Lemaitre-Tolman-Bondi universes as alternatives to dark energy: Does positive averaged acceleration imply positive cosmic acceleration?

We show that positive averaged acceleration $a_D$ obtained in Lemaitre-Tolman-Bondi (LTB) models through spatial averaging can require integration over a region beyond the event horizon of the central observer. We provide an example of a LTB model with positive $a_D$ in which the luminosity distance $D_L(z)$ does not contain information about the entire spatially averaged region, making $a_D$ unobservable. Since the cosmic acceleration $a^{\mathrm{FRW}}$ is obtained from fitting the observed luminosity distance to a Friedmann-Robertson-Walker (FRW) model we conclude that in general a positive $a_D$ in LTB models does not imply a positive $a^{\mathrm{FRW}}$.

Figure 1

The solid curve shows the light geodesic $T(r)$ and the dashed curve shows the bang function $t_b(r)$. For $r<0.6$ $t_b(r)$ is so small that it cannot be distinguished from the horizontal zero axis.

Figure 2

$k(r)$ is plotted for the model corresponding to row 2 of Table I from 20.

Figure 3

$t_b(r)$ is plotted for the model corresponding to row 2 of Table I from 20.

Figure 4

$R(T(r),r)$ and $R^{\prime }(T(r),r)$ are plotted for model 2 of 20. At about $r_{\mathrm{Hor}}=0.03298$ there is a singularity.