A Test of the Copernican Principle

The blackbody nature of the cosmic microwave background (CMB) radiation spectrum is used in a modern test of the Copernican principle. The reionized universe serves as a mirror to reflect CMB photons, thereby permitting a view of ourselves and the local gravitational potential. By comparing with measurements of the CMB spectrum, a limit is placed on the possibility that we occupy a privileged location, residing at the center of a large void. The Hubble diagram inferred from lines of sight originating at the center of the void may be misinterpreted to indicate cosmic acceleration. Current limits on spectral distortions are shown to exclude the largest voids which mimic cosmic acceleration. More sensitive measurements of the CMB spectrum could prove the existence of such a void or confirm the validity of the Copernican principle.

Figure 1

Illustrated is a cross-section through a model Universe with the observer (O) at the center of a void, in violation of the Copernican principle. CMB photons traveling in any direction may Thomson scatter off reionized gas. The observed spectrum will be a mixture of blackbody spectra with different (anisotropic) tempertures, producing a distorted blackbody. The yellow lines represent: incoming beams of unscattered, primary CMB photons (dashed); incoming beams of scattered photons (thin), and the observed beams (thick) for representative scattering centers with last-scattering surfaces represented by the dark circles. A is in the Doppler zone: Beams 1–3 experience the same SW temperature shift, introducing no anisotropy. However, gradients in the void gravitational potential cause the gas to move with respect to the CMB frame, so A sees a differential Doppler anisotropy, resulting in spectral distortions. B is in the reflection zone: B is at rest with respect to the CMB frame and sees no Doppler anisotropy. However, some of the incoming photons, e.g., beam 4, originate inside the void so there will be an anisotropic SW temperature shift, leading to spectral distortions.

Figure 2

The dependence of the spectral distortion, $u$, on the size of a Hubble bubble parameterized by $z_{\mathrm{edge}}$, is shown in units of $u_{\mathrm{scale}}\equiv {\tau }_{e0}\mathbf{(}{\Phi }_0/(3c^2){\mathbf{)}}^2(1+z_{\mathrm{edge}})/(\sqrt{1+z_{\mathrm{edge}}}-1)$. The thick curves show the various contributions to $u$. The overlap contribution is due to the Doppler and Sach-Wolfe cross term. The thin curves correspond to the case in which $z_{\mathrm{rei}}\to \infty$. For small voids the Doppler contribution dominates, and the value of $z_{\mathrm{rei}}$ is unimportant.

Figure 3

The test of the Copernican principle, in terms of constraints on the size and depth of a local, spherically-symmetric void, is shown. The blue shaded regions show the range of parameters excluded by the $u$-distortion test, whereas the red regions show the range of parameters compatible with the current SNe Hubble diagram data.