bailin d., love a. cosmology in gauge field theory and string theory

Preface xi 1.3 Einstein equations for a Friedmann–Robertson–Walker universe 51.4 Scale factor dependence of the energy density 7 1.8 Equilibrium thermodynamics in the expanding universe

Series Editor:

Professor Douglas F Brewer, MA, DPhil

Emeritus Professor of Experimental Physics, University of Sussex

COSMOLOGY IN GAUGE

FIELD THEORY AND STRING THEORY

University of London

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Preface xi

1.3 Einstein equations for a Friedmann–Robertson–Walker universe 51.4 Scale factor dependence of the energy density 7

1.8 Equilibrium thermodynamics in the expanding universe 171.9 Transition from radiation to matter domination 191.10 Cosmic microwave background radiation (CMBR) 21

2.3 The effective potential at finite temperature 33

2.6 Phase transitions in grand unified theories 48

2.8 Phase transitions in supergravity theories 55

3 Topological defects 65

3.5 Gravitational fields of local cosmic strings 74

3.9 Magnetic monopoles in grand unified theories 85

4.3 Out-of-equilibrium decay of heavy particles 96

4.7 Baryon-number non-conservation in the Standard Model 114

4.10 Phase transitions and electroweak baryogenesis 1294.11 Supersymmetric electroweak baryogenesis 132

5.3.1 Introduction: the strong CP problem and the axion solution 151

5.3.3 Astrophysical constraints on axions 159

Contents ix

6.2 Weakly interacting massive particles or WIMPs 175

6.4 Minimal supersymmetric standard model (MSSM) 179

6.6.1 Neutralino–nucleon elastic scattering 1886.6.2 WIMP annihilation in the sun or earth 189

7.2 Horizon, flatness and unwanted relics problems 195

8.3 D-term supergravity inflation 232

8.5 Thermal production of gravitinos by reheating 237

8.6.1 Inflaton decays before Polonyi field oscillation 2408.6.2 Inflaton decays after Polonyi field oscillation 244

10.4 Perturbative microstates in string theory 289

The new particle physics of the past 30 years, including electroweak theory,quantum chromodynamics, grand unified theory, supersymmetry, supergravityand superstring theory, has greatly changed our view of what may have happened

in the universe at temperatures greater than about 1015 K (100 GeV) Variousphase transitions may be expected to have occurred as gauge symmetries whichwere present at higher temperatures were spontaneously broken as the universecooled At these phase transitions topological defects, such as domain walls,cosmic strings and magnetic monopoles, may have been produced Varioustypes of relic particles are also expected These may include neutrinos withsmall mass and axions associated with the solution of the strong CP problem

in quantum chromodynamics If supersymmetry exists, there should also berelic supersymmetric partners of particles, some of which could be dark mattercandidates If the supersymmetry is local (supergravity) these will include thegravitino, the spin-32 partner of the graviton Insight may also be gained intothe observed baryon number of the universe from mechanisms for baryogenesiswhich arise in the context of grand unified theory and electroweak theory.Supersymmetry and supergravity theories may have scope to provide the particlephysics underlying the inflationary universe scenario that resolves such puzzles

as the extreme homogeneity and flatness of the observed universe Superstringtheory also gives insight into the statistical thermodynamics of black holes Inthe context of superstring theory, bold speculations have been made as to a period

of evolution of the universe prior to the big bang (‘pre-big-bang’ and ‘ekpyroticuniverse’ cosmology)

These matters, amongst others, are the subject of this book The book gives

a flavour of the new cosmology that has developed from these recent advances

in particle physics The aim has been to discuss those aspects of cosmology thatare most relevant to particle physics From some of these it may be possible touncover new particle physics that is not readily discernible elsewhere This is aparticularly timely enterprise, since, as has been noted by many authors, the recentdata from WMAP and future data expected from Planck mean that cosmologymay at last be regarded as precision science just as particle physics has been formany years

Copeland, Beatriz de Carlos, Mark Hindmarsh, George Kraniotis, Andrew Liddle,Andr´e Lukas and Paul Saffin for the particle and cosmological physics that wehave learned from them Special thanks also to Malcolm Fairbairn for helping

us with the diagrams Finally, we wish to thank our wives for their invaluableencouragement throughout the writing of this book

We intend to maintain an updated erratum page for the book at

http://www.pact.cpes.sussex.ac.uk/∼mpfg9/cosmobook.htm

David Bailin and Alexander Love

June, 2004

on the history of the universe when its temperature was greater than 1015 K Thiswill be studied in the context of the Friedman–Robertson–Walker solution of theEinstein equations of general relativity In this chapter, therefore, our first task

is the derivation of the field equations relating the scale factor R (t) that appears

in the metric to the energy density ρ and the pressure p that characterize the

(assumed homogeneous and isotropic) energy–momentum tensor This is done

in the following two sections In section 1.4 we show how, for a given equation

of state, energy–momentum conservation determines the scale dependence of theenergy density and pressure The standard solutions for the time dependence ofthe scale factor in a radiation-dominated universe, in a matter-dominated universe,and in a cosmological constant-dominated universe are presented in section 1.5;

we give an estimate of the age of the universe in the matter-dominated case insection 1.6 In section 1.7, we present the evidence that there is, in fact, a non-zero cosmological constant and discuss why its size is so difficult to explain Thediscussion of phase transitions and of relics that is given in later chapters alsorequires a description of the thermodynamics of the universe So in the followingtwo sections we describe the equilibrium thermodynamics of the expandinguniverse and derive the time dependence of the temperature in the various epochs

In section 1.10, we discuss briefly the ‘recombination’ of protons and electronsthat left the presently observed cosmic microwave background radiation Finally,the synthesis of the light elements that commenced towards the end of the firstthree minutes is discussed in section 1.11 The consistency of the predictedabundances with those inferred from the measured abundances determines theso-called baryon asymmetry of the universe, whose origin is discussed at length

in chapter 4

1.2 The Robertson–Walker metric

The standard description of the hot big bang assumes a universe which is

homogeneous and isotropic with a metric involving a single function R (t),

the ‘scale factor’ (or ‘radius’ of the universe) The appropriate metric is theRobertson–Walker metric

where the (time and spherical polar) coordinates(t, r, θ, φ), called the ‘comoving’

coordinates, are the coordinates of an observer in free fall in the gravitational

field of the universe The parameter k takes the values −1, 0, 1 corresponding

to a universe which has spatial curvature which is negative, zero or positive,respectively (This can be seen from the curvature scalar derived from the secondequality of (1.30) with a change in sign for Euclidean rather than Minkowski

space.) Units have been chosen in which the speed of light c is 1.

An immediate use of this metric is to calculate the size of regions of theuniverse that have been in causal contact (in the sense that there has been thepossibility of causal influence occurring between points within the region at some

time between the big bang at t = 0 and time t) Causal influences cannot occur over distances greater than the (proper) distance d H (t) that light has been able to

travel from the the big bang at t = 0 to the time t being studied This distance

is called the ‘particle horizon’ Without loss of generality, consider emission of

a light signal from coordinate(r, θ, φ) at t = 0 to coordinate (0, θ, φ) at time t

along the (radial) geodesic withθ and φ constant (It may be checked that this is

indeed a geodesic by using the coefficients of affine connection given in the next

section (exercise 1).) For a light beam, ds2= 0 and we have

The Robertson–Walker metric 3

We shall discuss the time dependence of the scale factor R (t) in the next section.

Equation (1.4) then allows us to calculate the particle horizon For example, when

received at r = 0 at (around) the present time t = t0 from a distant galaxy at

r = r1 Suppose that two adjacent crests of a light wave are received at t = t0

and t = t0+ t0 having been emitted from the distant galaxy at t = t1and

t = t1+ t1 Equation (1.3) applies but with appropriate modifications to thelimits of integration Thus,

Because the variation of R (t) on the time scale of an electromagnetic wave period

is very small, this equation may be approximated by

Equations (1.19) and (1.17), reinterpreted in terms of photons, mean that a photon

emitted at time t1 undergoes a redshifting of its wavelength as the universe

expands, such that its wavelength at time t0is increased by a factor R (t0)/R(t1).

Since the momentum (or energy) of the photon is inversely proportional to itswavelength, the momentum (or energy) of the photon is reduced by a factor

R (t1)/R(t0) as a result of the expansion of the universe This is often expressed

as energy of photons being redshifted away

When|t1− t0| is not too large, we can make the expansion

R (t1) = R(t0) + (t1 − t0) ˙R(t0) + 1

2(t1 − t0) 2¨R(t0) + · · ·

= R(t0)(1 + H0(t1− t0) − 1

2q0H02(t1 − t0) 2+ · · ·) (1.20)where

Einstein equations for a Friedmann–Robertson–Walker universe 5leading to

Then, after expanding 1/R(t) in (1.11) in powers of t − t0 , we may determine r1

as a function of z Expanding (1.11) gives

Here x i , i = 1, 2, 3, denotes the (spatial) coordinates (r, θ, φ) Equation (1.29)

is just the coefficients of affine connection for the three-dimensional subspace

(r, θ, φ) It is also straightforward to calculate the Ricci tensor R µν from thecofficients of affine connection (exercise 2) It has non-zero components

The Einstein equations for the Robertson–Walker metric, usually referred to

as the Friedman–Robertson–Walker (FRW) universe, are

always increasing Consequently, ignoring the effects of quantum gravity, there

was a past time when R was zero—the moment of the ‘big bang’.

Returning to the Friedmann equation (1.34) with zero cosmological constant,the universe is spatially flat when

Scale factor dependence of the energy density 7

where m Pis the Planck mass, and

M P  2.44 × 1018 GeV m P  1.22 × 1019 GeV. (1.40)Since the Hubble parameter varies with time, so doesρ c The density parameter

is defined as

c

(1.41)and measures the density as a fraction of the ‘critical’ densityρ c The currentvalue of , denoted by 0, has a value [1]

1.4 Scale factor dependence of the energy density

There is also conservation of the energy–momentum tensor to take into account:

where

D λ V µ = ∂ λ V µ +  λρ µ V ρ (1.44)

is the action of the covariant derivative D λ on a contravariant index Theµ = 0

component of (1.43) yields (exercise 3)

˙ρ + 3(ρ + p) ˙R

It is easy to see that this is just the first law of thermodynamics

for a comoving volume V ∝ R3(t).

The energy densityρ may be related to the scale factor R(t) once we have

the equation of state If this is of the form

Equation (1.50) may be understood as a constant number of massive particles

occupying a volume expanding as R3(t) as the universe expands Equation (1.49)

may be understood as the number density of photons (or other massless particles)

decreasing as R−3(t), as for massive matter but, in addition, the energy of each

photon decreasing as R−1(t) because of the redshifting of the photon energy

discussed in section 1.2 Another interesting case isw = −1, which gives

This may be interpreted as vacuum energy and allows us to incorporate thecosmological constant into the discussion without introducing it explicitly, if wewish

1.5 Time dependence of the scale factor

It is easy to solve the Friedmann equation (1.34) in the case of zero cosmological

constant and k = 0, a spatially flat universe Both of these assumptions arealways good approximations for sufficiently early times because, as discussed

in section 1.4, ρ ∝ R−4 for radiation domination and ρ ∝ R−3 for matter

domination Consequently, for a ‘big-bang’ universe with R → 0 as t → 0,

terms With the energy densityρ given by (1.48), the solution of (1.34) (provided

1.6 Age of the universe

We shall estimate the age of the universe in the case = 0 We shall also

assume a matter-dominated universe for the calculation This is a reasonable

Age of the universe 9approximation because, as can be seen from section 1.8, the universe was matter-dominated for most of its history First, rewrite the Friedmann equation (1.34) interms of the valueρ0of the energy densityρ today From (1.50),

ρ



R R0

−3

Thus, the Friedmann equation may be written as

 ˙R R0

where the parameter h is measured to have the value

Thus, the present age of the universe is

t0∼ 1010

1.7 The cosmological constant

In 1917, attempting to apply his general theory of relativity (GR) to cosmology,Einstein sought a static solution of the field equations for a universe filled withdust of constant density and zero pressure The general static solution of (1.34)and (1.36) has

k

With zero cosmological constant ( = 0), the only solution of these equations,

apart from an empty, flat universe, requires that either the energy densityρ or the

pressure p is negative It was this unphysical result that led him to introduce the

cosmological term Then the solution for pressureless dust is

The cosmological constant 11galaxies were observed, the presumption of a static universe could be abandonedand there was no need for a cosmological constant.

However, anything that contributes to the energy density of the vacuumρ

acts just like a cosmological constant This is because the Lorentz invariance

of the vacuum requires that the energy–momentum tensor in the vacuumT µν satisfies

Then, by inspection of (1.32), we see that the vacuum energy density contributes

8πG N ρ to the effective cosmological constant

Equivalently, we may regard the cosmological constant as contributing/8πG N

to the effective vacuum energy density

ρvac = ρ + 

8πG N = effM P2. (1.77)Thus, a cosmological constant is often referred to as ‘dark energy’, not to beconfused with dark matter which contributes to the non-vacuum energy density(and has zero pressure)

A priori, in any quantum theory of gravitation, we should expect the scale

of the vacuum energy density to be set by the Planck scale M P Since has

the dimensions of M2, it follows that we should have expected that/M2

P ∼ 1

We shall see that, in reality, the scale of any such energy density must be muchsmaller We noted in section 1.5 that the effect of the cosmological constant isnegligible at sufficiently early times, because the energy densityρ scales as a

negative power of R for radiation or matter domination Thus, the most stringent

bounds arise from cosmology when the expansion of the universe has diluted thematter energy density sufficiently From the observation that the present universe

The first evidence suggesting this came from measurements of the redshifts

of type Ia supernovae Such supernovae arise as remnants of the explosion ofwhite dwarfs which accrete matter from neighbouring stars Eventually the whitedwarf mass exceeds the Chandrasekhar limit and the supernova is born after theexplosion The intrinsic luminosity of such supernovae is considered to be aconstant That is, they are taken as standard candles and any variation in theirapparent luminosity as measured on earth must be explicable in terms of their

differing distances from the earth In a Euclidean space, the apparent luminosity l

of a source with intrinsic luminosity L at a distance D from the observer is given

In GR we must be more careful So consider the circular mirror, area A, of a

telescope at the origin, normal to the line of sight to a source at r1 Light emitted

from the source at time t1and arriving at the mirror at time t0is bounded by acone with solid angle

as explained in section 1.2, (see (1.18)) Also, photons emitted at time intervals of

δt1reach the mirror at time intervalsδt0 = δt1R(t0)/R(t1) Thus, the total power

P received at the mirror is given by

The cosmological constant 13

14 16 18 20 22

24

22 23 24 25

,

supernovae in the primary low-extinction subset The full line is the best-fit flat-universecosmology from the low-extinction subset, the broken and dotted lines represent theindicated cosmologies

Hence, for nearby supernovae the luminosity distance is proportional to theredshift of the source

Astronomers measure the apparent magnitude m of the various supernovae sources The difference m − M, where M ∼ −19.5, is the (assumed constant)

intrinsic magnitude of the source, is just the logarithm of the luminosity distance

So the apparent magnitude is predicted to be linear in ln z for small z This is consistent with the data for zº0.1, see figure 1.1 taken from [2] For more distant

supernovae the linear relationship between D L and z is distorted by quadratic terms depending on the present deceleration parameter q0of the universe Thedata for 0.7ºzº1 do display such a distortion, see figure 1.1 [2]

For an FRW universe, it follows from (1.36) and the definition (1.22) of q0

that, in general, the deceleration may be written as

for a universe with components labelled by i having energy density ρ i and

pressure p i ≡ w i ρ i; here i ≡ ρ i /ρ c where ρ c ≡ 3H2

0/8πG N is thecritical density In particular, for a universe with just (pressureless) matter and

Figure 1.2 68%, 90%, 95%, and 99% confidence regions for m and ...

[3] See, for example, Bailin D and Love A 1994 Supersymmetric Gauge Field Theory< /i>

and String Theory (Bristol: IOP) ch 1

[4] Weinberg S 1987 Phys Rev Lett 59... universe collapses to a point singularity in a time

• Weinberg S 1972 Gravitation and Cosmology: Principles and Applications

of the General Theory of Relativity (New... emerges as the low-energy limit of string theory The form

of the potential in a supergravity theory is given in section 2.8 The mainpoint to note is that, as in the case of global supersymmetry,