What is Inflation Good For?

1. What is Inflation Good For? Sean Carroll (w/ Heywood Tam) Caltech Inflation is supposed to make the state of the (early) universe seem “natural.” What does that really mean?

2. Consider the flatness problem. In an FRW universe dominated by “matter” with a-n, we define density parameters so the relative importance of curvature grows with time, Compared to conditions at the Planck scale, we have

3. So the flatness problem is: “If curvature were at all comparable to matter at early (Planckian) times, it would be enormously larger than matter today.” But … should curvature be comparable to matter early on?

4. So the flatness problem is: “If curvature were at all comparable to matter at early (Planckian) times, it would be enormously larger than matter today.” But … should curvature be comparable to matter early on? We might claim to have no idea -- so “comparable” is as likely as anything else. But that implicitly assumes a (flat) measure. Without assuming some sort of measure on the space of possibilities, we can never say a condition is “unnatural.”

5. This is a question of (a particular example of) classical mechanics. Evolution in GR with some set of fields can be thought of as Hamiltonian evolution in an n-dimensional phase space. This phase space comes equipped with the Liouville measure: phase space = {pa , qa}

6. This is a question of (a particular example of) classical mechanics. Evolution in GR with some set of fields can be thought of as Hamiltonian evolution in an n-dimensional phase space. This phase space comes equipped with the Liouville measure: Liouville’s Theorem: measure is conserved under time evolution. phase space = {pa , qa}

7. For cosmology we don’t care about the measure on phase space; we want a measure on the space of entire histories of the universe. phase space = {pa , qa} histories

8. For cosmology we don’t care about the measure on phase space; we want a measure on the space of entire histories of the universe. In fact, we want to compare different trajectories that satisfy the Hamiltonian constraint (H = 0), which is just the Friedmann equation. phase space = {pa , qa} histories H = 0 surface

9. t H For cosmology we don’t care about the measure on phase space; we want a measure on the space of entire histories of the universe. In fact, we want to compare different trajectories that satisfy the Hamiltonian constraint (H = 0), which is just the Friedmann equation. Gibbons, Hawking & Stewart (1986) showed how to do this. Just choose qn = H, pn = t. Then the induced measure on the space of trajectories is phase space = {pa , qa} histories H = 0 surface

10. t H Crucial feature: the measure on histories is local in phase space. To evaluate it, we integrate over a surface transverse to the Hamiltonian flow. phase space = {pa , qa} histories Transverse surface H = 0 surface

11. t H Crucial feature: the measure on histories is local in phase space. To evaluate it, we integrate over a surface transverse to the Hamiltonian flow. The result will be independent of what transverse surface is chosen. Thus, no choice of Hamiltonian will change the measure in between the early universe and today. Alternatively, no choice of Hamiltonian can turn “unnatural” configurations into “natural” ones. phase space = {pa , qa} histories Transverse surfaces H = 0 surface

12. Apply this to the flatness problem. Simple model: massless scalar field, with Coordinates include the lapse function , the scale factor , and the scalar field . Conjugate momenta are We can set and use the Friedmann equation to solve for . We can evaluate the measure on surfaces of constant , as a function of . We end up with

13. To make this more physically transparent, change variables from to . Then the measure becomes

14. To make this more physically transparent, change variables from to . Then the measure becomes Startling result: the measure diverges at = 0!

15. To make this more physically transparent, change variables from to . Then the measure becomes Startling result: the measure diverges at = 0! In the rigorous measure on trajectories, almost all Robertson- Walker universes are flat. The flatness problem does not exist. 

16. What about the horizon problem? (Don’t worry, it exists.) Consider a mode k of a linearized scalar perturbation. In terms of the Mukhanov-Sasaki variable v and its conjugate momentum pv, the Hamiltonian is Here, cs is the speed of sound, and -z”/z is a time-dependent mass2 that is negative in a matter-dominated universe. It’s just an harmonic oscillator with time-dependent mass. The measure is simply dpv dv. There are no divergences; nothing favors small perturbations. The horizon problem is real.

17. So what does inflation have to say about the horizon problem? We assume that our comoving patchevolvesautonomously -- as an approximately-closed system. This evolution is unitary (reversible), with a fixed space of states. The subset we might call “smooth states” -- those described by quantum field theory on a smooth spacetime background -- grows as the patch expands (LP <  < aH0-1.) At early times, the vast majority of states are not smooth states. They are wild, Planckian, stringy, etc. [cf. Mathur]

18. Inflation does not explain why the universe is relatively smooth; it assumes the universe was extremely smooth right from the start. Decompose the Hilbert space of fields in our current observable universe into long-wavelength and short-wavelength modes: If we choose the cutoff to be c = 0.1 Mpc (EI /1015 GeV)-4, then all short modes were trans-Planckian at the onset of inflation. They had to be in their vacuum state at the start, or inflation would not have occurred. H(in)= H(l)0 H(s). unique vacuum state for short- wavelength modes [e.g. Vachaspati & Trodden]

19. Think in terms of entropy. Inflation sounds simple: a tiny patch of smooth space, where potential energy dominates gradient energy. But that’s a lower-entropy (more finely-tuned) initial condition than conventional Big-Bang cosmology. HI-1 Sinflation ~ SdS ~ (MP/EI)4 ~ 1012 << SCMB ~ 1088 ! Entropy counts the number of microstates: S = k log A proto-inflationary patch is less likely to arise “randomly” as a configuration of the universe’s degrees of freedom than the usual hot Big Bang cosmology. [Penrose]

20. What inflation actually does Inflation doesn’t turn a large number of initial states into a small number of final states -- Liouville’s theorem still holds. all states of our observable degees of freedom [exp(10120)] states like our current universe [exp(10100)] What inflation does do is to create a class of smooth states -- those dominated by the potential of an “inflaton” field -- that naturally evolve into something like our current universe. Still need to explain how the universe got into such a state! H = H0 with inflation without inflation H = HI proto-inflationary states

21. Inflation does not: • explain flatness (there’s no need to) • explain homogeneity, isotropy (low entropy) • remove sensitive dependence on initial conditions.On the other hand, inflation does: • produce scale-free primordial density perturbations • create a lot of particles from almost nothing • specify a set of states that evolve into nice universes. • Inflation is definitely worth taking seriously. • But it does not remove the need for a theory of initial • conditions: it makes that need more urgent than ever!