Large-scale structure phenomenology of viable Horndeski theories

Phenomenological functions $\mathrm{\Sigma }$ and $\mu$, also known as $G_{\text{light}}/G$ and $G_{\text{matter}}/G$, are commonly used to parametrize modifications of the growth of large-scale structure in alternative theories of gravity. We study the values these functions can take in Horndeski theories, i.e., the class of scalar-tensor theories with second order equations of motion. We restrict our attention to models that are in broad agreement with tests of gravity and the observed cosmic expansion history. In particular, we require the speed of gravity to be equal to the speed of light today, as required by the recent detection of gravitational waves and electromagnetic emission from a binary neutron star merger. We examine the correlations between the values of $\mathrm{\Sigma }$ and $\mu$ analytically within the quasistatic approximation and numerically by sampling the space of allowed solutions. We confirm that the conjecture made in [L. Pogosian and A. Silvestri, Phys. Rev. D 94, 104014 (2016)], that $(\mathrm{\Sigma }-1)(\mu -1)\ge 0$ in viable Horndeski theories, holds very well. Along with that, we check the validity of the quasistatic approximation within different corners of Horndeski theory. Our results show that, even with the tight bound on the present-day speed of gravitational waves, there is room within Horndeski theories for nontrivial signatures of modified gravity at the level of linear perturbations.

Figure 1

The peak values and the standard deviation of the Gaussian prior imposed on the evolution of the Hubble parameter, $H(z)$. The fiducial expansion history corresponds to the Planck 2015 best fit $\mathrm{\Lambda }\mathrm{CDM}$ model [81]. The standard deviation is chosen to be wide enough to accommodate any tensions that may exist between different data sets.

Figure 2

Distributions of $\mathrm{\Sigma }$ and $\mu$ in GBD models, i.e., the scalar-tensor models with a canonical kinetic term, at representative values of $a$ and $k$. Shown are results obtained by numerically solving exact equations for cosmological perturbations (orange dots) and by using the quasistatic (black crosses) forms of $\mathrm{\Sigma }$ and $\mu$ given by Eqs. (8) and (9).

Figure 3

Same as in Fig. 2 but for the $H_S$ models, i.e., the subset of Horndeski models in which the speed of gravitational waves is the same as the speed of light at all redshifts.

Figure 4

Same as in Figs. 2 and 3, but for the full class of Horndeski models with the restriction on variation of the speed of gravitational waves imposed only at $z=0$.

Figure 5

Effects of imposing the stability conditions and observational priors on the $\mathrm{\Sigma }\text{−}\mu$ distribution in the $H_S$ model for $a=0.9$ and $k=0.1\mathrm{h}/\mathrm{Mpc}$. The three panels correspond to samples obtained in three different runs: sampling without any constraints (left panel), sampling with the stability constraints (middle), and sampling with both the stability constraints and observational priors (right). Each panel contains $10^4$ points. The impacts of stability and observational constraints shown here are representative of what happens at other redshifts and scales and in the other classes of models that we studied.