Show that ΩH^2 R^3= constant in a matter-dominated Universe....

Name: Problem 17, Chapter 5: An introduction to the science of cosmology
Uploaded: 2020-05-27T02:28:33-08:00
Duration: 2 min 14 s
Channel: Mark Mathison
Description: Show that ΩH^2 R^3= constant in a matter-dominated Universe.

Key Concepts

Density Parameter (?)

The density parameter ? is defined as the ratio of the actual matter density to the critical density needed for a flat universe. In a matter-dominated context, ? plays a pivotal role in linking the Friedmann equations to observable cosmic expansion. Its interrelation with the Hubble parameter and the scale factor facilitates the derivation of conserved quantities, such as the product ? H² R³ remaining constant over time.

Matter-Dominated Universe

In a matter-dominated universe, the overall energy density is predominantly due to non-relativistic matter. This results in the matter density scaling inversely with the volume (i.e., ? ? R?³, where R is the scale factor). This property is key to deriving various conservation relations in cosmology, as it establishes how matter content dilutes with the expanding universe.

Friedmann Equations

The Friedmann equations are central to cosmological models as they relate the expansion rate of the universe (via the Hubble parameter) to its energy content and spatial curvature. For a matter-dominated universe, these equations simplify since the pressure of non-relativistic matter is negligible, enabling a straightforward analysis of how quantities like the Hubble parameter, density parameter, and scale factor interrelate.

Hubble Parameter

The Hubble parameter, defined as H = (dR/dt)/R, quantifies the rate at which the universe expands. Its behavior in a matter-dominated universe is directly influenced by the matter density, and it evolves in proportion to the dilution of matter as the universe expands. This evolution is central to maintaining conservation conditions over time at cosmic scales.

Scale Factor

The scale factor R(t) represents the relative expansion of the universe over time and is a fundamental variable in cosmology. It indicates how distance scales change due to cosmic expansion and is a critical component in the relationship between matter density, the Hubble parameter, and the curvature of space.

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