Spatial and temporal tuning in void models for acceleration

There has been considerable interest in recent years in cosmological models in which we inhabit a very large, underdense void as an alternative to dark energy. A long-standing objection to this proposal is that observations limit our position to be very close to the void center. By selecting from a family of void profiles that fit supernova luminosity data, we carefully determine how far from the center we could be. To do so, we use the observed dipole component of the cosmic microwave background, as well as an additional stochastic peculiar velocity arising from primordial perturbations. We find that we are constrained to live within 80 Mpc of the center of a void—a somewhat weaker constraint than found in previous studies, but nevertheless a strong violation of the Copernican principle. By considering how such a Gpc-scale void would appear on the microwave sky, we also show that there can be a maximum of one of these voids within our Hubble radius. Hence, the constraint on our position corresponds to a fraction of the Hubble volume of order ${10}^{-8}$. Finally, we use the fact that void models only look temporarily similar to a cosmological-constant-dominated universe to argue that these models are not free of temporal fine-tuning.

Figure 1

Maximum temperature anisotropy, $|\Delta T|/T$, vs radial coordinate of the void center for a selection of profiles from the chain (with no outer shells), with the grayscale level indicating the relative likelihood of the fit to $\mathrm{CMB}+\mathrm{SN}$ data. Apart from where $\Delta T$ changes sign at $r/r_{\mathrm{LSS}}\simeq 0.8$, the magnitude of the effect always exceeds ${10}^{-3}$.

Figure 2

Distribution of dipole magnitude $v$ at radii 24 Mpc (green/leftmost curves) and 80 Mpc (red/rightmost curves) over the chain of void models, with random peculiar velocity (solid colors) and without (faint colors, apparently noisy due to the finite Markov-chain sampling). At each distance, curves are scaled to the same height. The addition of peculiar velocity is seen to significantly widen both distributions.

Figure 3

Fractional difference between luminosity distance in $\Lambda \mathrm{CDM}$ and in a void whose redshift-luminosity distance relation matches well to $\Lambda \mathrm{CDM}$ at the current epoch, plotted at different observation times. Both relations are recalculated at each observation time. The discrepancy exceeds 2% at $\pm 1\mathrm{Gyr}$, and increases at later or earlier times. The worse fit at low redshift stems from our constraint that the void profile be smooth at the center.

Figure 4

We simulate the $\mu$–$z$ relationship for $\Lambda \mathrm{CDM}$ at various times given by 2000 SNe uniformly distributed from $z=0.01–1.7$, with a Gaussian dispersion characterized by ${\sigma }_{\mu }=0.15$. We then calculate ${\chi }^2$ values at those times for an effective “fit” of this relationship to that of a void profile, averaged over 30 fiducial data sets. There is a clear preference for the current epoch, with times more than $\sim 1.5\mathrm{Gyr}$ into the past or future exhibiting severe deviations from $\Lambda \mathrm{CDM}$.