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 General Relativity and Gravitation University of Tokyo, Tokyo, Japan 13 November 2012 Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 1 / 33
B RIEF H ISTORY OF THE U NIVERSE Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 2 / 33
B RIEF H ISTORY OF THE U NIVERSE Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 2 / 33
B RIEF H ISTORY OF THE U NIVERSE Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 2 / 33
B RIEF H ISTORY OF THE U NIVERSE Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 2 / 33
A S IMPLE M ODEL OF I NFLATION ... Model with potential V ( φ ) V ( φ , χ ) = λ 4 φ 4 + g 2 2 φ 2 χ 2 is invariant under a − 2 g µν , g µν �→ φ a φ , �→ χ �→ a χ equation of state is 1 / 3 1.0 0.5 radiation 0.0 w -0.5 φ -1.0 0 20 40 60 80 100 120 Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 3 / 33
... E NDING VIA B ROAD P ARAMETRIC R ESONANCE Stability of Lame equation: Model with potential V ( φ , χ ) = λ 4 φ 4 + g 2 3.0 2 φ 2 χ 2 2.5 is invariant under 2.0 a − 2 g µν , g µν �→ 1.5 φ a φ , �→ χ �→ a χ 1.0 equation of state is 1 / 3 0.5 1.0 0.0 0.5 0 2 4 6 8 10 radiation 0.0 w κ 2 + g 2 � x ,2 − 1 / 2 �� -0.5 ( a χ k ) ′′ + λ cn 2 � ( a χ k ) = 0 -1.0 0 20 40 60 80 100 120 Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 3 / 33
D EVELOPMENT OF L INEAR I NSTABILITY g 2 /λ = 2.99 , κ 2 = 0 g 2 /λ = 3.01 , κ 2 = 0 Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 4 / 33
P ARAMETRIC R ESONANCE IS A G ENERIC F EATURE k + � κ 2 + q cos 2 ( τ ) � χ k = 0 k + � κ 2 + q cn 2 � τ ,2 − 1 / 2 �� χ k = 0 χ ′′ χ ′′ 5.5 4.0 5.0 3.5 κ 2 = k 2 / ( m 2 a 2 ) 4.5 0 ) κ 2 = k 2 / ( λ Φ 2 3.0 4.0 3.5 2.5 3.0 2.0 2.5 1.5 2.0 1.5 1.0 1.0 0.5 0.5 0.0 0.0 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 2 3 4 5 6 7 8 9 10 q = g 2 Φ 2 0 / ( m 2 a 3 ) q = g 2 /λ Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 5 / 33
DEFROST: A N EW 3D N UMERICAL S OLVER c 3 c 2 c 3 c 2 c 1 c 2 c 3 c 2 c 3 coefficient c 3 c 2 c 1 − c 0 8 12 6 1 degeneracy c 2 c 1 c 2 standard 0 0 1 6 c 1 c 0 c 1 1 1 0 4 isotropic A c 2 c 1 c 2 6 3 1 2 14 0 isotropic B 12 3 3 1 1 7 64 isotropic C c 3 c 2 c 3 30 10 15 15 c 2 c 1 c 2 c 3 c 2 c 3 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
F IELD E VOLUTION AS A C HAOTIC B ILLIARD Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 7 / 33
D ENSITY E VOLUTION [ V ( φ , χ ) = 1 2 g 2 φ 2 χ 2 ] 4 λφ 4 + 1 Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 8 / 33
D ENSITY E VOLUTION [ V ( φ , χ ) = 1 2 g 2 φ 2 χ 2 ] 2 m 2 φ 2 + 1 Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 9 / 33
D ENSITY E VOLUTION [ V ( φ , χ ) = 1 4 λχ 4 ] 2 m 2 φ 2 + 1 2 σφχ 2 + 1 Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 10 / 33
H ERE IS HOW INSTABILITY DEVELOPS ! Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 11 / 33
I N F ACT , I T ’ S Q UITE L OG -N ORMAL ! V ( φ , χ ) = 1 4 λφ 4 + 1 2 g 2 φ 2 χ 2 1.0 simulation data 0.9 log-normal fit 0.8 0.7 0.6 P( ρ ) 0.5 0.4 0.3 0.2 0.1 0.0 0.1 1 10 ρ /3H 2 Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 12 / 33
I N F ACT , I T ’ S U NIVERSALLY L OG -N ORMAL ! V ( φ , χ ) = 1 2 m 2 φ 2 + 1 2 g 2 φ 2 χ 2 1.0 simulation data 0.9 log-normal fit 0.8 0.7 0.6 P( ρ ) 0.5 0.4 0.3 0.2 0.1 0.0 0.1 1 10 ρ /3H 2 Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 13 / 33
I N F ACT , I T ’ S U NIVERSALLY L OG -N ORMAL ! V ( φ , ψ ) = 1 2 m 2 φ 2 + 1 2 σφχ 2 + 1 4 λχ 4 1.0 simulation data 0.9 log-normal fit 0.8 0.7 0.6 P( ρ ) 0.5 0.4 0.3 0.2 0.1 0.0 0.1 1 10 ρ /3H 2 Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 14 / 33
(N ON -T HERMAL ) S CALAR F IELD F IXED P OINT ? V ( φ , χ ) = 1 φ 2 + χ 2 − v 2 � 2 � 4 λ 1.00 0.75 0.50 radiation 0.25 dust w 0.00 -0.25 -0.50 -0.75 -1.00 0 100 200 300 400 500 t Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 15 / 33
T HIS S IMPLY CAN ’ T BE J UST A C OINCIDENCE ... 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
T HIS S IMPLY CAN ’ T BE J UST A C OINCIDENCE ... 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
A RE W E G OING TO E VER S EE A NY OF T HIS ? Thermal Bath Wipes Out Everything?! Scales Too Small?! W E SHOULD LOOK FOR THINGS THAT CAN SURVIVE THERMALIZATION : stable relics (primordial black holes) 1 decoupled fields (gravitational waves) 2 anomalies in expansion history (non-gaussianity) 3 Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 17 / 33
A RE W E G OING TO E VER S EE A NY OF T HIS ? Thermal Bath Wipes Out Everything?! Scales Too Small?! W E SHOULD LOOK FOR THINGS THAT CAN SURVIVE THERMALIZATION : stable relics (primordial black holes) 1 decoupled fields (gravitational waves) 2 anomalies in expansion history (non-gaussianity) 3 Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 17 / 33
A RE W E G OING TO E VER S EE A NY OF T HIS ? Thermal Bath Wipes Out Everything?! Scales Too Small?! W E SHOULD LOOK FOR THINGS THAT CAN SURVIVE THERMALIZATION : stable relics (primordial black holes) 1 decoupled fields (gravitational waves) 2 anomalies in expansion history (non-gaussianity) 3 Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 17 / 33
C AN P RIMORIDAL B LACK H OLES F ORM ? N O ... Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 18 / 33
C AN P RIMORIDAL B LACK H OLES F ORM ? N O ... 2 1 Ψ ∗ 10 3 0 -1 -2 -3 limits 95% -4 68% median -5 0 20 40 60 80 100 120 t Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 18 / 33
G RAVITATIONAL W AVES FROM P REHEATING ? Dufaux, Felder, Kofman & Navros (2009) Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 19 / 33
N ON -G AUSSIANITY FROM P REHEATING ? CMB and Reheating scales different by 50 + e-folds! Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 20 / 33
N ON -G AUSSIANITY FROM P REHEATING ? CMB and Reheating scales different by 50 + e-folds! Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 20 / 33
D IFFERENCES IN E XPANSION CAN M ODULATE CMB Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 21 / 33
M ODULATION C OMES FROM I SOCURVATURE M ODE � �� � �� = ��� g � �� � �� = ��� �� � g � �� = ��� �� � g � � H end � �� � �� = ��� g � �� k � �� � �� = ��� � k g �� �� � �� = ��� g �� �� �� �� � �� = ��� g �� �� ���� ��� � ln(k � H end ) Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 22 / 33
E VOLUTION D EPENDS ON I NITIAL I SOCON V ALUE Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 23 / 33
S YMPLECTIC I NTEGRATOR TO THE R ESCUE ! Problem: long-term oscillator evolution H = p 2 2 + q 4 4 4 th order Runge-Kutta Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 24 / 33
S YMPLECTIC I NTEGRATOR TO THE R ESCUE ! Problem: Solution: long-term oscillator evolution enforce energy conservation H = p 2 2 + q 4 e A t / 2 e B t e A t / 2 = e ( A + B ) t + O ( t 3 ) 4 4 th order Runge-Kutta 4 th order Symplectic Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 24 / 33
I SOCURVATURE M ODE C ONVERTS TO C URVATURE 12.0 10.0 δ N = ln( a end / a ref ) * 10 5 8.0 6.0 4.0 2.0 0.0 -2.0 0.1 1 10 100 ( χ ini / m pl ) * 10 7 Are these peaks Real? Yes... Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 25 / 33
I SOCURVATURE M ODE C ONVERTS TO C URVATURE 12.0 10.0 δ N = ln( a end / a ref ) * 10 8 8.0 6.0 4.0 2.0 0.0 -2.0 0.1 1 10 100 ( χ ini / m pl ) * 10 7 Are these peaks Real? Yes... Andrei Frolov (SFU) Primordial Non-Gaussianity from Preheating JGRG22 25 / 33
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