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Astron. Astrophys. 327, 1-7 (1997)

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1. General equations

In Paper I, we presented the basic equations governing the evolution of the large scale energy density structure in the prerecombination universe, when the effect of magnetic fields cannot be ignored. In the linear regime, the magnetic field configuration remains time-independent throughout this era, only growing with the expansion. The degree and the way this field structure affects the evolution of density inhomogeneities would depend on the particular pattern of the magnetic field lines. We will consider here a small cell in which the magnetic field has the simplest structure: a magnetic flux tube. Different sizes of the flux tube will be considered, for small, intermediate and large scales.

We will deal with the integration of Eq. (86) of Paper I

[EQUATION]

where: [FORMULA] is the relative energy density contrast, defined as [FORMULA], where [FORMULA] is the energy density and [FORMULA] the difference between its value within the inhomogeneity and its mean value in the Universe. As we are considering relativistic particles, we also have [FORMULA] where p is the hydrostatic pressure of the relativistic particles (either photons or any hot dark matter particles).

[FORMULA] is the second derivative with respect to the time-like variable [FORMULA].

[FORMULA] is defined as [FORMULA] where t is the time and [FORMULA] is the last time considered in this paper, close to equality. More specifically we have considered [FORMULA], or [FORMULA], before the acoustic epoch.

[FORMULA] is the Laplacian operator, when the spatial coordinates are [FORMULA] instead of [FORMULA], the usual comoving coordinates.

[FORMULA] is defined as [FORMULA], where K is the constant in the expansion law [FORMULA]. The value of K is [FORMULA] and therefore [FORMULA], where [FORMULA] are measured in seconds. The relation between [FORMULA] and d, the length measured in present-day-Mpc is [FORMULA].

X characterizes the magnetic energy density and is defined as

[EQUATION]

[FORMULA] would be the present magnetic field if no source and no loss other than expansion had taken place after the time period considered here. Because the post-recombination epoch is probably very complicated, concerning the evolution of magnetic fields, [FORMULA] has in practice no relation with present magnetic fields. [FORMULA] is exactly defined as [FORMULA]. We measure magnetic field strengths in [FORMULA], with the equivalence being 1 Gauss = [FORMULA].

[FORMULA] is the relativistic particle hydrostatic pressure at present. It is defined as [FORMULA] and we have chosen the value [FORMULA].

m also characterizes the magnetic field configuration

[EQUATION]

where

[EQUATION]

To integrate our basic equation it is then necessary to specify the field. In this paper we analyze the influence of a magnetic tube flux, i.e. [FORMULA] is given by

[EQUATION]

where r and [FORMULA] are measured with the same unit as [FORMULA]. For instance [FORMULA] is equivalent to a comoving-length of 0.28 Mpc . For our purposes, the solution very much depends on the value of [FORMULA] and we call it large scale if [FORMULA], intermediate scale, if [FORMULA], and small scale, if [FORMULA].

In the particular case of a flux tube:

[EQUATION]

where

[EQUATION]

and

[EQUATION]

For integrating, it is better to define another time-like variable as

[EQUATION]

Using this dimensionless time the basic equation becomes:

[EQUATION]

where now [FORMULA] and [FORMULA] are first and second derivatives with respect to [FORMULA].

From now on we will write simply t instead of [FORMULA] and [FORMULA] instead of [FORMULA] without any risk of confusion.

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© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998
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