A S IMPLE M ODEL OF I NFLATION ... Model with potential V ( ) V ( - - PowerPoint PPT Presentation

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A S IMPLE M ODEL OF I NFLATION ... Model with potential V ( ) V ( - - PowerPoint PPT Presentation

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P RIMORDIAL N ON -G AUSSIANITY FROM P REHEATING arXiv:0903.3407 (BFHK); arXiv:1004.3559 (REVIEW) Andrei Frolov Department of Physics Simon Fraser University & Lev Kofman, Dick Bond, Jonathan Braden, Zhiqi Huang (CITA) RESCEU Symposium on

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SLIDE 1

PRIMORDIAL NON-GAUSSIANITY

FROM PREHEATING

arXiv:0903.3407 (BFHK); arXiv:1004.3559 (REVIEW)

Andrei Frolov

Department of Physics Simon Fraser University

& Lev Kofman, Dick Bond, Jonathan Braden, Zhiqi Huang (CITA)

RESCEU Symposium on General Relativity and Gravitation University of Tokyo, Tokyo, Japan 13 November 2012

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 1 / 33

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SLIDE 2

BRIEF HISTORY OF THE UNIVERSE

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 2 / 33

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SLIDE 3

BRIEF HISTORY OF THE UNIVERSE

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 2 / 33

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SLIDE 4

BRIEF HISTORY OF THE UNIVERSE

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 2 / 33

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SLIDE 5

BRIEF HISTORY OF THE UNIVERSE

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 2 / 33

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SLIDE 6

A SIMPLE MODEL OF INFLATION...

Model with potential

V (φ,χ) = λ 4 φ4+ g 2 2 φ2χ2

is invariant under

g µν → a −2g µν, φ → aφ, χ → aχ

equation of state is 1/3

  • 1.0
  • 0.5

0.0 0.5 1.0 20 40 60 80 100 120 w

radiation Andrei Frolov (SFU)

Primordial Non-Gaussianity from Preheating JGRG22 3 / 33

φ V (φ)

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SLIDE 7

... ENDING VIA BROAD PARAMETRIC RESONANCE

Model with potential

V (φ,χ) = λ 4 φ4+ g 2 2 φ2χ2

is invariant under

g µν → a −2g µν, φ → aφ, χ → aχ

equation of state is 1/3

  • 1.0
  • 0.5

0.0 0.5 1.0 20 40 60 80 100 120 w

radiation

Stability of Lame equation:

2 4 6 8 10 0.0 0.5 1.0 1.5 2.0 2.5 3.0

(aχk)′′ +

  • κ2 + g 2

λ cn2 x,2−1/2 (aχk) = 0

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 3 / 33

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SLIDE 8

DEVELOPMENT OF LINEAR INSTABILITY

g 2/λ = 2.99, κ2 = 0 g 2/λ = 3.01, κ2 = 0

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 4 / 33

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SLIDE 9

PARAMETRIC RESONANCE IS A GENERIC FEATURE

χ′′

k + κ2 +q cos2(τ)χk = 0

χ′′

k + κ2 +q cn2 τ,2−1/2χk = 0

κ2 = k 2/(m 2a 2)

1 2 3 4 5 6 7 8 9 10 11 12 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

κ2 = k 2/(λΦ2

0)

1 2 3 4 5 6 7 8 9 10 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

q = g 2Φ2

0/(m 2a 3)

q = g 2/λ

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 5 / 33

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SLIDE 10

DEFROST: A NEW 3D NUMERICAL SOLVER

c0 c1 c1 c1 c1 c1 c1 c2 c2 c2 c2 c2 c2 c2 c2 c2 c2 c2 c2 c3 c3 c3 c3 c3 c3 c3 c3

coefficient

c3 c2 c1 −c0

degeneracy

8 12 6 1

standard

1 6

isotropic A

1 6 1 3

4

isotropic B

1 12 2 3 14 3

isotropic C

1 30 1 10 7 15 64 15

http://www.sfu.ca/physics/cosmology/defrost

[Fortran-90, 600 lines, very fast, instrumented for 3D]

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 6 / 33

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SLIDE 11

FIELD EVOLUTION AS A CHAOTIC BILLIARD

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 7 / 33

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SLIDE 12

DENSITY EVOLUTION [V (φ,χ) = 1

4 λφ4 + 1 2 g 2φ2χ2]

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 8 / 33

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SLIDE 13

DENSITY EVOLUTION [V (φ,χ) = 1

2 m 2φ2 + 1 2 g 2φ2χ2]

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 9 / 33

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SLIDE 14

DENSITY EVOLUTION [V (φ,χ) = 1

2 m 2φ2 + 1 2 σφχ2 + 1 4 λχ4]

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 10 / 33

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SLIDE 15

HERE IS HOW INSTABILITY DEVELOPS!

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 11 / 33

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SLIDE 16

IN FACT, IT’S QUITE LOG-NORMAL!

V (φ,χ) = 1 4 λφ4 + 1 2 g 2φ2χ2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.1 1 10 P(ρ) ρ/3H2 simulation data log-normal fit

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 12 / 33

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SLIDE 17

IN FACT, IT’S UNIVERSALLY LOG-NORMAL!

V (φ,χ) = 1 2 m 2φ2 + 1 2 g 2φ2χ2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.1 1 10 P(ρ) ρ/3H2 simulation data log-normal fit

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 13 / 33

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SLIDE 18

IN FACT, IT’S UNIVERSALLY LOG-NORMAL!

V (φ,ψ) = 1 2 m 2φ2 + 1 2 σφχ2 + 1 4 λχ4

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.1 1 10 P(ρ) ρ/3H2 simulation data log-normal fit

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 14 / 33

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SLIDE 19

(NON-THERMAL) SCALAR FIELD FIXED POINT?

V (φ,χ) = 1 4 λ

  • φ2 + χ2 − v 22
  • 1.00
  • 0.75
  • 0.50
  • 0.25

0.00 0.25 0.50 0.75 1.00 100 200 300 400 500 w t

dust radiation

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 15 / 33

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SLIDE 20

THIS SIMPLY CAN’T BE JUST A COINCIDENCE...

A universal characteristic of random scalar field evolution?! What’s Going On Here? Very tempting to blame scalar field turbulence!..

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 16 / 33

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SLIDE 21

THIS SIMPLY CAN’T BE JUST A COINCIDENCE...

A universal characteristic of random scalar field evolution?! What’s Going On Here? Very tempting to blame scalar field turbulence!..

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 16 / 33

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SLIDE 22

ARE WE GOING TO EVER SEE ANY OF THIS?

Thermal Bath Wipes Out Everything?! Scales Too Small?!

WE SHOULD LOOK FOR THINGS THAT CAN SURVIVE THERMALIZATION:

1

stable relics (primordial black holes)

2

decoupled fields (gravitational waves)

3

anomalies in expansion history (non-gaussianity)

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 17 / 33

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SLIDE 23

ARE WE GOING TO EVER SEE ANY OF THIS?

Thermal Bath Wipes Out Everything?! Scales Too Small?!

WE SHOULD LOOK FOR THINGS THAT CAN SURVIVE THERMALIZATION:

1

stable relics (primordial black holes)

2

decoupled fields (gravitational waves)

3

anomalies in expansion history (non-gaussianity)

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 17 / 33

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SLIDE 24

ARE WE GOING TO EVER SEE ANY OF THIS?

Thermal Bath Wipes Out Everything?! Scales Too Small?!

WE SHOULD LOOK FOR THINGS THAT CAN SURVIVE THERMALIZATION:

1

stable relics (primordial black holes)

2

decoupled fields (gravitational waves)

3

anomalies in expansion history (non-gaussianity)

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 17 / 33

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SLIDE 25

CAN PRIMORIDAL BLACK HOLES FORM? NO...

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 18 / 33

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SLIDE 26

CAN PRIMORIDAL BLACK HOLES FORM? NO...

  • 5
  • 4
  • 3
  • 2
  • 1

1 2 20 40 60 80 100 120 Ψ ∗ 103 t limits 95% 68% median

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 18 / 33

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SLIDE 27

GRAVITATIONAL WAVES FROM PREHEATING?

Dufaux, Felder, Kofman & Navros (2009)

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 19 / 33

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SLIDE 28

NON-GAUSSIANITY FROM PREHEATING?

CMB and Reheating scales different by 50+ e-folds!

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 20 / 33

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SLIDE 29

NON-GAUSSIANITY FROM PREHEATING?

CMB and Reheating scales different by 50+ e-folds!

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 20 / 33

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SLIDE 30

DIFFERENCES IN EXPANSION CAN MODULATE CMB

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 21 / 33

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SLIDE 31

MODULATION COMES FROM ISOCURVATURE MODE

  • ln(kHend)

k

k Hend

  • g

=

g

=

g

=

g

=

g

=

g

=

g

=

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 22 / 33

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SLIDE 32

EVOLUTION DEPENDS ON INITIAL ISOCON VALUE

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 23 / 33

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SLIDE 33

SYMPLECTIC INTEGRATOR TO THE RESCUE!

Problem:

long-term oscillator evolution

H = p 2 2 + q 4 4 4th order Runge-Kutta

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 24 / 33

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SLIDE 34

SYMPLECTIC INTEGRATOR TO THE RESCUE!

Problem: Solution:

long-term oscillator evolution enforce energy conservation

H = p 2 2 + q 4 4 e A t /2e B t e A t /2 = e (A+B)t +O(t 3) 4th order Runge-Kutta 4th order Symplectic

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 24 / 33

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SLIDE 35

ISOCURVATURE MODE CONVERTS TO CURVATURE

  • 2.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0 0.1 1 10 100 δN = ln(aend/aref) * 105 (χini/mpl) * 107

Are these peaks Real? Yes...

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 25 / 33

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SLIDE 36

ISOCURVATURE MODE CONVERTS TO CURVATURE

  • 2.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0 0.1 1 10 100 δN = ln(aend/aref) * 108 (χini/mpl) * 107

Are these peaks Real? Yes...

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 25 / 33

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SLIDE 37

PRIMORDIAL NON-GAUSSIANITY IS PRODUCED!

  • 2.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0 0.1 1 10 100 δN = ln(aend/aref) * 105 (χini/mpl) * 107

Φ( x) = ΦG( x) + FNL(χG)

very different from f NL parametrization!

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 25 / 33

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SLIDE 38

HERE IS A TRAJECTORY FROM THE PEAK

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 26 / 33

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SLIDE 39

A SIMPLE ANALYTICAL MODEL

  • 3.0
  • 2.0
  • 1.0

0.0 1.0 2.0 3.0

  • 10
  • 5

5 10 aχ

V

eff(φ,χ) = λ

4 φ4 + g 2 2

  • φ2 + 〈δφ2〉
  • χ2

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 27 / 33

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SLIDE 40

TIME EVOLUTION OF EFFECTIVE POTENTIAL

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 28 / 33

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SLIDE 41

THIS EXPLAINS WHY PEAKS ARE LOG-PERIODIC!

  • 2.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0 0.1 1 10 100 δN = ln(aend/aref) * 105 (χini/mpl) * 107 µ0T periodicity and its harmonics

φ(t + T ) = φ(t ) χ(t + T ) = χ(t )e µ0T

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 29 / 33

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SLIDE 42

LET’S EXPLORE THE PARAMETER SPACE...

11 10 9 8 7 ln(χ0/φ0)/µ0T 1.4 1.6 1.8 2.0 2.2 2.4 2.6 g2/λ

φ(t + T ) = φ(t ) χ(t + T ) = χ(t )e µ0T

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 29 / 33

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SLIDE 43

LET’S EXPLORE THE PARAMETER SPACE...

11 10 9 8 7 ln(χ0/φ0)/µ0T 1.4 1.6 1.8 2.0 2.2 2.4 2.6 g2/λ

φ(t + T ) = φ(t ) χ(t + T ) = χ(t )e µ0T

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 29 / 33

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SLIDE 44

LET’S EXPLORE THE PARAMETER SPACE...

11 10 9 8 7 ln(χ0/φ0)/µ0T 1.4 1.6 1.8 2.0 2.2 2.4 2.6 g2/λ

φ(t + T ) = φ(t ) χ(t + T ) = χ(t )e µ0T

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 29 / 33

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SLIDE 45

LET’S EXPLORE THE PARAMETER SPACE...

11 10 9 8 7 ln(χ0/φ0)/µ0T 1.4 1.6 1.8 2.0 2.2 2.4 2.6 g2/λ

φ(t + T ) = φ(t ) χ(t + T ) = χ(t )e µ0T

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 29 / 33

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SLIDE 46

... AND HERE’S HOW PEAKS SPLIT & MULTIPLY!

11 10 9 8 7 ln(χ0/φ0)/µ0T 1.4 1.6 1.8 2.0 2.2 2.4 2.6 g2/λ

φ(t + T ) = φ(t ) χ(t + T ) = χ(t )e µ0T

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 29 / 33

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SLIDE 47

... AND HERE’S HOW PEAKS SPLIT & MULTIPLY!

11 10 9 8 7 ln(χ0/φ0)/µ0T 1.4 1.6 1.8 2.0 2.2 2.4 2.6 g2/λ

φ(t + T ) = φ(t ) χ(t + T ) = χ(t )e µ0T

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 29 / 33

slide-48
SLIDE 48

... AND HERE’S HOW PEAKS SPLIT & MULTIPLY!

11 10 9 8 7 ln(χ0/φ0)/µ0T 1.4 1.6 1.8 2.0 2.2 2.4 2.6 g2/λ

φ(t + T ) = φ(t ) χ(t + T ) = χ(t )e µ0T

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 29 / 33

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SLIDE 49

... AND HERE’S HOW PEAKS SPLIT & MULTIPLY!

11 10 9 8 7 ln(χ0/φ0)/µ0T 1.4 1.6 1.8 2.0 2.2 2.4 2.6 g2/λ

φ(t + T ) = φ(t ) χ(t + T ) = χ(t )e µ0T

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 29 / 33

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SLIDE 50

... AND HERE’S HOW PEAKS SPLIT & MULTIPLY!

11 10 9 8 7 ln(χ0/φ0)/µ0T 1.4 1.6 1.8 2.0 2.2 2.4 2.6 g2/λ

φ(t + T ) = φ(t ) χ(t + T ) = χ(t )e µ0T

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 29 / 33

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SLIDE 51

... AND HERE’S HOW PEAKS SPLIT & MULTIPLY!

11 10 9 8 7 ln(χ0/φ0)/µ0T 1.4 1.6 1.8 2.0 2.2 2.4 2.6 g2/λ

φ(t + T ) = φ(t ) χ(t + T ) = χ(t )e µ0T

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 29 / 33

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SLIDE 52

... AND HERE’S HOW PEAKS SPLIT & MULTIPLY!

11 10 9 8 7 ln(χ0/φ0)/µ0T 1.4 1.6 1.8 2.0 2.2 2.4 2.6 g2/λ

φ(t + T ) = φ(t ) χ(t + T ) = χ(t )e µ0T

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 29 / 33

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SLIDE 53

PREHEATING CAN LEAVE A SIGNATURE IN CMB SKY!

the way inflation ends can lead to a new signal:

usual non-Gaussianity:

Φ( x) = ΦG( x) + f NLΦ2

G(

x)

new from preheating:

Φ( x) = ΦG( x) + FNL(χG) FNL can be a very non-trivial function:

  • 2.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0 0.1 1 10 100 δN = ln(aend/aref) * 105 (χini/mpl) * 107

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 30 / 33

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SLIDE 54

PREHEATING CAN LEAVE A SIGNATURE IN CMB SKY!

the way inflation ends can lead to a new signal:

usual non-Gaussianity:

Φ( x) = ΦG( x) + f NLΦ2

G(

x)

new from preheating:

Φ( x) = ΦG( x) + FNL(χG) FNL can be a very non-trivial function:

  • 2.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0 0.1 1 10 100 δN = ln(aend/aref) * 105 (χini/mpl) * 107

realization of excursion set can naturally give cold spots!

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 30 / 33

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SLIDE 55

HOW WOULD IT LOOK LIKE ON THE SKY?

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 31 / 33

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SLIDE 56

HOW WOULD IT LOOK LIKE ON THE SKY?

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 31 / 33

slide-57
SLIDE 57

HOW WOULD IT LOOK LIKE ON THE SKY?

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 31 / 33

slide-58
SLIDE 58

HOW WOULD IT LOOK LIKE ON THE SKY?

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 31 / 33

slide-59
SLIDE 59

HOW WOULD IT LOOK LIKE ON THE SKY?

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 31 / 33

slide-60
SLIDE 60

HOW WOULD IT LOOK LIKE ON THE SKY?

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 31 / 33

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SLIDE 61

ERR, THERE IS A COLD SPOT ON CMB SKY!..

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 32 / 33

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SLIDE 62

ERR, THERE IS A COLD SPOT ON CMB SKY!..

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 32 / 33

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SLIDE 63

ERR, THERE IS A COLD SPOT ON CMB SKY!..

0.00 0.25 0.50 0.75 1.00

  • 6σ -5σ -4σ -3σ -2σ -1σ

0σ 1σ 2σ 3σ 4σ 5σ 6σ SSG42 filter at 800.0 arcmin FWHM 105 peaks, coldest at (-0.14-4.73)σ significance 1:290.3 L1-SSG42-400 data Gaussian peak stats

  • 0.75
  • 0.50
  • 0.25

0.00 0.25 0.50 0.75 1.00 Kolmogorov deviation from Gaussian peak CDF

  • 5
  • 4
  • 3
  • 2
  • 4.87σ

log10 CDF

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 32 / 33

slide-64
SLIDE 64

POLARIZATION WILL TELL IF IT’S PRIMORDIAL!

Komatsu et. al. (2010)

Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 33 / 33