This is called the hyperbolic, or open, Robertson–Walker model. Notice one peculiar property. As the proper radial coordinate χ increases away from the origin, the circumferences of spheres increase as sinh χ . Since sinh χ > χ for all χ > 0, it follows that these circumferences increase more rapidly with proper radius than in flat space. For this reason this hypersurface is not realizable as a three-dimensional hypersurface in a fouror higher-dimensional Euclidean space. That is, there is no picture which we can easily draw such as that for the three-sphere. The space is call ‘open’ because, unlike for k = +1, circumferences of spheres increase monotonically with χ : there is no natural end to the space.

In fact, as we show in Exer. 8, § 12.6, this geometry is the geometry of a hypersurface embedded in Minkowski spacetime. Specifically, it is a hypersurface of events that all have the same timelike interval from the origin. Since this hypersurface has the same interval from the origin in any Lorentz frame (intervals are Lorentz invariant), this hypersurface is indeed homogeneous and isotropic.

Cosmological redshift as a distance measure

When studying small regions of the universe around the Sun, astronomers measure proper distances to stars and other objects and express them in parsecs, as we have seen, or in the multiples kpc and Mpc. But if the object is at a cosmological distance in a universe that is expanding, then what we mean by distance is a little ambiguous, due to the long time it takes light to travel from the object to us. Its separation from our location when it emitted the light that we receive today may have been much less than its separation at present, i.e. on the present hypersurface of constant time. Indeed, the object may not even exist any more: all we know about it is that it existed at the event on our past light-cone when it emitted the light we receive today. But between then and now it might have exploded, collapsed, or otherwise changed dramatically. So the notion of the separation between us and the object now is not as important as it might be for local measurements.

Instead, astronomers commonly use a different measurement of separation: the redshift z of the spectrum of the light emitted by the object, let us say a galaxy. In an expanding universe that follows Hubble’s Law, Eq. (12.1), the further away the galaxy is, the faster it is receding from us, so the redshift is a nice monotonic measure of separation: larger redshifts imply larger distances. Of course, as we noted in the discussion following Eq. (12.1), the galaxy’s redshift contains an element due to its random local velocity; over cosmological distances this is a small uncertainty, but for the nearby parts of the universe astronomers use conventional distance measures, mainly Mpc, instead of redshift.

To compute the redshift in our cosmological models, let us assume that the galaxy has a fixed coordinate position on some hypersurface at the cosmological time t at which it emits the light we eventually receive at time t0. Recall our discussion of conserved quantities in § 7.4: if the metric is independent of a coordinate, then the associated covariant component of momentum is constant along a geodesic. In the cosmological case, the homogeneity of the hypersurfaces ensures that the covariant components of the spatial momentum of the photon emitted by our galaxy are constant along its trajectory. Suppose that we place

ourselves at the origin of the cosmological coordinate system (since the cosmology is homogeneous, we can put the origin anywhere we like), so that light travels along a radial line θ = const., φ = const. to us. In each of the cosmologies the line-element restricted to the trajectory has the form

(To get this for the flat hypersurfaces, simply rename the coordinate r in the first part of Eq. (12.15) to χ .) It follows that the relevant conserved quantity for the photon is pχ . Now, the cosmological time coordinate t is proper time, so the energy as measured by a local observer at rest in the cosmology anywhere along the trajectory is −p0. We argue in Exer. 9, § 12.6, that conservation of pχ implies that p0 is inversely proportional to R(t). It follows that the wavelength as measured locally (in proper distance units) is proportional to R(t), and hence that the redshift z of a photon emitted at time t and observed by us at time t0 is given by

It is important to keep in mind that this is just the cosmological part of any overall redshift: if the source or observer is moving relative to the cosmological rest frame, then there will be a further factor of 1 + zmotion multiplied into the right-hand-side of Eq. (12.24).

We now show that the Hubble parameter H(t) is the instantaneous relative rate of expansion of the universe at time t:

˙

H(t) = R(t) . (12.22)

R(t)

Our galaxy at a fixed coordinate location χ is carried away from us by the cosmological expansion. At the present time t0 its proper distance d from us (in the constant-time hypersurface) is the same for each of the cosmologies when expressed in terms of χ :

It follows by differentiating this that the current rate of change of proper distance between the observer at the origin and the galaxy at fixed χ is

where H0 is the present value of the Hubble parameter. By comparison with Eq. (12.1), we

˙

see that this is just the present value of the Hubble parameter R/R. We show in Exer. 10, § 12.6 that this velocity is just v = z, which is what is required to give the redshift z,

we can substitute in the first term of the previous equation and get an expansion for H as a function of z:

Note that Eq. (12.28) can also be inverted to give the exact and very simple relation

Although cosmology is self-consistent only within a relativistic framework, it is nevertheless useful to ask how the expansion of the universe looks in Newtonian language. We imagine a spherical region uniformly filled with galaxies, starting at some time with radially outward velocities that are proportional to the distance from the center of the sphere. If we are not near the edge – and of course the edge may be much too far away for us to see today – then we can show that the expansion is homogeneous and isotropic about every point. The galaxies just fly away from one another, and the Hubble constant is the scale for the initial velocity: it is the radial velocity per unit distance away from the origin. The problem with this Newtonian model is not that it cannot describe the local state of the universe, it is that, with gravitational forces that propagate instantaneously, the dynamics of any bit of the universe depends on the structure of this cloud of galaxies arbitrarily far away. Only in a relativistic theory of gravity can we make sense of the dynamical evolution of the universe. This is a subject we will study below.

When light is redshifted, it loses energy. Where does this energy go? The fully relativistic answer is that it just goes away: since the metric depends on time, there is no conservation law for energy along a geodesic. Interestingly, in the Newtonian picture of the universe just described, the redshift is just caused by the different velocities of the diverging galaxies relative to one another. As the photon moves outward in the expanding cloud, it finds itself passing galaxies that are moving faster and faster relative to the center. It is not surprising that they measure the energy of the photon to be smaller and smaller as it moves outwards.

Cosmography : measures of distance in the universe

Cosmography refers to the description of the expansion of the universe and its history. In cosmography we do not yet apply the Einstein equations to explain the motion of the universe, instead we simply measure its expansion history. The language of cosmography is the language of distance measures and the evolution of the Hubble parameter.

By analogy with Eq. (12.1), we would like to replace t in Eq. (12.29) with distance. But what measure of distance is suitable over vast cosmological separations? Not coordinate distance, which would be unmeasurable. What about proper distance? The proper distance between the events of emission and reception of the light is zero, since light travels on null lines. The proper distance between the emitting galaxy and us at the present time is

also unmeasurable: in principle, the galaxy may not even exist now, perhaps because of a collision with another galaxy. To get out of this difficulty, let us ask how distance crept into Eq. (12.1) in the first place.

Distances to nearby galaxies are almost always inferred from luminosity measurements. Consider an object whose distance d is known, which is at rest, and which is near enough to us that we can assume that space is Euclidean. Then a measurement of its flux F leads to an inference of its absolute luminosity:

Alternatively, if L is known, then a measurement of F leads to the distance d. The role of d in Eq. (12.1) is, then, as a replacement for the observable (L/F)1/2.

Astronomers have used brightness measurements to build up a carefully calibrated cosmological distance ladder to measure the scale of the universe. For each step on this ladder they identify what is called a standard candle, which is a class of objects whose absolute luminosity L is known (say from a theory of their nature or from reliably calibrated distances to nearby examples of this object). As their ability to see to greater and greater distances has developed, astronomers have found new standard candles that they could calibrate from previous ones but that were bright enough to be seen to greater distances than the previous ones. The distance ladder starts at the nearest stars, the distances to which can be measured by parallax (independently of luminosity), and continues all the way to very distant high-redshift galaxies.

In the spirit of such measurements, cosmologists define the luminosity distance dL to any object, no matter how distant, by inverting Eq. (12.33):

The luminosity distance is often the observable that can be directly measured by astronomers: if the intrinsic luminosity L is known or can be inferred, then a measurement of its brightness F determines the luminosity distance. The luminosity distance is the proper distance the object would have in a Euclidean universe if it were at rest with respect to us, if it had an intrinsic luminosity L, and if we received an energy flux F from it. However, in an expanding cosmology this will not generally be the proper distance to the object.

We shall now find the relation between luminosity distance and the cosmological scales we have just introduced. Consider an object emitting with luminosity L at a time te. What flux do we receive from it at the later time t0? Suppose for simplicity that the object gives off only photons of frequency νe at time te. (This frequency will drop out in the end, so our result will be perfectly general.) In a small interval of time δte the object emits

photons in a spherically symmetric manner. To find the flux we receive, we must calculate the area of the sphere that these photons occupy at the time we observe them.

We place the object at the origin of the coordinate system, and suppose that we sit at coordinate position r in this system, as given in Eq. (12.13). Then when the photons reach our coordinate distance from the emitting object, the proper area of the sphere they occupy is given by integrating over the sphere the solid-angle part of the line-element in Eq. (12.13), which is R20r2d 2. The integration is just on the spherical angles and produces the area:

Now, the photons have been redshifted by the amount (1 + z) = R0/R(te) to frequency ν0:

Moreover, they arrive spread out over a time δt0, which is also stretched by the redshift:

The energy flux at the observation time t0 is thus Nhν0/(Aδt0), from which it follows that

From Eq. (12.34), we then find

To use this, we need to know the comoving source coordinate location r as a function of the redshift z of the photon the source emitted. This comes from solving the equation of motion of the photon. In this case, all we have to do is use Eq. (12.13) with ds2 = 0 (a photon world line) and d 2 = 0 (photon traveling on a radial line from its emitter to the observer at the center of the coordinates). This leads to the differential equation

where the last step follows from differentiating Eq. (12.21). This equation involves the curvature parameter k, but for small r and z the curvature will come into the solution only at second order. If we ignore this at present and work only to first order beyond the Euclidean relations, it is not hard to show that

If we can measure the luminosity distances and redshifts of a number of objects, then

˙

we can in principle measure H0. Measurements of this kind led to the discovery of the accelerating expansion of the universe (below).

Another convenient measure of distance is the angular diameter distance. This is based on another way of measuring distances in a Euclidean space: the angular size θ of an object