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

118 Thermal History of the Hot Big Bang Model

contributions: from the Big Bang until teq, and from teq to t0. Given that zeq 1 and, therefore, that teq t0, the former contribution is negligible compared with the second. It is therefore a good approximation to calculate t0 by putting Ωr = 0 in Equation (5.6.1) and taking the lower limit of integration to be t = 0. One will thus obtain the values derived in Section 2.4 for the age of a matter-dominated universe.

Bibliographic Notes on Chapter 5

The material in this chapter is very well established. The main results are discussed in Peebles (1971, 1993) and Weinberg (1972). A nice review article of retrospective interest is given by Harrison (1973).

Problems

1.The result (5.1.7) is obtained for photons by integrating over the Planck distribution appropriate for bosons. In the case of neutrinos (or other fermions), show that the number-density in thermal equilibrium at a temperature T0ν is

2.The Friedmann Equation (5.6.1) describing the evolution of a Universe containing only non-relativistic matter and photons can be written

Show that for any choice of Ω0 < 1 there is a value of Ω0r that makes the right-hand side a perfect square of a function of a. Obtain an exact solution for a(t) in such a case.

3.Show that, in a flat radiation-dominated Universe, the radiation temperature varies with time t as

T = At−1/2,

and obtain an expression for A in terms of physical quantities. Use your result to estimate the temperature at t = 1 second after the Big Bang.

6

The Very Early

Universe

6.1 The Big Bang Singularity

As we explained in Chapter 2, all homogeneous and isotropic cosmological models containing perfect fluids of equation of state p = wρc2, with 0 w 1, possess a singularity at t = 0 where the density diverges and the proper distance between any two points tends to zero. This singularity is called the Big Bang. Its existence is a direct consequence of four things: (i) the Cosmological Principle; (ii) the Einstein equations in the absence of a cosmological constant; (iii) the expansion of the Universe (in other words, (a/a)˙ 0 = H0 > 0); and (iv) the assumed form of the equation of state.

It it clear that the Big Bang might well just be a consequence of extrapolating deductions based on the theory of general relativity into a situation where this theory is no longer valid. Indeed, Einstein (1950) himself wrote:

The theory is based on a separation of the concepts of the gravitational field and matter. While this may be a valid approximation for weak fields, it may presumably be quite inadequate for very high densities of matter. One may not therefore assume the validity of the equations for very high densities and it is just possible that in a unified theory there would be no such singularity.

We clearly need new laws of physics to describe the behaviour of matter in the vicinity of the Big Bang, when the density and temperature are much higher than can be achieved in laboratory experiments. In particular, any theory of matter under such extreme conditions must take account of quantum e ects on a cosmological scale. The name given to the theory of gravity that replaces general

120 The Very Early Universe

relativity at ultra-high energies by taking these e ects into account is quantum gravity. We are, however, a very long way from being able to construct a satisfactory theory to accomplish this. It seems likely, however, that in a complete theory of quantum gravity, the cosmological singularity would not exist. In other words, the existence of a singularity in cosmological models based on the classical theory of general relativity is probably just due to the incompleteness of the theory. Moreover, there are ways of avoiding the singularity even without appealing to explicitly quantum-gravitational e ects and remaining inside Einstein’s theory of gravity.

Firstly, one could try to avoid the singularity by proposing an equation of state for matter in the very early Universe that is di erent to the usual perfect fluid with p/ρ > −13 . Let us begin by writing down Equation (1.10.3):

Recall that, if we have a perfect fluid satisfying

then the argument we gave in Section 2.1 based on the concavity of a(t) is no longer valid and the singularity can be avoided. Fluids with w < −13 in this way are said to violate the strong energy condition. There are various ways in which this condition might indeed be violated. For example, suppose we describe the contents of the Universe as an imperfect fluid, that is one in which viscosity and thermal conductivity are not negligible. The energy momentum tensor of such a fluid is no longer of the form (1.9.2); it must contain dependences on the coefficient of shear viscosity η, the coe cient of bulk viscosity ζ, and the thermal conductivity χ. The physical significance of the first two of these coe cients can be recognised by looking at the equation of motion (Euler equation) for a nonrelativistic fluid neglecting self-gravity:

One can demonstrate that in a Robertson–Walker metric the terms in η and χ must be zero because of homogeneity and isotropy: there can be no gradients in pressure or temperature. The terms in the bulk viscosity, however, need not be zero: their e ect upon the Friedmann equations is to replace the pressure p by an ‘e ective’ pressure p :

The resulting Equation (6.1.1) does not change in form, but one must replace p by p . Generally speaking, the bulk viscosity is expected to be negligible in non-relativistic fluids as well as ultra-relativistic ones. It need not be small in the intermediate regime, such as one obtains if there is a mixture of relativistic and non-relativistic fluids. With an appropriate expression for ζ (for example ζ = α ρ, with α = const. > 0, or ζ = const. > 0), one can obtain homogeneous and isotropic solutions to the Einstein equations that do not possess a singularity. In general, however, ζ has to be very small but non-zero; it is not trivial to come up with satisfactory models in which bulk viscosity is responsible for the absence of a singularity.

The Big Bang does not exist in many models with a non-zero cosmological constant, Λ > 0. As we shall see, the present value of Λ can be roughly bounded observationally

which is very small. The e ect of such a cosmological constant at very early times would be very small indeed, since its dynamical importance increases with time. A more realistic option is to interpret the cosmological constant as an e ective quantity related to the vacuum energy density of a quantum field; this can be a dynamical quantity and may therefore have been more important in the past than a true cosmological constant. For example, as we shall see in Chapter 7 when we discuss inflation, it is possible that the dynamics of the very early Universe is dominated by a homogeneous and isotropic scalar quantum field whose evolution is governed by the e ective classical Lagrangian

where the first term is ‘kinetic’ and the second is the ‘e ective potential’. To simplify Equation (6.1.7) and the following expressions, we have now adopted units in which c = = 1. The energy–momentum tensor for such a field is

where the ‘energy-density’ ρΦc2 and the ‘pressure’ pΦ are to be interpreted as e ective quantities (the scalar field is not a fluid), and are given by

In particular, if the kinetic term is negligible with respect to the potential term, the e ective equation of state for the field becomes

122 The Very Early Universe

The scalar field can therefore be regarded as behaving like a fluid with an equation- of-state parameter w = −1 (thus violating the strong energy condition) or as an e ective cosmological constant

The density ρΦ is zero or at least negligible today, but could have been the dominant dynamical factor in certain phases of the evolution of the Universe. It may also have been important in driving an epoch of inflation; see Chapter 7.

Whether the singularity is avoidable or not remains an open question, as does the question of what happens to the Universe for t < 0. It is reasonable to call this question the problem of the origin of the Universe: it is one of the big gaps in cosmological knowledge; some comments about the possible physics of the creation of the Universe are discussed in Sections 6.4 and 6.5.

6.2 The Planck Time

We have already mentioned that the theory of general relativity should be modified in situations where the density tends to infinity, in order to take account of quantum e ects on the scale of the cosmological horizon. In fact, Einstein himself believed that his theory was incomplete in this sense and would have to be modified in some way. When do we expect quantum corrections to become significant? Of course, in the absence of a complete theory (or indeed any theory) of quantum gravity, it is impossible to give a precise answer to this question. On the other hand, one can make fairly convincing general arguments that yield estimates of the timescales and energy scales where we expect quantum gravitational e ects to be large and where we should therefore distrust calculations based only upon the classical theory of general relativity. As we shall now explain, the limit of validity of Einstein’s theory in the Friedmann models is fixed by the Planck time which is of the order of 10−43 s after the Big Bang.

The Planck time tP is the time for which quantum fluctuations persist on the scale of the Planck length lP ctP. From these two scales one can construct a Planck mass, mP ρPl3P, where the Planck density ρP is of the order of ρP (GtP2)−1 (from the Friedmann equations). Starting from the Heisenberg uncertainty principle, in the form

124 The Very Early Universe

we gave above. Let us define the Compton time for a body of mass m (or of energy mc2) to be

this quantity represents the time for which it is permissible to violate conservation of energy by an amount ∆E mc2, as deduced from the uncertainty principle. For example one can create a pair of virtual particles of mass m for a time of order tC. Let us also define the Compton radius of a body of mass m to be

Obviously tC and lC both decrease as m increases. These scales are indicative of quantum physics.

On the other hand the Schwarzschild radius of a body of mass m is

this represents, to order of magnitude, the radius which a body of mass m must have so that its rest-mass energy mc2 is equal to its internal gravitational potential energy U Gm2/lS. General relativity leads to the conclusion that any particle (even a photon) cannot escape from a region of radius lS around a body of mass m; in other words, speaking purely in terms of classical mechanics, the escape velocity from a body of mass m and radius lS is equal to the velocity of light: c2/2 = Gm/lS. Notice, however, that in the latter expression we have taken the ‘kinetic energy’ per unit mass of a photon to be c2/2 as if it were a non-relativistic material. It is curious that the correct result emerges with these approximations. One can similarly define a Schwarzschild time to be the quantity

this is simply the time taken by light to travel a proper distance lS. A body of mass m and radius lS has a free-fall collapse time t (Gρ)−1/2, where ρ m/l3S , which is of order tS. Notice that tS and lS both increase, as m increases.

One can easily verify that for a mass equal to the Planck mass, the Compton and Schwarzschild times are equal to each other, and to the Planck time. Likewise, the relevant length scales are all equal. For masses m > mP, that is to say macroscopic bodies, we have tC < tS and lC < lS: quantum corrections are expected to be negligible in the description of the gravitational interactions between di erent parts of the body. Here we can describe the self-gravity of the body using general relativity or even, to a good approximation, Newtonian theory. On the other hand, for bodies with m < mP, i.e. microscopic entities such as elementary particles, we have tC > tS and lC > lS: quantum corrections will be important in a description of their self-gravity. In the latter case, one must use a theory of quantum gravity in place of general relativity or Newtonian gravity.

At the cosmological level, the Planck time represents the moment before which the characteristic timescale of the expansion τH t is such that the cosmological horizon, given roughly by lP, contains only one particle (see above) having lC lS. On the same grounds, as above, we are therefore required to take into account quantum e ects on the scale of the cosmological horizon.

It is also interesting to note the relationship between the Planck quantities given in Section 6.2 to known thermodynamical properties of black holes (Thorne et al. 1986). According to theory, a black hole of mass M, due to quantum e ects, emits radiation like a black body called Hawking radiation. The typical energy of photons emitted by the black hole is of order H kBT, where T is the black-body temperature given by the relation

The time needed for such a black hole to completely evaporate, i.e. to lose all its rest-mass energy Mc2 through such radiation, is of the order of

It is easy to verify that, if one extrapolates these formulae to the Planck mass mP, the result is that H(mP) mPc2 and τ(mP) tP. A black hole of mass mP therefore evaporates in a single Planck time tP by the emission of one quantum particle of energy EP.

These considerations show that quantum-gravitational e ects are expected to be important not only at a cosmological level at the Planck time, but also continuously on a microscopic scale for processes operating over distances of order lP and times of order tP. In particular, the components of a space–time metric gik will su er fluctuations of order |∆gik/gik| lP/l tP/t on a spatial scale l and a temporal scale t. At the Planck time, the fluctuations are of order unity on the spatial scale of the horizon, which is lP, and on the timescale of the expansion, which is tP. One could imagine the Universe at very early times might behave like a collection of black holes of mass mP, continually evaporating and recollapsing in a Planck time. This picture is very di erent from the idealised, perfect-fluid universe described by the Friedmann equations, and it would not be surprising if deductions from these equations, such as the existence of a singularity were found to be invalid in a full quantum description.

Before moving on to quantum gravity itself, let us return for a moment to the comments we made above about the creation of virtual particles. From the quantum point of view, a field must be thought of as a flux of virtual pairs of particles that are continually created and annihilated. As we explained above, the time for which a virtual particle of mass m can exist is of order the Compton time tC, and the distance it moves before being annihilated is therefore the Compton length, lC.

In an electrostatic field the two (virtual) particles, being charged, can be separated by the action of the field because their electrical charges will be opposite. If

126 The Very Early Universe

the separation achieved is of order lC, there is a certain probability that the pair will not annihilate. In a very intense electrical field, one can therefore achieve a net creation of pairs. From an energetic point of view, the rest-mass energy of the pair ∆E 2mc2 will be compensated by a loss of energy of the electric field, which will tend to be dissipated by the creation of particles. Such an e ect has been described theoretically, and can be observed experimentally in the vicinity of highly charged, unstable nuclei.

A similar e ect can occur in an intense, non-uniform gravitational field. One creates a pair of particles (similar to the process by which black holes radiate particles). In this case, separation of the particles does not occur because of opposite charges (the gravitational ‘charge’, which is the mass, is always positive), but because the field is not uniform. One finds that the creation of particles in this way can be very important, for example, if the gravitational field varies strongly in time, as is the case in the early stages of the expansion of the Universe, above all if the expansion is anisotropic. Some have suggested that such particle creation processes might be responsible for the origin of the high entropy of the Universe. The creation of pairs will also tend to isotropise the expansion.

6.4 Quantum Cosmology

We have explained already that there is no satisfactory theory of quantum gravity, and hence no credible formulation of quantum cosmology. The attempt to find such a theory is technically extremely complex and somewhat removed from the main thrust of this book, so here is not the place for a detailed review of the field. What we shall do, however, is to point out aspects of the general formulation of quantum cosmology to give a flavour of this controversial subject, and to give some idea where the di culties lie. The reader is referred to the reference list for more technical details.

The central concept in quantum mechanics is that of the wavefunction. To give the simple example of a single-particle system, one looks at ψ(x, t). Although the interpretation of ψ is by no means simple, it is generally accepted that the square of the modulus of ψ (for ψ will in general be a complex function) determines the probability of finding the particle at position x at time t. One popular formulation of quantum theory involves the concept of a ‘sum over histories’. In this formulation, the probability of the particle ending up at x (at some time t) is given by an integral over all possible paths leading to that space–time location, weighted by a function depending on the action, S(x, t), along the path. Each path, or history, will be a function x(t), so that x specifies the intersection of a given history with a time-like surface labelled by t. In fact, one takes

ψ(x, t) dx dt exp[iS(x , t )], (6.4.1)

where the integration is with respect to an appropriate measure on the space of all possible histories. The upper limit of integration will be the point in space–time

given by (x, t), and the lower limit will depend on the initial state of the system. The action describes the forces to which the particle is subjected.

This ‘sum-over-histories’ formalism is the one which appears the most promising for the study of quantum gravity. Let us illustrate some of the ideas by looking at quantum cosmology. To make any progress here one has to make some simplifying assumptions. First, we assume that the Universe is finite and closed: the relevant integrals appear to be undefined in an open universe. We also have to assume that the spatial topology of the Universe is fixed; recall that the topology is not determined in general relativity. We also assume that the relevant ‘action’ for gravity is the action of general relativity we discussed briefly in Chapters 1 and 3, which we here write as SE.

In fact, as an aside, we should mention that this is one of the big deficiencies in quantum gravity. There is no choice for the action of space–time coupled to matter fields which yields a satisfactory quantum field theory judged by the usual local standards of renormalisability and so on. There is no reason why the Einstein action SE should keep its form as one moves to higher and higher energies. For example, it has been suggested that the Lagrangian for general relativity might pick up terms of higher order in the Ricci scalar R, beyond the familiar L R. Indeed, second-order Lagrangian theories with L = −R/(16πG)+αR2 have proved to be of considerable theoretical interest because they can be shown to be conformally equivalent to general relativity with the addition of a scalar field. Such a theory could well lead to inflation (see Chapter 7 below), but would also violate the conditions necessary for the existence of a singularity. Some alternative cosmological scenarios based on modified gravitational Lagrangians have been discussed in Chapter 3. Since, however, we have no good reason in this context to choose one action above any other, we shall proceed assuming that the classical Einstein action is the appropriate one to take.

To have any hope of formulating cosmology in a quantum manner, we have to first think of the appropriate analogue to a ‘history’. Let us simplify this even further by dealing with an empty universe, i.e. one in which there are no matter or radiation fields. It is perhaps most sensible to think of trying to determine a wavefunction for the configuration of the Universe at a particular time, and in general relativity the configuration of such a Universe will be simply given by the 3-geometry of a space-like hypersurface. Let this geometry be described by a 3-metric hµν (x). In this case, the corresponding quantity to a history x(t) is just a (Lorentzian) 4-geometry, specified by a 4-metric gij, which induces the 3- geometry hµν on its boundary. In general relativity, the action depends explicitly on the 4-metric gij so it is clear that, when we construct an integral by analogy with (6.4.1), the space over which it is taken is some space of allowed 4-geometries. The required wavefunction will then be a function of hµν (x) and will be given by an integral of the form

Ψ[hµν (x)] = dgij exp[iS(gij)]. (6.4.2)

The wavefunction Ψ is therefore defined over the space of all possible 3- geometries consistent with our initial assumptions (i.e. closed and with a fixed