#**Part 1**#

There are quite a few common misconceptions about the expansion of the universe, even among professional physicists. I will try to clarify a few of these issues; for more information, I highly recommend the article
"[Expanding Confusion: common misconceptions of cosmological horizons and the superluminal expansion of the Universe][1]" from Tamara M. Davis and Charles H. Lineweaver.

This is going to be a lengthy post, packed with LaTeX, so I will split it into parts.

I will assume a standard ΛCDM-model, with
$$
\begin{align}
H_0 &= 67.3\;\text{km}\,\text{s}^{-1}\text{Mpc}^{-1},\\
\Omega_{R,0} &= 9.24\times 10^{-5},\\
\Omega_{M,0} &= 0.315,\\
\Omega_{\Lambda,0} &= 0.685,\\
\Omega_{K,0} &= 1 - \Omega_{R,0} - \Omega_{M,0} - \Omega_{\Lambda,0} = 0.
\end{align}
$$

The expansion of the universe can be described by a *scale factor* $a(t)$, which can be thought of as the length of an imaginary ruler that expands along with the universe, relative to the present day, i.e. $a(t_0)=1$ where $t_0$ is the present age of the universe.

From the standard equations, one can derive the *Hubble parameter*
$$
H(a) = \frac{\dot{a}}{a} = H_0\sqrt{\Omega_{R,0}\,a^{-4} + \Omega_{M,0}\,a^{-3} + \Omega_{K,0}\,a^{-2} + \Omega_{\Lambda,0}},
$$
such that $H(1)=H_0$ is the *Hubble constant*. [In a previous post][2], I showed that the age of the universe, as a function of $a$, is
$$
t(a) = \frac{1}{H_0}\int_0^a\frac{a'\,\text{d}a'}{\sqrt{\Omega_{R,0} + \Omega_{M,0}\,a' + \Omega_{K,0}\,a'^2 + \Omega_{\Lambda,0}\,a'^4}},
$$
which can be numerically inverted to yield $a(t)$, and consequently $H(t)$. It also follows that the present age of the universe is $t_0=t(1)=13.8$ billion years.

Now, another consequence of the Big Bang models is *Hubble's Law*,
$$
v_\text{rec}(t_\text{ob}) = H(t_\text{ob})\,D(t_\text{ob}),
$$
describing the relation between the *recession velocity* $v_\text{rec}(t_\text{ob})$ of a light source and its *proper distance* $D(t_\text{ob})$, at a time $t_\text{ob}$. In fact, this follows immediately from the definition of $H(t_\text{ob})$, since $v_\text{rec}(t_\text{ob})$ is proportional to $\dot{a}$ and $D(t_\text{ob})$ is proportional to $a$. 

However, it should be noted that this is a theoretical relation: neither $v_\text{rec}(t_\text{ob})$ nor $D(t_\text{ob})$ can be observed directly. The recession velocity is not a "true" velocity, in the sense that it is not an actual motion in a local inertial frame; clusters of galaxies are locally at rest. The distance between them increases as the universe expands, which can be expressed as $v_\text{rec}(t_\text{ob})$. Some cosmologists therefore prefer to think of $v_\text{rec}(t_\text{ob})$ as an *apparent* velocity, a theoretical quantity with little physical meaning.

A related quantity that *is* observable is the *redshift* of a light source, which is the cumulative increase in wavelength of the photons as they travel through the expanding space between source and observer. There is a simple relation between the scale factor and the redshift of a source, observed at a time $t_\text{ob}$:
$$
1 + z(t_\text{ob}) = \frac{a(t_\text{ob})}{a(t_\text{em})},
$$ 
such that the observed redshift of a photon immediately gives the time $t_\text{em}$ at which the photon was emitted.

**(continues)**

  [1]: http://arxiv.org/abs/astro-ph/0310808/ "Expanding Confusion: common misconceptions of cosmological horizons and the superluminal expansion of the Universe"
  [2]: http://physics.stackexchange.com/questions/63571/lookback-time-age-of-the-universe-calculations/63673#63673
  [3]: http://en.wikipedia.org/wiki/Luminosity_distance
  [4]: http://en.wikipedia.org/wiki/Angular_diameter_distance
  [5]: http://en.wikipedia.org/wiki/Comoving_distance