(a) Show that the mass inside the Hubble sphere MH grows as R^3 / 2 in a matter-dominated model.

Step 1: u000AIn a matteru002Ddominated model, the mass inside the Hubble sphere $M_{u005Cmathrm{H}}$ is give

(a) Show that the mass inside the Hubble sphere MH grows as...

(a) Show that the mass inside the Hubble sphere $M_{\mathrm{H}}$ grows as $R^{3 / 2}$ in a matter-dominated model. Find the corresponding rate of growth in the radiation-dominated case. (b) Perturbation $\delta \rho / \rho$ grow proportional to $R$ in the matter-dominated case and proportional to $R^2$ in the radiation-dominated case. For perturbations $(\delta \rho / \rho)_t$ outside the horizon at time $t$ show that in both radiation-dominated and matter-dominated models $$ \left(\frac{\delta \rho}{\rho}\right)_t=M_{\mathrm{H}}^{-2 / 3}\left(\frac{\delta \rho}{\rho}\right)_{\mathrm{H}} . $$

(a) Show that the mass inside the Hubble sphere $M_{\mathrm{H}}$ grows as $R^{3 / 2}$ in a matter-dominated model. Find the corresponding rate of growth in the radiation-dominated case.
(b) Perturbation $\delta \rho / \rho$ grow proportional to $R$ in the matter-dominated case and proportional to $R^2$ in the radiation-dominated case. For perturbations
$(\delta \rho / \rho)_t$ outside the horizon at time $t$ show that in both radiation-dominated and matter-dominated models
$$
\left(\frac{\delta \rho}{\rho}\right)_t=M_{\mathrm{H}}^{-2 / 3}\left(\frac{\delta \rho}{\rho}\right)_{\mathrm{H}} .
$$

Key Concepts

Cosmological Perturbations

Cosmological perturbations refer to small deviations from the perfectly homogeneous and isotropic background of the universe. They are key to understanding how structures, such as galaxies and clusters, originated and evolved. The growth rates of these perturbations depend on the underlying cosmological model and the dominant form of energy density (matter or radiation), with different scaling behaviors inside and outside the horizon. The study of these perturbations provides insight into the initial conditions and the evolution of the large-scale structure of the cosmos.

Radiation-Dominated Universe

In a radiation-dominated universe, the energy density is primarily contributed by relativistic particles, such as photons and neutrinos. The dynamics of the expansion differ from the matter-dominated case, leading to different scaling behaviors for quantities like the mass inside the Hubble sphere and the growth rates of density perturbations. Understanding this regime is essential for describing the early universe, particularly in the context of the cosmic microwave background and primordial nucleosynthesis.

Hubble Sphere

The Hubble sphere is a cosmological concept representing the region of the universe surrounding an observer within which objects recede at velocities less than or equal to the speed of light due to cosmic expansion. It is defined by the Hubble radius, which is the inverse of the Hubble parameter. This concept plays a critical role in understanding the causal structure and dynamic mass content of the universe at any given time.

Cosmological Expansion and Scale Factor

The scale factor, commonly denoted as R (or a(t)), quantifies the expansion of the universe in cosmological models. It relates the physical distance between points in space at different times and is directly connected to the redshift of distant objects. The behavior of the scale factor governs how various physical quantities, such as the mass inside the Hubble sphere and density perturbations, evolve over time in an expanding universe.

Matter-Dominated Universe

A matter-dominated universe is one in which the dynamics of cosmic expansion are primarily influenced by non-relativistic matter (typically cold dark matter and baryons). In such a universe, the scale factor grows as a specific power law with time, and the evolution of gravitational potentials and density perturbations follow distinct growth laws. This regime is crucial for understanding structure formation and the evolution of large-scale cosmic structures during certain epochs.

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