Gauging the cosmic microwave background

We provide a new derivation of the anisotropies of the cosmic microwave background (CMB), and find an exact expression that can be readily expanded perturbatively. Close attention is paid to gauge issues, with the motivation to examine the effect of super-Hubble modes on the CMB. We calculate a transfer function that encodes the behavior of the dipole, and examine its long-wavelength behavior. We show that contributions to the dipole from adiabatic super-Hubble modes are strongly suppressed, even in the presence of a cosmological constant, contrary to claims in the literature. We also introduce a naturally defined CMB monopole, which exhibits closely analogous long-wavelength behavior. We discuss the geometrical origin of this super-Hubble suppression, pointing out that it is a simple reflection of adiabaticity, and hence argue that it will occur regardless of the matter content.

Figure 1

A conformal spacetime diagram showing a light ray $\mathcal{O}$ emitted from the LSS on ${\Sigma }_{\mathrm{LS}}$ at event $E$ and received at $R$. A foliation ${\Sigma }_t$ is indicated together with its orthogonal congruence $u^{\mu }$, which is comoving on the LSS and at $R$. For clarity, orthogonal vectors are displayed as if the geometry was Euclidean.

Figure 2

A conformal spacetime diagram showing a light ray $\mathcal{O}$ emitted from the LSS at $E$. The arbitrary foliation ${\Sigma }_t$ is indicated together with its orthogonal congruence $u^{\mu }$. Also, the hypersurface ${\Sigma }_q$ orthogonal to the direction ${\tilde{u}}^{\mu }$ comoving with the plasma is indicated, together with the boost displacement $\delta t_B$, which takes ${\Sigma }_{t_E(n^{\mu })}$ into ${\Sigma }_q$.

Figure 3

Dipole transfer function $T_1(k)T(k)$ for a comoving observer (solid curve). For comparison, the monopole transfer functions for an observer on a uniform energy density slice, $T_0(k)T(k)$, and the transfer functions for $\ell =2$, 3, and 4 are also shown. Absolute values are plotted. The scales $k{r}_{\mathrm{LS}}={10}^2$ and $k{r}_{\mathrm{LS}}=1$ correspond roughly to the Hubble scales at last scattering and today, respectively.

Figure 4

Dipole transfer function $T_1(k)T(k)$ for a comoving observer. Absolute values of individual contributions from Sachs-Wolfe terms evaluated at emission, $E$, and at the observation point, $R$, as well as the line-of-sight ISW contribution, are indicated.

Figure 5

Monopole transfer function $T_0(k)T(k)$, for an observer on a uniform energy density slice. Absolute values of individual contributions from Sachs-Wolfe terms evaluated at emission, $E$, and at the observation point, $R$, as well as the line-of-sight ISW contribution, are indicated.