Learning to Inflate Tom Rudelius IAS Based on 1810.05159/hep-th - - PowerPoint PPT Presentation

Learning to Inflate

Tom Rudelius IAS

Based on 1810.05159/hep-th

Outline

• Machine Learning and hep-th
• Introduction to Inflation
• Learning to Inflate
• Algorithm
• Results
• Possible Next Steps

Machine Learning and hep-th

Machine Learning

?

Machine Learning Works Best When…

• Have clear measure of success

Example: Chess

Example: Self-Driving Cars

Machine Learning Works Best When…

• Have clear measure of success
• Have large, representative dataset on which to train

Example: Deciphering Digits

Example: Deciphering Digits

Advantages and Limitations of AI

• When it comes to string theory, we don’t understand

the large datasets very well

• Known string vacua are likely small, biased sample
• Don’t have a clear measure of “success”
• Need to find problems suitable for machine learning:

train a computer to do something we can already do and then do it “better” than we can

• Inflation is nice in this regard because we have some

experimental data, but more will come in. Best human models can explain data, but lack predictive power

Introduction to Inflation

Inflation

• Postulated period of exponential expansion in

early universe

t

a(t)

Inflation a ∝ eHt

Radiation Domination a ∝ t1/2

aend

a0

log aend/a0 ≈ 60

Inflation in Field Theory

• Inflation can be thought of as the theory of a ball

rolling down a hill with friction

V()

• ¨

φ + 3H ˙ φ + V,φ = 0

H2 = 1 3 ✓1 2 ˙ φ2 + V (φ) ◆ .

Slow-Roll Inflation

• Slow-roll parameters computed from shape of

potential

|¨ φ| ⌧ |3H ˙ φ|, |V,φ|

˙ φ2 ⌧ V (φ)

• Slow-roll approximation:

✏ = 1 2 ✓V 0() V () ◆2 , ⌘ = V 00() V ()

Inflation and Cosmic Observables

• Slow-roll parameters related to cosmic
• bservables
• * denotes evaluation at horizon crossing,

Tensor-to-scalar ratio Spectral index

r∗ ≈ 16✏∗

n∗

s − 1 ≈ 2⌘∗ − 6✏∗

log a∗/a0 ≈ 60

Duration of Inflation

• e-folds of inflation:

φ∗

V()

• φ

V (φ)

φend

Ne = Z φ∗

φend

d √ 2✏ = 60

Large-Field vs. Small-Field Inflation

φ

V (φ)

φ∗ − φend > 1

φend

φ∗

φ

V (φ)

φ∗ − φend < 1

φend

φ∗

Inflationary Potential

• Potential is key to slow-roll inflation
• Exact potential not known

Random Inflation

• String theory leaves open many possible

inflationary potentials, suggests statistical approach

• Different approaches to “random

inflation” (DBM, GRF, etc.) have been attempted

• Goal of this talk: introduce new approach,

motivated by machine learning

Learning to Inflate

The Algorithm

Initializing the Potential

• Consider Taylor-expanded potential with random

coefficients

• Set i.i.d.
• Set (maintain perturbativity)
• Set N = 15

ci ∼ N(0, 1)

˜ φ0 = 0.5

V (φ) = Λ4

v N

X

i=2

ci ✓ φ Λh ◆i := Λ4

v N

X

i=2

ci ˜ φi

Initializing the Potential

• For , potential will typically be very

steep, giving large-field model of inflation

φ

V (φ)

∆φ ∼ 15

Ne > 60

Main Idea

• Want model with more e-folds/smaller
• Treat coefficients as weights in a single-layer

neural network

• Use gradient ascent to maximize
• Can interpret results in two ways:
• 1. Minimizing for fixed
• 2. Maximizing for fixed

∆φ

ci

Λh

Ne

Λh

Ne

˜ Ne := Ne/Λ2

h

Results

The Learning Process

a) b) c) d) Λh = 35

Λh = 25

Λh = 22

Λh = 10

Hilltop vs. Inflection Point

90% 10%

Phenomenology of Hilltop Models

Successful phenomenology for large-field hilltop models

• Pheno. of Inflection Point Models

V (˜ φ) = W0 + W1(˜ φ − ˜ φinf) + 1 3!W3(˜ φ − ˜ φinf)3

• Inflection point potential takes the form
• Gradient ascent sends
• Even for small , eventually learns model with

good phenomenology

W1 → 0

Λh

Two-Field Models

a) b) c) d)

First learns single-field model, then multi-field

Two-Field Models

Can have several plateaus (not observed in single-field case)

Summary

• Used gradient ascent in space of Taylor

coefficients to maximize e-folds of inflation

• Found large-field hilltop models and small-field

inflection point models with viable phenomenology

• In the two-field case, learned genuine multi-field

models of inflation, with significant curvature

Possible Next Steps

• Non-trivial kinetic term—might it learn DBI

inflation, etc.?

• More fields, move beyond slow-roll slow-turn

approximation

• More sophisticated machine learning techniques/

network architecture

• Other reward functions
• Combine with other approaches to random

inflation