Intuition for why matter dominated expansion is faster than radiation domination? In a matter dominated universe $a_{\rm mat.}(t)\sim t^{2/3}$, while in a radiation dominated universe, $a(t)_{\rm rad.}\sim t^{1/2}$.  Therefore, a matter dominated universe is expanding more quickly, in the sense that $$\frac{H_{\rm mat.}(t)}{H_{\rm rad.}(t)}=\frac{4}{3}\ ,$$ where we're comparing at equal cosmological time since the big bang.
What is an intuitive explanation for this fact?
Naive intuition would indicate the opposite: radiation has pressure, while matter does not, so somehow the radiation should push the universe to expand more quickly.  Clearly, this is wrong, as the acceleration equation shows that more pressure decelerates the universe:
$$\frac{\ddot{a}}{a}\propto -(\rho+3p)$$
I've never had a good feel for why this is so.
 A: Both a matter-dominated and a radiation-dominated flat universe with no dark energy expand, but decelerate. The ratio of the respective Hubble parameters shows that the expansion rate is faster in a matter-dominated universe.
$H_{matter}(t) / H_{radiation}(t) = 4/3$ 
Apparent contradiction:
The $-$ sign in the second Friedmann equation causes a cosmological fluid with pressure (radiation) to be more effective in decelerating the universe than a fluid with no pressure (matter).  
Explanation:
Pressure does not help to counteract the deceleration due to gravity, as one would intuitively expects. The reason is that only a pressure gradient can induce a force. Since there is no pressure gradient in a homogeneous universe, pressure cannot help expand the universe.
A: Nonrelativistically, a uniform pressure does not cause acceleration or deceleration, because there are no pressure gradients. In fact, just like energy, the absolute value of pressure is not defined at all, in the sense that you can add a constant to the pressure everywhere, and nothing will change. 
In general relativity, energy and pressure are defined absolutely, and they both contribute to gravity. One way to see this is with the geodesic deviation equation, which states that the acceleration of geodesics separated by $x^\mu$ is
$$a^\mu = - R^\mu_{\ \ 0 0 \nu} x^\nu.$$
Assuming isotropy and averaging over all directions, the expansion rate of the volume $V$ of a ball of geodesics is 
$$\frac{d^2 V}{d t^2} = - R^\mu_{\ \ 0 0 \mu} = - R_{00}.$$
On the other hand, Einstein's equations give the Ricci tensor in terms of the stress-energy tensor, and state that
$$R_{00} = 4 \pi G (\rho + p_x + p_y + p_z).$$
Hence both pressure and energy contribute equally to gravitational attraction. This is the core physical content of the Einstein field equation; the whole equation can be recovered from this statement alone. So I can't give an intuitive explanation for why pressure appears with a positive sign, any  more than I can intuitively explain why Faraday's law is true -- it's our starting point. But given this, it's straightforward to see that radiation causes more attraction than matter, and that vacuum energy does the opposite.
