Gravitational Slip in the Parameterized Post-Newtonian Cosmology

TL;DR

The work develops a theory-agnostic Parameterized Post-Newtonian Cosmology (PPNC) to test gravity with cosmological data by deriving a comprehensive gravitational slip relation between the scalar potentials $\Phi$ and $\Psi$ on both small and large scales. On sub-horizon scales, the slip follows the familiar small-scale relation arising from the Poisson-like equations, while on super-horizon scales it is governed by the derivative of a redefined parameter $\hat{\gamma}$, via $\Phi-\Psi = - \frac{\hat{\gamma}'}{\hat{\gamma}}$, plus a gauge-like term. The authors implement a PSTF perturbation framework anchored in homogeneous Bianchi I cosmologies to connect large- and small-scale perturbations, and verify the construction in canonical scalar-tensor theories where the limits are recovered. They validate the framework with Planck, Cassini, and Mars ephemeris data using CLASS, finding results consistent with General Relativity and showing the large-scale slip has limited impact on current constraints, while enabling a direct mapping from $\Sigma$ constraints to PPN parameters and a unified gravity test across cosmic and Solar-System scales.

Abstract

A key signature of general relativity is that the two scalar potentials $Φ$ and $Ψ$, when expressed in the longitudinal gauge, are equal in the absence of fluids with anisotropic stress. This is often expressed by stating that their ratio, the "gravitational slip", is equal to unity. However, the equality of $Φ$ and $Ψ$ is typically broken in alternative theories of gravity. Observational constraints on the slip parameter are therefore of direct interest for testing Einstein's theory. In this paper we derive theory-independent expressions for the slip parameter on both large and small scales in Friedmann cosmologies, expressing it as a function of the post-Newtonian parameters. This is the final ingredient required for a complete parameterization of dust and dark energy-dominated cosmologies within the framework of Parameterized Post-Newtonian Cosmology (PPNC), which allows for the fully self-consistent modelling of cosmological observables without assuming any specific theory of gravity.

Figures (3)

• Figure 1: One and two-dimensional $0.3 \sigma$ smoothed posterior probabilities on the Hubble rate $100 \, h \equiv H_0/{\rm km}\space {\rm s}^{-1} {\rm Mpc}^{-1}$, the parameters $\bar{\alpha}$ and $\bar{\gamma}$, and the constant $c_1$. The time-dependence of the gravitational parameters is a power-law with $n=0.1$, and the data used is from Planck planck1planck2, the Cassini probe cassini, and the observations of the ephemeris of Mars mars. The grey band shows the $1\sigma$ interval from the Mars ephemeris constraint on $\dot{\alpha}(a_0)$ converted into a Gaussian prior on $\bar{\alpha}$.
• Figure 2: A comparison of the one and two-dimensional constraints available using Eq. (\ref{['larges']}) for the large-scale slip, compared to assuming the general relativistic relation $\Phi_{\rm L} =\Psi_{\rm L}$, for the parameters $\omega_c$, $H_0$, $\bar{\alpha}$, $\bar{\gamma}$ and $n$.
• Figure 3: Evolution of the gravitational slip on small ($k=6 \pi \, H_0 \times 10^2$), medium ($k=6 \pi \, H_0$) and large ($k=6 \pi \, H_0 \times 10^{-2}$) spatial scales, displayed as dotted, dashed and long-dashed lines respectively, for a scalar-tensor theory with $\omega=10$ and $\Omega_{\Lambda}=0.7$. The blue and red lines show the small and large-scale limits prescribed by Equation (\ref{['result']}) for the cases $k=6 \pi \, H_0 \times 10^2$ and $k=6 \pi \, H_0 \times 10^{-2}$.