Electrically charged curvaton
Michela D’Onofrioa,b , Rose Lernera,b , Arttu Rajantiec
a University of Helsinki, b Helsinki Institute of Physics, c Imperial College London
[JCAP - astro-ph/1207.1063]
COSMO 2013
Cambridge, 2-6 Sept 2013
Michela D’Onofrio - Helsinki University, HIP
Electrically charged curvaton
Introduction
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Inflation introduced to solve three problems of Standard Cosmology:
flatness, horizon, unwanted relics.
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Curvaton model:
inflaton φ drives the expansion;
curvaton σ produces curvature perturbations.
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During inflation: σ is subdominant and light.
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After inflation: σ decays and perturbations affect the Universe.
Michela D’Onofrio - Helsinki University, HIP
Electrically charged curvaton
Motivation
why do we want a charged curvaton?
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connect curvature perturbation to Standard Model;
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give U(1)-charge to curvaton;
−→ less free parameters!
−→ large coupling g 0 ≈ 0.36, interesting
curvaton–photon interactions;
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when curvaton decays, significant contribution to curvature
perturbation.
Michela D’Onofrio - Helsinki University, HIP
Electrically charged curvaton
Model
We assume the curvaton carries one unit of U(1) weak hypercharge Y = 1.
The Lagrangian is:
1
L φ = ∂µ φ∂µ φ − V (φ)
2
(1)
1
L σ = −m2 σ† σ − λ(σ† σ)2 − Fµν F µν + |(i∂µ − g 0 Aµ )σ|2
4
(2)
We obtain a curvaton e.o.m. which is exactly solvable only by
non-perturbative methods.
Michela D’Onofrio - Helsinki University, HIP
Electrically charged curvaton
Constraints on the Effective Potential
Due to the large value of g 0 ≈ 0.36, the potential gains quantum
corrections (Coleman-Weinberg)
Veff (σ) = m2 |σ|2 +
3 g0
|σ|2
4
|σ|
ln
64 π2
µ2
which have impact on the parameter space.
A curvaton must satisfy:
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vacuum stability
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shallow potential
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linearity
Michela D’Onofrio - Helsinki University, HIP
Electrically charged curvaton
Constraints on the Effective Potential
Veff
Ï VACUUM STABILITY:
increasing µ −→
Vacuum at σ = 0 must
otherwise tunneling to
Veff
be true vacuum,
false vacuum →
σ
spontaneously break
σ/H∗
U(1) symmetry, photon
massive.
Ï SHALLOW POTENTIAL :
00
),
in order to have a light curvaton (m2eff ≡ Veff
and Vσ ¿ Vφ .
Ï LINEARITY:
σ evolves linearly both during and after inflation.
Michela D’Onofrio - Helsinki University, HIP
Electrically charged curvaton
Curvaton dynamics
The background curvaton has e.o.m.:
σ̈ + 3H σ̇ + m2 σ = 0
with H(t) =
1
2t
(radiation-dominated epoch).
15
The curvaton evolves in time as:
µ
¶
3π
σ∗
σ(t) ≈
cos mt −
8
(mt)3/4
σ/H∗
10
5
20
-5
40
60
80
100
mt
-10
-15
Michela D’Onofrio - Helsinki University, HIP
Electrically charged curvaton
Possible evolution after Inflation
After the end of inflation, the curvaton is a homogeneous condensate that
oscillates in its potential. Its evolution depends on interactions with other
fields, which cause it to decay into curvaton particles.
- Interaction with a
thermal bath of photons
- Decay through
parametric resonance
T ¿m
late decay
T Àm
σ decays too quickly
T ¿m
linear
curvaton leftovers
nonlinear
T Àm
thermal bath
Michela D’Onofrio - Helsinki University, HIP
Electrically charged curvaton
Interaction with Thermal Bath
Curvaton–photon interaction: condensate −→ curvaton particles.
Γth ≈ 0.03 g 02 T
T ¿ m: particles are non-relativistic and decay at a very late time.
T À m: if inflaton decays into photons immediately after inflation
→ curvaton decays too quickly, and ζ too small (if chemical
equilibrium)
Viable model if there are almost no photons after inflation in thermal bg:
(i) φ decays to hidden sector;
(ii) φ decays late → φ4 -potential.
Michela D’Onofrio - Helsinki University, HIP
Electrically charged curvaton
Non-perturbative Decay
– Provided NO interaction with thermal bath:
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Curvaton produces photons non-perturbatively: parametric
resonance
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Gauge field in the curvaton background follows Mathieu equation
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Solutions are either oscillatory or growing
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Growing solution = energy transfer from curvaton to photon
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Type of solution given by instability plot
Michela D’Onofrio - Helsinki University, HIP
Electrically charged curvaton
Gauge field dynamics
The evolution of the gauge field follows Mathieu equation:
¡
¢
00
B (z, k) + Σk (z) + 2q(z) cos 2z B(z, k) = 0
where B(t, k) = a(t)1/2 A(t, k), z = mt and coefficients:
q(z)
≈
Σk (z)
≈
g 02 σ2∗
m2 z3/2
k2
3
+
+ 2q(z)
2mH∗ z 16z2
k is the comoving momentum, and we have inserted the curvaton
solution.
Michela D’Onofrio - Helsinki University, HIP
Electrically charged curvaton
Instability plot of Mathieu equation
- shaded regions = stable bands;
q(z)
- white regions = resonance bands with
exponentially growing solutions;
- the solid line shows Σ = 2q
⇒
k = 0;
- starting position and speed depend on m
Σ(z)
and σ∗ ;
- As modes with k > 0 follow a similar evolution, but on a lower line,
modes with higher k spend less time in the instability bands −→ weaker
resonance.
Michela D’Onofrio - Helsinki University, HIP
Electrically charged curvaton
amplification of Aµ
Amplification of gauge field
mt
Amplification of the gauge field A as a function of mt, for different σ∗ (H∗ )
and m(H∗ ). Upper line: huge amplification, resonance is nonlinear. Lower
line: linear evolution. Dashed line: nonlinearity condition.
Michela D’Onofrio - Helsinki University, HIP
Electrically charged curvaton
σ∗ /H∗
Constraints on Parameter Space
m/H∗
Allowed region for a viable curvaton model which produces ζ = 10−5 . The
size of the allowed region reduces as H∗ reduces. For H∗ . 108 GeV there
is no allowed parameter space.
Michela D’Onofrio - Helsinki University, HIP
Electrically charged curvaton
Constraints on Parameter Space
H∗ > 3 × 109 : regions 1, 2, 3 allowed
σ∗ /H∗
109 < H∗ < 3 × 109 : regions 2, 3
2 × 108 < H∗ < 109 : region 3
3
2
1
m/H∗
Allowed region for a viable curvaton model which produces ζ = 10−5 . The
size of the allowed region reduces as H∗ reduces. For H∗ . 108 GeV there
is no allowed parameter space.
Michela D’Onofrio - Helsinki University, HIP
Electrically charged curvaton
Constraints on Parameter Space
shallow potential constraint:
σ∗ /H∗
00
¿ H∗2
m2eff ≡ Veff
m < H∗
linearity constraint
m/H∗
Allowed region for a viable curvaton model which produces ζ = 10−5 . The
size of the allowed region reduces as H∗ reduces. For H∗ . 108 GeV there
is no allowed parameter space.
Michela D’Onofrio - Helsinki University, HIP
Electrically charged curvaton
Conclusions
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We explored the possibility of having a U(1)-charged curvaton.
Ï
We connected SM to inflation and reduced the number of free
parameters.
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Two different decay modes: interaction with thermal bath and
parametric resonance.
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The model is allowed, although parameter space is restricted by
theoretical and observational constraints.
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Non-perturbative calculations are needed to further investigate the
model.
Michela D’Onofrio - Helsinki University, HIP
Electrically charged curvaton