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

58 Alternative Cosmologies

take account of the non-conservation of the energy–momentum tensor through the famous ‘C-field’, via a term Cij

substituting the Robertson–Walker metric appropriate for a steady-state model in Equations (3.2.4) one obtains

Hoyle suggested that Cij should be given by

(as usual, the symbol ‘;’ stands for the covariant derivative), and the scalar field C is given by

The popularity of the steady-state universe took a nosedive with the discovery of the 3 K cosmic background radiation by Penzias and Wilson (1965), which has a natural explanation only within the framework of the hot Big Bang model. To reconcile the presence of the microwave background radiation with the steadystate theory it would be necessary to postulate the continuous creation not just of matter, but also of photons. Such a hypothesis appears even more unnatural than the creation of matter. An important development was also Sandage’s revision of the cosmological distance scale, which brought the ages of astronomical objects into rough agreement with the Hubble timescale, H0−1. Until recently, the last significant works in defence of the steady-state model were made by Hoyle and Narlikar in the late 1960s. More recently, however, a variant of this model called the ‘quasi-steady-state’ universe has been proposed. In this scenario, matter is created in chunks of cosmological scale, rather than individually in nucleons. These elaborations remind one of the epicycles used in an attempt to rescue the Earth-centred Solar System model; the steady-state model being advanced nowadays certainly shares none of the compelling simplicity of its predecessor.

Nevertheless, some ideas from the steady-state universe do live on in modern cosmology. In particular, many aspects of the inflationary universe scenario, such as the exponential expansion, are exactly the same as in the steady-state model. However, in the former case, the driving force is not particle creation but rather the vacuum energy of a scalar quantum field with e ective potential V(Φ) const.

3.3 The Dirac Theory

Dirac (1937, 1974) originated a novel approach to cosmology based on the consideration of dimensionless numbers constructed from fundamental physical quantities. For example, the dimensionless number

represents the ratio of the Coulomb force and the gravitational force between an electron and a proton;

is the ratio between the Compton wavelength and the Schwarzschild radius of a proton;

is roughly the ratio between the cosmological horizon distance (sometimes somewhat inaccurately called the ‘radius of the Universe’) and the classical electron radius. One must make a distinction between relations of the type (3.3.3) and similar expressions, such as

(mπ is the pion mass), which are between cosmological and microphysical quantities, and other relations which exist between either two cosmological or two microphysical quantities. For example,

ρ0m(cH−1)3

mp

0 1080 = (1040)2 (3.3.5)

represents the number of baryons within the cosmological horizon;

expresses the near-flatness of the Universe; and

represents the ratio between the number densities of photons and baryons. Relations like (3.3.5)–(3.3.7) can be explained within the framework of an adequate cosmological model such as the inflationary universe. The relations (3.3.1)–(3.3.4) cannot be explained in this manner, and must be thought about in some other way.

60 Alternative Cosmologies

There seem to be two possibilities: either they are essentially numerical coincidences, which occur because of some special property of the present epoch when we happen to be observing the Universe; or they have some deep physical significance which is yet to be elucidated. Arguments of the first type were advanced by Dicke in the 1960s, who explained that the present value of H0−1 in the Big Bang model must be constrained by the requirement that life must have had time to evolve. This requires at least a main sequence stellar lifetime to have passed. The horizon must therefore be large simply in order for us to have evolved, and the number of baryons it contains must also be large. In the second type of argument a deeper explanation, based on fundamental physics, must be sought of the relations such as Equations (3.3.5) to (3.3.7).

This second approach was adopted by Dirac in numerous writings between 1934 and 1974. His basic assumption was that the large dimensionless numbers that keep appearing in relations between microphysical and cosmological scales are connected by a simple relation in which the only dimensionless coe cients that appear are of order unity. For example, let the first terms in Equations (3.3.1) and (3.3.3) be R1 and R2, respectively, so that

If Equation (3.3.8) is valid at any cosmological epoch, given that H0 varies, then at least one of the relevant physical ‘constants’ – e, G, me, mp, c – must be time dependent. Dirac proposed two alternatives: either the charge of the electron or the constant of gravitation are variable. For simplicity, let us look at the second of these possibilities. From Equation (3.3.8) we obtain

G0 is the present value of the ‘constant’ of universal gravitation and t0 is the age of the Universe. We find that

too small compared with the nuclear timescale for stellar evolution which does not depend upon the assumption that G varies with time.

This result is bad news for the Dirac hypothesis. Nevertheless, Dirac’s idea has inspired many attempts to construct theories of gravitation with a variable G. The most complete and interesting example is the scalar–tensor theory of Brans and Dicke (1961), which we describe in the next section. It is noteworthy, however, that the large-number coincidences which were the inspiration for Dirac’s theory either became of secondary importance or were completely neglected in these alternatives. Nowadays it is generally accepted that the correct interpretation of the large-number coincidences is that due to Dicke, and that they are essentially consequences of the Weak Anthropic Principle which we shall discuss later, near the end of Chapter 7.

3.4 Brans–Dicke Theory

The Einstein equations of general relativity can be obtained by applying a variational principle to a Lagrangian of the form

where R is the scalar curvature and L is the Lagrangian action corresponding to the matter. In the Brans–Dicke theory, the appropriate gravitational Lagrangian is instead assumed to be

where ϕ is a scalar field and ω is a dimensionless coupling constant. Comparing Equation (3.4.2) with (3.4.1) shows that the inverse of the field ϕ plays the role of the gravitational constant G. From (3.4.2) we can derive the relation

where Tij is the energy–momentum tensor appropriate for L and, in the place of the Einstein equations, we get

62 Alternative Cosmologies

which, after introducing the Robertson–Walker metric to get the cosmological equations, give the following:

One can also show that, in the framework of a Newtonian approximation, the ‘constant’ in Newton’s law of gravitation is

The cosmological models which solve Equations (3.4.5)–(3.4.8) depend on the four quantities a0, a˙0, ϕ0 and ρ0, and the two parameters K (which takes the values 1, 0 or −1) and ω > 0. Recall that the Friedmann models depend only on three initial values and only one parameter K. The set of cosmological solutions to the Brans– Dicke theory therefore forms a family of solutions which is much larger than that of the Friedmann models. We shall not describe these solutions in any detail, though it is perhaps worth mentioning that the homogeneous and isotropic Brans– Dicke solutions also possess a singularity in the past. Just to give one example, however, consider the flat Universe (K = 0). The present matter density is given by

3H2 (4 + 3ω)(4 + 2ω)

ρ0m = 0 + 2 , (3.4.10) 8πG 6(1 ω)

Equations (3.4.10)–(3.4.12) all become identical to the Einstein–de Sitter case for

ω → ∞.

The mysterious relations (3.3.1)–(3.3.7) do not find an explanation in the framework of this theory, which was not formulated with that intention. The situation with respect to the observational implications of this theory is very complicated, given the large set of allowed models. Cosmological considerations (such as the age of the Universe, nucleosynthesis, etc.) do not place strong constraints on the

Brans–Dicke theory. The most important tests of the validity of this theory are those that involve the time-variation of G. There are various relevant observations: the orbital behaviour of Mercury and Venus; historical data about lunar eclipses; properties of fossils; stellar evolution (particularly the Sun); deflection of light by celestial bodies; the perihelion advance of Mercury. These observations together do not rule out the Brans–Dicke theory, but a rough limit on the parameter ω is obtained: ω > 500.

In recent years, interest in the Brans–Dicke theory as an alternative to general relativity has greatly diminished, but there has been a great deal of recent work on the behaviour of certain types of inflationary model which involve a scalar field with essentially the same properties as the Brans–Dicke field ϕ; these are usually called extended inflation models.

3.5 Variable Constants

One of the consequences of Brans–Dicke theory is that the Newtonian gravitational constant changes with time. In recent years this general framework has given rise to suggestions that other fundamental physical quantities may also not be constant. For example, the fine-structure constant α, given in SI units as

may change with time. The presence of e in this expression indicates that the parameter α measures the strength of the electromagnetic interaction. To have this strength change on a cosmological timescale we therefore need to introduce into the Lagrangian a term involving the electromagnetic field. In general the electromagnetic field is described by a tensor of the form

where Aµ is the usual vector potential that appears in Maxwell’s equations. The appropriate Lagrangian for electromagnetism can be seen to be

One way of building a model in which the coupling to electromagnetism changes is then to use a Lagrangian containing an extra term that couples some scalar field ψ to this in much the same way that the Brans–Dicke theory (3.4.2) couples a scalar field to the metric in order to change the strength of gravity. A possibility is to add a term like Lem exp(−2ψ). In this case the Einstein equations become

leading to changes in the cosmological equations and possible observational consequences in absorption line systems (e.g. Sandvik et al. 2002).

64 Alternative Cosmologies

However, interpreting this change as a change of α alone is not the only possibility. It is possible to use this general idea also to motivate models in which the speed of light c is also variable. The connection between variable c theories and variable α theories lies in (3.5.1). For example, given a variable α theory it is always possible to redefine units so that c and are constant and e varies. It is possible therefore to interpret the model described above as a variable c cosmology in which ψ is just some function of c or vice versa. Somewhat surprisingly, it is possible to make such a theory both covariant and Lorentz invariant (Mo at 1993; Magueijo 2000).

3.6 Hoyle–Narlikar (Conformal) Gravity

Another theory of gravitation that has given rise to interesting cosmological models was proposed by Hoyle and Narlikar in 1964; we shall hereafter call this the HN theory. The important di erence between HN theory and both general relativity and the Brans–Dicke theory mentioned above is that the latter are field theories, while the former is based on the idea of direct interparticle action. Mach’s Principle suggests the existence of action-at-a-distance by the following argument. The mass of an object mi according to Mach’s Principle is not entirely an intrinsic property of the object, but is due to the background provided by all the other objects in the Universe. Building on some ideas of Dirac at representing electromagnetism in a similar way and exploiting the notion of conformal invariance, Hoyle and Narlikar produced a theory of gravitation which, when expressed in the language of field theory, is identical to general relativity.

So what has been gained in this exercise? It seems that this theory provides no new predictions. In fact there are a number of subtle and interesting ways in which this theory di ers from general relativity. First, while the Einstein equations have valid solutions for an empty Universe, the HN equations in this case yield an indeterminate solution for the metric gik. This makes sense in light of Mach’s Principle: without a set of background masses against which to measure motion, the concept of a trajectory is meaningless. Second, the sign of the gravitational constant G is only fixed in general relativity by comparing its weak-field limit with Newtonian gravity. There is no a priori reason intrinsic to general relativity why G could not be negative. In HN theory, G is always positive. Likewise, there is no space for the cosmological constant Λ in the field equations of HN theory. Finally, we mention that in the HN cosmological solutions, redshift arises from the variation of particle masses with time.

The HN theory is an interesting physically motivated alternative to Einstein’s general relativity. While we assume throughout most of this book that GR is the correct law of gravity on cosmological scales, we still feel it is important to stress that there have been no compelling strong-field tests of Einstein’s theory. Alternatives like the HN theory have an important role to play in reminding us how di erent cosmology could be if Einstein’s theory turned out to be wrong!

Bibliographic Notes on Chapter 3

A wide-ranging review of alternatives to the Big Bang cosmology may be found in Ellis (1987). In the early 1990s there was an interesting sequence of review articles in Nature for and against the standard cosmology: see Arp et al. (1990) for the discontents and the riposte by Peebles et al. (1991). A nice review of anisotropic and inhomogeneous cosmologies is given by MacCallum (1993).

Problems

1.Prove that the largest possible group for a spatially homogeneous model is six dimensional.

2.What is special about h = −19 in the Bianchi classification?

3.Investigate the possible behaviour of the singularity as t → 0 in the Kasner solution.

4.Integrate Equation (3.1.8) to identify the third undetermined function in the Tolman–Bondi model and discuss its physical interpretation.

5.Identify the coordinate transformation that turns (3.1.16) into the Minkowski metric.

6.Is there an Olbers Paradox in the steady-state model?

4

Observational

Properties of

the Universe

4.1 Introduction

Our approach to cosmology so far has been almost entirely theoretical, apart from reference to the observational motivation for the Cosmological Principle which was essential in constructing the Friedmann models. We should now fill in some details on what is known about the bulk properties of our Universe, and how one makes measurements in cosmology. Before doing so, however, we take this opportunity to remind the reader of some simple background material from observational astronomy.

4.1.1Units

The standard unit of distance in astronomy is the parsec, which is defined as the distance at which the deflection of an object’s angular position on the sky in the course of the Earth’s orbital motion is one second of arc. (Note that, during half an orbit, the angular change is two arcseconds.) Alternatively and equivalently, one parsec is the distance of an object at which the semi-major axis of the Earth’s orbit around the Sun subtends an angle of one arcsecond at the object. It turns out that