Archive for proper distance

Thoughts on Cosmological Distances

Posted in The Universe and Stuff with tags , , , , , on July 18, 2019 by telescoper

At the risk of giving the impression that I’m obsessed with the issue of the Hubble constant, I thought I’d do a quick post about something vaguely related to that which I happened to be thinking about the other night.

It has been remarked that the two allegedly discrepant sets of measures of the cosmological distance scale seen, for example, in the diagram below differ in that the low values are global measures (based on observations at high redshift) while the high values of are local (based on direct determinations using local sources, specifically stars of various types).

The above Figure is taken from the paper I blogged about a few days ago here.

That is basically true. There is, however, another difference in the two types of determination: the high values of the Hubble constant are generally related to interpretations of the measured brightness of observed sources (i.e. they are luminosity distances) while the lower values are generally based on trigonometry (specifically they are angular diameter distances). Observations of the cosmic microwave background temperature pattern, baryon acoustic oscillations in the matter power-spectum, and gravitational lensing studies all involve angular-diameter distances rather than luminosity distances.

Before going on let me point out that the global (cosmological) determinations of the Hubble constant are indirect in that they involve the simultaneous determination of a set of parameters based on a detailed model. The Hubble constant is not one of the basic parameters inferred from cosmological observations, it is derived from the others. One does not therefore derive the global estimates in the same way as the local ones, so I’m simplifying things a lot in the following discussion which I am not therefore claiming to be a resolution of the alleged discrepancy. I’m just thinking out loud, so to speak.

With that caveat in mind, and setting aside the possibility (or indeed probability) of observational systematics in some or all of the measurements, let us suppose that we did find that there was a real discrepancy between distances inferred using angular diameters and distances using luminosities in the framework of the standard cosmological model. What could we infer?

Well, if the Universe is described by a space-time with the Robertson-Walker Metric (which is the case if the Cosmological Principle applies in the framework of General Relativity) then angular diameter distances and luminosity distances differ only by a factor of (1+z)2 where z is the redshift: DL=DA(1+z)2.

I’ve included here some slides from undergraduate course notes to add more detail to this if you’re interested:

The result  DL=DA(1+z)2 is an example of Etherington’s Reciprocity Theorem. If we did find that somehow this theorem were violated, how could we modify our cosmological theory to explain it?

Well, one thing we couldn’t do is change the evolutionary history of the scale factor a(t) within a Friedman model. The redshift just depends on the scale factor when light is emitted and the scale factor when it is received, not how it evolves in between. And because the evolution of the scale factor is determined by the Friedman equation that relates it to the energy contents of the Universe, changing the latter won’t help either no matter how exotic the stuff you introduce (as long as it only interacts with light rays via gravity).

In the light of the caveat I introduced above, I should say that changing the energy contents of the Universe might well shift the allowed parameter region which may reconcile the cosmological determination of the Hubble constant from cosmology with local values. I am just talking about a hypothetical simpler case.

In order to violate the reciprocity theorem one would have to tinker with something else. An obvious possibility is to abandon the Robertson-Walker metric. We know that the Universe is not exactly homogeneous and isotropic, so one could appeal to the gravitational lensing effect of lumpiness as the origin of the discrepancy. This must happen to some extent, but understanding it fully is very hard because we have far from perfect understanding of globally inhomogeneous cosmological models.

Etherington’s theorem requires light rays to be described by null geodesics which would not be the case if photons had mass, so introducing massive photons that’s another way out. It also requires photon numbers to be conserved, so some mysterious way of making photons disappear might do the trick, so adding some exotic field that interacts with light in a peculiar way is another possibility.

Anyway, my main point here is that if one could pin down the Hubble constant tension as a discrepancy between angular-diameter and luminosity based distances then the most obvious place to look for a resolution is in departures of the metric from the Robertson-Walker form.

Addendum: just to clarify one point, the reciprocity theorem applies to any GR-based metric theory, i.e. just about anything without torsion in the metric, so it applies to inhomogeneous cosmologies based on GR too. However, in such theories there is no way of defining a global scale factor a(t) so the reciprocity relation applies only locally, in a different form for each source and observer.

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:

t16_three_distances_4

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.