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

A general property of flat-universe models is that the expansion parameter a grows indefinitely with time, with constant deceleration parameter q0. The comments we made above about the role of pressure can be illustrated again by the fact that increasing w and, therefore, increasing the pressure causes the deceleration parameter also to increase. Conversely, and paradoxically, a negative value of w indicating behaviour similar to a cosmological constant corresponds to negative pressure (tension) but nevertheless can cause an accelerated expansion (see Section 2.8 later).

Note also the general result that in such models the age of the Universe, t0, is closely related to the present value of the Hubble parameter, H0.

2.3 Curved Models: General Properties

After seeing the solutions corresponding to flat models with Ωw = 1, we now look at some properties of curved models with Ωw ≠ 1. We begin by looking at the behaviour of these models at early times.

In (2.1.12) and (2.1.13) the term (1 − Ω0w) inside the parentheses is negligible with respect to the other term, while

During the interval 0 < a a , Equations (2.1.12) and (2.1.13) become, respectively,

40 The Friedmann Models

t

Figure 2.2 Evolution of the expansion parameter a(t) in an open model (Ω0 < 1), flat or Einstein–de Sitter model (Ω0 = 1) and closed model (Ω0 > 1).

It is interesting to note that, if t0 is taken to coincide with t , equation (2.3.6) implies

the parameter Ω0w(t) passes from a value very close to unity, at t t , to a value of 12 , for t = t , and to a value closer and closer to zero for t t .

2.3.2 Closed models

In models with Ωw > 1 (closed universes) there exists a time tm at which the derivative a˙ is zero. From (2.1.12), one can see that

After the time tm the expansion parameter decreases with a derivative equal in modulus to that holding for 0 a am: the curve of a(t) is therefore symmetrical around am. At tf = 2tm there is another singularity in a symmetrical position with respect to the Big Bang, describing a final collapse or Big Crunch.

In Figure 2.2 we show a graph of the evolution of the expansion parameter a(t) for open, flat and closed models.

2.4 Dust Models

Models with w = 0 have an exact analytic solution, even for the case where Ω ≠ 1 (we gave the solution for Ω = 1 in Section 2.2). In this case, Equation (2.1.12) becomes

2.4.1 Open models

For these models Equation (2.4.1) has a solution in the parametric form:

2.4.2Closed models

For these models Equation (2.4.1) has a parametric solution in the form of a cycloid:

The expansion parameter a(t) grows in time for 0 ϑ ϑm = π. The maximum value of a is

which we have obtained previously in (2.3.12), occurring at a time tm given by

The curve of a(t) is symmetrical around tm, as we have explained before. One can obtain an expression for t0 from Equations (2.4.6) and (2.4.7). The result is

which is a particular case of Equation (2.3.5). We can therefore define a look-back time by

This represents the time elapsed since the emission of a signal which arrives now, at t0, with a redshift z. In other words, the time it has taken light to reach us from a source which we observe now at a redshift z.

2.5 Radiative Models

The models with w = 13 also have simple analytic solutions for Ωr ≠ 1 (the solution for Ωr = 1 was given in Section 2.2). Equation (2.1.12) can be written in the form

2.5.1 Open models

For t tr or, alternatively, (a ar ), where

Equation (2.5.2) shows that the behaviour of a(t) takes the form of an undecelerated expansion

One can also find the present cosmic time by putting a = a0 in this equation:

44 The Friedmann Models

2.5.2 Closed models

In this case Equation (2.5.2) shows that there is a maximum value of a at

The function a(t) is symmetrical around tm. One obtains an expression for t0 by putting a = a0 in (2.5.2); the result is

There obviously also exists another solution of Equation (2.5.2), say t0, obtained by reflecting t0 around tm; at this time a(t˙ 0) < 0.

2.5.3 General properties

The formula analogous to (2.4.16) is, in this case,

For z 1 and Ω0rz 1, Equation (2.5.9) yields

which is, again, a particular case of (2.3.5).

2.6 Evolution of the Density Parameter

In most of the expressions derived so far in this chapter, the quantity that appears is Ωw0 or, in the special case of w = 0, just Ω0. This is simply because we have chosen to parametrise the solutions with the value of Ω at the time t = t0. However, it is very important to bear in mind that Ω is a function of time in all these models. If we instead wish to calculate the density parameter at an arbitrary redshift z, the relevant expression is

which will be useful later on, particularly in Chapter 7. Notice that if Ω0w > 1, then Ωw(z) > 1 for likewise, if Ω0w < 1, then Ωw(z) < 1 for all z; on the other hand, if Ω0w = 1, then Ωw(z) = 1 for all time. The reason for this is clear: the expansion cannot change the sign of the curvature parameter K. also worth noting that, as z tends to infinity, i.e. as we move closer and closer to the Big Bang, Ωw(z) always tends towards unity.

These results have already been obtained in di erent forms in the previous parts of this chapter: one can summarise them by saying that any universe with Ωw ≠ 1 behaves like an Einstein–de Sitter model in the vicinity of the Big Bang. We shall come back to this later when we discuss the flatness problem, in Chapter 7.

2.7 Cosmological Horizons

Consider the question of finding the set of points capable of sending light signals that could have been received by an observer up to some generic time t. For simplicity, place the observer at the origin of our coordinate system O. The set of points in question can be said to have the possibility of being causally connected with the observer at O at time t. It is clear that any light signal received at O by the time t must have been emitted by a source at some time t contained in the interval between t = 0 and t. The set of points that could have communicated with O in this way must be inside a sphere centred upon O with proper radius

In (2.7.1), the generic distance c dt travelled by a light ray between t and t + dt has been multiplied by a factor a(t)/a(t ), in the same way as one obtains the relative proper distance between two points at time t. In (2.7.1), if one takes the lower limit of integration to be zero, there is the possibility that the integral diverges because a(t) also tends to zero for small t. In this case the observer at O can, in principle, have received light signals from the whole Universe. If, on the other hand, the integral converges to a finite value with this limit, then the

46 The Friedmann Models

spherical surface with centre O and radius RH is called the particle horizon at time t of the observer. In this case, the observer cannot possibly have received light signals, at any time in his history, from sources which are situated at proper distances greater than RH(t) from him at time t. The particle horizon thus divides the set of all points into two classes: those which can, in principle, have been observed by O (inside the horizon), and those which cannot (outside the horizon). From (2.1.12) and (2.7.1) we obtain

The integral in (2.7.2) can be divergent because of contributions near to the Big Bang, when a(t) is tending to zero. At such times, the second term in the square brackets is negligible compared with the first, and one has

which is finite and which also vanishes as a(t) tends to zero. It can also be shown that

The solution (2.7.4) is valid exactly in any case if Ωw = 1; interesting special cases are RH(t) = 3ct for the flat dust model and RH(t) = 2ct for a flat radiative model.

For reference, the integral in (2.7.2) can be solved exactly in the case w = 0 and Ω0 ≠ 1. The result is

in the cases Ω0 < 1 and Ω0 > 1, respectively. The previous analysis establishes that there does exist a particle horizon in Friedmann models with equation-of- state parameter 0 w 1. Notice, however, that in a pure de Sitter cosmological model, which expands exponentially and lasts forever, there is no particle horizon because the integral (2.7.1) is not finite. We shall return to the nature of these horizons and some problems connected with them in Chapter 7.

We should point out the distinction between the cosmological particle horizon and the Hubble sphere, or speed-of-light sphere, Rc. The radius of the Hubble sphere, the Hubble radius, is defined to be the distance from O of an object moving with the cosmological expansion at the velocity of light with respect to O. This can be seen very easily to be

by virtue of the Hubble expansion law. One can see that, if p > −13 ρc2, the value of Rc coincides, at least to order of magnitude, with the distance to the particle horizon, RH. For example, if Ωw = 1, we have

One can think of Rc as being the proper distance travelled by light in the characteristic expansion time, or Hubble time, of the universe, τH, where

The Hubble sphere is, however, not the same as the particle horizon. For one thing, it is possible for objects to be outside an observer’s Hubble sphere but inside his particle horizon. It is also the case that, once inside an observer’s horizon, a point stays within the horizon forever. This is not the case for the Hubble sphere: objects can be within the Hubble sphere at one time t, outside it sometime later, and, later still, they may enter the sphere again. The key difference is that the particle horizon at time t takes account of the entire past history of the observer up to the time t, while the Hubble radius is defined instantaneously at t. Nevertheless, in some cosmological applications, the Hubble sphere plays an important role which is similar to that of the horizon, and is therefore often called the e ective cosmological horizon. We shall see the importance of the Hubble sphere when we discuss inflation, and also the physics of the growth of density fluctuations. It also serves as a reminder of the astonishing fact that the Hubble law in the form (1.4.6) is an exact relation no matter how large the distance at which it is applied. Recession velocities greater than the speed of light do occur in these models as when the proper distance is larger than

Rc = c/H0.

There is yet another type of horizon, called the event horizon, which is a most useful concept in the study of black holes but is usually less relevant in cosmology. The event horizon again divides space into two sets of points, but it refers to the future ability of an observer O to communicate. The event horizon thus separates those points which can emit signals that O can, in principle, receive at some time in the future from those that cannot. The mathematical definition is the same as in (2.7.1) but with the limits of the integral changed to run from t to either tmax, which is either tf (the time of the Big Crunch) in a closed model, or t = ∞ in a flat or open model. The radius of the event horizon is given by

The event horizon does not exist in Friedmann models with −13 < w < 1, but does exist in a de Sitter model.