Local and nonlocal measures of acceleration in cosmology

Current cosmological observations, when interpreted within the framework of a homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) model, strongly suggest that the Universe is entering a period of accelerating expansion. This is often taken to mean that the expansion of space itself is accelerating. In a general spacetime, however, this is not necessarily true. We attempt to clarify this point by considering a handful of local and nonlocal measures of acceleration in a variety of inhomogeneous cosmological models. Each of the chosen measures corresponds to a theoretical or observational procedure that has previously been used to study acceleration in cosmology, and all measures reduce to the same quantity in the limit of exact spatial homogeneity and isotropy. In statistically homogeneous and isotropic spacetimes, we find that the acceleration inferred from observations of the distance-redshift relation is closely related to the acceleration of the spatially averaged universe, but does not necessarily bear any resemblance to the average of the local acceleration of spacetime itself. For inhomogeneous spacetimes that do not display statistical homogeneity and isotropy, however, we find little correlation between acceleration inferred from observations and the acceleration of the averaged spacetime. This shows that observations made in an inhomogeneous universe can imply acceleration without the existence of dark energy.

Figure 1

Schematic representation of a line-of-sight through a universe with alternating expanding vacuum regions (clear), and collapsing dust regions (grey).

Figure 2

Magnitude of sources in the spherical collapse model, minus the magnitude they would have in pure vacuum. Observations made by a single observer, from the center of the dust and vacuum regions, are displayed as solid red (lower at $z=1$) and blue lines, respectively. The dashed black line is the same quantity in the Buchert averaged model.

Figure 3

Upper Panel: Magnitude of sources in the spherical collapse model, minus the magnitude they would have in pure vacuum. The solid red line is the mean observed magnitude, obtained by spatially averaging distance-redshift relations on a surface of $t=$ constant. The pink bands are $1\sigma$ and $2\sigma$ confidence regions. The black dashed line shows the corresponding quantity in the Buchert averaged model. The short blue line is the distance modulus for a FLRW model with deceleration parameter $q=⟨q_{\Theta }⟩$. Lower Panel: A difference plot of the red line and dashed line, from the panel above. The solid black line corresponds to a model with regions that are a quarter of the size of those used in the upper panel. The pink (wide) and grey (narrow) bands are $1\sigma$ confidence regions for the two different models.

Figure 4

Dependence of the Buchert average deceleration parameter, $q_{\mathcal{D}}$ (dashed black line), and spatially-averaged local volume deceleration parameter, $⟨q_{\Theta }⟩$ (solid red line), on the parameters of the spherical collapse model. Shown is the dependence on the Hubble rates in the collapsing and vacuum regions, $h_I$ and $h_{II}$, and the comoving region sizes, ${\chi }_I$ and ${\chi }_{II}$.

Figure 5

The same quantities that were plotted in Fig. 2, but using the Kasner-EdS model.

Figure 6

The same quantities that were plotted in Fig. 3, but using the Kasner-EdS model.

Figure 7

Density and expansion rates as a function of radius in an LTB void model. The model has spatial curvature of the form given in Eq. (53), with $A_k=-3.82\times {10}^{-8}{\mathrm{Mpc}}^{-2}$ and $w_k=1800\mathrm{Mpc}$. Shown are the density, $\rho$ (solid black), transverse Hubble rate, $H_1$ (blue dashed), and radial Hubble rate, $H_2$ (blue dash-dot). All of the curves are normalized to their values in the asymptotically-homogeneous region ($\rho =9.2\times {10}^{10}{\mathrm{M}}_{\odot }{\mathrm{Mpc}}^{-3}$, $H_1=H_2=57.7{\mathrm{kms}}^{-1}{\mathrm{Mpc}}^{-1}$).

Figure 8

Distance moduli in the LTB model. The red curves (uppermost) are for averaging domains of size $r_{\mathcal{D}}=1000\mathrm{Mpc}$, and the blue curves for $r_{\mathcal{D}}=3000\mathrm{Mpc}$. The solid lines are the mean distance moduli found by spatially averaging the monopole of the observed distance-redshift relation, with accompanying $1\sigma$ and $2\sigma$ confidence bands. The dashed lines are the distance moduli for the distance-redshift relation in the Buchert averaged spacetime. The dotted and dot-dashed lines are the distance moduli for a series expansion of the FLRW $d_A(z)$ relation, with deceleration parameters $q=⟨q_{\Theta }⟩$ and $q=⟨q_{\mathrm{KS}}⟩$, respectively. The curves have all had their zero-points shifted to match at $z=0$.

Figure 9

Distance modulus curves for an off-center observer in the same model, at $r=915\mathrm{Mpc}$. The dotted blue lines show the distance modulus as a function of redshift for radial lines of sight looking out of and into the void (upper and lower curves, respectively). The dashed blue line is the distance modulus for the monopole of the distance-redshift relation, calculated using the dipole approximation. The solid black line is the relation for an observer at the center of symmetry.

Figure 10

Spatial averages of various deceleration parameters as a function of averaging domain radius, $r_{\mathcal{D}}$. The local volume (dash-dotted green line), Kristian-Sachs (dotted blue), and Buchert average (dashed black) deceleration parameters have rather similar values. A close-up of these curves is shown in the inset so that they can be distinguished. The thick, solid yellow line is the deceleration parameter found by fitting an FLRW model to the distance-redshift relation for a wide range of redshifts, $z\le 1$. The narrower solid red line shows the same measure, but for a fit to only $z\le 0.1$.