Re: Theoretical limit on distance measurement with doppler shift?

Hi Tiago,

Thank you for your interesting question. This type of astronomy isn't really my field, but I'll try to answer your question as well as I can.

Being able to do a distance determine by the Doppler shift is essentially a two-stage process. First, the Doppler shift of the light is determined [1]. From this Doppler shift, the velocity of the object from which the light originated with respect to us can be determined. The second step is relating this velocity to a distance. Because of the way the universe expands, it is assumed that the distance is proportional to the velocity [2].

The formula for the Doppler shift is given by [3]:

t = l (c-v)

with t the time, which is the inverse of the frequency, l the wavelength, c the speed of light, and v the velocity. Basically, the maximum distance you could find is the distance at which t reaches 0. This is the distance at which v = c. Basically, if the source moves away with a relative speed greater than the speed of light, light won't catch up, and you won't see it. This cutoff velocity does not depend on the wavelength.

The universe expands, and because this expansion is intrinsic [2], two objects may indeed increase their distance faster than the speed of light. One way of looking at this is not that objects are moving, but that space is moving with the objects in it. When the relative distance between the objects becomes so large that rate of expansion becomes equal to or larger than the speed of light, they become invisible, and this is the maximum distance that can be measured in any way - including the Doppler shift. The radius of this sphere is called the Hubble sphere, and it is given by [4]:

r_HS = c H₀^-1

This is about 13.8 billion light years. From the first equation, it follows that the point where the observed frequency is equal to one divided by the age of the universe is just at the outer edge of this sphere. The observable universe is in fact larger than this [5]. The reason is that the universe has expanded since the observed light was emitted; the object seen is further away now what it as in the past. In principle, the observable distance in the universe is three times larger than the Hubble sphere [5].

I hope this answers your question.

Regards,

Bart Broks

1. http://en.wikipedia.org/wiki/Doppler_effect
2. http://en.wikipedia.org/wiki/Metric_expansion_of_space
3. http://en.wikipedia.org/wiki/Relativistic_Doppler_effect
4. http://en.wikipedia.org/wiki/Hubble%27s_law
5. http://www.astro.ucla.edu/~wright/cosmology_faq.html#DN