Expanding Universe So an expanding universe has the metric $$ds^2 = g_{\mu \nu} dx^\mu dx^\nu = -dt^2 + a(t) ( dr^2+r^2 d\Omega^2)$$ where $d\Omega^2 = d\theta^2 + \sin^2\theta d\varphi^2$ as the usual spherical coordinates. 
Specifically, for $a(t) = e^{Ht},$ aka the exponentially expanding universe, we see that there is an $R_{max}$ which after that, you cannot send signals with observers (I think it's called the de Sitter horizon). 
Special relativity requires that a physical particle stays within its light cone at all times, but in this model light follows the 'null' trajectory, i.e. $ds^2=0$ or $$dt=a(t) dr$$ This produces an ever narrowing and thinning out light cone as time goes by. My question is regarding this, which I am kind of confused by. If the light cones are getting infinitely narrow and sharper as lots of time goes by, does this mean that the particle's range where it can move gets infinitesimally small to the point where it can't move at all? Because the light cones y-axis is time and x-axis is position (or $r$), so if it's getting thinner and thinner, that means that $r$ for the particle (and thus its wordline) is getting smaller and smaller. What are the implications of that, or maybe I am just misunderstanding? Thank you.
 A: This is essentially the same as the narrowing of the light cones that happens as you approach the event horizon of a Schwarzschild black hole, and it occurs for the same reason i.e. the coordinate velocity of light tends to zero as you approach the horizon.
There is nothing physically interesting in this. It is a result of the coordinates we are using. If you use different coordinates, e.g. Eddington-Finkelstein coordinates for the black hole or comoving coordinates for the FLRW metric then the narrowing of the light cones does not occur.
A: To carry forwards with John Rennie's response let us divide the metric by $dt^2$
$$
\left(\frac{ds}{dt}\right)^2~=~1~-~a(t)\left(\frac{dr}{dt}\right)^2
$$
and with the generalized Lorentz gamma factor $\Gamma~=~\left(\frac{dt}{ds}\right)^2$ means we have
$$
\Gamma~=~\frac{1}{\sqrt{1~-~a(t)\left(\frac{dr}{dt}\right)^2}}
$$
This factor explodes at $t~=~0$ and is problematic.
We need to consider the velocity, which pertains to some galaxy. The FLRW equation gives
$$
\left(\frac{\dot a}{a}\right)^2~=~H~=~\frac{8\pi\Lambda}{3c^2}
$$
And we then have for the velocity of a galaxy $v~=~Hd$, the standard Hubble relation, that in greater generality
$$
v/c~=~z~\simeq~H\frac{a(t_0)}{a(t)}
$$
This reduces the gamma factor to $\Gamma~=~(1~-~a(t_0)^{-2})^{-1/2}$, which becomes unity as time becomes large. This means relativistic effects attenuate away
I derived some basis cosmic $101$ expansion physics in this post using just Newton's law. It is fairly remarkable that for the FLRW factor $k~=~0$ or for long enough time so the $k/a(t)^2$ term is negligible and not too close to early events such as inflation that basic Newtonian mechanics captures $90\%$ of expansionary cosmology. This is another example of this.
What about redshifting? That can be entirely accounted for by the expansion of space, or the scale factor $a(t)$, so the volume changes and the wave length expands. The result is that energy density $\rho~\propto~t^{-4}$ and wavelength $\lambda~\propto~\lambda_0a(t)$. So these can be seen entirely without relativistic effects.
What then is happening? It is the case that light cones are narrowing in, which happens with light cones as one approaches the Schwarzschild singularity. However, in this case the relative velocity of massive particles is also increasing. This has the effect of cancelling out much observed physics form the narrowing of light cones. From the perspective of any observer the motion of other galaxies is Newtonian(like) and relativistic effects are not abundantly apparent. Also unlike the Schwarzschild case this de Sitter physics has complete isotropy, which hides away much of the relativistic effects. 
Cosmologies in the Petrov classification are type O, which means there are no Killing vectors and isometries. They are then Weyl flat, which is much the same as saying relativistic physics of cosmologies is "masked," at least for time sufficiently long after the big bang.
