Mixing of blackbodies: Increasing our view of inflation to 17 e-folds with spectral distortions from Silk damping
Silk damping in the early Universe, before and during recombination, erases anisotropies in the cosmic microwave background (CMB) on small scales. This power, which disappears from anisotropies, appears in the monopole as -type, -type and -type distortions. The observation of the CMB spectral distortions will thus make available to us the information about the primordial power spectrum on scales corresponding to the comoving wavenumbers increasing our total view of inflation, when combined with CMB anisotropies, to span e-folds. These distortions can be understood simply as mixing of blackbodies of different temperatures and the subsequent comptonization of the resulting distortions.
1 A view of inflation spanning 17 e-folds
Successes of CMB experiments such as COBE [cobe, cobedmr], WMAP [wmap], SPT [spt], ACT [act] and many others111see NASA LAMBDA website for a complete list, http://lambda.gsfc.nasa.gov/product/expt/ has ushered in the era of precision cosmology. Combination of CMB anisotropies and Lyman- constraints from SDSS [ly2006, ssm2006] gives information about the primordial anisotropies on scales from the horizon size today, comoving wavenumber , to scales of , a view of inflation spanning e-folds.
Any energy released into CMB at redshifts is very quickly thermalized by the combined action of double Compton scattering and bremsstrahlung, which create photons at low frequencies, and Compton scattering which redistributes these photons[sz1970, dd1982] to maintain Bose-Einstein spectrum with occupation number , where is the chemical potential parameter, dimensionless frequency , h is Planck’s constant, is Boltzmann’s constant. The parameter is driven to zero because of the creation of photons by double Compton scattering and bremsstrahlung. At redshifts photon creation becomes very inefficient and any energy injected is comptonized to -type (), intermediate (-type)[ks2012b] between and -type () and -type[zs1969] () distortions. Photon diffusion damps the perturbations in the primordial plasma on small scales [silk, Peebles1970, kaiser], wavenumbers getting damped at and at . Observations of -type and -type distortions will thus allow us to measure the amplitude and spectral index of the primordial power spectrum on these very small scales, see Fig. 1. -type distortions due to Silk damping are indistinguishable and are of much smaller amplitude compared to those created later during reionization and are thus not a very useful probe of primordial power spectrum [cks2012], hence the gap in Fig. 1 at . Also shown are the current limits from COBE [cobe] which constrained at level to and . Proposed experiment Pixie [pixie] will be able to detect the spectral distortions for the WMAP values of the amplitude of primordial power spectrum[wmap] and spectral index . Spectral distortion can thus deliver to us additional e-folds, reaching , giving us a view of inflation spanning a total of 17 e-folds when combined with CMB anisotropies.
2 Silk damping as mixing of blackbodies
CMB spectral distortions from Silk damping were previously calculated by Refs. \refcitesz1970b,hss94,daly1991 who, however, overestimated the energy going into spectral distortions. The calculation of spectral distortions becomes straightforward once we realize that photon diffusion in the early Universe mixes photons from parts of the Universe with different temperature, see right panel in Fig. 2. We thus expect a -type distortion from the mixing of blackbodies [zis1972, cs2004, cks2012, ksc2012b] which can subsequently comptonize into and -type distortions[ks2012b] (at ). This evolution of -type into -type and -type distortions is shown in Fig. 2. The energy going into spectral distortions is easily calculated by doing a Taylor series expansion of
the blackbody photon occupation number with position dependent temperature up to second order and doing an ensemble average. The result for energy in , and -type distortions (), , is [cks2012, ksc2012b]
where are the multipole moments of spherical harmonic decomposition of temperature anisotropies, and is the initial power spectrum. Before recombination the above expression is easily calculated using the tight coupling solutions[hs1995]. In particular, as a check, it should be noted that the oscillating part of the monopole (, where is the sound horizon) and dipole () terms in the tight coupling limit in the radiation dominated era have same amplitude but are out of phase by . The total energy in the sound waves is conserved and it is only the kinetic and internal energy parts which oscillate, as expected. There is, thus, no need to do the traditional ’averaging over an oscillation’ when calculating the sound wave dissipation. 222Thanks to Nail Inogamov for discussion on this aspect. Finally, the shape of the -type distortion depends on the spectral index of the primordial power spectrum and thus provides a way to measure on small scales.[ks2012b]