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We hereby derive the Newtonian metric potentials for the fourth-derivative gravity including the one-loop logarithm quantum corrections. It is explicitly shown that the behavior of the modified Newtonian potential near the origin is improved respect to the classical one, but this is not enough to remove the curvature singularity in $r=0$. Our result is grounded on a rigorous proof based on numerical and analytic computations., 10 pages, 5 figures. V2: references added; matches the version to appear in PRD

We determine the complete space-time metric from the bootstrapped Newtonian potential generated by a static spherically symmetric source in the surrounding vacuum. This metric contains post-Newtonian parameters which can be further used to constrain the complete underlying dynamical theory. For values of the post-Newtonian parameters within experimental bounds, the reconstructed metric appears very close to the Schwarzschild solution of General Relativity in the whole region outside the event horizon. The latter is however larger in size for the same value of the mass compared to the Schwarzschild case., Comment: 27 pages, 3 figures, minor changes to match the accepted version in PRD

We consider Einstein gravity with the addition of $R^2$ and $R^{\mu \nu} R_{\mu \nu}$ interactions in the context of effective field theory, and the corresponding scattering amplitudes of gravitons and minimally-coupled heavy scalars. First, we recover the known fact that graviton amplitudes are the same as in Einstein gravity. Then we show that all amplitudes with two heavy scalars and an arbitrary number of gravitons are also not affected by these interactions. We prove this by direct computations, using field redefinitions known from earlier applications in string theory, and with a combination of factorisation and power-counting arguments. Combined with unitarity, these results imply that, in an effective field theory approach, the Newtonian potential receives neither classical nor quantum corrections from terms quadratic in the curvature., Comment: 15 pages

Gravity gradient is known as a serious systematic effect in atomic tests of the universality of free fall, where the initial central position and velocity of atoms need to be exactly controlled. In this paper, we study quantum free fall with high-order gravity gradients. It is shown that the cubic terms in the Newtonian potential shall generate a new phase shift in atom interferometers, which depends on the position and velocity uncertainties of the incident atoms. We further investigate the nonclassicality of free fall and show that, due to the cubic potential, the gravitational Wigner equation in phase space of position and velocity is different from the classical Liouville equation. There exists a mass-dependent correction in the dynamical equation regardless of initial state. Nevertheless, this is just a quantum mechanical effect of microparticles, which does not violate equivalence principle that inertia mass is equal to the gravitational mass., Comment: Minor revision for v2

Horndeski theory is the most general scalar-tensor theory retaining second-order field equations, although the action includes higher-order terms. This is achieved by a special choice of coupling constants. In this paper, we investigate the thick brane system in reduced Horndeski theory, especially the effects of the nonminimal derivative coupling and the cubic Galileon term on thick brane. First, the equations of motion are presented and a set of analytic background solutions are obtained. Then, the effects of these two terms on the thick brane are analyzed. It is found that the background solutions become asymmetric with the presence of the cubic Galileon term. However, the energy density of the thick brane is still symmetric with or without this term and splits with the presence of the nonminimal derivative coupling. The stability of the thick brane under tensor perturbation is also considered. It is shown that the tachyon is absent and the graviton zero mode can be localized on the brane. The localized graviton zero mode recovers the four-dimensional Newtonian potential and the presence of the nonminimal derivative coupling results in a splitting of its wave function. Besides, the presence of the cubic Galileon term results in the asymmetry of the effective potential and zero mode of the graviton. The correction of the massive graviton Kaluza-Klein modes to the Newtonian potential is also analyzed briefly.

In this paper we consider spherically symmetric interior spacetimes filled by anisotropic fluids in the context of Ho\v{r}ava gravity and Einstein-aether theory. We assume a specific non-static configuration of the aether vector field and show that the field equations admit a family of exact analytical solutions which can be obtained if one of the two metric coefficients is assigned. We study as an illustrative example the case in which the metric of the interior spacetime reproduces the Newtonian potential of a fluid sphere with constant density., Comment: v1: 8 pages; v2: 9 pages. Matches published version in PRD

We analyze a modified $f(R)$ theory of gravity in the Palatini formulation, when a Holst term endowed with a dynamical Immirzi field is included. We study the basic features of the model, especially in view of eliminating the torsion field via the Immirzi field and the scalar-tensor degrees of freedom of the $f(R)$ model. The main task of this study is the investigation of the morphology of the gravitational wave polarization when their coupling to a circle of test particles is considered. We first observe that the dynamics of the scalar mode of the $f(R)$ Lagrangian is frozen out, since its first order term identically vanishes. This allows a detailed characterization of the linearized theory, which outlines the emergence of a modified Newtonian potential in the static limit, and when time independence is relaxed a standard gravitational wave plus the scalar wave associated to the Immirzi field. Investigating the effect of the coupling of this scalar-tensor wave on a circle of test particles, we arrive to define two effective gravitational polarizations, corresponding to an equivalent phenomenological wave, whose morphology is anomalous with respect the standard case of general relativity. In fact, the particle circle suffers modifications as it was subjected to modified plus and cross modes, whose specific features depend on the model free parameters and are, in principle, detectable via a data analysis procedure.

Ongoing and upcoming cosmological surveys will significantly improve our ability to probe the equation of state of dark energy, ${w}_{\mathrm{DE}}$, and the phenomenology of large scale structure. They will allow us to constrain deviations from the $\mathrm{\ensuremath{\Lambda}}$ cold dark matter predictions for the relations between the matter density contrast and the weak lensing and the Newtonian potential, described by the functions $\mathrm{\ensuremath{\Sigma}}$ and $\ensuremath{\mu}$, respectively. In this work, we derive the theoretical prior for the joint covariance of ${w}_{\mathrm{DE}}$, $\mathrm{\ensuremath{\Sigma}}$ and $\ensuremath{\mu}$, expected in general scalar-tensor theories with second order equations of motion (Horndeski gravity), focusing on their time-dependence at certain representative scales. We employ Monte Carlo methods to generate large ensembles of statistically independent Horndeski models, focusing on those that are physically viable and in broad agreement with local tests of gravity, the observed cosmic expansion history and the measurement of the speed of gravitational waves from a binary neutron star merger. We identify several interesting features and trends in the distribution functions of ${w}_{\mathrm{DE}}$, $\mathrm{\ensuremath{\Sigma}}$ and $\ensuremath{\mu}$, as well as in their covariances; we confirm the high degree of correlation between $\mathrm{\ensuremath{\Sigma}}$ and $\ensuremath{\mu}$ in scalar-tensor theories. The derived prior covariance matrices will allow us to reconstruct jointly ${w}_{\mathrm{DE}}$, $\mathrm{\ensuremath{\Sigma}}$ and $\ensuremath{\mu}$ in a nonparametric way.

Alternative theories of gravity may serve to overcome several shortcomings of the standard cosmological model but, in their weak field limit, general relativity must be recovered so as to match the tight constraints at the Solar System scale. Therefore, testing such alternative models at scales of stellar systems could give a unique opportunity to confirm or rule them out. One of the most straightforward modifications is represented by analytical f(R)-gravity models that introduce a Yukawa-like modification to the Newtonian potential thus modifying the dynamics of particles. Using the geodesics equations, we have illustrated the amplitude of these modifications. First, we have integrated numerically the equations of motion showing the orbital precession of a particle around a massive object. Second, we have computed an analytic expression for the periastron advance of systems having their semimajor axis much shorter than the Yukawa-scale length. Finally, we have extended our results to the case of a binary system composed of two massive objects. Our analysis provides a powerful tool to obtain constraints on the underlying theory of gravity using current and forthcoming data sets.

In the standard approach to studying cosmological structure formation, the overall expansion of the Universe is assumed to be homogeneous, with the gravitational effect of inhomogeneities encoded entirely in a Newtonian potential. A topic of ongoing debate is to what degree this fully captures the dynamics dictated by general relativity, especially in the era of precision cosmology. To quantitatively assess this, we directly compare standard N-body Newtonian calculations to full numerical solutions of the Einstein equations, for cold matter with various magnitude initial inhomogeneities on scales comparable to the Hubble horizon. We analyze the differences in the evolution of density, luminosity distance, and other quantities defined with respect to fiducial observers. This is carried out by reconstructing the effective spacetime and matter fields dictated by the Newtonian quantities, and by taking care to distinguish effects of numerical resolution. We find that the fully general relativistic and Newtonian calculations show excellent agreement, even well into the nonlinear regime. They only notably differ in regions where the weak gravity assumption breaks down, which arise when considering extreme cases with perturbations exceeding standard values., Comment: 17 pages, 14 figures; revised to match PRD version

It was proposed that if a higher-derivative gravity is renormalizable it implies necessarily a finite Newtonian potential at the origin, but the reverse of this statement is not true. Here, we show that the reverse is true when taking into account the van Dam-Veltman-Zakharov discontinuity, which states that the theory obtained from the massive one by taking a zero mass limit is not equivalent to the theory obtained in the zero mass case. The surviving degree of freedom in the zero mass limit is an extra scalar that does not affect the light bending angle but affects the Newtonian potential. This asserts that in order to make the singularity cancellation the number of massive ghost and healthy tensors matches with that of massive ghost and healthy scalars.

We study an effective quantum description of the static gravitational potential for spherically symmetric systems up to the first post-Newtonian order. We start by obtaining a Lagrangian for the gravitational potential coupled to a static matter source from the weak field expansion of the Einstein-Hilbert action. By analysing a few classical solutions of the resulting field equation, we show that our construction leads to the expected post-Newtonian expressions. Next, we show that one can reproduce the classical Newtonian results very accurately by employing a coherent quantum state and modifications to include the first post-Newtonian corrections are considered. Our findings establish a connection between the corpuscular model of black holes and post-Newtonian gravity, and set the stage for further investigations of these quantum models., Comment: 26 pages, 4 figures. Typos corrected, references and clarifications added

In the work of Hohmann et al. [Phys. Rev. D 88, 084054 (2013)], the authors worked out the parametrized post-Newtonian (PPN) parameters $\ensuremath{\gamma}$ and $\ensuremath{\beta}$ of a scalar-tensor theory with an arbitrary coupling function and a generic potential, and they found that these two PPN parameters depend on the radial distance $r$ from the Sun, $\ensuremath{\gamma}(r)$ and $\ensuremath{\beta}(r)$. Based on the assumption that measurements on the PPN parameters can be characterized by the shortest distance to the Sun, the authors obtained their best constraints on the model parameters of the scalar-tenor theory by light deflection observation and the Cassini tracking experiment. However, as the authors stated, this approach might not be rigorous. In the present work, we physically model astronomical observations and physical experiments by calculating the null and timelike geodesics in the scalar-tensor theory. We show that, contrary to the results in the previous work, the light deflection and the Cassini tracking cannot distinguish the scalar-tensor theory from general relativity. We also investigate the additional advances in perihelia caused by the largest correction of the scalar field on the Newtonian potential. Since this correction has a Yukawa-like form, we obtain very much improved lower bounds on the model parameters by using current upper limits on the Yukawa parameters.

In the restricted four-body problem consisting of the Earth, the Moon and the Sun as the primaries and a spacecraft as the planetoid, we take into account the solar perturbation in the description of the motion of a spacecraft in the vicinity of the stable Earth-Moon libration points L4 and L5 both in the classical regime and in the context of effective field theories of gravity. We then evaluate the location of all Lagrangian points in the Earth-Moon system within the framework of general relativity. For the points L4 and L5, the corrections of coordinates are of order a few millimeters. After that, we set up a scheme where the theory which is quantum corrected has as its classical counterpart the Einstein theory, instead of the Newtonian one. By virtue of the effective-gravity correction to the longdistance form of the potential among two point masses, all terms involving the ratio between the gravitational radius of the primary and its separation from the planetoid get modified. Within this framework, for the Lagrangian points of stable equilibrium, we find quantum corrections of order two millimeters, whereas for Lagrangian points of unstable equilibrium we find quantum corrections below a millimeter. Finally, general relativity corrections to Newtonian position of collinear Lagrangian points turn out to be below the millimiter, whereas on stable equilibrium points they are of order of a few millimiters., 41 pages, 9 figures. Comments in the very end of Sec. VI have been amended and shortened. The last typos have been corrected

We revisit the calculation of matter quantum effects on the graviton self-energy on a flat Minkowski background, with the aim to acquire a deeper understanding of the mechanism that renders the graviton massless. To this end, we derive a low-energy theorem which directly relates the radiative corrections of the cosmological constant to those of the graviton mass to all orders in perturbation theory. As an illustrative example, we consider an Abelian Higgs model with minimal coupling to gravity and show explicitly how a suitable renormalization of the cosmological constant leads to the vanishing of the graviton mass at the one-loop level. In the same Abelian Higgs model, we also calculate the matter quantum corrections to the Newtonian potential and present analytical formulae in terms of modified Bessel and Struve functions of the particle masses in the loop. We show that the correction to the Newtonian potential exhibits an exponential fall-off dependence on the distance $r$, once the non-relativistic limit with respect to the non-zero loop mass is carefully considered. For massless scalars, fermions and gauge bosons in the loops, we recover the well-known results presented in the literature., Comment: 31 pages, 9 figures, typos in some equations eliminated, version to be submitted to the journal

We present the first calculation of an intrinsically relativistic quantity in fully non-linear cosmolog- ical large-scale structure studies. Traditionally, non-linear structure formation in standard {\Lambda}CDM cosmology is studied using N-body simulations, based on Newtonian gravitational dynamics on an expanding background. When one derives the Newtonian regime in a way that is a consistent ap- proximation to the Einstein equations, a gravito-magnetic vector potential - giving rise to frame dragging - is present in the metric in addition to the usual Newtonian scalar potential. At leading order, this vector potential does not affect the matter dynamics, thus it can be computed from Newtonian N-body simulations. We explain how we compute the vector potential from simulations in {\Lambda}CDM and examine its magnitude relative to the scalar potential. We also discuss some possible observable effects., Comment: 5 pages, 3 figurs

One of the great puzzles of modern cosmology is the dark matter problem. Its existence is inferred by analyzing the rotational curve of some spiral galaxies and the dynamics of galactic clusters @1#. With the introduction of dark matter, the discrepancies obtained using only luminous matter can be avoided. Nevertheless, the origin of dark matter and also its properties are not well understood. Although feasible, the dark matter hypothesis demands the existence of yet nondetected particles. Moreover, its nature and properties are a matter of controversy. Besides dark matter, alternative gravitational theories were proposed by some authors @2,3# in a tentative way to explain these observational discrepancies. The results of these theories applied in astrophysical situations may be used to constrain or even avoid some problems resulting from the use of standard gravitational theory. These modifications range from changing Newton’s second law @3# to modifications in general relativity @4–9#. Among several theories proposed as substitutes of general relativity, the very interesting ones are those with non-Newtonian limit in the weak field approximation. In one class of theories a Yukawa-like term is added to standard the Newtonian potential. The introduction of this term in the classical Newtonian gravitational potential has several origins: scale invariance @4#, Brans-Dicke theory with finite mass scalar field @5#, and generalized Einstein action with quadratic terms @6,8,9#. The main reason for developing these theories was the search for possible candidates to a viable quantum gravity. Nevertheless, with the detection of some disturbances in the galactic dynamics, some authors have tried to explain this with nonstandard gravitational theories. In particular, Sanders @2# used a gravitational potential with a Yukawa term to explain some features of galactic rotational curves. The difficulty with these modified theories is that the analysis of the observational data is inconclusive @1#. So the preferred explanation for galactic dynamics is the presence of dark matter. Therefore it would be important to search other observational situations to test the possible existence of a Yukawa-like term in the gravitational potential. In this work we analyze an idealized homogeneous stellar system which is perturbed due to the passage of a massive intruder star. We calculate the influence of the Yukawa term in the scattering of the massive star and the relaxation time of this system. For this purpose, we use an approach developed by Saslaw @10#, in which the scattering is described using a collisionless Boltzmann equation. The advantage of this method is that we can perform a fully analytical approach to the subject, and determine the influence of Yukawa-like term in the final results.

Starting from the equations of modified gravity hydrodynamics, we derive the equ tions of motion governing linear, adiabatic, radial perturbations of stars in scalar-tensor theories. There are two new features: first, the eigenvalue equation for the period of stellar oscillations is modified such that the eigenfrequencies are always larger than predicted by General Relativity. Second, the General Relativity condition for stellar instability is altered so that the adiabatic index can fall below 4/3 before unstable modes appear. Stars are more stable in modified gravity theories. Specialising to the case of chameleon-like theories, we investigate these effects numerically using both polytropic Lane-Emden stars and models coming from modified gravity stellar structure simulations. The change in the oscillation period can be as large as 50% and the critical adiabatic index for instability falls by a composition dependent amount of order 10^(-1). By solving the new equation for Cepheid models, it is found that the change in the inferred distance using the period-luminosity relation can be up to three times larger than if one had only considered the modified equilibrium structure. We discuss the implications of these results for recent and up-coming astrophysical tests and estimate that previous methods can produce new constraints such that the modifications are screened in regions of Newtonian potential of order 10^(-8)., Version 3: Updated to reflect published version. Mainly typos but the discussion of certain approximations has been expanded. Results and content unchanged from previous versions

We study the complete cosmological evolution of dark matter (DM) density fluctuations for DM particles that decoupled being ultrarelativistic during the radiation dominated era which is the case of keV scale warm DM (WDM). The new framework presented here can be applied to other types of DM and in particular we extend it to cold DM (CDM). The collisionless and linearized Boltzmann-Vlasov equations (B-V) for WDM and neutrinos in the presence of photons and coupled to the linearized Einstein equations are studied in detail in the presence of anisotropic stress with the Newtonian potential generically different from the spatial curvature perturbations. We recast this full system of B-V equations for DM and neutrinos into a system of coupled Volterra integral equations. These Volterra-type equations are valid both in the radiation dominated (RD) and matter dominated (MD) eras during which the WDM particles are ultrarelativistic and then nonrelativistic. This generalizes the so-called Gilbert integral equation only valid for nonrelativistic particles in the MD era. We succeed to reduce the system of four Volterra integral equations for the density and anisotropic stress fluctuations of DM and neutrinos into a system of only two coupled Volterra equations. The kernels and inhomogeneities in these equations are explicitly given functions. Combining the Boltzmann-Vlasov equations and the linearized Einstein equations constrain the initial conditions on the distribution functions and gravitational potentials. In the absence of neutrinos the anisotropic stress vanishes and the Volterra-type equations reduce to a single integral equation. These Volterra integral equations provide a useful and precise framework to compute the primordial WDM fluctuations over a wide range of scales including small scales up to k ~ 1/5 kpc., 26 pages, 1 figure. To appear in Phys Rev D

In this work we explore the consequences that a nonminimal coupling between geometry and matter can have on the dynamics of perfect fluids. It is argued that the presence of a static, axially symmetric pressureless fluid does not imply a Minkowski space-time like as is in general relativity. This feature can be attributed to a pressure mimicking mechanism related to the nonminimal coupling. The case of a spherically symmetric black hole surrounded by fluid matter is analyzed, and it is shown that under equilibrium conditions the total fluid mass is about twice that of the black hole. Finally, a generalization of the Newtonian potential for a fluid element is proposed and its implications are briefly discussed.

We study the flavor-changing quark-graviton vertex that is induced at the one-loop level when gravitational interactions are coupled to the standard model. Because of the conservation of the energy-momentum tensor the corresponding form factors turn out to be finite and gauge-invariant. Analytical expressions of the form factors are provided at leading order in the external masses. We show that flavor-changing interactions in gravity are local if the graviton is strictly massless while if the graviton has a small mass long-range interactions inducing a flavor-changing contribution in the Newton potential appear. Flavor-changing processes with massive spin-2 particles are also briefly discussed. These results can be generalized to the case of the lepton-graviton coupling., 21 pages, 2 figures. New results and two references added. To be published in PRD

Several recent theories suggest that light moduli or particles in "large" extra dimensions could mediate macroscopic forces exceeding gravitational strength at length scales below a millimeter. Such new forces can be parameterized as a Yukawa-type correction to the Newtonian potential of strength $\alpha$ relative to gravity and range $\lambda$. To extend the search for such new physics we have improved our apparatus utilizing cryogenic micro-cantilevers capable of measuring attonewton forces, which now includes a switchable magnetic force for calibration. Our most recent experimental constraints on Yukawa-type deviations from Newtonian gravity are more than three times as stringent as our previously published results, and represent the best bound in the range of 5 - 15 microns, with a 95 percent confidence exclusion of forces with $|\alpha| > 14,000$ at $\lambda$ = 10 microns., Comment: 12 pages, 9 figures, accepted for publication in PRD. Minor changes, replaced and corrected Figs 4,5,8

We consider the problem of gravitational forces between point particles on the branes in a five dimensional (5D) Randall-Sundrum model with two branes (at $y_1$ and $y_2$) and $S^1/Z_2$ symmetry of the fifth dimension. The matter on the branes is viewed as a perturbation on the vacuum metric and treated to linear order. In previous work \cite{ad} it was seen that the trace of the transverse part of the 4D metric on the TeV brane, $f^T(y_2)$, contributed a Newtonian potential enhanced by $e^{2\beta y_2} \cong 10^{32}$ and thus produced gross disagreement with experiment. In this work we include a scalar stabilizing field $\phi$ and solve the coupled Einstein and scalar equations to leading order for the case where $\phi_{0}^2/M_{5}^3$ is small and the vacuum field $\phi_{0}(y)$ is a decreasing function of $y$. $f^T$ then grows a mass factor $e^{-\mu r}$ where however, $\mu$ is suppressed from its natural value, $\mathcal{O}(M_{Pl})$, by an exponential factor $e^{-(1+\lambda_b)\beta y_2}$, $\lambda_b > 0$. Thus agreement with experiment depends on the interplay between the enhancing and decaying exponentials. Current data eliminates a significant part of the parameter space, and the Randall-Sundrum model will be sensitive to any improvements on the tests of the Newtonian force law at smaller distances., Comment: 22 pages, Fig.1 added

Brane-World models suggest modification of Newton's law of gravity on the 3-brane at submillimeter scales. The brane-world induced corrections are in higher powers of inverse distance and appear as additional terms with the Newtonian potential. The average inter-particle distance in white dwarf and neutron stars are $10^{-10} cms$ and $10^{-13} cms$ respectively, and therefore, the effect of submillimeter corrections needs to be investigated. We show, by carrying out simple manybody calculations, that the mass and mass-radius relationship of the white dwarf and neutron stars are not effected by submillimeter corrections. However, our analysis shows that the correction terms in the effective theory give rise to force akin to surface tension in normal liquids., 15 pages, no figures, explanations of pedagogical nature removed, typos corrected, final version to appear in Phys. Rev. D

The general formalism for calculating the Newtonian potential in fine-tuned or critical Randall-Sundrum braneworlds is outlined. It is based on using the full tensor structure of the graviton propagator. This approach avoids the brane-bending effect arising from calculating the potential for a point source. For a single brane, this gives a clear understanding of the disputed overall factor 4/3 entering the correction. The result can be written on a compact form which is evaluated to high accuracy for both short and large distances., 12 pages, LaTeX2e with RevTeX4, 3 postscript figures; Minor corrections, references updated

Using a parametrized expansion of the solar metric to second order in the Newtonian potential we calculate the trajectories of photons scattered by the Sun. In general the photon is scattered in two directions orthogonal to its initial momentum because of the anisotropic features of the solar field. We calculate both components of the deflection as measured by an observer within the solar field.

Using a parametrized expansion of the solar metric to second order in the Newtonian potential, we calculate the relativistic delay in the round-trip travel time of a radar signal reflected from a nearby planet. We find that one second-order contribution to the delay is on the order of ten nanoseconds, which is comparable to the uncertainties in present-day experiments involving the Viking spacecraft.

We revisit the question of viability of bigravity cosmology as a candidate for dark energy. In the context of the low energy limit model, where matter couples to a single metric, we study linear perturbations around homogeneous and isotropic backgrounds to derive the Poisson’s equation for the Newtonian potential. Extending to second order perturbations, we identify the Vainshtein radius below which nonlinear scalar self-interactions conspire to reproduce general relativity on local scales. We combine all of these results to determine the parameter space that allows a late-time de Sitter attractor compatible with observations and a successful Vainshtein mechanism. We find that the requirement of having a successful Vainshtein mechanism is not compatible with the existence of cosmological solutions at early times.