Hooshyar Assadullahi- Primordial non-Gaussianity from two curvaton decays

8/3/2019 Hooshyar Assadullahi- Primordial non-Gaussianity from two curvaton decays

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Primordial non-Gaussianity

from two curvaton decays

Hooshyar Assadullahi (ICG. Portsmouth)

Phys. Rev. D 76,103003(arxiv:0708.0223v1)

H. Assadullahi, J. Valiviita, D. Wands

1th of July-Sheffield university



Outline:

•Primordial curvature perturbation.

•Curvaton scenario.

•Non-Gausianity with one curvaton.

•Non-Gausianity with two curvatons.

•Conclusion.



Curvature perturbation (first order):

•The gauge invariant, used to describe the large-scale curvature perturbation is:

ρ

δρ ψ ζ

& H −−≡

•It is curvature perturbation on uniform density gauge, and the densityperturbation on uniform curvature slices.

Generally curvature perturbation is constant if:

1.Consider large scale

2.Pressure is only function of density (Baryotropic)

3.Non-interacting system



Single field inflation:

•In inflation models, the inflaton field plays two roles: makesthe universe expand enough and generates primordialdensity perturbation.

•If the inflaton made primordial density perturbations, introduceextra fine tuning into the model: for example, inflaton masscould be large (m H), without considering perturbations.

Curvaton scenario:

perturbations in some field other than theinflaton could be responsible for theprimordial density perturbation ⇒ curvaton

Wands, Lyth :Phys. Lett. B 524, 5

Moroi and T. Takahashi, Phys. Lett. B 522, 215K. Enqvist and M. S. Sloth, Nucl. Phys. B626, 395

≈



Generating curvature perturbation by curvaton:

•The curvaton field is supposed to be light during inflation, with small effectivemass, << H

•At end of inflation the energy of the inflaton field is converted into radiation4−

∝ar ρ

•After inflation H is a decreasing function of time. Finally and curvatonstarts to oscillate.

•The curvaton field behaves like pressure less matter

• so

• Decay rate( ) of curvaton is negligible as long as

• Curvaton decays some time before nucleosynthesis.

3−∝ aσ ρ

a

r

∝

ρ

ρ σ

→

22 H m >σ

1− H

Γ > H Γ

σ m

Γ =dec H



Curvaton perturbations:

•The curvaton field is supposed to be practically free during inflation, so it’spower spectrum at Hubble exit is:

π δσ

2)(2

1

∗≈

H k P

•The energy density in oscillating field after inflation is:

),(~),( 22 t xmt x σ ρ σ σ =

•The perturbation of the curvaton field is:

σ

δσ

ρ

δρ ζ

σ

σ σ

3

2

3

1≈=



•Considering two fields (radiation and curvaton) the total curvature

perturbation will be:

σ σ ζ

ρ ρ ζ

ρ ρ ζ

&&

&& += r

r tot

•Curvaton scenario corresponds to the case where the density

perturbation in the radiation produced at the end of inflation isnegligible:

0≈r ζ

σ

σ

σ σ

σ σ

σ σ ζ ρ ρ

ρ ζ

ρ ρ

ρ ζ

r r r

tot p H p H

p H

43

3

)(3)(3

)(3

+=

+−+−

+−=⇒

Before

curvato

n decay

Aftercurvaton

decayσ ζ ζ r tot = decay

r

r )43

3(

ρ ρ

ρ

σ

σ

+=⇒

•If the curvaton completely dominates the energy density before itdecays we have r = 1, otherwise r < 1.



Non-Gaussianity:•The simplest (possible) non-Gaussianity is caused by the square of the first

order perturbation.

•Up to second order the perturbation can be written:

2

115

3ζ ζ ζ NL f +=

•The primordial power spectrum:

•The primordial bispectrum is:

)()()2( 21

3

1

3

)()( 21

k k k Pk k

rrrr += δ π ζ ζ ζ

•The bispectrum vanishes for a purely Gaussian distribution

•The bispectrum relative to the power spectrum is commonly parameterized interms of : NL f

]2[5

6)( )()(321 21 permPP f k k k B k k NL +=++

)()()2( 321

3

321

3

)()()( 321

k k k k k k Bk k k

rrrrrr ++++= δ π ζ ζ ζ



•In the sudden decay approximation, the bi-spectrum calculation leads to:

Non-Gausianity with one curvaton (M. Sasaki , J.Valiviita & D. Wands(Phys. Rev. D 74,

103003))

r g

gg

r r f NL

6

5

3

5)1(

1

4

5)(

2'

''

−−+=

Where g describes the evolution of the curvaton between Hubble-exit duringinflation and the beginning of the field oscillations:

)(∗

= σ σ σ gosc

•If we consider (r<<1) and linear evolution( ):0''=g

r

f NL4

5=

• The non-linearity becomes large only if the transfer of curvaton to curvatureperturbation is inefficient (curvaton decays before it dominates the universe).



Non-Gausianity with two curvatons: (H. Assadullahi, J. Valiviita , D. Wands Phys. Rev. D 76,103003)

•The primordial curvature perturbation after both curvatons decay can be written up to

second order in terms of the first order curvatons perturbations:

a H Γ =•We assume that the curvaton a decays first when followed by the decay of the

curvaton b when , ( ).b H Γ = ab Γ <Γ

)1()1(

2

)1(

2

)1()1()1()2(2)1(222

1

2

1

2

1

2

1bababa E DC B A ζ ζ ζ ζ ζ ζ ζ ζ ζ ++++=+=

•We also find the relation between power spectra of each curvaton fieldand power spectrum of curvature perturbation:

ζ ζ β

P B A

Pa 222

1+

=ζ ζ

β β P

B AP

b 222

2

+=

Where is the ratio between power spectrum of and : β aζ bζ

ab PP ζ ζ β

2=

•If we use all above relations we get:

permsk k k k Pk P B A

ABE D BC A

k k k ++++

++

= )()2)(()()(

2

1

)()()( 321

)3(3

212222

2242

321

rrrrrr

δ π β

β β

ζ ζ ζ ζ ζ



So we can derive for two curvatons: NL f

2222

2422

)(

21

6

5

B A

DB EABCA

f NL β

β β

+

++

=

•At first order we have:inbbinaa r r ,, ζ ζ ζ +=

)34(3

)334(3

)334(3

])1(3[])3()1[(])1(3[])3)(1[(

2

01

01

1111121

11112

br bb

bar bb

bar aa

abbaabbb

abaaba

f

f

f

f f f f f f f r f f f f f r

Ω+ΩΩ=

Ω+Ω+ΩΩ=

Ω+Ω+ΩΩ=

+−++−=

+−+−=

at first decay

at first decay

at second decay

Where:

•We can find A,B,C,D from second order calculation which are complicated

functions of energy density of curvatons at their decay time.



NL f in various limits:

I:Single curvaton limits:

•If density of first or second curvaton is always zero ( or ):0=a f 021 == bb f f

NL NL f r

r

f =−−=

6

5

3

5

4

5(single)

II: Simultaneous decay of two curvatons:

•If second curvaton decays at the same time as the first curvaton, we get:

21121 )31( babbb f f f +=⇒Ω=Ω

NL NL f f = single11

65)( ba f f −

•In the limiting case where the second curvaton is homogeneous ( = 0): β

•When the first curvaton is homogeneous ( → ∞) we get: β

NL NL f f =single

11

6

5)( ab f f −



•In either case the example of two curvatons which decay at the same time does not

reduce exactly to the case of a single curvaton.

•The minimum value for (for any value of ) is still that found for a single curvaton: NL f β

)min(4

5)min( NL NL f f =−=

(Single)

III:Both curvatons subdominate at decay:

•If both curvatons decay before they dominate the energy density( ) wecan simplify the non-Gausianity parameter:

1,, 211 <<bba f f f

2222

343

)(4

5

ba

ba NL

r r

r r f

β

β

+

+=

•Like for one curvaton, is big if both curvatons are subdominant when they decay.

•But if the last curvaton is homogenous ( = 0), it dilutes the effect of first curvaton andwe can have big non-Gausianity even when both curvatons dominate before they decay.(this is a novel effect that has not been discovered with one curvaton)

11)(),()()](1)[(,1

2121 >>⇒≅⇒−∝∝ NLa

a

NL f b f a f if b f a f r r f

NL f

β



General case:

when ( )

NL f

NL

f ba

PPζ ζ

= 1= β



Conclusion:

•The curvaton is the proposal for how primordial density perturbation may

arise from vacuum fluctuations of a light scalar field that does not dominateduring inflation.

•Detection of primordial non-Gaussianity could provide support for a curvaton-type scenario.

•With one or two curvatons fields the non-Gausianity becomes large if thetransfer of curvaton to curvature perturbation is inefficient.

•With two curvaton we can have big non-Gausianity even when the two fieldsdominate before they decay, if the second curvaton is homogeneous. In thiscase the second curvaton dilutes the first order perturbation.

•If we take seriously the multiplicity of scalar fields in the early universe, weshould consider more than one field can contribute to the primordial densityperturbation.

•It should be straightforward to extend our analysis to three or more curvatons.



When NL f 0= β

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Hooshyar Assadullahi- Primordial non-Gaussianity from two curvaton decays