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A&A 501, 29–47 (2009)DOI: 10.1051/0004-6361/200809840c© ESO 2009

Astronomy&

Astrophysics

Pre-recombinational energy release and narrow featuresin the CMB spectrum

J. Chluba1 and R. A. Sunyaev1,2

1 Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching bei München, Germanye-mail: [email protected]

2 Space Research Institute, Russian Academy of Sciences, Profsoyuznaya 84/32, 117997 Moscow, Russia

Received 25 March 2008 / Accepted 27 February 2009

ABSTRACT

Energy release in the early Universe (z <∼ 2 × 106) should produce a broad spectral distortion of the cosmic microwave background(CMB) radiation field, which can be characterized as y-type distortion when the injection process started at redshifts z <∼ 5×104. Herewe demonstrate that if energy was released before the beginning of cosmological hydrogen recombination (z ∼ 1400), closed loops ofbound-bound and free-bound transitions in H i and He ii lead to the appearance of (i) characteristic multiple narrow spectral featuresat dm and cm wavelengths; and (ii) a prominent sub-millimeter feature consisting of absorption and emission parts in the far Wientail of CMB spectrum. The additional spectral features are generated in the pre-recombinational epoch of H i (z >∼ 1800) and He ii(z >∼ 7000), and therefore differ from those arising due to normal cosmological recombination in the undisturbed CMB blackbodyradiation field. We present the results of numerical computations including 25 atomic shells for both H i and He ii, and discuss thecontributions of several individual transitions in detail. As examples, we consider the case of instantaneous energy release (e.g., due tophase transitions), and exponential energy release (e.g., because of long-lived decaying particles). Our computations show that becauseof possible pre-recombinational atomic transitions the variability in the CMB spectral distortion increases when comparing with thedistortions arising in the normal recombination epoch. The amplitude of the spectral features, both at low and high frequencies,directly depends on the value of the y-parameter, which describes the intrinsic CMB spectral distortion resulting from the energyrelease. The time-dependence of the injection process also plays an important role, for example leading to non-trivial shifts in thequasi-periodic pattern at low frequencies along the frequency axis. The existence of these narrow spectral features would provide anunique way to separate y-distortions caused by pre-recombinational (1400 <∼ z <∼ 5 × 104) energy release from those arising in thepost-recombinational era at redshifts z <∼ 800.

Key words. atomic processes – radiation mechanisms: general – cosmic microwave background – early Universe –cosmology: theory

1. Introduction

Measurements completed using data acquired with theCobe/Firas instrument have proven that the spectrum of thecosmic microwave background (CMB) is close to being a per-fect blackbody (Fixsen et al. 1996) of thermodynamic tempera-ture T0 = 2.725 ± 0.001 K (Mather et al. 1999; Fixsen & Mather2002). However, from the theoretical point of view, deviationsof the CMB spectrum from that of a pure blackbody are not onlypossible but even inevitable if, for example, energy was releasedin the early Universe (e.g., due to viscous damping of acous-tic waves, or annihilation or decay of particles). For very earlyenergy release (5× 104 <∼ z <∼ 2× 106), the resulting spectral dis-tortion can be characterized as a Bose-Einstein μ-type distortion(Sunyaev & Zeldovich 1970b; Illarionov & Syunyaev 1975a,b),while for energy release at low redshifts (z <∼ 5×104), the distor-tion is close to being a y-type distortion (Zeldovich & Sunyaev1969).

The most robust observational limits to these types ofdistortions are |y| ≤ 1.5 × 10−5 and |μ| ≤ 9.0 × 10−5

(Fixsen et al. 1996). Due to rapid technological developments,

improvements in these limits by a factor of ∼50 in principle mayhave been possible already several years ago (Fixsen & Mather2002), and some efforts are being made to determine the abso-lute value of the CMB brightness temperature at low frequen-cies using the balloon-borne experiment Arcade (Kogut et al.2004, 2006; Fixsen et al. 2009). However, today even a factor of∼1000 is probably within reach for absolute measurement of theCMB spectrum (Mather 2007), in principle bringing us down toy ∼ 10−8−10−7.

Also in the post-recombinational epoch (z <∼ 800), y-typespectral distortions caused by different physical mechanismsshould be produced. As an example, when performing mea-surements of the average CMB spectrum (e.g., with wide-anglehorns or as achieved by Cobe/Firas), all clusters of galax-ies, hosting hot intergalactic gas, due to the thermal SZ-effect(Sunyaev & Zeldovich 1972b), should contribute to the inte-grated value of the observed y-parameter. Similarly, supernovaremnants at high redshifts (Oh et al. 2003), or shock waves aris-ing due to large-scale structure formation (Sunyaev & Zeldovich1972a; Cen & Ostriker 1999; Miniati et al. 2000) should lead toa contribution to the overall y-parameter. For its possible valuetoday, we only have the upper limit determined by Cobe/Firas

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30 J. Chluba and R. A. Sunyaev: Pre-recombinational energy release and narrow features in the CMB spectrum

and lower limits derived by estimating the total contribution ofall clusters in the Universe (Markevitch et al. 1991; da Silvaet al. 2000; Roncarelli et al. 2007). These lower limits exceedy ∼ 10−6, and it is possible that the contributions to the totalvalue of y because of early energy release are comparable to orexceed those coming from the low redshift Universe.

Several detailed analytical and numerical studies for vari-ous energy injection histories and mechanisms can be foundin the literature (e.g., Zeldovich & Sunyaev 1969; Sunyaev &Zeldovich 1970b,c,a; Illarionov & Syunyaev 1975a,b; Zeldovichet al. 1972; Chan et al. 1975; Danese & de Zotti 1982; Ostriker& Thompson 1987; Daly 1991; Burigana et al. 1991b,a, 1995;Hu & Silk 1993a,b; Hu et al. 1994; Salvaterra & Burigana 2002;Burigana & Salvaterra 2003; Chluba & Sunyaev 2004). Two im-portant conclusions can be drawn from these all studies: (i) thearising spectral distortions are always very broad and practicallyfeatureless; and (ii) due to the absence of narrow spectral fea-tures, distinguishing between the different injection histories isextremely difficult. This implies that if one would find a y-typespectral distortion in the average CMB spectrum, then it is prac-tically impossible to determine whether the energy injection oc-curred just before, during, or after the epoch of cosmologicalrecombination. Furthermore, these measurement require an ab-solute calibration and cross-calibration of the instrument, like forCobe/Firas.

In this paper, we show that the pre-recombinational emis-sion within the bound-bound and free-bound transition of atomichydrogen and helium should leave multiple narrow features(Δν/ν ∼ 10−30%) in the CMB spectrum, that might become ob-servable at cm, dm, and sub-mm wavelengths (see Sect. 5). Asfor example discussed in Sunyaev & Chluba (2007), such kindof measurement could be performed differentially in frequency,without the requirement of an absolute calibration. This could inprinciple open a way to directly distinguish between pre- andpost-recombinational y-distortions and even shed light on thetime-dependence of the energy injection process. We also findthat the pre-recombinational emission produces a broad contin-uum spectrum, which close to the maximum of the CMB black-body contributes very little but can reach ∼10% of the intrin-sic y-distortion at low frequencies (ν ∼ 1 GHz). Althoughthis continuum has a different spectral behavior from that ofa y-distortion, observationally the narrow spectral features arepossibly more interesting, since it should be easier to extractthem by employing a differential observing strategy.

How does this work? At redshifts well before the epoch ofHe iii → He ii-recombination (z >∼ 8000), the total number ofCMB photons is unaffected by atomic transitions if the intrin-sic CMB spectrum is represented by a pure blackbody. Thisis because the atomic emission and absorption processes bal-ance each other in full thermodynamic equilibrium. However, atlower redshifts (z <∼ 8000), due to the expansion of the Universe,the medium became sufficiently cold to allow the formationof neutral atoms. The transition to the neutral state is associ-ated with the release of several additional photons per baryon(e.g., ∼5 photons per hydrogen atom, Chluba & Sunyaev 2006),even within a pure blackbody, ambient, CMB radiation field.Refining early estimates (Zeldovich et al. 1968; Peebles 1968;Dubrovich 1975; Dubrovich & Stolyarov 1995, 1997), the spec-tral distortions arising during hydrogen recombination (800 <∼z <∼ 1800), He ii → He i-recombination (1600 <∼ z <∼ 3000),and He iii → He ii-recombination (4500 <∼ z <∼ 7000) withina pure blackbody ambient radiation field have been computedin detail (Rubiño-Martín et al. 2006; Chluba & Sunyaev 2006;Chluba et al. 2007a; Rubiño-Martín et al. 2008). It was also

emphasized that measuring these distortions in principle mayopen another independent way to determine the temperature ofthe CMB monopole, the specific entropy of the Universe, and theprimordial helium abundance, well before the first appearance ofstars (e.g., Sunyaev & Chluba 2008; Chluba & Sunyaev 2008b;Sunyaev & Chluba 2007).

However, the intrinsic CMB spectrum deviates from a pureblackbody (e.g., due to early energy injection, as explainedabove), then full equilibrium is perturbed, and the small imbal-ance between emission and absorption in atomic transitions canlead to a net change in the number of photons, even prior tothe epoch of recombination, in particular owing to loops start-ing and ending in the continuum (Lyubarsky & Sunyaev 1983).These loops attempt to diminish the maximal spectral distortionsand produce several new photons per absorbed one. In this paper,we attempt to demonstrate how the cosmological recombinationspectrum is affected by an intrinsic y-type CMB spectral distor-tion. We investigate the cases of single instantaneous energy in-jection (e.g., due to phase transitions) and for exponential energyinjection (e.g., because of long-lived decaying particles). Thereis no principle difficulty in performing the calculations for moregeneral injection histories, also including μ-type distortions, ifnecessary. However, this still requires a slightly more detailedstudy, which will be left for a future paper.

We also estimate the corrections due to free-free absorptionand electron scattering. At observing frequency ν >∼ 1 GHz,the free-free process is not important for y-distortions appear-ing at redshifts z <∼ 105. However, the broadening caused byelectron scattering must be taken into account for features ap-pearing at redshifts z >∼ 6000. The recoil effect is mainly im-portant for the Lyman-series features, but can otherwise beneglected. We also checked the effect of intrinsic y- and μ-distortions on the CMB temperature and polarization powerspectra but found no significant effect for the current upper lim-its |y| ≤ 1.5× 10−5 and |μ| ≤ 9.0× 10−5 imposed by Cobe/Firas(Fixsen et al. 1996).

In Sect. 2, we provide a short overview of the thermaliza-tion of CMB spectral distortions after early energy release, andprovide formulae that we used in our computations to describey-type distortions. In Sect. 3, we present explicit expressions forthe net bound-bound and free-bound rates in a distorted ambientradiation field. We then derive some estimates of the expectedcontributions to the pre-recombinational signals coming fromprimordial helium in Sect. 4. Our main results are presented inSect. 5, where we discuss a few simple cases (Sects. 5.1 and 5.2)to gain some level of understanding. We support our numericalcomputations by several analytic considerations in Sect. 5.1.1and Appendix B. In Sect. 5.3, we then discuss the results forour 25 shell computations of hydrogen and He ii. First we con-sider the dependence of the spectral distortions on the value of y(Sect. 5.3.1), where Figs. 5 and 6 play the main role. Then inSects. 5.3.2 and 5.3.3, we investigate the dependence of the spec-tral distortions on the injection redshift and history, where we areparticularly interested in changes in the low frequency variabil-ity of the signal (see Figs. 9 and 11). We present our conclusionsin Sect. 6.

2. CMB spectral distortions after energy releaseAfter any energy release in the Universe, the thermody-namic equilibrium between matter and radiation will ingeneral be perturbed, and in particular, the distribution ofphotons will deviate from that of a pure blackbody. Thecombined action of Compton scattering, double Compton

J. Chluba and R. A. Sunyaev: Pre-recombinational energy release and narrow features in the CMB spectrum 31

emission1 (Lightman 1981; Thorne 1981; Pozdniakov et al.1983, Chluba et al. 2007b), and bremsstrahlung will attempt torestore full equilibrium, but, depending on the injection redshift,may not fully succeed. Using the approximate formulae givenin Burigana et al. (1991b) and Hu & Silk (1993a), for the pa-rameters within the concordance cosmological model (Spergelet al. 2003; Bennett et al. 2003), one can distinguish betweenthe following cases for the residual CMB spectral distortionsarising from a single energy injection, δργ/ργ � 1, at heatingredshift zh:

(I) zh < zy ∼ 6.3 × 103: compton scattering is unable to es-tablish full kinetic equilibrium of the photon distributionwith the electrons. Photon-producing processes (mainlybremsstrahlung) can only restore a Planckian spectrumat very low frequencies. Heating results in a Comptony-distortion (Zeldovich & Sunyaev 1969) at high frequen-cies, as in the case of the thermal SZ effect, for y-parametery ∼ 1

4 δργ/ργ.

(II) zy < zh < zμ ∼ 2.9 × 105: compton scattering can estab-lish partial kinetic equilibrium of the photon distributionwith the electrons. Photons produced at low frequencies(mainly by bremsstrahlung) diminish the spectral distor-tion close to their initial frequency, but cannot strongly up-scatter. The deviations from a blackbody represent a mix-ture of a y-distortion and a μ-distortion.

(III) zμ < zh < zth ∼ 2 × 106: compton scattering can establishfull kinetic equilibrium between the photon distributionand the electrons after a very short time. Low-frequencyphotons (mainly by double Compton emission) upscatterand slowly reduce the spectral distortion at high frequen-cies. The deviations from a blackbody can be described bya Bose-Einstein distribution with a frequency-dependentchemical potential, which is constant at high and vanishesat low frequencies.

(IV) zth < zh: both Compton scattering and photon productionprocesses are extremely efficient, restoring practically anyspectral distortion caused by heating, eventually producinga pure blackbody spectrum with slightly higher tempera-ture Tγ than before the energy release.

For the case (I) and (III), it is possible to approximate the dis-torted radiation spectrum analytically. There is no principle diffi-culty in computing numerically the time-dependent solution forthe radiation field after energy release (e.g., Hu & Silk 1993a) formore general cases, if necessary. However, here we are partic-ularly interested in demonstrating the main difference betweenthe additional radiation caused by atomic transitions in hydrogenand helium before the true epoch of recombination, and for lateenergy release, in which a y-type spectral distortion is formed.We therefore only distinguish between cases (I) and (III), andassume that the transition between these two cases occurs atzμ,y ≈ zμ/4

√2 ∼ 5.1× 104, i.e., the redshift at which the energy-

exchange timescale equals the expansion timescale (Sunyaev& Zeldovich 1980; Hu & Silk 1993a). The effects of intrinsicμ-type spectral distortions will be considered in a future work,so below we restrict ourselves to energy injection at redshiftsz <∼ 50 000.

1 Because of the huge excess in the number of photons to baryons(Nγ/Nb ∼ 1.6×109), the double Compton process is the dominant sourceof new photons at redshifts zdc >∼ 3 × 105−4 × 105, while at z <∼ zdc

bremsstrahlung is more important.

2.1. Compton y-distortion

For energy release at low redshifts, the Compton process is nolonger able to establish full kinetic equilibrium between theCMB photons and electrons. If the temperature of the radiationis lower than the temperature of the electrons, photons are up-scattered. For photons initially distributed according to a black-body spectrum of temperature Tγ, the efficiency of this processis determined by the Compton y-parameter,

y =

∫kB(Te − Tγ)

mec2Ne σT dl , (1)

where σT is the Thomson cross-section, dl = c dt, Ne is theelectron number density, and Te is the electron temperature. Fory � 1, the resulting intrinsic distortion in the photon occupationnumber of the CMB is approximately given by (Zeldovich &Sunyaev 1969)

Δnγ = yx ex

(ex − 1)2

[x

ex + 1ex − 1

− 4

], (2)

where x = hν/kBTγ is the dimensionless frequency.For computational reasons, it is convenient to intro-

duce the frequency-dependent chemical potential produced bya y-distortion, which can be obtained with

μ(x) = ln

(1 + nγ

nγ

)− x

y�1↓≈ −y x

[x

ex + 1ex − 1

− 4

], (3)

where npl(x) = 1/[ex−1] is the Planckian occupation number andnγ = npl + Δnγ. For x → 0 and y � 1, one finds μ(x) ≈ 2xy. Forx 1 one has μ(x) ≈ − ln[1+y x2], or μ(x) ≈ −y x2 for 1 � x �√

1/y. Comparing with a blackbody spectrum of temperature Tγ,for y > 0 a deficit of photons exists at low frequencies, whilethere is an excess at high frequencies. In particular, the spectraldistortion changes sign at xy ∼ 3.8.

2.1.1. Compton y-distortion from decaying particles

If all the energy is released at a single redshift, zi <∼ zμ,y ∼ 50 000,then after a very short time a y-type distortion forms, where they-parameter is approximately given by y ∼ 1

4 δργ/ργ.However, when the energy release is caused by decaying

unstable particles of sufficiently long lifetimes, tX, then theCMB spectral distortion accumulates as a function of redshift.In this case, the fractional energy-injection rate is given byδρ̇γ/ργ ∝ e−t(z)/tX/(1+z), so that the time-dependent y-parametercan be computed as

y(z) = y0 ×∫ ∞

zdz′ e−t(z′)/tX/H(z′)(1 + z′)2

∫ ∞0

dz′ e−t(z′)/tX/H(z′)(1 + z′)2, (4)

where y0 =14 δργ/ργ is related to the total energy release, and

H(z) is the Hubble expansion factor. We note that y(z) is a rathersteep function of redshift, which increases sharply about the red-shift, zX, at which t(z) ≡ tX.

3. Atomic transitions in a distorted ambient CMBradiation field

3.1. Bound-bound transitions

Using the occupation number of photons, nγ = 1/[ex+μ−1], withfrequency-dependent chemical potential μ(x), one can express


the net rate connecting two bound atomic states i and j in theconvenient form

ΔRi j = pi jAi j Ni exi j+μi j

exi j+μi j − 1

[1 − gi

g j

N j

Nie−[xi j+μi j]

], (5)

where pi j is the Sobolev-escape probability (e.g., see Seageret al. 2000), Ai j is the Einstein-A-coefficient of the transitioni → j, Ni and gi are the population and statistical weight of theupper and Nj and g j of the lower hydrogen level, respectively.Furthermore, we have introduced the dimensionless frequencyxi j = hνi j/kT0(1 + z) of the transition, where T0 = 2.725 Kis the present CMB temperature (Fixsen & Mather 2002), andμi j = μ(xi j).

3.2. Free-bound transitions

For the free-bound transitions from the continuum to the boundatomic states i, one has

ΔRci = Ne Nc αi − Ni βi , (6)

where Nc is equal to the number density of free protons, Np, inthe case of hydrogen, and the number density of He iii nuclei,NHe iii, in the case of helium. The recombination coefficient, αi,and photoionization coefficient, βi, are given by the integrals

αi =8πc2

f̃i(Te)∫ ∞

νic

ν2 σi(ν) ex+μ(x)+(xic−x)/ρ

ex+μ(x) − 1dν (7a)

βi =8πc2

∫ ∞

νic

ν2σi(ν)ex+μ(x) − 1

dν , (7b)

where xic = hνic/kTγ is the dimensionless ionization frequency,ρ = Te/Tγ is the ratio of the photon to electron tempera-ture, σi is the photoionization cross-section for the level i, and

f̃i(Te) =gi

2

[h2

2πmekTe

]3/2 ≈ gi

2 × 4.14 × 10−16 T−3/2e cm3. In full

thermodynamic equilibrium, the photon distribution is given bya blackbody with Tγ = Te. As expected, in this case one findsfrom Eq. (7) that αeq

i ≡ f̃i(Te) ehνic/kTe βeqi .

4. Expected contributions from helium

The number of helium nuclei is only ∼8% of the number ofhydrogen atoms in the Universe. Compared to the radiation re-leased by hydrogen, one therefore naively expects a small addi-tion of photons due to atomic transitions in helium. However, ata given frequency the photons due to He ii have been releasedat redshifts about Z2 = 4 times higher than for hydrogen, whenboth the number density of particles and the temperature of themedium was higher. The expansion of the Universe then was alsofaster. As we show below, these circumstances make the contri-butions from helium comparable to those from hydrogen, whereHe ii plays a far more important role than He i.

4.1. Contributions due to He II

The speed at which atomic loops can be passed through is de-termined by the effective recombination rate to a given level i,since the bound-bound rates are always much faster. To esti-mate the contributions to the CMB spectral distortion by He ii,we compute the change in the population of level i due to di-rect recombinations to that level over a very short time inter-val Δt � texp ∼ 1/H(z), i.e. ΔNi ≈ Ne Ncα

He iii Δt. Since we

consider the pre-recombinational epoch of helium, everything iscompletely ionized, so that ΔNi(z) ∝ (1 + z)6αHe ii

i (z)Δt.Because all the bound-bound transition rates in He ii are a

factor of 16 higher than for hydrogen, the relative importance ofthe different channels to lower states should remain the same asin hydrogen2. Therefore, one can assume that the relative num-ber of photons, fi j, emitted in the transition i→ j per additionalelectron on the level i is similar to that for hydrogen at a factorof 4 lower redshift, i.e., f He ii

i j (4z) ≈ f H ii j (z).

If we want to know how many of the emitted photons areobserved in a fixed frequency interval Δν today (zobs = 0), wemust also consider that at higher redshift the expansion of theUniverse is faster. Hence, the redshifting of photons by a givenintervalΔν is accomplished in a shorter time interval. For a giventransition, these are related by Δt = − Δz

H(zem)[1+zem] =1+zem

H(zem) νi jΔν,

where νi j is the transition frequency and zem the redshift of emis-sion. Here it is important that Δν = −νi jΔz/[1 + zem]2. Then thechange in the number of photons produced by emission in thetransition i→ j today should be proportional to

ΔNγ(νi j) ∼ fi j(zem) Ne Nc αi

H(zem) (1 + zem)3

(1 + zem)νi j

Δν , (8)

where zem is the redshift of emission, and the change in the vol-ume element due to the expansion of the Universe is taken intoaccount by the factor of (1 + zem)3. This now must be comparedwith the corresponding change in the number of photons emit-ted in the same transition by hydrogen, but at about 4 times loweremission redshift.

For hydrogenic atoms with charge Z the recombinationrate (including stimulated recombination) within the ambientCMB blackbody, scales as (Kaplan & Pikelner 1970)

αi ∝ Z4

T 3/2

∫ ∞

hνi/kT

dxx2∝ Z2

T 1/2, (9)

where νi is the ionization frequency of the level i, and T isthe temperature of the plasma. It was assumed that hνi � kT .Therefore, one finds that αHe ii

i (4T )/αH ii (T ) ∼ 2. One also has

Ne(4z) NHe ii(4z)/[Ne(z) Np(z)] ∼ 8% × 46, and, assuming radi-ation domination, H(z)/H(4z) ∼ 1/16. Furthermore, the fac-tor of 1/[1 + z]3 in Eq. (8) leads to 1/43. Hence, we findΔNHe iiγ (νHe ii

i j , 4zem)/ΔNH iγ (νH i

i j , zem) ∼ 8% × 43 2/16 ∼ 64%. We

note that (1 + zem)/νH ii j ≡ (1 + 4zem)/νHe ii

i j .This estimate shows that prior to the epoch of He iii →

He ii recombination the release of photons by helium shouldbe by a factor of ∼8 higher than hydrogen at about 4 timeslower redshift! Therefore, one expects that at a given frequencyin the Rayleigh-Jeans part of the CMB blackbody, the pre-recombinational emission originating in He ii is already com-parable to the contributions from hydrogen. As we show below(see Sect. 5.3.1), this is in good agreement with the results of ournumerical calculations.

4.2. Contributions due to He I

In the case of neutral helium, the highly excited levels are ba-sically hydrogenic. Therefore, one does not expect any ampli-fication of the emission within loops prior to its recombina-tion epoch. Furthermore, the total period during which neutralhelium can contribute significantly is limited to the redshift

2 Even the factors due to stimulated emission in the ambient blackbodyradiation field are the same!


range starting at the end of He iii → He ii recombination, say1600 <∼ z <∼ 6000. Therefore, neutral helium is typically inactiveover a wide range of redshifts.

Still there could be some interesting features appearing inconnection with the fine-structure transitions, which even withinthe standard computations lead to strong negative features in theHe i recombination spectrum (Rubiño-Martín et al. 2008). Thespectrum of neutral helium, especially at high frequencies, isalso more complicated than for hydrogenic atoms, so that somenon-trivial features might arise. We leave this problem to futurework, and focus on the contributions of hydrogen and He ii.

5. Results for intrinsic y-type CMB distortions

Here we discuss the results for the changes in the recombi-nation spectra of hydrogen and He ii for different values ofthe y-parameter. We use a modified version of the code devel-oped for computations of the standard recombination spectrum(Rubiño-Martín et al. 2006; Chluba et al. 2007a), and numeri-cally solve the pre-recombinational problem in the presence ofan intrinsic y-distortion. Some of the computational details andthe formulation of this problem can be found in Appendix A.

5.1. The 2 shell atom

To understand the properties of the numerical solution and alsoto check the correctness of our computations, we first consid-ered the problem including only a small number of shells. Ifwe take 2 shells into account, we deal with only a few atomictransitions, namely the Lyman- and Balmer-continuum, and theLyman-α line. In addition, one expects that during the recom-bination epoch of the considered atomic species (here H i orHe ii) the 2s-1s-two-photon decay channel will also contribute,but very little before that time.

In Fig. 1, we show the spectral distortion, ΔIν, including2 shells in our computations for different transitions as a func-tion of redshift3. It was assumed that energy was released in asingle injection at zi = 50 000, producing y = 10−5. All showncurves were computed using the δ-function approximation forthe line-profiles (Kholupenko et al. 2005; Wong et al. 2006;Rubiño-Martín et al. 2006), in which the distortion is given byΔIi j(z) = hc

4π ΔRi j(z)/H(z)[1+z]3, where ΔRi j is the net rate of thetransition i → j. This approximation is insufficient when one isinterested in computing the true spectral distortions in frequencyspace, since for the free-bound contribution the recombinationallines are very broad (e.g., see Chluba & Sunyaev 2006). In ad-dition, one should include the line-broadening due to electronscattering as explained in Appendix A.4 and the correct map-ping from z→ ν.

We note that in the δ-function approximation for the line pro-files one has ΔIi j(z) ≡ f (z)ΔRi j(z), where f (z) depends only onredshift and not on the considered transition. Therefore, consid-ering ΔIi j(z) is a convenient way of studying the redshift depen-dence of the net rates between different levels.

Looking at Fig. 1, one can see that prior to the recom-bination epoch of the considered species one can find pre-recombinational emission and absorption in the Lyman- and

3 This is a convenient representation of the spectrum, when one is in-terested in the time-dependence of the photon release, rather than theobserved spectral distortion in frequency space. To obtain the latter, inthe δ-function approximation for the line-profile one simply has to plotthe presented curves as a function of ν = νi j/(1 + z), where νi j is therest-frame frequency of the considered transition.

500 103

104 50000

z

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

ΔIν

[ 1

0-26 J

m-2

s-1

Hz-1

sr-1

]

HI Ly-αHI Ly-cHI Balmer-cHI 2s-1sHI Ly-α, analytic

y = 10-5

nmax

= 2

Feature due toHeIII-HeII recombination

Lyman-continuum becomesoptically thick

Lines from normalHI recombination

4000 104 50000

z

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

ΔIν

[ 1

0-26 J

m-2

s-1

Hz-1

sr-1

]

HeII Ly-αHeII Ly-cHeII Balmer-cHeII 2s-1sHeII Ly-α, analytic

y = 10-5

nmax

= 2


Lines from normalHeII recombination

Fig. 1. Spectral distortion, ΔIν, including 2 shells into our computationsfor different transitions as a function of redshift and for y = 10−5.All shown curves were computed using the δ-function approximationfor the intensity. The upper panel shows the results for hydrogen, thelower those for He ii. In both cases the analytic approximation for theLyman-α line based on Eqs. (B.2) and (B.7), and including the es-cape probabilities in the Lyman-α line and Lyman-continuum, are alsoshown. Note that in the chosen representation the Lyman-α line andBalmer-continuum coincide at high redshifts. This is because their pho-tons are emitted simultaneously and in equal amount as parts of thesame atomic loop (c→ 2p→ 1s→ c).

Balmer-continuum, and the Lyman-α line, which would becompletely absent for y = 0. As expected, during the pre-recombinational epochs the 2s-1s-two-photon transition is notimportant. This is because the 2s-1s transition is simply unableto compete with the ∼108 times faster Lyman-α transition whilethe latter remains optically thin.

Summing the spectral distortions caused by the continua,one finds cancellation of the redshift-dependent emission at alevel close to our numerical accuracy (relative accuracy <∼10−4

for the spectrum). This is expected because of electron numberconservation: in the pre-recombinational epoch the overall ion-ization state of the plasma is not affected significantly by thesmall deviations of the background radiation from full equilib-rium. Therefore, all electrons that enter an atomic species willleave it again, in general via another route to the continuum.This implies that

∑i ΔRci = 0, which is a general property of

the solution in the pre-recombinational epoch. This can also beconcluded from Fig. 1, since in the δ-function representation the

http://dexter.edpsciences.org/applet.php?DOI=10.1051/0004-6361/200809840&pdf_id=1


Balmer-continuum line prior to the recombination epoch cancelsthe Lyman-continuum line at each redshift.

If we look at the Lyman- and Balmer-continuum in thecase of hydrogen, we can see that at redshifts z >∼ 30 000,electrons enter by the Lyman-continuum, and leave by theBalmer-continuum, while in the redshift range 2000 <∼ z <∼30 000 the opposite is true (see Sect. 5.1.1 for a detailed ex-planation). As expected, for z >∼ 2000 the Lyman-α transitionexactly follows the Balmer-continuum, since every electron thatenters the 2p-state and then reaches the ground level, also mustpass through the Lyman-α transition. Using the analytic solutionfor the Lyman-α line given in the Appendix B, we find excellentagreement with the numerical results until the true recombina-tion epoch is entered at z <∼ 2000.

In the case of He ii for the considered range of redshifts, thepre-recombinational emission (z >∼ 7000) is always generated inthe loop c → 2p → 1s → c. We again find excellent agreementwith the analytic solution for the Lyman-α line. We note thatfor He ii the total emission in the pre-recombinational epoch ismuch higher than in the recombination epoch at z ∼ 6000 (seediscussion in Sect. 4). The height of the maximum is even com-parable to the H i Lyman-α line.

As one can see in Fig. 1, at high redshift all transitions be-come weaker. This because the rest-frame frequencies of all linesare in the Rayleigh-Jeans part of the CMB spectrum, where theeffective chemical potential of the y-distortion (see Sect. 2.1) de-creases as4 μ(x) ≈ 2x y. This implies that at higher redshift alltransitions are more and more within a pure blackbody ambientradiation field. On the other hand, the effective chemical poten-tial increases towards lower redshift, so that also the strengthof the transitions increases. However, at z <∼ 3000 in the caseof hydrogen, and z <∼ 11 000 for He ii, the escape probabilityin the Lyman-continuum (see Appendix A.1 and Eq. (B.3) forquantitative estimates) begins to decrease significantly, so thatthe pre-recombinational transitions cease. For a 2-shell atom thisis because, the only loop that can operate is via the Lyman-continuum. The maximum in the pre-recombinational Lyman-αline forms because of this rather sharp transition to the opticallythick region of the Lyman-continuum (see also Sect. 5.1.1 formore details).

5.1.1. Analytic description of the pre-recombinationalLyman-α line

One can understand the behavior of the numerical solution forthe spectral distortions in more detail using our analytic descrip-tion of the Lyman-α line given in Appendix B.

In Fig. 2, we show the comparison of different analytic ap-proximations with the full numerical result. The curves labeled“analytic Ia” (dotted line) and “analytic Ib” (boxes) are bothbased on the formulae in Eqs. (B.2) and (B.7). In their deriva-tion, we assumed that the population of the 2p-state evolves un-der quasi-stationary conditions, where the free electron fractionand proton number density are given by the Recfast-solution.For the approximation “analytic Ia”, we did not include theescape probabilities in the H i Lyman-α line and H i Lyman-continuum, while the approximation “analytic Ib” also in-cludes the escape probabilities described in Appendix B.1.1.A comparison of these curves indicates that for the shape ofthe distortion at z <∼ 3000, the escape probabilities are very

4 Or more correctly μ(x) ≈ 7.4 x y if one also takes into account thedifference in the photon and electron temperature Te ≈ Tγ[1 + 5.4 y](see Appendix A.1).

2000 3000 4000 6000 104 30000 50000

z

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ΔIν

[ 1

0-26 J

m-2

s-1

Hz-1

sr-1

]

numerical solutionanalytic Iaanalytic Ibanalytic IIanalytic III

y = 10-5

nmax

= 2



Pre-Recombinational HI Lyman-α line

Fig. 2. Analytic representation of the pre-recombinational H i Lyman-αspectral distortions for the 2 shell atom and y = 10−5. See text for ex-planations. Note that at z >∼ 3000 the curves labeled “analytic Ia” and“analytic Ib” coincide.

important. However, although at this redshift the Sobolev opti-cal depth in the H i Lyman-α line is roughly a factor of 14 higherthan the optical depth in the H i Lyman-continuum, the deriva-tion of Eq. (B.8) shows that the H i Lyman-α escape probabilityplays only a secondary role in the pre-recombinational era.

Given the formulae in Appendix B.1.2, the spectral distortioncan be written in the form ΔIν(z) = F(z)×Δ. The factor F(z) de-scribes the normalization of the line (see Eq. (B.8)), also includ-ing the effect of the escape probabilities in the H i Lyman-α lineand continuum. One does not expect a strong change in the lineshape when using simple equilibrium values in its evaluation.For simplicity, we assume below that F is given by Eq. (B.10),which should be accurate to a level of ∼20% for the consideredredshift range.

The term, Δ, which is related to the imbalance of emis-sion and absorption, should allow us to understand when andwhy the Lyman-α line becomes negative. In the most radicalapproximation (see Appendix B.1.2 for details), one can useΔ ≈ μ21+μ2 pc−μ1 sc, as derived in Eq. (B.14b), so that we obtainthe curve labeled “analytic II”. One can clearly see that this ap-proximation represents the global behavior, but fails to explainthe Lyman-α absorption at z >∼ 30 000. For this approximation,the Lyman-α line should always be in emission, even at high red-shift, since with μ(x) ≈ 2yx[1− x2/12] in the limit of small x, wehave Δ ≈ 3 y x3

1sc/32 > 0. This approximation indicates that tolowest order the main reason for emission in the Lyman-α lineis the deviation of the effective chemical potential from zero atthe Lyman-α resonance, and the Lyman- and Balmer-continuumfrequency.

If we also account for the higher order terms of the line im-balance Δ according to Eq. (B.16), then we obtain the curvelabeled “analytic III”, which is close to the full solution andalso reproduces its high redshift behavior, starting at a slightlyhigher redshift (z ∼ 40 000 instead of z ∼ 30 000). This ismainly caused by the approximations in the integrals (B.15) overthe photoionization cross-sections (in particular M−1). However,if one evaluates these integrals more accurately, one does notrecover the full solution exactly, since the free-bound Gaunt-factors are neglected; only when these factors are included andwe evaluate the factor F correctly can we again obtain the fullnumerical result.



500 103

104 50000

z

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

ΔIν

[ 1

0-26 J

m-2

s-1

Hz-1

sr-1

]

HI Ly-αHI Ly-βHI Balmer-αHI Ly-cHI Balmer-cHI Paschen-cHI 2s-1s

y = 10-5

nmax

= 3



Lines from normalHI recombination

4000 104 50000

z

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

ΔIν

[ 1

0-26 J

m-2

s-1

Hz-1

sr-1

]

HeII Ly-αHeII Ly-βHeII Balmer-αHeII Ly-cHeII Balmer-cHeII Paschen-cHeII 2s-1s

y = 10-5

nmax

= 3


Lines from normalHeII recombination

Fig. 3. Spectral distortion, ΔIν, including 3 shells into our computationsfor different transitions as a function of redshift and for y = 10−5. Forall shown curves we used the δ-function approximation to compute theintensity. The upper panel shows the results for hydrogen, the lowerthose for He ii.

5.2. The 3-shell atom

If one considers 3 shells, the situation becomes more compli-cated, since more loops connecting to the continuum are pos-sible. Looking at Fig. 3, again we find that the sum over alltransition in the continua vanishes at redshifts prior to the truerecombination epoch of the considered species. At z <∼ 3000 inthe case of hydrogen and z <∼ 11 000 for He ii, the escape proba-bility in the Lyman-continuum becomes small. For 2 shells, thisstopped the pre-recombinational emission until the true recombi-nation epoch of the considered atomic species was entered (seeFig. 1). However, for 3 shells electrons can leave the 1s-levelvia the Lyman-β transition, and reach the continuum through theBalmer-continuum. For both hydrogen and He ii, one can alsosee that the emission in the Lyman-α line stops completely whenthe Lyman-continuum is fully blocked. In this situation, only theloop c → 3 → 2 → c via the Balmer-continuum operates. Onlywhen the main recombinational epoch of the considered speciesis entered, is the Lyman-α line reactivated.

In Fig. 4, we sketch the main atomic loops in hydrogen andHe iiwhen including 3 shells. For y = 10−5, in the case of hydro-gen the illustrated Lyman-continuum loops work in the redshiftrange 2000 <∼ z <∼ 30 000, while the Balmer-continuum loopworks for 1600 <∼ z <∼ 2000. In the case of He ii, one finds that

��

��

n=1

n=2

n=3

Continuum��

��

n=1

n=2

n=3

Continuum

Fig. 4. Sketch of the main atomic loops for hydrogen and He ii whenincluding 3 shells. The left panel shows the loops for transitions thatare terminating in the Lyman-continuum. The right panel shows thecase, when the Lyman-continuum is completely blocked, and unbal-anced transitions are terminating in the Balmer-continuum instead.

8000 <∼ z <∼ 1.2 × 105 and 6200 <∼ z <∼ 8000 for the Lyman-and Balmer-continuum loops, respectively. It is clear that in ev-ery closed loop, one energetic photons is destroyed and at leasttwo photons are generated at lower frequencies. Including moreshells will open the possibility of generating more photons perloop, simply because electrons can enter through highly excitedlevels and then preferentially cascade to the lowest shells viaseveral intermediate levels, leaving the atomic species taking thefastest available route back to the continuum. We discuss thissituation below in more detail (see Sect. 5.3.1).

Figures 1 and 3 both show that the pre-recombinational linesare emitted in a typical redshift rangeΔz/z ∼ 1, while the signalsfrom the considered recombinational epoch are released withinΔz/z ∼ 0.1−0.2. For the pre-recombinational signal, the ex-pected line-width is Δν/ν ∼ 0.6−0.7. However, the overlap ofseveral lines, especially at frequencies where emission and ab-sorption features nearly coincide, and the asymmetry of the pre-recombinational line profiles, still leads to more narrow spectralfeatures with Δν/ν ∼ 0.1−0.3 (see Sect. 5.3, Fig. 6).

We also note that in all cases the true recombination epochis not affected significantly by the small y-distortion in the am-bient photon field. There the deviations from Saha-equilibriumbecause of the recombination dynamics dominate over thosedirectly related to the spectral distortion, and in particular thechanges in the ionization history are tiny.

5.3. The 25-shell atom

In this section, we discuss the results for our 25-shell computa-tions. Given the large amount of transitions, it is more efficient tolook directly at the spectral distortion as a function of frequency.However, following the approach of Sect. 5.1, we checked thatthe basic properties of the first few lines and continua as a func-tion of redshift qualitatively do not change in comparison tothe previous cases. In particular, only the Lyman- and Balmer-continuum become strongly negative, again for similar redshiftranges as for 2 or 3 shells. In overall absorption, the other con-tinua (n > 2) play no important role, although about 10% of allloops do end there (see Sect. 5.3.1 for more details).

We note that for the spectral distortion now the impact ofelectron scattering must be considered and the free-bound con-tribution must be computed using the full differential, photoion-ization cross-section (see Appendix A.4).




1 10 100 1000 3000ν [ GHz ]

10-29

10-28

10-27

10-26

ΔIν

[J

m-2

s-1

Hz-1

sr-1

]

y = 10-5

y = 10-6

y = 10-7

y = 0

HI bound-bound-spectra

zin

= 40000n

max = 25

negative feature

1 10 100 1000 3000ν [ GHz ]

10-30

10-29

10-28

10-27

10-26

ΔIν

[J

m-2

s-1

Hz-1

sr-1

]

y = 10-5

y = 10-6

y = 10-7

y = 0

HeII bound-bound-spectra

zin

= 40000n

max = 25

negative feature

1 10 100 1000 3000ν [ GHz ]

10-29

10-28

10-27

10-26

ΔIν

[J

m-2

s-1

Hz-1

sr-1

]

y = 10-5

y = 10-6

y = 10-7

y = 0

HI free-bound-spectra

zin

= 40000n

max = 25

negative feature

1 10 100 1000 3000ν [ GHz ]

10-30

10-29

10-28

10-27

10-26

ΔIν

[J

m-2

s-1

Hz-1

sr-1

]

y = 10-5

y = 10-6

y = 10-7

y = 0

HeII free-bound-spectra

zin

= 40000n

max = 25

negative feature

1 10 100 1000 3000ν [ GHz ]

10-29

10-28

10-27

10-26

ΔIν

[J

m-2

s-1

Hz-1

sr-1

]

y = 10-5

y = 10-6

y = 10-7

y = 0

HI bb+fb-spectra

zin

= 40000n

max = 25

negative feature

1 10 100 1000 3000ν [ GHz ]

10-30

10-29

10-28

10-27

10-26

ΔIν

[J

m-2

s-1

Hz-1

sr-1

]

y = 10-5

y = 10-6

y = 10-7

y = 0

HeII bb+fb-spectra

zin

= 40000n

max = 25

negative feature

Fig. 5. Contributions to the H i (left panels) and He ii (right panels) recombination spectrum for different values of the initial y-parameter. Energyinjection was assumed to occur at zi = 4 × 104. In each column the upper panel shows the bound-bound signal, the middle the free-bound signal,and the lower panel the sum of both. Thin red lines represent the overall negative parts of the signals. Note that the 2s-1s-two-photon decaycontribution is not shown, since it does not lead to any significant pre-recombinational signal.

5.3.1. Dependence of the distortion on the value of y

In Fig. 5, we show the contributions to the recombination spec-trum for different values of the initial y-parameter. In addition,Fig. 6 shows some of the main contributions to the total hydro-gen and He ii spectral distortion in more detail.

Bound-bound transitions

Focusing first on the contributions of bound-bound transitions,one can see that the standard recombination signal due to hy-drogen is not strongly affected when y = 10−7, whereas the he-lium signal already changed notably. Increasing the value of yin both cases leads to an increase in the overall amplitude of the



500 600 700 800 900 1000 2000 3000ν [ GHz ]

-2

-1

0

1

2

ΔIν

[ 1

0-26 J

m-2

s-1

Hz-1

sr-1

]

total spectrumLyman-α-lineLyman-series for 3≤n≤10Balmer-series for 4≤n≤10Lyman-continuumBalmer-continuumsum of shown contributions

HI bb+fb-spectra

y = 10-5

nmax

= 25z

i = 40000

Ly-n

Lyc

Lyα

pre

pre

rec

pre

recHc

500 600 700 800 900 1000 2000 3000ν [ GHz ]

-2

-1

0

1

2

ΔIν

[ 1

0-26 J

m-2

s-1

Hz-1

sr-1

]

total spectrumLyman-α-lineLyman-series for 3≤n≤10Balmer-series for 4≤n≤10Lyman-continuumBalmer-continuumsum of shown contributions

HeII bb+fb-spectra

y = 10-5

nmax

= 25z

i = 40000

Ly-n

Lyc

Lyα

pre

pre

rec

pre

Hc

& Hn

100 200 300 400 500ν [ GHz ]

-1

-0.5

0

0.5

1

1.5

2

ΔIν

[ 1

0-26 J

m-2

s-1

Hz-1

sr-1

]

total spectrumLyman-α-lineBalmer-αBalmer-series for 4≤n≤10Paschen-αLyman-continuumBalmer-continuumsum of shown contributions

HI bb+fb-spectra

y = 10-5

nmax

= 25z

i = 40000

Hα

pre

Pα

prerec

100 200 300 400 500ν [ GHz ]

-1

-0.5

0

0.5

1

1.5

2

ΔIν

[ 1

0-26 J

m-2

s-1

Hz-1

sr-1

]

total spectrumLyman-α-lineBalmer-αBalmer-series for 4≤n≤10Paschen-αLyman-continuumBalmer-continuumsum of shown contributions

HeII bb+fb-spectra

y = 10-5

nmax

= 25z

i = 40000

Hα

pre

Pα

pre

rec

1 2 3 4 5 6 10 20ν [ GHz ]

10-30

10-29

10-28

10-27

ΔIν

[ J

m-2

s-1

Hz-1

sr-1

]

total spectrumn → 10n → 9 + n → 10 + n → 11 HI bb+fb-spectra

y = 10-5

nmax

= 25z

i = 40000

11 → 10

12 → 10

13 → 10

rec

rec

recpre

pre

pre

1 2 3 4 5 6 10 20ν [ GHz ]

10-30

10-29

10-28

10-27

ΔIν

[ J

m-2

s-1

Hz-1

sr-1

]

total spectrumn → 10n → 9 + n → 10 + n → 11 HeII bb+fb-spectra

y = 10-5

nmax

= 25z

i = 40000

11 → 10

12 → 10

13 → 10

rec

rec

rec

pre

pre

pre

Fig. 6. Main contributions to the H i (left panels) and He ii (right panels) spectral distortion at different frequencies for energy injection at zi =40 000 and y = 10−5. We have also marked those peaks coming (mainly) from the recombination epoch (“rec”) and from the pre-recombinationepoch (“pre”) of the considered atomic species. Note that the 2s-1s-two-photon decay contribution is not shown, since it does not lead to anysignificant pre-recombinational signal.

distortion at low frequencies, and a large rise in the emission andvariability at ν >∼ 100 GHz. For H i at low frequencies, the levelof the signal increased roughly by a factor of 5 when increasingthe value of y from 0 to 10−5, while for He ii the increase is abouta factor of 40. This indicates that in the pre-recombinational

epoch He ii indeed behaves in a way similar to hydrogen, butwith an amplification ∼8 (see Sect. 4).

At high frequencies, a strong emission-absorption featureappears in the range ν ∼ 500−1600 GHz, which is completelyabsent for y = 0. For y = 10−5 from peak to peak, this feature



exceeds the normal Lyman-α distortion (close to ν ∼ 1750 GHzfor H i and ν ∼ 1680 GHz for He ii) by a factor of ∼5 for H i, andabout 30 for He ii. The absorption part is caused mainly by thepre-recombinational Lyman-β, -γ, and -δ transitions, while theemission part is dominated by the pre-recombinational emissionin the Lyman-α line (also see Fig. 6 for more detail).

We note that in the case of He ii most of the recombina-tional Lyman-α emission (ν ∼ 1750 GHz) is completely wipedout by the pre-recombinational absorption in the higher Lyman-series, while for H i only a small part of the Lyman-α lowfrequency wing is affected. This is possible only because thepre-recombinational emission in the He ii Lyman-series is sostrongly enhanced, compared to the signal produced during therecombinational epoch.

Free-bound transitions

Considering the free-bound contributions, one can again see thatthe hydrogen signal changes much less with increasing value ofy than in the case of He ii. In both cases, the variability in thefree-bound signal decreases at low frequencies, while at high fre-quencies a strong and broad absorption feature appears, which ismainly due to the Lyman-continuum. For y = 10−5, this absorp-tion feature even completely erases the Balmer-continuum con-tribution appearing during the true recombination epoch of theconsidered species. It is 2 times stronger than the H i Lyman-αline from the recombination epoch, and in the case of He ii itexceeds the normal He ii Lyman-α line by more than one orderof magnitude.

However, apart from the absorption feature at high frequen-cies the free-bound contribution becomes practically feature-less when reaching y = 10−5. This is because of the strongoverlap of different lines from the high redshift part, since thecharacteristic width of the recombinational emission increasesas Δν/ν ∼ kTe/hνic (see middle panels in Fig. 5). In addition,the photons are released in a broader range of redshifts (seeSects. 5.1 and 5.2), leading to a decrease in the contrast betweenthe spectral features from the recombinational epoch.

Total distortion

In the total spectra (see lower panels in Fig. 5), one can alsoclearly see a strong absorption feature at high frequencies, whichis mainly associated with the Lyman-continuum and Lyman-series for n > 2 (see Fig. 6 also). For y = 10−5, in the caseof hydrogen it exceeds the Lyman-α line from recombination(ν ∼ 1750 GHz) by a factor of ∼4 at ν ∼ 1250 GHz, while forHe ii it is even stronger by a factor of ∼20, reaching ∼80% of thecorresponding hydrogen feature. Checking the level of emissionat low frequencies, as expected (see Sect. 4), one can find thatHe ii indeed contributes about 2/3 to the total level of emission.

As illustrated in the upper panels of Fig. 6, the emission-absorption feature at high frequencies is caused by the overlapbetween the pre-recombinational Lyman-α line (emission) andthe combination of the higher pre-recombinational Lyman-seriesand Lyman-continuum (absorption). At intermediate frequen-cies (middle panels), the main spectral features are due to theBalmer-α, the pre-recombinational Balmer-series from n > 3,and the Paschen-α transition, with some additional broad con-tributions to the overall amplitude of the bump originating inhigher continua.

The lower panels of Fig. 6 show, as an example, the sepa-rate contributions to the bound-bound series for the 10th shell.

One can notice that for hydrogen the recombinational andpre-recombinational emission are of similar amplitude, whilefor He ii the pre-recombinational signal is more than one or-der of magnitude stronger (see Sect. 4 for an explanation). Inboth cases, the pre-recombinational emission is much broaderthan the recombinational signal, again mainly due to the time-dependence of the photon emission process (see Sects. 5.1and 5.2), but to some extent also because of electron scattering.

Number of additional photons and loops

Using the free-bound spectrum, one can also estimate the effec-tive number of loops5, Nloop, involved in the production of allphotons seen as additional CMB spectral distortion. The easiestway to compute this number is to consider the free-bound spec-trum of each level using the δ-function approximation. In thisway, one avoids the overlap between the emission and absorp-tion contributions at different redshifts, which would lead to asmall underestimation of Nloop. In the case of full equilibrium,one should find that Nloop = 0.

By computing the number of photons that can be seen asoverall free-bound emission, one obtains the number of times anelectron has entered an uncompensated loop via the consideredlevel. On the other hand, by computing the number of photonsthat can be seen as overall bound-free absorption, one can de-termine the number of times that an uncompensated transitionended at that level. By evaluating the sum over the number ofphotons that can be seen as overall free-bound emission from alllevels, one should find Nloop + 1 photons per nucleus. Similarly,the total number of photons absorbed in all bound-free transi-tions should beNloop photons per nucleus. The difference of ∼1γper nucleus is due the fact that at the end of the recombinationalepoch of each atomic species practically all electrons have beencaptured, without returning to the continuum afterwards. In asimilar way, one can also identify the partial contribution of eachcontinuum.

In Table 1, we provide a few examples, comparing also withthe number of photons emitted for y = 0. One can see that the ef-fective number of loops per nucleus varies roughly in proportionto the values of y, i.e.,NH i

loop ∼ 3.3 × [y/10−5] and NHe iiloop ∼ 31 ×

[y/10−5]. If one considers a lower injection redshift, the pro-portionality constant should decrease, since the loops will beactive over a shorter period. When also including more shells,Nloop should increase because there are more channels throughwhich the electrons can enter the atoms. Furthermore, the effec-tive number of loops per nucleus is about one order of magnitudelarger for He ii than for hydrogen. As explained in Sect. 4, this iscaused by the amplification of transitions for hydrogenic heliumat high redshift.

When comparing with the number of photons absorbed inthe Lyman-continuum, one can also see that in practically allshown cases ∼90% of all loops should end there. By com-puting this more carefully for hydrogen assuming y = 10−5,we found that ∼85% of the pre-recombinational loops end inthe Lyman-continuum, about 4.2% in the Balmer-continuum,∼0.4% in the Paschen-continuum, and the remaining ∼10.4% isdistributed more or less evenly over all the other continua, withno strong drop toward higher shells. For helium, we found simi-lar numbers.

5 This number can be non-integer, since in general only a fraction ofall electrons that are captured by the nuclei really run through uncom-pensated loops during the pre-recombinational epoch.


Table 1. Approximate number of photons and loops per nucleus of theconsidered species for zi = 4 × 104 and different values of y.

y = 0 y = 10−7 y = 10−6 y = 10−5

NH iLy−c 0 −0.028 −0.28 −2.75

NH iLy−α 0.42 0.47 0.92 5.37

NH ibb 2.49 2.59 3.45 12.03

NH iloop 0 0.032 0.33 3.27

NHe iiLy−c 0 −0.27 −2.67 −26.6

NHe iiLy−α 0.55 1.06 5.68 51.8

NHe iibb 2.48 3.49 12.5 103

NHe iiloop 0 0.30 3.06 31.0

1 10 100 1000 3000ν [ GHz ]

-2

-1

0

1

2

ΔIν

[ 1

0-26 J

m-2

s-1

Hz-1

sr-1

]

z = 40000z = 10000z = 8000z = 6000z = 4000z = 3000Signal for y = 0

HI bb+fb-spectra

y = 10-5

nmax

= 25

1 10 100 1000 3000ν [ GHz ]

-2

-1

0

1

2

ΔIν

[ 1

0-26 J

m-2

s-1

Hz-1

sr-1

]

z = 40000z = 20000z = 15000z = 10000z = 12000z = 8000Signal for y = 0

HeII bb+fb-spectra

y = 10-5

nmax

= 25

Fig. 7. H i (upper panel) and He ii (lower panel) recombination spectrafor different energy injection redshifts.

If we considered the total number of photons per nucleusemitted in the bound-bound transitions and subtracted the num-ber of photons emitted for y = 0, we were able to estimate theloop-efficiency, εloop, or number of bound-bound photons gen-erated per loop prior to the recombination epoch. For hydrogen,one finds εloop ∼ 2.9−3.0, while for He ii one has εloop ∼ 3.2−3.3.Similarly, one obtains a loop efficiency of εloop ∼ 1.7−1.8for both the H i and He ii Lyman-α lines. As expected thesenumbers are rather independent of the value of y, since theyshould reflect an atomic property. They should also be rather

1 10 50ν [ GHz ]

0

0.2

0.4

0.6

0.8

1

ΔIν

[ 1

0-26 J

m-2

s-1

Hz-1

sr-1

]

z = 40000z = 15000z = 10000z = 8000z = 4000Signal for y = 0

HI + HeII bb+fb-spectra

y = 10-5

nmax

= 25

50 100 1000 3000ν [ GHz ]

-4

-2

0

2

4

ΔIν

[ 1

0-26 J

m-2

s-1

Hz-1

sr-1

]

z = 40000z = 15000z = 10000z = 8000z = 4000Signal for y = 0


y = 10-5

nmax

= 25

Fig. 8. Total H i + He ii recombination spectra for different energy in-jection redshifts. The upper panel shows details of the spectrum at low,the lower at high frequencies.

independent of the injection redshift, which mainly affects thetotal number of loops and thereby the total number of emittedphotons. However, the loop efficiency should still increase whenincluding more shells in the computation.

5.3.2. Dependence of the distortion on the redshift of energyinjection

To understand how the pre-recombinational emission dependson the redshift at which the energy was released, in Fig. 7we show a compilation of different cases for the total H i andHe ii signal. In Fig. 8, we also present the sum of both in moredetail.

Features at high frequencies

For all shown cases, the absolute changes in the curves arestrongest at high frequencies (ν >∼ 100 GHz). One can alsoidentify a rather broad bump at 100 GHz <∼ ν <∼ 400 GHz, fol-lowed by an emission-absorption feature in the frequency range500 GHz <∼ ν <∼ 1600 GHz. In particular the strength and posi-tion of this emission-absorption feature depends strongly on theredshift of energy injection.




1 10 50ν [ GHz ]

-40

-30

-20

-10

0

10

20

30

40

ΔT [

nK

]

zi = 4000

Signal for y = 0

HI & HeII bb+fb-spectra

y = 10-5

nmax

= 25

1 10 50ν [ GHz ]

-15

-10

-5

0

5

10

15

ΔT [

nK

]

zi = 8000

Signal for y = 0


y = 10-5

nmax

= 25

1 10 50ν [ GHz ]

-15

-10

-5

0

5

10

15

ΔT [

nK

]

zi = 15000

Signal for y = 0


y = 10-5

nmax

= 25

1 10 50ν [ GHz ]

-15

-10

-5

0

5

10

15

ΔT [

nK

]

zi = 40000

Signal for y = 0


y = 10-5

nmax

= 25

Fig. 9. Comparison of the variable component in the H i + He ii bound-bound and free-bound recombination spectra for single energy injection atdifferent redshifts. In all cases the computations were performed including 25 shells and y = 10−5. The blue dashed curve in all panel shows thevariability in the normal H i + He ii recombination spectrum (equivalent to energy injection below z ∼ 800 or y = 0) for comparison.

For the broad high-frequency signature, it is more importantthat the variability itself varies, rather than the overall amplitudeincreases. For example, in the frequency range 100 GHz <∼ ν <∼400 GHz, the normal recombinational signal has ∼2 spectral fea-tures, while for injection at zi ∼ 4000 approximately 4 featuresare visible, which in this case only come from hydrogen becauseat z ∼ 4000 He ii is already completely recombined. We alsonote that neutral helium should add some signal, which was notincluded here. Nevertheless, we expect that this contribution isnot strongly amplified as in the case of He ii (see Sect. 4), andhence should not increase the total signal by more than 10−20%.

Variability at low frequencies

Focusing on the spectral distortions at low frequencies, the over-all level of the distortion in general increases for higher redshiftsof energy release. However, there is also some change in thevariability of the spectral distortions. To study this variability inmore detail, in each case we performed a smooth spline fit ofthe total H i + He ii recombination spectrum and then subtractedthis curve from the total spectrum. The remaining modulationof the CMB intensity was then converted into variations in the

CMB brightness temperature using the relation ΔT/T0 ≈ ΔI/Bν,where Bν is the blackbody intensity at temperature T0 = 2.725 K.

Figure 9 shows the results of this procedure in several cases.It is most striking that the amplitude of the variable componentdecreases with increasing energy injection redshift. This can beunderstood as follows: we have seen in Sects. 5.1 and 5.2 thatfor very early energy injection most of the pre-recombinationalemission is expected to be produced at z ∼ 3000 for hydro-gen, and z ∼ 11 000 for helium, (i.e., the redshifts at which theLyman-continuum of the considered atomic species becomes op-tically thick) with a typical line-width Δν/ν ∼ 1. In this case, thetotal variability of the signal is mainly due to the non-trivial su-perposition of many broad neighboring spectral features. Mostimportantly, little variability will be added by the high redshiftwing of the pre-recombinational lines and in particular the be-ginning of the injection process. This is because (i) at high z theemission is much smaller (cf. Figs. 1 and 3); and (ii) electronscattering broadens lines significantly, smoothing any broad fea-ture even more (see Appendix A.4).

On the other hand, when the energy injection occurs atlower redshift, this increases the variability of the signal be-cause (i) electron scattering in the case of single momentaryenergy release does not smooth the step-like feature due to the



1 10 50ν [ GHz ]

-15

-10

-5

0

5

10

15

ΔT [

nK

]

nmax

= 100

nmax

= 25

HI & HeII bb+fb-spectray = 0

Fig. 10. Comparison of the variable component in the standard (y = 0)H i + He ii bound-bound and free-bound recombination spectrum fornmax = 100 and 25.

beginning of the injection process as strongly; and (ii) the totalemission amplitude and hence the step-like feature increases (seeFigs. 1 and 3). When the injection occurs close to the redshift atwhich the Lyman-continuum is optically thick (see Sect. 5.1),i.e., where the pre-recombinational emission has an extremum,this should produce a strong increase in the variability. On theother hand, for energy injection well before this epoch the atomictransitions produce an increase in the overall amplitude of thedistortions rather than the variability.

This can also be seen in Fig. 9, where for zi = 4000 the vari-able component is ∼3−8 times larger than the normal recom-binational signal with a peak-to-peak amplitude of ∼50−70 nKinstead of ∼10−15 nK at frequencies around ∼1.5 GHz. Evenfor zi >∼ 15 000, the amplitude of the variable component isstill 1.5−2 times larger than in the case of standard recombi-nation, but it practically does not change anymore when goingto higher injection redshifts. For zi ∼ 11 000, one expects a sim-ilarly strong increase in the variability as for zi ∼ 4000, but thistime due to He ii. In addition to the change in amplitude of thevariable component, in all cases the signal is shifted with re-spect to the normal recombinational signal. These shifts shouldalso make it easier to distinguish between the spectral signaturesfrom pre-recombinational energy release and those produced bynormal recombination.

It is important to mention that the total amplitude of the vari-able component should still increase when more shells are in-cluded in the computation. As shown in Chluba et al. (2007a),for y = 0 in particular the overall level of recombinational emis-sion at low frequencies depends strongly on the completeness ofthe atomic model. Similarly, the variability in the recombinationspectrum changes. This is illustrated in Fig. 10, where we com-pare the variability in the H i + He ii recombination spectrum for25 shells (y = 0), with that obtained in our 100-shell computa-tions (Chluba et al. 2007a; Rubiño-Martín et al. 2008). As onecan see, at low frequencies (ν ∼ 1−3 GHz) the amplitude of thevariable component increases by more than a factor of 2 when100 shells are included, reaching a peak-to-peak amplitude of∼30 nK. This is due to the fact that for a more complete atomicmodel additional electrons are able to pass through a particulartransition between highly excited states.

1 10 50ν [ GHz ]

-40

-30

-20

-10

0

10

20

30

40

ΔT [

nK

]

zi = 4000

zX

= 4000


y = 10-5

nmax

= 25

1 10 50ν [ GHz ]

-15

-10

-5

0

5

10

15

ΔT [

nK

]

zi = 8000

zX

= 8000


y = 10-5

nmax

= 25

Fig. 11. Comparison of the variable component in the H i +He ii bound-bound and free-bound recombination spectra for single energy injection(black solid curves) and energy injection due to long-lived decayingparticles with different lifetimes (red dashed-dotted curves). In all casesthe computations were performed including 25 shells and a maximaly-parameter y = 10−5.

5.3.3. Dependence of the distortion on the energy injectionhistory

Until now we have only considered cases of single momen-tary energy injection. However, physically this may not be veryrealistic, since most of the possible injection mechanisms re-lease energy over a broader range of redshifts. The discussion inSect. 5.3.2 has also shown that for single injection a large part ofthe variability can be attributed to the onset of the energy release.Therefore it is important to investigate the potential signatures ofother injection mechanisms.

For the signals under discussion, long-lived, decaying parti-cles are the most interesting. In Sect. 2.1.1, we have given somesimplified analytic description of this problem. In Fig. 11, weshow the variable component for the CMB spectral distortiondue to the presence of hydrogen and He ii at low frequencies,for both single injection and energy release from long-lived de-caying particles. For zX = 4000, it is clear that the variability issignificantly smaller than in the case of single energy injection.This is because the onset of the atomic transitions is much moregradual than in the case of single injection. However, one shouldmention that for energy injection due to decaying particles the




effective y-parameter at z = 4000 remains only ∼1/3 of its max-imal value, so that the level of variability cannot be compared di-rectly with the case of single energy injection. Nevertheless, thestructure of the variable component still depends non-triviallyon the effective decay redshift, so that one in principle should beable to distinguish between the different injection scenarios.

Similarly, one could consider the case of annihilating parti-cles. However, here energy is effectively released at higher red-shift6 and also over a much broader redshift interval. In this case,one has to follow the evolution in the CMB spectrum causedby this heating mechanism from an initial μ-type distortion to apartial y-type distortion in more detail. One can also expect thatthe redistribution of photons via electron scattering will becomemuch more important (see Appendix A.4), and that the free-freeprocess will strongly alter the number of photons emitted viaatomic transitions (see Appendix A.5). In addition, collisionalprocesses may become significant, in particular those leading totransitions among different bound-bound levels, or to the contin-uum, since they are not associated with the emission of photons.This problem will be considered in a future work.

6. Discussion and conclusions

In the previous sections, we have shown in detail how intrinsicy-type CMB spectral distortions modify the radiation releasedby atomic transitions in primordial hydrogen and He ii at highredshift. We presented the results of numerical computations in-cluding 25 atomic shells for both H i and He ii, and discussed thecontributions of several individual transitions in detail (e.g., seeFig. 6), by also taking the broadening of lines due to electronscattering into account. As examples, we investigated the caseof instantaneous energy release (Sect. 5.3.2) and exponential en-ergy release (Sect. 5.3.3) due to long-lived, decaying particles,separately.

Our computations indecated that several additional photonsare released during the pre-recombinational epoch, which interms of number can strongly exceed those from the recombina-tional epoch (see Sect. 5.3.1). The number of loops per nucleusscales roughly in proportion to the values of y, i.e.,NH i

loop ∼ 3.3 ×[y/10−5] and NHe ii

loop ∼ 31 × [y/10−5] for hydrogen and He ii, re-spectively, where about 3 photons per loop are effectively emit-ted in the bound-bound transition.

Because of the non-trivial overlap of broad neighboring pre-recombinational lines (from bound-bound and free-bound tran-sitions), rather narrow (Δν/ν ∼ 0.1−0.3) spectral features ap-pear on top of a broad continuum, which both in shape andamplitude depend on the time-dependence of the energy injec-tion process and the value of the intrinsic y-type CMB distor-tion. At high frequencies (ν ∼ 500−1600 GHz), an emission-absorption feature forms, which is completely absent for y = 0,and is mainly due to the superposition of pre-recombinationalemission in the Lyman-α line, and the higher Lyman-series andLyman continuum.

Looking at Fig. 12, it becomes clear that this absorp-tion feature (close to ν ∼ 1400 GHz) in all showncases even exceeds the intrinsic y-distortion. For y =10−5, it even reaches ∼10% of the CMB blackbody inten-sity. Unfortunately, it appears in the far Wien-tail of theCMB spectrum, where the cosmic infrared forming galaxies is

6 This conclusion depends also on the temperature/energy dependenceof the annihilation cross-section. We assumed s-wave annihilation (e.g.see McDonald et al. 2001).

1 10 100 1000 3000ν [ GHz ]

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

ΔIν /

Bν

y = 10-5

y = 10-6

y = 10-7

y = 0


zin

= 40000

nmax

= 25

negative feature

possible level of emission for n=100

y = 10-5

y = 10-6

y = 10-7

Fig. 12. Spectral distortions relative to the CMB blackbody spectrum,Bν. The thin blue curves show the absolute value of a y-type distortion.At low frequencies we indicate the expected level of emission whenincluding more shells in our computations.

dominant (Fixsen et al. 1998; Lagache et al. 2005). Still one mayhope to be able to extract such spectral features in the future.

One should emphasize that all the discussed additional pre-recombinational spectral distortions are in general small in com-parison to the intrinsic y-distortion. As Fig. 12 shows, the ampli-tude of the additional distortions is typically well below 1% ofthe CMB y-distortion. However, at low frequencies (ν ∼ 1 GHz)the additional distortions should reach ∼10% of the intrinsicy-distortion, and at even lower frequency may also exceed it. Butin this context, it is more important that due to the processes dis-cussed here new narrow spectral features appear, of an uniquevariability (e.g. see Fig. 9), even stronger than that of the re-combinational lines from standard recombination (y = 0). Thisvariability is very hard to mimic by any astrophysical foregroundor systematic problem with the instruments. As emphasized ear-lier for the cosmological recombination spectrum (Sunyaev &Chluba 2008, 2007), this may allow us to measure them dif-ferentially, also making use of the fact that the same signal iscoming from practically all directions on the sky. For intrinsicy-distortions, direct differential measurements are much harder,since its spectrum is so broad. Furthermore, as pointed out in theintroduction, by measuring the narrow spectral features underdiscussion one could in principle distinguish between pre- andpost-recombinational energy release, an observation that cannotbe easily achieved by measuring directly the average y-distortionof the CMB.

We also note that in addition to the average y-parameterone could include possible knowledge on the angular depen-dence of the CMB spectrum. The y-distortion from resolvedSZ-clusters is a trivial example and their signals can certainlybe removed. However, the contributions from the warm-hot-interstellar-medium should be more uniform and should varyat very different angular scales. The presence of halos of an-nihilating (∝N2

dm) or decaying (∝Ndm) dark matter at high red-shifts (z >∼ 20) will also lead to a slowly varying, angular-dependent, post-recombinational y-distortion. Because of theirdifferent dependencies on the dark matter number density Ndm,these contributions will influence different angular scales at dif-ferent redshifts. This angular dependence in principle wouldprovide another way to separate pre- and post-recombinationalenergy release. However, one should also consider that the



pre-recombinational energy release may not be uniform, sincefor example, phase transitions in different parts of the Universeoccur at different redshifts. This makes the problem more in-volved and a detailed analysis of this possibility is beyond thescope of this paper.

We have also pointed out that there is no principle difficultyin computing the spectral distortions due to pre-recombinationalatomic transitions in H i and He ii for more general energy injec-tion histories, if required. In particular very early injection, in-volving μ-type distortions, may be interesting, since stimulatedemission could strongly amplify the emission at low frequen-cies and hence the total number of emitted photons per atom.However, to treat this problem one has to follow the detailedevolution in the CMB spectral distortion produced by the injec-tion process (see e.g. Hu & Silk 1993a). The effect of electronscattering on the distortions caused by the pre-recombinationalatomic transitions, in particular because of the recoil effect, mustalso be treated more rigorously. Simple estimates also show (seeAppendix A.5) that at low frequencies the modifications due tofree-free absorption becomes significant. Furthermore, one mustaccount for collisional processes, since they should become im-portant at very high redshift, even for shells with low n.

An additional difficulty arises because at both very lowand very high frequencies, the back-reaction of the pre-recombinational distortion on the ambient radiation field maynot be negligible (see Fig. 12). This may also affect the detailsof the results presented here, although our main conclusionsshould not change. The inter-species feedback and reprocess-ing of photons (e.g. γHe ii → He i and γHe → H i) may alsoproduce some differences, close to the beginning of the pre-recombinational epochs of each atomic species. Furthermore,for more accurate predictions of the positions of the narrow fea-tures one should account for the background-induced, stimulatedelectron-scattering, which was discussed by Chluba & Sunyaev(2008a). We defer all of these problems to another paper.

Acknowledgements. The authors wish to thank J. A. Rubiño-Martín for usefuldiscussions. We also wish to thank the referee for his comments and sugges-tions. In addition, we are grateful for discussions on experimental possibili-ties with J. E. Carlstrom, D. J. Fixsen, A. Kogut, L. Page, M. Pospieszalski,A. Readhead, E. J. Wollack and especially J. C. Mather. We would also like tothank J. Carlstrom and J. Ostriker for their comments and interest in the problem,and C. Thompson for additional discussion on superconducting strings. Also RSis very glad that he had the chance to work with Y. E Lyubarsky on the earlyideas related to this problem.

Appendix A: Computational details

In this section, we outline the most important changes that hadto be made to our multi-level code (Chluba et al. 2007a) to allowfor an intrinsic y-type CMB spectral distortion. We also mentionseveral approximations that can be used to ease the numericalintegration of the coupled system of rate equations for the dif-ferent populations of electrons in the energy levels of hydrogenand helium.

A.1. Formulation of the problem

Details about the formulation of the cosmological recombinationproblem for a pure blackbody CMB radiation field are given inSeager et al. (2000), Rubiño-Martín et al. (2006), and referencestherein. We shall use the same notation as in Rubiño-Martínet al. (2006). The main difference with respect to the stan-dard recombination computation is caused by the possibility ofa non-blackbody ambient radiation field, which affects the net

bound-bound and free-bound rates as explained in Sect. 3. Thetemperature of the electrons in general is also no longer equalto the effective temperature of the photons, as we discuss below(see Sect. A.3). Since we consider only small intrinsic spectraldistortions, all the modification to the solution for the level pop-ulations are rather small, and most of the differences will appearonly as pre-recombinational emission due to atomic transitions,but with practically no net effect on the ionization history.

One additional modification is related to the Lyman-continuum. As was realized earlier (Zeldovich et al. 1968;Peebles 1968), during the recombination epochs photons can-not escape from the Lyman-continuum. However, at high red-shift the number of neutral atoms is very small, so the Lyman-continuum becomes optically thin. To include the escape ofphotons in the Lyman-continuum, we follow the analytic de-scription of Chluba & Sunyaev (2007), in which an approxi-mation of the escape probability in the Lyman-continuum wasgiven by

PLy−cesc (z) ≈ 1

1 + τescc, (A.1)

with τescc =

cσ1sc N1s

HkBTehνc

. Here σ1sc is the threshold photoioniza-tion cross-section of the 1s-state, N1s is the number density ofatoms in the ground state, and νc is the threshold frequency. Fora standard cosmology, the H i Lyman-continuum becomes opti-cally thin above z ∼ 3000−4000, while for He ii this occurs atz >∼ 12 000−16 000. As our computations show, it is crucial toinclude this process, since at high redshift almost all loops beginor terminate in the Lyman-continuum (see Sect. 5).

A.2. High redshift solution

At high redshift, well before the true recombination epoch of theconsidered atomic species, one can simplify the problem by re-alizing that the ionization degree does not change significantly.Although the inclusion of intrinsic CMB spectral distortion pro-duces some small changes in the populations with respect to theSaha values, the total number of electrons captured by protonsand helium nuclei is tiny compared to the total number of freeelectrons. Therefore, one can neglect the evolution equation forthe electrons, until the true recombination epoch is entered. ForH i, we used this simplification until z ∼ 3500, while for He iiwefollow the full system below z ∼ 20 000. Before we simply usedthe Recfast-solution for Ne (Seager et al. 1999, 2000). In sev-eral different cases, we checked that these settings do not affectthe spectra.

Furthermore, we note that at high redshift for n > 2 we usedthe variable ΔNi = Ni − N2s

i instead of Ni, since ΔNi/Ni be-comes so small. Here N2s

i is the expected population of level i inBoltzmann-equilibrium relative to the 2s-level. We then changedback to the variable Ni at sufficiently low redshifts.

A.3. Recombination and photoionization rates

The computation of the photoionization and recombination ratesfor many levels is rather time-consuming. In an earlier versionof our code (Chluba et al. 2007a), we tabulated the recombina-tion rates for all levels before the actual computation and useddetailed balance to infer the photoionization rates. This treat-ment is possible as long as the photon and electron temperaturedo not depart significantly from each other, and when the back-ground spectrum is given by a blackbody. Here we now gener-alize this procedure accounting for the small difference between


the electron and photon temperatures, in particular at low red-shift (z <∼ 800), and allowing for non-blackbody ambient photondistributions.

At high redshift (z >∼ 3000), the electron temperature is al-ways equal to the Compton equilibrium temperature (Zeldovich& Sunyaev 1969):

T eqe =

kBTγh

∫x4nγ(x) dx

4∫

x3nγ(x) dx(A.2)

within the given ambient radiation field. Because of the ex-tremely high specific entropy of the Universe (there are ∼1.6 ×109 photons per baryon), this temperature is reached on a muchshorter timescale than the redistribution of photons via Comptonscattering requires. For a μ-type distortion, T eq

e is always close tothe effective photon temperature, whereas for a y-type distortionwith y � 1 one has Te ≈ Tγ[1 + 5.4 y] (Illarionov & Syunyaev1975a). This simplifies matters, since there is no need to solvethe electron temperature evolution equation, and the photoion-ization and recombination rates can therefore be precalculated.

At redshifts below z ∼ 3000, we solve for the electron tem-perature accounting for the non-blackbody ambient radiationfield. In this case, the photon temperature inside the term dueto the Compton interaction must be replaced by T eq

e as given byEq. (A.2), such that the temperature evolution equation reads

∂Te

∂z=

κC T 4γ

H(z)[1 + z]Xe

1 + fHe + Xe[Te − T eq

e ] +2Te

1 + z, (A.3)

where κC = 4.91 × 10−22 s−1 K−4.Since for small intrinsic CMB spectral distortions, the cor-

rection to the solution for the temperature of the electrons israther small, it is always possible to use the standard Recfastsolution for Te as a reference. Tabulating both the photoion-ization and recombination rates, and their first derivatives withrespect to the ratio of the electron to photon temperature ρ =Te/Tγ, it is possible to approximate the exact rates to high accu-racy using first-order Taylor polynomials. To save memory, weonly consider all these rates in some range of redshifts aroundthe current point in the evolution and then update them fromtime to time. At high redshift, we typically used 200 points perdecade in logarithmic spacing. At low redshift (z <∼ 5000), weuse 2 points per Δz = 1. Another improvement can be achievedby rescaling the reference solution for Te with the true solutionwhenever the tabulated rates are updated. With these settings,we found excellent agreement with the full computation but atsignificantly lower computational cost.

A.4. Inclusion of electron scattering

As mentioned by Dubrovich & Stolyarov (1997) and shown inmore detail by Rubiño-Martín et al. (2008), the broadening dueto the scattering of photons by free electrons must be included inthe computation of the He ii recombination spectrum. Similarly,one has to account for this effect, when computing the spec-tral distortions arising from higher redshifts. Here we only con-sider redshifts z <∼ 5 × 104, and hence the electron scatteringCompton-y-parameter7

ye(z) =∫ z

0

kTe

mec2

c NeσT

H(z′)(1 + z′)dz′ ≈ 4.8 × 10−11 [1 + z]2 (A.4)

7 Note that ye differs from y as defined in Eq. (1), since it describes theredistribution of some photon over frequency because of electron scat-tering rather than the global energy exchange with the ambient black-body radiation field.

is lower than ∼0.12, so that the line-broadening due to theDoppler effect is significant (Δν/ν|Doppler ∼ 0.58), but still mod-erate in comparison with the width of the quasi-continuous spec-tral features produced at high redshift (see Sects. 5.1 and 5.2).However, already at z <∼ 2.5 × 104 one has ye <∼ 0.03, such thatΔν/ν|Doppler <∼ 0.29.

Regarding the line-shifts caused by the recoil effect, onefinds that they are not very important, since even for theH i Lyman-continuum one has Δν/ν|recoil <∼ −0.14 at z <∼ 5× 104.Although for the He ii Lyman-continuum the shifts due to therecoil-effect is a factor of four higher, we shall not include it inour results. One therefore expects that at frequencies ν >∼ 57 GHzthe presented distortions may still be modified due to this pro-cess, but we will consider this problem in a future paper.

For the bound-bound spectrum we follow the procedure de-scribed in Rubiño-Martín et al. (2008), where the resulting spec-tral distortion at observing frequency ν for one particular transi-tion is given by (see also Zeldovich & Sunyaev 1969)

ΔIi j(ν)∣∣∣Doppler

=

∫ν3

ν30

ΔIi j(ν0)√4πye

× e− (ln[ν/ν0]+3ye)2

4yedν0ν0· (A.5)

Here ΔIi j(ν0) denotes the spectral distortion for the consideredtransition evaluated at frequency ν0 and computed without theinclusion of electron scattering (e.g., see Rubiño-Martín et al.2006), but accounting for the non-blackbody ambient radiationfield. We note that ye(z) has to be calculated starting at theemission redshift zem = νi j/ν0 − 1, where νi j is the transitionfrequency.

For the spectral distortion resulting from the free-bound tran-sitions, one in addition has to include the frequency-dependenceof the photoionization cross-section. We shall neglect the linebroadening because of electron scattering for the moment. Then,following Chluba & Sunyaev (2006) and using the definitions ofSect. 3.2, in the optically thin limit the spectral distortion of theCMB at observing frequency ν due to direct recombinations tolevel i is given by

ΔIic(ν) =2 hν3

c2

∫ ∞

zt

nγ(νz, z)cNi σi(νz)

H(z)(1 + z)

×[

Ne Np

Nif̃i(Te) exz+μ(xz)+[xic−xz]/ρ − 1

]dz, (A.6)

where νz = ν (1 + z), 1 + zt = νic/ν and xz = hνz/kTγ ≡ hν/kT0.Furthermore, nγ(νz, z) denotes the intrinsic CMB occupationnumber at redshift z including the spectral distortion, evaluatedat frequency νz. For the Lyman-continuum, one would in addi-tion multiply the integrand of Eq. (A.6) by PLy−c

esc (z) to obtain theapproximate solution for the resulting distortion.

To include the broadening because of scattering by electrons,one has to solve the 2-dimensional integral

ΔIic(ν)|Doppler =2 hν3

c2

∫ ∞

0dz

∫Δn(ν̃z, z)√

4πye

e− (ln[ν/ν̃]+3ye)2

4yedν̃ν̃, (A.7)

where ν̃z = ν̃ (1 + z) and

Δn(ν, z) = nγ(ν, z)cNi σi(ν)

H(z)(1 + z)

×[Ne Np

Nif̃i(Te) ex+μ(x)+[xic−x]/ρ − 1

]. (A.8)

In the numerical evaluation of these integrals, it is advisable touse knowledge about the integrand, since otherwise they mayconverge very slowly.


A.5. Estimate regarding the free-free process

The free-free optical depth, τff , is given by

τff(x, z, zf) =∫ zf

zKff(x, z′)

Ne σT c dz′

H(z′)(1 + z′), (A.9)

Here Ne is the free electron number density and H(z) is theHubble factor, which in the radiation-dominated era (z >∼ 3300)is given by H(z) ≈ 2.1× 10−20 [1+ z]2 s−1. The free-free absorp-tion coefficient Kff(x, z) is given by

Kff(x, z) =αλ3

e

2π√

6π

[1 − e−x]

x3 θ7/2γNb g

H+ff (x, θγ) , (A.10)

where λe = h/mec = 2.426 × 10−10 cm is the Compton wave-length of the electron, α ≈ 1/137 is the fine structure con-stant, gH+

ffis the free-free Gaunt factor for hydrogen. We also

introduced the dimensionless temperature of the photon fieldθγ = kBTγ/mec2 ≈ 4.6 × 10−10 [1 + z]. For simplicity, weagain assume that Te ≡ Tγ. Furthermore, we approximated theHe++ free-free Gaunt factor by gHe++

ff≈ 4gff,p and assumed that

z >∼ 8000, since in the considered frequency range most of thefree-free absorption occurs well before He iii → He ii recombi-nation (see below).

Using the condition τff ≈ 1, one can estimate the fre-quency xff(z) below which one expects free-free absorption tobecome important. Since at z >∼ 8000, all the atoms are ion-ized, the number density of free electrons is given by Ne =(1 − Yp/2) Nb ≈ 2.2 × 10−7 (1 + z)3 cm−3. Here we usedNb,0 = 2.5 × 10−7 cm−3 as the present-day baryon numberdensity. For xff � 1, one then finds that τff(x, z = 0, zem) ≈9.6×10−7 [gff/5]

√1 + zem x−2, where zem is the redshift of emis-

sion. Here we are only interested in photons that can be observedat x >∼ 0.02, i.e., ν >∼ 1 GHz today. At this frequency τff >∼ 1 forzem >∼ 1.7×105. Below this redshift one can neglect the free-freeprocess in the computation of the bound-bound and free-boundspectra. However, a more complete treatment will be presentedin a future paper.

Assuming that zem ∼ 8000, one finds that τff <∼ 0.21 forx >∼ 0.02. This justifies the approximations made above, sincethe contributions to the free-free optical depth coming from z <∼8000 are not very large.

Appendix B: Analytic solution

B.1. The 2-shell atom

Including only 2 shells, one can analytically derive the solu-tion for the Lyman-α line under quasi-stationary evolution of thepopulations. For this, we need to determine the net radiative rate,ΔRLyα = A21(1+n21) N2p[1−wN1s/N2p×n21/(1+n21)], for w = 3and n21 = nγ(ν21, Tγ). The ratio Λ = n21/(1 + n21) is determineddirectly by the given ambient radiation field including the spec-tral distortion. Since the distortions are assumed to be small, wecan use A21(1 + n21) N2p ≈ A21(1 + neq

21)Neq2p for the term in front

of the brackets. Here neq21 and Neq

2p are equilibrium values for thephoton occupation number and the 2p-population, respectively.Therefore, we only have to determine the ratio ξ = wN1s/N2p tocompute the Lyman-α line intensity analytically.

We shall first consider the situation for hydrogen at highredshift (z >∼ 3000−4000). There the escape probability in theH i Lyman-α line and the H i Lyman-continuum are close tounity. Therefore, the 2s-1s-two-photon transition does not play

any important role in defining the number density of atomsin the ground state. Furthermore, one can assume that the2s-population is always in Saha-equilibrium with the continuum,and hence N2s ≈ NeNp α2s/β2s where even Np ≈ NH, since thetotal fraction of neutral atoms is tiny.

For the 1s- and 2p-states, the rate equations read

NeNp α1s − ξβ1s

wN2p + A21(1 + n21) N2p[1 − ξΛ] ≈ 0 (B.1a)

NeNp α2p − β2p N2p − A21(1 + n21) N2p[1 − ξΛ] ≈ 0 , (B.1b)

where we have substituted N1s = ξN2p/w and Λ = n21/(1 + n21).Solving this system with8 Np ≈ NH for ξ, one finds

ξ =α1sβ2p + A21(α1s + α2p)[1 + n21]

α2pβ1s/w + A21(α1s + α2p) n21· (B.2)

With the appropriate replacements of terms, the same expressioncan be used to compute the He ii Lyman-α line.

B.1.1. Including the Lyman-α and continuum escape

To include the escape probability in the Lyman-α line, P21, andthe Lyman-continuum, P1c, one should simply replace A21 →P21 A21, α1s → P1cα1s and β1s → P1cβ1s, where the es-cape probabilities can be computed using equilibrium valuesfor N1s and N2p. As long as the 2s-1s-two-photon transitioncan be neglected, this yields an accurate approximation for theLyman-α line (cf. Sects. 5.1).

Around the region where the Lyman-continuum becomes op-tically thick (z ∼ 3000 for H i and z ∼ 11 000 for He ii), forsimple estimates one can use

τescLy−c ≈

{7.2 × 10−24 ex1s [1 + z]4 for H i3.1 × 10−24 ex1s [1 + z]7/2 for He ii

(B.3)

τescLyα≈

{3.0 × 10−19 ex1s [1 + z]3 for H i5.0 × 10−19 ex1s [1 − e−3x1s/4] [1 + z]5/2 for He ii ,

(B.4)

where x1s ≈ 5.79 × 104 Z2[1 + z]−1.

B.1.2. More approximate behavior

To understand the solution for the H i Lyman-α line, we nowturn to the corresponding intensity as a function of redshift (e.g.Rubiño-Martín et al. 2006). This yields

ΔIν =h c4π

ΔRLyα(z)

H(z)[1 + z]3=

h c4π

A21(1 + n21) N2p[1 − ξΛ]

H(z)[1 + z]3· (B.5)

Using the approximation (B.2) for ξ, one can then find that

1 − ξΛ ≈ α2pβ1s/w (1 + n21) − α1sβ2p n21

[α2pβ1s/w + A21(α1s + α2p) n21](1 + n21)· (B.6)

With this, one then has

ΔIν ≈ h c4π

A21 N2p

H(z)[1 + z]3

α2pβ1s (1 + n21)

α2pβ1s + w A21(α1s + α2p) n21

×[1 − w α1sβ2p

α2pβ1se−(x21+μ21)

]· (B.7)

Here we used x21 = hν21/kTγ and (1 + n21)/n21 = ex21+μ21 , withthe frequency-dependent chemical potential μ21 = μ(x21).

8 Note that even if one (more correctly) uses Np = NH−N1s−N2s−N2p

in Eq. (B.1), the solution for ξ does not change.


First factor

We can now simplify the expression in Eq. (B.7) when real-izing that except for the term inside brackets, in the case ofsmall intrinsic CMB spectral distortions, one can just use equi-librium values. At high redshifts one has H(z) ∝ (1 + z)2.Furthermore Neq

2p ≈ 3 Neq1s e−x21 ≈ 3 Ne NH α

eq1s e−x21/β

eq1s and βeq

i =

αeqi e−xic/ f̃i(Te). To high accuracy, one also finds αeq

2p ≈ αeq1s/3 and

α2pβ1s

w� A21(α1s + α2p) n21, so that

F(z) =h c4π

A21 N2p

H(z)[1 + z]3

α2pβ1s (1 + n21)

α2pβ1s + w A21(α1s + α2p) n21

≈ h c4π

3 Ne NH

H(z)[1 + z]3

αeq2p

4 + τescLy−c

∝ (1 + z)1/2

4 + τescLy−c

· (B.8)

Here we have also included the escape probabilities in theLyman-α line and continuum as explained in Appendix B.1.1.We note that the Lyman-α escape probability drops out of theexpression, so that only the Lyman-continuum escape probabil-ity strongly affects the pre-recombinational line shape. We havealso used αeq

i =8πc2 f̃i(Te) exic Ieq

i , where the integral

Ieqi =

∫ ∞

νic

ν2 σi(ν)ex − 1

dν ≈ σi(νic) ν3ic M−1(xic) , (B.9a)

and we have assumed that σi(ν) ≈ σi(νic)ν3icν3

. The integral Mi(x)is defined and discussed in Appendix C. For the 2p-state, onehas σ2p(ν2pc) ν32pc ≈ 7.54 × 1027 Z4 cm−2 s−3.

We checked the scaling of F numerically and found that

F(z) ≈ 5.6 × 10−26 (1 + z)1/2

1 + τescLy−c/4

J m−2 s−1 Hz−1 sr−1 (B.10)

to within <∼20% accuracy in the important redshift range.

Second factor

Using the definitions of αi and βi as given in Sects. 3.1 and 3.2,one finds directly w α1sβ2p

α2pβ1se−(x21+μ21) ≡ e−μ21 G(z) with

G(z) =

⟨n eμ(x)+(x1sc−x)Δρ/ρ

⟩1s〈 n 〉2p⟨

n eμ(x)+(x2pc−x)Δρ/ρ⟩

2p〈 n 〉1s

· (B.11)

Here Δρ = 1 − ρ and we have introduced the notation

〈 f (ν) 〉i =∫ ∞

xic

ν2σi(ν) f (ν) dν (B.12)

for the average of some function f (ν) over the photoionizationcross-section of level i.

In full thermodynamic equilibrium, one has Geq(z) ≡ 1, aproperty that can be verified using Eq. (B.11) with μ = 0 andρ = 1, since then

⟨n eμ(x)+(xic−x)Δρ/ρ

⟩i≡

⟨npl

⟩i. Therefore, we

can write G = 1 + ΔG. Using 〈 f 〉i = 〈 f eq 〉i + 〈Δ f 〉i for smallintrinsic CMB distortions (i.e. 〈Δ f 〉i/〈 f 〉i � 1), one finds

ΔG ≈⟨

npl [μ − μρ1s]⟩

1s⟨npl

⟩1s

−⟨

npl [μ − μρ2p]⟩

2p⟨npl

⟩2p

, (B.13)

where μρi = (x− xic)Δρ/ρ. Combining these expressions, we thenhave

1 − w α1sβ2p

α2pβ1se−(x21+μ21) ≈ μ21 +

⟨npl μ

⟩2p⟨

npl

⟩2p

−⟨

npl μ⟩

1s⟨npl

⟩1s

−⟨

npl μρ2p

⟩2p⟨

npl

⟩2p

+

⟨npl μ

ρ1s

⟩1s⟨

npl

⟩1s

(B.14a)

≈ μ21 + μ2pc − μ1sc , (B.14b)

To lowest order, Eq. (B.14b) shows that the main reason for theemission in the Lyman-α line is the deviation of the effectivechemical potential from zero at the Lyman-α resonance, and theLyman- and Balmer-continuum frequency. However, the aver-ages over the photoionization cross-section still lead to somenotable corrections, so that the small difference in the electronand photon temperature also plays a role.

If we again use the Kramers-approximation for the photoion-

ization cross-section, σi(ν) ≈ σi(νic)ν3icν3

, looking at Eq. (3) for μin the case of a small y-type distortion, one can write⟨

npl

⟩i≈ κi M−1(xic) (B.15a)⟨

npl μ⟩

i≈ κi y [4 M0(xic) − S (xic)] (B.15b)

⟨npl μ

ρi

⟩i≈ κi Δρ

ρ[M0(xic) − xic M−1(xic)] , (B.15c)

where κi = const. and the integrals S and M0 are defined inAppendix C. Keeping only the leading order terms, we have

1 − w α1sβ2p

α2pβ1se−(x21+μ21)

≈ −y x1sc

[6.3 − 0.9375 x1sc + 4.7 e−x1sc − 1.175 e−x1sc/4

]

+ y x21

[9.4 − x21

ex21 + 1ex21 − 1

]· (B.16a)

It is important to mention that this is still a rather rough approx-imation, since by already applying the Kramers-formula to thephotoionization cross-section, we have introduced some signifi-cant simplification. However, this approximation may be usefulfor simple estimates.

Appendix C: Some integrals

C.1. Integrals Mi

In the evaluation of the recombination and photoionization rates,integrals of the form Mi =

∫ ∞xic

xi dx/[ex − 1] appear. Below wenow discuss those of importance to us here.

C.1.1. Integral M−1

For i = −1, one can write

∫ ∞

xic

dxx[ex − 1]

=

∞∑k=1

Ei(kxic)

hνickTγ↓≈ e−xic

xic(C.1a)

hνic≤kTγ↓≈ 1

xic

⎡⎢⎢⎢⎢⎣1 − 11 − 6γ12

xic −x2

ic

12+

xic

2ln(xic)

⎤⎥⎥⎥⎥⎦,(C.1b)

where γ ≈ 0.5772 is the Euler constant and we have madeuse of the exponential integral Ei(x) =

∫ ∞x

e−t dt/t. In the limit


hνic ≤ kTγ, the given approximation is more accurate than 1%.For hνic ≥ kTγ, the first five terms in the full sum also yieldsimilar accuracy.

Since xic ≈ 5.79 × 104 Z2n−2[1 + z]−1, it is clear that at z <∼1.45 × 104 Z2 both Lyman- and Balmer-continuum are still inthe exponential tail of the CMB blackbody. In the redshift range1.45 × 104 Z2 <∼ z <∼ 5.79 × 104 Z2, the Lyman-continuum is stillin the exponential tail of the CMB, while the Balmer-continuumis already in the Rayleigh-Jeans part of the spectrum. Only atz >∼ 5.79 × 104 Z2 can one use the low frequency expansion ofEq. (B.9) in both cases. However, to within <∼30% one may alsoapply Eq. (C.1a) to the entire range.

C.1.2. Integral M0

For i = 0, one can write

M0 =

∫ ∞

xic

dx/[ex − 1] =∞∑

k=1

∫ ∞

xic

e−kx dx =∞∑

k=1

e−kxic/k

= xic − ln(exic − 1) , (C.2)

which for xic <∼ 1 can be approximated as M0 ≈ xic

2 −x2

ic

24 − ln(xic),while for xic 1 one has M0 ≈ e−xic [1 + e−xic/2].

C.2. Integral S

In the evaluation of the recombination and photoionization rates,one also encounters S (x) =

∫ ∞xic

dx x ex+1[ex−1]2 . The first part of

this integral, ∝xex/[ex − 1]2, can be directly taken yielding∫ ∞xic

dx xex/[ex − 1]2 = xicexic/[exic − 1]− ln(exic − 1). Introducing

the polylogarithm Lin(x) =∑∞

k=1 xk/kn and realizing x/[ex−1]2 =∑∞k=1 k x e−(k+1) x, one can find

S (xic) = xicexic + 1exic − 1

+ xic(1 − xic)

− (2 − xic) ln(exic − 1) − Li2(e−xic)hνic>∼kTγ↓≈

m≈5∑k=1

2k − 1k2

[1 + k xic] e−kxic (C.3a)

hνic<∼kTγ↓≈ 2 − π

2

6+ xic −

x2ic

6− 2 ln(xic) , (C.3b)

The given approximations are accurate to <∼1%.

ReferencesBennett, C. L., Halpern, M., Hinshaw, G., et al. 2003, ApJS, 148, 1Burigana, C., & Salvaterra, R. 2003, MNRAS, 342, 543Burigana, C., Danese, L., & de Zotti, G. 1991a, ApJ, 379, 1Burigana, C., Danese, L., & de Zotti, G. 1991b, A&A, 246, 49Burigana, C., de Zotti, G., & Danese, L. 1995, A&A, 303, 323Cen, R., & Ostriker, J. P. 1999, ApJ, 514, 1Chan, K. L., Grant, C., & Jones, B. J. T. 1975, ApJ, 195, 1Chluba, J., & Sunyaev, R. A. 2004, A&A, 424, 389Chluba, J., & Sunyaev, R. A. 2006, A&A, 458, L29Chluba, J., & Sunyaev, R. A. 2007, A&A, 475, 109

Chluba, J., & Sunyaev, R. A. 2008a, A&A, 488, 861Chluba, J., & Sunyaev, R. A. 2008b, A&A, 478, L27Chluba, J., Rubiño-Martín, J. A., & Sunyaev, R. A. 2007a, MNRAS, 374, 1310Chluba, J., Sazonov, S. Y., & Sunyaev, R. A. 2007b, A&A, 468, 785da Silva, A. C., Barbosa, D., Liddle, A. R., & Thomas, P. A. 2000, MNRAS,

317, 37Daly, R. A. 1991, ApJ, 371, 14Danese, L., & de Zotti, G. 1982, A&A, 107, 39Dubrovich, V. K. 1975, SvA Lett., 1, 196Dubrovich, V. K., & Stolyarov, V. A. 1995, A&A, 302, 635Dubrovich, V. K., & Stolyarov, V. A. 1997, Astron. Lett., 23, 565Fixsen, D. J., & Mather, J. C. 2002, ApJ, 581, 817Fixsen, D. J., Cheng, E. S., Gales, J. M., et al. 1996, ApJ, 473, 576Fixsen, D. J., Dwek, E., Mather, J. C., Bennett, C. L., & Shafer, R. A. 1998, ApJ,

508, 123Fixsen, D. J., Kogut, A., Levin, S., et al. 2009, ArXiv e-printsHu, W., & Silk, J. 1993a, Phys. Rev. D, 48, 485Hu, W., & Silk, J. 1993b, Phys. Rev. Lett., 70, 2661Hu, W., Scott, D., & Silk, J. 1994, ApJ, 430, L5Illarionov, A. F., & Syunyaev, R. A. 1975a, SvA, 18, 413Illarionov, A. F., & Syunyaev, R. A. 1975b, SvA, 18, 691Kaplan, S. A., & Pikelner, S. B. 1970, The interstellar medium (Cambridge:

Harvard University Press)Kholupenko, E. E., Ivanchik, A. V., & Varshalovich, D. A. 2005, Gravitation and

Cosmology, 11, 161Kogut, A., Fixsen, D. J., Levin, S., et al. 2004, ApJS, 154, 493Kogut, A., Fixsen, D., Fixsen, S., et al. 2006, New Astron. Rev., 50, 925Lagache, G., Puget, J.-L., & Dole, H. 2005, ARA&A, 43, 727Lightman, A. P. 1981, ApJ, 244, 392Lyubarsky, Y. E., & Sunyaev, R. A. 1983, A&A, 123, 171Markevitch, M., Blumenthal, G. R., Forman, W., Jones, C., & Sunyaev, R. A.

1991, ApJ, 378, L33Mather, J. 2007, Nuovo Cimento B Serie, 122, 1315Mather, J. C., Fixsen, D. J., Shafer, R. A., Mosier, C., & Wilkinson, D. T. 1999,

ApJ, 512, 511McDonald, P., Scherrer, R. J., & Walker, T. P. 2001, Phys. Rev. D, 63, 023001Miniati, F., Ryu, D., Kang, H., et al. 2000, ApJ, 542, 608Oh, S. P., Cooray, A., & Kamionkowski, M. 2003, MNRAS, 342, L20Ostriker, J. P., & Thompson, C. 1987, ApJ, 323, L97Peebles, P. J. E. 1968, ApJ, 153, 1Pozdniakov, L. A., Sobol, I. M., & Syunyaev, R. A. 1983, Astrophys. Space

Phys. Rev., 2, 189Roncarelli, M., Moscardini, L., Borgani, S., & Dolag, K. 2007, MNRAS, 378,

1259Rubiño-Martín, J. A., Chluba, J., & Sunyaev, R. A. 2006, MNRAS, 371, 1939Rubiño-Martín, J. A., Chluba, J., & Sunyaev, R. A. 2008, A&A, 485, 377Salvaterra, R., & Burigana, C. 2002, MNRAS, 336, 592Seager, S., Sasselov, D. D., & Scott, D. 1999, ApJ, 523, L1Seager, S., Sasselov, D. D., & Scott, D. 2000, ApJS, 128, 407Spergel, D. N., Verde, L., Peiris, H. V., et al. 2003, ApJS, 148, 175Sunyaev, R. A., & Zeldovich, Y. B. 1970a, Ap&SS, 7, 3Sunyaev, R. A., & Zeldovich, Y. B. 1970b, Ap&SS, 7, 20Sunyaev, R. A., & Zeldovich, Y. B. 1970c, Comments Astrophys. Space Phys.,

2, 66Sunyaev, R. A., & Zeldovich, Y. B. 1972a, A&A, 20, 189Sunyaev, R. A., & Zeldovich, Y. B. 1972b, Comments Astrophys. Space Phys.,

4, 173Sunyaev, R. A., & Zeldovich, I. B. 1980, ARA&A, 18, 537Sunyaev, R. A., & Chluba, J. 2007, Nuovo Cimento B Serie, 122, 919Sunyaev, R. A., & Chluba, J. 2008, in Frontiers of Astrophysics: A Celebration

of NRAO’s 50th Anniversary, ed. A. H. Bridle, J. J. Condon, & G. C. Hunt,ASP Conf. Ser., 395, 35

Thorne, K. S. 1981, MNRAS, 194, 439Wong, W. Y., Seager, S., & Scott, D. 2006, MNRAS, 367, 1666Zeldovich, Y. B., & Sunyaev, R. A. 1969, Ap&SS, 4, 301Zeldovich, Y. B., Kurt, V. G., & Syunyaev, R. A. 1968, Zh. Eksper. Teoretich.

Fiz., 55, 278Zeldovich, Y. B., Illarionov, A. F., & Syunyaev, R. A. 1972, Sov. J. Exp. Theor.

Phys., 35, 643

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