## Non-Solutions to the Hubble Constant Problem…

A rather pugnacious paper by George Efstathiou appeared on the arXiv earlier this week. Here is the abstract:

This paper investigates whether changes to late time physics can resolve the `Hubble tension’. It is argued that many of the claims in the literature favouring such solutions are caused by a misunderstanding of how distance ladder measurements actually work and, in particular, by the inappropriate use of distance ladder H0 priors. A dynamics-free inverse distance ladder shows that changes to late time physics are strongly constrained observationally and cannot resolve the discrepancy between the SH0ES data and the base LCDM cosmology inferred from Planck.

For a more detailed discussion of this paper, see Sunny Vagnozzi’s blog post. I’ll just make some general comments on the context.

One of the reactions to the alleged “tension” between the two measurements of H_{0} is to alter the standard model in such a way that the equation of state changes significantly at late cosmological times. This is because the two allegedly discrepant sets of measures of the cosmological distance scale (seen, for example, in the diagram below taken from the paper I blogged about a while ago here) 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).

That is basically true. There is, however, another difference in the two types of distance determination: the high values of the Hubble constant are generally related to interpretations of the measured brightness of observed sources (i.e. they are based on 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-spectrum, 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:

*D*.

_{L}=D_{A}(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 *D _{L}=D_{A}(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 this, the fact there are significant numbers of theorists pushing for such things as interacting dark-energy models to engineer late-time changes in expansion history is indeed a bit perplexing.

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. 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.

All of this begs the question of whether or not there is real tension in the H_{0} measures. I certainly have better things to get tense about. That gives me an excuse to include my long-running poll on the issue:

March 18, 2021 at 11:38 am

“the high values of the Hubble constant are generally related to interpretations of the measured brightness of observed sources (i.e. they are based on luminosity distances) while the lower values are generally based on trigonometry (specifically they are angular diameter distances)”That is true. However, as you note, if those two types of distances don’t give the same result, then something really strange must be going on, as the reciprocity theorem applies to quite a large range of cosmological models.

Probably more relevant is the physical scale of the angles involved. (Luminosity distance also involves angles, but with the apex at the other end.). The global measurements are concerned with objects much larger than the size of galaxies, while the supernova measurements are concerned with objects about the size of the Solar System. The lensing stuff is somewhere in-between.

In a lumpy universe, there can be less matter in the beam on small angular scales than on large ones. However, this can’t be the explanation, because it goes in the wrong direction. If supernovae were seen essentially through empty beams, then one would overestimate their proper distance (objects in the mirror are closer than they appear), and the proper distance is the relevant distance for the Hubble constant. However, overestimating the distance would mean underestimating the Hubble constant.

It doesn’t seem likely that the beams to supernovae are over-dense. In a flux-limited sample, one will preferentially observe magnified objects (more matter in the beam, a type of weak lensing affecting the brightness), but that doesn’t seem to play a role in the observations.

March 18, 2021 at 6:48 pm

I prefer the gauge theory of gravity due to Tom Kibble (J Math Phys 2, 212; 1961) and put into the more transparent mathematical language of Clifford Algebra by Anthony Lasenby et al in the 1990s. This gauges the Dirac equation wrt transformations comprising the Poincare group and finds a covariant derivative. The resulting gauge fields together give GR locally in vacuo but have torsion in the presence of aligned fermion spin density. In this theory the relation between spin and torsion is purely algebraic, containing no differential terms. Wherever torsion appears in the analogue of the Friedmann equations, it can therefore be eliminated algebraically in favour of spin. As a result, terms quadratic in the spin density supplement the pressure and density terms in the original Friedmann equations of GR. These terms do not vanish in a homogeneous isotropic universe even though the bulk spin vector is zero. But the spin-spin interaction is O(Gh^2) and incredibly weak, which is why it has not been observed to date. Whether it has consequences for estimates of the Hubble constant may depend on whether dark stuff is of this type.

March 18, 2021 at 7:26 pm

It would have been nice if someone had predicted the tension, its sign, and its magnitude before it was known. At least intuitively, such a theory would be more credible than one explaining things already known.

March 18, 2021 at 8:01 pm

They have done, Phillip! I’m not sure what you mean by “before it was known”. It is too weak to be observable in the foreseeable future.

March 18, 2021 at 9:31 pm

I was referring to the current Hubble tension.

March 18, 2021 at 9:43 pm

Got it!

March 19, 2021 at 1:59 pm

Also https://arxiv.org/abs/1911.06456

March 19, 2021 at 2:46 pm

I’m pretty sure that the answer is that or something similar to that.

I’m pretty sure that it’s a typo, but if the authors were known to be grammar obsessives, “peculiar velocity corrections” would have perhaps been a sly criticism.

March 19, 2021 at 2:48 pm

After a year on the arXiv, it was revised recently, but still no information is to where it might be published.