Inverse chirp signals from stellar core collapse in massive scalar - - PowerPoint PPT Presentation

Inverse chirp signals from stellar core collapse in massive scalar tensor gravity

Ulrich Sperhake

DAMTP, University of Cambridge arXiv: 1708.03651 [gr-qc], 1903.09704 [gr-qc]

STAG Research Centre Gravity Seminar University of Southampton, 12 Dec 2019

1

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 690904, from H2020-ERC-2014-CoG Grant No. ”MaGRaTh" 646597, from NSF XSEDE Grant No. PHY-090003 and from STFC Consolidator Grant No. ST/L000636/1.

C Moore, M Agathos, R Rosca, D Gerosa, C Ott

Overview

Introduction and motivation Theoretical framework Massive scalar-tensor gravity Massive self interacting scalar-tensor gravity Observation strategies Conclusions

• 1. Introduction and motivation

Do we need a theory beyond GR?

When asked what he would do if Eddington’s mission failed… But we have reasons to search for “beyond GR” Renormalization: Requires, e.g., higher curvature terms.

GR is low-energy limit of more fundamental theory

Dark energy: Why is so small and why Dark matter: “Neptune” or “Vulcan” ?

→ Λ ρdark ∼ ρmat

Scalar tensor theory of gravity

Scalars appear naturally in extra-dimensional theories Scalars prominent in cosmology ST theory well-posed; fairly well understood mathematically No-hair theorems limit potential of black-hole spacetimes

Matter: Neutron stars, core-collapse

Best example of smoking gun to date:

Spontaneous scalarization Damour & Esposito-Farese PRL 1993

Collapse studies in massless case

Novak PRD 1998/1999 Novak & Ibanez ApJ 2000, Gerosa+ CQG 2016

⇒

Core-collapse scenario to 0th order

Massive stars: Core compressed from to

to

Released gravitational energy:

in neutrinos, in outgoing shock, explosion

Explosion mechanism: still uncertainties… Failed explosions lead to BH formation

“Collapsar”: possible engine for long-soft GRBs

All of this handled for us by Woosley & Heger Phys.Rept. 2007

Initial pre-collapse profile

MZAMS = 8 . . . 100 M ∼ 1 500 km ∼ 15 km ∼ 1010 g/cm3 & 1015 g/cm3 O(1053) erg ∼ 99 % ∼ 1051 erg

→

• 2. Theoretical framework

Theoretical framework

Action Energy momentum tensor:

Einstein frame: conformal metric

Spherical symmetry: Equations (gravity): Equations (matter): HRSC

GR1D code O’Connor & Ott CQG 2009

¯ gµν = F(ϕ) gµν S = 1 16π Z dx4√−¯ g [ ¯ R − 2¯ gµν∂µϕ ∂νϕ − 4V (ϕ)] + Sm[ψm, ¯ gµν/F(ϕ)] Tαβ = ρhuαuβ + Pgαβ d¯ s2 = ¯ gµνdxµdxν = −Fα2dt2 + FX2dr2 + r2dΩ2 uα = 1 √ 1 − v2 [α−1, vX−1, 0, 0] ∂rα = . . . , ∂rX = . . . ∂t∂tϕ = . . . (ρ, h, v) ↔ (D, Sr, τ) ⇒

Theoretical framework

Action Energy momentum tensor:

Einstein frame: conformal metric

Spherical symmetry: Equations (gravity): Equations (matter): HRSC

GR1D code O’Connor & Ott CQG 2009

¯ gµν = F(ϕ) gµν S = 1 16π Z dx4√−¯ g [ ¯ R − 2¯ gµν∂µϕ ∂νϕ − 4V (ϕ)] + Sm[ψm, ¯ gµν/F(ϕ)] Tαβ = ρhuαuβ + Pgαβ d¯ s2 = ¯ gµνdxµdxν = −Fα2dt2 + FX2dr2 + r2dΩ2 uα = 1 √ 1 − v2 [α−1, vX−1, 0, 0] ∂rα = . . . , ∂rX = . . . ∂t∂tϕ = . . . (ρ, h, v) ↔ (D, Sr, τ) ⇒

Coupling function

F(ϕ)

Potential

V (ϕ)

Equation of state

h P

The coupling function and potential

Coupling function:

determine all modifications at 1st PN order

F(ϕ) = e−2α0ϕ−β0ϕ2 V (ϕ) = 1

2µ2ϕ2

Mass introduces characteristic frequency

Here typically:

Potential for a massive non-interacting scalar field

α0, β0 µ µ = 10−14 eV ⇔ ω∗ = 15.2 s−1

ω∗ = µc2 ~

Equation of state

Pressure: “cold” + “thermal” contribution: Hybrid EOS for cold part: Internal energy from 1st law: Thermal pressure: Parameters:

from continuity at

P = Pc + Pth Pc = ( K1ρΓ1 if ρ ≤ ρnuc K2ρΓ2 if ρ > ρnuc

✏c =   

K1 Γ1−1⇢Γ1−1

if ⇢ ≤ ⇢nuc

K2 Γ2−1⇢Γ2−1 + E3

if ⇢ > ⇢nuc

Pth = (Γth − 1)⇢(✏ − ✏th) Γ1 = 1.3 , Γ2 = 2.5 , Γth = 1.35 K1 = 4.9345 × 1014 [cgs] , ρnuc = 2 × 1014 g cm−3 K2 , E3 ρ = ρnuc

The coupling function and potential

Coupling function, potential:

F(ϕ) = e−2α0ϕ−β0ϕ2 V (ϕ) = 1

2µ2ϕ2

−6 −4 −2 2 4 6

β0

10−4 10−3 10−2

α0

PSRJ0348+0432 PSR J1738+0333 Cassini Runs

Free parameters:

µ, α0, β0, Γ1, Γ2, Γth

Only for !! Here:

Ramazanoglu & Pretorius PRD 2016

µ . 10−19 eV µ[eV] ∈ [10−15, 10−12]

Convergence test

For Using points Discretization error:

µ = 10−14 eV, α0 = 10−4, β0 = −20 Γ1 = 1.3, Γ2 = 2.5, Γth = 1.35 N1 = 5000, N2 = 10000, N3 = 20000 ∼ 5 %

• 3. Massive ST gravity

Waveforms ``close to’’ the source

For

µ = 10−14 eV, α0 = 10−2, β0 = −20 Γ1 = 1.3, Γ2 = 2.5, Γth = 1.35

massless case; fairly insensitive to parameters; dispersion!

rϕ

1 2 3 t [s]

• 3×10

5

• 2×10

5

• 1×10

5

rexϕ [cm]

Fiducial α0 = 10

• 4

α0 = 1 µ = 3x10

• 14eV

Γ2 = 3 α0 = 1, β0 = -10

0.1 0.2 0.3 0.4 0.5

• 2×10

5

• 1×10

5

Waveforms ``far from’’ the source

LIGO will observe the

above scalar profiles after they propagate to large distances

In the massless case

this is almost trivial

In the massive case

things are more complicated: signals propagate with dispersion

ϕ(t; r) = 1 r ϕ(t − r; rextract)

Waveforms ``far from’’ the source

Far from the source, scalar dynamics are governed by the

flat-space Klein-Gordon wave equation

Easier to work with the radially rescaled field As the signal propagates outwards:

• Low frequencies are suppressed
• High frequency power spectrum is unaffected
• Signal spreads out in time
• High frequencies arrive earlier than low frequencies
• Signal becomes increasingly oscillatory

∂2

t ϕ r2ϕ + ω2 ∗ϕ = 0

σ ≡ rϕ

The scalar field mass has a natural frequency ω∗ = c2µ/~

A toy model

Flat space Klein-Gordon: Collapsing star a source abruptly switched on. Approximate with

where

Solve with Green’s function:

(@2

t r2)' + µ2' = 4⇡%

≈

%(t; x) = '∗Wτ(t)(3)(x)

Wτ(t) = (

2 πarctan

t

τ

• if t ≥ 0

if t < 0

'(x) = Z G(x; x0) %(x0) d4x

⇒ . . . ⇒

ϕ(t; r) = ϕ⇤ " Wτ(t − r) r − Z tr µJ1(µ p t − t0)2 − r2) Wτ(t0) p (t − t0)2 − r2 # if t ≥ r else 0

“massless” term Dispersive tail

A toy model

Result for Signal increasingly oscillatory at larger

µ = ϕ∗ = τ = 1

r

Waveforms ``far from’’ the source

Signals become more oscillatory as they propagate outwards In the large-distance limit the stationary phase approximation

applies analytic expression for the time domain signal

Signals have a characteristic “inverse chirp” lasting many years

→

Ω(t) = ω∗ q 1 − d

t

2

Frequency for Amplitude A(t) = r 2 π (Ω2 − ω2

∗)3/4

ω∗ √ d

• ˜

σ ⇥ Ω(t) ⇤

• <latexit sha1_base64="E7+cuRtkzBGPa/o9RPiA2Xy70I=">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</latexit><latexit sha1_base64="E7+cuRtkzBGPa/o9RPiA2Xy70I=">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</latexit><latexit sha1_base64="E7+cuRtkzBGPa/o9RPiA2Xy70I=">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</latexit><latexit sha1_base64="E7+cuRtkzBGPa/o9RPiA2Xy70I=">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</latexit>

t > d

Phase φ(t) = p Ω2 − ω2

∗ d − Ωt − π

4 + Arg n ˜ σ ⇥ Ω(t) ⇤o

Waveforms ``far from’’ the source

Signals become more oscillatory as they propagate outwards In the large-distance limit the stationary phase approximation

applies analytic expression for the time domain signal

Signals have a characteristic “inverse chirp” lasting many years

→

≈

1010 s ∼ 300 yr

Amplitude const.

Frequency universal Over long times!!!

• 4. Massive self-interacting ST gravity

Self interacting scalar fields

Coupling function:

determine all modifications at 1st PN order

F(ϕ) = e−2α0ϕ−β0ϕ2

All are dimensionless.

Mass introduces characteristic frequency Here typically:

Potential for a self-interacting field

α0, β0 V (ϕ) = 1 2µ2ϕ2 ✓ 1 + λ1 ϕ2 2 + λ2 ϕ4 3 + . . . + λn ϕ2n n + 1 ◆ , λn > 0 λi µ µ = 10−14 eV ⇔ ω∗ = 15.2 s−1

ω∗ = µc2 ~

Collapse to NS with self interaction

Varying quartic term : Wave signal highly robust!

λ1 = 0 . . . 106

Same observation for λ2, λ3

α0 = 102 , β0 = −20 , MZAMS = 41 M , Z = Z4

Collapse to BH with self interaction

Varying quartic term : Wave signal highly robust!

λ1 = 0 . . . 106

Same observation for λ2, λ3

α0 = 103 , β0 = −5 , MZAMS = 39 M , Z = Z4

GW Propagation with self interaction

Far-field wave equation is now non-linear: SPA no longer applicable Numerically evolve signal to light seconds with

O(102) λi 6= 0

∆σ [cm]

• 10000

10000

σ [cm]

λ1 = 0 λ1 = 10

10

• 0.5

0.5

u = t - r [s]

∆σ [cm]

• 50

50

∆σ [cm]

0.2 0.4 0.6 0.8 1

• 5000

5000 50 50.2 50.4 50.6 50.8 90 90.2 90.4 90.6 90.8 91 λ1 = 10

6

λ1 = 10

10

λ1 = 10

8

Need to see non-negligible effects!!

λ1 = 1010

• 5. Observation strategies

Detection with LIGO-Virgo

Burst signals: For light scalars and short

distances , the pulse does not disperse significantly; will look like a burst

Continuous wave signal: for heavier scalars, long dispersion

turns pulse into a quasi-monochromatic signal capture using standard directed CW searches, assuming EM counterpart; e.g. SN1987A, Kepler1604

Stochastic background:

• Many quiet sources very long duration (superposed)
• Cosmological redshift mass variation smeared low- cutoff
• Characteristic “bump” in background, peaking at
• Well in reach for aLIGO/AdVirgo stochastic searches

GWs from core-collapse in ST gravity may fall into 3 classes:

(µ < 10−20 eV) (10 kpc) < 1 s

→

+ + → f ∼ ω∗

Conclusions

We have simulated stellar core collapse in massive ST theory Spontaneous scalarization occurs as in massless case, but

effect can be more dramatic because the scalar mass “screens” the effect of the scalar, allowing larger values of to be compatible with binary pulsar observations

Signals propagate with dispersion, signals can last for years to

centuries at distances

Signals can show up in LIGO/Virgo burst, CW or stochastic searches GW generation + propagation very robust to self interaction terms

α0, β0 kpc