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 .