Archive for luminosity distance

The Scales of Things

Posted in Maynooth, The Universe and Stuff with tags , , , , , on October 9, 2022 by telescoper

A few people have asked me why I needed such extravagant equipment (ping-pong balls, a torch and a metre-ruler) in my lecture on Thursday night.

I did only use one ping pong ball in the talk but I found the local budget shop Eurosaver only sells them in packs of twelve (for the princely sum of €3) so I now have plenty of spares. The metre ruler was borrowed from the Department of Experimental Physics (who have expertise using sophisticated measurement devices) and returned on Friday morning. The torch was procured from Tesco along with two batteries.

One of the things I wanted to do in my lecture was to explain some of the difficulties about measuring cosmological distances. I started by holding up a ping pong ball (radius 2cm) and asking if the ping pong ball were the Sun (radius 7 × 108 m), on the same scale how far away would be the nearest other star (Proxima Centauri)?

To cut a long story short – and you can do the arithmetic yourself – the answer surprises most people who haven’t seen this demonstration before. It’s not the back of the lecture theatre, nor is it the town centre, nor the next town. It’s 1200 km away. That’s as far from Maynooth as, say, Geneva, or Copenhagen. The distances between stars is huge, even in the relatively dense part of a Galaxy, such as where the Sun is situated. The Universe is very big and very empty, even in the places that look crowded.

The torch and the metre rule were used to demonstrate two ways of possibly measuring astronomically large distances. I had a student stand up at the back of the theatre holding the metre rule. I explained that I could measure the distance to the student using geometry by measuring the angle subtended by the ruler if I knew its length (which I do). This is the principle behind the angular diameter distance; the metre rule is called a “standard rod”.

The torch is used to illustrate the luminosity distance. If I knew its power output I could measure the intensity of light using a lightmeter and infer the distance from that using the fact that it follows an inverse-square law. The torch is thus a “standard candle”.

Of course in cosmology we don’t have perfectly standard rods or candles but we can apply the principle of the angular diameter distance to features in the galaxy distribution or the cosmic microwave background or gravitational lenses and supernovae can provide us with accurate luminosity distances.

There are additional complications. Objects at large distances are receding with the Hubble expansion so light from them is redshifted, affecting their apparent luminosity. Einstein’s theory of general relativity allows for the possibility that light rays don’t travel in straight lines either (because space is curved), affecting the angular diameters. That means the two methods don’t necessarily give the same distance unless these factors are taken into account.

Leave a comment »

Thoughts on Cosmological Distances

Posted in The Universe and Stuff with tags , , , , , on July 18, 2019 by telescoper

At the risk of giving the impression that I’m obsessed with the issue of the Hubble constant, I thought I’d do a quick post about something vaguely related to that which I happened to be thinking about the other night.

It has been remarked that the two allegedly discrepant sets of measures of the cosmological distance scale seen, for example, in the diagram below differ in that the low values are global measures (based on observations at high redshift) while the high values of are local (based on direct determinations using local sources, specifically stars of various types).

The above Figure is taken from the paper I blogged about a few days ago here.

That is basically true. There is, however, another difference in the two types of determination: the high values of the Hubble constant are generally related to interpretations of the measured brightness of observed sources (i.e. they are luminosity distances) while the lower values are generally based on trigonometry (specifically they are angular diameter distances). Observations of the cosmic microwave background temperature pattern, baryon acoustic oscillations in the matter power-spectum, and gravitational lensing studies all involve angular-diameter distances rather than luminosity distances.

Before going on let me point out that the global (cosmological) determinations of the Hubble constant are indirect in that they involve the simultaneous determination of a set of parameters based on a detailed model. The Hubble constant is not one of the basic parameters inferred from cosmological observations, it is derived from the others. One does not therefore derive the global estimates in the same way as the local ones, so I’m simplifying things a lot in the following discussion which I am not therefore claiming to be a resolution of the alleged discrepancy. I’m just thinking out loud, so to speak.

With that caveat in mind, and setting aside the possibility (or indeed probability) of observational systematics in some or all of the measurements, let us suppose that we did find that there was a real discrepancy between distances inferred using angular diameters and distances using luminosities in the framework of the standard cosmological model. What could we infer?

Well, if the Universe is described by a space-time with the Robertson-Walker Metric (which is the case if the Cosmological Principle applies in the framework of General Relativity) then angular diameter distances and luminosity distances differ only by a factor of (1+z)2 where z is the redshift: DL=DA(1+z)2.

I’ve included here some slides from undergraduate course notes to add more detail to this if you’re interested:

The result DL=DA(1+z)2 is an example of Etherington’s Reciprocity Theorem. If we did find that somehow this theorem were violated, how could we modify our cosmological theory to explain it?

Well, one thing we couldn’t do is change the evolutionary history of the scale factor a(t) within a Friedman model. The redshift just depends on the scale factor when light is emitted and the scale factor when it is received, not how it evolves in between. And because the evolution of the scale factor is determined by the Friedman equation that relates it to the energy contents of the Universe, changing the latter won’t help either no matter how exotic the stuff you introduce (as long as it only interacts with light rays via gravity).

In the light of the caveat I introduced above, I should say that changing the energy contents of the Universe might well shift the allowed parameter region which may reconcile the cosmological determination of the Hubble constant from cosmology with local values. I am just talking about a hypothetical simpler case.

In order to violate the reciprocity theorem one would have to tinker with something else. An obvious possibility is to abandon the Robertson-Walker metric. We know that the Universe is not exactly homogeneous and isotropic, so one could appeal to the gravitational lensing effect of lumpiness as the origin of the discrepancy. This must happen to some extent, but understanding it fully is very hard because we have far from perfect understanding of globally inhomogeneous cosmological models.

Etherington’s theorem requires light rays to be described by null geodesics which would not be the case if photons had mass, so introducing massive photons that’s another way out. It also requires photon numbers to be conserved, so some mysterious way of making photons disappear might do the trick, so adding some exotic field that interacts with light in a peculiar way is another possibility.

Anyway, my main point here is that if one could pin down the Hubble constant tension as a discrepancy between angular-diameter and luminosity based distances then the most obvious place to look for a resolution is in departures of the metric from the Robertson-Walker form.

Addendum: just to clarify one point, the reciprocity theorem applies to any GR-based metric theory, i.e. just about anything without torsion in the metric, so it applies to inhomogeneous cosmologies based on GR too. However, in such theories there is no way of defining a global scale factor a(t) so the reciprocity relation applies only locally, in a different form for each source and observer.

16 Comments »