**3.3. The effect on the electron thermodynamics**

In the above we discussed how the phase-space of electrons and positrons is modified by the presence of strong magnetic fields and how this effect changes the weak processes rates. The consequences of the variation of the electron phase-space, however, go well beyond that effect. Electron and positron thermodynamics functions will also be affected. In fact, by applying the prescription (3.10), we find that the number density, the energy density and the pressure of the electron-positron gas are now given by

(3.26) | |

(3.27) | |

(3.28) |

where

(3.29) |

is the Fermi-Dirac distribution function, and
*E*_{n}(*p*_{z}) is given by
(3.9). As for the case of the weak processes rates, it is
possible to show that Eqs. (3.26-3.28) reduce to the
their well know standard from in the limit *B*
0 (see
e.g. Ref. [98]).

Numerical computations
[94]
show that, for small *T*,
_{e}
grows roughly linearly with *B* when *B* >
*B*_{c}. This effect is
mainly due to - 1) the reduction, for each Landau level, of the area
occupied by the cyclotron motion of the electron in plane perpendicular
to the field; 2) the growth of the energy
gap among the lowest Landau level and the *n* > 0 levels, which
produces an overpopulation of the lowest Landau level. The first
effect is the dominant one. The net number density and the pressure of
the electron-positron gas follow a similar behaviour. As we
already noted in Sec. 3.1, the energy cost of
producing the electron-positron pairs excess cannot be paid by the
magnetic field which is supposed to be static. Rather, the "power bill" is
paid from the heat-bath, or better, from its photon component
[96,
99].
Especially in the context of BBN this
point is quite relevant since the energy transfer from the photons
to the lowest Landau level of the electron-positron gas will
affect the expansion rate of the Universe, its cooling rate and
the effective baryon-to-photon ratio
[94,
95,
98]. We start
discussing the first two effects.
We observe that the growth of the electron and positron energy
density, due to the presence of the magnetic field, gives rise
to a faster expansion rate of the Universe.
This point was first qualitatively discussed by Matese and
O'Connell [87]
and recently analyzed in more detail by
Grasso and Rubinstein
[94].
The time-temperature
relation will also be modified. The relevance of latter effect has
been first showed by Kernan, Starkman and Vachaspati
[98]
by solving numerically the relation

(3.30) |

where _{em}
_{e}
+ _{} and
*p*_{em}
*p*_{} + *p*_{e} are the energy
density and the pressure of the electromagnetic component of the
primordial heat-bath. In agreement with our previous
considerations, Eq. (3.30) has been obtained by
imposing energy conservation of the electromagnetic component plasma.

For small values of the ratio *eB*/*T*^{2}, the most
relevant effect of the magnetic field enters in the derivative
*d*_{em} /
*dT*_{} that is smaller than the free field value.
This effect can be interpreted as a delay in the electron-positron
annihilation time induced by the magnetic field. This will give
rise to a slower entropy transfer from the electron-positron pairs
to the photons, then to a slower reheating of the heat bath. In
fact, due to the enlarged phase-space of the lowest Landau level
of electrons and positrons, the equilibrium of the process
*e*^{+} *e*^{-} <->
is shifted
towards its left side. Below we
will discuss as this effect has a clear signature on the deuterium
and ^{3}He predicted abundances. Another point of interest is that
the delay in the *e*^{+} *e*^{-} annihilation
causes a slight decrease in
the *T*_{} /
*T* ratio with respect to the canonical value,
(4/11)^{1/3}
[98].

The delay in the entropy transfer from the *e*^{+}
*e*^{-} gas to the heat-bath induces also an increment in
the value of baryon-to-photon ratio
. In the
absence of other effects a
larger value of
would induce smaller relic abundances of
Deuterium and ^{3}He. This effect, was first predicted in the
Refs. [94,
95],
Furthermore, it is interesting to observe
that in the case the primordial magnetic field is inhomogeneous
and it is confined in finite volume regions where its strength
exceed the cosmic mean value (e.g. flux tubes), this effect may give
rise to spatial variation in the relic element abundances.