Effects of inhomogeneities on apparent cosmological observables: “fake” evolving dark energy

Abstract

Using the exact Lemaitre–Bondi–Tolman solution with a non-vanishing cosmological constant Λ, we investigate how the presence of a local spherically symmetric inhomogeneity can affect apparent cosmological observables, such as the deceleration parameter or the effective equation of state of dark energy (DE), derived from the luminosity distance under the assumption that the real space-time is exactly homogeneous and isotropic. The presence of a local underdensity is found to produce apparent phantom behavior of DE, while a locally overdense region leads to apparent quintessence behavior. We consider relatively small large scale inhomogeneities which today are not linear and could be seeded by primordial curvature perturbations compatible with CMB bounds. Our study shows how observations in an inhomogeneous ΛCDM universe with initial conditions compatible with the inflationary beginning, if interpreted under the wrong assumption of homogeneity, can lead to the wrong conclusion about the presence of “fake” evolving dark energy instead of Λ.

Appendix: Calculating the density contrast

In the text, we have carried out all our calculations in the coordinates (η,r) since this allows one to take full advantage of the existence of an analytical solution. But if we are interested in the radial profile of a quantity on a fixed time-slice t=constant, we need to go back to the coordinates (t,r). Below we carry this out for the density contrast, δ=(ρ(t,r)−ρ(t,∞))/ρ(t,∞), where our LTB model is assumed to approach a flat FLRW universe as r→∞.

We need to introduce the inverse of the function defined in Eq. (14), i.e., we need to express η as a function of (t,r), η=v(t,r), from

(A.1)

such that

$$ u\bigl(v(t,r),r\bigr)=t. $$

(A.2)

The value of v(t,r) can be evaluated numerically by solving for x the equation

$$ u(x,r)=t. $$

(A.3)

The function η=v(t ₀,r) thus obtained is plotted for the different models in Figs. 4 and 13. As can be seen, η=v(t ₀,r) varies substantially in the region of inhomogeneity, while it levels off to a constant far from the inhomogeneity.

Fig. 4

η=v(t ₀,r) in units of $H_{0}^{-1}$ for inhomogeneity of types I⁻ and I⁺

The energy density in the coordinates (t,r) is given by

$$ \rho(t,r)=\frac{2\, \partial_rM}{R^2(t,r)\partial_r R(t,r)}. $$

(A.4)

But since the analytical solution is given in terms of η we have another expression:

$$ \rho(\eta,r)=\frac{2\, \partial_r M}{ a(\eta,r)^2 r^2 [\partial_r (a(\eta,r)r) - a^{-1}\partial_{\eta} (a(\eta,r) r) \partial_r t] }. $$

(A.5)

Then the density contrast on the hypersurface t=t ₀ is given by

$$ \delta(t_0,r)=\frac{\rho(\eta(t_0,r),r) -\rho(\eta(t_0,\infty),\infty)}{\rho(\eta(t_0,\infty)),\infty )}, $$

(A.6)

where

(A.7)

The density contrast is plotted in Fig. 5 for type I^± inhomogeneities and Fig. 14 for type II^± inhomogeneities.

Fig. 5

δ(t ₀,r) as a function of r for inhomogeneity of types I⁻ and I⁺

Fig. 6

H ^app(z) for inhomogeneity of types I⁻ and I⁺

Fig. 7

q ^app(z) for inhomogeneity of types I⁻ and I⁺

Fig. 8

$w^{\mathrm{app}}_{\mathrm{DE}}(z)$ for inhomogeneity of types I⁻ and I⁺

Fig. 9

Om ^app(z) for inhomogeneity of types I⁻ and I⁺

Fig. 10

The same as Fig. 1 but for inhomogeneity of types II⁻ and II⁺

Fig. 11

The same as Fig. 2 but for inhomogeneity of types II⁻ and II⁺

Fig. 12

The same as Fig. 3 but for inhomogeneity of types II⁻ and II⁺

Fig. 13

The same as Fig. 4 but for inhomogeneity of types II⁻ and II⁺

Fig. 14

The same as Fig. 5 but for inhomogeneity of types II⁻ and II⁺

Fig. 15

The same as Fig. 6 but for inhomogeneity of types II⁻ and II⁺

Fig. 16

The same as Fig. 7 but for inhomogeneity of types II⁻ and II⁺

Fig. 17

The same as Fig. 8 but for inhomogeneity of types II⁻ and II⁺

Fig. 18

The same as Fig. 9 but for inhomogeneity of types II⁻ and II⁺

Fig. 19

$\delta D_{L}(z)=[D_{L}^{\varLambda \mathrm{CDM}}(z)-D_{L}(z)]/D^{\varLambda \mathrm{CDM}}_{L}(z)$, the relative difference between the luminosity distances, for model II⁺. As can be seen, the large scale inhomogeneities considered have a very small effect