Effects of the field-space metric on Spiral Inflation

Transcription

1 Effects of the field-space metric on Spiral Inflation Josh Erlich College of William & Mary digitaldante.columbia.edu Miami 2015 December 20, 2015

2 The Cosmic Microwave Background Planck collaboration

3 Composition of the Universe

4 26%: Dark Matter Composition of the Universe

5 26%: Dark Matter Composition of the Universe 69%: Dark Energy

6 26%: Dark Matter Composition of the Universe 69%: Dark Energy 5%: Known Stuff

7

8 Polarization from Last Scattering Different spectrum from hot and cold region leads to net polarization.

9 Cosmological Inflation Inflation solves a number of cosmological puzzles, including: Horizon Problem: How could regions of the universe that were never in causal contact have come to be so homogeneous? Flatness Problem: How did the universe come to be so flat at the Big Bang?

10 Cosmological Inflation Inflation solves a number of cosmological puzzles, including: Horizon Problem: How could regions of the universe that were never in causal contact have come to be so homogeneous? Inflation s answer: They didn t. Flatness Problem: How did the universe come to be so flat at the Big Bang?

11 Cosmological Inflation Inflation solves a number of cosmological puzzles, including: Horizon Problem: How could regions of the universe that were never in causal contact have come to be so homogeneous? Inflation s answer: They didn t. Flatness Problem: How did the universe come to be so flat at the Big Bang? Inflation s answer: Inflation drives the universe to be flat.

12 Hubble parameter H ȧ a Slow-Roll Inflation: The Basics FRW equations describe expansion as a function of pressure and density of fluids in the universe. ȧ 2 = 1 a 3 k a 2 ä a = 1 6 ( +3p) acceleration if p < 1 3

13 Slow-Roll Inflation: The Basics (Linde; Liddle, Lyth) From definition of H: ä a = H2 (1 ) where Ḣ H 2 Requirement for accelerated expansion: < 1 Hence, inflation ends when. =1

14 Slow-Roll Inflation: The Basics Uniform scalar field coupled to gravity in flat FRW universe: Einstein s equations H 2 = V ( ) 3 2 Scalar field equation Using results on previous slides: +3H + V 0 ( )=0 = H 2

15 Slow-Roll Inflation: The Basics Slow-roll inflation requires 1 2 V ( ) Inflation typically ends too quickly unless V 0 ( ), 3H 1 where H Slow-roll Conditions

16 Slow-Roll Inflation: The Basics In the slow-roll regime, the slow-roll parameters can be expressed in terms of the shape of the potential around the value of the inflaton field: M 2 P 2 V 0 ( ) V 2 M 2 P V 00 ( ) V

17 Slow-Roll Inflation: The Basics In the slow-roll regime, the slow-roll parameters can be expressed in terms of the shape of the potential around the value of the inflaton field: M 2 P 2 V 0 ( ) V 2 M 2 P V 00 ( ) V Reduced Planck Mass M P = GeV

18 Slow-Roll Inflation: The Basics Number of e-folds: N( i ) ln a end = 1 Z p a i 2 i end d p Solving the horizon and flatness problem typically requires the number of e-folds during inflation to be larger than 40-60, but is sensitive to the post-inflationary dynamics.

19 Cosmological Perturbations The Cosmic Microwave Background carries information about the primordial gravitational and field perturbations produced during inflation. Curvature perturbations: hr k R k 0i = 16 5 k 3 (k + k 0 ) Scalar tilt: n s 1+ d ln d ln k Running of scalar tilt: 2 s 2 s(k) n r dn s d ln k (Similar definitions for tensor perturbations)

20 Cosmological Perturbations Spectrum of perturbations depends on shape of the inflaton potential: 2 s(k) = 1 H MP 2 2 t (k) = 2 H 2 2 MP 2 1 k=khor k=k hor r 2 t 2 s = 16 (at time of horizon crossing)

21 The Lyth Bound is related to the rate of change of the inflaton with respect to the number of e-folds. Z NCMB r r r rcmb = M P dn N M N end 8 8 P change in inflaton typically O(1) r 1/ MP Problem: If r>0.01 then the inflaton is super-planckian during inflation.

22 Inflationary Observables In single-field slow-roll inflation, observables may be calculated in terms of slow-roll parameters: = M 2 P 2 V 0 V 2, = M 2 P 2 s V 00 V, 1 V 24 2 M P n s =1+2 ( CMB ) 6 ( CMB ) n t = 2 ( CMB ) r = 16 ( CMB ) CMB = M 4 P V 0 V 000 n r = 16 ( CMB ) ( CMB ) 24 ( CMB ) 2 2 ( CMB ) V 2

23 Canonical example: Single-Field chaotic inflation with monomial potential V ( )= p n s =0.96! N = 49 4 p + 25 With N=60, p=20/7! r =0.188 (Excluded)

24 Planck 2015 Measured Observables

25 Axions in Inflation Axions are popular inflaton candidates because they can give naturally flat potentials. Natural inflation (Freese, Frieman, Olinto ) Little inflatons (Kaplan, Weiner ) Pseudonatural inflation (Arkani-Hamed, Cheng, Creminelli, Randall -2003) Aligned natural inflation (Kim, Nilles, Peloso ) N-flation (Dimopoulos, Kachru, McGreevy, Wacker )... Single-axion potential: V ( ) = 4 [1 cos( /f)]

26 Axion Monodromy Inflation If primordial gravitational waves are discovered with current sensitivity of CMB polarization measurements, then the Lyth bound causes challenges with effective field theory. There are several ways to get around the Lyth bound. A popular one is axion monodromy inflation. (Silverstein,Westfall,McAllister ) If there is a monodromy as the axion changes by 2 f, then a super-planckian shift in the axion field might be replaced with a smaller sub-planckian shift of some other field.

27 Dante s Inferno Berg, Pajer, Sjors ; similar to Kim, Nilles, Peloso Two axions r, θ - with canonical kinetic terms Typically large power in tensor modes. Breaks discrete shift symmetry, f r f Trajectory r f r f 1/n

28 Dante s Waterfall Carone, JE, Sensharma, Wang Inflation ends when trajectory becomes unstable, as in hybrid inflation (Linde ). Small power in tensor modes. symmetry-breaking type potential

29 Dante s Waterfall JE, Carone, Sensharma, Wang r θ Contour plot of potential, with trench and solution to field equations.

30 Spiral Inflation McDonald ; Barenboim, Park Complex scalar = r p 2 e i single axion θ, not canonically normalized same as Dante s Inferno potential p g L = p g apple 1 2 (@ µr) r2 (@ µ ) 2 V (r, )

31 What general statements can be made about the effect of nontrivial kinetic terms? e.g. does field-space curvature tend to increase/decrease power in tensor modes?

32 From several fields to one Spiral/Dante-type models behave like single-field models as long as the trench is steep enough to prevent isocurvature fluctuations. (Some experts disagree.) Field-space metric Suppose we know the trajectory in field space, parameter I along the trajectory. a (I), with Choose I to satisfy

33 From several fields to one Single-field effective description The field I plays the role of a canonically normalized inflaton field. The field-space metric only affects V(I).

34 Mass matrix for spiral inflation JE, Olsen, Wang To calculate the slow-roll parameters we can use the single-field description, but some authors prefer to diagonalize a mass matrix to determine V (I). Instantaneous inflaton direction in polar coordinates: c r c

35 Mass matrix for spiral inflation Note: This is not the same as the matrix of pairs of covariant derivatives on V. That would not take into account the changing direction of the inflaton.

36 Summary Inflation models which differ only in the form of kinetic terms are common in the literature. By a field redefinition, these terms can be absorbed in the potential (at least locally), but at the same time they can motivate unusual potentials, and can render a model non-viable. This is what happens in certain classes of spiral inflation models.