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Does Bose-Einstein Condensation of CMB Photons Cancel Μ Distortions Created by Dissipation of Sound Waves in the Early Universe

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Does Bose-Einstein Condensation of CMB Photons Cancel Μ Distortions Created by Dissipation of Sound Waves in the Early Universe A&A 540, A124 (2012) Astronomy DOI: 10.1051/0004-6361/201118194 & c ESO 2012 Astrophysics Does Bose-Einstein condensation of CMB photons cancel µ distortions created by dissipation of sound waves in the early Universe? R. Khatri1,R.A.Sunyaev1,2,3, and J. Chluba4 1 Max Planck Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany e-mail: [email protected] 2 Space Research Institute, Russian Academy of Sciences, Profsoyuznaya 84/32, 117997 Moscow, Russia 3 Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540, USA 4 Canadian Institute for Theoretical Astrophysics, 60 St George Street, Toronto, ON M5S 3H8, Canada Received 3 October 2011 / Accepted 12 February 2012 ABSTRACT The difference in the adiabatic indices of photons and non-relativistic baryonic matter in the early Universe causes the electron temperature to be slightly lower than the radiation temperature. Comptonization of photons with colder electrons results in the transfer of energy from photons to electrons and ions, owing to the recoil effect (spontaneous and induced). Thermalization of photons with a colder plasma results in the accumulation of photons in the Rayleigh-Jeans tail, aided by stimulated recoil, while the higher frequency final initial spectrum tries to approach Planck spectrum at the electron temperature Tγ = Te < Tγ ; i.e., Bose-Einstein condensation of photons occurs. We find new solutions of the Kompaneets equation describing this effect. No actual condensate is, in reality, possible since the process is very slow and photons drifting to low frequencies are efficiently absorbed by bremsstrahlung and double Compton processes. The spectral distortions created by Bose-Einstein condensation of photons are within an order of magnitude (for the present range of allowed cosmological parameters), with exactly the same spectrum but opposite in sign, of those created by diffusion damping of the acoustic waves on small scales corresponding to comoving wavenumbers 45 < k < 104 Mpc−1. The initial perturbations on these scales are completely unobservable today due to their being erased completely by Silk damping. There is partial cancellation of these two distortions, leading to suppression of μ distortions expected in the standard model of cosmology. The net distortion depends on the scalar power index ns and its running dns/dlnk, and may vanish for special values of parameters, for example, for a running spectrum with, ns = 1, dns/dlnk = −0.038. We arrive at an intriguing conclusion: even a null result, non-detection of μ-type distortion at a sensitivity of 10−9, gives a quantitative measure of the primordial small-scale power spectrum. Key words. radiation mechanisms: thermal – scattering – cosmic background radiation – cosmological parameters – cosmology: theory – early Universe 1. Introduction expansion of the Universe thus makes the matter cool faster than the radiation, while Comptonization tries to maintain matter and In the early Universe we have two strongly interacting fluids: radiation at the same temperature by transferring energy from ra- cosmic microwave background (CMB) radiation and plasma diation to matter. As a result, the energy of radiation decreases, (=ions + electrons). Electrons and CMB are strongly coupled while the number of photons is conserved (neglecting absorption due to the Compton process with the rate of energy transfer from of photons by bremsstrahlung and double Compton). This ef- −20 4 −1 CMB to electrons given by Rcompton ∼ 1.4 × 10 (1 + z) s . fect was discussed by Zeldovich et al. (1968)andPeebles (1968) This is much faster than the expansion rate , H(z) ∼ 2.1 × and leads to a decrease in the temperature of plasma compared −20 2 −1 1 10 (1 + z) s , and keeps Tγ − T Tγ at redshifts to the radiation temperature at z 200, when Comptonization e ffi z 200, where Tγ is the effective temperature of radiation becomes ine cient because of depletion of electrons from re- (to be defined below) and Te is the electron temperature. Ions combination. It also drives the 21-cm line spin temperature to and electrons exchange energy via Coulomb collisions at a rate below the radiation temperature, raising the possibility of 21-cm −9 3/2 −1 Rcoulomb ∼ 3 × 10 (1 + z) s (Lifshitz & Pitaevskii 1981) absorption of CMB photons by neutral hydrogen. This transfer keeping Te − Ti Te,whereTi is the temperature of ions. of energy from CMB to matter happens at all redshifts, as long Thus we have Tγ ≈ Te ≈ Ti to high precision at z > 200. as there is significant free electron number density, and results Non-relativistic matter has an adiabatic index of 5/3, while the in deviations in CMB spectrum from the blackbody. Recently, adiabatic index of radiation is 4/3. Adiabatic cooling due to the Chluba & Sunyaev (2012) have numerically studied these de- viations from the blackbody in the CMB spectrum. Below we demonstrate that the problem of spectral deviations of CMB due 1 We give the formula that is valid during radiation domination to loss of energy to plasma, in the case of small spectral devi- just for illustration. The Hubble rate remains much lower than the y Comptonization rate at z > 200 and than the Coulomb rate at even ations and small Compton parameter , has a simple analytic lower redshifts. solution, when photon production and destruction is neglected. Article published by EDP Sciences A124, page 1 of 10 A&A 540, A124 (2012) The resulting flow of photons towards low frequencies as the each other, individually resulting in spectral distortions with spectrum tries to approach the Planck spectrum is in fact the opposite signs but the same spectral shape in any part of the Bose-Einstein condensation of photons. The Bose-Einstein con- CMB spectrum. densation of photons unfortunately does not progress very far The cooling of electrons and the corresponding spectral dis- in reality, because low-frequency photons are efficiently ab- tortions are easy to calculate and only depend on the standard sorbed by bremsstrahlung and double Compton processes, and cosmological parameters, such as baryon to photon number den- the Compton process freezes out as the electron density falls ow- sity nB/nγ and helium fraction, which decide the amount of en- ing to the expansion of the Universe and recombination. ergy losses by the CMB, the Hubble constant (H0), and densities Early papers about energy release in the early Universe were of constituents of the Universe, which in turn determine the ex- concerned with exotic sources such as annihilation of matter pansion rate and thus the efficiency of Comptonization. On the and antimatter, primordial turbulence, decay of new unstable other hand, the energy released by dissipation of sound waves particles, unwinding of topological defects like domain walls, crucially depends on the power in small-scale fluctuations, and and cosmic strings. Experiments beginning with COBE FIRAS thus the spectral index (ns) in the standard cosmological model, (Fixsen et al. 1996) have been unable to find any significant dis- in addition to the other well-measured parameters of the stan- tortions in the CMB from blackbody. New proposals like Pixie dard cosmological model (Komatsu et al. 2011). It is interest- (Kogut et al. 2011) are demonstrating that a tremendous increase ing to note that for a spectrum with constant ns the energy re- in the sensitivity is possible in the future experiments. Pixie is lease due to dissipation exceeds the energy losses to adiabatic proposed to be 1000 times more sensitive than COBE FIRAS. cooling of baryons, and there is net heating of electrons. For a There is hope that this is still not the last word, and even higher primordial spectrum with a running spectral index, the role of sensitivity might become possible in the future. Under these cir- Comptonization with colder electrons and Bose-Einstein con- cumstances we decided to check what minimum levels of devia- densation can become dominant, with the spectral distortions tions from blackbody spectrum are expected in the standard cos- changing sign, and there can be net cooling of electrons and a mological model (Komatsu et al. 2011). In this paper we only corresponding decrease of CMB entropy per baryon (specific consider the spectral distortions arising before the end of the entropy). Additional increase in entropy during the recombina- dark ages and beginning of reionization. We demonstrate that, in tion epoch due to superposition of blackbodies, free streaming the absence of decay and annihilation of new unknown particles as well as Silk damping and the second-order Doppler effect or any other new physics beyond the standard model, there are only produce y-type distortions and can be distinguished from only three key reasons for significant global spectral distortions the μ distortions created in the earlier epoch. in the CMB. 1. Bose-Einstein condensation of CMB photons due to the dif- ference in the adiabatic indices of non-relativistic plasma and 2. Thermodynamic equilibrium in the early Universe photons. We can obtain the energy losses due to adiabatic cooling 2. The energy release due to dissipation of sound waves and of baryons under the assumption that Compton scattering, initial perturbations due to Silk damping (Silk 1968)andthe ff bremsstrahlung, and double Compton scattering can maintain second-order Doppler e ect due to non-zero peculiar veloc- full thermodynamic equilibrium between the electrons and pho- ity of electrons and baryons in the CMB rest frame, both tons. This is true to high accuracy at z > 106 while at lower leading to the superposition of blackbodies (Zeldovich et al. redshifts, although bremsstrahlung and double Compton scatter- 1972) in the electron rest frame. ing cannot help in restoring a Planck spectrum at all frequen- 3. The cosmological recombination radiation from hydrogen cies, Comptonization keeps the electron temperature equal to and helium (Dubrovich 1975; Peebles 1968; Zeldovich et al.
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