Cosmology. The Origin and Evolution of Cosmic Structure - Coles P., Lucchin F

50 The Friedmann Models

Problems

1.For a universe with k = 0 and in which (a/a0) = (t/t0)n, where n < 1, show that the coordinate distance of an object seen at redshift z is

For n = 23 deduce that the present proper distance to a quasar at redshift z = 5 is

where H0 is present value of the Hubble constant.

2.Consider a dust model in the limit Ω0 → 0. On the one hand, this is an example of an open Friedmann model which has negatively curved spatial sections. On the other hand, being undecelerated and purely kinematic, it ought to be described by special relativity, which is described by the flat metric of Minkowski space. Can these two views be reconciled?

3.By substituting in (2.4.1), show that the parametric open solution given by (2.4.2) and (2.4.3) does indeed solve the Friedmann equation. Repeat the exercise for the closed solution (2.4.6) and (2.4.7).

4.A closed Friedmann universe contains a single perfect fluid with an equation of state of the form p = wρc2. Transforming variables to conformal time τ using dt = a(t) dτ, show that the variable y = a(1+3w)/2 is described by a simple harmonic equation as a function of τ. Hence argue that all closed Friedmann models with a given equation of state have the same conformal lifetime.

5.Calculate the present proper distance to the event horizon in a de Sitter model described by (1.2.14). What is the radius of the Hubble sphere in this case? Is there a particle horizon in this model?

6.A flat matter-dominated (Einstein–de Sitter) universe is populated with galaxies at various proper distances l from an observer at the origin. The distance of these galaxies increases with cosmological proper time in a manner described by the Hubble law. If the galaxies emit light at various times te, calculate the locus of points in

the l–te plane that lie on the observer’s past light cone (i.e. those points that emit

light at te that can be detected at t = t0 by the observer). Show that the maximum proper distance of a galaxy on this locus is lmax = 49 ct0.

3

Alternative

Cosmologies

Most of this book is devoted to a survey of the standard (Big Bang) cosmology and its consequences for the large-scale structure of the Universe. We nevertheless feel it is important to mention some non-standard cosmologies as illustrations of how di erent world models can behave. Some of these alternative cosmologies have been important in the past, during the development of modern cosmology as an observational science. Others are more recent speculations about how the Big Bang model may be a ected by developments in fundamental physics. Although there are good grounds for believing that the standard cosmology is basically correct, one should never close one’s eyes to the possibility that it may turn out to be wrong and that one of the non-standard alternatives may be a better or more complete description of reality. We have not the space, however, to give a panoramic view of all possible alternative cosmologies so we shall concentrate on a few which are of particular historical or contemporary interest and confine ourselves to brief remarks upon them. Those readers not interested in this material may skip this chapter at a first reading.

Before proceeding, we should remind the reader that the fundamentals of the standard Big Bang model are essentially the theory of general relativity, the expanding Universe and the Cosmological Principle. These basic assumptions allow the flexibility to incorporate the models of Einstein, de Sitter and Lemaître characterised by Λ ≠ 0 in Section 1.11 within this standard framework. These models are of historical interest as well as sharing many of the modern ‘inflationary’ cosmologies constructed using a scalar field whose vacuum energy essentially plays the role of a time-varying cosmological constant. We discuss inflation in more detail in Chapter 7.

52 Alternative Cosmologies

3.1 Anisotropic and Inhomogeneous Cosmologies

The Cosmological Principle plays such an important role in the development of the Friedmann models that it is well worth looking at the consequences of relaxing the assumptions of homogeneity and isotropy. One motivation for this is that the Universe is neither homogeneous nor isotropic. In the standard cosmology, however, variations in density are treated as perturbations of a Friedmann model. This means that structure-formation theory is inherently approximate. It would be nice to be able to solve Einstein’s equations exactly for lumpy models, but this is extremely di cult except in cases of special symmetry. Indeed, only a few exact anisotropic or inhomogeneous cosmological solutions are known. We shall discuss a few examples here, just to give an idea of the di erent behaviour one might expect.

3.1.1 The Bianchi models

The first class of non-standard models we discuss are spatially homogeneous but anisotropic. In the Friedmann models the constant time surfaces upon which the matter density is constant are surfaces of constant cosmological proper time. We can give a more general definition of homogeneity by requiring that all comoving observers see essentially the same version of cosmic history. In mathematical terms this means that there must be some symmetry that relates what the Universe looks like as seen by observer A to what is seen in a coordinate system centred on any other observer B. The possible symmetries can be classified into classes usually called the Bianchi types, although there is one peculiar solution of the Einstein equations, called the Kantowski–Sachs solution, that does not fit into this scheme.

The Bianchi classification is based on the construction of spacelike hypersurfaces upon which it is possible to define at least three independent vector fields, ξα (α and other Greek indices run from 1 to 3), that satisfy the constraint

This is called Killing’s equation and the vectors that satisfy it are called Killing vectors. The commutators of the ξα are defined by

where the Cαβδ are called structure constants. These are antisymmetric, in the sense that,

The components of the metric, gij, describing a Bianchi space are invariant under the isometry generated by infinitesimal translations of the Killing vector fields. In other words, the time-dependence of the metric is the same at all points. The

Table 3.1 The Bianchi types shown in terms of the number of arbitrary constants needed to specify the model on a given constant time surface in vacuum r and with a perfect fluid s.

Einstein equations relate the energy–momentum tensor Tij to the derivatives of gij, so if the metric is invariant under a given set of operations, then so are the physical properties encoded by Tij.

The set of n Killing vectors will have some n-dimensional group structure, say Gn, that depends on the properties of the structure constants and this is used to classify all spatially homogeneous cosmological models. The most useful form of this classification proceeds as follows. On any particular spacelike hypersurface, the Killing vector basis can be chosen so that the structure constants can be decomposed as

where Hαβγ is the total antisymmetric tensor and δβα is the Kronecker delta. The tensor nαβ is diagonal with entries, say, n1, n2, and n3. The vector aα = (a, 0, 0) for some constant a. All the parameters a and nα can be normalised to be ±1 or zero. If an2n3 = 0, then n2 and n3 can be set to ±1 and a is then conventionally taken to be |h|, where h is a parameter used in the classification. The possible combinations of n1 and a then fix the Bianchi types, which can also be described in terms of the number of arbitrary functions needed to specify the solution in vacuum (r) or in the presence of a perfect fluid (s) as shown in Table 3.1. The ‘most general’ anisotropic models are therefore those that have the largest number of free functions, or free parameters on each hypersurface.

The Friedmann models form special cases of the Bianchi types. These have G6 symmetry groups with G3 subgroups. The flat Friedmann model is a special case of either Bianchi I or Bianchi VII0, the open Friedmann model is a special case of types V or VIIh and the closed model belongs to type IX.

General solutions of the Einstein equations are only known for some special cases of the Bianchi types, which demonstrates the di culty of finding meaningful

54 Alternative Cosmologies

exact solutions in situations of restricted symmetry. There is, however, one very- well-known example which is a useful illustration of the sort of behaviour one can obtain. This solution, called the Kasner solution, belongs to Bianchi type I. The metric in this case has a relatively simply form:

Substituting this metric into the Einstein Equations (1.2.20) (with Λ = 0 and a perfect fluid with pressure p and density ρ) yields

in which a3 = X1X2X3. Note that this emerges from the diagonal part of the Einstein equations so the summation convention does not apply in Equation (3.1.6). One also obtains

˙

This is easy to interpret: the spatial sections expand at a rate Xi/Xi in each direction. The mean rate of expansion is just

In the neighbourhood of an observer at the centre of a coordinate system xi, fluid particles will move with some velocity ui. In general,

where ωij is the rate of rotation: in more familiar language, the vorticity vector ωi = Hijkωjk, which is just the curl of ui. The tensor θij can be decomposed into a diagonal part and a trace-free part according to

where σii = 0. In this description θ, σij and ωij, respectively, represent the expansion, shear and rotation of a fluid element.

More complicated Bianchi models have non-zero rotation. We can further rewrite Equation (3.1.6) in the form of evolution equations for

where the Σi are constants such that Σ1 + Σ2 + Σ3 = 0. The Kasner solution itself

is for a vacuum p = ρ = 0, which has a particularly simple behaviour described by Xi = Aitp, where p1 + p2 + p3 = p12 + p22 + p32 = 1. Notice that in general these models possess a shear that decreases with time. They therefore tend to

behave more like a Friedmann model as time goes on. Their behaviour as t → 0 is, however, quite complicated and interesting.

There is one other particularly interesting case to mention before we leave this discussion. The mix-master universe of Misner (1968) we mentioned in Chapter 1 is of Bianchi type IX.

3.1.2 Inhomogeneous models

Before the formulation of general relativity and the discovery of the Hubble expansion, which is describable by the Friedmann models founded on Einstein’s theory, most astronomers imagined the Universe to be infinite, eternal, static and Euclidean. The distribution of matter within the Universe was likewise assumed to be more or less homogeneous and static. It is worth mentioning at this point that the discovery that galaxies were actually external and comparable in size with the Milky Way was made only a few years or so before Hubble’s discovery of the expansion of the Universe.

It is nevertheless noteworthy that, beginning in the last century, there were a number of prominent supporters also of the hierarchical cosmology, according to which the material contents of the Universe are distributed in a hierarchical manner reminiscent of the modern concept of a fractal. In such a model, the mean density of matter on a scale r varies with scale as ρ(r) r−γ, where γ is some constant γ 2. In this way the mean density of the Universe tends to zero on larger and larger scales. On the other hand, the velocity induced by the hierarchical fluctuations varies with scale according to v2(r) = Gρ(r)r2 r2−γ const. The idea of a fractal Universe still has its adherents today, although the evidence we have from the extreme isotropy of the cosmic microwave background suggests that the Universe is homogeneous and isotropic on scales greater than a few hundred Mpc.

3.2 The Steady-State Model

The model of the steady-state universe is now primarily of historical interest. In the past, however, from its original conception by Bondi, Gold and Hoyle in 1948 it was for many years a compelling rival to the Big Bang. Indeed it is ironic that Hoyle, a bitter opponent of the Big Bang, was the man who actually gave that model its name. He meant the term ‘Big Bang’ to be derogatory, but the term stuck.

The theory of the steady-state universe is based on the Perfect Cosmological Principle, according to which the universe must appear identical (at least in some average sense) when viewed from any point, in any direction and at any time. This is clearly a stronger version of the usual Cosmological Principle, which applies to spatial positions only. A particular consequence of this principle is that the Hubble constant really has to be constant in time:

from this relationship one can immediately deduce that the universe is expanding exponentially:

It is worth mentioning one immediate conundrum arising from this requirement. Although, as we have seen, it is di cult to measure the Hubble parameter unambiguously, most observations do seem to suggest a value of H0−1, which is at least within an order of magnitude of the ages of the oldest objects we can see. In a steady-state universe this is a surprise. There is no reason a priori why the age of the matter at a particular spatial location should bear any relation at all to the value of H0−1. The steady-state universe was partly motivated by the fact that, in the 1940s, the ‘best’ observational estimates of the Hubble constant were very large: H0 300 km s−1 Mpc−1. With this value, the ages of the oldest stars are much larger than H0−1, which is a powerful argument against the Big Bang. Modern estimates of H0 are much lower and have blunted most of the force of this argument.

One can demonstrate, starting from the perfect Cosmological Principle, that the curvature parameter K which appears in the Robertson–Walker metric must be zero, and that the spatial sections in this model must therefore be flat. One consequence of Equation (3.2.2) is that, if the Universe is to look the same to all observers at all times, there must be a continuous creation of matter, in such a way that the mean density of particles remains constant. This creation must take place at a rate

It has never been clear exactly how this matter can be created, though it has been suggested that creation events might be responsible for driving active galactic nuclei. Hoyle’s idea was to postulate a modification of the Einstein equations to