Abstract
Primordial density perturbations at astronomically large scales are necessary to seed the large scale structure formation in the Universe. In this chapter, we present the theory of gravitational instability in non-relativistic Newtonian theory and in General Relativity. The evolution of the density perturbations is regulated by the cosmological parameters and therefore the comparison between theoretical predictions and astronomical data is today a powerful tool to test the Standard Model of cosmology. The evolution of perturbations in modified gravity is also briefly discussed.
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Notes
- 1.
The assumption that \(\rho _b = const\) is technically essential because it allows to reduce the differential equations governing the evolution of perturbations to algebraic ones by Fourier transformation. For time dependent \(\rho (t)\), as it takes place in cosmology, one has either to find analytical solutions of an ordinary differential equation for \(\delta \rho _k (t)\) or to solve it numerically.
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Problems
Problems
12.1
Evaluate numerically the evolution of perturbations in the time dependent background (12.17) and (12.18).
12.2
Check that for \(P_b = w \rho _b\), with \(w = 0, 1/3\), and \(-1\), \(\rho _b\) indeed satisfies the first Friedmann equation (4.9).
12.3
Check that the covariant conservation of \(T_{\mu \nu }\) (4.12) in conformal time becomes
12.4
Show that during a quasi-de Sitter regime, namely in a period dominated by a vacuum-like energy, matter fluctuations decrease as the cube of the scale factor.
12.5
Solve Eq. (12.135) numerically for \(y = y_0 \cos (\varOmega _1 \tau )\) for different \(y_0\) and \(\varOmega _1\) to observe the effects of parametric resonance and anti-friction (mentioned in Sect. 12.3.3).
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Bambi, C., Dolgov, A.D. (2016). Density Perturbations. In: Introduction to Particle Cosmology. UNITEXT for Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48078-6_12
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