Parametrizing modified gravity for cosmological surveys

One of the challenges in testing gravity with cosmology is the vast freedom opened when extending General Relativity. For linear perturbations, one solution consists in using the effective field theory of dark energy. Even then, the theory space is described in terms of a handful of free functions of time. This needs to be reduced to a finite number of parameters to be practical for cosmological surveys. We explore in this article how well simple parametrizations, with a small number of parameters, can fit observables computed from complex theories. Imposing the stability of linear perturbations appreciably reduces the theory space we explore. We find that observables are not extremely sensitive to short time-scale variations and that simple, smooth parametrizations are usually sufficient to describe this theory space. Using the Bayesian information criterion, we find that using two parameters for each function (an amplitude and a power-law index) is preferred over complex models for 86% of our theory space.

Figure 1

Representation of the method used in this paper. We will compute observables (galaxy and weak lensing power spectra) from complex EFT of DE functions after imposing stability conditions. They will be fitted to simplified versions of these functions. Then, we will use model comparison tools to assess whether the complexity (i.e., the size of the circle) of the original functions is actually transferred to the observables or if simple functions are enough capture the physical features. In this diagram, this essentially corresponds to comparing the size of the dark blue and the orange circles. Moreover, we want the orange circle on the right to be the same size as the middle one; i.e., the simple functions have to explain the whole observable space.

Figure 2

Comparisons of the distribution of redshift evolution of ${\alpha }_{\mathrm{T}}$ (top left), ${\alpha }_{\mathrm{B}}$ (top right), ${\alpha }_{\mathrm{M}}$ (bottom left), and ${\mu }_{\mathrm{eff}}$ and ${\mu }_{\text{light}}$ (bottom right) for $\{{\alpha }_{\mathrm{B},0},{\alpha }_{\mathrm{T},0},{\alpha }_{\mathrm{M},0}\}=\{-0.3,1,-0.9\}$. The lines delimit where 95% of the curves resides. In dark purple is the case in which there is no restriction on $c_s^2\alpha$. In orange is the parameters taken only if the corresponding $c_s^2\alpha$ is positive. Finally, in light green is the case in which $c_s^2\alpha >0$ and ${\sigma }_8$, as well as ${\mu }_{\text{light}}{\sigma }_8$, are within 25% of the $\mathrm{\Lambda }\mathrm{CDM}$ value. The black dotted-dashed lines correspond to the $\{{\alpha }_{\mathrm{X}}\}$ evolving according to Eq. (10), while the red dashed lines are for a quintic Galileon model with a the same set $\{{\alpha }_{\mathrm{X},0}\}$, fixed by the model parameters (see the main text for details).

Figure 3

We plot the relative difference of ${\mu }_{eff}$ and ${\mu }_{light}$ w.r.t. their LCDM values (dashed) compared to the relative difference in the corresponding observables (same color, full line), respectively the galaxy power spectrum and the $C_{ij}$ averaged over $\ell$. The five colors are for five different random realizations of Eq. (8). For weak lensing, the crosses indicate the center of the redshift bins. While ${\mu }_{\mathrm{eff}}$ and ${\mu }_{\text{light}}$ vary on a short time scale and by significant amounts (more than 200% for ${\mu }_{\mathrm{eff}}$), the observables are much smoother, and the deviations are much smaller.

Figure 4

Histogram of the BIC values for “${\mathrm{\Omega }}_{\mathrm{m}}$” (light purple), “1i” (green), and “3i” (dark orange), compared to the true value (pink line). The last one corresponds to a perfect fit ${\chi }^2=0$ with Eq. (8) but is penalized in the BIC due to its high number of parameters. Most (93%) of the time, at least one simpler model has a significantly lower BIC ($\mathrm{\Delta }\mathrm{BIC}<-10$) than true, meaning it is preferred over the complex model. In the corner, we show the percentage of times when a given model has the lowest BIC.

Figure 5

Evolution of the $\{{\alpha }_{\mathrm{X}}\}$ as a function of redshift for the true model (pink), “${\mathrm{\Omega }}_{\mathrm{m}}$” (purple), “1i” (green), and “3i” (orange). The dashed lines represent a case in which the simple parametrizations all failed at reconstructing the observables, while they all succeeded for the full lines. The bottom right panel shows the relative difference in the observables between the simple models and the true ones in a percentage. The crosses indicate the center of the redshift bins in the weak lensing measurement.