Cosmological magnetic fields: their generation, evolution and observation

Abstract

We review the possible mechanisms for the generation of cosmological magnetic fields, discuss their evolution in an expanding Universe filled with the cosmic plasma and provide a critical review of the literature on the subject. We put special emphasis on the prospects for observational tests of the proposed cosmological magnetogenesis scenarios using radio and gamma-ray astronomy and ultra-high-energy cosmic rays. We argue that primordial magnetic fields are observationally testable. They lead to magnetic fields in the intergalactic medium with magnetic field strength and correlation length in a well defined range.

We also state the unsolved questions in this fascinating open problem of cosmology and propose future observations to address them.

Appendices

Appendix A: Viscosities and Reynolds numbers

In this work the Reynolds number of a given scale which is inversely propositional to the viscosity plays an important role since it defines the time when the velocity field and with it the magnetic field on the given scale is damped into heat. We closely follow the treatment of Caprini et al. (2009c).

1.1 A.1 Kinematic viscosity

The kinematic viscosity is given by

$$ \nu=\frac{\eta}{\rho+p}, $$

(154)

where η is the shear viscosity. The kinematic viscosity characterizes the diffusion of transverse momentum due to collisions, and is given roughly by the mean free path $\ell_{\rm mfp}$ of the particles. In this appendix $\ell_{\rm mfp}$ is the physical mean free path, while $\lambda_{\rm mfp}$ appearing in the main text is the comoving mean free path. The relation is simply $\ell_{\rm mfp} =a\lambda_{\rm mfp}$.

A more precise expression for the shear viscosity is, see Weinberg (1971),

$$ \eta=\frac{4}{15} \frac{\pi^2}{30} g_* T^4 \ \ell_{\rm mfp} \quad \mbox {so~that}\quad\nu=\frac{\ell_{\rm mfp}}{5} . $$

(155)

The largest viscosity comes from the weakest interactions. However, non-interacting particles do not contribute to the viscosity. For this reason simple analytical approximations to the viscosity have unphysical jumps whenever a species decouples from the plasma.

Estimates from kinetic theory show that the shear viscosity of highly relativistic particles, T≫m, behaves as (to leading-log accuracy):

$$ {\eta}= C \frac{T^3}{g^4 \log g^{-1}}, $$

(156)

where g is the appropriate coupling constant (depending on the temperature and the length scale at which one wants to compute the Reynolds number) and C is a numerical coefficient that can only be obtained from a detailed analysis.

At temperatures larger than the electroweak phase transition, neutrino interactions are not suppressed. The shear viscosity is dominated by right-handed lepton transport and is given by Arnold et al. (2000)

$$ {\eta} \approx \biggl(\frac{5}{2} \biggr)^3 \zeta(5)^2 \biggl(\frac {12}{\pi } \biggr)^5 \frac{3/2}{9\pi^2+224 (5+1/2)} \frac{T^3}{g'^4\log g'^{-1}}, $$

(157)

where g′ is the hypercharge coupling. This leads to

$$ \nu( T\gtrsim100 \mbox{ GeV}) \approx\frac{21.6}{T} . $$

(158)

After electroweak symmetry breaking, neutrino interactions are suppressed by a factor (T/M _W )⁴. In this regime, neutrinos have the longest mean free path and dominate the viscosity. We use Heckler and Hogan (1993)

$$ \ell_{\rm mfp}\approx\bigl(3 G_F^2 T^5\bigr)^{-1} , $$

(159)

leading to

$$ \nu( T\lesssim100 \mbox{ GeV}) \approx4.9 \times10^8 \ \frac {\mbox {GeV}^4}{T^5} . $$

(160)

At temperatures smaller than 100 MeV, after the QCD phase transition, the remaining relativistic particles in the cosmic plasma are electron/positrons, neutrinos and photons and the neutrino mean free path increases to

$$ \ell_{\rm mfp}\approx\frac{10}{9} \bigl(G_F^2 T^5\bigr)^{-1} $$

(161)

such that

$$ \nu( T\lesssim100 \mbox{ MeV}) \approx1.6 \times10^9 \ \frac {\mbox {GeV}^4}{T^5} . $$

(162)

The neutrino mean free path determines the viscosity until neutrinos decouple at T∼1.4 MeV, after which photons take over. Below 1 MeV, when electrons and positrons annihilate and the remaining electrons become non-relativistic, the viscosity can be approximated by

$$ \nu(T<0.5 \mbox{ MeV}) \approx(\sigma_T n_e)^{-1} \simeq0.5\times 10^{-22} \mbox{ GeV}^{-1} \biggl( \frac{ \mbox{ GeV}}{T} \biggr)^3 . $$

(163)

After neutrino decoupling, the viscosity drops by about 30 orders of magnitude and the Reynolds number increases correspondingly. Therefore, all scales on which turbulence is maintained until T∼1 MeV will the remain turbulent until decoupling, T∼1 eV.

But even if turbulence is lost before neutrino decoupling, as long as the magnetic field survives, it will become turbulent again after T∼1 MeV and we expect equipartition between the magnetic field and the velocity field to be re-established.

Summing up all the results we find

$$ \nu(T) \approx \left \{ \begin{array}{l@{\quad}l} 21.6\text{ GeV}^{-1} (\frac{ \text{ GeV}}{T} )& \text{if } T\gtrsim100 \text{ GeV}\\ 4.9 \times10^8 \text{ GeV}^{-1} (\frac{ \text{ GeV}}{T} )^5 & \text{if } 100 \text{ GeV}> T\gtrsim100 \text{ MeV}\\ 1.6 \times10^9 \text{ GeV}^{-1} (\frac{ \text{ GeV}}{T} )^5 & \text{if } 100 \text{ MeV}> T\gtrsim1 \text{ MeV}\\ 0.5\times10^{-22} \text{ GeV}^{-1} (\frac{ \text{ GeV}}{T} )^3 & \text{if } 0.5\text{ MeV}> T\gtrsim1 \text{ eV} . \end{array} \right . $$

(164)

The unphysical jumps come from regions where our approximations are invalid. Nevertheless, at neutrino decoupling viscosity is significantly reduced and turbulence resumes on the relevant scales. This is different after matter and radiation equality since then the Alfvén speed decays and the coupling of the magnetic field to the velocity field soon becomes negligible.

The evolution of ν with temperature is plotted in Fig. 20.

Fig. 20

Evolution of the kinematic viscosity ν(T) as a function of temperature for T>1 eV. The unphysical discontinuities and kinks come from our crude approximation

1.2 A.2 Magnetic diffusivity

Here we derive expressions for the magnetic diffusivity also called resistivity for relativistic electrons in the cosmic plasma with temperatures 1 MeV <T<100 GeV. Again, we follow the treatment of Caprini et al. (2009c).

To determine the magnetic diffusivity, we derive an expression for the conductivity σ(T), which is the inverse of the diffusivity. The Lorentz force acting on an electron is

$$m_e\frac{du^\mu}{d\tau} = eF^{\mu\nu}u_\nu . $$

If we average this equation over a fluid element containing many electrons, the magnetic field term is sub-dominant. Even though the electrons are highly relativistic, the average fluid velocity is small. Furthermore $\gamma=1/\sqrt{1-v_{e}^{2}} \simeq T/m_{e}$ is nearly constant and we may neglect the contribution dγ/dτ from du ⁱ/dτ=d(γv ⁱ)/dτ above. With dτ=γ ⁻¹  dt=(m _e /T) dt, this yields the following equation for the mean velocity of the electron fluid:

$$\frac{d\mathbf{v}}{dt} = \frac{e}{T} {\bf E} . $$

If we denote the collision time for the electrons by t _c , they can acquire velocities of the order $\mathbf{v}\simeq\frac{e}{T} {\bf E} t_{c}$ between successive collisions. Hence the current is

$${\bf J} \simeq en_e{\bf v} \simeq t_c \frac{e^2n_e}{T}{\bf E}\equiv \sigma{\bf E} $$

so that the conductivity becomes

$$\sigma= t_c\frac{e^2n_e}{T} . $$

We now derive an estimate for t _c from Coulomb interactions. For a strong collision between the electron and another charged particle we need an impact parameter b such that e ²/b>E _e ≃T. Hence the cross section becomes σ _t ∼πb ²≃πe ⁴/T ² (this simple argumentation neglects the Coulomb logarithms which enhance the cross section by ln(1/α _min) where α _min is the minimal deflection angle (see Landau and Lifschitz 1990). With v _e =1 the time between collisions is therefore t _c =1/(σ _t n _e )≃T ²/(πe ⁴ n _e ) and

$$ \sigma\simeq\frac{T}{\pi e^2} . $$

(165)

Note that this result is independent of the electron density. This is physically sensible as n _e enhances the current on the one hand but it reduces in the same way the collision time.

With (165) we obtain for the magnetic diffusivity

$$ \frac{1}{\sigma} \simeq\frac{e^2(T)}{4T} \simeq \frac{10^{-1}}{T} - \frac{10^{-2}}{T} . $$

(166)

The first value applies close to T∼100 GeV, where α=e ²/4π∼0.1, while the second value corresponds to low energies, T∼1 MeV. For non-relativistic electrons we obtain the standard result for the conductivity by simply replacing T by the electron mass and multiplication by v ³≃(m _e /T)^3/2 so that (Spitzer 1978)

$$ \frac{1}{\sigma} \simeq\frac{e^2m_e^{1/2}}{T^{3/2}} . $$

(167)

1.3 A.3 Reynolds numbers and Prandl number

The kinematic Reynolds number is given by

$$ {\mathrm{R}_k}(T) = \frac{v_K\lambda_K}{\tilde{\nu}(T)} , $$

(168)

where ν is the kinematic viscosity, $v_{K} = \sqrt {k_{K}^{3}P_{v}(k_{K})}/(2\pi^{2})$ is the mean velocity which is roughly the velocity at the integral scale λ _K =2π/k _K , and $\tilde{\nu}= \nu /a$, see Sect. 4.

Correspondingly, the magnetic Reynolds number is defined by

$$ {\mathrm{R}_m}(T) = \frac{v_A\lambda_B}{1/\tilde{\sigma}(T)} . $$

(169)

Inserting the resistivity from Eq. (166) and the kinematic viscosity from Eqs. (160) or (162), assuming equipartition so that v _A =v _K and λ _B =λ _K , we obtain for the Prandl number

$$ {\mathrm{P}_m} \equiv\frac{{\mathrm{R}_m}(k,T)}{{\mathrm{R}_k}(k,T)} = \nu(T)\sigma (T)\simeq10^{12} \biggl(\frac{\rm GeV}{T} \biggr)^4 . $$

(170)

This number is larger than 1 for all temperatures 1 MeV<T≪100 GeV where the derivation applies.

The non-linearities in the Euler and induction equation are stronger than the damping term whenever the Reynolds numbers are larger than unity. In this regime MHD turbulence develops.

1.4 A.4 The Prandl number at very high energy

Finally let us consider the situation at very high temperature assuming that all particle interactions are given by the same coupling strength g ² and all particles are relativistic and in thermal equilibrium. This approximation is roughly valid above the electroweak scale. (We neglect strong interactions in this picture.) The cross section then is of the order of σ _c ≃g ⁴ T ⁻² and

$$ t_c=\lambda_{\rm mfp} = (\sigma_c n)^{-1} \simeq\frac{1}{g^4 T} \simeq \nu , \quad T> 100 \mbox{ GeV} . $$

(171)

This qualitatively reproduces Eq. (158). For the conductivity we have with the same approximations

$$ \sigma= t_c\frac{g^2n}{T} \simeq\frac{T}{g^2} , \quad T> 100\mbox { GeV} . $$

(172)

In this case the Prandl number becomes

$$ P \simeq g^{-2} \simeq10 , \quad T> 100\mbox{ GeV} . $$

(173)

It will be important for our discussions that at the electroweak phase transition T∼100 GeV, both, the kinetic viscosity and the magnetic diffusivity are actually of the same order. At significantly lower temperatures, the magnetic diffusivity is always much smaller than the kinetic viscosity. This is due to the fact that the kinetic viscosity is governed by the most weakly interacting particles, the neutrinos while the conductivity is of course determined by the stronger electromagnetic interactions.

Appendix B: Maxwell’s equation in curved spacetimes

We consider the 4-velocity u ^μ with u _μ u ^μ=−1 in an arbitrary curved spacetime and define the electric and magnetic fields as in Sect. 2, Eqs. (3),

$$ E_\mu= F_{\mu\nu}u^\nu , \qquad B_\mu= \epsilon_{\mu\nu\gamma }F^{\nu \gamma}/2 , $$

(174)

such that

$$ F_{\mu\nu} = u_\mu E_\nu-u_\nu E_\mu+ \epsilon_{\mu\nu\gamma }B^\gamma . $$

(175)

We define the expansion rate θ, the shear σ _μν , the vorticity ω _νμ and the acceleration a _μ of the 4-velocity u ^μ by

$$\begin{aligned} \theta =& u^\mu_{;\mu} , \qquad\sigma_{\mu\nu} = \frac {1}{2} \biggl(u_{\mu ;\nu} + u_{\nu;\mu} - \frac{1}{3} \theta h_{\mu\nu} \biggr) , \end{aligned}$$

(176)

$$\begin{aligned} \omega_{\mu\nu} =& \frac{1}{2} (u_{\mu;\nu} - u_{\nu;\mu} ) , \qquad a_\mu= u^\nu u_{\mu;\nu} . \end{aligned}$$

(177)

Here h _μν =g _μν +u _μ u _ν is the projector to the tangent space normal to u.

In terms of these quantities the homogeneous Maxwell equations, F _(μν,α)=0, become (see Barrow et al. 2007)

$$\begin{aligned} h_{\mu\nu}u^\alpha{B^{\nu}}_{;\alpha} + \epsilon_{\mu\nu\gamma }E^{\nu;\gamma} =& \biggl( \sigma_{\mu\nu} + \omega_{\mu\nu} +\frac{1}{3} \theta h_{\mu\nu} \biggr)B^{\nu} -\epsilon_{\mu\nu \alpha }a^\nu E^\alpha, \end{aligned}$$

(178)

$$\begin{aligned} {B^{\nu}}_{;\nu} =& \epsilon_{\mu\nu\alpha} \omega^{\nu\alpha }E^\mu . \end{aligned}$$

(179)

The 3-component ϵ-tensor is given in terms of the totally antisymmetric tensor η _βμνα by ϵ _μνα =u ^β η _βμνα , see Sect. 2.

Introducing the 4-current j _μ , the charge density ρ _e =−u ^μ j _μ and the 3-current J _μ =j _μ −ρ _e u _μ we obtain for the inhomogeneous Maxwell equations, F ^μν _;ν =j ^μ,

$$\begin{aligned} {E^{\nu}}_{;\nu} =& \rho_e - \epsilon_{\mu\nu\alpha}\omega ^{\nu\alpha }B^\mu , \end{aligned}$$

(180)

$$\begin{aligned} -h_{\mu\nu}u^\alpha{E^{\nu}}_{;\alpha} + \epsilon_{\mu\nu\gamma }B^{\nu;\gamma } =&- \biggl( \sigma_{\mu\nu} + \omega_{\mu\nu} +\frac{1}{3} \theta h_{\mu\nu} \biggr)E^{\nu} -\epsilon_{\mu\nu \alpha }a^\nu B^\alpha+J^\mu . \end{aligned}$$

(181)

The energy momentum tensor of the Maxwell field in terms of E, B and u is

$$\begin{aligned} T_{\mu\nu}^{\rm(em)} =& -F_{\mu\alpha}{F^\alpha}_\nu- \frac {1}{4}F_{\alpha \beta}F^{\alpha\beta}g_{\mu\nu} \\ =& \frac{1}{2}\bigl(E^2 + B^2 \bigr)u_\mu u_\nu+ \frac{1}{2}\bigl(E^2 + B^2\bigr)h_{\mu \nu} -E_\mu E_\nu-B_\mu B_\nu+ P_\mu u_\nu+ u_\mu P_\nu , \end{aligned}$$

(182)

where P _μ =ϵ _μαβ E ^α B ^β is the Poynting vector, i.e. the energy flux seen by an observer with 4-velocity u. The energy density is $\rho^{\rm (em)} = T_{\mu\nu}^{\rm(em)}u^{\mu}u^{\nu}= (E^{2} + B^{2})/2$ and $T_{\mu }^{\rm(em) \mu} =0$ as we expect it from the electromagnetic energy momentum tensor.

In a Friedmann Universe, for a comoving observer with u=a ⁻¹ ∂ _t , these equations simplify considerably. Since ω _μν =σ _μν =a _μ =0 and $\theta= 3\dot{a}/a^{2} = 3H$, we obtain with (B _μ )=a(0,B) and (E _μ )=a(0,E).

$$\begin{aligned} \partial_t\bigl(a^2 \mathbf{B}\bigr)+ a^2\mathbf{\nabla}\wedge\mathbf{E} =& 0, \end{aligned}$$

(183)

$$\begin{aligned} \mathbf{\nabla}\cdot\mathbf{B} =& 0 , \end{aligned}$$

(184)

$$\begin{aligned} \mathbf{\nabla}\cdot\mathbf{E} =& a\rho_e , \end{aligned}$$

(185)

$$\begin{aligned} -\partial_t \bigl(a^2\mathbf{E}\bigr) +a^2 \mathbf{\nabla}\wedge\mathbf {B} =& a^3\mathbf{J} . \end{aligned}$$

(186)

Note that B and E scale like 1/a ². In terms of the rescaled quantities a ² B, a ² E, and a ³ J, the Maxwell equations assume the same form as in Minkowski space, the expansion factor can be ‘scaled out’.