## Faster Than The Speed of Light?

Posted in The Universe and Stuff with tags , , , , , on January 5, 2015 by telescoper

Back to the office after starting out early to make the long journey back to Brighton from Cardiff, all of which went smoothly for a change. I’ve managed to clear some of the jobs waiting for me on my return from the Christmas holidays so thought I’d take my lunch break and write a quick blog post. I hasten to add, however, that the title isn’t connected in any way with the speed of this morning’s train, which never at any point threatened causality.

What spurred me on to write this piece was an exchange on Twitter, featuring the inestimable Sean Carroll who delights in getting people to suggest physics for him to explain in fewer than three tweets. It’s a tough job sometimes, but he usually does it brilliantly. Anyway, the third of his tweets about the size of the (observable universe), and my rather pedantic reply to it, both posted on New Year’s Day, were as follows:

I thought I’d take the opportunity to explain in a little bit more detail how and why it can be that the size of the observable universe is significantly larger than what one naively imagine, i.e. (the speed of light) ×(time elapsed since the Big Bang) = ct, for short. I’ve been asked about this before but never really had the time to respond.

Let’s start with some basic cosmological concepts which, though very familar, lead to some quite surprising conclusions.  First of all, consider the Hubble law, which I will write in the form

$v=HR$

It’s not sufficiently widely appreciated that for a suitable definition of the recession velocity $v$ and distance $R$, this expression is exact for any velocity, even one much greater than the speed of light! This doesn’t violate any principle of relativity as long as one is careful with the definition.

Let’s start with time. The assumption of the Cosmological Principle, that the Universe is homogeneous and isotropic on large scales, furnishes a preferred time coordinate, usually called cosmoloogical proper time, or cosmic time, defined in such a way that observers in different locations can set their clocks according to the local density of matter. This allows us to slice the four-dimensional space-time of the Universe into three spatial dimensions of one dimension of time in a particularly elegant way.

The geometry of space-time can now be expressed in terms of the Robertson-Walker metric. To avoid unnecessary complications, and because it seems to be how are Universe is, as far as we can tell, I’ll restrict myself to the case where the spatial sections are flat (ie they have Euclidean geometry). This the metric is:

$ds^{2}=c^{2}dt^{2} - a^{2}(t) \left[ d{r}^2 + r^{2}d\Omega^{2} \right]$

Where $s$ is a four-dimensional interval $t$ is cosmological proper time as defined above, $r$ is a radial coordinate and $\Omega$ defines angular position (the observer is assumed to be at the origin). The function $a(t)$ is called the cosmic scale factor, and it describes the time-evolution of the spatial part of the metric; the coordinate $r$ of an object moving with the cosmic expansion does not change with time, but the proper distance of such an object evolves according to

$R=a(t)r$

The name “proper” here relates to the fact that this definition of distance corresponds to an interval defined instantaneously (ie one with $dt=0$). We can’t actually measure such intervals; the best we can do is measure things using signals of some sort, but the notion is very useful in keeping the equations simple and it is perfectly well-defined as long as you stay aware of what it does and does not mean. The other thing we need to know is that the Big Bang is supposed to have happened at $dt=0$ at which point $a(t)=0$ too.

If we now define the proper velocity of an object comoving with the expansion of the Universe to be

$v=\frac{dR}{dt}=\left(\frac{da}{dt} \right)r = \left(\frac{\dot{a}}{a}\right) R = HR$

This is the form of the Hubble law that applies for any velocity and any distance. That does not mean, however, that one can work out the redshift of a source by plugging this velocity into the usual Doppler formula, for reasons that I hope will become obvious.

The specific case $ds=0$ is what we need here, as that describes the path of a light ray (null geodesic); if we only follow light rays travelling radially towards or away from the origin, the former being of greatest relevance to observational cosmology, then we can set $d\Omega=0$ too and find:

$dr =\frac{cdt}{a(t)}$

Now to the nub of it. How do we define the size of the observable universe? The best way to answer this is in terms of the particle horizon which, in a nutshell, is defined so that a particle on the particle horizon at the present cosmic time is the most distant object that an observer at the origin can ever have received a light signal from in the entire history of the Universe. The horizon in Robertson-Walker geometry will be a sphere, centred on the origin, with some coordinate radius. The radius of this horizon will increase in time, in a manner that can be calculated by integrating the previous expression from $t=0$ to $t=t_0$, the current age of the Universe:

$r_p(t_0)=\int_{0}^{t_0} \frac{cdt}{a(t)}.$

For any old cosmological model this has to be integrated by solving for the denominator as a function of time using the Friedmann equations, usually numerically. However, there is a special case we can do trivially which demonstrates all the salient points. The matter-dominated Einstein- de Sitter model is flat and has the solution

$a(t)\propto t^{2/3}$

so that

$\frac{a(t)}{a(t_0)} = \left(\frac{t}{t_0}\right)^{2/3}$

Plugging this into the integral and using the above definitions we find that in this model the present proper distance of an object on our particle horizon is

$R_p = 3ct_{0}$

By the way, some cosmologists prefer to use a different definition of the horizon, called the Hubble sphere. This is the sphere on which objects are moving away from the observer according to the Hubble law at exactly the velocity of light. For the Einstein-de Sitter cosmology the Hubble parameter is easily found

$H(t)=\frac{2}{3t} \rightarrow R_{c}= \frac{3}{2} ct_{0}.$

Notice that velocities in this model are always decaying, so in it the expansion is not accelerating but decelerating, hence my comment on Twitter above. The apparent paradox therefore has nothing to do with acceleration, although the particle horizon does get a bit bigger in models with, e.g., a cosmological constant in which the expansion accelerates at late times. In the current standard cosmological model the radius of the particle horizon is about 46 billion light years for an age of 13.7 billion years, which is just 10% larger than in the Einstein de Sitter case.

There is no real contradiction with relativity here because the structure of the metric encodes all the requirements of causality. It is true that there are objects moving away from the origin at proper velocities faster than that of light, but we can’t make instantaneous measurements of cosmological distances; what we observe is their redshifted light. In other words we can’t make measurements of intervals with $dt=0$ we have to use light rays, which follow paths with $ds=0$, i.e. we have to make observations down our past light cone. Nevertheless, there are superluminal velocities, in the sense I have defined them above, in standard cosmological models. Indeed, these velocities all diverge at $t =0$. Blame it all on the singularity!

This figure made by Mark Whittle (University of Virginia) shows our past light cone in the present standard cosmological model:

If you were expectin the past light cone to look triangular in cross-section then you’re probably thinking of Minkowski space, or a representation involving coordinates chosen to resemble Minkowski space. Cosmological If you look at the left hand side of the figure, you will find the world lines of particles moving with the cosmic expansion labelled by their present proper distance which is obtained by extrapolating the dotted lines until they intersect a line parallel to the x-axis running through “Here & Now”.  Where we actually see these objects is not at their present proper distance but at the point in space-time where their world line intersects the past light cone.  You will see that an object on the particle horizon intersected our past light cone right at the bottom of the figure.

So why does the light cone look so peculiar? Well, I think the simplest way to explain it is to say that while the spatial sections in this model are flat (Euclidean) the four-dimensional geometry is most definitely curved. You can think of the bending of light rays shown in the figure as a kind of gravitational lensing effect due to all the matter in the Universe. I’d say that the fact that the particle horizon has a radius larger than $ct$ is not because of acceleration but the curvature of space-time, an assertion consistent with the fact that the only familiar world model in which this effect does not occur is the (empty) purely kinemetic Milne cosmology, which is based entirely on special relativity.

## The World as a Beach

Posted in Biographical, The Universe and Stuff with tags , , , , , , on April 10, 2012 by telescoper

Well, as some of you will have noticed, I’ve been offline over the long weekend. There’s no internet connection – not one that I could get to work, anyway – at the residence I’m staying in and I couldn’t be bothered to traipse all the way up the hill to the department in the pouring rain to connect from my office. Hence the first gap in my postings this year. I don’t suppose anyone minds that much. Anyway, here are a few pictures and random thoughts from the weekend.

Here’s a picture of the residence, by the way. It’s called Kopano, although when I previously stayed it was called Driekoppen. The old name was a relic of the days of slavery – three slaves were tortured and executedin public  after rebelling against the terrible conditions they were held in. Their heads were displayed on pikes nearby, hence the name which means “Three Heads”. This was in 1724. I’m not surprised that the end of apartheid brought a change in the name, although keeping it as it was would have served as a reminder of South Africa’s terrible past. One shouldn’t  become obsessed by events that took place such a long time ago, but neither should one forget them.

Good Friday was a very Good Friday indeed, starting with a lovely breakfast and a walk on the beach in Muizenberg. Apparently this is something of a surfer’s paradise but, as I said, I didn’t have an internet connection so couldn’t join in. Also, they have sharks here. I mean big ones. Great White ones, as  a matter of fact. None showed up while I was there, though, and in any case I was only paddling along the shoreline. It may not be obvious from the picture, but it was pretty hot. Almost 30 degrees.

I was watching a chap surfing while we walked along and it reminded me of the post I did a while ago about teaching analogies. Standing on a beach looking out towards the horizon is a bit like doing cosmology. Off in the far distance everything looks smooth; the waves on the surface are much lower in amplitude than the depth of the sea out there, so everything evolves linearly and is quite easy to understand. That’s like looking back in time at the early Universe imprinted on the cosmic microwave background. Nearer to the shore, however, the waves become non-linear because their height is comparable to, or larger than, the depth of the water. These waves evolve in a non-linear way producing, breaking on the beach to produce foam and spray, just as the primordial waves collapse to form galaxies and the foam of large-scale structure when their self-gravity becomes sufficiently strong.

That’s enough of that, I think.

Unfortunately, the weather changed for the worse over the rest of the Easter weekend and torrential rain kept me from doing much on Saturday or Sunday. The finishing section of the  Two Oceans Marathon, which ended on the UCT campus on Saturday, was like a quagmire. As you can see from the picture, I reached the line well in front of the pack. About two days in front, actually. I took this as they were building the stands and hospitality tents a few days before the race.

Anyway, the good side of the bad weather was that I got quite a lot of work done, catching up on things I have let slip for far too long. I also exhausted the reading material I brough with me, so will have to find a good bookshop in the next day or two. Well, that’s about enough for now. I hope to continue regular dispatches from now on until I return to Blighty  next week.

## The Hawking Paradox on BBC iPlayer

Posted in The Universe and Stuff with tags , , , on August 17, 2011 by telescoper

I just heard at lunchtime that a TV programme I was in was recently repeated on BBC4 and is consequently now available on BBC i Player, so I thought I’d advertise it on here.  I didn’t see the broadcast myself, because I scarcely watch TV these days.

The programme was originally made for the BBC TV series Horizon and first broadcast in the UK in 2005. You’ll find yours truly in a couple of places, when I was working at the University of Nottingham and had more hair. In fact I got quite a bit of stick, from some people at a certain University I used to attend, for being insufficiently reverential in my comments about Stephen Hawking but, for what it’s worth, I stand by everything I said. I do admire him enormously as a physicist, but I think his very genuine contributions are sometimes lost in the cult that has developed around him.

Anyway, I thought the programme turned out relatively well but you can watch it yourself by clicking here and form your own opinion!

## Hard Decisions, Easy Targets

Posted in Science Politics, The Universe and Stuff with tags , , , , , , on January 25, 2011 by telescoper

Just back from a day trip to London – at the Institute of Physics to be precise – to wrap up the proceedings of this years protracted STFC Astronomy Grants Panel (AGP) business. The grant letters have already gone out, so no real decisions were made relating to the current round, but we did get the chance to look at a fairly detailed breakdown of the winners and losers. Perhaps more significantly we also discussed issues relating to the implementation of the brand new system which will be in place for 2011/12.

I’m not exactly sure at the moment how much of what we discussed is in the public domain, so I won’t write anything about the meeting here. Tomorrow there is a meeting of the RAS Astronomy Forum at which department representatives will also be briefed about these issues. I will, however, in due course, on as much information as I can through this blog in case there is anyone out there who doesn’t hear it via the Forum.

Not being able to blog about AGP business, I thought I’d comment briefly on a couple of recent things that sprang to mind on the train journey into London. Last night there was a programme in the BBC series Horizon called Science under Attack, presented by Nobel laureate Sir Paul Nurse. I didn’t watch all of it, but I was fortunate (?) enough to catch a segment featuring a chap called James Delingpole, whom I’d never heard of before, but who apparently writes for the Daily Torygraph.

My immediate reaction to his appearance on the small screen was to take an instant dislike to him. This is apparently not an uncommon response, judging by the review of the programme in today’s Guardian. I wouldn’t have bothered blogging about this at all had I wanted to indulge in an ad hominem attack on this person, but he backed up his “unfortunate manner” by saying some amazing things, such as

It’s not my job to sit down and read peer-reviewed papers, because I don’t have the time; I don’t have the expertise

Yet he feels qualified to spout off on the subject nevertheless. The subject, by the way, was climate change. I’m sure not even the most hardened climate skeptic would want Mr Delingpole on their side judging by his performance last night or, apparently, his track-record.

Anyway, this episode reminded me of another egregious example of uninformed drivel that appeared in last week’s Times Higher. This was a piece purporting to be about the limits of mathematical reasoning by another person who is quite new to me, Chris Ormell, who appears to have some academic credentials, if only in the field of philosophy.

I’m still amazed that such a pisspoor article could have made it through the Times Higher’s editorial procedures but more worrying still is the ract that Ormell is himself the editor of a journal, called Prospero, which is “a journal of new thinking of philosophy for education”. The last thing education needs is a journal edited by someone so sloppy that he can’t even be bothered to acquire a basic understanding of his subject matter.

What’s in common between these stories is, however, in my opinion, much more important than the inadequate scientific understanding of the personalities involved. Rubbishing the obviously idiotic, which is quite easy to do, may blind us to the fact that, behind all the errors, however badly expressed it may be, people like this may just have a point. Too often the scientific consensus is portrayed as fact when there are clearly big gaps missing in our understanding. Of course falsehoods should be corrected, but what science really needs to go forward is for bona fide scientists to be prepared to look at the technical arguments openly and responsibly and be candid about the unknowns and uncertainties. Big-name scientists should themselves be questioning the established paradigms and be actively exploring alternative hypotheses. That’s their job. Closing ranks and stamping on outsiders is what makes the public suspicious, not reasoned argument.

In both climatology and cosmology there are consensus views. Based on what knowledge I have, which is less in the former case than in the latter, both these views are reasonable inferences but not absolute truths. In neither case am I a denier, but in both cases I am a skeptic. Call me old-fashioned, but I think that’s what a scientist should be.

Posted in The Universe and Stuff with tags , , , on July 3, 2010 by telescoper

I found this on Youtube. The programme was made for the BBC TV series Horizon and first broadcast in the UK in 2005. You’ll find yours truly in a couple of places, when I was working at the University of Nottingham and had more hair. In fact got a bit of stick, from some people at a certain University I used to attend, for being insufficiently reverential in my comments about Stephen Hawking but, for what it’s worth, I stand by everything I said. I do admire him enormously as a physicist, but I think his very genuine contributions are sometimes lost in the cult that has developed around him.

Anyway, I thought the programme turned out relatively well. Horizon has gone steadily downhill since 2005, obviously because I haven’t been involved…

It’s in 5 parts so if you want to watch all of it, you will need to click through to the next at the end of each segment.

## Dark Horizons

Posted in Cosmic Anomalies, The Universe and Stuff with tags , , , , , , on March 21, 2010 by telescoper

Last Tuesday night I gave a public lecture as part of  Cardiff University’s contribution to National Science and Engineering Week. I had an audience of about a hundred people, although more than half were students from the School of Physics & Astronomy rather than members of the public. I’d had a very full day already by the time it began (at 7pm) and I don’t mind admitting I was pretty exhausted even before I started the talk. I’m offering that as an excuse for struggling to get going, although I think I got better as I got into it. Anyway, I trotted out the usual stuff about the  Cosmic Web and it seemed to go down fairly well, although I don’t know about that because I wasn’t really paying attention.

At the end of the lecture, as usual, there was a bit of time for questions and no shortage of hands went up. One referred to something called Dark Flow which, I’ve just noticed, has actually got its own wikipedia page. It was also the subject of a recent Horizon documentary on BBC called Is Everything we Know about the Universe Wrong? I have to say I thought the programme was truly terrible, but that’s par for the course for Horizon these days I’m afraid. It used to be quite an interesting and informative series, but now it’s full of pointless special effects, portentous and sensationalising narration, and is repetitive to the point of torture. In this case also, it also portrayed a very distorted view of its subject matter.

The Dark Flow is indeed quite interesting, but of all the things that might threaten the foundations of the Big Bang theory this is definitely not it. I certainly have never lost any sleep worrying about it. If it’s real and not just the result of a systematic error in the data – and that’s a very big “if” – then the worst it would do would be to tell us that the Universe was a bit more complicated than our standard model. The same is true of the other cosmic anomalies I discuss from time to time on here.

But we know our standard model leaves many questions unanswered and, as a matter of fact, many questions unasked. The fact that Nature may present us with a few surprises doesn’t mean the whole framework is wrong. It could be wrong, of course. In fact I’d be very surprised if our standard view of cosmology survives the next few decades without major revision. A healthy dose of skepticism is good for cosmology. To some extent, therefore, it’s good to have oddities like the Dark Flow out in the open.

However, that shouldn’t divert our attention from the fact that the Big Bang model isn’t just an arbitrary hypothesis with no justification. It’s the result of almost a century of  vigorous interplay between theory and observation, using an old-fashioned thing called the scientific method. That’s probably too dull for the producers of  Horizon, who would rather portray it as a kind of battle of wills between individuals competing for the title of next Einstein.

Anyway, just to emphasize the fact that I think questioning the Big Bang model is a good thing to do, here is a list of fundamental questions that should trouble modern cosmologists. Most of them are fundamental,  and we do not have answers to them.

Is General Relativity right?

Virtually everything in the standard model depends on the validity of Einstein’s general theory of relativity (or theory of general relativity…). In a sense we already know that the answer to this question is “no”.

At sufficiently high energies (near the Planck scale) we expect classical relativity to be replaced by a quantum theory of gravity. For this reason, a great deal of interest is being directed at cosmological models inspired by superstring theory. These models require the existence of extra dimensions beyond the four we are used to dealing with. This is not in itself a new idea, as it dates back to the work of Kaluza and Klein in the 1920s, but in older versions of the idea the extra dimensions were assumed to be wrapped up so small as to be invisible. In “braneworld models”, the extra dimensions can be large but we are confined to a four-dimensional subset of them (a “brane”). In one version of this idea, dubbed the Ekpyrotic Universe, the origin of our observable universe lies in the collision between two branes in a higher-dimensional “bulk”. Other models are less dramatic, but do result in the modification of the Friedmann equations at early times.

It is not just in the early Universe that departures from general relativity are possible. In fact there are many different alternative theories on the market. Some are based on modifications of Newton’s gravitational mechanics, such as MOND, modifications of Einstein’s theory, such as the Brans-Dicke theory, as well as those theories involving extra dimensions, such as braneworld theory, and so on

There remain very few independent tests of the validity of Einstein’s theory, particularly in the limit of strong gravitational fields. There is very little independent evidence that the curvature of space time on cosmological scales is related to the energy density of matter. The chain of reasoning leading to the cosmic concordance model depends entirely this assumption. Throw it away and we have very little to go on.

What is the Dark Energy?

In the standard cosmology, about 75% of the energy density of the Universe is in a form we do not understand. Because we’re in the dark about it, we call it Dark Energy. The question here is twofold. One part is whether the dark energy is of the form of an evolving scalar field, such as quintessence, or whether it really is constant as in Einstein’s original version. This may be answered by planned observational studies, but both of these are at the mercy of funding decisions. The second part is to whether dark energy can be understood in terms of fundamental theory, i.e. in understanding why “empty space” contains this vacuum energy.  I think it is safe to say we are still very far from knowing how vacuum energy on a cosmological scale arises from fundamental physics. It’s just a free parameter.

What is the Dark Matter?

Around 25% of the mass in the Universe is thought to be in the form of dark matter, but we don’t know what form it takes. We do have some information about this, because the nature of the dark matter determines how it tends to clump together under the action of gravity. Current understanding of how galaxies form, by condensing out of the primordial explosion, suggests the dark matter particles should be relatively massive. This means that they should move relatively slowly and can consequently be described as “cold”. As far as gravity is concerned, one cold particle is much the same as another so there is no prospect for learning about the nature of cold dark matter (CDM) particles through astronomical means unless they decay into radiation or some other identifiable particles. Experimental attempts to detect the dark matter directly are pushing back the limits of technology, but it would have to be a long shot for them to succeed when we have so little idea of what we are looking for.

Did Inflation really happen?

The success of concordance cosmology is largely founded on the appearance of “Doppler peaks” in the fluctuation spectrum of the cosmic microwave background (CMB). These arise from acoustic oscillations in the primordial plasma that have particular statistical properties consistent owing to their origin as quantum fluctuations in the scalar field driving a short-lived period of rapid expansion called inflation. This is strong circumstantial evidence in favour of inflation, but perhaps not strong enough to obtain a conviction. The smoking gun for inflation is probably the existence of a stochastic gravitational wave background. The identification and extraction of this may be possible using future polarisation-sensitive CMB studies even before direct experimental probes of sufficient sensitivity become available. As far as I am concerned, the jury will be out for a considerable time.

Despite these gaps and uncertainties, the ability of the standard framework to account for such a diversity of challenging phenomena provides strong motivation for assigning it a higher probability than its competitors. Part of this  is that no other theory has been developed to the point where we know what predictions it can make. Some of the alternative  ideas  I discussed above are new, and consequently we do not really understand them well enough to know what they say about observable situations. Others have adjustable parameters so one tends to disfavour them on grounds of Ockham’s razor unless and until some observation is made that can’t be explained in the standard framework.

Alternative ideas should be always explored. The business of cosmology, however,  is not only in theory creation but also in theory testing. The great virtue of the standard model is that it allows us to make precise predictions about the behaviour of the Universe and plan observations that can test these predictions. One needs a working hypothesis to target the multi-million-pound investment that is needed to carry out such programmes. By assuming this model we can make rational decisions about how to proceed. Without it we would be wasting taxpayers’ money on futile experiments that have very little chance of improving our understanding. Reasoned belief  in a plausible working hypothesis is essential to the advancement of our knowledge.

Cosmologists may appear a bit crazy (especially when they appear on TV), but there is method in their madness. Sometimes.