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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

Equation 5   (7.5)

where sigma, phi, theta 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 sigma is dimensionless, while the distance between galaxies expands according to the scale length curlyR(t).

Because they follow the galaxies as the Universe expands, sigma, phi, theta are called comoving coordinates. The origin sigma = 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 sigma = 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; curlyR(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 sigma >> 1, the formula for Delta 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, theta, z) and (R + Delta R, theta + Delta theta , z + Delta z) as Delta s2 = Delta R2 + R2 Delta theta2 + Delta z2. The equation R2 + z2 = curlyR2 describes a sphere of radius curlyR: show that if our points lie on this sphere, then the distance between them is

Equation 6   (7.6)

where sigma = R / curlyR. When k = 1 in Equation 7.5, any surface of constant phi is the surface of a sphere of radius curlyR(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 sigma, theta, phi expands proportionally to curlyR(t). Since d propto s, Equation 7.5 tells us that the two systems are carried away from each other at a speed

Equation 7   (7.7)

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 tau along a path through space and time connecting the events, given by integrating

Equation 8   (7.8)

Light rays always travel along paths of zero proper time, Delta tau = 0. If we place ourselves at the origin of coordinates, then the light we receive from a galaxy at comoving distance sigmae has followed the radial path

Equation 9   (7.9)

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:

Equation 10   (7.10)

Suppose that lambdae is the wavelength of the emitted radiation; then the following wavecrest sets off later, by a time Delta te = c lambdae. We receive it with wavelength lambdaobs, at time Delta t = c lambdaobs after the previous crest. But the galaxy's comoving position sigmae, and the integral on the right of Equation 7.10, have not changed; so the left side also stays constant:

Equation 11   (7.11)

as long as Delta t << curlyR(t) / dotcurlyR (t). The wavelength grows along with the scale length curlyR(t), while the frequency, momentum, and energy of each photon decay proportionally to 1 / curlyR(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 curlyR(t); once we know this, Equation 7.10 tells us the comoving distance sigmae 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 rho(t); we take r >> curlyR(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:

Equation 12   (7.12)

Our sphere of matter is expanding along with the rest of the Universe, so its radius r(t) propto curlyR(t). The mass m of the cloud cancels out, giving

Equation 13   (7.13)

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: rho (t) curlyR3 (t) is constant. Multiplying by dotcurlyR (t) tells us how the kinetic energy decreases as the sphere expands:

Equation 14   (7.14)

where the time t0 refers to the present day. Integrating, we have

Equation 15   (7.15)

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 16   (7.16)

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 ~ rho cs2, and can safely neglect the pressure term. But for radiation, and particles moving almost at the speed of light, pressure is important: p approx rho c2 / 3, where rho is now the energy density divided by c2. For any mixture of matter and radiation, the term rho + 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 rhoVAC = Lambda / 8 pi G, and negative pressure or tension pVAC = - Lambda c2 / 8 pi G. The vacuum energy is now very small, but there are reasons to believe that very early, at 10-34 s ltapprox t ltapprox 10-32 s, rhoVAC might have been much larger than the density of matter or radiation. During this period, curlyR(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 curlyR(t) propto exp (t sqrt[Lambda/3]).

Since rho(t) curlyR2 (t) decreases as curlyR(t) grows, in a closed Universe with k = 1 the right side of Equation 7.15 becomes negative at large curlyR. But dotcurlyR2 cannot be negative, so the distance between galaxies does not grow forever; curlyR(t) attains some maximum before shrinking again. In an open Universe with k leq 0, there is no such limit; expansion continues indefinitely and curlyR(t) grows without bound. In the borderline case k = 0, Equation 7.16 requires that the density rho is equal to the critical value

Equation 17   (7.17)

At the present day, the critical density rhocrit (t0) = 3.3 x 1011 h2 Msun 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 Omega (t) as

Equation 18   (7.18)

and writing Omega0 for its present-day value. Equation 7.15 then becomes

Equation 19   (7.19)

If the Universe is closed, with k = 1, then Omega (t) > 1 and the density always exceeds the critical value, while if k = -1, we always have Omega (t) < 1. If the density is now equal to the critical value, so that Omega0 = 1, then Omega (t) = 1 at all times. Most astronomers would agree that 0.05 ltapprox Omega0 ltapprox 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 rhor c2 of a gas of photons decreases as curlyR-4 (t): the number per unit volume is proportional to 1 / curlyR3 (t), while by Equation 7.11 the energy of each photon falls as 1 / curlyR(t). As expansion proceeds, the density rhom of matter decreases more slowly, since rhom (t) propto curlyR-3 (t). So at late times, its energy density rhom (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 rhor c2 = 4.2 x 10-13 erg cm-3. From Equation 1.24, the matter density rhom = 1.9 x 10-29 Omega0 h2 g cm-3. Show that the time teq, when the energy density rhom c2 was equal to that in radiation, corresponds to redshift zeq approx 40 000 Omega0 h2. If the neutrinos nue, nuµ, nutau have masses mnu << 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 approx 24 000 Omega0 h2.

To measure the expansion of the Universe relative to the present day, we define the dimensionless scale factor a(t) ident curlyR(t) / curlyR (t0). Using Equation 7.19 to rewrite curlyR (t0) in terms of H0 and Omega0, Equation 7.15 becomes

Equation 20   (7.20)

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 rho propto a-3, and from Equation 1.28, 1 + z = 1 / a(t); so Equation 7.20 reads

Equation 21   (7.21)

If the density is exactly at the critical value, with Omega0 = 1 and k = 0, we have adot propto a1/2, and a(t) propto t2/3.

Since the Universe is not completely empty, Omega0 > 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) propto t2/3. Thus as t -> 0 in the early Universe, space is nearly flat and Omega -> 1. If the density is below critical, with Omega (t) < 1, then at late times the second term of Equation 7.21 takes over, giving a(t) propto 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

Equation 22   (7.22)

We believe that Omega0 gtapprox 0.05; so the Universe expanded with a(t) propto 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 Omega = 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 rho propto curlyR-4 (t), show that when a(t) is small or k = 0, Equation 7.20 implies adot propto 1 / a(t); hence curlyR(t) propto 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 sigma = 0 by time t originates within our horizon at comoving radius sigmaH, defined by

Equation 23   (7.23)

At early times the Universe was radiation dominated, with curlyR(t) propto t1/2, and we can disregard the curvature k; show that then, curlyR(t) sigmaH = 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 curlyR sigmaH (teq) would subtend about 3° on the sky. By the present, this region has expanded to ~ 13 (Omega0 h2)-1 Mpc, and contains a mass MH approx 1015 (Omega0 h2)-2 Msun 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.)

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