Effects of Rayleigh scattering on the CMB and cosmic structure

During and after recombination, in addition to Thomson scattering with free electrons, photons also couple to neutral hydrogen and helium atoms through Rayleigh scattering. This coupling influences both cosmic microwave background (CMB) anisotropies and the distribution of matter in the Universe. The frequency dependence of the Rayleigh cross section breaks the thermal nature of CMB temperature and polarization anisotropies and effectively doubles the number of variables needed to describe CMB intensity and polarization statistics, while the additional atomic coupling changes the matter distribution and the lensing of the CMB. We introduce a new method to capture the effects of Rayleigh scattering on cosmological power spectra. Rayleigh scattering modifies CMB temperature and polarization anisotropies at the $\sim 1\%$ level at 35 GHz (scaling $\propto {\nu }^4$), and modifies matter correlations by as much as $\sim 0.3\%$. We show the Rayleigh signal, especially the cross-spectra between the thermal (Rayleigh) $E$-polarization and Rayleigh (thermal) intensity signal, may be detectable with future CMB missions even in the presence of foregrounds, and how this new information might help to better constrain the cosmological parameters.

Figure 1

The comoving opacity as a function of comoving time. The black (solid) line is for Thomson scattering while the blue (large dashed), red (small dashed), green (dot dashed) and brown (dotted) lines are for Rayleigh scattering at frequencies 857, 545, 353, and 217 GHz respectively.

Figure 2

The matter two-point correlation function, $r^2\xi (\overrightarrow{r})$, as a function of the distance between two overdensities for our fiducial cosmological parameters.

Figure 3

The percentage change in the matter correlation function due to Rayleigh scattering for our fiducial cosmological parameters.

Figure 4

The redshift evaluation of a narrow Gaussian-shaped adiabatic density fluctuation in real space. The blue (solid) and red (dashed) lines are respectively the baryon and photon density waves. At very early times [panel (a)], baryons and photons are tightly coupled and their density waves travel together. As time goes by [panels (b)–(d)], they decouple, photons free stream to us and baryons cluster around the initial overdensity and in a shell at about 150 Mpc radius.

Figure 5

The percentage change in physical baryon density fluctuations in real space due to Rayleigh scattering at different redshifts. The blue (solid), red (dashed), green (dot dashed) and brown (dotted) lines correspond to redshifts 0, 100, 500 and 1050 respectively.

Figure 6

The total visibility function as a function of conformal time for several frequencies. The black (solid), red (dotted), blue (dot dashed) and green (dashed) lines are the total visibility functions for frequencies 0, 545, 700 and 857 GHz respectively. The total photon visibility function shifts toward later times with increasing frequency.

Figure 7

The cross-correlation temperature power spectrum $C_l^{TT(2r,2r^{\prime })}$ of the ${\mathrm{\Theta }}_{Il}^{(2r)}$ and ${\mathrm{\Theta }}_{Il}^{(2r^{\prime })}$ intensity coefficients for the ${\nu }^0$, ${\nu }^4$ and ${\nu }^6$ spectral distortions.

Figure 8

The cross-correlation temperature power spectrum $C_l^{EE(2r,2r^{\prime })}$ of the ${\mathrm{\Theta }}_{El}^{(2r)}$ and ${\mathrm{\Theta }}_{El}^{(2r^{\prime })}$ $E$-polarization coefficients for the ${\nu }^0$, ${\nu }^4$ and ${\nu }^6$ spectral distortions.

Figure 9

Shown is a fractional measure $\delta C_l^{XY}/\sqrt{C_l^{XX}{C}_l^{YY}}$ of the change $\delta C_l^{XY}$ in (lensed) scalar CMB anisotropy spectra due to Rayleigh scattering. The blue (solid), red (dashed), green (dotted) and purple (dot dashed) are for the temperature, $E$ polarization, $B$ polarization from lensing and Temperature-E polarization cross-correlation spectra respectively.

Figure 10

The eight nonzero power spectra in the Rayleigh-distorted CMB covariance matrix as a function of l. The first eigenvalues of intensity and polarization are almost proportional to the primary thermal signal and the second eigenvalues of intensity and $E$ polarization are purely Rayleigh signals which are uncorrelated to the first eigenvalues.

Figure 11

The eight nonzero elements of the Rayleigh-distorted CMB covariance matrix (blue, solid) and their signal-to-noise ratio at each $l$ (red, dashed) as well as the accumulative signal-to-noise ratio for the PRISM-like experiment. Note that the signal-to-noise ratio for the temperature-polarization cross power spectrum can be negative at some $l$ values due to anticorrelation of the temperature and polarization. However the accumulative signal to noise added in quadrature is always positive.

Figure 12

Accumulative signal-to-noise ratios for the eight nonzero elements of the CMB covariance matrix. The blue (solid), red (dashed), green (dotted) and black (dot dashed) lines are the signal-to-noise ratios respectively for a PRISM-like experiment. For case I: improved foregrounds removal method. For case II: improved detector noise, and for case III which combines cases I and II.

Figure 13

The biases and constraints on cosmological parameters that could potentially occur if one ignores the Rayleigh signal. The blue contours are the $1\sigma$ and $2\sigma$ constraints on parameters using only the primary signal centred at the fiducial value of the parameters. The red, green, orange and black dots represent the bias introduced by ignoring the Rayleigh signal respectively in a PRISM-like experiment, case I (improving foreground removal), case II (reducing detector noise) and case III (combination of both).

Figure 14

The $2\sigma$ constraints on cosmological parameters by considering both the primary and Rayleigh signal. The smallest and darkest contour represents the cosmic-variance limited case. The lighter contours show the case III, case II, case I and the PRISM-like experiment respectively as we go from smallest-darkest to largest-lightest contours. Note that the largest contours essentially delineate the conventional (primary only) cosmic-variance limit, and smaller contours represent an improvement in parameter constraints beyond this limit.