Constraining hybrid inflation models with WMAP threeyear results

1
Constraining hybrid inflation models with WMAP
three-year results

• António Cardoso
• (with David Wands)
• Institute of Cosmology Gravitation
• University of Portsmouth

2
Motivation

• Reconsider the simplest models of quadratic
  hybrid inflation (the original hybrid and
  inverted hybrid models) in light of the WMAP
  three-year results
• Study the possibility of obtaining a spectral
  index of primordial density perturbations smaller
  than one from these models
• The original hybrid inflation model naturally
  predicts ns gt 1 in the false vacuum dominated
  regime, but it is also possible to have ns lt 1
  when the quadratic term dominates (Copeland et
  al., astroph/9401011). We are therefore
  interested in whether there is also an
  intermediate regime compatible with the latest
  constraints
• Investigate whether there is a regime where the
  scalar field value during the last 50 e-folds of
  inflation is less than the Planck scale, because
  such models are easier to construct within
  supergravity.

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The hybrid inflation model
Bassett et al., astroph/0507632
We assume that stays at the origin while
rolls down from an initially large (positive)
value until it reaches the critical value ,
after which becomes unstable.
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The inverted hybrid inflation model
The inflaton starts with an initial small
(positive) value and rolls down towards larger
values.
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Dynamics
We use the slow-roll aproximation for our
calculations
Therefore we can write the number of e-folds as
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Dynamics
For the hybrid inflation potential we have
Inflation will end either with the instability or
because of the end of slow-roll, as in the single
field case, depending on which occurs first. When
the end of slow-roll occurs before the inflaton
reaches we identify the end of inflation
with the condition
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Dynamics
We assume that cosmological scales exit the
Hubble scale at least 50 e-folds before the end
of inflation. In practice the actual number of
e-folds is dependent upon the details of
reheating at the end of inflation. Given that
can be made arbitrarily small by suitable choice
of ?/? we leave aaa, the value of when
cosmological scales leave the Hubble scale, as a
free parameter to be determined by observations,
subject only to the restriction
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Perturbations
Power spectrum of scalar curvature perturbations
For each value of M and m, we calculate the value
of such that the amplitude of PR is
consistent with the WMAP three-year results
Spectral tilt
Running
Tensor to scalar ratio
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Results for the hybrid inflation model
r
r
ns
ns
WMAPSDSS
WMAP
68 and 95 confidence level contours from
Spergel et al., astro-ph/0603449.
10-5 M 10-2 10-8 m 10-4
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Results for the hybrid inflation model
r
as
as
F50
r
r
F50
E
Red 0.945 lt ns lt 0.955 Green 0.965 lt ns lt
0.975 Blue 0.985 lt ns lt 0.995
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Results for the inverted hybrid inflation model
r
r
ns
ns
WMAPSDSS
WMAP
68 and 95 confidence level contours from
Spergel et al., astro-ph/0603449.
10-6 M 10-2 10-10 m 10-4
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Results for the inverted hybrid inflation model
r
as
as
F50
r
r
Red 0.945 lt ns lt 0.955 Green 0.965 lt ns lt
0.975 Blue 0.985 lt ns lt 0.995
E
F50
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Conclusions

• There is an intermediate regime for the original
  hybrid inflation model compatible with ns lt 1,
  but as one approaches the false vacuum dominated
  regime the tensor-to-scalar ratio, r, and the
  running of the spectral tilt, as, become large
  (with a positive running), which is in
  contradiction with the WMAP three-year results
• There is a lower bound on the allowed values of
  r, which puts strong constraints on the original
  hybrid model
• It is difficult to get smaller than the
  Planck scale in this model because decreasing
  requires us to increase the vacuum energy
  scale, M, and this increases the tensor-to-scalar
  ratio, r
• The inverted hybrid inflation model is in much
  better agreement with the WMAP three-year results
  than the original hybrid inflation model when we
  consider the false vacuum dominated regimes,
  because there is no lower bound on the allowed
  values of r
• In the inverted hybrid model there is no problem
  obtaining small values for aa.