## Hubble’s Constant – A Postscript on w

Last week I posted about new paper on the arXiv (by Wong et al.) that adds further evidence to the argument about whether or not the standard cosmological model is consistent with different determinations of the Hubble Constant. You can download a PDF of the full paper here.

Reading the paper through over the weekend I was struck by Figure 6:

This shows the constraints on H_{0} and the parameter w which is used to describe the dark energy component. Bear in mind that these estimates of cosmological parameters actually involve the simultaneous estimation of several parameters, six in the case of the standard ΛCDM model. Incidentally, H_{0} is not one of the six basic parameters of the standard model – it is derived from the others – and some important cosmological observations are relatively insensitive to its value.

The parameter *w* is the equation of state parameter for the dark energy component so that the pressure *p *is related to the energy density *ρc ^{2}* via

*p=wρc*. The fixed value

^{2}*w=-1*applies if the dark energy is of the form of a cosmological constant (or vacuum energy). I explained why here. Non-relativistic matter (dominated by rest-mass energy) has

*w=0*while ultra-relativistic matter has

*w=1/3.*

Applying the cosmological version of the thermodynamic relation for adiabatic expansion “dE=-pdV” one finds that *ρ ∼ a ^{-3(1+w)}* where

*a*is the cosmic scale factor. Note that

*w=-1*gives a constant energy density as the Universe expands (the cosmological constant); w=0 gives

*ρ ∼ a*, as expected for `ordinary’ matter.

^{-3}As I already mentioned, in the standard cosmological model *w* is fixed at *w=-1* but if it is treated as a free parameter then it can be added to the usual six to produce the Figure shown above. I should add for Bayesians that this plot shows the posterior probability *assuming a uniform prior* on *w*.

What is striking is that the data seem to prefer a* very low value* of *w. *Indeed the peak of the likelihood (which determines the peak of the posterior probability if the prior is flat) appears to be off the bottom of the plot. It must be said that the size of the black contour lines (at one sigma and two sigma for dashed and solid lines respectively) suggests that these data aren’t really very informative; the case *w=-1* is well within the 2σ contour. In other words, one might get a slightly better fit by allowing the equation of state parameter to float, but the quality of the fit might not improve sufficiently to justify the introduction of another parameter.

Nevertheless it is worth mentioning that if it did turn out, for example, that *w=-2* that would imply *ρ ∼ a ^{+3}*, i.e. an energy density that increases steeply as

*a*increases (i.e. as the Universe expands). That would be pretty wild!

On the other hand, there isn’t really any physical justification for cases with *w<-1* (in terms of a plausible model) which, in turn, makes me doubt the reasonableness of imposing a flat prior. My own opinion is that if dark energy turns out not to be of the simple form of a cosmological constant then it is likely to be too complicated to be expressed in terms of a single number anyway.

Postscript to this postscript: take a look at this paper from 2002!

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July 15, 2019 at 3:35 pm

Worth pointing out that the orange blob in the above Figure looks like “CMB + H0liCOW only”, which looks fine with w around -1.2.

However, that w would lead to fairly serious tension with PanTHEON and similar supernova samples, and non-negligible tension with BAOs as well, so lower w is not really a panacea but just shunts the tension somewhere else.

There is an interesting paper by Aylor et al, arXiv:1811.00537, which makes the case that if the H0 tension is “real”, the solution is more likely to be in high-z changes (such as extra neutrino-like species and N_eff ) which lead to a shorter sound horizon length, rather than purely low-z modifications e.g. dark energy, curvature etc.

July 15, 2019 at 10:48 pm

One of the most depressing aspects of my PhD was the realisation that any w function was never going to have more than a couple of parameters constrained, if that.

July 17, 2019 at 3:19 pm

“On the other hand, there isn’t really any physical justification for cases with w<-1 (in terms of a plausible model) which, in turn, makes me doubt the reasonableness of imposing a flat prior."Indeed. A top-hat (or boxcar) prior with an infinitely sharp cutoff at -1 would be more appropriate. One has to introduce physically motivated restrictions. Some early attempts at modelling the

m–zrelation for type Ia supernovae claimed a best fit for negative values of Omega. 😐“My own opinion is that if dark energy turns out not to be of the simple form of a cosmological constant then it is likely to be too complicated to be expressed in terms of a single number anyway.”Right. And, as far as I know, there is no physical motivation for the usual parameterization in terms of w_0 and w_a or whatever.

July 17, 2019 at 3:29 pm

Indeed. Using a series expansion of w(z) or w(a) is barmy.