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 deviations 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 difference 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. the effective radiation temperature (cf. Eq. (16)) to high pre- 1968). cision until z ∼ 200. In the thermodynamic equilibrium, en- We omit the distortions caused by the cosmological recombi- tropy is conserved, and we can use this to calculate the energy nation process (e.g., see Chluba & Sunyaev 2006; Sunyaev & losses due to the expansion of the Universe without referring to Chluba 2009, for more details and references therein) in the dis- Comptonization, since it is not really important which physical cussion below. They have narrow features and different contin- process is responsible for the equilibrium. uum spectrum and can be distinguished from y and μ distortions We start with the thermodynamic relation, from the first two mechanisms. Cosmological recombination radiation can, however, become comparable to or larger than those TγdS = dU + PdV − MdN, (1) discussed here (Chluba & Sunyaev 2012), if the distortions from the first and second processes partially cancel. We also note that where S is the entropy, U total thermal energy, P pressure, V for any experiment with finite beam size there will be a mix- volume, M chemical potential, and N the number of particles. Tγ ing of blackbodies in the beam owing to the temperature fluc- is the common temperature of photons, ions, and electrons. We tuations on the last scattering surface on scales smaller than can ignore the last term for ions and electrons since their number the beam size, leading to inevitable y distortions of magnitude 2 − − is fixed after big bang nucleosynthesis ends . For photons as well ∼(ΔT/T)2 ∼ 10 9−10 10 (Chluba & Sunyaev 2004). we can ignore the last term if we assume that their chemical In this paper we find new solutions for the Kompaneets potential is 0, which is true in full thermodynamic equilibrium. equation describing the CMB spectral distortions arising from With this assumption we can write the equation for total entropy process 1 described above. A second mechanism results in a per baryon, σ = s/nB,wheres is the entropy density and nB is y-distortion and heating of the electrons at low redshifts. At high redshifts Comptonization of CMB on the hotter electrons 2 During recombination, the number of particles changes and the converts the y-type distortion to a μ-type distortion (Sunyaev departure from thermodynamic equilibrium also becomes significant. & Zeldovich 1970b; Illarionov & Sunyaev 1975b). We demon- The following calculations are therefore strictly valid only before strate below that these two sources of distortions work against recombination.

A124, page 2 of 10 R. Khatri et al.: Does Bose-Einstein condensation of CMB photons cancel μ distortions... the baryon number density: In the Universe there are also inevitable processes like the dissipation of acoustic waves and SZ effect from reionization, d(E/n ) + Pd(1/n ) dσ = B B (2) which would lead to a normal SZ effect of similar or greater Tγ magnitude. Due to the additivity of small spectral distortions, ff 4 3 these e ects will cancel the YBEC distortions, and this dimin- E = a Tγ + Nn k Tγ (3) R 2 B B ishes the hope that this Bose-Einstein condensation will ever be 1 observed. Nevertheless it is important to stress that in the stan- 4 6 P = aRTγ + NnBkBTγ. (4) dard cosmology there was an epoch 10 > z > 1000 when Bose- 3 Einstein condensation of CMB photons due to the difference in Here aR is the radiation constant and N the number of non- the adiabatic indices of matter and radiation was able to create a relativistic particles per baryon, so that NnB = neb.Wenow peculiar deviation of CMB spectrum from the blackbody. integrate Eq. (2) to obtain ⎛ ⎞ 3 ⎜ 3/2 ⎟ 4aRTγ ⎜ Tγ ⎟ 3. Alternative direct calculation of energy losses σ = + NkB ln ⎝⎜ ⎠⎟ , (5) 3nB nBC in CMB where C is an arbitrary constant of integration, which is not im- We would like to remind the reader that adiabatic cooling of portant for our calculation. The first term above is the contribu- baryons leads to the cooling of radiation, but Comptonization tion from photons, and it is clear that in the absence of second itself conserves the number of photons. We can write the equa- ∝ + term, Tγ (1 z). The second term is the contribution of non- tion for the evolution of average thermal energy density EB in relativistic particles and causes the temperature to drop slightly baryons before recombination as faster in order to conserve the entropy per baryon. We can write Tγ = T(1 + t), where T = Ti(1 + z)/(1 + zi) is the background dEB 3 dneb 3 dTe + ≡ kBTe + kBneb · (8) temperature proportional to 1 z,andt 1 is the fractional de- dz 2 dz 2 dz viation from this law. We can take the initial deviation t(zi) = 0at initial redshift zi and then calculate the subsequent energy losses The derivative in the first term on the right-hand side, dneb/dz = to adiabatic cooling of baryons at later redshifts. Thus we can 3neb/(1 + z), is just the decrease in number density of particles write σ, correct to leading order in t,as (ions and electrons) with redshift z due to the Hubble expansion, = ρ /μ ρ μ 3 3 3/2 with neb b mol,where b is the baryon mass density and mol 4aRT 4aRT (3t) T 3NkBt the mean molecular weight. The change in temperature has con- σ = + + NkB ln + · (6) 3nB 3nB nBC 2 tributions from adiabatic cooling and also from energy gained from CMB by Comptonization: Equating the above to the initial entropy σi = σ(zi) with t ≡ 0, T ≡ Ti, we get upon solving for t (and ignoring the last term dTe 2Te 2 in Eq. (6) compared to the second term) = − S Compton (9) dz 1 + z 3kBneb 3NnBkB 1 + z t = ln = σ − / + 8a T 3 1 + z where S Compton 4kBEγne T(Tγ Te) mecH(z)(1 z)istheen- R i ergy transfer rate per unit volume from radiation to baryons by E 1 + z = B ln Compton scattering, ne is the number density of free electrons, 4Eγ 1 + z σ k i T the Thomson scattering cross section, B the Boltzmann con- . × −10 + stant, Eγ the energy density of radiation, c the speed of light, me = −5 9 10 1 zi ln + (7) the mass of electron, and H(z) the Hubble parameter. That red- 4 1 z shift decreases with increasing time, so the terms with “+”sign 4 3 are cooling terms and terms with “−” sign are heating terms. The with ΔE/E = 4t,sinceEγ ∝ T ,andEB = kBnebTe.We 2 change in photon energy density can therefore be written as note that t is negative since zi > z and there is net energy loss. ln(z /z ) ∼ 10 for the redshift range of interest, 200 < max min adiabatic Compton < × 6 μ y dEγ dEγ dEγ z 2 10 , where a mixture of and -type distortions are = + produced. This result for the total energy extraction is consis- dz dz dz tent with the estimate obtained by Chluba (2005)andChluba & 4Eγ = + , Sunyaev (2012). + S Compton (10) To estimate the magnitude of total distortions, μ and y-type, 1 z = = × 6 we take zmax zi 2 10 , since at greater redshifts spectral where the first term is just the adiabatic cooling due to the ex- distortions are rapidly destroyed by bremsstrahlung and dou- pansion of the Universe. We can estimate the energy transfer, ble Compton processes and Comptonization. The latter redis- = ∝ + = S Compton, by noting that the baryon temperature Te Tγ 1 z tributes photons over the whole spectrum. For zmin 200 we ∼ = . + Δ / = . × −9 to a high accuracy until z 200, where Tγ 2 725(1 z)Kis get E E 5 4 10 . This transfer of energy from radia- the CMB temperature. Using this to evaluate the total derivative tion to baryons due to a difference in their adiabatic indices of electron temperature dTe/dz on the left-hand side of Eq. (9) results in an inevitable distortion and Bose-Einstein condensa- and the first term on the right-hand side, we get tion of photons. We show below that the distortion has a magnitude of Y ∼ (1/4) ΔE/E ∼ 10−9. We define parameter BEC 3 2Tγ 3 dTγ Y and prove this result below. The corresponding deviation = − BEC S Compton kBneb + kBneb of the electron temperature from the equilibrium temperature of 2 1 z 2 dz E electrons in the radiation field is described by Eq. (16)andis = B · (11) −12 6 + (Tγ − Te)/Tγ ≈ 10 at z = 10 growing to ∼0.01 at z = 500. 1 z

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Thus we have fractional rate at which energy is lost by radiation found (Zeldovich & Sunyaev 1969). By approximating3 n+n2 ≈ to baryons −∂n/∂x in Eq. (14)weget Compton − ∂ ,y ∂ ∂ ,y (dEγ/dz) S E /Eγ . × 10 n(x ) 1 4 n(x ) = Compton = B = 5 9 10 · = x , (17) (12) Δ∂y x2 ∂x ∂x Eγ Eγ (1 + z) 1 + z

where Δ ≡ Te/TBB − 1. This particular form of equation em- We can now calculate the total fractional energy losses of radi- phasizes that the Doppler term dominates over recoil and gives ation that contribute to the spectral distortions between redshifts rise to the spectral distortion by boosting low-energy photons to zmin and zmax, higher frequency. The above equation can be reduced to the diffusion or heat equation (Zeldovich & Sunyaev 1969) by chang- Δ zmin dEγ E = ing variables, for which the Green’s function is well known. E z Eγ For small distortions, the solution takes a particularly simple max z Compton n min dz dEγ form and can be found by substituting BB on the right-hand side = of Eqs. (14)or(17): Eγ dz zmax + xex ex + 1 −10 1 zmax = − . × · Δn ≡ n(x, YSZ) − nBB(x) = YSZ x − 4 , (18) 5 9 10 ln (13) x − 2 x − 1 + zmin (e 1) e 1 = y Δ y We have integrated dE/E, since immediate distortions are pro- wherewehavedeﬁnedYSZ d . The SZ parameter 0 = portional to ΔE/E, and the distortions can be added linearly if is related to fractional energy release by the formula YSZ they are small. This is the same result as Eq. (7). (1/4)ΔEγ/Eγ. This can be obtained by integrating the above spectrum over all frequencies.

4. Kompaneets equation 5. New solution of Kompaneets equation The interaction of radiation with electrons through Compton for Comptonization of photons with colder scattering or Comptonization is described by the Kompaneets plasma equation (Kompaneets 1956) in the Fokker-Planck approxima- tion, when the energy transfer in each scattering is small com- The Kompaneets equation describes, to lowest order, the compared to temperature, and the incoming photon distribution is petition between the Doppler boosting of low-energy photons wide compared to the width of the scattering kernel: to high energies and the down scatter of high-energy photons to low energies by recoil and stimulated recoil. The initial radiation ∂n 1 ∂ T ∂n spectrum is blackbody with temperature TBB. If the electrons are = x4 n + n2 + e , = ff 2 (14) also at the same temperature, Te TBB, then these two e ects ∂y x ∂x TBB ∂x cancel each other out exactly. If Te > TBB, the Doppler boosting becomes stronger than recoil and we have the normal SZ effect in where we have defined the Compton parameter the limit of small y parameter. If Te < TBB then Doppler boosting is weaker and we have a net movement of photons from high z k σ n T y(z, z ) = − dz B T e BB , (15) to low energies, which is the complete opposite of the SZ effect. max m c H(1 + z) zmax e But since at Te = TBB the two effects exactly balance each other out, at linear order the spectral distortions for Te < TBB would which is convenient to use instead of time or redshift. We start be exactly the same as for Te > TBB, just with the opposite sign. our calculation at the reference redshift zmax. One can further- For Te < TBB, we can write the approximate Kompaneets more introduce the Compton equilibrium electron temperature in equation by approximating ∂n/∂x ≈−(n + n2)inEq.(14), a radiation field (Levich & Sunyaev 1971; Zel’Dovich & Levich ∂ ∂ 1970) n = − Te 1 4 + 2 . 1 2 x n n (19) ∂y TBB x ∂x eq + 4 Te n(1 n)x dx = · (16) Analytic solutions can be obtained by the standard method of T 3 BB 4 nx dx characteristics for the recoil (n) and induced recoil (n2)terms individually, and they consist of photons moving down the fre- Here TBB is a reference temperature which is equal to the ra- quency axes at a speed proportional to x2 for the recoil term diation blackbody temperature if the initial radiation field is a (Arons 1971; Illarionov & Syunyaev 1972) and at a speed pro- = ν/ Planck spectrum, and x h kBTBB is the dimensionless fre- portional to the x2n for the induced recoil term (Zel’Dovich ν quency with the frequency of photons. The equations written & Levich 1969; Syunyaev 1971). To get the linear solution, in this way factor out the expansion of the Universe and are we can just substitute n(x) = nBB(x) on the righthand side of applicable to cosmology, as well as non-expanding astrophysi- Eq. (19) and define the parameter for the amplitude of distortion y cal systems. This equation and the Kompaneets equation form a in the case T < T as Y = (1 − T /T )dy,theresult coupled system to be solved simultaneously. e BB BEC 0 e BB is Eq. (18) with YSZ = −YBEC. We should emphasize that the The three terms in the inner brackets of Eq. (14) describe linear solution, Eq. (18), can be arrived at in a straightforward recoil (n), induced recoil (n2), and the Doppler effect of the ther- way by just substituting n(x) = nBB(x) on the righthand side of mal motion of electrons (Te/TBB∂n/∂x). For y 1 and an initial x blackbody spectrum with temperature TBB, nBB = 1/(e − 1), a 3 For a Bose-Einstein distribution with constant chemical potential this simple first-order correction to the blackbody spectrum can be expression is an identity.

A124, page 4 of 10 R. Khatri et al.: Does Bose-Einstein condensation of CMB photons cancel μ distortions...

νGHz νGHz 1 10 100 1000 1 10 100 1000 4e-26 1e-26 -8 CMBx10 8e-27 -9 3e-26 YSZ=YBEC=10

) ) 6e-27 -1 -1 4e-27

Hz 2e-26 Hz

-1 -1 2e-27

ster 1e-26 -SZ ster 0 -2 -2 -2e-27 0 -9 (Wm (Wm μ=-3x10

ν -4e-27ν I I μ -9

Δ Δ =3x10 -6e-27 -10 -1e-26 YSZ=5.5x10 +SZ -8e-27 Recombination -2e-26 -1e-26 0.01 0.1 1 10 0.01 0.1 1 10 x=hν/kT x=hν/kT

Fig. 1. Difference in intensity from the blackbody radiation for normal Fig. 3. Comparison of positive and negative μ-type distortions expected −9 −9 SZ effect with YSZ = 10 and negative SZ effect with YBEC = 10 . in the early Universe (to be calculated in later sections) with the y-type distortions before reionization and the cosmological recombi- νGHz nation spectrum (taken from Chluba & Sunyaev 2006; Rubiño-Martín 1 10 100 1000 et al. 2008) for illustration. 4e-26 CMBx10-8 3e-26 |μ|=5.6x10-9 where Tμ is the temperature to which the spectrum relaxes ) / Δ / Δ / = -1 for an initial energy addition loss of E E with Tμ T −9 Hz 2e-26 0.64ΔE/E = ±2.5 × 10 (Illarionov & Sunyaev 1975a), and -1 ΔE/E = ±4 × 10−9 is the energy addition/loss that gives rise ster 1e-26 -μ −9 μ 5 -2 to SZ distortion of 10 . The chemical potential is given by μ = 2.2ΔT/T = ±5.6 × 10−9. This is the spectrum that an ini- 0 (Wm −9

ν , = y I tial spectrum with YBEC YSZ 10 will approach at high at Δ μ ∼ −9 -1e-26 +μ x 10 . The frequency at which the distortion crosses zero is at x = 2.19 compared to x = 3.83 for the SZ distortion -2e-26 in Fig. 1. 0.01 0.1 1 10 The y-type and μ-type distortions expected in the early x=hν/kT Universe (calculated in the later sections) are compared with the cosmological recombination spectrum (Rubiño-Martín et al. Fig. 2. Difference in intensity from the blackbody radiation for μ dis- μ ff hν/k Tμ+μ 2006)inFig.3. Clearly type distortions have a di erent spec- tortion defined by the equation with n(ν) = 1/(e B − 1). Tμ is the temperature to which the spectrum relaxes for an initial energy tral shape than the recombination radiation (both from hydro- −9 addition/loss of ΔE/E with ΔTμ/T = 0.64ΔE/E = ±2.5 × 10 . gen (Chluba & Sunyaev 2006) and helium (Rubiño-Martín et al. ΔE/E = ±4 × 10−9 is the energy addition/loss that gives rise to SZ 2008)) and y-type distortions and can be distinguished from the distortion of 10−9. The dimensionless chemical potential μ is given by last two. This is very important because the information in the μ = 2.2ΔT/T = ±5.6 × 10−9. μ type distortions about the early Universe physics can be ex- tracted. On the other hand, the y-type distortions from the early Universe get swamped by the much larger y-type distortions the full Kompaneets equation, Eq. (14), without any restriction from the low redshifts and the two contributions are difficult to on Te. However the dominant physical effects when Te > TBB separate. The μ distortions expected from the early Universe also (Doppler) and when Te < TBB (recoil and induced recoil) are have higher magnitude than the recombination spectrum in the completely different and that in equilibrium they balance each Rayleigh-Jeans part of the spectrum but has no quasi-periodic other out exactly gives us the same mathematical form of the structure like the cosmological recombination radiation. linear solution in both cases. This flow of photons toward lower frequencies as the spec- In the Rayleigh-Jeans part of the spectrum we now have trum tries to approach the Planck spectrum due to recoil, an increase in the brightness temperature of ΔT/T =+2YBEC, and induced recoil is Bose-Einstein condensation of photons which is independent of frequency and thus maintains the (Illarionov & Sunyaev 1975a; Landau & Lifshitz 1980). We Rayleigh-Jeans shape of the spectrum. In the Wien part we have show the evolution of the spectrum (solution of the coupled 2 a decrease in intensity ΔI/I =Δn/n ≈−x YBEC. Figure 1 system of Eqs. (14)and(16)) starting with the initial distor- ff −5 shows the di erence in intensity from the blackbody radiation tion given by Eq. (18) with −YSZ = YBEC = 10 in Fig. 4. ff = −9 for normal SZ e ect with YSZ 10 and the negative SZ ef- In the Rayleigh-Jeans region intensity Iν ∝ Teff and therefore −9 fect with YBEC = 10 . Figure 2 shows the spectrum that would ΔT/T =ΔI/I. The initial evolution is similar to that of a spec- y be achieved at a high value of the Compton parameter .Itisa trum with positive YSZ with the photon distribution approaching ν/ +μ Bose-Einstein spectrum with the dimensionless chemical poten- a Bose-Einstein spectrum defined by n(ν) = 1/(eh kBTe − 1) ν/ +μ tial4 μ defined by the equation with n(ν) = 1/(eh kBTμ − 1), with negative μ (marked “-ve BE” in the figure). We should

4 This definition has a sign difference with respect to the usual defi- 5 These relationships can be easily derived using photon number and nition of the chemical potential in thermodynamics (Landau & Lifshitz energy conservation and requiring that the final spectrum have the equi- 1980)usedinEq.(1). librium Bose-Einstein distribution.

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0.001 1 -ve +ve y=0 10 y=0.02 0.0001 y=0.1 0.01

FIRAS Blackbody Surface 1 y=1.0 1e-05 y=5.0 1e-06 0.0001 0.1 y=10.0 y=100.0 final initial 1e-07 μ /μ initial μ

-ve BE / 0.01 E/E Pixie 1e-06 Δ 1e-08 final 0.001 1e-09 1e-08 μ 10-10 T/T|=|x/Log(1/n+1)-1| 0.0001 1e-10 Δ | 1e-10 1e-11 μ-distortion Blackbody Photosphere 1e-05 +ve -ve 1e-12 1e-12 1e-06 1e+06 1e+07 1e-05 0.0001 0.001 0.01 0.1 1 10 100 energy injection redshift (zi) x

Fig. 4. Evolution of initial spectrum at y = 0givenbyEq.(18) with Fig. 5. Regions of the blackbody photosphere probed by different exper- −5 −5 −YSZ = YBEC = 10 . |ΔT/T|≡|Teff/T − 1| is plotted, where Teff is iments, COBE FIRAS with sensitivity μ = 10 , Pixie with sensitivity the temperature of a blackbody spectrum with the corresponding in- μ = 10−8 and a hypothetical experiment with sensitivity of μ = 10−10. tensity at a particular frequency. Bose-Einstein spectrum defined by Shaded regions below the curves are allowed and inaccessible to these ν/ +μ n(ν) = 1/(eh kBTe − 1) with negative μ is labeled “-ve BE”, red experiments. We also plot the ratio of final μ distortion today to the dashed, double-dotted line in the figure. The distortions are positive in initial μ distortion as a function of energy injection redshift. We de- the Rayleigh-Jeans region (labeled “+ve”) on the left side and negative fine the blackbody surface as the energy injection redshift such that final initial 6 in the Wien region (labeled “-ve”) on the right side of the figure. μ /μ = 1/e, zbb = 1.98 × 10 .

6. Bose-Einstein condensation of CMB photons emphasize that there is no singularity in the actual solutions of in the early Universe the Kompaneets equation plotted above. The singularity is just in the mathematical formula, which correctly describes the spec- In the early Universe, before z ∼ 2 × 106 double Compton and trum at high frequencies, x |μ|. The actual spectrum deviates Comptonization destroy any spectral distortions and maintain from the Bose-Einstein spectrum near the singularity (positive the Planck spectrum of CMB. For small distortions due to sin- everywhere in the Rayleigh-Jeans region) and can be described gle, quasi-instantaneous episode of energy release, we can write by a chemical potential decreasing in magnitude with decreas- the ratio of final-to-initial μ as an exponential function of red- ing frequency. The evolution at y>1 is therefore very different shift defined by the square root of the product of Comptonization from the positive YSZ case. For x |μ| the spectrum is the Bose- and absorption rates (Sunyaev & Zeldovich 1970b). For a double Einstein spectrum at the electron temperature. For x |μ| there Compton process, this formula gives (Danese & de Zotti 1982) is an excess of photons compared to the Bose-Einstein spec- final trum which grows with time, a feature of Bose-Einstein con- μ − / 5/2 G(z) = ≈ e (zi zdc) , (21) densation. The fractional change in temperature ΔT/T takes a μinitial very simple form in the Rayleigh-Jeans region (x 1) for a 6 Bose-Einstein spectrum and can be understood as follows. It is where zdc ≈ 1.98 × 10 defines the “surface of the blackbody given, in terms of frequency referred to the electron temperature, photosphere”. We call this function G(z) the blackbody visibil- xe ≡ hν/kTe,by ity function for spectral distortions. We call the region z > zdc where the initial μ distortion can be reduced by a factor of more ΔT xe −μ than e the “blackbody photosphere”. Thus inside the blackbody = − 1 = · (20) ffi T x + μ x + μ photosphere, a Planckian spectrum can be established e ciently. e e Figure 5 shows how the μ distortion at high redshifts decreases |μ| Δ / = −μ/ due to the double Compton and Compton scattering. The regions For xe 1wehave T T xe. Thus the frac- ff tional temperature deviation is positive for negative μ and nega- of the blackbody photosphere allowed by di erent experiments tive for the positive μ.Forx greater than and close to |μ|,μ< 0 are shaded. e At redshifts 105 < z < 2 × 106,wehavetheComptonpa- the spectral distortions can become very large and exceed unity. y> Since the high-frequency spectrum is in equilibrium at the elec- rameter 1, and the Bose-Einstein spectrum with a negative . 6 tron temperature by y = 10, the subsequent evolution is very chemical potential is established at x 0 01 . At lower frequen- slow, and |μ| decreases while the electron temperature increases cies, bremsstrahlung and double Compton create a Planck spectrum corresponding to the electron temperature. At z < 104 the slowly as the photon distribution approaches a Planck spectrum y< . with the extra photons accumulating at low frequencies; i.e., Compton parameter 0 01 and the distortions created can be Bose-Einstein condensation happens. The low-frequency spec- described by the linear solution of Eq. (18). Thus the net distor- trum approaches the stationary solution of Kompaneets equation tions created in the early Universe owing to adiabatic cooling of with only the induced recoil term, n(x) ∝ 1/x2, with a contin- baryons are a linear superposition of distortions corresponding to = uous flow of photons towards x 0(Syunyaev 1971). Then, 6 Strictly speaking, a Bose-Einstein spectrum is established at y ∼ few / − − . × −5 − . × −5 Te T 1 just increases from 2 558 10 to 2 555 10 as can be seen from Fig. 4 However at y ∼ 0.25−1 the spectrum is in going from y = 10 to y = 100 for the chosen energy losses − already very close to Bose-Einstein at high frequencies. We use this to of ΔE/E = 4 × 10 5.Theeffect discussed below in the real divide the energy release into μ-type and y-type estimates. Exact results − Universe is of magnitude ∼10 9. are presented elsewhere (Chluba et al. 2012).

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0.01 which themselves do not survive. Later during recombination the z=1.5x106 second-order Doppler effect due to non-zero electron/baryon ve- 5 0.001 z=4.6x10 locity in the CMB rest frame also becomes important, but at that z=1.5x105 0.0001 4 y ff z=5.9x10 time only -type distortions similar to the SZ e ect arise, and 4 1e-05 z=1.2x10 μ distortions cannot be created. We can estimate the amount of z=1760 μ distortions created by dissipation of acoustic waves following 1e-06 z=840 z=220 Hu et al. (1994) by calculating how the total power in the den- 1e-07 sity fluctuations changes with time because of photon diffusion. 1e-08 The comoving energy density in acoustic waves in the photon- 1e-09 baryon plasma (neglecting the baryon energy density) is given T/T|=|x/Log(1/n+1)-1| = ρ 2 δ 2 ρ Δ by Q γc γ(x) ,where γ is the comoving photon energy | s 1e-10 2 ∼ / δ density, cs 1 3 is sound speed squared, γ is the photon den- 1e-11 -ve +ve -ve sity perturbation at position x, and angular brackets denote the 1e-12 ensemble average: 0.0001 0.001 0.01 0.1 1 10 x 3 2 d k δγ(x) = Pγ(k), (22) Fig. 6. Evolution of spectral distortions in CMB in ΛCDM cosmol- (2π)3 ogy. The values of total Compton parameter y = y(100, z) for differ- = . × 6,y = = . × 5,y = 2 i i ent curves are (z 1 5 10 109), (z 4 6 10 10), where Pγ(k) =Δγ(k)Pγ(k), and Pγ(k) is the initial power spec- 5 4 4 (z = 1.5 × 10 ,y= 1), (z = 5.9 × 10 ,y= 0.16), (z = 1.2 × 10 ,y= trum. It was shown by Mukhanov & Chibisov (1981)thatpri- . = ,y = × −5 = ,y = . × −8 0 005), (z 1760 3 10 ), (z 840 1 8 10 )and mordial fluctuations from inflation can have a spectrum deviat- (z = 220,y= 1.2 × 10−10). Reference spectrum is the blackbody CMB = 7 ing from the scale-invariant Harrison-Zeldovich spectrum, with spectrum at z 10 . At low frequencies the spectrum is dominated by < > double Compton and bremsstrahlung which maintain Planck spectrum spectral index ns 1. In the cyclic ekpyrotic models ns 1is at the electron temperature. The zero point at low frequencies moves also possible (Lehners et al. 2007). WMAP gives the constraint to the right at low redshifts because bremsstrahlung dominates over the on curvature perturbation in comoving gauge, ζ. This is related Compton scattering at higher and higher frequencies and absorbs pho- to the gravitational perturbation ψ in the radiation era (assuming tons, although both are slower than the expansion rate. neutrinos are free streaming) by the relation ψ = ζ/(2/5Rν+1.5), where Rν = ρν/(ργ + ρν) ≈ 0.4, ρν is the neutrino energy density δi = − ψ i = / / + ff y and γ 2 (Ma & Bertschinger 1995). Thus Pγ 4 (2 5Rν di erent values of , and a few of them are shown in Fig. 6.We − + 1 / . 2 = . = π2/ 3 / ns 1 2 r(ln k k0) show the evolution of actual spectrum (Eqs. (14)and(16)) in- 1 5) Pζ 1 45Pζ and Pζ (Aζ2 k )(k k0) , −1 −9 cluding the effects of bremsstrahlung and double Compton pro- k0 = 0.002 Mpc , Aζ = 2.4×10 (Keisler et al. 2011; Komatsu = / cesses (Burigana et al. 1991; Chluba & Sunyaev 2012; Hu & et al. 2011), r dns dlnk is the running of the index. The trans- Silk 1993; Illarionov & Sunyaev 1975a,b)intheΛCDM cos- fer function for modes well inside the horizon before recombi- mology with WMAP parameters (Komatsu et al. 2011). The ref- nation is given by = 7 erence spectrum is the blackbody CMB spectrum at z 10 . − 2/ 2 Δ ≈ k kD At x < 0.01 the spectrum is dominated by bremsstrahlung and γ 3cos(krs)e (23) double Compton processes that destroy the condensing low- ff frequency photons. The effect of bremsstrahlung compared to and the di usionscaleby(Kaiser 1983; Weinberg 2008) Comptonization becomes stronger with decreasing redshift and 1 ∞ c(1 + z) R2 16 can be felt at successively higher frequencies. = dz + (24) 4 2 6H(1 + R)n σ 1 + R 15 At z 5 × 10 the spectral distortions in the x 0.01 re- kD z e T / 2 + 2 gion can relax to a Bose-Einstein spectrum due to high value d(1 kD ) = − c(1 z) R + 16 , of parameter y. Contributions from lower redshifts are described (25) dz 6H(1 + R)neσT 1 + R 15 by the analytic solution of Eq. (18). The distortions created at 4 z < 10 are just the negative of the usual SZ effect and would where R ≡ 3ρb/4ργ, ρb is the baryon energy density. ff 2 be completely overwhelmed by the SZ e ect from reionization Replacing cos (krs) with its average value over an oscillation 2 at z ∼ 10 which is expected to be YSZ = (kBTe/mec )τri ∼ of 1/2, we get the energy release per unit redshift −6 × . = −7 τ ∼ . 10 0 1 10 ,where ri 0 1 is the optical depth to the ∼ 4 / − 3 dΔ2 last scattering surface due to reionization and Te 10 Kis dQ dz 1 d k i γ = Pγ(k) the average electron temperature during reionization. The dis- ργ 3 (2π)3 dz tortions created by energy loss at z > 105 can, however, be de- 3 2 5 d k 2 2 d(1/k ) stroyed only by energy injection at z > 10 since at lower red- i 2 −2k /kD D = 3 Pγ(k)k e shift Comptonization cannot create μ-type distortions. (2π)3 dz . 2 4 3Aζ d(1/kD ) − 2/ 2 = ns 2k kD − dkk e 7. Energy release from dissipation of acoustic kns 1 dz 0 waves during radiation domination . 2 4 3Aζ d(1/kD ) − + / + n + 1 = (3 ns) 2 ns 1Γ · − 2 kD (26) In the standard cosmology the main source of energy injec- ns 1 dz 2 k0 tion at high redshifts is the dissipation of acoustic waves in the baryon-photon plasma (Sunyaev & Zeldovich 1970b) due For a running index, the above integration must be done numer- to Silk damping (Silk 1968). Sunyaev & Zeldovich (1970a) ically. We are interested in radiation-dominated epoch where μ 1/2 2 first proposed using the resulting spectral distortions to mea- distortions are generated, and we have H(z) = H0Ωr (1 + z) 3 3 sure the spectral index and power in the small-scale fluctuations, and ne(z) = (nH0 + 2nHe0)(1 + z) ≡ ne0(1 + z) . In addition we

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−1/2 3/2 2 −4 have kD = A (1 + z) and d(1/kD )/dz = −3AD(1 + z) ,and

D Blackbody Edge the above equation simpliﬁes to HeII->HeI HeIII->HeII / 13Aζ + dQ dz −(3+n )/2 ns 1 (1−ns)/2 (3n −5)/2 n =1.0 = − s Γ + s γ 1e-08 s − 2 A (1 z) ργ ns 1 2 D k0 ns=0.96 where ns=1.0,run=-0.038 1e-09 ns=1.0,run=-0.045 8c 10 2 free AD = = 5.92 × 10 Mpc . (27) streaming cooling / Planck 135H Ω1 2n σ

0 r e0 T G(z) dE/dln(1+z)/E y μ The redshift and ns dependence obtained above matches that of −1 4 1e-10 Hu et al. (1994). The diffusion scale kD = 46 Mpc at z = 5×10 4 −1 6 and kD = 10 Mpc at z = 2 × 10 . Thus we are probing the primordial fluctuations on very small scales that are not accessible 100 1000 10000 100000 1e+06 in any other way. redshift (z) = Thus for a Harrison-Zeldovich spectrum with ns 1 Fig. 7. Sketch of fractional rate of energy release due to Silk damping and free streaming for different initial power spectra. Also shown for / − . × −9 dQ dz = 7 8 10 comparison is the rate of energy loss due to adiabatic cooling of bary- ργ 1 + z onic matter. = × 4 z 5 10 Table 1. Energy injection in μ distortions during 5 × 104 < z < 2 × Q = × 6 − z 2 10 = . × 8. 6 ff ρ 2 9 10 (28) 10 for di erent initial power spectra without running compared with γ energy losses due to Bose-Einstein condensation. We note the surprising fact that this redshift dependence is ex- Δ / actly the same as for the energy losses due to adiabatic cooling ns E E Eq. (13), as already pointed out by Chluba & Sunyaev (2012). 1.07 6.8 × 10−8 −8 For the currently preferred value of ns = 0.96 we have 1.04 4.7 × 10 1.02.9 × 10−8 / − × −8 . . × −8 dQ dz = 1 10 0 96 1 8 10 . . . × −8 ργ (1 + z)1 06 0 92 1 1 10 −9 4 BEC −2.2 × 10 Qz=5×10 z=2×106 = . × −8. ρ 1 8 10 (29) γ Table 2. Energy injection in μ distortions during 5 × 104 < z < 2 × 106 for different initial power spectra with running compared with energy Figure 7 compares the rate of energy release multiplied by the losses due to Bose-Einstein condensation. blackbody visibility function, G(z)(1 + z)dQ/dz/Eγ for different initial power spectra with the energy losses due to the adi- ns dns/dlnk ΔE/E abatic cooling of baryons, (1 + z)S /Eγ obtained by solv- Compton −9 ing Eq. (9) for the standard recombination history (Seager et al. 1.07 –0.05 2.2 × 10 . . × −9 2000) calculated using the effective multilevel approach and 1 07 –0.035 5 7 10 . . × −8 taking recent corrections into account (Ali-Haïmoud & Hirata 1 07 –0.02 1 6 10 1.04 –0.05 1.6 × 10−9 2011; Chluba & Thomas 2011). The spike due to increase . . × −9 7 1 04 –0.035 4 1 10 in Silk damping during recombination is clearly noticeable . 1.04 –0.02 1.1 × 10−8 The smaller spikes due to HeII and HeI are also noticeable. 1.0 –0.05 1.1 × 10−9 Recombination of each species leads to a decrease in the num- 1.0 –0.038 2.2 × 10−9 ber of particles, hence to adiabatic losses. But the photon diffu- 1.0 –0.035 2.6 × 10−9 sion length, and the associated energy release, increases owing 1.0 –0.02 6.9 × 10−9 to a decrease in the electron fraction. The transition from radia- BEC – −2.2 × 10−9 tion to matter domination results in unnoticeable change in the slope of the curves. Even after recombination photons continue Notes. Energy injection values ≤ BEC value in magnitude are shown in to mix on horizon scales because of free streaming, while the red (in the electronic edition). energy losses from Comptonization drop sharply as a result of depletion of electrons. We switch to the free-streaming solution also footnote 7). The y distortion due to mixing of blackbodies described in Appendix A when diffusion length 2π/kD becomes equal to the comoving particle horizon. In the blackbody photo- during free streaming after recombination and up to reionization, < < sphere region, the blackbody visibility function G(z) makes the 20 z 800, calculated using the result in the Appendix A,is Δ / ∼ −10 ffi curves drop sharply. Global energy release and y-type distortions E E 10 . These are, however, di cult to separate from y during the free streaming epoch after recombination are due to much the larger thermal and non-linear Doppler distortion ∼ −7 the superposition of blackbody spectra on horizon scales. The from reionization of YSZ 10 . ff energy injected into y-type distortion during recombination in Energy releases for di erent power spectra, including the −8 ones with running spectral index, are summarized in Tables 1 the peak (800 < z < 1500) is ΔE/E ∼ 10 for ns = 0.96 (see and 2. The energy losses from adiabatic cooling of baryons dur- − 7 The actual energy injected in the recombination peak would be much ing the same time period is ∼2.2 × 10 9. This equals the en- less because of the transition to free streaming. ergy release, for example, for a spectral index of ns = 1.0,

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3.5e-08 excess signal at 3 GHz by ARCADE (Fixsen et al. 2011) exper- 3e-08 iment, which is not completely explained by the current galactic and extragalactic emission models; nevertheless achievement of 2.5e-08 a goal of 10−9 in sensitivity, although challenging, seems possi- 2e-08 ble in the near future.

µ 1.5e-08 1e-08 8. Conclusions 5e-09 Comptonization of CMB photons with slightly cooler electrons results in spectral distortions that then evolve (in the absence of 0 any other physical process) towards the Bose-Einstein conden- -5e-09 sate solution, which is the equilibrium solution and consists of a 0.5 0.6 0.7 0.8 0.9 1 blackbody spectrum with zero chemical potential, with the extra nS photons accumulating at zero frequencies (Illarionov & Sunyaev

Fig. 8. μ = 1.4ΔE/E as a function of spectral index ns without running. 1975a; Landau & Lifshitz 1980). This is because the radiation has more photons than can be accommodated in a Bose-Einstein 1e-08 spectrum with non-negative chemical potential at the electron temperature. However, in reality, bremsstrahlung and double 8e-09 Compton scattering destroy extra photons at low frequencies. Thus in practice we have the high-frequency spectrum evolv- 6e-09 ing towards a blackbody with the extra photons slowly moving ns=1.05 down in frequency to be eaten up by bremsstrahlung and dou- 4e-09

µ 1.02 ble Compton scattering. This is what is seen in the numerical solution in Fig. 6. 2e-09 1.0 The difficulty of conserving photon number is the reason 0 Bose-Einstein condensation (or accumulation of photons at low frequencies) is not observed in astrophysical systems. Bose- -2e-09 Einstein condensation of photons has only been achieved in the laboratory very recently by Klaers et al. (2010). In the real -0.1 -0.09 -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 Universe we also have inevitable sources of energy injection Running into the CMB, such as the SZ effect after reionization, which Fig. 9. μ = 1.4ΔE/E as a function of running of the spectral index r for creates distortions of similar magnitude but with opposite signs canceling and overwhelming this effect. The situation is much three values of spectral index ns. more interesting for μ distortions, which can be created only at high redshifts and thus have no foregrounds at lower redshifts. dn /dlnk = −0.038 for a running spectrum. The net μ distor- In standard cosmology, positive μ distortions are created by dis- s ff tion as a function of spectral index ns without running is shown sipation of acoustic waves on small scales due to photon di u- in Fig. 8 and for n = 1.0, 1.02, 1.05 as a function of the running sion. For the currently allowed values of cosmological param- s μ of the index in Fig. 9. For low values of ns andhighnegative eters, there is a possibility that the positive distortions from values of running, Bose-Einstein condensation dominates, and photon diffusion can almost exactly cancel the negative μ dis- net distortion approaches a constant value of −3×10−9. A detec- tortions from Bose-Einstein condensation of CMB, leading to a −9 μ net distortion that is much smaller. Bose-Einstein condensation tion or non-detection at a sensitivity of 10 of -type distortion − is thus a very sensitive probe of primordial power spectrum on can even dominate, leading to a net distortion μ ∼−3 × 10 9 small scales. Given the importance of the effective heating rate to which Silk damping contributes, but only as a small perturba- for the amplitude of the net distortion, it is important to carry out tion. Nevertheless, we must emphasize that the photon number a more careful and refined calculation of the problem, which we conserving Comptonization, along with differences in adiabatic present elsewhere (Chluba et al. 2012). indices of radiation and matter, creates a unique system in the Over the years a number of ground- and balloon-based ex- early Universe in which photons can begin to evolve towards a periments that aimed at measuring the absolute CMB brightness Bose-Einstein condensate. It is a remarkable coincidence that a have been conducted before COBE (Johnson & Wilkinson 1987; completely unrelated physical process, diffusion damping of pri- Levin et al. 1992)andafterit(Bersanelli et al. 1994;Fixsen mordial perturbations, can produce distortions of almost exactly et al. 2011, 2004; Raghunathan & Subrahmnayan 2000; Staggs the same or greater magnitude, leading to the suppression of this et al. 1996; Zannoni et al. 2008). The best limits on CMB μ-type unique effect in astrophysics. We thus arrive at an intriguing conclusion: even a null result, non-detection of μ-type distortion at spectral distortions from these experiments are at the level of − 10−4−10−5 (Fixsen et al. 1996; Gervasi et al. 2008; Seiffert et al. a sensitivity of 10 9, rather than just placing a upper limit, ac- 2011). Achieving a sensitivity of 10−9 would require subtract- tually gives a quantitative measure of the primordial small-scale ing foregrounds due to synchrotron emission, free free emission, power spectrum. The importance of improving the experimental dust emission, and spinning dust emission at the same precision sensitivity to reach this critical value cannot be overemphasized. level. The proposed experiment Pixie (Kogut et al. 2011)aims to achieve this goal by using 400 effective channels between fre- Appendix A: Erasure of perturbations due to free quencies of 30 GHz to 6 THz. Their simulations indicate that an streaming accuracy of 1nK in foreground subtraction is achievable. There is, however, some uncertainty in our understanding of the fore- We can derive the solution for erasure if perturbations on hori- grounds and possible systematics as indicated by an observed zon scales due to free streaming as follows. We start with the

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Boltzmann hierarchy, ignoring quadrupole and higher order mo- Dodelson, S. 2003, Modern cosmology Scott Dodelson (Amsterdam: Netherlands Academic Press) ments. We deﬁne multipole ( ) moments of the temperature Δ / ≡ Θ= − + P ˆ. Θ P Dubrovich, V. K. 1975, Sov. Astron. Lett., 1, 196 anisotropy T T ( i) (2 1) (k nˆ) ,where Fixsen, D. J., Cheng, E. S., Gales, J. M., et al. 1996, ApJ, 473, 576 is the Legendre polynomial, and kˆ.nˆ the angle between pho- Fixsen, D. J., Kogut, A., Levin, S., et al. 2004, ApJ, 612, 86 ton momentum and comoving wavenumber. In the limit of zero Fixsen, D. J., Kogut, A., Levin, S., et al. 2011, ApJ, 734, 5 Thomson optical depth we get Gervasi, M., Zannoni, M., Tartari, A., Boella, G., & Sironi, G. 2008, ApJ, 688, 24 Hu, W., & Silk, J. 1993, Phys. Rev. D, 48, 485 dΘ0 + kΘ1 = 0 Hu, W., Scott, D., & Silk, J. 1994, ApJ, 430, L5 dη Illarionov, A. F., & Syunyaev, R. A. 1972, Sov. Ast., 16, 45 Illarionov, A. F., & Sunyaev, R. A. 1975a, Sov. Ast., 18, 413 dΘ1 k kψ − Θ0 = , Illarionov, A. F., & Sunyaev, R. A. 1975b, Sov. Ast., 18, 691 dη 3 3 Johnson, D. G., & Wilkinson, D. T. 1987, ApJ, 313, L1 Kaiser, N. 1983, MNRAS, 202, 1169 where η = dt/a is the conformal time. Using the fact that in Keisler, R., Reichardt, C. L., Aird, K. A., et al. 2011, ApJ, 743, 28 the matter-dominated era the potential is suppresses by a fac- Klaers, J., Schmitt, J., Vewinger, F., & Weitz, M. 2010, Nature, 468, 545 torof9/10 compared to primordial value on large scales, we Kogut, A., Fixsen, D. J., Chuss, D. T., et al. 2011, JCAP, 07, 025 Θ = − ψ/ Komatsu, E., Smith, K. M., Dunkley, J., et al. 2011, ApJS, 192, 18 have 0 2 3 (see for example Dodelson 2003) and look- Kompaneets, A. S. 1956, Zh. Eksp. Teor. Fiz., 31, 876 i dηω ing for a solution√ with time dependence of type Θ ∝ e Landau, L. D., & Lifshitz, E. M. 1980, Statistical physics, Part. 1 (Oxford: ω = / Butterworth-Heinemann) we get ik 6 Thus free√ streaming damps the perturba- − η / − . η Lehners, J.-L., McFadden, P., Turok, N., & Steinhardt, P. J. 2007, Phys. Rev. D, tions by a factor of e d k 6 = e 0 4 d k. Finally we want to 76, 103501 mention that this solution is very approximate, and in principle Levich, E. V., & Sunyaev, R. A. 1971, Sov. Ast., 15, 363 we should also take higher modes into account. For example, Levin, S., Bensadoun, M., Bersanelli, M., et al. 1992, ApJ, 396, 3 Lifshitz, E. M., & Pitaevskii, L. P. 1981, Physical Kinetics (Oxford: Butterworth- −0.3 dηk on going up to = 3 we get for the damping factor e . Heinemann) Also for free streaming we have super position of blackbod- Ma, C.-P., & Bertschinger, E. 1995, ApJ, 455, 7 ies, and the y distortion is given by (Chluba & Sunyaev 2004) Mukhanov, V. F., & Chibisov, G. V. 1981, Sov. J. Exp. Theor. Phys. Lett., 33, = / Θ2 = / δ2 532 YSZ 1 2 1 32 γ , and so the equivalent energy release Peebles, P. J. E. 1968, ApJ, 153, 1 2 is ΔE/E = 4YSZ = 1/8 δγ . Raghunathan, A., & Subrahmnayan, R. 2000, JApA, 20, 1 Rubiño-Martín, J. A., Chluba, J., & Sunyaev, R. A. 2006, MNRAS, 371, 1939 Acknowledgements. We would like to thank Matias Zaldarriaga for important Rubiño-Martín, J. A., Chluba, J., & Sunyaev, R. A. 2008, A&A, 485, 377 remarks on the manuscript. We would also like to thank Yacine Ali-Haïmoud for Seager, S., Sasselov, D. D., & Scott, D. 2000, ApJS, 128, 407 ﬀ careful reading and comments on the manuscript. Sei ert, M., Fixsen, D. J., Kogut, A., et al. 2011, ApJ, 734, 6 Silk, J. 1968, ApJ, 151, 459 Staggs, S. T., Jarosik, N. C., Wilkinson, D. T., & Wollack, E. J. 1996, ApJ, 458, References 407 Syunyaev, R. A. 1971, Sov. Ast., 15, 190 Ali-Haïmoud, Y., & Hirata, C. M. 2011, Phys. Rev. D, 83, 043513 Sunyaev, R. A., & Chluba, J. 2009, Astron. Nachr., 330, 657 Arons, J. 1971, ApJ, 164, 437 Sunyaev, R. A., & Zeldovich, Y. B. 1970a, Ap&SS, 9, 368 Bersanelli, M., Bensadoun, M., de Amici, G., et al. 1994, ApJ, 424, 517 Sunyaev, R. A., & Zeldovich, Y. B. 1970b, Ap&SS, 7, 20 Burigana, C., Danese, L., & de Zotti, G. 1991, A&A, 246, 49 Weinberg, S. 2008, Cosmology (Oxford: Oxford University Press) Chluba, J. 2005, Ph.D. Thesis, LMU München Zannoni, M., Tartari, A., Gervasi, M., et al. 2008, ApJ, 688, 12 Chluba, J., & Sunyaev, R. A. 2004, A&A, 424, 389 Zel’Dovich, Y. B., & Levich, E. V. 1969, Sov. J. Exp. Theor. Phys., 28, 1287 Chluba, J., & Sunyaev, R. A. 2006, A&A, 458, L29 Zel’Dovich, Y. B., & Levich, E. V. 1970, Sov. J. Exp. Theor. Phys. Lett., 11, 35 Chluba, J., & Thomas, R. M. 2011, MNRAS, 412, 748 Zeldovich, Y. B., & Sunyaev, R. A. 1969, Ap&SS, 4, 301 Chluba, J., & Sunyaev, R. A. 2012, MNRAS, 419, 1294 Zeldovich, Y. B., Kurt, V. G., & Syunyaev, R. A. 1968, Zh. Eksp. Teor. Fiz., 55, Chluba, J., Khatri, R., & Sunyaev, R. A. 2012, MNRAS, submitted 278 [arXiv:1202.0057] Zeldovich, Y. B., Illarionov, A. F., & Syunyaev, R. A. 1972, Sov. J. Exp. Theor. Danese, L., & de Zotti, G. 1982, A&A, 107, 39 Phys., 35, 643

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