4.4 Reconstructing the global curvature at different redshifts

Clarkson et al. [240Jump To The Next Citation Point] presented an observational test for the Copernican principle which relies on the consistency relation between expansion rate and angular diameter distance. Here we discuss the implications for Euclid.

Let us recall that the angular diameter distance in a FLRW model can be written as:

( ∘ ------∫ ) --1--------1----- (0) z ′-H0--- DA (z) = 1 + z ∘ ---(0)sin − ΩK dz H (z′) . (4.4.1 ) H0 − ΩK 0
where (0) ΩK is the curvature parameter today.

We can invert the last equation to obtain an expression for the curvature parameter that depends on the Hubble parameter H and comoving angular diameter distance D (z) = (1 + z)DA (z) only, see [240]:

(0) [H (z)D ′(z)]2 − 1 ΩK = -------------2---- (4.4.2 ) [H0D (z)]
where here the prime refers to the derivative with respect the redshift. Then Eq. (4.4.2View Equation) tells us how the curvature parameter can be measured from the distance and the Hubble rate observations, in a model-independent way.

The idea is then to measure the curvature parameter Ω (0) K at different redshifts. Let us consider again Eq. (4.4.2View Equation); if we are in a FLRW universe then (0) ΩK should be independent of redshift, i.e., its derivative with respect to z should be zero

(0) dΩ-K-- 𝒞 (z ) = dz = 0. (4.4.3 )
If it happens that 𝒞 (z) ⁄= 0 even at a single redshift then this means the large-scale universe is not homogeneous.

A possible test to measure Ω (0K) at various redshifts is provided by baryon acoustic oscillations. Observing the features of BAO in the galaxy power spectrum in both angular (orthogonal to the line of sight L ⊥) and radial direction (along the line of sight L ∥) allows us to measure with a great accuracy both DA (z) and H (z), respectively.

If the geometry is not FLRW, then the standard BAO will be deformed in three different ways:

  1. The sound horizon scale, which is the characteristic ruler, will be different in the ⊥ and ∥ directions and it will be also different from that for the FLRW universe.
  2. Even if the sound horizon were isotropic at decoupling, the subsequent expansion in the ⊥ and ∥ directions will be different just because they will be governed by two distinct Hubble parameters: H ⊥ and H∥.
  3. The redshift distortion parameter will be different because it will depend on the background expansion.

Also the growth factor will be modified, perhaps in a scale dependent way. If the true underlying model is radically inhomogeneous, but we assume a FLRW in interpreting the observations, the derived cosmological parameters will be biased (or unphysical) and the parameters derived from BAO data will be different from those measured by SN Ia and/or lensing. As argued also in different contexts, a mismatch on the value of one of more parameters may indicate that we are assuming a wrong model.

We show here the sensitivity that can be reached with an experiment like Euclid for the curvature parameter (0) Ω K (Amendola and Sapone, in preparation). We choose a redshift survey with a depth of z = 1.6 and consider different redshift bins.

In Figure 53View Image we show the first 1σ absolute errors on the curvature parameter for different redshift bins that can be obtained measuring the Hubble parameter and the angular diameter distance. In obtaining these errors we used Fisher-based forecasts for the radial and angular BAO signal following [815Jump To The Next Citation Point, 338], as discussed in Section 1.7.3.

The sensitivity that can be reached with an experiment like Euclid is extremely high; we can measure the curvature parameter better than 0.02 at redshift of the order of z ≃ 1. This will allow us to discriminate between FLRW and averaged cosmology as for example illustrated in Figure 54View Image.

View Image

Figure 53: Relative errors on ΩK for our benchmark survey for different redshifts.
View Image

Figure 54: Left: same as Figure 53View Image but now with superimposed the prediction for the Lemaître–Tolman–Bondi model considered by [380]. Right: zoom in the high-redshift range.

An alternative to measuring the global curvature is to measure the shear of the background geometry. If there is a large inhomogeneous void then a congruence of geodesics will not only expand but also suffer shear [382Jump To The Next Citation Point]. The amount of shear will depend on the width and magnitude of the transition between the interior of the void and the asymptotic Einstein–de Sitter universe. Normalizing the shear w.r.t. the overall expansion, one finds [382]

[ ] H⊥-(z) −-H-||(z) ---------1-−-H-||(z)∂z-(1-+-z)DA-(z)------------ 𝜀 = 2H ⊥ + H || ≃ ( [ ] ). (4.4.4 ) 3H ||(z)DA (z) + 2 1 − H ||(z )∂z (1 + z )DA (z )
Clearly, in homogeneous FRW universes the shear vanishes identically since H ⊥ = H || = H. Also note that the function H ||(z)DA (z) is nothing but the Alcock–Paczynski factor, which is normally used as a geometric test for the existence of vacuum energy in ΛCDM FRW models.


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