A non-perturbative study of the evolution of cosmic magnetised sources

Abstract

We undertake a hydrodynamical study of a mixture of tightly coupled primordial radiation, neutrinos, baryons, electrons and positrons, together with a gas of already decoupled dark matter WIMPS and an already existing “frozen” magnetic field in the infinite conductivity regime. Considering this cosmic fluid as the source of a homogeneous but anisotropic Bianchi I model, we describe its interaction with the magnetic field by means of suitable equations of state that are appropriate for the particle species of the mixture between the end of the leptonic era and the beginning of the radiation-dominated epoch. Fulfilment of observational bounds on the magnetic field intensity yields a “near FLRW” (but strictly non-perturbative) evolution of the geometric, kinematic and thermodynamical variables. This evolution is roughly comparable to the weak field approximation in linear perturbations on a spatially flat FLRW background of sources in which the frozen magnetic fields are coherent over very large supra-horizon scales. Our approach and results may provide interesting guidelines in potential situations in which non-perturbative methods are required to study the interaction between magnetic fields and the cosmic fluid.

Appendices

Appendix 1: Local kinematic variables

The kinematics of local fluid elements can be described through covariant objects defined by the 4-velocity field $u^\alpha$. For a Kasner metric in the comoving frame endowed with a normal geodesic 4-velocity, the only non-vanishing kinematic parameters are the expansion scalar, $\theta$, and the shear tensor $\sigma _{\alpha \beta }$:

$$\begin{aligned} \theta =u^{\alpha }\,_{;\alpha },\quad \sigma _{\alpha \beta }=u_{(\alpha ;\beta )}-\frac{\theta }{3}h_{\alpha \beta }, \end{aligned}$$

(49)

where $h_{\alpha \beta }=u_{\alpha }u_{\beta }+g_{\alpha \beta }$ is the projection tensor and rounded brackets denote symmetrization. For the Kasner metric these parameters take the form:

$$\begin{aligned}&\theta =\frac{\dot{a}_{1}}{a_{1}}+\frac{\dot{a}_{2}}{a_{2}}+\frac{\dot{a}_{3}}{a_{3}}, \end{aligned}$$

(50)

$$\begin{aligned}&\sigma ^{\,\,\alpha }_{\beta }=\hbox {diag}\,\left[ \sigma ^{\,\,x}_{x},\sigma ^{\,\,y}_{y},\sigma ^{\,\,z}_{z},0\right] =\hbox {diag}\,\left[ \varSigma _{1},\varSigma _{2},\varSigma _{3},0\right] , \end{aligned}$$

(51)

where:

$$\begin{aligned} \varSigma _{\mathrm{a}}= & {} \frac{2}{3}\frac{\dot{a}_{\mathrm{a}}}{a_{\mathrm{a}}}-\frac{1}{3}\frac{\dot{a}_{\mathrm{b}}}{a_{\mathrm{b}}}- \frac{1}{3}\frac{\dot{a}_{\mathrm{c}}}{a_{\mathrm{c}}}, \quad \mathrm {a}\ne \mathrm {b}\ne \mathrm {c}\,\left( \mathrm {a},\mathrm {b},\mathrm {c}=1,2,3\right) . \end{aligned}$$

(52)

The geometric interpretation of these parameters is straightforward: $\theta$ represents the isotropic rate of change of the 3-volume of a fluid element, while $\sigma ^{\,\,\alpha }_{\beta }$ describes its rate of local deformation along different spatial directions given by its eigenvectors. Since the shear tensor is traceless: $\sigma ^{\,\,\alpha }_{\alpha }=0$, it is always possible to eliminate any one of the three quantities $\left( \varSigma _{1},\varSigma _{2},\varSigma _{3}\right)$ in terms of the other two. We choose to eliminate $\varSigma _{1}$ as a function of $\left( \varSigma _{2},\varSigma _{3}\right)$.

Finally, the substitution of (50) and (51) into the Einstein equations, allow us to obtain a first order system of differential equations.

Appendix 2: Particles

At temperature values in the range $100 \,\hbox {MeV}>T>m_e$ neutrons and protons are free in chemical equilibrium. The equilibrium is possible because of reactions transforming neutrons into protons and vice versa [89]:

$$\begin{aligned} \nu _e n \leftrightarrow e^- p \, , \quad \overline{\nu }_e p \leftrightarrow e^+ n \,, \end{aligned}$$

(53)

which implies the following variation of their numbers:

$$\begin{aligned} \frac{n_n}{n_p}=e^{-\frac{\varDelta M}{T}}, \quad \varDelta M = m_n - m_p\, . \end{aligned}$$

(54)

On the other hand, for temperature values less than $m_e$ the neutrons decay freely. Hence the densities of neutrons and protons satisfy

$$\begin{aligned} \dot{n}_n= & {} - \theta n_n - \varGamma n_n \,. \end{aligned}$$

(55)

$$\begin{aligned} \dot{n}_p= & {} - \theta n_p + \varGamma n_n \,, \end{aligned}$$

(56)

where $\varGamma =1/\tau _n$ is the decay rate ($\tau _n \approx 881.5\, \hbox {s}$). Also, from the charge neutrality we have $n_{e^-}=n_p$.

Finally, since we assume during the whole evolution a baryons/photon ratio $\eta = 5\times 10^{-5}$, we have

$$\begin{aligned} \frac{n_b}{n_\gamma }=\frac{n_n+n_p}{n_\gamma }=\eta , \end{aligned}$$

(57)

where $n_n$, $n_p$ and $n_\gamma$ are respectively the numbers density of neutrons, protons and photons. In this way we can roughly estimate the neutron and proton concentrations, a necessary task to obtain their rest energy. Since we are not interested in calculating element abundances, this rough approximated result is sufficient for our purposes. To improve these calculations the magnetic field should be included in the analysis [19], but this is beyond the scope of the present paper.

1.1 Appendix 2.1: Nucleosynthesis constraints on light elements

Since standard calculations of cosmological nucleosynthesis assume an FLRW universe with a radiation equation of state, it is worthwhile commenting on the effects of considering an anisotropic universe model on the primordial production of $^4 \hbox {He}$. As discussed in [91], the time dependence of the radiation density is very important in determining the helium abundance. This yields the following bound for the shear eigenvalues in a Bianchi I model during the nucleosynthesis: $\hbox {Y}_{\hbox {p}}<0.26$ requires $\left( \sigma /H\right) <0.2$ . In the previous equations $\hbox {Y}_{\hbox {p}}$ and $\hbox {H}$ denote the primordial mass fraction of $^4 \hbox {He}$ and expansion scalar, respectively, and $\sigma ^2=\left( \varSigma _1^{\,2}+\varSigma _2^{\,2}+\varSigma _3^{\,2}\right) /2$. However, for the above considered models with magnetic fields such that $\langle B_0 \rangle \lesssim 10^{-6} \hbox {G}$, we have that $\left( \sigma /H\right) \ll 1$ remains valid throughout the nucleosynthesis process (notice that we assumed the anisotropy of the fluid to be caused only by the magnetic field).