\in {\\mathcal{C}^{{N_\\text{SBS}} \\times {N_k}}}$ are the realizations of fast fading channels and their components are always assumed to be i.i.d. Rayleigh flat-fading random variables $\\mathcal{N}(0,1)$,\n ${{\\boldsymbol{D}}_{kk}}$, ${{\\boldsymbol{D}}_{0k}}$ and ${{\\boldsymbol{D}}_{lk}} \\in {\\mathcal{C}^{{N_k}\\times {N_k}}}$ are the diagonal matrix whose elements represent large scale fading factors between the $k$-th SBS and the users,\n ${{\\boldsymbol{p}}_l}$ and ${\\boldsymbol{x}}_l^{}$ are the transmit power and the transmit data symbol of the users served by the $l$-th SBS, respectively,\nThe diagonal components of ${{\\boldsymbol{D}}_{00}}$, ${{\\boldsymbol{D}}_{l0}}$, ${{\\boldsymbol{D}}_{kk}}$, ${{\\boldsymbol{D}}_{0k}}$ and ${{\\boldsymbol{D}}_{lk}}$ have the form of \\({\\beta } = \\varphi \\varsigma \/d^\\alpha \\), where \\(\\varphi \\) is a constant related to carrier frequency and antenna gain, \\({d}\\) is the distance between the BS and the corresponding user, \\(\\alpha \\) is the path loss exponent, \\(\\varsigma \\) represents the shadow fading with logarithmic normal distribution \\(10{\\log _{10}}\\varsigma \\sim \\mathcal{CN}(0,{\\sigma ^2})\\).\n\n {We assume} that the maximum-ratio combining (MRC) received matrices are adopted at the MBS and SBSs.\n Then, they can be written as:\n\\begin{equation}\\begin{split}\\label{MRC MBS}\n{\\boldsymbol{A}}_{kk}^{} = \\frac{{{\\boldsymbol{G}}_{kk}^{}}}{{\\left\\| {{\\boldsymbol{G}}_{kk}^{}} \\right\\|}},k=0,1,...,K\n\\end{split}\\end{equation}\nwhere $\\left\\| \\cdot \\right\\|$ represents the $L2$-norm.\n The received signal at the MBS or SBSs after the MRC received matrices can be written as:\n\\begin{equation}\\begin{split}\\label{MRC SBS}\n{{\\boldsymbol{z}}_k}={\\boldsymbol{A}}_{kk}^{T}{{\\boldsymbol{y}}_k},k=0,1,...,K\n\\end{split}\\end{equation}\n\nCorrespondingly, the received signal to interference and noise ratio (SINR) of the user served by the MBS on the $i$-th subcarrier can be written as:\n\\begin{equation}\\begin{split}\\label{Macro SINR}\n\\gamma_{0i}^{} = \\frac{{p_{0i}^{}{{\\left| {{\\boldsymbol{a}}_{00i}^H{\\boldsymbol{g}}_{00i}^{}} \\right|}^2}}}{{\\sum\\limits_{l = 1}^K {p_{li}^{}{{\\left| {{\\boldsymbol{a}}_{00i}^H{\\boldsymbol{g}}_{l0i}^{}} \\right|}^2} + {{\\left\\| {{\\boldsymbol{a}}_{00i}^{}} \\right\\|}^2}{N_0}} }}\n\\end{split}\\end{equation}\nand the received SINR of the user served by the $k$-th SBS on the $i$-th subcarrier can be written as:\n\\begin{multline}\\label{Smallcell SINR}\n\\gamma_{ki}^{} = \\frac{{p_{ki}^{}{{\\left| {{\\boldsymbol{a}}_{kki}^H{\\boldsymbol{g}}_{kki}^{}} \\right|}^2}}}{{p_{0i}^{}{\\left| {{\\boldsymbol{a}}_{kki}^H{\\boldsymbol{g}}_{0ki}^{}} \\right|}^2}+{\\underset{l \\ne k}{\\sum\\limits_{l = 1}^{K}} {p_{li}^{}{{\\left| {{\\boldsymbol{a}}_{kki}^H{\\bf{g}}_{lki}^{}} \\right|}^2} + {{\\left\\| {{\\boldsymbol{a}}_{kki}^{}} \\right\\|}^2}{N_0}} }},\n\\\\\nk=1,2,...,K\n\\end{multline}\nwhere $p_{0i}^{}$ and $p_{ki}^{}$ are the $i$-th diagonal elements of ${{\\boldsymbol{p}}_0}$ and ${{\\boldsymbol{p}}_k}$,\n${\\boldsymbol{a}}_{00i}^{}$ and ${\\boldsymbol{a}}_{kki}^{}$ are the $i$-th column vectors of ${\\boldsymbol{A}}_{00}^{}$ and ${\\boldsymbol{A}}_{kk}^{}$,\n${\\boldsymbol{g}}_{00i}^{}$, ${\\boldsymbol{g}}_{l0i}^{}$, ${\\boldsymbol{g}}_{kki}^{}$, ${\\boldsymbol{g}}_{0ki}^{}$ and ${\\boldsymbol{g}}_{lki}^{}$ are the $i$-th column vectors of ${\\boldsymbol{G}}_{00}^{}$, ${\\boldsymbol{G}}_{l0}^{}$, ${\\boldsymbol{G}}_{kk}^{}$, ${\\boldsymbol{G}}_{0k}^{}$ and ${\\boldsymbol{G}}_{lk}^{}$, respectively.\n\nThe data transmit rate of the user on the $i$-th subcarrier served by the MBS or the $k$-th SBS can be expressed as\n\\begin{equation}\\begin{split}\\label{Smallcell R}\nr_{ki}^{} = {\\log _2}(1 + \\gamma_{ki}^{}), k = 0,1,...,K\n\\end{split}\\end{equation}\n\nThe power consumption model of the user on the $i$-th subcarrier served by the MBS or the $k$-th SBS can be expressed as\n\\begin{equation}\\begin{split}\\label{Smallcell Psum}\nP_{\\text{sum},ki}^{} = p_{ki}^{} + p_{c,ki}^{}, k=0,1,...,K\n\\end{split}\\end{equation}\nwhere $p_{c,ki}$ is the static circuit power at user equipment side.\n$p_{ki}^{}$ is set to be zero if there is no user on the $i$-th subcarrier served by the MBS or the $k$-th SBS.\n\n\n\\newtheorem{lemma}{Lemma}\n\\newtheorem{theorem}{Theorem}\nWith the rapid deployment of wireless communication technology, energy consumption has increased dramatically, which makes it imperative to achieve higher EE.\nWe define the EE of the user on the $i$-th subcarrier served by the MBS or the $k$-th SBS as follows,\n\\begin{equation}\\begin{split}\\label{System EE}\n{\\rm{EE}}_k^{i} ={\\frac{{r_{ki}}}{P_{\\text{sum},ki}^{}} },k=0,1,...,K\n\\end{split}\\end{equation}\nThe EE of the two-tier network based on \\cite{Jiang} can be defined as\n\\begin{equation}\\begin{split}\\label{Overal EE}\n{\\rm{EE}}{\\bf{ = }} \\sum\\limits_{k = 0}^K \\sum\\limits_{i = 1}^N{\\rm{EE}}_k^{i}\n\\end{split}\\end{equation}\nThis definition is based on the sum EE of all the users in the two-tier network rather than the ratio of the sum network throughput to the sum network power consumption.\nThis is because neither powers of different users nor their EE in the two-tier network can be shared.\n Therefore, we focus on the EE of the users in uplink two-tier networks and the overall EE is optimized on condition that the EE of each user is optimal.\n\n\n\n\n\\section{The distributed power control algorithm}\n{The SBSs can be randomly deployed either by the operators or by the users in the two-tier networks with small cells and massive MIMO.\nIn addition, the computational complexity of the EE optimization increases with the number of the subcarriers and the number of cells for the multi-user and multi-cell scenario.\nThus, the centralized power control algorithm is not appropriate for the self-organized small cells, and distributed power control algorithm is needed in the two-tier networks.\nTherefore, we propose a distributed energy efficient power control algorithm in this section to decrease the computational complexity greatly.\n}\n\n\\subsection{ Formulation of the Distributed Energy Efficient Power Control Algorithm}\n{The distributed power control algorithm is implemented by decoupling the EE optimization into two steps.\nIn the first step, we propose to decompose the process of the energy efficient power control. In the second step, multiple power control games based on EGT are formulated.\n}\n\\subsubsection{ Decomposing of the Energy Efficient Power Control}\nA brute force approach to solve the optimization problem related to (\\ref{Overal EE}) has an exponential complexity with respect to the\nnumber of subcarriers and the number of cells in the two-tier\nnetwork in the two-tier network.\n{\nTherefore, we propose to decompose the process of the energy efficient power control by dividing the users into groups.\nThe users on the same subcarrier are assigned to the same group.\nThus, the users can be divided into $N$ groups $\\{g_1,g_2,...,{g_N}\\}$, and the users on the $i$-th subcarrier are in the group $g_i\\in \\{g_1,g_2,...,{g_N}\\}$.\nThen, the original optimization problem can be decomposed into $N$ independent optimization subproblems, and each group can optimize its own EE, respectively.\nThe EE of the group $g_i$ can be expressed as\n\\begin{equation}\\begin{split}\\label{Group EE}\n{\\rm{EE}}_i{\\bf{ = }} \\sum\\limits_{k = 0}^K {\\rm{EE}}_k^{i}\n\\end{split}\\end{equation}\n}\n\n\n\\subsubsection{ Formulation of the Energy Efficient Power Control Games based on EGT}\n{\nOn the one hand, the direct method to solve the optimization problem related to (\\ref{Group EE}) has an exponential complexity with respect to the number of cells.\nTo further reduce the complexity, we propose a distributed power control scheme from a game theory perspective.\nThus, each user can optimize its own EE as defined in (\\ref{System EE}), respectively.\n}\n\n\nOn the other hand, the fairness problem among the cells in the two-tier networks is concerned with the widespread deployment of the SBSs.\nThe fairness problem exists not only between the macrocell and the small cells, but also among the small cells.\nFurthermore, the performance of some users may be very poor by using the traditional game theory, which exacerbates the fairness problem.\nTherefore, we propose to formulate the power control game by using EGT \\cite{Niy}.\nIn the EGT-based power control games, each player selects a strategy which gives a higher payoff than the average payoff.\nThus, the proposed distributed power control algorithm can improve the fairness among the users in the two-tier networks.\n\n\nWe formulate $N$ power control games $\\{G_1,G_2,...,{G_N}\\}$ for the groups $\\{g_1,g_2,...,{g_N}\\}$ based on EGT.\nThe users on the same subcarrier are assigned to the same power control game.\nFor each game ${G_i} \\in\\{G_1,G_2,...,{G_N}\\}$, the EGT-based power control game includes four main factors as follows.\n\n\nPlayers: For the EGT-based power control game ${G_i}$, its players are all the users on the $i$-th subcarrier. And the number of the players in the game ${G_i}$ is $K+1$.\n\nActions: We define the action set for each player as\n${A_i} = \\{ {a_{i,1}},{a_{i,2}},...,{a_{i,L}}\\}$ which includes all the possible transmit power strategies for each player.\nEach player selects a suitable transmit power strategy from the strategy set ${A}$.\n\nPopulation: In the context of the EGT-based power control game, the set of players also constitutes the population.\nWe define $k_{i,a}$ as the number of users selecting strategy $a \\in {A_i}$ in the game ${G_i}$.\nThen, the proportion of the population in the game ${G_i}$ choosing action $a$ is given by\n\\begin{equation}\\begin{split}\\label{SINR}\n{x_{i,a}} = \\frac{{{k_{i,a}}}}{K+1}\n\\end{split}\\end{equation}\nWe can see that $\\sum_{a \\in {A}}x_{i,a}=1$.\n\n\n{Payoff function: The payoff of each player is determined by its EE.\nLet $\\mathcal{K}_{i,a}$ denote the set of the users in the game ${G_i}$ selecting the strategy $a$, and\n$\\pi_{i,a}$ the payoff function of each player.\nThen, $\\pi_{i,a}$ can be expressed as\n\\begin{equation}\\begin{split}\\label{payoff}\n{ \\pi_{i,a}} = \\frac{\\sum\\nolimits_{k\\in\\mathcal{K}_{i,a}}{\\rm{EE}}_k^{i}}{(K+1){x_{i,a}}}\n\\end{split}\\end{equation}\n}\n\n\\subsubsection{ The Distributed Power Control Algorithm}\nThe proposed distributed power control algorithm is presented in Algorithm 1.\n{The distributed power control algorithm is implemented in two steps.\nIn the first step, we decompose the process of the energy efficient power control by dividing the users into $N$ groups.\nThe users on the same subcarrier are in the same group.\nIn the second step, $N$ power control games are formulated for each group based on EGT.}\nFor each power control game ${G_i} \\in\\{G_1,G_2,...,{G_N}\\}$, all players initially play a randomly selected strategy.\nThen, the players compare their payoffs with the average payoff of the population and select a strategy which would give a higher payoff than the average one.\nIf there is no transmit power strategy making its payoff higher than the average one, the user's strategy remains unchanged.\n{\nThen, the rest users continue to select the proper strategies in accordance with the above method.\nIf the strategies of all the users remain unchanged, the power control process will be terminated and the proposed algorithm will be convergent.\nA sub-optimal solution to the maximization of EE can be attained by the proposed algorithm, while the fairness among the users can be greatly improved.\nIn addition, the proposed algorithm has a linear complexity with respect to the number of subcarriers and the number of cells in comparison with the brute force approach which has an exponential complexity.\n}\n\n\\begin{algorithm}[!t]\n\\doublespacing\n\\caption{The Distributed Power Control Algorithm}\n{\\renewcommand\\baselinestretch{1}\\selectfont\n\\begin{itemize}\n\\item [1)] All the users in the two-tier network randomly choose an initialized transmit power strategy.\n\\item [2)] Assign the users to $N$ groups , and the users on the same subcarrier are in the same group.\n\\item [3)] Formulate $N$ power control games $\\{G_1,G_2,...,{G_N}\\}$ for each group.\n\\item [4)] For each game $G_i$, \\textbf{loop}\n\\begin{itemize}\n \\item [5)] The user in the game $G_i$ computes the payoff ${\\pi_{ki}}$ and sends it to the centralized controller.\n\n \\item [6)] The centralized controller computes the average payoff\n \\[\\bar \\pi_i = \\sum\\limits_{k = 0}^K {{\\pi _{ki}}}\/({K+1})\\]\n and broadcasts it back to the users.\n\n \\item [7)] \\textbf{if} ${\\pi_{ki}}\\leq\\bar \\pi_i$, \\textbf{then}\n \\begin{itemize}\n \\item [8)] \\textbf{if} the selected transmit power strategy is not the last one, \\textbf{then}\n \\begin{itemize}\n \\item [9)]the user selects another transmit power strategy.\n \\end{itemize}\n \\item [10] \\textbf{else}\n \\begin{itemize}\n \\item [11)]the transmit power strategy remains unchanged.\n \\end{itemize}\n \\item [12)] \\textbf{end if}\n \\end{itemize}\n\n \\item [13)] \\textbf{end if}\n\\end{itemize}\n\\item [14)] \\textbf{end loop} for the users in the game $G_i$.\n\\end{itemize}\n\\par}\n\\end{algorithm}\n\n\n\n\\subsection{ Replicator Dynamics and Evolutionary Equilibrium}\nIn the considered EGT-based power control game, the strategy adaptation process and population state evolution of the players can be modeled by using a set of ordinary differential equations called replicator dynamics. For each games ${G_i} \\in\\{G_1,G_2,...,{G_N}\\}$, the replicator dynamics can be defined as follows:\n\\begin{equation}\\begin{split}\\label{replicator dynamics}\n{\\dot x_{i,a}} = {x_{i,a}}({\\pi _{i,a}} - {\\bar \\pi_{i}})\n\\end{split}\\end{equation}\nwhere ${\\pi _{i,a}}$ is the payoff of the individuals in the game ${G_i}$ choosing strategy $a$ and the replicator dynamics satisfy the condition of $\\sum_{a}\\dot x_{i,a}=0$,\n{ and ${\\bar \\pi_i}$ is the average payoff of the entire population, which can be expressed as}\n\\begin{equation}\\begin{split}\\label{entier average payoff}\n\\bar \\pi_i = \\sum\\limits_{a \\in A} {{\\pi _{i,a}}{x_{i,a}}}\n\\end{split}\\end{equation}\n For the EGT-based power control game ${G_i}$, the average payoff is given by\n\\begin{equation}\\begin{split}\\label{game average payoff}\n\\bar \\pi_i = \\sum\\limits_{k = 0}^K {{\\pi _{i,k}}}\/({K+1})\n\\end{split}\\end{equation}\nThe replicator dynamics are important for the evolutionary game since they can provide information about the population.\nIt is also useful to investigate the speed of convergence of strategy adaptation to attain the solution to the game.\nBased on this replicator dynamics of the players in the two-tier network, the number of players choosing strategy $a$ increases if their payoff is above the average payoff.\n\nWe consider the evolutionary equilibrium as the solution to the EGT-based power control games, which is defined as the fixed points of the replicator dynamics.\nWe can obtain the equilibrium of the EGT-based power control games through population evolution.\nThe speed of strategy adaptation of each power control game is zero (i.e., $\\dot x_{i,a}=0$) at the evolutionary equilibrium, where no player deviates to gain a higher payoff.\n{\nTo evaluate the stability of the evolutionary equilibrium,\nthe eigenvalues of the Jacobian matrix corresponding to the\nreplicator dynamics need to be evaluated. The evolutionary\nequilibrium in the proposed power control algorithm is stable\nif all the eigenvalues have a negative real part \\cite{Kuz}.\n}\n\n\\subsection{ Complexity Analysis}\nIn this section, we compare the complexity of the proposed distributed power control algorithm with the centralized power control algorithm.\nIt can be seen that there are $L^{(K+1)N}$ strategies to solve the centralized optimization problem with respect to (\\ref{Overal EE}).\nWe can describe the computational complexity as $\\mathcal{O}(L^{(K+1)N})$.\n{We can optimize the EE of the users on the same subcarrier individually after assigning the users on the same resource to the same group.\n}\nThe computational complexity is $\\mathcal{O}(L^{(K+1)})$ to solve the optimization problem with respect to (\\ref{Group EE}).\nThen, the computational complexity can be reduced to $\\mathcal{O}(NL^{(K+1)})$ to solve the corresponding optimization problem.\n{In order to further decrease the computational complexity, we propose to formulate multiple power control games for each group.\n}\nThus, each user can optimize its own EE, respectively.\nFor each game ${G_i} \\in\\{G_1,G_2,...,{G_N}\\}$, its players are all the users on the $i$-th subcarrier.\nThe action set for each player in the game ${G_i}$ can be described as\n${A_i} = \\{ {a_{i,1}},{a_{i,2}},...,{a_{i,L}}\\}$ which includes all the possible transmit power strategies for each user.\nThen, the computational complexity of the game $G_i$ is $\\mathcal{O}((K+1)L)$. Correspondingly, the computational complexity of the distributed power control algorithm is $\\mathcal{O}((K+1)LN)$.\nIt can be seen that the proposed algorithm has a\nlinear complexity with respect to the number of subcarriers and\nthe number of cells in comparison with the brute force approach\nwhich has an exponential complexity.\nNote here that lower complexity will contribute further to the reduction of energy consumption in baseband processing.\n\\section{Simulation Results}\nIn this section, the performance of the proposed distributed power control algorithm is evaluated via simulations.\nIn the simulations, we consider a two-tier networks including an MBS with 128 antennas and several SBSs with 4 antennas each.\nAssume that the number of the users in each cell is the same for the sake of simplicity.\nLet $N_u$ denote the number of the users in each cell.\nThe simulation parameters are shown in Table I.\n\n\n\\begin{table}[!t]\n\\centering\n\\caption{Simulation parameters}\n\\begin{tabular}{|c|c|}\n\\hline\nParameter & Value\\\\\n\\hline\nThe radius of the macrocell & 1000m\\\\\n\\hline\nThe radius of the small cells & 100m\\\\\n\\hline\nPath-loss exponent $\\alpha$ & 3.8\\\\\n\\hline\nNoise spectral density & -194dBm\/Hz\\\\\n\\hline\nAntennas number of the MBS, \\({N_\\text{MBS}}\\) & 128\\\\\n\\hline\nAntennas number of the SBS, \\({N_\\text{SBS}}\\) & 4\\\\\n\\hline\nVariance of log-normal shadow fading $\\sigma ^2$& 10dB\\\\\n\\hline\nFactor $\\varphi$& 1\\\\\n\\hline\nConstant power per user \\(p_{c,ki}^{}\\) &0.01W\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nIn Fig. \\ref{fig4}, we show the evolution of the proposed distributed algorithm with \\(K=2\\) and different $N_u$.\nThe EE in the figure is the average payoff.\nIt can be observed that the proposed distributed algorithm converges within several iterations for all the considered \\(N_u\\).\nIt can also be observed from the figure that the EE increases with $N_u$ by using the proposed EGT-based algorithm.\nThe reason is that different users in the same cell do not crosstalk each other and the users can be allocated the power that are more suitable for them.\nThen, multiuser diversity can be exploited.\n\n\nIn Fig. \\ref{fig2}, we show the performance comparison between the proposed distributed algorithm and the NGT based algorithm in \\cite{Liang}.\nFig. \\ref{1111} depicts the EE versus the number of iterations with $N_u=6$ by using the proposed distributed algorithm.\nFig. \\ref{2222} shows the EE versus the iterations with $N_u=6$ by using the NGT-based algorithm.\nIt can be observed that the EE of each cell approximately approaches the average EE by using the proposed distributed algorithm, while there are quite great differences among the cells by using the NGT-based algorithm.\nIt can be readily seen that the proposed distributed algorithm can greatly improve the fairness among the cells.\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.44\\textwidth]{{2}}\n\\caption{EE versus the number of iterations with \\(K=2\\).\n\\label{fig4}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\subfigure[The EGT-based algorithm]{\\label{1111}\n\\includegraphics[width=0.2\\textwidth]{{3}}}\n\\subfigure[The NGT-based algorithm]{\\label{2222}\n\\includegraphics[width=0.22\\textwidth]{{4}}}\n\\caption{EE versus the number of iterations with different power control algorithm.}\n\\label{fig2}\n\\end{figure}\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.44\\textwidth]{{5}}\n\\caption{EE versus the noise with \\(K=2\\).\n\\label{fig5}\n\\end{figure}\n\nIn Fig. \\ref{fig5}, we show the EE versus the noise power $N_0$ with $K=2$ for different $N_u$.\nIt is obvious that the EE decreases with $N_0$ and the EE increases with $N_u$ by using the proposed EGT-based algorithm.\n\n\n\\section{Conclusions}\n{In this paper, we have proposed a distributed energy efficient power control algorithm for the multi-user and multi-cell scenario in the uplink two-tier networks with small cells and massive MIMO.\nOn the one hand, the computational complexity has been greatly reduced from an exponential complexity to a linear complexity by using the proposed distributed power control algorithm.\nOn the other hand, the fairness among the cells has been remarkably improved by using EGT.\n}\n\n\n\\section*{Acknowledgments}\nThis work was supported in part by the National Basic Research Program of China (973 Program 2012CB316004),\nthe National 863 Project (2015AA01A709),\nand the Natural Science Foundation of China (61221002).\n\n\n\n\n\\bibliographystyle{IEEEtran}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe Asymptotic Giant Branch (hereafter AGB) phase lies at the end\nof the active life for Low- and Intermediate-Mass Stars (M $<$ 8\nSolar Masses); a detailed description can be found in \\cite[Busso\net al. (1999)]{Busso_etal99} and references herein. Stars in this\nphase lose mass very effectively; stellar winds deeply affect\ntheir evolution, and are fundamental for the C-enrichment of the\nInterstellar Medium. Moreover, AGB winds account for about 70$\\%$\nof all the matter returned after stellar evolution (\\cite[Sedlmayr\n1994]{Sedlmayr94}) and provide the starting conditions for the\nformation of planetary nebulae. Radiation from cool dust particles\nin the infrared (IR) normally dominates the energy distribution of\nAGB stars: this fact is due to their strong stellar winds building\nup huge dusty circumstellar envelopes.\n\nOur quantitative knowledge of crucial chemical and physical\nparameters of AGB sources is unfortunately still poor. Among the\nuncertain issues we emphasize in particular the mass loss rate and\nthe distance, hence also the luminosity. These facts have hampered\nfor decades our capability of satisfactorily describing the\nphysics of these dust-enshrouded variable objects, despite their\nevolution is based on the two best known nucleosynthesis phases,\nnamely H and He burning.\n\n\\section{A Study of Galactic AGBs}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=4in]{guandalini_fig1.eps}\n\\caption{ISO-SWS1 spectra of 2 C-rich evolved AGB stars\n(\\cite[Guandalini et al. 2006]{Guandalini_etal06}).}\n \\label{fig1}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[t]\n\\begin{center}\n{\\includegraphics[width=2.3in,\nheight=2.3in]{guandalini_fig2a.eps}\\includegraphics[width=2.3in,\nheight=2.3in]{guandalini_fig2b.eps}}\n\\caption{Bolometric Corrections for samples of C-rich (left) and\nS-type (right) AGB stars.}\n \\label{fig2}\n\\end{center}\n\\end{figure}\n\nWe are performing an analysis of Galactic AGB stars looking for\ncorrelations between their basic parameters (bolometric\nluminosity, mass loss rate, etc... ) and observable quantities.\nOur main aim is that of establishing observationally-based\ncriteria permitting a more quantitative determination of mass loss\nrates and of luminosities for the various types of AGB stars, thus\nproviding general rules to be adopted in the improvement of\nstellar codes. Extensive samples of well studied Galactic AGB\nstars (M-type, S-type, C-rich) have been collected. They are large\nenough (several hundreds of sources per type) that conclusions on\nthem have a good statistical significance. They also have detailed\nand accurate Spectral Energy Distributions (SEDs) at near- and\nmid-IR wavelengths and, very often, reliable measurements of mass\nloss rates and distances. The first results of this ongoing\nresearch have been already published for carbon-rich and S-type\nstars (\\cite[Guandalini et al. 2006]{Guandalini_etal06},\n\\cite[Busso et al. 2007]{Busso_etal07}, \\cite[Guandalini \\& Busso\n2008]{GuaBus08}).\n\n\\section{The Importance of Infrared Observations}\n\nAn extended wavelength coverage is fundamental in the study of the\nevolutionary phases of AGB stars. In order to show this,\nFig.\\,\\ref{fig1} displays the SEDs of evolved AGBs (usually\nMiras), which emit a large part of their flux at mid-IR\nwavelengths. As a consequence, both near and mid-IR observations\nsources are necessary to obtain good estimates of the apparent\nbolometric magnitudes, either by physically integrating the SEDs,\nor by applying pre-calibrated, reliable bolometric corrections\n(B.C.). Examples of such corrections are presented in\nFig.\\,\\ref{fig2} as a function of infrared colours (see\n\\cite[Guandalini et al. 2006]{Guandalini_etal06}, \\cite[Guandalini\n\\& Busso 2008]{GuaBus08}). Once mid-IR wavelengths and bolometric\ncorrections have been properly included, an example of the\n(absolute) HR diagrams that can be obtained is shown in\nFig.\\,\\ref{fig3}. The straight dashed line illustrates how, at\nleast for Mira variables, a rather well defined relation emerges\nbetween the absolute bolometric magnitude and a near-to-mid\ninfrared color (this last also directly linked to the extension\nand temperature of the circumstellar envelope).\n\n\\section{AGB Stars and Magellanic Clouds}\n\n\\begin{figure}[t]\n\\begin{center}\n \\includegraphics[width=3.7in, angle=-90]{guandalini_fig3.eps}\n \\caption{HR diagram of a sample of S-type stars (\\cite[Guandalini\n\\& Busso 2008]{GuaBus08}).}\n \\label{fig3}\n\\end{center}\n\\end{figure}\n\nThe study of the AGB phase through the analysis of Galactic\nsources presents many problems. As an example we can remember the\nlarge difficulties in estimating their distances. AGB sources from\nthe Magellanic Clouds will therefore be fundamental in our\nunderstanding of the AGB evolution, because for them the problem\nof obtaining reliable distances is avoided by the good knowledge\nof the distance modulus. The above fact allows a good estimate for\nthe Luminosity in each photometric band, so that our bolometric\ncorrections would provide the bolometric magnitudes directly.\nMoreover, AGB sources of the two Magellanic Clouds have different\nvalues of the metallicity, both with respect to each other and\nwith respect to the Galaxy. This will allow us to study global\nproperties (Luminosity, mass loss rate, ratio of the number of C\nstars to M giants) also as a function of the chemical composition.\nOur aim is therefore to extend the analysis, which has been almost\ncompleted for Galactic AGB stars, to the Magellanic Clouds and to\nclose-by Dwarf Spheroidal Galaxies. For doing this, important\ntools will be offered by the exploitation of Antarctic Infrared\nAstronomy, as offered by the Italo- Spanish robotic telescope\nIRAIT (\\cite[Tosti et al. 2006]{Tosti_etal06}).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nSpeech synthesis consists of creating an artificial voice through algorithms. This technology can be used in a wide range of cases like voice assistants, audio-books, tutorials, translation, making disabled people's lives easier, etc. At the beginning, speech synthesis consisted of simply recording someone's voice, speaking thousands of words and stitching them together on-demand later on. Although this method behaves and it has been extensively used, the results are not optimal.\n\nOne of the problems that arises when dealing with speech synthesis is that the results tend to sound ``robotic''. The generated voices are flat and do not contain emotional content. Fortunately, speech synthesis is a prominent area of research and the rise of deep learning has made possible huge progress. However, the current state of the art methods are not faultless either. We can differentiate two main drawbacks, the computational needs \\cite{tacotron2} and the control over the generated voice. \n\nCurrent state of the art end-to-end methods for speech synthesis use a two-stage approach where the first model takes raw text as input and generates a spectrogram and the second one (so-called vocoder) transforms the spectrogram into an audio waveform. In this paper we focus only on the text-to-spectrogram area of speech synthesis. More specifically we will create a text-to-Mel spectrogram model with an easy prosody control system using deep learning. To do so, we study several available datasets and we investigate how the use of labelled data can improve the results. We modify a state of the art model to accept labels (emotions) and it is also combined with adversarial training to improve the quality of the model. To that end, in this paper we explore the use of Generative Adversarial Networks (GANs) \\cite{gan} to add emotions as input of text-to-speech (TTS) models. \n\nHowever, defining what emotions are is a difficult task since it is fluid in meaning \\cite{describing-emotional-states}. The Oxford dictionary defines it as ``a strong feeling such as love or anger, or strong feelings in general'', a very general description. This also excludes key areas that are important when emotions are related to speech like arousal or attitude. Theoretically, it makes sense to say that full-blown emotions cannot be found on speech since pure emotions make people speechless or incoherent \\cite{describing-emotional-states} \\cite{science-of-emotion}. There is much controversy when trying to define what emotions actually are \\cite{emotion-regulation}. Furthermore, applying the same stimulus to different individuals is not guaranteed to evoke the same response \\cite{emotion-elicitation}. As a compromised, in this work we implemented a model which considers five different emotions (namely, anger, disgust, fear, happiness, sadness and neutral) with different intensity levels. \n\nIn essence, the work presented in this paper includes three core novelties. First of all, as above-mentioned, a new model combining a text-to-Mel spectrogram model with an adversarial training approach is proposed. Secondly, this model is implemented taking the emotional content into consideration. To that end, the architecture incorporates emotion-related cues as input. Finally, with the aim of enhancing the training convergence of our architecture, a novel training strategy, based on guided attention loss, is introduced.\n\nThe remainder of this paper is structured as follows. Firstly, we explore the related work in Section \\ref{sec:related-work}. Section \\ref{sec:methodology} explains the methodology and datasets used in this work. Then, in Section \\ref{sec:experiments} the experiment conducted are described and their results are summarized in Section \\ref{sec:results}. Finally, we include the conclusions in Section \\ref{sec:conclusions}.\n\n\\section{Related work}\n\\label{sec:related-work}\nBefore the deep learning revolution we are living right now, TTS systems consisted of very complicated language models that relied on huge datasets and human knowledge of the language. These models could be divided into two sections, a linguistic processor and a waveform generator. The linguistic processor takes raw text as input and generates features like phonetic strings, syllabification, and prosody. This part is a mix of human-created resources (text normalization, pronunciation dictionaries) and computer-based ones (to deal with the pronunciation of words not found in the dictionaries). The waveform generator receives the information extracted by the linguistic processor and generates the end raw audio waveform \\cite{tts-decade-progress}.\n\nIn that regard, WaveNet \\cite{wavenet} was a breakthrough on audio synthesis thanks to its deep learning approach. It is able to generate speech from linguistic features and even model music. The main drawback of this model is that it is very slow. Nonetheless, it sparked the deep learning revolution in the vocoder's field. Parallel WaveNet \\cite{oord2017parallelwavenet} improved WaveNet by adding parallel feed-forward processing and using probability density distillation. The idea of probability density distillation is similar to GANs. In this case, instead of a discriminator, there is a pre-trained WaveNet model acting as a teacher and a Parallel WaveNet acting as the student. The latter is trained to replicate the output of the teacher. This translates into a faster model without losing significant inference quality. In a similar vein, Clarinet \\cite{ping2018clarinet}, a fully convolutional network using deep voice 3 as the encoder-decoder structure, becomes the first text-to-wave model. WaveGLOW \\cite{prenger2018waveglow} is a flow-based model that combines the techniques from GLOW \\cite{kingma2018glow} and WaveNet. It is as fast as Clarinet but with a simpler architecture, that makes it easier to train and to replicate the results. Parallel WaveGAN \\cite{yamamoto2019parallelwavegan} uses Generative Adversarial Networks (GANs) to train the model. Compared to Clarinet, Parallel WaveGAN training and inference are 4.82 and 1.96 times faster respectively. Lastly, MelGAN \\cite{kumar2019melgan}, a fully convolutional model that generalizes to unseen speakers, runs more than twice faster than real-time on CPU, and is 10 times faster than WaveGLOW. It does not use any noise as input to the generator and uses weight normalization. It also includes three different discriminators at different scales.\n\nDeep learning also arrived to the text-to-spectrogram domain to improve quality and training speed. There are two main models families, based on Deep Voice \\cite{deepvoice} and based on Tacotron \\cite{tacotron}. The first version of Deep Voice follows similar architecture to traditional models but replaces every part with neural networks. It consists of five blocks: grapheme-to-phoneme model, segmentation model, phoneme duration model, fundamental frequency model, and audio synthesis model. Deep Voice 2 \\cite{deepvoice2} introduced trainable speaker embeddings to the model and proves that a single neural TTS system can learn hundreds of unique voices from less than half an hour of data per speaker. Even if Deep Voice 3 \\cite{deepvoice3} keeps the name, it is a completely new architecture, a sequence-to-sequence (seq2seq) fully convolutional model with an attention mechanism. \n\nFor its part, Tacotron is a seq2seq model with an attention mechanism since its first version. It transforms raw text into a linear-scale spectrogram and applies the Griffin-Lim reconstruction to output audio waveforms. Tacotron 2 \\cite{tacotron2} improves the previous version using WaveNet as vocoder instead of Griffin-Lim, Mean Square Error (MSE) for output loss instead of log-likelihood and adding a ``stop token'' layer to predict when the output sequence is completed. It also predicts Mel spectrograms (explained in Section \\ref{sec:methodology}) instead of linear-scale spectrograms. Having both systems separated (seq2seq + WaveNet) allows for independent training, as long as both are trained in Mel-spectrograms sharing the same characteristics.\n\nVery soon, researchers wanted to have more control over the synthesized voice (prosody control \\cite{prosody-effect}). This attempt started by adding control to select different speaker's voices, age, and gender \\cite{huang2021}. To that end, ``style tokens'' were introduced in Tacotron. They are latent variables that capture prosodic information not present in the text, while no data annotation was required, neither global nor local. This style tokens can be obtained in different manners: with context-based RNN attention \\cite{latent-style-factors}, with a variational autoencoder \\cite{vae} \\cite{leglaive2020}, with the attention mechanism \\cite{attention-all-you-need} (either self-attention \\cite{yang2020} or multi-head attention \\cite{style-tokens}) or with embedding like the ``Capacitron'' \\cite{embed-cap-exp}. Recently, very impressive results have been shown in the use of unlabelled data, such as making a model sing when it has never seen anything similar, like Mellotron \\cite{mellotron} or a prosody control of very high quality like Flowtron \\cite{flowtron}.\n\n\\section{Methodology}\n\\label{sec:methodology}\nGenerative Adversarial Networks (GANs) have shown impressive results in the domain of content generation, while Tacotron 2 has proven to be very effective in creating human-like speech synthesis. Therefore, in this work, we combine Tacotron 2 with GANs to create a new model, GANtron.\nGANtron is composed of a generator, a sequence-to-sequence model with an encoder-decoder structure, and a discriminator. The encoder receives the input text and transforms it into a latent space. The decoder translates the output of the encoder into a mel-spectrogram using the attention mechanism. It also accepts labels as input either in the encoder or in the decoder. The discriminator is trained to differentiate between real spectrograms and those created by the generator.\n\n\\subsection{Mel spectrogram}\nA spectrogram is a representation of the frequencies that are present in a signal and how it varies over time. The problem with spectrograms is that they do not take into consideration human hearing capabilities. On the Hertz scale, two sounds that have the same distance, for example sounds with frequencies 100Hz and 600Hz or 10,000Hz and 10,500Hz, do not sound equally different to humans. This phenomenon is corrected with the Mel spectrogram. To that end, the Mel scale partitions the Hertz scale into bins\nand transforms each bin into a corresponding bin in the Mel scale. Hence, a Mel spectrogram is simply a spectrogram where the Y-axis is the Mel scale instead of the Hertz scale.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{images\/GANtron\/GANtron.png}\n \\caption{GANtron architecture.}\n \\label{fig:GANtron}\n\\end{figure}\n\n\\subsection{Tacotron 2}\nThe paper describing Tacotron 2 divides it into two models, a modification of Tacotron and a modification of WaveNet \\cite{tacotron2}. Tacotron is a seq2seq feature prediction network where the input is text and the prediction is a Mel-spectrogram. Wavenet translates the spectogram to waveforms. WaveNet is out of the scope of this work, and therefore, when we mention Tacotron 2 further in this paper, we only refer to the modification of Tacotron (the seq2seq model).\n\nThe encoder takes the input text as a list of characters and learns a character embedding. The embedding is sent to a stack of convolutional layers that model short-term context. Finally, it uses a bidirectional Long-Short Term Memory (LSTM) to generate the hidden encoded feature representations. Then, the decoder predicts the spectrogram one frame at a time, using the previous time step that is first processed by a Pre-Net. The outcome of the LSTM layers is sent to two different linear projection layers. The first is charged with predicting a ``stop token'' that indicates the end of the generation and the second predicts the Mel-spectrogram. To improve the result, the predicted Mel-spectrogram is processed by five convolutional layers as a residual Post-Net. \n\n\n\\subsection{GANtron} \nIn this work, we propose a novel approach, trying to combine the strenght of two state of the art approaches. In the introduced model, we expand Tacotron 2 to take advantage of its power to extract meaningful features of seq2seq nature. With that in mind, we apply generative adversarial networks, well-known for its ability to generate realistic samples in an unsupervised fashion. Therefore, the initial seq2seq architecture from Tacotron 2 is modified to include a Generator and a Discriminator that compete with each other during the training process (see Figure \\ref{fig:GANtron}). \n\nIt is worth noting that Tacotron 2 was trained for 500 thousand steps and, in our experiments with a Tesla P100-16GB, training for two days achieved around 65 thousand steps. This constraint creates a big computational challenge and the results achieved will probably have a reduced quality when compared to the original Tacotron 2. To mitigate this issue by increasing the speed at which the models' attention mechanism would converge, we propose the integration of a guided attention loss \\cite{guidedattn}. \n\nFinally, the last novelty of the proposed approach is the introduction of emotional content in the text-to-speech architecture. For this purpose, we also investigate the use of datasets containing emotional content. Finally, this information is considered at different levels with the aim of exploring the effect of this emotional content in this framework, either using labels representing the emotions or not (refer to Section \\ref{sec:trained_models} for further information).\n\n\\subsubsection{Generator}\nIn the case of text-to-speech, it is not possible to have a generator that only uses noise as input. In our architecture, the generator is a modification of the seq2seq model of Tacotron 2, adapted to have extra input conditions, called ``style tokens''. They are n-dimensional vectors used in concatenation with the input. When multiple speakers are used, then the speaker ID is the first value of the style token. The other values, depending on the experiment, are either random noise or labels representing emotions, in both cases values between 0 and 1 (see Section \\ref{sec:trained_models}). Depending on which version, the number of parameters differs, ranging \\~28M (only labels) to \\~32M (labels and noise of size 512). GANtron has been designed so that the style tokens can be used either in the encoder or in the decoder. In the former, they are concatenated to the output of the character embedding layer. In the latter, they are concatenated to the output of the encoder.\n\n\\subsubsection{Discriminator}\n\\label{sec:discriminator}\nThe discriminator is a key part of the GAN architecture, it competes with the Generator, trying to differentiate between real and fake Mel-spectrograms. To improve the learning stability and to avoid mode collapse, WGAN \\cite{wasserstein} has been implemented for the loss calculation. Furthermore, we have developed two different discriminators to investigate which one gives better results.\n\nThe first discriminator is composed of 1D convolutional layers. This allows processing the whole spectrogram in one go, no matter its size. To decide if a spectrogram is real or not, we take the average of the output for each frame. It has been designed in such a way that multiple frames can be packed together (windows). This discriminator will be referred to as ``convolutional discriminator''. This model consists of 4 $Conv1d+Dropout(0.5)+Tanh$ blocks, and a fifth one with only a $Conv1d$. The dimension of the first layer is calculated as: $window\\_size*n\\_mels$. In our case it would be $20*80=1600$, then reduced to 1024. The other blocks use 512 nodes and the last one is reduced to 80. The total number of parameters for this architecture is \\~12M.\n\nThe second discriminator is composed of linear dense layers. For this reason, the input must be of fixed size. We process the whole spectrogram with a moving window, allowing overlapping between windows in a random fashion, with a maximum value allowed (notice that this randomness is not present in the first approach). To decide whether a spectrogram is real or not, the same procedure applies, taking the average of every processed window. We refer to this discriminator as ``linear discriminator''. In this case, we have 3 blocks, going the fourth layer from 512 (instead of 80) to 1. The total number of parameters for this architecture is \\~1M.\n\n\\subsubsection{Loss function}\nThe loss used in this work is a combination of several loss functions performed as follows:\n\\begin{equation}\nG_{loss} = Mel_{loss} + Gate_{loss} + Wasserstein_{loss} + Attn_{loss}\n\\end{equation}\nwhere:\n\\begin{equation}\n\\begin{split}\nMel_{loss} = MSE(mel_{(postnet-output)}, mel_{target}) \\\\ + MSE(mel_{output}, mel_{target})\n\\end{split}\n\\end{equation}\n\n\\begin{equation}\nGate_{loss} = BCE(gate_{out}, gate_{target}) \n\\end{equation}\n\nRegarding the guided attention loss ($Attn_{loss}$), we use the one proposed in \\cite{guidedattn}.\n\n\\subsubsection{Generator-discriminator training}\nIn connection with the iterative training of the generator and discriminator, it is noteworthy that finding a point of equilibrium between them is one of the challenges of the GANs training process. One of the critical points is to select a proper training ratio between the two networks. To choose the best discriminator-generator update ratio, a hyper-parameter search was run. The results of our experiments showed that training the generator twice after every discriminator update led the best results in our approach.\n\n\n\\subsection{Datasets}\n\\label{sec:datasets}\nIn this work, for training purposes, two datasets are used, each one with a different purpose. Firstly, a dataset intended to train the speech generation compose of several recorded hours is considered. This dataset, the LJ Speech \\cite{ljspeech17} was also used by Tacotron 2 as it only needs to have audio files and their corresponding text. Secondly, a datasets labelled with emotions apart from the related text is needed. To that end, we considered VESUS \\cite{VESUS}. This dataset has more audio files and more speakers and different utterances than similar options such as CREMA-D \\cite{CREMA-D}, IEMOCAP \\cite{IEMOCAP}, RAVDESS \\cite{RAVDESS} or SAVEE \\cite{SAVEE}. Table \\ref{tab:datasets} includes a summary of the main characteristics of these datasets. Moreover, VESUS' authors did a phonetic comparison of the script and showed that their dataset is well-balanced. It has labels for each file given by annotators. After the normalization we concluded that this is the dataset with more emotional charge and the most balanced and therefore, it was chosen for the emotional factor of our model. \n\nAdditionally, two extra datasets, namely CREMA-D and RAVDESS, are used for evaluation purposes. These dataset are concatenated with VESUS samples in order to investigate whether the generated samples contain emotional content (see Section \\ref{sec:comparison} for further details). \n\n\n\n\n\\begin{table}[]\n\\centering\n\\caption{Publicly available datasets}\n\\label{tab:datasets}\n\\begin{tabular}{l|c|c|c|c|c}\n\\multicolumn{1}{c|}{Dataset} & \\#speakers & \\#unique & \\#files & \\#emotions & Labels? \\\\ \n\\multicolumn{1}{c|}{} & & utterances & & & \\\\\n\\hline\nCREMA-D & 91 & 12 & 7,442 & 6 & Y \\\\\nIEMOCAP & 10 & 8,068 & 10,039 & 9 & Y \\\\\nRAVDESS & 24 & 2 & 1,044 & 8 & N \\\\\nSAVEE & 4 & 87 & 480 & 7 & N \\\\\nVESUS & 10 & 253 & 12,593 & 5 (+1) & Y \n\\end{tabular}\n\\end{table}\n\n\\begin{figure}[h]\n \\begin{subfigure}[b]{0.48\\textwidth}\n \\includegraphics[width=\\textwidth]{images\/guided-attention\/no-guidance-3nvuq5ol-alignment-30000.png}\n \\caption\n }\n \\label{fig:no-warm-up}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.48\\textwidth}\n \\centering\n \\includegraphics[width=\\linewidth ]{images\/guided-attention\/5k-guidance-y1y7qbqp-alignment-10000.png}\n \\caption\n }\n \\label{fig:warm-up-10k}\n \\end{subfigure}\n \\caption{Attention alignment at 30k steps, without guided attention warm-up (a) and at 10k steps, with guided attention warm-up stopped at 5k steps (b).}\n \\label{fig:alignment}\n\\end{figure}\n\n\\subsection{Trained models}\n\\label{sec:trained_models}\n \n\n\nBefore the explanation of the models, some information regarding the style tokens is needed. To evaluate the importance of the input parameters and the training procedure, in this work we defined three alternative style tokens. Firstly, only noise is established as the most basic input to be used. It does not consider any emotional content. Secondly, a contextual style tokens is defined. In this case, as we are using, as labelled data, the VESUS dataset (see Section \\ref{sec:datasets}), the style token is defined as a vector containing six values. Five of these values are the possible emotions considered in the dataset (namely, anger, fear, happiness, sadness and neutral). Additionally, whenever a model uses a dataset (or combination of datasets) that have more than one speaker, we need to include the speaker ID. There are two reasons for the inclusion of this information. Firstly to avoid introducing noise. If the model does not have the information of which speaker the spectrograms corresponds to, it will try to learn a voice that averages all of them. Secondly, to add more prosody control, being able to generate different voices with the same model. This speaker ID then becomes an extra value of the style token, increasing the initial number of tokens (emotions) by one. Finally, a combination of both, noise and contextual labels is considered as style token.\n\n\nBased on the style tokens previously described and the applied training, four configurations were evaluated\n\\begin{itemize}\n \\item \\textbf{Baseline model}: following the classical GAN architecture, we use only noise as style token. It is trained using only the LJ Speech dataset (refer to Section \\ref{sec:datasets} for further details) with two variants, the style token being either as input to the encoder or to the decoder.\n \\item \\textbf{Expressive baseline}: this model is similar to the previous one but in this case, the VESUS dataset is included as part of the training set. However, in this model, the labels are not used but, since VESUS has more emotional content, we hypothesize that it can help the model to better grasp emotions.\n \\item \\textbf{Labelled model}: as in MelGAN, we created a GAN architecture where only the condition, without noise, is utilized as input. In this case, we use LJ Speech along with a labelled dataset (again, VESUS). Since LJ Speech has no labels, it was decided to assign a vector of zeros as contextual information.\n \\item \\textbf{Complete model}: the last variation combines the previous two models, using LJ Speech and VESUS as datasets as well but in this case, considering labels and noise as style token.\n\\end{itemize}\n\n\n\\section{Experiments}\n\\label{sec:experiments}\nIn this section, the experiments conducted to evaluate the four models presented in Section \\ref{sec:trained_models} are described.\n\n\\subsection{Data preparation and visualization information}\nTo train and evaluate the models, we split each dataset randomly into train, evaluation, and test sets, with proportions 85\\%, 5\\% and 10\\% respectively. Additionally, it is worth mentioning that the figures in this paper that correspond to the training phase show the validation losses instead of its training counterparts. This way, we ensure that the model does not suffer from overfitting.\n\n\\subsection{Guided Attention}\nThe most important part of sequence-to-sequence models that use attention is the alignment. The faster the network learns the alignment, the faster the training converges. Therefore, we implemented a guided attention loss where we compare the alignment with a ``straight alignment'' (similar to a multi-variable Gaussian matrix). In our experiments, we identified that it was very effective but if used for too long, it becomes an impediment for the network's training. To tackle this constraint, we introduce the concept of a \\textit{guided attention warm-up}. With this strategy, we guide the attention for only \\textit{X} steps and then we let the network refine it. Experimentally, we found that the best value for \\textit{X} was around 5k. We show a side by side comparison in Figure \\ref{fig:alignment} where the speed improvement is noticeable. For visualization purposes, the Y axis has been transposed in both graphs.\n\n\n\\subsection{Model architecture and hyperparameters selection}\nTo ensure a fair comparison of the models, we conducted experiments to find a common architecture. For this case, we used only the LJ Speech dataset, 88 nodes of input random noise, and the same discriminator with the dimension of each layer equal to 512. We compared two techniques used to help with the training stability, namely WGAN (Wasserstein GAN) and WGAN-GP (WGAN with gradient penalty instead of weight clipping). WGAN proved to be more stable in our experiments. We also used hyperparameters tuning to find the best combination of them.\n\n\\subsection{Evaluation strategy}\nEvaluating text-to-speech models in an objective way is not trivial. Different datasets will have different losses, different models have different ways of calculating the losses, and training the models takes a very long time. That is why most of the published papers are compared with a subjective Mean Opinion Score (MOS). However, in this paper, MOS or a similar (subjective) metric was not utilized since the main objective of the presented model is not to obtain the highest TTS quality but to tune the generated speech with emotions. Moreover, as far as we know, there is no standard objective metric to evaluate this.\nHence, an alternative evaluation schema was applied. This strategy relies in a simple idea: six groups of samples are created, each one with a different style token, and for each group we inference 1,000 files. Then, we train a classifier that tries to separate them into those groups, using only generated files. The more the classifier can differentiate them, the better we will consider the GANtron model. The way each group is created differs depending on which type of style token we are using and the objective. When using only noise, each group is assigned a token where each value is random. When using only labels we have five groups with one strong emotion (which value is establish randomly between 0.5 and 1.0) and the rest of the values in the token set to zero.\nAdditionally, with the aim of simulating a class with no clear emotional content, a sixth group is also created. In this case, each position of the vector (corresponding to possible emotions) is established with random values form 0 to 1. \nRegarding the case of using the combination of noise and labels, two approaches are evaluated. For this evaluation, an splitting of 85-5-10\\% (training-validation-test) was applied. Further details are provided in Section \\ref{exp:complete_models}.\n\n\n\n\\begin{figure*}\n \\centering\n \\begin{subfigure}[b]{0.45\\linewidth}\n \\includegraphics[width=\\linewidth]{images\/evaluate_models\/vanilla_GANTron.png}\n \\caption{Baseline model: Vanilla GANtron.}\n \\label{fig:LJ-noise}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.45\\linewidth}\n \\includegraphics[width=\\linewidth]{images\/evaluate_models\/expressive_baseline.png}\n \\caption{Expressive baseline model (with Linear Discriminator).}\n \\label{fig:LJ-VESUS-noise-lD}\n \\end{subfigure}\n \n \\begin{subfigure}[b]{0.45\\linewidth}\n \\includegraphics[width=\\linewidth]{images\/evaluate_models\/labelled_model.png}\n \\caption{Labelled model.}\n \\label{fig:LJ-VESUS-labels}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.45\\linewidth}\n \\includegraphics[width=\\linewidth]{images\/evaluate_models\/complete_model.png}\n \\caption{Complete model.}\n \\label{fig:LJ-VESUS-noise-labels}\n \\end{subfigure}\n \\caption{Accuracy of the different models' configurations using, as style token and dataset respectively, only noise and LJ Speech dataset (a), only noise and LJ Speech and VESUS datasets (b), only contextual information and LJ Speech and VESUS datasets (c), and noise and contextual information and LJ Speech and VESUS datasets (d).}\n \\label{fig:accuracy-models}\n\\end{figure*}\n\n\n\\subsection{Baseline model, the Vanilla GANtron}\nThis is the most basic version of GANtron, where we only use one dataset to train (LJ Speech) and we add noise as style token. In this model, we study the effect that the noise size (128 or 512) and the placement of the style token (input of the encoder or the decoder) have on the performance of the model with the linear discriminator. \n\n\\subsection{Expressive baseline model}\n\\label{exp:expressive}\nThe most important difference between VESUS and LJ Speech datasets is that the former has only 253 phrases but in 12,593 files. Therefore, the model will be forced to learn that, for the same utterance, there are many different ways of expressing it. The second difference is that the LJ Speech dataset only has one speaker while VESUS has 10. We hypothesize that the combination of this information helps the model grasp better the emotional content of the audios. Since LJ Speech does not have any annotation regarding the emotions, we set a vector of zeros for the relevant labels. Since we want to test how much can the model generalize the emotions, we test using the LJ speaker\n\n\n\n\\subsection{Labelled GANtron model}\n\\label{exp:labels_only}\nAs a step toward promoting the emotional content, we take advantage of the labeled dataset VESUS dataset, utilizing this information as style token. In this experiment we compare the performance of the models using the linear and the convolutional discriminators. In this test, we included six groups evaluated as a classification task. The first five groups have one strong emotion (values between 0.5 and 0.8 in the relevant field) and the rest set to 0, the sixth group has random values for each emotion (intended to simulate a class with no clear emotional content).\n\n\n\\subsection{Complete model}\n\\label{exp:complete_models}\nIn the previous sections we investigate whether GANtron can generate audio files based on style tokens that were either noise or labels. Therefore, as the next step, we decided to combine them as a single style token. In this experiment we examine not only the accuracy but also, investigate which component (noise or labels) is more important in the new style token. For this purpose we conducted an ablation study applying the same model twice, one where each group has a fixed noise (as in Experiment \\ref{exp:expressive}) and the labels of each file are randomized, and another where each group has a fixed label (as in Experiment \\ref{exp:labels_only}) and random noise for each file. The results are also compare with a control groups using only noise.\n\n\n\\subsection{Comparison of distributions}\\label{sec:comparison}\nThe previous experiments are intended to show whether our models can recreate audios with emotional content. However, we also need to ensure that the generated samples keep the emotional content in the original ones. To evaluate this point, a classifier was trained to differentiate among five emotions, namely anger, fear, happiness, sadness and neutral. This classify is firstly trained with only VESUS data (12,593 files) and subsequently with VESUS and 6,000 generated files using GANtron (increasing the size of the dataset by around 50\\%). To classify the audio files, MEL-spectrograms from 80 frames were used. To avoid bias, after every prediction, the starting frame for the 80-frames window is randomized. Additionally, to verify its consistency, we performed the same test using a combination of three datasets, VESUS, CREMA-D and RAVDESS (denoted as 3DS in Figure \\ref{fig:distribution-check}). These two new datasets have the same five emotions as VESUS along with some additional ones that, for consistency reasons, are discarded in this experiment.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{images\/data_augm_vs_no.png}\n \\caption{Models trained with GANtron data augmentation vs original data.}\n \\label{fig:distribution-check}\n\\end{figure}\n\n\n\\section{Results and discussion}\n\\label{sec:results}\nRegarding the Vanilla GANtron model, Figure \\ref{fig:LJ-noise} shows the comparison of the four possible combinations (two noise sizes and two possible placements). It is clear that the effect of the style token gets lost if used as input in the encoder. It is also interesting to note that a higher noise dimensionality achieves better performance when used as input of the decoder. The best model (noise size 512, input in the decoder) achieved only a 52\\% accuracy. Considering that humans were able to recognize the emotions of the VESUS dataset only with 65\\% accuracy and 41\\% on CREMA-D, this performance can be considered satisfactory. Nonetheless, listening to the audios we found that the difference between them was far from the intended outcome. \n\nFigure \\ref{fig:LJ-VESUS-noise-lD} shows how using a more expressive dataset in the Vanilla GANtron increases the accuracy by 10 percentage points. We also present a control group where each file uses different random noise and we still assign them to six groups. We can see how it achieves an accuracy of 17\\%, what is expected as a random classification, showing that the classifier technique is reliable. \n\nBoth linear and convolutional models achieved very high accuracy (96 and 90\\% respectively) on experiment \\ref{exp:labels_only} (see Figure \\ref{fig:LJ-VESUS-labels}). Unfortunately, 18,03\\% of the files created with the convolutional model were erroneous (0\\% on the linear model). Meaning that the attention mechanism failed and the inference stopped only because it reached the maximum number of steps allowed. Therefore the use of the convolutional discriminator was discarded.\n\nFigure \\ref{fig:LJ-VESUS-noise-labels} shows how the model where the labels were forced with random noise achieves very high accuracy. We conclude that the noise in this model is useful to avoid overfitting. Furthermore, it is not affecting the distribution of the classes of the generated samples. \n\nWe proved that the performance is very similar when having augmentation and not (see Figure \\ref{fig:distribution-check}), meaning that the distribution of the GANtron data must be the same as (or very similar to) the original. If the distributions were distanced from each other, the accuracy would drop as it happens when VESUS-based generated samples are used together with CREMA-D and RAVDESS samples (denoted as 3DS). Furthermore, it is important to remember that the speaker in the generated data is different from VESUS and still the performance of the classifier does not drop significantly.\n\nIt is worth stressing that the primary objective of this work is to generate reliable text-to-speech outputs with emotional content. As Audio-Emotion Recognition usually suffers from a lack of data, we considered that a data augmentation schema would be appropriate to evaluate whether the generated samples rely on a distribution close to the original samples. However, we do not intend this model as a domain adaption technique but to help with regularization.\n\n\\section{Conclusion}\n\\label{sec:conclusions}\nIn this paper, we present a novel text-to-spectrogram model pursuing a threefold objective: 1) improve speech synthesis by proposing a new text-to-speech architecture combining the advantages of the Tacotron 2 model and GANs; 2) the incorporation of emotional content into the proposed model to generate more human-like outcomes; and 3) the improvement of the training process with the use of a novel guided attention warm-up that greatly increases the training speed.\nAdditionally, as intermediate step in an effort to find the most appropriate architecture, several experiments have been conducted considering different parameters such as the kind of input used or certain characteristics of it such as the noise size or the placement to input the emotional information. Finally, different experiments were established to assess the validity of the results obtained.\n\nDuring this process, each experiment built on the knowledge gathered on the previous ones to test a new hypothesis. We started with a model that used a non-labelled dataset and noise as style token. Then, a second dataset, this time with more emotional content, was used but the style tokens is kept as noise. However, this version achieved better results based on our metrics, proving that GANtron can generalize the emotions with a speaker even when labels are not provided. Furthermore, we showed that we can use noise in the style tokens together with emotions to avoid overfitting. Finally, a model that generates audio files with enough emotional content that can be used for data augmentation of emotion classifiers is proposed. Additionally, for validation purposes, we assess the results of our experiments in a quantitative manner, proving that the generated samples (Mel spectrograms) lie in the same distribution as the original ones.\n\nSome of the main constraints of this and similar works is the data available. Although plenty of dataset have been made accessible in the last years, the challenges related to the annotation of the emotional content represent a limitation when training the algorithms. When exploring five well-known emotional datasets (VESUS, CREMA-D, IEMOCAP, RAVDESS and SAVEE), we identified that they all lack expressiveness and have a dominance of the neutral state with respect to the other emotions. Furthermore, when humans were asked to annotate these datasets and majority voting was applied, around 40 to 60 \\% (depending on the dataset) of the annotations had a different emotion than the intended by the actors. For these reasons we believe that the creation of a labeled dataset with high quality and higher intensity ratings would improve the results.\n\nIt is worth noting some important future works. Regarding the outcomes of the proposed text-to-speech architecture, even though the quantitative results are promising, the qualitatively assessment of the results would be advisable.\nFurthermore, the research on using adversarial neural networks on text-to-speech synthesis could be expanded. We believe that we have not unlocked the full potential of these networks. A potential modifications could be a change on the loss function used to train the architecture. We would like to explore the possibility of reducing the importance to current loss (a combination of the Wassesstein loss and the loss used by the Tacotron 2 part) using an extra loss. To this effect, the integration of an emotion classifier in the training loop as part of the loss calculation, similar to the work done in other areas with the perceptual loss, especially in the field of computer vision \\cite{yin2021} \\cite{athanasiadis2020audio}, could lead to a potential improvement.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nTheories of gravity, alternative to Einstein's General Relativity\n(GR), have been proposed to cure the problems of the standard\ncosmological model and, above all, because they arise in\nquantizations of gravity. These alternative gravitational\ntheories constitute at least an attempt to formulate a\nsemi-classical scheme in which GR and its most successful\nfeatures can be recovered. One of the most fruitful approaches\nthus far has been that of {\\it Extended Theories of Gravity}\n(ETGs), which have become a paradigm in the study of the\ngravitational interaction. ETGs are based on corrections and\nextensions of Einstein's theory. The paradigm consists,\nessentially, of adding higher order curvature invariants and\/or\nminimally or non-minimally coupled scalar fields to the dynamics;\nthese corrections emerge from the effective action of quantum\ngravity \\cite{odintsov}.\n\nFurther motivation to modify GR arises from the problem of fully\nimplementing Mach's principle in a theory of gravity, which\nleads\none to contemplate a varying gravitational coupling. Mach's\nprinciple states that the local inertial frame is determined by\nthe average motion of distant astronomical objects \\cite{bondi}.\nThis fact would imply that\nthe gravitational coupling here and now is determined by the\ndistant distribution of matter, and it can be scale-dependent\nand related to some scalar field. As a consequence, the concept\nof ``inertia'' and the Equivalence Principle have to be revised.\nBrans-Dicke theory \\cite{BD} constituted the first consistent\nand complete theory alternative to Einstein's GR. Brans-Dicke\ntheory incorporates a variable gravitational coupling strength\nwhose dynamics are governed by a scalar field non-minimally\ncoupled to the geometry, which implements Mach's principle\nin the gravitational theory \\cite{BD, cimento,sciama}.\n\n\nIndependent motivation for extending gravity comes from the fact\nthat every unification scheme of the fundamental interactions,\nsuch as Superstring, Supergravity, or Grand Unified Theories\nexhibit effective actions containing non-minimal couplings to the\ngeometry or higher order terms in the curvature invariants. These\ncontributions are one-loop or higher loop corrections in the\nhigh-curvature regime approaching the full, and still unknown,\nquantum gravity regime \\cite{odintsov}. Specifically, this scheme\nwas adopted in the study of quantum field theory on curved\nspacetime and it was found that interactions between quantum\nscalar fields and background geometry, or gravitational\nself-interactions, yield such corrections to the Einstein-Hilbert\nLagrangian \\cite{birrell}. Moreover, it has been realized that\nthese corrective terms are inescapable in the effective action of\nquantum gravity close to the Planck energy \\cite{vilkovisky}. Of\ncourse, all these approaches do not constitute a full quantum\ngravity theory, but are needed as working schemes toward it.\n\n\nIn summary, higher order terms in the invariants of the\nRiemann tensor, such as\n$R^{2}$, $R^{\\mu\\nu} R_{\\mu\\nu}$,\n$R^{\\mu\\nu\\alpha\\beta}R_{\\mu\\nu\\alpha\\beta}$, $R \\,\\Box R$, or $R\n\\,\\Box^{k}R$, and non-minimal coupling terms between scalar\nfields and geometry such as $\\phi^{2}R$, have to be added to the\neffective gravitational Lagrangian when\nquantum corrections are introduced. These terms occur also in\nthe effective Lagrangian of string or Kaluza-Klein theories\nwhen a mechanism of compactification of extra spatial dimensions\nis used \\cite{veneziano}.\n\n\n\nFrom a conceptual point of view, there is no {\\it a priori}\nreason to restrict the gravitational Lagrangian to a\nlinear function of the Ricci scalar $R$ minimally coupled with\nmatter \\cite{francaviglia}. Furthermore, the idea has been\nproposed that there are no exact laws of physics, in the sense\nthat the effective Lagrangians describing physical interactions\ncould be stochastic functions at the microscopic level. This\nproperty would imply that local gauge invariances and the\nassociated conservation hold only in the low energy limit and\nthe fundamental constants of physics can vary \\cite{ottewill}.\n\nBesides fundamental physics motivations, all these theories\nhave been the subject of enormous attention in cosmology due to\nthe fact that they naturally exhibit an inflationary behaviour\nwhich can overcome the shortcomings of the GR-based\nstandard cosmological model. The cosmological scenarios arising\nfrom ETGs seem realistic and capable of reproducing\nobservations of the the cosmic\nmicrowave background (CMB)\n \\cite{starobinsky,kerner,la}. It has been\nshown that, by means of conformal transformations, the\nhigher order and non-minimally coupled terms\ncan be related to Einstein gravity with one\nor more scalar fields minimally coupled to9 gravity\n\\cite{teyssandier,maeda,wands1,wands,gottloeber}.\n\n\n\nHigher order terms always appear as contributions of even order\nin the field equations. For example, the term $R^{2}$ produces\nfourth order equations \\cite{ruzmaikin}, $R \\ \\Box R$ gives sixth\norder equations \\cite{gottloeber,sixth}, $R \\,\\Box^{2}R$ eighth\norder equations \\cite{eight}, and so on. By means of a conformal\ntransformation, any second order derivative term corresponds to a\nscalar field.\\footnote{The dynamics of these scalar fields are\ngoverned given by a second order Klein-Gordon-like equation.}.\nFourth-order gravity corresponds to Einstein gravity with one\nscalar field, sixth-order gravity to Einstein gravity with two\nscalar fields, {\\em etc.} \\cite{gottloeber,schmidt1} It is also\npossible to show that $f(R)$ gravity is equivalent not only to a\nscalar-tensor theory, but also to GR plus an ideal fluid\n\\cite{cno}. This feature becomes interesting if multiple\ninflationary events are desired, because an early inflationary\nstage could select very large scale structures (observed as\nclusters of galaxies today), while a later inflationary epoch\ncould select smaller scale structures (observed as galaxies\ntoday) \\cite{sixth}, with each inflationary era corresponding to\nthe dynamics of a scalar field. Finally, these extended schemes\ncould naturally solve the graceful exit problem bypassing the\nshortcomings of known inflationary models \\cite{la,aclo}.\n\n\nIn addition to the revision of standard cosmology at early\nepochs with the concept of inflation, a new approach is\nnecessary also at late epochs. ETGs could play a fundamental\nrole also in this context. In fact, the increasing bulk of data\naccumulated in the past few years have nurtured a new\ncosmological model referred to as the\n{\\it Concordance Model}. The Hubble diagram of type Ia Supernovae\n(hereafter SNeIa) measured by both the Supernova Cosmology\nProject \\cite{SCP} and\nthe High-$z$ Team \\cite{HZT} up to redshifts $z \\sim 1$, has been\nthe first piece of evidence that the universe is\ncurrently undergoing a\nphase of accelerated expansion. Balloon-born\nexperiments, such as BOOMERanG \\cite{Boomerang} and MAXIMA\n\\cite{Maxima}, have detected the first and second\npeak in the anisotropy spectrum of the CMB\nradiation indicating that the geometry of the\nuniverse is spatially flat. In conjunction with\nconstraints on the matter density\nparameter $\\Omega_M$ coming from galaxy clusters,\nthese data indicate that the universe is dominated by an\nunclustered fluid with negative pressure, generically dubbed\n{\\it dark energy}, which is able to drive the accelerated\nexpansion. This picture has been further strengthened by the\nrecent precise measurements of the CMB spectrum obtained by the\nWMAP\nexperiment \\cite{WMAP,hinshaw,hinshaw1}, and by the extension of\nthe SNeIa Hubble diagram to redshifts higher than one\n\\cite{Riess04}.\nAn overwhelming flood of papers has appeared following\nthis observational evidence,\npresenting a great variety of models trying\nto explain this phenomenon. The simplest explanation\nis the well known cosmological constant $\\Lambda$\n\\cite{LCDMrev}. Although it is the best fit to most of the\navailable astrophysical data \\cite{WMAP}, the $\\Lambda$CDM model\nfails in explaining why the inferred value of $\\Lambda$ is so tiny\n(120 orders of magnitude smaller) in comparison with\nthe typical\nvacuum energy values predicted by particle physics and why its\nenergy density is comparable to the matter density today (the\n{\\it coincidence problem}).\n\nAs a tentative solution, many authors have replaced the\ncosmological constant with a scalar field rolling down its\npotential and giving rise to the model referred to as {\\it\nquintessence} \\cite{QuintRev,tsu1}. Even when successful in\nfitting\nthe data, the quintessence approach to dark energy is still\nplagued by the coincidence problem since the dark energy and\nmatter densities evolve differently and reach comparable values\nfor a very limited portion of the cosmic evolution coinciding\nat the present era. To be more precise, the quintessence dark\nenergy\nis tracking matter and evolves in the same way for a long time.\nBut then, at late times, somehow it has to change its behavior\nfrom tracking the dark matter to dominating as a\ncosmological constant. This is the coincidence problem of\nquintessence.\n\nMoreover, the origin of this quintessence scalar field is unknown,\nleaving a great uncertainty on the choice of the scalar field\npotential. The subtle and elusive nature of dark energy has led\nmany authors to look for completely different scenarios able to\ngive a quintessential behavior without the need for exotic\ncomponents. To this end, it is worth stressing that the\nacceleration of the universe only calls for a dominant component\nwith negative pressure, but does not tell us anything about the\nnature\nand the number of cosmic fluids filling the universe. This\nconsideration suggests that it could be possible to explain\nthe accelerated expansion by introducing a single cosmic fluid\nwith an equation of state causing it to act like dark matter at\nhigh densities and dark energy at low densities. An attractive\nfeature of these models, usually referred to as {\\it Unified Dark\nEnergy} (UDE) or {\\it Unified Dark Matter} (UDM) models, is that\nsuch an approach naturally solves, al least phenomenologically,\nthe coincidence problem. Interesting examples are the\ngeneralized Chaplygin gas \\cite{Chaplygin}, the tachyon field\n\\cite{tachyon} and the condensate cosmology \\cite{Bassett}. A\ndifferent class of UDE models has been proposed \\cite{Hobbit}\nin which a single fluid is considered: its energy density scales\nwith\nthe redshift in such a way that the radiation-dominated era, the\nmatter era, and the accelerating phase can be naturally\nachieved. These models are very\nversatile since they can be interpreted both in the framework of\nUDE models and as a two-fluid scenario with dark matter and scalar\nfield dark energy. The main advantage of this approach is that a\nsuitable generalized equation of state can be always obtained and\nobservational data can be fitted.\n\nThere is a yet different way to address the problem of the\ncosmic acceleration. As stressed in \\cite{LSS03}, it is possible\nthat the observed acceleration is not the manifestation of another\ningredient of the cosmic pie, but rather the first signal of a\nbreakdown of our understanding of the laws of gravitation in the\ninfrared limit. From this point of view, it is tempting to\nmodify the\nFriedmann equations to see whether it is possible to fit the\nastrophysical data with models comprising only standard\nmatter. Interesting examples of this kind are the Cardassian\nexpansion \\cite{Cardassian} and DGP gravity \\cite{DGP}. In the\nsame framework it is possible to find alternative\nschemes in which a quintessential behavior is obtained by taking\ninto\naccount effective models coming from fundamental physics\nand giving\nrise to generalized or higher order gravity actions\n\\cite{curvature} (see \\cite{odinoj} for a comprehensive review).\nFor instance, a cosmological constant term may be recovered as a\nconsequence of a non-vanishing torsion field, leading to a\nmodel consistent with both the SNeIa Hubble diagram and\nSunyaev-Zel'dovich data of galaxy clusters\n\\cite{torsion}. SNeIa data could also be efficiently fitted\nincluding higher order curvature invariants in the gravitational\nLagrangian \\cite{curvfit,camfr}. These alternative models provide\nnaturally a\ncosmological component with negative pressure whose origin is\nrelated to the cosmic geometry, thus overcoming\nthe\nproblems linked to the physical significance of the scalar field.\n\n\nThe large number of cosmological models which constitute viable\ncandidates to explain the observed accelerated expansion is\nevident from this short overview. On the one hand, this\noverabundance of models signals the fact that only a\nlimited number of cosmological tests are available to\ndiscriminate between competing\ntheories and, on the other hand, it shows that we are facing an\nurgent degeneracy problem. It is useful to remark that both the\nSNeIa Hubble\ndiagram and the angular size-redshift relation of compact\nradio sources \\cite{AngTest} are distance-based probes of\ncosmological models, so systematic errors and biases could be\niterated. From this point of view, it is interesting to search for\ntests based on time-dependent observables. For example, one can\ntake into account the {\\it lookback time} to\ndistant objects since this quantity can discriminate between\ndifferent cosmological models. The lookback time is\nobservationally estimated as the difference between the present\nage of the universe and the age of a given object at redshift\n$z$. Such an estimate is possible if the object is a galaxy\nobserved in more than one photometric band since its color is\ndetermined by its age as a consequence of stellar evolution. It is\nthus possible to get an estimate of the galaxy age by measuring\nits magnitude in different bands and then using stellar\nevolutionary codes to choose the model that best reproduces the\nobserved colors.\n\nComing to the weak-field-limit approximation, which essentially\nmeans considering Solar System scales, ETGs are expected to\nreproduce GR which, in any case, is firmly tested only in this\nlimit \\cite{Will}. This fact is a matter of debate since several\nrelativistic theories do not reproduce exactly the Einsteinian\nresults in the Newtonian approximation but, in some sense,\ngeneralize them. As first noticed by Stelle\n\\cite{stelle}, an $R^2$-theory gives rise to Yukawa-like\ncorrections in the Newtonian potential. This feature could have\ninteresting physical consequences; for example, certain authors\nclaim\nto explain the flat rotation curves of galaxies by using such\nterms \\cite{sanders}. Others \\cite{mannheim} have shown that a\nconformal theory of gravity is nothing but a fourth-order\ntheory containing such terms in the Newtonian limit. Besides,\nan apparent, anomalous, long-range acceleration in the data\nanalysis of the Pioneer 10\/11, Galileo, and\nUlysses spacecrafts could be framed in a general theoretical\nscheme by taking into\naccount corrections to the Newtonian potential \\cite{anderson}.\n\nIn general, any relativistic theory of gravitation yields\ncorrections to the Newtonian and post-Newtonian (PPN) potentials\n({\\em e.g.}, \\cite{schmidt}) which test the theory \\cite{Will}.\nFurthermore, the\nnewborn {\\it gravitational lensing astronomy} \\cite{ehlers} is\ngenerating additional tests of gravity over small, large, and\nvery large scales which soon will provide direct measurements for\nthe variation of the Newtonian coupling \\cite{krauss}, the\npotential\nof galaxies, clusters of galaxies and several other features of\nself-gravitating systems. Such data, very likely, will be\ncapable of confirming or ruling\nout the physical consistency of GR or of any ETG. In summary, the\ngeneral features of ETGs are that the Einstein field equations\nare modified in two ways: $i)$ the geometry can be\nnon-minimally coupled to some scalar field, and\/or $ii)$ higher\nthan second order derivatives of the metric appear. In the\nfirst case we deal with scalar-tensor theories of\ngravity; in the second case we have higher order theories.\nCombinations of non-minimally coupled and higher order\nterms can emerge as contributions to effective Lagrangians;\nthen we have higher order-scalar-tensor theories of\ngravity.\n\nFrom the mathematical point of view, the problem of reducing\ngeneralized theories to an Einstein-like form has been\nextensively discussed. Under suitable regularity\nconditions on the Lagrangian and using a Legendre\ntransformation on the metric, higher order theories take the\nform of GR in which one or more scalar field(s)\nsource of the gravitational field (see, {{\\rm e}.g.},\n\\cite{francaviglia,sokolowski,ordsup,magnano-soko}). On the other\nhand, as discussed above, the mathematical\nequivalence between models with variable gravitational coupling\nand Einstein gravity has been studied using suitable\nconformal transformations \\cite{dicke,nmc}. A debate on the\nphysical meaning of these conformal\ntransformations seems to be ongoing (\\cite{faraoni} and\nreferences therein). Several authors\nclaim a physical difference between Jordan frame\n(higher order theories and\/or variable gravitational couplings)\nsince there is experimental and observational evidence\nsuggesting that the Jordan frame is better suited for\nmatching solutions and data. Others state that the true physical\nframe is the Einstein one according to the energy theorems\n\\cite{magnano-soko}. However, the discussion is open and no\ndefinitive conclusion seems to have been reached. The problem\nbecomes more involved at the semiclassical and quantum level, and\nshould be faced from a more general point of view---the\nPalatini approach to gravity could be useful to this goal.\n\n\nThe\nPalatini approach to gravitational theories was first\nintroduced and analyzed by Einstein himself\n\\cite{palaeinstein}, but was named as a\nconsequence of an historical misunderstanding\n\\cite{buchdahl,frafe}.\n\nThe fundamental idea of the Palatini formalism is to consider the\ntorsion-free connection $\\Gamma^{\\mu}_{\\alpha\\beta}$ entering\nthe definition of the Ricci tensor, to be independent of the\nspacetime metric $g_{\\mu\\nu}$. The Palatini\nformulation of the standard Einstein-Hilbert theory turns out\nto be equivalent\nto the purely metric theory. This property follows from the\nfact that the field equations for the connection\n$\\Gamma^{\\mu}_{\\alpha\\beta}$, considered\nto be independent of the metric, produce the Levi-Civita\nconnection of the metric $g_{\\mu\\nu}$. As a consequence, there is\nno reason to impose the Palatini variational principle instead of the metric\nvariational\nprinciple in\nthe Einstein Hilbert theory. However, the situation\nchanges if we consider the ETGs, which\ndepend on functions of the curvature invariants (such as $f(R)$\ntheories)\nor couple non-minimally to some scalar field. In these\ncases the\nPalatini and the metric variational principles provide different\nfield equations and the theories thus derived differ\n\\cite{magnano-soko,FFV}. The relevance of the Palatini approach\nfor cosmological applications in\nthis framework has been recently demonstrated\n\\cite{curvature,odinoj,palatinifR}.\n\n\nFrom the physical point of view, considering the metric\n$g_{\\mu\\nu}$ and the connection $\\Gamma^{\\mu}_{\\nu\\alpha}$\nas independent fields means to decouple the metric structure of\nspacetime and its geodesic structure (the\nconnection $\\Gamma^{\\mu}_{\\alpha\\beta}$, in general, is not the\nLevi-Civita\nconnection of $g_{\\mu\\nu}$). The\ncausal structure of spacetime is governed by $g_{\\mu\\nu}$ while\nthe spacetime trajectories of particles\nare governed\nby $\\Gamma^{\\mu}_{\\alpha\\beta}$. This decoupling enriches the\ngeometric structure of spacetime and\ngeneralizes the purely metric formalism. This metric-affine\nstructure of spacetime is naturally translated, by means of the\nPalatini field equations, into a bi-metric structure of\nspacetime. Besides the physical metric $g_{\\mu\\nu}$, another\nmetric $\\tilde{g}_{\\mu\\nu}$ appears. This new metric is\nrelated, in the case of\n$f(R)$ gravity, to the connection. The\nconnection $\\Gamma^{\\mu}_{\\alpha\\beta}$ turns out to be the\nLevi-Civita connection of $ \\tilde{g}_{\\mu\\nu}$ and provides\nthe geodesic structure of spacetime.\n\nFor non-minimally coupled interactions in\nthe gravitational Lagrangian in scalar-tensor theories, the new\nmetric $ \\tilde{g}_{\\mu\\nu}$ is related to the non-minimal\ncoupling; $\\tilde{g}_{\\mu\\nu} $ can be related to a\ndifferent geometric and physical aspect of the gravitational\ntheory. Thanks to the Palatini formalism, the non-minimal\ncoupling and the scalar field, entering\nthe evolution of the gravitational fields, are separated from the\nmetric structure of spacetime. The situation mixes when we\nconsider the case of higher order-scalar-tensor theories. Due to\nthese features, the Palatini approach could contribute to\nclarify the physical meaning of conformal transformations\n\\cite{ACCF}.\n\n\\vspace{3.mm}\n\nIn this review paper, without claiming for completeness, we want\nto give a survey on the formal and phenomenological aspects of\nETGs in metric and Palatini approaches, considering the\ncosmological and astrophysical applications of some ETG models.\nThe layout is the following. The field equations for generic ETGs\nare derived in Sec.\\ref{due}. Specifically, we discuss metric,\nPalatini and metric-affine approaches. In Sec. \\ref{tre} the e\nquivalence of metric and Palatini $f(R)$ gravities with\nBrans-Dicke theories are discussed. In Sec. \\ref{quattro} we\nintroduce theoretical and experimental viability of\n$f(R)$-gravity. Briefly, we discuss on the correct cosmological\ndynamics and on the instabilities for a particular case of $f(R)$.\nAfter we discuss the precence of ghost fields and the wealk field\nlimit for metric approach. Finally we consider the growth of\ncosmological perturbations and the Chauchy problem. Cosmological\napplications are considered in Sec.\\ref{cinque}-\\ref{sei}. We show\nthat dark energy and the dark matter can be addressed as\n\"curvature effects\", if ETGs (in particular $f(R)$ theories) are\nconsidered. We work out some cosmological models comparing the\nsolutions with data coming from\n observational surveys. As further result in Sec. \\ref{sette}. , we show that also the\nstochastic cosmological background of gravitational waves can be\n\"tuned\" by ETGs. This fact could open new perspective also in the\nproblems of detection of gravitational waves which should be\ninvestigated not only in the standard GR-framework. Discussion and conclusions\nare drawn in Sec.\\ref{otto}.\n\n\\section{The three versions of $f(R)$ gravity}\n\\label{due}\n\nIn this survey we focus on $f(R)$ gravity (see\n\\cite{review} for a more comprehensive discussion and a list of\nreferences, and \\cite{otherreviews} for short introductions to\nthe\nsubject). In these theories\nthe Einstein-Hilbert action\\footnote{Here $R$ is the Ricci\ncurvature\nof the metric tensor $g_{\\mu\\nu} $, which has\ndeterminant $g$, $G$ is Newton's\nconstant, and $\\kappa \\equiv 8\\pi G$. We mostly follow the\nnotations of Ref.~\\cite{Wald}.}\nequation}{{\\beta}{equation}\nS_{EH}=\\frac{1}{2\\kappa}\\int d^4x \\, \\sqrt{-g} \\,\nR+S^{(m)} equation}{\\end{equation}{equation}\nis modified to\n\\begin{equation} \\label{actionmetric}\nS=\\frac{1}{2\\kappa} \\int d^4x \\, \\sqrt{-g} \\, f(R)+S^{(m)} \\;,\n\\end{equation}\nwhere $f(R)$ is a non-linear function of its argument\n and $S^{(m)}$ is the matter part of the action.\n Actually, there are two variational\nprinciples that one can apply to the Einstein-Hilbert action\nin\norder to derive Einstein's equations: the standard metric\nvariation and a less standard variation dubbed Palatini\nvariation. In the latter the metric and the connection are\nassumed to be independent variables and one varies the action\nwith respect to both of them, under the important\nassumption that the matter action does not depend on the\nconnection. The choice of the variational principle is usually\nreferred to as a formalism, so one can use the terms metric\n(or second order) formalism and Palatini (or first order)\nformalism. However, even though both\nvariational principles lead to the same field equation for an\naction whose Lagrangian is linear in $R$, this is no longer\ntrue for a more general action. Therefore, it is intuitive that\nthere will be two version of $f(R)$-gravity, according to which\nvariational principle or formalism is used.\nIndeed this is the case: $f(R)$-gravity in the metric formalism\nis called {\\em metric $f(R)$-gravity} and $f(R)$-gravity in the\nPalatini formalism is called {\\em Palatini $f(R)$-gravity}.\n\nFinally, there is actually even a third version of $f(R)$-gravity:\n {\\em metric-affine $f(R)$-gravity}. This comes about if one\nuses the Palatini variation but abandons the assumption that\nthe matter action is independent of the connection. Clearly,\nmetric affine $f(R)$-gravity is the most general of these\ntheories and reduces to metric or Palatini $f(R)$-gravity if\nfurther assumptions are made. In this section we will present\nthe actions and field equations of all three versions of $f(R)$-\ngravity and point out their difference. We will also clarify\nthe physical meaning behind the assumptions that discriminate\nthem.\n\nThen brefly has we show above three versions of $f(R)$-gravity have been studied:\\\\\nequation}{{\\beta}{itemize}\n\\item metric (or second order) formalism;\\\\\n\\item Palatini (or first order) formalism;\nequation}{\\end{equation}{itemize}\n and\n equation}{{\\beta}{itemize}\n\\item metric-affine gravity.\\\\\nequation}{\\end{equation}{itemize}\n\nThese families of theories are discussed in the following.\n\n\n\\subsection{Metric $f(R)$ gravity}\n\n\nIn the metric formalism the action is\n\\begin{equation} \\label{metricaction2}\nS_{metric}=\\frac{1}{2\\kappa}\\int d^4 x \\, \\sqrt{-g} \\,\nf(R)+S^{(m)} ,\n\\end{equation}\nand its variation with respect to $g^{\\mu\\nu}$ yields, after some\nmanipulations and modulo surface terms,\nthe field equation\n\\begin{equation} \\label{metricfieldeqs}\nf'(R)R_{\\mu\\nu}-\\frac{f(R)}{2} \\,\ng_{\\mu\\nu}=\\nabla_\\mu\\nabla_\\nu f'(R) -g_{\\mu\\nu} \\Box\nf'(R) +\\kappa\\, T_{\\mu\\nu} \\;,\n\\end{equation}\nwith a prime denoting differentiation with respect to $R$, $\\nabla_\\mu$ is the covariant derivative associated with the\n Levi-Civita connection of the metric, and $\\Box\\equiv\n \\nabla^\\mu\\nabla_\\mu$. Fourth\norder derivatives of the metric appear in the first two terms\non the right hand side, justifying the alternative name ``fourth\norder gravity'' used for this class of theories.\n\n\n\nBy taking the trace of eq.~(\\ref{metricfieldeqs}) one obtains\n\\begin{equation} \\label{tracemetric}\n3\\Box f'(R)+Rf'(R)-2f(R)=\\kappa \\, T \\;,\n\\end{equation}\nwhere $T\\equiv {T^{\\alpha}}_{\\alpha}$ is the trace of the\nenergy-momentum tensor of matter. This second order\ndifferential equation for $f'(R)$ is qualitatively different from\nthe trace of the Einstein equation $R=-\\kappa \\, T$ which,\ninstead, constitutes an algebraic relation between $T$ and\nthe Ricci scalar, displaying the fact that $f'(R)$ is a\ndynamical (scalar) degree of freedom of the theory.\nThis is already an indication that the field equations of $f(R)$\ntheories will admit a larger variety of solutions than\nEinstein's theory. As an\nexample, we mention here that the Jebsen-Birkhoff's theorem,\nstating that the\nSchwarzschild solution is the unique spherically symmetric vacuum\nsolution, no longer holds in metric $f(R)$ gravity.\nWithout going into details, let us stress that $T=0$ no longer\nimplies that $R=0$, or is even constant.\nEq.~(\\ref{tracemetric}) will turn out to be very useful in\nstudying various aspects of $f(R)$ gravity, notably its\nstability and weak-field limit. For the moment, let us use it\nto make some remarks about maximally symmetric solutions. Recall\nthat maximally\nsymmetric solutions lead to a constant Ricci scalar. For $R={\\rm\nconstant}$ and $T_{\\mu\\nu}=0$, eq.~(\\ref{tracemetric}) reduces to\n \\begin{equation}\n\\label{metftr}\nf'(R)R-2f(R)=0,\n\\end{equation}\n which, for a given $f$, is an algebraic equation in $R$. If $R=0$ is\na root of this equation and one takes this root, then eq.~(\\ref{metricfieldeqs})\nreduces to $R_{\\mu\\nu}=0$ and the maximally symmetric solution is\nMinkowski spacetime. On the other hand, if the root of\neq.~(\\ref{metftr}) is $R=C$, where $C$ is a constant, then\neq.~(\\ref{metricfieldeqs}) reduces to $ R_{\\mu\\nu}= g_{\\mu\\nu} C\/4 $ and\nthe\nmaximally symmetric solution is de Sitter or anti-de Sitter\nspace\ndepending\non the sign of $C$, just as in GR with a cosmological\nconstant.\nAnother issue that should be stressed is that of energy\nconservation. In metric $f(R)$ gravity the matter is minimally\ncoupled to the metric. One can, therefore, use the usual arguments based on the invariance of the action under diffeomorphisms of the spacetime manifold\n[coordinate transformations $x^{\\mu}\\rightarrow\nx'^{\\mu}=x^{\\mu}+\n\\xi^{\\mu} $ followed by a pullback,\nwith the field $\\xi^{\\mu} $ vanishing on the boundary of the\nspacetime region considered, leave the physics unchanged, see\n\\cite{Wald} to show that $T_{\\mu\\nu}$ is\ndivergence-free. The\nsame can be done at the level of the field equations: a ``brute\nforce'' calculation reveals that the left hand side of\n eq.~(\\ref{metricfieldeqs}) is divergence-free (generalized Bianchi\nidentity) implying that $\\nabla_\\mu T^{\\mu\\nu}=0$.\n\n\n\n\nThe field equation~(\\ref{metricfieldeqs}) can be rewritten\nas form of Einstein\nequations with an effective stress-energy tensor to the right hand side. Specifically, as\n\n\\begin{equation}\nG_{\\mu\\nu}=\\kappa \\left( T_{\\mu\\nu}+T_{\\mu\\nu}^{(eff)} \\right)\n\\end{equation}\nwhere\n\\begin{equation}\\label{effectivetensor}\nT_{\\mu\\nu}^{(eff)}=\\frac{1}{\\kappa} \\left[ \\frac{ f(R)-Rf'(R)}{2}\\,\ng_{\\mu\\nu}+\\nabla_\\mu \\nabla_\\nu f'(R)-g_{\\mu\\nu} \\Box f'(R) \\right]\n\\end{equation}\nis an effective energy-momentum tensor constructed with geometric\nterms. Since $T_{\\mu\\nu}^{(eff)}$ is only a formal\nenergy-momentum tensor, it is not expected to satisfy any\nof the energy\nconditions deemed reasonable for physical matter, in particular\nthe effective energy density cannot be\nexpected to be positive-definite. An effective\ngravitational coupling $G_{eff}\\equiv G\/f'(R)$ can be defined in\na way analogous to scalar-tensor gravity. It is apparent that\n$f'(R)$ must be positive for the graviton to carry\npositive kinetic energy.\n\n\n\nMotivated by the recent cosmological observations, we adopt the\nspatially flat Friedmann-Lemaitre-Robertson-Walker (FLRW)\nmetric to describe the universe,\n\\begin{equation}\nds^2=-dt^2 +a^2(t) \\left( dx^2+dy^2+dz^2 \\right) \\;,\n\\end{equation}\nwhere $a$ is the scale factor.\nThen, the field equations of metric $f(R)$ cosmology become\nequation}{{\\beta}{eqnarray}\n& & H^2=\\frac{\\kappa}{3f'(R)} \\left[\n\\rho^{(m)}+\\frac{Rf'(R)-f(R)}{2}-3H \\dot{R} f''(R) \\right] \\;,\\\\\n&&\\nonumber \\\\\n&& 2\\dot{H}+3H^2= -\\frac{\\kappa}{f'(R)} \\left[\nP^{(m)}+ f'''(R) \\left( \\dot{R}\\right)^2 +2H\\dot{R}\nf''(R)+\\ddot{R}f''(R) \\right. \\nonumber \\\\\n&& \\nonumber \\\\\n&& \\left. +\\frac{ f(R)-Rf'(R) }{2} \\right] \\;,\nequation}{\\end{equation}{eqnarray}\nwhere $H \\equiv \\dot{a}\/a$ is the Hubble\nparameter and an overdot denotes differentiation with respect to\nthe comoving\ntime $t$. The corresponding phase space is a 2-dimensional curved\nmanifold embedded in a 3-dimensional space and with a rather\ncomplicated structure \\cite{deSouzaFaraoni}.\n\n\\subsection{Palatini $f(R)$ gravity}\nIn the Palatini version of $f(R)$ gravity, both the metric\n$g_{\\mu\\nu}$ and the connection\n$\\Gamma^\\mu_{\\nu\\gamma}$ are regarded as independent variables.\nIn other words, the connection is not the metric connection of\n$g_{\\mu\\nu}$. While in GR the metric and Palatini variations\nproduce the same field equations ({\\em i.e.}, the Einstein\nequations), for non-linear Lagrangians one obtains\ntwo different sets of field equations.\\footnote{By imposing\nthat the metric and Palatini variations generate the\nsame field equations, Lovelock gravity is\nselected \\cite{Exirifard}. GR is a special case of Lovelock\ntheory.}\n\n\nPalatini $f(R)$ gravity was proposed as an alternative to dark\nenergy, on the same footing as metric $f(R)$ models. The original\nmodel advanced for this purpose was based on the specific\nform $f(R)=R-\\mu^4\/R$ \\cite{palatinifR}.\n\n\nThe Palatini action is \\begin{equation}\\label{actionPalatini}\nS_{Palatini}=\\frac{1}{2\\kappa}\\int d^4 x \\, \\sqrt{-g} \\, f(\n\\tilde{R}) +S^{(m)}\\left[ g_{\\mu\\nu}, \\psi^{(m)} \\right] \\;, \\end{equation}\nwhere a distinction needs to be made between two different Ricci\ntensors contained in the theory. $R_{\\mu\\nu}$ is constructed from\nthe metric connection of the (unique) physical metric\n$g_{\\mu\\nu}$, while $\\tilde{R}_{\\mu\\nu}$ is the Ricci tensor of\nthe non-metric connection $\\Gamma^\\mu_{\\nu\\gamma}$ and defines\nthe scalar $\\tilde{R}\\equiv g^{\\mu\\nu}\\tilde{R}_{\\mu\\nu}$. The\nmatter part of the action does not depend explicitly from the\nconnection $\\Gamma^{\\mu}_{\\alpha\\beta}$, but only from the metric\nand the matter fields, which we collectively label as\n$\\psi^{(m)}$.\n\n\n\nBy varying the Palatini action~(\\ref{actionPalatini}) one\nobtains the field equation\n\\begin{equation} \\label{Palatinifieldeq1}\nf'(\\tilde{R}) \\tilde{R}_{\\mu\\nu}-\\frac{ f(\\tilde{R})}{2} \\,\ng_{\\mu\\nu}=\\kappa \\,\nT_{\\mu\\nu} \\;,\n\\end{equation}\nin which no second covariant derivative of $f'$ appears, in\ncontrast with eq.~(\\ref{metricfieldeqs}). An\nindependent variation with respect to the connection yields\n\\begin{equation} \\label{Palatinifieldeq2}\n\\tilde{\\nabla}_\\sigma \\left( \\sqrt{-g} \\, f'(\\tilde{R}) g^{\\mu\\nu}\n\\right)-\\tilde{\\nabla}_\\sigma \\left( \\sqrt{-g} \\, f'(\\tilde{R})\ng^{\\sigma(\\mu}\\right) \\delta^{\\nu)}_\\gamma =0 \\;,\n\\end{equation}\nwhere $\\tilde{\\nabla}_\\gamma$ denotes the covariant derivative\nassociated to the (non-metric) connection\n$\\Gamma^{\\mu}_{\\alpha\\beta}$.\n\n\nBy tracing eqs.~(\\ref{Palatinifieldeq1}) and\n(\\ref{Palatinifieldeq2}) we obtain\n\\begin{equation} \\label{Palatinitrace}\nf'(\\tilde{R}) \\tilde{R} -2f(\\tilde{R})=\\kappa \\, T\n\\end{equation}\nand\n\\begin{equation} \\label{Palatinifieldeq3}\n\\tilde{\\nabla}_\\gamma \\left( \\sqrt{-g} \\, f'(\\tilde{R})\ng^{\\mu\\nu} \\right)=0 \\;,\n\\end{equation}\nrespectively. Eq.~(\\ref{Palatinifieldeq3}) is interpreted as\nstating that\n$\\tilde{\\nabla}_\\gamma$ is the covariant derivative of the\n``new'' metric tensor\n\\begin{equation}\n\\tilde{g}_{\\mu\\nu}\\equiv f'( \\tilde{R}) g_{\\mu\\nu}\n\\end{equation}\nconformally related to $g_{\\mu\\nu}$. Eq.~(\\ref{Palatinitrace})\nis an algebraic\n(or trascendental, according to the functional form of $f(R)$)\nequation for $f'(\\tilde{R})$, not a differential equation\ndescribing its evolution. Therefore, $f'(R)$ is a non-dynamical\nquantity, in contrast to what happens in metric $f(R)$ gravity.\nThe lack of dynamics has consequences which are discussed below.\nIt is possible to eliminate the non-metric connection\nfrom the field equations by rewriting them as\nequation}{{\\beta}{eqnarray}\n&& G_{\\mu\\nu}=\\frac{\\kappa}{f'}\\, T_{\\mu\\nu}-\\frac{1}{2}\\left(\nR-\\frac{f}{f'}\\right) g_{\\mu\\nu} +\\frac{1}{f'}\n\\left(\\nabla_\\mu\\nabla_\\nu\n-g_{\\mu\\nu}\\Box \\right) f'\\nonumber\\\\\n&&\\nonumber \\\\\n&& -\\frac{3}{2(f')^2} \\left[\n\\nabla_\\mu f' \\nabla_\\nu\nf' -\\frac{1}{2} g_{\\mu\\nu} \\nabla_\\gamma f' \\nabla^\\gamma f'\n\\right] \\;.\n\\label{Palatinireformulated}\nequation}{\\end{equation}{eqnarray}\n\n\\subsection{Metric-affine $f(R)$ gravity}\n\nThe third family of $f(R)$ theories, metric-affine $f(R) $\ngravity \\cite{metricaffine}, is characterized by the fact\nthat also the matter part of the action depends explicitly on\nthe connection $\\Gamma$, as described by the action\n\\begin{equation}\nS_{affine}=\\frac{1}{2\\kappa}\\int d^4 x \\, \\sqrt{-g} \\, f\\left(\n\\tilde{R} \\right) +S^{(m)}\\left[ g_{\\mu\\nu},\n\\Gamma^\\mu_{\\nu\\gamma}, \\psi^{(m)} \\right]\n\\;.\n\\end{equation}\n$\\Gamma^{\\mu}_{\\alpha\\beta}$ is possibly a non-symmetric\nconnection, which would\nlead to torsion associated with matter and to a reincarnation\nof torsion theories. The latter were introduced in view of\nelementary particles, rather than cosmology, by coupling the spin\nof elementary particles to the torsion.\nThe study of metric-affine $f(R)$ gravity has not been\ncompleted yet, in particular its cosmological\nconsequences have not been fully elucidated. It is for this\nreason that our discussion will be limited to metric\nand Palatini $f(R)$ gravity in what follows.\n\n\n\\section{Equivalence of metric and Palatini $f(R)$ gravities with\nBrans-Dicke theories}\n\\label{tre}\nIn the same way that one can make variable redefinitions in\nclassical mechanics in order to bring an equation describing a\nsystem to a more attractive, or easy to handle, form (and in a\n very similar way to changing coordinate systems), one can also\n perform field redefinitions in a field theory, in order to\nrewrite the action or the field equations.\n\nThere is no unique prescription for redefining the fields of a\ntheory. One can introduce auxiliary fields, perform\nrenormalizations or conformal transformations, or even simply\nredefine fields to one's convenience.\nIt is important to mention that, at least within a classical\nperspective such as the one followed here, two theories are\nconsidered to be dynamically equivalent if, under a suitable\nredefinition of the\ngravitational and matter fields, one can make their field equations\ncoincide. The same statement can be made at the level of the\naction. Dynamically equivalent theories give exactly the same\nresults when\ndescribing a dynamical system which falls within the purview of\nthese theories. There are clear advantages in exploring the\ndynamical\nequivalence between theories: we can use results already\nderived for one theory in the study of another, equivalent,\ntheory.\n\n\n\nThe term `{\\it `dynamical equivalence}'' can be considered misleading\nin classical gravity. Within a classical perspective, a theory\nis fully\ndescribed by a set of field equations. When we are referring to\ngravitation theories, these equations describe the\ndynamics of gravitating systems. Therefore, two dynamically\nequivalent theories\nare actually just different representations of the same\ntheory (which also makes it clear that all allowed\nrepresentations can be used on an equal footing).\n\nThe issue of distinguishing between truly different theories and\ndifferent representations of the same theory (or dynamically\nequivalent theories) is an intricate one. It has serious\nimplications\nand has been the cause of many misconceptions in the past, especially\nwhen conformal transformations are used in order to redefine the\nfields ({\\em e.g.,~}the Jordan and Einstein frames in\nscalar-tensor\ntheory).\nIn what follows, we review the equivalence between metric and\nPalatini $f(R)$ gravity with specific theories within the\nBrans-Dicke class with a potential.\n\nMetric $f(R)$ gravity is equivalent to an $\\omega=0$ Brans-Dicke\ntheory\\footnote{The\nBrans-Dicke action for general values of the Brans-Dicke\nparameter $\\omega$ is\n$ S_{BD} = \\frac{1}{2\\kappa} \\int d^4x \\, \\sqrt{-g} \\left[\n\\phi R -\\frac{\\omega}{\\phi} \\, \\nabla^\\gamma\\phi\n\\nabla_{\\gamma}\\phi -V(\\phi)\n\\right] +S^{(m)} $.} when $f''(R) \\neq 0$\n\\cite{BD}, while Palatini modified gravity\nis equivalent to one with $\\omega=-3\/2$. The equivalence\nhas been rediscovered several times over the years, often in the\ncontext of particular theories \\cite{STequivalence}.\n\n\\subsection{Metric formalism}\n\nIt has been noticed quite early that metric quadratic gravity\ncan be cast into the form of a Brans-Dicke theory and it did\nnot take long for these results to be extended to more\ngeneral actions which are functions of the Ricci scalar of the\nmetric . Let us\npresent this equivalence in some detail.\n\nWe will work at the level of the action but the same approach\ncan be used\nto work directly at the level of the field equations. We begin with\nmetric $f(R)$ gravity.\nLet $f''(R) $ be non-vanishing and consider the\naction~(\\ref{actionmetric}); by using the\nauxiliary scalar field $\\phi =R$, it is easy to see that the\naction\n\\begin{equation} \\label{equivalentmetric}\nS=\\frac{1}{2\\kappa} \\int d^4 x \\, \\sqrt{-g} \\left[ \\psi( \\phi)R\n-V(\\phi)\n\\right] +S^{(m)}\n\\end{equation}\nwith\n\\begin{equation}\n\\psi(\\phi) = f'(\\phi) \\;, \\;\\;\\;\\;\\;\\;\nV(\\phi)=\\phi f'(\\phi)-f(\\phi)\n\\end{equation}\nis equivalent to the previous one. It is trivial that\n(\\ref{equivalentmetric}) reduces to~(\\ref{actionmetric}) if\n$\\phi=R$. Vice-versa, the variation of~(\\ref{equivalentmetric})\nwith respect to $g^{\\mu\\nu}$ yields\n\\begin{equation}\nG_{\\mu\\nu}=\\frac{1}{\\psi}\\left( \\nabla_\\mu\\nabla_\\nu \\psi-\ng_{\\mu\\nu}\\Box\\psi -\\frac{V}{2}\\, g_{\\mu\\nu}\n\\right)+\\frac{\\kappa}{\\psi} \\, T_{\\mu\\nu} \\;.\n\\end{equation}\nThe variation with respect to $\\phi$, instead, gives us\n\\begin{equation}\nR\\, \\frac{d\\psi}{d\\phi} -\\frac{dV}{d\\phi}=\\left( R-\\phi\n\\right)f''(\\phi)=0 \\;,\n\\end{equation}\nfrom which it follows that $\\phi=R$ because $f''\\neq 0$. The\nscalar field $\\phi=R$ is clearly a dynamical quantity which\nobeys the trace equation\n\\begin{equation}\n3f''(\\phi)\\Box \\phi+3f'''(\\phi)\\nabla^{\\alpha}\\phi\\nabla_{\\alpha}\n\\phi +\\phi\nf'(\\phi) -2f(\\phi) =\\kappa \\, T\n\\end{equation}\nand is massive. Its mass squared\n\\begin{equation}\nm_{\\phi}^2=\\frac{1}{3} \\left( \\frac{f_0'}{f_0 ''}-R_0 \\right)\n\\end{equation}\nis computed in the analysis of small perturbations of de Sitter\nspace (here a zero subscript denotes quantities evaluated at\nthe constant curvature $R_0$ of the de Sitter background).\nIt is convenient to consider, instead of $\\phi$, the scalar\n$ \\psi \\equiv f'(\\phi)$ obeying the evolution equation\n\\begin{equation}\n3\\Box \\psi +2 U(\\psi) -\\psi\\, \\frac{dU}{d\\psi}=\\kappa \\, T \\;,\n\\end{equation}\nwhere $ U(\\psi)=V(\\phi(\\psi))-f(\\phi(\\psi)) $.\n\n\nTo summarize, metric $f(R)$ gravity contains a scalar degree\nof freedom and the action\n\\begin{equation}\nS=\\frac{1}{2\\kappa} \\int d^4 x \\, \\sqrt{-g} \\left[ \\psi R -U(\\psi)\n\\right] +S^{(m)} \\;,\n\\end{equation}\nis identified as an $\\omega=0$ Brans-Dicke theory. This theory\n(``massive dilaton gravity'') was introduced in the\n1970's in order to generate a Yukawa term in the Newtonian limit\n\\cite{OHanlon72}, and then abandoned. The assumption\n$f''\\neq 0$ is interpreted as the requirement of\ninvertibility of the change of variable $R\\rightarrow \\psi(R)$.\n\n\n\\subsection{Palatini formalism}\nIn Palatini modified gravity the equivalence with\na Brans-Dicke theory is discovered in a way similar to\nthat of the metric formalism. Beginning with the\naction~(\\ref{actionPalatini}) and defining $\\phi\n\\equiv \\tilde{R}$\nand $\\psi \\equiv f'(\\phi)$, it is seen that, apart from an\nirrelevant\nboundary term, the action\ncan be rewritten as\n\\begin{equation} \\label{equivalentPalatini}\nS_{Palatini}=\\frac{1}{2\\kappa}\\int d^4x \\, \\sqrt{-g} \\left[ \\psi R\n+\\frac{3}{2\\psi} \\, \\nabla^\\gamma \\psi\\nabla_\\gamma \\psi -V(\\psi) \\right]\n+S^{(m)}\n\\end{equation}\nin terms of the metric $g_{\\mu\\nu}$ and its\nRicci tensor $R_{\\mu\\nu}$. Here we have used the property that,\nsince $\\tilde{g}_{\\mu\\nu}=\\psi \\, g_{\\mu\\nu}$, the\nRicci curvatures of $g_{\\mu\\nu}$ and\n$\\tilde{g}_{\\mu\\nu}$ satisfy the relation\n\\begin{equation}\n\\tilde{R}=R +\\frac{3}{2\\psi}\n\\nabla^\\gamma\\psi \\nabla_{\\gamma}\\psi-\\frac{3}{2} \\Box \\psi \\;.\n\\end{equation}\nThe action~(\\ref{equivalentPalatini}) is easily identified as a\nBrans-Dicke theory with Brans-Dicke parameter $\\omega=-3\/2$.\n\n\\section{Theoretical and experimental viability of $f(R)$\ngravity}\n\\label{quattro}\n\nIn order to be acceptable, $f(R)$ theories should not only\nreproduce the current acceleration of the universe, but they\nmust also satisfy the constraints imposed by Solar System and\nterrestrial experiments on relativistic gravity, and they must\nobey certain minimal requirements for theoretical viability.\nMore precisely, these families of theories must:\n\nequation}{{\\beta}{itemize}\n\n\\item possess the correct cosmological dynamics;\n\\item be free from instabilities and ghosts;\n\\item attain the correct Newtonian and post-Newtonian limits;\n\\item originate cosmological perturbations compatible with the\nobservations of the CMB and with large\nscale structure surveys;\nand\n\\item possess a well-formulated and well-posed initial value problem.\nequation}{\\end{equation}{itemize}\n\n\n\nIf a single one of these criteria is not met the theory should\nbe regarded as unviable. In the following we examine how\n$f(R)$ gravity performs with regard to these criteria.\n\n\n\\subsection{Correct cosmological dynamics}\nAccording to the tenets of standard cosmology, an acceptable\ncosmological model must contain an early inflationary era (or\npossibly another mechanism) solving the horizon, flatness, and\nmonopole problems and generating density perturbations, followed\nby a radiation- and then a matter-dominated era. The present\naccelerated epoch then begins, possibly explained by $f(R)$\ngravity. The future universe usually consists of an eternal de\nSitter attractor, or ends in a Big Rip singularity \\cite{abdalla}.\nSmooth transitions between different eras are necessary. The exit\nfrom the radiation era, in particular, was believed to be\nimpossible in many models \\cite{Amendolaetal}, but this proved to\nbe not true. In fact, exit from the radiation or any era can be\nobtained as follows. In the approach dubbed ``designer $f(R)$\ngravity'' in \\cite{otherreviews}, the desired expansion history of\nthe universe can be obtained by specifying the desired scale\nfactor $a(t)$ and integrating an ordinary differential equation\nfor the function $f(R)$ that produces the chosen $a(t)$\n\\cite{designerf(R)}. In general, the solution to this ODE is not\nunique and can assume a form that appears rather contrived in\ncomparison with simple forms adopted in most popular models.\n\n\n\\subsection{Instabilities}\n\nThe choice $f(R)=R-\\mu^4\/R$ with $\\mu\\sim H_0\\sim\n10^{-33}$~eV is again the prototypical example model to discuss\ninstabilities. Shortly after it was advanced as an explanation\nof the cosmic acceleration, this model was found to suffer\nfrom the pernicious ``Dolgov-Kawasaki'' instability\n\\cite{DolgovKawasaki}. This type of instability\nwas later shown to be common to any metric $f(R)$ theory with\n$f''(R)<0$ (\\cite{mattmodgrav})\nand the extension\nto even more general gravitational theories has been discussed\n\\cite{Zerbini}. Let us parametrize the deviations\nfrom GR as\n\\begin{equation}\nf(R)=R+\\epsilon \\varphi(R)\n\\end{equation}\nwith $\\epsilon >0$ a small constant with the\ndimensions of a mass squared and $\\varphi$ dimensionless. The\ntrace equation for the Ricci scalar $R$ becomes\n\\begin{equation}\n\\Box R+\\frac{\\varphi '''}{\\varphi ''} \\, \\nabla^\\gamma R\n\\nabla_\\gamma R +\\left( \\frac{\\epsilon \\varphi ' -1}{3\\epsilon\n\\varphi ''} \\right) R =\\frac{\\kappa \\, T}{3\\epsilon \\varphi\n''}+\\frac{2\\varphi}{3\\varphi ''}\n\\;.\n\\end{equation}\nBy expanding around a de Sitter background and\nwriting the metric {\\em locally} as\n\\begin{equation} \\label{localdS}\ng_{\\mu\\nu}=\\eta_{\\mu\\nu}+h_{\\mu\\nu} \\;,\n\\end{equation}\nand the scalar $R$ as\n\\begin{equation}\nR=-\\kappa\\, T +R_1 \\;,\n\\end{equation}\nwith $R_1$ a perturbation, the first order trace equation\ntranslates into the\ndynamical equation for $R_1$\n\\begin{equation}\n\\ddot{R}_1 -\\nabla^2 R_1 -\\frac{2\\kappa \\varphi '''}{\\varphi ''}\n\\, \\dot{T}\\dot{R}_1+\\frac{2\\kappa \\varphi '''}{\\varphi ''} \\,\n\\vec{\\nabla}T \\cdot \\vec{\\nabla}R_1 + \\frac{1}{3\\varphi ''}\\left(\n\\frac{1}{\\epsilon}-\\varphi ' \\right) R_1=\\kappa \\, \\ddot{T}-\\kappa\n\\nabla^2 T -\\frac{ \\left( \\kappa T \\varphi^2 +2\\varphi \\right)}{3\\varphi\n''} \\;.\n\\end{equation}\nThe expression containing $\\epsilon^{-1}$ dominates the last term\non the left hand side, giving the effective mass squared of\n$R_1$\n\\begin{equation}\nm^2 \\simeq \\frac{1}{3\\epsilon \\varphi ''} \\;.\n\\end{equation}\nTherefore, the theory is stable if $f''(R)>0$ and unstable if\n$f''(R)<0$. Strictly speaking, GR is excluded by the assumption\n$f''\\neq 0$, but the well-known stability of this case can easily\nbe included by writing the stability criterion for metric $f(R)$\ngravity as $f'' \\geq 0$.\n\nTo go back to the example model of \\cite{DolgovKawasaki}\n$f(R)=R-\\mu^4\/R$, this is unstable because\n$f''<0$. The small scale $\\mu$ determines the time scale for the\nonset of this instability as $\\sim 10^{-26}$~s\n\\cite{DolgovKawasaki}, making this an\nexplosive instability.\n\n\n\nA physical interpretation of this stability criterion is the\nfollowing \\cite{myinterpretation}: the\neffective gravitational coupling is $G_{eff}=G\/f'(R)$ and, if\n$dG_{eff}\/dR=-f''G\/(f')^2>0$ (corresponding to $f''<0$), then\n$G_{eff}$ increases with $R$ and a large curvature causes gravity\nto become stronger and stronger, which in turn\ncauses a larger $R$, in a positive feedback loop. If\ninstead $dG_{eff}\/dR<0$, then a negative feedback stops the\ngrowth of the\ngravitational coupling.\n\n\nWhat about Palatini $f(R)$ gravity? Since this formalism\ncontains only second order\nfield equations and the trace equation $ f'( \\tilde{R}) \\tilde{R}\n-2f(\\tilde{R})=\\kappa \\, T$ is not a differential equation but\nrather a non-dynamical equation, as noted above, there is no\nDolgov-Kawasaki instability \\cite{SotiriouPalatiniinstab}.\n\n\nThe discussion of metric $f(R)$ instabilities\npresented above is based on the\nlocal expansion~(\\ref{localdS}) and, therefore, is\nlimited to short wavelength modes (compared to the\ncurvature radius). However, it can be extended to the longest\nwavelengths in the case of a de Sitter background\n\\cite{mydS}. This extension requires a more complicated formalism\nbecause long modes introduce inhomogeneities and are affected by\nthe notorious gauge-dependence problems of cosmological\nperturbations. A covariant and gauge-invariant formalism is\nneeded here. One proceeds by assuming that the background space\nis de Sitter and by considering the general action\n\\begin{equation}\nS=\\int d^4 x \\, \\sqrt{-g} \\, \\left[ \\frac{f \\left(\\phi, R\n\\right)}{2} -\\frac{\\omega(\\phi)}{2}\\, \\nabla^\\gamma \\phi\n\\nabla_\\gamma \\phi -V(\\phi) \\right]\n\\end{equation}\ncontaining $f(R)$ and scalar-tensor gravity as special cases,\nand mixtures of them. The field equations originating from this\naction become, in a FLRW background space,\nequation}{{\\beta}{eqnarray}\n& & H^2= \\frac{1}{3f'}\n\\left(\\frac{\\omega}{2}\\,\n\\dot{\\phi}^2+\\frac{Rf'-f}{2}+V-3H\\dot{f}\\right)\n\\;, \\\\\n&&\\nonumber \\\\\n&& \\dot{H}=\\frac{-1}{2f'} \\left( \\omega \\dot{\\phi}^2\n+ \\ddot{f'}-H\\dot{f'} \\right) \\;, \\\\\n&&\\nonumber \\\\\n&& \\ddot{\\phi}+3H\\dot{\\phi} +\\frac{1}{2\\omega}\\left(\n\\frac{d\\omega}{d\\phi}\n\\dot{\\phi}^2 -\\frac{\\partial f}{\\partial \\phi} +2 \\,\n\\frac{dV}{d\\phi}\n\\right)=0 \\;.\nequation}{\\end{equation}{eqnarray}\nde Sitter space is a solution of the field equations provided\nthat the conditions\n\\begin{equation}\n6H_0^2 f_0'-f_0 +2V_0=0 \\;, \\;\\;\\;\\;\\;\\;\\;\\;\nf_0'=2V_0' \\;,\n\\end{equation}\nare satisfied. An analysis of inhomogeneous perturbations of\nsmall amplitude and arbitrary wavelengths \\cite{mydS} using\nthe covariant and gauge-invariant Bardeen-Ellis-Bruni-Hwang\nformalism \\cite{Bardeen} in Hwang's version\n\\cite{Hwang} for alternative gravitational theories yields the\nstability condition in the zero momentum limit\n\\begin{equation} \\label{stabilitydS}\n\\frac{ (f_0')^2 -2f_0 f_0''}{f_0' f_0''} \\geq 0 \\;,\n\\end{equation}\nThis is the stability condition of de Sitter space\nin metric $f(R)$~gravity with respect to {\\em inhomogeneous}\nperturbations and coincides\nwith the corresponding stability condition with respect to {\\em\nhomogeneous} perturbations \\cite{myinterpretation}.\n\n\nThe\nequivalence between metric $f(R)$ gravity and an $\\omega=0$\nBrans-Dicke theory holds also at the level of perturbations;\ndoubts advanced to this regard have now\nbeen resolved. The stability condition\nof de Sitter space with respect to inhomogeneous perturbations\nin $\\omega=0$ Brans-Dicke theory\nis given again by eq.~(\\ref{stabilitydS}), while that for\nstability with respect\nto homogeneous perturbations is\n\\begin{equation}\n\\frac{ (f_0')^2 -2f_0 f_0''}{f_0'} \\geq 0 \\;.\n\\end{equation}\nThis inequality is again equivalent to~(\\ref{stabilitydS})\nif stability against {\\em local} perturbations ({\\em i.e.},\n$f_0''>0$) is also required. Hence, metric $f(R)$ gravity and\n$\\omega=0$ Brans-Dicke theory are equivalent also\nwith regard to perturbations.\n\n\nBeyond the linear approximation, metric $f(R)$ theories have been\nshown to be susceptible to non-linear instability, potentially\nthreatening the possibility of constructing models of relativistic\nstars in strong $f(R)$ gravity. Inside compact objects with\nspherical symmetry, a singularity could develop if $R$ becomes\nlarge \\cite{Frolovetc}. Avoiding this singularity requires some\ndegree of fine-tuning. Various authors have contended that this\nproblem can be cured by adding, for example, a quadratic term $\n\\alpha R^2$ to the action as first \\cite{abdalla,bamba}. This\nproblem needs further study, since it could be the biggest\nchallenge left for metric $f(R)$ theories.\n\n\n\n\\subsection{Ghost fields}\nGhosts are massive states of negative norm which ruin unitarity\nand appear frequently in attempts to quantize\nEinstein's theory. Fortunately, $f(R)$ gravity theories are free\nof ghosts. More general ETGs of the form $f\\left( R,\nR_{\\mu\\nu}R^{\\mu\\nu},\nR_{\\mu\\nu\\gamma\\sigma} R^{\\mu\\nu\\gamma\\sigma}, ...\n\\right) $, in general, are plagued by the presence of ghosts. A\npossible exception under certain conditions\nstudied in \\cite{GBghosts} is provided by theories in which\nthe extra terms are restricted to appear in the Gauss-Bonnet\ncombination ${\\cal\nG}=R^2-4R_{\\mu\\nu}R^{\\mu\\nu}+\nR_{\\mu\\nu\\gamma\\sigma}R^{\\mu\\nu\\gamma\\sigma}$, as in $f=f\\left(\nR, {\\cal G}\n\\right)$. Then, the field equations reduce to second order\nequations without ghosts \\cite{ Comelli05,\nNavarroVanAcoleyen06}.\n\n\n\\subsection{The weak-field limit for metric $f(R)$ gravity}\n\nAfter errors and omissions in the early treatments of the\nweak-field limit of metric and Palatini modified gravity, a\nsatisfactory discussion of the particular model $f(R)=R-\\mu^4\/R$\nin the metric formalism appeared \\cite{Chibaetal06}, followed\nby the generalization to arbitrary forms of the function $f(R)$\n\\cite{Olmo07, CSE}.\n\nOne studies the PPN parameter\n$\\gamma$ which is constrained by light deflection experiments in\nthe Solar System. The goal consists of finding the weak-field\nsolution of the field equations and, using this solution,\ncomputing the parameter\n$\\gamma$. A static, spherically symmetric, non-compact body which\nconstitutes a perturbation of a\nbackground de Sitter universe is considered, as described by the\nline element\n\\begin{equation}\nds^2=-\\left[ 1+2\\Psi(r)-H_0^2r^2 \\right] dt^2 +\\left[ 1+2\n\\Phi(r) +H_0^2 r^2 \\right] dr^2 +r^2 d\\Omega^2\n\\end{equation}\nin Schwarzschild coordinates, with $d\\Omega^2$ being the line\nelement on the unit 2-sphere. $\\Psi$ and $\\Phi$ are\npost-Newtonian potentials with small amplitudes, {\\em\ni.e.}, $\\left|\\Psi (r) \\right| ,\n\\left|\\Phi (r) \\right| <<1$, and small\n(non-cosmological)\nscales such that $H_0r <<1$ are considered. The\nRicci scalar is expanded around the\nconstant curvature of the background de Sitter space as\n$R(r)=R_0+R_1$. The PPN parameter $\\gamma$ is then given by $\n\\gamma =-\\Phi(r) \/ \\Psi(r) $ \\cite{Will}. The analysis relies\nupon three assumptions \\cite{CSE}:\n\nequation}{{\\beta}{enumerate}\n\\item $f(R)$ is analytical at $R_0$;\n\\item $mr<<1$, where $m$ is the effective mass of the scalar\ndegree of freedom of the theory. In other words, this scalar\nfield\nis assumed to be light and with a range larger than the size\nof the Solar System (there are no\nexperimental constraints on scalars\nwith range $m^{-1} <0.2$~mm).\n\\item The matter composing the spherical body has negligible\npressure, $P\\simeq 0$ and $ T=T_0+T_1 \\simeq -\\rho$.\nequation}{\\end{equation}{enumerate}\n\nWhile it is easy to satisfy the first and the last assumptions,\nthe second one is more tricky, as discussed below. The trace\nequation~(\\ref{tracemetric}) turns into \\begin{equation}\n\\nabla^2 R_1 -m^2 R_1 =\\frac{-\\kappa \\rho}{3f_0''} \\;,\n\\end{equation}\nregulating the Ricci scalar perturbation, where\n\\begin{equation} \\label{cicci}\nm^2 =\\frac{ (f_0')^2 -2f_0f_0''}{3f_0' f_0''}\n\\end{equation}\nis the effective mass squared of the scalar, which\nreproduces the expression derived in the gauge-invariant\nstability analysis of de Sitter space and in propagator\ncalculations.\n\n\nIf $mr<<1$, the solution of the linearized field equations is\nequation}{{\\beta}{eqnarray}\n&& \\Psi(r)=\\frac{-\\kappa M}{6\\pi f_0'}\\, \\frac{1}{r} \\;, \\\\\n&&\\nonumber \\\\\n&& \\Phi(r) =\\frac{\\kappa M}{12 \\pi f_0'} \\, \\frac{1}{r} \\;,\nequation}{\\end{equation}{eqnarray}\nand the PPN parameter $\\gamma$ is given by\n\\begin{equation}\n\\gamma =\\frac{-\\Phi(r)}{\\Psi(r)}=\\frac{1}{2} \\;.\n\\end{equation}\nThis value manifestly violates the experimental bound\n\\cite{BertottiIessTortora}\n\\begin{equation}\n\\left| \\gamma -1 \\right| <2.3 \\cdot 10^{-5} \\;.\n\\end{equation}\nThis violation would mark the demise of metric $f(R)$ gravity\nwere it not for the fact that the second assumption necessary to\nperform this calculation is usually not satisfied. In fact, $mr$\nfails to be smaller than unity due to\nthe {\\em chameleon effect}. This effect consists of a\ndependence of the effective mass $m$ on the spacetime curvature\nor, alternatively, on the matter density of the surroundings. The\nscalar degree of freedom can have a short range (for example, $m\n> 10^{-3}$~eV, corresponding to a range $\\lambda\n< 0.2$~mm) at Solar System densities, escaping the experimental\nconstraints, and have a long range at cosmological\ndensities, which allows it to have an effect on the\ncosmological dynamics \\cite{NavarroVanAcoleyen06, Faulkneretal06}.\nWhile the chameleon effect may seem a form of fine-tuning, one\nshould bear in mind that $f(R)$ gravity is complicated and the\neffective range does indeed depend on the\nenvironment. The chameleon mechanism is not arranged, but is\nbuilt into the theory and is well-known and\naccepted in quintessence models, in which it was originally\ndiscovered \\cite{chameleon}. It has been studied for many forms\nof the function $f(R)$ which pass the observational tests. For\nexample, the model\n\\begin{equation} \\label{Faulknermodel}\nf(R)=R-\\left(1-n \\right) \\mu^2 \\left( \\frac{R}{\\mu^2} \\right)^n\n\\end{equation}\nis compatible with the PPN limits if $ \\mu \\sim\n10^{-50}$~eV$\\sim10^{-17} H_0$ \\cite{Faulkneretal06}. To\nunderstand how this model can work it is sufficient to note that\na correction $\\sim R^n$ to the\nEinstein-Hilbert\nLagrangian $R$ with $n<1$ will eventually dominate as\n$R\\rightarrow 0^{+}$. The model~(\\ref{Faulknermodel}) agrees with\nthe experimental data but could be essentially\nindistinguishable from a dark energy model. Discriminating\nbetween dark energy and $f(R)$ models, or between modified\ngravity scenarios should be possible on the basis of the\ngrowth history of cosmological\nperturbations.\n\n\n\\subsection{Growth of cosmological perturbations}\n\nSince the spatially homogeneous and isotropic FLRW metric solves\nthe field equations of many gravitational theories, the\nexpansion history of the universe by itself cannot\ndiscriminate between various ETGs. However, the growth of\nstructures depends on the theory of gravity considered and has\nthe potential to achieve this goal.\nA typical study is that of Ref.~\\cite{SongHuSawicki06}; these\nauthors postulate an expansion history $a(t)$ characteristic of\nthe $\\Lambda$CDM model and find that vector\nand tensor modes are not affected by $f(R)$ corrections to Einstein\ngravity, to lowest order, and can be neglected, whereas\nscalar modes do depend on the theory chosen.\nIn~\\cite{SongHuSawicki06} the\nstability condition $f''(R)>0$ discussed above for scalar\nperturbations is also recovered. It is found\nthere that $f(R)$ corrections lower the large\nangle anisotropies of the cosmic microwave background and produce\ncorrelations between cosmic microwave background and\ngalaxy surveys which are different from those obtained in dark\nenergy models. A rigorous and mathematically self-consistent approach to the problem of cosmological perturbations in $f(R)$-gravity as been developed using covariant and gauge-invariant quantities in \\cite{sante1,sante2,sante3}).\n\n\nThe study of structure formation in modified gravity is still uncomplete\n and, most of the times, is carried out within\nspecific $f(R)$ models. Insufficient attention has been paid\nto the fact that some of these models are already ruled out\nbecause they contradict the weak-field limit or the stability\nconditions. A similar situation is found in Palatini models\nwhich, for this reason, will not be discussed\nhere with regard to their weak-field limit and cosmological\nperturbations.\n\n\\subsection{The initial value problem}\n\nA physical theory is required to make predictions and,\ntherefore, it must have a well-posed Cauchy problem. GR\nsatisfies this requirement for ``reasonable'' forms of matter\n\\cite{Wald}. The well-posedness of the initial value problem for\nvacuum $f(R)$ gravity was briefly discussed for special metric\nmodels a long time ago\n\\cite{Noakes}. Owing to the\nequivalence between $f(R)$ gravity and\nscalar-tensor gravity when\n$f''(R)\\neq 0$, the initial value\nproblem of $f(R)$ gravity is reduced to the one\nfor Brans-Dicke gravity with $\\omega=0$ or $ -3\/2$.\n The Cauchy problem was shown to be well-posed for\nparticular scalar-tensor theories\nin \\cite{Cocke, Noakes} but a general analysis has\nbeen completed only relatively recently \\cite{Salgado,\nSalgado2}. A separate treatment, however, was necessary\nfor $\\omega=0, -3\/2 $ Brans-Dicke theory.\n\n\nWe begin by defining the basic concepts employed: a system of\n$3+1$ equations is said to be {\\em well-formulated}\nif it can be written as a\nsystem of equations of only first order in\nboth temporal and spatial derivatives. Assume that this\nsystem can be cast in the full first order form\nequation}{{\\beta}{equation}\n\\partial_t \\, \\vec{u} + M^i \\nabla_i \\vec{u}=\\vec{S}\\left(\n\\vec{u}\\right) ,\nequation}{\\end{equation}{equation}\nwhere $\\vec{u}$ collectively denotes the fundamental variables\n$h_{ij}, K_{ij}$, {\\em etc.} introduced below, $M^i$ is\ncalled the {\\em characteristic matrix} of the system, and\n$\\vec{S}\\left( \\vec{u} \\right)$ describes source terms and\ncontains only the fundamental variables but not their\nderivatives. Then, the initial\nvalue formulation is {\\em well-posed} if the system of PDEs is {\\em\nsymmetric hyperbolic} ({\\em\ni.e.}, the matrices $M^i$ are symmetric) and {\\em strongly\nhyperbolic} if $ s_iM^i$ has a real set of\neigenvalues and a complete\nset of eigenvectors for any 1-form $s_i$, and obeys some\nboundedness conditions \\cite{Solin}.\n\n\nTo summarize the results of \\cite{TremblayFaraoni}, the Cauchy\nproblem for metric $f(R)$ gravity is well-formulated and is\nwell-posed in vacuo and with ``reasonable'' forms of matter ({\\em\ni.e.}, perfect fluids, scalar fields, or the Maxwell field). For\nPalatini $f(R)$ gravity, instead, the Cauchy problem is\nwell-formulated \\cite{ijggmp} but not well-posed in general, due\nto the presence of higher derivatives of the matter fields in the\nfield equations and to the fact that it is impossible to eliminate\nthem \\cite{TremblayFaraoni}. However, as it was remarked in\n\\cite{Cauchycomments}, the Cauchy problem for Palatini is still\nwell-posed in vacuo and when the trace of the matter\nenergy-momentum tensor vanishes or it is a constant. On the other\nhand, it is possible to show the well-formulation and the\nwell-position as soon as the source of the field equations is\nperfect-fluid matter \\cite{Cauchycomments}.\n\n\n\n\n\nAs an alternative, the Brans-Dicke theory equivalent to\nPalatini $f(R)$ gravity can be mapped into its Einstein frame\nrepresentation. In this conformal frame the redefined Brans-Dicke\nfield couples minimally to gravity and non-minimally to matter\n\\cite{FaraoniTremblay} and the non-dynamical\nrole of this scalar is even more obvious\n\\cite{FaraoniTremblay}.\n\n\n\nThe problems with Palatini $f(R)$ gravity manifest themselves\nfrom a completely different angle when one tries to\nmatch static interior and exterior solutions with spherical\nsymmetry \\cite{BarausseSotiriouMiller}.\\footnote{Other problems\nof Palatini $f(R)$ gravity were reported and discussed\nin~\\cite{Kaloper, PalatiniPLB}.}\nThe field equations are second order PDEs for the metric\ncomponents and, since $f$ is a function of $\\tilde{R}$, which\nin turn is an algebraic function of $T$ due to\neq.~(\\ref{Palatinitrace}), the right hand side of\neq.~(\\ref{Palatinireformulated}) contains second derivatives\nof $T$. Now, $T$ contains derivatives of the matter fields up to\nfirst order, hence eq.~(\\ref{Palatinireformulated}) contains\nderivatives of the matter\nfields up to third order. This property is very different from\nthe familiar situation of GR and most of its extensions, in\nwhich the field equations contain only\nfirst order derivatives of the matter fields. A consequence of\nthis dependence on lower order derivatives of the matter\nfields is that, in these theories the metric is generated by an\nintegral over the\nmatter sources and discontinuities in the matter fields and\ntheir derivatives are not accompanied by unphysical\ndiscontinuities of the metric. In\nPalatini $f(R)$ gravity, instead, the algebraic dependence\nof the metric on the matter fields creates unacceptable\ndiscontinuities in the metric and singularities in the\ncurvature, which were discovered in\n\\cite{BarausseSotiriouMiller}. Both the failure\nof the initial value problem and the presence of curvature\nsingularities with matter fields can be\nascribed to the non-dynamical nature of the scalar degree of\nfreedom and to the fact that the latter is related algebraically\nto $T$. A possible cure consists of modifying the gravitational\nsector of the Lagrangian in such a way that the order\nof the field equations is raised.\n\n\\section{Dark energy as curvature}\n\\label{cinque}\nLet us now show, by some straightforward arguments, how\n$f(R)$-gravity can be related to the problem of dark energy.\n The\nfield equations~(\\ref{metricfieldeqs}) may be recast in the\nEinstein-like form\nequation}{{\\beta}{equation}\\label{5}\nG_{\\mu \\nu} = R_{\\mu\\nu}-\\frac{1}{2}g_{\\mu\\nu}R =\nT^{(eff)}_{\\mu\\nu}+T_{\\mu\\nu}\/f^\\prime(R)\nequation}{\\end{equation}{equation}\nwith $T^{(eff)}$ given by eq.~(\\ref{effectivetensor}) and in\nwhich matter couples non-minimally to the geometry through\nthe term $1\/f^\\prime(R)$. As noted above, the appearance of\n$f^\\prime(R)_{;\\mu\\nu}$ in $T_{\\mu\\nu}^{(eff)}$ makes\neq.~(\\ref{5}) a fourth order equation (unless $f(R) = R$, in\nwhich case the\ncurvature stress\\,-\\,energy tensor $T^{(eff)}_{\\alpha \\beta}$\nvanishes identically and~(\\ref{5}) reduces to the\nsecond order Einstein equation). As is clear from\neq.~(\\ref{5}), the curvature stress-energy tensor\n$T_{\\mu\\nu}^{(eff)} $ formally plays the role of a\nsource in the field equations and its effect is the same\nas that of an effective fluid of\npurely geometrical origin. However, one can also consider\nthe Palatini approach \\cite{FFV, metricaffine}, in which the\nEinstein equations can still be rewritten as effective Einstein\nequations containing a fluid of geometric origin.\n\n\nIn principle, the scheme outlined above provides all the\ningredients needed to tackle the dark side of the universe.\nDepending on the scale considered, the effective curvature\nfluid can play the role of both dark matter and dark energy.\nFrom the cosmological point of view, in the standard framework of\na spatially flat homogenous and isotropic universe, the\ncosmological dynamics are determined by the energy budget\nthrough the Friedmann equations. In particular, the cosmic\nacceleration is achieved when the right hand side\nof the acceleration equation remains positive. In units\nin which $8 \\pi G = c = 1$ this means\nequation}{{\\beta}{equation}\n\\frac{\\ddot{a}}{a} = - \\frac{1}{6} \\left ( \\rho_{tot} + 3\nP_{tot} \\right ) \\ , \\label{eq: fried2}\nequation}{\\end{equation}{equation}\nwhere the subscript $tot$ denotes the sum of the curvature fluid\nand the matter contributions to the energy density and pressure.\nThe acceleration condition $\\ddot{a}>0$ for a dust-dominated\nmodel is\nequation}{{\\beta}{equation}\n\\rho_{eff} + \\rho_M + 3P_{eff} < 0\n\\end{equation}\nor\n\\begin{equation}\nw_{eff} < - \\frac{\\rho_{tot}}{3 \\rho_{eff}} \\;.\n\\label{eq: condition}\nequation}{\\end{equation}{equation}\nThen, the effective quantities\nequation}{{\\beta}{equation}\n\\rho_{eff} = \\frac{8}{f'(R)} \\left \\{ \\frac{1}{2} \\left [ f(R) -\nR f'(R) \\right ] - 3 H \\dot{R} f''(R) \\right \\} \\ , \\label{eq:\nrhocurv}\nequation}{\\end{equation}{equation}\nand\nequation}{{\\beta}{equation}\nw_{eff} = -1 + \\frac{\\ddot{R} f''(R) + \\dot{R} \\left [ \\dot{R}\nf'''(R) - H f''(R) \\right ]} {\\left [ f(R) - R f'(R) \\right ]\/2 -\n3 H \\dot{R} f''(R)} \\label{eq: wcurv}\nequation}{\\end{equation}{equation}\nplay a key role in determining the dynamics of the universe.\nTo gain insight into the dynamics, one can begin by neglecting\nordinary matter and studying the power-law form $f(R) = f_0\nR^n$ (with $n$ a real number), which represents a straightforward\ngeneralization of Einstein's GR corresponding the $n=1$ limit.\nThis choice yields power-law solutions for the scale\nfactor $a(t)$ which provide a good fit to the SNeIa\ndata and are in good agreement with the estimated age of the\nuniverse in the range $1.366 < n < 1.376$ \\cite{curv-ijmpd}. The\nsame kind of analysis can be carried out in the presence of\nordinary matter, but in this case, numerical solution of\nthe field equations is required. Then, it is still possible\nto confront the Hubble flow described by such a model with the\nHubble diagram of SNeIa.\nequation}{{\\beta}{figure}\n\\centering\\resizebox{7.5cm}{!}{\\includegraphics{bf-ObhSq.eps}}\n\\caption{The Hubble diagram of twenty radio galaxies together\nwith the ``gold\" SNeIa sample is plotted versus the redshift $z$,\nas suggested in \\cite{daly}. The best-fit curve corresponds to\nthe $f(R)$ gravity model without dark matter. \\label{fig:SNeIa}}\nequation}{\\end{equation}{figure}\nThe fit to the data is remarkably good (see\nFig.~\\ref{fig:SNeIa}) improving the $\\chi^2$ value and it fixes\nthe best-fit value at $n=3.46$ if the\nbaryons contribute to the energy density by $\\Omega_b \\approx\n0.04$, in agreement with the prescriptions if Big Bang\nnucleosynthesis. The inclusion of dark matter\ndoes not modify the fit appreciably,\nsupporting the assumption that dark matter is not essential in\n this model. From the evolution of\nthe Hubble parameter in terms of redshift, one can even calculate\nthe age of the universe $t_{univ}$. The best-fit value $n=3.46$\nprovides\n$t_{univ}\\approx 12.41$ Gyr. Of course,\n$f(R)\\,=\\,f_0\\,R^n$ gravity represents only a toy model\ngeneralization of Einstein's theory. Here we only suggest\nthat several cosmological and astrophysical results can be well\nreproduced in the realm of a power-law extended gravity model.\nThis approach allows flexibility in the value of the exponent\n$n$, although it would be preferable to determine a model capable\nof working at various scales. Furthermore, we do not expect to\nbe able to reproduce the entire cosmological phenomenology by\nmeans of a simple power-law model, which is not\nsufficiently versatile \\cite{Amendolaetal}. For example, it can\nbe easily demonstrated that this model fails when\nit is analyzed with respect to its ability of providing the\ncorrect evolutionary conditions for the perturbation spectra of\nmatter overdensities \\cite{pengjie}. This point is typically\nregarded as one of the most important arguments suggesting\nthe need for darm matter. If one wants to discard this\ncomponent, it is crucial to match the observational results\nrelated to the large-scale structure of the universe with the\nCMB. These carry the imprints of the initial\nmatter spectrum at late times\nand at early times, respectively. It is important that the\nquantum\nspectrum of primordial perturbations, which provide the seeds of\nmatter perturbations, can be recovered in the\nframework of $R^n$ gravity. In fact, the model $f(R)\\,\\propto R+\nR^2$ can represent a viable model with respect to CMB data and\nis a good candidate for cosmological inflation. To obtain the\nmatter power spectrum suggested by this model, we resort to the\nequation for the matter contrast obtained in Ref.~\\cite{pengjie}\nfor fourth order gravity. This equation can be deduced in the\nNewtonian conformal gauge for the perturbed metric\n\\cite{pengjie}\nequation}{{\\beta}{equation}\\label{metric-pert}\nds^2\\,=- \\left(\n1+2\\psi \\right) dt^2\\,+ \\,a^2\n\\left( 1+2\\phi \\right) \\Sigma_{i\\,=1}^3(dx^i)^2\\,.\nequation}{\\end{equation}{equation}\nIn GR, it is $\\phi\\,=\\,-\\psi$ because there is no anisotropic\nstress; in\ngeneral, this relation breaks down in ETGs and\nthe non-diagonal components of the field equations yield new\nrelations between the potentials $\\phi$ and $\\psi$. In $f(R)$\ngravity, due\nto the non-vanishing $f_{R;i;j}$ with $i\\,\\neq\\,j$, the\n$\\phi-\\psi$ relation becomes scale-dependent. Instead of the\nperturbation equation for the matter contrast $\\delta$, we provide\nhere its evolution in terms of the growth index\n$ s\\equiv \\,d\\ln{\\delta}\/d\\ln{a}$, a quantity\nmeasured at $z\\sim 0.15$:\nequation}{{\\beta}{equation} \\label{growind}\ns'(a)-\\frac{s(a)^2}{a}+\n\\left[\\frac{2}{a}+\n\\frac{1}{a}E'(a)\\right] s(a)\n-\\frac{1-2Q}{2-3Q}\\cdot\\frac{3\\Omega_m\\,a^{-4}}{n\\,E(a)^2\n\\tilde{R}^{n-1}}\\,=\\,0\\,,\nequation}{\\end{equation}{equation}\nwhere $E(a)\\,=\\,H(a)\/H_0$, $\\tilde{R}$ is the dimensionless\nRicci scalar, and\nequation}{{\\beta}{equation}\\label{Q}\nQ\\,=\\,-\\frac{2f_{RR}\\,c^2\\,k^2}{f_R\\,a^2}\\,.\nequation}{\\end{equation}{equation}\nFor $n=1$, eq.~(\\ref{Q}) gives the\nordinary growth index relation of the Standard Cosmological\nModel. It is clear from eq.~(\\ref{growind}) that the latter\nsuggests a dependence of the growth index on the scale which is\ncontained in the corrective term $Q$ and that this dependence\ncan be safely neglected when\n$Q\\rightarrow0$. In the most general case one can\nresort to the limit $aH\\,<\\,k\\,<\\,10^{-3}h\\,Mpc^{-1}$ in which\neq.~(\\ref{growind}) is a good approximation, and non-linear\neffects on the matter power spectrum can be neglected.\n\n\nBy studying numerically eq.~(\\ref{growind}) one obtains the\n evolution of the growth index in term of the scale factor.\nAssuming, for simplicity, the initial condition\n$ s(a_{ls})\\,=\\,1$ at the last scattering surface as in the case\nof matter domination, the results are summarized in\nFig.~\\ref{fig: grwf}, which displays the evolution of the\ngrowth index in $R^n$ gravity and in the $\\Lambda$CDM model.\nequation}{{\\beta}{figure}\n{\\includegraphics{grwfaR.eps}} \\caption{The evolution of\nthe growth index $f$ in terms of the scale factor. The left\npanel corresponds to modified gravity, in the case\n$\\Omega_m\\,=\\,\\Omega_{bar}\\,\\sim 0.04$, for the SNeIa best fit\nmodel with $n\\,=\\,3.46$. The right panel shows the\nsame evolution in the $\\Lambda$CDM model. In the case\nof $R^n$ gravity it is shown also the dependence on the scale\n$k$. The three cases\n$k\\,=\\,0.01,\\ 0.001$,and $0.0002$ have been eamined, and only\nthe last of these three cases revelas a very small deviation\nfrom the leading behavior.\n\\label{fig: grwf}}\nequation}{\\end{equation}{figure}\n\n\nIn the case of $\\Omega_m\\,=\\,\\Omega_{bar}\\,\\sim 0.04$, one can\nobserve a strong disagreement between the expected rate of the\ngrowth index and the behavior induced by power-law fourth order\ngravity models. This negative result is evident in the\npredicted value of $ s(a_{z\\,=\\,0.15})$, which has been\nobservationally estimated by the analysis of the correlation\nfunction for 220,000 galaxies in the 2dFGRS dataset at the\nsurvey effective depth $z\\,=\\,0.15$. The observational result\nsuggests $ s\\,=\\,0.58\\pm0.11$ \\cite{lahav}, while our model gives\n$ s(a_{z\\,=\\,0.15})\\,\\sim\\,0.117\\ (k\\,=\\,0.01),\\ 0.117\\\n(k\\,=\\,0.001),\\ 0.122\\ (k\\,=\\,0.0002)$. Although this result seems\nfrustrating with respect to the underlying idea of discarding the\ndark components in the cosmological dynamics, it does not give\nsubstantial improvement in the case of $R^n$ gravity model\nplus dark matter. In fact, it is possible to show that, even in\nthis case, the growth index prediction is far from being in\nagreement with the $\\Lambda$CDM model and again, at the\nobservational scale $z\\,=\\,0.15$, there is not enough growth of\nperturbations to match the observed large scale structure. In\nthis case one obtains $ s(a_{z\\,=\\,0.15})\\,\\sim\\,0.29\\\n(k\\,=\\,0.01),\\ 0.29\\\n(k\\,=\\,0.001),\\ 0.31\\ (k\\,=\\,0.0002)$, {\\em i.e.}, values which\nare substantially increased with respect to the previous case\nbut still very far from the experimental estimate. No\nsignificantly different results are obtained if one varies the\npower $n$ (of course, for $n\\rightarrow 1$, one recovers\nthe standard behavior if a cosmological constant is\nadded to the model). These results seem to suggest that an\nextended gravity model incorporating a simple power-law of the\nRicci scalar, although cosmologically relevant at late times, is\nnot a viable description of the cosmic evolution at all\nscales. Such a scheme seems too simple to account for the entire\ncosmological phenomenology. In \\cite{pengjie} a gravity\nLagrangian considering an exponential correction to the Ricci\nscalar $f(R)\\,=\\,R\\,+\\,A\\exp(-B\\,R)$ (with $A,\\ B$ constants)\nproduces more interesting results and exhibits a grow factor rate\nin agreement with the observational results at least in\nthe dark matter case. To corroborate this point of view, one has\nto consider that when $f(R)$ is chosen starting\nfrom observational data in an inverse approach as in\n\\cite{mimicking}, the reconstructed Lagrangian is a\nnon-trivial polynomial in the Ricci scalar. This result\nsuggests that the whole cosmological phenomenology\ncan be accounted only by a suitable non-trivial function of\nthe Ricci scalar rather than a simple power-law. The results\nobtained in the study of the matter power spectra for\nsimple $R^n$ gravity do not invalidate the general approach.\n\n\n\\section{Dark matter as curvature}\n\\label{sei}\nThe results obtained at cosmological scales motivate further\nanalysis of $f(R)$ theories from the phenomenological point of\nview. One wonders whether the curvature fluid which works as dark\nenergy could also play the role of effective dark matter,\nproviding an opportunity to reproduce the observed astrophysical\nphenomenology using only visible matter (see for a discussion\n\\cite{otherreviews}). It is well known that, in the low energy\nlimit, higher order gravity implies a modified gravitational\npotential, which will play a fundamental role in our discussion.\nBy considering a spherical mass distribution with mass $m$ and\nsolving the vacuum field equations for a Schwarzschild-like\nmetric, one obtains the modified gravitational potential of the\ntheory $f(R)=f_0 R^n $ \\cite{mnras}\nequation}{{\\beta}{equation}\n\\Phi(r) = - \\frac{G m}{2r} \\left [ 1 + \\left ( \\frac{r}{r_c}\n\\right )^{\\beta} \\right ] \\;, \\label{eq: pointphi}\nequation}{\\end{equation}{equation}\nwhere\nequation}{{\\beta}{equation}\n\\beta = \\frac{ 12n^2 -7n - 1 - \\sqrt{ 36n^4 + 12n^3 - 83n^2 + 50n\n+ 1 } }{ 6n^2 - 4n + 2 } \\;. \\label{eq: bnfinal}\nequation}{\\end{equation}{equation}\nThis potential corrects the ordinary Newtonian potential with a\npower-law term. The correction becomes\nimportant on scales larger than\n$r_c$ and the value of this threshold constant depends\nessentially on the mass of the system. The corrected\npotential~(\\ref{eq: pointphi}) reduces to the\nstandard Newtonian potential $\\Phi \\propto 1\/r$ for $n=1$, as\nfollows from the inspection of eq.~(\\ref{eq: bnfinal}).\n\nThe result~(\\ref{eq: pointphi}) deserves some comments. As\ndiscussed in detail in~\\cite{mnras}, we have assumed\nspherical symmetry in the the weak-field approximation\nof the field equations, which leads to a corrected Newtonian\npotential due to the strong non-linearity of higher order\ngravity. Note that Birkhoff's theorem does not hold, in general,\nin $f(R)$ gravity, and that spherically symmetric\nsolutions different from the Schwarzschild one exist in\nthese ETGs \\cite{noether}.\n\n\nThe generalization of eq.~(\\ref{eq: pointphi}) to extended\nsources is achieved by dividing the latter into infinitesimal\nmass elements and integrating the potentials generated by\nthese individual elements. An integral over the\nmass density of the system is calculated, taking care of possible\nsymmetries of the mass distribution \\cite{mnras}. Once the\ngravitational potential has been computed, one can evaluate the\nrotation curve $v_c^2(r)$ and compare it with the\nastronomical data. For\nextended systems, one typically must resort to numerical\ntechniques, but the main effect may be illustrated by the rotation\ncurve for the point-like situation, that is,\nequation}{{\\beta}{equation}\nv_c^2(r) = \\frac{G m}{2r} \\left [1 + (1 - \\beta) \\left (\n\\frac{r}{r_c} \\right )^{\\beta} \\right ] \\ . \\label{eq: vcpoint}\nequation}{\\end{equation}{equation}\nIn comparison with the Newtonian result $v_c^2 = G\nm\/r$, the corrected rotation curve is modified by the addition of\nthe second term on the right hand side of eq.~(\\ref{eq:\nvcpoint}). For $0 <\\, \\beta \\,< 1$, the\ncorrected rotation curve is higher than the Newtonian one. Since\nmeasurements of the rotation curves of spiral galaxies signal\ncircular velocities larger than predicted by the observed\nluminous mass and Newtonian potential, the above result\nsuggests the possibility that the modified gravitational\npotential of fourth order gravity may fill the gap between theory\nand observations without the need for additional dark matter.\n\n\nThe corrected rotation curve vanishes asymptotically as\nin the Newtonian case, while it is usually claimed that observed\nrotation curves are flat ({\\em i.e.}, asymptotically constant).\nActually, observations do not probe $v_c$ up to infinite radii\nbut only show a flat rotation curve (within the\nuncertainties) up to the last measured\npoint. The possibility that $v_c$\ngoes to zero at infinity is by no means excluded. In order to\ncheck observationally this result, we have considered a sample\nof low surface brightness (LSB) galaxies with well measured HI\nand H$\\alpha$ rotation\ncurves extending far beyond the visible edge of the system. LSB\ngalaxies are known to be ideal\ncandidates to test dark matter models because of their\nhigh gas content, which allows the rotation curves to be well\nmeasured and\ncorrected for possible systematic errors by comparing 21~cm HI\nline emission with optical H$\\alpha$ and ${\\rm [NII]}$ data.\nMoreover, these galaxies are supposed to be\ndominated by dark matter, so fitting their rotation curves\nwithout this elusive component would support ETGs as\nalternatives to dark matter.\n\nOur sample contains fifteen LSB galaxies with data on the\nrotation curve, the surface mass density of the gas component and\n$R$-band disk photometry extracted from a larger sample\nselected by de Blok \\& Bosma \\cite{BlokBosma}. We assume that\nstars\nare distributed in an infinitely thin and circularly symmetric\ndisk with surface density $\\Sigma(r) = \\Upsilon_\\star\nI_0$exp${(-r\/r_d)}$, where the central surface luminosity $I_0$\nand\nthe disk scale length $r_d$ are obtained from fitting to the\nstellar photometry. The gas surface density has been obtained by\ninterpolating the data over the range probed by HI measurements\nand extrapolated outside this range. When fitting the\ntheoretical rotation curve, there are three quantities to be\ndetermined, namely the stellar mass-to-light ($M\/L$) ratio\n$\\Upsilon_{\\star}$, and the theory parameters $\\left( \\beta,\nr_c \\right) $. It is\nworth stressing that, while fit results for different galaxies\nshould provide the same value of $\\beta$, $r_c$ is related to\none of the integration constants in the field equations. As such,\nthis quantity is not universal and its value must be\ndetermined on a galaxy by galaxy basis. However, it is\nexpected that galaxies with similar\nmass distributions\nhave similar values of $r_c$ so that the scatter in $r_c$ must\nreflect the scatter in the circular velocities. In order\nto match the model with the data, we perform a likelihood analysis\nfor each galaxy, using as fitting parameters $\\beta$,\n$\\log{r_c}$ (with $r_c$ in kpc) and the gas mass\nfraction\\footnote{This is related to the $M\/L$ ratio\nby $\\Upsilon_{\\star} = [(1 - f_g) M_{g}]\/(f_g L_d)$, where w\n$M_g = 1.4 M_{HI}$ is the gas (HI + He) mass, and $M_d =\n\\Upsilon_{\\star} L_d$ and\n$L_d = 2 \\pi I_0 r_d^2$ are the disk total mass and luminosity,\nrespectively.}\n$f_g$. As it is evident from the results of the different\nfits, the experimental data are successfully fitted by the model\n\\cite{mnras}. In particular, from a purely phenomenological\npoint of view and leaving aside for the moment other\nviability criteria, from the best fit range $\\beta=0.80\\pm\n0.08$, one can conclude that $R^n$ gravity with $2.3 < n <5.3$\n(best fit value\n$n=3.2$ which overlaps well the above-mentioned range of $n$\nfitting SNeIa Hubble diagram) can be a good candidate for solving\nthe missing matter problem in LSB galaxies without dark matter.\nequation}{{\\beta}{figure}\n\\centering\\resizebox{7.5cm}{!}{\\includegraphics{ngc4455.eps}}\n\\centering\\resizebox{7.5cm}{!}{\\includegraphics{ngc5023.eps}}\n\\caption{Best-fit theoretical rotation curve superimposed to the\ndata for the LSB galaxy NGC 4455 (left) and NGC 5023 (right). To\nbetter show the effect of the correction to the Newtonian\ngravitational potential, we report the total rotation curve\n$v_c(r)$ (solid line), the Newtonian one (short dashed) and the\ncorrection term (long dashed).\\label{fig: lsb1}}\nequation}{\\end{equation}{figure}\n\nAt this point, one wonders whether a link may be found\nbetween $R^n$ gravity and the standard approach based on dark\nmatter haloes since both theories fit equally well the same data.\nAs a matter of fact, it is possible to define an {\\it effective\ndark matter halo} by imposing that its rotation curve equals the\ncorrection term to the Newtonian curve induced by $R^n$ gravity.\nMathematically, one can split the total rotation curve derived\nfrom $R^n$ gravity as $v_c^2(r) = v_{c, N}^2(r) + v_{c,\ncorr}^2(r)$, where the second term is the correction.\nConsidering, for simplicity a spherical halo embedding a thin\nexponential disk, we may also write the total rotation curve as\n$v_c^2(r) = v_{c,\ndisk}^2(r) + v_{c, DM}^2(r)$ with $v_{c, disk}^2(r)$ the Newtonian\ndisk rotation curve and $v_{c, DM}^2(r) = G M_{DM}(r)\/r$ the dark\nmatter one, $M_{DM}(r)$ being its mass distribution. Equating the\ntwo expressions yields\nequation}{{\\beta}{equation}\nM_{DM}(\\eta) =\nM_{vir}\\left(\\frac{\\eta}{\\eta_{vir}}\n\\right)\\frac{2^{\\beta-5}\\eta^{-\\beta}_c(1-\\beta)\n\\eta^{\\frac{\\beta-5}{2}}{\\cal I}_0(\\eta)-{\\cal V}_d(\\eta)}\n{2^{\\beta-5}\\eta^{-\\beta}_c(1-\\beta)\\eta^{\\frac{\\beta-5}{2}}{\\cal\nI}_0(\\eta_{vir})-{\\cal V}_d(\\eta_{vir})} \\label{eq: mdm}\nequation}{\\end{equation}{equation}\nwith $\\eta = r\/r_d$ and $\\Sigma_0 = \\Upsilon_{\\star} i_0$,\n${\\cal\nV}_d(\\eta)\\,=\\,I_0(\\eta\/2)K_0(\\eta\/2)\\times\nI_1(\\eta\/2)K_1(\\eta\/2)$.\\footnote{Here $I_l$ and $K_l$, with\n$l\\,=\\,1,2$ are the Bessel functions of first and second type,\nrespectively.}\nMoreover,\nequation}{{\\beta}{equation} {\\cal{I}}_0(\\eta, \\beta) =\n\\int_{0}^{\\infty}{{\\cal{F}}_0(\\eta, \\eta', \\beta) k^{3 - \\beta}\n\\eta'^{\\frac{\\beta - 1}{2}} {\\rm e}^{- \\eta'} d\\eta'}\n\\;, \\label{eq:\ndeficorr}\nequation}{\\end{equation}{equation}\nwhere ${\\cal{F}}_0$ depends only on the geometry of the\nsystem and the subscript ``$vir$\" indicates virial quantities.\nEq.~(\\ref{eq: mdm}) defines the mass profile of an effective\nspherically symmetric dark matter halo whose ordinary rotation\ncurve provides the part of the corrected disk rotation curve\nresulting from the addition of the curvature correction to\nthe gravitational potential. Clearly, from a phenomenological\npoint of view there is no way to\ndistinguish this dark halo model from $R^n$ gravity.\n\n\nHaving assumed spherical symmetry for the mass distribution, it is\nstraightforward to compute the mass density for the effective\ndark halo\nas $\\rho_{DM}(r) = (1\/4 \\pi r^2) dM_{DM}\/dr$. The most interesting\nfeature of the density profile is its asymptotic behavior\nquantified by the logarithmic slope $\\alpha_{DM} =\nd\\ln{\\rho_{DM}}\/d\\ln{r}$, which can be computed only numerically\nas\na function of $\\eta$ for fixed values of $\\beta$ (or $n$). As\nexpected, $\\alpha_{DM}$ depends explicitly on $\\beta$, while\n$(r_c, \\Sigma_0, r_d)$ enter indirectly through $\\eta_{vir}$. The\nasymptotic values at the centre and at infinity ($\\alpha_0$ and\n$\\alpha_{\\infty}$, respectively) are of particular\ninterest. $\\alpha_0$ almost vanishes and, in the\ninnermost regions, the density is approximately constant. Indeed,\n$\\alpha_0 = 0$ is the value corresponding to models with\nan isothermal sphere as the inner core. It is well\nknown that galactic rotation curves are typically best-fitted by\ncored dark halo models. Moreover, the outer asymptotic\nslope lies between $-3$ and $-2$, values typical of most dark\nhalo models in the literature. In\nparticular, for $\\beta = 0.80$ one finds $(\\alpha_0,\n\\alpha_{\\infty}) = (-0.002, -2.41)$, values which are quite\nsimilar to those in the Burkert model, $(0, -3)$. This empirical\nmodel has been proposed to fit the LSB and dwarf galaxies\nrotation curves. The values of $(\\alpha_0, \\alpha_{\\infty})$\nfound for the best-fit effective dark halo therefore suggest a\npossible theoretical motivation for Burkert-like models.\nBy construction, the properties\nof the effective dark matter halo are closely related to the disk\nproperties, hence some correlation between the\ndark halo\nand the disk parameters is expected. In this regard, exploiting\nthe relation between the virial mass and the disk parameters,\none obtains the relation for the Newtonian virial velocity\n$V_{vir} = G M_{vir}\/r_{vir}$\nequation}{{\\beta}{equation}\nM_d \\propto \\frac{(3\/4 \\pi \\delta_{th} \\Omega_m\n\\rho_{crit})^{\\frac{1 - \\beta}{4}} r_d^{\\frac{1 + \\beta}{2}}\n\\eta_c^{\\beta}}{2^{\\beta - 6}\n (1 - \\beta) G^{\\frac{5 - \\beta}{4}}} \\frac{V_{vir}^{\\frac{5 -\n\\beta}{2}}}{{\\cal{I}}_0(V_{vir}, \\beta)} \\label{eq: btfvir} \\ .\nequation}{\\end{equation}{equation}\nWe have checked numerically that eq.~(\\ref{eq: btfvir}) may be\nwell approximated by $M_d \\propto V_{vir}^{a}$. This relation has\nthe same formal structure of the baryonic Tully-Fisher (BTF)\nrelation $M_b \\propto V_{flat}^a$ where $M_b$ is the total (gas\nplus stars) baryonic mass and $V_{flat}$ is the circular\nvelocity on the flat part of the observed rotation curve. In\norder to test whether the BTF can be explained by the\neffective dark matter halo proposed, we should look for a\nrelation between $V_{vir}$ and $V_{flat}$. Such a relation\ncannot be derived\nanalytically because the estimate of $V_{flat}$ depends on\nthe peculiarities of the observed rotation curve, such as how\nfar\nit extends, and the uncertainties\non the outermost points. For given values of the disk parameters,\nwe simulated theoretical rotation curves for some values\nof $r_c$ and measured $V_{flat}$ finally choosing the fiducial\nvalue for $r_c$ that gives a value of $V_{flat}$ as close as\npossible to the measured one. Inserting the relation thus found\nbetween $V_{flat}$ and $V_{vir}$ into eq.~(\\ref{eq: btfvir}) and\naveraging over different simulations, we finally obtain\nequation}{{\\beta}{equation}\n\\log{M_b} = (2.88 \\pm 0.04) \\log{V_{flat}} + (4.14 \\pm 0.09) \\;,\n\\label{eq: btfour}\nequation}{\\end{equation}{equation}\nwhile a direct fit to the observed data gives \\cite{ssm}\nequation}{{\\beta}{equation}\n\\log{M_b} = (2.98 \\pm 0.29) \\log{V_{flat}} + (3.37 \\pm 0.13) \\ .\n\\label{eq: btfssm}\nequation}{\\end{equation}{equation}\nThe slope of the predicted and observed BTF are in good\nagreement, lending further support to our approach. The\nzero point\nis markedly different from the predicted one, being significantly\nlarger than the observed one. However, both relations fit the\ndata with a similar scatter. A discrepancy in\nthe zero point can be due to our approximate treatment of the\neffective halo which does not take into account the gas component.\nNeglecting this term, we should increase the effective halo mass\nand hence $V_{vir}$ which affects the relation with $V_{flat}$\nleading to a higher than observed zero point. Indeed, the larger\n$M_g\/M_d$, the more the points deviate from our predicted BTF thus\nconfirming our hypothesis. Given this caveat, we can conclude\nwith some confidence that $R^n$ gravity offers a theoretical\nfoundation even for the empirically found BTF relation.\n\nAlthough the results outlined here pertain to the\nsimplistic choice $f(R)\\,=\\,f_0R^n$ of fourth order gravity,\nthey are nevertheless interesting. The incompatibility\nof this model with the correct matter power spectrum\nsuggests that a more complicated\nLagrangian is needed to reproduce the entire dark sector\nphenomenology at all scales, but it has been shown that\nETGs allow one to approach important\nissues in cosmological and astrophysical phenomenology. We have\nseen that ETGs can reproduce the SNeIa Hubble\ndiagram without dark matter and predict the age of the universe.\nThe modification of the gravitational potential\narising in higher order gravity could constitute a\nfundamental ingredient in interpreting the flatness of\nthe rotation curves of LSB galaxies.\nFurthermore, if one considers the model parameters selected by\nthe fit of the observational data of LSB rotation curves, it is\npossible to construct a phenomenological analog of the dark\nmatter halo with shape similar to that of the Burkert\nmodel. Since the latter has been empirically introduced to\naccount for the dark matter distribution in LSB\nand dwarf galaxies, this result provides a theoretical\nmotivation of the Burkert model.\n\n\nBy investigating the relation between dark halo and disk\nparameters, a relation has been deduced between $M_d$ and\n$V_{flat}$, which reproduces the baryonic Tully-Fisher law.\nExploiting the relation between the virial mass and the disk\nparameters, one can obtain a relation for the virial velocity\nwhich can be satisfactorily approximated as $M_d \\propto\nV_{vir}^{a}$. This result is also intriguing because it\nprovides a theoretical interpretation of another phenomenological\nrelation. Although not definitive, these\nphenomenological aspects of $f(R)$ point to a potentially\ninteresting avenue of research and support the quest for\na unified view of the dark side of the universe.\n\n\n\\section{Massive scalar modes of $f(R)$ gravitational waves}\n\\label{sette}\nAs we have seen, a pragmatic point of view could be to\n``reconstruct'' the suitable theory of gravity starting from data.\nThe main issues of this ``inverse '' approach is matching\nconsistently observations at different scales and taking into\naccount wide classes of gravitational theories where ``ad hoc''\nhypotheses are avoided. In principle, as discussed in the previous\nsection, the most popular dark energy cosmological models can be\nachieved by considering $f(R)$ gravity without considering unknown\ningredients. The main issue to achieve such a goal is to have at\ndisposal suitable datasets at every redshift. In particular, this\nphilosophy can be taken into account also for the cosmological\nstochastic background of gravitational waves (GW) which, together\nwith CMBR, would carry, if detected, a huge amount of information\non the early stages of the Universe evolution.\nIn this section we discuss the cosmological background of\ngravitational waves (GWs) in generic $f(R)$ theories. The\nachievement of detecting massive modes or selecting\n$f(R)$-signatures in the stochastic background could be the final\nway to retain or rule out such theories with respect to GR. GWs\nare perturbations $h_{\\mu\\nu}$ of the metric which transform as\n3-tensors. In GR, the equations ruling the propagation of GWs in\nthe transverse-traceless gauge are\nequation}{{\\beta}{equation}\n\\square h_{i}^{j}=0\\label{eq: 1}\\,,\nequation}{\\end{equation}{equation}\nwhere Latin indexes run from~1 to~3. We want to derive\nthe analog of eq.~(\\ref{eq: 1}) for a generic $f(R)$ theory\ndescribed by the action~(\\ref{actionmetric}). The linearized\ntheory in vacuo ($\\mathcal{S}^{(m)}=0$) is considered below, so\nthat\nequation}{{\\beta}{equation}\n\\mathcal{S}=\\frac{1}{2k}\\int d^{4}x\\sqrt{-g}f(R)\\label{eq:2}\\,.\nequation}{\\end{equation}{equation}\nUsing a conformal transformation, the scalar degree of\n freedom $f'(R)$ of metric $f(R)$ gravity appears as the\nconformal factor in\nequation}{{\\beta}{equation}\n\\widetilde{g}_{\\mu\\nu}=e^{2\\Phi}g_{\\mu\\nu} \\, \\qquad\n\\qquad e^{2\\Phi}=f'(R)\\,.\\label{eq:3}\nequation}{\\end{equation}{equation}\nThe conformally equivalent Einstein-Hilbert action is\nequation}{{\\beta}{equation}\n\\mathcal{\\widetilde{S}}=\n\\frac{1}{2k}\\int d^{4}x \\sqrt{-\\tilde{g}} \\, \\left[\\widetilde{R}+\n\\mathcal{L}\\left(\\Phi\\mbox{,}\n\\Phi_{\\mbox{;}\\mu}\\right)\\right] \\;, \\label{eq:4}\nequation}{\\end{equation}{equation}\nwhere $ \\mathcal{L} \\left(\\Phi\\mbox{,} \\Phi_{\\mbox{;}\n\\mu}\\right)$\nis the scalar field Lagrangian obtained using the\nrelation\nequation}{{\\beta}{equation}\n\\widetilde{R}=\ne^{-2\\Phi}\\left(R-6\\square\\Phi-\n6\\Phi_{;\\delta}\\Phi^{;\\delta}\\right)\\label{eq:6}\nequation}{\\end{equation}{equation}\nbetween the Ricci curvatures of the conformally related metrics\n$g_{\\mu\\nu}$ and $ \\tilde{g}_{\\mu\\nu}$.\nThe equation for the gravitational waves is now\nequation}{{\\beta}{equation}\n\\widetilde{\\square}\\tilde{h}_{i}^{j}=0 \\;, \\label{eq:7}\nequation}{\\end{equation}{equation}\nwhere\nequation}{{\\beta}{equation}\n\\widetilde{\\square}=e^{-2\\Phi}\\left(\\square+\n2\\Phi^{;\\lambda}\\partial_{;\\lambda}\\right)\n\\,.\\label{eq:9}\nequation}{\\end{equation}{equation}\nSince scalar and tensor modes are decoupled, we have\nequation}{{\\beta}{equation}\n\\widetilde{h}_{i}^{j}=\\widetilde{g}^{lj}\\delta\\widetilde{g}_{il}=e^{-2\\Phi}g^{lj}e^{2\\Phi}\\delta\ng_{il}=h_{i}^{j} \\;, \\label{eq:8}\nequation}{\\end{equation}{equation}\nwhich means that $h_{i}^{j}$ is conformally invariant. As a\nconsequence, the plane wave amplitudes\n$h_{i}^{j}(t)=h(t)e_{i}^{j}\\exp(ik_{m}x^{m}),$ where $e_{i}^{j}$\nis the polarization tensor, are the same in both metrics, a fact\nthat is important in the following.\n\n\nIn a FLRW background, eq.~(\\ref{eq:7}) becomes\nequation}{{\\beta}{equation}\n\\ddot{h}+\n\\left(3H+2\\dot{\\Phi}\\right) \\dot{h}+k^{2}a^{-2}h=0\\label{eq:10}\nequation}{\\end{equation}{equation}\nwhere $k$ is the wave number and $h$ is the amplitude.\nThe solutions of this equation are linear combinations of\nBessel functions. Several primordial mechanisms generating GWs\nare possible. In principle, one could seek for contributions\ndue to all known high-energy processes in the early phases of\nthe cosmic history.\n\nHere we consider the background of GWs generated during\ninflation, which is strictly related to the dynamics of the\ncosmological model. In particular, one can assume that the main\ncontribution to this background comes from the\namplification of vacuum fluctuations at the transition between the\ninflationary phase and the radiation era. However, we can assume\nthat the GWs generated as zero-point fluctuations during\ninflation undergo adiabatically damped $(\\sim 1\/a)$ oscillations\nuntil they reach the Hubble radius $H^{-1}$. This is the particle\nhorizon for the growth of perturbations. Any previous\nfluctuation is smoothed away by the inflationary expansion. The\nGWs freeze out for $a\/k\\gg H^{-1}$ and re-enter the horizon\nafter reheating. The re-entry in the Friedmann era\ndepends on the spatial scale of the GWs. After re-entry, GWs\nare in principle detectable due the Sachs-Wolfe effect that\nthey induce on the CMB temperature anisotropy\n$\\bigtriangleup T\/T$ at decoupling. If $\\Phi$ is the\ninflaton field, then $\\dot{\\Phi}\\ll H$ during inflation.\nBy using the conformal time $\\eta$ defined by $d\\eta=dt\/a$,\neq.~(\\ref{eq:10}) becomes\nequation}{{\\beta}{equation}\nh''+2\\frac{\\chi'}{\\chi}h'+k^{2}h=0 \\;, \\label{eq:16}\nequation}{\\end{equation}{equation}\nwhere $\\chi=ae^{\\Phi}$ and a prime now denotes differentiation\nwith respect to $\\eta$. Inside the radius $H^{-1}$, it is\n$k\\eta\\gg 1$. Since there are no gravitons in the\ninitial vacuum state, only negative-frequency modes appear and\nthe solution of eq.~(\\ref{eq:16}) is\nequation}{{\\beta}{equation}\nh=k^{1\/2}\\sqrt{2\/\\pi}\\frac{1}{aH}C\\exp(-ik\\eta)\\,,\\label{eq:18}\nequation}{\\end{equation}{equation}\nwhere $C$ is the amplitude parameter. At the first horizon\ncrossing $aH=k$, the averaged amplitude\n$A_{h}=(k\/2\\pi)^{3\/2}\\left|h\\right|$ of the perturbation is\nequation}{{\\beta}{equation}\nA_{h}=\\frac{C}{2\\pi^{2}} \\,.\\label{eq:19}\nequation}{\\end{equation}{equation}\nWhen the scale $a\/k$ becomes larger than the Hubble radius\n$H^{-1}$, the growing mode freezes. It can be shown that\nthe upper limit $\\bigtriangleup T\/T \\lesssim A_{h} $ is valid\nsince other effects\ncan contribute to the background anisotropy. From this\nconsideration, it is clear that the only relevant quantity is the\ninitial amplitude $C$ in eq.~(\\ref{eq:18}), which is conserved\nuntil re-entry. This amplitude depends on the fundamental\nmechanism that generates the perturbations. Inflation\nproduces perturbations as zero-point energy\nfluctuations, a mechanism which depends on the gravitational\ninteraction and $(\\bigtriangleup T\/T)$ further constrains\nthe theory of gravity.\nConsidering a single graviton\nin the form of a monochromatic wave, its zero-point amplitude is\nobtained from the canonical commutation relation\nequation}{{\\beta}{equation}\n\\left[h(t,x),\\,\\pi_{h}(t,y)\\right]=i\\delta^{3}(x-y) \\label{eq:20}\nequation}{\\end{equation}{equation}\nat fixed time $t$, where the amplitude $h$ is the field and\n$\\pi_{h}$ is the conjugate momentum operator. The Lagrangian for\nthe $h$-quantity is\nequation}{{\\beta}{equation}\n\\widetilde{\\mathcal{L}}=\n\\frac{1}{2}\\sqrt{-\\widetilde{g}} \\, \\widetilde{g}^{\\mu\\nu}\nh_{;\\mu}h{}_{;\\nu} \\label{eq:21}\nequation}{\\end{equation}{equation}\nin the conformally rescaled FLRW metric $\\widetilde{g}_{\\mu\\nu}$,\nwhere the amplitude $h$ is conformally invariant. This\nLagrangian leads to\nequation}{{\\beta}{equation}\n\\pi_{h}=\\frac{\\partial\\widetilde{\\mathcal{L}}}{\n\\partial\\dot{h}}=e^{2\\Phi}a^{3}\\dot{h}\\label{eq:22}\nequation}{\\end{equation}{equation}\nand eq.~(\\ref{eq:20}) becomes\nequation}{{\\beta}{equation}\n\\left[h(t,x),\\,\\dot{h}(y,y)\\right]\n=i\\frac{\\delta^{3}(x-y)}{a^{3}e^{2\\Phi}} \\;.\\label{eq:23}\nequation}{\\end{equation}{equation}\nThe fields $h$ and $\\dot{h}$ can be expanded in terms of\ncreation and annihilation operators. The commutation relations in\nconformal time are\nequation}{{\\beta}{equation}\n\\left[hh'^{*}-h^{*}h'\\right]=\\frac{i(2\\pi)^{3}}{a^{3}e^{2\\Phi}}\n\\,.\\label{eq:26}\nequation}{\\end{equation}{equation}\nEqs.~(\\ref{eq:18}) and~(\\ref{eq:19}) yield\n$ C=\\sqrt{2}\\pi^{2}He^{-\\Phi}$, where $H$ and $\\Phi$ are\ncalculated at the first horizon crossing and, using\n$e^{2\\Phi}=f'(R)$, the relation\nequation}{{\\beta}{equation}\nA_{h}=\\frac{H}{\\sqrt{2f'(R)}} \\label{eq:27}\nequation}{\\end{equation}{equation}\nis found to hold for a generic $f(R)$ theory. This result\ndeserves some discussion. Clearly, the GW amplitude produced\nduring inflation depends on the theory of gravity\nwhich, if different from GR, contains extra degrees of freedom\nwhich could be probed by the Sachs-Wolfe effect. This effect\ncould be combined with other constraints on the GW background\nif ETGs are probed independently at other scales\n\\cite{tuning, SFM}.\n\nequation}{{\\beta}{figure}\nequation}{{\\beta}{tabular}{|c|c|}\n\\hline\n\\includegraphics[scale=0.7]{plot1}&\n\\includegraphics[scale=0.7]{plot2}\\tabularnewline\n\\hline\n\\includegraphics[scale=0.7]{plot5}&\n\\includegraphics[scale=0.7]{plot6}\\tabularnewline\n\\hline\n\\includegraphics[scale=0.7]{plot7}&\n\\includegraphics[scale=0.7]{plot9}\\tabularnewline\n\\hline\nequation}{\\end{equation}{tabular}\n\\caption {The evolution of the GW amplitude for a few power-law\nchoices of the scale factor $a(t)\\sim t^s$, the scalar\nfield $\\phi\\sim t^m$, and the function $f(R)\\sim R^n$.\nThe horizontal (time) and vertical (amplitude) scales depend on\nthe cosmological background providing a signature of the\nmodel.}\n\\label{fig:1}\nequation}{\\end{equation}{figure}\n\nWe are by now familiar with the trace of the field equations\nequation}{{\\beta}{equation}\n3\\square f'(R)+Rf'(R)-2f(R)=0,\\label{eq: KG}\nequation}{\\end{equation}{equation}\nand, using the identifications \\cite{SCF}\nequation}{{\\beta}{equation}\nequation}{{\\beta}{array}{ccccc}\n\\Phi\\rightarrow f'(R) & &\n\\textrm{and } & & \\frac{dV}{d\\Phi}\n\\rightarrow\\frac{2f(R)-Rf'(R)}{3}\nequation}{\\end{equation}{array}\\label{eq: identifica}\nequation}{\\end{equation}{equation}\nthe Klein-Gordon equation for the effective scalar field $\\Phi$\nequation}{{\\beta}{equation}\n\\square\\Phi=\\frac{dV}{d\\Phi} \\label{eq: KG2}\nequation}{\\end{equation}{equation}\nfollows. Linearizing around a constant curvature\nbackground corresponding to $\\Phi=\\Phi_{0}$,\nassuming that $V$ has a minimum at $\\Phi_{0}$ \\cite{SCF},\nand expanding as in\nequation}{{\\beta}{equation}\nV\\simeq\\frac{1}{2}\\alpha\\delta\n\\Phi^{2} \\;, \\;\\;\\;\\;\\;\\; \\frac{dV}{d\\Phi}\\simeq m^{2}\\delta\\Phi\n];,\n\\label{eq: minimo}\nequation}{\\end{equation}{equation}\nwhere the constant $m$ has the dimensions of a mass, yields\nequation}{{\\beta}{equation}\nequation}{{\\beta}{array}{c}\ng_{\\mu\\nu}=\\eta_{\\mu\\nu}+h_{\\mu\\nu} \\;, \\\\\n\\nonumber \\\\\n\\Phi=\\Phi_{0}+\\delta\\Phi \\;,\nequation}{\\end{equation}{array}\\label{eq:\nlinearizza}\nequation}{\\end{equation}{equation}\nto first order in $h_{\\mu\\nu}$ and $\\delta\\Phi$. If\n$\\widetilde{R}_{\\mu\\nu\\rho\\sigma}$,\n$\\widetilde{R}_{\\mu\\nu}$, and $\\widetilde{R}$ are the linearized\nquantities corresponding to $R_{\\mu\\nu\\rho\\sigma}$,\n$R_{\\mu\\nu}$, and $R$, then the linearized field equations are\nequation}{{\\beta}{equation}\nequation}{{\\beta}{array}{c}\n\\widetilde{R}_{\\mu\\nu}-\n\\frac{1}{2} \\eta_{\\mu\\nu} \\widetilde{R} =\n\\partial_{\\mu}\\partial_{\\nu}h_{f} -\\eta_{\\mu\\nu}\\square h_{f}\n\\;, \\\\\n\\\\{}\\square h_{f}=m^{2}h_{f} \\;,\nequation}{\\end{equation}{array}\n\\label{eq:linearizzate1}\nequation}{\\end{equation}{equation}\nwhere\nequation}{{\\beta}{equation}\nh_{f}\\equiv\\frac{\\delta\\Phi}{\\Phi_{0}} \\;.\\label{eq:\ndefinizione}\nequation}{\\end{equation}{equation}\nThe curvature tensor $\\widetilde{R}_{\\mu\\nu\\rho\\sigma}$ and\neqs.~(\\ref{eq:linearizzate1})\nare left invariant under the gauge transformations\nequation}{{\\beta}{equation}\nequation}{{\\beta}{array}{c}\nh_{\\mu\\nu}\\rightarrow h'_{\\mu\\nu}=h_{\\mu\\nu}\n-\\partial_{(\\mu}\\epsilon_{\\nu)} \\;, \\\\\n\\\\\\delta\\Phi\\rightarrow\\delta\\Phi'\n=\\delta\\Phi \\;.equation}{\\end{equation}{array}\\label{eq: gauge}\nequation}{\\end{equation}{equation}\nBy introducing\nequation}{{\\beta}{equation}\n\\bar{h}_{\\mu\\nu}\\equiv\nh_{\\mu\\nu}-\\frac{h}{2}\\eta_{\\mu\\nu}+ \\eta_{\\mu\\nu}h_{f}\\label{eq:\nridefiniz}\nequation}{\\end{equation}{equation}\nand considering the gauge vector\n$\\epsilon^{\\mu}$ given by\nequation}{{\\beta}{equation}\n\\square\\epsilon_{\\nu}=\\partial^{\\mu}\n\\bar{h}_{\\mu\\nu} \\;,\\label{eq:lorentziana}\nequation}{\\end{equation}{equation}\nthe Lorenz gauge\nequation}{{\\beta}{equation}\n\\partial^{\\mu}\\bar{h}_{\\mu\\nu}=0 \\label{eq:\ncondlorentz}equation}{\\end{equation}{equation}\ncan be chosen. In this gauge the field equations assume the form\nequation}{{\\beta}{equation}\n\\square\\bar{h}_{\\mu\\nu}=0 \\;, \\label{eq: ondaT}\nequation}{\\end{equation}{equation}\nequation}{{\\beta}{equation}\n\\square h_{f}=m^{2}h_{f} \\;.\\label{eq: ondaS}equation}{\\end{equation}{equation}\nThe solutions of eqs.~(\\ref{eq: ondaT}) and~(\\ref{eq: ondaS}) are\nthe plane waves\nequation}{{\\beta}{equation}\n\\bar{h}_{\\mu\\nu}=\nA_{\\mu\\nu}(\\overrightarrow{p})\n\\exp(ip^{\\alpha}x_{\\alpha})+ \\mbox{c.c.} \\;, \\label{eq:\nsolT}\nequation}{\\end{equation}{equation}\nequation}{{\\beta}{equation}\nh_{f}=a(\\overrightarrow{p})\\exp(iq^{\\alpha}x_{\\alpha})\n+\\mbox{c.c.} \\;,\n\\label{eq: solS}equation}{\\end{equation}{equation}\nwith\nequation}{{\\beta}{equation}\nequation}{{\\beta}{array}{ccc}\nk^{\\alpha}\\equiv(\\omega,\\overrightarrow{p})\n & & \\omega=p\\equiv|\\overrightarrow{p}|\\\\\n\\\\q^{\\alpha}\\equiv(\\omega_{m},\\overrightarrow{p})\n & & \\omega_{m}=\\sqrt{m^{2}+p^{2}}\\;.\nequation}{\\end{equation}{array}\\label{eq:\nkeq}equation}{\\end{equation}{equation}\nEqs.~(\\ref{eq: ondaT}) and~(\\ref{eq: solT}) are the\nwave equation for standard GR and its\ngravitational wave solutions, respectively, whereas\neqs.~(\\ref{eq: ondaS}) and~(\\ref{eq: solS})\nare the wave equation and its solution for the massive\nscalar mode of $f(R)$ gravity (cf. \\cite{SCF,BCDF}). The\ndispersion relation for the modes of the\nmassive\nfield $h_{f}$ is non-linear. ``Ordinary'' ({\\em i.e.}, GR)\ntensor modes $\\bar{h}_{\\mu\\nu}$ propagate at the speed of light\n$c$, but the dispersion law (the second of eqs.~(\\ref{eq: keq}))\nfor the scalar modes $h_{f}$ is that of a massive field\nwave packet \\cite{SCF,BCDF}. The group velocity of a\nwave packet of $h_{f}$ centered on $\\overrightarrow{p}$ is\nequation}{{\\beta}{equation}\n\\overrightarrow{v_{G}}=\n\\frac{\\overrightarrow{p}}{\\omega}\\;,\n\\label{eq: velocita' di\ngruppo}equation}{\\end{equation}{equation}\nwhich is the velocity of a massive particle with mass $m$\nand momentum $\\overrightarrow{p}$. The second of eqs.~(\\ref{eq:\nkeq}) in conjunction with eq.~(\\ref{eq: velocita' di gruppo})\nyields\nequation}{{\\beta}{equation}\nv_{G}=\\frac{\\sqrt{\\omega^{2}-m^{2}}}{\\omega} \\label{eq:\nvelocita' di gruppo 2}\nequation}{\\end{equation}{equation}\nand a wave packet propagates at constant speed if\nequation}{{\\beta}{equation}\nm=\\sqrt{(1-v_{G}^{2})} \\, \\omega \\;.\n\\label{eq: relazione\nmassa-frequenza}\nequation}{\\end{equation}{equation}\nThe Lorenz gauge is preserved by gauge trasformations with\n$\\square\\epsilon_{\\nu}=0$; this gauge imposes the\ntransversality condition $k^{\\mu}A_{\\mu\\nu}=0$ for the tensor\nmodes, but not for the field $h_{\\mu\\nu}$ which contains a\nscalar mode, as seen from eq.~(\\ref{eq: ridefiniz}), or\nequation}{{\\beta}{equation}\nh_{\\mu\\nu}=\\bar{h}_{\\mu\\nu}-\n\\frac{\\bar{h}}{2}\\eta_{\\mu\\nu}+ \\eta_{\\mu\\nu}h_{f}.\\label{eq:\nridefiniz 2}\nequation}{\\end{equation}{equation}\nWere the scalar mode massless, one could impose that\nequation}{{\\beta}{equation}\nequation}{{\\beta}{array}{c}\n\\square\\epsilon^{\\mu}=0 \\;, \\\\\n\\\\\\partial_{\\mu}\\epsilon^{\\mu}=\n-\\frac{\\bar{h}}{2}+h_{f} \\;,\nequation}{\\end{equation}{array}\\label{eq: gauge2}equation}{\\end{equation}{equation}\nthus obtaining a transversal ``total'' field. However, as is\nclear from the previous sections, we are dealing with a massive\nscalar mode and transversality is impossible. By applying\nd'Alembert's operator to the second of eqs.~(\\ref{eq:\ngauge2}) and using eqs.~(\\ref{eq: ondaT}) and~(\\ref{eq: ondaS}),\nit follows that\nequation}{{\\beta}{equation}\n\\square\\epsilon^{\\mu}=m^{2}h_{f} \\;,\\label{eq:\ncontrasto}equation}{\\end{equation}{equation}\nin contrast with the first of eqs.~(\\ref{eq: gauge2}).\nSimilarly, it is shown that a linear\nrelation between the tensorial modes $\\bar{h}_{\\mu\\nu}$ and the\nmassive scalar $h_{f}$ cannot exist. Thus, a gauge in wich\n$h_{\\mu\\nu}$ is purely spatial cannot be chosen, {\\em i.e.}, it\nis impossible to impose $h_{\\mu0}=0,$ see eq.~(\\ref{eq:\nridefiniz 2}). However, the traceless gauge condition can\nbe imposed on $\\bar{h}_{\\mu\\nu}$,\nequation}{{\\beta}{equation}\nequation}{{\\beta}{array}{c}\n\\square\\epsilon^{\\mu}=0 \\;,\\\\\n\\\\\\partial_{\\mu}\\epsilon^{\\mu}=\n-\\frac{\\bar{h}}{2} \\;,equation}{\\end{equation}{array}\n\\label{eq: gauge\ntraceless}\nequation}{\\end{equation}{equation}\nimplying that\nequation}{{\\beta}{equation}\n\\partial^{\\mu}\\bar{h}_{\\mu\\nu}=0 \\;.\\label{eq:\nvincolo}equation}{\\end{equation}{equation}\nThe gauge transformations\nequation}{{\\beta}{equation}\nequation}{{\\beta}{array}{c}\n\\square\\epsilon^{\\mu}=0 \\;, \\\\\n\\\\\\partial_{\\mu}\\epsilon^{\\mu}=0 \\;, equation}{\\end{equation}{array}\\label{eq: gauge\n3}equation}{\\end{equation}{equation}\npreserve the gauge $\\partial_{\\mu}\\bar{h}^{\\mu\\nu}=0$,\n$\\bar{h}=0$. By choosing $\\overrightarrow{p}$ along the\n$z$-direction, a gauge can be chosen in which only $A_{11}$,\n$A_{22}$, and $A_{12}=A_{21}$ are different from zero, with\nthe condition $\\bar{h}=0$ providing $A_{11}=-A_{22}$. The\nsubstitution of these\nequations into eq.~(\\ref{eq: ridefiniz 2}) then yields\nequation}{{\\beta}{equation}\nh_{\\mu\\nu}(t,z)=A^{+}(t-z)e_{\\mu\\nu}^{(+)}\n+A^{\\times}(t-z)e_{\\mu\\nu}^{(\\times)}+h_{f}(t-v_{G}z)\\eta_{\\mu\\nu}\n\\;.\\label{eq: perturbazionetotale}equation}{\\end{equation}{equation}\nThe term $A^{+}(t-z)e_{\\mu\\nu}^{(+)}+A^{\\times}(t-z)e_{\\mu\\nu}^{\n(\\times)}$ describes the two standard polarizations of\ntensor gravitational waves familiar from GR, while the term\n$h_{f}(t-v_{G}z)\\eta_{\\mu\\nu}$ is the massive scalar field\ncharacteristic of $f(R)$ gravity. As expected, the scalar $f'(R$)\ngenerates a third massive polarization for gravitational\nwaves which is absent in GR.\n\n\\section{Conclusions}\n\\label{otto}\n\nLet us emphasize once more that we regard $f(R) $ gravity\ntheories not as definitive theories, but rather as toy\nmodels and proofs of principle that modifying gravity at\nlarge scales can explain the\nobserved acceleration of the universe without the need to\nadvocate exotic dark energy. This hope has stimulated a very\nintense activity among theoreticians (\\cite{review} and\nreferences therein).\n\nTo summarize the status of modified\ngravity, let us note that metric $f(R)$ gravity models exist\nthat pass all the observational and theoretical constraints (see,\n{\\em e.g.}, the Starobinsky model \\cite{SongHuSawicki06} $\nf(R)=R+\\lambda R_0 \\left[ \\frac{1}{\\left( 1+ \\frac{R^2}{R_0^2}\n\\right)^n }-1 \\right] $).\nThe viable models require the chameleon mechanism in order to\npass the weak-field limit tests.\n\n\nAll metric $f(R)$ theories must satisfy the condition $ f''(R)\n\\geq 0$ to avoid the Dolgov-Kawasaki local instability. This is\na condition on short-wavelength modes. The stability\ncondition~(\\ref{stabilitydS}) is valid for arbitrary wavelengths,\nbut is restricted to de Sitter space (which is, anyway, an\nadiabatic approximation for slowly expanding FLRW spaces). An\nimportant open problem is\nwhether curvature singularities appear, in general, in\nrelativistic strong field stars.\n\n\n\n\nAs far as the Palatini formalism is concerned, the\ncentral idea of this version of modified gravity is to\nregard the torsion-free connection $\\Gamma^{\\mu}_{\\alpha\\beta}$\nas a quantity\nindependent of the spacetime metric $g_{\\mu\\nu}$. The\nPalatini formulation of the standard Hilbert-Einstein theory is\n equivalent to the purely metric theory, as a consequence of the\nfact that the field equations for the connection give the\nLevi-Civita connection of the metric $g_{\\mu\\nu}$. Therefore,\nthere is no reason to impose the Palatini variational principle,\ninstead of the metric variational principle, in the\nEinstein-Hilbert theory. However, the situation is difefrent in\nETGs containing non-linear functions of the curvature invariants,\nsuch as $f(R)$, or non-minimally coupled scalars. In these cases,\nthe Palatini and the metric variational principle yield different\nfield equations and different physics \\cite{magnano-soko,FFV}. The\nrelevance of the Palatini approach for cosmological applications\nhas been amply demonstrated\n\\cite{curvature,odinoj,palatinifR,palatinicam1}. However, Palatini\n$f(R)$ theories could have some problems due to the fact that\nthey could contain non-dynamical scalar field and the initial\nvalue problem could be ill-posed. In any case, when the trace of\nthe matter energy-momentum tensor vanishes identically or it is a\nconstant, and when it can be recast in a perfect-fluid form, the\nCauchy problem results well-formulated and well-posed.\n\n\nMetric-affine gravity has not been developed in\nsufficient detail to assess its viability according to all the\ncriteria presented here, and its cosmological consequences are\nessentially unexplored.\n\n\nIt seems fair to say that $f(R)$ theories of gravity can help to\nprogress in our understanding of the peculiarities of GR in the\nwider landscape of relativistic theories of gravity.\nFurthermore, these theories point out important aspects of\ngeneralizations of GR, and, from a phenomenological point of view,\nconstitute viable alternatives to dark energy models in explaining\nthe cosmic acceleration, and to dark matter in reproducing\ndynamical features as the galactic rotation curves or the halo of\nclusters of galaxies \\cite{salzano}. Finally, it is possible to\n\"tune\" the stochastic background of GWs and this occurrence could\nconstitute a further cosmological test capable of confirming or\nruling out ETGs once data from interferometers, like VIRGO, LIGO\nand LISA, will be available (see \\cite{corda} for a discussion on\nthis topic). At present, no definite prediction sets $f(R)$\ntheories apart from dark energy and other models once and for all,\nbut it is hoped that progress will me made in this direction.\n\n\n\n\\section*{Acknowledgements}\n\nSC and MD acknowledge V. Cardone, M. Francaviglia, A. Troisi and\nS. Vignolo for discussions and some common results used in this\nreview paper. VF acknowledges financial support from Bishop's\nUniversity and the Natural Sciences and Engineering Research\nCouncil of Canada (NSERC).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{INTRODUCTION}\n\n\nThe discovery of strong inverse Compton components\nin $X$ and $\\gamma$-ray emission from jets in active galactic \nnuclei (hereafter AGN) for a wide range of spatial scales\n(e.g., Collmar 2001 for review) enables us to probe \nquantitatively the energetics of relativistic jets. The \nkinetic power of non-thermal electrons has been estimated \nby various authors both for inner core jets (i.e., blazars)\n(e.g., Kino, Takahara and Kusunose 2002, hereafter KTK;\nKusunose, Takahara and Kato 2003)\nand large scale jets (e.g., \nTavecchio et al. 2000; \nLeahy and Gizani 2001, 2002; \nKataoka et al. 2003).\nHowever, the material content of relativistic jets is not \neasily constrained by observations since the emission is \ndominated by that from non-thermal electrons and probably positrons\nand it is difficult to directly constrain thermal matter \ncontent. \nHence, the plasma composition in AGN jets, whether normal \nproton-electron ($e\/p$) plasma \nor electron-positron pairs ($e^{\\pm}$) is \na dominant composition, is \nstill a matter of open issue \n(e.g., Reynolds et al. 1996; \nCelotti, Kuncic, Rees and Wardle 1998; \nWardle et al. 1999; Hirotani et al. 1999;\nSikora and Madejski 2000;\nRuszkowski and Begelman 2002;\nKino and Takahara 2004, hereafter KT04). \nThis problem prevents us from estimating\nthe total mass and energy flux ejected from a\ncentral engine.\n\nTo constrain invisible matter content such as thermal electron-positron\npairs and\/or protons co-existing with non-thermal electrons, \ndynamical considerations are indispensable.\nIn KT04, we proposed a new procedure \nto constrain the invisible thermal plasma component\nin classical FR II radio sources.\nWe used the fact that the mass and energy densities of \nthe sum of thermal and non-thermal particles \nare larger than those of non-thermal electrons which \nare determined by observations.\nHere we apply the same technique to the inner core jets of \nAGNs (i.e., blazars) \nbased on the internal shock model.\nThe internal shock model is\nbelieved to be most plausible \nto explain the production of high energy photons \nand time variabilities in blazars.\nIt has been widely applied also \nto the prompt emission of gamma-ray bursts (hereafter GRBs)\n(e.g.,\nRees 1978; \nRees and Meszaros 1994; \nKobayashi, Piran and Sari 1997;,\nDaigne and Mochkovitch 1998; \nGhisellini 1999;\nSpada, Ghisellini, Lazzati and Celotti 2001).\nIt is worth to note that\nrecently Ghisellini et al. (2005)\nproposed a structured jet model cosisting of a fast spine \nsurrounded by a slowly moving layer\nfor explaining VLBI scale radio blobs.\nAt the present, however, it is not evident where\nis the acceleration site of electrons in the structured jet model. \nThis is one of the prime issues which should be answered. \nInternal shocks are potentially the building blocks of \nthe spine part of the structured jet. \nWhereas we recognize the importance of the\ndetailed structure of jets, \nas a first step \nwe focus on the physical condition of the \nflow based on the simple internal shock model.\n\n\n\n\nThe methodology of constraining \nthe invisible plasma content in the emission region is as follows.\nAs mentioned above,\na lower limit to the total mass density \n(sum of non-thermal electrons and invisible plasma) \nis restricted by the definition that the mass density\nof total plasma should be smaller than that of \nthe non-thermal electrons. \nThe mass density of non-thermal electrons can be \nestimated by multi-frequency observations.\nFor this purpose, in \\S \\ref{sec:shock}\nwe review the shock dynamics of two colliding shells. \nNote that we do not use the simple\ntwo point-mass approximation \n(e.g., Piran 1999; Lazzati et al. 1999;\nZhang and M{\\' e}sz{\\' a}ros 2004 for review)\nbut employ the exact shock dynamics throughout this work.\nThis makes outcomes more accurate.\nIn \\S \\ref{sec:NT}\nwe briefly review the previous results \non the amount of non-thermal electrons \nbased on KTK.\nIn \\S \\ref{sec:invisible},\nwe constrain on the amount of total mass density.\nAs for the upper limit,\nwe use the constraint that bremsstrahlung emission from \nthermal electron (and positron) component should \nnot exceed the observed $\\gamma$-ray emission. \nWe postulate synchrotron self-Compton (SSC) \nemission dominance in the $\\gamma$-ray band which is\nsupported by the observed \ncorrelations between TeV$\\gamma$-ray and X-ray \nin TeV blazars\n(e.g., Takahashi et al. 1996, 2000; \nCatanese et al. 1997; Maraschi et al. 1999). \nWe can thus bracket the amount of total mass density\nin the emission region from below and above. \nIn this way we apply this method to\n the archetypal TeV blazar Mrk 421.\nIn \\S \\ref{sec:dissipation},\nwe further estimate the shock dissipation rate of the \ncolliding cold shells.\nThe dissipation rate\nis a widely discussed quantity in literatures concerning gamma-ray bursts\n(e.g., Lazzati, Ghisellini and Celotti 1999; Piran 1999). \nThe shock dissipation is believed to be \nthe ultimate source of heating and accelerating particles.\nSummary and discussion are in \n\\S \\ref{sec:summary}.\n\n\n\n\n\\section{Key features of this work}\n\nThe key features of this work are\nbriefly summarised here in advance.\nThe existence of copious amount of invisible plasma \nis predicted from a qualitative consideration.\n\n\\subsection{Existence of invisible plasma content}\n\n\n\n\nAs mentioned in the introduction,\nwe constrain on the amount of invisible plasma content \nby introducing the dynamical considerations.\nThe point is that\nwe divide total mass and energy densities into \ntwo components, i. e., those of non-thermal electrons and those\nof the other invisible components. \nThe comparison of these obtained quantities enables\nus to constrain on the amount of invisible plasma and\nthis is a new attempt compared with the previous works.\n\n\nBearing this in mind, \nnext we show a quantitative consideration which derives\nthe existence of invisible plasma in colliding shells\nof blazar jets.\nLet us discuss a collision between \na pair of equal mass-density shells for instance.\nIn the comoving frame of one shell, the particles of the other shell \nare coming in with a relative bulk Lorentz factor $\\Gamma_{ij}$\n(see in \\S 3 for details) of a few at most (shown in Table 1). \nWhen only pair $e^{\\pm}$ plasma are \npresent and all of them are accelerated, then \nthe average Lorentz factor of non-thermal electrons\n$\\langle\\gamma_{e}\\rangle$ is expected to be \n$\\langle\\gamma_{e}\\rangle\\approx\\Gamma_{ij}$.\nThis is too small to account for observed blazar spectra \n $\\langle\\gamma_{e}\\rangle\\approx 300$ (this is the case of Mrk 421) \nobtained by KTK. Therefore only a fraction of \nthe pair $e^{\\pm}$ should be accelerated, the ratio\nof \nthe rest mass density of \nnon-thermal electrons \nto that of total plasma is about\n$\\Gamma_{ij}\/\\langle\\gamma_{e}\\rangle$.\nSimilarly we can discuss the case for shells with pure $e\/p$ plasma makeup. \nIf all of the dissipated energy goes into the electron acceleration, \nthen we have $\\langle\\gamma_{e}\\rangle \\approx (m_{p}\/m_{e})\\Gamma_{ij}$. \nThis is too large to account for the spectra\nand it requires a limited fraction of electrons being accelerated.\nThus, invisible plasma is qualitatively expected when the \ninternal shock is responsible for\nthe production of non-thermal electrons.\nIn this paper, we will quantitatively\nexplore the amount of invisible plasma in jets.\n\n\n\n\n\\subsection{Why we use shock dynamics?}\n\n\nIn the previous studies, \nthe colliding shells have been approximately\nmodeled as the simple two-point-mass collision \n(e.g., Piran 1999; Lazzati et al. 1999;\nZhang and M{\\' e}sz{\\' a}ros 2004).\nThe reason why we use the shock dynamics instead of the \ntwo-point-mass model is as follows.\nWhen one try to derive the mass density from the mass,\none eventually needs to know lengths and velocities\nand they can be consistently obtained by the shock model.\nHence the shock analysis is the best way\nfor investigating the invisible plasma content in jets.\n\n\n\n\n\n\n\n\\section{Shocks in colliding shells}\\label{sec:shock}\n\nHere we review the relativistic shock jump conditions.\nWe use one-dimensional shock dynamics \nof a pair of colliding shells to apply the \nstandard internal shock model to blazars. \nSuppose the situation in which a rapid shell overtakes a \npreviously ejected slow shell.\nThere are four characteristic regions designated by\n(1) unshocked slow shell,\n(2) shocked slow shell,\n(3) shocked rapid shell, and\n(4) unshocked rapid shell.\nThese regions are separated by \nthe forward shock (FS),\nthe contact discontinuity (CD), and\nthe reverse shock (RS).\nIn this paper, we use the terminology \nof {\\it regions} $i$ ($i$=1, 2, 3, and 4) and\n{\\it position of discontinuity}\n$i$ ($i$=FS, CD, and RS) where \nFS, CD, and RS stand for the forward shock front, \ncontact discontinuity,\nand reverse shock front, respectively.\nThe fluid velocity and Lorentz factor in the region $i$ \nmeasured in the interstellar medium (hereafter ISM) frame \nare expressed as\n$v_{i}(=\\beta_{i}c)$ and $\\Gamma_{i}$, respectively.\nThe relative velocity and Lorentz factor of the fluid $i$ \nmeasured in the frame $j$ are denoted by \n$v_{ij}(=-v_{ji}=\\beta_{ij}c=-\\beta_{ji}c)$ \nand $\\Gamma_{ij}(=\\Gamma_{ji})$, respectively. \nRest mass density, pressure, and internal energy density are \nexpressed as\n$\\rho_{i}$,\n$P_{i}$, and \n$e_{i}$,\nrespectively. As for the equation of state (EOS), we take\n$P_{i}=(\\hat{\\gamma}_{i}-1)(e_{i}-\\rho_{i}c^2)$,\nwhere $\\hat{\\gamma}_{i}$ is the adiabatic index.\nWe sometimes use the subscripts s and r \ninstead of $1$ and $4$, such as \n$\\Gamma_{1}=\\Gamma_{\\rm s}$ and \n$\\Gamma_{4}=\\Gamma_{\\rm r}$.\n\n\nIn the limit of strong shock, \nwith the assumption of cold upstream ($P_1=0$),\nthe jump conditions for the forward shock \nare written as follows\n(Blandford \\& McKee 1976):\n\\begin{eqnarray}\\label{eq:FS}\n\\Gamma_{\\rm FS1}^{2}\n=\\frac{(\\Gamma_{\\rm 12}+1)[\\hat{\\gamma}_{2}(\\Gamma_{\\rm 12}-1)+1]^{2}}\n{\\hat{\\gamma}_{2}(2-\\hat{\\gamma}_{2})(\\Gamma_{\\rm 12}-1)+2},\\nonumber \\\\\ne_{2}=\\Gamma_{\\rm 12}\\rho_{2} \\, ,\n\\qquad\n\\frac{\\rho_{2}}{\\rho_{1}}=\n\\frac{\\hat{\\gamma}_{2}\\Gamma_{12}+1}{\\hat{\\gamma}_{2}-1} \\, ,\n\\end{eqnarray}\nwhere $\\Gamma_{12}=\\Gamma_{1}\\Gamma_{2}(1-\\beta_{1}\\beta_{2})$, and\n$\\Gamma_{\\rm FS1}$\nis the Lorentz factor of forward shock measured \nin the rest frame of the unshocked slow shell. \nIn the relativistic limit, the adiabatic index is \n${\\hat \\gamma}_{2}=4\/3$. \nUsing the same assumptions as in the forward shock,\nthe jump conditions for the reverse shock are given by:\n\\begin{eqnarray}\\label{eq:RS}\n\\Gamma_{\\rm RS4}^{2}\n=\\frac{(\\Gamma_{\\rm 34}+1)[\\hat{\\gamma}_{3}(\\Gamma_{\\rm 34}-1)+1]^{2}}\n {\\hat{\\gamma}_{3}(2-\\hat{\\gamma}_{3})(\\Gamma_{\\rm 34}-1)+2},\\nonumber \\\\\ne_{3}=\\Gamma_{\\rm 34}\\rho_{3} \\, ,\n\\qquad\n\\frac{\\rho_{3}}{\\rho_{4}}=\n\\frac{\\hat{\\gamma}_{3}\\Gamma_{34}+1}{\\hat{\\gamma}_{3}-1} \\, ,\n\\end{eqnarray}\nwhere $\\Gamma_{34}=\\Gamma_{3}\\Gamma_{4}(1-\\beta_{3}\\beta_{4})$,\nand $\\Gamma_{\\rm RS4}$\nis the Lorentz factor of the reverse shock measured in \nthe rest frame of the unshocked rapid shell. \nThe equality of pressure and velocity across the contact discontinuity \ngives\n\\begin{eqnarray}\nP_{2}=P_{3} ,\n\\qquad\n\\Gamma_{2}=\\Gamma_{3} \\, .\n\\end{eqnarray}\n\nAfter the shocks break out the shells,\n$\\Gamma_{2}=\\Gamma_{3}$ is not satisfied\nbecause a rarefaction wave changes the \ndensity and velocity profiles \n(e.g., Kino, Mizuta and Yamada 2004, hereafter KMY).\nWe do not treat the rarefaction waves for simplicity, \nconcentrating on the major duration before shock \nbreakout.\nIt may be useful \nto rewrite the pressure balance along the CD as\n\\begin{eqnarray}\\label{eq:ratiorho}\n\\frac{\\rho_{4}}{\\rho_{1}}\n=\\frac{(\\hat{\\gamma}_{2}\\Gamma_{12}+1)(\\Gamma_{12}-1)}\n{(\\hat{\\gamma}_{3}\\Gamma_{34}+1)(\\Gamma_{34}-1)} \\ .\n\\end{eqnarray}\n\nIn general, \nthe number of physical quantities\nin each region is $3$, \n$\\rho_{i}$, $P_{i}$ (or $e_{i}$), and $v_{i}$.\nForward and reverse shock speeds \n(i.e., $v_{\\rm FS}$ and $v_{\\rm FS}$) \nare two other quantities.\nIn all, there are $3\\times 4+2=14$ physical quantities. \nNote that $P_i$ and $e_i$ are connected with EOS. \nThe total number of the jump conditions \nis $3+3+2=8$.\nHence, given $3+3=6$ upstream quantities \nfor each shock, \nwe can obtain the remaining $8$ downstream \nquantities by using $8$\njump conditions. It is to be noted that the absolute value of \nthe rest mass density is irrelevant to the shock dynamics \nsince the shock dynamics is linear with respect to the mass desnity. \nThen, actually we need to specify 5 quantities if we \ngive the density ratio $\\rho_4\/\\rho_1$.\n\n\nFor a specific case for TeV blazars,\nwe here impose the following two conditions;\n(i) the unshocked shells are cold, \ni.e., $P_{1}=P_{4}=0$, \n(ii) the Lorentz factor of the \nshocked regions $\\Gamma_{3}(=\\Gamma_{2})$ \nis identified as that of the \nemission region obtained by the \nobserved broadband spectra. \nFurther, we examine the following \nthree cases for the ratio $\\rho_{\\rm r}\/\\rho_{\\rm s}$; \n(a) the energy of bulk motion of the rapid shell \n($E=\\Gamma mc^{2}$) \nequals to that of the slow one in the ISM frame\n(we refer to it as ``equal energy (or $E$) case''),\n(b) the mass of the rapid shell ($m=\\rho \\Gamma\\Delta$)\nequals to that of the slow one \n(hereafter we call it ``equal mass (or $m$) case''), and \n(c) the rest mass density of the rapid shell equals to that of the slow one \n(hereafter we call it ``equal rest mass density (or $\\rho$) case'').\nHere, $\\Delta$ denotes the thickness of the shell \nmeasured in ISM frame. \nThese choices are based on the conjecture that \nthe ejecta from the ``central engine'' is likely \nto have a correlation with each other (e.g., NP02; KMY).\nHereafter, we assume that the widths of two shells are the \nsame in the ISM frame, that is $\\Delta_{\\rm r}\/\\Delta_{\\rm s}=1$ \n(e.g., NP02, Spada et al. 2001).\nNote that in the case of \n$\\Delta_{\\rm r}=\\Delta_{\\rm s}$\nand $\\Gamma_{\\rm r} >\\Gamma_{\\rm s}$,\n$\\rho_{\\rm s}$ is always\nlarger than $\\rho_{\\rm r}$ for equal $E$ and equal $m$ cases.\n\nThus, we give $4$ quantities, $P_1$, $P_4$, $\\Gamma_2=\\Gamma_3$\nand one relation between the rapid and slow shells, \n$\\rho_4\/\\rho_1$ depending on cases (a) \nthrough (c) described above.\nAs a remaining quantity, the Lorentz factor of the \nrapid shell\n$\\Gamma_{4}$ is treated as a free parameter.\nAlthough we do not specify the absolute value of $\\rho$, \nwe treat the abosolute value in actual applications. \nIt is compared with \nthat of non-thermal electrons in the shocked regions as described \nin \\S \\ref{sec:invisible}. \nThe absolute value of the rest mass density comes into play \nwhen two-body processes \nsuch as bremsstrahlung emission is used to obtain the upper limit \nof $\\rho$.\nWe will properly discuss these points.\n\nIn the following sections,\nwe focus on the values of \n(i) the value of $\\Gamma_{1}$ and $\\Gamma_{4}$,\n(ii) $e_{3}$ and\/or $\\rho_{3}$,\nas a tool to examine the physical quantities of \ninvisible matter content.\n\n\n\\section{Amount of non-thermal electrons}\\label{sec:NT}\n\n\\subsection{Number and energy densities}\n\nBased on the detection of inverse Compton emission in \n$\\gamma$-ray band,\nthe number and energy densities of \nthe non-thermal (hereafter ``NT'')\nelectrons $n_{e}^{\\rm NT}$ and $e_{e}^{\\rm NT}$ \nin shocked regions can be determined\nby the comparison of \nthe observed broadband spectrum and \nthe theoretical one. Although the minimum Lorentz factor \nof relativistic electrons \nis not definitely determined and affects mainly \nthe number density $n_{e}^{\\rm NT}$, \nwe regard that low energy electrons below $\\gamma_{e,\\rm min}$ \nconstitute thermal electrons. \nConsidering the observed flat number spectrum of electrons, \nfixing $\\gamma_{e,\\rm min}=10$ does not cause any major problem with \n$n_{e}^{\\rm NT}$. \n\nHere, we briefly quote the resultant \n$n_{e}^{\\rm NT}$ and $e_{e}^{\\rm NT}$\nobtained in KTK.\nHereafter, \nwe omit the subscript expressing the \nregions $i(=2,3)$ for simplicity.\nFor clearness of the following argument,\nwe define that $n_{e}^{\\rm NT}$ and $e_{e}^{\\rm NT}$\nalso include NT positrons when they exist. \nThe quantity $n_{e}^{\\rm NT}$ is written as\n$n_{e}^{\\rm NT}\\equiv\n\\int^{\\infty}_{\\gamma_{e,\\rm min}}\nn_{e}(\\gamma_{e})d\\gamma_{e}$,\nwhile $e_{e}^{\\rm NT}$ is given by\n$e_{e}^{\\rm NT}=\n\\langle\\gamma_{e}\\rangle\nn_{e}^{\\rm NT}m_{e}c^{2}$, \nwhere \n$n_{e}(\\gamma_{e})$ and \n$\\langle\\gamma_{e}\\rangle$ are the energy spectrum and \nthe average Lorentz factor of NT electrons, respectively.\nBy a detailed comparison of the SSC model with\nobserved broadband spectrum of Mrk 421,\nwe obtained $n_{e}^{\\rm NT}$ as\n\\begin{eqnarray} \\label{eq:??}\nn_{e}^{\\rm NT}\n\\simeq\n11\\times\n\\left(\n\\frac{\\gamma_{e,\\rm min}}{10}\n\\right)^{-0.6} \\ \\rm cm^{-3}.\n\\end{eqnarray}\nHere, we adopt the index of injected electrons for Mrk 421 \nas $s=1.6$ \n(e.g., Mastichiadis \\& Kirk 1997; Kirk \\& Duffy 1999)\nand the case of $\\gamma_{e,\\rm min}=10$ was examined in KTK.\nThe best choice of the size of the emission region is \n$R=2.8 \\times 10^{16}{\\rm cm}$ with an order of magnitude \nuncertainty. Thus, the corresponding uncertainty of $n_{e}^{\\rm NT}$ \namounts to two orders of magnitude; for smaller $R$, larger \n$n_{e}^{\\rm NT}$ should be adopted. \nBut, as far as the the shock dynamics is concerned, only \nthe density ratio plays a role, therefore we adopt the above value \nas the canonical one. \n\nAs for the average energy of NT electrons, we obtained \n\\begin{eqnarray} \\label{eq:uacc}\ne_{e}^{\\rm NT}\/n_{e}^{\\rm NT}\n=\\langle\\gamma_{e}\\rangle m_{e}c^{2}\n\\simeq\n3.1\\times 10^{2} m_{e}c^{2}.\n\\end{eqnarray}\nSince for $s=1.6$, \nelectrons near the cooling break energy $\\sim \\gamma_{e,\\rm br}$\ncarry most part of the kinetic energy\nand $e_{e}^{\\rm NT}$ has a weak dependence on $\\gamma_{e,\\rm min}$\nprovided that $\\gamma_{e,\\rm min}$ is smaller than\n$\\gamma_{e,\\rm br}\\sim 10^{4}$.\nNote that the case of \n$\\gamma_{e,\\rm min}\\sim 10^{4}$ is ruled out for Mrk 421\nsince the case does not fit the EGRET data (KTK).\nTherefore, Eq. (\\ref{eq:uacc}) is justified in any case \nfor Mrk 421.\n\n\n\n\n\\subsection{Forward and reverse shocks}\n\nTo clarify whether the observed non-thermal \nemission comes mainly from FS or from RS region, \nthe typical frequency of non-thermal synchrotron radiation \nand internal energy density in each region are examined here.\n\nAccording to the standard diffusive shock acceleration, \nthe acceleration time scale is estimated as (e.g., Drury 1983) \n$t_{\\rm acc}=(2\\pi\\gamma_{e} m_{e}c\\xi)\/(eB)$ \nwhere \n$\\xi=\\lambda\/r_{g}$\nis a parameter related to the amplitude of \nmagnetic fluctuations, $\\lambda$ and $r_{g}$ are\nthe mean free path for \nthe scattering of electrons and Larmor radius, respectively.\nHere the shock speed is taken to be $c$. \nOn the other hand,\nthe synchrotron cooling time is given by\n$t_{\\rm syn}=\n(6\\pi m_{e}c^{2})\/(\\sigma_{T}\\gamma_{e}cB^{2})$.\nThe maximum Lorentz factor of \nthe non-thermal electrons is evaluated as\n$\\gamma_{\\rm max}\\propto B^{-1\/2}$\nby using the condition of $t_{\\rm acc}=t_{\\rm syn}$ \nat $\\gamma_{\\rm max}$\nwith the assumption \nthat $\\xi$ in FS and RS regions takes the same value. \nHence,\nthe characteristic synchrotron photon energy is given by\n$h\\nu_{\\rm syn,o,max\n\\propto \\Gamma_{i}B\\gamma_{\\rm max}^{2}={\\rm const.}$,\nHence, the value $\\nu_{\\rm syn,o,max}$ \nin FS region and RS regions is the same. \n\n\n\nThe total internal energy of NT electrons\nin FS and RS regions may be discussed as follows.\nIf $\\Gamma_{21}\\gg1$ and $\\Gamma_{43}\\gg1$\nare satisfied, we have\n$e_{2}+P_{2}\\simeq e_{3}+P_{3}$. In the actual case of blazars,\n$\\Gamma_{ij}$ is close to order of unity and we have \n$P_{2}=P_{3}$.\nThus, \nthe energy densities of regions 2 and 3 are similar and \nthe internal energy \nis controlled by the comoving shell widths.\nSince $\\Gamma_{43}\\ge\\Gamma_{21}$ is always \nsatisfied, co-moving length of RS region is \nlarger than that of FS region \nin the case of $\\Delta_{\\rm r}=\\Delta_{\\rm s}$ \n(e.g., Kobayashi and Sari 2001; NP02; KMY). \nThus, the radiation from RS dominates over that from FS region. \nBased on this consideration, \nwe focus on RS dominated case in this paper.\nHereafter, we omit the subscript $3$ for simplicity. \n\n\n\n\\section{Constraints on the amount of invisible plasma}\\label{sec:invisible}\n\n\n\n\n\n\\subsection{Lorentz factors of cold shells}\\label{subsec:coldshell}\n\n\nIt is hard to estimate the \nbulk Lorentz factors of cold shells\nsimply because they are invisible. \nHowever, by using the value of Lorentz factor of shocked shell which\ncorresponds to the beaming factor of the emission region, \nwe can constrain on the \nLorentz factors of the cold shells.\nHere, we consider the range from 3 to 100 for $\\Gamma_{\\rm r}$\nand $\\Gamma_{\\rm s}$. \nFollowing \nBegelman, Rees \\& Sikora (1994),\nwe consider the upper limit of the \nLorentz factors of the emission region as \n$\\Gamma_{\\rm r,max}= 100$,\nwhile\nas for the lower limit\nwe employ $\\Gamma_{\\rm s,min}=3$ \nbased on Wardle \\& Aaron (1997). \nHere, we exclude cases of very\nweak collisions with \n$\\Gamma_{\\rm r}\/\\Gamma_{\\rm s}<2$\nas in NP02. \nAs for the adiabatic index in Eq. (\\ref{eq:ratiorho}),\nwe approximate $\\hat{\\gamma}_{i}=4\/3$ for $\\Gamma_{ij}>2$, otherwise\n$\\hat{\\gamma}_{i}=5\/3$ for simplicity (e.g., Kirk and Duffy 1999).\n\n\nFor the TeV blazar Mrk 421, we have already \nobtained $\\Gamma_2=\\Gamma_{3}=12$ by the observed\nmulti-frequency spectrum (KTK). \nHence, \nEq. (\\ref{eq:ratiorho}) is solvable for\n$\\Gamma_{\\rm s}$ given $\\rho_{\\rm r}\/\\rho_{\\rm s}$ \nand $\\Gamma_{\\rm r}$. \nQualitatively, a faster $\\Gamma_{\\rm r}$ requires a slower \n$\\Gamma_{\\rm s}$ to attain the\nsame value of $\\Gamma_{3}$.\nThus, minimun value of $\\Gamma_{\\rm r}$ corresponds to \n$\\Gamma_{\\rm r}\/\\Gamma_{\\rm s}=2$, while the maximum value of\n$\\Gamma_{\\rm r}$ corresponds to $\\Gamma_{\\rm r}=100$ \nor $\\Gamma_{\\rm s}=3$. \n\n\n\n\n\n\n\nIn Table \\ref{table:1},\nwe show \nthe minimum and maximum values of \n$\\Gamma_{\\rm r}$ and $\\Gamma_{\\rm s}$ and \nthe corresponding relative Lorentz factors \n$\\Gamma_{12}$ and\n$\\Gamma_{34}$ which control the shock heating of the downstreams \n(see Eqs. (\\ref{eq:FS}) and (\\ref{eq:RS})).\nFrom this, we see that\nthe range of $\\Gamma_{ij}$ lies between 1.03 and 4.2. \nIn other words, a mildly relativistic shock is realized\nin the case of Mrk 421.\nWe also note that our adopted value of $\\gamma_{e,\\rm min}=10$ \nis a reasonable choice with the assumption that \n$\\gamma_{e,\\rm min}$ should be a few times larger than $\\Gamma_{ij}$.\nThe corresponding value of \n$\\Gamma_{\\rm r}\/\\Gamma_{\\rm s}$ is found as\n\\begin{eqnarray} \n2<\\Gamma_{\\rm r}\/\\Gamma_{\\rm s}\\le 16.0 && \n({\\rm equal} \\ \\rho), \\nonumber \\\\\n2<\\Gamma_{\\rm r}\/\\Gamma_{\\rm s}\\le 19.5 && \n({\\rm equal} \\ m), ~{\\rm and} \\nonumber \\\\\n2<\\Gamma_{\\rm r}\/\\Gamma_{\\rm s}\\le 11.7 && \n({\\rm equal} \\ E) ,\n\\end{eqnarray} \nrespectively.\n\n\n\\input{table1}\n\n\n\n\n\\subsection{Total mass density}\\label{subsec:massdensity}\n\n\n\\subsubsection{Lower limit of $\\rho$}\\label{sec:lower-rho}\n\nIn \\ref{subsec:coldshell}, we show that\nshock is at most mildly relativistic\nthough each shell moves at a relativistic speed.\nAs a consequence, dissipation efficiency \nis relatively small and\n$\\langle\\gamma_{e}\\rangle \\gg \\Gamma_{34}$ is satisfied.\nTherefore $e_{e}^{\\rm NT}\/e\n=\\langle\\gamma_{e}\\rangle\\rho_{e}^{\\rm NT}\/\n\\Gamma_{34}\\rho<1$ gives a tighter constraint than \n$\\rho_{e}^{\\rm NT}\/\\rho<1$.\nBy rewriting the condition of \n$e_{e}^{\\rm NT}\/e<1$, the lower limit of \nthe total mass density is given by \n\\begin{eqnarray}\\label{eq:lower}\n\\frac\n{\\rho} \n{\\rho_{e}^{\\rm NT}}\n&>& \\frac\n{\\langle\\gamma_{e}\\rangle}\n{\\Gamma_{34}}\n\\simeq \\frac{3.1\\times 10^{2}}{\\Gamma_{34}} .\n\\end{eqnarray}\nHere we omit the subscript of region number $i=3$\nfor the various densities for thumbnail writing.\nFrom this \nwe directly see that\nin order to accelerate electrons up to \n$\\langle\\gamma_{e}\\rangle\\sim 3.1\\times10^{2}$ in the framework of standard\ninternal shock model, where only \na small available shock dissipation energy \n$\\Gamma_{34}\\sim a \\ few$ is realized,\nthe invisible mass density at least about 100 times the rest mass density \nof NT electrons is definitely required. \nIn other words,\nwe need a loading of baryons and\/or a thermal pair plasma. \nIt is worth to note the effects of an uncertainty with\n$\\langle\\gamma_{e}\\rangle$. \nWe estimated the uncertainty range as \n$ 2.3\\times 10^{2}\n< \\langle\\gamma_{e}\\rangle <\n4.3\\times 10^{2}$ (KTK).\nThe uncertainty simply leads to a shift of \nlower limit curve by the same factor.\nSince it causes only a small change on the resultant value,\nwe focus on the best-fit case in this work \nfor simplicity.\n\n\n\n\n\n\n\\subsubsection{Upper limit of $n_{e}^{\\rm T}$}\\label{sec:upper-n}\n\nHere, we constrain the upper limit of the number density of \nthermal electrons $n_{e}^{\\rm T}$.\nAs mentioned in the Introduction,\nit is widely accepted that\nobserved GeV and TeV $\\gamma$-rays are \nSSC dominated.\nIn the MeV range, bremsstrahlung radiation by the\nthermal electrons with temperature \n$\\Theta_{e}\\equiv kT_{e}\/m_{e}c^{2}\n\\sim \\Gamma_{34} \\sim $ a few MeV \nis expected \nif adequate amount of thermal electrons exist in the\nemission region.\nAt the moment,\nwe do not have any observational evidence for \nthe bremsstrahlung in MeV band.\nAt the same time, it is fair to note that\nobservation in MeV range itself is a challenging area\n(e.g., Takahashi et al. 2003).\nHere we estimate the upper limit of the number density of\nthermal electrons\nby assuming the observed bolometric luminosity of\nbremsstrahlung $L_{{\\rm brem,o}}$ should be lower \nthan that of SSC $L_{\\rm ssc,o}$ which is estimated as\n$ L_{{\\rm ssc,o}}=\n7 \\times 10^{44}$ erg s$^{-1}$ (KTK).\n\n\nFor $e^{+}$$e^{-}$ plasma content,\nwe employ \nEqs. (21) and (22)\nof Svensson (1982) which express\nthe emissivity of relativistic $e^{+}$$e^{-}$ \nbremsstrahlung $\\epsilon_{\\rm brem}(\\Theta_{e},n_{e}^{\\rm T})$\nwhere $n_{e}^{\\rm T}$ is\nthe number density of thermal electrons.\nNote that these expressions do not include the bremsstrahlung\nbetween electron-electron and positron-positron\nand the limit will be severer by a factor of $\\sim 2$\nif we include them.\nThen, the condition of \n$ L_{\\rm ssc,o}>L_{\\rm brem,o}$\nis rewritten as\n\\begin{eqnarray}\\label{eq:upper1}\nn_{e}^{\\rm T}\n&<& 9.7 \\times 10^{2} \n\\left[\\Theta_{e}^{1\/2}(1+1.7\\Theta_{e}^{1.5})\n\\right]^{-1\/2} \n\\rm \\ cm^{-3} \\quad (\\Theta_{e}<1) \\nonumber \\\\\n&<& 5.7 \\times 10^{2} \n\\left[\n\\Theta_{e}(\\ln(1.1\\Theta_{e})+5\/4\n\\right]^{-1\/2} \n\\rm \\ cm^{-3}\\quad (\\Theta_{e}\\ge1) . \\nonumber \\\\\n\\end{eqnarray}\nThe \nbolometric luminosity\nof the optically-thin bremsstrahlung is estimated by\n$L_{{\\rm brem,o}}\n=(4\\pi R^{3}\/3) \\Gamma_{3}^{4} \n\\epsilon_{\\rm brem}$ \nwith the emission size \n$R=2.8\\times 10^{16}\\rm cm$ \nand the Lorentz factor\n$\\Gamma_{3}=12$ as obtained by the broadband spectral fitting\nof Mrk 421 (KTK).\nThe electron temperature is evaluated by \n$(\\hat{\\gamma}_3-1)\\Theta_{e}=\\Gamma_{34}-1$.\nThe upper limit turns out to be about a thousand times \nlarger than the mass density of non-thermal electrons. \nIt is consistent with and relatively close to\nthe required lower limit of the\nmass density by Eq. (\\ref{eq:lower}).\nThis upper limit depends on the adopted value of $R$, \nand it is proportional to $R^{-3\/2}$. \nConsidering that $n_e^{\\rm NT}$ is roughly proportional \nto $R^{-2}$, the ratio of this upper limit to $n_e^{\\rm NT}$\nonly has a weak dependence on $R$.\n\n\nSimilarly, in the case of electron-proton \n(hereafter $e\/p$) plasma content,\nwe can rewrite the condition of \n$ L_{\\rm ssc,o}>L_{\\rm brem,o}$\nas\n\\begin{eqnarray}\\label{eq:upper2}\nn_{e}^{\\rm T}\n&<& 9.5 \\times 10^{2} \n\\left[\\Theta_{e}^{1\/2}(1+1.78\\Theta_{e}^{1.34})\n\\right]^{-1\/2} \\rm \\ cm^{-3} \n (\\Theta_{e}<1) \\nonumber \\\\\n&<& 9.6 \\times 10^{2} \n\\left[\n\\Theta_{e}(\\ln(1.1\\Theta_{e}+0.42)+3\/2\n\\right]^{-1\/2} \n\\rm \\ cm^{-3} \\nonumber \\\\ \n&& (\\Theta_{e}\\ge1) \n\\end{eqnarray}\nwith Eqs. (17) and (18)\nof Svensson (1982).\nNote that electron-electron \nbremsstrahlung is not considered in these equations.\nIt is clear that the \nupper limit of $\\rho$ in this case is \n$m_{p}\/m_{e}$ times larger than $n_{e}^{\\rm T}m_e$.\n\n\n\nLastly, let us check \nthe timescale of $e^{\\pm}$ pair \nannihilation $t_{\\rm ann}$.\nIt is evaluated as $t_{\\rm ann}\n\\simeq \\Theta_{e}^{2}\/(n_{e}\\sigma_{T}c)\n\\simeq 6 \\times 10^{10}\\Theta_{e}^{2}\n(n_{e} \/10^{3} \\rm cm^{-3})^{-1}$ sec.\nHence we see that\nthe annihilation time scale is \nmuch longer than the dynamical time scale \n$t_{\\rm dyn}\\equiv \\sqrt{3}R\/c\n\\approx 2\\times 10^{6}(R\/10^{16}~{\\rm cm})~{\\rm sec}$.\nTherefore $e^{\\pm}$ pair \nannihilation is not effective in this situation. \n\n\n\n\n\n\n\n\n\n\n\\subsubsection{Allowed range of $\\rho$}\\label{sec:allowed}\n\nWe thus obtained the upper and lower limits on \n$\\rho\/\\rho_e^{\\rm NT}$ and the results are shown \nin the plane of mass density of invisible plasma and \n$\\Gamma_{\\rm r}\/\\Gamma_{\\rm s}$\nin the cases of \n``equal $\\rho$'', \n``equal $m$'', and\n``equal $E$''\nin Figs. \n\\ref{fig:eqrho},\n\\ref{fig:eqmass}, and\n\\ref{fig:eqE}, respectively.\nThey are obtained by solving Eq. (\\ref{eq:ratiorho})\nand inserting $\\Gamma_{34}$\ninto Eqs. (\\ref{eq:lower}), (\\ref{eq:upper1}), and (\\ref{eq:upper2}).\nThe qualitative features are the same for these three cases, \nalthough different in quantitative detail.\nSumming up in advance, the most important result is that a\nlarge amount of mass density of invisible plasma \nis required in the emission region.\nAs the value of $\\Gamma_{r}\/\\Gamma_{s}$ increases, \nthe value of $\\Gamma_{34}$ becomes larger\nand the lower limit on the invisible mass density \n($\\rho\/\\rho^{\\rm NT}_{e}$) \nreduces. \nBelow we discuss two extreme cases of different plasma content.\nOne is the case of the jet \nwith pure $e^{\\pm}$ pair plasma content,\nwhilst\nthe other is the jet made of pure $e\/p$ plasma. \n\n\n\n\nFor pure $e^{\\pm}$ pair jet,\nthe resultant total mass density \nnormalized by $\\rho_{e}^{\\rm NT}$ is \n\\begin{eqnarray}\\label{eq:e-rho}\n2\\times 10^{2}<\\rho\n\/\\rho^{\\rm NT}_{e}<2\\times 10^{3} && ({\\rm equal} ~ \\rho) \\nonumber \\\\ \n7\\times 10^{1}<\\rho\n\/\\rho^{\\rm NT}_{e}<2\\times 10^{3} && ({\\rm equal} ~ m)\\nonumber \\\\ \n6\\times 10^{1}<\\rho\n\/\\rho^{\\rm NT}_{e}<2\\times 10^{3} && ({\\rm equal} ~ E) .\n\\end{eqnarray}\nFor the jets consisting of pure $e^{\\pm}$ plasma, \nthe predicted $\\rho\/\\rho^{\\rm NT}_{e}$ is \nconstrained in a narrow range around 100-1000\nas shown in Figs. \\ref{fig:eqrho},\n\\ref{fig:eqmass}, and \\ref{fig:eqE}. \nThe number density fractions of the shock accelerated \n$e^{\\pm}$ pairs are directly obtained as \n$\n\\rho^{\\rm NT}_{e}\/\\rho=n^{\\rm NT}_{e}\/(n^{\\rm T}_{e}+n^{\\rm NT}_{e})\n\\sim\n10^{-3}-10^{-2}$.\nThis seems a reasonable result since the number of accelerated \nparticles is expected to be a small fraction of the thermal pool. \n\n\n\n\n\nIn the case of pure $e\/p$ content, the allowed range\nof $\\rho\/\\rho^{\\rm NT}_{e}$\nare found to be\n\\begin{eqnarray}\\label{eq:p-rho}\n2\\times 10^{2}<\\rho\n\/\\rho^{\\rm NT}_{e}<3\\times 10^{6} && ({\\rm equal} ~ \\rho), \\nonumber \\\\ \n7\\times 10^{1}<\\rho\n\/\\rho^{\\rm NT}_{e}<3\\times 10^{6} && ({\\rm equal} ~ m), ~{\\rm and} \\nonumber \\\\ \n6\\times 10^{1}<\\rho\n\/\\rho^{\\rm NT}_{e}<3\\times 10^{6} && ({\\rm equal} ~ E),\n\\end{eqnarray}\nrespectively.\nThe maximum values\nof $\\rho\/\\rho^{\\rm NT}_{e}$ are about $m_{p}\/m_{e}$\ntimes larger than those \nin the case of pure $e^{\\pm}$ pair content.\n\n\n\n\n\n\n\\subsection{Allowed range of $e\/e_{e}^{\\rm NT}$}\n\n\nAs shown above, the lower and upper limit of \n$\\rho\/\\rho^{\\rm NT}_{e}$ have been obtained in \\ref{sec:lower-rho}\nand \\ref{sec:upper-n}, respectively.\nBy using the obtained $\\rho\/\\rho^{\\rm NT}_{e}$ \nshown in \\ref{sec:allowed}, \nwe can estimate \n$e\/e^{\\rm NT}_{e}=\\Gamma_{34}\\rho\/\n\\langle\\gamma_{e}\\rangle\\rho^{\\rm NT}_{e}$.\nFor the case of pure $e^{\\pm}$ content,\nsince the allowed range of $\\rho\/\\rho^{\\rm NT}_{e}$ \nis narrow, the corresponding\n$e\/e^{\\rm NT}_{e}$ is also well\nconstrained as\n\\begin{eqnarray}\n12$.\n\n\n\n\nOn the contrary, for pure $e\/p$ content,\nthe energetics relevant to thermal electrons \nand NT and thermal protons is all quite uncertain.\nBased on Eq. (\\ref{eq:p-rho}) and Table 1,\nwe can derive\n\\begin{eqnarray}\n13$,\n$\\Gamma_{\\rm r,max}<100$, and\n$\\Gamma_{\\rm r}\/\\Gamma_{\\rm s}>2$\nbased on the literatures \n(Wardle and Aarons 1997;\nBegelman, Rees and Sikora 1994;\nNP02),\nwe find that the values of $\\Gamma_{\\rm r}\/\\Gamma_{\\rm s}$\nfor Mrk 421 are limited in the ranges of \n$2<\\Gamma_{\\rm r}\/\\Gamma_{\\rm s}<16$ (equal $\\rho$),\n$2<\\Gamma_{\\rm r}\/\\Gamma_{\\rm s}<19.5$ (equal $m$), and \n$2<\\Gamma_{\\rm r}\/\\Gamma_{\\rm s}<11.7$ (equal $E$),\nrespectively.\nAs mentioned in Kirk and Duffy (1999), a very hard\ninjection index of $s\\sim 1.6$ observed in Mrk 421 well agrees \nwith this mildly relativistic \nshock regime (See Fig. 3 in their paper). \nHence we conclude that mildly relativistic shocks\ntake place in Mrk 421 \nfrom the analysis of \nthe observed spectrum and the internal shock dynamics.\n\n\n\n\n(2) \n{\\it The mass density of invisible\nplasma is much heavier than that of non-thermal electrons.}\n\nUsing the condition that the mass and energy densities\nof non-thermal electrons should be lower than those\nof the total ones, we derive the \nlower limit of total mass density at the shocked region.\nSince the relative Lorentz factor between the shocked and unshocked regions\nis expected to be a few (in Table 1), \ncopious amount of mass density \nof invisible plasma is inevitably required.\nThe upper limit of $n^{\\rm T}_{e}$ is constrained by\nthe condition that the luminosity of bremsstrahlung emission \nshould be smaller than the observed $\\gamma$-ray luminosity\nwhich is well explained by\nthe synchrotron-self-Compton emission.\nCombining them, the allowed ranges of\n$\\rho\/\\rho_{e}^{\\rm NT}$ \nfor pure $e^{\\pm}$ pair content are found as\n$2\\times 10^{2}<\\rho\/\\rho^{\\rm NT}_{e}<2\\times 10^{3}({\\rm equal} ~ \\rho)$,\n$7\\times 10^{1}<\\rho\/\\rho^{\\rm NT}_{e}<2\\times 10^{3}({\\rm equal} ~ m)$, and \n$6\\times 10^{1}<\\rho\/\\rho^{\\rm NT}_{e}<2\\times 10^{3}({\\rm equal} ~ E)$,\nrespectively.\nFor pure $e\/p$ plasma content, \nthe upper limit of $\\rho\/\\rho_{e}^{\\rm NT}$ turns out to be\n$3\\times 10^{6}$. \n\n\nAlthough the specific index $s=1.6$ for Mrk 421\nis discussed here, we emphasize that the value $s<2$ is \ncommon character for TeV blazars as\nthey indeed display the smaller $s$ \nthan 2 (Kirk and Duffy 1999 for review).\nFor instance, the choice of $s=2.2$ leads to the\nsynchrotron emission with $\\nu F_{\\nu}\\propto \\nu^{0.4}$.\nSuch a soft spectrum significantly conflicts \nwith the observed synchrotron emission in blazars \n(e. g., Fossati et al. 1998).\n\n\n\n\n(3) \n{\\it Electron acceleration efficiency in the shocked\nregion is evaluated.}\n\n\nOnce $\\rho\/\\rho^{\\rm NT}_{e}$ is bounded as shown \nin Figs. 1, 2, and 3, \nwe can obtain the electron acceleration efficiency \nas\n$e\/e^{\\rm NT}_{e}=\n\\Gamma_{34}\\rho\/\\langle\\gamma_{e}\\rangle\\rho^{\\rm NT}_{e}$\nfor given $\\Gamma_{34}$.\nSince the allowed value of $\\rho\/\\rho_{\\rm NT}$ \nis in the narrow range for the case of pure $e^{\\pm}$ content, \nwe obtain the electron acceleration efficiency as\n$1 < e\/e^{\\rm NT}_{e} < 7$.\nCorrespondingly \nthe total kinetic power of the emission region $L_{\\rm kin}$ \nresides in the range $1