# Cosmological horizons

## Particle horizon and the Hubble sphere

### Problem 1: particle horizon in one-component Universe

Calculate the particle horizon for a Universe with dominating

b) dust,

c) matter with state equation $p=w\rho$.

### Problem 2: particle horizon and the age of the Universe

Show the the comoving particle horizon equals to the age of the Universe in conformal time.

Show that, if ultrarelativistic matter is dominating in the matter content of a spatially flat Universe ($k=0$), its particle horizon coincides with the Hubble radius.

### Problem 4: comoving Hubble

Find the comoving Hubble radius $R_{H}/a$ as function of the scale factor for a spatially flat Universe that consists of one component with equation of state $p=w\rho$.

### Problem 5: Hubble radius and particle horizon

Express the comoving particle horizon $L_{p}/a$ through the comoving Hubble radius $R_{H}/a$ for the case of domination of a substance with state parameter $w$.

### Problem 6: observed Universe

Show that in an open Universe filled with dust the number of observed galaxies increases with time.

### Problem 7: particle horizon and curvature scale

Show that even in early Universe the scale of particle horizon is much less than the curvature radius, and thus curvature does not play significant role within the horizon.

### Problem 8: the observable portion of the Universe

Estimate the ratio of the volume enclosed by the Hubble sphere to the total volume of the closed Universe.

## FLRW in terms of proper distances

The following five problems are based on work by F. Melia (arXiv:0711.4181, arXiv:0907.5394).

Standard cosmology is based on the FLRW metric for a spatially homogeneous and isotropic three-dimensional space, expanding or contracting with time. In the coordinates used for this metric, $t$ is the cosmic time, measured by a comoving observer (and is the same everywhere), $a(t)$ is the expansion factor, and $r$ is an appropriately scaled radial coordinate in the comoving frame.

F.Melia demonstrated the usefulness of expressing the FRLW metric in terms of an observer-dependent coordinate $R=a(t)r$, which explicitly reveals the dependence of the observed intervals of distance, $dR$, and time on the curvature induced by the mass-energy content between the observer and $R$; in the metric, this effect is represented by the proximity of the physical radius $R$ to the cosmic horizon $R_{h}$, defined by the relation $R_{h}=2G\,M(R_h).$ In this expression, $M(R_h)$ is the mass enclosed within $R_h$ (which terns out to be the Hubble sphere). This is the radius at which a sphere encloses sufficient mass-energy to create divergent time dilation for an observer at the surface relative to the origin of the coordinates.

### Problem 9: cosmic horizon in a flat Universe

Show that in a flat Universe $R_{h}=H^{-1}(t)$.

### Problem 10: FLRW in terms of proper distances

Represent the FLRW metric in terms of the observer-dependent coordinate $R=a(t)r$.

### Problem 11: divergences and the horizon

Show, that if we were to make a measurement at a fixed distance $R$ away from us, the time interval $dt$ corresponding to any measurable (non-zero) value of $ds$ must go to infinity as $r\to R_h$.

### Problem 12: horizon only increases

Show that $R_h$ is an increasing function of cosmic time $t$ for any cosmology with $w>-1$.

### Problem 13: simple examples

Using FLRW metric in terms of the observer-dependent coordinate $R=a(t)r$, consider specific cosmologies:

a) the De Sitter Universe;

b) a cosmology with $R_h =t$, ($w=-1/3$);

c) radiation dominated Universe ($w=1/3$);

d) matter dominated Universe ($w=0$).