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Astron. Astrophys. 326, 13-22 (1997)

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7. Density perturbations

Let us define the relative perturbed density as:

[EQUATION]

([FORMULA] is not so defined by other authors. Some others adopt [FORMULA] which becomes in this case [FORMULA] times the value of our [FORMULA]). The final purpose is to deduce [FORMULA], i.e. how density inhomogeneities evolve, under the action of gravity and an unperturbed underlying magnetic configuration. To find the differential equation for [FORMULA], we must perform some algebra because our set of equations is still rather large and complicated. We must first choose which equations will be handled, because as stated before, they are not all independent. Our unknowns are [FORMULA], [FORMULA], [FORMULA], as [FORMULA] has already been determined as a function of time. We therefore have ten unknowns. The Einstein field equations are also made up of ten equations, and from them the equations of motion and energy are derivable. On the other hand, Maxwell's equations have already been used to deduce the evolution of the magnetic field and the electrical current. Though we could therefore use only the Einstein field equations, the final objective is straightforwardly reached with the equation of motion (54), the equation of energy balance (49), and Eqs. (62) and (63) from the whole set of Einstein field equations, selected because they do not contain the tensor [FORMULA] but just its trace h. The unknowns are now [FORMULA], [FORMULA] and h. We still have more equations than unknowns. Let us rewrite the equations with the introduction of "present day" quantities, with the subindex 0.

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

Here [FORMULA] and [FORMULA] would actually be the present values of p and [FORMULA] if our equations were valid for all subsequent periods. The law [FORMULA] is valid for photons in the Acoustic and Post-Recombination eras. However [FORMULA] has recently been affected by non-linear effects and probably does not coincide with the present magnetic field. While [FORMULA] and [FORMULA] are constant, [FORMULA] and [FORMULA] are not. Neither [FORMULA] nor [FORMULA] represent actual present values, but they are preferred as variables because in the variations of [FORMULA] and [FORMULA] the effect of pure expansion is suppressed. We then have:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

As suggested by the equations themselves, we now change the temporal and spatial variables, and use the definition (64) of [FORMULA]. As new spatial variables we will use

[EQUATION]

where K is the constant in the expansion law

[EQUATION]

in the radiation-dominated era. Its value is

[EQUATION]

and [FORMULA] is the cosmological scale factor at the last time in the period considered in this paper, which is close to Equality. For [FORMULA] we have adopted the value [FORMULA]. It should be noted that [FORMULA] are comoving coordinates measured in seconds. They now coincide with present coordinates. (It is easily obtained that [FORMULA]). However [FORMULA] is dimensionless

[EQUATION]

where d is the distance in Mpc.

As a temporal variable let us choose

[EQUATION]

and therefore

[EQUATION]

where [FORMULA] is the last time of the period considered here, close to Equality, corresponding to [FORMULA], i.e. [FORMULA]. [FORMULA] is a time variable increasing backwards from future to past. We then introduce some time independent functions defining the magnetic pattern:

[EQUATION]

[EQUATION]

[EQUATION]

where [FORMULA] is the gradient using [FORMULA] instead of [FORMULA]. In the unit system we are using [FORMULA]. In this way all the magnitudes of the different quantities involved are close to unity and the equations become extremely simple.

We also change the nomenclature, so that a point over a quantity's symbol now means its derivative with respect to [FORMULA], instead of with respect to t. Thus we have

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

Taking [FORMULA] in Eq. (84) and [FORMULA] in equation (85) and subtracting, inserting this [FORMULA] in Eq. (82) to obtain [FORMULA], calculating [FORMULA] in Eq. (83), taking into account the obtained functions [FORMULA] and [FORMULA] in Eq. (84), we obtain a differential equation containing only [FORMULA] as a variable, [FORMULA]

[EQUATION]

This is our basic equation, an elliptic linear second order differential equation with variable coefficients. The complete discussion may have complications arising from its elliptic nature (if we prefer not to choose a boundary condition for [FORMULA]). The following section contains a preliminary insight.

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© European Southern Observatory (ESO) 1997

Online publication: April 20, 1998
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