Structure formation in the early universe is a key problem in modern cosmology. It is believed that galaxies,
clusters and all large-scale structures observed today originated from random sources of primordial
inhomogeneities (density contrast) in the early universe, amplified by the expansion of the universe.
Theories of structure formation based on general relativity theory have been in existence for
over 60 years [27, 241, 242] (see, e.g., [17, 284
, 300]), long before the advent of inflationary
cosmology [233
, 245
, 269]. But the inflation paradigm [2, 140, 243, 244] provided at least two major
improvements in the modern theory of cosmological structure formation [19, 141, 159, 270
]:
Stochastic gravity provides a sound and natural formalism for the derivation of the cosmological
perturbations generated during inflation. In [316] it was shown that the correlation functions that follow
from the Einstein–Langevin equation, which emerges in the framework of stochastic gravity, coincide with
that obtained with the usual quantization procedures [270
] when both the metric perturbations and
the inflaton fluctuations are linearized. Stochastic gravity, however, can naturally deal with
the fluctuations of the inflaton field even beyond the linear approximation. In Section 7.4 we
will enumerate possible advantages of the stochastic-gravity treatment of this problem over
the usual methods based on the quantization of the linear cosmological and linear inflaton
perturbations.
We should point out that the equivalence at the linearized level is proved in [316] directly from the field
equations of the perturbations and by showing that the stochastic and the quantum correlations are both
given by identical expressions. Within the stochastic gravity framework an explicit computation of the
curvature perturbation correlations was performed by Urakawa and Maeda [353
]. A convenient
approximation for that computation, used by these authors, leads only to a small discrepancy with the
usual approach for the observationally relevant part of the spectrum. We think the deviation from
the standard result found for superhorizon modes would not arise if an exact calculation were
used.
Here we illustrate the equivalence with the conventional approach with one of the simplest chaotic
inflationary models in which the background spacetime is a quasi de Sitter universe [315, 316
].
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