We discuss the correspondence between the DGP brane cosmology and 5D Ricci-flat cosmology by letting their metrics equal each other. By this correspondence, a specific geometrical property of the arbitrary integral constant I in DGP metric is given and it is related to the curvature of 5D bulk. At the same time, the relation of arbitrary functions $μ$ and $ν$ in a class of Ricci-flat solutions is obtained from DGP brane metric.
We consider the imbalance of matter and antimatter by using a gravitational baryogenesis mechanism in the background of Dvali-Gabadadze-Porrati (DGP) brane cosmology. By taking into account a flat Friedmann-Lemaitre-Robertson-Walker (FLRW) metric in the DGP brane model, we find that for a radiation dominated universe, $w = 1/3$, the ratio of baryon number density to entropy from the gravitational baryogenesis is not zero, contrary to ordinary general relativity. Also, we study the ratio of baryon number density to entropy against the observational constraints in DGP cosmology.
In this article we study an intermediate inflationary model in the context of Dvali-Gabadadze-Porrati (DGP) cosmology caused by a tachyon scalar field. Considering slow-roll inflation we discuss the dynamics of the Universe. Using perturbation theory, we estimate some of the model parameters numerically and compare them with the recent results from Planck satellite.
We investigated the effects of bulk scalar field in the braneworld cosmological scenario. The Friedmann equations and acceleration condition in presence of the bulk scalar field for a zero tension brane and cosmological constant are studied. In DGP model the effective Einstein equation on the brane is obtained with bulk scalar field. The rescaled bulk scalar field on the brane in the DGP model behaves as an effective four dimensional field, thus standard type cosmology is recovered. In present study of the DGP model, the late-time accelerating phase of the universe can be explained .
The varying speed of light (VSL) has been used in cosmological models in which the physical constants vary over time. On the other hand, the Dvali, Gabadadze and Porrati (DGP) brane world model, especially its normal branch has been extensively discussed to justify the current cosmic acceleration. In this article we show that the normal branch of DGP in VSL cosmology leads to a self-accelerating behavior and therefore can interpret cosmic acceleration. Applying statefinder diagnostics demonstrate that our result slightly deviates ΛCDM model.
We study cosmological implications of a Lorentz invariance violating DGP-inspired braneworld scenario. A minimally coupled scalar field and a single, fixed-norm, Lorentz-violating timelike vector field within an interactive picture provide a wide parameter space which accounts for late-time acceleration and transition to phantom phase of the scalar field.
This paper is aimed at investigating a 5D holographic dark energy in DGP-BRANE cosmology by employing a combination of Sne Ia, BAO and CMB observational data to fit the cosmological parameters in the model. We describe the dynamic of a FRW for the normal branch ($ε$ = +1) of solutions of the induced gravity brane-world model. We take the matter in 5D bulk as holographic dark energy that its holographic nature is reproduced effectively in 4D. The cosmic evolution reveals that the effective 4D holographic dark energy behaves as quintessence while taking into account the 4D cold dark matter resul…
A proposal to study the effect of interaction in an agegraphic dark energy model in DGP brane-world cosmology is presented in this manuscript. After explaining the details, we proceed to apply the dynamical system approach to the model to analyze its stability. We first, constrain model parameters with a variety of independent observational data such as cosmic microwave background anisotropies, baryon acoustic oscillation peaks and observational Hubble data. Then, we obtain the critical points related to different cosmological epochs. In particular, we conclude that in the presence of interact…
In this work we have investigated the emergent scenario of the universe described by Loop quantum cosmology model, DGP brane model and Kaluza-Klein cosmology. Scalar field along with barotropic fluid as normal matter is considered as the matter content of the universe. In Loop quantum cosmology it is found that the emergent scenario is realized with the imposition of some conditions on the value of the density of normal matter in case of normal and phantom scalar field. This is a surprising result indeed considering the fact that scalar field is the dominating matter component. In case of Tach…
The dynamics of Dvali-Gabadadze-Porrati Cosmology (DGP) braneworld with an anisotropic brane is studied. The Friedmann equations and their solutions are obtained for two branches of anisotropic DGP model. The late time behavior in DGP cosmology is examined in the presence of anisotropy which shows that universe enters a self-accelerating phase much later compared to the isotropic case. The acceleration conditions and slow-roll conditions for inflation are obtained.
Evolution of the density parameter in the anisotropic DGP braneworld model is studied. The role of shear and cross-over scale in the evolution of $Ω_ρ$ is examined for both the branches of solution in the DGP model. The evolution is modified significantly compared to the FRW model and further it does not depend on the value of $γ$ alone. Behaviour of the cosmological density parameter $Ω_ρ$ is unaltered in the late universe. The study of decceleration parameter shows that the entry of the universe into self accelerating phase is determined by the value of shear. We also obtain an estimate of t…
In this paper we consider a $Z_2$ symmetrical 3-brane embedded in a 5-dimensional spacetime. We study the effective Einstein equation and acceleration condition in presence of the quintom dark energy fluid as the bulk matter field. It is shown that the time-dependent bulk quintom field induces a time-dependent cosmological constant on the brane. In the framework of the DGP model, the effective Einstein equation is obtained in two different cases: i) where the quintom field is considered as the bulk matter field and the brane is empty and, ii) where the quintom dark energy is confined on the br…
In this proposal we study the problem of the virial mass discrepancy in the context of DGP brane gravity. DGP model is a kind of brane-world model such that the corrections to the usual gravity occurred in the large distance limit relative to the distance scale of the model defined as a ratio of the brane Planck scale to the bulk Planck scale. The extra dimension of this model is noncompact. This model is composed with an Einstein-Hilbert action in 5 dimensions plus an induced 4D action guarantees the existence of gravity on the brane. The importance of this model is that it can explain the se…
We solve for the behaviour of cosmological perturbations in the Dvali-Gabadadze-Porrati (DGP) braneworld model using a new numerical method. Unlike some other approaches in the literature, our method uses no approximations other than linear theory and is valid on large scales. We examine the behaviour of late-universe density perturbations for both the self-accelerating and normal branches of DGP cosmology.
We consider a DGP brane scenario where a scalar field is present on the brane through the introduction of a scalar potential, itself motivated by the notion of modified gravity. This theory predicts that the mass appearing in the gravitational potential is modified by the addition of the mass of the scalar field. The cosmological implications that such a scenario entails are examined and shown to be consistent with a universe expanding with power-law acceleration.
We present an holographic brane-world model of the Dvali-Gabadadze-Porrati (DGP) scenario with and without a Gauss-Bonnet term (GB) in the bulk. We show that an holographic dark energy component with the Ricci scale as the infra-red cutoff can describe the late-time acceleration of the universe. In addition, we show that the dimensionless holographic parameter is very important in characterising the DGP branches, and in determining the behaviour of the Ricci dark energy as well as the asymptotic behaviour of the brane. On the one hand, in the DGP scenario the Ricci dark energy will exhibit a p…
We consider a DGP inspired brane scenario where the action on the brane is augmented by a function of the Ricci scalar, ${\cal L}(R)$. The cosmological implications that such a scenario entails are examined for $R^{n}$ and shown to be consistent with a universe expanding with power-law acceleration. It is shown that two classes of solutions exist for the usual FRW metric and small Hubble radii. When the Hubble radius becomes larger, we either have a transition to a fully 5D regime or to a self-inflationary solution which produces a late accelerated expansion such that the radius becomes a func…
We consider a five dimensional DGP-brane scenario endowed with a non-minimally coupled scalar field within the context of Brans-Dicke theory. This theory predicts that the mass appearing in the gravitational potential is modified by the addition of the mass of the effective intrinsic curvature on the brane. We also derive the effective four dimensional field equations on a 3+1 dimensional brane where the fifth dimension is assumed to have an orbifold symmetry. Finally, we discuss the cosmological implications of this setup, predicting an accelerated expanding universe with a value of the Brans…
We analyse a DGP brane filled with a k-essence field and assume the k-field evolving linearly with the cosmic time of the brane. We then solve analytically the Friedmann equation and deduce the different behaviour of the brane at the low and the high energy regimes. The asymptotic behaviour can be quite different involving accelerating branes, big bangs, big crunches, big rips or quiescent singularities. The latter correspond to a type of sudden singularity.
In this paper, we use Statefinder hierarchy method to distinguish between bimetric theory of massive gravity, galileon modified gravity and DGP models applied to late time expansion of the universe. We also carry out comparison between bimetric and DGP models using Statefinder pairs {r, s} and {r, q}. We show that statefinder diagnostic can differentiate between ΛCDM and above mentioned cosmological models of dark energy, and finally show that Statefinder S2 is an excellent discriminant of ΛCDM and modified gravity models.
In this thesis we investigate certain cosmological brane world models of the Randall-Sundrum type. The models are motivated by string theory but we focus on the phenomenology of the cosmology. Two models of specific interest are the Dvali-Gabadadze-Porrati (DGP, induced-gravity) model, where the brane action is modified, and the Gauss-Bonnet model where the bulk action is modified. Both of these modifications maybe motivated by string theory. We provide a brief review of Randall-Sundrum models and then consider the Kaluza-Klein modes on Minkowski and de Sitter branes, in both the two and one b…
We study Dvali-Gabadadze-Porrati braneworlds with finite thickness. In respect to standard (thin) DGP Friedmann equation, finite thickness of the brane causes a subtle modification of the cosmological equations that can lead to significant physical consequences. The resulting cosmology is governed by two length scales which are associated with the brane thickness and with the crossover length respectively. In this setup both early inflation and late-time acceleration of the expansion are a consequence of the 5D geometry. At early times, as well as at late times, 5D effects become dominant (gra…
We investigate observational constraints on the normal branch of the warped DGP braneworld cosmology by using observational data from Type Ia Supernovae (SNIa), Baryon Acoustic Oscillations (BAO), Cosmic Microwave Background (CMB) and Baryon Gas Mass Fraction of cluster of galaxies. The best fit values of model free parameters are: $Ω_m=0.240^{+0.050}_{-0.130}$ and $Ω_{r_c}=0.000^{+0.014}$ at $1σ$ confidence interval by using Gold sample SNIa+CMB shift parameter+BAO+Gas mass fraction of baryons in cluster of galaxies. The results for essence sample SNIa combined with CMB shift parameter, BAO a…
The DGP brane-world model provides a simple alternative to the standard LCDM cosmology, with the same number of parameters. There is no dark energy - the late universe self-accelerates due to an infrared modification of gravity. We compute the joint constraints on the DGP model from supernovae, the cosmic microwave background shift parameter, and the baryon oscillation peak in the SDSS luminous red galaxy sample. Flat DGP models are within the joint 2 sigma contour, but the LCDM model provides a significantly better fit to the data. These tests are based on the background dynamics of the DGP m…
A model universe is proposed in the framework of 5-dimensional noncompact Kaluza-Klein cosmology which is not Ricci flat. The 4D part as the Robertson-Walker metric is coupled to conventional perfect fluid, and its extra-dimensional part is coupled to a dark pressure through a scalar field. It is shown that neither early inflation nor current acceleration of the 4D universe would happen if the non-vacuum states of the scalar field would contribute to 4D cosmology.
In the standard picture of cosmology it is predicted that a phase transition, associated with chiral symmetry breaking after the electroweak transition, has occurred at approximately 10 μseconds after the Big Bang to convert a plasma of free quarks and gluons into hadrons. We consider the quark-hadron phase transition in a DGP brane world scenario within an effective model of QCD. We study the evolution of the physical quantities useful for the study of the early universe, namely, the energy density, temperature and the scale factor before, during, and after the phase transition. Also, due to …
We discuss the impact of Lorentz violation on the cosmology. Firstly, we show that the Lorentz violation affects the dynamics of the chaotic inflationary model and gives rise to an interesting feature. Secondly, we propose the Lorentz violating DGP brane models where the Lorentz violating terms on the brane accelerate the current universe. We conjecture that the ghost disappears in the Lorentz violating DGP models.
We derive the modified Friedmann equations for a generalization of the Dvali-Gabadadze-Porrati (DGP) model in which the brane has one additional compact dimension. The main new feature is the emission of gravitational waves into the bulk. We study two classes of solutions: First, if the compact dimension is stabilized, the waves vanish and one exactly recovers DGP cosmology. However, a stabilization by means of physical matter is not possible for a tension-dominated brane, thus implying a late time modification of 4D cosmology different from DGP. Second, for a freely expanding compact directio…
We find an exact solution of Kerr-Newman-de Sitter type on the braneworld(4D) of the DGP model. When a constant 4D Ricci scalar is assumed, only zero(flat) and a positive(de-Sitter) values satisfy the Hamiltonian constraint equation coming from the extra dimension. With a Z_2-symmetry across the brane and a stationary and axisymmetric metric ansatz on the brane, we solve the constraint equation exactly in the Kerr-Schild form with de-Sitter background. In the de-Sitter background this Kerr-Schild solution is well behaved under Boyer-Lindquist transformation: the constraint equation is preserve…
In the present work we have studied some features of the generalized Brans-Dicke (BD) model in which the scalar field is allowed to couple non-minimally with matter sector. Extended holographic Ricci dark energy (EHRDE) has been considered in the above framework of BD cosmology. Some restrictions have been derived for the BD parameter $ω$ and a stronger matter-chameleon coupling has been observed with expansion of the universe. In this framework, the equation of state parameter of EHRDE has behaved like quintom. Also, we have reconstructed the potential and coupling function for BD model for t…
We analyze the background cosmology for an extension of the DGP gravity with Gauss-Bonnet term in the bulk and $f(R)$ gravity on the brane. We investigate implications of this setup on the late-time cosmic history. Within a dynamical system approach, we study cosmological dynamics of this setup focusing on the role played by curvature effects. Finally we constraint the parameters of the model by confrontation with recent observational data.
In this letter, cosmology of a simple NMDC gravity with $ξR ϕ_{,μ}ϕ^{,μ}$ term and a free kinetic term is considered in flat geometry and in presence of dust matter. A logarithm field transformation $ϕ' = μ\ln ϕ$ is proposed phenomenologically. Assuming slow-roll approximation, equation of motion, scalar field solution and potential are derived as function of kinematic variables. The field solution and potential are found straightforwardly for power-law, de-Sitter and super-acceleration expansions. Slow-roll parameters and slow-roll condition are found to depend on more than one variable. At l…
In this article we study a constant-roll inflationary model in the context of the DGP brane-world cosmology caused by a quintessence scalar field. We determine an analytical solution for the Friedman equation coupled to the equation of motion of the scaler field. The evolution of the primordial scalar and tensor perturbations is also studied. To check the viability of the model we use numerical approaches and plot some figures. Our results for the scalar spectral index and the tensor to scaler ratio show good consistency with observations for given values of model parameters.
The purpose of this paper is to study the Tsallis agegraphic dark energy with an interaction term between dark energy and dark matter in the DGP brane-world scenario. For this, we assume some initial conditions to obtain the dark energy density, deceleration, dark energy EoS, and total EoS parameters. Then, we analyze the statefinder parameters, $ω'{}_{DE}-ω_{DE}$ plots, and classical stability features of the model. The results state that the deceleration parameter provides the phase transition from decelerated to accelerated phase. The $ω_{DE}$ graphs show the phantom behavior, while the $ω_…
In this work, we studied the Power Law and the Logarithmic Entropy Corrected versions of the Ricci Dark Energy (RDE) model in a spatially non-flat universe and in the framework of Hořava-Lifshitz cosmology. For the two cases containing non-interacting and interacting RDE and Dark Matter (DM), we obtained the exact differential equation that determines the evolutionary form of the RDE energy density parameter. Moreover, we obtained the expressions of the deceleration parameter $q$ and, using a parametrization of the equation of state (EoS) parameter $ω_D$ as $ω_D\left(z\right)=ω_0+ω_1 z$, we de…
In the framework of noncompact Kaluza-Klein theory, we investigate a $(4+1)$-dimensional universe consisting of a $(4+1)$ dimensional Robertson-Walker type metric coupled to a $(4+1)$ dimensional energy-momentum tensor. The matter part consists of an energy density together with a pressure subject to 4D part of the $(4+1)$ dimensional energy-momentum tensor. The dark part consists of just a dark pressure $\bar{p}$, corresponding to the extra-dimension endowed by a scalar field, with no element of dark energy. It is shown that for a flat universe, coupled with the non-vacuum states of the scala…
We revisit spatially flat FLRW cosmology in light of recent advances in standard model relativistic fluid dynamics. Modern fluid dynamics requires the presence of curvature-matter terms in the energy-momentum tensor for consistency. These terms are linear in the Ricci scalar and tensor, such that the corresponding cosmological model is referred to as ``Ricci cosmology''. No cosmological constant is included, there are no inflaton fields, bulk viscosity is assumed to be zero and we only employ standard Einstein gravity. Analytic solutions to Ricci cosmology are discussed, and we find that it is…
We provide a consistency relation between cosmological observables in general relativity without relying on the equation of state of dark energy. The consistency relation should be satisfied if general relativity is the correct theory of gravity and dark energy clustering is negligible. As an extension, we also provide the DGP counterpart of the relation.
Recently, a new cosmological framework, dubbed Ricci Cosmology, has been proposed. Such a framework has emerged from the study of relativistic dynamics of fluids out of equilibrium in a curved background and is characterised by the presence of deviations from the equilibrium pressure in the energy-momentum tensor which are due to linear terms in the Ricci scalar and the Ricci tensor. The coefficients in front of such terms are called the second order transport coefficients and they parametrise the fluid response to the pressure terms arising from the spacetime curvature. Under the preliminary …
We have investigated some bouncing models in the framework of an extended gravity theory where the usual Ricci scalar in the gravitational action is replaced by a sum of the Ricci scalar and a term proportional to the trace of the energy momentum tensor. The dynamical parameters of the model are derived in most general manner. We considered two bouncing scenarios through an exponential and a power law scale factor. The non singular bouncing models also favour a late time cosmic speed up phenomenon. The dynamical behaviour of the equation of state parameter is studied for the models. It is obse…
We consider the dynamics of a cosmological substratum of pressureless matter and holographic dark energy with a cutoff length proportional to the Ricci scale. Stability requirements for the matter perturbations are shown to single out a model with a fixed relation between the present matter fraction $Ω_{m0}$ and the present value $ω_{0}$ of the equation-of-state parameter of the dark energy. This model has the same number of free parameters as the $Λ$CDM model but it has no $Λ$CDM limit. We discuss the consistency between background observations and the mentioned stability-guaranteeing paramet…
We discuss the averaging hypothesis tacitly assumed in standard cosmology. Our approach is implemented in a "3+1" formalism and invokes the coarse graining arguments, provided and supported by the real-space Renormalization Group (RG) methods. Block variables are introduced and the recursion relations written down explicitly enabling us to characterize the corresponding RG flow. To leading order, the RG flow is provided by the Ricci-Hamilton equations studied in connection with the geometry of three-manifolds. The properties of the Ricci-Hamilton flow make it possible to study a critical behav…
It is shown that if the timelike eigenvector of the Ricci tensor be hypersurface orthogonal so that the space time allows a foliation into space sections then the space average of each of the scalar that appear in the Raychaudhuri equation vanishes provided the strong energy condition holds good. This result is presented in the form of a singularity theorem.
We explore the cosmological implications provided by the geodetic brane gravity action corrected by an extrinsic curvature brane term, describing a codimension-1 brane embedded in a 5D fixed Minkowski spacetime. In the geodetic brane gravity action, we accommodate the correction term through a linear term in the extrinsic curvature swept out by the brane. We study the resulting geodetic-type equation of motion. Within a Friedmann-Robertson-Walker metric, we obtain a generalized Friedmann equation describing the associated cosmological evolution. We observe that, when the radiation-like energy …
The field equations in FRW background for the so called C-theories are presented and investigated. In these theories the usual Ricci scalar is substituted with $f(\mathcal{R})$ where $\mathcal{R}$ is a Ricci scalar related to a conformally scaled metric $\hat{g}_{μν} = \mathcal{C}(\mathcal{R})g_{μν}$, where the conformal factor itself depends on $\mathcal{R}$. It is shown that homogeneous perturbations of this Ricci scalar around general relativity FRW background of a large class of these theories are either inconsistent or unstable.
Cosmology of f(Q,L_m) gravity with Holographic Ricci Dark Energy: Early-Time Inflation and Late-Time Acceleration and RGUP Corrected Observables
Amity University Kolkata（阿米提大学加尔各答分校）；Ahram Canadian University, Faculty of Engineering, Basic Science Department（艾哈兰加拿大大学工程学院基础科学系）；The Egyptian Center for Theoretical Physics (ECTP)（埃及理论物理中心）
We consider the cosmology of the Ricci-tensor-squared gravity in the Palatini variational approach. The gravitational action of standard general relativity is modified by adding a function f(R^abR_ab) to the Einstein-Hilbert action, and the Palatini variation is used to derive the field equations. A general method of obtaining the background and first-order covariant and gauge-invariant perturbation equations is outlined. As an example, we consider the cosmological constraints on such theories arising from the supernova type Ia and cosmic microwave background observations. We find that the bes…
The two-dimensional renormalization group acting as the Ricci flow $Λ\frac{\partial}{\partialΛ} g_{μν} = R_{μν}$ produces a specific 1+3 dimensional space-time metric which describes an expanding universe that starts with a big bang $a \sim t^{\scriptscriptstyle 1/\sqrt3}$ then decelerates until $z=0.2$ then accelerates until ending at $t_{\max}=1.6\,t_{H}$ with a big blowup $a \sim (t_{\max}-t)^{\scriptscriptstyle -1/\sqrt3}$. The only free parameters are the overall time scale and the value of the present time $t_{0}$. These are fixed by the Hubble constant $H_{0}=t_{H}^{-1}$ and the present…
We study the long distance behaviour of brane theories with quasi-localized gravity. The 5D effective theory at large scales follows from a holographic renormalization group flow. As intuitively expected, the graviton is effectively four dimensional at intermediate scales and becomes five dimensional at large scales. However in the holographic effective theory the essentially 4D radion dominates at long distances and gives rise to scalar anti-gravity. The holographic description shows that at large distances the GRS model is equivalent to the model recently proposed by Dvali, Gabadadze and Por…
Braneworld scenarios consider our observable universe as a brane embedded in a 5D space, named bulk. In this work, I derive the field equations of a braneworld model in a generalized theory of gravitation, namely $f(R,T)$ gravity, with $R$ and $T$, representing the Ricci scalar and the trace of the energy-momentum tensor, respectively. The cosmological parameters obtained from this approach are in agreement with recent constraints from Supernovae Ia data combined with baryon acoustic oscillations and cosmic microwave background observations, favouring such an alternative description of the uni…
We analytically derive the covariant form of the Riemann (curvature) tensor for homogeneous Metric-Affine Cosmologies. That is, we present, in a Cosmological setting, the most general covariant form of the full Riemann tensor including also its non-Riemannian pieces which are associated to spacetime torsion and non-metricity. Having done so we also compute a list of the curvature tensor by-products such as Ricci tensor, homothetic curvature, Ricci scalar, Einstein tensor etc. Finally we derive the generalized Metric-Affine version of the usual Gauss-Bonnet density in this background and demons…
The Newtonian Lagrangian perturbation theory is a widely used framework to study structure formation in cosmology in the nonlinear regime. We review a general-relativistic formulation of such a perturbation approach, emphasizing results on already developed extensive formalism including among other aspects: the non-perturbative modeling of Ricci and Weyl curvatures, gravitational waves and pressure-supported fluids. We discuss subcases of exact solutions related to Szekeres Class II and, as exact average model, Ricci-flat LTB models. This latter forms the basis of a generalization that we then…
Multidimensional cosmological model describing the evolution of one Einstein space of non-zero curvature and n Ricci-flat internal spaces is considered. The action contains several dilatonic scalar fields and antisymmetric forms. When forms are chosen to be proportional of volume forms of p-brane submanifolds of internal space manifold, the Toda-like Lagrange representation is obtained. Wheeler-De Witt equation for the model is presented. The exact solutions in classical and quantum cases are obtained when dimensions of p-branes and dilatonic couplings obey some orthogonality conditions.
A theory of gravitation is proposed, modeled after the notion of a Ricci flow. In addition to the metric an independent volume enters as a fundamental geometric structure. Einstein gravity is included as a limiting case. Despite being a scalar-tensor theory the coupling to matter is different from Jordan-Brans-Dicke gravity. In particular there is no adjustable coupling constant. For the solar system the effects of Ricci flow gravity cannot be distinguished from Einstein gravity and therefore it passes all classical tests. However for cosmology significant deviations from standard Einstein cos…
We assume the cosmological dark sector to consist of pressureless matter and holographic dark energy with a cutoff length proportional to the Ricci scale. The requirement of separate energy-momentum conservation of the components is shown to establish a relation between the matter fraction and the (necessarily time-dependent) equation-of-state parameter of the dark energy. Focusing on intrinsically adiabatic pressure perturbations of the dark-energy component, the matter perturbations are found as linear combinations of the total energy-density perturbations of the cosmic medium and the relati…
