MODERN COSMOLOGY

204 Dark matter and particle physics

As for the redshiftZNRat which gravitinos become non-relativistic, it
corresponds to the epoch at which their temperature becomesm 3 / 2 /3, that is:

ZNR

(

g∗(Tψμd)

g∗S(T 0 )

) 1 / 3

m 3 / 2 / 3

T 0

= 4. 14 × 106 ×

(

g∗(Tψμd)

100

) 1 / 3 (

m 3 / 2

1keV

)

, (5.44)

whereT 0 = 2 .726 K is the temperature of the CMB at present time. OnceZNRis
known, one can estimate the free-streaming length until the epoch of the matter–
radiation equality,λFS, which represents a quantity of crucial relevance for the
formation of large-scale cosmic structures.
The free-streaming length for the thermal gravitinos is about 1 Mpc (for
ZNR ∼ 4 × 106 ) which, in turn, corresponds to∼ 1012 M , if it is required
to provide a density parameter close to unity. This explicitly shows that light
gravitinos are actually WDM candidates. We also note that, takingh= 0 .5, the
requirement of not overclosing the universe turns intom 3 / 2 ≤200 eV.
However, critical density models with pure WDM are known to suffer from
serious troubles [41]. Indeed, a WDM scenario behaves much like CDM on scales
aboveλFS. Therefore, we expect in the light gravitino scenario that the level of
cosmological density fluctuations on the scale of galaxy clusters (∼ 10 h− 01 Mpc)
to be almost the same as in CDM. As a consequence, the resulting number density
of galaxy clusters is predicted to be much larger than what is observed [42].
This is potentially a critical test for any WDM-dominated scheme, the
abundance of high-redshift galaxies having been already recognized as a non-
trivial constraint for several DM models. It is, however, clear that quantitative
conclusions on this point would at least require the explicit computation of the
fluctuation power spectrum for the whole class of WDM scenarios.

5.7 Dark energy,CDM and xCDM or QCDM

The expansion of the universe is described by two parameters, the Hubble
constantH 0 and the deceleration parameterq 0 :

(i) H 0 ≡R ̇(t 0 )/R(t 0 ),whereR(t 0 )is the scale factor,t 0 the age of the universe

at present epoch, and we have

H 0 = 65 ±5kms−^1 Mpc−^1 (h= 0. 65 ± 0. 05 ); (5.45)

(ii) q 0 ≡−(R ̈(t 0 )/H 02 )R(t 0 )states whether the universe is accelerating or

decelerating.q 0 is related to
 0 as follows

q 0 =

0

2

+

3

2

∑

i

iwi (5.46)