Implications of neutrino species number and summed mass measurements in cosmological observations

The effective massless degrees of freedom $N_{\rm eff}$ determines the cosmic energy density, hence controls the speed of cosmic expansion. This quantity at nucleo-synthesis ($\sim 100$ seconds since the big-bang) is measurable by comparing measured light element abundances with theoretical calculation, while the same quantity at later epochs after recombination ($\sim 4\times 10^{5}$ years after the bang) is measured by Planck and CMB-S4 observations. The quantity $N_{\rm eff}$ at these two epochs is sensitive function of ⁴He abundance $Y_{p}$. The allowed region in the $(Y_{p},N_{\rm eff})$ plane from observations is usually presented by contour maps. The important fact is that correlations given by the derivative signs of $dN_{\rm eff}/dY_{p}$ are opposite at two epochs, positive at nucleo-synthesis and negative at recombination. This makes it possible to determine both of $N_{\rm eff}$ and $Y_{p}$ at high precision.

The primordial ⁴He abundance $Y_{p}$ is around 0.25. Reference [10] on nucleo-synthesis cites a conservative bound $N_{\rm eff}<4$, but also mentions a more restrictive bound $N_{\rm eff}<3.2$. The correlation with $Y_{p}$ at nucleo-synthesis gives a straight line in the $(N_{\rm eff},Y_{p})$ plane, as reviewed in [9], ranging in $N_{\rm eff}=2.2\sim 3.4$. The final status of Planck 2018 + BAO observations gives an impressive upper bound $N_{\rm eff}<3.33$ at 95% CL [4].

Not only neutrinos, but also other stable light relics left behind from the early thermal history give additional contribution, extra $\Delta N_{\rm eff}=0.03\sim 0.05$ per a single relic. Examples are axion, dark photon and sterile neutrino. Contribution from a spin-less stable light relic is $\Delta N_{\rm eff}\sim 0.027$. This quantity can also be negative if light relics are unstable and decay during two epochs. We shall assume that there is no such relic, or even if there is, their cumulative contribution is restricted at most by $|\Delta N_{\rm eff}|<0.028$ (CMB-S4 target value), and concentrate on diluted Dirac $\nu_{R}$ which may behave in a similar way to a light relic.

Assume that $\nu_{R}$ in the Dirac theory is thermally abundant at early cosmological epochs, and calculate the amount of dilution. After $\nu_{R}$ decoupling the universe goes through many annihilation events of thermal anti-particles, thereby reheating the universe. Electroweak interaction is far stronger than interactions in BSM, hence left-handed $\nu_{L}$ is reheated, but $\nu_{R}$ is not. This gives rise to difference of $\nu_{L}$ and $\nu_{R}$ effective temperatures. Note that even after $\nu_{R}$’s decouple and become non-thermal, one can assign their effective temperature. This is because the entropy conservation of $a^{3}s(T)$ (with $a(t)$ the time-dependent cosmic scale factor) holds at nearly instantaneous reheating processes. The entropy density $s(T)$ is proportional to the massless degrees of freedom $g_{*}(T)$ in radiation-dominated epoch. Hence one can readily calculate the temperature ratio before and after rehearing events by counting respective species number contributing to $g_{*}$.

Assuming that the standard electroweak phase transition took place in the Big Bang Universe, it is straightforward to evaluate $g_{*}$ in the era up to the electroweak phase transition in the standard model.

Fermions are counted by the weight $7/8$ for one spin state and bosons by the weight $1$. Counting $g_{*}(T)$ gives the dilution factor in terms of number density ratio, $n(\nu_{R})/n(\nu_{L})=43/427=0.101$, since the electroweak phase transition.

We summarize $g_{*}$ factors relevant to $\nu_{R}$ dilution in the following table, Table(1) in which $g_{*}^{\rm i}$ indicates the species number at cosmological events i.

It is convenient in the rest of analysis to relate the extra $\Delta N_{\rm eff}$ to the extra relativistic degrees of freedom $\Delta g_{*}$,

where $g_{*}^{\rm EW}=427/4$ is the species number of electroweak theory, and $g_{*}^{\nu_{L}}=43/4$. Note that $\Delta g_{*}$ can be negative, implying that $\nu_{R}$ decoupling may occur below the electroweak temperature $T^{\rm EW}\sim 250\,$GeV.

Alternatively, one can relate $\Delta N_{\rm eff}$ upper bound or observed value to required $\Delta g_{*}$ value, using the formula (1).

Let us first recall what happens in the standard electroweak theory based on the gauge group SU(2)${}_{L}\times$U(1), when one introduces the right-handed neutrino $\nu_{R}$. It is known that introduction of SU(2)${}_{L}\times$U(1) singlet $\nu_{R}$ opens the possibility of Higgs boson ($h$) coupling proportional to doublet bi-linear fermion $(\bar{\nu_{L}},\bar{l}_{L})\nu_{R}$ which, after the spontaneous electroweak gauge symmetry breaking, gives the Dirac-type neutrino mass. Inevitable process $\nu_{L}h\rightarrow\nu_{R}$ from thermal $\nu_{L},h$ produces $\nu_{R}$, but with a negligible amount of $\nu_{R}$ ($\Delta N_{\rm eff}\ll 0.05$) if neutrino masses are less than of order 100 eV [10]. There seems nothing wrong with this scheme, but this minimum extension does not give any clue to many outstanding problems of particle physics such as generation of the baryon asymmetry. We need a new theoretical framework of massive Dirac-type neutrino.

A natural scheme is grand unified gauge theories (GUT), but it is sufficient to first think of an intermediate step of subgroup unification towards GUT. We find it most natural to analyze $\nu_{R}$ dilution in the left-right symmetric extension of the standard electroweak theory, SU(2)${}_{R}\times$SU(2)${}_{L}\times$U(1) gauge theory [11], as also made in [7]. The Higgs system consists of irreducible representations, $(2,2)\,,(2,1)\,,(1,2)$ of SU(2)${}_{R}\times$SU(2)_L group. Regarding all species of particle as massless, there are 18 extra $g_{*}$ in addition to the standard one $427/4$ and three right-handed neutrinos $21/4$.

Some details of this doublet left-right symmetric model (DLRSM) are given in Appendix.

The Boltzmann equation for the number density, obtained after integrating the distribution function $f(\vec{p})$ in the phase space (space volume times the momentum-space volume $d^{3}p/(2\pi)^{3}$ in spatially homogeneous universe), describes time evolution in the expanding universe. Consider a number density of $f$ species $n_{f}$. The equations is

with $H(T)$ the Hubble rate $\sqrt{\frac{8\pi G_{N}}{3}\rho_{r}}$ ($\rho_{r}$ the energy density of effectively massless particles, $\pi^{2}g_{*}T^{4}/30$). When the Hubble rate $H(T)$ compared with right-hand side (RHS) is small, thermal equilibrium is realized with vanishing RHS of (2) (meaning the positive production and the negative destruction balance). This thermal equilibrium is ended at decoupling temperature $T_{d}^{f}$ that is determined by equating the rate to the Hubble rate, $H(T_{d}^{f})=\Gamma_{R}$. One may use for $\nu_{R}$ annihilation rate $\Gamma_{R}$

Momenta of initial and final particles, assumed massless, are denoted $p_{i}\,,q_{i}\,,i=1,2$, respectively. Lorentz-invariant squared amplitude ${\cal R}(s,t)$ summed over spin states is written in terms of invariant variables, $s=(p_{1}+p_{2})^{2}\,,t=(p_{1}-q_{1})^{2}$. They are calculated below. $n_{R}(T)=3\zeta(3)(2s_{p}+1)T^{3}/4\pi^{2}$ is the number density of initial thermal $\nu_{R}$ ($s_{p}=1/2$ is its spin).

One may separately confirm that thermalization condition is satisfied by considering inverse processes. Since we treat all initial and final fermions as massless (actually much lighter than gauge bosons $W_{R},Z_{R}$) inverse $\nu_{R}$ production process has equal rate to the annihilation rate. Thus, time integrated quantity $n_{R}\Gamma_{R}$ gives thermally summed $\nu_{R}$ number density.

Relevant annihilation processes of right-handed neutrino $\nu_{R}$ (conjugate processes of $\bar{\nu}_{R}$ production to be included as well) in LR symmetric models occur via two-body colliding processes of many kinds. Typical Feynman diagrams are depicted in Fig(1). In order to calculate rate, we classify individual contributions as

with $f=q\,,l$ being quarks and charged lepton. We note that process involving light higgs pair is absent, because a neutral scalar field has no vector current that may couple to $Z_{R}$. Feynman rules for amplitude and rate calculation can readily be extracted using formulas in Appendix.

A part of these contributions to annihilation rates have been calculated in the literature. Their substantial part is missing in the literature, and we shall cover and add all relevant contributions. The decoupling temperature $T_{d}$ at which two rates are equal is related to right-handed gauge coupling masses and gauge coupling $g_{R}$. Remarkably, we shall be able to provide analytic results for important quantities we need.

SU(2)${}_{R}\times$SU(2)${}_{L}\times$U(1) gauge theory gives contributions from $(a)\sim c)$ listed in (5) $\sim$ (9), some of them depicted in Fig(1). Invariant squared amplitudes ${\cal R}(s,t)$ after spin summation consist of coupling factors and dynamical parts given in terms of $s,t$ variables. Listed contributions have different $s,t$ dependence. It is sufficient to calculate ${\cal R}(s,t)$ in the temperature range $T\ll$ gauge boson masses, hence four-Fermi approximation is excellent, with the common strength factor $G_{R}$,

We used notations $M_{W}\,,M_{Z}$ for $W_{R},Z_{R}$ masses, to distinguish from ordinary electroweak gauge bosons $m_{W},m_{Z}$, and $g_{R}\,,\theta_{R}$ are gauge coupling constant and mixing angle in the SU(2)_R sector. Flavor dependent coupling factors are further multiplied to squared Fermi constant;

Contributions $C^{\alpha}_{\varphi}$ in squared amplitudes arise from $\alpha-$type of diagrams in Fig(1) and those of $\varphi-$type of exchanged gauge bosons. For instance, $C^{aq}_{2t}$ is from squared $t-$channel exchange of Fig(1a) in which leptons $l$ are replaced by relevant quarks $q$.

The last coupling $C_{st}$ arises from interference contribution of t-channel $W_{R}$-exchange (a) diagram and s-channel $Z_{R}$-exchange (b) diagram for two-body process of $\nu_{R}\bar{\nu}_{R}\rightarrow l_{R}\bar{l}_{R}\,,l=e,\mu,\tau$. As explained in Appendix, a value $\sin^{2}\theta_{R}=0.299$ should be used. Using these couplings, squared invariant amplitudes are given by

The last integral over final states in (4) is to be calculated in general coordinate frames. If one replaces this by the total cross section in the center-of-mass frame, the important part of asymmetric collisions between initial fermion pairs is lost.

The angular integration over final state variables is done in Lorentz-invariant manner, which gives trivial angular integrations, $\int(s+t)^{2}=\int t^{2}=\frac{1}{3}s^{2}\,,\int s^{2}=s^{2}$. Thus, the annihilation cross section is

Using $s=2p_{1}p_{2}(1-\cos\theta_{12})$, the angular integral in the initial state gives $4\pi^{2}\int_{0}^{4p_{1}p_{2}}ds/p_{1}p_{2}$. Finally, integration over initial thermal fermions leads to

using the value of $x=\sin^{2}\theta_{R}=0.299$, given in Appendix.

We presented results using exact Fermi-Dirac (FD) distribution function for fundamental fermions, quarks and leptons. In Appendix, we also derive results taking the approximate Maxwell-Boltzmann (MB) distribution, which gives analytic results that differ by

The result of annihilation rate divided by $n_{f}$ is $0.9948$ times FD value, two values being surprisingly close to each other within 1 % difference.

Some comments on thermal distribution function may be in order. After copious production right-handed neutrinos scatter with thermal particles (via $Z_{R}$ exchange) and their energies are redistributed. Thus, $\nu_{R}$ thermal distribution function is realized with zero chemical potential. Thermal gauge bosons $W_{R}$ can produce $\nu_{R}$, but they decouple much earlier, resulting in no further thermal $\nu_{R}$ production.

We have also calculated $\nu_{R}$ production rates in a much more simplified approximation of using thermally averaged values in the invariant total cross section $\sigma v$. Using the averaged $\bar{t}=-s/2\,,\bar{t^{2}}=s^{2}/3$, one has the thermally averaged $\sigma v=\overline{{\cal S}}/4p_{1}p_{2}$ given by

This gives $\Gamma_{R}\approx 6\zeta(3)G_{R}^{2}\bar{s}I/\pi^{3}$. The thermal average of $s$ is $\bar{s}=2\bar{p}^{2}=1.814\,T^{2}$, hence one derives $\Gamma_{R}\sim 1.79G_{R}^{2}T^{5}$, which grossly differs from the more precise result (17). This estimate confirms the importance of asymmetric collision in rate calculations. Note that one is not allowed to take the center of mass frame in the comoving frame of thermal universe, since head-on collisions do not necessarily occur and collisions with angles do occur usually. Thus, one has to perform thermal average over angular configurations. The invariant variables $s=2E_{1}E_{2}(1-\cos\theta_{12})\,,t=-2E_{1}E_{2}(1-\cos\theta_{13})$ of a massless $(1,2)$ pair collision $(1,2)\rightarrow(3,4)$ must be averaged using thermal distribution functions of fermions, $1/(e^{E_{i}/T}+1)$.

$\nu_{R}$ production rate $\Gamma_{R}$ may be compared to the Hubble rate,

The decoupling temperature $T_{d}^{\nu_{R}}$ is defined by equating two rates: $\Gamma_{R}(T_{d}^{\nu_{R}})=H(T_{d}^{\nu_{R}})$ gives

Using the cited value $\sim 550$MeV of Planck-BAO limit [5], one may derive a limit on $G_{R}^{-1/2}>42\,$TeV $(g_{*}^{\nu_{R}}/g_{*}^{\rm EW})^{-1/8}$.

The decoupling temperature of right-handed neutrinos is discussed in the literature [5, 6, 7, 8]. Among them, Ref. [7] treats of the DLRSM as in the present work. It is difficult to compare our decoupling temperature with that in Ref. [7], since it has not given details of how to calculate $\nu_{R}$ annihilation rate. We, however, obtain a considerably smaller decoupling temperature than that of Ref. [7] by a few orders of magnitude.

$\nu_{R}$ contributes to an extra $\Delta N_{\rm eff}$ given by

with $g_{*}^{\nu_{L}}=10.75$ (species number at left-handed neutrino decoupling). If $\nu_{R}$ decoupling occurs below the electroweak scale 250 GeV, one has $\Delta N_{\rm eff}>0.14$ since $g_{*}(T^{\rm EW})/g_{*}^{\nu_{L}}=9.93$, a value slightly smaller than Planck limit, but much larger than CMB-S4 target value 0.028. One can eliminate $g_{*}^{\nu_{R}}$ in favor of measurable $\Delta N_{\rm eff}$, to give

Calculations so far are based on that thermal environment consists of fundamental quarks and leptons. This picture is valid at temperatures above $\sim 200\,$MeV at which hadronization occurs and quarks are incorporated into hadrons, namely baryons and mesons. This restricts the applicable region of unification scale to

For $G_{R}^{-1/2}$ outside this region the predicted $\Delta N_{\rm eff}$ is grossly inconsistent with Planck observations. The LR symmetric unification, if the Majorana option is disfavored, thus appears at energy scale above $O(10^{4})\,$ GeV, far beyond the electroweak scale.