dark matter production during reheating [-5pt] (by example)

XVIIMultiDarkConsoliderWorkshop

27/01/2021

Dark Matter Production During Reheating(by example)

Marcos A. G. GarcíaIFT-UAM

2011.13458 with G. Ballesteros and M. Pierre

2012.10756 with K. Kaneta, Y. Mambrini and K. Olive2006.03325 with Y. Mambrini, K. Olive and S. Verner2004.08404 with K. Kaneta, Y. Mambrini and K. Olive1806.01865 with M. Amin

Instituto de

Física

Teór icaUAM-CSIC

Is a spin-32 dark matter particle the missing piece in the puzzle?

1. ModelBuilding

2. Reheating

3. Freeze-in

4. Constraints

Described by Rarita-Schwinger Lagrangian

L03/2 = −1

2 Ψµ

(iγµρν∂ρ + m3/2γ

µν)Ψν

with γµν = γ[µγν] and γµνρ = γ[µγνγρ]

Not a new idea: the gravitino in supergravity is a well-known non-thermal relic. ForWIMP-like models see

• Z. H. Yu et al., Nucl. Phys. B 860 (2012), 115• R. Ding et al., JCAP 05 (2013), 028• N. D. Christensen et al., Eur. Phys. J. C 73 (2013) no.10, 2580• K. G. Savvidy and J. D. Vergados, Phys. Rev. D 87 (2013) 075013

For these there’s a Z2 symmetry to make it stable.

Is a spin-32 dark matter particle the missing piece in the puzzle?

1. ModelBuilding

2. Reheating

3. Freeze-in

4. Constraints

Described by Rarita-Schwinger Lagrangian

L03/2 = −1

2 Ψµ

(iγµρν∂ρ + m3/2γ

µν)Ψν

with γµν = γ[µγν] and γµνρ = γ[µγνγρ]

Consider instead a minimal set-up,

L = LSM + L03/2 + L0

νR + yH νLνR +MR2 νc

RνR

+ i α12MP

νRγµ[γρ, γσ]ΨµFρσ + i α2

2MPiσ2(DµH)∗LΨµ + h.c.

with usual mixing relations

m1 ≃ y2v2

2MR, m2 ≃ MR , tan θ ≃ yv√

2MR

Decays1. ModelBuilding

2. Reheating

3. Freeze-in

4. Constraints

α1 dominates

×

3/2

γ, Z

ν1

νR

3/2

γ, Z

H

ν1

νR

τ2b3/2 ≃ 1.6 × 1029

(10−2

y α1

)2 ( MR1014 GeV

)2 (104 GeVm3/2

)3

s

τ3b3/2 ≃ 5.6 × 1028

(10−2

y α1

)2 ( MR1014 GeV

)2 (104 GeVm3/2

)5

s

α2 dominates

3/2

H

ν

3/2

ℓ (ν)

W (Z)

3/2

ν

f

f

H

3/2

ℓ (ν)

f ′ (f)

f

W (Z)

τ3/2

1028s ≃

14.8(

10−7

α2

)2 (1 GeVm3/2

)3, m3/2 > mH

0.6(

10−3

α2

)2 (1 GeVm3/2

)5.28, me < m3/2 < mW

4.8(

10−3

α2

)2 (1 GeVm3/2

)5, m3/2 < me

10−3 10−1 10 103

m3/2 [GeV]

10−15

10−10

10−5

1

Γ3/

2/Γ

2b

τ3/2

1028s ≃

14.8(

10−7

α2

)2 (1 GeVm3/2

)3, m3/2 > mH

0.6(

10−3

α2

)2 (1 GeVm3/2

)5.28, me < m3/2 < mW

4.8(

10−3

α2

)2 (1 GeVm3/2

)5, m3/2 < me

Decays1. ModelBuilding

2. Reheating

3. Freeze-in

4. Constraints

α1 dominates

×

3/2

γ, Z

ν1

νR

3/2

γ, Z

H

ν1

νR

τ2b3/2 ≃ 1.6 × 1029

(10−2

y α1

)2 ( MR1014 GeV

)2 (104 GeVm3/2

)3

s

τ3b3/2 ≃ 5.6 × 1028

(10−2

y α1

)2 ( MR1014 GeV

)2 (104 GeVm3/2

)5

s

α2 dominates

10−3 10−1 10 103

m3/2 [GeV]

10−15

10−10

10−5

1

Γ3/

2/Γ

2b

τ3/2

1028s ≃

14.8(

10−7

α2

)2 (1 GeVm3/2

)3, m3/2 > mH

0.6(

10−3

α2

)2 (1 GeVm3/2

)5.28, me < m3/2 < mW

4.8(

10−3

α2

)2 (1 GeVm3/2

)5, m3/2 < me

Production (via scatterings)1. ModelBuilding

2. Reheating

3. Freeze-in

4. Constraints

α1 dominates

Bµ

3/2

ν1

H

νR

3/2

ν1H

Bµ

νR

3/2

Hν1

Bµ

νR

σ(s) =11α2

1y2s2

72πm23/2M 2

RM 2P

α2 dominates

σ(s) =α2

2s9216πm2

3/2M 2P

× (639g2 + 87g′2 + 144h2t + 32h2

τ )

Production (via inflaton decay)1. ModelBuilding

2. Reheating

3. Freeze-in

4. Constraints

Assume LΦ ⊃ yνΦνRνR. Via α1,

MR ≪ mΦ:

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R

R

Φ

νR

B

3/2

MR ≫ mΦ:

Φ

νR

νR

νL

H

H

Φ

νR

νR

H

ν1

ν1

Φ

νR

νR

B

3/2

3/2

(via α2 are 2-loop suppressed)

Reheating1. ModelBuilding

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2. Reheating

3. Freeze-in

4. Constraints

After inflation, the Universe is reheated through the decay of the inflaton Φ

V(Φ)

Φ

k = 2 4 6

V(Φ) = λM 4P

[√6 tanh

(Φ√6MP

)]kΦ≪MP−→ λ

Φk

M k−4P

(R. Kallosh and A. Linde, JCAP 07 (2013), 002)

ρΦ + 3H(ρΦ + PΦ) = 03H 2M 2

P = ρΦ

where

ρΦ =12 Φ

2 + V(Φ)

PΦ =12 Φ

2 − V(Φ)

Inflaton oscillation1. ModelBuilding


2. Reheating

3. Freeze-in

4. Constraints

Φ2.

t

|Φ|3

t

Φ4.

t

Over one oscillation

⟨Φ2⟩ ≃ ⟨ΦV ′(Φ)⟩y⟨PΦ⟩ =

k − 2k + 2 ⟨ρΦ⟩

Inflaton oscillation1. ModelBuilding


2. Reheating

3. Freeze-in

4. Constraints

Φ2.

t

|Φ|3

t

Φ4.

t

∼matter

ρΦ = ρend

(a

aend

)− 6kk+2

a ∝ tk+23k

∼ radiation

Decay of the inflaton1. ModelBuilding


2. Reheating

3. Freeze-in

4. Constraints

ρΦ + 3(

2kk + 2

)HρΦ = −ΓΦ(t)ρΦ

ρR + 4HρR = ΓΦ(t)ρΦ3M 2

PH 2 = ρΦ + ρR

yΦ

f

f

ΓΦ =y2

8πmΦ(t) ,

m2Φ ≡ ∂2

ΦV(Φ) ∝ ρk−2

kΦ

10-4

0.001 0.010 0.100 1 10

0.0

0.2

0.4

0.6

0.8

1.0

ρ/ρ

end

t/treh

Φ

R

2k = 4



2. Reheating

3. Freeze-in

4. Constraints

ρΦ + 3(

2kk + 2



PH 2 = ρΦ + ρR

µΦ

b

b

ΓΦ =µ2

8πmΦ(t),

m2Φ ≡ ∂2

ΦV(Φ) ∝ ρk−2

kΦ

10-4

0.001 0.010 0.100 1 10

0.0

0.2

0.4

0.6

0.8

1.0

ρ/ρ

end

t/treh

Φ

R

4k = 2



2. Reheating

3. Freeze-in

4. Constraints

ρΦ + 3(

2kk + 2



PH 2 = ρΦ + ρR

yΦ

f

f

ΓΦ =y2

8πmΦ(t) ,

m2Φ ≡ ∂2

ΦV(Φ) ∝ ρk−2

kΦ

10 104 107 1010

a/aend

103

106

109

1012

T[G

eV]

k = 2

3

4

y = 10−5

T =

(30ρRπ2g∗

)1/4

∝ a− 32

k−3k+4

Treh

Tmax



2. Reheating

3. Freeze-in

4. Constraints

ρΦ + 3(

2kk + 2



PH 2 = ρΦ + ρR

yΦ

f

f

ΓΦ =y2

8πmΦ(t) ,

m2Φ ≡ ∂2

ΦV(Φ) ∝ ρk−2

kΦ

10 103 105 107

a/aend

109

1010

1011

1012

1013

〈p〉[G

eV]

hard

soft

y = 10−5

k = 2

ΓΦtth ≃ α−16/5SM

(ΓΦm2

Φ

M3P

)2/5Tth

Freeze-in during reheating1. ModelBuilding


2. Reheating

3. Freeze-in

4. Constraints

For the out-of-equilibrium process i + j + · · · → Ψ+ a + b + · · · ,

∂f3/2

∂t − H|p|∂f3/2

∂|p| ≃ 12p0

∫ gad 3pa(2π)32pa0

gbd 3pb(2π)32pb0

· · · gid 3pi(2π)32pi0

gjd 3pj

(2π)32pj0· · ·

× (2π)4δ(4)(p + pa + pb + · · · − pi − pj − · · · )

× |M|2i+j+···→Ψ+a+b+···fi fj · · ·

Inflaton decay Φ → Ψ+Ψ1. ModelBuilding


2. Reheating

3. Freeze-in

4. Constraints

∂f3/2

∂t − H|p|∂f3/2

∂|p| ≃ 12p0

∫ g3/2d 3k(2π)32k0

d 3P(2π)32P0

(2π)4δ(4)(P − p − k)

× 2α14y2

νm2Φ

9π4M4pm43/2

[5 − 6 ln

(M 2

Rm2

Φ

)]2

(2π)3nΦ(t)δ(3)(P)

0.0 0.2 0.4 0.6 0.8 1.0q

0.0

0.5

1.0

q2f 3/2(q

)

0 1 2 3 4q

0.0

0.5

1.0f3/2 ∝

(mΦ

2p

)3/2θ(mΦ/2 − p) f3/2 ∝ q−3/2e−0.74q2

q ∝(

Treh

mΦ

)pco

T0

t ≪ treh t ≫ treh

Inflaton decay Φ → Ψ+Ψ1. ModelBuilding


2. Reheating

3. Freeze-in

4. Constraints

∂f3/2

∂t − H|p|∂f3/2

∂|p| ≃ 12p0

∫ g3/2d 3k(2π)32k0

d 3P(2π)32P0

(2π)4δ(4)(P − p − k)

× 2α14y2

νm2Φ

9π4M4pm43/2

[5 − 6 ln

(M 2

Rm2

Φ

)]2

(2π)3nΦ(t)δ(3)(P)

Ω3/2h2 ≃ 0.1(

α1

1.1 × 10−8

)4( mΦ

3 × 1013 GeV

)5(0.15 eVm1

)2

×(

104 GeVm3/2

)3( Treh

1010 GeV

)×

(ln(M2R/m2

ϕ)− 5/6)2

ln2(M2R/m2

ϕ).

DM production from non-quadratic inflaton decay → work in progress!

Scatterings H + ν → Ψ+ B1. ModelBuilding


2. Reheating

3. Freeze-in

4. Constraints

∂f3/2

∂t − H|p|∂f3/2

∂|p| ≃ 12p0

∫2d 3p′

(2π)32p′0

d 3k1

(2π)32k01

2d 3k2

(2π)32k02(2π)4δ(4)(p + p′ − k1 − k2)

×

(−8

3α1

2y2

m23/2M 2

RM 2P

s2t)

1ek1/T + 1

1ek2/T − 1

0 2 4 6 8 10 12 14 16q

0

1

2

3

4

5

q2f 3/2

(q) f3/2 ∝ q0.21e−0.06q2.04

q ∝ pco

T0



2. Reheating

3. Freeze-in

4. Constraints

∂f3/2

∂t − H|p|∂f3/2

∂|p| ≃ 12p0

∫2d 3p′

(2π)32p′0

d 3k1

(2π)32k01

2d 3k2

(2π)32k02(2π)4δ(4)(p + p′ − k1 − k2)

×

(−8

3α1

2y2

m23/2M 2

RM 2P

s2t)

1ek1/T + 1

1ek2/T − 1

Ω3/2h2 ≃ 0.1(

α1

1.1 × 10−3

)2(427/4greh

)3/2( Treh

1010 GeV

)5

×( m1

0.15 eV

)(1014 GeVMR

)(104 GeV

m3/2

)

(quadratic inflaton potential)



2. Reheating

3. Freeze-in

4. Constraints

∂f3/2

∂t − H|p|∂f3/2

∂|p| ≃ 12p0

∫2d 3p′

(2π)32p′0

d 3k1

(2π)32k01

2d 3k2

(2π)32k02(2π)4δ(4)(p + p′ − k1 − k2)

×

(−8

3α1

2y2

m23/2M 2

RM 2P

s2t)

1ek1/T + 1

1ek2/T − 1

Ω3/2h2 ≃ 0.1(

α1

2 × 10−3

)2(427/4greh

)3/2( Treh

1010 GeV

)5

×( m1

0.15 eV

)(1014 GeVMR

)(104 GeV

m3/2

)(Tmax

Treh

)10/3

(quartic inflaton potential, ϕ → f f )



2. Reheating

3. Freeze-in

4. Constraints

∂f3/2

∂t − H|p|∂f3/2

∂|p| ≃ 12p0

∫2d 3p′

(2π)32p′0

d 3k1

(2π)32k01

2d 3k2

(2π)32k02(2π)4δ(4)(p + p′ − k1 − k2)

×

(−8

3α1

2y2

m23/2M 2

RM 2P

s2t)

Brν(

24π2ΓΦtnΦ

m3Φ

)2( m2Φ

4k1k2

)3/2

θ(mΦ2 − k1)θ(

mΦ2 − k2)

0.0 0.5 1.0 1.5q

0

50

100

150

200

q2f 3/2

(q) f3/2 ∝ q−3/2e−2.5q2.6

q ∝ α32/25SM

(Treh

mΦ

)7/15 pco

T0



2. Reheating

3. Freeze-in

4. Constraints

∂f3/2

∂t − H|p|∂f3/2

∂|p| ≃ 12p0

∫2d 3p′

(2π)32p′0

d 3k1

(2π)32k01

2d 3k2

(2π)32k02(2π)4δ(4)(p + p′ − k1 − k2)

×

(−8

3α1

2y2

m23/2M 2

RM 2P

s2t)

Brν(

24π2ΓΦtnΦ

m3Φ

)2( m2Φ

4k1k2

)3/2

θ(mΦ2 − k1)θ(

mΦ2 − k2)

Ω3/2h2 ≃ 0.1(

α1

1.1 × 10−3

)2(0.030αSM

)16/5 ( m1

0.15 eV

)( greh

427/4

)7/10(104 GeVm3/2

)×(

1014 GeVMR

)(mΦ

3 × 1013 GeV

)14/5( Treh

1010 GeV

)19/5( B1

7 × 10−4

)

Thermalization in non-quadratic reheating not known yet

Constraints: ΩDM + γ + ν1. ModelBuilding


2. Reheating

3. Freeze-in

4. Constraints

Thermal

Non-thermal

1

10−4

α1 = 10−8

τ3/2 < τexp

Brν = 10−4

−6 −4 −2 0 2 4 6log10(m3/2/GeV)

5

6

7

8

9

10

11

12

log 1

0(T

reh/G

eV)

FERMI 130 GeV

IceCube PeV

Scattering

T reh=

1010 GeVT reh

=10

5 GeV

τ3/2 < τexpΩ3/2h2 > 0.1

−4 −2 0 2 4 6log10(m3/2/GeV)

−12

−10

−8

−6

−4

−2

0

log 1

0α

1

Inflaton decay

Constraints: ΩDM + γ + ν + Lyman-α1. ModelBuilding


2. Reheating

3. Freeze-in

4. Constraints

Thermal

Non-thermal

1

10−4

α1 = 10−8

Ly-α

excluded τ3/2 < τexp

Brν = 10−4

−6 −4 −2 0 2 4 6log10(m3/2/GeV)

5

6

7

8

9

10

11

12

log 1

0(T

reh/G

eV)

FERMI 130 GeV

IceCube PeV

Scattering

T reh=

1010 GeV

τ3/2 < τexp

Ly-α excluded

−4 −2 0 2 4 6log10(m3/2/GeV)

−12

−10

−8

−6

−4

−2

0

log 1

0α

1

Inflaton decay

For further details, see Mathias’ talk!

dark matter production during reheating [-5pt] (by example)

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