# Hubble sphere

The Hubble radius is the proper distance $R_H (t)=c H^{-1}(t)$. The sphere of this radius is called the Hubble sphere. From definition, the Hubble recession "velocity" of a comoving observer on the Hubble sphere is $v(R_H)=H R_H =c$ and equal to the speed of light $c$. This is true, of course, at the same moment of time $t$, for which $H (t)$ is taken.

### Problem 1

All galaxies inside the Hubble sphere recede subluminally (slower than light) and all galaxies outside recede superluminally (faster than light). This is why the Hubble sphere is sometimes called the "photon horizon". Does this mean that galaxies and their events outside the photon horizon are permanently hidden from the observer's view? If that were so, the photon horizon would also be an event horizon. Is this correct?

### Problem 2

Show, by the example of static universe, that the Hubble sphere does not coincide with the boundary of the observable Universe.

### Problem 3

Estimate the ratio of the volume enclosed by the Hubble sphere to the full volume of a closed Universe. </div>

### Problem 4

Show that in a spatially flat Universe ($k=0$), in which radiation is dominating, the particle horizon coincides with the Hubble radius. </div>

### Problem 5

Find the dependence of comoving Hubble radius $R_H /a$ on scale factor in a flat Universe filled with one component with the state equation $\rho=w p$. </div>

### Problem 6

Express the comoving particle horizon through the comoving Hubble radius for the case of domination of a matter component with state parameter $w$. </div>

### Problem 7

Show that $$\frac{dR_H }{dt}=c(1+q),\label{dRHdt}$$ where $$q=-\frac{\ddot{a}/a}{H^{2}}$$ is the deceleration parameter.

### Problem 8

Show that $$\frac{dL_p}{dt}=1+\frac{L_p}{R_H}.$$

### Problem 9

Show that in the Einstein-de Sitter Universe the relative velocity of the Hubble sphere and galaxies on it is equal to $c/2$. </div>

### Problem 10

Find $a(t)$ in a universe with constant positive deceleration parameter $q$.

### Problem 11

Show that in universes of constant positive deceleration $q$, the the ratio of distances to the particle and photon horizons is $1/q$. </div>

### Problem 12

Show that the Hubble sphere becomes degenerate with the particle horizon at $q=1$ and with the event horizon at $q=-1$. </div>

### Problem 13

Show that if $q$ is not constant, comoving bodies can be inside and outside of the Hubble sphere at different times. But not so for the observable universe; once inside, always inside.