Wednesday, October 28, 2015

conditions for cosmic horizons

This week's CQW is a guest post by Phillip Helbig:

An important concept in classical cosmology (for the purposes of this post defined as cosmology within the framework of Friedmann-Lemaître cosmological models) is that of horizons.  The particle horizon is the spherical surface from which radiation from the big bang is just reaching us now or, alternatively, due to symmetry, the sphere which is now just reached by radiation emitted from our location at the big bang. This is the same as the "observable universe".  The event horizon is the sphere beyond which radiation emitted now will never reach us or, alternatively, due to symmetry, the sphere which will ever just be reached by radiation emitted from our location now.  (Note for experts: I am discussing spatial horizons defined at the current cosmic time.  In the literature, one also finds discussion of horizons defined in space-time.  Somewhat confusing is when the particle horizon is described as a two-dimensional surface in space and the event horizon as a three dimensional surface in space-time.  There is no inconsistency,  but one must be careful when comparing different papers on this topic.)

The particle horizon exists if the integral
\begin{equation}
  \int_{0}^{t_0} \frac{\mathrm{d}t}{R(t)}
\end{equation}
is finite; $t$ is cosmic time ($t=0$ corresponds to the big bang if there is one), $t_0$ is the current epoch, and $R$ is the scale factor.

The event horizon exists if the integral
\begin{equation}
  \int_{t_0}^{\infty} \frac{\mathrm{d}t}{R(t)}
\end{equation}
is finite. One can of course work out $R(t)$ for given values of the cosmological parameters $\Omega$ (density parameter) and $\Lambda$ (cosmological constant), compute the integral, and check whether it is finite in order to determine whether the corresponding horizon exists for the cosmological model in question.

However, there are simple qualitative descriptions which give the necessary and sufficient conditions for each type of horizon.  What are they?

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