The Green functions

The previous results show that the linear density field can be written in general as and

The actual growing and decaying modes can then be obtained by inverting this system. For instance, for an EdS background, one gets

with similar results for the velocity divergence. Following Scoccimarro (1998), this result can be encapsulated in a simple form after one introduces the doublet Ta(k, r),

where a is an index whose value is either 1 (for the density component) or 2 (for the velocity component). The linear growth solution can now be written as

where ga is the Green function of the system. It is usually written with the following time variable:

(not to be mistaken for the conformal time). For an EdS universe, one has explicitly

We will see in the following that, provided the doublet Ta is properly defined, this form remains practically unchanged for any background.

The general background case

For a general background, it is fruitful to extent the definition of the doublet to


Defining 6 = —6(x, n)/f+ and for the time variable n, the linearized motion equations read

which can be rewritten as with

In general, the formal solution of this system can be written in terms of a Green function gj3(n, По)- The latter satisfies the differential equation

with the condition

where Sj3 is the identity matrix. It should be noted that the Green function can be written formally in terms of the particular solutions of the systems. For instance, if one considers the growing and decaying solution uj+)(n) and uj ц), the Green function can be written as

where the constants cba)(n0) are set such that (2.39) is satisfied.

With the definition (2.32) of Фа, the growing and decaying modes are, to a (surprisingly) good approximation, given by

This is due to the fact that Dm/f+ « 1 in most of the regimes and models that we consider.

As a result, in practice, we will always use the form (2.31) for the Green function.

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