Lecture 12

• Inflation –Horizon problem –Flatness problem –Monopole problem • What causes inflation

Physical Cosmology 2011/2012 Inflation

What is inflation good for?

Inflation solves

1. horizon problem 2. flatness problem 3. monopole problem (relic particles)

Physical Cosmology 2011/2012 What is inflation? Essential property of inflation is R  0

Phase of rapid expansion due to a large cosmological constant.

2 2 The Λ term in the Friedmann  R  c    equation dominates  R  3

dR c2 rearrange  dt R 3

 c2  integrate R  exp t  3 

Physical Cosmology 2011/2012 I The Horizon problem t0 dt r  c Recall particle horizon from lecture 7 hor  0 R(t)

t0 dt Proper size of horizon d  r R(t)  R(t)c H hor  0 R(t) We did a calculation to show that the horizon at the last scattering surface subtends

hor 1.7deg on the sky for an Einstein-de Sitter Universe

Physical Cosmology 2011/2012 Let’s do the calculation again in an even simpler way t0 dt d  r R(t )  R(t )c H hor 0 0  0 R(t) Assume a matter-dominated universe R  t 2/3 t0 dt so d  t 2/3  3ct H 0  2/3 0 0 t

5 5 dH  410 pc size of horizon at t0  410 years

Two points dH apart at z~1100 will now be separated by a distance (1100+1)  dH ~ 400Mpc because the universe has expanded.

Physical Cosmology 2011/2012 t=t0, dH ~ 40,000 pc 0 z~1100

We are at 0. D 0 400Mpc θ t=13.7 billion years (today) and z=0

13.3109 years D  4105 pc  13.3109 pc 4105 years 400 Mpc so   1.8 13000 Mpc Physical Cosmology 2011/2012 Any two points on the sky at z~1100, that are separated by more than 1.8 degrees cannot have had any causal contact when they emitted the CMB photons that we eventually detect. So why is the CMB so isotropic? This is the “horizon problem”. Inflation fixes this problem because the universe expands so much that a small region that is in causal contact ends up being bigger than the observable universe.

End of inflation Time Region that eventually ends up being our observable universe Original small region within Start of inflation particle horizon Physical Cosmology 2011/2012 The Horizon Problem II The Flatness Problem

8G kc2 c2 H 2     -[1] 3 R2 3 8G kc2 c2 1     3H 2 R2 H 2 3H 2 kc2 1      mat R2 H 2  k c2  1  where      R2 H 2 mat  Equation [1]  H2  ρ when the k and Λ terms are small (i.e. when radiation or matter dominates).

Physical Cosmology 2011/2012 so when radiation - dominated R2 H 2  t 1 and when matter - dominated R2 H 2  t 2/3  when radiation - dominated  1  t  when matter - dominated  1  t 2/3 In both cases |Ω-1| increases with time so the situation is unstable. If Ω is close to unity today then it must have been extremely close to unity in the past. This fine tuning argument is called the “flatness problem”. d d R Inflation  R  0  R  0  HR  0 (H  ) dt dt R so R2 H 2 increases with time and therefore k c2   1  decreases with time H 2 R2  stable  inflation solves the flatness problem Physical Cosmology 2011/2012 Need either tot,0 = 1 or tiny deviation from tot,0 = 1 in early Universe:

Friedmann equation: 8 G R 2   R2  k c2 3 tot If curvature and energy term were of similar order at the GUT scale, inflation will make the curvature arbitrarily small compared to the density term if it lasts for many e-foldings ()tot  vac  const.

Physical Cosmology 2011/2012 prediction k  0

for most inflationary models Can havek  0 in more contrived models of inflation with two subsequent periods of inflation. These are models which attempt to explain why Ω is close to unity but not actually equal to unity.

During inflation matter and radiation energy density also become arbitrarily small compared to the vacuum energy density.

Physical Cosmology 2011/2012 III Monopole Problem (see Coles and Lucchin) At the GUT phase transition one monopole should form inside each particle horizon.

If the GUT transition occurs at 1015 GeV this corresponds to a monopole space density comparable to the baryon density nmonopole  nbar

and monopoles would have a mass 11 mmonopole 10 kg This would correspond to

16 monopole 10

Physical Cosmology 2011/2012 Inflation solves the monopole problem because the particle horizon (which contains one monopole) becomes so big at the end of inflation that our observable universe contains too few monopoles to be detected.

However, it’s important that inflation ends at a temperature that’s low enough that the monopoles are not created again.

The amount of inflation that’s needed to solve all three problems is about 60 e-foldings (i.e. universe expands by a factor of e60).

Physical Cosmology 2011/2012 R 4G  3p  recall       R 3  c2  3p  R  0 (i.e. we have inflation) if    0 c2 1 p  wc2  w   3 w  1 corresponds to the cosmological constant  case

What can give us such an equation of state and what causes it to end? Need to turn to particle physics. Look for a specific particle physics phase transition which is controlled by a scalar field.

Physical Cosmology 2011/2012 What is responsible for inflation?

Simple versions of inflation are described by a scalar field with kinetic energy potential energy 2 1  2 c  2   V () 1  2 p  2   V () The phase transition can be depicted as moving between two equilibrium states of the potential. There is considerable freedom in choosing Φ and the shape of the potential curve. It’s all a bit ad hoc and there are many many variants of inflation.

Physical Cosmology 2011/2012 I T > TGUT II Inflation  2 V(T)   V ()

2 c2G vacc vac

 III IV

Reheating T < TGUT

V(T)  2 ~ V ()

Physical Cosmology 2011/2012 During inflation the scale scale factor inflation factor grows rapidly while matter density and temperature decreases rapidly.

At the end of inflation the vacuum energy density is first

temperature transformed into kinetic energy of the scalar field which is then transformed into heat and particles (reheating). t

Physical Cosmology 2011/2012 Inflationary theory candidates

• The Higgs Field • GUTs • Supersymmetry • String theory • Brane Theory

Physical Cosmology 2011/2012

How does inflation end?

Phase transition Reheating

Matter and radiation which have been cooled by the inflationary expansion are reheated at the end of inflation. If that phase transition were sudden, the energy density of radiation and matter would stay basically unchanged. This is an effect of the strange “equation of state” of vacuum energy.

2 p    c  vac  const.

Physical Cosmology 2011/2012 Inhomogeneities from inflation

During inflation, quantum fluctuations produce inhomogeneities in the energy density

     which grow under gravity   and seed structure formation at the present epoch.

Physical Cosmology 2011/2012 But:

The value of  necessary to drive inflation at the GUT scale is some factor 1080 bigger than  today

Why?

Most cosmologists see this as the biggest unsolved problem in cosmology.

Physical Cosmology 2011/2012