but gravitation can act in the extra dimensions too. This would lead to a modification of the inverse-square-law of Newtonian gravity on short distances, on scales comparable to a relevant length-scale in the surrounding space. All we can say is that this scale must be smaller than about a millimeter, from experiments on the inverse square law. But there are many decades between a millimeter and the Planck scale, and new physics might be waiting to be discovered anywhere in between (Maartens 2004).

The new physics could take many different forms. Some kind of collision with another brane might have triggered the Big Bang. A nearby extra brane might have a parallel world of stars and galaxies, interacting with us only through gravity: shadow matter. There might be extra amounts of gravitational radiation, due either to shadow matter or to unusual brane-related initial conditions at the Big Bang.

Although these ideas sound like science fiction, they are firmly grounded in model theories, which are deliberate over-simplifications of the full equations of string theory, and which involve deliberate choices of the values of certain constants in order to get these strange effects. They should be treated as neither predictions nor idle speculation, but rather as harbingers of the kind of revolutionary physics that a full quantum theory of gravity might bring us. Experimental hints, from high-precision physics, or observational results, perhaps from gravitational waves or from cosmology, might at any time provide key clues that could point the way to the right theory.

12.5 F u r t h e r re a d i n g

The literature on cosmology is vast. In the body of the chapter I have given the principal references to original results, so I list here some recommended books on the subject.

Standard cosmology is treated in great detail in Weinberg (1972). Cosmological models in general relativity become somewhat more complex when the assumption of isotropy is dropped, but they retain the same overall features: the Big Bang, open vs. closed. See Ryan and Shepley (1975). A well-balanced introduction to cosmology is Heidmann (1980).

Early discussions on physical cosmology that remain classics include Peebles (1980), Liang and Sachs (1980), and Balian et al. (1980). More modern is Liddle (2003).

An important current research area is into inhomogeneous cosmologies. See MacCallum (1979). Another subject closely allied to theoretical cosmology is singularity theory: Geroch and Horowitz (1979), Tipler et al. (1980). See also the stimulating article by Penrose (1979) on time asymmetry in cosmology.

For greater depth on physical cosmology, see the excellent text by Mukhanov (2005). For a different point of view on ‘why’ the universe has the properties it does, see the book by Barrow and Tipler (1986) on the anthropic principle.

For popular-level cosmology articles written by research scientists, see the Einstein Online website: http://www.einstein-online.info/en/.

10Show from Eq. (12.24) that the relationship between velocity and cosmological redshift for a nearby object (small light-travel-time to us) is z = v, as we would expect for an

object with a recessional velocity v.

11(a) Prove Eq. (12.29) and deduce Eq. (12.30) from it.

(b)Fill in the indicated steps leading to Eq. (12.31).

12Derive Eq. (12.42) from Eq. (12.31). Derive Eq. (12.44) from Eq. (12.40).

13Astronomers usually do not speak in terms of intrinsic luminosity and flux. Rather, they use absolute and apparent magnitude. The (bolometric) apparent magnitude of a star is defined by its flux F relative to a standard flux Fs:

where Fs = 3 × 10−8 J m−2 s−1 is roughly the flux of visible light at Earth from the brightest stars in the night sky. The absolute magnitude is defined as the apparent magnitude the object would have at a distance of 10 pc:

Using Eq. (12.42), with Eq. (12.27), rewrite Eq. (12.34) in astronomer’s language as:

Astronomers call this the redshift-magnitude relation.

14(a) For the Robertson–Walker metric Eq. (12.13), compute all the Christoffel symbolsμαβ . In particular show that the nonvanishing ones are:

(b)Using these Christoffel symbols, show that the time-component of the divergence of the stress-energy tensor of the cosmological fluid is

˙

T0α ;α = ρ˙ + 3(ρ + p) R . (12.72)

R

(c) By multiplying this equation by R3, derive Eq. (12.46).

15Show from Eq. (12.49) that if the radiation has a black-body spectrum of temperature T, then T is inversely proportional to R.

16Use the Christoffel symbols computed in Exer. 14 above to derive Eq. (12.50).

¨

17 Use Eq. (12.46) and the time-derivative of Eq. (12.54) to derive Eq. (12.55) for R. Make sure you use the fact that p = −ρ .

18 In this chapter we saw that the negative pressure (tension) of the cosmological constant is responsible for accelerating the universe. But is this a contradiction to ordinary physics? Does a tension pull inward, not push outward? Resolve this apparent contradiction by showing that the net pressure force on any local part of the universe is zero. Refer to the discussion at the end of § 4.6.

19Assuming the universe to be matter-dominated and to have zero cosmological constant,

show that at times early enough for one to be able to neglect k in Eq. (12.54), the scale factor evolves with time as R(t) t2/3.

20Assume that the universe is matter dominated and find the value of ρ that permits the universe to be static.

(a)Because the universe is matter-dominated at the present time, we can take ρm(t) = ρ0[R0/R(t)]3 where the subscript ‘0’ refers to the static solution we are looking for. Differentiate the ‘energy’ equation Eq. (12.54) with respect to time to find the

dynamical equation governing a matter-dominated universe:

Set this to zero to find the solution

ρ = 12 ρ0.

For Einstein’s static solution, the cosmological constant energy density has to be half of the matter energy density.

(b)Put our expression for ρm into the right-hand-side of Eq. (12.54) to get an energylike expression which has a derivative that has to vanish for a static solution. Verify that the above condition on ρ does indeed make the first derivative vanish.

(c)Compute the second derivative of the right-hand-side of Eq. (12.54) with respect to R and show that, at the static solution, it is positive. This means that the ’potential’ is a minimum and Einstein’s static solution is stable.

21 Explore the possible futures and histories of an expanding cosmology with negative cosmological constant. You may wish to do this graphically, by drawing figures analogous to Fig. 11.1. See also Fig. 12.4.

22(Parts of this exercise are suitable only for students who can program a computer.) Construct a more realistic equation of state for the universe as follows.

(a)Assume that, today, the matter density is ρm = m × 10−27 kg m−3 (where m is of order 1) and that the cosmic radiation has black-body temperature 2.7 K. Find the

ratio ε = ρr/ρm, where ρr is the energy density of the radiation. Find the number of photons per baryon, εmpc2/kT.

(b)Find the general form of the energy-conservation equation, T0μ,μ = 0, in terms of ε(t) and m(t).

(c)Numerically integrate this equation and Eq. (12.54) for = 0 back in time from

exchange of energy between matter and radiation. Do the integration for m = 0.3, 1.0, and 3.0. Stop the integration when the radiation temperature reaches Ei/26.7 k, where Ei is the ionization energy of hydrogen (13.6 eV). This is roughly the temperature at which there are enough photons to ionize all the hydrogen: there is roughly a fraction 2 × 10−9 photons above energy Ei when kT = Ei/26.7, and this is roughly the fraction needed to give one such photon per H atom. For each m, what is the value of R(t)/R0 at that time, where R0 is the present scale factor? Explain this result. What is the value of t at this epoch?

k = +1 universe?

25Estimate the times earlier than which our uncertainty about the laws of physics prevents us drawing firm conclusions about cosmology as follows.

(a)Deduce that, in the radiation-dominated early universe, where the curvature term depending on the curvature constant k (0, 1, −1) is negligible, the temperature T

behaves as (from now on, k is Boltzmann’s constant)

T= βt−1/2, β = (45 3/32π 3)1/4k−1.

(b)Assuming that our knowledge of particle physics is uncertain for kT > 103 GeV, find the earliest time t at which we can have confidence in the physics.

(c)Quantum gravity is probably important when a photon has enough energy kT to form a black hole within one wavelength (λ = h/kT). Show that this gives kT h1/2. This is the Planck temperature. At what time t is this an important worry?