These are the slides of the talk I gave on discriminating between models of inflation using space based gravitational wave detectors, at KEK in Tskuba University, Japan.

1. A Further
Discriminatory
Signature of Inflation
Laila Alabidi
Yukawa Institute of Theoretical Physics
talk presented at KEK
19th of March 2012

2. Based on
various papers with D.H. Lyth (Lancaster U), James
Lidsey and Ian Huston (Queen Mary, University of
London).
and most recently
01203.???? (very soon we hope!)
with
Kazunori Kohri (KEK), Misao Sasaki (YITP),Yuuiti
Sendouda (Hirosaki U.)

3. CMB

4. Some parameter
definitions
Scale factor
distance distance
dt = a(t)d0
at time t at initial time
a
˙ derivative w.r.t.
Hubble parameter H= a
time
Conformal time dτ 2 = a2 dt2
a H
Conformal Hubble parameter H= a = a
derivative
w.r.t. conformal time

5. 0s
98
Inflation: brief review
:1
e
Pr
“Cosmological Principle”-- the universe
has to be both homogeneous and
isotropic, simply because it seemed to
be!
Original matter/radiation
perturbations which sourced the
evolution of structure were put in by
hand.

6. Inflation: a brief
80s
19
review
t:
s
Po
Paradigm: the near exponential expansion of the
universe at t=10-37s
Effect: the comoving Hubble Horizon decreases.
d 1
dt aH 0
Result: explanation of:
Universal homogeneity, causality and
isotropy.
Origin of structure: allows for a quantum to
classical transition of vacuum fluctuations.

7. The basic problem

8. A
ll
‘m ac
s!
a i hie
el
od n’ v
ai e t
m
m he
y
an
s!
M

9. Which is the correct
model?

10. Overview
Inflation: the parameters.
Inflation models I.
Observation: the bounds.
Results I: from CMB and related data sets.
Going beyond the CMB constraints: the
induced gravitational waves.
Inflation models II.
Results II: prospects from gravitational wave
experiments.

11. Inflation: the
requirements
Inflation requires an accelerated rate of change of expansion
a0
¨
recall the acceleration equation? energy density
a
¨ 4πG
=− (ρ + 3P )
a 3 pressure
negative ρ is nonsensical, but negative pressure is not
a scalar field!
potential scalar field
1 ˙2 1 ˙2
ρ = φ + V (φ) P = φ − V (φ)
2 2
Just considering canonical kinetic terms for now!

12. Inflation: slow roll
¨ ˙ dV (φ)
φ + 3H φ + =0
dφ
and we can define
2
}
1 V
=
2 V
V
η= 1
V
V V
ξ=
V2
a ‘ is a derivative with respect
to the “inflaton” field φ

13. Number of e-folds
Time re-parametrisation
A measure of how long inflation lasts
V
ae
N = ln ai = Hdt V,φ dφ
Start time is the time that scales of cosmological interest leave
the horizon

14. A brief introduction
to perturbations

15. background
φ = φ0 + δφ perturbation
quantity
imprint themselves on the background (via the Einstein
equations) generating what we call the curvature perturbation
ζ(φ)
the characteristics and future evolution of which is
determined by the model of inflation

16. Inflation: the
observational
parameters

17. Spectrum and Spectral
scale on whichIndex the ‘spectrum’
perturbation is defined
2π 2 (3)
ζ(k)ζ(k ) = k3 δ (k − k )Pζ (k)
1 V
Pζ = 24π2
The ‘s pectral index’ defines the scale dependance of the
spectrum P
d ln ζ
ns − 1 = d ln k
ns = 1 + 2η − 6
The scale dependance of which is called the ‘running’
dns 2 2
ns = d ln k ns = 16η − 24 − 2ξ

18. Gravitational waves
Traceless, transverse part of the perturbed spatial metric
with a spectrum
H 2
Pgrav = 8 2π
The ratio to the scalar spectrum is defined as
Pgrav
Pζ ≡r
In terms of slow roll parameters
r = 16

19. Not really an
observable but...

20. Primordial Black Holes
The spectrum on small scales has not been ‘well’ measured
If spectrum very large (0.03) then PBHs will form.
Have an ‘upper’ bound due to astrophysical constraints
Pζend 10 −2

21. PBH pre-requisites
For an enhanced spectrum towards the end of
inflation:
Pζe → 10−2
Then a decreasing slow roll parameter is required:
→0
and a running of the spectral index:
n 0
s

22. Classification of single
field models
“Small” field vs. “Large” field models
∆φ = φend − φ∗
“Observable” gravitational waves generated up to 4-efolds
after horizon exit
∆φ ∼ 0.5Mpl r ∼ 0.1
Gravitational waves on the order of 0.1 will be detectable soon!

23. Small field models

24. Tree level potential
p p0, inflaton rolls away from the origin
φ
V = V0 1−
µ
Taylor expand about the vacuum, then
assume one of the p’s dominates.
Logarithmic potential
2
gsφ
V = V0 1+ ln
2π Q p0 logarithmic and exponentialtowards the
p0, logarithmic, towards the origin
p0, inflaton rolls inflaton rolls , inflaton
Dvali, Shafi Schaefer (1994) origin
rolls towards the origin
Exponential potential
V = V0 1 − e−qφ/Mpl
Dimopoulos, Lazarides, Lyth Ruiz de Austri (2003) Stewart (1995); Lazarides Panagiotakopoulos (1995)
2 p−1
1 − ns = *not so if μmpl
N p−2

25. Hilltop-type inflation models
p q
V = V0 (1 + ηp φ − ηq φ )
Kohri et al (2007)
The Running Mass Model End of
2
µ0
+ A0 2 Inflation
V = V0 1 − φ Scales
2
leave
A0
+ 2
φ 2
horizon
2(1 + α ln φ)
Stewart (1996)
These models have bet ween 3 and 4 independent parameters
so they can be ‘fit’ to data
these are the only 2 models which can lead to PBH
Kohri et al (2007) and Drees et al (2011)
formation.

26. Large field models

27. Monomial potential
α
V ∝φ
4α 2+α
r= , 1 − ns =
N 2N
Linde 1983 Silverstein Westphal 2008
α positive integer: chaotic
α 1: monodromy
Sinusoidal potential
Hilltop regime
V0 2 |η0 |
V = 1 + cos φ
2 Mpl
16|η0 | Chaotic regime
r = 2N |η |
e 0 − 1
2|η0 |N Freese, Frieman Olinto 1990; Adams, Bond, Freese, Frieman Olinto 1993
e +1
1 − ns = 2|η0 | 2|η |N
e 0 −1

28. Observational
bounds

29. 95
Observational Constraints %
c. l
Spectral index (r and ns’=0)
0.939 ns 0.987
Tensor fraction (ns’=0)
r 0.24
Running
−0.084 ns 0.017

30. Results

31. +p
2
gs φ −p φ
φ V = V0 1 −
V = V0 1 + ln V = V0 1 −
2π Q µ µ
p=0 p-∞
p0
V = V0 1 − e −qφ/mpl
p0

32. α
V ∝φ
N=30 N=47 N=61
Axion Monodromy
McAllister, Silverstein Westphal (2008)
η=0
Planck sensitivity

33. 3
AP
M
W
0
Multifield/Chaotic Inflation
1
Natural Inflation Brane Inflation
2
tensor desert
log(r)
3
Original Hybrid Model
4
Modular Inflation (p=2)
5 Modular Inflation (p3) Mutated Hybrid Inflation
6
0.9 0.95 1 1.05
n

34. Degenaracies still
exist, so what next?

35. Future data may alleviate some of the
degeneracy, e.g. PLANCK or CMBPOL. But
not fully.
Further signatures of inflation models.
Be more philosophical and ask what
makes a model ‘natural’.

37. The Data Fantasy
•Space based detectors of gravitational waves, DECIGO and
LISA.
• DECIGO and LISA cover a frequency range of approx. 10-3 to
101.5 Hz.
•This corresponds to scales which leave the horizon at the
end of inflation.
•Funding has not been approved yet but ...
•Proposals and white papers for these projects have
appeared on the arXiv!

38. Induced
Gravitational Waves

39. Anatomy of Induced Gravitational
Waves
These are induced by scalar perturbations entering the horizon
after inflation
(2)
ds2 = 2 (1)
a (τ ) − 1 + 2Φ + 2Φ (2)
dτ + 2Vi dτ dxi + · · ·
2
(1) (2) 1
+ 1 − 2Φ − 2Φ δij + hij dxi dxj
2
contains 1st
and 2nd order
terms
Einstein Eqns

40. Get the equation of motion:
2
hij + 2Hhij − ∇ hij = −4Sij source term
just 2nd order
Sij = 4Φ∂i ∂j Φ + 2∂i Φ∂j Φ + · · ·
4
+ ∂i (Φ + HΦ)∂j (Φ + HΦ)
3H2 (1 + w)
Where the Φ is a first order quantity given by
6(1 + w) 1 2
Φk + Φk + wk Φk = 0
1 + 3w τ
Note: the source term depends only on the (1) scale and
(2) epoch of re-entry (w, the equation of state).

41. Fourier transform
calculate spectra
the equation
2
2π
hk hk = 3 δ(k + k )Ph (k, τ )
k
Spectrum of induced gravitational waves
scalar spectra
integral over all k-space
from inflation
Ph (k, τ ) ∼ ˜
dk ˜ ˜
dµPζ (k)Pζ (|k − k|)I1 (τ )I2 (τ )
cosine of angle integrals over
between modes conformal time
independent of inflation
model
The time integrals are model independent, and require
only Φ and thus the equation of state.
Ananda et al (2007), Baumann et al (2007), Acquaviva et al (2003), Saito et al (2009)

42. ˜
k
Time integrals
v= y = 1 + v 2 − 2vµ
k
Matches results found on Maple
x = kτ and previous Authors results
asymptote to
zero!

43. Which Models are we
looking at?

44. Scales
V model picture
leave
End of
horizon
Inflation
ε decreases and the
scalar spectrum increases
field

45. Selection Criteria:
Hilltop-Model
V = V0 (1 + ηp φp − ηq φq )
Pre-set couplings to {0,1}.
Reject values that violate WMAP.
Reject values which require super-
planckian evolution.
Unique values then selected which
maximise the spectrum after N e-folds.

46. Selection criteria:
Running Mass
µ +A 2
A 0 0
0 2
V = V0 1 − φ + 2
φ2
2 2(1 + α ln φ)
Less intense criteria: pick values which
satisfy WMAP.
Evolve until just before the spectrum
hits the PBH bound.
Terminate inflation and evaluate N.

47. Spectra

48. PBH Bound PBH Bound
fractional powers
p=2, q=2.2,2.3,2.5,2.7,2.9
55
60
N=
integral powers
N=
{p,q}={2,3},{2,4},{3,4},{4,5}
Note: integral powers are already strongly constrained at
the pivot scale

49. Running Mass Model
PBH Bound ns’=0.005
ns’=0.002
0.007ns’0.012
lead to PBHs
with
20 28
10 g MBH 10
compatible
with Dark Matter

50. Results: the Induced
Gravitational

51. Present Day Spectrum
The Energy density (per logarithmic
interval) is:
1 dρGW energy density
of GW
ΩGW (k, τ ) =
ρc d log f critical
energy density
This is related to the primordial spectrum as:
2 transfer function
(2) a(τ )k 2
ΩGW = t (k, τ )Ph (k)
aeq keq
For scales kkeq this gives us the relation:
(2) 1
ΩGW = Ph (k)
1 + ze Baumann et al (2007)

52. The Running Mass Model
PBH Bound ns’=0.002
ns’=0.005
0.007ns’0.012
Models are
terminated at
N=64
DM N=43
29N39

53. For comparison ...
DE
BBO
CIG O
The spectra of primordial gravitational waves look
very different!
this image is taken from Kuroyanagi et al (2010)

54. Hilltop models
lines=fractional q, stars and circles = integral q
N=
55
60
N=

55. Conclusions
Hilltop model: generates potentially obser vable induced
GWS
reasonable e-folds.
also p=2,q=3 within the reach of BBO/DECIGO.
Running Mass:
within the reach of BBO/DECIGO for N50.
DM production measurable by LISA.

56. Thank You
ありがとうございます！