It is amazing that the universe has been such a good conductor. Therefore, a seed magnetic field can hardly be generated in it. Conversely, seed magnetic fields were kept frozen and preserved in the conducting plasma state of the universe if they were generated by a whatever mechanism in the early universe once. There has been a lot of proposals for generating magnetic fields in the cosmic plasma via astrophysical magnetogenesis. However, the best way is thought that to generate primordial magnetic fields from photon quantum fluctuations
in a vacuum state for avoiding the high conductance. Unfortunately, this thought failed due to the conformal invariance of electromagnetism.
In this chapter, we use the axial coupling mechanism that assuming the EM field couples to a pseudo-scalar inflaton. We try to find the possibility of generating a significant PMF outside the Hubble radius. By the spinodal instability and a fast-roll regime, copious PMFs can be generated in the inflationary epoch. However, the inflationary process does not evolve in accordance with the constraint of the energy density of the magnetic field ρB on any viable inflationary magnetogenesis mechanism (Fig.
2.3). The energy deposited in the magnetic field
at N < 2 increases the ratio of ρB/ρφ up to about 54 orders greater than unity. The boost in the magnetic energy density due to the conversion of the inflationary kinetic energy by the spinodal instability make ρB≫ ρ
φ at most times along the history of cosmic inflation.Therefore, we are going to try to use a different method to generate the magnetic field.
Figure 2.2: The spectrum of the ratio of the magnetic energy density to the photon energy density during the inflation. The coupling constant is α = 50.8. The peak mode at q = 425.5 is corresponding to a comoving distance of about 10 Mpc.
Figure 2.3: The evolution of the energy density ratio ρB within the inflationary epoch.
As in Fig. 2.2the peak mode crossing out the horizon at q∼ 425 corresponds to a comoving distance of about 10 Mpc. Apparently, ρB≫ ρφ for most of the time in the history of the inflationary expansion.
Chapter 3
Inflationary dilaton-axion magnetogenesis
We have assumed that the EM field couples to a pseudoscalar inflaton φ to generate a sizable PMF outside the Hubble radius, and tried to boost the PMF by the spinodal instability in the pre-inflationary regime. However, too much magnetic power will be produced in such an axial coupling mechanism that violate the energy constraint, which is required by the inflaton field driving the de Sitter like expansion. In this chapter, we will combine both the spinodal instability in the axial coupling mechanism and the dilaton electromagnetism.
3.1 Dilaton-axion electromagnetism
From Eq. (2.6), we consider
S =
∫
d
4x √
−g
[−
14
F
µνF
µν− α
4f
φ ˜ F
µνF
µν ],
(3.1)where Fµν = ∂µ
A
ν− ∂
νA
µ, and α = e2/(4π) is the fine structure constant now. Here use the spatially flat FRW metric Eq. (2.7). After rescaling the vector potential by the couplingconstant, Aµ
→ eA
µ,the action (2.10) becomesWe adopt a time-dependent coupling constant I(η) = I(σ(η)) = 1/e2, which is described by a time function with a dilaton field σ. Inspired by extra-dimension theories, we assume that the energy scale is related to the reduced Planck mass by f = Mp/S(η). In many string theory constructions, this relation has a simple geometrical origin. The factor S is determined by a combination of the size of the compactification manifold, the string length, and the string coupling [32]. The factor S may depend on time, as extra dimension and string parameters may evolve over time. We define a time function J(η) = Sφ/(4πMp) for monitoring the axial effect on the processes of magnetogenesis. The action can be rewritten by the pair of coupling functions I(η) and J(η) as
And then we can get the wave equation for the vector potential from the action (3.3):
∂
2A ⃗
To get the equation of motion for the mode functions, we decompose the gauge field ⃗
A(η, ⃗ x)
by Eqs. (2.10) and (3.4):Therefore, the energy density of the produced EM fields is dominated by the vacuum expec-tation value, electromagnetism, this means EM mode functions A± in Eq. (3.5) behaved as simply plane
wave solutions. It is the so-called dilaton electromagnetism as dictated by the conformal invariance of the action that limited the growth of EM fields in inflation. We will review the efficiency of each mechanism in following contexts shortly, which generated PMF by breaking the conformal invariance of the EM field in inflation.
In the strong EM coupling case, i.e., I(η) < 1, the coupling constant is inversely propor-tional to I, and it will become to an order one number at the end of inflation. The PMF can be produced strong enough with extremely large effective coupling constant in the beginning of inflation, but the extremely strongly coupled EM theory should not be trusted at all. And in the weak EM coupling case, i.e., I(η) > 1, no sizeable PMF can be produced [33]. The e-folding N is defined by N (t) = ∫t
0
H(t
′)dt′, where dt = adη, and we find that our inflation ends at N≃ 60. Since we control the production of the PMF ρ
EM≪ ρ
ϕ+ ργ during inflation, we can neglect the backreaction from EM field on the inflationary background. We do not specify any model for the dynamics of the dilaton or axion during inflation to make a more general scenario of inflationary magnetogenesis. Therefore, we define thatI(η) =
[
a(η
f)a(η)
]n
, J
′(η) = cH(η)ap(η), (3.7)where n, p, and c are constant parameters within time intervals, and ηf denotes the time at the end of inflation. Also, we set I(η) = 1 and J′(η) = 0 when η
≥ η
f, so the EM theory becomes Maxwellian electromagnetism after inflation.Then we insert I(η) and J′(η) in Eq. (3.7) into mode function Eq. (3.5). We set the initial conditions at η = ηi to solve for the mode functions:
A
±(ηi, k) =
1Then we calculate the spectral magnetic energy density,
dρ
Bd ln k
=k
5I
4π2a
4end(
|A
+|
2+|A
−|
2),
(3.9)and its ratio to the energy density of the thermal background at the end of inflation,
By these equations we can find the relation of the parameters and the possibility of the generating of the magnetic field via astrophysical magnetogenesis [34]. At first, we set n = 1, therefore the electric energy density will dominate over the magnetic energy density during this weak EM coupling regime. The other constants are specified as c = 0.5 and p = 0.064, which leads the electric and magnetic energy densities to grow exponentially near N (t)
∼ 35.
We change n to n =
−3.1 after N ∼ 35 to avoid over-production, and adopt a wavenumber
cutoff by assuming that c = 0 for q = k/H(ηi) > 300. The strong EM coupling leads to the electric fields damp out while the magnetic fields frozen. The wavenumber cutoff prevents the rapid production of high k-modes from affecting the backreactions, here we just ignore these modes. The result is showing in Fig.3.1. Because of the wavenumber cutoff we got the
Figure 3.1: Left panel: Solid line shows the time evolution of the electromagnetic energy density over the total energy density during inflation, ρEM/(ρϕ+ ργ). The time is counted with e-folding number N (t) from the beginning of inflation. The dashed (dotted) line denotes the electric (magnetic) component. The horizontal dotted-dashed line denotes the conservation of energy. Right panel: Solid line shows the ratio of the spectral energy density of primordial magnetic fields to the thermal background ργ at the end of inflation versus the dimensionless comoving wavenumbers q. The horizontal dotted-dashed line denotes the constraint of magnetic energy density. The mode with q = 1 corresponds to the size of the present universe while the spectrum peaks at q = 300 on a comoving scale larger than Mpc.