Calculating travel times on cosmological scales

[mistergrinch]

OK I'm still new to cosmology and this question is bugging me: if a photon is emitted toward me from a distant galaxy at a distance D, how do I calculate the time it takes to reach me? I assume the universe is expanding exponentially with a DeSitter scale factor a = exp(Ht), (H = Hubble constant). Is there a simple way to do this calculation?

I tried using special relativity velocity addition to add -c + Hx, where Hx = speed of expanding space at a distance x. This gives me dx/dt = (-c + H*x) / (1 - H*x/c) = -c. I.e. t = D/c. Is this correct? Intuitively it seems like the expansion of space would increase the time it takes for the photon to reach me.

 

[Chalnoth]

OK I'm still new to cosmology and this question is bugging me: if a photon is emitted toward me from a distant galaxy at a distance D, how do I calculate the time it takes to reach me? I assume the universe is expanding exponentially with a DeSitter scale factor a = exp(Ht), (H = Hubble constant). Is there a simple way to do this calculation?

I tried using special relativity velocity addition to add -c + Hx, where Hx = speed of expanding space at a distance x. This gives me dx/dt = (-c + H*x) / (1 - H*x/c) = -c. I.e. t = D/c. Is this correct? Intuitively it seems like the expansion of space would increase the time it takes for the photon to reach me.

Well, the way you can do this is as follows. The light travel time can, first, simply be written as follows:

$$t = \int_{t_a}^{t_b} dt$$

This is obviously true, since $t_a$ is the time the light ray left the object, $t_b$ is the time the light ray arrived, and the results is $t_b - t_a$, which is the light travel time. The difficulty is that we don't know what $t_a$ or $t_b$ are. However, we can perform a useful change of variables:

$$dt = {dt \over da} da = {da \over {aH}}$$

So now we can find the time it took for the universe to expand from some $a_1$ to some other size $a_2$. Unfortunately, this also isn't terribly useful, as it's not very obvious how this connects to light travel time. We can get there, however, through a second change of variables:

$$a = {1 \over 1+z}$$
$$da = {-dz \over (1+z)^2}$$

Here $z$ is the redshift, so with this change of variables, we can say how long it took light to arrive from some far-away object if it was redshifted by a factor of $z$. This gives us the integral:

$$t = \int_0^z {dz' \over (1+z')H(z')$$

Does that help?

 

[marcus]

..., how do I calculate the time it takes to reach me? I assume the universe is expanding exponentially with a DeSitter scale factor a = exp(Ht), (H = Hubble constant)...

That is a good exercise! Chalnoth showed how. You probably realize that H(t) changes over time so that the way you set the problem up it is not realistic---only gives good answers for comparatively short travel times. But someone else reading the post might not know that, and might imagine that you can actually calculate in general that way.

I expect you are already familiar with the various online calculators that are used to get travel times on cosmological scales.

If anyone is not, here is one such calculator. It is not the perfect tool for the job but you can make do with it:
http://www.astro.ucla.edu/~wright/CosmoCalc.html
or simply google "wright calculator"

the initial distance, the proper distance the galaxy was when the light was emitted and started on its journey, is labeled "angular size distance".

===================
Here is a variant that might be a bit easier to use:
http://www.astro.ucla.edu/~wright/DlttCalc.html

You want the relation between angular size distance (= initial distance) and travel time.
Put in your guess about the travel time, press the button to get it to calculate, see what the initial distance is, and then refine your guess.

I know it doesn't sound very elegant, but it works. Maybe someone else knows a calculator that does this directly---so you don't have to work backwards.

 

[marcus]

Grinch, there is an intriguing side aspect to the problem. The way you pose it, there is an ambiguity: no one unique answer.

If the initial distance of the emitter is, say, 41 million lightyears, then the travel time could be either short---say just a few percent over 41 million years---or it could be long, namely around 13.7 billion years.

According to the standard cosmo model, the CMB radiation was emitted by matter that was (at the time of emission) 41 million lightyears from us---from the matter that eventually became us.

But it took almost the whole age of expansion to reach us. The whole 13.7 billion years minus 380,000 years, which is still around 13.7 billion years.

So if you want to start with an initial distance (at the moment of emission) and calculate a travel time from that, there is bound to be a certain ambiguity. Just a savory detail to make the business more interesting :-D

 

[mistergrinch]

Thank you, these replies are very helpful. Chalnoth, how do you get the relation a = 1/(1+z)?

 

[Chalnoth]

Thank you, these replies are very helpful. Chalnoth, how do you get the relation a = 1/(1+z)?

In an expanding universe, the wavelength of light is linear with the expansion: expand the universe by a factor of two, and the wavelength doubles.

By convention, redshift is defined such that:

$$\lambda_{obs} = (1+z)\lambda_{emit}$$

So you should be able to see that for a photon that began when the universe was half the size it is now ($a = 0.5$), the observed redshift will be $z = 1$, which is the same as saying that the observed wavelength is twice the emitted wavelength.

 

[mistergrinch]

OK I see the derivation at Wikipedia: http://en.wikipedia.org/wiki/Redshift#Mathematical_derivation

1 + z = a_now/a_then

I take it by convention a_now = 1? I see also that wavelength is linear with expansion from the derivation, but this doesn't seem obvious to me. Is there a simple, intuitive way to see that this is true?

 

[Chalnoth]

I take it by convention a_now = 1?

Yes, this is correct. Sorry that I missed that point.

Granted, this isn't the only possible convention, but it is a very common one in cosmology.

I see also that wavelength is linear with expansion from the derivation, but this doesn't seem obvious to me. Is there a simple, intuitive way to see that this is true?

Well, I don't know of an intuitive way to arrive at the exact result, but maybe this is a reasonably intuitive way of arriving at the fact that the wavelength must expand as the universe expands.

Instead of a photon, consider a particle moving against the background of an expanding universe. This particle moves at a constant velocity. But as it moves, it is continually moving towards objects that are moving away from it at a faster rate. Thus, as it traverses the universe, despite not changing its velocity, its velocity compared to the local matter decreases.

A similar thing must happen with light waves as it catches up to new parts of the universe moving away from it faster. Obviously, the speed of light relative to the local matter doesn't fall, but its momentum does. And that corresponds to an increase in wavelength.

To get that it is exactly the case that the wavelength is stretched by the expansion, I don't know of a way other than some more detailed calculations. You can do it, for instance, by looking at the way the energy density of a gas of photons changes as the universe expands (the energy density goes as $a^{-4}$, with a factor of $a^{-3}$ that stems from the drop in the number density of photons, with the remaining factor of $a^{-1}$ coming from the drop in energy of each individual photon which increases the wavelength).

 

[mistergrinch]

OK I think I understand the photon case, but what about doing this calculation for a particle? I'm guessing there's no simple derivation and we have to solve some complicated Euler-Lagrange equations?

 

[Chalnoth]

OK I think I understand the photon case, but what about doing this calculation for a particle? I'm guessing there's no simple derivation and we have to solve some complicated Euler-Lagrange equations?

You mean, how would you calculate how the expansion of the universe acts as a sort of friction on particles moving with respect to the background? Well, I don't think it'd be all that difficult. If you assume for a moment that the particle moves at constant velocity with respect to wherever it was emitted, you just need to calculate how far it has moved after a given time ($d = vt$), then calculate the local recession velocity ($v_r = Hd$), and subtract that recession velocity from the particle velocity to get the particle's velocity with respect to the local matter.

This won't be entirely accurate, because the other stuff in the universe will add an acceleration to the particle. Dark energy, for example, will cause it to speed up over time, turning the distance calculation into an integral. But at least that gives you a very rough idea of how it might be done.

 

[Tanelorn]

Were any particles at all produced at the time of the CMBR photons? If so could they still be detected?

 

[Chalnoth]

Were any particles at all produced at the time of the CMBR photons? If so could they still be detected?

Well, the CMB photons were...but no, at the time the CMB photons were emitted, the temperature of our universe was in the range of 3000K, which is too low to produce any particles but photons.

Now, earlier in the universe, when the temperatures were much higher, lots of particle/anti-particle pairs were being produced (and annihilated).