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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
Dong-Gang Wang Leiden University
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
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horizon-exit
decoherences t r e t c h
recombination
reionization
clustering
The success of single field slow-roll inflation
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✦ 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]
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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]
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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
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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:
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The stretching effect of the hyperbolic geometry
11
The simplest inflation model as an example
Stretching flattens your potential
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Stretching! Stretching!! Stretching!!!
12
An original random potential on the
Poincare disk
The stretched potential with canonical field
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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?
✓
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The angular dependence of the potential
15
A brief review of multi-field inflation
Intro to multi-field inflation
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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
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curvature modes
isocurvature modes
perturbations along the trajectory
perturbations orthogonal to the trajectory {
turning trajectory ⌦ 6= 0
⇣̇ =2⌦p2✏
�coupled evolution
⇣ = H⇡
��
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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
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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?
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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
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Rolling in the deep?
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Rolling on the ridge
25
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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
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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…
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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
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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!
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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
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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
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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
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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
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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
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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
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Multi-field 𝜶-attractors: preheating
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
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Multi-field 𝜶-attractors: preheating
37
analytical approach — scaling behaviour
�end ' 3p↵
Hend ⇠p↵
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Multi-field 𝜶-attractors: preheating
38
Floquet analysis
Krajewski, Turzynski,︎ Wieczorek 2018
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Multi-field 𝜶-attractors: preheating
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↵)
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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
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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