During hundred years of General Relativity (GR), many significant gravitational phenomena have been predicted and discovered. General Relativity is still the best theory of gravity. Nevertheless, some (quantum) theoretical and (astrophysical and cosmological) phenomenological difficulties of modern gravity have been motivation to search more general theory of gravity than GR. As a result, many modifications of GR have been considered. One of promising recent investigations is Nonlocal Modified Gravity. In this article we present a brief review of some nonlocal gravity models with their cosmological solutions, in which nonlocality is expressed by an analytic function of the d'Alembert-Beltrami operator $\Box$. Some new results are also presented.
We consider some cosmological aspects of nonlocal modified gravity with $Λ$ term, where nonlocality is of the type $R \mathcal{F}(\Box) R$. Using ansatz of the form $\Box R = r R +s,$ we find a few a(t) nonsingular bounce cosmological solutions for all three values of spatial curvature parameter k. We also discuss this modified gravity model from F(R) theory point of view.
Despite many nice properties and numerous achievements, general relativity is not a complete theory. One of actual approaches towards more complete theory of gravity is its nonlocal modification. We present here a brief review of nonlocal gravity with its cosmological solutions. In particular, we pay special attention to two nonlocal models and their nonsingular bounce solutions for the cosmic scale factor.
The recent nonlocal generalization of Einstein's theory of gravitation reduces in the Newtonian regime to a nonlocal and nonlinear modification of Poisson's equation of Newtonian gravity. The nonlocally modified Poisson equation implies that nonlocality can simulate dark matter. Observational data regarding dark matter provide limited information about the functional form of the reciprocal kernel, from which the original nonlocal kernel of the theory must be determined. We study this inverse problem of nonlocal gravity in the linear domain, where the applicability of the Fourier transform meth…
A nonlocal gravity model (2.1) was introduced and considered recently [49], and two exact cosmological solutions in flat space were presented. The first solution is related to some radiation effects generated by nonlocal dynamics on dark energy background, while the second one is a nonsingular time symmetric bounce. In the present paper we investigate other possible exact cosmological solutions and find some the new ones in nonflat space. Used nonlocal gravity dynamics can change background topology. To solve the corresponding eqations of motion, we first look for a solution of the eigenvalue …
We consider a new modified gravity model with nonlocal term of the form $R^{-1} \mathcal{F}(\Box) R. $ This kind of nonlocality is motivated by investigation of applicability of a few unusual ansätze to obtain some exact cosmological solutions. In particular, we find attractive and useful quadratic ansatz $\Box R = q R^{2}.$
A non-local modified gravity model with an analytic function of the d'Alembert operator that has been proposed as a possible way of resolving the singularities problems in cosmology is considered. We show that the anzats that is usually used to obtain exact solutions in this model provides a connection with $f(R)$ gravity models.
The Weinberg no-go theorem for cosmological constant and nonlocal gravity
Università degli Studi di Napoli “Federico II”（那不勒斯费德里科二世大学）；Scuola Superiore Meridionale（南方高等学院）；INFN Sezione di Napoli（意大利国家核物理研究所那不勒斯分部）；Van Swinderen Institute for Particle Physics and Gravity, University of Groningen（格罗宁根大学范斯温德恩粒子物理与引力研究所）
This paper is devoted to a simple nonlocal de Sitter gravity model and its exact vacuum cosmological solutions. In the Einstein-Hilbert action with $Λ$ term, we introduce nonlocality by the following way: $R - 2 Λ= \sqrt{R-2Λ}\ \sqrt{R-2Λ} \to \sqrt{R-2Λ}\ F(\Box)\ \sqrt{R-2Λ} ,$ where ${F} (\Box) = 1 + \sum_{n= 1}^{+\infty} \big( f_n \Box^n + f_{-n} \Box^{-n} \big) $ is an analytic function of the d'Alembert-Beltrami operator $\Box$ and its inverse $\Box^{-1}$. By this way, $R$ and $Λ$ enter with the same form into nonlocal version as they are in the local one, and nonlocal operator $F(\Box)$…
We consider nonlocal modified Einstein gravity without matter, where nonlocal term has the form $P(R) F(\Box) Q(R)$. For this model, in this paper we give the derivation of the equations of motion in detail. This is not an easy task and presented derivation should be useful to a researcher who wants to investigate nonlocal gravity. Also, we present the second variation of the related Einstein-Hilbert modified action and basics of gravity perturbations.
A general class of nonlocal cosmological models is considered. A new method for solving nonlocal Friedmann equations is proposed, and solutions of the Friedmann equations with nonlocal operator are presented. The cosmological properties of these solutions are discussed. Especially indicated is $p$-adic cosmological model in which we have obtained nonsingular bouncing solution and string field theory tachyon model in which we have obtained full solution of nonlocal Friedmann equations with $w=-1$ at large times. The possibility of obtaining realistic value of cosmological constant from nonlocal…
Even if the fundamental action of gravity is local, the corresponding quantum effective action, that includes the effect of quantum fluctuations, is a nonlocal object. These nonlocalities are well understood in the ultraviolet regime but much less in the infrared, where they could in principle give rise to important cosmological effects. Here we systematize and extend previous work of our group, in which it is assumed that a mass scale $Λ$ is dynamically generated in the infrared, giving rise to nonlocal terms in the quantum effective action of gravity. We give a detailed discussion of concept…
We find nonperturbative solutions of a nonlocal scalar field equation, with cubic or exponential potential on a cosmological background. The former case corresponds to the lowest level effective tachyon action of cubic string field theory. While the well known Minkowski solution is wildly oscillating, due to Hubble friction its cosmological counterpart describes smooth rolling towards the local minimum of the potential.
We have recently presented a manifestly local and general coordinate invariant formulation of a nonlocal approach to the cosmological constant problem which has been proposed by Carroll and Remmen. In this article, based on our formulation, we investigate some quantum aspects of this approach to the cosmological constant problem. In particular, we explicitly evaluate quantum effects to the effective cosmological constant from matter fields at one-loop level, and show that the effective cosmological constant receives radiative corrections and is not stable against quantum corrections so that it…
Nonlocal cosmological models with quadratic potentials are considered. We study the action with an arbitrary analytic function F(\Box_g), which has both double and simple roots. The formulae for nonlocal energy-momentum tensor, which correspond to double roots, have been obtained. The way to find particular solutions for nonlocal Einstein equations in the case when F(\Box_g) has both simple and double roots has been proposed. One and the same functions solve the initial nonlocal Einstein equations and the obtained local Einstein equations.
We discuss the cosmological implications of nonlocal modifications of general relativity containing tensorial structures. Assuming the presence of standard radiation- and matter-dominated eras, we show that, except in very particular cases, the nonlocal terms contribute a rapidly growing energy density. These models therefore generically do not have a stable cosmological evolution.
We present a manifestly local and general coordinate invariant formulation of a nonlocal approach to the cosmological constant problem which has been recently proposed by Carroll and Remmen. To do that, we need to introduce a topological term involving a new 3-form gauge field. The equations of motion for this new 3-form gauge field lead to a constant Lagrange multiplier parameter and the resulting action becomes equivalent to that of Carroll and Remmen. In our formulation, nonlocal informations are encoded via the procedure of taking the space-time average at the stage of the equations of mot…
A nonlocal gravity model which does not assume the existence of a new dimensional parameter in the action and includes a function $f(\Box^{-1} R)$, with $\Box$ the d'Alembertian operator, is studied. By specifying an exponential form for the function f and including a matter sector with a constant equation of state parameter, all available power-law solutions in the Jordan frame are obtained. New power-law solutions in the Einstein frame are also probed. Furthermore, the relationship between power-law solutions in both frames, established through conformal transformation, is substantially clar…
In this paper we consider modification of general relativity extending $R - 2 Λ$ by nonlocal term of the form $\sqrt{R-2Λ}\, \mathcal{F}(\Box)\, \sqrt{R-2Λ} ,$ where $\mathcal{F}(\Box)$ is an analytic function of the d'Alembert operator $\Box$. We have found some exact cosmological solutions of the corresponding equations of motion without matter and with $Λ\neq 0$. One of these solutions is $ a (t) = A \, t^{\frac{2}{3}} \, e^{\fracΛ{14} t^2} ,$ which imitates properties similar to an interplay of the dark matter and the dark energy. For this solution we calculated some cosmological parameter…
We consider nonlocal modification of the Einstein theory of gravity in framework of the pseudo-Riemannian geometry. The nonlocal term has the form $\mathcal{H}(R) \mathcal{F}(\Box)\mathcal {G}(R)$, where $\mathcal{H}$ and $\mathcal{G}$ are differentiable functions of the scalar curvature $R,$ and $ \mathcal{F}(\Box)= \displaystyle \sum_{n =0}^{\infty} f_{n}\Box^{n}$ is an analytic function of the d'Alambert operator $\Box .$ Using calculus of variations of the action functional, we derived the corresponding equations of motion. The variation of action is induced by variation of the gravitation…
We consider some aspects of nonlocal modified gravity, where nonlocality is of the type $R \mathcal{F}(\Box) R$. In particular, using ansatz of the form $\Box R = c R^γ,$ we find a few $R(t)$ solutions for the spatially flat FLRW metric. There are singular and nonsingular bounce solutions. For late cosmic time, scalar curvature R(t) is in low regime and scale factor a(t) is decelerated. R (t) = 0 satisfies all equations when k = -1.
In this paper, we investigate a nonlocal modification of general relativity (GR) with action $S = \frac{1}{16πG} \int [ R- 2Λ+ (R-4Λ) \, \mathcal{F}(\Box) \, (R-4Λ) ] \, \sqrt{-g}\; d^4x ,$ where $\mathcal{F} (\Box) = \sum_{n=1}^{+\infty} f_n \Box^n$ is an analytic function of the d'Alembertian $\Box$. We found a few exact cosmological solutions of the corresponding equations of motion. There are two solutions which are valid only if $Λ\neq 0, \, k = 0,$ and they have not analogs in Einsten's gravity with cosmological constant $Λ$. One of these two solutions is $ a (t) = A \, \sqrt{t} \, e^{\f…
Nonlocal massive gravity can provide an interesting explanation for the late-time cosmic acceleration, with a dark energy equation of state $w_{\rm DE}$ smaller than $-1$ in the past. We derive the equations of linear cosmological perturbations to confront such models with the observations of large-scale structures. The effective gravitational coupling to nonrelativistic matter associated with galaxy clusterings is close to Newton's gravitational constant $G$ for a mass scale $m$ slightly smaller than today's Hubble parameter $H_0$. Taking into account the background expansion history as well …
We study the cosmological consequences of a recently proposed nonlocal modification of general relativity, obtained by adding a term $m^2R\,\Box^{-2}R$ to the Einstein-Hilbert action. The model has the same number of parameters as $Λ$CDM, with $m$ replacing $Ω_Λ$, and is very predictive. At the background level, after fixing $m$ so as to reproduce the observed value of $Ω_M$, we get a pure prediction for the equation of state of dark energy as a function of redshift, $w_{\rm DE}(z)$, with $w_{\rm DE}(0)$ in the range $[-1.165,-1.135]$ as $Ω_M$ varies over the broad range $Ω_M\in [0.20,0.36]$. …
There are many different phenomenological models describing the cosmological dark energy and accelerating Universe by choosing adjustable functions. In this paper we consider a specific model of scalar tachyon field which is derived from the NSR string field theory and study its cosmological applications. We find that in the effective field theory approximation the equation of state parameter w<-1, i.e. one has a phantom Universe. It is shown that due to nonlocal effects there is no quantum instability that the usual phantom models suffer from. Moreover due to a flip effect of the potential th…
Semiclassical contributions to the gravitational action include various terms with zero, two, and four derivatives of the metric as well as nonlocal form factors for these terms. Contributions to some of these terms could be confused with others on a specific metric background or for a particular gauge fixing. We present a critical analysis of the recent works where the tensor structure and the number of derivatives in the action of gravity were not properly taken into account. Taking these relevant aspects into account, we show that although some contributions owing to the quantum fluctuation…
We construct a model in which the cosmological constant is canceled from the gravitational equations of motion. Our model relies on two key ingredients: a nonlocal constraint on the action, which forces the spacetime average of the Lagrangian density to vanish, and a dynamical way for this condition to be satisfied classically with arbitrary matter content. We implement the former condition with a spatially-constant Lagrange multiplier associated with the volume form and the latter by including a free four-form gauge field strength in the action. These two features are enough to remove the cos…
We study lump solutions in nonlocal toy models and their cosmological applications. These models are motivated by a description of D-brane decay within string field theory framework. In order to find cosmological solutions we use the simplest local approximation keeping only second derivative terms in nonlocal dynamics. We study a validity of this approximation in flat background where time lump solutions can be written explicitly. We work out the validity of this approximation. We show that our models at large time exhibit the phantom behaviour similar to the case of the string kink.
The one-loop effective action for non-minimal scalar modified gravity on de Sitter background with a constant scalar field is found. The corresponding induced cosmological constant is evaluated. It is shown that quantum effects in non-minimal modified gravity may induce an early-time de Sitter universe even in the situation when such solution does not occur on the classical level. Classical reconstruction of the theory is presented in such a way that the resulting theory has a cosmological solution unifying early-time inflation with late-time acceleration.
We have recently presented a manifestly local and general coordinate invariant formulation of a nonlocal approach to the cosmological constant problem. In this article, we investigate quantum effects from both matter and gravitational fields in this formulation. In particular, we pay our attention to the gravitational loop effects and show that the effective cosmological constant is radiatively stable even in the presence of the gravitational loop effects in addition to matter loop effects. For this purpose we need to add the $R^2$ term and the corresponding topological action as an total acti…
It is well known that string theories naturally compactify on anti-de Sitter spaces, and yet cosmological observations show no evidence of a negative cosmological constant in the early Universe's evolution. In this letter we present two simple nonlocal modifications of the standard Friedmann cosmology that can lead to observationally viable cosmologies with an initial (negative) cosmological constant. The nonlocal operators we include are toy models for the quantum cosmological backreaction. In Model I an initial quasiperiodic oscillatory epoch is followed by inflation and a late time matter e…
A nonlocal gravity model, which does not assume the existence of a new dimensional parameter in the action and includes a function $f(\Box^{-1} R)$, with $\Box$ the d'Alembertian operator, is considered. The model is proven to have de Sitter solutions only if the function f satisfies a certain second-order linear differential equation. The de Sitter solutions corresponding to the simplest case, an exponential function f, are explored, without any restrictions on the parameters. If the value of the Hubble parameter is positive, the de Sitter solution is stable at late times, both for negative a…
We have recently constructed a manifestly local formulation of a nonlocal approach to the cosmological constant problem which can treat with quantum effects from both matter and gravitational fields. In this formulation, it has been explicitly shown that the effective cosmological constant is radiatively stable even in the presence of the gravitational loop effects. Since we are naturally led to add the $R^2$ term and the corresponding topological action to an original action, we make use of this formulation to account for the late-time acceleration of expansion of the universe in case of the …
We study cosmological solutions in nonlocal teleparallel gravity or $f(T)$ theory, where $T$ is the torsion scalar in teleparallel gravity. This is a natural extenstion of the usual teleparallel gravity with nonlocal terms. In this work the phase space portrait proposed to describe the dynamics of an arbitrary flat, homogeneous cosmological background with a number of matter contents, both in early and late time epochs. The aim was to convert the system of the equations of the motion to a first order autonomous dynamical system and to find fixed points and attractors using numerical codes. For…
The evidence for an accelerating Hubble expansion appears to have confirmed the heuristic prediction, from causal set theory, of a fluctuating and ``ever-present'' cosmological term in the Einstein equations. A more concrete phenomenological model incorporating this prediction has been devised and tested, but it remains incomplete. I will review these developments and also mention a possible consequence for the dimensionality of spacetime.
Università di Napoli “Federico II”（那不勒斯费德里科二世大学）；Scuola Superiore Meridionale（南方高等学院）；INFN Sezione di Napoli（意大利国家核物理研究所那不勒斯分部）；Van Swinderen Institute, University of Groningen（格罗宁根大学范斯温德伦研究所）
A nonlocal gravity model is considered which does not assume the existence of a new dimensional parameter in the action and includes a function $f(\Box^{-1} R)$, with $\Box$ the d'Alembertian operator. Using a reconstruction procedure for the local scalar-tensor formulation of this model, a function f is obtained for which the model exhibits power-law solutions with the Hubble parameter $H=n/t$, for any value of the constant n. For generic n - namely except for a few special values which are characterized and also specifically studied - the corresponding function f is a sum of exponential func…
Cosmological perturbations are considered in $f(T)$ and in scalar-torsion $f(φ)T$ teleparallel models of gravity. Full sets of linear perturbation equations are accurately derived and analysed at the relevant limits. Interesting features of generalisations to other teleparallel models, spatially curved backgrounds, and rotated tetrads are pointed out.
This work is my contribution to the proceedings of the conference "SFT2010 -- the Third International Conference on String Field Theory and Related Topics". We discuss properties of nonlocal tachyons and nonlocal SFT inspired cosmology.
The cosmological reconstruction scheme for modified $F(R)$ gravity is developed in terms of e-folding (or, redshift). It is demonstrated how any FRW cosmology may emerge from specific $F(R)$ theory. The specific examples of well-known cosmological evolution are reconstructed, including $Λ$CDM cosmology, deceleration with transition to phantom superacceleration era which may develop singularity or be transient. The application of this scheme to viable $F(R)$ gravities unifying inflation with dark energy era is proposed. The additional reconstruction of such models leads to non-leading gravitati…
Nonlocal cosmological models derived from String Field Theory are considered. A new method for constructing rolling tachyon solutions in the FRW metric in two field configuration is proposed and solutions of the Friedman equations with nonlocal operator are presented. The cosmological properties of these solutions are discussed.
In this paper we investigate cosmological dynamics on the normal branch of a DGP-inspired scenario within a phase space approach where induced gravity is modified in the spirit of $f(R)$-theories. We apply the dynamical system analysis to achieve the stable solutions of the scenario in the normal DGP branch. Firstly, we consider a general form of the modified induced gravity and we show that there is a standard de Sitter point in phase space of the model. Then we prove that this point is stable attractor only for those $f(R)$ functions that account for late-time cosmic speed-up.
We investigate the features of the cosmological expansion history described by a recent model of gravity characterised by two nonlocally interacting metrics. We perform a detailed analysis of the dynamical system formed by the field equations and we find no stable critical points at finite and infinite distance. Nonetheless, we show that even if the universe does not evolve towards a de Sitter attractor, the effective equation of state parameter $ω_{\rm eff}$ always tends to $-1$, independently from the value of the free parameter $m^2$, which characterises the nonlocality of the theory. We al…
In modified gravity the propagation of gravitational waves (GWs) is in general different from that in general relativity. As a result, the luminosity distance for GWs can differ from that for electromagnetic signals, and is affected both by the dark energy equation of state $w_{\rm DE}(z)$ and by a function $δ(z)$ describing modified propagation. We show that the effect of modified propagation in general dominates over the effect of the dark energy equation of state, making it easier to distinguish a modified gravity model from $Λ$CDM. We illustrate this using a nonlocal modification of gravit…
We study cosmological constraints on the various accelerating models of the universe using the time evolution of the cosmological redshift of distant sources. The important characteristic of this test is that it directly probes the expansion history of the universe. In this work we analyze the various models of the universe which can explain the late time acceleration, within the framework of General Theory of Relativity (GR) (XCDM, scalar field potentials) and beyond GR (f(R) gravity model).
We address the stability issue of Ricci-flat and maximally symmetric spacetimes in nonlocal gravity to all perturbative orders in the gravitational perturbation. Assuming a potential at least cubic in curvature tensors but quadratic in the Ricci tensor, our proof consists on a mapping of the stability analysis in nonlocal gravity to the same problem in Einstein-Hilbert theory. One of the consequences is that only the graviton field can propagate and the theory is ghost-free at all perturbative orders. All the results known in Einstein gravity in vacuum with or without a cosmological constant c…
The quasi-static solutions of the matter density perturbation in various dark energy models and modified gravity models have been investigated in numerous papers. However, the oscillating solutions in those models have not been investigated enough so far. In this paper, the oscillating solutions, which have a possibility to unveil the difference between the models of the late-time accelerated expansion of the Universe, are also mentioned by using appropriate approximations.
Earlier constructed a simple nonlocal de Sitter gravity model has a cosmological solution in a very good agreement with astronomical observations. In this paper, we continue the investigation of the nonlocal de Sitter model of gravity, focusing on finding an appropriate solution for the Schwarzschild-de Sitter metric. We succeeded to solve the equations of motion in a certain approximation. The obtained approximate solution is of particular interest for examining the possible role of non-local de Sitter gravity in describing the effects in galactic dynamics that are usually attributed to dark …
Very recently, Verlinde considered a theory in which space is emergent through a holographic scenario, and proposed that gravity can be explained as an entropic force caused by changes in the information associated with the positions of material bodies. Then, motivated by the Debye model in thermodynamics which is very successful in very low temperatures, Gao modified the entropic force scenario. The modified entropic force (MEF) model is in fact a modified gravity model, and the universe can be accelerated without dark energy. In the present work, we consider the cosmological constraints on t…
Along this review, we focus on the study of several properties of modified gravity theories, in particular on black-hole solutions and its comparison with those solutions in General Relativity, and on Friedmann-Lemaitre-Robertson-Walker metrics. The thermodynamical properties of fourth order gravity theories are also a subject of this investigation with special attention on local and global stability of paradigmatic f(R) models. In addition, we revise some attempts to extend the Cardy-Verlinde formula, including modified gravity, where a relation between entropy bounds is obtained. Moreover, a…
A nonlocal quantum gravity theory is presented which is finite and unitary to all orders of perturbation theory. Vertex form factors in Feynman diagrams involving gravitons suppress graviton and matter vacuum fluctuation loops by introducing a low-energy gravitational scale, Lambda_{Gvac} < 2.4 X 10^{-3} eV. Gravitons coupled to non-vacuum matter loops and matter tree graphs are controlled by a vertex form factor with the energy scale, Lambda_{GM} < 1-10 TeV. A satellite Eotvos experiment is proposed to test a violation of the equivalence principle for coupling of gravitons to pure vacuum ener…
We study the condition for the consistency of the Gödel metric with the dynamical Chern-Simons modified gravity. It turns out to be that this compatibility can be achieved only if the cosmological constant is variable in the space.
A string field theory(SFT) nonlocal model of the cosmological dark energy providing w<-1 is briefly surveyed. We summarize recent developments and open problems, as well as point out some theoretical issues related with others applications of the SFT nonlocal models in cosmology, in particular, in inflation and cosmological singularity.
We construct the covariant nonlocal action for recently suggested long-distance modifications of gravity theory motivated by the cosmological constant and cosmological acceleration problems. This construction is based on the special nonlocal form of the Einstein-Hilbert action explicitly revealing the fact that this action within the covariant curvature expansion begins with curvature-squared terms.
We construct the covariant nonlocal action for recently suggested long-distance modifications of gravity theory motivated by the cosmological constant and cosmological acceleration problems. This construction is based on the special nonlocal form of the Einstein-Hilbert action explicitly revealing the fact that this action within the covariant curvature expansion begins with curvature-squared terms. Nonlocal form factors in the action of both quantum and brane-induced nature are briefly discussed. In particular, it is emphasized that for certain class of quantum initial value problems nonlocal…
Modified Newtonian dynamics (MOND) can be obtained by modifying the entropic formulation of gravity, this is achieved by considering the quantum statistical nature of the degrees of freedom on the holographic screen. Through this frame work, we find some constraints on a cosmological extension for MOND, with no additional auxiliary fields. The connections between MOND to conformal gravity and Rindler force gravity are examined. These two alternative gravity theories are subsequently considered as possible cosmological extensions of MOND.
Based on the ultraviolet asymptotic freedom of nonlocal quantum gravity, we show that the trans-Planckian energy regime is unattainable in laboratory experiments. As physical implications, it turns out that the violation of causality, typical of nonlocal field theories, can never be detected in particle accelerators, while the asymptotic freedom of the theory provides an elegant solution to the so called trans-Planckian cosmological problem.
In this paper we address the issue of exploring some cosmological scenarios in modified Einstein gravity through non-dynamical (auxiliary) fields. We found that all scenarios are controlled by a specific parameter associated with an auxiliary field. We explore the emergence of inflationary, radiation, matter and dark energy dominated regimes. Furthermore, an interesting possibility such as the emergence of a self-tuning mechanism to the cosmological constant problem in the radiation dominated era is also discussed.
Classical versions of the Big Bang cosmological models of the universe contain a singularity at the start of time, hence the time variable in the field equations should run over a half-line. Nonlocal string field theory equations with infinite number of derivatives are considered and an important difference between nonlocal operators on the whole real line and on a half-line is pointed out. We use the heat equation method and show that on the half-line in addition to the usual initial data a new arbitrary function (external source) occurs that we call the daemon function. The daemon function g…
In Randall-Sundrum-type brane-world cosmologies, the dynamical equations on the three-brane differ from the general relativity equations by terms that carry the effects of imbedding and of the free gravitational field in the five-dimensional bulk. We derive the covariant nonlinear dynamical equations for the gravitational and matter fields on the brane, and then linearize to find the perturbation equations on the brane. The local energy-momentum corrections are significant only at very high energies. The imprint on the brane of the nonlocal gravitational field in the bulk is more subtle, and w…
