Non-Gaussianity: from Inflation to CMB anisotropies

1
Non-Gaussianity from Inflation to CMB
anisotropies

• Nicola Bartolo
• Dipartimento di Fisica G. Galilei, Padova, Italy
  

2
Plan of the talk
Large-scale CMB anisotropies and primordial
NG (second-order radiation transfer function for
large-scales)

CMB Anisotropies on all scales Boltzmann
equations at second-order

Conclusions

- B.
N., Matarrese S., Riotto A., Komatsu E., 2004
Phys. Rept. 402, 103 - B.
N., Matarrese S., Riotto A., 2004, Phys. Rev.
Lett. 93, 231301 - B. N.,
Matarrese S., Riotto A., 2005, JCAP 0508, 010
- B. N., Matarrese S., Riotto A.,
2005, JCAP 0605, 24 - B. N.,
Matarrese S., Riotto A., 2006, JCAP 0606, 24
- B. N., Matarrese S., Riotto A.,
2006, astro-ph/0610110
Based on
3
Motivations
Planck and future experiments may be sensitive to
non-Gaussianity (NG) at the level of second- or
higher order perturbation theory
Crucial to provide accurate theoretical
predictions for the statistics of the CMB
anisotropies
Fundamental a) what is the relation between
primordial NG and
non-linearities in the CMB anisotropies on
different angular
scales? b) what is the
non-linear dynamics of the baryon-photon
fluid and CDM gravity near
recombination?
4
Large-angular scales
3 main effects a) Sachs-Wolfe effect
b) Early and late Integrated
Sachs-Wolfe c) Tensor
contributions
In this limit we do not really need to solve the
Boltzmann equations. The relevant effects comes
from gravity (at the last scattering surface and
from the scattering surface to the observer).
5
Second-order Sachs-Wolfe effect
B. N., Matarrese, Riotto, Phys. Rev. Lett. 2004
gravitational potential at emission
specifies the level of primordial non-Gaussianity
which depends on the particular scenario for the
generation of perturbations
Such an expression allows to single out the
non-linearities of large-scale CMB anisotropies
at last scattering . Evolving them inside the
horizon requires the full radiation transfer
function at second-order in the perturbations
6
Fully non-Linear Sachs Wolfe effect
?
Non-linear perturbations
a la Salopek-Bond (1990) definition of the
non-linear ? quantity
?
Non-perturbative extension of the Sachs-Wolfe
effect
N.B., S.
Matarrese, A. Riotto JCAP 2005
Linear order
Second order
see expression in N.B. , Komatsu, Matarrese,
Riotto (2004)
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From primordial NG to CMB anisotropies

• Evaluate the non-Gaussianity generated during
  inflation (or
• immediately after as in the curvaton
  scenario) primordial input

• After inflation, follow the evolution of the
  non-linearities on large
• scales by matching the conserved quantity
  ?(2) to the initial input,
• plus Einstein equations at second-order
  non-linearities in the
• gravitational potentials

?? The non-linearities in the gravitational
potentials are then transferred to the
observable ?T/T fluctuations additional
non-linearities are acquired
8
Sachs-Wolfe effect a compact formula
N.B., Matarrese, Riotto, Phys. Rev. Lett. (2004)
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Extracting the non-linearity parameter fNL
Connection between theory and observations
This is the proper quantity measurable by CMB
experiments, via the phenomenological analysis
by Komatsu and Spergel (2001)
k k1 k2
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Primordial non-Gaussianity

• Standard single field models (Maldacena (2002),
  Acquaviva, Bartolo, Matarrese A.R. (2002),
  Lidsey and Seery (2004))
• Multiple field models (Kofman (1990), Bartolo et
  al., Bernardeau Uzan (2002), Rigopoulos,
  Shellard and Van Tent (2003), Rigopoulos,
  (2003), Lyth, Wands, Malik, Vernizzi (2004))
• Curvaton models and or modulated perturbations
  (Lyth, Ungarelli and Wands (2002) Bartolo,
  Matarrese and A.R. (2004) Lyth, Rodriguez, Malik
  (2005))
• From preheating (Enqvist et al. (2004), Barnaby
  and Cline 2006))
• Ghost (Arkani-Hamed et al. (2003)) and DBI
  inflation
• 
  (Silverstein and Tong (2003))

may be either tiny, or moderate or very large
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Inflation models and fNL
N. B., E. Komatsu, S. Matarrese and A.Riotto.,
Phys. Rept. 2004
comments
fNL(k1,k2)
model
K universal, goes to zero in squeezed limit
Standard inflation
-1/6 K(k1,k2)
r (rs/r)decay
-11/6 - 5r/6 5/4r K(k1, k2)
curvaton
modulated
I - 5/2 5G / (12 aG1) I 0 (minimal
case)
1/12 I K(k1, k2)
order of magnitude estimate of the absolute value
may be large ?
multi-field inflation
unconventional inflation set-ups
second-order corrections not included
Warm inflation
typically 10-1
post-inflation corrections not included
ghost inflation
- 140 b a-3/5
post-inflation corrections not included
- 0.1 g2
DBI
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Angular decomposition
The linear and non-linear parts of the
temperature fluctuations correspond to a linear
Gaussian part and a non-Gaussian contribution
At linear order
Initial fluctuations
Linear radiation transfer function
Ex Linear Sachs-Wolfe
Linear ISW
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2nd-order radiation transfer function on
large-scales
Express the observed CMB anisotropies in terms of
the quadratic curvature perturbations
with the gravitational potential at
last scattering, and are convolutions

with kernels
and
generalize the radiation transfer

function at second-order
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2nd-order transfer functions for the Sachs-Wolfe
Primordial NG
Non-linear evolution of the gravitational
potentials after Inflation and additional
2nd-order corrections to temperature anisotropies
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2nd-order transfer functions for the late ISW
growth suppression factor
Primordial NG
Non-linear evolution of the gravitational
potentials after inflation
Additional second-order corrections to
temperature anisotropies (ISW)2
Expression for 2nd-order Early ISW, vector and
tensor modes available as well in B. N.,
Matarrese S., Riotto A., 2005, JCAP 0605
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On large scales NG NG from gravity
(universal) NG primordial

• Gravity itself is non-linear
• Non-linear (second-order) GR perturbations in the
  standard cosmological model introduce some order
  unity NG
• ? we would be in trouble if NG turned out to be
  very close
• to zero
• such non-linearities have a non-trivial form
• their computation ? core of the (large-scale)
  radiation
• transfer function at second-order

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WHAT ABOUT SMALLER SCALES ? Aim - have a
full radiation transfer function at second-order
for all scales - in
particular compute the CMB
anisotropies generated by the non-linear
dynamics of the photon-baryon fluid for
subhorizon modes at recombination
(acoustic oscillations at second-order)
Remember crucial to extract information from
the bispectrum are the
scales of acoustic peaks according to the
phenomenological analysis of Komatsu
and Spergel (2001)
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2nd-order CMB Anisotropies on all scales
Apart from gravity account also for a) Compton
scattering of photons
off
electrons
b) baryon velocity terms v


Boltzmann equation for photons
Collision term
Gravity effects
Boltzmann equations for baryons and CDM
Einstein equations
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Metric perturbations
Poisson gauge
?i and ?ij second-order vector and tensor modes.
Examples using the geodesic equation for the
photons
Redshift of the photon (Sachs-Wolfe and ISW
effects)
Direction of the photons changes due to
gravitational potentials Lensing effect (it
arises at second-order)
PS Here the photon momentum is
(
quadri-momentum vector)
20
Photon Boltzmann equation
Expand the distribution function in a linear and
second-order parts around the zero-order
Bose-Einstein value
Left-hand side
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The collision term Cf
Up to recombination photons are tightly coupled
to electrons via Compton scatterings e(q) ?(p) ?
e(q) ?(p). The collision term governs small
scale anisotropies and spectral distorsions
?
?
Important also for secondary scatterings
reionization, kinetic and thermal
Sunyaev-Zeldovich and Ostriker-Vishniac effects
22
The 2nd-order brightness equation
with optical depth
Source term
23
Hierarchy equations for multipole moments
and
Expand the temperature anisotropies in multipole
moments
System of coupled differential equations
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Integral Solution
and
One can derive an integral solution in term of
the Source
Important
The main information is contained in the Source,
which contains peculiar effects from the
non-linearity of the perturbations
?
?
The integral solution is a formal solution (the
source contains second-order moments up to l2),
but still more convinient than solving the whole
hierarchy
25
CMB anisotropies generated at recombination
KEY POINT Extract all the effects generated at
recombination (i.e. Isolate from the Source all
those terms ? optical depth ? )
Visibility function sharply peaks at
recombination epoch ?
26
Yields anisotropies generated at recombination
due to the non-linear dynamics of the
photon-baryon fluid
27
Boltzmann equations for massive particles
The Source term requires to know the evolution of
baryons and CDM
Left-hand side

just extend to a massive particle with mass m and
energy E(m2q2)1/2
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Momentum continuity equation
Photon velocity
2nd-order velocity
Quadrupole moments of photon distribution
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Acoustic oscillations at second-order
In the tight coupling limit the energy and
momentum continuity equations for photons and
baryons reduce to
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The quadrupole moment at recombination
Two important differences w .r. s to the linear
case
2) At second-order the quadrupole of the photons
is no longer suppressed in the tight coupling
limit
Similar term analyzed by W. Hu in ApJ 529 (2000)
in the context of reionazation
31
Non-linear dynamics at recombination
Modes entering the horizon during the matter
epoch (?-1 lt k lt ?-1EQ)
Acoustic oscillations of primordial
non-Gaussianity
Non-linear evolution of gravity
32
Linear vs. full radiation transfer function
primordial non-Gaussianity is transferred
linearly Radiation Transfer function at first
order
Non-linear evolution of gravity the core of the
2nd order transfer function
(how these contributions mask the primordial
signal? how do they fit into the analysis of
the bispectrum?) Numerical analysis in
progress (N.B., Komatsu, Matarrese, Nitta, Riotto)
(from Komatsu Spergel 2001)
33
Modes entering the horizon during radiation epoch
(k gt ?-1EQ )
In this case the driving force is the quadrupole
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Modes entering the horizon during radiation
epoch (II) The Meszaros effect
Around the equality epoch ?EQ Dark Matter starts
to dominate
Consider the DM perturbations on subhorizon
scales during the radiation epoch
Meszaros effect
This allows to fix the gravitational potential
at ? gt?EQ through the Poisson equation and to
have a more realistic and accurate analytical
solutions for the acoustic oscillations from
the equality onwards
35
Meszaros effect at second-order
Combining the energy and velocity continuity
equations of DM
for a R.D. epoch
Solution
Initial conditions
36
Meszaros effect at second-order
Dark Matter density contrast on subhorizon scales
for ? lt ?EQ
Can be used for two pourposes
37
Second-order transfer function

• First step calculation of the full 2-nd
  order radiation transfer function on large scales
  (low-l), which includes
• NG initial conditions
• non-linear evolution of gravitational potentials
  on large scales
• second-order SW effect (and second-order
  temperature fluctuations on last-scattering
  surface)
• second-order ISW effect, both early and late
• ISW from second-order tensor modes (unavoidably
  arising from non-linear evolution of scalar
  modes), also accounting for second-order tensor
  modes produced during inflation
• Second step solve Boltzmann equation at
  2-nd for the photon, baryon and CDM fluids,which
  allows to follow CMB anisotropies at 2-nd order
  at all scales
• this includes both scattering and
  gravitational secondaries, like
• Thermal and Kinetic Sunyaev-Zeldovich effect
• Ostriker-Vishniac effect
• Inhomogeneous reionization
• Further gravitational terms, including
  gravitational lensing (both by scalar and tensor
  modes), Rees-Sciama effect, Shapiro time-delay,
  effects from second-order vector (i.e.
  rotational) modes, etc.
• In particular we have computed the
  non-linearities at recombination

(Bartolo, Matarrese A.R. 2005)
38
Conclusions
Up to now a lot of attention focused on the
bispectrum of the curvature Perturbation ?.
However this is not the physical quantity which
is observed is (the CMB anisotropy)
?
Need to provide an accurate theoretical
prediction of the CMB NG in terms of the
primordial NG seeds ? full second-order
radiation transfer function at all scales
?
Future techniques (predicted angular dependence
of fNL, extensive use of simulated NG CMB maps,
measurements of polarization and use of
alternative statistical estimators ) might help
NG detection down to fNL1 need to compute
exactly the predicted amplitude and shape of CMB
NG from the post-inflationary evolution of
perturbations.
?