Weaker yet again: mass spectrum-consistent cosmological constraints on the neutrino lifetime

Precision measurements of the cosmic microwave background (CMB) anisotropies and the large-scale matter distribution have in the past two decades yielded some of the tightest constraints on particle physics. The most widely known amongst these are upper limits on the neutrino mass sum, which at $\sum m_{\nu}\lesssim 0.12$ eV for a one-parameter extension of the standard $\Lambda$CDM model Planck:2018vyg ,¹¹195% credible limit derived from a 7-parameter fit to the data combination Planck TT,TE,EE+lowE+lensing+BAO Planck:2018vyg . supersedes at face value current laboratory $\beta$-decay end-point spectrum measurements by a factor of 30 Aker:2019uuj . Of lesser prominence albeit equal significance are CMB constraints on the neutrino lifetime.

Relativistic invisible neutrino decay occurring on time scales of CMB formation leave an interesting signature on the CMB anisotropies. In the cosmological context, Standard-Model neutrinos decouple at temperatures of $T\sim 1$ MeV and free-stream thereafter. Free-streaming could be inhibited, however, if non-standard collisional processes such as neutrino decay and especially its concurrent inverse decay—the latter of which is kinematically feasible only in the limit of an ultra-relativistic neutrino ensemble—should become efficient at $T\lesssim 1$ MeV. At CMB formation times, the absence of neutrino free-streaming prevents the transfer of power from the monopole (density) and dipole (velocity divergence) fluctuations of the neutrino fluid to higher multipole moments (e.g., anisotropic stress) Hannestad:2004qu . Such a suppression of higher-order anisotropies manifests itself predominantly in the CMB primary anisotropies as an enhancement of power at multipoles $\ell\gtrsim 200$ in the CMB temperature power spectrum, which can be exploited to place constraints on the non-standard interaction(s) responsible for the phenomenon Hannestad:2005ex ; Basboll:2008fx ; Cyr-Racine:2013jua ; Archidiacono:2013dua ; Escudero:2019gfk ; Barenboim:2020vrr ; Lancaster:2017ksf ; Oldengott:2014qra ; Oldengott:2017fhy ; Kreisch:2019yzn ; Forastieri:2019cuf ; Barenboim:2019tux ; Venzor:2022hql .

Following this line of argument, measurements of the Planck mission (2018 data release) currently constrain the rest-frame neutrino lifetime to $\tau_{0}\gtrsim(4\times 10^{5}\to 4\times 10^{6})\ (m_{\nu}/50\,{\rm meV})^{5}$ s, assuming invisible decay of a parent neutrino not exceeding $m_{\nu}\simeq 0.2$ eV in mass into a massless particles Barenboim:2020vrr .²²2It is also possible to invoke invisible neutrino decay in the non-relativistic limit as a means to relax cosmological neutrino mass bounds. See, e.g., Refs. Lorenz:2021alz ; Abellan:2021rfq for recent studies. The rest-frame lifetimes required are typically of order $0.01\to 0.1$ of the universe’s age, leading to a phenomenology that differs completely from the ultra-relativistic case considered in this work. This lower bound is to be compared with laboratory limits obtained from analyses of the neutrino disappearance rate in JUNO and KamLAND+JUNO, $\tau_{0}/m_{\nu}>7.5\times 10^{-11}$ (s/eV) and $\tau_{0}/m_{\nu}>1.1\times 10^{-9}$ (s/eV) for a normal and an inverted mass ordering respectively Porto-Silva:2020gma , as well as astrophysical limits from solar neutrinos $\tau_{0}/m_{\nu}\gtrsim 10^{-5}\to 10^{-3}$ (s/eV) SNO:2018pvg ; Funcke:2019grs , SN1987A $\tau_{0}\gtrsim(1\to 10)$ s Kachelriess:2000qc , and IceCube $\tau_{0}/m_{\nu}\gtrsim 2.4\times 10^{3}$ (s/eV) Song:2020nfh .

Central to the extraction of $\tau_{0}$ from cosmological data is the neutrino anisotropy loss rate or transport rate $\Gamma_{\rm T}$, particularly how this rate relates to $\tau_{0}$ itself and other fundamental properties of the particles participating in the interaction. Some of us have recently considered this issue via a rigorous computation of the transport rate from the full Boltzmann hierarchies incorporating the two-body decay $\nu_{H}\leftrightarrow\nu_{l}+\phi$ and its inverse process induced by a Yukawa interaction Barenboim:2020vrr . For decay into massless daughter particles, Ref. Barenboim:2020vrr finds a transport rate $\Gamma_{\rm T}\sim(m_{\nu}/E_{\nu})^{5}(1/\tau_{0})$ under the assumption of linear response. This result is in contrast to the rate $\Gamma_{\rm T}\sim(m_{\nu}/E_{\nu})^{3}(1/\tau_{0})$ employed in previous analyses Hannestad:2005ex ; Basboll:2008fx ; Archidiacono:2013dua ; Escudero:2019gfk , which is based upon a heuristic but (as we shall show) incorrect random walk argument.

While representing a considerable step forward, the result of Ref. Barenboim:2020vrr is nonetheless incomplete in that the mass difference between the parent neutrino $\nu_{H}$ and daughter neutrino $\nu_{l}$ must respect the neutrino mass spectra allowed by neutrino oscillations experiments. Global three-flavour analyses of oscillation data currently find the squared mass differences to be deSalas:2020pgw ; Esteban:2020cvm ; Capozzi:2017ipn

	$\displaystyle\Delta m^{2}_{21}$	$\displaystyle=$	$\displaystyle 7.50^{+0.22}_{-0.20}\times 10^{-5}~{}{\rm eV}^{2},$		(1)
	$\displaystyle|\Delta m^{2}_{31}|_{\rm N}$	$\displaystyle=$	$\displaystyle 2.55^{+0.02}_{-0.03}\times 10^{-3}~{}{\rm eV}^{2},$		(2)
	$\displaystyle|\Delta m^{2}_{31}|_{\rm I}$	$\displaystyle=$	$\displaystyle 2.45^{+0.02}_{-0.03}\times 10^{-3}~{}{\rm eV}^{2},$		(3)

where $|\Delta m^{2}_{31}|_{\rm N}$ applies to a normal mass ordering (NO; $m_{3}>m_{2}>m_{1}$), and $|\Delta m^{2}_{31}|_{\rm I}$ to an inverted mass ordering (IO; $m_{2}>m_{1}>m_{3}$). Relative to the benchmark parent neutrino mass of $0.2$ eV, these squared mass differences are clearly small enough that the assumption of a massless daughter neutrino may become untenable and consequently alter the relationship between the transport rate $\Gamma_{\rm T}$ and the rest-frame decay rate $\tau_{0}$ away from $\Gamma_{\rm T}\sim(m_{\nu}/E_{\nu})^{5}(1/\tau_{0})$.

In this work, we improve upon the original calculation of Ref. Barenboim:2020vrr to explicitly allow for a massive daughter neutrino $\nu_{l}$. We furthermore extend the computation of the anisotropic stress (i.e., multipole $\ell=2$) damping rate to $\ell>2$ multipole moments—for compatibility with the neutrino Boltzmann hierarchy Ma:1995ey solved in the CMB code class Blas:2011rf —and put the generalisation of the calculation to other interaction structures (besides Yukawa) on firmer grounds. We find that, in the presence of a massive $\nu_{l}$, a consistent transport rate $\Gamma_{\rm T}$ is suppressed by an extra phase space factor proportional to $(\Delta m_{\nu}^{2}/m_{\nu}^{2})^{2}$ in the limit of a small squared mass difference between the parent and daughter neutrinos, i.e., $\Gamma_{\rm T}\sim(\Delta m_{\nu}^{2}/m_{\nu}^{2})^{2}(m_{\nu}/E_{\nu})^{5}(1/\tau_{0})$, and this factor is in addition to the phase space suppression ($\sim\Delta m_{\nu}^{2}/m_{\nu}^{2}$) already inherent in the lifetime $\tau_{0}$.

Using a general version of this result valid for all $\Delta m_{\nu}^{2}$, we fit the Planck 2018 CMB data to derive a new set of cosmological constraints on the neutrino lifetime as functions of the lightest neutrino mass (i.e., $m_{1}$ in NO and $m_{3}$ in IO) that are consistent with mass difference measurements. Readers interested in a quick number can jump directly to Fig. 3 where the new constraints on $\tau_{0}$ are presented. Here, we note that for a parent neutrino of $m_{\nu}\lesssim 0.1$ eV, the new phase space suppression factor weakens the CMB constraint on its lifetime by as much as a factor of 50 if $\Delta m_{\nu}^{2}$ corresponds to the atmospheric mass gap and by five orders of magnitude if the solar mass gap, compared with naïve limits that do not account for the daughter neutrino mass.

The paper is organised as follows. We begin with a description of the physical system in Sect. 2. We then introduce the basic equations of motion for the individual $\nu_{H}$, $\nu_{l}$ and $\phi$ component in Sect. 3, before condensing them into a single set of effective equations of motion for the combined neutrino+$\phi$ system in Sect. 4. Using standard statistical inference techniques presented in Sect. 5, we derive in Sect. 6 new CMB constraints on the neutrino lifetime that are consistent with the experimentally determined neutrino mass splittings. We conclude in Sect. 7. Four appendices contain respectively full expressions of the collision integrals (A), computational details of the $\ell\geq 2$ effective damping rates (B), a revised random walk picture explaining the $\propto(m_{\nu}/E_{\nu})^{5}$ dependence of the transport rate and why the old $\propto(m_{\nu}/E_{\nu})^{3}$ rate is incorrect (C), and summary statistics of our parameter estimation analyses (D).

We consider the two-body decay of a parent neutrino $\nu_{H}$ into a massive daughter neutrino $\nu_{l}$ and a massless particle $\phi$, whose transition probability is described by a Lorentz-invariant squared matrix element $|{\cal M}|^{2}$. Previously in Ref. Barenboim:2020vrr we had assumed $\phi$ to be a scalar and the decay described by a Yukawa interaction, motivated by Majoron models Gelmini:1980re ; Chikashige:1980ui ; Schechter:1981cv in which a new Goldstone boson of a spontaneously broken global $U(1)_{B-L}$ symmetry couples to neutrinos. Another possibility arises within gauged $L_{i}-L_{j}$ models, where a new vector boson plays the role of the massless state Dror:2020fbh ; Ekhterachian:2021rkx ; Chen:2021qaf ; Alonso-Alvarez:2021pgy . Models of neutrinos decaying on cosmological timescales can be found in Refs. Gelmini:1983ea ; Joshipura:1992vn ; Akhmedov:1995wd ; Escudero:2020ped . Here, in order to keep our analysis as general as possible, apart from the quantum statistical requirement that $\phi$ must be boson, we shall leave its nature and the nature of the interaction unspecified.

Because cosmological observables do not measure neutrino spins, it is not possible to define a reference direction in the decaying neutrino’s rest-frame. Thus, for an ensemble of neutrinos with randomly aligned spins, we can effectively consider the decay to be isotropic in the decaying neutrino’s rest-frame, with a decay rate $\Gamma_{\rm dec}^{0}\equiv 1/\tau_{0}$ related to the squared matrix element via

$$\Gamma^{0}_{\rm dec}=\frac{\Delta(m_{\nu H},m_{\nu l},m_{\phi})}{16\pi m_{\nu H}^{3}}|{\cal M}|^{2},$$

(4)

where

$$\Delta(x,y,z)\equiv\sqrt{\left[x^{2}-\left(y+z\right)^{2}\right]\left[x^{2}-\left(y-z\right)^{2}\right]}\\$$

(5)

is the Källén function, and it is understood that $|{\cal M}|^{2}$ needs to be evaluated at particle energies and momenta consistent with conservation laws.

In the cosmological context, the decay process can be expected to come into effect around scale factors $a\sim a_{\rm dec}$, where $a_{\rm dec}$ is defined via $\Gamma_{\rm dec}(a_{\rm dec})=H(a_{\rm dec})$, with $\Gamma_{\rm dec}=\left\langle m_{\nu H}/E_{\nu H}\right\rangle\Gamma_{\rm dec}^{0}$ the ensemble-averaged Lorentz-boosted decay rate. We are interested in the decay of $\nu_{H}$ occurring (i) up to the time of recombination but substantially after neutrino decoupling, i.e., $10^{-10}\ll a_{\rm dec}\lesssim 10^{-3}$, and (ii) while the bulk of the $\nu_{H}$ ensemble is ultra-relativistic. Two observations are in order:

1. 1.

   The ultra-relativistic requirement guarantees that the inverse decay process $\nu_{l}+\phi\to\nu_{H}$ is kinematically allowed and kicks in on roughly the same time scale as $\nu_{H}$ decay, enabling $\nu_{H}$, $\nu_{l}$, and $\phi$ to form a coupled system whose combined anisotropic stress can be suppressed through collisional damping. The same requirement also effectively limits the parent neutrino mass to the ballpark $m_{\nu H}\lesssim 3T_{\nu}(a_{\rm rec})\sim 3\times(4/11)^{1/3}T_{\gamma}(a_{\rm rec})\ \sim 0.2$ eV, if the phenomenology of anisotropic stress loss is to remain unambiguously applicable up to the recombination era when the temperature of the photon bath drops to $T_{\gamma}(a_{\rm rec})\sim 0.1$ eV.

2. 2.

   Since decay and inverse decay occur by construction only after neutrino decoupling, our scenario does not generate an excess of radiation. Rather, the $(\nu_{H},\nu_{l},\phi)$-system merely shares the energy that was in the original $\nu_{H}$ and $\nu_{l}$ populations. Consequently, in the time frame of interest cosmology is unaffected at the background level, and the phenomenology of the $(\nu_{H},\nu_{l},\phi)$-system vis-à-vis the CMB primary anisotropies is at leading order limited to anisotropic stress loss.

These observations tell us that, as far as the CMB primary anisotropies are concerned, the $(\nu_{H},\nu_{l},\phi)$-system can be modelled at leading order as a single massless fluid whose energy density is equivalent to that of the original $\nu_{H}$ and $\nu_{l}$ populations at $a<a_{\rm dec}$. What remains to be specified is the rate at which the fluid’s anisotropic stress and higher-order multipole moments are lost via decay and inverse decay; this is the subject of Sects. 3 and 4.

A complete set of evolution equations for the individual $\nu_{H}$, $\nu_{l}$, and $\phi$ phase space density under the influence of gravity and decay/inverse decay has been presented in Ref. Barenboim:2020vrr . We briefly describe the relevant ones below.

We work in the synchronous gauge defined by the invariant ${\rm d}s^{2}=a^{2}(\tau)\left[-{\rm d}\tau^{2}+(\delta_{ij}+h_{ij}){\rm d}x^{i}{\rm d}x^{j}\right]$, and split the Fourier-transformed phase space density of species $i$ into a homogeneous and isotropic component and a perturbed part, i.e., $f_{i}(\mathbf{k},\mathbf{q},\tau)=\bar{f}_{i}(|\mathbf{q}|,\tau)+F_{i}(\mathbf{k},\mathbf{q},\tau)$, where $\tau$ is conformal time, $\mathbf{q}\equiv|\mathbf{q}|\hat{q}$ the comoving 3-momentum, and $\mathbf{k}\equiv|\mathbf{k}|\hat{k}$ is the Fourier wave vector conjugate to the comoving spatial coordinates $\mathbf{x}$ of the line element.

Because for the scenario at hand the $(\nu_{H},\nu_{l},\phi)$-system behaves at leading order essentially as a single massless fluid whose energy density remains constant in the time frame of interest, the evolution of the individual homogeneous phase space density $\bar{f}_{i}(|\mathbf{q}|,\tau)$ is not a main concern. In fact, in Sect. 4 we shall approximate all $\bar{f}_{i}$’s with equilibrium distributions in order to derive closed-form expressions for the effective damping rates.

Of greater interest to us is the perturbed component $F_{i}(\mathbf{k},\mathbf{q},\tau)$. Following standard practice Ma:1995ey , we decompose $F_{i}$ in terms of a Legendre series,

	$\displaystyle F_{i}(|\mathbf{k}|,|\mathbf{q}|,x)$	$\displaystyle\equiv\sum_{\ell=0}^{\infty}(-{\rm i})^{\ell}(2\ell+\!1)F_{i,\ell}(|\mathbf{k}|,|\mathbf{q}|)P_{\ell}(x),$		(6)
	$\displaystyle F_{i,\ell}(|\mathbf{k}|,|\mathbf{q}|)$	$\displaystyle\equiv\frac{{\rm i}^{\ell}}{2}\int_{-1}^{1}\mathrm{d}x\,F_{i}(|\mathbf{k}|,|\mathbf{q}|,x)P_{\ell}(x),$

where $x\equiv\hat{k}\cdot\hat{q}$, and $P_{\ell}(x)$ is a Legendre polynomial of degree $\ell$. Then, the equations of motion for the Legendre moments ${F}_{i,\ell}$ can be written as

	$\displaystyle\dot{F}_{i,0}$	$\displaystyle=-\frac{|\mathbf{q}||\mathbf{k}|}{\epsilon_{i}}F_{i,1}+\frac{1}{6}\frac{\partial\bar{f}_{i}}{\partial\ln|\mathbf{q}|}\dot{h}+\left(\frac{{\rm d}f_{i}}{{\rm d}\tau}\right)_{C,0}^{(1)},$		(7)
	$\displaystyle\dot{F}_{i,1}$	$\displaystyle=\frac{|\mathbf{q}||\mathbf{k}|}{\epsilon_{i}}\left(-\frac{2}{3}F_{i,2}+\frac{1}{3}F_{i,0}\right)+\left(\frac{{\rm d}f_{i}}{{\rm d}\tau}\right)_{C,1}^{(1)},$
	$\displaystyle\dot{F}_{i,2}$	$\displaystyle=\frac{|\mathbf{q}||\mathbf{k}|}{\epsilon_{i}}\left(-\frac{3}{5}F_{i,3}+\frac{2}{5}F_{i,1}\right)$
		$\displaystyle\qquad-\frac{\partial\bar{f}_{i}}{\partial\ln|\mathbf{q}|}\left(\frac{2}{5}\dot{\eta}+\frac{1}{15}\dot{h}\right)+\left(\frac{{\rm d}f_{i}}{{\rm d}\tau}\right)_{C,2}^{(1)},$
	$\displaystyle\dot{F}_{i,\ell>2}$	$\displaystyle=\,\frac{|\mathbf{k}|}{2\ell+1}\frac{|\mathbf{q}|}{\epsilon_{i}}\left[\ell F_{i,\ell-1}-(\ell+1)F_{i,\ell+1}\right]$
		$\displaystyle\qquad\qquad+\left(\frac{{\rm d}f_{i}}{{\rm d}\tau}\right)_{C,\ell}^{(1)}.$

Here, $\cdot\equiv\partial/\partial\tau$, $\epsilon_{i}(|\mathbf{q}|)\equiv(|\mathbf{q}|^{2}+a^{2}m_{i}^{2})^{1/2}$ is the comoving energy of the particle species $i$ of mass $m_{i}$, $h\equiv\delta^{ij}h_{ij}(\mathbf{k},\tau)$ and $\eta=\eta(\mathbf{k},\tau)\equiv-k^{i}k^{j}h_{ij}/(4k^{2})+h/12$ are the trace and traceless components of $h_{ij}$ in Fourier space respectively, and $\left({\rm d}f_{i}/{\rm d}\tau\right)_{C,\ell}^{(1)}$ represents the $\ell$th Legendre moment of the linear-order Boltzmann collision integral.

The linear-order decay/inverse decay collision integrals have been derived in Ref. Barenboim:2020vrr assuming a Yukawa interaction. However, following the arguments of Sect. 2 and using the fact that the squared matrix element is Lorentz-invariant, it is straightforward to generalise the integrals of Ref. Barenboim:2020vrr to any interaction structure. The same argument also allows us to recast the collision integrals in favour of the rest-frame decay rate $\Gamma_{\rm dec}^{0}$ as a fundamental parameter (over the squared matrix element $|{\cal M}|^{2}$). The interested reader can find the full expressions in Eqs. (38)–(40) in A.

The equations of motion (7) together with the collision integrals (38)–(40) can be programmed directly into and solved by a CMB code such as class Blas:2011rf . In practice, however, numerical solutions particularly in the ultra-relativistic limit can be highly non-trivial, because of the vastly different time scales at play: the basic decay time scale $1/\Gamma_{\rm dec}$ and the emergent transport time scale $1/\Gamma_{\rm T}\sim(E_{\nu}/m_{\nu})^{4}(1/\Gamma_{\rm dec})$. In Sect. 4 we show how these equations of motion can be condensed in the ultra-relativistic limit into a single set of equations for the combined $(\nu_{H},\nu_{l},\phi)$-system under certain reasonable simplifying assumptions.

We are interested in the evolution of perturbations in the combined $(\nu_{H},\nu_{l},\phi)$-system in the ultra-relativistic limit. To this end we define the integrated Legendre moment,

	$\displaystyle{\cal F}_{\ell}(|\mathbf{k}|)\equiv$	$\displaystyle\frac{(2\pi^{2}a^{4})^{-1}}{\bar{\rho}_{\nu\phi}}\sum_{i=\nu_{H},\nu_{l},\phi}g_{i}$		(8)
		$\displaystyle\quad\quad\times\int{\rm d}|\mathbf{q}|\;|\mathbf{q}|^{2}\epsilon_{i}\left(\frac{|\mathbf{q}|}{\epsilon_{i}}\right)^{\ell}F_{i,\ell}(|\mathbf{q}|),$

where $g_{i}$ is the internal degree of freedom of the $i$th particle species, and $\bar{\rho}_{\nu\phi}$ represents the total background energy density of the $(\nu_{H},\nu_{l},\phi)$-system which we take to be ultra-relativistic, i.e., $\bar{\rho}_{\nu\phi}\propto a^{-4}$, and equal to the energy density in the pre-decay $\nu_{H}$ and $\nu_{l}$ populations.

Summing and integrating the equations of motion, Eq. (7), as per the definition (8) and taking the ultra-relativistic limit everywhere except the collision terms, we obtain

	$\displaystyle\dot{\cal F}_{0}$	$\displaystyle=-|\mathbf{k}|{\cal F}_{1}-\frac{2}{3}\dot{h},$		(9)
	$\displaystyle\dot{\cal F}_{1}$	$\displaystyle=\frac{|\mathbf{k}|}{3}\left({\cal F}_{0}-2{\cal F}_{2}\right),$
	$\displaystyle\dot{\cal F}_{2}$	$\displaystyle=\frac{|\mathbf{k}|}{5}\left(2{\cal F}_{1}-3{\cal F}_{3}\right)+\frac{4}{15}\dot{h}+\frac{8}{5}\dot{\eta}+\left(\frac{{\rm d}{\cal F}}{{\rm d}\tau}\right)_{C,2},$
	$\displaystyle\dot{\cal F}_{\ell>2}$	$\displaystyle=\,\frac{|\mathbf{k}|}{2\ell+1}\left[\ell{\cal F}_{\ell-1}-(\ell+1){\cal F}_{\ell+1}\right]+\left(\frac{{\rm d}{\cal F}}{{\rm d}\tau}\right)_{C,\ell},$

where

	$\displaystyle\left(\frac{\mathrm{d}{\cal F}}{\mathrm{d}\tau}\right)_{C,\ell}$	$\displaystyle\equiv\,\frac{(2\pi^{2}a^{4})^{-1}}{\bar{\rho}_{\nu\phi}}\sum_{i=\nu_{H},\nu_{l},\phi}g_{i}$		(10)
		$\displaystyle\times\int{\rm d}|\mathbf{q}|\;|\mathbf{q}|^{2}\epsilon_{i}\left(\frac{|\mathbf{q}|}{\epsilon_{i}}\right)^{\ell}\left(\frac{\mathrm{d}f_{i}}{\mathrm{d}\tau}\right)_{C,\ell}^{(1)}(|\mathbf{q}|)$

are the effective collision integrals constructed from the individual collision integrals (38)–(40). Note that the effective collision integrals for $\ell=0,1$ integrated moments must vanish exactly because of energy and momentum conservation in the $(\nu_{H},\nu_{l},\phi)$-system.

Having derived the effective equations of motion in a way consistent with neutrino oscillation data, we wish now to put constraints on the neutrino rest-frame lifetime $\tau_{0}$ using cosmological data.

Before we describe our data analysis, observe that the effective collision integral (13) can be broken down in terms of its dependence on the scale factor $a$ into

$$\left(\frac{{\rm d}{\cal F}}{{\rm d}\tau}\right)_{C,\ell}=-\alpha_{\ell}\,a^{6}\,Y\,\mathscr{F}\left(aX\right){\cal F}_{\ell}.$$

(25)

Here,

$$X\equiv\frac{m_{\nu H}}{T_{0}}\simeq 298\,\left(\frac{m_{\nu H}}{0.05~{}{\rm eV}}\right)\,\left(\frac{T_{\nu,0}}{T_{0}}\right)$$

(26)

is an effective mass parameter, while

	$\displaystyle Y$	$\displaystyle\equiv\,\frac{C}{12}\,\frac{(a^{4}\bar{\rho}_{\nu H})}{(a^{4}\bar{\rho}_{\nu\phi})}\,\Phi\left(\frac{m_{\nu l}}{m_{\nu H}}\right)\left(\frac{m_{\nu H}}{T_{0}}\right)^{5}\Gamma^{0}_{\rm dec}$		(27)
		$\displaystyle\simeq 6.55\,C\times 10^{10}\,\left(\frac{(a^{4}\bar{\rho}_{\nu H})}{(a^{4}\bar{\rho}_{\nu\phi})}\Big{/}\frac{1}{3}\right)\left(\frac{T_{\nu,0}}{T_{0}}\right)^{5}$
		$\displaystyle\hskip 65.44133pt\times\Phi\left(\frac{m_{\nu l}}{m_{\nu H}}\right)\,\left(\frac{m_{\nu H}}{0.05~{}{\rm eV}}\right)^{5}\,\Gamma^{0}_{\rm dec}$

is an effective transport rate, with $C=1$ for the one parent/one daughter scenario (“one decay channel”), and $C=2$ if we wish to use the three-state to two-state approximation to describe the case of two parents/one daughter or one parent/two daughters (“two decay channels”). Both $X$ and $Y$ contain only quantities that are taken to be constant in time in our analysis (see Sect. 4.3), and together these two effective parameters determine completely the phenomenology of the effective collision integral (13).