2016/12/06 Rikkyo U seminar

Recent developments
in open Inflation
Daisuke Yamauchi
Faculty of Engineering, Kanagawa Univ.

2016/12/06 Rikkyo U seminar

Recent developments
in open Inflation
Inflation withDaisukeYamauchi
quantumtunneling
Faculty of Engineering, Kanagawa Univ.

Part 1INTRODUCTION

Inflation
 Almost scale invariant spectrum
 Highly Gaussian fluctuations
 Spatially quite flat

[Planck2015]

Excellent agreement with the prediction ofthe simplest inflationary model!

potential
Today’stopic

Slow-roll inflation
ReheatingField

Neighborhood we live in
It is quite possible that our part of the Universe appeared
as a result of quantum tunneling after trapped in one ofmany metastable vacua.

De Sitterexpansion
Quantumtunneling

Our “pocket”
Universe
Slow-rollinflation

Whole universe
Pocketuniverse

Eternal inflating region

Pocketuniverse

[Bousso+Polchinski (2000), Susskind (2003), Douglas (2003), Kachru+ (2003),…]String theorists consider…
O(10500) metastable dS states with

L. Susskind
A. Lindedifferent properties depending on
vev of scalar fields, compactifications,…
2009@Stanford U.

“String landscape” picture
The inflationary multiverse becomes divided into
an exponentially large number of different
exponentially large “pocket universes” with different
laws of low-energy physics operating in each of them.

Is there any way to know what kindof neighborhood we live in ?

Revisit “Open Inflation” !

Plan
1.Introduction (finish)

2.Open inflation
3.Recent developments
3.1. Effect of rapid-rolling3.2. CMB asymmetry
4.Future prospects

Part 2

OPEN INFLATION–BACKGROUND EVOLUTION–

[Jensen+Steinhardt(1984)]

Two possibilities

MPl=1 unitV(σ)
VwallCurvatureof potential
V,σσ |V,σσ|<<Vwall → Hawking-Moss tunneling |V,σσ|>>Vwall → Coleman-de Luccia tunnelingσ[Hawking+Moss(1982)]

Hawking-Moss (HM) tunneling Tunneling to a top of the barrier due to the quantum fluctuations
 Too large density perturbations unless e-folds>>60 [Linde(1995)]→ If the final transition is HM, we can not see any deviationfrom the standard prediction of slow-roll inflation.

V(σ)

HMtransition

rolling
down|V,σσ|<<Vσ[Coleman+de Luccia(1980)]

Coleman-de Luccia (CDL) tunneling Tunneling mediated by an O(4)-symmetric solution ofEuclidean Einstein-scalar equations.
 Reflecting the symmetry of a tunneling process,the region inside the bubble becomes an open FLRW universe!

V(σ)|V,σσ|>>V

CDLtransitionσ[Coleman+de Luccia(1980)]

O(4) symmetric CDL bounce solutiontE
tE=-π/2

True
False
vacuum vacuumtE=π/2

Bubble wall @tE=tw
 Note that an O(4)-sym solution is expected to be most likely to occur(at least in the case of vacuum decay without gravity).

[Coleman+de Luccia(1980)]

Background evolution in CDL
 Basic equations-V(σ)

 Boundary conditionσ[Coleman+de Luccia(1980)]

Background evolution in CDLa(tE)

σ(tE)

tE=te～π/2tE=ti ～-π/2-V(σ)σ

Geometry after CDL

Bubble wall

are obtained by analytic continuation of an CDL solution
from Euclidean to Lorentzian:-T2 + R2 = 1/H2
Hyperbolic

Bubble nucleation

TE2 + R2 = 1/H2
Spherical

[Sasaki+Tanaka+Yamamoto(1994), Bucher+Goldhaver+Turok(1995),…]

X2+Y2+Z2+U2+TE2=H-2
TE=cos[tE]cos[rE]U=sin[tE]
Z=cos[tE]sin[rE]cos[θ]X=cos[tE]sin[rE]sin[θ]cos[φ]Y=cos[tE]sin[rE]sin[θ]sin[φ]TE  i T

tE = -i tR+π/2 = tC
rE = -i rR = -i rC+π/2

X2+Y2+Z2+U2-T2=H-2
T=sinh[tR]cosh[rR]U=cosh[tR]Z=sinh[tR]sinh[rR]cos[θ]X=sinh[tR]sinh[rR]sin[θ]cos[φ]Y=sinh[tR]sinh[rR]sin[θ]sin[φ]T=sinh[rC]cos[tC]
U=sin[tC]
Z=cosh[rC]cos[tC]cos[θ]X=cosh[rC]cos[tC]sin[θ]cos[φ]Y=cosh[rC]cos[tC]sin[θ]sin[φ]Geometry after CDL
are obtained by analytic continuation of an CDL solution
from Euclidean to Lorentzian:

Lorentzian
Euclidean

False
True
vacuum vacuum

Geometry after CDL
are obtainedcontinuation
Open-slicebyofanalytic
de Sitter[ O(3,1) symof] an CDL solution
from Euclidean to Lorentzian:

Lorentzian
Euclidean

CDL solution
sym ]False [ O(4)True
vacuum vacuum

Truevacuum

False
vacuum
Bubblewall

Bubble nucleation

Bubble wall

Geometry after CDL

Bubble nucleationde Sitter Penrose-like diagram

Open-slice of de Sitter [ O(3,1) sym ][Gott III (1982), Got III+Statler (1984),Sasaki+Tanaka+Yamamoto+Yokoyama (1993), …]Open Inflation

potential

Trapped in one of metastable vacua→ CDL tunneling
Reflecting the symmetry of a tunneling process,the region inside the bubble becomes open universe!
slow-roll inflation
reheating

local minimum

Field

global minimum

Ex. 1 : Simplest polynomial potential
 |V,σσ|<4H2 : HM tunneling [Linde (1999)]
 The condition for CDL and subsequent slow-roll inflation
are not easily satisfied at the same time !

V(σ)

HMtransition

slow-rollinflation
CDLσ[Linde+Linde+Mezhlumian (1995)]

Ex. 2 : Simple two-field model Naturally/easily realized in the landscapeσ : heavy field → false vacuum decay
φ : light field → starts rolling after FV decayV(σ,φ) = Vtunnel(σ)+m2φ2/2
 Too large perturbations from supercurvature modeof φ unless e-folds>>60 [Sasaki+Tanaka(1996)]

Vtunnel(σ)
CDLslow-rollinflation

σLargeHubblefrictionφ[Sugimura+DY+Sasaki, 1110.4773]

Ex. 3 : Multi-field tunneling and inflation|V,σσ|>>4H2single-field
simplest polynomial
potential

multi-fieldtunnelingmass potential

light field : φheavy field : σ

Fast-rolling down
→ slow-roll inflation

[Sugimura+DY+Sasaki, 1110.4773]

Ex. 3 : Tunneling rate ～exp(-B)
Mass of lightfield φ at TV

effective
multi-CDL single-CDL

HM (multi) - (single)

Multi-field dynamics tends toincrease the tunneling-rate (?).

Part 2

OPEN INFLATION–FLUCTUATIONS–Quantization
Step 1
We need to find the reduced action that contains only
the physical degree of freedom.
Ex) Simple scalar fieldQuantization
Step 2

Bubble wall

We need to find a complete set of functions which obey
the field equation and which are regular on the lowerhemisphere.
A set of all modes which can be Klein-Gordon normalized
on a Cauchy surface
Bubble nucleation

Quantization
Step 3We promote the physical degree of freedom to operator
and expand it by mode functions {σN, σN} as

Mode function σN plays the role ofpositive frequency functions!

[For massive U(1) field, DY+Fujita+Mukohyama, 1402.2784]

Difficulty in quantization
The surfaces which respect the maximal sym. (tR=const.)
are not the Cauchy surface of the whole spacetime.
We need to work in the center region (rC=const.), where
the background configuration is spatially inhomogenous.

TruevacuumFalse
vacuum
Bubblewall

rC=const.
Cauchy surface

[For massive U(1) field, DY+Fujita+Mukohyama, 1402.2784]

Difficulty in quantization
The surfaces which respect the maximal sym. (tR=const.)
are not the Cauchy surface of the whole spacetime.
We need to work in the center region (rC=const.), where
the background configuration is spatially inhomogenous.

Truevacuum

False
vacuum
Bubblewall

rC=const.
Cauchy surface(Discrete) supercurvature mode
There could appear a set of modes which have finite KG norms
on Cauchy surfaces, but which cannot be quantized on the open
chart because of the divergent KG norms on the open chart.

Normal modes
Continuous
Discretenormalsupercurvature
modes
mode

wave number

Supercurvature
mode

rC=const.
Cauchy surface(Discrete) supercurvature mode
There could appear a set of modes which have finite KG norms
on Cauchy surfaces, but which cannot be quantized on the openchart because of the divergent KG norms on the open chart.

× (for m＞3H/2)〇 (for m≦3H/2)

 Canonical scalar field[Sasaki+Tanaka+Yamamoto (1995)]

 Tensor perturbations

×kSC2 ≃ (2m2/3H2) kcurv2[Tanaka+Sasaki (1997)]

 U(1) gauge field×[DY+Fujita+Mukohyama, 1402.2784]

 Metric pert.+scalar field[Garriga+Montes+Sasaki+Tanaka(1999)]? (model-depend)Evolution inside bubble
log L

H-1
Curvaturedominated era

Inflationary era

log aafter bubble
nucleation

Power spectrum for scalar/tensor
Supercurvature mode

k3P(k)/2π2tensor
with rapid-rolling

Tensor enhancement[DY+Linde+Naruko+Sasaki+Tanaka(2011)]

scalartensor
k3kSC kcurv

k3

k3

Scalar suppression[Linde+Sasaki+Tanaka(1999)
White+Zhang+Sasaki(2014)]～2kcurvk(若干の適当さを含みます)

Part 3

RECENT DEVELOPMENTS 1—SIGNALS FROM STRINGY INFLATION—

Classification for open inflation
Inflation
Tunneling
Curvature Supercurvaturedriver (m<H) driver (m>H)
pert. ζmode?

σσσσ → 〇/×σ

ρσσ→〇φρσσ→〇φρχσ→〇note・Linde(1998)・Linde+Sasaki+Tanaka(1999)
・Garriga+Montes+Sasaki+Tanaka(1999)
・DY+(2011)
・Linde+Mezhlumian(1995)
・Sasaki+Tanaka(1996)[・tunneling sol :Sugimura+DY+Sasaki(2011)]

・Kanno+Sasaki+Tanaka(2013)
・Brnes+Domenech+Sasaki+Takahashi(2016)[・Sugimura+DY+Sasaki(2012)]

Requirements
for a single-field stringy modelV(σ)
Vlandscape

Vlandscape=O(Mpl4) >> Vinf
The universe will most likely tunnel to a point
where the energy scale is still very high andhave a large hierarchy between energy scales.
Ex.) Stringy inflation Vinf=O(10-30MPl4)[Kallosh+Linde(2014)]

Hinf2<<MPl2
VinfσRequirements
for a single-field stringy modelV(σ)
Vlandscape|V,σσ|>>V/MPl2
Slow-roll inflation does not start immediately
after the CDL tunneling, and there must be
some intermediate stage of rapid rolling down.
[Linde(1998), Linde+Sasaki+Tanaka(1999),
DY+Linde+Naruko+Sasaki+Tanaka, 1105.2674]

VinfσTensor-type perturbations
may not be suppressed at all !

Memory of false vacuum may remain in
the perturbations on the curvature scale!
 Note : Scalar perturbations may be suppressedby the velocity during RAPID ROLL phase…

[DY+Linde+Naruko+Sasaki+Tanaka,1105.2674]

Primordial tensor spectrum

A toy model :V(φ) ∝φ2 exp[(φ2-φR2)/φR2]
curvature scale

Normalized tensor spectrum

Keeps the memory of the high energy density inlarge angular scales, and can be strongly red-tilted.

wave number normalized by curvature scale

[DY+Linde+Naruko+Sasaki+Tanaka,1105.2674]

Primordial tensor spectrum

Normalized temperature
anisotropies : l(l+1)Cl/2π

Keeps the memory of the high energy density inlarge angular scales, and can be strongly red-tilted.

(ΩK)lA toy model :V(φ) ∝φ2 exp[(φ2-φR2)/φR2]

multipole lPart 4ASYMMETRYFROM OPEN INFLATION

[Zarei+DY+Hirouzjahi+Ohta+Naruko+Yamaguchi, work in progress]

Polarizations from open inflation
 Spatially openness : ΩK=10-2-10-4

implement in
CAMB code A complete set of mode functions
 (Continuous) normal modes (Discrete) supercurvature modes(ΩK)llow ll (l+1) Cl [BB]l (l+1) Cl [TT]CMB temperature + polarizations
Large enhancement
due to rapid-rolling?

???
low l

[Planck23 (2013)]

CMB anomalies
ΔT(n)=(1+A p・n)ΔTiso(n)A=0.07±0.02
Power asymmetry

Dipole modulation

[DY+Zarei+Hirouzjahi+Ohta+Naruko+Yamaguchi, work in progress]

Modulation due to supercurvature[see also Kanno+Sasaki+Tanaka(2013), Bernes+Domenech+Sasaki+Takahashi(2016)]

We will treat the supercurvature mode as a nonstochastic quantity
and we can only observe one realization in our Hubble patch.
The modulation of the continuous spectrum
Eternal inflating region

Hubble patchNormal modes
Supercurvature
mode

Normal modes

Supercurvaturemode

[DY+Zarei+Hirouzjahi+Ohta+Naruko+Yamaguchi, work in progress]

Modulation due to supercurvature[see also Kanno+Sasaki+Tanaka(2013), Bernes+Domenech+Sasaki+Takahashi(2016)]

We will treat the supercurvature mode as a nonstochastic quantity
and we can only observe one realization in our Hubble patch.
The modulation of the continuous spectrum

Spatial modulation due to SC mode!
 Power asymmetry : A = ΔPR/2PR

 Quadrupole : a20

[DY+Zarei+Hirouzjahi+Ohta+Naruko+Yamaguchi, work in progress]

Modulation due to supercurvature[see also Kanno+Sasaki+Tanaka(2013), Bernes+Domenech+Sasaki+Takahashi(2016)]

We will treat the supercurvature mode as a nonstochastic quantity
and we can only observe one realization in our Hubble patch.
The modulation of the continuous spectrum
Hemispherical power asymmetry!

[DY+Zarei+Hirouzjahi+Ohta+Naruko+Yamaguchi, work in progress]

Modulation due to supercurvature[see also Kanno+Sasaki+Tanaka(2013), Bernes+Domenech+Sasaki+Takahashi(2016)]

We will treat the supercurvature mode as a nonstochastic quantity
and we can only observe one realization in our Hubble patch.
The modulation of the continuous spectrum

Question…

Hemispherical power asymmetry!
 Does the supercurvature mode with the large
amplitude really exist in the realistic open
inflationary scenario?

Future prospects
Multifield tunneling and supercurvature mode
 Background solution [Sugimura+DY+Sasaki (2011)]
 Quantization including the effect of the metric perturbations
should be taken into account.

 would rescue the simplest two field model [Tanaka+Sasaki(1996)]

L=(Mpl2/2)R-Gab(φ)dφa・dφb/2-V(φ)
supercurvature modes?

Future prospects
Higher-order correlations
 Bispectrum from scalar normal modes in exact dS[Sugimura+Komatsu (2013)]

Bζsubcurv≃ Bζusual + BζNBD (squeezed limit, subcurvature limit)Bζusual  (1-ns)Pζ(klong)Pζ(kshort)

BζNBD  (kshort/klong) exp(-π kshort) Pζ(klong)Pζ(kshort)enhancement

suppression

 Extension to the dynamical background is needed.
 Contributions from supercurvature modes@k~ksuperlong?

Summary
We are already testing the model of inflationin the context of cosmic/string landscape !