This is a short talk I gave at the National Astronomy Meeting 2011 in Llandudno, Wales.

1. Second Order Perturbations During
Inflation Beyond Slow-roll
Ian Huston
Astronomy Unit, Queen Mary, University of London
In Collaboration with Karim Malik (QMUL)
arXiv:1103.0912 and 0907.2917 (JCAP 0909:019)
Software available at http://pyflation.ianhuston.net

2. Faucher-Gigure et al., Science 2008

3. perturbations
Long review: Malik & Wands 0809.4944
Short technical review: Malik & Matravers 0804.3276

4. Separate quantities into
background and
perturbation.

5. ϕ(t, x) = ϕ0(t) + δϕ1(t, x)
1
+ δϕ2(t, x)
2
+ ...

6. δGµν = 8πGδTµν
⇓
Eqs of Motion

7. code():
Papers: 1103.0912, 0907.2917
Software: http://pyflation.ianhuston.net
2nd order equations: Malik astro-ph/0610864, JCAP

8. Non-linear processes:
Non-Gaussianity of CMB
Vorticity generation (See Adam’s poster)
Magnetic field generation
2nd order Gravitational waves

9. Other Approaches:
δN formalism
Lyth, Malik, Sasaki a-ph/0411220, etc.
In-In formalism
Maldacena a-ph/0210603, etc.
Moment transport equations
Mulryne, Seery, Wesley 0909.2256, 1008.3159

10. pyflation():
python & numpy
parallel
open source
Following Salopek et al. PRD40 1753, Martin &
Ringeval a-ph/0605367

11. Single field slow roll
Single field full equation
Multi-field calculation

12. k j qj δϕ1(q i)δϕ1(k i − q i)d3q

13. Bump potential
1 ϕ − ϕb
Vb (ϕ) = m2 ϕ2 1 + c sech
2 d
×10−5
10−2
Full Bump Potential
Half Bump Potential
Zero Bump Potential
3.1
−1/2
−1/2
10−3
k 3/2 |δϕ1 |/MPL
k 3/2 |δϕ1 |/MPL
3.0
2.9
10−4
2.8 Full Bump Potential
Half Bump Potential
Zero Bump Potential
10−5 2.7
60 50 40 30 20 10 0 57 56 55 54 53
Nend − N Nend − N

14. Source term
δϕ2 (k i ) + 2Hδϕ2 (k i ) + Mδϕ2 (k i ) = S(k i )
10−1
Full Bump Potential
10 −3 Half Bump Potential
Zero Bump Potential
10−5
10−7
−2
|S|/MPL
10−9
10−11
10−13
10−15
60 50 40 30 20 10 0
Nend − N

15. Second order perturbation
Full Bump Potential
Half Bump Potential
10−5 Zero Bump Potential
−2
|δϕ2 (k)|/MPL
10−7
10−9
60 50 40 30 20 10 0
Nend − N

16. Second order perturbation
×10−7
Full Bump Potential
2.60 Half Bump Potential
Zero Bump Potential
2.55
2.50
−2
|δϕ2 (k)|/MPL
2.45
2.40
2.35
2.30
2.25
57 56 55 54 53
Nend − N

17. Features Inside and Outside the Horizon
10−5 Sub-Horizon Bump
Super-Horizon Bump
Standard Quadratic Potential
10−7
−2
|S|/MPL
10−9
10−11
10−13
61 60 59 58 57 56 55 54
Nend − N

18. Features Inside and Outside the Horizon
1.04
Sub-Horizon Bump
1.02 Super-Horizon Bump
Standard Quadratic Potential
1.00
|δϕ2 (k)|/|δϕ2quad |
0.98
0.96
0.94
0.92
0.90
70 60 50 40 30 20 10 0
Nend − N

19. Future Plans
Three-point function of δϕ
Multi-field equation
Tensor & Vorticity similarities

20. Summary
Perturbations seed structure
Non-linear regime observationally
interesting
Numerically intensive calculation
Code available now
(http://pyflation.ianhuston.net)

21. i i 2 i 2 8πG 2 8πG i
δϕ2 (k ) + 2Hδϕ2 (k ) + k δϕ2 (k ) + a V,ϕϕ + 2ϕ0 V,ϕ + (ϕ0 ) V0 δϕ2 (k )
H H
1 3 3 3 i i i 16πG i i 2 i i
+ d pd qδ (k − p − q ) Xδϕ1 (p )δϕ1 (q ) + ϕ0 a V,ϕϕ δϕ1 (p )δϕ1 (q )
(2π)3 H
8πG 2
2 i i i i
+ ϕ0 2a V,ϕ ϕ0 δϕ1 (p )δϕ1 (q ) + ϕ0 Xδϕ1 (p )δϕ1 (q )
H
4πG 2 ϕ X
0 i i i i i
−2 Xδϕ1 (k − q )δϕ1 (q ) + ϕ0 δϕ1 (p )δϕ1 (q )
H H
4πG i i 2 8πG i i
+ ϕ0 δϕ1 (p )δϕ1 (q ) + a V,ϕϕϕ + ϕ0 V,ϕϕ δϕ1 (p )δϕ1 (q )
H H
1 3 3 3 i i i 8πG pk q k i i i
+ d pd qδ (k − p − q ) 2 δϕ1 (p ) Xδϕ1 (q ) + ϕ0 δϕ1 (q )
(2π)3 H q2
 
2 16πG i i 4πG 2 ϕ
0 pi qj kj ki
+p δϕ1 (p )ϕ0 δϕ1 (q ) + p q l −  ϕ δϕ (ki − q i )ϕ δϕ (q i )
l 0 1 0 1
H H H k2
X 4πG 2 p q l p q m + p2 q 2
l m i i i
+2 ϕ0 δϕ1 (p ) Xδϕ1 (q ) + ϕ0 δϕ1 (q )
H H k2 q 2
4πG q 2 + pl q l i i l i i
+ 4X δϕ1 (p )δϕ1 (q ) − ϕ0 pl q δϕ1 (p )δϕ1 (q )
H k2
4πG pl q l pm q m
2 ϕ
0 i i i i
+ Xδϕ1 (p ) + ϕ0 δϕ1 (p ) Xδϕ1 (q ) + ϕ0 δϕ1 (q )
HH p2 q 2
 
ϕ0 pl q l + p2 2 i i q 2 + pl q l i i
+ 8πG  q δϕ1 (p )δϕ1 (q ) − δϕ1 (p )δϕ1 (q )
H k2 k2
4πG 2 kj k pi pj
i i i i
+ 2 Xδϕ1 (p ) + ϕ0 δϕ1 (p ) Xδϕ1 (q ) = 0
H k2 p2