Appendix

A Lorentz Vectors and Tensors

Consider a coordinate system ;;;1', fL = 0,1,2,3 and another coordinate system x(x). The differential dxIL is given by the chain rule

(A.l)

Any object that transforms like dxIL is called a "contravariant 4-vector", i.e.

VIL = axIL vex axe> '

(A.2)

where V is the vector in the x system and V is the vector in the x system. The best examples of contravariant vectors are the 4-velocity, v lL = dxIL /dT, and the 4-momentum, plL = rnv lL . These two are obviously contravariant 4-vectors because dxIL is a contravariant 4-vector and dT is an invariant. (It is defined by the clock reading in the clock's rest-frame.)

A second kind of vector arises naturally when one considers the 4-gradient of a scalar (Lorentz invariant) function ¢. By the chain rule again, this object transforms as

fJ¢ (A.3)

axIL axIL axC> .

Any object that transforms like a 4-gradient is called a "covariant 4-vector", l.e.

(A.4)

Tensors are just generalizations of vectors. For example, a contravariant rank-2 tensor, TILV, transforms as

TILv = axIL axv Te>r] axe> ax(3 ,

while a covariant rank-2 tensor, TILv , transforms as

axC> a;;/3 _ TILv = -a -a Ue>(3. XIL XV

(A.5)

(A.6)

276 Appendix

The energy-momentum tensor of a system of particles (4.30) is an example of a contravariant tensor. On the other hand, the energy-momentum tensor of a scalar field is more naturally written as a covariant tensor (4.70). Since we will want to put the two in the same Friedmann equation, we will need a way to change covariant tensors into contravariant tensors. This will be done soon.

In this book we are mostly concerned with Lorentz transformations and we will now concentrate on this simple class of transformations. As shown in Exercise 3.1, the transformations coefficients oi;o. lox!' of Lorentz boosts and rotations must be constants satisfying

ox" ox).. 7],,).. oi;o. oi;(3 = 7]0.(3 • (A. 7)

Comparison with (A.6) shows that 7]!'v is an invariant rank-2 covariant tensor. It is easy to show that 7]!'v = 7]!'v is an invariant rank-2 contravariant tensor:

0.(3 ox" ox).. _ ,,).. (A ) 7] oi;o. oi;(3 - 7] .8

The rank of a tensor can be changed by contraction. For example, it is easy to show that the 4-divergence of a Lorentz vector, oV!' lox!', is a Lorentz invariant. Similarly, the 4-divergence of a Lorentz tensor, oT!'v lox!' = VV, is a Lorentz vector.

A quantity with vanishing 4-divergence gives a local Lorentz invariant conservation law, e.g.

0. 0 -.L-V.j=O, at (A. g)

meaning that the time derivative of the "density" (j0) is equal to the diver­gence of the "current" (j). If there is a closed surface over which the current vanishes, the total "charge" within the surface is conserved:

(A.I0)

We will now show that it is possible to change a contravariant 4-vector into a covariant 4-vector by the simple operation

(A.11)

To verify that the objects obtained by this operation are indeed contravectors we transform from the i; system:

oi;o. - oi;o. - (3 oi;o. ox" ox).. - (3

oxv Vo. oxv 7]o.(3V oxv 7]")..oi;o. oi;(3 V , (A.12)

where in the second form we have used (A. 7). The first and third factors on the right combine to give a Kronecker J

(A.13)

B Natural Units 277

so we find !l-a ux ---;::;-- Va uxV

(A.14)

Comparing the extreme right with the extreme left, we see that Vv as defined by (A.l1) is, indeed a covariant 4-vector.

Equation (A.l1) can be generalized to tensors in the obvious way, e.g.

(A.15)

B Natural Units

In this book we often use "natural units" meaning that in formulas we omit factors of n, e and k. This is basically an ink-saving convention because there is no information contained in such factors. Consider two quantities, A and B, which have the dimensions

[AJ

[BJ

lengthl massm timen .

length>' mass" timeV •

(B.l)

(B.2)

We can generally find unique exponents x, y and z such that the following equation is dimensionally correct

A = eX nY B Z , (B.3)

where x, y and z are the solutions of the simultaneous equations:

l = x + 2y + AZ m = y+j.Lz n = -x-y+vz. (B.4)

Since the exponents are unique, we can drop the factors of e and n and write (B.3) as

A = B Z • (B.5)

If we have used the correct z in this formula, x and y can be found by dimensional analysis and the "dimensionally correct" (B.3) recovered.

We can obviously generalize this procedure for formulas involving sums of terms. The dimension of temperature can also be added since temperatures can be converted to energies by multiplying by k.

The most common use of natural units in this book involves relativistic expressions where the factors of e are omitted, e.g.

dT2 = dt2 - dx2 ---+ dT2 = de - dx2e-2 ,

and

E2 = p2 + m 2 ---+ E2 = p 2 e2 + m 2 e4

H - 1 eo·

(B.6)

(B.7)

(B.8)

278 Appendix

Of special interest are "dimensionless" quantities, e.g. velocities

v

c

and gravitational potentials

10-3 (B.9)

(B.IO)

In quantum formulas, we usually set quantities equal to some power of an energy (or temperature). Energies, masses and momenta clearly have z = 1 as in (B.7). Of more interest is the fact that lengths and times have z = -1, e.g.

A 27r

A = 27rnc

(B.ll) - -+ --Ey Ey

W = Ey -+ W = Eyln. (B.I2)

The fact that lengths have z = -1 means that cross-sections have z = - 2

87r 2 87r CTT = 3 a m;2 -+ CTT = 3 a 2 (mec2 )-2 (nc)2 . (B.I3)

Similarly, particle densities have z = 3:

n = n = 2.4 3 1 -;2 (kT) (nc)3 (B.I4)

n 2 (~~) 3/2 exp( -miT)

(mC2kT)3/2 2 1

n = 2 ----:z;;:- exp( -mc IkT) (ncp . (B.I5)

Note that in this last non-relativistic formula, the factors of c cancel as ex­pected.

Energy densities have z = 4

27r2 P = _T4

30 (B.I6)

Finally, the Friedmann equation has z = 2

H2 _ 87r3 T4 -+ H2 = 87rG p - 45 m~J 3 '

(B.I7)

where the Planck mass mpJ is defined by mpJ = (ncIG)1/2.

G= (B.I8)

C Standard Particles and Beyond 279

C Standard Particles and Beyond

The known particles listed in Table 6.2 can scatter elastically and inelastically with each other. The rules that tell us how to calculate the reactions rates constitute the "standard model of particle physics" [8-11,20]. In this model, most of the allowed reactions can be assembled from primitive three-particle reactions of the form

fermion + boson +-+ fermion' (C.l)

and

fermion + antifermion +-+ boson. (C.2)

(There are also three-body reactions of the form (boson + boson +-+ boson) but they are not important for the reactions considered here.) The only al­lowed three-particle reactions are those that conserve electric charge and, in the case of quark-gluon interactions, "color". Examples of some assembled reactions are shown in Figs. C.l and C.2.

The Standard Model is a quantum field theory that provides the rules for turning each picture of Figs. C.l and C.2 into amplitudes. Rates for a given reaction are found by summing over all amplitudes for the same initial and final particles and then squaring the sum. When the as-yet undiscovered "Higgs boson" is added to the known particles, the standard model gives well defined rates for all physical processes, i.e. the theory is renormalizable.

Amongst other things, each "vertex" of a fundamental three particle re­action is associated with a coupling constant that gives the strength of the interaction. In the standard model, all coupling constants are within an order of magnitude of the electric charge. A diagram with n vertices therefore gives an amplitude of order an.

When cross-sections are calculated by summing amplitudes, the effective coupling constants appear to evolve with energy. All effective constants are expected to become equal to each other at the "Grand Unification" energy scale, estimated to be of order 1015 to 1016 GeV. A more complete theory containing heavy particles is expected to become manifest at these energies. Quantum gravity effects should become important at the Planck scale '" 1019 GeV.

Below the deconfinement temperature, '" 400 MeV, quarks and gluons in­teract in a way that confines them in hadrons. Baryons, e.g. protons and neu­trons, are bound states of three quarks and associated gluons while mesons, e.g. 7r±, 7r0 , are bound states of quarks and antiquarks. Above the deconfine­ment temperature, quarks and gluons are believed to be liberated and behave as normal particles.

Most cosmologically interesting reactions involve "electroweak" interac­tions where the bosons in the fundamental reactions are y, ZO or W±. The fundamental three-particle interactions are of two types, "neutral current interactions" where the boson is y or ZO and "charged current interactions

280 Appendix

where the boson is W±. Reactions due to the exchange of W or Z bosons are said to be due to "weak" interactions because in the low-energy limit, the amplitudes are inversely proportional to the square of the heavy boson mass and are thus "small."

Neutral current reactions mostly appear in elastic scattering and particle­antiparticle annihilation. In the standard model, they all obey the rule that the two fermions must be identical, i.e.

e-ZO +--7 e but not e-Zo +--7 p,- ,

e+e- +--7 ZO but not e+ p,- +--7 ZO .

(C.3)

(C.4)

Charged current reactions change fermions into other types of fermions e.g.:

(C.5)

Charged current interactions can also annihilate fermions with antifermions:

(C.6)

Because W± have integer charge, there are no reactions changing leptons into quarks.

Diagrams for the annihilation of electron-positron pairs and for neutrinos are shown in Fig. C.l.

Reactions for neutrons and protons can be treated by including "spectator quarks" in the diagram. Examples are shown in Fig. C.2.

The rules for turning the diagrams into amplitudes are given in the stan­dard texts [8-11]. Besides the coupling constants for each vertex, an am­plitude contains kinematic factors as dictated by quantum field theory. For example, in the low-energy neutrino interactions important for cosmology at T '" MeV, the presence of the W± and ZO give factors of mvJ '" mz2 in the amplitudes. Combined with the coupling constants, this gives squared amplitudes of order a 2 /m~ '" G~ where GF is the Fermi coupling constant. Cross-sections and decay rates are obtained by multiplying by factors that are related to the total phase space for the final state particles. The forms of these factors can often be guessed by dimensional analysis. Cross-sections have dimensions of energy-2 so G~ must be multiplied by the square of an energy. For example the cross-section for veve annihilation is

a = G~E;m (! + 2sin2 0 + 4sin4 0 ) 121T 2 w w, (C.7)

while that for v~v~ and vrvr annihilation is

a = G~E;m (! - 2sin2 0 + 4sin4 0 ) 121T 2 w w (C.8)

In these two expressions, sin2 Ow '" 0.23 is one of the parameters of the standard model.

C Standard Particles and Beyond 281

Ve VI! V't e+

Z

e Ve VI! V't

Ve e

Iw Ve e+

In the range me « Ev « mp the cross-section for Ven -+ e-p is

a = G:E; [1 + 3gi] cos2 0c rv 9 X 1041 (lO~eV ) 2 cm2 ,

where cos2 Oc rv 0.98 and gA rv 1.2.

(C.9)

Decay rates have dimension of energy so G~ must be multiplied by the fifth power of an energy. For example, the neutron decay rate is

(C.lO)

where Q = mn - mp - me is the energy release per decay. Supersymmetric extensions of the standard model are cosmologically in­

teresting because they provide a well-motivated nonbaryonic dark matter candidate. In these theories, each fermion (boson) in Table 6.2 is paired with a heavy, as-yet undiscovered, partner that is a boson (fermion). For example the spin 1 photon, 'Y is paired with a heavy spin 1/2 "photino" y. Each spin 1/2 quark, q is paired with a heavy spin 0 "squark" q.

282 Appendix

ve __________ ,-__________ e

w

d ________ -+ _________ u u------------------------------ u d ------------------------------ d

d ______________________________ d u ------------------------------ u

d u

w e

Fig. C.2. Diagrams for the reactions Ven --+ e-p and n --+ pe-ve .

The interactions of the supersymmetric particles can be found by tak­ing the fundamental three-particle interactions of the normal particles and turning two of the particles into their supersymmetric partners. For example:

(C.11)

where e is the "select ron. " The diagrams for photino annihilation to quarks and photino scattering

on quarks are shown in Fig. C.3. The lightest supersymmetric particle (LSP), called generically X, is ex­

pected to be a linear combination of the photinos, the Zino (partner of ZO) and the higgsinos (partners of the Higgs bosons). The relic abundance of this particle is determined by its annihilation cross-section (Sect. 6.6). As with neutrinos, the annihilation cross-section will contain a factor (12, a factor m-4

for the exchanged particle and a kinematic factor that turns out to be mi:

u(v/c) rv

(12 2

4 m X· me (C.12)

D Magnitudes 283

:=JJ--ii _

q

q

Fig. C.3. Diagrams for the reactions yy -t qq and yq -t yq. The first would con­tribute to the photino annihilation diagram, determining the photino relic abun­dance. The second determines the photino elastic scattering cross-section on nuclei.

For a given mx it is then possible to find an appropriate me such that the annihilation cross-section is of the correct magnitude ((J"v / C rv 10-37 cm2 , see Sect. 6.6) to provide the cosmological dark matter.

D Magnitudes

Because of the wide range of stellar and galactic luminosities, astronomers generally give luminosities on a logarithmic scale of "absolute magnitude." For an object of total luminosity L, the absolute bolometric luminosity is defined as

iVhol == -2.5Iog(L/LG ) +4.76, (D.l)

where LG is the solar luminosity and 4.76 is the solar magnitude. We note that because of the minus sign bright stars have small magnitudes. The choice of a factor 2.5 may seem strange but it has the characteristic of giving simple expressions for both large luminosity differences (5 magnitudes is a factor 100 in luminosity) and small luminosity differences (i1L/ L rv i1NI).

It is very difficult to measure bolometric magnitudes and astronomers generally observe with a filter that makes them sensitive to a particular spec­tral band or "color." The most widely used filters are described in Table D.l. For an object of luminosity Lc in the band C, the absolute magnitude "~lc is defined by

Mc == -2.5Iog(Lc/LcG) + MCG C = U, B, V, R, I .... (D.2)

284 Appendix

Table D.l. Standard filters in the UBVRI system used for ground-based observa­tions [1]. Except for certain infrared bands, wavelengths between radio (,\ ~ 1 cm) and near-infrared (,\ ~ 1000 nm) are strongly absorbed by the atmosphere and ob­servations are done from aircraft, balloons or satellites. Satellite observations are also required for short wavelengths between ,\ ~ 300 nm and those of Te V pho­tons. The last column gives the band's Galactic interstellar absorption (extinction coefficient) relative to that of the V band.

filter (,\) .:::1'\ M>'8 Ac/Av (nm) (FWHM) (Sun)

U (ultraviolet) 365 66 5.61 1.531

B (blue) 445 94 5.48 1.324

V (visible) 551 88 4.64 1.0

R (red) 658 138 4.42 0.748

I (infrared) 806 149 4.08 0.482

J 1200 213 3.64 0.282

K 2190 390 3.28 0.112

bolometric CXJ 4.76

The difference between two magnitudes, e.g. lvIE - ]tlv , is a "color index" of an object. By convention, a color index is always the shorter wavelength magnitude minus the longer wavelength magnitude. Because of the inverse log scale, a blue (hot) star has small color indices.

As with luminosities, astronomers generally give fluxes on a logarithmic scale of "apparent magnitude." For a star with flux cPe (outside the Earth's atmosphere) in the band C, the apparent magnitude me is defined as

me == C = -2.5 log cPe +ae C = U,B, V,R,I,

where ae is a constant. If there is no interstellar absorption, we have cPe = Le/(47rR2) where R is the distance to the star. This implies that m = M + 5 log R + constant. The constant ae in the definition of me is chosen so that, in the absence of absorption, the apparent magnitude is equal to the absolute magnitude if R = 10 pc:

me = Me + 5log(d/10pc) + Ae C = U,B, V,R,I.

The "extinction coefficient" , Ae , takes into account interstellar or intergalac­tic absorption. The absorption length, corresponding to Av rv 1 is typically rv 1 kpc in the galactic plane, but varies greatly according to the line-of-sight

E Useful Formulas and Numbers 285

because of discrete absorbing clouds. Perpendicular to the galactic plane, the absorption of an extragalactic source is typically 10% (Av rv 0.1).

We note that in the absence of a wavelength-dependent absorption, the color index of an object is equal to its apparent color index, e.g. mB - mv = B - V = MB - Mv. In reality, absorption is stronger at short wavelengths than long so in the presence of absorption, an object is "reddened". Table D.1 gives typical relative absorption for the interstellar medium of the Milky Way.

E Useful Formulas and Numbers

• Friedmann equation for the scale factor a(t):

(0,)2 _ 87rGp H2 ( n) A-2 ~ - -3- + 0 1 - J£T a , (E.1)

where DT is the present-day total density divided by the critical density

p(ao) DT = 3H5I87rG'

and where

a(t) = a(t) . ao

• Friedmann equation for the present epoch:

~ = Ho (DR a- 4 + DMa-3 + DA + (1- DT)a-2)1/2 , a

(E.2)

(E.3)

(E.4)

where DR, DM , and DA are the present-day contributions of relativistic matter, non-relativistic matter and vacuum energy:

(E.5)

• Friedmann equation during the radiation epoch:

. (8 G 2) 1/2 ( T ) 2 1/2 a 7r 7r 4 -1 9E - = -9E(T)-T rv 0.65s -- (-) a 3 30 1 MeV 10

(E.6)

where 9E is the effective number of spin states (Fig. 6.1). • The radial coordinate X of an object of redshift z:

lao da x(z) = 2 .

ao/(l+z) a (a/a) (E.7)

286 Appendix

• X(z) for z « 1:

aOXl(z) = aOrl(z) = H01z [1_I~qOz+ ... ]

• Luminosity and angular distances:

aor(z)(1 + z)

aor(z )/(1 + z) ,

where the radial coordinate r is

{sin X = X - X3 /6 + ...

r = X 3/ sinh X X + X 6 + ...

{ fh > 1 fh = 1 . fh < 1

Table E.1. Selected phY8icai constants, adapted from [20].

quantity symbol value

speed of light in vacuum e 2.99792458 x 108 ms- 1

Planck constant li 1.054571596(82) x 10-34 J s

(E.8)

(E.g)

(E.I0)

(E.ll)

conversion constant lie 1.973269602(77) x 10- 7 eV m conversion constant 27rlie 1.24 x 103 eV nm

Fine structure constant ct 1/137.03599976(50) Thomson cross-section O"T 0.665245854(15) x 10-28 m 2

Gravitational constant GN (= G) 6.673(10) X 10- 11 m 3 kg- 1s- 2

Planck maS8 mpJ = Jlie/G 1.2210(9) x 1019 GeV /c2

Fermi coupling constant G F /(lie)3 1.16639(1) x 1O- 5 GeV- 2

electron mass rnc 0.510998902(21) MeV /e2

proton mass lnp 938.271998(38) MeV /e 2

neutron-proton Llm mn -mp 1.293318(9) MeV / e2

deuteron mass mel 1875.612762(75) MeV /e2

Boltzmann constant k 1.3806503(24) x 1O-23 JK- 1

8.617342(15) x 1O-5 eVK- 1

E Useful Formulas and Numbers 287

Table E.2. Selected astrophysical and cosmological quantities, adapted from [20]. Ho and all densities refer to the present epoch.

quantity symbol value

astronomical unit AU 1.49597870660(20) x 1011 m parsec pc 3.085677 580 7( 4) x 1016 m

=3.262 ... ly solar mass M0 1.9889(30) x 1030 kg

= 1.189 X 1057 mp solar luminosity L0 3.846(8) x 1026 W S-l

2.40 X 1045 eV S-l

solar equatorial radius R0 6.961 x 108 m

Hubble expansion rate Ho 70h70 kms-1Mpc-1 100h km S-l Mpc-1

h70 = 1.0 ± 0.15 h = 0.7± 0.1

Hubble time tH = Hal 1.40 h701 x 1010 yr 4.41 h701 x 1017 s

Hubble distance dH = CH01 4280 h701 M pc 1.32 h701 x 1026 m

Critical density pc = 3Hg /87rG 0.92h~01O-26 kg m-3

5.16h~0109 eV /c2 m-3

1.36h~01011 M0 Mpc-3 CBR temperature Ty 2.725 ± 0.001 K

kTy (2.348 ± 0.002) x 10-4 eV CBR energy density py 0.26038 (Ty/2.725)4 eV cm-3

fly 5.06 h702 x 10-5

CBR number density ny 410.50 (Ty/2.725)3 cm-3

neutrinos (+antineutrinos) nv = (3/11)ny 111.95 (Ty/2.725)3 cm-3 number density per species

baryons number density nb = TIny "-' 5 x 10- 1Ony "-' 0.2m-3

density flb "-' 0.04h702

non-relativistic matter flM "-' 0.3 vacuum energy flA "-' 0.7 matter-radiation equality

scale factor iieq = 1. 68 fly / flM 0.85 X 10-4 /(flMh~o) CBR temperature kTeq 2.8 flMh~o eV

recombination scale factor Urec 1/1100 CBR temperature kTrec 0.26 eV

288 Appendix

F Solutions and Hints for Selected Exercises

Chapter 1

1.2 The Universe is expanding today because it was expanding yesterday (see (1.53)). It was expanding yesterday because .....

It will be difficult to get an ultimate explanation since it will require knowledge of the physics that was in charge of things at the Beginning.

Chapter 2

2.1 ¢ rv 100 m-2 S-l / z2. ¢galaxy rv 10-2 ¢star.

2.2 nstarlight rv 5 X 10-7 nCBR. np--+4He rv 5 X 10-3 np.

2.3 Compton scattering dominates with a mean free path of order 103dH .

2.5 It is possible to count the number of galaxies with a redshift less than z. The volume of the corresponding space is V = (47r /3)z3d~ ex: h"io3 so the measured number density is ex: h~o.

2.6 You should get a reasonable value of Ho with this method.

2.8 NCC 5033 is a typical spiral galaxy so you should get numbers compa­rable to those in the text for typical spiral galaxies.

Chapter 3

3.3 The rocket is in free-fall after lift-off and therefore follows a geodesic. The rocket's clock must, therefore, measure a longer elapsed time than the ground clock which is not in free-fall. On the other hand, an airplane is not in free-fall so the answer depends on its velocity and altitude.

3.8 To order z2, we find d1 = dA < do < dL . The distances differ in the coef­ficient of the z2 term so they differ by rv 10% at z rv 0.1 or R rv 430h"io1 Mpc.

3.9 If a comoving observer sees the explorer leave with velocity v, then

dR dX cit = a dt = v. (F.1)

The derivative is with respect to t since this is the time measured by the comoving clock. Using the metric, we then find

dX v

dt dT J1- v2 .

(F.2)

(F.3)

For an empty universe (a ex: t), the most distant galaxy that can be reached has a redshift of z = v / c (not too surprising). For a critical universe (a ex: t 2/ 3 ), the most distant galaxy that can be reached has a redshift of z = 2v/c. A simple Newtonian argument can explain the extra distance in a critical universe.

F Solutions and Hints for Selected Exercises 289

3.12 B measures the length of the rod by radar, emitting two photons at t = - L /2 and receiving the echo at t = L /2. B sees a Lorentz contraction because, by postulate (3.5), B's clock measures t' = ±O.5LV1 - (32 and therefore concludes that the bar has a length L' = LVI - (32.

Chapter 4

4.1 For ii = 0 and for dr = 0 (for a comoving clock), we have

dT = dt + (1/2)(aJa)2 (ar)2dt = dt + (1/2)v2dt, (F.4)

where v = H R is the Hubble velocity of the co-moving clock. This gives

dT 2 dt = 2/ rv dT(l - v /2) .

1 + v 2 (F.5)

This is as predicted by special relativity where we expect the time dt mea­sured by the moving clock to be smaller than the time dT measured by the stationary clock by a factor \11 - v2 rv 1 - v2 /2.

To second order, the trajectory of the photon in Fig. 4.1 between ta and t is governed by

a(t)x rv it dt' [1 - H(t)(t' - t)] = (t - ta) + H(t) (t - ta)2 . (F.6) ta 2

For the trajectory between t and tb we have

H(t) 2 a(t)x rv (tb - t) - -(tb - t) .

2

Taking the sum we find

a(t)x rv (tb ; td + Hit) [(t - t a )2 - (tb _ t)2] .

(F.7)

(F.8)

Since, to first order, both (t - ta) and (tb - t) are equal to a(t)x, the second term vanishes to order X2 so we have

a(t)r rv (tb ; h) + O(r3) , (F.9)

where we use the fact that X = r + O(r3). Taking the difference between the two trajectories we have

H(t) [ 2 2] o = tb + ta - 2t - -2- (tb - t) + (t - ta) (F.lO)

Using (t - ta) rv (tb - t) rv a(t)r, we find

tb ; ta = t + (1/2)aar2 + O(r3) , (F.11)

which is equivalent to (4.11).

290 Appendix

4.3 For a non-relativistic ideal gas, E '" m and (p2/2m) = (3/2)T from which it follows that p = nT « mT = p. For a relativistic ideal gas, p = E from which it follows that p = p/3.

Chapter 5

5.1 For .aT = .aM = 0, we find

aCt) ex: t qo = 0 aOXI(z) = Holln(1 + z) . (F.12)

Using ao = HOI for an empty universe, we find

( ) 1 + z + (1 + Z)-l rl z = 2 . (F.13)

For .aT = .aM = 1, the minimum angular size occurs for z = 1.25.

5.2 The equation should be a good approximation as long as there are no other relativistic species, i.e. for T « me' After the neutrinos start to become non-relativistic, the equation is still a good approximation because the .aM term dominates in any case.

For tree we can make the approximation that the universe is matter­dominated in which case the age is just (2/3) the Hubble time at that epoch:

tree = (2/3)Hol .a~1/2a~1; '" 2.6 h~05 yr . (F.14) .aM

For teq , we can neglect neither the radiation nor the matter (at teq ) so it is necessary to do a non-trivial integral:

_1(1.68.ay)3/211 xdx 4 ( 0.3 )2 teq = Ho n2 ~ '" 5 x 10 yr n h2 .

HM 0 Y 1 + X J tM 70 (F.15)

Neglecting the matter would have given the correct order of magnitude and the correct dependence on .aM h?o.

5.4 to'" (2/3)Ho l (1 + .aA /3 + ... ) 5.12 Absorption by interstellar matter would not resolve Olbers' paradox because the matter would heat up until it reached a temperature at which it would radiate blackbody photons at the same rate as it absorbed starlight.

The modern calculation gives a flux per unit solid angle of

d¢ 3 J rNXI = noL (to dt l a(h) d.a = noaoL 41Td~ 41T 10 ao

noL to da = 41T 10 ao(a/a)'

(F.16)

where we have used dL = aorl(1 + z) and dX = dt/a. Using the Friedmann equation to evaluate a/a we find

F Solutions and Hints for Selected Exercises 291

d¢ (3/5) noLto to = (2/3)Hol DM = DT = 1 (F .17) dD 47f

d¢ (1/2) noLto to = HOI DM = DT = O. (F.18) dD 47f

In both cases the correct calculation adds only a numerical factor to the naive answer calculated assuming that the stars have been burning for a time to. The fact that the factors are less than unity is due to the redshift.

In the inflationary model, we find

ainf ( )4

p(ao) = Pv -;;- . (F.19)

We see that the energy density falls as a -4, as expected.

Chapter 6

6.1 It is important to factor out the physical parameters, e.g.:

9 J 47fp3dp gT4 (Xl x 3dx p(T, It = 0) = (27f)3 exp(p/T) ± 1 = 27f2 Jo exp(x) ± 1 . (F.20)

The integrals give only the numerical factors in Table 6.l. For a neutral relativistic gas of electrons and positrons, the potential

energy per particle is of order a ~ 1/137 times the kinetic energy per particle. The gas should be ideal to a good approximation.

The particle-antiparticle asymmetry for a relativistic gas in thermal equi­librium is of order (n - n)/n ~ It/T.

6.4 The number of remaining interactions is

JOC 1= da r(t)dt = r-(./ ) . II a, a a a

(F.21)

To quickly get the answer, it is a good idea to evaluate a/a with a Friedmann equation normalized at aI, e.g.

~ ~ TT, [DR (0,) (:,r\r . (F.22)

Substituting this into the integral, we find that the number of interactions is just r(tdHl1 times a numerical factor of order unity. Since r(tdHl1 « I, this proves the conjecture.

6.5 Numerically, the photon scattering rate (always dominated by Compton scattering) is equal to the expansion rate at T ~ 0.236eV. The recombination rate is equal to the expansion rate at T ~ 0.215eV. A fraction ~ 3 X 10-5 of the electrons remain free.

More reali"tic calculations including all the states of the hydrogen atom give a recombination time of T ~ 0.26eV.

292 Appendix

6.6 The last annihilations take place at T '" 10 ke V. A photon of E = 510 keY needs about 10 collisions to reach a reasonably thermal energy of E '" 30keV. The time to do this, '" lO(ne O'Tc)-l, is much less than the Hubble time at this epoch.

6.8 The minimum mass is of order (mw/a)y'mw/mp].

Chapter 7

7.6 The universe must be reionized before a '" 1/30 in order to have an optical depth of order unity. If this is the case, between recombination and reionization, photons travel a distance of the order of the Hubble distance at reionization. Photons scattered into a given line of sight will therefore have originated at recombination from a region of angular scale given by the Hubble distance at reionization.

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Index

D problem 168,171 DCDM 2,11 DM 12, 38, 72, 73, 156 - and CBR anisotropies 269 - and structure formation 31,72 - from age of universe 153 - from cluster M/ L 76 - from cluster baryon fraction 71,76 DR 24 - and CBR anisotropies 269 DT 2,9, 38, 72 - evolution 30, 168 - from CBR anisotropies 74, 268 Db 2,10,71 - and CBR anisotropies 268 DA 2, 15,38, 72, 156 - and CBR anisotropies 269 - from age of universe 153 - from supernovae 73 Dvis 10 (78 236,238,239 to 17

absorption (interstellar) 43,284 acoustic oscillations 240, 265 adiabatic density fluctuations 224 - and CBR anisotropies 247,265,268 age of the Universe 153 angular distance 108,156 anthropic principle 37,251 antimatter 10,209 axions 62

baryogenesis 209 baryon-to-photon ratio, 'T] 37,71,268 - and baryogenesis 209,217 - and isocurvature fluctuations 224 - and nucleosynthesis 198, 201-203,

205,206,215,219 - and sound speed 266 baryonic dark matter 62, 255 bias parameter 236

big bang 8 black holes 12,49,62 Boltzmann equation 25,186 Boomerang 74,159,161,261,268 Bose-Einstein statistics 182, 190 bottom-up structure formation 255 bouncing universe 153 bremsstrahlung 55,71,82-84,190 brown dwarfs 10 bulk flows 240

C and CP violation 210 Cepheid variable stars 46,47,50,52,

55,68,75,76,79 chemical equilibrium 179, 180, 191,

192,197,199 chemical potential 182,190,209,211,

214 classical horizon 167 clump giants 43, 45 clusters of galaxies 6,54,67,75 - and filaments 226 - and hot dark matter 255 - baryon fraction 55,71 - evolution 72 - formation 226 - gravitational lensing by 69,114,116 - number density evolution 236, 238,

239,254 - number density fluctuations 225,

236,237 - SZ effect 69, 264 - velocity dispersion 231,249 - X-ray emission 82 COBE 4,13,70,248,261,268 cold dark matter (CDM) 11,59,206,

222 - and CBR anisotropies 265 - and Silk damping 224, 256 - evolution of fluctuations 223, 224,

226

300 Index

- spectrum of fluctuations 236,237, 252,253

color-magnitude diagram 43-45, 155 comoving coordinates 87,93,95 cosmic background radiation (CBR)

12,70,74,177 and causality 163, 165

- and SZ effect 69 - anisotropies 73,74,159-161,206,

223,224,248,249,256,259,261-268 - dipole 239 - temperature evolution 23, 180 cosmological constant 15 cosmological principle 9 critical density 9 curvature epoch 28 curved space versus curved space-time

143

dark matter - in clusters of galaxies 55 - in galaxies 4,5,51,52 deceleration parameter qo 28, 104,

137,161,162 - and expanding photosphere 175 - 'and luminosity distance 107 - and number counts 175 - and SZ effect 175 - and type Ia supernovae - correlation with distance

156

73.157 105,106,

density fluctuation spectrum, Llk 222, 235,237 evolution for k- 1 > d H 246

- in CDM models 252 primordial spectrum 247

deuterium 201,202,206,218 distance ladder 46

effective number of spin states 184, 185

Einstein equations 143, 144. 149 Einstein tensor 143 electroweak interactions 279 empty universe 142 energy conservation 125, 132. 133 energy-momentum tensor 129,148 entropy 180,183,185,195,210,214 equation of state 132, 151 event horizon 168, 176

Fermi constant 280 Fermi-Dirac statistics 182, 190 filter systems 284

fine-tuning 36, 169 free-streaming 222,223,254 freely falling coordinates 86,91,126,

129 freeze-out 180,214,215 - and entropy production 211,212 - and free energy 180

electron-positron 193, 194 neutrino 197,219 neutron 200 nuclear 202 positron 196,217

- wimps 207,208 Friedmann equation 19-21,28,134,

136, 144, 147

galactic halos 4,51,52 galactic rotation curves 4,34,51,52,

76,249 galaxy number counts 175 geodesics 90,93, 103, 108-111 globular clusters 154, 155 grand unified theories (GUTs) 33,35,

168,170,209,210,217 gravitational collapse - dissipationless 226 - dissipative 226 gravitational lensing 55,57,111,114,

116,160 - microlensing 63 - on large-scale structure 239 - time delay 69, 115 - weak 116 gravitational redshift 112 Gunn-Peterson effect 172

helium 10,39,40,74,200-202,205 High-Z Supernova Search 157-159 Hipparcos 42,44 horizon 162, 176 horizon problem hot dark matter Hubble

162, 171 254

constant Ho 6, 17,67 - diagranl 6, 157, 158 - distance 8

law 6, 8, 104, 106 radius entry 222,231,234,244-246, 250,252

- fluctuation amplitude at LlH 248 - radius exit 244-246, 249

Space Telescope, HST 48, 68 - time 8, 153, 156

inertial frames 88, 91 inflation 156, 165, 169, 170 - and fh 168, 169 - and causality 163, 166

and CBR anisotropies 268 - and gravitational waves 269 - and homogeneity 167 - and Hubble exit 245 - and particle physics 172

and primordial fluctuations 249-251

- duration of 166,167 - evolution of inhomogeneities 246 isocurvature fluctuations 224, 233,

256,266,268

Jeans wavelength 241

kinetic equilibrium 179, 216

large-scale structure 56 last-scattering surface 262 lepton number asymmetry 210 Liouville equation 25, 186, 214 Lorentz transformations 89, 118, 121,

275 luminosity distance 106,156 Lyman-a forest 50, 202

machos 62 Magellanic Clouds 43,45,63,66,68,

79 magnetic monopoles 35 magnitudes 283 main sequence 40,43 mass-to-light ratio 4,51,52,55,56,76 matter epoch 28 matter-radiation equality, aeq 24,246 maxima 74,161,261,268 metric 85,90 molecular gas as dark matter 65

natural units 8, 277 neutrino oscillations 15 neutrinos 12, 13,46,61,71,196,254

and aeq 24 - as dark matter 12 - density 23 - masses 15 - number of species 219 - wrong helicity 217 neutron stars 46,62 neutrons 199-201,219

Index 301

Olbers' paradox 176

parallax 42,43 particle horizon 168 particle-antiparticle asymmetry 194,

197,209 peculiar velocity 6,232,239,240 phase transitions 16, 170, 185 phase-space distribution 25,26, 182,

186,214,270 Planck mass 16, 278, 279 power spectrum 236 pressure 132, 148, 182 - negative 132 primordial nucleosynthesis 10,71,198 proper distance 140 proper time 85

quasi-stellar objects (QSOs), quasars 49,69,172,202,204

quintessence 16,36

radiation epoch 28 reaction rate 179,187 recombination 12, 32, 162, 215, 262,

263 redshift 7,23,24 reduced scale factor &(t) 17 reheating 171 reionization 172 relic particle densities 180,195,206 Ricci tensor 149 Riemann tensor 139, 149 Robertson-Walker metric 99,102

Sachs-Wolfe effect 258 Saha equation 202,215,219 scalar field 16,36, 137, 170 scale factor a(t) 17,21,22,29 scale-invariant fluctuation spectrum

224,248,249,251,261,264,265,268 Schwarzschild metric 111 Silk damping 256 sound speed 241,243,266,273 spherical collapse 31,226 standard candles 46,47,68,157 standard model of particle physics

279 steady-state universe 156 stellar evolution 39 Sunyaev-Zel'dovich (SZ) effect 69,71,

175,264 Supernova Cosmology Project 48,

157-159

302 Index

supernovae 46,52,68,73,157 - expanding photosphere 69, 175 supersymmetry 11,35,59,61,207,281 synchronous gauge 246

thermal equilibrium 179, 182, 188, 190 tilted fluctuation spectrum 253, 254 top-down structure formation 255 Tully-Fisher relation 52,55,68,131 twin paradox 89

vacuum epoch 28 violent relaxation 226 virial theorem 76, 231 virialization 226,230,231,251

weak interactions 280 white dwarfs 47,62,154 wimps 11,206,241 - detection 59

X-rays 55,70,71,76,82-84,239

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