Probing Inflation with CMB Polarization
Daniel Baumann
High Energy Theory GroupHarvard University
Chicago, July 2009
“How likely is it that B-modes exist at the r=0.01 level?”
Daniel Baumann
High Energy Theory GroupHarvard University
Chicago, July 2009
the organizers
“I don’t know!”
Daniel Baumann
High Energy Theory GroupHarvard University
Chicago, July 2009
“What can we learn from a B-mode detection at the r=0.01 level?”
Daniel Baumann
High Energy Theory GroupHarvard University
Chicago, July 2009
Based on
CMBPol Mission Concept Study: Probing Inflation with CMB PolarizationDaniel Baumann, Mark G. Jackson, Peter Adshead, Alexandre Amblard, Amjad Ashoorioon, Nicola Bartolo, Rachel Bean, Maria Beltran, Francesco de Bernardis, Simeon Bird, Xingang
Chen, Daniel J. H. Chung, Loris Colombo, Asantha Cooray, Paolo Creminelli, Scott Dodelson, Joanna Dunkley, Cora Dvorkin, Richard Easther, Fabio Finelli, Raphael Flauger, Mark P.
Hertzberg, Katherine Jones-Smith, Shamit Kachru, Kenji Kadota, Justin Khoury, William H. Kinney, Eiichiro Komatsu, Lawrence M. Krauss, Julien Lesgourgues, Andrew Liddle, Michele
Liguori, Eugene Lim, Andrei Linde, Sabino Matarrese, Harsh Mathur, Liam McAllister, Alessandro Melchiorri, Alberto Nicolis, Luca Pagano, Hiranya V. Peiris, Marco Peloso, Levon Pogosian, Elena Pierpaoli, Antonio Riotto, Uros Seljak, Leonardo Senatore, Sarah Shandera, Eva Silverstein, Tristan Smith, Pascal Vaudrevange, Licia Verde, Ben Wandelt, David Wands,
Scott Watson, Mark Wyman, Amit Yadav, Wessel Valkenburg, and Matias Zaldarriaga
White Paper of the Inflation Working Group
arXiv: 0811.3919
1. B-modes2. Non-Gaussianity
Outline
1. Classical Dynamics of Inflation
2. Quantum Fluctuations from Inflation
3. Current Observational Evidence
PART 1: THE PRESENT
PART 2: THE FUTURE
Part 1:THE PRESENT
InflationThe Quantum Origin of Structure in the Early Universe
Inflation Guth (1980)
ds2 = dt2 ! a(t)2 dx2
A period of accelerated expansion
sourced by
a > 0
a nearly constant energy density
and negative pressure
a
a= !!
6(1 + 3w) > 0 " w < !1
3
a
a= (H2 + H) > 0 ! H =
!!
3" const.
Inflation Guth (1980)
A period of accelerated expansion
solves the horizon and flatness problems
! = 0
! != 0
i.e. explains why the universe is so large and old!
creates a nearly scale-invariant spectrum of primordial fluctuations
horizonH!1
‣ stretches microscopic scales to superhorizon sizes
‣ correlates spatial regions over apparently acausal distances
time
The Shrinking Hubble SphereThe comoving “horizon” shrinks during inflation
and grows after inflation
today
time
comoving scales
inflation reheating hot big bang
(aH)!1
(aH)!1 ! a12 (1+3w)
w > !13
w < !13
“Goodbye and Hello-again”
horizon exit horizon re-entry
sub-horizonsuper-horizon
sub-horizon
large-scale correlations can be set up causally!
today
time
comoving scales
inflation reheating hot big bang
(aH)!1
k!1
Conditions for Inflation
w < !13
" d(aH)!1
dt< 0 " a > 0
negative pressure
shrinking comoving horizon
accelerated expansion
crucial for the generation of perturbations
at the heart of the solution of the horizon and flatness problems
and
What is the Physics of the Inflationary Expansion?
We don’t know!
This is why we are here!
Classical DynamicsParameterize the decay of the inflationary energy by a scalar field Lagrangian
A flat potential drives acceleration
slow-roll conditions
! !M2
pl
2
!V !
V
"2
! !M2pl
V !!
V
!, |"| < 1Linde (1982)
Albrecht and Steinhardt (1982)
L =12(!")2 ! V (")
“clock”
inflation
end of inflation
Quantum DynamicsQuantum fluctuations lead to a local time delay in the end of inflation and density fluctuations after reheating
reheating
!"
Zeta in the Sky
reheating
!"
!
!"
!T
inflaton fluctuations
curvature perturbationson uniform density hypersurfaces
density perturbations
CMB anisotropies
Zeta in the Sky
• gauge-invariant
• freeze on super-horizon scales!
curvature perturbationson uniform density hypersurfaces
ds2 = dt2 ! a2(t) e2!(t,x) dx2
Zeta in the Sky
!!k!k!" = (2")3 #(k + k!) P!(k)
two-point correlation function !!(x)!(x!)"
power spectrum
curvature perturbationon uniform density hypersurfaces
ds2 = dt2 ! a2(t) e2!(t,x) dx2
evaluated at horizon crossing
DB: TASI Lectures on Inflation (2009)
today
time
comoving scales
inflation reheating hot big bang
(aH)!1
k!1
The Inverse Problem
DB: TASI Lectures on Inflation (2009)
sub-horizon
today
time
comoving scales
zero-point fluctuations
inflation reheating hot big bang
(aH)!1
!kk!1
The Inverse Problem
DB: TASI Lectures on Inflation (2009)
sub-horizon
todayhorizon exit
time
comoving scales
zero-point fluctuations
inflation reheating hot big bang
(aH)!1
!!k!k!"
!kk!1
The Inverse Problem
DB: TASI Lectures on Inflation (2009)
super-horzionsub-horizon
todayhorizon exit
time
comoving scales
zero-point fluctuations
inflation reheating hot big bang
(aH)!1
!!k!k!"! # 0
!kk!1
The Inverse Problem
observed
predicted by inflation
DB: TASI Lectures on Inflation (2009)
super-horzionsub-horizon
transfer function
CMBrecombination today
projection
horizon exit
time
comoving scales
horizon re-entry
zero-point fluctuations
inflation reheating hot big bang
(aH)!1
C!!T!!k!k!"! # 0
!kk!1
The Inverse Problem
predicted by inflation
DB: TASI Lectures on Inflation (2009)
super-horzionsub-horizon
transfer function
CMBrecombination today
projection
horizon exit
time
comoving scales
horizon re-entry
zero-point fluctuations
inflation reheating hot big bang
(aH)!1
C!!T!!k!k!"! # 0
!kk!1
The Inverse Problemobserved
Prediction from Inflation
!2s !
k3
2!2P!(k) =
!H
2!
"2 !H
"
"2
!" !"! #de Sitter fluctuations conversion
Scalar Fluctuations
evaluated at horizon crossing k = aH
Prediction from Inflation
!2s !
k3
2!2P!(k) =
!H
2!
"2 !H
"
"2
Scalar Fluctuations
!2s = Ask
ns!1
scale-dependence
ns ! 1 = 2! ! 6"e.g. slow-roll inflation
how the power is distributed over the scales is determined by the expansion history during inflation
H(t)
Prediction from Inflation
!2s !
k3
2!2P!(k) =
!H
2!
"2 !H
"
"2
Scalar Fluctuations
Different shapes for the inflationary potential
lead to slightly different predictions!
reheating 2!f0
Observations
Observational EvidenceFlatnessInflation predicts WMAP sees1
!k = 0 (±10!5) !0.0181 < !k < 0.0071
1WMAP 5yrs.+BAO: 95% C.L.
Observational EvidenceScalar FluctuationsInflation predicts WMAP seespercent-level deviations from ns = 1
ns = 0.963+0.014!0.015
2.5! ns = 1away from
Observational EvidenceScalar FluctuationsInflation predicts WMAP sees
Gaussian and Adiabatic Fluctuations
Gaussian and Adiabatic Fluctuations
Observational EvidenceScalar FluctuationsInflation predicts WMAP sees
Correlations of Superhorizon Fluctuations
Correlations of Superhorizon Fluctuations at Recombination
Multipole moment
(!+
1)C
TE
!/2
"[µ
K2]
tem
pera
ture
-pol
ariz
atio
n cr
oss-
corr
elat
ion
! > 1!
Fingerprints of the Early UniverseWe have only just begun to probe the fluctuations
created by inflation:
The next decade of experiments will be tremendously exciting!
Tensor Fluctuationsgravitational waves
not yet detected
Scalar Fluctuationsdensity fluctuations
detected• hints of scale-dependence
• first constraints on Gaussianity and Adiabaticity• superhorizon nature confirmed
Part 2:THE FUTURE
How can we probe the Physical Origin of Inflation?
How do we constrain the Inflationary Action?
“like in all of physics we ultimately want to know the action”
• field content• potential, kinetic terms• interactions• symmetries• couplings to gravity• etc.
L = f!(!")2, (!#)2,!", . . .
"! V (", #)
How do we constrain the Inflationary Action?
• minimal models: single-field slow-roll
L =12(!")2 ! V (")
Gaussian fluctuations
all information in power spectra
ScalarsAs
ns
!s
shapeV !
V,
V !!
V,
V !!!
V
Tensors At
scale!
V
“like in all of physics we ultimately want to know the action”
How do we constrain the Inflationary Action?
• minimal models: single-field slow-roll
• non-minimal models: - non-minimal coupling to gravity- multiple fields- higher-derivative interactions
fluctuations can be non-Gaussian and non-adiabatic
“like in all of physics we ultimately want to know the action”
information beyond the power spectra bispectrum, trispectum, etc.
How do we constrain the Inflationary Action?
What do different parameter regimes for observables teach us
about high-energy physics?
e.g.r > 0.01 f local
NL > 1large tensors large non-Gaussianity
“like in all of physics we ultimately want to know the action”
(this talk) (the coffee break)
Primordial Gravitational Waves
Tensors
Besides scalar fluctuations inflation produces tensor fluctuations:
ds2 = dt2 ! a2(t)(1 + hij)dxidxj
gravitational wavesmassless gravitons
de Sitter fluctuations of any light field
robust prediction of inflation!
!2t (k) =
8M2
pl
!H
2!
"2
TensorsThe tensor-to-scalar ratio
r ! !2t
!2s
is model-dependent because scalars are!
!2s (k) =
!H
2!
"2 !H
"
"2
strong model-dependence!
TensorsThe tensor-to-scalar ratio
r ! !2t
!2s
is model-dependent because scalars are!
The prediction for tensors is simple and the same in all models!
In contrast,
!2t ! H2
Detecting Tensors via B-modes
E- and B-modes
E-mode
• Polarization is a rank-2 tensor field.
• One can decompose it into a divergence-like E-mode and a vorticity-like B-mode.
B-mode
The B-mode TheoremB-modes are only created by gravitational
waves (tensor modes not scalar modes!)
E < 0
E > 0
B < 0
B > 0
! hijtensorsscalars
Seljak and Zaldarriaga, Kamionkowski et al.
The B-mode Theorem
Seljak and Zaldarriaga, Kamionkowski et al.
! hij
Challinor
B-modes are only created by gravitational waves (tensor modes not scalar modes!)
The B-mode Theorem
So far only upper-limits, but many experiments now in operation or in planning.
Detection of primordial B-modes is often considered a smoking gun of inflation.
Komatsu et al. (2008)WMAP 5 yrs.
r < 0.22tensor-to-scalar ratio
Seljak and Zaldarriaga, Kamionkowski et al.
Superhorizon B-modes
1 23 4
2
1
1
2
-
-
CB(!) [mK2]
! [!]
DB and Zaldarriaga
superhorizon at recombination
unambiguous signature of inflation!
like the TE test for superhorizon scalarsSpergel and Zaldarriaga
B-modes as a Probe of High-Energy Physics
Energy Scale of InflationTensors measure the energy scale of inflation
Einf ! V 1/4 = 1016GeV! r
0.01
"1/4
Single most important data point about inflation!
If we observe tensors it proves that inflation occurred at the GUT-scale!
r > 0.01
Energy Scale of Inflation
Einf ! V 1/4 = 1016GeV! r
0.01
"1/4
Single most important data point about inflation!
r > 0.01It is hard to overstate the importance of such a result for the high-energy physics community which currently only has two indirect clues about physics at that scale:
1. Apparent Unification of Gauge Couplings2. Lower Bound on the Proton Lifetime
Tensors measure the energy scale of inflation
The Lyth Bound
!2t (k) =
8M2
pl
!H
2!
"2
dNe ! Hdt = d ln a
tensor-to-scalar ratio
where
!2s (k) =
!H
2!
"2 !H
"
"2
tensors scalars
r ! !2t
!2s
= 8!
d!
dNe
1Mpl
"2
The Lyth Boundtensor-to-scalar ratio
!!
Mpl!
! r
0.01
"1/2
field evolution over 60 e-foldsr ! !2t
!2s
= 8!
d!
dNe
1Mpl
"2
The Lyth Bound
!!
Mpl!
! r
0.01
"1/2
If we observe tensors it proves that the inflaton field moved over
a super-Planckian distance!r > 0.01
!!!Mpl
The Lyth Bound
i.e. we require a smooth potential over a range !!!Mpl
few !Mpl
But, in an effective field theory with cutoff
we generically don’t expect a smooth potential over a super-Planckian range
!
! < Mpl
EMpl
Ms
Msusy
MX
MY
MKK
H
m!
!
low-energy EFT
UV-completione.g. string theory
Effective Field Theory
Effective Field Theory EMpl
Ms
Msusy
MX
MY
MKK
H
m!
!
low-energy EFT
UV-completione.g. string theory
• integrate out heavy fields M > !If we know the complete theory:
V (!, ") ! Ve!(!)
Effective Field Theory EMpl
Ms
Msusy
MX
MY
MKK
H
m!
!
low-energy EFT
UV-completione.g. string theory
• integrate out heavy fields M > !
• low-energy effective potential receives (computable) corrections
If we know the complete theory:
!V =O!
M !!4
Effective Field Theory EMpl
Ms
Msusy
MX
MY
MKK
H
m!
!
low-energy EFT
UV-completione.g. string theory
• integrate out heavy fields M > !
• parameterize our ignorance about the UV, i.e. add all corrections consistent with symmetries.
Wilson
!V =!
!
O!
"!!4
If we know the complete theory:
• low-energy effective potential receives (computable) corrections
Typically, we don’t know the complete theory:
Large-Field Inflation
UV sensitivity of inflation is especially strong in any model with
observable gravitational waves
Large-Field Inflation
1. No Shift Symmetry in the UV
we generically don’t expect a smooth potential over a super-Planckian range
!
Ve!(!) =12m2!2 +
14"!4 +
!!
p=1
"p!4
"!
!
#2p
Effective Field Theory with Cutoff ! < Mpl
Large-Field Inflation
If the action is invariant under
! ! ! + const.
then the dangerous corrections are forbidden by symmetry
2. Shift Symmetry in the UV
Ve!(!) =12m2!2 +
14"!4 +
!!
p=1
"p!4
"!
!
#2p
Large-Field Inflation
! ! ! + const.
Ve!(!) =12m2!2
few !Mpl
e.g.
2. Shift Symmetry in the UV
With such a shift symmetry chaotic inflation is “technically natural”
Large-Field Inflation2. Shift Symmetry in the UV
Seeing B-modes would show that the inflaton field respected a shift symmetry up to the Planck scale!
We know fields with that property:
axions1
1 The first controlled large-field models using axions have recently been constructed in string theory.
Large-Field vs. Small-Field
Small-Field!!!Mpl
Primordial B-modes undetectable
• The Patch Problem
homogeneous patch
L > H!1
physical horizonH!1For inflation to start the inflaton
field has to be homogeneous over a distance that is a few times the horizon size at that time!
H!1 ! r!1/2Since
this seems to be a bigger problem for small-field (low-r) models.
Small-Field!!!Mpl
Primordial B-modes undetectable
• The Patch Problem
• The Overshoot Problem
For inflation to start the field has to reach the flat part of the potential with small speed.
! ! V 1/2
overshoot
This problem is worse for small-field (low-r) models.
Small-Field!!!Mpl
Primordial B-modes undetectable
• The Patch Problem
• The Overshoot Problem
• “Fine-Tuned” PotentialsBoyle, Steinhardt and Turok (2005)
!end !cmb
" = 1 " ! 1
For single-field models the transition
within 60 e-folds requires “fine-tuned” potentials!
r = 16 ! ! 1 " ! = 1
Small-Field!!!Mpl
Primordial B-modes undetectable
• The Patch Problem
• The Overshoot Problem
• “Fine-Tuned” Potentials
These qualitative fine-tuning problems of small-field inflation are hard to quantify!
Typically, the arguments run into the ill-understood measure problem!
Small-Field!!!Mpl
Primordial B-modes undetectable
• The Patch Problem
• The Overshoot Problem
• “Fine-Tuned” Potentials
Large-FieldPrimordial B-modes detectable!
!!!Mpl
• Potentials are “Simple” Functions
e.g.V (!) =
12m2!2
but are corrections under control?
Small-Field!!!Mpl
Primordial B-modes undetectable
• The Patch Problem
• The Overshoot Problem
• “Fine-Tuned” Potentials
Large-FieldPrimordial B-modes detectable!
!!!Mpl
• Attractor Solutions
• Potentials are “Simple” Functions
Small-Field!!!Mpl
Primordial B-modes undetectable
• The Patch Problem
• The Overshoot Problem
• “Fine-Tuned” Potentials
Large-FieldPrimordial B-modes detectable!
!!!Mpl
• Improved Initial Conditions
• Potentials are “Simple” Functions
• Attractor Solutions
Small-Field!!!Mpl
Primordial B-modes undetectable
• The Patch Problem
• The Overshoot Problem
• “Fine-Tuned” Potentials
Large-FieldPrimordial B-modes detectable!
!!!Mpl
• Potentials are “Simple” Functions
• Attractor Solutions
• Improved Initial Conditions
A convincing argument that the tensor amplitude has to be large does NOT exist!
HOWEVER, there is also no convincing argument that it has to be small!
Small-Field!!!Mpl
Primordial B-modes undetectable
• The Patch Problem
• The Overshoot Problem
• “Fine-Tuned” Potentials
Large-FieldPrimordial B-modes detectable!
!!!Mpl
My prediction:
Before we (theorists) come up with such an argument, YOU will have DETECTED B-modes!
• Potentials are “Simple” Functions
• Attractor Solutions
• Improved Initial Conditions
Conclusions
“How likely is it that B-modes exist at the r=0.01 level?”
The Wagner Principle
50%
“How likely is it that B-modes exist at the r=0.01 level?”
The Wagner Principle
50% If something can either happen or not happen the chances are 50-50.
black holes at the LHC destroying the Earth?
Walter Wagner
More seriously:
i.e. flatness and homogeneity of the universe and a primordial spectrum of nearly scale-invariant, Gaussian and adiabatic scalar fluctuations.
Inflation has been remarkably successful at explaining all current observations
However, in some sense this may be viewed as a “postdiction” (debatable)
(We knew the universe was homogeneous and had small density fluctuations)
i.e. flatness and homogeneity of the universe and a primordial spectrum of nearly scale-invariant, Gaussian and adiabatic scalar fluctuations.
Tensor modes are a robust prediction of inflation.
Seeing their B-mode signature would be a remarkable achievement.
It would teach us a great deal about the physics of inflation and rule out all alternative theories.
Inflation has been remarkably successful at explaining all current observations
A B-mode detection would teach us a great deal about the physics of inflation:
1. Inflation occured!
2. It happened near the GUT-scale!
3. The inflaton field moved over a super-Planckian distance!
3’. Its potential was controlled by a shift symmetry.i.e. the inflaton was something like an axion
Thank you for your attention!
There is more to be learnt from Scalars!
Non-Gaussianity and Non-Adiabaticity probe further details of the inflationary action:
field content, interactions, symmetries, ...
I didn’t have time to describe this, but see:
CMBPol Mission Concept Study: Probing Inflation with CMB Polarization
White Paper of the Inflation Working Group
arXiv: 0811.3919