Holography for the Pseudo-Conformal Universe

Kurt Hinterbichler (Perimeter Institute)

KH, Justin Khoury arXiv:1106.1428 (JCAP 070 1011) KH, James Stokes, Mark Trodden arXiv:1408.1955 (JHEP 073 1014)

Cosmo Cruise, Sept. 8, 2015

Inflation

Smoothness, flatness, monopole problems: Other possible components with w > -1 are emptied out

3M2PH

2 = ⇢V � 3M2P k

a2+

⇢m,0

a3+

⇢r,0a4

+⇢A,0

a6

curvature matterradiation anisotropies

! 0constant

Inflation ≈ exponential expansion ≈ de Sitter space

ds

2 = �dt

2 + a(t)2dx2 a(t) = eHt,

Driven by vacuum energy with w ≈ -1

3M2PH

2 = ⇢V

⇢ ⇠ a�3(1+w)

constant

,

Dominant paradigm for the very early universe

Inflation is rooted in symmetries

⌧ = 0

⌧ ! �1

a(⌧) =1

H(�⌧)

de Sitter space is maximally symmetric: There are 10 Killing vectors:

• 3 spatial translations and 3 spatial rotations, forming iso(3)

D = ⌧@⌧ + x

i@i

Ki = 2xi⌧@⌧ ���⌧

2 + x

2�@i + 2xix

j@j

• Plus a dilation and 3 special generators:

Symmetry algebra is: so(4,1)

Symmetry accounts for the observed scale invariance of fluctuations, and relations among observables

Strong coupling: dS/CFT realization of inflation

⌧ = 0

⌧ ! �1

CFT lives at future infinity of dS

• Interested in bulk cosmology: boundary theory is a mathematical trick to calculate bulk observables

boundary CFT correlators cosmological correlation functions

Pseudo-conformal scenarioPseudo-Conformal Framework

• Non-inflationary scenario

• Gravity is relatively unimportant: spacetime is approximately flat

• More symmetric than inflation: CFT with so(4,2) symmetry

• Spontaneously broken: so(4,2) → so(4,1)

• Essential physics is fixed by the symmetry breaking pattern, independently of the specific

realization or microphysics

• Many possible realizations:

⇡

Rubakov’s U(1) modelGalilean Genesis Creminelli, Nicolis & Trincherini, 1007.0027

V. Rubakov, 0906.3693; 1007.3417; 1007.4949; 1105.6230

...

{ model�4 KH, Justin Khoury arXiv:1106.1428

V (�)

�

�(t) ⇠ 1(�t)

MPl

Figure 1: Sketch of the scalar potential for the simplest realization of our mechanism. The potential

is well approximated by a negative quartic form along the solid curve. Higher-dimensional operators

such as O(�6/M2

Pl

) become important along the dotted curve, stabilizing the potential.

16

Simplest example: negative quartic

where M is the single component of the mass matrix. From analogous computations in inflation

it is well-known that the spectrum will be scale invariant provided that M ⌧ 1. Indeed, in this

regime, the solution for the mode functions, assuming the standard adiabatic vacuum, is given by

�k ⇠ e�ikt

p2k

✓1� i

kt

◆, (35)

where once again the mode normalization is not fixed. The long-wavelength spectrum for � is

therefore scale invariant,

k3/2|�k| ' constant . (36)

Hence, under very general assumptions, a conformal weight 0 field acquires a Harrison-Zeldovich

spectrum in our background (6).

An immediate corollary of the above derivation is that the growing mode solution for � is

a constant. Thus perturbations in this field are amplified, but only to a particular finite value.

This is a key di↵erence from other non-inflationary multi-field mechanisms. In the New Ekpyrotic

scenario [30, 32, 33], for instance, the amplification of scale invariant perturbations in a second field

relies on a tachyonic instability [34]. The background solution is therefore unstable to unbounded

growth along this field direction, though it was shown in [32] that an earlier phase of evolution

can bring the field arbitrarily close to the desired trajectory. Similarly, the general two-field

non-inflationary mechanisms of [72] are also tachyonically unstable.

3 An Example: Negative Quartic Potential

The simplest realization of our mechanism relies on a canonical scalar field � of conformal weight 1

rolling down a negative �4 potential [51]. The � part of the action reads

L� = �1

2(@�)2 +

�

4�4 + . . . (37)

This transforms to a total derivative under (3), hence the theory is conformally invariant. Of

course this is only true at the classical level, and we will discuss radiative corrections shortly.

With � > 0, corresponding to negative potential energy, it is easy to see from the beta function

that this theory is in fact asymptotically free.

Although the potential is unbounded from below as it stands, we envision that higher-

dimensional operators — denoted by the ellipses — regularize the potential at large values of �.

14

Classically, this is a conformal field theory, with a field of weight � = 1

KH, Justin Khoury arXiv:1106.1428

p� =2

�t4, w = 1

• Solution has zero energy

• Pressure is non-zero and positive (satisfies the NEC, infinite equation of state)

• Solution is an attractor

⇢� = 0

There is a solution where the field rolls down the negative quartic potential:

� =

p2p

�(�t), t 2 (�1, 0)

• Symmetry breaking pattern: so(4,2) → so(4,1)

Adding gravityCouple minimally to gravity (breaks conformal invariance at 1/MP level):

S =

Zd4x

p�g

✓M

2Pl

2R� 1

2g

µ⌫@µ�@⌫�+

�

4�

4

◆

Solution has zero energy ⇒ spacetime approximately flat Solve the Friedman equations in powers of 1/MP

�(t) ⇡p2p

�(�t)a(t) ⇡ 1� 1

6�t2M2P

H(t) ⇡ 1

3�t3M2P

, ,

Solution is a slowly contracting universe

Approximation is valid in the range

tend ⇠ � 1p�MPl

, �end ⇠ MPl .

�1 < t < tend

⇢� ⇡ 1

3�2t6M2P

, p� ⇡ 2

�t4, w ⇡ 6�t2M2

P

w goes from ≫1 to O(1) as t ranges from -∞ to tend

The field forms a very stiff fluid

Solution to flatness, smoothness problems

There is now a scalar field component with extremely stiff equation of state w ≫ 1

3M2PH

2 = �3M2P k

a2+

⇢m,0

a3+

⇢r,0a4

+⇢A,0

a6+ ⇢�

⇠ 1

t6rapidly increasing

Homogeneous energy density of the scalar washes out everything else

Similar to ekpyrotic cosmology (contracting universe with w >> 1)Khoury, Ovrut, Steinhardt, Turok (2001);Gratton, Khoury, Steinhardt, Turok (2003);Erickson, Wesley, Steinhardt, Turok (2004).

≈ constant

General framework

Start with any CFT with scalar primary operators:

These need not be fundamental fields or degrees of freedom, and a conformal invariant stable ground state need not exist.

�I , I = 1, . . . , N. conformal weight �I

KH, Justin Khoury arXiv:1106.1428

Dynamics must be such that the operators get a VEV:

�I(t) =cI

(�t)�I,

VEV preserves an so(4,1) subgroup of so(4,2):

Symmetry breaking pattern for pseudo-conformal scenario is: so(4,2) → so(4,1)

Distinguishable from inflation?

• Detailed predictions (spectral index, non-gaussianity, etc...) will depend on the realization.

• Pseudo-conformal scenario is more symmetric than inflation

• Symmetries → Ward identities → constraints/relations on observables

• Gravity waves:

Spacetime is not doing very much

Primordial gravity wave amplitude is exponentially suppressed

Predicts scalar/tensor ratio: r ~ 0

If Bicep successors find anything, theory is ruled out

KH, Austin Joyce, Justin Khoury arXiv:1202.6056

AdS/CFT realization

• Realization in a true (strongly coupled) CFT

• Cosmological application of AdS/CFT where we are interested in the boundary.

KH, Stokes, Trodden (arXiv:1408.1955)

AdS/CFT generalities

ratio (for example the interpretation of the result reported by BICEP2 [14]) would provide a crucial

test of the pseudo-conformal proposal.

In this paper, we will be interested in constructing a bulk dual to a CFT in the pseudo-conformal

phase. In the case of dS/CFT and duals to inflation, the physics of interest is in the bulk and the

dual is the boundary CFT. Here, by contrast, the physics of interest is the non-gravitational CFT

on the boundary, and the dual is the bulk gravitational theory.

If a CFT possesses a gravitational dual, then the conformal vacuum corresponds to empty AdS,

and other states of the CFT Hilbert space correspond to activating configurations of fields in the

bulk. These field configurations can break the so(2, 4) isometry group of AdS5

to a subgroup, which

in turn breaks the isometry group of the spacetime through gravitational backreaction. At large

values of the radial coordinate ⇢, however, the warp factor should approach ⇠ e

⇢/R, corresponding

to restoration of the full conformal group at high energies. In general the broken/unbroken isome-

tries of the asymptotically anti-de Sitter background map to the corresponding broken/unbroken

conformal generators of the field theory. It follows that to implement the pseudo-conformal mech-

anism in AdS/CFT, we need a spacetime with the isometries of so(1, 4), which are enhanced to

so(2, 4) at the boundary. Geometrically, this corresponds to a domain-wall spacetime asymptoting

to anti-de Sitter space, where the domain wall is foliated by four-dimensional de Sitter slices. Since

the isometry group of dS4

is so(1, 4), this realizes the required breaking pattern. In the limit in

which backreaction is ignored, and near the boundary, this should revert to AdS5

in the de Sitter

foliation.

In Section 2 we will first consider the simpler case of a probe scalar, ignoring backreaction, in

which the background profile for the scalar should be preserved by a so(1, 4) subgroup of so(2, 4).

We find exact solutions of the wave equation for any value of the mass of the scalar. We then

generalize in Section 3 to the fully interacting Einstein-scalar theory and obtain the background

equations of motion for domain-wall spacetimes which have the appropriate symmetries, providing

an explicit solution for the case of a massless scalar. We determine the VEV of the operators dual

to the metric and scalar field and show that they have the correct form (1.2) to break the conformal

group of the boundary field theory to a de Sitter subalgebra. We review the appropriate holographic

renormalization machinery in Appendix A, in which we calculate the exact renormalized one-point

functions for arbitrary scalar source configurations and boundary metric. We describe the analytic

continuations which relate the our setup to the interface CFTs in Appendix C.

2 Probe Scalar Limit

As a warmup for the full back-reacted problem, we first consider a probe scalar on AdS5

. The

coordinates adapted for working with a dual Minkowski CFT are those of the Poincare patch, in

which AdS5

is foliated by Minkowski slices parametrized by x

µ = (t, xi),

ds

2 =1

z

2

�dz

2 + ⌘

µ⌫

dx

µ

dx

⌫

�. (2.1)

3

Here the radial coordinate is z, which ranges over (0, 1), with 0 the boundary and 1 deep in the

bulk. We have set the AdS radius to unity. The Killing vectors for AdS5

in these coordinates are

P

µ

= �@µ

, (2.2)

L

µ⌫

= x

µ

@

⌫

� x

⌫

@

µ

, (2.3)

K

µ

= x

2

@

µ

� 2x

µ

x

⌫

@

⌫

+ z

2

@

µ

� 2x

µ

z@

z

, (2.4)

D = �x

µ

@

µ

� z@

z

. (2.5)

The first two sets of Killing vectors are the generators of the 4D Poincare subalgebra iso(1, 3)

preserved by constant z slices. Going to the boundary at z = 0, the Killing vectors become the

generators of the 4D conformal group so(2, 4), which is the statement that the isometries of anti-de

Sitter act on the boundary as conformal transformations.

A configuration of a bulk scalar � of mass m will have an expansion near the boundary of the

form

�(x, z) = z

��⇥�

0

(x) + O(z2)⇤+ z

�+⇥

0

(x) + O(z2)⇤, (2.6)

where

�± = 2 ±p

4 + m

2

. (2.7)

(There can be additional logarithmic terms ifp

4 + m

2 is an integer.) The coe�cient �0

(x) is a

source term in the lagrangian of the CFT which sources an operator O of dimension �+

, and the

coe�cient 0

(x) determines the vacuum expectation value (VEV) of this operator [15],

hOi = (2�+

� 4) 0

. (2.8)

We are interested in the case in which there is a VEV of the form (1.2) in the absence of sources,

so that the symmetries are spontaneously broken by the VEV rather than explicitly broken by a

source. Thus we seek configurations for which 0

/ 1/(�t)�+ and �0

= 0.

The field profile we want must preserve the D, P

i

, L

ij

and K

i

conformal generators, which form

the unbroken de Sitter so(1, 4) subalgebra of interest [11]. We seek the most general bulk scalar

field configuration which preserves these symmetries. Preservation of the spatial momentum and

rotations, P

i

and L

ij

, implies that the scalar depends only on t and z,

� = �(z, t) . (2.9)

Demanding invariance under D = �z@

z

� x

µ

@

µ

, gives

z@

z

�+ t@

t

� = 0 , (2.10)

which implies that the field is a function only of the ratio z/t,

� = �(z/t) . (2.11)

4

Here the radial coordinate is z, which ranges over (0, 1), with 0 the boundary and 1 deep in the

bulk. We have set the AdS radius to unity. The Killing vectors for AdS5

in these coordinates are

P

µ

= �@µ

, (2.2)

L

µ⌫

= x

µ

@

⌫

� x

⌫

@

µ

, (2.3)

K

µ

= x

2

@

µ

� 2x

µ

x

⌫

@

⌫

+ z

2

@

µ

� 2x

µ

z@

z

, (2.4)

D = �x

µ

@

µ

� z@

z

. (2.5)

The first two sets of Killing vectors are the generators of the 4D Poincare subalgebra iso(1, 3)

preserved by constant z slices. Going to the boundary at z = 0, the Killing vectors become the

generators of the 4D conformal group so(2, 4), which is the statement that the isometries of anti-de

Sitter act on the boundary as conformal transformations.

A configuration of a bulk scalar � of mass m will have an expansion near the boundary of the

form

�(x, z) = z

��⇥�

0

(x) + O(z2)⇤+ z

�+⇥

0

(x) + O(z2)⇤, (2.6)

where

�± = 2 ±p

4 + m

2

. (2.7)

(There can be additional logarithmic terms ifp

4 + m

2 is an integer.) The coe�cient �0

(x) is a

source term in the lagrangian of the CFT which sources an operator O of dimension �+

, and the

coe�cient 0

(x) determines the vacuum expectation value (VEV) of this operator [15],

hOi = (2�+

� 4) 0

. (2.8)

We are interested in the case in which there is a VEV of the form (1.2) in the absence of sources,

so that the symmetries are spontaneously broken by the VEV rather than explicitly broken by a

source. Thus we seek configurations for which 0

/ 1/(�t)�+ and �0

= 0.

The field profile we want must preserve the D, P

i

, L

ij

and K

i

conformal generators, which form

the unbroken de Sitter so(1, 4) subalgebra of interest [11]. We seek the most general bulk scalar

field configuration which preserves these symmetries. Preservation of the spatial momentum and

rotations, P

i

and L

ij

, implies that the scalar depends only on t and z,

� = �(z, t) . (2.9)

Demanding invariance under D = �z@

z

� x

µ

@

µ

, gives

z@

z

�+ t@

t

� = 0 , (2.10)

which implies that the field is a function only of the ratio z/t,

� = �(z/t) . (2.11)

4

source for in the dual theoryO expectation value in the dual theory: hOi ⇠ 0

,

ρ = 0ρ = π/2

θ = π

τ = π

τ = −π

τ = 0

Figure 25: Poincare patch of AdS3.

X

0 = r cosh�

X

1 =pr

2 � R2 sinh t

X

2 =pr

2 � R2 cosh t

X

3 = r sinh� cos ✓1

X

4 = r sinh� sin ✓1

cos ✓2

... (6.18)

Here t,� 2 (�1,1), r 2 (R,1), ✓i parametrize an n � 3 sphere. For n = 3 we losethe ✓’s and we have X3 = r sinh�. For n = 2 we lose � as well, and we have X0 = r.The Rindler patch is shown in global coordinates in figures 26 and 27.

The metric is

ds

2 = ��r

2 � R2

�d⌧

2 +R2

r

2 � R2

dr

2 + r

2

d�

2 + r

2 sinh2

� d⌦2

(n�3)

. (6.19)

6.7 Foliations

• AdS

n can be foliated with AdS

n�1 time-like slices (E1 for n = 2), given byintersecting the planes X2 = const. with the hyperbola. The adapted metric is

36

τ = 0

τ = π

τ = −π

ρ = −π/2 ρ = π/2

t → −∞

t → ∞

u → ∞

u → 0

t = 0

Figure 24: Poincare patch of AdS2 in the Penrose diagram. The change of coordinatesis u

R = cos ⇢cos ⌧+sin ⇢ ,

tR = sin ⌧

cos ⌧+sin ⇢

6.6 Rindler (BTZ) coordinates

Rindler coordinates cover the region X

0

> R, X2

> 0, known as the Rindler

patch

35

Poincare coordinates:

z ! 1z ! 0

The dual region

t = 0

t = 1

t = �1⌘ = �1

⌘ = 0

z=

0

z=1

⇢=

1

⇢=0

t = 0

t = �1

t = 1

• Dual AdS state should be a configuration which is constant on a foliation of AdS5 by dS4 leaves:

KH, Stokes, Trodden (2014)

-∞

0

Bulk solutions

This is now automatically invariant under the spatial special conformal generators, K

i

,

K

i

� = �2x

i(z@

z

+ t@

t

)� = 0 . (2.12)

Thus, a profile �(z/t) is the most general one which preserves the required so(1, 4) symmetries.

It will be useful to work in coordinates adapted to the unbroken so(1, 4) symmetries. Define a

radial coordinate ⇢ 2 (0, 1) and a time coordinate ⌘ 2 (�1, 0) by the relations

t = ⌘ coth ⇢, z = (�⌘) csch ⇢,

⇢ = cosh�1

(�t)

z

�, ⌘ = �

pt

2 � z

2

. (2.13)

In these coordinates, the metric becomes

ds

2 = d⇢

2 + sinh2

⇢

�d⌘

2 + d~x

2

⌘

2

�, (2.14)

which we recognize as the foliation of AdS5

by inflationary patch dS4

slices. These coordinates

cover the region t < 0, (�t) > z, which is the bulk causal diamond associated with the time

interval t = (�1, 0). This is the region we expect to be dual to the pseudo-conformal scenario1.

The boundary is approached as ⇢ ! 1, and the line (�t) = z is approached as ⇢ ! 0. Near

the boundary, the coordinate ⌘ corresponds with t, and ⇢ approaches e

⇢ ' 2 (�t)

z

. The region and

coordinates are illustrated in Figure 2.

We now consider the scalar wave equation in these coordinates. The equation of motion for a

minimally coupled real scalar field of mass m on curved space is

(⇤(5) � m

2)� = 0 . (2.15)

In the de Sitter adapted coordinates, a scalar configuration with the desired profile (2.11) is one

which depends only on the ⇢ coordinate. The wave equation then reduces to

�

00(⇢) + 4 coth ⇢ �

0(⇢) � m

2

�(⇢) = 0 . (2.16)

The general solution is

�(⇢) = C

+

e

�p4+m

2⇢

⇣p4 + m

2 + coth ⇢

⌘csch2

⇢

p4 + m

2

+C�e

p4+m

2⇢

⇣p4 + m

2 � coth ⇢

⌘csch2

⇢

3 + m

2

, m

2 6= �3, �4 ,

�(⇢) = C

+

csch3

⇢ + C� csch3

⇢(sinh 2⇢ � 2⇢) , m

2 = �3 ,

�(⇢) = C

+

csch2

⇢(⇢ coth ⇢ � 1) + C� csch2

⇢ coth ⇢ , m

2 = �4 , (2.17)

1Note that this region does not satisfy the criterion proposed in [16], which suggests that a full duality may require

non-local operators in an essential way. See also [17–20].

5

This is now automatically invariant under the spatial special conformal generators, K

i

,

K

i

� = �2x

i(z@

z

+ t@

t

)� = 0 . (2.12)

Thus, a profile �(z/t) is the most general one which preserves the required so(1, 4) symmetries.

It will be useful to work in coordinates adapted to the unbroken so(1, 4) symmetries. Define a

radial coordinate ⇢ 2 (0, 1) and a time coordinate ⌘ 2 (�1, 0) by the relations

t = ⌘ coth ⇢, z = (�⌘) csch ⇢,

⇢ = cosh�1

(�t)

z

�, ⌘ = �

pt

2 � z

2

. (2.13)

In these coordinates, the metric becomes

ds

2 = d⇢

2 + sinh2

⇢

�d⌘

2 + d~x

2

⌘

2

�, (2.14)

which we recognize as the foliation of AdS5

by inflationary patch dS4

slices. These coordinates

cover the region t < 0, (�t) > z, which is the bulk causal diamond associated with the time

interval t = (�1, 0). This is the region we expect to be dual to the pseudo-conformal scenario1.

The boundary is approached as ⇢ ! 1, and the line (�t) = z is approached as ⇢ ! 0. Near

the boundary, the coordinate ⌘ corresponds with t, and ⇢ approaches e

⇢ ' 2 (�t)

z

. The region and

coordinates are illustrated in Figure 2.

We now consider the scalar wave equation in these coordinates. The equation of motion for a

minimally coupled real scalar field of mass m on curved space is

(⇤(5) � m

2)� = 0 . (2.15)

In the de Sitter adapted coordinates, a scalar configuration with the desired profile (2.11) is one

which depends only on the ⇢ coordinate. The wave equation then reduces to

�

00(⇢) + 4 coth ⇢ �

0(⇢) � m

2

�(⇢) = 0 . (2.16)

The general solution is

�(⇢) = C

+

e

�p4+m

2⇢

⇣p4 + m

2 + coth ⇢

⌘csch2

⇢

p4 + m

2

+C�e

p4+m

2⇢

⇣p4 + m

2 � coth ⇢

⌘csch2

⇢

3 + m

2

, m

2 6= �3, �4 ,

�(⇢) = C

+

csch3

⇢ + C� csch3

⇢(sinh 2⇢ � 2⇢) , m

2 = �3 ,

�(⇢) = C

+

csch2

⇢(⇢ coth ⇢ � 1) + C� csch2

⇢ coth ⇢ , m

2 = �4 , (2.17)

1Note that this region does not satisfy the criterion proposed in [16], which suggests that a full duality may require

non-local operators in an essential way. See also [17–20].

5

dS-slice coordinate bulk scalar field mass

2 integration constants (2nd order equation)

Changing back to the Poincare coordinates z, t and using the asymptotic relation e

⇢ ⇠ (�t)

z

, we

have

�(⇢) ' C

+

z

�+

1

(�t)�++ O(z2)

�+ C�z

��

1

(�t)��+ O(z2)

�. (2.19)

(We have again absorbed unimportant constants into C±, and there are also terms logarithmic in

z in the cases m

2 = �3, �4). Comparing with (2.6) we see that this configuration has �0

= C�(�t)

�� ,

0

= C+

(�t)

�+. Since we are interested in spontaneous breaking for which there is no source, �

0

= 0,

we must fix C� = 0, after which we obtain a spontaneously generated VEV proportional to 0

,

hOi / C

+

(�t)�+. (2.20)

We thus have configurations which correctly reproduce the time-dependent vacuum expectation

value required to break so(2, 4) ! so(1, 4), where �+

is the scaling dimension of the dual operator,

defined over the causal diamond of the region t 2 (�1, 0) of the Poincare patch. This is precisely

what is required to realize the pseudo-conformal mechanism.

A feature generic to these solutions is that they diverge as ⇠ 1/⇢

3 as ⇢ ! 0, which means the

scalar is blowing up and back-reaction is becoming important as we approach the line (�t) = z

in the Poincare patch. This is dual to the approach to t ! 0 in the boundary, where the pseudo-

conformal phase ends and additional physics must kick in to reheat to a traditional radiation

dominated universe. We now turn to the fully back-reacted case with dynamical bulk gravity.

3 Including Back-Reaction

We now consider a scalar field minimally coupled to Einstein gravity

S =

Zd

5

x

p�G

1

2R[G] + 6 � 1

2G

MN

@

M

�@

N

�� V (�)

��Z

d

4

x

p�gK . (3.1)

G

MN

is the bulk metric, and we have a Gibbons-Hawking term on the boundary [22, 23] which

depends on the boundary metric and extrinsic curvature. We have set the radius of AdS5

to unity

and absorbed the overall factors of the Planck mass into the definition of the action. We seek a

metric which respects the unbroken dS4

isometries and approaches AdS5

at the boundary. The

appropriate ansatz is thus a domain wall spacetime in which the wall is foliated by dS4

slices,

ds

2 = d⇢

2 + e

2A(⇢)

�d⌘

2 + d~x

2

⌘

2

�, � = �(⇢) . (3.2)

We are interested in asymptotically AdS5

solutions, for which the scale factor A(⇢) ! ⇢+ const. as

⇢ ! 1. We demand that the scalar potential has an extremum at � = 0 with the value V (0) = 0

into which the scalar flows as ⇢ ! 1,

V =1

2m

2

�

2 + O(�3) . (3.3)

7

sourceExpectation value

Spontaneous symmetry breaking ➔ no source ➔ set

Changing back to the Poincare coordinates z, t and using the asymptotic relation e

⇢ ⇠ (�t)

z

, we

have

�(⇢) ' C

+

z

�+

1

(�t)�++ O(z2)

�+ C�z

��

1

(�t)��+ O(z2)

�. (2.19)

(We have again absorbed unimportant constants into C±, and there are also terms logarithmic in

z in the cases m

2 = �3, �4). Comparing with (2.6) we see that this configuration has �0

= C�(�t)

�� ,

0

= C+

(�t)

�+. Since we are interested in spontaneous breaking for which there is no source, �

0

= 0,

we must fix C� = 0, after which we obtain a spontaneously generated VEV proportional to 0

,

hOi / C

+

(�t)�+. (2.20)

We thus have configurations which correctly reproduce the time-dependent vacuum expectation

value required to break so(2, 4) ! so(1, 4), where �+

is the scaling dimension of the dual operator,

defined over the causal diamond of the region t 2 (�1, 0) of the Poincare patch. This is precisely

what is required to realize the pseudo-conformal mechanism.

A feature generic to these solutions is that they diverge as ⇠ 1/⇢

3 as ⇢ ! 0, which means the

scalar is blowing up and back-reaction is becoming important as we approach the line (�t) = z

in the Poincare patch. This is dual to the approach to t ! 0 in the boundary, where the pseudo-

conformal phase ends and additional physics must kick in to reheat to a traditional radiation

dominated universe. We now turn to the fully back-reacted case with dynamical bulk gravity.

3 Including Back-Reaction

We now consider a scalar field minimally coupled to Einstein gravity

S =

Zd

5

x

p�G

1

2R[G] + 6 � 1

2G

MN

@

M

�@

N

�� V (�)

��Z

d

4

x

p�gK . (3.1)

G

MN

is the bulk metric, and we have a Gibbons-Hawking term on the boundary [22, 23] which

depends on the boundary metric and extrinsic curvature. We have set the radius of AdS5

to unity

and absorbed the overall factors of the Planck mass into the definition of the action. We seek a

metric which respects the unbroken dS4

isometries and approaches AdS5

at the boundary. The

appropriate ansatz is thus a domain wall spacetime in which the wall is foliated by dS4

slices,

ds

2 = d⇢

2 + e

2A(⇢)

�d⌘

2 + d~x

2

⌘

2

�, � = �(⇢) . (3.2)

We are interested in asymptotically AdS5

solutions, for which the scale factor A(⇢) ! ⇢+ const. as

⇢ ! 1. We demand that the scalar potential has an extremum at � = 0 with the value V (0) = 0

into which the scalar flows as ⇢ ! 1,

V =1

2m

2

�

2 + O(�3) . (3.3)

7

Changing back to the Poincare coordinates z, t and using the asymptotic relation e

⇢ ⇠ (�t)

z

, we

have

�(⇢) ' C

+

z

�+

1

(�t)�++ O(z2)

�+ C�z

��

1

(�t)��+ O(z2)

�. (2.19)

(We have again absorbed unimportant constants into C±, and there are also terms logarithmic in

z in the cases m

2 = �3, �4). Comparing with (2.6) we see that this configuration has �0

= C�(�t)

�� ,

0

= C+

(�t)

�+. Since we are interested in spontaneous breaking for which there is no source, �

0

= 0,

we must fix C� = 0, after which we obtain a spontaneously generated VEV proportional to 0

,

hOi / C

+

(�t)�+. (2.20)

We thus have configurations which correctly reproduce the time-dependent vacuum expectation

value required to break so(2, 4) ! so(1, 4), where �+

is the scaling dimension of the dual operator,

defined over the causal diamond of the region t 2 (�1, 0) of the Poincare patch. This is precisely

what is required to realize the pseudo-conformal mechanism.

A feature generic to these solutions is that they diverge as ⇠ 1/⇢

3 as ⇢ ! 0, which means the

scalar is blowing up and back-reaction is becoming important as we approach the line (�t) = z

in the Poincare patch. This is dual to the approach to t ! 0 in the boundary, where the pseudo-

conformal phase ends and additional physics must kick in to reheat to a traditional radiation

dominated universe. We now turn to the fully back-reacted case with dynamical bulk gravity.

3 Including Back-Reaction

We now consider a scalar field minimally coupled to Einstein gravity

S =

Zd

5

x

p�G

1

2R[G] + 6 � 1

2G

MN

@

M

�@

N

�� V (�)

��

Zd

4

x

p�gK . (3.1)

G

MN

is the bulk metric, and we have a Gibbons-Hawking term on the boundary [22, 23] which

depends on the boundary metric and extrinsic curvature. We have set the radius of AdS5

to unity

and absorbed the overall factors of the Planck mass into the definition of the action. We seek a

metric which respects the unbroken dS4

isometries and approaches AdS5

at the boundary. The

appropriate ansatz is thus a domain wall spacetime in which the wall is foliated by dS4

slices,

ds

2 = d⇢

2 + e

2A(⇢)

�d⌘

2 + d~x

2

⌘

2

�, � = �(⇢) . (3.2)

We are interested in asymptotically AdS5

solutions, for which the scale factor A(⇢) ! ⇢+ const. as

⇢ ! 1. We demand that the scalar potential has an extremum at � = 0 with the value V (0) = 0

into which the scalar flows as ⇢ ! 1,

V =1

2m

2

�

2 + O(�3) . (3.3)

7

Near-boundary asymptotics In Poincare coords:

VEV:

KH, Stokes, Trodden (2014)

Challenges and future work

• Calculate higher point functions holographically (consistency relations)

• Requires matching onto a standard radiation dominated cosmology (may require NEC violation at some stage)

• Can reheating/matching be described holographically?

• How are the scale invariant perturbations of the CFT fields imprinted onto the adiabatic mode?

Brandenberger, Wang 1206.4309

Brandenberger, Davis, Perreault 1105.5649

KH, James Stokes, Mark Trodden (arXiv:1505.05513)