## 7. Density perturbationsLet us define the relative perturbed density as: ( is not so defined by other authors. Some
others adopt which becomes in this case
times the value of our
). The final purpose is to deduce
, i.e. how density inhomogeneities evolve,
under the action of gravity and an unperturbed underlying magnetic
configuration. To find the differential equation for
, 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 ,
, , as
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
but just its trace Here and would
actually be the present values of As suggested by the equations themselves, we now change the temporal and spatial variables, and use the definition (64) of . As new spatial variables we will use where in the radiation-dominated era. Its value is and is the cosmological scale factor at the last time in the period considered in this paper, which is close to Equality. For we have adopted the value . It should be noted that are comoving coordinates measured in seconds. They now coincide with present coordinates. (It is easily obtained that ). However is dimensionless where As a temporal variable let us choose where is the last time of the period considered here, close to Equality, corresponding to , i.e. . is a time variable increasing backwards from future to past. We then introduce some time independent functions defining the magnetic pattern: where is the gradient using instead of . In the unit system we are using . 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
, instead of with respect to Taking in Eq. (84) and in equation (85) and subtracting, inserting this in Eq. (82) to obtain , calculating in Eq. (83), taking into account the obtained functions and in Eq. (84), we obtain a differential equation containing only as a variable, 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 ). The following section contains a preliminary insight. © European Southern Observatory (ESO) 1997 Online publication: April 20, 1998 |