Curvature versus distances: Testing the FLRW cosmology

We test the FLRW cosmology by reconstructing in a model-independent way both the Hubble parameter $H(z)$ and the comoving distance $D(z)$ via the most recent Hubble and Supernovae Ia data. In particular we use data binning with direct error propagation, the principal component analysis, the genetic algorithms and the Padé approximation. Using our reconstructions we evaluate the Clarkson et al. test known as ${\mathrm{\Omega }}_K(z)$, whose value is constant in redshift for the standard cosmological model, but deviates otherwise. Using present data, we find good agreement with the expected values of the standard cosmological model within the experimental errors, which are, however, large enough to allow for alternative cosmologies. A discrimination between the models may be possible for more precise future data. Finally, we provide forecasts, exploiting the baryon acoustic oscillation measurements from the Euclid survey.

Figure 1

Comoving distances computed directly from the 580 SnIa of the Union2.1 compilation [45] by using Eq. (11).

Figure 2

The mean Hubble function $\overline{H}$, comoving distance $\overline{D}$, and derivative of the comoving distance $D_{,z}$ computed with the binning of Secs. 3b1 (orange thin error bars) and 3b2 (black thick error bars). We omit to show horizontal error bars in order not to make the plot too crowded and confusing.

Figure 3

The curvature parameter ${\mathrm{\Omega }}_K$ obtained by using $H(z)$ data from passively evolving galaxies and SnIa data and applying to them the binning procedure described in Sec. 3b1. For comparison, we also show results from the second binning procedure, described in Sec. 3b2, as red thin error bars. The light-blue long-dashed line corresponds to the case of a flat FLRW universe: ${\mathrm{\Omega }}_K(z)=0$. On the left panel we show our full results, while on the right panel we focus on the region inside the yellow dashed box, i.e. on redshifts where constraints are best: $z>0.12$. We omit to show horizontal error bars in order not to make the plot too crowded and confusing.

Figure 4

The curvature parameter ${\mathrm{\Omega }}_K$ obtained by using $H(z)$ data from passively evolving galaxies and SnIa data and applying to them the binning procedure described in Sec. 3b2. The dashed light-blue long-dashed line corresponds to the case of a flat FRW universe. On the left panel we show our full results, while on the right panel we focus on the region inside the yellow dashed box, i.e. on redshifts where constraints are best: $z>0.2$. We omit to show horizontal error bars in order not to make the plot too crowded and confusing.

Figure 5

We show the deceleration parameter for the two data sets, $H(z)$ (left panel) and SnIa (right panel), using the PCA technique. As an example, we also report the deceleration parameter of the $\mathrm{\Lambda }\mathrm{CDM}$ model (red dashed line).

Figure 6

We show the curvature parameter computed by combining the two data sets, $H(z)$ and SnIa, using the PCA technique. The red dashed line refers to the flat FLRW model.

Figure 7

The reconstruction of ${\mathrm{\Omega }}_K(z)$ from the GA. In both cases the gray region is the $1\sigma$ error and the solid line the best-fit. Clearly, it is consistent with ${\mathrm{\Omega }}_K=0$.

Figure 8

We plot the ${\mathrm{\Omega }}_K$ as a function of redshift (blue dashed line) and the superposed error region (orange region).

Figure 9

Error bars for the curvature parameter using the Euclid survey only.

Figure 10

Error bars for the curvature parameter using Euclid and adding SnIa with: $\mathrm{N}=100$ and $\mathrm{N}=1000$, left and right, respectively. In the right panel we also plotted, as an example, three non-FLRW models. The blue solid line corresponds to the curvature parameter in the LTB model. The brown dashed line indicates the Tardis model as in Fig. 12(a) of [25] and is courtesy of Mikko Lavinto and Syksy Räsänen. The green dot-dashed line corresponds to the timescape scenario, see [16, 63] and is courtesy of David Wiltshire.

Figure 11

Example of a binning scheme. The derivations assume that the point of interest has $z\in (z_{i-1},z_i]$, so that $q(z)=q_i$.