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 H₀ 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, H₀ 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 ρc² via p=wρc². The fixed value 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^-3, as expected for `ordinary’ matter.

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!

Follow @telescoper

Advertisements

Hubble’s Constant – The Tension Mounts!

There’s a 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. The abstract is here:

You can download a PDF of the full paper here.

You will that these measurements, based on observations of time delays in multiply imaged quasars that have been  gravitationally lensed, give higher values of the Hubble constant than determinations from, e.g., the Planck experiment.

Here’s a nice summary of the tension in pictorial form:

And here are some nice pictures of the lensed quasars involved in the latest paper:

 

It’s interesting that these determinations seem more consistent with local distance-scale approaches than with global cosmological measurements but the possibility remains of some unknown systematic.

Time, methinks, to resurrect my long-running poll on this!

Please feel free to vote. At the risk of inciting Mr Hine to clog up my filter with further gibberish,  you may also comment through the box below.

Follow @telescoper

 

Thirty Years as a Doctor!

A chance discovery while rummaging around in my filing cabinet reminded me that today is the anniversary of a momentous event. What I found was this:

It’s the programme of the summer Graduation Ceremony in 1989 at which I formally received my DPhil (Doctor of Philosophy). As you will see that was precisely thirty years ago today!

I actually submitted my thesis the previous summer (either at the end of August or start of September 1988) but had to wait a few months for the examination, which I think was in December.  By the time I had done my corrections (mainly typographical errors) the next available date for the degree to be formally conferred was in July 1989 so that’s when I officially got doctored. I was actually still in Brighton at the time, as had started work as a postdoctoral researcher soon after I had submitted my thesis.

Here’s my thesis:

In those days they actually printed the thesis title in the programme, alongside the graduand’s name in the case of DPhil degrees.

It’s normal practice for people to assume the title of Doctor as soon as they have passed the viva voce examination but although I’ve never objected to that,  I’ve always been a bit unsure of the legality. Probably one doesn’t actually have a doctorate until it is conferred (either at a ceremony or in absentia).

Anyway, here is a picture of me (aged 26!)  emerging from the Brighton Centre wearing the old-style Sussex doctoral gown just after I received my DPhil:

Unfortunately the University of Sussex decided a while ago to change the style of its academic dress recently to something a bit more conventional and as far as I know it’s not possible to obtain the old-style gowns any more. They also changed the title DPhil to PhD because it confused potential students, especially those not from the UK.

My first degree came from Cambridge so I had to participate in an even more archaic ceremony for that institution. The whole thing is done in Latin there (or was when I graduated) and involves each graduand holding a finger held out by their College’s Praelector and then kneeling down in front of the presiding dignitary, who is either the Vice-Chancellor ot the Chancellor. I can’t remember which. It’s also worth mentioning that although I did Natural Sciences (specialising in Theoretical Physics), the degree I got was Bachelor of Arts. Other than that, and the fact that the graduands had to walk to the Senate House from their College through the streets of Cambridge,  I don’t remember much about the actual ceremony.

I was very nervous for that first graduation. The reason was that my parents had divorced some years before and my Mum had re-married. My Dad wouldn’t speak to her or her second husband. Immediately after the ceremony there was a garden party at my college, Magdalene, at which the two parts of my family occupied positions at opposite corners of the lawn and I scuttled between them trying to keep everyone happy. It was like that for the rest of the day and I have to say it was very stressful. A few years later I got my doctorate from the University of Sussex, at the Brighton Centre on the seafront. It was pretty much the same deal again with the warring family factions, but I enjoyed the whole day a lot more that time. And I got to wear the funny gown.

Follow @telescoper

Father Callan and the Induction Coil

Historically speaking, Maynooth is more strongly associated with theology than with science but I thought I’d mention here one famous pioneering physicist, who happened also to be a Roman Catholic priest, who spent his working life in these parts.

Father Nicholas Callan (or, more formally, The Reverend Professor Nicholas Joseph Callan) was born in County Louth in 1799 went to the seminary of St Patrick’s College, Maynooth, in 1816 to train as a priest. During his time as a seminarian Callan studied ‘Natural Philosophy’ and became interested in experiments involving electricity. In 1823 Callan was ordained as a priest, and went to Rome in 1826 to obtain his doctorate in Divinity. At the time Italy was a centre for research into electricity and here Callan became familiar with the work of the Italian physicist Alessandro Volta who had developed the world’s first battery. Callan returned to Maynooth where he was made chair of Natural Philosophy, a post he would hold until his death in 1864.

Callan is most famous for inventing the induction coil (in 1836). By connecting two copper wire coils to a battery and electromagnet and then interrupting the current he was able to generate much larger voltages than could be obtained from batteries alone. His 1837 version that used a clock mechanism to interrupt the current 20 times a second is estimated to have produced 60,000 volts – the largest artificially generated charge at that time. It is said that his induction coil could produce sparks 15″ long, which must have been fun to watch.

Callan’s biggest induction coil, unfinished at the time of his death, can be found in the National Science Museum of Ireland (which is in Maynooth). This was one of the largest in the world at the time. The iron core is 109 cm long. The secondary windings are 53 cm in diameter and consist of about 50 km of iron wire insulated with beeswax. They were made in three separate rings separated by air gaps, so wires carrying large voltage differences would not lie adjacent to each other, reducing the risk of the insulation breaking down. At the left end is a vibrating mercury ‘contact breaker’ in the primary circuit, actuated by the magnetic field in the primary, which interrupted the primary current to generate potentials of over 200,000 volts.

Sadly Callan’s work was forgotten for quite a period after his death – experimental electromagnetism was not a priority for St Patrick’s College at this time – for which reason the invention of the induction coil has often been attributed to Heinrich Ruhmkorff who made his first device (independently) about 15 years after Callan. More recently, however, Callan’s achievements have been more widely recognized and in 2000 the Irish government issued a stamp in his honour.

The Callan Building

Nicholas Callan was laid to rest in the College Cemetery at Maynooth in 1864. The Callan Building (above) on the North Campus of the present-day Maynooth University is named in his honour.

Follow @telescoper

Cosmology with the Minimal Spanning Tree

There’s a nice paper on the arXiv (by Naidoo et al) with the abstract:

The code mentioned at the end can be found here.

The appearance of this paper gives me an excuse to mention that I actually wrote a paper (with Russell Pearson) on the use of the Minimal (or Minimum) Spanning Tree (MST) to analyze galaxy clustering way back in 1995.

Here’s how we described the Minimal Spanning Tree in that old paper:

Strictly speaking , we used the Euclidean Minimum Spanning Tree in which the total length of the lines connecting a set of points in a tree is minimized. In general cases a weight can be assigned to each link that is not necessarily defined simply by the length. Here is visual illustration (which I think we drew by hand!)

You can think of the MST as a sort of pre-processing technique which accentuates linear features in a point process that might otherwise get lost in shot noise. Once one has a tree (pruned and/or separated as necessary) one can then extract various statistical properties in order to quantify the pattern present.

Way back in 1995 there were far fewer datasets available to which to apply this method and it didn’t catch on at the time. Now, with  ever-increasing availability of spectroscopic redshift surveys maybe its time has come at last! I look forward to playing with the Python code in due course!

Follow @telescoper

 

Wolfram Alpha and the Principle of Astrogeometry

Regular readers of this blog (both of them) may remember the basically tedious and offensive but occasionally (accidentally) hilarious troll who keeps attempting to post comments like this:

I thought you might like to see the result of feeding the expression found in the above rant into Wolfram Alpha:

This is exactly the expression described above but produces nothing like the claimed value of the Hubble constant, and it’s in the wrong units too.

Update for the benefit of the extremely hard of thinking (especially Mr Hine):

π²¹ ≈ 2.75 × 10¹⁰ (dimensionless).

One parsec = 3.086 ×10¹⁶ m so one Megaparsec is 3.086 ×10²² m. Hence 2 × `a Mpc’ × c ≈ 2 × 3.086 ×10²² m × 3 × 10⁸ m s^-1 ≈ 1.83 × 10³¹ m² s^-1.

Thus the full expression is obtained by dividing this by the value for π²¹ obtained above giving a value approximately 6.7× 10²⁰ m² s^-1 as demonstrated by Wolfram Alpha.

The correct value for the Hubble constant is about 2.2 × 10⁻¹⁸ s⁻¹.

 

UPDATE: It’s interesting how the Megaparsec appears in the numerator in Mr Hine’s expression, but magically transfers to the denominator as far as the units are concerned:

ANOTHER UPDATE:

I think I may have cracked it. I believe Mr Hine’s calculation involves using light-years instead of Mpc or SI (for some reason) the calculation is in which case the calculation becomes:

π²¹ ≈ 2.75 × 10¹⁰ (dimensionless) as before

One parsec = 3.26 light years so one Megaparsec is 3.26 ×10⁶ million light years. Hence 2 × `a Mpc’ × c ≈ 2 × 3.26 ×10⁶ m × 3 × 10⁵ m s^-1, using c in km/s.

When divided by the value of π²¹ this gives a number around 71 (I couldn’t be bothered with the extra decimal places).

However, although it is a number around 71 the units are then km/s times light years, not the correct units which are km/s divided by Megapersecs. The fact that the number comes out close to 70 is just a numerical artefact of Mr Hine’s basic misunderstanding of units and dimensions. In other words, it’s gibberish. I know you’ll all be shocked by this revelation, but it’s true.

Follow @telescoper

New Publication at the Open Journal of Astrophysics!

In a blog I posted just a couple of day ago I mentioned that there were a number of papers about to be published by the Open Journal of Astrophysics and, to show that I wasn’t making that up, the first of the latest batch has just appeared. Here is how it looks on the site!

There are thirteen authors altogether (from Oxford, Liverpool, Edinburgh, Leiden, British Columbia, Zurich and Munich); the lead other is Elisa

You can find the accepted version on the arXiv here. This version was accepted after modifications requested by the referee and editor.

This is another one for the `Cosmology and Nongalactic Astrophysics’ folder. We would be happy to get more submissions from other areas of astrophysics. Hint! Hint!

A few people have asked why the Open Journal of Astrophysics is not yet listed in the Directory of Open Access Journals. The answer to that is simple: to qualify for listing a journal must publish a minimum of five papers in a calendar year. Since OJA underwent a failure long hiatus after publishing its first batch of papers we haven’t yet qualified. However, this new one means that we have now published five papers so have reached the qualifying level.  I’ll put in the application as soon as I can, but will probably wait a little because we have a bunch of other papers coming out very soon to add to that number.

P.S. Please note that we now have an Open Journal of Astrophysics Facebook page where you can follow updates from the Journal should you wish..

Follow @telescoper