Gravitational Waves from Inflation and Primordial Black Holes Marco - - PowerPoint PPT Presentation

Gravitational Waves from Inflation and Primordial Black Holes

Marco Peloso, University of Padua

• GW from axion inflation
• GW from primordial black holes (PBH)
• Characterization of the stochastic GW background (SGWB)

Barnaby, Bartolo, Bertacca, De Luca, Domcke, Figueroa, Franciolini, Garc ´ ıa-Bellido, Lewis, Matarrese, Nardini, Pajer, Pieroni, Racco, Ricciardone, Riotto, Sakellariadou, Sorbo, Tasinato, Unal

− ∝ CMB modes produced at N ≃ 60 before the end of inflation, when a ≃ e−60aend

Development of GW interferometers

• pens a new window on much

smaller scales

λ ≃ 103 − 109 km We can probe N = 15 − 28 CMB and Large Scale Structure in excellent agreement with inflation.

However, only probe λ ≃ 102 − 105 Mpc

time matter/radiation wavelength horizon inflation They only probe

N = 56 − 63

≃ − Smaller scales / later times es essentially unprobed.

We give time in terms of e − foldings : a ∝ eHt = e−N

GW as a probe of inflation

N = 0 ↔ f ≃ 108 Hz

• 18.0
• 14.0
• 10.0
• 6.0
• 2.0

2.0 6.0 10.0 Log[f]

• 16.0
• 14.0
• 12.0
• 10.0
• 8.0
• 6.0

Log[h0

2ΩGW]

scale-invariant (RD modes) (MD modes)

ΩGW ∝ 1/k2

r e d t i l t e d ( q u a s i

• )

s c a l e

• i

n v a r i a n t ( R D m

• d

e s ) aLIGO LISA

Quantum

Fluctuations

GW production during inflation h′′

ij + 2a′′

a hij + k2 hij = 2 M2

p

T TT

ij

Amplification of vacuum modes from

inflationary expansion guaranteed

signal, but too small for present and next generation detectors

• D. Figueroa

Several mechanisms result in sourced GW during inflation.

Subject to the same limits as vacuum modes at CMB scales

Signal must be blue to be visible at interferometers

axion inflation Natural property in

Axion inflation

Freese, Frieman, Olinto ’90, . . .

Main theoretical difficulty

is to keep the potential flat against radiative corrections

2 4 6

Φ

0.5 1.0 1.5

V

• Coupling to matter invariant under φ → φ + constant

Coupling to fermions : ∆L = ∂µφ f ¯ ψγ5γµψ

to gauge fields : ∆L = φ f Fµν ˜ F µν ≡ −4∂µφ f ϵµναβAν∂αAβ

∂µ φ ψ ∂µ φ ψ φ ψ φ ψ

∆V = 0

Loops with these couplings

do not modify the potential

Turner, Widrow ’88 Garretson, Field, Carroll ’92 Anber, Sorbo ’06

Originally studied for magnetogenesis. Here, generic U(1) φF ˜ F breaks parity, ̸= results for two polarizations

+ left handed

d − right handed

Vector production from φ f F ˜ F

• ∂2

∂τ 2 + k2 ∓ ak ˙ φ f

• A± (τ, k) = 0

10 100 1000 10000 100000 1e+06 1e+07 1e+08 1e+09 0.01 0.1 1 10 100 1/H4 d (E2+B2/2) / d log k a H / k = 5 + vacuum

• One tachyonic helicity at horizon crossing
• Then diluted by expansion

Physical ρ in one mode

• Max amplitude A+ ∝ e ˙

φ

δA δA δφ

Barnaby, MP ’10

Planck ’15

through inverse decay. These modes are highly non-gaussian.

• The produced A+ modes source inflaton perturbations δφ

This imposes f > ∼ 1016 GeV

• recall L ⊃ −φF ˜

F f

• Bartolo et al ’16; LISA cosmology WG

˙ φ grows during inflation (inflation ends because ˙ φ too large) ⇒ Blue GW and potentially visible at interferometers

Cook, Sorbo ’11; Barnaby, Pajer, MP’11; Domcke, Pieroni, Bin´ etruy ’16; . . .

Signal is chiral hL ≫ hR and highly non-Gaussian, ⟨h3⟩ ∼ ⟨h2⟩ 3/2

Sorbo ’11 Barnaby, MP ’10

• The amplified gauge fields also produce GW, though A+A+ → hL

V (φ) = 1 2 m2 φ2

• V (φ) = V0
• 1 − e

−

2 3 φ

2

• V (φ) = V0
• 1 −

φ

v

42

• V (φ) = V0
• 1 −

φ

v

32 Domcke, Pieroni, Bin´ etruy ’16

Due to ∝ e ˙

φ, signal very sensitive to the inflaton potential

10-15 10-10 10-5 100 105 10-29 10-24 10-19 10-14 10-9 10 20 30 40 50 60

f[Hz]

GW h2

N

N = number of e-folds before the end of inflation when a mode is

• produced. Different experiments probe different ranges of V (φ)

V (φ) from shift symmetry

AL AL δφ hL

,

• As in all mechanisms of GW from inflation,

the key difficulty is to produce observable GW without overproducing density perturbations

• For a monomial V (φ), PBH bounds prevent GW from being observable

Linde, Mooij, Pajer ’13

ΞCMB 1.66 PBH limit

60 50 40 30 20 10

N

1011 109 107 105 0.001 0.1

PΖ

ΞCMB 1.66

LISA aLIGO 30 20 10

1014 1010 106 0.01 100 f Hz 1018 1015 1012 109 106

GW h2

N ∼ 15 − ln

• f

100 Hz

• at aLIGO and LISA
• Due to ∝ e ˙

φ, significant differences from a minor change of V

Φ1 Φ2

9 8 7 6 5 4 4 5 6 7 8 9

Φ Mp V M3 Mp

ΞCMB 2.41 Ξmax 4.43

PBH limit

60 50 40 30 20 10

N

1011 109 107 105 0.001 0.1

ΞCMB 2.41 Ξmax 4.43 LISA aLIGO 30 20 10

1014 1010 106 0.01 100 f Hz 1018 1015 1012 109 106

GW h2

Garcia-Bellido, do, MP, Unal ’16

H

! d

Matter / Radiation Inflation

If sufficiently large, at horizon re-entry, the perturbation collapses to form a Primordial Black Hole (PBH)

ΞCMB 2.41 Ξmax 4.43

PBH limit

60 50 40 30 20 10

N

1011 109 107 105 0.001 0.1

• Mechanism for a peaked distribution of PBH

M

λ

A significant fraction of the mass in the horizon collapses

into the PBH. So, parametrically, λ ↔ MPBH

• PBH and PBH-DM long standing idea
• Recent interest due to lack of detection of particle

candidates, and LIGO / VIRGO events

• Zel’dovich, Novikov ’67

Hawking ’71; Carr ’75; Chapline ’75

Bird et al ’16; 16; Sasaki et al ’16 16; Clesse, Garc ´ ıa-Bellido ’16;

PBH dark matter

Capela, Pshirkov, Tinyakov ’13

nuclear physics Limits from capture from NS and WD

astrophysical abundance, and on not shown due to uncertainty in DM

Montero-Camacho, Fang, Vasquez,

z, Silva, Hirata ’19

• 2 windows, one at ∼ 10−12M⊙, and (possibly) one at ∼ 10−100M⊙.

10-18 10-15 10-12 10-9 10-6 10-3 100 103 10-6 10-5 10-4 10-3 10-2 10-1 100 1015 1018 1021 1024 1027 1030 1033 1036

E G γ

• b

k g HSC EROS/MACHO OGLE SNe CMB UFD

and Inomata et al ’17 ∼ Credit: G. Franciolini, update

• f Carr, Kuhnel, Sandstad ’16

Katz, Kopp, Sibiryakov, Xue ’18

Cut on HSC and on limits from femtolensing of γ-ray bursts. Schwarzschild radiusPBH < λγ

1) during inflation, by the same source that produced δρ 2) by δρ at horizon re-entry after inflation

• Whenever δρ present GW produced

PBH ← enhanched δρ → GW

Standard gravitational interaction:

ζ ζ hi = ⇒ hi ζ ζ hi

• Mechanism 2 is unavoidable and model-independent
• Technical (but important !) point. Power spectrum

δρ2 controls

• the amount of GW. Full statistics of δρ relevant for PBH abundance.

PΖ Χ2 PΖ Gaussian

1022 1026 1030 1034 107 105 0.001 0.1

• g

PΖ

Stronger constraint on Pδρ sourced in axion infaltion (non-gaussian statistics) ⇒ Fewer GW

M

λ

PTA SKA

• 12

GaussInd Χ2Prim

1011 1010 109 108 107 106 1016 1014 1012 1010 108 106 fHz GWh2

SKA LISA GaussInd Χ2Prim

106 104 0.01 1 1014 1012 1010 108 fHz GWh2

fGW ∼ 1 λ ∼ 3 mHz

• 10−12 M⊙

M

M ∼ 10 M⊙ ⇒ fGW ∼ nHz PTA! M ∼ 10−12 M⊙ ⇒ fGW ∼ mHz LISA!

From axion inflation Gaussian δρ Garc ´ ıa-Bellido, MP, Unal ’17

• Spectral shape ΩGW (f)
• Net Polarization

ΩGW,λ

• Statistics

Ωn

GW

• Directionality

ΩGW (⃗ x)

SGWB from cosmological sources superimposed with astrophysical one.

Potential observables to disentangle them

Measurement of the SGWB

101 102 103 f (Hz) 10−10 10−9 10−8 10−7 10−6 10−5 ΩGW

O1 (2σ) O1+O2 (2σ) Design (2σ) ΩBBH+BNS+NSBH ΩBBH+BNS (Median) ΩBBH+BNS (Poisson)

Current LIGO bounds

Measurement of GW polarization

−3 −2 −1 1 2 3 10

−16

10

−14

10

−12

10

−10

10

−8

• 2nd Gen. H1−L1

2nd Gen. H1−L1−V1−K1 3rd Gen.

Crowder, Namba, Mandic, c, Mukohyama, MP ’12

• Amplitude needed to detect ΩGW

and exclude ΩGW,R = ΩGW,L at 2σ Assume ΩGW,L = Ωα

• f

100 Hz

α

and ΩGW,R = 0

One more motivation for an Australian detector !

Antipodes LL-P

100 200 300 400

• 0.15
• 0.10
• 0.05

0.00 0.05

f [Hz] Δ ℳ

∆tdetector i

t ∆tdetector j t

• =
• d

f f

• Mij,R (f) PGW,R (f) + Mij,L (f) PGW,L (f)
• ∆M = MR − ML measure of chirality

maximized for anti-podal detectors

Measurement of GW polarization at LISA / ET

hL hR

Two GWs related by a mirror symmetry produce the same response in a planar detector. Cannot detect net circular polarization of an isotropic SGWB hR hL ⃗ vd

Isotropy in any case broken by peculiar motion of the solar system. Assumption, vd ≃ 10−3 as CMB Domcke, Garc ´ ıa-Bellido, MP, Pieroni Ricciardone, Sorbo, Tasinato ’19

(one order of magnitude greater than estimate in Seto ’06)

SNRLISA ≃ vd 10−3 ΩGW,R − ΩGW,L 1.2 · 10−11

• T

3 years

• Measurement at LISA: X, Y, Z ≡ time delays at the vertices

Z X Y

Detector response function : signal2 ∼ Rλ (k) ⟨hλ (k) hλ (k)⟩

• Rλ, has opposite sign for the two helicities, and ∝ cosine of the angle

between the direction of the dipole and the normal to the LISA plane Rλ =

• d (angles on the sky) λ × F [θ, φ]

Earth Sun 1 AU (150 million km) 19 – 23° 60° 2 . 5 m i l l i

• n

k m

• Correlation

(2X − Y − Z) ∗ (Z − Y ) vanishes if PR = PL

Ecliptic coordinates

CMB dipole LISA Normal

Non-G, angular anisotropies, and a probe of the large scale structure

• f the Universe

h

h ζL ≡ δρ/ρ

Production mechanism & propagation imprint anisotropies, ρGW (⃗ x) ∝ ˙ hij ˙ hij

• Treatment as CMB
• Alba, Maldacena ’15; Contaldi ’16; Cusin, Pitrou, Uzan ’17; Jenkins,

Sakellariadou ’18; Bartolo, Bertacca, Matarrese, MP, Ricciardone, Riotto, Tasinato ’19 Tasinato ’19

ρGW =

• ℓm

aGW

ℓm Yℓm

Angular power spectrum

⟨aℓm a∗

ℓ′m′⟩ = Cℓ δℓℓ′ δmm′

Bispectrum (non-G)

⟨aℓ1m1 aℓ2m2 aℓ3m3 ⟩ ∝ bℓ1ℓ2ℓ3

This is ρ3

GW

• h3

not observable

Cℓ,in (f) 4π =

• dk

k Pin (f, k) j2

ℓ (k t0) f ∼ mHz observed GW frequency k ∼ H0 ∼ (10 billion yrs)−1 scale of anisotropie

Cℓ do not depend on f (initial thermal state)

Power in initial condition. Can depend on f - different from CMB, where

Bartolo et al ’19

Anisotopies from the production mechanism

k ↔ 1 |δ⃗ x| ↔ ℓ

Cℓ (f)

10-15 10-10 10-5 1 105 1 2 3 4

f Hz

ℱ(f)

δρGW (f, k) ρGW ∝ F (f) δ ˙ φ (k)

For instance, in axion inflation ˙ φ (t) + δ ˙ φ (t, ⃗ x) → PGW + δPGW

Cℓ,S + Cℓ,T 4π =

• dk

k

• Pζ (k) Tscalar + Ph (k) Ttensor
• bℓ1,ℓ2ℓ3 ≃ 2 fNL [Cℓ1 Cℓ2 + Cℓ1 Cℓ3 + Cℓ2 Cℓ3]

“local” scalar NG,

Anisotopies from the propagation

Large scale density

ty and tensor anisotropies h (f)

G, δρ ∼ δρg + fNL δρ2

g

∼ New probe of large scale anisotropies (like CMB photons)

Bispectrum from 2nd order interactions. Already a first order, due

to propagation, induced by the non-Gaussianity of δρ. At large scales

Anisotropies & non-G at the production - GW in models with PBH Very large GW signal @LISA in models of PBH-DM. Is is isotropic ? Is it Gaussian?

h

ζ

ρGW

ζ + ζ → h ρGW ∼ ˙ h2 ∼ ζ4

• ζ is a short-scale mode, that

generates GW of f ∼ mHz today

10-5 10-4 10-3 10-2 10-1 10-15 10-14 10-13 10-12 10-11 10-10 10-9 10-8 10-7 10-6 10-15 10-14 10-13 10-12 10-11 10-10 10-9 10-8 10-7

LISA

∼ In presence of scalar non-G, a long mode ζL modulates the power of ζ on small scales,

ρGW

h

ρGW

fNL fNL

ζL

Bartolo et al ’19

• Greater scalar fNL leads to greater anisotropies and non-G of GW.
• −11.1 ≤ fNL ≤ 9.3 , at 95% C.L.

Planck ’19

• Isocurvature constraints impose a tighter limit on fNL for PBH-DM

Observing a bump at LISA, with h significant anisotropy and non-G indicates that the PBH constitute

ute only a small fraction of the DM

2.5×10-4 5.×10-4 7.5×10-4 1.×10-3 1.25×10-3

, slight change in ΩGW.

Slight change of Pζ → large change of fPBH,

Anisotropies can differentiate

te between these two cases.

10-5 10-4 10-3 10-2 10-1 10-15 10-14 10-13 10-12 10-11 10-10 10-9 10-8 10-7 10-6 10-15 10-14 10-13 10-12 10-11 10-10 10-9 10-8 10-7

LISA

fPBH = 1 fPBH = 10−5

Conclusions

• Signal from inflation only if blue
• Probe of PBH (possibly, PBH DM)

SKA LISA GaussInd Χ2Prim

106 104 0.01 1 1014 1012 1010 108 fHz GWh2

• SGWB characterization

2.5×10-4 5.×10-4 7.5×10-4 1.×10-3 1.25×10-3