Anisotropic non-Gaussianity
arXiv:0812.0264
Mindaugas Karčiauskas
work done with
Konstantinos Dimopoulos
David H. Lyth
Density perturbations
● Primordial curvature perturbation – a unique
window to the early universe;
● Origin of structure <= quantum fluctuations;
● Usually light, canonically normalized scalar
fields => statistical homogeneity and isotropy;
● Statistically anisotropic perturbations from
the vacuum with a broken rotational symmetry;
● The resulting
observable.
is anisotropic and may be
Statistical homogeneity and
isotropy
● Density perturbations – random
fields;
● Density contrast:
;
● Multipoint probability distribution
function
:
● Homogeneous if the same under translations of all
● Isotropic if the same under spatial rotation;
;
Statistical homogeneity and
isotropy
● Assume statistical homogeneity;
● Two point correlation function
● Isotropic if
for
;
● The isotropic power spectrum:
● The isotropic bispectrum:
Statistical homogeneity and
isotropy
● Two point correlation function
● Anisotropic if
even for
● The anisotropic power spectrum:
● The anisotropic bispectrum:
;
Isotropic
Random Fields
with Statistical
Anisotropy
- preferred direction
Present Observational
Constrains
● The power spectrum of the curvature perturbation:
& almost scale invariant;
● Non-Gaussianity from WMAP5
(Komatsu et. al. (2008)):
● No tight constraints on anisotropic contribution yet;
● Anisotropic power spectrum can be parametrized as
● Present bound
(Groeneboom, Eriksen (2008));
● We have calculated
of the anisotropic curvature
perturbation - new observable.
Origin of Statistically
Anisotropic Power Spectrum
● Homogeneous and isotropic vacuum => the
statistically isotropic perturbation;
● For the statistically anisotropic perturbation <=
a vacuum with broken rotational symmetry;
● Vector fields with non-zero expectation value;
● Particle production => conformal invariance of
massless U(1) vector fields must be broken;
● We calculate
for two examples:
● End-of-inflation scenario;
● Vector curvaton model.
δN formalism
● To calculate
we use
formalism
(Sasaki, Stewart (1996); Lyth, Malik, Sasaki (2005));
● Recently in was generalized to include vector
field perturbations (Dimopoulos, Lyth, Rodriguez (2008)):
where
,
, etc.
End-of-Inflation Scenario:
Basic Idea
Linde(1990)
End-of-Inflation Scenario:
Basic Idea
- light scalar field.
Lyth(2005);
Statistical Anisotropy at the
End-of-Inflation Scenario
- vector field.
Yokoyama, Soda (2008)
Statistical Anisotropy at the
End-of-Inflation Scenario
●
●
●
●
●
Physical vector field:
Non-canonical kinetic function
Scale invariant power spectrum =>
Only the subdominant contribution;
Non-Gaussianity:
where
,
- slow roll parameter
;
;
Curvaton Mechanism: Basic Idea
● Curvaton
(Lyth, Wands (2002); Enquist, Sloth (2002)):
● light scalar field;
● does not drive inflation.
Curvaton
Inflation
HBB
Vector Curvaton
● Vector field acts as the curvaton field
(Dimopoulos (2006));
● Only a small contribution to the perturbations
generated during inflation;
● Assuming:
● scale invariant perturbation spectra;
● no parity braking terms;
● Non-Gaussianity:
where
Estimation of
● In principle isotropic perturbations are possible
from vector fields;
● In general power spectra will be anisotropic =>
must be subdominant (
);
● For subdominant contribution
can be
estimated on a fairly general grounds;
● All calculations were done in the limit
● Assuming that
one can show that
;
Conclusions
● We considered anisotropic contribution to the power
spectrum and
● calculated its non-Gaussianity parameter
.
● We applied our formalism for two specific examples:
end-of-inflation and vector curvaton.
● .
is anisotropic and correlated with the amount and
direction of the anisotropy.
● The produced non-Gaussianity can be observable:
● Our formalism can be easily applied to other known
scenarios.
● If anisotropic
is detected => smoking gun for vector
field contribution to the curvature perturbation.