Relativistic effective degrees of freedom and quantum statistics of neutrinos

TL;DR

This paper derives analytical formulas for the relativistic effective degrees of freedom considering neutrinos with mixed quantum statistics and shows that lepton asymmetries can influence these degrees.

Contribution

It provides new analytical expressions for $g_\ast$ with non-pure fermionic neutrinos and explores their dependence on lepton flavor asymmetries.

Findings

$g_\ast$ can be larger with fermionic neutrinos than bosonic ones under certain conditions.

Lepton flavor asymmetries significantly affect the effective degrees of freedom.

Analytical expressions facilitate better understanding of neutrino contributions in cosmology.

Abstract

Analytical expressions of the relativistic effective degrees of freedom $g_\ast$ with non-pure fermionic neutrinos are presented. A semi-analytical study is performed to show that $g_\ast$ with pure fermionic neutrinos may be greater than $g_\ast$ with pure bosonic neutrinos for non-vanishing lepton flavor asymmetries.

Equations

$$f_i=\frac{g_i}{e^{(E-{\mu }_i)/T}+{\kappa }_i},$$

$$n_{i-\bar{i}}$$

$${\xi }_i=\frac{{\mu }_i}{T},F_s(x)\equiv \frac{{\mathrm{Li}}_s(-x)}{-x},$$

$${\rho }_{i+\bar{i}}={\rho }_i+{\rho }_{\bar{i}}\simeq \frac{3g_i{T}^4}{{\pi }^2}\left(2F_4({\kappa }_i)+{\xi }_i^2{F}_2({\kappa }_i)\right).$$

$${\mu }_u+{\mu }_{\ell }$$

$${\mu }_u={\mu }_c={\mu }_t,{\mu }_d={\mu }_s={\mu }_b,$$

$$sQ$$

$$sB$$

$$s{L}_{\ell }$$

$$0=-({\xi }_e+{\xi }_{\mu }+{\xi }_{\tau })+4{\xi }_u-3{\xi }_d,$$

$$\frac{3sB}{T^3}=2{\xi }_u+3{\xi }_d,$$

$$\frac{3s{L}_{\ell }}{T^3}={\xi }_{\ell }+\frac{6}{{\pi }^2}{F}_2({\kappa }_{\nu }){\xi }_{{\nu }_{\ell }},$$

$${\xi }_u$$

$${\xi }_d$$

$${\xi }_{{\nu }_{\ell }}$$

$${\xi }_{\ell }$$

$$\rho =\sum _i{\rho }_i=\frac{{\pi }^2{T}^4}{30}{g}_{\ast }.$$

$$g_{\ast }(\xi ,{\kappa }_{\nu })=g_{\ast }({\xi }_i=0,{\kappa }_{\nu }=1)+\Delta g_{\ast },$$

$$g_{\ast }({\xi }_i=0,{\kappa }_{\nu }=1)=\sum _{b=bosons}{g}_b+\frac{7}{8}\sum _{f=fermions}{g}_f,$$

$$\Delta g_{\ast }$$

$$\Delta g_{\ast }({\kappa }_{\nu }=1)=\frac{50}{529{\pi }^2}{\left(\frac{s}{T^3}\right)}^2\left(873B^2-162BL+362L^2+1587\sum _{\ell }{L}_{\ell }^2\right),$$

$$\Delta g_{\ast }({\kappa }_{\nu }=-1)=\frac{45}{289{\pi }^2}{\left(\frac{s}{T^3}\right)}^2\left(529B^2-78BL+140L^2+867\sum _{\ell }{L}_{\ell }^2\right)+\frac{3}{4},$$

$$\Delta g_{\ast }({\kappa }_{\nu }=1)\simeq 1.65\times 10^4{L}^2,\Delta g_{\ast }({\kappa }_{\nu }=-1)\simeq 1.31\times 10^4{L}^2+\frac{3}{4}.$$

Full text

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PACS Nos.14.60.St:98.80.Cq

1 Introduction

We understand the neutrinos obey purely Fermi-Dirac statistics on the analogy of the electrons[1]; however, they may possess mixed statistics [2, 3, 4, 5, 6, 7, 8]. Dolgov, et.al. studied the effects of continuous transition from Fermi-Dirac to Bose-Einstein statistics of neutrinos and discussed the possible modification of the big bang nucleosynthesis [9]. J.I and T.K estimated the relativistic degrees of freedom with non-pure fermionic neutrinos in the early universe by numerical calculations [11, 10].

In this letter, to complement our previous numerical studies [11, 10], we show analytical expressions of the relativistic effective degrees of freedom in the early universe with non-pure fermionic neutrinos, and perform a semi-analytical study by using these analytical expressions.

2 Analytical expressions

**Net number density and net energy density: ** The distribution function is given by [9]

[TABLE]

where $g_{i}$, $E$, $\mu_{i}$ and $T$ denote the number of internal degrees of freedom, energy, chemical potential and temperature, respectively. The Fermi-Bose parameter $\kappa_{i}$ describes the continuous transition from Fermi-Dirac $\kappa_{i}=1$ distribution to Bose-Einstein $\kappa_{i}=-1$ distribution via Maxwell-Boltzmann $\kappa_{i}=0$ distribution.

The net number density of particle species $i$ is obtained as [12, 13]

[TABLE]

where

[TABLE]

and ${\rm Li}_{s}(x)$ denotes the polylogarithm function. We note that $F_{2}(1)=\pi^{2}/12$, $F_{4}(1)=7\pi^{4}/720$ are obtained for pure fermions and $F_{2}(-1)=\pi^{2}/6$, $F_{4}(-1)=\pi^{4}/90$ are obtained for pure bosons.

Similarly, the net energy density is obtained as

[TABLE]

**Chemical potentials: ** The beta-decay of down quark via weak interactions $d\rightarrow u+\ell+\bar{\nu}_{\ell}$ provides

[TABLE]

and the relation of $\mu_{i}=\mu_{\bar{i}}$ and $\mu_{\gamma}=\mu_{W}=\mu_{Z}=\mu_{g}=\mu_{H}=0$ are appropriate. With the following assumptions [14]

[TABLE]

there are only five independent chemical potentials and we take these as $\mu_{u},\mu_{d},\mu_{\nu_{e}},\mu_{\nu_{\mu}},\mu_{\nu_{\tau}}$. These five independent chemical potentials are uniquely determined by the following five conservation laws [14]

[TABLE]

where $Q$, $B$ and $L_{\ell}$ denote electric charge, baryon number and lepton flavor number of the universe, respectively.

For electrically neutral universe, $Q=0$, from Eqs. (2), (7), (8) and (9), the following coupled equations for chemical potentials up to $\mathcal{O}(\xi_{i}^{2})$ are obtained

[TABLE]

where we assume $m_{W}<T<m_{t}$, $\kappa_{\nu}=\kappa_{\nu_{e}}=\kappa_{\nu_{\mu}}=\kappa_{\nu_{\tau}}$ and take $\kappa_{i}=1$ for the fermions in the standard model except neutrinos. From Eqs.(5) and (10), the five independent $\xi_{i}$ ($i=u,d,\nu_{e},\nu_{\mu},\nu_{\tau}$) as well as $\xi_{\ell}$ are analytically determined

[TABLE]

where $L=\sum_{\ell=e,\mu,\tau}L_{\ell}$.

Relativistic effective degrees of freedom: The relativistic effective degrees of freedom for energy density, $g_{*}$, is defined by [13, 14]

[TABLE]

Similarly, for entropy density, $g_{\ast s}$ is defined by $s=\sum_{i}s_{i}=\frac{2\pi^{2}T^{3}}{45}g_{\ast s}$. From Eqs.(12), (4) and (11), the relativistic effective degrees of freedom for energy density is obtained as

[TABLE]

where

[TABLE]

denotes the well-known relativistic effective degrees of freedom for vanishing chemical potentials ($\xi_{i}=0$) and for pure fermionic neutrinos ($\kappa_{\nu}=1$) [13]. The effects on $g_{\ast}$ from the non-vanishing chemical potentials and non-pure fermionic neutrinos are estimated as

[TABLE]

Similarly, the relativistic effective degrees of freedom for entropy density is expressed as $g_{*s}(\xi,\kappa_{\nu})=g_{*s}(\xi_{i}=0,\kappa_{\nu}=1)+\Delta g_{*s}$. For $T\sim 100$GeV, we obtain $g_{*s}(\xi_{i}=0,\kappa_{\nu}=1)=g_{*}(\xi_{i}=0,\kappa_{\nu}=1)$ and $\Delta g_{*s}=\Delta g_{*}$.

3 Discussions and summary

With the vanishing chemical potential, the equilibrium energy density of pure bosonic particle is larger than it of pure fermionic particle [13]. One may expect that the relation $g_{*}^{\rm FD}<g_{*}^{\rm BE}$ is guaranteed where $g_{\ast}^{\rm FD}$ denotes $g_{\ast}$ with pure fermionic neutrinos and $g_{\ast}^{\rm BE}$ denotes $g_{\ast}$ with pure bosonic neutrinos. However, in our previous numerical studies[11, 10], we have shown that this relation is not always satisfied with non-vanishing lepton flavor asymmetries in the early universe [15, 16, 17, 18, 19, 20, 21, 22].

We complement our previous numerical studies [11, 10] by semi-analytical calculations. From Eqs. (11) and (15), we obtain

[TABLE]

for pure fermionic neutrinos, and

[TABLE]

for pure bosonic neutrinos. For the sake of simplicity, we assume $L_{\ell}=L/3$ and $L\gg B\sim 0$, and use very rough estimation of $s/T^{3}=2\pi^{2}g_{*s}/45=2\pi^{2}g_{*}/45\sim 44$ with $T=100$ GeV. In this case, we obtain

[TABLE]

and $\Delta g_{*}(\kappa_{\nu}=1)\gtrsim\Delta g_{*}(\kappa_{\nu}=-1)$ as well as $g_{*}^{\rm FD}\gtrsim g_{*}^{\rm BE}$ for $L\gtrsim 0.015$.

We comment on possible application of our results in a cosmological context. In the leptogenesis scenario [23], the baryon-photon ratio in the universe $\eta_{B}$ is related to the lepton asymmetry $Y_{L}$ via $\eta_{B}\propto Y_{L}\propto g_{\ast}^{-1}$. Thus, the lepton number in the early universe yields change of the baryon-photon ratio. More detailed analysis will be found in our future study.

In summary, analytical expressions of the relativistic effective degrees of freedom with non-pure fermionic neutrinos are presented. A semi-analytical study has been performed to complement our previous numerical studies which show that the relation of $g_{*}^{\rm FD}\gtrsim g_{*}^{\rm BE}$ may be allowed with non-vanishing lepton flavor asymmetries.