Phase Correlations and Cosmic Structure

Posted in Biographical, The Universe and Stuff with tags , , , on July 9, 2022 by telescoper

I’m indebted to a friend for tipping me off about a nice paper that appeared recently on the arXiv by Franco et al. with the title First measurement of projected phase correlations and large-scale structure constraints. The abstract is here:

Phase correlations are an efficient way to extract astrophysical information that is largely independent from the power spectrum. We develop an estimator for the line correlation function (LCF) of projected fields, given by the correlation between the harmonic-space phases at three equidistant points on a great circle. We make a first, 6.5σ measurement of phase correlations on data from the 2MPZ survey. Finally, we show that the LCF can significantly improve constraints on parameters describing the galaxy-halo connection that are typically degenerate using only two-point data.

I’ve worked on phase correlations myself (with a range of collaborators) – you can see a few of the papers here. Indeed I think it is fair to say I was one of the first people to explore ways of quantifying phase information in cosmology. Although I haven’t done anything on this recently (by which I mean in the last decade or so), other people have been developing very promising looking approaches (including the Line Correlation Function (LCF) explored in the above paper. In my view there is a lot of potential in this approach and as we await even more cosmological data and hopefully more people will look at this in future. In my opinion we still haven’t found the optimal way to exploit phase information statistically so there’s a lot of work to be done in this field.

Anyway, I thought I’d try to explain what phase correlations are and why they are important.

One of the challenges we cosmologists face is how to quantify the patterns we see in, for example, galaxy redshift surveys. In the relatively recent past the small size of the available data sets meant that only relatively crude descriptors could be used; anything sophisticated would be rendered useless by noise. For that reason, statistical analysis of galaxy clustering tended to be limited to the measurement of autocorrelation functions, usually constructed in Fourier space in the form of power spectra; you can find a nice review here.

Because it is so robust and contains a great deal of important information, the power spectrum has become ubiquitous in cosmology. But I think it’s important to realize its limitations.

Take a look at these two N-body computer simulations of large-scale structure:

The one on the left is a proper simulation of the “cosmic web” which is at least qualitatively realistic, in that in contains filaments, clusters and voids pretty much like what is observed in galaxy surveys.

To make the picture on the right I first  took the Fourier transform of the original  simulation. This approach follows the best advice I ever got from my thesis supervisor: “if you can’t think of anything else to do, try Fourier-transforming everything.”

Anyway each Fourier mode is complex and can therefore be characterized by an amplitude and a phase (the modulus and argument of the complex quantity). What I did next was to randomly reshuffle all the phases while leaving the amplitudes alone. I then performed the inverse Fourier transform to construct the image shown on the right.

What this procedure does is to produce a new image which has exactly the same power spectrum as the first. You might be surprised by how little the pattern on the right resembles that on the left, given that they share this property; the distribution on the right is much fuzzier. In fact, the sharply delineated features  are produced by mode-mode correlations and are therefore not well described by the power spectrum, which involves only the amplitude of each separate mode.

If you’re confused by this, consider the Fourier transforms of (a) white noise and (b) a Dirac delta-function. Both produce flat power-spectra, but they look very different in real space because in (b) all the Fourier modes are correlated in such away that they are in phase at the one location where the pattern is not zero; everywhere else they interfere destructively. In (a) the phases are distributed randomly.

The moral of this is that there is much more to the pattern of galaxy clustering than meets the power spectrum…

Cosmology Talks: Volker Springel on GADGET-4

Posted in The Universe and Stuff with tags , , , , , , , on May 18, 2021 by telescoper

It’s time I shared another one of those interesting cosmology talks on the Youtube channel curated by Shaun Hotchkiss. This channel features technical talks rather than popular expositions so it won’t be everyone’s cup of tea but for those seriously interested in cosmology at a research level they should prove interesting.

In this talk from a couple of months ago  Volker Springel discusses Gadget-4 which is a parallel computational code that combines cosmological N-body and SPH code and is intended for simulations of cosmic structure formation and calculations relevant for galaxy evolution and galactic dynamics.

The predecessor of GADGET-2 is probably the most used computational code in cosmology; this talk discusses what new ideas are implemented in GADGET-4 to improve on the earlier version and what new features it has.  Volker also explains what happened to GADGET-3!

The paper describing Gadget-4 can be found here.

With the Cosmic Web in Mind..

Posted in Astronomy Lookalikes, The Universe and Stuff with tags , , , , , on November 23, 2019 by telescoper

Some time ago I posted one of my Astronomy Look-alikes about the remarkable similarity between the structure of the human brain and that revealed by computer simulations of the large-scale structure of the Universe:

I wonder whether this means that the Cosmic Web is really just all in the mind?

Anyway I just came across an article by Franco Vazza and Alberto Fenetti that takes the comparison between brain cells (among other things) and the Cosmic Web a bit further, including a look at the corresponding power spectra:

The main point to take from this picture is that many naturally occurring patterns have approximately power-law power spectra, at least over a limited range of scales. However, as I have pointed out before on this blog, the power spectrum on its own does not really quantify pattern in any meaningful way. Here for example are two patterns with exactly the same power spectrum:

The point is that the power spectrum does not contain any information about the phase correlations of the Fourier modes, which are important in quantifying localised features. For further discussion of this issue, see here.

That was the Science Week that was..

Posted in Books, Talks and Reviews, The Universe and Stuff with tags , , , on November 15, 2019 by telescoper

So, as advertised, this morning I gave a talk mainly to school students as part of Science Week Ireland on the subject of the cosmic web. This was a similar talk to the one I gave at DIAS a couple of weeks ago.

There was a slight confusion about rooms but we did eventually get everyone into the right lecture theatre and weren’t too late getting started. The audience was about 140, so the room was very full and most of them didn’t fall asleep. I had a nice chat afterwards with a group of them and they seemed to have enjoyed it. Anyway, in case anyone is interested here are my slides. Most of them are recycled from previous versions of this talk.

Following this morning’s exertions we had lovely seminar after lunch by Wyn Evans of Cambridge about the stellar dynamics of the Milky Way and the wonders of Gaia and soon will be going to dinner.

New Publication at the Open Journal of Astrophysics!

Posted in Open Access, The Universe and Stuff with tags , , , , , , on June 26, 2019 by telescoper

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..

The Most Ancient Heavens

Posted in Art, Biographical, Poetry, The Universe and Stuff with tags , , , , , , , , on March 21, 2019 by telescoper

So here I am, in that London, getting ready for the start of a two-day conference at the Royal Astronomical Society on cosmology, large-scale structure, and weak gravitational lensing, to celebrate the work of Professor Alan Heavens, on (or near) the occasion of his 60th birthday. Yes, it is a great name for an astronomer.

I was honoured to be invited to give a talk at this meeting, though my immediate reaction when I was told about was But he can’t be sixty! He’s only a few years older than me…oh.’ I gather I’m supposed to say something funny after the conference dinner tomorrow night too.

Courtesy of alphabetical order it looks like I’m top of the bill!

Anyway, I’ve known Alan since I was a research student, i.e. over thirty years, and we’re co-authors on 13 papers (all of them since 2011). I’m looking forward to the HeavensFest not only for the scientific programme (which looks excellent) but also for the purpose of celebrating an old friend and colleague.

Just to clear up a couple of artistic points.

First, the title of the meeting, The Most Ancient Heavens, is taken from Ode to Duty by William Wordsworth.

Second, the image on the conference programme shown above is a pastiche of The Creation of Alan Adam which is part of the ceiling of the Sistine Chapel painted by Michelangelo di Lodovico Buonarroti Simoni, known to his friends as Michelangelo. Apparently he worked flat out painting this enormous fresco. It was agony but the ecstasy kept him going. I’ve often wondered (a) who did the floor of the Sistine Chapel and (b) how could Michelangelo create such great art when it was so clearly extremely cold? Anyway, I think that is a picture of Alan at high redshift on the far right, next to the man with beard who at least had the good sense to wear a nightie to spare his embarrassment.

Anyway, that’s all for now. I must be going. Time for a stroll down to Piccadilly.

Update: you can find a bunch of pictures of this conference here.

New Publication at the Open Journal of Astrophysics!

Posted in Open Access, The Universe and Stuff with tags , , , , , , , , , on March 20, 2019 by telescoper

It’s nice to be able to announce that the Open Journal of Astrophysics has just published another paper. Here it is!

It’s by Darsh Kodwani, David Alonso and Pedro Ferreira from a combination of Oxford University and Cardiff University.

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!

P.S. A few people have asked why the Open Journal of Astrophysics is not 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 year. Since OJA underwent a failure long hiatus after publishing its first batch of papers we don’t yet qualify. However, so far in 2019 we have published four papers and have several others in the pipeline. We will reach the qualifying level soon and when we do I will put in the application!

Subaru and Cosmic Shear

Posted in The Universe and Stuff with tags , , , , , , on February 15, 2019 by telescoper

Up with the lark this morning I suddenly remembered I was going to do a post about a paper which actually appeared on the arXiv some time ago. Apart from the fact that it’s a very nice piece of work, the first author is Chiaki Hikage who worked with me as a postdoc about a decade ago. This paper is extremely careful and thorough, which is typical of Chiaki’s work. Its abstract is here:

The work described uses the Hyper-Suprime-Cam Subaru Telescope to probe how the large-scale structure of the Universe has evolved by looking at the statistical effect of gravitational lensing – specifically cosmic shear – as a function of redshift (which relates to look-back time). The use of redshift binning as demonstrated in this paper is often called tomography. Gravitational lensing is sensitive to all the gravitating material along the line of sight to the observer so probes dark, as well as luminous, matter.

Here’s a related graphic:

The article that reminded me of this paper is entitled New Map of Dark Matter Spanning 10 Million Galaxies Hints at a Flaw in Our Physics. Well, no it doesn’t really. Read the abstract, where you will find a clear statement that these results do not show significant evidence for discordance’. Just a glance at the figures in the paper will convince you that is the case. Of course, that’s not to say that the full survey (which will be very much bigger; the current paper is based on just 11% of the full data set) may not reveal such discrepancies, just that analysis does not. Sadly this is yet another example of misleadingly exaggerated science reporting. There’s a lot of it about.

Incidentally, the parameter S8 is a (slightly) rescaled version of the more familiar parameter σ8  – which quantifies the matter-density fluctuations on a scale of 8 h-1 Mpc – as defined in the abstract; cosmic shear is particularly sensitive to this parameter.

Anyway, if this is what can be done with just 11%, the full survey should be a doozy!

Wave Mechanics and Large-scale Structure

Posted in Books, Talks and Reviews, The Universe and Stuff with tags , , , on May 24, 2017 by telescoper

I thought I’d share the slides I used for the short talk I gave last Thursday at the Osservatorio Astronomico di Bologna, on the topic of Wave Mechanics and Large-scale Structure. I’ve posted about the general idea underpinning this workhere, and here are some links to references with more details of the cosmological setting, including a couple of papers by myself and Chris Short on some of whose old slides I based the talk.

I had a few problems with the movies during the actual talk, and they probably don’t work in this embedded version. There are a few formatting errors in the slideshare version too, but hopefully you can figure out what’s going on!

The Fractal Universe, Part 2

Posted in History, The Universe and Stuff with tags , , , , , , on June 27, 2014 by telescoper

Given the recent discussion in comments on this blog I thought I’d give a brief update on the issue of the scale of cosmic homogeneity; I’m going to repeat some of the things I said in a post earlier this week just to make sure that this discussion is reasonable self-contained.

Our standard cosmological model is based on the Cosmological Principle, which asserts that the Universe is, in a broad-brush sense, homogeneous (is the same in every place) and isotropic (looks the same in all directions). But the question that has troubled cosmologists for many years is what is meant by large scales? How broad does the broad brush have to be? A couple of presentations discussed the possibly worrying evidence for the presence of a local void, a large underdensity on scale of about 200 MPc which may influence our interpretation of cosmological results.

I blogged some time ago about that the idea that the Universe might have structure on all scales, as would be the case if it were described in terms of a fractal set characterized by a fractal dimension $D$. In a fractal set, the mean number of neighbours of a given galaxy within a spherical volume of radius $R$ is proportional to $R^D$. If galaxies are distributed uniformly (homogeneously) then $D = 3$, as the number of neighbours simply depends on the volume of the sphere, i.e. as $R^3$, and the average number-density of galaxies. A value of $D < 3$ indicates that the galaxies do not fill space in a homogeneous fashion: $D = 1$, for example, would indicate that galaxies were distributed in roughly linear structures (filaments); the mass of material distributed along a filament enclosed within a sphere grows linear with the radius of the sphere, i.e. as $R^1$, not as its volume; galaxies distributed in sheets would have $D=2$, and so on.

We know that $D \simeq 1.2$ on small scales (in cosmological terms, still several Megaparsecs), but the evidence for a turnover to $D=3$ has not been so strong, at least not until recently. It’s just just that measuring $D$ from a survey is actually rather tricky, but also that when we cosmologists adopt the Cosmological Principle we apply it not to the distribution of galaxies in space, but to space itself. We assume that space is homogeneous so that its geometry can be described by the Friedmann-Lemaitre-Robertson-Walker metric.

According to Einstein’s theory of general relativity, clumps in the matter distribution would cause distortions in the metric which are roughly related to fluctuations in the Newtonian gravitational potential $\delta\Phi$ by $\delta\Phi/c^2 \sim \left(\lambda/ct \right)^{2} \left(\delta \rho/\rho\right)$, give or take a factor of a few, so that a large fluctuation in the density of matter wouldn’t necessarily cause a large fluctuation of the metric unless it were on a scale $\lambda$ reasonably large relative to the cosmological horizon $\sim ct$. Galaxies correspond to a large $\delta \rho/\rho \sim 10^6$ but don’t violate the Cosmological Principle because they are too small in scale $\lambda$ to perturb the background metric significantly.

In my previous post I left the story as it stood about 15 years ago, and there have been numerous developments since then, some convincing (to me) and some not. Here I’ll just give a couple of key results, which I think to be important because they address a specific quantifiable question rather than relying on qualitative and subjective interpretations.

The first, which is from a paper I wrote with my (then) PhD student Jun Pan, demonstrated what I think is the first convincing demonstration that the correlation dimension of galaxies in the IRAS PSCz survey does turn over to the homogeneous value $D=3$ on large scales:

You can see quite clearly that there is a gradual transition to homogeneity beyond about 10 Mpc, and this transition is certainly complete before 100 Mpc. The PSCz survey comprises “only” about 11,000 galaxies, and it relatively shallow too (with a depth of about 150 Mpc),  but has an enormous advantage in that it covers virtually the whole sky. This is important because it means that the survey geometry does not have a significant effect on the results. This is important because it does not assume homogeneity at the start. In a traditional correlation function analysis the number of pairs of galaxies with a given separation is compared with a random distribution with the same mean number of galaxies per unit volume. The mean density however has to be estimated from the same survey as the correlation function is being calculated from, and if there is large-scale clustering beyond the size of the survey this estimate will not be a fair estimate of the global value. Such analyses therefore assume what they set out to prove. Ours does not beg the question in this way.

The PSCz survey is relatively sparse but more recently much bigger surveys involving optically selected galaxies have confirmed this idea with great precision. A particular important recent result came from the WiggleZ survey (in a paper by Scrimgeour et al. 2012). This survey is big enough to look at the correlation dimension not just locally (as we did with PSCz) but as a function of redshift, so we can see how it evolves. In fact the survey contains about 200,000 galaxies in a volume of about a cubic Gigaparsec. Here are the crucial graphs:

I think this proves beyond any reasonable doubt that there is a transition to homogeneity at about 80 Mpc, well within the survey volume. My conclusion from this and other studies is that the structure is roughly self-similar on small scales, but this scaling gradually dissolves into homogeneity. In a Fractal Universe the correlation dimension would not depend on scale, so what I’m saying is that we do not live in a fractal Universe. End of story.