Viscosity in a lepton-photon universe

Lars Husdal

doi:10.1007/s10509-016-2847-4

Astrophysics and Space Science

August 2016, 361:263 | Cite as

Viscosity in a lepton-photon universe

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Lars HusdalEmail author

Original Article

First Online: 12 July 2016

Received: 23 May 2016

Accepted: 27 June 2016

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Abstract

We look at viscosity production in a universe consisting purely of leptons and photons. This is quite close to what the universe actually look like when the temperature was between $10^{10}~\mbox{K}$ and $10^{12}~\mbox{K}$ (1–100 MeV). By taking the strong force and the hadronic particles out of the equation, we can examine how the viscous forces behave with all the 12 leptons present. By this we study how shear- and (more interestingly) bulk viscosity is affected during periods with particle annihilation. We use the theory given by Hoogeveen et al. from 1986, replicate their 9-particle results and expanded it to include the muon and tau particles as well. This will impact the bulk viscosity immensely for high temperatures. We will show that during the beginning of the lepton era, when the temperature is around 100 MeV, the bulk viscosity will be roughly 100 million times larger with muons included in the model compared to a model without.

Keywords

Viscous cosmology Shear viscosity Bulk viscosity Lepton era Relativistic kinetic theory

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Notes

Acknowledgements

Thanks to my supervisor Kåre Olaussen for giving me the project, and for helping me throughout this work. His help has been invaluable for understanding the concepts and the underlying mathematics. Thanks also to Iver Brevik for introducing us to the field of viscous cosmology, and for fruitful discussions on this paper.

Appendix A: Coefficients occurring in the linear equations

We begin be rewriting the equations for shear and bulk viscosities

$$\begin{aligned} \eta_{s} &= \frac{1}{10} cn \biggl( \frac{k_{\text{B}}T}{c} \biggr)^{3} \sum_{k} c_{k} \gamma_{k}, \end{aligned}$$

(A.1)

$$\begin{aligned} \eta_{v} &= nk_{\text{B}}T \sum_{k} a_{k} \alpha_{k}, \end{aligned}$$

(A.2)

The $c_{k}$ and $a_{k}$ coefficients are solutions to the linearized relativistic Boltzmann (transport) equation for the shear and bulk cases are given as follows:

$$\begin{aligned} &c_{k} = \frac{1}{n} \biggl(\frac{k_{\text{B}}T}{c} \biggr)^{2} c_{kl}^{-1} \gamma_{k}, \end{aligned}$$

(A.3)

$$\begin{aligned} & a_{k} = \frac{1}{n} a_{kl}^{-1} \alpha_{k}, \end{aligned}$$

(A.4)

where

$$\begin{aligned} &c_{kl}=x_{k} \Biggl( x_{l} c'_{kl} + \delta_{kl} \sum_{m=1}^{N} x_{m} c''_{km} \Biggr), \end{aligned}$$

(A.5)

$$\begin{aligned} & a_{kl}=x_{k} \Biggl( x_{l} a'_{kl} + \delta_{kl} \sum_{m=1}^{N} x_{m} a''_{km} \Biggr) \end{aligned}$$

(A.6)

are matrix elements related to the differential cross-sections in the frame of the center of momentum of the system. The primes and double primes are defined as

$$\begin{aligned} c'_{kl}={}&\frac{1}{3} \biggl(\frac{k_{\text{B}}T}{c} \biggr)^{4} \bigl(-40I^{1}_{12110}-80I^{1}_{13110}+6I^{2}_{21000} \\&{} +12I^{2}_{22000}+8I^{2}_{23000} \bigr), \end{aligned}$$

(A.7)

$$\begin{aligned} c''_{kl}={}&\frac{1}{3} \biggl( \frac{k_{\text{B}}T}{c} \biggr)^{4} \bigl(40I^{1}_{12200}+80I^{1}_{13200}+6I^{2}_{21000} \\&{} +12I^{2}_{22000}+8I^{2}_{23000} \bigr), \end{aligned}$$

(A.8)

$$\begin{aligned} a'_{kl} ={}& {-} 2I_{12000}, \end{aligned}$$

(A.9)

$$\begin{aligned} a''_{kl} ={}& 2I_{12000} = -a'_{kl}. \end{aligned}$$

(A.10)

Here, as in the Hoogeveen article, the particle indexes $k$ and $l$ are suppressed on the right sides of the equations. The $10+2$ I-integrals described above are collision integrals, and have dimension $\sigma c$ [$\mbox{cm}^{3}/\mbox{s}$] and are defined by van Erkelens and van Leeuwen (1980):

$$\begin{aligned} &I^{\mu}_{i j \kappa \lambda h}(k,l) \\ &\quad = \biggl( \frac{c\pi}{z_{k}^{2}z_{l}^{2} K_{2}(z_{k}) K_{2}(z_{l})} \biggr) \biggl( \frac{c}{k_{\text{B}}T} \biggr)^{2i-j+\kappa+\lambda+6} \\ &\qquad {}\times \int_{(m_{k}+m_{l})c}^{\infty}dP_{kl} g_{kl}^{2i+2} P_{kl}^{-j+3}\bigl(g^{2}_{kl}+m_{k}^{2}c^{2} \bigr)^{\kappa/2} \\ &\phantom{\qquad \times \int_{(m_{k}+m_{l})c}^{\infty}}\times \bigl(g^{2}_{kl}+m_{l}^{2}c^{2} \bigr)^{\lambda/2} \\ &\qquad {}\times K_{j+h} \biggl( \frac{cP_{kl}}{k_{\text{B}}T} \biggr) \int_{0}^{\pi}\bigl(1-\cos^{\mu}\theta_{kl}\bigr) \frac{d\sigma_{kl}}{d\varOmega_{\text{CM}}} (P_{kl}, \theta_{kl} ) \\ &\phantom{\qquad \times K_{j+h} \biggl( \frac{cP_{kl}}{k_{\text{B}}T} \biggr) \int_{0}^{\pi}}\times \sin \theta_{kl} d\theta_{kl}, \end{aligned}$$

(A.11)

where

$$ g_{kl}=P_{kl}^{-1} \sqrt{ \tfrac{1}{4}\bigl(P_{kl}^{2}-m_{k}^{2} c^{2}-m_{l}^{2} c^{2} \bigr)^{2} - m_{k}^{2}m_{l}^{2}c^{4} } $$

(A.12)

is the invariant relative momentum.

Appendix B: Weak interaction cross-sections

The weak interaction cross-sections are given in Appendix C in Hoogeveen et al. (1986a). However we have included the tau particles in our model to give us the differential cross-sections for 68 (compared to 45) elastic collisions

$$ k+l \rightarrow k+l, $$

(B.1)

where

$$\begin{aligned} k &= \nu_{e}, \overline{\nu}_{e}, \nu_{\mu}, \overline{\nu}_{\mu}, \nu_{\tau}, \overline{ \nu}_{\tau}, \end{aligned}$$

(B.2)

$$\begin{aligned} l &= \nu_{e}, \overline{\nu}_{e}, \nu_{\mu}, \overline{\nu}_{\mu}, \nu_{\tau}, \overline{ \nu}_{\tau}, e^{-}, e^{+}, \mu^{-}, \mu^{+}, \tau^{-}, \tau^{+}. \end{aligned}$$

(B.3)

The weak interaction cross-sections from Eq. (A.11) is given as:

$$ \frac{d\sigma_{kl}}{d\varOmega_{\text{CM}}} = \frac{G_{\text{F}}^{2} s}{32\pi^{2} \hbar^{4}c^{2}} Sf(s,t), $$

(B.4)

where $S$ is equal to the product of the usual symmetry factor $1/n_{\text{f}}!$, where $n_{\text{f}}$ is the number of identical particles in the final state, and a factor 2 for each incoming neutrino, so e.g. for $e^{+}e^{+} \rightarrow e^{+}e^{+}$ we have $S=(1\times1)/2!$ and for $\nu_{e}\nu_{e} \rightarrow \nu_{e}\nu_{e}$ we have $S=(2\times2)/2!$. $f(s,t)$ is a collisions function for the two incoming particles and is given by:

$$\begin{aligned} f(s,t)={}&(v+a)^{2} \biggl(\frac{s-m^{2}c^{2}}{s} \biggr)^{2} \\&{} +(v-a)^{2} \biggl(\frac{s+t-m^{2}c^{2}}{s} \biggr)^{2} \\&{} +\bigl(v^{2}-a^{2} \bigr)\frac{2m^{2}c^{2}t}{s^{2}}, \end{aligned}$$

(B.5)

with the $a$ and $v$ matrices are related to the neutral current coupling (Hoogeveen et al. 1986a), basically how neutrinos couples to the other particles. They are listen in Table 3. The $s$ and $t$ being the Mandelstam variables:

Table 3

$v$ and $a$ matrices. w⁺ and w⁻ is abbreviation for $\sin ^{2} \theta _{\text{W}}$ +½and $\sin ^{2} \theta _{\text{W}}$ -½, respectively. The electromagnetic contribution is so small it can be ignored. The crossection $\sigma_{\gamma \nu}$ is zero

(a) a matrix

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(b) v matrix

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$$\begin{aligned} &s \equiv (p_{k}+p_{l})^{2} = P^{2}_{kl}, \end{aligned}$$

(B.6)

$$\begin{aligned} & t \equiv \bigl(p_{k}-p'_{k} \bigr)^{2}, \end{aligned}$$

(B.7)

where $s$ is the square of the center-of-mass energy (invariant mass) for the two incoming particles $k$ and $l$. $t$ is the square of the four-momentum transfer, and is related to the scattering angle in the center of momentum system, being defined below. $p'_{k}$ is the four-momentum of particle $k$ after collision, such that

$$\begin{aligned} t ={}& m_{k}^{2}c^{2}+m_{l}^{2}c^{2}-P_{kl}^{2} \\&{} + \frac{(P_{kl}^{2}+m_{k}^{2}c^{2}-m_{l}^{2}c^{2})(P_{kl}^{2}-m_{k}^{2}c^{2}+m_{l}^{2}c^{2})}{2P_{kl}^{2}} \\ &{}+ \frac{[P_{kl}^{2}-(m_{k}c+m_{l}c)^{2}][P_{kl}^{2}-(m_{k}c-m_{l}c)^{2}]}{2P_{kl}^{2}}\cos \theta_{kl}, \end{aligned}$$

(B.8)

with $\theta_{kl}$ being the scattering angle of the CM-system.

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Lars Husdal
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1.Department of PhysicsNorwegian University of Science and TechnologyTrondheimNorway

Viscosity in a lepton-photon universe

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Acknowledgements

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