Archive for cosmological principle

Gamma-Ray Bursts and the Cosmological Principle

Posted in Astrohype, Bad Statistics, The Universe and Stuff with tags , , , on September 13, 2015 by telescoper

There’s been a reasonable degree of hype surrounding a paper published in Monthly Notices of the Royal Astronomical Society (and available on the arXiv here). The abstract of this paper reads:

According to the cosmological principle (CP), Universal large-scale structure is homogeneous and isotropic. The observable Universe, however, shows complex structures even on very large scales. The recent discoveries of structures significantly exceeding the transition scale of 370 Mpc pose a challenge to the CP. We report here the discovery of the largest regular formation in the observable Universe; a ring with a diameter of 1720 Mpc, displayed by 9 gamma-ray bursts (GRBs), exceeding by a factor of 5 the transition scale to the homogeneous and isotropic distribution. The ring has a major diameter of 43° and a minor diameter of 30° at a distance of 2770 Mpc in the 0.78 < z < 0.86 redshift range, with a probability of 2 × 10−6 of being the result of a random fluctuation in the GRB count rate. Evidence suggests that this feature is the projection of a shell on to the plane of the sky. Voids and string-like formations are common outcomes of large-scale structure. However, these structures have maximum sizes of 150 Mpc, which are an order of magnitude smaller than the observed GRB ring diameter. Evidence in support of the shell interpretation requires that temporal information of the transient GRBs be included in the analysis. This ring-shaped feature is large enough to contradict the CP. The physical mechanism responsible for causing it is unknown.

The so-called “ring” can be seen here:
ring_Australia

In my opinion it’s not a ring at all, but an outline of Australia. What’s the probability of a random distribution of dots looking exactly like that? Is it really evidence for the violation of the Cosmological Principle, or for the existence of the Cosmic Antipodes?

For those of you who don’t get that gag, a cosmic antipode occurs in, e.g., closed Friedmann cosmologies in which the spatial sections take the form of a hypersphere (or 3-sphere). The antipode is the point diametrically opposite the observer on this hypersurface, just as it is for the surface of a 2-sphere such as the Earth. The antipode is only visible if it lies inside the observer’s horizon, a possibility which is ruled out for standard cosmologies by current observations. I’ll get my coat.

Anyway, joking apart, the claims in the abstract of the paper are extremely strong but the statistical arguments supporting them are deeply unconvincing. Indeed, I am quite surprised the paper passed peer review. For a start there’s a basic problem of “a posteriori” reasoning here. We see a group of objects that form a map of Australia ring and then are surprised that such a structure appears so rarely in simulations of our favourite model. But all specific configurations of points are rare in a Poisson point process. We would be surprised to see a group of dots in the shape of a pretzel too, or the face of Jesus, but that doesn’t mean that such an occurrence has any significance. It’s an extraordinarily difficult problem to put a meaningful measure on the space of geometrical configurations, and this paper doesn’t succeed in doing that.

For a further discussion of the tendency that people have to see patterns where none exist, take a look at this old post from which I’ve taken this figure which is generated by drawing points independently and uniformly randomly:

pointaI can see all kinds of shapes in this pattern, but none of them has any significance (other than psychological). In a mathematically well-defined sense there is no structure in this pattern! Add to that difficulty the fact that so few points are involved and I think it becomes very clear that this “structure” doesn’t provide any evidence at all for the violation of the Cosmological Principle. Indeed it seems neither do the authors. The very last paragraph of the paper is as follows:

GRBs are very rare events superimposed on the cosmic
web identified by superclusters. Because of this, the ring is
probably not a real physical structure. Further studies are
needed to reveal whether or not the Ring could have been
produced by a low-frequency spatial harmonic of the large-
scale matter density distribution and/or of universal star
forming activity.

It’s a pity that this note of realism didn’t make it into either the abstract or, more importantly, the accompanying press release. Peer review will never be perfect, but we can do without this sort of hype. Anyway, I confidently predict that a proper refutation will appear shortly….

P.S. For a more technical discussion of the problems of inferring the presence of large structures from sparsely-sampled distributions, see here.

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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:

correlations

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:

homogeneity

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):

chicken

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

chicken

Even better examples are provided by the following snapshot:

IMG-20140626-00352

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:

IMG-20140626-00353

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.

IMG-20140625-00351

The Zel’dovich Universe – Day 2 Summary

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

IMG-20140624-00349

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:

IMG-20140624-00350

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.

The Importance of Being Homogeneous

Posted in The Universe and Stuff with tags , , , , , , , , on August 29, 2012 by telescoper

A recent article in New Scientist reminded me that I never completed the story I started with a couple of earlier posts (here and there), so while I wait for the rain to stop I thought I’d make myself useful by posting something now. It’s all about a paper available on the arXiv by Scrimgeour et al. concerning the transition to homogeneity of galaxy clustering in the WiggleZ galaxy survey, the abstract of which reads:

We have made the largest-volume measurement to date of the transition to large-scale homogeneity in the distribution of galaxies. We use the WiggleZ survey, a spectroscopic survey of over 200,000 blue galaxies in a cosmic volume of ~1 (Gpc/h)^3. A new method of defining the ‘homogeneity scale’ is presented, which is more robust than methods previously used in the literature, and which can be easily compared between different surveys. Due to the large cosmic depth of WiggleZ (up to z=1) we are able to make the first measurement of the transition to homogeneity over a range of cosmic epochs. The mean number of galaxies N(<r) in spheres of comoving radius r is proportional to r^3 within 1%, or equivalently the fractal dimension of the sample is within 1% of D_2=3, at radii larger than 71 \pm 8 Mpc/h at z~0.2, 70 \pm 5 Mpc/h at z~0.4, 81 \pm 5 Mpc/h at z~0.6, and 75 \pm 4 Mpc/h at z~0.8. We demonstrate the robustness of our results against selection function effects, using a LCDM N-body simulation and a suite of inhomogeneous fractal distributions. The results are in excellent agreement with both the LCDM N-body simulation and an analytical LCDM prediction. We can exclude a fractal distribution with fractal dimension below D_2=2.97 on scales from ~80 Mpc/h up to the largest scales probed by our measurement, ~300 Mpc/h, at 99.99% confidence.

To paraphrase, the conclusion of this study is that while galaxies are strongly clustered on small scales – in a complex `cosmic web’ of clumps, knots, sheets and filaments –  on sufficiently large scales, the Universe appears to be smooth. This is much like a bowl of porridge which contains many lumps, but (usually) none as large as the bowl it’s put in.

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?

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 this particularly one does 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.

P.S. And I might add that the importance of the Scrimgeour et al paper to me personally is greatly amplified by the fact that it cites a number of my own articles on this theme!

Cosmic Clumpiness Conundra

Posted in The Universe and Stuff with tags , , , , , , , , , , , , , , on June 22, 2011 by telescoper

Well there’s a coincidence. I was just thinking of doing a post about cosmological homogeneity, spurred on by a discussion at the workshop I attended in Copenhagen a couple of weeks ago, when suddenly I’m presented with a topical hook to hang it on.

New Scientist has just carried a report about a paper by Shaun Thomas and colleagues from University College London the abstract of which reads

We observe a large excess of power in the statistical clustering of luminous red galaxies in the photometric SDSS galaxy sample called MegaZ DR7. This is seen over the lowest multipoles in the angular power spectra Cℓ in four equally spaced redshift bins between 0.4 \leq z \leq 0.65. However, it is most prominent in the highest redshift band at z\sim 4\sigma and it emerges at an effective scale k \sim 0.01 h{\rm Mpc}^{-1}. Given that MegaZ DR7 is the largest cosmic volume galaxy survey to date (3.3({\rm Gpc} h^{-1})^3) this implies an anomaly on the largest physical scales probed by galaxies. Alternatively, this signature could be a consequence of it appearing at the most systematically susceptible redshift. There are several explanations for this excess power that range from systematics to new physics. We test the survey, data, and excess power, as well as possible origins.

To paraphrase, it means that the distribution of galaxies in the survey they study is clumpier than expected on very large scales. In fact the level of fluctuation is about a factor two higher than expected on the basis of the standard cosmological model. This shows that either there’s something wrong with the standard cosmological model or there’s something wrong with the survey. Being a skeptic at heart, I’d bet on the latter if I had to put my money somewhere, because this survey involves photometric determinations of redshifts rather than the more accurate and reliable spectroscopic variety. I won’t be getting too excited about this result unless and until it is confirmed with a full spectroscopic survey. But that’s not to say it isn’t an interesting result.

For one thing it keeps alive a debate about whether, and at what scale, the Universe is homogeneous. The 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?

At our meeting a few weeks ago, Subir Sarkar from Oxford pointed out that the evidence for cosmological homogeneity isn’t as compelling as most people assume. I blogged some time ago about an alternative 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.

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. I will do a “Part 2” to that post before long, but I’m waiting for some results I’ve heard about informally, but which aren’t yet published, before filling in the more recent developments.

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 is not so strong. The point is, however, at what scale would we say that homogeneity is reached. Not when D=3 exactly, because there will always be statistical fluctuations; see below. What scale, then?  Where D=2.9? D=2.99?

What I’m trying to say is that much of the discussion of this issue involves the phrase “scale of homogeneity” when that is a poorly defined concept. There is no such thing as “the scale of homogeneity”, just a whole host of quantities that vary with scale in a way that may or may not approach the value expected in a homogeneous universe.

It’s even more complicated than that, actually. 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 to perturb the background metric significantly. Even the big clumps found by the UCL team only correspond to a small variation in the metric. The issue with these, therefore, is not so much that they threaten the applicability of the Cosmological Principle, but that they seem to suggest structure might have grown in a different way to that usually supposed.

The problem is that we can’t measure the gravitational potential on these scales directly so our tests are indirect. Counting galaxies is relatively crude because we don’t even know how well galaxies trace the underlying mass distribution.

An alternative way of doing this is to use not the positions of galaxies, but their velocities (usually called peculiar motions). These deviations from a pure Hubble flow are caused by lumps of matter pulling on the galaxies; the more lumpy the Universe is, the larger the velocities are and the larger the lumps are the more coherent the flow becomes. On small scales galaxies whizz around at speeds of hundreds of kilometres per second relative to each other, but averaged over larger and larger volumes the bulk flow should get smaller and smaller, eventually coming to zero in a frame in which the Universe is exactly homogeneous and isotropic.

Roughly speaking the bulk flow v should relate to the metric fluctuation as approximately \delta \Phi/c^2 \sim \left(\lambda/ct \right) \left(v/c\right).

It has been claimed that some observations suggest the existence of a dark flow which, if true, would challenge the reliability of the standard cosmological framework, but these results are controversial and are yet to be independently confirmed.

But suppose you could measure the net flow of matter in spheres of increasing size. At what scale would you claim homogeneity is reached? Not when the flow is exactly zero, as there will always be fluctuations, but exactly how small?

The same goes for all the other possible criteria we have for judging cosmological homogeneity. We are free to choose the point where we say the level of inhomogeneity is sufficiently small to be satisfactory.

In fact, the standard cosmology (or at least the simplest version of it) has the peculiar property that it doesn’t ever reach homogeneity anyway! If the spectrum of primordial perturbations is scale-free, as is usually supposed, then the metric fluctuations don’t vary with scale at all. In fact, they’re fixed at a level of \delta \Phi/c^2 \sim 10^{-5}.

The fluctuations are small, so the FLRW metric is pretty accurate, but don’t get smaller with increasing scale, so there is no point when it’s exactly true. So lets have no more of “the scale of homogeneity” as if that were a meaningful phrase. Let’s keep the discussion to the behaviour of suitably defined measurable quantities and how they vary with scale. You know, like real scientists do.