Cosmological Distances applet
where a(t) is the scale factor of the universe. (All the models I look at are homogeneous and isotropic and I ignore the non-radial space coordinates, and I normalise everything so that speed of light=1 and Hubble constant=1) The form of a(t) is related to the matter, radiation and cosmological constant within the universe by Einstein's equations of general relativity. I consider four simple cosmological models, corresponding to four expressions for a(t). These are
- Steady State: a(t)=eHt
- Empty: a(t)=t
- Radiation Dominated, (critical density): a(t)=t½
- Matter Dominated (critical density): a(t)=t2/3
The simplest way of obtaining a distance from the parameter χ is to multiply it by the scale factor a(t). This gives the Comoving distance. Note that sometimes the term comoving distance is applied to the parameter χ on its own, but I prefer to have distances which increase with the expansion of the universe. Hence the distance will always depend on both χ and t.
Next imagine that we have sent an electromagnetic pulse which reaches the object at time t and is reflected back to us. This allows us to calculate the Radar distance. If we knew the size of an object, the angle which it takes up in our field of view allows a calculation of distance . This defines the Angular Diameter distance. Likewise if we knew the absolute luminosity we could use the luminosity we see and inverse square law to obtain the distance - this is the Luminosity distance. The Transverse Comoving distance is that we would calculate from the observed transverse speed of an object if we knew the absolute speed relative to a the comoving object (an example here might be a jet emitted from a galaxy). Finally, the Light travel time is simply the time of observation of a signal minus the time the signal was emitted.
Now we get on to the question of when the object was at the given distance. The interesting thing here is that most distances are taken to be of where the object is when the light we see was emitted, but the Comoving distance is generally taken to be where it is when it is observed (i.e. now). One of the reasons for writing this applet was to investigate this difference. Just to be awkward - that is to highlight that 'now' doesn't have a unique definition at large distances - I've included a different type of 'now', that which you would calculate from sending a signal and getting one back - well I thought that if I had radar distance I ought to have radar time as well. Note that this concept of time isn't used in cosmology - unlike distance, there's little reason to use any time coordinate other that the local time since the big bang. There are thus 3 choices for the time used, plus the 'usual for dist.' option which disables the other choices and gives the distance at emission for all distance types except for comoving distance, when the 'comoving now' is used.
The 'larger range' option increases the range of redshift displayed - this can be useful to get a better idea of the asymptotic behaviour of the curves.
This selection of distance definitions certainly doesn't exhaust all of those which have been proposed at some time. I may expand the applet at a later time to include some others. Also I might add in more choices of cosmological model - I realise that I omit the consensus (0.3,0.7) model of the universe, but the four that I chose are computationally simple.
This applet was inspired by the diagrams on Ned Wright's website and his Cosmological distance calculator. The formulae used in calculating distances can be found in his paper A Cosmology Calculator for the World Wide Web and David Hogg's paper Distance measures in cosmology
You should also take a look at the iCosmos calculator
So what have I learned from writing this applet? Well it took me some time to get my head around the changing of redshift with time, so that the z which I use in a formula isn't necessarily the z value on the graph (which is the redshift as observed now). I also realised that the angular diameter distance might be more useful than is generally acknowledged. For the critical density universes it matches the comoving distance, provided, that is, that you match the times for the two distance measures. For the empty universe it matches the 'special relativistic' notion of distance.
There were several occasions on which I had to correct my calculations, and although I am reasonably confident that they are now correct, there might still be some errors (I'm still a bit suspicious at how the curves for luminosity distance at various times can cross each other). If you think you have found such an error, or want to comment on any other aspect of the applet, then please contact me at
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