MODERN COSMOLOGY

56 An introduction to the physics of cosmology

Figure 2.8.A plot of transfer functions for various models. For adiabatic models,Tk→ 1
at smallk, whereas the opposite is true for isocurvature models. A number of possible
matter contents are illustrated: pure baryons; pure CDM; pure HDM; MDM (30% HDM,
70% CDM). For dark-matter models, the characteristic wavenumber scales proportional
to
h^2. The scaling for baryonic models does not obey this exactly; the plotted cases
correspond to
=1,h= 0 .5.

We consider the following cases:

(1) adiabatic CDM;

(2) adiabatic massive neutrinos (one massive, two massless); and

(3) isocurvature CDM; these expressions come from Bardeenet al (1986;

BBKS).

Since the characteristic length-scale in the transfer function depends on the
horizon size at matter–radiation equality, the temperature of the CMB enters.
In these formulae, it is assumed to be exactly 2.7 K; for other values, the
characteristic wavenumbers scale∝T−^2. For these purposes massless neutrinos
count as radiation, and three species of these contribute a total density that is 0.68
that of the photons.

(1) Tk=

ln( 1 + 2. 34 q)

2. 34 q

[ 1 + 3. 89 q+( 16. 1 q)^2 +( 5. 46 q)^3 +( 6. 71 q)^4 ]−^1 /^4

(2) Tk=exp(− 3. 9 q− 2. 1 q^2 )

(3) Tk=( 5. 6 q)^2 ( 1 +[ 15. 0 q+( 0. 9 q)^3 /^2 +( 5. 6 q)^2 ]^1.^24 )−^1 /^1.^24.

The case of mixed dark matter (MDM: a mixture of massive neutrinos and CDM)
is more complex. See Pogosyan and Starobinksy (1995) for a fit in this case.