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)
Outline 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
Inflation: a phenomenological success Planck all sky map, 2015
Inflation: a phenomenological success Primordial fluctuations are adiabatic, super horizon at recombination, almost scale-invariant, Gaussian. l ( l + 1) C l / 2 π [ µK 2 ] Planck 2015 data The simplest models, single-field slow-roll, economically explain all current data
The Planck ns-r plane r = 16 ✏ ? n s − 1 = 2 ⌘ ? − 6 ✏ ?
Beyond toy-models? • So far, merely phenomenological description. • Inflation is sensitive to the physics at the Planck scale - eta-problem m 2 φ Why is the inflaton so light? H 2 ⌧ 1 η ⇡ like the Higgs hierarchy problem uv � H 2 m 2 φ ∼ Λ 2 - multiple (heavy) fields in ultraviolet completions of slow-roll single-field inflation do not decouple - Inflation is not in general predictive without reheating
UV-sensitivity of inflation L = − 1 O δ ( φ ) X 2( ∂φ ) 2 − V 0 ( φ ) + Λ δ − 4 δ Corrections to the low-energy Slow-roll action effective action Unless symmetry forbids it, ∆ V = cV 0 ( φ ) φ 2 presence of terms of the form Λ 2 ◆ 2 ✓ M P φ ∼ c V 0 ∆ m 2 Λ 2 ∼ c H 2 Λ Wilson coefficient ∆ η & 1 c ∼ O (1) Sensitivity of inflation to + Planck-suppressed operators Λ . M P
Outline 1. Inflation 2. General mechanism of the geometrical destabilization 3. Minimal realization and fate of the instability 4. Premature end of inflation
Basic idea Realistic inflationary models have fields which live in an internal space with curved geometry. Initially neighboring geodesics tend to fall away from each other in the presence of negative curvature. This effect applies during inflation, it easily overcomes the effect of the potential, and can destabilize inflationary trajectories.
Basic mechanism Renaux-Petel, Turzynski, September 2016 PRL Editors’ Highlight V ( φ 1 , φ 2 ) Simplest ‘realistic’ models (hope): Light inflaton + Extra heavy fields φ 2 Effective single-field dynamics (valley with steep walls) φ 1
Basic mechanism Renaux-Petel, Turzynski, September 2016 PRL Editors’ Highlight V ( φ 1 , φ 2 ) More realistic: Light inflaton + Extra heavy fields + φ 2 Curved field space Geometrical instability φ 1
Basic mechanism Renaux-Petel, Turzynski, September 2016 PRL Editors’ Highlight V ( x 1 , x 2 ) Simple analogy: - Position of a charged particle - Electric force x 2 - Surface geometry Geometrical instability x 1
Multifield Lagrangian ✓ ◆ Z − 1 d 4 x √− g 2 G IJ ( φ K ) ∂ µ φ I ∂ µ φ J − V ( φ I ) S = 1. A curved field space is generic Top-down (e.g. supergravity), or bottom-up (EFT) Field space curvature ∼ 1 /M 2 2. A priori, M can lie anywhere between H and Mp Pl = − 2 R field space M 2 Example: alpha-attractors 3 α
Linear perturbation theory D t D t Q I + 3 H D t Q I + k 2 a 2 Q I + M I J Q J = 0 Sasaki, Stewart, 95 Q I = fluctuations of field I in flat gauge D t A I = ˙ A I + Γ I JK ˙ φ J A K Mass matrix: ✓ a 3 ◆ 1 φ K ˙ φ I ˙ KLJ ˙ ˙ M I J = V I ; J − R I φ L − D t φ J a 3 M 2 H Pl Riemann curvature tensor cf geodesic deviation equation of the field space metric
Two-field models (simplicity) super-Hubble evolution of the entropic field Q s + 3 H ˙ ¨ Q s + m 2 s (e ff ) Q s = 0 Effective entropic mass squared: Gordon et al, 2000 m 2 ≡ V ; ss s (e ff ) H 2 + 3 ⌘ 2 ⊥ + ✏ R field space M 2 Pl H 2 bending ‘geometrical’ Hessian contribution contribution contribution
Geometrical destabilization m 2 ≡ V ; ss s (e ff ) H 2 + 3 ⌘ 2 ⊥ + ✏ R field space M 2 Pl H 2 bending ‘geometrical’ Hessian contribution contribution contribution 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.
Geometrical destabilization R field space < 0 Necessary condition (2-field): generically Pl ∼ ( M Pl /M ) 2 R field space M 2 � 1 (string scale, Let us consider M = O (10 − 2 , 10 − 3 ) M Pl KK scale, for instance GUT scale...) The effective mass V ; ss becomes tachyonic when: Even for H 2 ∼ 100 ✏ → ✏ c = 10 − 4 10 − 2 or
Outline 1. Inflation 2. General mechanism of the geometrical destabilization 3. Minimal realization and fate of the instability 4. Premature end of inflation
Minimal realization 1 + 2 χ 2 ✓ ◆ L = − 1 − V ( φ ) − 1 2( ∂χ ) 2 − 1 2( ∂φ ) 2 2 m 2 h χ 2 M 2 • Slow-roll model of inflation, with inflaton φ χ with • Heavy field m 2 h � H 2 • Simple dimension 6 operator suppressed by a mass scale of new physics M � H
Minimal realization 1 + 2 χ 2 ✓ ◆ L = − 1 − V ( φ ) − 1 2( ∂χ ) 2 − 1 2( ∂φ ) 2 2 m 2 h χ 2 M 2 • Generally expected from the effective theory point of view (respect approximate shift-symmetry of inflaton) • Terms linear in chi absent for consistency (or Z2 symmetry), and higher-orders in chi suppressed near the inflationary valley • 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 1 + 2 χ 2 ✓ ◆ L = − 1 − V ( φ ) − 1 2( ∂χ ) 2 − 1 2( ∂φ ) 2 2 m 2 h χ 2 M 2 • Apparently benign high-energy correction (small correction to the kinetic term) but ... R field space ' � 4 for χ ⌧ M M 2 m 2 ◆ 2 = m 2 ✓ M Pl s (e ff ) along h χ = 0 H 2 − 4 ✏ ( t ) H 2 M • The inflationary trajectory becomes unstable after ✏ → ✏ c
Similarity with the eta-problem O i [ φ I , ∂φ I , . . . ] X L e ff [ φ I ] = L l [ φ I ] + c i Λ δ i − 4 i Corrections to the low-energy Slow-roll action effective action ∆ L = c ( ∂φ ) 2 χ 2 Unless symmetry forbids it, presence of terms of the form Λ 2 ◆ 2 χ ∼ c ( @� ) 2 ✓ M P ∆ m 2 ∼ c ✏ H 2 Λ 2 Λ Geometrical destabilization of inflation Λ ⌧ M P ✏ c ⌧ 1 Modified reheating ✏ c ∼ 1 Λ ' M P
Fate of the instability? Rapid and efficient growth of super-Hubble entropic fluctuations Example: P Q s ( k, N ) Starobinsky potential P Q s ( k, N c ) m h = 10 H c 10 7 10 7 M = 10 − 2 M Pl 10 5 Numerical resolution (linear theory) 10 3 1000 Theoretical modeling (early time): 10 N − N c 1 m 2 c η c ( N − N c ) 3 1 h 496.5 497.0 497.5 498.0 498.5 499.0 H 2 3 ∼ e 0 1 2 3
Fate of the instability? • Backreaction of fluctuations on background trajectory? Non-perturbative phenomenon • Similar to hybrid inflation (but different kinetic origin and kinetic effects). • Tachyonic preheating, possible production of primordial black holes, inflating topological defects … Challenging! Work in progress
Fate of the instability? Inhomogeneities dominate Inhomogeneities are shut off OR Premature end of inflation Second phase of inflation - Universal bound on curvature scale RP , Turzynski, 1510.01281, PRL RP , Turzynski, 1510.01281 - Modified ranking of inflationary models Works to appear 1706.01835 JCAP RP , Turzynski, Vennin
Outline 1. Inflation 2. General mechanism of the geometrical destabilization 3. Minimal realization and fate of the instability 4. Premature end of inflation
A universal bound on the field space curvature • 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: ✓ H ? ◆ 2 ◆ 2 m 2 ✓ M P 1 + h A s = > 4 ✏ ? H 2 8 ⇡ 2 ✏ ? M P M ? m 2 CMB normalization s (e ff ) ? > 0 M 1 1 1 extends to any > ⌘ ' 5500 p 2 π 2 A s (2-)field model ⇣ ⇣ ⌘ H ? m h m h and any dynamics H ? H ? • Models with a lower value of the curvature scale generate a universe with more structure than ours, so they are excluded!
(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)
Premature end of inflation V ( ϕ ) Λ 4 1.0 0.8 0.6 0.4 Window on the potential probed by ✏ ∼ 1 observable modes 0.2 ϕ 1 2 3 4 5 6 7 Mp Standard end of inflation
Premature end of inflation New end of V ( ϕ ) inflation Λ 4 ✏ c ⌧ 1 1.0 New observational 0.8 window 0.6 0.4 Window on the potential probed by ✏ ∼ 1 observable modes 0.2 ϕ 1 2 3 4 5 6 7 Mp Standard end of inflation
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