Lecture 1 Inflation Horizon problem Flatness problem Monopole problem What causes inflation Physical Cosmology 11/1
Inflation What is inflation good for? Inflation solves 1. horizon problem. flatness problem 3. monopole problem (relic particles) Physical Cosmology 11/1
What is inflation? Essential property of inflation is R Phase of rapid expansion due to a large cosmological constant. The Λ term in the Friedmann equation dominates R R c 3 rearrange dr R c 3 dt integrate R exp c 3 t Physical Cosmology 11/1
I The Horizon problem Recall particle horizon from lecture 7 r hor c t dt R( t) Proper size of horizon d H r hor R( t) R( t) c t dt R( t) We did a calculation to show that the horizon at the last scattering surface subtends hor 1.7 deg on the sky for an Einstein-de Sitter Universe Physical Cosmology 11/1
Let s do the calculation again in an even simpler way d H hor R( t ) Assume a matter-dominated universe r R( t ) c t dt R( t) R t / 3 so d H t t /3 dt t /3 3ct d H 41 5 pc size of horizon at t 41 5 years Two points d H apart at z~11 will now be separated by a distance (11+1) d H ~ 4Mpc because the universe has expanded. Physical Cosmology 11/1
d H ~ 4, pc t=t, z~11 4Mpc D t=13.7 billion years (today) and z= We are at. θ D 41 5 pc 9 13.31 years 5 41 years 13.31 9 pc so 4 Mpc 13 Mpc 1.8 Physical Cosmology 11/1
Any two points on the sky at z~11, 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. Time Original small region within particle horizon Physical Cosmology 11/1 Start of inflation End of inflation Region that eventually ends up being our observable universe
The Horizon Problem
II The Flatness Problem H 8G 3 kc R c 3 -[1] 1 8G 3H kc R H c 3H 1 mat kc R H 1 R k c H where mat Equation [1] H ρ when the k and Λ terms are small (i.e. when radiation or matter dominates). Physical Cosmology 11/1
so when radiation - dominated and when matter - dominated when radiation - dominated Physical Cosmology 11/1 R H t 1 t 1 /3 /3 when matter - dominated 1 t 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 R HR (H ) dt dt R so R H increases with time and therefore k c 1 decreases with time H R stable inflation solves the flatness problem R H t
Need either tot, = 1 or tiny deviation from tot, = 1 in early Universe: R Friedmann equation: 8 G 3 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 ( const. ) tot vac tot R k c Physical Cosmology 11/1
prediction k for most inflationary models k Can have 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 11/1
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 1 15 GeV this corresponds to a monopole space density comparable to the baryon density nmonopole n bar and monopoles would have a mass mmonopole 1 11 This would correspond to 16 monopole 1 kg Physical Cosmology 11/1
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 6 e-foldings (i.e. universe expands by a factor of e 6 ). Physical Cosmology 11/1
recall R R 4G 3 3p c R (i.e. we have inflation) if 3p c p wc w 1 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 11/1
What is responsible for inflation? Simple versions of inflation are described by a scalar field with p c 1 1 kinetic energy V ( ) V ( ) potential energy 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 11/1
T > T GUT I Inflation II V(T) V ( ) vac c c G vac Reheating III T < T GUT IV V(T) ~ V ( ) Physical Cosmology 11/1
inflation scale factor During inflation the scale factor grows rapidly while matter density and temperature decreases rapidly. t temperature At the end of inflation the vacuum energy density is first transformed into kinetic energy of the scalar field which is then transformed into heat and particles (reheating). Physical Cosmology 11/1
Inflationary theory candidates The Higgs Field GUTs Supersymmetry String theory Brane Theory Physical Cosmology 11/1
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. p c vac const. Physical Cosmology 11/1
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 11/1
But: The value of necessary to drive inflation at the GUT scale is some factor 1 8 bigger than today Why? Most cosmologists see this as the biggest unsolved problem in cosmology. Physical Cosmology 11/1