Evolution of the cosmic microwave background

We discuss the time dependence and future of the cosmic microwave background (CMB) in the context of the standard cosmological model, in which we are now entering a state of endless accelerated expansion. The mean temperature will simply decrease until it reaches the effective temperature of the de Sitter vacuum, while the dipole will oscillate as the sun orbits the Galaxy. However, the higher CMB multipoles have a richer phenomenology. The CMB anisotropy power spectrum will for the most part simply project to smaller scales, as the comoving distance to last scattering increases, and we derive a scaling relation that describes this behavior. However, there will also be a dramatic increase in the integrated Sachs-Wolfe contribution at low multipoles. We also discuss the effects of tensor modes and optical depth due to Thomson scattering. We introduce a correlation function relating the sky maps at two times and the closely related power spectrum of the difference map. We compute the evolution both analytically and numerically, and present simulated future sky maps.

Figure 1

Our spheres of last scattering at time $\eta$ (inner) and ${\eta }^{\prime }$ (outer). The group of vertical lines indicates the crests of a mode $\mathit{k}$, and hot spots will be observed at the intersections of those crests with the spheres. One wave length of the mode will span angle $\theta$ at $\eta$ but a smaller angle ${\theta }^{\prime }$ at ${\eta }^{\prime }$.

Figure 2

CMB power spectrum $\ell (\ell +1)C_{\ell }/(2\pi )$ as a function of multipole $\ell$ and scale factor of observation $a_{\mathrm{obs}}$, as calculated from our modified version of CAMB for our fiducial $\Lambda \mathrm{CDM}$ model. We also plot the analytic scaling of the first acoustic peak (thick dashed line) predicted by Eq. (29).

Figure 3

Anisotropy power spectra for scalars (top pair of curves) and for tensors (bottom pair, arbitrary scale), calculated using our modified version of CAMB for today (dashed curves) and for the asymptotic future (solid curves).

Figure 4

Anisotropy power spectrum for $a_{\mathrm{obs}}=2.0$ calculated with our modified version of CAMB (solid line) and using the scaling relation Eq. (29) and analytical ISW approximation Eq. (42) (dashed line). We set $n_S=1$ for these calculations.

Figure 5

Absolute value of the difference in the CMB power spectra $\ell (\ell +1)|\delta C_{\ell }|/(2\pi )$ between $a_{\mathrm{obs}}=1$ and $a_{\mathrm{obs}}^{\prime }=1+\delta a$. The solid curves were calculated from our modified version of CAMB, and from top to bottom denote $\delta a=0.01$, 0.001, and ${10}^{-4}$. The dashed curve was calculated using the analytical expression Eq. (45) for $\delta a={10}^{-4}$. We used $n_S=1$ for these calculations.

Figure 6

Two source modes, ${\mathit{k}}_1$ and ${\mathit{k}}_2$, with the same wave number $\tilde{k}$. Diagonal lines indicate troughs (solid lines) and crests (dashed lines). The observer is towards the bottom. The solid horizontal line indicates the position of the LSS at initial time $\eta$. The dotted horizontal lines indicate the position of the LSS at the first later time that produces perfect anticorrelations with the initial time, so that hot spots (solid circles) line up with cold spots (open circles) and vice versa. Mode ${\mathit{k}}_1$, which corresponds to a smaller observed $\ell$, reaches anticorrelation before mode ${\mathit{k}}_2$ (since $\delta x_1<\delta x_2$). A mode $\mathit{k}$ parallel to the LSS would never reach anticorrelation, while a mode parallel to the line of sight would correspond to $\ell =0$ (in the flat-sky approximation).

Figure 7

Difference map angular power spectrum $D_{\ell }^{\eta {\eta }^{\prime }}$ calculated from our modified version of CAMB (solid lines) for the two times corresponding to $a_{\mathrm{obs}}=1$ and $a_{\mathrm{obs}}^{\prime }=1+\delta a$, for the cases (top to bottom) $\delta a=0.1$, 0.01, 0.001, and ${10}^{-4}$. Also shown is the analytical result (dashed curve) calculated using Eq. (69) for $\delta a={10}^{-4}$.

Figure 8

Normalized correlation function ${\overline{C}}_{\ell }^{\eta {\eta }^{\prime }}$ between the CMB sky observed today ($a=1$) and at $a=1+\delta a$, calculated with our modified version of CAMB, for $\delta a=0.001$ (top panel, solid curve), 0.01 (top panel, dashed curve), 0.03 (bottom panel, solid curve), and 0.1 (bottom panel, dashed curve). Anticorrelations are seen to develop as $\delta a$ increases, before the correlation function decays to zero for large $\delta a$.

Figure 9

Distribution from which the $a_{\ell m}$’s are drawn. Here, $\eta$ corresponds to $a_{\mathrm{obs}}=1$ and ${\eta }^{\prime }$ to $a_{\mathrm{obs}}=2.0$, and contours show the $2\sigma$ error ellipse. The distribution of $a_{2m}$ is shown by the solid contour, which has a correlated set of variances $C_2(\eta )$ and $C_2({\eta }^{\prime })$. The distribution of $a_{5m}$, shown by the dotted contour, has a smaller variance at both times and these are much less correlated.

Figure 10

Simulated map realizations (top and left panels) and difference map (relative to $a_{\mathrm{obs}}=1$) (right panels) for, from top to bottom, $\delta a=0$, $\delta a={10}^{-3}$ (corresponding to a 13 Myr interval), $\delta a={10}^{-2}$ (130 Myr), $\delta a=0.1$, and $\delta a=1.0$. Note the vastly different power scales between the sky and difference maps. The maps presented here are for a patch of sky covering $\sim 1000$ square degrees.

Figure 11

Simulated map realization for the asymptotic future.