Spectral distortion in a radially inhomogeneous cosmology

The spectral distortion of the cosmic microwave background blackbody spectrum in a radially inhomogeneous space-time, designed to exactly reproduce a $\Lambda \mathrm{CDM}$ expansion history along the past light cone, is shown to exceed the upper bound established by COBE-FIRAS by a factor of approximately 3700. This simple observational test helps uncover a slew of pathological features that lie hidden inside the past light cone, including a radially contracting phase at decoupling and, if followed to its logical extreme, a naked singularity at the radially inhomogeneous big bang.

DOI: http://dx.doi.org/10.1103/PhysRevD.88.103502

Figure 1

The causal structure of our model space-time. Here and now is point O. The cuspy surface is the constant-energy-density decoupling surface. The intersection of the past light cone of O with the decoupling surface yields the last scattering surface. One possible line of sight to last scattering is shown as the line OA. However, photons that originate within our past light cone may scatter off free electrons after reionization at point S and merge with photons in our line of sight. Because the temperature of photons that originate on the last scattering surface of S at B, for example, is in general different from those originating at A, the mixture produces a distortion of the line-of-sight blackbody.

Figure 2

The redshift and present-day proper distance are shown as a function of coordinate distance out to a redshift z=2 (left) and z=1100 (right) on the past light cone. The panel on the left is identical to Fig. 2 of KL [11]. (Note that the vertical axis is logarithmic in the second panel.)

Figure 3

The metric functions α, β and derivatives are shown as functions of redshift out to z=2 (left) and z=1100 (right). All quantities are shown in units of H₀². The panel on the left is identical to Fig. 3 of KL [11].

Figure 4

The integration path is shown for solving the differential equations for the metric variables R, R^′ along geodesics inside the past light cone. The procedure outlined in Appendix  is followed to obtain α, β and R, R^′ along the past light cone, OA. Following a geodesic path from S to B requires R, R^′ at each step. At a given value of r, we can evolve Eqs. (5, 6) down from the past light cone to the desired value of t, as illustrated by the dashed line with the arrowhead. (This is a similar recipe as illustrated in Fig. 4 of Ref. 12.)

Figure 5

Light rays that contribute to the spectral distortion of the CMB are shown in this space-time diagram. Radially directed light rays that merge with the past light cone at a redshift z are shown as solid lines. The past light cone is indicated by thick dashed lines. The bottom thick dashed line shows the inhomogeneous big bang surface. Thin dashed lines show surfaces of constant density, with values 1.5, 3, 300 times the present-day density at the origin.

Figure 6

The temperature T_LS of secondary photons that scatter into the past light cone at a redshift z_sc from a direction σ, relative to the temperature T_CMB of primary photons traveling on the past light cone, are shown. Photons at ∼15°, 180° are substantially hotter than unscattered photons, despite the fact that all photons originate from the same constant-density surface. This is a consequence of a negative radial expansion rate on a portion of this surface, which leads to a strong blueshifting of the photon energies.

Figure 7

A space-time diagram showing the surface where the radial contraction turns to expansion (H_∥=0; thick red dot-dashed line) and the constant energy density surface at decoupling (thin black line). The same scattered geodesics as in Fig. 5 are also shown. All lines are shown with r≥0. The decoupling contour and the big bang surface (thick dashed blue lines) cannot be distinguished by eye in this figure.

Figure 8

(left) The spatial curvature and density in units of the anisotropic expansion rate along the past light cone. The dashed line shows the density in units of the radial expansion rate. (right) The ratio of the shear to the expansion, (H_⊥-H_∥)/(2H_⊥+H_∥), as a measure of the anisotropy of the expansion.