Liam McAllisterCornell

APC/PI/Solvay Workshop

Cosmological Frontiers in Fundamental Physics

APC, June 16, 2011

Nishant Agarwal, Rachel Bean, L.M., Gang Xu, 1103.2775

Sohang Gandhi, L.M., Stefan Sjörs, 1106.0002

Based on:

I. Background and motivation: random potentials from

Planck-suppressed interactions

II. Computing the inflaton potential by perturbing around a

local Calabi-Yau geometry

III. Statistical predictions from an ensemble of models

• Inflation provides a beautiful causal mechanism to

generate the observed CMB anisotropies and distribution

of large-scale structure.

• Inflation is sensitive to Planck-scale physics: Planck-

suppressed operators generically make critical

contributions to the dynamics.

• This provides a remarkable opportunity to probe aspects of

the ultraviolet completion of gravity through observation.

• To make meaningful use of this connection, we can:

– Compute these Planck-suppressed contributions to the inflaton

action in string theory

– Search for characteristic properties: “what kind of inflation is natural

in string theory?” (contrast “can I realize inflation of type X in string

theory”).

0

For small inflaton excursions, , one must control

corrections with .

For large inflaton excursions, , one must control

an infinite series of corrections, with arbitrarily large Δ.

In general, the contributions of interest arise from

integrating out massive fields that are part of the

ultraviolet completion of gravity.

In string theory, the contributing massive fields notably

include stabilized moduli.

So we should consider the spectrum and couplings of

moduli in string theory.

• String compactifications typically include many moduli. If

massless, these are very problematic.

• Significant progress in the past decade: in flux

compactifications, many moduli obtain masses.

want: light inflaton, Planck-mass moduli

MP

Mφ

Mmoduli

H

• String compactifications typically include many moduli. If

massless, these are very problematic.

• Significant progress in the past decade: in flux

compactifications, many moduli obtain masses.

want: light inflaton, Planck-mass moduli

MP

Mφ

Mmoduli

unnatural

H

• String compactifications typically include many moduli. If

massless, these are very problematic.

• Significant progress in the past decade: in flux

compactifications, many moduli obtain masses.

hard to achieve

want: light inflaton, Planck-mass moduli

MP

Mφ

Mmoduli

unnatural

H

• String compactifications typically include many moduli. If

massless, these are very problematic.

• Significant progress in the past decade: in flux

compactifications, many moduli obtain masses.

• But, these masses are finite!

hard to achieve

want: light inflaton, Planck-mass moduli get: most masses near H

MP

Mφ

Mmoduli

unnatural

MP

H

Mφ

Mmoduli

H

• Thus, string theory strongly motivates scenarios involving

multiple light fields whose potentials are controlled by

Planck-suppressed contributions.

• With some effort we can compute the form of the inflaton

action, but computing the Wilson coefficients is hopeless in

general. At best we can make statistical statements.

• The resulting models generically involve a number of light

fields in a complicated potential.

• Key question: when inflation arises in this context, what are

its characteristic properties? What can we learn without

knowing the details of the Wilson coefficients?

• Suppose we construct an ensemble of NxN matrices by

drawing the entries of each matrix from a given

distribution W .

• For large N, we can predict the spectrum of eigenvalues

to excellent accuracy, and the result is independent of W.

• Suppose we construct an ensemble of NxN matrices by

drawing the entries of each matrix from a given

distribution W .

• For large N, we can predict the spectrum of eigenvalues

to excellent accuracy, and the result is independent of W.

2 2

2

1( ) 4

2

e.g., suppose A is an NxN matrix whose entries Aij

are drawn from a Gaussian distribution with mean

zero and variance 2/N.

Let M = A + AT .

Then the spectrum of M is

• the distribution W need not be Gaussian

• the mean of W need not be small

• the entries can have some correlations

General result:

If M is a matrix whose constituent entries are drawn from any

distribution with appropriately bounded moments, then in the

large N limit the spectrum of M approaches that of a matrix

whose constituent entries have a Gaussian distribution with

mean zero.

This sort of universality is a strong motivation for seeking

simple inflationary dynamics in a suitable large N limit.

Z.D.Bai

mass

pro

babili

ty

300N

In N-flation, one considers hundreds of axions whose collective

excitation drives inflation.

Each individual axion mass depends on many details of the stabilized

compactification, but the spectrum of masses is simple!

Easther & L.M., 2005

Komatsu et al., 2010

Dimopoulos, Kachru, McGreevy, Wacker, 2005

Arguably simplest model: Gaussian random landscape.

– Have used weak-coupling (integer) dimensions

– Wilson coefficients can be drawn from some

distribution, e.g. a Gaussian.

– The special case J=2 was covered by RMT.

• At strong coupling, operator dimensions can change

substantially. Should we restrict to integers? Does it

matter?

• Also, we do not have a solid prior for the distribution – does

it matter whether it is Gaussian?

• Are all possible terms included with equal weight?

Our approach:

• Work in a string theory construction where we can explicitly

compute the form of the potential.

• „Experimentally‟ check for dependence on the distribution.

• Although N will not be very large, we will find universal

behavior.

String theory motivates considering many light fields

governed by a complicated potential induced by Planck-

suppressed couplings.

We will study a specific example of this sort and search for

simplifications when the number of terms in the potential

is large.

First step: identify a model where we can obtain the

structure of the potential (but not the Wilson coefficients).

Sohang Gandhi, L.M., Stefan Sjörs, 1106.0002

• In principle we‟d like to choose a compact model with

explicitly stabilized moduli, and directly compute the

effective theory up to (say) Δ=8 contributions.

• This would require a tour de force combining , gs

corrections with unprecedented understanding of the

moduli potential.

• Idea: study the effective theory of a local region of a

stabilized compactification, and use 10d locality to

simplify the calculation.

• Key simplification: in type IIB string theory, the best-

understood methods of moduli stabilization yield 10d

solutions that can be formulated in systematic expansion

around conformally Calabi-Yau backgrounds.

• Now we want to incorporate corrections that encode the

effects of moduli stabilization.

• So we need to find new solutions in perturbation theory

around the conformally Calabi-Yau background.

• The small parameter is the weak breaking of the ISD

condition, equivalent to the smallness of the moduli

potential in string units.

• We find a remarkable simplification!

Consider k scalar functions of a variable r,

and a background

Gandhi, L.M., Sjörs, 1106.0002

The type IIB equations of motion expanded

around any conformally Calabi-Yau

background are triangular, to any order!

(SUSY is not required.)

So in principle we can solve iteratively.

Borokhov and Gubser, 2002

Bena, Graña, Halmagyi 2009 et seq.

Dymarsky 2011

Gandhi, L.M., Sjörs, 1106.0002

Triangularity applied to ODEs for linearized

perturbations of Klebanov-Strassler:

The type IIB equations of motion expanded

around any conformally Calabi-Yau

background are triangular, to any order!

(SUSY is not required.)

So in principle we can solve iteratively.

Background: a noncompact warped CY cone with ISD fluxes, e.g. KS.

Goal: finite, warped, non-SUSY region attached to compact bulk.

We obtain separable solutions: we give all Green‟s functions in terms of

angular harmonics on the Sasaki-Einstein base. For T1,1 the harmonics

are known in closed form.

The same Green‟s functions apply at any order, so we can obtain a

solution to any desired order by a purely algebraic procedure.

cf. Ceresole, Dall‟Agata, D‟Auria 1999

• Now we use these tools to study the

particular local geometry that is relevant

for D3-brane inflation.

CY orientifold, with

fluxes and nonperturbative W

(KKLT 2003)

warped throat

(e.g. Klebanov-Strassler)

anti-D3-brane

D3-brane

Dvali&Tye 1998

Dvali,Shafi,Solganik 2001

Burgess,Majumdar,Nolte,Quevedo,Rajesh,Zhang 2001

Kachru, Kallosh, Linde, Maldacena, L.M., Trivedi, 2003

CY orientifold, with

fluxes and nonperturbative W

(KKLT 2003)

warped throat

(e.g. Klebanov-Strassler)

anti-D3-brane

D3-brane

Note: the inflaton potential is NOT just the Coulomb

term! Planck-suppressed effects from moduli

stabilization are dominant!

• We found the supergravity solution corresponding to the

most general bulk contributions and read off the structure of

the potential.

Baumann, Dymarsky, Kachru, Klebanov, L.M., 2008, 2009, 2010

Gandhi, L.M., Sjörs, 2011

We will keep the 333 terms with Δ<7.8.

We draw the coefficients ci from a Gaussian distribution,

generating an ensemble of potentials. (We checked that the

choice of distribution is not significant.)

We draw a potential from the ensemble, choose a random

initial condition (with appropriately bounded kinetic energy),

and find the full six-field evolution of the homogeneous

background. Repeat 7 x 107 times.

We stare at the results and try to identify “robust observables”,

i.e. quantities that depend very weakly on the details of the

distribution.

Typical successful trajectories. The angles evolve initially

and then settle into an angular minimum.

cf. Spinflation, Easson et al., 2007

Total number of e-folds, Ne

P(N

e)

x 1

05

model parameters measure

Freivogel, Kleban, Rodriguez Martinez and Susskind, 2005

model parameters measure

• When multiple fields are active during inflation,

computing the perturbations is subtle.

– Bending trajectories induce conversion of

isocurvature to curvature outside the horizon

– Prospect of interesting non-Gaussianity

• As a first step, we can identify trajectories that

bend so little that a single-field treatment is

applicable. Use

• Worthwhile to do the full multifield analysis!

Groot Nibbelink and van Tent, 2001

Gordon, Wands, Bassett and Maartens, 2001

Total number of e-folds

• Overshooting the inflection point is rather rare; one can start far

above it without difficulty.

• We find attractor behavior in the angular directions.

• Characteristic of successful examples: a funnel towards an inflection

point, which need not be parallel to the radial direction.

• We find 60+ e-folds of inflation approximately once in 103 trials.

• Restricting (for now) to cases that are effectively single-field when

the CMB is produced, we find models consistent with all data

approximately once in 105 trials, provided we set the overall scale in

the right range.

• No tensors: r < 10-12.

• DBI inflation did not occur by chance.

• The probability of N e-folds is proportional to N-3.

• String theory strongly motivates considering inflation

models with multiple light fields whose potential is

controlled by Planck-suppressed contributions.

• We studied a particular case with N=6 fields (D3-brane

inflation).

• To compute the structure of the inflaton potential, we

developed a toolkit for finding solutions in perturbation

theory around conformally Calabi-Yau backgrounds.

• Drawing the Wilson coefficients from various

distributions, we constructed ensembles of potentials.

• We then studied the typical behavior in these

ensembles.

• Idea: when there are many fields and many terms in the

potential, central limit behavior will take over, yielding a

sort of universality.

• Predictivity comes at the cost of statistical insensitivity to

ultraviolet physics.

• We found scaling behavior: the probability of Ne e-folds of

inflation is a power law, Ne-3 .

• We did not yet study multifield effects in the

perturbations, but there is plausibly a rich story involving

bending trajectories and transients.

• To do: extension to more general, higher-dimensional

systems.