Kinematic Sunyaev-Zel’dovich effect as a test of general radial inhomogeneity in Lemaître-Tolman-Bondi cosmology

The apparent accelerating expansion of the Universe, determined from observations of distant supernovae, and often taken to imply the existence of dark energy, may alternatively be explained by the effects of a giant underdense void if we relax the assumption of homogeneity on large scales. Recent studies have made use of the spherically symmetric, radially inhomogeneous Lemaître-Tolman-Bondi (LTB) models to derive strong constraints on this scenario, particularly from observations of the kinematic Sunyaev-Zel’dovich effect which is sensitive to large-scale inhomogeneity. However, most of these previous studies explicitly set the LTB “bang time” function to be constant, neglecting an important freedom of the general solutions. Here we examine these models in full generality by relaxing this assumption. We find that although the extra freedom allowed by varying the bang time is sufficient to account for some observables individually, it is not enough to simultaneously explain the supernovae observations, the small-angle CMB, the local Hubble rate, and the kinematic Sunyaev-Zel’dovich effect. This set of observables is strongly constraining, and effectively rules out simple LTB models as an explanation of dark energy.

Figure 1

Upper panel: If $t_B^{\prime }>0$ then a surface exists with $a_2=0$, corresponding to the occurrence of a shell crossing singularity (dotted line). Beyond this, the formal solution for the energy density gives negative values (hatched area). Lower panel: If $t_B^{\prime }<0$, regions with $H_2<0$ form (hatched area). The solid lines correspond to the surfaces $t=t_B$.

Figure 2

An example of the relative velocity with respect to the CMB rest frame, $\beta (z)$, that would be inferred by a central observer in an example giant void with constant bang time.

Figure 3

Marginalized likelihood for the parameters $S$ and $H_{*}$, found using WMAP 7-year data and a modified version of CosmoMC [63]. Shaded regions show the 68%, 95%, and 99.7% confidence regions.

Figure 4

Marginalized likelihoods for a Gaussian void model with varying bang time. Shaded regions show the 68%, 95%, and 99.7% confidence regions. Here we use $H_0=100h\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$. See the text for other definitions.

Figure 5

Upper panel: Velocity with respect to the CMB rest frame, $\beta (z)$, for models with constant bang time (dashed blue line) and nonconstant bang time (solid red line). Both models have the same spatial curvature $k(r)$. The data points are the upper limits for the nine clusters used in 35. The solid red curve has a bang time fluctuation that is the sum of two Gaussians. Lower panel: Normalized density as a function of $r$ on a hypersurface of constant $t$ for the same two models.

Figure 6

The value of $\Delta T/T$ as a function of redshift for the best fitting models to the supernova, CMB, and $H_0$ data sets, from Sec. 4b. The corresponding value of $\beta$ is shown on the right-hand axis. The median and 68% and 95% confidence intervals are shown as the black line, and the light gray and dark gray bands, respectively. The actual distribution is bimodal, and here we show only the models with low $\Delta T/T$. Even for low redshifts, $\beta$ is a large fraction of the speed of light.

Figure 7

The 68% and 95% likelihood contours in the space of $\Delta T/T$ and $H_0=100h\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$, for off-center observers at $z=0.05$, 0.10, 0.15, and $z=0.20$, for the MCMC sample constrained by $\mathrm{SN}+\mathrm{CMB}+H_0$ data.

Figure 8

The 68% and 95% likelihood contours in the space of $\Delta T/T$ and $H_0=100h\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$, for off-center observers at $z=0.15$, 0.20, and 0.55, for the MCMC sample constrained by $\mathrm{SN}+\mathrm{CMB}+H_0+\mathrm{kSZ}$ data. The $\Delta T/T$ are much lower than for the MCMC sample in Fig. 7, but the Hubble rate is too low to be considered consistent with observations.

Figure 9

Upper panel: Bang time function $t_B(r)$ of the LTB model found in 8 which reproduces the Hubble rate $H(z)$ and luminosity distance $d_L(z)$ of $\Lambda \mathrm{CDM}$. Lower panel: Velocity with respect to the frame where the CMB is isotropic, $\beta (z)$, for the same LTB model. $|\beta (z)|$ rapidly becomes very large ($\beta \approx -c$ for $z\gtrsim 0.6$) and produces a very poor fit to the kSZ data.