Excursion in hyperbolic space - Leiden Observatorywdgang/hyper.pdf · Intro to multi-field inflation Dong-Gang Wang Leiden University!17 They can be eﬀectively described by a single

Dong-Gang Wang

王东刚

Lorentz Institute + Leiden Observatory

APCTP workshop “New perspectives on cosmology”

based on arXiv: 1711.09478 with A. Achucarro, R. Kallosh, A. Linde and Y. Welling arXiv: 1803.09911 with A. Linde, Y. Welling, Y. Yamada and A. Achucarro

arXiv: 1810.02804 with O. Iarygina, E. Sfakianakis and A. Achucarro

Excursion in hyperbolic space

Dong-Gang Wang Leiden University

Outline

✦ Inflationary cosmology in a nutshell✦ The story of curved field space

✦ Intro to inflationary alpha-attractors

✦ A brief review of multi-field inflation

✦ Multi-field alpha-attractors in the hyperbolic space

Universality from “rolling on the ridge”

Hypernatural inflation

Universality in preheating

✦ Outlook

2

Cosmic Inflation

Exponential stretching of the Universe and all

inhomogeneities

An homogeneous, isotropic and flat initial condition for the Hot Big Bang

a(t) ⇠ eHt

t ⇠ 10�35s

e60

quasi de Sitter Expansion

Guth 1982; Linde 1982;

Starobinsky 1980;

Sato 1981; Fang 1980;

…….

Single field slow-roll inflation


as the Standard Model of the very early universe

potential dominated expansion

long enough duration of

inflation

Many different models: chaotic inflation, hilltop inflation, natural inflation, Higgs inflation, Starobinsky inflation…

[Pic. by Baumann]

Quantum fluctuations => Cosmic structures

!5Dong-Gang Wang Leiden University

horizon-exit

decoherences t r e t c h

recombination

reionization

clustering

The success of single field slow-roll inflation


✦ Test the single field slow-roll predictions by CMB observations — nearly scale-invariant power spectrum for curvature perturbation with a slightly red tilt — small tensor-to-scalar ratio — small non-Gaussianity — small isocurvature modes

[Planck 2018]

Dong-Gang Wang Leiden University 7

Everything is okay for the single field slow-roll inflation paradigm, but maybe not okay for specific models…

✏ ⌧ ⌘ plateau-like potentials

[Planck 2018]

Inflationary 𝜶-attractors

[Escher 1960]

[Kallosh, Linde, Roest 2013]


The story of the curved field space

9

The hyperbolic field space

Kahler potential from Supergravity and String Theory

Coset Space Non-Linear Sigma Model from EFT

In General Relativity, the spacetime can be curved, such as the spatial metric for an open Universe

Similarly, in Quantum Field Theory, the field space may also be curved

SO(2,1)/SO(2)

K = �3↵ ln(1� ZZ̄) moduli space of complex scalar moduli field

ds2 =dr2

1� r2+ r2(d✓2 + sin2 ✓d�2) = d�2 + sinh2 �(d✓2 + sin2 ✓d�2)

= 3↵d⇢2 + ⇢2d✓2

(1� ⇢2)2ds2

very common in UV theories:

ds2 = Gabd�ad�b


The mechanism of 𝜶-attractors

10

[Kallosh, Linde, Roest 2013]

Start with the simplest chaotic inflation model:

Introduce the hyperbolic field space to modify the kinetic sector:

Switch to the canonically normalized field variable:

The potential becomes:


The stretching effect of the hyperbolic geometry

11

The simplest inflation model as an example

Stretching flattens your potential


Stretching! Stretching!! Stretching!!!

12

An original random potential on the

Poincare disk

The stretched potential with canonical field


Universal predictions from 𝜶-attractors

13

large 𝜑 approximation (𝜑>>1, 𝜌→1)

For ↵ . O(1)

Carrascoa, Kallosh, Linde 2015

B-mode targets for CMB-S4, AliCPT and next generation

experiments

Choice of the radial potential almost does not matter, as long as it is non-singular at the boundary.

R = � 2

3↵

Starobinsky & Higgs Inflation

Essentially, 𝜶-attractors are multi-field inflation models.

How about the axion field besides dilaton/inflaton?

✓


The angular dependence of the potential

15

A brief review of multi-field inflation

Intro to multi-field inflation


They can be effectively described by a single scalar field only if:

1) there is no coupling between inflaton and other fields which are stabilized;

2) the extra fields are heavy and can be integrated out

From the perspective of fundamental realization, ALL the inflation models are essentially multi-field.

field space metric potential

isocurvature perturbations primordial non-Gaussianity

Perturbations in multi-field inflation


curvature modes

isocurvature modes

perturbations along the trajectory

perturbations orthogonal to the trajectory {

turning trajectory ⌦ 6= 0

⇣̇ =2⌦p2✏

�coupled evolution

⇣ = H⇡

��


Mass scales in multi-field inflation

19

Hessian of the potential

curvature of the field space

turning rate

entropy mass

Multi-field inflation in a nutshell


MH

light M<<H

medium M~H

heavy M>>H

no interaction with inflation

local shape of non-Gaussianity;

Example: curvaton models

mixing with inflaton

intermediate shapes of non-Gaussianity;

Example: quasi-single field inflation

reduced cs

equilateral shape of non-Gaussianity;

Example: effective single field inflation

Chen & Wang 2010; Arkani-Hamed & Maldacena 2015

Achucarro, Gong, Hardeman, Palma & Patil 2011; 2012

Enqvist & Sloth 2001, Lyth & Wands 2002, Moroi & Takahashi 2001

Do not go gentle into that “good” regime!

How about light fields m<<H coupled to inflaton?

large isocurvature perturbations

large local non-Gaussianity

(which is more common in the fundamental perspective)

Do we still have universal predictions in multi-field 𝜶-attractors?


Multi-field 𝜶-attractors: the model

23

the axion-dilaton system:

supergravity Lagrangian

A natural choice of the potential without hierarchies

Achucarro, Kallosh, Linde, WDG & Welling 2017

Dong-Gang Wang Leiden University 24

Rolling in the deep?


Rolling on the ridge

25


Multi-field 𝜶-attractors: the background dynamics

26

equations of motion

The stream of ϕ and θ fields on the potential with random angular dependence

“rolling on the ridge”

the magic of hyperbolic geometry


Another explanation: Geometrical Stabilisation?

27

Hessian of the potential

curvature of the field space

turning rate

The hyperbolic geometry stabilises the “rolling on

the ridge” trajectory

the entropy mass

If we consider a radial trajectory…


Multi-field 𝜶-attractors: the background dynamics

28

Then it returns to single field? No

Inflation is NOT along a geodesic

sinh(p2')✓̇ ⇠ '̇

Curvature and isocurvature perturbations are coupled

physical angular velocity

the dimensionless turning rate is nonzero


Multi-field 𝜶-attractors: perturbations

29

1) Renormalization of the slow-roll parameter;

2) Super-horizon growth of curvature modes sourced by isocurvature modes

Two nontrivial multi-field effects

P⇣

cancellation!


Multi-field 𝜶-attractors: perturbations

30

primordial non-Gaussianity

δN formalism:

Typically multi-field inflation models predict O(1) squeezed bispectrum, becasue of the superhorizon growth of curvature perturbations.

squeezed triangle (local)

Planck 1σ constraint:

Maldacena’s consistency relation


Multi-field 𝜶-attractors: rolling on the ridge in E-model

31

half-plane variable

E-model

kinetic term

Linde, WDG, Welling, Yamada, Achucarro 2018


Multi-field 𝜶-attractors: non-monotonic potential

32

hypernatural inflation

Linde, WDG, Welling, Yamada, Achucarro 2018

hyperbolic stretching of the Mexican-hat potential with pseudo-Goldstone (axion) modification

The axion decay constant fa can be exponentially large

natural inflation regime intermediate regime 𝜶-attractor regime

swampland??

A pedagogical lesson for the UV completion of inflation


single field inflationexcessive isocurvature

perturbations OR

large local non-Gaussianity

string theory in 10D spacetime

4D effective field theory with many light fields

string compactification

KKLT LVS …

stabilize all extra fields

multi-field inflation?

string inflation as an example

A pedagogical lesson for the UV completion of inflation


multi-field inflation on an axion-dilation system can be compatible with current observational constraints without the

need to stabilize all other fields besides the inflaton.

Achucarro, Copeland, Iarygina, Palma, WDG & Welling 1901.0xxxx [to appear soon!]

Shift-symmetric orbital inflation

✦ Only one degree of freedom (isocurvature) is responsible for observed curvature perts;

✦ The phenomenology mimics the one of single field inflation:

Small isocurvature perturbations

Small local non-Gaussianity


Multi-field 𝜶-attractors: preheating

35

Carrasco, Kallosh, Linde, Roest 2015

A simpler version of T-model

particle production after inflation

reheating — perturbative decay

preheating — non-perturbative parametric resonance



36

non-perturbative particle production of isocurvature modes

parametric resonance caused by the oscillating inflaton

tychyonic resonance caused by the negative Ricci curvature

Krajewski, Turzynski,︎ Wieczorek 2018lattice simulation shows very efficient preheating for n=3/2 and very small 𝜶



37

analytical approach — scaling behaviour

�end ' 3p↵

Hend ⇠p↵



38

Floquet analysis

Krajewski, Turzynski,︎ Wieczorek 2018



39

Another kind of universality

n=3/2

The floquet exponents are almost the same

�0/p↵

oscillations per Hubble time scale as

the amplification per Hubble time grows as O(1/

p↵)

!/H ⇠ O(1/p↵)


The role of hyperbolic geometry: a bigger picture

40

Geometrical destabilization — Renaux-Petel and Turzynski 2015

Shift-symmetric orbital inflation— Achucarro, Copeland, Iarygina, Palma, DGW, Welling, 2019

Achucarro, Palma, DGW, Welling, in preparationHyperinflation

— Brown 2017; Mizuno, Mukohyama 2017 Ambient inflation

— Klein, Roest, Stefanyszyn 2017 Angular inflation

— Christodoulidis, Sfakianakis, Roest 2018 ……

the axion-dilaton system:

To be continued!also see Valeri’s talk on alpha-attractors and late-time Universes


Take home messages

✦ 𝜶-attractors are a class of models with hyperbolic field geometry, which can give universal predictions and are favored by observations;

✦ Due to the magic of hyperbolic geometry, the multi-field 𝜶-attractors yield the same universal predictions as the single-field case;

✦ Hypernatural inflation provides the first realization of natural inflation in supergravity, and demonstrates interesting phenomenology;

✦ The tachyonic instability caused by hyperbolic geometry results in universality in the preheating of alpha-attractors

✦ To be continued: more interesting explorations going on in the curved moduli space of inflation…

41

Happy 𝜶-attractors Day!

Leiden, The Netherlands

Documents

Excursion in hyperbolic space - Leiden Observatorywdgang/hyper.pdf · Intro to multi-field inflation Dong-Gang Wang Leiden University!17 They can be eﬀectively described by a single