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7.2 Expansion of a homogeneous Universe
Because the cosmic background radiation is highly uniform, we infer that the Universe is isotropic - it is the same in all directions. We believe that on a large scale the cosmos is also homogeneous - it would look much the same if we lived in any other galaxy. Then, it can be shown that the length s of a path linking any two points at time t is given by integrating the expression
where , , are spherical polar coordinates in an expanding curved space. Apart from their small peculiar speeds, galaxies remain at points with fixed values of those coordinates. The coordinate is dimensionless, while the distance between galaxies expands according to the scale length (t).
Because they follow the galaxies as the Universe expands, , , are called comoving coordinates. The origin = 0 looks like a special point, but in fact it is not. Just as at the Earth's poles where lines of longitude converge, the curvature here is the same as everywhere else, and we can equally well take any point to be = 0. The constant k specifies the curvature of space. For k = 1, the Universe is closed, with positive curvature and finite volume, analogous to the surface of a sphere; (t) is the radius of curvature. If k = -1, we have an open Universe, a negatively curved space of infinite volume, while k = 0 describes familiar unbounded flat space. Near the origin, where >> 1, the formula for s is almost the same for all values of k; on a small enough scale, curvature does not matter. If we look at a tiny region, the relationships among angles, lengths, and volumes will be the same as they are in flat space.
Problem 7.7: In ordinary three-dimensional space, using cylindrical polar coordinates we can write the distance between two nearby points (R, , z) and (R + R, + , z + z) as s2 = R2 + R2 2 + z2. The equation R2 + z2 = 2 describes a sphere of radius : show that if our points lie on this sphere, then the distance between them is
where = R / . When k = 1 in Equation 7.5, any surface of constant is the surface of a sphere of radius (t).
Further reading: For further discussion of cosmology in curved spacetime, see Chapters 6 and 7 of M.V. Berry, 1989, Principles of Gravitation and Cosmology, 2nd edition (Institute of Physics Publishing, London).
According to general relativity, Hubble's law is just one symptom of the expansion of curved space; the distance d between galaxies with fixed comoving coordinates , , expands proportionally to (t). Since d s, Equation 7.5 tells us that the two systems are carried away from each other at a speed
here H(t) is the Hubble parameter, which presently has the value H0.
Relativity tells us that the distance between two events happening at different times and in different places depends on the motion of the observer. But all observers will measure the same proper time along a path through space and time connecting the events, given by integrating
Light rays always travel along paths of zero proper time, = 0. If we place ourselves at the origin of coordinates, then the light we receive from a galaxy at comoving distance e has followed the radial path
it covers less comoving distance per unit of time as the scale of the Universe grows. We can integrate this equation for a wavecrest that sets off at time te, arriving at our position at the present time t0:
Suppose that e is the wavelength of the emitted radiation; then the following wavecrest sets off later, by a time te = c e. We receive it with wavelength obs, at time t = c obs after the previous crest. But the galaxy's comoving position e, and the integral on the right of Equation 7.10, have not changed; so the left side also stays constant:
as long as t << (t) / (t). The wavelength grows along with the scale length (t), while the frequency, momentum, and energy of each photon decay proportionally to 1 / (t). The measured redshift z of a distant galaxy tells us how much expansion has taken place since the time te when its light was emitted. This is the cosmological redshift of Equation 1.28:
To describe processes in the expanding Universe, we can use redshift as a substitute for time: z(t) is the redshift of light emitted at time t, reaching us now at time t0. The time corresponding to a given redshift depends on the function (t); once we know this, Equation 7.10 tells us the comoving distance e from which the light would have started.
The rate at which the Universe expands is set by the gravitational pull of matter and energy within it. We first use Newtonian physics to calculate the expansion, and then discuss how general relativity modifies the result. Consider a small sphere of radius r, at a time t when our homogeneous Universe has density (t); we take r >> (t), so that we can neglect the curvature of space. Everything is symmetric about the origin r = 0, so we appeal to Newton's first theorem in Section 3.1: the gravitational force at radius r is determined only by the mass M (< r) within the sphere. If our sphere is large enough that gas pressure forces are much smaller than the pull of gravity (see Section 7.4 below), then Equation 3.20 gives the force on a gas cloud of mass m at that radius:
Our sphere of matter is expanding along with the rest of the Universe, so its radius r(t) (t). The mass m of the cloud cancels out, giving
the higher the density, the more strongly gravity slows the expansion.
Nothing enters or leaves our sphere, so the mass within it does not change: (t) 3 (t) is constant. Multiplying by (t) tells us how the kinetic energy decreases as the sphere expands:
where the time t0 refers to the present day. Integrating, we have
where k is a constant of integration. Although we derived it using Newtonian theory, Equation 7.15 is also correct in general relativity, which tells us that the constant k is the same one as in Equation 7.15. Since the pressure p of a gas contributes to its energy and hence to its gravitational force, general relativity amends Equation 7.13 to read
Equation 7.15 and 7.16 describe the Friedmann models, telling us how the contents of the Universe determine its expansion. For cool matter, where the sound speed cs << c, we have p ~ cs2, and can safely neglect the pressure term. But for radiation, and particles moving almost at the speed of light, pressure is important: p c2 / 3, where is now the energy density divided by c2. For any mixture of matter and radiation, the term + 3p / c2 must be positive, so the expansion always slows down. If the Universe enters a contracting phase, the collapse speeds up as time goes on.
The inflation theory postulates a vacuum energy, contributing density VAC = / 8 G, and negative pressure or tension pVAC = - c2 / 8 G. The vacuum energy is now very small, but there are reasons to believe that very early, at 10-34 s t 10-32 s, VAC might have been much larger than the density of matter or radiation. During this period, (t) inflated, growing exponentially by a factor ~ e100. The almost uniform cosmos that we now observe would have resulted from the expansion of a tiny near-homogeneous region. Because this patch was so small, the curvature of space within it would be negligible; hence devotees of inflation expect our present Universe to be nearly flat, with k = 0.
Problem 7.8: By substituting into Equation 7.16, show that during inflation, expansion proceeds according to (t) exp (t sqrt[/3]).
Since (t) 2 (t) decreases as (t) grows, in a closed Universe with k = 1 the right side of Equation 7.15 becomes negative at large . But 2 cannot be negative, so the distance between galaxies does not grow forever; (t) attains some maximum before shrinking again. In an open Universe with k 0, there is no such limit; expansion continues indefinitely and (t) grows without bound. In the borderline case k = 0, Equation 7.16 requires that the density is equal to the critical value
At the present day, the critical density crit (t0) = 3.3 x 1011 h2 M Mpc-3: see Equation 1.24. We can measure the mass content of the Universe as a fraction of the critical density, defining the density parameter (t) as
and writing 0 for its present-day value. Equation 7.15 then becomes
If the Universe is closed, with k = 1, then (t) > 1 and the density always exceeds the critical value, while if k = -1, we always have (t) < 1. If the density is now equal to the critical value, so that 0 = 1, then (t) = 1 at all times. Most astronomers would agree that 0.05 0 1; our real Universe is unlikely to be much denser than the critical value.
In its early stage, the Universe was radiation dominated. It was extremely hot, and its energy was almost entirely due to radiation and relativistic particles; these are particles moving close to light speed, so that their energy, momentum and pressure are related in the same way as for photons. The energy density r c2 of a gas of photons decreases as -4 (t): the number per unit volume is proportional to 1 / 3 (t), while by Equation 7.11 the energy of each photon falls as 1 / (t). As expansion proceeds, the density m of matter decreases more slowly, since m (t) -3 (t). So at late times, its energy density m (t) c2 exceeds that in radiation. Since the time teq of matter-radiation equality, about a million years after the Big Bang, the Universe has been matter dominated.
Problem 7.9: The cosmic background radiation is now a blackbody of temperature T = 2.73 K: show that its energy density r c2 = 4.2 x 10-13 erg cm-3. From Equation 1.24, the matter density m = 1.9 x 10-29 0 h2 g cm-3. Show that the time teq, when the energy density m c2 was equal to that in radiation, corresponds to redshift zeq 40 000 0 h2. If the neutrinos e, µ, have masses m << kB Teq / c2, where Teq is the temperature at the time teq, then at earlier times they are relativistic. The energy density of `radiation' is increased by a factor of 1.68, and equalization is delayed until zeq 24 000 0 h2.
To measure the expansion of the Universe relative to the present day, we define the dimensionless scale factor a(t) (t) / (t0). Using Equation 7.19 to rewrite (t0) in terms of H0 and 0, Equation 7.15 becomes
Most of the structure of galaxy clusters and voids that we see today developed after the Universe became matter dominated. In this phase, the density falls as a-3, and from Equation 1.28, 1 + z = 1 / a(t); so Equation 7.20 reads
If the density is exactly at the critical value, with 0 = 1 and k = 0, we have a1/2, and a(t) t2/3.
Since the Universe is not completely empty, 0 > 0. Hence the first term on the right of Equation 7.21 will be large when a(t) is small, soon after the Big Bang; we again have a(t) t2/3. Thus as t -> 0 in the early Universe, space is nearly flat and -> 1. If the density is below critical, with (t) < 1, then at late times the second term of Equation 7.21 takes over, giving a(t) t. Expansion then proceeds almost at a constant speed, barely slowed by the gravity of matter. Using the redshift z to specify time indirectly, in a matter-dominated Universe we can write
We believe that 0 0.05; so the Universe expanded with a(t) t2/3 from the time of matter-radiation equality until at least z ~ 18.
Problem 7.10: Use Equation 7.21 to show that if = 1 in a matter-dominated Universe, H(t) = 2 / 3 t, so that the time t0 since the Big Bang is two-thirds of our simple estimate tH = 1 / H0 in Equation 1.22
Problem 7.11: While blackbody radiation and relativistic particles provide most of the energy density, so -4 (t), show that when a(t) is small or k = 0, Equation 7.20 implies 1 / a(t); hence (t) t1/2, and H(t) = 1/2 t.
Problem 7.12: Even if the cosmos has infinite volume, we can observe only a finite portion. From Equation 7.10, light reaching us at = 0 by time t originates within our horizon at comoving radius H, defined by
At early times the Universe was radiation dominated, with (t) t1/2, and we can disregard the curvature k; show that then, (t) H = 2 c t. Explain why only points within this distance can exchange signals or particles before time t. (At the time of matter-radiation equality, a patch of diameter H (teq) would subtend about 3° on the sky. By the present, this region has expanded to ~ 13 (0 h2)-1 Mpc, and contains a mass MH 1015 (0 h2)-2 M in neutrons and protons. Little time elapsed between teq and recombination, when the matter became neutral, and photons of the cosmic background radiation could stream freely toward us. So it is somewhat surprising that the cosmic microwave background has almost the same spectrum across the whole sky. Inflation theory would allow large-scale uniformity to be established before expansion began to proceed according to Equation 7.20.)