Parametrized post-Newtonian Parameters Research Articles

PPN meets EFT of dark energy: post-Newtonian approximation in higher-order scalar-tensor theories

We study the post-Newtonian limit of higher-order scalar-tensor theories that are degenerate in the unitary gauge. They can be conveniently described by the effective field theory (EFT) of dark energy. We determine all the parametrized post-Newtonian (PPN) parameters in terms of the EFT of dark energy parameters. Experimental bounds on the PPN parameters are then translated to constraints on the EFT parameters. We present a Lagrangian of a unitary degenerate higher-order scalar-tensor theory characterized by a single function of the kinetic term of the scalar field whose PPN parameters have the same values as in general relativity.

Multi-scalar theories of gravity with direct matter couplings and their parametrized post-Newtonian parameters

We study theories of gravity including, in addition to the metric, several scalar fields in the gravitational sector. The particularity of this work is that we allow for direct couplings between these gravitating scalars and the matter sector, which can generally be different for the source and the probe of gravity, in addition to the universal interactions generated by the Jordan frame metric. The weak gravity regime of this theory, which would describe solar-system experiments, is studied using the parametrized post-Newtonian (PPN) formalism. We derive the expression of the ten parameters of this formalism. Among them, ζ 3 and ζ 4 are modified with respect to their values in the theories without direct couplings. This fact holds even after eliminating the direct couplings between the gravitating scalars and the energy density of the source, by redefinition of the Jordan frame. All other PPN parameters are insensitive to the direct couplings once in the correctly identified Jordan frame. When direct couplings are different for the source and the probe of gravity, they make non-relativistic probes deviate from the geodesics of the PPN metric in this frame, already at Newtonian order. Such couplings would thus be directlydetectable and would have been excluded by experiments.This shows that, contrary to the claims in the recent literature, it is impossible to screen the presence of gravitating scalars relying only on a curved target space and direct couplings to matter.

Unscreening of f(R) gravity near the galactic center black hole: Testability through pericenter shift below S0-2’s orbit

General Relativity (GR) has been tested extensively in the solar system and is being tested in the new environment of the Galactic Center (GC) black hole where the dimensionless gravitational potential ([Formula: see text]) is 100 times stronger than the one encountered in solar system. Therefore, the neighborhood of the GC black hole is a naive opportunity to test modified theories of gravity. In this work, effect of [Formula: see text] gravity near the black hole is studied. The difference of pericenter shift between GR and [Formula: see text] gravity is studied for compact orbits having semi-major axis equal to and below [Formula: see text] au (S0-2 like orbits). In a model-dependent approach, we choose [Formula: see text] (power law gravity) model which is cosmologically motivated and study the deviation in orbital pericenter shift for both zero spin and non-zero spin of the black hole. It is found that effect of [Formula: see text] gravity becomes prominent for compact orbits. In model-independent approach to [Formula: see text] gravity with the generic scalaron fields ([Formula: see text]), we extract the parameters of [Formula: see text] gravity from the current bounds on Parametrized Post-Newtonian (PPN) parameters ([Formula: see text]) near the GC black hole. The screening of [Formula: see text] gravity is also investigated for these bounds on PPN parameters. It has been found that sufficiently massive scalarons ([Formula: see text][Formula: see text]eV) are completely screened but light and intermediate mass scalarons ([Formula: see text][Formula: see text]eV and [Formula: see text][Formula: see text]eV) are unscreened towards S0-2 like orbits as well as in the orbit of the newly discovered short period star S4716 ([Formula: see text][Formula: see text]au). The possibility of detection of the [Formula: see text] gravity effects due to these unscreened scalarons is forecasted with existing and upcoming astrometric capabilities of Extremely Large Telescopes (ELTs).

Post-Newtonian parameters of ghost-free parity-violating gravities

We investigate the slow-motion and weak-field approximation of the general ghost-free parity-violating (PV) theory of gravity in the parametrized post-Newtonian (PPN) framework and derive the perturbative field equations, which are modified by the PV terms of this theory. The complete PPN parameters are obtained by solving the perturbative field equations. We find that all the PPN parameters are exactly the same as those in general relativity, except for an extra parameter κ, which is caused by the new curl-type term in the gravitomagnetic sector of the metric in this theory. We calculate the precession effects of gyroscopes in this theory and constrain the model parameters by the observations of the Gravity Probe B experiment.

Is It Possible to See the Breaking Point of General Relativity near the Galactic Center Black Hole? Consideration of Scalaron and Higher-dimensional Gravity

Abstract The Galactic center black hole is a putative laboratory to test general relativity (GR) and constrain its alternatives. f(R) scalaron gravity is an interesting alternative to GR and has tremendous prospects for astrophysics and fundamental physics near the black hole. In this work, we search for breaking points of GR through estimation of pericenter shift of stellar orbits with semimajor axis a = (45–1000) au. The black hole spin is taken as the maximum χ = 0.99, and orbital eccentricity is taken as e = 0.9. We work with theoretical scalaron field amplitude and coupling, predicted by Kalita, and also consider the constraints reported by Hees et al. The scalaron mass is taken in the range (10−22–10−17) eV. It is found that GR suppresses scalaron gravity at all orbital radii for the theoretical values of scalaron field coupling predicted by Kalita. Breaking point arises only for higher scalaron coupling resulting from the Hees et al. observations within a few tens of au to a = 1000 au. We also estimate the pericenter shift with a power-law potential V(r) ∼ 1/r 2 arising in five-dimensional gravity and obtain allowed ranges of the five-dimensional Planck mass through existing bounds on the parameterized post-Newtonian parameters coming from the orbits of S-2, S-38, and S-55. The breaking point for GR arises for a five-dimensional Planck mass of about 104 GeV. Constraint on this parameter, expected from the astrometric capabilities of existing and upcoming large telescopes, is also presented.

Post-Newtonian limit of generalized symmetric teleparallel gravity

In this article we analyze the post-Newtonian approximation of a generalization of the symmetric teleparallel gravity with the help of the parameterized post-Newtonian (PPN) formalism. This class of theories is based on a free function of the five independent quadratic contractions of the non-metricity tensor. By calculating the PPN metric of these theories, we can restrict the Taylor coefficients of the free function with the help of the PPN parameters and their observational bounds. We find two families of theories whose PPN parameters are identical to those of general relativity, and thus in full agreement with observations. For three further families, we find that only the PPN parameters $\beta$ and $\gamma$ deviate, but can be brought arbitrarily close to their general relativity values by an appropriate choice of the Lagrangian, so that also these families contain candidate theories which agree with observations. The remaining theories either possess no well-defined solution of the post-Newtonian field equations, or possess a post-Newtonian limit which exceeds the form assumed in the PPN formalism.

Special dyon-like black hole solution in the model with two Abelian gauge fields and two scalar fields

Dilatonic black hole dyon-like solution in the gravitational 4d model with two scalar fields and two 2-forms, governed by two 2-dimensional dilatonic coupling vectors obeying , i = 1, 2, is found. Some physical parameters of the solutions are obtained: gravitational mass, scalar charge, Hawking temperature, black hole area entropy and parametrized post-Newtonian (PPN) parameters β and γ. The PPN parameters do not depend on the coupling vectors . A bound on the gravitational mass is proved.

PPN rotation curves in static distributions with spherical symmetry

From a parametrized post-Newtonian (PPN) perspective, we address the question of whether or not the new degrees of freedom represented by the PPN potentials can lead to significant modifications in the dynamics of galaxies in the direction of rendering dark matter obsolete. Here, we focus on the study of rotation curves associated with spherically symmetric configurations. The values for the post-Newtonian parameters, which help us to classify the different metric theories of gravity, are tightly constrained, mainly by Solar System experiments. Such restrictions render the modifications of gravitational effects, with respect to general relativity (GR), to be insignificant, making attempts to find alternative metrical theories rather fruitless. However, in recent years, metric theories characterized by screening mechanisms have become popular, due to the fact that they lead to the possibility of modifications in larger scales than the Solar System while retaining the success of GR on it, allowing for violations of the constraints of the post-Newtonian parameters. In such a context, we consider here two kinds of solutions for field equations: (i) Vacuum solutions (i.e., when no matter fields are present) and (ii) fields in the presence of a polytropic distribution of matter. For case (i), we find that the post-Newtonian corrections do not lead to modifications significant enough to be considered an alternative to the dark matter hypothesis. In case (ii), we find that for a wide range of values for the PPN parameters $\ensuremath{\gamma}$, $\ensuremath{\beta}\ensuremath{\le}1$, $\ensuremath{\xi}\ensuremath{\ge}0$, ${\ensuremath{\alpha}}_{3}$, ${\ensuremath{\zeta}}_{1}$, and ${\ensuremath{\zeta}}_{2}$, the need for dark matter is unavoidable in order to find flat rotation curves. It is only for theories in which ${\ensuremath{\zeta}}_{3}g0$, or $\ensuremath{\beta}g1$, or $\ensuremath{\xi}l0$ that some resemblance of flat rotation curves is found. But the latter two require some direct fine-tuning of the screening radius ${r}_{c}$, while ${\ensuremath{\zeta}}_{3}g0$ implies the most sound modifications. The latter suggests, at least for the models considered, that these are the only theories (consistent with the usual PPN approach) capable of replacing dark matter as a possible explanation for the dynamics of galaxies.

PPN Motion of S-Stars Around Sgr A*

Parametrized Post-Newtonian (PPN) equations of motion for several S-stars nearest to the Galactic Center compact relativistic object SgrA* are considered. The effect of the orbital periods difference between Newtonian and Post-Newtonian cases is taken into account. The best fit PN orbit of S2 has a period which is 16 days longer than Newtonian one. The PPN parameters $\beta_\text{PPN}$ and $\gamma_\text{PPN}$ are measured. Bayesian sampling is used to fit the trajectories of the PPN laws of motion. Posterior estimates of $\beta_\text{PPN}$ and $\gamma_\text{PPN}$ are $0.97^{+0.42}_{-0.65}$ and $0.81^{+0.46}_{-0.60}$ respectively. The result confirms General Relativity prediction for the Post-Newtonian equations of motion in the conditions of orbital motions in vicinity of the SgrA*.

Relativistic Effects in Orbital Motion of the S-Stars at the Galactic Center

The Galactic Center star cluster, known as S-stars, is a perfect source of relativistic phenomena observations. The stars are located in the strong field of relativistic compact object Sgr A* and are moving with very high velocities at pericenters of their orbits. In this work we consider motion of several S-stars by using the Parameterized Post-Newtonian (PPN) formalism of General Relativity (GR) and Post-Newtonian (PN) equations of motion of the Feynman’s quantum-field gravity theory, where the positive energy density of the gravity field can be measured via the relativistic pericenter shift. The PPN parameters β and γ are constrained using the S-stars data. The positive value of the Tg00 component of the gravity energy–momentum tensor is confirmed for condition of S-stars motion.

Gauge-invariant approach to the parametrized post-Newtonian formalism

We present an approach to the parametrized post-Newtonian (PPN) formalism which is based on gauge-invariant higher order perturbation theory. This approach divides the components of the metric perturbations into gauge-invariant quantities, which carry information about the physical system under consideration, and pure gauge quantities, which describe the choice of the coordinate system. This separation generally leads to a simplification of the PPN procedure, since only the gauge-invariant quantities appear in the field equations and must be determined by solving them. Another simplification arises from the fact that the gauge-invariant approach supersedes the necessity to first choose a gauge for solving the gravitational field equations and later transforming the obtained solution into the standard PPN gauge, as it is conventionally done in the PPN formalism, whose standard PPN gauge is determined only after the full solution is known. In addition to the usual metric formulation, we also present a tetrad formulation of the gauge-invariant PPN formalism. To illustrate their practical application, we demonstrate the calculation of the PPN parameters of a well-known scalar-tensor class of theories.

A New Method to Test the Einstein’s Weak Equivalence Principle

Abstract The Einstein’s weak equivalence principle (WEP) is one of the foundational assumptions of general relativity and some other gravity theories. In the theory of parametrized post-Newtonian (PPN), the difference between the PPN parameters γ of different particles or the same type of particle with different energies, Δγ, represents the violation of WEP. Current constraints on Δγ are derived from the observed time delay between correlated particles of astronomical sources. However, the observed time delay is contaminated by other effects, such as the time delays due to different particle emission times, the potential Lorentz invariance violation, and none-zero photon rest mass. Therefore, current constraints are only upper limits. Here, we propose a new method to test WEP based on the fact that the gravitational time delay is direction-dependent while others are not. This is the first method that can naturally correct other time-delay effects. Using the time-delay measurements of BASTE gamma-ray burst sample and the gravitational potential of local super galaxy cluster Laniakea, we find that the constraint on Δγ of different energy photons can be as low as 10−14. In the future, if more gravitational wave events and fast radio bursts with much more precise time-delay measurements are observed, this method can give a reliable and tight constraint on WEP.

Post-Newtonian parameters γ and β of scalar-tensor gravity for a homogeneous gravitating sphere

We calculate the parameters $\gamma$ and $\beta$ in the parametrized post-Newtonian (PPN) formalism for scalar-tensor gravity (STG) with an arbitrary potential, under the assumption that the source matter is given by a non-rotating sphere of constant density, pressure and internal energy. For our calculation we write the STG field equations in a form which is manifestly invariant under conformal transformations of the metric and redefinitions of the scalar field. This easily shows that also the obtained PPN parameters are invariant under such transformations. Our result is consistent with the expectation that STG is a fully conservative theory, i.e., only $\gamma$ and $\beta$ differ from their general relativity values $\gamma = \beta = 1$, which indicates the absence of preferred frame and preferred location effects. We find that the values of the PPN parameters depend on both the radius of the gravitating mass source and the distance between the source and the observer. Most interestingly, we find that also at large distances from the source $\beta$ does not approach $\beta = 1$, but receives corrections due to a modified gravitational self-energy of the source. Finally, we compare our result to a number of measurements of $\gamma$ and $\beta$ in the Solar System. We find that in particular measurements of $\beta$ improve the previously obtained bounds on the theory parameters, due to the aforementioned long-distance corrections.

Post-Newtonian parameter γ and the deflection of light in ghost-free massive bimetric gravity

We consider the parametrized post-Newtonian (PPN) limit of ghost-free massive bimetric gravity with two mutually non-interacting matter sectors coupled to the two metrics. Making use of a gauge-invariant differential decomposition of the metric perturbations, we solve the field equations up to the linear PPN order for a static, point-like mass source. From the result we derive the PPN parameter $\gamma$ for spherically symmetric systems, which describes the gravitational deflection of light by visible matter. By a comparison to its value measured in the solar system we obtain bounds on the parameters of the theory. We further discuss the deflection of light by dark matter and find an agreement with the observed light deflection by galaxies. We finally speculate about a possible explanation for the observed distribution of dark matter in galactic mergers such as Abell 520 and Abell 3827.

Dilatonic dyon-like black hole solutions in the model with two Abelian gauge fields

Dilatonic black hole dyon-like solutions in the gravitational 4d model with a scalar field, two 2-forms, two dilatonic coupling constants lambda _i ne 0, i =1,2, obeying lambda _1 ne - lambda _2 and the sign parameter varepsilon = pm 1 for scalar field kinetic term are considered. Here varepsilon = - 1 corresponds to a ghost scalar field. These solutions are defined up to solutions of two master equations for two moduli functions, when lambda ^2_i ne 1/2 for varepsilon = - 1. Some physical parameters of the solutions are obtained: gravitational mass, scalar charge, Hawking temperature, black hole area entropy and parametrized post-Newtonian (PPN) parameters beta and gamma . The PPN parameters do not depend on the couplings lambda _i and varepsilon . A set of bounds on the gravitational mass and scalar charge are found by using a certain conjecture on the parameters of solutions, when 1 +2 lambda _i^2 varepsilon > 0, i =1,2.

Parameterized post-Newtonian approximation in a teleparallel model of dark energy with a boundary term

We study the parameterized post-Newtonian approximation in teleparallel model of gravity with a scalar field. The scalar field is non-minimally coupled to the scalar torsion as well as to the boundary term introduced in Bahamonde and Wright (Phys Rev D 92:084034 arXiv:1508.06580v4 [gr-qc], 2015). We show that, in contrast to the case where the scalar field is only coupled to the scalar torsion, the presence of the new coupling affects the parameterized post-Newtonian parameters. These parameters for different situations are obtained and discussed.

Parameterized post-Newtonian cosmology

Einstein’s theory of gravity has been extensively tested on solar system scales, and for isolated astrophysical systems, using the perturbative framework known as the parameterized post-Newtonian (PPN) formalism. This framework is designed for use in the weak-field and slow-motion limit of gravity, and can be used to constrain a large class of metric theories of gravity with data collected from the aforementioned systems. Given the potential of future surveys to probe cosmological scales to high precision, it is a topic of much contemporary interest to construct a similar framework to link Einstein’s theory of gravity and its alternatives to observations on cosmological scales. Our approach to this problem is to adapt and extend the existing PPN formalism for use in cosmology. We derive a set of equations that use the same parameters to consistently model both weak fields and cosmology. This allows us to parameterize a large class of modified theories of gravity and dark energy models on cosmological scales, using just four functions of time. These four functions can be directly linked to the background expansion of the universe, first-order cosmological perturbations, and the weak-field limit of the theory. They also reduce to the standard PPN parameters on solar system scales. We illustrate how dark energy models and scalar-tensor and vector-tensor theories of gravity fit into this framework, which we refer to as ‘parameterized post-Newtonian cosmology’ (PPNC).

Post-Newtonian parameters and cosmological constant of screened modified gravity

Screened modified gravity (SMG) is a kind of scalar-tensor theories with screening mechanisms, which can generate screening effect to suppress the fifth force in high density environments and pass the solar system tests. Meanwhile, the potential of scalar field in the theories can drive the acceleration of the late universe. In this paper, we calculate the parameterized post-Newtonian (PPN) parameters $\gamma$ and $\beta$, the effective gravitational constant $G_{\rm eff}$ and the effective cosmological constant $\Lambda$ for SMG with a general potential $V$ and coupling function $A$. The dependence of these parameters on the model parameters of SMG and/or the physical properties of the source object are clearly presented. As an application of these results, we focus on three specific theories of SMG (chameleon, symmetron and dilaton models). Using the formulae to calculate their PPN parameters and cosmological constant, we derive the constraints on the model parameters by combining the observations on solar system and cosmological scales.

Solar System tests of a scalar-tensor gravity with a general potential: Insensitivity of light deflection and Cassini tracking

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.

Parametrized post-Newtonian limit of Horndeski’s gravity theory

We discuss the parametrized post-Newtonian (PPN) limit of Horndeski's theory of gravity, also known under the name generalized G-inflation or ${\mathrm{G}}^{2}$-inflation, which is the most general scalar-tensor theory of gravity with at most second-order field equations in four dimensions. We derive conditions on the action for the validity of the post-Newtonian limit. For the most general class of theories consistent with these conditions we calculate the PPN parameters $\ensuremath{\gamma}(r)$ and $\ensuremath{\beta}(r)$, which in general depend on the interaction distance $r$ between the gravitating mass and the test mass. For a more restricted class of theories, in which the scalar field is massless, we calculate the full set of PPN parameters. It turns out that in this restricted case all parameters are constants and that the only parameters potentially deviating from observations are $\ensuremath{\gamma}$ and $\ensuremath{\beta}$. We finally apply our results to a number of example theories, including Galileons and different models of Higgs inflation.