PHYSICAL REVIEW D 92, 125027 (2015) Effect of neutrino rest mass on ionization equilibrium freeze-out
E. Grohs,1 G. M. Fuller,1 C. T. Kishimoto,1,2 and M. W. Paris3 1Department of Physics, University of California, San Diego, La Jolla, California 92093, USA 2Department of Physics, University of San Diego, San Diego, California 92110, USA 3Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA (Received 31 December 2014; published 23 December 2015) We show how small neutrino rest masses can increase the expansion rate near the photon decoupling epoch in the early Universe, causing an earlier, higher temperature freeze-out for ionization equilibrium compared to the massless neutrino case. This yields a larger free-electron fraction, thereby affecting the photon diffusion length differently than the sound horizon at photon decoupling. This neutrino-mass and recombination effect depends strongly on the neutrino rest masses. Though below current sensitivity, this effect could be probed by next-generation cosmic microwave background experiments, giving another observational handle on neutrino rest mass.
DOI: 10.1103/PhysRevD.92.125027 PACS numbers: 98.80.Cq, 14.60.Lm, 26.35.+c, 98.70.Vc
The history of the early Universe is a history of freeze- Universe, ending at aγd, the epoch of photon decoupling at outs, where reaction rates fall below the Hubble expansion a redshift z ¼ 1090.43 [11]. In the analysis to follow,P we 1 rate. We point out here that the energy density associated ignore the small dependence of the value of aγd on mν . with neutrino rest mass results in a subtle increase in the We should note that Eq. (2) is approximate, and a more expansion rate at photon decoupling. This causes an earlier, complete analysis would include effects beyond the tight- higher temperature epoch for the freeze-out of ionization coupling approximation [12]. equilibrium. The physics of this freeze-out and its relation The photon diffusion length rd depends on the number ðfreeÞ to observations of the cosmic microwave background density of free electrons ne . The free-electron fraction, ðfreeÞ ðtotalÞ (CMB) is a well-studied issue [1–8]. The effect we Xe ≡ ne =ne , parametrizes the free-electron number consider, easily derivable with existing CMB analysis tools density. To evolve Xe, our recombination network uses the [9], has been mentioned [10] but not computed quantita- Saha equation to treat recombination onto He III and tively. We find that these neutrino rest-mass-induced coupled Boltzmann equations [13,14] to treat other recom- changes in CMB observables are below the sensitivity of bination and ionization processes associated with H II, He current methods used to observe and analyze the CMB II, and He III. We employ a recombination reaction network data. The effects, however, may be within the reach of the which is similar to, but independent of, the code RECFAST next generation of precision CMB observations coupled [15]. Figure 1 shows a calculation of the free-electron with a self-consistent computational approach. fraction as a function of scale factorP ratio a=a0 (≡1 at Here we focus on the influence of the recombination current epoch), where we have taken mν ¼ 0. Effects history on CMB observables, in particular the sound due to reionization processes at low redshift, z ∼ Oð1Þ are horizon rs and the photon diffusion length rd. The earlier neglected. The free-electron fraction of Fig. 1, evolved ionization freeze-out caused by neutrino rest mass affects through the photon decoupling epoch, shows the freeze-out the deduced radiation energy density in a perhaps unex- from ionization equilibrium. The first drop from the initial −4 pected way. The quantities rs and rd are given in terms of value of Xe ¼ 1 near a=a0 ≃ 2 × 10 is a consequence of integrals over the scale factor a [11], the recombination onto He III. Z PIn Fig. 2 we plot the change inPXe for nonzero valuesP of aγd da 1 m m ¼ 0 m r ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð Þ ν relative to the case with ν . Nonzero ν s 2 1 X 0 a H 3ð1 þ RÞ Phas a discernible effect on the freeze-out of e. A larger mν implies a larger Hubble rate giving an earlier epoch Z 2 16 for Xe freeze-out. In a study of the expansion rate during aγd da 1 R þ ð1 þ RÞ r2 ¼ π2 15 ; ð Þ recombination, Ref. [16] observed that scaling the Hubble d 2 2 2 0 a H aneðaÞσT 6ð1 þ RÞ rate affects the recombination history. Here we build on this where H ¼ HðaÞ is the Hubble expansion rate, σT is the 1This is not entirely self-consistent but we will demonstrate Thomson cross section, neðaÞ is the free-electron number that a consistent treatment changes the decoupling redshift from density, and RðaÞ ≡ 3ρb=ð4ργÞ is a ratio involving the z ¼ 1090 to z ¼ 1091. The associated change in aγd has ρ ρ baryon rest mass and photon energy densities, b and γ, negligible effect on rs and rd; this is similar to the finding in respectively. The integrals span the early history of the Ref. [6].
1550-7998=2015=92(12)=125027(6) 125027-1 © 2015 American Physical Society E. GROHS et al. PHYSICAL REVIEW D 92, 125027 (2015)
FIG. 1 (color online). The free-electron fraction, Xe,isgivenas FIG. 2 (color online). The relative change in the free-electron a function of scale factor ratio, a=a0 (≡1 at current epoch), and δX ¼ ΔX =X redshift, z (at top). The primordial helium mass fraction is taken fraction, e e e given as a function of scale factor ratio, a=a0, and redshift, z, (at top). The primordial helium mass fraction to be YP ¼ 0.242. Photon decoupling, denoted by the shaded, vertical bar, corresponds to the epoch as determined by Ref. [11]. and vertical bar areP identical to Fig. 1. Each curve corresponds to a different nonzero mν. The curves are in equal incrementsP of ΔΣmν ¼ 0.2 eV, starting with the smallest changeP for mν ¼ 0.2 eV and ending with the largest change for mν ¼ 1.0 eV. argument to explicitly consider the role of neutrino rest mass on recombination.P The curve describing the largest change corresponds to mν P¼ 1.0 eV, whereas the small- radiation energy density inferred from observations of the m ¼ 0 2 N~ est change corresponds to ν . eV; consecutive CMB, which we shall term eff. The theoretical definition ΔΣm ¼ 0 2 N curves are spaced by ν . eV. of eff arises from the familiar parametrization of radiation ρ TðaÞ Considering Eqs. (1) and (2) we note that both quantities, energy density rad in terms of the photon temperature rs and rd depend on the expansion history HðaÞ. Only the at decoupling Tγ ≡ TðaγdÞ given by diffusion length, however, depends explicitly on the n ðaÞ 7 4 4=3 π2 recombination history e , which is itself dependent ρ ¼ 1 þ NðthÞ T4: ð Þ on the expansion history. This recombination effect rad 8 11 eff 15 γ 4 changes rd and this change is opposite to the effect of N the change of that due directly to the Hubble expansion. We adorn eff with a superscript (th) to distinguish the N A scaling analysis, similar to that of Ref. [6], demon- theoretical version of eff,aninput parameter in public N strates the approximate relation between the sound horizon, Boltzmann codes [9], from the CMB inferred value of eff, N~ the diffusion length, and the Hubble rate. Consider a scale eff described below. Calculations which include non- H → λH NðthÞ ¼ transformation to the Hubble rate, , and the equilibrium processes in the early Universe suggest eff r r 3 046 N~ corresponding alteration to s and d. If we neglect the . [1,3,4,17]. We determine eff by computing the dependence of RðaÞ and neðaÞ on λ we have sound horizon, rs, and the photon diffusion length, rd,at the photon decoupling epoch, as follows. 1 1 rs 1 Active neutrinos decouple from the plasma with ultra- rs ∝ and rd ∝ pffiffiffi ⇒ ∝ pffiffiffi : ð3Þ λ λ rd λ relativistic kinematics. As their occupation probabilities are comoving invariants thereafter, the energy density of non- m These relations suggest that a larger Hubble rate (λ > 1) degenerate neutrinos with rest masses νi and neutrino T results in a smaller value of the ratio rs=rd. However, we temperature ν is show below that when the dependence of the recombination Z X d3p history neðaÞ on the expansion HðaÞ is taken into account ρνðmν ;TνÞ¼ Eifνðp; TνÞð5Þ r =r i ð2πÞ3 s d increases. i The radiation energy density is not directly measured by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi observation of the CMB. References [6] and [8], however, Z 2 2 1 X ∞ p þ mν have shown that the ratio of the sound horizon to the photon ¼ dpp2 i ; ð Þ 2 6 2π p=Tν diffusion length at the photon decoupling epoch is sensitive i 0 e þ 1 to the radiation energy density. Consequently, we distin- NðthÞ ν guish between eff , the input parameter that determines where the sum is over active neutrino mass eigenstates i, the radiation energy density in Eq. (4), and a measure of and the second expression follows from an assumption of
125027-2 EFFECTS OF NEUTRINO REST MASS ON… PHYSICAL REVIEW D 92, 125027 (2015)
P N~ m FIG. 3 (color online). The sound horizon rs and diffusion FIG. 4 (color online).P eff as a function of ν. The r NðthÞ m ∼0 06 length d as a function of eff . minimum value for ν is . eV for the normal mass hierarchy.
decoupled neutrinos, ignoring the small distortion due to larger Hubble rate results in a larger ratio of rs=rd; see irreversible neutrino transport effects. With this Figs. 4–6 and discussion below. assumption, the neutrino energy density for a given mass In order to investigate physics beyond the standard eigenstate is given by the product of the ultrarelativistic models of particle physics and cosmology, we have for- Fermi-Dirac occupation probabilities, fν ¼ mulated a self-consistent approach that is not constrained to −1 ðexpðp=TνÞþ1Þ , with theq energyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dispersion relation minimal extensions to the standard model. To do so, we 2 2 simulate the early Universe from weak decoupling through of a massive particle, Ei ¼ p þ mν . Therefore, the i Big Bang Nucleosynthesis (BBN) to photon decoupling energy density in the presence of a massive-neutrino using the BURST code [18]. This treatment self-consistently species is larger than in the massless case and becomes incorporates binned, general, momentum occupation prob- increasingly significant at later times; it does not scale as abilities for each of six neutrino species (νe, ν¯e, νμ, ν¯μ, ντ, T4 ∼ a−4 as in Eq. (4). and ν¯τ) and a Boltzmann treatment of neutrino scattering, Using Eq. (6) with neutrinoP masses taken as described absorption and emission processes to evolve the early above for a given value of mν and the known relations ρ ρ r r Universe. for γ and b we compute the quantities s and d from In this treatment, neutrinos decouple from the γ, e Eqs. (1) and (2), respectively. We determine the inferred 1 ≲ T ≲ 3 N N~ NðthÞ plasma at high temperatures, MeV, with ultra- measure of eff, eff, by choosing eff in Eq. (4) to r =r relativistic kinematics [19,20] as expected. The computa- reproduce this ratio of s d, which is a monotonically Y NðthÞ tion of the helium abundance P in this treatment however decreasing, invertible function of eff ; see Fig. 3. The Y N~ NðthÞ is more nuanced than in the standard cosmology. Here P is quantity eff reduces to for massless, decoupled ðthÞ eff not simply a function of N and ωb. Assuming zero neutrinos. There is clearly no requirement that N~ be eff eff lepton numbers and an adopted world-average neutron independent of scale factor a. It has been motivated here by lifetime of 886 s, our calculations give a 4He primordial the need to characterize massive neutrinos but it is also mass fraction YP ¼ 0.242 taking the baryon number applicable to nonstandard cosmologies with constituents 2 Ωbh ≡ ωb ¼ 0.022068 from the Ref. [11] best fit. This that may be far from equilibrium. Using the ratio rs=rd in is consistent with the observationally inferred primordial the determination of N~ avoids any reference to the eff helium abundance [21,22]. Although we take the neutrinos angular diameter distance to last scattering and, therefore, to decouple in weak eigenstates, i.e. flavor states, we write dependence on the dark energy equation of state [6].In contrast to Ref. [6] we do not change the value of the their occupation probabilities in the mass eigenbasis. Since we are assuming the neutrinos have identical thermal primordial helium abundance, YP, to keep θd fixed. This Δðr =r Þ ∼−ΔN~ r =r spectra with zero-chemical potential in the weak eigen- fact and s d eff (since s d is monotonically decreasing with N~ ) means that a larger Hubble rate would states, we can use the same occupation probabilities for the eff p imply a larger value of N~ . In fact, as suggested in Fig. 2, mass eigenstates at a given momentum [19,23]. eff Before considering the effect of the neutrino mass on the dependence of neðaÞ on the increased energy density N~ from the neutrino rest mass dominates the explicit depend- eff, the deduced radiation energy density, we estimate its HðaÞ NðthÞ ence on in Eq. (2). This has the consequence that a effect on eff . The cosmological constraint (at the level of
125027-3 E. GROHS et al. PHYSICAL REVIEW D 92, 125027 (2015) P 2σ) on the sum of the lightP neutrino masses is mν ≤ 0.23 eV [11]. If we take mν to be at this upper limit and assume degenerate mass eigenvalues, each neutrino has an associated mass ∼0.08 eV. We see that the neutrino rest masses and temperatures at photon decoupling (Tγ ≈ 0.2 eV, Tν ≈ 0.15 eV) are coincidentally at the same scale, meaning that neutrinos can not be treated either as pure matter or pure radiation. An individual neutrino has an average momentum of ∼0.5 eV at photon decoupling. As a consequence, we expect fractional corrections to the relativistic neutrino energy density stemming from neutrino rest mass to be ∼m2=2p2 ∼ 0.01, with a concomitant NðthÞ ∼3 0 01 ∼ þ0 03 change to eff of × . . . If we were to nonphysically classify the entire neutrino energy density ð Þ ρ N th ~ ~ into rad the corresponding change to eff would be FIG. 5 (color online). The change in N , ΔN , is given as a ð Þ ð Þ P eff eff th th 5 11 2=3 3 mi 2 a=a z ΔN ≡ N − 3 ≃ 2 ð Þ ð Þ . We arrive then function of scale factor ratio, 0, and redshift, , (at top). The eff eff 7π 4 i¼1 Tγ ð Þ P primordial helium mass fractionP and vertical bar are identical to ΔN th ≃ 0 04 m ¼ 0 23 m ΔN~ ΔN~ at eff . for ν . eV, a change consistent Fig. 1. For each value of ν, eff is initially positive. eff with the simple kinematic estimate above, and not to be becomes negative once the recombination histories of Fig. 2 ΔNðthÞ ≈ 0 046 differ from the massless case. confused with eff . stemming from nonequili- P brium neutrino scattering and quantum-electrodynamics ~ ΔN ðaÞ is a rapidly decreasing function of mν. We note effects inherent in Refs. [1–3]. P eff P ~ that for mν ¼ 0.23 eV, the preferred upper limit from Figure 4 shows N versus mν. Contrary to the eff ΔN~ ¼ −0 005 expectation of increased N~ in our previous scaling and Ref. [11], we find eff . . This effect is certainly eff below present sensitivities of CMB observations. Next- estimate arguments,P we observe a monotonic decrease in N~ m generation CMB measurements, however, aspire to percent eff with increasing ν. We denote this phenomenon the ν level accuracy in determinations of the relativistic energy neutrino mass/recombination ( MR) effect. For illustrative ν ΔN~ density [24]. The exquisite sensitivity of the MR effect on purposes in Fig. 5, we treat eff as a quantity to be ~ ΔN ðaγdÞ suggests that it may be an important component determined at any epoch, although N is only observed at eff eff in future precision determinations of cosmological photon decoupling. The neutrino rest mass has no dis- N~ a=a parameters. cernible effect on eff at early epochs, at small 0.At −4 As mentioned earlier, we do not constrain aγd to maintain larger values of a=a0ð∼5 × 10 Þ, the extra energy density a uniform optical depth τðaγdÞ, from the neutrino rest masses produces a larger N~ in eff Z accordance with Eq. (3). If we were to extrapolate this a0 da τða Þ¼ an ðaÞσ ≡ 1; ð Þ evolution trend to the epoch of photon decoupling, we γd a2H e T 7 ΔN~ > 0 ν aγd would find a value of eff . The MR effect inter- ΔN~ < 0 P venes to modify this extrapolation and results in eff when comparing different values for mν. Note that this a at γd. definition of τðaγdÞ does not include reionization effects on ΔN~ Each evolution curve for eff in Fig. 5 corresponds to neðaÞ. We should emphasize that each curve in Figs. 2 X ðaÞ Pan evolution curve for e inP Fig. 2 for various values of and 5 is calculated using the same value for the scale factor m m −4 ν. The smallest value of ν produces the smallest of last scattering aγd ¼ 9.162 × 10 (corresponding to X ΔN~ change in e, which subsequentlyP changes eff the least. z ¼ 1090.43). This is not self-consistent, strictly speaking, Conversely, the largest value of mν produces the largest but we have verified that the effect on N~ , due to the X ΔN~ eff change in e, which changes eff the most. From the differences in ne and H, is negligible. If we impose the curves in Fig. 5, it is clear that the effect of neutrino rest constraintP in Eq. (7), we find aγd decreases by a few parts in 4 mass in producing a higher Xe at freeze-out overwhelms the 10 for mν ¼ 0.23 eV, which has an insignificant effect N~ N~ effect of the extra energy density, thereby decreasing eff at on eff . photon decoupling, i.e. at a ¼ aγd, the vertical bar in Figs. 2 Up to this point in the present analysis, we have not and 5. considered variation of the primordial helium mass fraction There are several interesting features to note in Fig. 5. YP since BBN occurs at high-enough temperatures that the ΔN~ ¼ 0 Each curve in Fig. 5 goes through eff near the value neutrinos are effectively massless. If we consider, however, −4 Pa=a0 ∼ ð7.65 0.10Þ × 10 . The larger the value of cosmological parameters that affect YP, we can examine the m ΔN~ ðaÞ ν, the higher the curvature of the function eff . dependence of ionizationP freeze-out (and subsequent alter- a=a ΔN~ ¼ 0 N~ m Y For values of 0 above which eff , the slope of ation of eff) on both ν and P simultaneously. A
125027-4 EFFECTS OF NEUTRINO REST MASS ON… PHYSICAL REVIEW D 92, 125027 (2015) N~ Fig. 5. Note that the evolution of eff shown in this figure does not reflect a kinematical evolution of the radiation energy density with a massive component. The trends evidenced in this figure are a consequence of the νMR effect. Self-consistency is a primary motivation for defining the N~ radiation energy density parameter eff in terms of the ratio r =r NðthÞ s d; it generalizes the eff parameter to the massive N~ neutrino case. Further, eff is defined for general energy densities and nonequilibrium distribution functions where N~ the neutrino temperature is undefined. Moreover, eff makes no assumption regarding the underlying cosmologi- N~ cal model. We use eff to relate the sound horizon and photon diffusion length to predictions made by the standard ð Þ P cosmological model via the parameter N th . −ΔN~ m eff FIG. 6 (color online). Contours of constant eff in the ν The νMR effect is an example of a recurring phenome- ω vs b parameter space. The shaded, vertical bar corresponds to non in cosmology: an increase in the expansion rate leads to the 1σ error for ωb [11]. an earlier epoch of freeze-out. This effect was revealed in N~ the present context by using eff to infer the cosmic direct way to vary YP is to consider changes to the baryon radiation energy content from observable CMB data, rather number ωb. In the range of values of ωb that we’re ðthÞ than treating N as an input. The procedure we describe interested in, YP is a monotonically increasing function eff here differs from that adopted by the public Boltzmann of ωb. P ΔN~ m codes. CAMB [9], for example, includes options to evolve Figure 6 shows a contour plot of eff in the ν ω massive neutrino energy density through the epoch of versus b parameter space; contours correspond to constant NðthÞ −ΔN~ ω recombination andP requires eff to be provided as an input. values of eff. Varying b requires new computations m Y X ðaÞ Depending on ν, the concomitant changes in ion- Pof P from BBN and e from recombination. Changing N~ ization equilibrium and eff discussed here may be within mν requires a new computation of Xe but no new the sensitivity of the next generation CMB experiments computation of YP. As a consequence, we compute ω when polarization effects are taken into account [11, BBN with the BURST code only once for a given b, – ω 24 26]. CMB precision is planned to be increased to the Pand compute the recombinationP history for each pair ( b, N~ ∼ 1% m m jΔN~ j eff level which would probe both massive active ν). Holding ν fixed, the change in eff ω neutrinos and other possible components of dark radiation. increases with increasing b due to the different recombi- Scenarios with sterile neutrinos and other very weakly nation histories effecting a change in rd. Note that r coupled light massive species with masses larger than those the change in s does not completely compensate for the associated with the active neutrinos could enhance the change in rd. The shaded vertical region in Fig. 6 is the 1σ ω effects discussed here. However, depending on their masses range of b given by Ref. [11], but we explore a larger and their flavor mixing with active species, sterile neutrinos range in thePωb parameter space to illustrate the dependence N~ m ω could have number densities and energy spectra which of eff on ν and b. An interesting feature of these – ω differ from those of active neutrinos [27 32], complicating curves is theirP increasing curvature with decreasing b and the analysis given here. increasing mν. This is a consequence ofP an enhancement of the νMR effect with increasing ωb:as mν increases, N~ ω ACKNOWLEDGMENTS the change in eff is faster for higher values of b. We have discussed two ways in which neutrino rest mass We would like to acknowledge the Institutional affects measurable quantities at photon decoupling. First, Computing Program at Los Alamos National Laboratory neutrino rest mass drives an earlier recombination freeze- for use of their HPC cluster resources. This work was out resulting in a higher free-electron fraction. Second, this supported in part by National Science Foundation Grant effect is enhanced with increasing ωb stemming from a self- No. PHY-1307372 at UC San Diego, by the Los Alamos consistently calculated recombination history. As radiation National Laboratory Institute for Geophysics, Space energy density is not a directly measurable quantity, we use Sciences and Signatures Subcontract No. 257842, and observable quantities to indirectly arrive at the radiation the National Nuclear Security Administration of the U.S. energy density. For this purpose, we choose the ratio of the Department of Energy at Los Alamos National Laboratory sound horizon to the photon diffusion length. Photon under Contract No. DE-AC52-06NA25396. We thank J. J. diffusion is sensitive to the recombination history, which Cherry, Amit Yadav, and Lloyd Knox for helpful discus- requires a Boltzmann-equation treatment. We find a non- sions. We would also like to thank the anonymous referees ΔN~ trivial evolution of eff with scale factor, as shown in for their useful comments.
125027-5 E. GROHS et al. PHYSICAL REVIEW D 92, 125027 (2015) [1] A. D. Dolgov, S. H. Hansen, and D. V. Semikoz, Nucl. Phys. [17] G. Mangano, G. Miele, S. Pastor, T. Pinto, O. Pisanti, and B543, 269 (1999). P. D. Serpico, Nucl. Phys. B729, 221 (2005). [2] J.-L. Cambier, J. R. Primack, and M. Sher, Nucl. Phys. [18] E. Grohs, G. M. Fuller, C. T. Kishimoto, and M. W. Paris, J. B209, 372 (1982). Cosmol. Astropart. Phys. 05 (2015) 017. [3] R. E. Lopez and M. S. Turner, Phys. Rev. D 59, 103502 [19] G. M. Fuller and C. T. Kishimoto, Phys. Rev. Lett. 102, (1999). 201303 (2009). [4] G. Mangano, G. Miele, S. Pastor, and M. Peloso, Phys. Lett. [20] A. D. Dolgov, M. V. Sazhin, and Y. B. Zeldovich, Basics of B 534, 8 (2002). Modern Cosmology (Editions Frontieres, Gif-Sur-Yvette, [5] M. Shimon, N. J. Miller, C. T. Kishimoto, C. J. Smith, G. M. 1990). Fuller, and B. G. Keating, J. Cosmol. Astropart. Phys. 5 [21] Y. I. Izotov and T. X. Thuan, Astrophys. J. Lett. 710, L67 (2010) 037. (2010). [6] Z. Hou, R. Keisler, L. Knox, M. Millea, and C. Reichardt, [22] E. Aver, K. A. Olive, R. L. Porter, and E. D. Skillman, J. Phys. Rev. D 87, 083008 (2013). Cosmol. Astropart. Phys. 11 (2013) 017. [7] J. Birrell, C.-T. Yang, P. Chen, and J. Rafelski, Phys. Rev. D [23] A. D. Dolgov S. H. Hansen, S. Pastor, S. T. Petcov, G. G. 89, 023008 (2014). Raffelt, and D. V. Semikoz, Nucl. Phys. B632, 363 (2002). [8] Z. Hou, C. L. Reichardt, K. T. Story, B. Follin, R. Keisler, [24] K. Abazajian et al., Astropart. Phys. 63, 66 (2015). K. A. Aird, B. A. Benson, L. E. Bleem, J. E. Carlstrom, [25] P. A. R. Ade et al. The Polarbear Collaboration, Astrophys. C. L. Chang et al., Astrophys. J. 782, 74 (2014). J. 794, 171 (2014). [9] C. Howlett, A. Lewis, A. Hall, and A. Challinor, J. Cosmol. [26] J. E. Carlstrom and Spt Collaboration, IAU Special Session Astropart. Phys. 04 (2012) 027. 2, 34 (2003). [10] J. Lesgourgues and S. Pastor, Adv. High Energy. Phys. [27] K. Abazajian, N. F. Bell, G. M. Fuller, and Y. Y. Y. Wong, 2012, 608515 (2012). Phys. Rev. D 72, 063004 (2005). [11] Planck Collaboration, Astron. Astrophys. 571, A16 [28] C. T. Kishimoto, G. M. Fuller, and C. J. Smith, Phys. Rev. (2014). Lett. 97, 141301 (2006). [12] W. Hu and N. Sugiyama, Astrophys. J. 471, 542 (1996). [29] J. Hamann, S. Hannestad, G. G. Raffelt, I. Tamborra, and [13] P. J. E. Peebles, Astrophys. J. 153, 1 (1968). Y. Y. Y. Wong, Phys. Rev. Lett. 105, 181301 (2010). [14] Y. B. Zeldovich, V. G. Kurt, and R. A. Syunyaev, Zh. Eksp. [30] J. Hamann, S. Hannestad, G. G. Raffelt, and Y. Y. Y. Wong, Teor. Fiz. 55, 278 (1968). J. Cosmol. Astropart. Phys. 9, (2011) 034. [15] S. Seager, D. D. Sasselov, and D. Scott, Astrophys. J. 523, [31] C. Dvorkin, M. Wyman, D. H. Rudd, and W. Hu, Phys. Rev. L1 (1999). D 90, 083503 (2014). [16] O. Zahn and M. Zaldarriaga, Phys. Rev. D 67, 063002 [32] M. Cirelli, G. Marandella, A. Strumia, and F. Vissani, Nucl. (2003). Phys. B708, 215 (2005).
125027-6