Testing backreaction effects with observations

In order to quantitatively test the ability of averaged inhomogeneous cosmologies to correctly describe observations of the large-scale properties of the Universe, we introduce a smoothed template metric corresponding to a constant spatial curvature model at any time, but with an evolving curvature parameter. This metric is used to compute quantities along an approximate effective light cone of the averaged model of the Universe. As opposed to the standard Friedmann model, we parametrize this template metric by exact scaling properties of an averaged inhomogeneous cosmology, and we also motivate this form of the metric by results on a geometrical smoothing of inhomogeneous cosmological hypersurfaces. The purpose of the paper is not to demonstrate that the backreaction effect is actually responsible for the dark energy phenomenon by explicitly calculating the effect from a local model of the geometry and the distribution of matter, but rather to propose a way to deal with observations in the backreaction context, and to understand what kind of generic properties have to hold in order for a backreaction model to explain the observed features of the Universe on large scales. We test our hypothesis for the template metric against supernova data and the position of the cosmic microwave background peaks, and infer the goodness of fit and parameter uncertainties. We find that averaged inhomogeneous models can reproduce the observations without requiring an additional dark energy component (though a volume acceleration is still needed), and that current data do not disfavor our main assumption on the effective light cone structure. We also show that the experimental uncertainties on the angular diameter distance and the Hubble parameter from baryon acoustic oscillations measurements—forseen in future surveys like the proposed EUCLID satellite project—are sufficiently small to distinguish between a Friedmann-Lemaître-Robertson-Walker template geometry and the template geometry with consistently evolving curvature.

Figure 1

Evolutions of the coordinate distance $cr/H_{{\mathcal{D}}_{\mathbf{0}}}$ (a), the redshift (b), and the ratio between the FLRW redshift to the redshift, $1/(a_{\mathcal{D}}(1+z_{\mathcal{D}}))$ (c), in the averaged model as functions of the effective scale factor, in an averaged effective model with $n=-1$, ${\Omega }_m^{{\mathcal{D}}_{\mathbf{0}}}=0.3$, and $H_{{\mathcal{D}}_{\mathbf{0}}}=70\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$ (dashed lines). In (a) and (b), the solid lines represent a FLRW model with the same set of parameters. (d) represents $1/(a_{\mathcal{D}}(1+z_{\mathcal{D}}))$ for the best-fit averaged model described in Sec. 4b.

Figure 2

Supernovae and CMB constraints in the $({\Omega }_m^{{\mathcal{D}}_{\mathbf{0}}},n)$ plane for the averaged effective model (filled ellipses) and for a standard flat FLRW model with a quintessence field with constant equation of state $w=-(n+3)/3$ (black ellipses). The disk and diamond represent the absolute best-fit models, respectively, for the standard FLRW model and the averaged effective model.

Figure 3

(a) Evolution of ${\kappa }_{\mathcal{D}}$ for the best-fit averaged effective model (diamond in Fig. 2) with $n=0.12$, ${\Omega }_m^{{\mathcal{D}}_{\mathbf{0}}}=0.38$, and $H_0=78.54\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$. (b) Evolution of ${\Omega }_m^{\mathcal{D}}$ (dashed line) and ${\Omega }_X^{\mathcal{D}}$ (solid line) for the same model as in (a).

Figure 4

Probability distribution function for the difference of scalings $n-m$, marginalized over all other parameters. The vertical line indicates the case $n=m$ corresponding to assumption (26).

Figure 5

Evolution of $C_{\mathcal{D}}$ as a function of $1+z_{\mathcal{D}}$ for an averaged effective model with scaling backreaction whose parameters are $n=0.12$, ${\Omega }_m^{{\mathcal{D}}_{\mathbf{0}}}=0.38$, and $H_{{\mathcal{D}}_{\mathbf{0}}}=78.54\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$ (absolute best-fit model represented by the diamond in Fig. 2).

Figure 6

(a) Evolution of ${\Omega }_k(z_{\mathcal{D}})$ as a function of redshift for the absolute best-fit averaged model represented by the diamond in Fig. 2. The error bars are obtained using the EUCLID satellite project measure uncertainties. One can see that all positively curved FLRW models (${\Omega }_{k,0}<0$) and only highly negatively curved FLRW models (${\Omega }_{k,0}>0.5$) can be excluded by the estimation of ${\Omega }_k(z_{\mathcal{D}})$. (b) Evolution of the coordinate distance for the best-fit averaged model (solid line), for a $\Lambda \mathrm{CDM}$ model with ${\Omega }_{m,0}=0.277$, ${\Omega }_{\Lambda }=0.735$, and $H_0=73\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$ (dashed line), and for the FLRW model with the same parameters as the best-fit averaged model (dash-dotted line). The error bars are still obtained using the EUCLID expectations. (c) Evolution of the Hubble parameter $H/H_0$ for the best-fit averaged model (solid line), the FLRW model with the same parameters as the averaged best-fit model (dash-dotted line), and for the same $\Lambda \mathrm{CDM}$ model as in the central panel (dashed line). The error bars are still obtained using the EUCLID expectations.