\documentclass[preprint,notfootinbib]{revtex4} \usepackage{graphicx} \graphicspath{{c:/user/dell/desktop/papermgb/plots}} \usepackage{epsfig} \usepackage{amssymb} \usepackage{color} \usepackage{amsmath} \usepackage{lipsum} \usepackage[font={small}]{caption} %\usepackage{pinlabel} \usepackage{graphics} \newcommand{\affilmark}[1]{\rlap{\textsuperscript{\itshape#1}}} \newcommand{\CC}{\Lambda} \newcommand{\rL}{\rho_{\CC}} \newcommand{\rLo}{\rho_{\CC 0}} \newcommand{\ZPE}{V_{\rm ZPE}} \newcommand{\rLV}{\rho_{\CC\rm vac}} \newcommand{\rLb}{\rho_{\CC}^{({\rm b})}} \newcommand{\rLbo}{\rho_{\CC}^{(0)}} \newcommand{\rLbu}{\rho_{\CC}^{(1)}} \newcommand{\rVu}{\rho_{\rm vac}^{(1)}} \newcommand{\Gb}{G^{({\rm b})}} \newcommand{\Gbo}{G^{(0)}} \newcommand{\Gbu}{G^{(1)}} \newcommand{\aeq}{a_{\rm eq}} \newcommand{\ah}{\hat{a}} \newcommand{\Tgz}{T_{\gamma 0}} \hyphenation{cos-mo-lo-gists un-na-tu-ral-ly in-te-gra-ting ne-gli-gi-ble e-xis-ten-ce con-vin-cing des-crip-tion ma-xi-mum ra-dia-tion em-bo-died} \usepackage{epsf} \usepackage{psfrag} \usepackage{epstopdf} \usepackage{graphicx} \usepackage{caption} \usepackage{epstopdf} %\usepackage{dcolumn}% Align table columns on decimal point %\usepackage{bm}% bold math %\usepackage[brazil]{babel} %\usepackage[latin1]{inputenc} %\usepackage{shortvrb} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{epsfig} \baselineskip=24pt \begin{document} \title{ Perturbation Theory in Dark Energy universe} \author{ Z.Molavi , A.Khodam-Mohammadi }}} %\affiliation{Department of Physics, Faculty of Science, Bu-Ali Sina %University, Hamedan 65178, Iran} \begin{abstract} The huge amount of data from the sky gave us the chance to answer the main questions of cosmology and astrophysics. Questions like what is the reason of accelerated expansion of the universe? what is the origin of structures and so on. The most simple theory which answerd these questions was $\Lambda$CDM. However this models suffers from some problems . Theorists used to do investments to solve them. Many alternative models have been proposed for that. Mainly these models are categorized in two group. the first insist on changing the gravity, named modified gravity models. second introduce an exotic kind of matter which have diminated the universe and runs the expansion accelerately. The important note is that many of the models of two group have a good consistansy with the data. So haow we can detect between modified gravity model and DE models. Our important tool is structure formation which is originated in perturbations of components of universe. the perturbation theory is applied in theorytical physics subjects and is the important part of them. In this thesis we introduce perturbed dark energy models and study the structure formation as the observable of perturbations. In current work, the consistency of some dynamical dark energy models based on Gauss-Bonnet invariant, ${\cal G}$, is studied with cosmological data sets. The models are modified form of Gauss Bonnet dark energy, MGB-DE and two other versions which are interacting MGB and MGB + $n_0$. The energy density of proposed models are combinations of powers of the Hubble rate, H, and its time derivative. To inquire the performance of MGB dark energy models, we have used data analyzing methods and numerical solutions, in both background and perturbed levels, based on recent observational data from SNIa, Baryon Acoustic Oscillations (BAO), Hubble parameter, CMB data, and structure formation data surveys. Employing joint datasets and comparing the results to those of LCDM, it is demonstrated that MGB like models describe the cosmic evolution properly. MGB-DE is more consistent to late universe than early universe in statistical view point. \end{abstract} \maketitle % \noindent Key words: cosmology, dark energy, Gauss %Bonnet, observables PACS numbers: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} \noindent Entering the era of precision cosmology, scientists faced huge amount of data, received by several surveys from the mysterious sky. The accurate astrophysical data from distant Ia supernovae [1], [2] cosmic microwave background anisotropy [3], and large scale galaxy surveys , reveals that the universe is nearly spatially flat and is definitely passing an accelerating expansion phase. This is one of the most fundamental concepts in theoretical cosmology and particle physics.\\ During last decades, quite high number of models have been presented in this context. These models are mainly categorized in two classes. The first insists on modifying and extending the gravity itself, named modified gravity. Modified gravity models assume that, the present accelerating epoch is due to geometric effects and corresponds to modify General Relativity, by modifying the Einstein-Hilbert action. Modification of GR, subsequently, leads to new formulation in gravity. The models in this class are$ f(R)$ and $f(T)$ gravity [4], scalar-tensor theories [5], braneworld models [6] Gauss-Bonnet gravity , and so on.\\ Other category is based on presence of an exotic component in stress energy tensor, with sufficiently negative-pressure. This fluid which is known as dark energy, accounts for roughly 75 percent of the universe energy density today. Big variety of dark energy models are proposed, nevertheless the nature and mechanism of dark energy is not known yet. One of the most famous models, vastly used in literature, is cold dark matter plus a cosmological constant named ($\Lambda$CDM) model. It explains the scenario of acceleration of the universe and has an acceptable compatibility with recent observational data [7] [8], [9], [10]. However this model suffers from distinct problems; Fine tuning and coincidence. This made theorists seek for alternative models instead. Actually any offered dark energy model must entail all aspects of quantum theory, particle physics and general relativity. One condition is holographic principle, according to which, the entropy of a system scales not with its volume but with its surface area.The motivation for this, was first arisen from Bekenstein's entropy bound, $S\leq\pi M_p^2 L^2 $, from which it is implied that in entropies well below this bound, quantum field theory fails.\ Imposing a relation between UV and IR cut-offs, as indicated in, conciliated this problem. This relationship was established by using the limit set by black hole formation, that is $L^3\Lambda^4\leq\pi M_p^2 L^2$ where $ M_p$ is the reduced Planck Mass.\\ The studied model in present paper is proposed in[11] as GB dark energy. It complies the Holographic principle and obeys the above bound for black hole formation. The related energy density is proportional to the Gauss-Bonnet (GB) 4-dimensional invariant, ${\cal G}$, in such a way that it has the valid dimension of energy density . This invariant is used in corrections of low energy string gravity [12]. Moreover it is employed in dark energy context with different forms in the action, like coupled to some scalar field, used in modified theories [13], [14] or as modified dark energy models as in . GB DE energy density of is composed of powers of Hubble parameter and its derivative. Dynamic DE models composed of terms like $\dot{H},H\dot{H},H^{2}$, etc. are studied in and the role of terms like $H^{3}$, $\dot{H}H^{2}$ and $H^{4}$ in the evolution of early universe has been investigated . In this work, we study the cosmic evolution of Gauss Bonnet dark energy universe and it's compatibility with the observations through data analysis and the represented results. Current paper is organized as follows: In Sec.II we review the Gauss-Bonnet universe, the background and perturbed equations. In Sec.III we proceed to data analysis and the results of these methods for our model and at the last section we have some discussions. \section{ The Gauss-Bonnet universes} مدل گاوس بونه کیهان‌شناسی، علم مطالعه جهان به صورت یک کل است. علی‌رغم پیچیدگی‌های عظیم این سیستم، می‌توان مدل نسبتاً ساده‌ای برای توصیف دینامیک آن در مقیاس بزرگ ساخت. در این مقیاس‌ها برهمکنش بین اجزاء کیهان توسط قوانین گرانش تعیین می‌شود که به بهترین نحو توسط نظریه نسبیت عام توضیح داده می‌شود. بی‌شک مطرح شدن نسبیت عام توسط اینشتین ما را قادر کرد به یک تئوری قابل سنجش درکیهان برسیم. فهمیدن این حقیقت که جهان در حال انبساط است و در گذشته بسیار گرم و چگال بوده، سوالات دیرین بشر مانند اینکه چرا ما اینجا هستیم؟ چگونه به اینجا رسیدیم؟ و بسیاری سوالات دیگر را به شکل جدیدتری تبدیل کرد، از قبیل اینکه عناصر از کجا آمده‌اند؟ چرا جهان همگن و هموار است؟ کهکشان‌ها چگونه در این فضای هموار به وجود آمده‌اند؟ و... این سوالات و بسیاری دیگر پرسش‌هایی کمی هستند که با دانش فیزیک بنیادی و درک درست از شرایط در کیهان اولیه پاسخ داده شده و از آن مهم‌تر با مشاهدات نجومی قابل تطبیق و آزمودن هستند \subsection{Background equations} The GB dark energy with the energy density is introduced as \begin{equation}\label{eq1} \rho_{d}=\alpha\cal G \end{equation} here $\alpha$ is a dimensionless parameter and ${\cal G}$ is the 4-dimensional Gauss-Bonnet invariant that is defined as \begin{equation}\label{eq2} \mathcal{G}=R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\eta\gamma}R^{\mu\nu\eta\gamma} \end{equation} It's easy to see that for the flat FRW background, $ds^2=-dt^2+a(t)^2\sum^{3}_{i=1}(dx^i)^2$, the GB dark energy density (\ref{eq1}) could be written as \begin{equation}\label{eq3} \rho_{d}=24\alpha\left(H^4+H^2\dot{H}\right) \end{equation} we call it Gauss-Bonnet dark energy density or GB-DE here after. In absence of matter, the Friedmann equation with the energy density given by (\ref{eq2}), in the flat FRW background takes the form \begin{equation} H^2=\frac{\kappa^2}{3}\rho_{\Lambda}=8\alpha \kappa^2 H^2\left(H^2+\dot{H}\right) \end{equation}\label{eq3} where $\kappa^2=8\pi G=M_p^{-2}$. After simplifying \begin{equation} \frac{dH}{dt}+H^2-\frac{1}{8\alpha\kappa^2}=0 \end{equation}\label{eq4} This solution gives the asymptotic behavior of the Hubble parameter at late times. Similar equation has been obtained in for a model of scalar field with non-minimal derivative couplings. The equation has the solution \begin{equation} H(t)=\frac{1}{(8\alpha\kappa^2)^{1/2}}\tanh\left[\frac{1}{\sqrt{8\alpha\kappa^2}}(t-t_0)\right] \end{equation}\label{eq5} This asymptotic solution describes a bouncing universe that at far future approaches the de Sitter solution $H|_{t\rightarrow \infty}=(\frac{1}{8\alpha\kappa^2})^{1/2}$. It was also studied in a scalar-tensor model with auxiliary scalar field. In terms of the variable $x=\log a$ and the scaled Hubble parameter $\tilde{H}=H/H_0$, the Eq. (\ref{eq5}) becomes \begin{equation} \frac{1}{2}\frac{d\tilde{H}^2}{dx}+\tilde{H}^2-\tilde{A}=0 \end{equation} where $\tilde{A}=1/(8\alpha\kappa^2 H_0^2)$. Solving this equation with the initial condition $\tilde{H}^2|_{x=0}=1$, gives the solution \begin{equation} \tilde{H}^2=\tilde{A}+(1-\tilde{A}) e^{-2x} \end{equation} Note that the second term behaves as $a^{-2}$, producing an effect similar to that of the spatial curvature. Note also that $\tilde{A}$ should satisfy $\tilde{A}\leq 1$, since otherwise it would give unphysical results (i.e. $\tilde{H}^2<0$ in the past). Let's draw some conclusions. If we assume that $\tilde{A}\sim 1$ (it can be compared with the parameter of the energy density associated with the cosmological constant, which is about 0.7), then from the definition of $\tilde{A}$ follows \begin{equation} \alpha=\frac{M_p^2}{8\tilde{A}H_0^2}\approx\frac{M_p^2}{8H_0^2} \end{equation}\label{eq8} replacing back this value into the expression for the holographic density (\ref{eq2}), we give \begin{equation} \rho_{\Lambda}\approx 3M_p^2 H_0^{-2}\left(H^4+H^2\dot{H}\right) \end{equation}\label{eq9} giving the right magnitude of the current DE density $\rho_{\Lambda_0}\sim M_p^2 H_0^2$. So we don't need to involve from the beginning the Planck mass, and indeed the black hole bound, to obtain a probably interesting alternative for the holographic DE density. Compared with the usual saturated formula for the holographic density the present proposal could be interpreted as non saturated one. In this way the Friedmann equation and the initial condition take care about the current appropriate value for the dark energy density density. The models studied here are based on GB-DE and are listed below \subsection*{Modified Gauss-Bonnet Dark energy}\ The first model to be studied is an extended form of GB-DE, named modified Gauss-Bonnet Dark energy or MGB-DE [13] with the density of the following form \begin{equation}\label{eq4} \rho_{d}=\gamma H^4+\beta H^2\dot{H} \end{equation} For a single component universe (in the absence of matter), the Friedmann equation with the energy density given by (\ref{eq2}), in the flat FRW background takes the form \begin{equation}\label{eq5} \beta\frac{dH}{dt}+\gamma H^2-\frac{3}{\kappa^2}=0 \end{equation} we assume that $\kappa^2=8\pi G=M_p^{-2}$. With suitable initial condition, this equation is solved and discussed in[14].\\ In presence of matter(baryonic and dark) the Friedmann equation becomes non-linear and does not have exact solution. Adding the matter term $\rho_m=(\rho_c+\rho_b)=\rho_{m0}a^{-3}$, the Friedmann equation reads \begin{equation} a \tilde \beta E^3 E' - E^2 + \tilde\gamma E^4 +\frac{ \Omega_m}{a^3}=0\ \label{eqc1} \end{equation} the equation is written in terms of scale factor.\ We used $\tilde{\gamma}=\kappa^2H_0^2\gamma/3$, $\tilde{\beta}=\kappa^2H_0^2\beta/3$ and $\Omega_{m0}=\kappa^2\rho_{m0}/(3H_0^2)$ ($\Omega_m=\kappa^2\rho_m/(3H^2)$). $E=H/H_0$ is scaled Hubble parameter. With the initial condition, $E(1)=1$, and different amounts of parameters, this equation could be solved numerically.\\Since we are interested in late universe data or their mixture with those of background solutions in recent time, we can neglect radiation in the evolution equations. \subsection*{The Interacting MGB model}\ We introduce interacting MGB DE as the second model. Dark energy models in GR, suffer from the coincidence problem referred to energy density orders of dark matter and dark energy. This problem could be solved by assuming continuous energy exchange between dark sectors. The signature of non-gravitational interaction term $\bar Q_c$, in the continuity equations, shows the direction of energy transfer \begin{eqnarray} \dot{\bar{\rho}}_c+3H\bar{\rho}_c&=& \bar{Q}_c \,, \label{rhocb}\\ \dot{\bar{\rho}}_d+3H(1+w)\bar{\rho}_d&=& \bar{Q}_d \,, \label{rhoxb} \end{eqnarray} Here $w=\bar{P}_d/\bar{\rho}_d$ and $ \bar{Q}_d= -\bar{Q}_c$. The $\bar Q_c~(\bar Q_d)$ as the rate of energy density transfer is usually introduced as \begin{equation} \label{Q} \bar Q_c=-\left(\Gamma_c\bar{\rho}_c+\Gamma_d\bar{\rho}_d \right) \,, \end{equation} where $\Gamma_A$($\Gamma_c$ or $\Gamma_d$) are constant energy density transfer rates and show the decay of dark matter to dark energy, or vice versa (Baryons (b) and photons ($\gamma$) are not coupled to dark energy). We are interested in the special case $\Gamma_d=0$ and choose $\Gamma_c=3H\xi^2\rho_c$. Hence from the continuity equation, the dark matter density is \begin{equation} {\bar{\rho}}_c={\rho_0 }_c a^{-3(1-\xi^2)} \end{equation} so the Eq.\ref{eqc1} changes to \begin{equation} a \tilde \beta E^3 E' - E^2 + \tilde\gamma E^4 +\frac{ \Omega_b}{a^3}+\frac{ \Omega_c}{a^{3(1-\xi^2)} }=0\;.\label{eqcint1} \end{equation} \subsection*{ MGB with a constant}\ For the third model we consider the MGB with an arbitrary constant like the approach in[15] and . For this, we add the constant $n_0$ to the Eq.\ref{eqc1} directly. In the recent era, as $H(z)$ becomes smaller, this case tends to $ \Lambda$LCDM. \begin{equation} a \tilde \beta E^3 E' - E^2 + \tilde\gamma E^4 +\frac{ \Omega_m}{a^3}+n_0=0\;.\label{eqccc} \end{equation} \subsection{The linear perturbed equations} In perturbation theory, we consider a perturbed space-time that is close to the background space-time. This means that there exists a coordinate system on the perturbed space-time, where its metric can be written as \begin{equation} g_{\mu\nu}=\bar{g}_{\mu\nu}+\delta g_{\mu\nu} \end{equation} here $\bar{g}_{\mu\nu}$ is the metric of the background. Thus metric perturbations are divided into a scalar, vector and a tensor part, which do not couple to each other in first-order perturbation theory and evolve independently. Scalar perturbations are of special importance. They couple to density and pressure perturbations and cause gravitational instabilities. This make overdensities grow and become more overdense. The outcome is formation and growth of Large Scale Structure, from small initial perturbations. In order to study the linear perturbation theory, we start with perturbation equations. In the perturbed FRW universe, with scalar perturbations and in absence of anisotropic stress the line element is \begin{equation}\label{eq:line-element} ds^2=-(1+2\Phi)dt^2+a^2(t)(1-2\Psi)d\vec{x}^2\;, \end{equation} $\Phi$ and $\Psi$ are metric perturbations known as the Bardeen potentials. Perturbations in density(matter or energy) and pressure are \begin{equation} \rho=\bar{\rho}+\delta \rho \end{equation} \begin{equation} p=\bar{p}+\delta p \end{equation} where $\bar{p}$ and $\bar{\rho}$ are pressure and density of background. The perturbed energy momentum tensor is \begin{equation} T^\mu_{\;\;\nu}=\bar{T}^\mu_{\;\;\nu}+\delta T^\mu_{\;\;\nu} \end{equation} The DE component is expected to be smooth and we consider perturbations only on the matter component of the cosmic fluid. The energy-momentum continuity equation needs $ T^\mu_{\;\;\nu;\mu}=0$. In absence of interaction between dark matter and dark energy and in Fourier space this equation leads to \begin{equation} \dot{\delta_{\rm m}}=(1+\omega_{\rm m})(3 \dot{\Psi}+\frac{k}{a} \theta_{\rm m}) \end{equation} \begin{equation} \dot{\theta_{\rm m}}+(1-3 \omega_{\rm m}) H{\theta_{\rm m}}=\frac{k}{a} (\Phi+\frac{\omega_{\rm m}}{1+\omega_{\rm m}} \delta_{\rm m}) \end{equation} in which $\delta_m(=\delta \rho_m/\rho_m)$ is dark matter density contrast and $\theta_{\rm m}$ is the divergence of velocity field. We are interested in the case of non-relativistic fluid $ (\omega_{\rm m}=0)$ and scales much smaller than Hubble radius $(k\gg aH)$. So that the above equations result into a second order differential equation, for evolution of matter density contrast. In terms of scale factor it reads \begin{equation} \delta^{\prime\prime}_{\rm m}+\left(\frac{3}{a}+\frac{E^{\prime}}{E}\right)\delta^{\prime}_{\rm m}- \frac{3}{2a^2 E^2}\Omega_{\rm m}\delta_{\rm m}=0 \;. \label{eqc2} \end{equation} For coupled MBG dark energy, the changes are exhibited in the background evolution equations, in $\Omega_{m}$ term.\\ Solving the system of equations (\ref{eqc1}) and (\ref{eqc2}) gives the evolution of density contrast for the models. In order to study structure formation and compare models with data, it is needed to use some definitions. The first concept is the growth rate function defined with the following equation \begin{equation} \label{faa} f(a)=\frac{d\ln \delta_{m}}{d\ln a}\;. \end{equation}\ The observable we need to measure in structure formation context, is $f\sigma_8$, in which, $\sigma_8$ is \begin{equation} \label{sig8} \sigma_{8}(a)=\sigma_{8,0}\frac{ \delta_{m}(a)}{\delta_{0}}\;. \end{equation} where $\delta_{0}$ is the density contrast in $a=1$.\\ Another important quantity we can refer to is the $\gamma$-index.This index is related to matter perturbations and is defined via $f(z)\simeq \Omega_m (z)^{\gamma(z)}$, so the growth index $\gamma(z)$ can be written as \begin{equation} \label{eq:gamma} \gamma(z)\cong\frac{\ln\f(z)}{\ln\Omega_m(z)}\ \end{equation} \section{Observational constraints} In this section we use data analyzing methods in order to find the best fit values of the parameters in background and perturbed level for MGB-DE universe. To study the expansion history and the growth rate of structures we ought to define some observables at first. The most important are the background expansion indicators such as distance modulus of Supernovae Type Ia, Hubble parameter, Baryon acoustic oscillations (BAO) and CMB power spectrum. The observable related to perturbation growth rate of structures is $ f\sigma _8 $ data and is taken into account correspondingly .\\ The respective parameters to be defined are: parameters of the MGB-DE models, $ \tilde \beta$, $ \tilde \gamma$,$\xi$,$n_0$ plus usual cosmological parameters like current matter and baryon density parameters, $\Omega_{\rm m}^0$ , $\Omega_{\rm b}^0$ and $h=H_{0}/100$ (normalized Hubble constant). \\ Available observational data sets, used for these calculations are: distance modulus of Supernovae Type Ia, Baryon acoustic oscillations (BAO), Hubble evolution data, growth rate data f$\sigma 8$ and WMAP data for CMB which will be explained; \subsection{observables} The main evidences for cosmic accelerated expansion are Supernovae. Measuring the luminosity distance of these objects not only gives useful information about history of early universe but also constrain model parameters in low and intermediate redshifts confidently. Referred catalogue is the SnIa distance module from Union 2.1 sample , which includes 580 SnIa over the redshift range $0