Holographic Cosmology Mark Trodden Beyond Inflation? University of Pennsylvania

Workshop: Status and Future of Inflationary Theory! University of Chicago, August 22-24, 2014 Questions

Haven’t been thinking about inflation too much in recent years, but BICEP2 claim has rekindled my interest, like most people. Raises a number of old questions. ! Some are initial conditions questions (I’m surprised to see many people with related questions here):

• To what extent should we worry about the initial conditions for inflation? • How should we picture space, time, dimensions and field content prior to inflation? • Within an effective field theory treatment, it may be that inflation requires homogeneity over a large patch, or other unlikely conditions, to begin. • This can make the probability low. • If we have a UV complete theory (such as string theory) can we address this question? Are there non-statistical answers? (Do we understand accelerating solutions even?) • Possibly inflation itself can overcome this with sufficient volume enhancement. • Expect (simple arguments; Dvali; Arkani-Hamed et al.) that there are limits on the classical number of efolds. Does this affect this initial condition calculation? • Do we need eternal inflation?

Holographic Cosmology Beyond Inflation Mark Trodden, U. Penn Eternal Inflation?

V( )

V2

V1

• Is eternal inflation a done-deal? Do we trust the naive intuition that it happens? • In the usual picture, it involves a quantum fluctuation in the inflaton, up the potential, over an extended region. • When is this within the semiclassical approximation? (See Creminelli, Dubovsky, Nicolis, Senatore & Zaldarriaga arXiv:0802.1067 [hep-th]). • If not, is there a way to understand it within a Quantum Gravity model, such as string theory? (Has implications beyond just the early universe). • Recent evidence to the contrary in 1+1d (Martinec & Moore) • How might we answer this generally?

Holographic Cosmology Beyond Inflation Mark Trodden, U. Penn Other Models?

• What do we really know about the space of theories that can simultaneously

• Solve the flatness and horizon problems?

• Generate the observed spectrum of density perturbations?

• Make predictions for upcoming measurements (such as B-modes)

Motivated by these questions, been trying to learn more about frameworks in which might be useful to ask them. One possibility: holographic cosmology (using AdS/CFT).

Holographic Cosmology Beyond Inflation Mark Trodden, U. Penn Possibilities for the Early Universe

• When we think about early universe, we extrapolate GR coupled to sources back into epochs of increasingly high curvature, and decreasing control.

• It is remarkable that we have fascinating possibilities to address our problems, and explain structure, such as inflation, in the regime in which we know how to calculate.

• Understanding the space of alternatives is difficult if we restrict ourselves to our perturbative, weakly-coupled description.

• Do we need the entire theory of quantum gravity to expand possibilities?

• One thing we’ve learned is that, in some cases, there exists a weak/strong coupling duality between gravity and field theory (AdS/CFT). Is there a way to use this to learn more?

• Obviously many people have considered this. One concrete proposal is to use the domain wall/cosmology connection (McFadden & Skenderis).

Holographic Cosmology Beyond Inflation Mark Trodden, U. Penn Domain Wall Cosmology

Holographic RG Flow

The domain wall/cosmology The AdS/CFT correspondence correspondence

QFT (CFT) Cosmology

Here, “domain wall” just means “having a radially-dependent profile in the bulk”.

Holographic Cosmology Beyond Inflation Mark Trodden, U. Penn Domain Wall Cosmology

In cosmology, used to seeing the metric 2 2 2 i j ds = ⌘dz + a(z) ijdx dx =(z)

where ⌘ = 1 and z representing the time coordinate. If we instead consider ⌘ =+1 , then this describes a (Euclidean) “domain wall” (and if we made one of the x-coords timelike, it would be a Lorentzian one)

1 S = ⌘ d4x g R +(@)2 + V () | | 22  Z p So, for every flat FRW solution with potential V there exists a related domain-wall solution with potential −V

This holds, for both background solutions, and linear perturbations. These solutions their properties can then be, in principle, related to behavior of boundary field theory.

For inflationary cosmologies, has been used to relate power spectra and holographic 2-point functions (and, ultimately, higher-point functions). [McFadden & Skenderis]

Holographic Cosmology Beyond Inflation Mark Trodden, U. Penn For Example

Asymptotically AdS The domain wall/cosmology The AdS/CFT correspondence correspondence Domain Walls

Asymptotically dS CFT w/ dual to scalar Cosmology having a VEV

Different asymptotics & bulk slicings could probe (all?) other possible cosmologies

Holographic Cosmology Beyond Inflation Mark Trodden, U. Penn Inflation

We think of inflation as a solution to GR coupled to a fluid or scalar field so that 3M 2 H = ⇢,⇢ cst ds2 = dt2 + eHtd~x2 P ⇡ with H approximately constant

If it was precisely de Sitter, then would have symmetry of dS group. But because quasi-dS, this is broken to the Poincaré algebra so(1, 4) iso(3) !

4 We use the fact that the spaces dS 4 , H , AdS 4 are closely related

Holographic Cosmology Beyond Inflation Mark Trodden, U. Penn Holographic Cosmology

2 2 H 2 i j ds = ( d⌘ + ijdx dx ) dS4 ⌘2 These can be mapped into each other via L2 2 2 i j L=iH,z=i⌘ x1=ix0 ds 4 = (dz + ijdx dx ) 2 2 2 H 2 ds ds 4 ds z dS4 ! H ! AdS4 L2 ds2 = (dz2 + ⌘ dxµdx⌫ ) AdS4 z2 µ⌫

Corresponding symmetries O(1, 4) O(1, 4) O(2, 3) ! !

This allows us to model quasi-dS by a non-unitary Euclidean CFT deformed by relevant operators. In this picture, slow roll-parameters get related to field theory parameters; e.g. 2 2 =2MP ✏ Thus, in the boundary QFT (and hence in the bulk) this realizes the required pattern

so(1, 4) iso(3) !

Holographic Cosmology Beyond Inflation Mark Trodden, U. Penn Can we Extend to Other Models?

[K. Hinterbichler, J. Stokes and M.T., arXiv:1408.1955 [hep-th]] Are there dualities like this for any possible evolution of early universe? Might provide alternative way to probe uniqueness of inflation as solution to questions of early cosmos.

First example, is there a bulk (gravitational) dual to the Pseudo-Conformal Universe? [K. Hinterbichler & J. Khoury] Works by SSB of conformal group in 4D to subalgebra isomorphic to dS isometries. Postulate early universe described by a CFT w/ scalar primary operators w/conformal OI dimensions I in Minkowski space ! Assume operators develop t-dep VEVs of form cI I =

SSB pattern means spectator fields w/ vanishing conformal dimension acquire scale-invariant spectra

Holographic Cosmology Beyond Inflation Mark Trodden, U. Penn For duals to inflation, physics of interest is in bulk and dual is boundary CFT. Here, physics of interest is non-gravitational CFT on boundary, and dual is bulk gravitational theory. ! We’re constructing bulk dual to a CFT in the pseudo-conformal phase.

[arXiv:0909.0518]

Holographic Cosmology Beyond Inflation Mark Trodden, U. Penn Holography of the PCU

We’re looking for a gravitational dual that realizes the breaking so(2, 4) so(1, 4) ! Important point is, of course, that generators of so (2 , 4) and generators of AdS 5 are in one-to-one correspondence. Start simply, with CFT in conformal vacuum. We know gravity dual is exact AdS space 2 2 2r µ ⌫ r ds = dr + e ⌘µ⌫ dx dx ,L=1,z= e Killing vectors are P = @ µ µ L = x @ x @ µ⌫ µ ⌫ ⌫ µ 2 ⌫ 2 Kµ = x @µ +2xµx @⌫ z @µ +2xµz@z µ D = x @µ + z@z so(2, 4) iso(1, 3) If we turn on a constant VEV in the field theory, then this will break ! Bulk invariance under remaining symmetries then implies: P = L =0 = = (r) µ µ⌫ )

Holographic Cosmology Beyond Inflation Mark Trodden, U. Penn Domain Wall Cosmology for the PCU

So a constant VEV corresponds to a domain wall spacetime 2 2 2A(r) µ ⌫ ds = dr + e ⌘µ⌫ dx dx ,= (r) This will be asymptotically AdS if A ( r ) r as r ! ! 1 Consider a VEV that breaks so(2, 4) so(1, 4) ! There is a dS subalgebra of AdS, spanned by D, Pi,Lij,Ki P = L = D =0 = = (x) x = t/z i ij ) Metric can then be written dx2 dx dt 1 ds2 = 2 + (1 x2)dt2 + x2 dxidxj x2 xt t2 ij Which can be diagonalized as ⇥ ⇤ d⌘2 + dxidxj ds2 = d⇢2 +sinh2 ⇢ ij ⌘2 

Which is AdS 5 in the dS 4 slicing

Holographic Cosmology Beyond Inflation Mark Trodden, U. Penn Penrose Diagrams

t = 1 Global AdS cylinder; dS slice coordinate region is bounded from above by lightcone running downward from t=0, and from below by slanted ellipse, also marking lower boundary of Poincaré patch (upper slanted ellipse is upper boundary of Poincaré patch).

= 1 z t = 1 =0 z

⌘ =0 t =0

=0 ⇢ 1

t =0 = ⇢

⌘ = t = 1 1

t = 1 2d slice down axis of AdS cylinder; thin lines are Poincaré lines of constant z and t, thick lines de Sitter slice lines of constant ⇢ and ⌘

Holographic Cosmology Beyond Inflation Mark Trodden, U. Penn Check

Killing vectors for AdS 5 in the dS 4 slicing

p = @ i i j = x @ x @ , ij i j j i k = ~x 2@ +2x xj@ + ⌘2@ +2x ⌘@ i i i j i i ⌘ i d = x @i + ⌘@⌘ 1 K = @ + coth ⇢@ + ⌘ ⇢ ⌘

1 2 2 2 2 i K = ( ⌘ + ~x )@⇢ +(⌘ + ~x ) coth ⇢@⌘ +2⌘ coth ⇢x @i ⌘ 1 K = x @ + x coth ⇢@ + ⌘ coth ⇢@ i ⌘ i ⇢ i ⌘ i

pi ,jij ,ki ,dgenerate so(1, 4)

Ansatz preserves dS isometries = (⇢)= p = j = k = d =0 ) i ij i

Holographic Cosmology Beyond Inflation Mark Trodden, U. Penn Including Back-Reaction

More generally, a metric consistent with the unbroken symmetries is 2 2 2A(⇢) 2 ds = d⇢ + e dsdS4 ,= (⇢) Where asymptotic AdS requires A(⇢) log sinh ⇢ as ⇢ ! !1 We want a domain wall with dS 4 leaves in asymptotic AdS 5 Action is 1 1 1 S = d5xp g R gMN@ @ V () + d4xphK 4 2 M N 2 Z  Z Give scalar potential a negative extremum at the origin 1 V f V = 3+ m22 + (3) -3.00 2 O -3.02 Background fields then obey -3.04H L -3.06 @V -3.08 00(⇢)+4A0(⇢)0(⇢)= -3.10 @ -3.12 -3.14 2 2A(⇢) 1 2 f 3A0(⇢) 3e = 0(⇢) V -0.2 0.2 0.4 0.6 0.8 2

Holographic Cosmology Beyond Inflation Mark Trodden, U. Penn Simplest thing we could do is merely to choose V=-3. EOMs become 4A(⇢) 2 2A(⇢) 8A(⇢) 0(⇢)=ce ,A0(⇢) =1+e + be

Metric can be written as b = c2/6 du2 1 d⌘2 + d~x2 ds2 = + u = e 2A(⇢) 4u2(1 + u + bu4) u ⌘2  Asymptotically: 4⇢ (⇢) ↵ + e ' f r A r 10 2 8 H L H L 1 6 r 4 0.5 1.0 1.5 2.0 2.5 3.0 -1 2

r -2 0.0 0.5 1.0 1.5 2.0 2.5 3.0

Holographic Cosmology Beyond Inflation Mark Trodden, U. Penn 1-Point Functions etc.

The holographic dictionary ! • There exists a 1-1 correspondence: local gauge invariant operators of boundary QFT bulk modes. • Bulk metric corresponds to energy momentum tensor of boundary theory. • Bulk scalar fields correspond to scalar operators in boundary theory. • Can obtain correlation functions of gauge invariant operators from asymptotics of bulk solutions.

Want to determine VEVs of the stress energy tensor and the operator dual to the scalar field. ! Correlation functions of gauge invariant operators can be extracted from asymptotics of bulk solutions. ! Or, given correlation functions of dual operators, can reconstruct asymptotic solutions.

To perform holographic computations need some knowledge of structure of asymptotic solutions of field equations.

Holographic Cosmology Beyond Inflation Mark Trodden, U. Penn Fefferman-Graham Expansion

Any asymptotically AdS space admits expansion of form

dz2 + g (x, z)dxµdx⌫ ds2 = µ⌫ z2 g (x, z)=g (x)+z2g (x)+z4g (x)+h (x)z4 log z2 + . µ⌫ (0)µ⌫ (2)µ⌫ (4)µ⌫ (4)µ⌫ ··· 4 (x, z)= (x)+z (x)+ (massless scalar) (0) (4) ··· The GKPW prescription is

d4x (x) (x) I [ ] e O 0 = e string 0 h R i on-shell Dual operator marginal =2+ 4+m2 m=0 =4 Holographic renormalizationp (most of the paper) gives 1-point functions =(4 2) , T = g hOi (4) h µ⌫ i (4)µ⌫

Holographic Cosmology Beyond Inflation Mark Trodden, U. Penn How Does This Work?

Can put our metric du2 1 d⌘2 + d~x2 ds2 = + 4u2(1 + u + bu4) u ⌘2  in FG form. e.g. for b non-zero define

4 t 2 2 2 2 1+u + bu = + bf1(x)+ (b ) ⌘ = t z + bzf2(x)+ (b ) pt2 z2 O O p p with 2 4 2 2 2 1 xC1 2+9x 6x 6x 1+x log 1+ x2 f1(x)= + (1 x2)3/2 8x ( 1+x2)7/2 ⇥ ⇤ 2 4 3 log 1+ 1 1 2 17 42x + 24x 4iC1 1 x2 f2(x)= 1+x +4C2 + 4 log 1 + 2 3 2 2 2 4 6( 1+x ) 1+x x ⇥1+x ⇤! p 

Then transformed metric is of FG form to leading order in b

dz2 1+ (b2) + 1+bc (x)+ (b)2 dt2 + 1+bc (x)+ (b)2 d~x2 2 O 3 O 4 O ds = 2 ⇥ ⇤ ⇥ z ⇤ ⇥ ⇤

Holographic Cosmology Beyond Inflation Mark Trodden, U. Penn FG Form Again

Form is fixed by unbroken O(1,4) symmetry b z 8 b z 8 g (z/t)= 1+ + (z/t)10 g (z/t)=1 + (z/t)10 tt 8 t O 11 8 t O ⇣ ⌘ ⇣ ⌘ T =0 h µ⌫ i Scalar equation is solved in FG coordinates by c z 4 = + (z/t)6 (0) 4 t O ⇣ ⌘ Setting =4 yields c = 4 = hOi (4) t4 cI which is correct form required by O(1,4) symmetry: I = hO i tI

cI [N.B. similar to interface CFT’s which break O (2 , 4) O (2 , 3) using I = ] ! hO i yI

Holographic Cosmology Beyond Inflation Mark Trodden, U. Penn Conclusions & Comments

• Have shown how to get a gravitational dual to one suggested alternative to inflation. • Should also be possible to calculate other correlators (doing that now)

(x) (y) Tµ⌫ (x) (y) T (x)T (y) hO O i h O i h µ⌫ ⇢ i

• Ward identities (Justin’s talk) have been computed, should be able to check them. • Admits a supergravity embedding. • Could consider holographic dual to turning on 4D gravity in boundary theory. • Dynamical gravity could be incorporated in boundary by moving boundary slightly into bulk, to a cut-off surface. • Local counter-terms now regarded as finite kinetic terms for graviton. • Was just an example of how dualities extend beyond the usual application to inflation. A hope is that they might provide a route to understanding other possible early universe models. Thank you for listening… … and a big thank you to our organizers See poster by James Stokes at COSMO-2014

Holographic Cosmology Beyond Inflation Mark Trodden, U. Penn