sylas said:
That is a really clear and concise statement. I'm going to steal it for future use, with your permission!
Certainly :)
sylas said:
It depends on how the expansion rate develops. In a matter dominated universe, or an empty (constant expansion) universe, then no; photons from arbitrarily far away will reach us.
Quite right. I addressed this point in my previous post, but was just a bit short for the sake of brevity here. What I said is accurate if the correct explanation for the accelerated expansion is dark energy or something similar. Otherwise...it depends.
sylas said:
But in the current consensus model, with dark energy and subcritical matter, I think you may be correct... but I am not sure. I have not tried to prove it one way or the other. Marcus might know...
Yes, actually, this one I'm certain about. I'm not used to thinking in terms of recession velocity, hence my previous mistake. But this is a standard result for de Sitter cosmology (which our universe will asymptotically approach if dark energy = cosmological constant): light can only travel a finite distance in comoving coordinates in de Sitter space.
To see this, consider the following. Take the FRW metric, neglecting the angular terms, and just taking the radial distance:
ds^2 = c^2 dt^2 - a(t)^2 dr^2
A light ray will move along a null geodesic, where ds^2 = 0, so we can get the path of light in the radial direction simply:
c dt = a(t) dr
If I want to know how far light has traveled in comoving coordinates, then, I integrate:
r = c \int_{t_1}^{t_2} \frac{dt}{a(t)}
So, for instance, if I want to ask how far, in comoving coordinates, a photon gets in infinite time, then I integrate:
r = c \int_{t_0}^{\infty} \frac{dt}{a(t)}
However, we typically don't work in these coordinates, so it is useful to change variables. The first change will be to use the definition of the Hubble parameter to change the integration from an integration in time to an integration over the scale factor:
H \equiv \frac{\dot{a}}{a}
dt = da \frac{dt}{da} = \frac{da}{a H}
This turns our integral into:
r = c \int_1^{\infty} \frac{da}{a^2 H}
Now, obviously this is only going to work for a universe that expands infinitely into the future (as only if the expansion continues indefinitely will a\rightarrow\infty as t\rightarrow\infty), but if the universe starts to collapse obviously all points will come in causal contact regardless.
So, then, if you know your integrals, you should be aware that if the Hubble parameter H approaches a constant value, then the above integral approaches a finite value. Specifically, if H is constant, the integral is:
r = \frac{c}{H} \left[\frac{-1}{a}\right]_1^\infty = \frac{c}{H}
Which is the statement that in a de Sitter universe (one where we only have a cosmological constant), we reach the conclusion that an object with a recession velocity equal to the speed of light or greater today is emitting photons that we will never see, and we are emitting photons that will never reach them.
Of course, we're not in a perfectly de Sitter universe, so this isn't actually true: H is decreasing with time, which corresponds to a somewhat larger distance at which things will eventually communicate. Using \Omega_m = 0.27 and \Omega_\Lambda = 0.73, I get r = 1.12 \frac{c}{H_0}. So objects currently receding up to about 12% higher than the speed of light are emitting photons that we will detect at some point. But beyond that, we can never see those photons (and bear in mind that this is assuming the cosmology is accurate, which is by no means certain).
the farther past z = 1.7 you go.
