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]:

- The sources: Instead of a classical white-noise source arbitrarily specified, the seeds of structures of the new theory are from quantum fluctuations, which obey equations derivable from the dynamics of the inflaton field, which is responsible for driving inflation.
- The spectrum: The almost scale-invariant spectrum (masses of galaxies as a function of their scales) has a more natural explanation from the almost exponential expansion of the inflationary universe than from the power-law expansion of the FRW universe in the traditional theory.

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].

7.1 The model

7.2 Einstein–Langevin equation for scalar metric perturbations

7.3 Correlation functions for scalar metric perturbations

7.4 Summary and outlook

7.2 Einstein–Langevin equation for scalar metric perturbations

7.3 Correlation functions for scalar metric perturbations

7.4 Summary and outlook

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