## 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.

## The Zel’dovich Universe – Day 4 Summary

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

And on the fourth day of this meeting about “The Zel’dovich Universe”  we were back to a full schedule (9am until 7.30pm) concentrating on further studies of the Cosmic Web. We started off with a discussion of the properties of large-scale structure at high redshift. As someone who’s old enough to remember the days when “high redshift” meant about z~0.1 the idea that we can now map the galaxy distribution at redshifts z~2. There are other measures of structure on these huge scales, such as the Lyman alpha forest, and we heard a bit about some of them too.

The second session was about “reconstructing” the Cosmic Web, although a more correct word have been “deconstructing”. The point about this session is that cosmology is basically a backwards subject. In other branches of experimental science we set the initial conditions for a system and then examine how it evolves. In cosmology we have to infer the initial conditions of the Universe from what we observe around us now. In other words, cosmology is an inverse problem on a grand scale.  In the context of the cosmic web, we want to infer the pattern of initial density and velocity fluctuations that gave rise to the present network of clusters, filaments and voids. Several talks about this emphasized how proper Bayesian methods have led to enormous progress in this field over the last few years.

All this progress has been accompanied by huge improvements in graphical visualisation techniques. Thirty years ago the state of the art in this field was represented by simple contour plots, such as this (usually called the Cosmic Chicken):

You can see how crude this representation is by comparing it with a similar plot from the modern era of precision cosmology:

Even better examples are provided by the following snapshot:

It’s nice to see a better, though still imperfect,  version of the chicken at the top right, though I found the graphic at the bottom right rather implausible; it must be difficult to skate at all with those things around your legs.

Here’s another picture I liked, despite the lack of chickens:

Incidentally, it’s the back of Alar Toomre‘s head you can see on the far right in this picture.

The afternoon was largely devoted to discussions of how the properties of individual galaxies are influenced by their local environment within the Cosmic Web. I usually think of galaxies as test particles (i.e. point masses) but they are interesting in their own right (to some people anyway). However, the World Cup intervened during the evening session and I skipped a couple of talks to watch Germany beat the USA in their final group match.

That’s all for now. Tonight we’ll have the conference dinner, which is apparently being held in the “House of Blackheads” on “Pikk Street”. Sounds like an interesting spot!

## The Zel’dovich Universe – Day 3 Summary

Posted in History, The Universe and Stuff with tags , , , , , , on June 26, 2014 by telescoper

Day Three of this meeting about “The Zel’dovich Universe” was slightly shorter than the previous two, in that it finished just after 17.00 rather than the usual 19.00 or later. That meant that we got out in time to settle down for a beer in time the World Cup football. I watched an excellent game between Nigeria and Argentina, which ended 3-2 to Argentina but could have been 7-7. I’ll use that as an excuse for writing a slightly shorter summary.

Anyway we began with a session on the Primordial Universe and Primordial Signatures led off by Alexei Starobinsky (although there is some controversy whether his name should end -y or -i). Starobinsky outlined the theory of cosmological perturbations from inflation with an emphasis on how it relates to some of Zel’dovich’s ideas on the subject. There was then a talk from Bruce Partridge about some of the results from Planck. I’ve mentioned already that this isn’t a typical cosmology conference, and this talk provided another unusual aspect in that there’s hardly been any discussion of the BICEP2 results here. When asked about at the end of his talk, Bruce replied (very sensibly) that we should all just be patient.

Next session after coffee was about cosmic voids, kicked off by Rien van de Weygaert with a talk entitled “Much Ado About Nothing”, which reminded me of the following quote from the play of the same name:

“He hath indeed better bettered expectation than you must expect of me to tell you how”

The existence of voids in the galaxy distribution is not unexpected given the presence of clusters and superclusters, but they are interesting in their own right as they display particular dynamical evolution and have important consequences on observations. In 1984, Vincent Icke proved the so-called “Bubble Theorem” which showed that an isolated underdensity tends to evolve to a spherical shape.Most cosmologists, including myself, therefore expected big voids to be round, which turns out to be wrong; the interaction of the perimeter of the void with its surroundings always plays an important role in determining the geometry. Another thing that sprang into my mind was a classic paper by Simon White (1979) with the abstract:

We derive and display relations which can be used to express many quantitative measures of clustering in terms of the hierarchy of correlation functions. The convergence rate and asymptotic behaviour of the integral series which usually result is explored as far as possible using the observed low-order galaxy correlation functions. On scales less than the expected nearest neighbour distance most clustering measures are influenced only by the lowest order correlation functions. On all larger scales their behaviour, in general, depends significantly on correlations of high order and cannot be approximated using the low-order functions. Bhavsar’s observed relation between density enhancement and the fraction of galaxies included in clusters is modelled and is shown to be only weakly dependent on high-order correlations over most of its range. The probability that a randomly placed region of given volume be empty is discussed as a particularly simple and appealing example of a statistic which is strongly influenced by correlations of all orders, and it is shown that this probability may obey a scaling law which will allow a test of the small-scale form of high-order correlations.

The emphasis is mine. It’s fascinating and somewhat paradoxical that we can learn a lot about the statistics of where the galaxies are fom the regions where galaxies are not.

Another thing worth mentioning was Paul Sutter’s discussion of a project on cosmic voids which is a fine example of open science. Check out the CosmicVoids website where you will find void catalogues, identification algorithms and a host of other stuff all freely available to anyone who wants to use them. This is the way forward.

After lunch we had a session on Cosmic Flows, with a variety of talks about using galaxy peculiar velocities to understand the dynamics of large-scale structure. This field was booming about twenty years ago but which has been to some extent been overtaken by other cosmological probes that offer greater precision; the biggest difficulty has been getting a sufficient number of sufficiently accurate direct (redshift-independent) distance measurements to do good statistics. It remains a difficult but important field, because it’s important to test our models with as many independent methods as possible.

I’ll end with a word about the first speaker of this session, the Gruber prize winner Marc Davis. He suffered a stroke a few years ago which has left him partly paralysed (down his right side). He has battled back from this with great courage, and even turned it to his advantage during his talk when he complained about how faint the laser pointer was and used his walking stick instead.

## The Zel’dovich Universe – Day 2 Summary

Posted in History, The Universe and Stuff with tags , , , on June 25, 2014 by telescoper

Day Two of this enjoyable meeting involved more talks about the cosmic web of large-scale structure of the Universe. I’m not going to attempt to summarize the whole day, but will just mention a couple of things that made me reflect a bit. Unfortunately that means I won’t be able to do more than merely mention some of the other fascinating things that came up, as phase-space flip-flops and one-dimensional Origami.

One was a very nice review by John Peacock in which he showed that a version of Moore’s law applies to galaxy redshift surveys; since the first measurement of the redshift of an extragalactic object by Slipher in 1912, the number of redshifts has doubled every 2-3 years ago. This exponential growth has been driven by improvements in technology, from photographic plates to electronic detectors and from single-object spectroscopy to multiplex technology and so on. At this rate by 2050 or so we should have redshifts for most galaxies in the observable Universe. Progress in cosmography has been remarkable indeed.

The term “Cosmic Web” may be a bit of a misnomer in fact, as a consensus may be emerging that in some sense it is more like a honeycomb. Thanks to a miracle of 3D printing, here is an example of what the large-scale structure of the Universe seems to look like:

One of the issues that emerged from the mix of theoretical and observational talks concerned the scale of cosmic homogeneity. 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.

The discussion of a fractal universe is one I’m overdue to return to. In my previous post I left the story as it stood about 15 years ago, and there have been numerous developments since then, not all of them consistent with each other. I will do a full “Part 2” to that post eventually, but in the mean time I’ll just comment that current large surveys, such as those derived from the Sloan Digital Sky Survey, do seem to be consistent with a Universe that possesses the property of large-scale homogeneity. If that conclusion survives the next generation of even larger galaxy redshift surveys then it will come as an immense relief to cosmologists.

The reason for that is that the equations of general relativity are very hard to solve in cases where there isn’t a lot of symmetry; there are just too many equations to solve for a general solution to be obtained. If the cosmological principle applies, however, the equations simplify enormously (both in number and form) and we can get results we can work with on the back of an envelope. Small fluctuations about the smooth background solution can be handled (approximately but robustly) using a technique called perturbation theory. If the fluctuations are large, however, these methods don’t work. What we need to do instead is construct exact inhomogeneous model, and that is very very hard. It’s of course a different question as to why the Universe is so smooth on large scales, but as a working cosmologist the real importance of it being that way is that it makes our job so much easier than it would otherwise be.

PS. If anyone reading this either at the conference or elsewhere has any questions or issues they would like me to raise during the summary talk on Saturday please don’t hesitate to leave a comment below or via Twitter using the hashtag #IAU308.

## Fly through of the GAMA Galaxy Catalogue

Posted in The Universe and Stuff with tags , , , , , , , , on March 13, 2014 by telescoper

When I’m struggling to find time to do a proper blog post I’m always grateful that I work in cosmology because nearly every day there’s something interest to post. I’m indebted to Andy Lawrence for bring the following wonderful video to my attention. It comes from the Galaxy And Mass Assembly Survey (or GAMA Survey for short), a spectroscopic survey of around 300,000 galaxies in a region of the sky comprising about 300 square degrees;  the measured redshifts of the galaxies enable their three-dimensional positions to be plotted. The video shows the shape of the survey volume before showing what the distribution of galaxies in space looks like as you fly through. Note that the galaxy distances are to scale, but the image of each galaxy is magnified to make it easier to see; the real Universe is quite a lot emptier than this in that the separation between galaxies is larger relative to their size.

## A Three-dimensional Map of the Early Universe

Posted in The Universe and Stuff with tags , , , , on August 14, 2013 by telescoper

I found this video via a web page describing the FastSound project, which is surveying galaxies in the Universe which are at such a huge distance that we are seeing them as they were over nine billion years ago. Using the Subaru Telescope‘s impressive new Fiber Multi-Object Spectrograph (FMOS). This project is “work in progress”. The survey so far contains only 1,100 galaxies, but while that is small by the standards of a modern redshift survey, and will in fact still only comprise about 5000 galaxies when complete, what is amazing about it is that the galaxies are at such enormous distances. Even using a telescope with an 8.2 metre primary mirror, this survey will take another year or so to be completed.

A survey of a representative region of the Universe at such high redshift allows astrophysicists to test theories of the growth of the large-scale structure of the Universe. In the standard cosmology, these form by a process of gravitational instability: small irregularities in the distribution of matter get amplified by the action of gravity to become large structures such as galaxies and galaxy clusters. Comparing the level of clustering at early times with that observed around us today allows us to check whether this growth matches theoretical predictions. There should be much less clumpiness earlier on if the theoretical picture is right.

I began my PhD DPhil at the University of Sussex in 1985, working on the large-scale structure of the Universe. Coincidentally, the largest redshift survey available at that time, the CfA1 Survey, also contained 1,100 galaxies – as displayed in the famous “stick man map”:

The galaxies mapped out in that survey, however, are all (relatively speaking) in our back yard: none is further than a few hundred million light years away…

## An Integral Appendix

Posted in Biographical, Cute Problems, The Universe and Stuff with tags , , , , , , on August 7, 2013 by telescoper

After the conference dinner at the Ripples in the Cosmos meeting in Durham I attended recently, a group of us adjourned to the Castle bar for a drink or several. I ended up chatting to one of the locals, Richard Bower, mainly on the subject of beards. I suppose you could call it a chinwag. Only later on did  we get onto the subject of a paper we had both worked on a while ago. It was with some alarm that I later realized that the paper concerned was actually published twenty years ago. Sigh. Where did all that time go?

Anyway, Richard and I both remembered having a great time working on that paper which turned out to be a nice one, although it didn’t exactly set the world on fire in terms of citations. This paper was written before the standard “concordance” (LCDM) cosmology was firmly established and theorists were groping around for ways of reconciling observations of the CMB from the COBE satellite with large-scale structure in the galaxy distribution as well as the properties of individual galaxies. The (then) standard model (CDM with no Lambda) struggled to satisfy the observational constraints, so in typical theorists fashion we tried to think of a way to rescue it. The idea we came up with was “cooperative galaxy formation”, as explained in the abstract:

We consider a model in which galaxy formation occurs at high peaks of the mass density field, as in the standard picture for biased galaxy formation, but is further enhanced by the presence of nearby galaxies. This modification is accomplished by assuming the threshold for galaxy formation to be modulated by large-scale density fluctuations rather than to be spatially invariant. We show that even a weak modulation can produce significant large-scale clustering. In a universe dominated by cold dark matter, a 2 percent – 3 percent modulation on a scale exceeding 10/h Mpc produces enough additional clustering to fit the angular correlation function of the APM galaxy survey. We discuss several astrophysical mechanisms for which there are observational indications that cooperative effects could occur on the scale required.

I have to say that Richard did most of the actual work on this paper, though all four authors did spend a lot of time discussing whether the idea was viable in principle and, if so, how we should implement it mathematically. In the end, my contribution was pretty much limited to the Appendix, which you can click to make it larger if you’re interested.

As is often the case in work of this kind, everything boiled down to evaluating numerically a rather nasty integral. Coincidentally, I’d come across a similar problem in a totally different context a few years previously when I was working on my thesis and therefore just happened to know the neat trick described in the paper.

Two things struck me looking back on this after being reminded of it over that beer. One is that a typical modern laptop is powerful enough to evaluate the original integral without undue difficulty, so if this paper had been written nowadays we wouldn’t have bothered trying anything clever; my Appendix would probably not have been written. The other thing is that I sometimes hear colleagues bemoaning physics students’ lack of mathematical “problem-solving” ability, claiming that if students haven’t seen the problem before they don’t know what to do. The problem with that complaint is that it ignores the fact that many problems are the same as things you’ve solved before, if only you look at them in the right way. Problem solving is never going to be entirely about “pattern-matching” – some imagination and/or initiative is going to required sometimes- but you’d be surprised how many apparently intractable problems can be teased into a form to which standard methods can be applied. Don’t take this advice too far, though. There’s an old saying that goes “To a man who’s only got a hammer, everything looks like a nail”. But the first rule for solving “unseen” problems has to be to check whether you might in fact already have seen them…