A possible solution to the dark energy problem is that Einstein's theory of general relativity is modified. A suite of models have been proposed that, in general, are unable to predict the correct amount of large scale structure in the distribution of galaxies or anisotropies in the Cosmic Microwave Background. It has been argued, however, that it should be possible to constrain a general class of theories of modified gravity by focusing on properties such as the growing mode, gravitational slip and the effective, time varying Newton's constant. We show that assuming certain physical requirements such as stability, metricity and gauge invariance, it is possible to come up with consistency conditions between these various parameters. In this paper we focus on theories which have, at most, 2nd derivatives in the metric variables and find restrictions that shed light on current and future experimental constraints without having to resort to a (as yet unknown) complete theory of modified gravity. We claim that future measurements of the growth of structure on small scales (i.e. from 1-200 h^{-1} Mpc) may lead to tight constraints on both dark energy and modified theories of gravity.
We present the most general parametrisation of models of dark energy in the form of a scalar field which is explicitly coupled to dark matter. We follow and extend the Parameterized Post-Friedmannian approach, previously applied to modified gravity theories, in order to include interacting dark energy. We demonstrate its use through a number of worked examples and show how the initially large parameter space of free functions can be significantly reduced and constrained to include only a few non-zero coefficients. This paves the way for a model-independent approach to classify and test interac…
With the increasing wealth of high-quality astronomical and cosmological data and the manifold departures from General Relativity in principle conceivable, the development of generalized parametrization frameworks that unify gravitational models and cover a wide range of length scales and a variety of observational probes to enable systematic high-precision tests of gravity has been a stimulus for intensive research. A review is presented here for some of the formalisms devised for this purpose, covering the cosmological large- and small-scale structures, the astronomical static weak-field reg…
We study the spherical collapse in the Parametrized Post-Friedmannian (PPF) scheme. We use a general form of the PPF parameter related to the Poisson equation and found the equations to solve that includes a non-trivial fifth force coming from the convolution of the modified gravity term in the k-space. In order to compute a concrete model, we use the parametrization proposed by Bertschinger and Zukin. The equations of the spherical collapse are solved assuming a Gaussian density profile and we show there is no shell crossing before reaching the turn around point. We show that the fifth force …
We study the post-Newtonian limit in the teleparallel equivalent of General Relativity with a scalar field which non-minimally couples to gravity. The metric perturbation is obtained from the vierbein field expansion with respect to the Minkowski background. Due to the structure of the teleparallel gravity Lagrangian, the potential of the scalar field shows no effect to the parametrized post-Newtonian parameters, and compatible results with Solar System observations are found.
We explore the signatures of quintessence and modified gravity theories in the relativistic description of galaxy clustering within a parametrized post-Friedmann framework. For this purpose, we develop a calibration method to consistently account for horizon-scale effects in the linear parametrized Post-Friedmann perturbations of minimally and nonminimally coupled scalar-tensor theories and test it against the full model-specific fluctuations. We further study the relativistic effects in galaxy clustering for the normal and self-accelerating branches of the Dvali-Gabadadze-Porrati braneworld m…
We study the parametrized post-Newtonian (PPN) limit of higher-derivative-torsion Modified Teleparallel Gravity. We start from the covariant formulation of modified Teleparallel Gravity by restoring the spin connection of the theory. Then, we perform the post-Newtonian expansion of the tetrad field around the Minkowski background and find the perturbed field equations. We compute the PPN metric for the higher-order Teleparallel Gravity theories which allows us to show that at the post-Newtonian limit this more general class of theories are fully conservative and indistinguishable from General …
We study the effects of dark energy (DE) anisotropic stress on features of the matter power spectrum (PS). We employ the Parametrized Post-Friedmannian (PPF) formalism to emulate an effective DE, and model its anisotropic stress properties through a two-parameter equation that governs its overall amplitude ($g_0$) and transition scale ($c_g$). For the background cosmology, we consider different equations of state to model DE including a constant $w_0$ parameter, and models that provide thawing (CPL) and freezing (nCPL) behaviors. We first constrain these parameters by using the Pantheon, BAO, …
Dark energy might directly interact with cold dark matter. However, in such a scenario, an early-time large-scale instability occurs occasionally, which may be due to the incorrect treatment for the pressure perturbation of dark energy as a nonadiabatic fluid. To avoid this nonphysical instability, we establish a new framework to correctly calculate the cosmological perturbations in the interacting dark energy models. Inspired by the well-known parametrized post-Friedmann approach, the condition of the dark energy pressure perturbation is replaced with the relationship between the momentum den…
Light degrees of freedom that modify gravity on cosmological scales must be "screened" on solar system scales in order to be compatible with data. The Vainshtein mechanism achieves this through a breakdown of classical perturbation theory, as large interactions involving new degrees of freedom become important below the so-called Vainshtein radius. We begin to develop an extension of the Parameterized Post-Newtonian (PPN) formalism that is able to handle Vainshteinian corrections. We argue that theories with a unique Vainshtein scale must be expanded using two small parameters. In this Paramet…
Dark energy might interact with dark matter in a direct, non-gravitational way, which can help remedy several theoretical defects. In order to find out the properties of interacting dark energy models, it is necessary to investigate the cosmological perturbations in detail. However, due to the improper use of pressure perturbation of dark energy, a large-scale instability at the early stage occurs occasionally. In recent years, parametrized post-Friedmann approach has been used to calculate the perturbation equations in the interacting dark energy scenario. Under this framework, the dark energ…
There possibly exists some direct, non-gravitational coupling between dark energy and dark matter. This possibility should be seriously tested by using observations, which requires us to understand such a scenario from the aspects of both expansion history and growth of structure. It is found that once calculating the perturbations in the interacting dark energy (IDE) scenario, for most cases the curvature perturbation on superhorizon scales is divergent, which is a catastrophe for the IDE cosmology. We found a solution to this issue, which is to establish an effective theory to treat the dark…
We apply the interacting parametrized post-Friedmann (IPPF) method to a coupled dark energy model where the interaction is proportional to dark matter density at background level. In doing so, we perform a Markov Chain Monte-Carlo analysis which combines several cosmological probes including the cosmic microwave background (WMAP9+Planck) data, baryon acoustic oscillation (BAO) measurements, JLA sample of supernovae, Hubble constant (HST), and redshift-space distortion (RSD) measurements through the ${\rm f}σ_{8}{\rm (z)}$ data points. The joint observational analysis of ${\rm Planck+WP+JLA+BAO…
Viable modifications of gravity on cosmological scales predominantly rely on screening mechanisms to recover Einstein's Theory of General Relativity in the Solar System, where it has been well tested. A parametrisation of the effects of such modifications in the spherical collapse model is presented here for the use of modelling the modified nonlinear cosmological structure. The formalism allows an embedding of the different screening mechanisms operating in scalar-tensor theories through large values of the gravitational potential or its first or second derivatives as well as of linear suppre…
Based on a mass-selected sample of galaxy-scale strong gravitational lenses from the SLACS, BELLS, LSD and SL2S surveys and using a well-motivated fiducial set of lens-galaxy parameters we tested the weak-field metric on kiloparsec scales and found a constraint on the post-Newtonian parameter $γ= 0.995^{+0.037}_{-0.047}$ under the assumption of a flat $Λ$CDM universe with parameters taken from \textit{Planck} observations. General relativity (GR) predicts exactly $γ=1$. Uncertainties concerning the total mass density profile, anisotropy of the velocity dispersion and the shape of the light-pro…
Dark energy can modify the dynamics of dark matter if there exists a direct interaction between them. Thus a measurement of the structure growth, e.g., redshift-space distortions (RSD), can provide a powerful tool to constrain the interacting dark energy (IDE) models. For the widely studied $Q=3βHρ_{de}$ model, previous works showed that only a very small coupling ($β\sim\mathcal{O}(10^{-3})$) can survive in current RSD data. However, all these analyses had to assume $w>-1$ and $β>0$ due to the existence of the large-scale instability in the IDE scenario. In our recent work [Phys. Rev. D 90, 0…
In this review we present a thoroughly comprehensive survey of recent work on modified theories of gravity and their cosmological consequences. Amongst other things, we cover General Relativity, Scalar-Tensor, Einstein-Aether, and Bimetric theories, as well as TeVeS, f(R), general higher-order theories, Horava-Lifschitz gravity, Galileons, Ghost Condensates, and models of extra dimensions including Kaluza-Klein, Randall-Sundrum, DGP, and higher co-dimension braneworlds. We also review attempts to construct a Parameterised Post-Friedmannian formalism, that can be used to constrain deviations fr…
We test anisotropic dark energy models with the 7-year WMAP temperature observations data. In the presence of imperfect sources, due to large-scale gradients or anisotropies in the dark energy field, the CMB sky will be distorted anisotropically on its way to us by the ISW effect. The signal covariance matrix then becomes nondiagonal for small multipoles, but at $\ell \gtrsim 20$ the anisotropy is negligible. We parametrize possible violations of rotational invariance in the late universe by the magnitude of a post-Friedmannian deviation from isotropy and its scale dependence. This allows to o…
We embed linear and nonlinear parametrisations of beyond standard cosmological physics in the halo model reaction framework, providing a model-independent prescription for the nonlinear matter power spectrum. As an application, we focus on Horndeski theories, using the Effective Field Theory of Dark Energy (EFTofDE) to parameterise linear and quasi-nonlinear perturbations. In the nonlinear regime we investigate both a nonlinear parameterised-post Friedmannian (nPPF) approach as well as a physically motivated and approximate phenomenological model based on the error function (Erf). We compare t…
We study the black hole's shadow for Schwarzschild - de Sitter and Kerr - de Sitter metrics with the contribution of the cosmological constant Λ. Based on the reported parameters of the M87* black hole shadow we obtain constraints for the $Λ$ and show the agreement with the cosmological data. It is shown that, the coupling of the Λ-term with the spin parameter reveals peculiarities for the photon spheres and hence for the shadows. Within the parametrized post-Newtonian formalism the constraint for the corresponding Λ-determined parameter is obtained.
A New Paradigm for Testing Gravity: Theory-Independent Constraints Using Data From All Astrophysical and Cosmological Scales
Jodrell Bank Centre for Astrophysics, University of Manchester（曼彻斯特大学乔德雷尔银行天体物理学中心）；Department of Physics and Astronomy, Queen Mary University of London（伦敦大学玛丽女王学院物理与天文系）
We compute the parametrized post-Newtonian parameter $γ$ in the case of a static point source for multiscalar-tensor gravity with completely general nonderivative couplings and potential in the Jordan frame. Similarly to the single massive field case $γ$ depends exponentially on the distance from the source and is determined by the length of a vector of non-minimal coupling in the space of scalar fields and its orientation relative to the mass eigenvectors. Using data from the Cassini tracking experiment, we estimate bounds on a general theory with two scalar fields. Our formalism can be utili…
n-DBI gravity explicitly breaks Lorentz invariance by the introduction of a unit time-like vector field, thereby giving rise to an extra (scalar) degree of freedom. We look for observational consequences of this mode in two setups. Firstly, we compute the parametrized post-Newtonian (PPN) expansion of the metric to first post-Newtonian order. Surprisingly, we find that the PPN parameters are exactly the same as in General Relativity (GR), and no preferred-frame effects are produced. In particular this means that n-DBI gravity is consistent with all GR solar system experimental tests. We discus…
Modified gravity theories generically predict a violation of Lorentz invariance, which may lead to a modified dispersion relation for propagating modes of gravitational waves. We construct a parametrized dispersion relation that can reproduce a range of known Lorentz-violating predictions and investigate their impact on the propagation of gravitational waves. A modified dispersion relation forces different wavelengths of the gravitational wave train to travel at slightly different velocities, leading to a modified phase evolution observed at a gravitational-wave detector. We show how such corr…
We study two large classes of alternative theories, modifying the action through algebraic, quadratic curvature invariants coupled to scalar fields. We find one class that admits solutions that solve the vacuum Einstein equations and another that does not. In the latter, we find a deformation to the Schwarzschild metric that solves the modified field equations in the small coupling approximation. We calculate the event horizon shift, the innermost stable circular orbit shift, and corrections to gravitational waves, mapping them to the parametrized post-Einsteinian framework.
We conclude the study of the Post-Minkowskian linearization of ADM tetrad gravity in the York canonical basis for asymptotically Minkowskian space-times in the family of non-harmonic 3-orthogonal gauges parametrized by the York time ${}^3K(τ, \vec σ)$ (the inertial gauge variable, not existing in Newton gravity, describing the general relativistic remnant of the freedom in clock synchronization in the definition of the instantaneous 3-spaces). As matter we consider only N scalar point particles with a Grassmann regularization of the self-energies and with a ultraviolet cutoff making possible t…
We work in the framework of a simple vector-tensor theory. The parametrized post-Newtonian approximation of this theory is identical to that of general relativity. Our attention is focused on cosmology. In an homogeneous isotropic universe, it is proved that the energy density, $ρ_{A}$, of the vector field $A$, and its pressure, $p_{A}$, do not depend on time, and also that the equation of state is $ρ_{A} = -p_{A}$. This means that, in the theory under consideration, there is a cosmological constant, which is not vacuum energy, but the dark energy of the cosmic vector field $A$, whose evolutio…
We investigate the imprint of dark energy on the kinetic Sunyaev-Zel'dovich (kSZ) angular power spectrum on scales of $\ell=1000$ to $10000$, and find that the kSZ signal is sensitive to the dark energy parameter. For example, varying the constant $w$ by 20\% around $w=-1$ results in a $\gtrsim10\%$ change on the kSZ spectrum; changing the dark energy dynamics parametrized by $w_a$ by $\pm0.5$, a 30\% change on the kSZ spectrum is expected. We discuss the observational aspects and develop a fitting formula for the kSZ power spectrum. Finally, we discuss how the precise modeling of the post-rei…
