Gravitational waves Part I ICTP, 18-22 June 2018 Literature Some - - PowerPoint PPT Presentation

Stanislav (Stas) Babak.

AstroParticule et Cosmologie, CNRS (Paris)

Gravitational waves Part I

ICTP, 18-22 June 2018

Literature

2

Some figures in these lectures are borrowed from theses books

LSC+Virgo, Phys. Rev. Lett. 116, 221101 (2016), LSC+Virgo, Phys. Rev. X 6, 041015 (2016), LSC+Virgo, Phys.Rev.Lett. 116 241102 (2016), LSC+Virgo, Phys.Rev. X6 041014 (2016), LSC+Virgo, Phys. Rev. Lett. 118, 221101 (2017), Berti et al., Class.Quantum Grav. 32, 243001 (2015), LSC+VIRGO, ArXiV: 1805.11579, S. Khan+, Phys.Rev. D93 (2016) 044007, LSC+Virgo arXiv: 1805.11579 , LIGO_Virgo, Astrophys.J. 848 (2017) L12, LIGO+Virgo Phys.Rev.Lett. 119 (2017) 161101, Babak+ Phys.Rev. D95 (2017) 103012, LISA consortium arXiv:1702.00786, Klein+ Phys.Rev. D93 (2016) 024003, LISA consortium, arXiv:1305.5720, Amaro-Seoane+ Class.Quant.Grav. 29 (2012) 124016 .

Lecture 2

3

Binary system Modelling GW signal from binary black holes Detecting GW signals with ground based observatories

Generation of GWs

4

Gravitational waves: can we attach stress energy tensor? Yes, but it is defined as a quantity averaged over several (GW) wavelengths.

T GW

αβ

= 1 16 hh+,αh+,β + h×,αh×,βi

dE dt = −1 5 ⌦ ... Mij ... Mij ↵ dSj dt = −2 5✏jkl D ¨ Mki ... Mli E

Energy loss (energy flux): shrinking of binary orbit Angular momentum loss: circularization of a binary Levi-Civita antisymmetric symbol

Binary system

5

Consider binary system on a circular orbit.

m1 > m2, M = m1 + m2

Use Kepler’s law

ω = r M a3

x1 = m2 M a cos ωt, y1 = m2 M a sin ωt x2 = −m1 M a cos ωt, y2 = −m1 M a sin ωt

Ijk = Z T 00xjxkd3x = Z [(~ x − ~ x1)m1 + (~ x − ~ x2)m2] xjxkd3x = m1xj

1xk 1 + m2xj 2xk 2

¨ Ixx = −¨ Iyy = −2µ (Mω)2/3 cos 2ωt, ¨ Ixy = ¨ Iyx = −2µ (Mω)2/3 sin 2ωt,

µ = m1m2/M

ˆ eθ = ˆ ex cos θ − ˆ ez sin θ, ˆ eφ = ˆ ey.

Iθθ = Ixx cos2 θ, Iφφ = Iyy, Iθφ = Ixy cos θ.

h+ = hθθ = −2

• 1 + cos2 θ

µ R(Mω)2/3 cos [2ω(t − R) − φ0] h× = hθφ = −4 cos θ µ R(Mω)2/3 sin [2ω(t − R) − φ0]

polarization basis

Binary system

6

h+ = hθθ = −2

• 1 + cos2 θd

µ R(Mω)2/3 cos [2ω(t − R) − φ0] h× = hθφ = −4 cos θd µ R(Mω)2/3 sin [2ω(t − R) − φ0]

m1

x y z

~ k

θd

φ0

~ L

Inclination: angle between orb. angular momentum and propagation direction ( ), alternatively angle between L and direction to the source.

θd

ι = π − θd

Distance to the source: luminosity distance R = DL

~ J = ~ L + ~ S1 + ~ S2

GW emission strongest if face on/off, and weakest if the source is edge-on If masses are not spinning: the total angular momentum is orbital angular momentum If masses are spinning, then the total angular momentum: if spins have arbitrary orientation (not aligned with the orbital angular momentum) - the orbit precesses (L rotates around J) due to spin-orbital coupling:

~ L

˙ ~ L ∝ ~ Ω × ~ L

Dominant harmonic: 2 x orbital freq. (circular), there are harminics 1,3,4,…x

• rbital freq. but lower in amplitude

Binary system

7

Lum = −dEGW dt = 1 5 ⌦ ... Mij ... Mij ↵ = 32 5 η2(Mω)10/3

Etot = m1m2 a + m1(ωr1)2 2 + m2(ωr2)2 2 = −m1m2 2a .

˙ Etot = m1m2 2a2 ˙ a = −32 5 η2(Mω)10/3.

Loss of energy due to GWs Total energy of the binary system Balance equation a = 256 5 ηM 3(tc − t) 1/4 , πf = 256 5 ηM 5/3(tc − t) −3/8

φ(N)

• rb =

Z 2πf(t) dt = −2  1 5Mc (tc − t) 5/8 + φc

∆t = 5 256M 5/3η (πf)8/3

This equation can be easily integrated

η = µ M = m1m2 M 2

Binary system

a = 256 5 ηM 3(tc − t) 1/4

πf = 256 5 ηM 5/3(tc − t) −3/8

The orbit shrinks as t —> tc The frequency depend on the “chirp mass” Mc = Mη3/5

GW (and orbital) frequency grows with time and infinite at tc (approach breaks down)

φ(N)

• rb =

Z 2πf(t) dt = −2  1 5Mc (tc − t) 5/8 + φc

The phase depends on the chirp mass: Mc is the best measured parameter

d f GW dt = 96 5π M 5/3

c

(πf)11/3

Very strong dependance on the frequency: very slow evolution for the broad orbits If m1 m2,

M 5/3

c

⇡ m2 m1 m5/3

1

the frequency evolution could be sloweven if the orbit is relativistic (extreme mass ratio inspiral EMRI)

Binary system

9

∆t = 5 256M 5/3η (πf)8/3

9

time to coalescence (merger) starting from freq. f LIGO/VIRGO: operates on the ground, freq range 30-2000 Hz. take f=40 Hz, NS-NS system each mass 1.4 solar mass: ~20sec take f = 30 Hz, BH-BH system each 30 solar mass: ~0.32sec ∆t

∆t

LISA (space based detector) will operate in freq. range 0.1- 100 mHz take f=0.1mHz, M = m1 + m2 = 106M,

∆t ≈ 35 days/η < 1 year

Post-Newtonian iterations: we plug back to Einstein equations the evolving orbit and linear solution for the GW, solve at the next order

Φ = Φ0 + ΦN + ✏2Φ1P N + ✏3Φ1.5P N + ✏4Φ2P N + ...

In the near zone the curvature is comparable to the GW wavelength: scattering (backscattering) of the GWs on the Newtonian potential.

Binary system

10

Waveform (GW signal) in the frequency domain

h+(t) = A+ cos Φ(t), h× = A× sin Φ(t)

GW, leading order in amplitude

˜ h(f) = Z h(t)e−2πiftdt

Ψ(f) = 2πftc − φ0 − π 4 + 3 4(8πMcf)−5/3 + ....(Mf)−5/3 + ...(Mf)−1 + ...(Mf)−2/3

Fourier transformation If amplitude is slowly evolving, monotoneous function of time The phase in freq. domain Amplitude and the dominant term in phase depend on Mc only (other terms depend

• n total mass and mass ratio.

Amplitude is higher at low frequencies (early inspiral, slow frequency evolution, many cycles.

˜ h+(f) = (1 + cos2 ι) r 5 6 1 4π2/3 M 5/6

c

DL f −7/6eiΨ(f), ˜ h+(f) = 2i cos ι r 5 6 1 4π2/3 M 5/6

c

DL f −7/6eiΨ(f)

GW signal

11

[Credits: SXS collaboration]

Detected GW signals from binaries

12

Modelling GW signal

13

GW signal from binary stsem can be conventionally split into three parts: inspiral (Post- Newtonian decomposiotion, merger (numerical relativity), ringdown (BH perturbation)

Modelling GW signal

14

• 20
• 10

10 20 X / M

• 20
• 10

10 20 Y / M EOB trajectory EOB ISCO EOB light ring EOB horizon

• 0.4
• 0.2

0.2 0.4 DL/M Reh22 0.2 0.4 Mω22

• 250
• 200
• 150
• 100
• 50

50 (t − R*) / M 0.1 0.2 0.3 0.4 2MΩ light ring ISCO

EOB dynamics inspiral-merger

[Plot: courtesy of A. Taracchini]

The inspiral part of the signal could be pushed to smaller separations using smart resummation (Effective One Body approach) and then stiched to the ringdown

Modelling GW signal

15

Waveform constructed in the frequency domain Uses Post-Newtonian results for the early evolution (inspiral) of a binary Precession is added by rotation taken from the Post-Newtonian evolution Very fast to generate

Phenomenological model: The signal is not that complicated if separated in amplitude and phase: Can construct analytic fit to the numerical data + using analytic solution in limiting parts

[Khan+, PRD 2016]

Detection of GWs

16

The basic principle used in detecting GWs is the geodesic deviation equation in the time dependent field of GWs

Detection of GWs

17

We use local inertial frame located at the beam splitter In case of LIGO/VIRGO

λGW L, ωGW L ⌧ 1

The whole detector can be covered (with high accuracy) by the LIF In this frame (LIF of the beam splitter) the coordinate distance is the proper distance:

ˆ x = Lx(t) = ✓ 1 + 1 2h+(t) ◆ Lx(t0) ˆ y = Ly(t) = ✓ 1 − 1 2h+(t) ◆ Ly(t0)

similar for x polarization

Detection of GWs

18

Since we have covered the whole detector by LIF, the propagation of the laser light as it was in the flat spacetime

φl

x = −νl(ˆ

t − ˆ x) + O(νlL0h(ωL0)2)

Frequency of the laser phase of the laser The phase shift between round trip along x and y For the ground based detectors we can choose the LIF which covers the whole detector, so that the spacetine is flat with high accuracy. In this frame the laser light propagation is as in flat s/t, the mirrors are moving under time varying tidal forces of GWs —> change in the optical path. λGW L, ωGW L ⌧ 1

∆φl = φl

x − φl y = −2νl(Ly − Lx) = 2νl(L(0) x

− L(0)

y

+ L(0)h+(t))

Detection of GWs

19

We cannot set the LIF covering the whole detector, but we can introduce LIF for the background curvature . This is the case for LISA and PTA. We use TT-gauge: λGW ≈ L, ωGW L ≈ 1

gµν = ηµν + hT T

µν + O(h2)

Consider the wave propagating in z-direction

ds2 = dt2 + [1 + h+(t − z)]dx2 + [1 − h+(t − z)]dy2 + dz2

Need to consider equation of propagation of e/m signal

gαβφl

,αφl ,β = 0

dˆ x2 = gxxdx2

Mirrors are moving along geodesics. In TT frame the coordinate distance does not change (but the proper distance does) The phase difference for the round trip is H(t) = Z t h(t0)dt0

∆φl = φl|x − φl|y = −νl  −2(L(0)

x

− L(0)

y ) + 1

2H(t − 2L(0)

x ) + 1

2H(t − 2L(0)

y ) − H(t)

• <latexit sha1_base64="h0+WFXT4fqYdAoQUfoJLR92JTGU=">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</latexit><latexit sha1_base64="h0+WFXT4fqYdAoQUfoJLR92JTGU=">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</latexit><latexit sha1_base64="h0+WFXT4fqYdAoQUfoJLR92JTGU=">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</latexit><latexit sha1_base64="h0+WFXT4fqYdAoQUfoJLR92JTGU=">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</latexit>

ωGW L ⌧ 1

∆φl ≈ 2νl(L(0)

x

− L(0)

y

+ L(0)h+(t))

LIGO/VIRGO

20

Consider the long wavelength approximation: suitable for LIGO/VIRGO. The responds of a single detector:

h(t) = 1 2(ˆ n(1)

i hijˆ

n(1)

j

− ˆ n(2)

i hijˆ

n(2)

j )

strain: differential change in the arm length unit vectors along arms We can decompose the GW signal in the polarization basis

hij = ✏ij

+h+ + ✏ij ×h×

h(t) = F+h+ + F×h×

F+ = 1 2(ˆ n(1)

i ✏ij +ˆ

n(1)

j

− ˆ n(2)

i ✏ij +ˆ

n(2)

j )

similar for x polarization

F+, F×

antenna beam function, it depends on the sky positions of the source and polarization angle

F+ = 1 2 cos(2ψ)(1 + cos2 θs) cos(2φs) − sin(2ψ) cos θs sin(2φs), F× = 1 2 sin(2ψ)(1 + cos2 θs) cos(2φs) + cos(2ψ) cos θs sin(2φs)

polarization angle sky position of GW source

LIGO/VIRGO

21

Substitute the GW signal from a binary system we get for the strain:

h = 2µ AI DL (Mωorb)2/3 cos(2φorb(t) + φI)

A = q (1 + cos2 ι)2F 2

+ + 4 cos2 ιF 2 ×

+ x unpolarized If the GW signal is short (in time), A~const: degeneracy in the parameter space.

A DL = 1 Deff

Only effective distance can be measured. Sky position of a short signal (transient) can be measured only by a network of detectors. detector index

LIGO/VIRGO

22

Triangulation: measuring the time of arrival of the signal at each interferometer - depends on the sky position 2 detectors - circle in the sky 3 detectors - two points in the sky Amplitude and phase consistency - helps! Global network of GW

• bservatories.

LIGO/VIRGO

23

Sensitivity

Seismic noise T h e r m a l n

• i

s e Q u a n t u m s h

• t

n

• i

s e

LIGO/VIRGO

24

∆L = 1 2hGW L

h ≈ 10−22, → L = 4km, → ∆L ≈ 10−16cm