The geometrical destabilization of inflation Sbastien Renaux-Petel - - PowerPoint PPT Presentation

The geometrical destabilization of inflation

Sébastien Renaux-Petel

CNRS - Institut d’Astrophysique de Paris

YITP , Kyoto University, Gravity and Cosmology 2018 13th February

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 758792)

1. Inflation 2. General mechanism of the geometrical destabilization 3. Minimal realization and fate of the instability 4. Premature end of inflation

Outline

1. Inflation

2. General mechanism of the geometrical destabilization 3. Minimal realization and fate of the instability 4. Premature end of inflation

Outline

Planck all sky map, 2015

Inflation: a phenomenological success

l(l + 1)Cl/2π [µK2]

Planck 2015 data

The simplest models, single-field slow-roll, economically explain all current data

Inflation: a phenomenological success

Primordial fluctuations are adiabatic, super horizon at recombination, almost scale-invariant, Gaussian.

r = 16✏? ns − 1 = 2⌘? − 6✏?

The Planck ns-r plane

• eta-problem

η ⇡ m2

φ

H2 ⌧ 1

Why is the inflaton so light?

m2

φ ∼ Λ2 uv H2

like the Higgs hierarchy problem

• multiple (heavy) fields in ultraviolet completions of

slow-roll single-field inflation do not decouple

• Inflation is not in general predictive without reheating
• So far, merely phenomenological description.
• Inflation is sensitive to the physics at the Planck scale

Beyond toy-models?

Unless symmetry forbids it, presence of terms of the form ∆V = cV0(φ) φ2 Λ2 L = −1 2(∂φ)2 − V0(φ) + X

δ

Oδ(φ) Λδ−4

Corrections to the low-energy effective action Slow-roll action

∆m2

φ ∼ c V0

Λ2 ∼ c H2 ✓MP Λ ◆2 ∆η & 1 Sensitivity of inflation to Planck-suppressed operators Wilson coefficient c ∼ O(1) Λ . MP +

UV-sensitivity of inflation

1. Inflation

• 2. General mechanism of the

geometrical destabilization

3. Minimal realization and fate of the instability 4. Premature end of inflation

Outline

Realistic inflationary models have fields which live in an internal space with curved geometry. This effect applies during inflation, it easily overcomes the effect of the potential, and can destabilize inflationary trajectories. Initially neighboring geodesics tend to fall away from each other in the presence of negative curvature.

Basic idea

V (φ1, φ2)

φ1

φ2

Renaux-Petel, Turzynski, September 2016 PRL Editors’ Highlight

Basic mechanism

Effective single-field dynamics Light inflaton + Extra heavy fields Simplest ‘realistic’ models (hope): (valley with steep walls)

V (φ1, φ2)

φ1

φ2

More realistic: Geometrical instability

Light inflaton + Extra heavy fields + Curved field space

Basic mechanism

Renaux-Petel, Turzynski, September 2016 PRL Editors’ Highlight

V (x1, x2)

x2

x1

Simple analogy:

• Position of a

charged particle

• Electric force
• Surface geometry

Geometrical instability

Basic mechanism

Renaux-Petel, Turzynski, September 2016 PRL Editors’ Highlight

• 1. A curved field space is generic

S = Z d4x√−g ✓ −1 2GIJ(φK)∂µφI∂µφJ − V (φI) ◆

Multifield Lagrangian

Top-down (e.g. supergravity), or bottom-up (EFT) Field space curvature

• 2. A priori, M can lie anywhere between H and Mp

Rfield spaceM 2

Pl = − 2

3α Example: alpha-attractors

∼ 1/M 2

DtAI = ˙ AI + ΓI

JK ˙

φJAK

DtDtQI + 3HDtQI + k2 a2 QI + M I

JQJ = 0

Sasaki, Stewart, 95

Riemann curvature tensor

• f the field space metric

cf geodesic deviation equation

Mass matrix:

QI = fluctuations of field I in flat gauge M I

J = V I ;J − RI KLJ ˙

φK ˙ φL − 1 a3M 2

Pl

Dt ✓a3 H ˙ φI ˙ φJ ◆

Linear perturbation theory

¨ Qs + 3H ˙ Qs + m2

s(eff)Qs = 0

Effective entropic mass squared:

Hessian contribution bending contribution ‘geometrical’ contribution m2

s(eff)

H2 ≡ V;ss H2 + 3⌘2

⊥ + ✏ Rfield spaceM 2 Pl

super-Hubble evolution

• f the entropic field

Gordon et al, 2000

Two-field models (simplicity)

When the geometrical contribution is negative and large enough, it can render the entropic fluctuation tachyonic, even with a large mass in the static vacuum, with potentially dramatic observational consequences.

Hessian contribution bending contribution ‘geometrical’ contribution m2

s(eff)

H2 ≡ V;ss H2 + 3⌘2

⊥ + ✏ Rfield spaceM 2 Pl

Geometrical destabilization

Rfield spaceM 2

Pl ∼ (MPl/M)2

Let us consider for instance

M = O(10−2, 10−3)MPl

Even for V;ss H2 ∼ 100 The effective mass becomes tachyonic when: ✏ → ✏c = 10−4

• r

10−2

(string scale, KK scale, GUT scale...)

1

generically

Geometrical destabilization

Rfield space < 0

Necessary condition (2-field):

1. Inflation 2. General mechanism of the geometrical destabilization

• 3. Minimal realization and

fate of the instability

4. Premature end of inflation

Outline

L = −1 2(∂φ)2 ✓ 1 + 2 χ2 M 2 ◆ − V (φ) − 1 2(∂χ)2 − 1 2m2

hχ2

• Slow-roll model of inflation, with inflaton
• Heavy field
• Simple dimension 6 operator suppressed by a mass

scale of new physics

m2

h H2

φ

χ with

M H

Minimal realization

L = −1 2(∂φ)2 ✓ 1 + 2 χ2 M 2 ◆ − V (φ) − 1 2(∂χ)2 − 1 2m2

hχ2

• Does correspond to lots of models in the literature, in which it

is sometimes said : «chi is stabilized by a large mass» so let us put chi=0 (consistently with the equations of motion)

Minimal realization

• Terms linear in chi absent for consistency (or Z2 symmetry),

and higher-orders in chi suppressed near the inflationary valley

• Generally expected from the effective theory point of view

(respect approximate shift-symmetry of inflaton)

L = −1 2(∂φ)2 ✓ 1 + 2 χ2 M 2 ◆ − V (φ) − 1 2(∂χ)2 − 1 2m2

hχ2

• Apparently benign high-energy correction (small

correction to the kinetic term) but ... Rfield space ' 4 M 2 for χ ⌧ M

m2

s(eff)

H2 = m2

h

H2 − 4 ✏(t) ✓MPl M ◆2

along

χ = 0

• The inflationary trajectory becomes

unstable after ✏ → ✏c

Minimal realization

Unless symmetry forbids it, presence of terms of the form

Corrections to the low-energy effective action Slow-roll action

∆L = c(∂φ)2 χ2 Λ2 ∆m2

χ ∼ c(@)2

Λ2 ∼ c ✏H2 ✓MP Λ ◆2

Leff[φI] = Ll[φI] + X

i

ci Oi[φI, ∂φI, . . .] Λδi−4

Λ ' MP Λ ⌧ MP ✏c ∼ 1 ✏c ⌧ 1

Geometrical destabilization of inflation Modified reheating

Similarity with the eta-problem

Fate of the instability?

Rapid and efficient growth of super-Hubble entropic fluctuations

Numerical resolution (linear theory) Theoretical modeling (early time):

496.5 497.0 497.5 498.0 498.5 499.0 10 1000 105 107

103 107

N − Nc

1 2 3

PQs(k, N) PQs(k, Nc)

1

Starobinsky potential

mh = 10Hc M = 10−2MPl

Example:

∼ e

1 3 m2 h H2 c ηc(N−Nc)3

• Tachyonic preheating, possible production of primordial

black holes, inflating topological defects …

• Similar to hybrid inflation (but different kinetic origin

and kinetic effects).

• Backreaction of fluctuations on background trajectory?

Non-perturbative phenomenon Challenging! Work in progress

Fate of the instability?

• Universal bound on curvature scale
• Modified ranking of inflationary models

Second phase of inflation Premature end of inflation Inhomogeneities dominate

Fate of the instability?

Inhomogeneities are shut off

1706.01835 JCAP RP , Turzynski, Vennin RP , Turzynski, 1510.01281, PRL RP , Turzynski, 1510.01281 Works to appear

OR

1. Inflation 2. General mechanism of the geometrical destabilization 3. Minimal realization and fate of the instability

4. Premature end of inflation

Outline

• With an abrupt end of inflation, let us simply write that the

extra field had a positive mass at Hubble exit for the pivot scale:

M H? > 1 p2π2As 1 ⇣

mh H?

⌘ ' 5500 1 ⇣

mh H?

⌘

• Models with a lower value of the curvature scale generate a

universe with more structure than ours, so they are excluded!

extends to any (2-)field model and any dynamics

m2

h

H2

?

> 4✏? ✓MP M ◆2 As = 1 8⇡2✏? ✓ H? MP ◆2

+

CMB normalization

m2

s(eff)? > 0

A universal bound on the field space curvature

(Non)-decoupling and the field space curvature scale

(CMB normalization impossible)

(Non)-decoupling and the field space curvature scale

Strong selection criterion on high-energy interactions above H! Model-independent information about field space geometry, important in high-energy physics!

(CMB normalization impossible)

1 2 3 4 5 6 7 ϕ Mp 0.2 0.4 0.6 0.8 1.0 V(ϕ) Λ4

Standard end

• f inflation

Window on the potential probed by

• bservable modes

✏ ∼ 1

Premature end of inflation

1 2 3 4 5 6 7 ϕ Mp 0.2 0.4 0.6 0.8 1.0 V(ϕ) Λ4

Standard end

• f inflation

Window on the potential probed by

• bservable modes

✏ ∼ 1 ✏c ⌧ 1

New end of inflation New observational window

Premature end of inflation

1 2 3 4 5 6 7 ϕ Mp 0.2 0.4 0.6 0.8 1.0 V(ϕ) Λ4

Standard end

• f inflation

✏ ∼ 1 ✏c ⌧ 1

New end of inflation New observational window

Quite dramatic impact on

• bservables!

Premature end of inflation

1 2 3 4 5 6 7 ϕ Mp 0.2 0.4 0.6 0.8 1.0 V(ϕ) Λ4

Standard end

• f inflation

✏ ∼ 1 ✏c ⌧ 1

New end of inflation New observational window

Generic trend:

• Smaller amplitude of gravitational waves
• Closer to scale invariance

Premature end of inflation

Starobinsky potential

mh = 10Hc

Example:

Observational predictions

ns

✶ ▽ ▲ △

single-field =- =- =-

✶ ✶ ▽ ▽ ▲ ▲ △ △

0.92 0.94 0.96 0.98 1. 0.001 0.01 0.1

0.92 0.94 0.96 0.98 1.00 0.001 0.01 0.1

r

L

• w

e r M

2017 constraints

• Different classes of inflationary models are affected differently

by a premature end of inflation (hilltop, inflection points, plateau, large-field…)

• Effects is degenerate with theoretical uncertainties about

reheating

• Need for a full Bayesian analysis, consistently scanning over M

>> H and reheating parameters

arXiv:1706.01835 RP , Turzynski, Vennin, JCAP

Geometrical destabilization, premature end

• f inflation and Bayesian model selection

Bayesian evidences ln(E/ESI) and best fits ln(Lmax/Lmax

SI )

• 5
• 2.5 -1 0

0.0

• 3.06
• 5.23
• 3.08
• 1.58
• 1.99
• 5.08
• 6.45
• 2.71
• 2.34

SI LFI2 LFI4 NI SFI2 SIPEI LFIPEI

2

LFIPEI

4

NIPEI SFIPEI

2

• 5
• 2.5 -1 0
• 0.87
• 3.69

<-11.12 <-10.72

• 2.15
• 2.46
• 2.82
• 2.79

SFI4 CWIf MSSMIo RIPIo SFIPEI

4

CWIPEI

f

MSSMIPEI

• RIPIPEI
• Plateau

Large field Quadratic hilltop Quartic hilltop Inflection point

Reassessing the status of inflationary models

Bayesian evidences ln(E/ESI)

without premature end with premature end

Bayesian evidences ln(E/ESI) and best fits ln(Lmax/Lmax

SI )

• 5
• 2.5 -1 0

0.0

• 3.06
• 5.23
• 3.08
• 1.58
• 1.99
• 5.08
• 6.45
• 2.71
• 2.34

SI LFI2 LFI4 NI SFI2 SIPEI LFIPEI

2

LFIPEI

4

NIPEI SFIPEI

2

• 5
• 2.5 -1 0
• 0.87
• 3.69

<-11.12 <-10.72

• 2.15
• 2.46
• 2.82
• 2.79

SFI4 CWIf MSSMIo RIPIo SFIPEI

4

CWIPEI

f

MSSMIPEI

• RIPIPEI
• Plateau

Large field Quadratic hilltop Quartic hilltop Inflection point

Reassessing the status of inflationary models

Bayesian evidences ln(E/ESI)

without premature end with premature end

m2φ2 one of the worst models!

Inflection points models back in the game!

• Study of concrete models in the literature
• Similar discussion in N-field models, with (N-1) threats of tachyonic

instabilities, and the Ricci scalar replaced by relevant sectional curvatures

• Even more dramatic impact on models with masses of order the

Hubble parameter (typical in susy)

• Links with constraints on primordial non-Gaussianities
• Constraints on the internal geometry of HEP models, including string

compactifications, rare!

Perspectives and generalizations

In generic inflationary models in high-energy physics, there is the threat of an instability, so far overlooked, that:

• can prematurely end inflation (new mechanism)
• dramatically impacts observables
• modifies the interpretation of observations in terms of

fundamental physics (and hence the observational status

• f models)
• constrain HEP in a unique manner

Summary

• The geometrical destabilization can qualitatively change our

vision of inflation (e.g. landscapes ‘with trivial field space geometry for simplicity’ may not capture the correct physics)

• As important as the eta problem
• Exciting perspectives: new theoretical developments needed

Conclusion