cherenkov-radiationfaster-than-lightrefractionspecial-relativityspeed-of-light
While the speed of light in vacuum is a universal constant ($c$), the speed at which light propagates in other materials/mediums may be less than $c$. This is obviously suggested by the fact that different materials (especially in the case of transparent ones) have a particular refractive index.
But surely, matter or even photons can be accelerated beyond this speed in a medium?
If so, what are the effects?
I think you may be confusing group velocity and phase velocity. The thing that we usually call the "speed of light" refers to the group velocity - how fast light can carry information from A to B.
When you look closely at the waves in a medium other than vacuum (which is "no medium"), a packet of finite duration can be thought of as having a mixture of frequencies present. If the medium is dispersive, not all those frequencies travel at the same speed. When you put them all back together, it looks as though there is a wave traveling faster than $c$. However, when you look at the envelope, it always travels at a speed $\le c$.
More on this at this link
Yes, the index of refraction of air does depend on the density of the air, usually expressed in terms of the air pressure rather than the density.
This effect limits the accuracy of displacement measurements by interferometry, particularly when measuring the displacement of a moving object which is producing turbulence (air pressure variations) in the air around it.
The fractional content of water vapor and CO2 in the air also affect the index of refraction measurably.
From some brief web research, there are widely accepted fitting formulas for these effects from Edlen (1966) updated in 1994 by Birch and Downs; and by Ciddor (1996). A presentation from the Canadian National Research Council gives formulas based on Edlen, Birch, and Downs:
Sadly, the individual terms (particularly $x$, $\sigma$, and $f$) are not fully explained, so you'll have to work out exactly what they mean or go back to the primary sources for an explanation.
The US's NIST provides an online calculator based on Ciddor, and some helpful instructions. I also found a page where you can download Python code for calculating the refractive index based on Ciddor.
I don't find any simple formula that gives just the sensitivity of index to pressure, but from the NIST page it seems that a difference in air pressure of approximately 0.4 kPa (standard air pressure being 101.325 kPa) produces an index of refraction change of about 1 ppm (this number likely slightly variable depending on wavelength, temperature, air composition, etc).
Best Answer
Speed of light is indeed lower when light propagates through materials (as opposed to vacuum). This doesn't mean that individual photons go slower but rather that the apparent speed of light pulse is lower due to interactions with atoms of the material. So in this case it is possible for some objects to go "faster than light" and indeed very similar effect to sonic boom, called Cherenkov radiation, appears.
Note that for most materials the apparent speed of light is still huge (of the order of speed of light in vacuum) so you need very energetic particles to generate Cherenkov radiation. So this effect is mainly relevant for high-energy particle physics, astrophysics and nuclear physics.