Gravitational Waves Induced by Scalar Perturbations during a - - PowerPoint PPT Presentation

Gravitational Waves Induced by Scalar Perturbations

during a gradual Transition from an Early Matter era to the Radiation era Keisuke Inomata

Institute for Cosmic Ray Research (ICRR), The University of Tokyo

Collaborators: K.Kohri(KEK), T.Nakama(HKUST), T.Terada(KEK) arXiv: 1904.12878 (and 1904.12879 for Terada’s talk )

Note: My talk is strongly related to Terada’s talk (right after mine).

Keisuke Inomata (ICRR, The Univ. of Tokyo) arXiv: 1904.12878 (& 1904.12879)

Overview of our two talks

We revisit the effect of an early matter dominated era on the gravitational waves induced by scalar perturbations.

0.001 0.0050.010 0.0500.100 0.500 1 0.001 10.000 105 109 1013

Previous works gradual reheating sudden reheating Our works

As a result, we find that the induced GWs strongly depend on how quickly the reheating occurs.

Pζ = AsΘ(k − 30/ηeq)Θ(kmax − k), kmax = 450/ηR

Terada’s talk My talk

2/18

Outline

• Introduction
• eMD effects in a gradual reheating scenario
• Summary

Keisuke Inomata (ICRR, The Univ. of Tokyo) arXiv: 1904.12878 (& 1904.12879)

GWs induced by scalar perturbations

At second order, scalar perturbations can induce GWs.

ds2 = a2  −(1 + 2Φ)dη2 + ✓ (1 − 2Ψ)δij + 1 2hij ◆ dxidxj

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Metric perturbations:

Scalar perturbations (related to curvature perturbations) Tensor perturbations (GWs)

E.o.M for GWs:

hλ00

k

+ 2Hhλ0

k + k2hλ k = 4Sλ(k, η)

Sλ(k, η) = Z d3k0 (2π)3 eλlm(ˆ k)k0

lk0 m [2Φk0(η)Φkk0(η)

+ 4 3(1 + w) ✓ Φk0(η) + Φ0

k0(η)

H ◆ ✓ Φkk0(η) + Φ0

kk0(η)

H ◆

Source term from second order scalar perturbations

(Ananda et al 2006, Baumann et al. 2007)

4/18

Keisuke Inomata (ICRR, The Univ. of Tokyo) arXiv: 1904.12878 (& 1904.12879)

Motivations for the induced GWs

The future and current GW projects can investigate the power spectrum of curvature perturbations. (See also Byrnes, Cole, & Patil 2018)

(Inomata, Nakama, 2018)

The frequencies of the induced GWs correspond to the scale of the scalar perturbations inducing GWs. We can investigate small-scale (scalar) perturbations using the induced GWs!

Since small-scale perturbations enter the horizon early, the induced GW could be a good probe of the early Universe.

What happens to the GWs if there is an early matter dominated era in the early Universe? Main question of this work

(λ . 1 Mpc)

Small scale Large scale

5/18

Keisuke Inomata (ICRR, The Univ. of Tokyo) arXiv: 1904.12878 (& 1904.12879)

The history of the Universe

Inflation era Early matter dominated (eMD) era Radiation dominated (RD) era (Late) matter dominated era (from z~3400) Dark energy dominated era (from z~0.5) Reheating We focus on this!

An eMD era can be realized by an inflaton oscillation

• r a curvaton oscillation.

6/18

Keisuke Inomata (ICRR, The Univ. of Tokyo) arXiv: 1904.12878 (& 1904.12879)

Results in previous works

(Assadullahi, Wands, 2009, Alabidi, Kohri, Sasaki, Sendaouda, 2013, Kohri, Terada, 2018)

During an eMD era, the transfer function of the gravitational potential (Φ), the source of GWs, does not decay even on subhorizon because of the growing matter perturbations. There are works discussing the effects of an eMD era on the induced GWs.

0.1 0.5 1 5 10 50 100 10-5 10-4 0.001 0.010 0.100 1 10

During eMD era During RD era

(x = kη)

Horizon entry

Induced GWs are enhanced?

(Alabidi et al., 2013)

7/18

Keisuke Inomata (ICRR, The Univ. of Tokyo) arXiv: 1904.12878 (& 1904.12879)

Implicit assumptions in previous works

In previous works, the authors implicitly assume the reheating occurs suddenly and stop the calculation at the reheating.

(Assadullahi, Wands, 2009, Alabidi, Kohri, Sasaki, Sendaouda, 2013)

• 1. The transition can occur gradually in some realistic situations.
• 2. Even if the transition occurs suddenly, we need to care about the GWs

induced after the transition. We revisit the effects of an eMD era on the induced GWs. We consider two cases:

• 1. Gradual reheating, whose time scale is comparable to the Hubble time scale at

that time. (My talk)

• 2. Sudden reheating, whose time scale is much shorter than the Hubble time

scale at that time. (Terada’s talk)

What we do in this work

However,

8/18

Keisuke Inomata (ICRR, The Univ. of Tokyo) arXiv: 1904.12878 (& 1904.12879)

Evolutions of perturbations

Time Horizon scale Physical length Subhorizon Superhorizon Inflation era eMD era RD era Reheating Perturbation scales To spotlight the effects of an eMD era, we focus on the perturbations entering horizon during eMD era (kR < k < kmax).

k=kmax k=kR kR<k<kmax

(log a)

9/18

Outline

• Introduction
• eMD effects in a gradual reheating scenario
• Summary

Keisuke Inomata (ICRR, The Univ. of Tokyo) arXiv: 1904.12878 (& 1904.12879)

ρm : ρr : δ : θ (≡ ikjvj) :

Gradual reheating

We assume that the dominant field during an eMD era (inflaton or curvaton) decays to radiation with a constant decay rate Γ. The situation is similar to the decaying dark matter scenario. Therefore we use the following equations when we calculate the evolutions.

ρ0

m = −(3H + aΓ)ρm,

ρ0

r = −4Hρr + aΓρm

Energy density of radiation Energy density of matter

E.o.M in decaying DM scenario (Poulin, Serpico, Lesgourgues, 2018)

Energy density perturbation Velocity divergence

δ0

m = −θm + 3Φ0 − aΓΦ,

θ0

m = −Hθm + k2Φ,

δ0

r = −4

3(θr − 3Φ0) + aΓρm ρr (δm − δr + Φ), θ0

r = k2

4 δr + k2Φ − aΓ3ρm 4ρr ✓4 3θr − θm ◆

(a3ρm ∝ e−Γt)

11/18

0.1 0.2 0.5 1 2 5 0.0 0.2 0.4 0.6 0.8 1.0

Keisuke Inomata (ICRR, The Univ. of Tokyo) arXiv: 1904.12878 (& 1904.12879)

Perturbations in the gradual reheating

The perturbations that entered the horizon during an eMD era gets suppressed during the gradual reheating.

12/18

Keisuke Inomata (ICRR, The Univ. of Tokyo) arXiv: 1904.12878 (& 1904.12879)

Suppression during the gradual reheating

w=P/ρ

1/3

t

Early Middle Late Start of reheating End of reheating

δr

δm

The radiation produced in the early or middle stage of the reheating moves from the birth place by the end of the reheating. Perturbations are erased! At the start of the reheating At the end of the reheating Gradual reheating This also means that the smaller-scale perturbations are erased more.

13/18

Keisuke Inomata (ICRR, The Univ. of Tokyo) arXiv: 1904.12878 (& 1904.12879) 10-4 0.001 0.010 0.100 1 10-9 10-7 10-5 0.001 0.100 10 1000

Results

kmax = 450/ηR = 225kR (kR = aH(ηR) = 2/ηR)

The induced GWs are suppressed compared to the previous results.

Pζ = A Θ(k − 30/ηeq)Θ(kmax − k)

GWs induced by curvature perturbations with

This work (gradual reheating) Previous works

A damping of tensor perturbation occurs during the gradual transition. Main reason

k = kR

14/18

Keisuke Inomata (ICRR, The Univ. of Tokyo) arXiv: 1904.12878 (& 1904.12879)

Why does the damping occur?

hλ

k ' Adec

adec a eikη

hλ00

k + 2Hhλ0 k + k2hλ k = 4Sλ k

hλ

k ' 4Sλ k

k2

Behavior of Effective E.o.M & solution for tensor prtb.

hλ00

k + 2Hhλ0 k + k2hλ k = 4Sλ k

Gradual reheating Time Sλ

k 00 < k2Sλ k

Sλ

k 00 > k2Sλ k

Decays gradually Remains constant Oscillates due to radiation pressure

Sλ

k(∼ O(Φ2, Φ02/H2))

hk decays following the decay of Sk (Freely propagating GWs)

15/18

Keisuke Inomata (ICRR, The Univ. of Tokyo) arXiv: 1904.12878 (& 1904.12879)

Effective E.o.M & solution for tensor prtb.

Why does the damping occur?

hλ

k ' Adec

adec a eikη

hλ00

k + 2Hhλ0 k + k2hλ k = 4Sλ k

hλ

k ' 4Sλ k

k2

Behavior of

hλ00

k + 2Hhλ0 k + k2hλ k = 4Sλ k

Gradual reheating Time Sλ

k 00 < k2Sλ k

Sλ

k 00 > k2Sλ k

Decays gradually Remains constant Oscillates due to radiation pressure (Freely propagating GWs)

Sλ

k(∼ O(Φ2, Φ02/H2))

This is the main reason for the suppression of GWs in the gradual reheating scenario.

hk decays following the decay of Sk

16/18

Outline

• Introduction
• eMD effects in a gradual reheating scenario
• Summary

Keisuke Inomata (ICRR, The Univ. of Tokyo) arXiv: 1904.12878 (& 1904.12879)

Summary of this talk (though we have one more talk)

As a result, we find that the induced GWs are suppressed compared to the previous results. We revisit the effect of an early matter dominated era on the gravitational waves induced by scalar perturbations. We carefully take into account the behaviors of the perturbations that enter the horizon during an eMD era in the gradual reheating scenario. 18/18 Our result could be important when we discuss PBHs produced during an eMD era.

(Harada, Yoo, Kohri, Nakano, Jhingan, 2016)

Backup

Keisuke Inomata (ICRR, The Univ. of Tokyo) arXiv: 1904.12878 (& 1904.12879)

Perturbations in gradual reheating

k2Φ ' 3 2H2 ✓ ρm ρtot δm + ρr ρtot δr ◆

On subhorizon scale, During eMD era, δm (∝ a) grows

• n subhorizon unlike δr .

Even after ηeq, Φ is mainly determined by ρmδm for a while. We naively expect

Fitting formula

(a(ηR)Γ = H(ηR) = 2/ηR)

If we take ηR = 0.83ηeq, the fitting formula fits the numerical result during the exponential decay very well!

Φ ∝ ρm ∝ exp ✓ − Z η d¯ η aΓ ◆

Φfit ≡ exp ✓ − Z η d¯ ηa(¯ η)Γ ◆ = 8 > > < > > : exp ✓ − 2

3

⇣

η ηR

⌘3◆ (η < ηR) exp ✓ −2 ✓⇣

η ηR

⌘2 −

η ηR + 1 3

◆◆ (η ≥ ηR)

20/18

Keisuke Inomata (ICRR, The Univ. of Tokyo) arXiv: 1904.12878 (& 1904.12879)

Formulas we use to calculate induced GWs

Ph(η, k) 4 ∞ dv 1+v

|1−v|

du 4v2 (1 + v2 u2)2 4vu 2 I2(v, u, kη)Pζ(kv)Pζ(ku)

Energy density of the induced GWs Power spectrum of the induced GWs

Summary of the formulas we use:

(Kohri, Terada, 2018) ΩGW(η, k) = ρGW(η, k) ρtot(η) = 1 24 ✓ k a(η)H(η) ◆2 Ph(η, k)

I(u, v, x, xR) = Z xR d¯ x ✓ 1 2(x/xR) − 1 ◆ ✓ ¯ x xR ◆2 kGeMD→RD

k

(η, ¯ η)f(u, v, ¯ x, xR) + Z x

xR

d¯ x ✓2(¯ x/xR) − 1 2(x/xR) − 1 ◆ kGRD

k

(η, ¯ η)f(u, v, ¯ x, xR) ≡IeMD(u, v, x, xR) + IRD(u, v, x, xR)

f(u, v, ¯ x, xR) =3

• 2(5 + 3w)Φ(u¯

x)Φ(v¯ x) + 4H1(Φ0(u¯ x)Φ(v¯ x) + Φ(u¯ x)Φ0(v¯ x)) + 4H2Φ0(u¯ x)Φ0(v¯ x)

• 25(1 + w)

Here, we assume that the scale factor is given as, (because this fits the numerical result well)

(x = kη, xR = kηR)

aapp(η) aapp(ηR) = 8 < : ⇣

η ηR

⌘2 (η < ηR) 2 η

ηR − 1

(η ≥ ηR)

Then we can express I as

(Φ(x → 0) = 1, Φ0(x) = ∂Φ(x)/∂η)

21/18

Keisuke Inomata (ICRR, The Univ. of Tokyo) arXiv: 1904.12878 (& 1904.12879)

What causes eMD era ?

One scenario: oscillation of inflaton

Inflation occurs the vacuum energy of the inflaton. During the oscillation of an inflaton, the Universe behaves as MD era. The oscillation amplitude decays due to the expansion and the decay to radiation.

Radiation

RD era begins.

During the oscillation

(Other scenario: oscillation of curvaton) Chaotic inflation

hρi = ⌧1 2 ˙ φ2 + m2 2 φ2

• = m2 hφ2i

hPi = ⌧1 2 ˙ φ2 m2 2 φ2

• = 0

MD era !

Time average

(w = P/ρ = 0)

22/18

Keisuke Inomata (ICRR, The Univ. of Tokyo) arXiv: 1904.12878 (& 1904.12879)

Scalar perturbation in Cosmology

Scalar perturbations induce the perturbations of energy density (𝜀𝜍/𝜍). Scalar perturbations are origins of many things!

(Credit: ESA and the Planck Collaboration)

Large Scale Structure CMB perturbations

Examples

(Credit: Millennium Simulation Project)

Galaxies

(Credit: NASA and ESA)

Scalar perturbations originate from vacuum fluctuations of an inflaton during the inflation era. From the observations, we already know the amplitude of the scalar perturbations. Pζ = 2.1 × 10−9

(Planck 2018)

(δρ/ρ ∼ 10−5)

Pζ = 9 4PΦ

curvature perturbation scalar perturbation

23/18

Keisuke Inomata (ICRR, The Univ. of Tokyo) arXiv: 1904.12878 (& 1904.12879)

Foregrounds

In future, stochastic GWs of astrophysical origin or cosmological origins that are different from the induced GWs, such as those from phase transition, may be detected. In that case, the constraints would become weak. Foreground example:

GWs from supermassive-black-hole binaries

Around PTA target frequency

GWs from galactic white-dwarf binaries

Around LISA target frequency

GWs from neutron star binaries

Around BBO target frequency 24/18

Keisuke Inomata (ICRR, The Univ. of Tokyo) arXiv: 1904.12878 (& 1904.12879)

Background in gradual reheating

The evolutions of background quantities.

0.1 0.5 1 5 10 10-5 0.001 0.100 10 1000 105

ρm = ρr at η = ηeq

wfit = 1 3 1 − exp −0.7 ✓ η ηeq ◆3!!

The fitting formula for w:

w ≡ P/ρ

The fitting formula for a: In the above figure, we take ηR = 0.83ηeq .

aapp(η) aapp(ηR) = 8 < : ⇣

η ηR

⌘2 (η < ηR) 2 η

ηR − 1

(η ≥ ηR)

25/18