Primordial non-Gaussianity in the cosmological perturbations - CBPF

A given comoving scale λ is therefore projected on the last-scattering surface sky on an angular scale
λ
θ , (158)
(τ0 − τls)
where we have neglected tiny curvature effects. Consider now that the scale λ is of the order of
the comoving sound horizon at the time of last-scattering, λ ∼ csτls, where cs 1/ √ 3 is the sound
velocity at which photons propagate in the plasma at the last-scattering. This corresponds to an
angle
τls
θ cs
(τ0 − τls)
τls
cs
τ0
, (159)
where the last passage has been performed knowing that τ0 ≫ τls. Since the universe is MD from the
time of last-scattering onwards, the scale factor has the following behaviour: a ∼ T −1 ∼ t 2/3 ∼ τ 2 ,
where we have made use of the relation (139). The angle θHOR subtended by the sound horizon on
the last-scattering surface then becomes
θHOR cs
1/2 T0
Tls
∼ 1 ◦ , (160)
where we have used Tls 0.3 eV and T0 ∼ 10 −13 GeV. This corresponds to a multipole ℓHOR
ℓHOR = π
θHOR
200. (161)
From these estimates we conclude that two photons which on the last-scattering surface were
separated by an angle larger than θHOR, corresponding to multipoles smaller than ℓHOR ∼ 200 were
not in causal contact. On the other hand, from Fig. 6 it is clear that small anisotropies, of the
same order of magnitude δT/T ∼ 10 −5 are present at ℓ ≪ 200. We conclude that one of the striking
features of the CMB fluctuations is that they appear to be noncausal. Photons at the last-scattering
surface which were causally disconnected have the same small anisotropies! The existence of particle
horizons in the standard cosmology precludes explaining the smoothness as a result of microphysical
events: the horizon at decoupling, the last time one could imagine temperature fluctuations being
smoothed by particle interactions, corresponds to an angular scale on the sky of about 1 ◦ , which
precludes temperature variations on larger scales from being erased.
To account for the small-scale lumpiness of the universe today, density perturbations with
horizon-crossing amplitudes of 10 −5 on scales of 1 Mpc to 10 4 Mpc or so are required. As can
be seen in Fig. 5, in the standard cosmology the physical size of a perturbation, which grows as the
scale factor, begins larger than the horizon and relatively late in the history of the universe crosses
inside the horizon. This precludes a causal microphysical explanation for the origin of the required
density perturbations.
From the considerations made so far, it appears that solving the shortcomings of the standard
Big Bang theory requires two basic modifications of the assumptions made so far:
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