Post-Newtonian Approximation

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Post-Newtonian gravity and gravitational-wave astronomy Polarization waveforms in the SSB reference frame Relativistic binary systems Eﬀective one-body formalism

Piotr Jaranowski

Faculty of Physcis, University of Bialystok,Poland

01.07.2013

P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw Post-Newtonian gravity and gravitational-wave astronomy Polarization waveforms in the SSB reference frame Relativistic binary systems Eﬀective one-body formalism

1 Post-Newtonian gravity and gravitational-wave astronomy

2 Polarization waveforms in the SSB reference frame

3 Relativistic binary systems Leading-order waveforms (Newtonian binary dynamics) Leading-order waveforms without radiation-reaction effects Leading-order waveforms with radiation-reaction effects Post-Newtonian corrections Post-Newtonian spin-dependent effects

4 Effective one-body formalism EOB-improved 3PN-accurate Hamiltonian Usage of Padéapproximants EOB flexibility parameters

1 Post-Newtonian gravity and gravitational-wave astronomy

2 Polarization waveforms in the SSB reference frame

4 Effective one-body formalism EOB-improved 3PN-accurate Hamiltonian Usage of Padéapproximants EOB flexibility parameters

P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw Relativistic binary systems exist in nature, they comprise compact objects: neutron stars or black holes. These systems emit gravitational waves, which experimenters try to detect within the LIGO/VIRGO/GEO600 projects.

Post-Newtonian gravity and gravitational-wave astronomy Polarization waveforms in the SSB reference frame Relativistic binary systems Eﬀective one-body formalism

We will concentrate in this lecture to the most important application of post-Newtonian approximation to general relativity: relativistic two-body problem, i.e. the problem of ﬁnding motion and gravitational radiation of selfgravitating relativistic systems which consist of two extended bodies. Relativistic binary systems exist in nature, they comprise compact objects: neutron stars or black holes. These systems emit gravitational waves, which experimenters try to detect within the LIGO/VIRGO/GEO600 projects.

Black-hole binary: Inspiral—merger—ringdown

As the result of emission of gravitational waves, the size of the binary orbit decreases and the binary components move faster, consequently leading to emission of gravitational waves with increasing amplitude and frequency —producing a gravitational-wave chirp signal.

y 0

-5

-10

-15 -15 -10 -5 0 5 10 15 x

P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw For construction of templates (waveforms) one has to know time evolution of the source of gravitational waves (i.e. compact binary system in the rest of this lecture).

Post-Newtonian gravity and gravitational-wave astronomy Polarization waveforms in the SSB reference frame Relativistic binary systems Eﬀective one-body formalism

To detect in noise of a detector weak gravitational waves (coming e.g. from a coalescing compact binary system) by means of matched ﬁltering method, one has to construct a discrete bank of templates (waveforms) parametrized by possible values of the GW signal’s parameters (e.g. masses and spins of binary components). To ensure the succesful detection, bank of templates has to cover the signal’s parameter space densly enough.

For construction of templates (waveforms) one has to know time evolution of the source of gravitational waves (i.e. compact binary system in the rest of this lecture).

P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw (Blanchet et al., Phys. Rev. D 81, 064004 (2010))

PN expansion: 0th order—Newtonian gravity; nPN order—corrections of order v 2n Gm n c ∼ rc2 to the Newtonian gravity. m Perturbation approach: 1 1. m2

Post-Newtonian gravity and gravitational-wave astronomy Polarization waveforms in the SSB reference frame Relativistic binary systems Eﬀective one-body formalism

Diﬀerent approaches for solving relativistic two-body problem: numerical relativity (breakthrough in 2005); approximate ‘analytical’ methods: post-Newtonian expansion , perturbation-based self-force approach.

(Blanchet et al., Phys. Rev. D 81, 064004 (2010))

PN expansion: 0th order—Newtonian gravity; nPN order—corrections of order v 2n Gm n c ∼ rc2 to the Newtonian gravity. m Perturbation approach: 1 1. m2

P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw For GW signals coming from coalescence of (especially) spinning binary black holes (BBHs) with arbitrary mass ratios, due to limitations in available computing power, it will not be possible in the nearest future to construct bank of templates based purely on numerical results.

Post-Newtonian gravity and gravitational-wave astronomy Polarization waveforms in the SSB reference frame Relativistic binary systems Eﬀective one-body formalism

Construction of bank of templates requires multiple integration of partial diﬀerential equations (numerical relativity), ordinary diﬀerential equations (approximate ‘analytical’ methods).

P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw An accurate equal-mass, non-spinning 8 orbit BBH simulation takes 200 000 CPU hours. ∼ The computational cost of a BBH simulation scales with the mass ratio: an accurate 1:10 mass ratio, non-spinning 8 orbit simulation would thus take 2 000 000 CPU hours. ∼ (I.Hinder, Class. Quantum Grav. 27, 114004 (2010))

Post-Newtonian gravity and gravitational-wave astronomy Polarization waveforms in the SSB reference frame Relativistic binary systems Eﬀective one-body formalism

Construction of bank of templates requires multiple integration of partial diﬀerential equations (numerical relativity), ordinary diﬀerential equations (approximate ‘analytical’ methods).

For GW signals coming from coalescence of (especially) spinning binary black holes (BBHs) with arbitrary mass ratios, due to limitations in available computing power, it will not be possible in the nearest future to construct bank of templates based purely on numerical results.

Construction of bank of templates requires multiple integration of partial diﬀerential equations (numerical relativity), ordinary diﬀerential equations (approximate ‘analytical’ methods).

An accurate equal-mass, non-spinning 8 orbit BBH simulation takes 200 000 CPU hours. ∼ The computational cost of a BBH simulation scales with the mass ratio: an accurate 1:10 mass ratio, non-spinning 8 orbit simulation would thus take 2 000 000 CPU hours. ∼ (I.Hinder, Class. Quantum Grav. 27, 114004 (2010)) P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw Detection of GW signals (and extraction their parameters) from coalescing BBH by advanced LIGO/VIRGO detectors —the most promising waveforms are hybrid waveforms: PN (early inspiral) + numerical (late inspiral, merger, ringdown) or waveforms based on eﬀective one-body formalism.

Post-Newtonian gravity and gravitational-wave astronomy Polarization waveforms in the SSB reference frame Relativistic binary systems Eﬀective one-body formalism

Gravitational-wave induced evolution of BBH computed numerically and by means of the PN approximations agree very well in the region, where the coalescing objects are suﬃciently far away.

Detection of GW signals (and extraction their parameters) from coalescing BBH by advanced LIGO/VIRGO detectors —the most promising waveforms are hybrid waveforms: PN (early inspiral) + numerical (late inspiral, merger, ringdown) or waveforms based on eﬀective one-body formalism.

Eﬀective one-body (EOB) formalism is a formalism, based on approximate PN and BH perturbation theory results, which allows to model ‘analytically’ the whole coalescence of a BBH system, from its adiabatic inspiral up to vibrations of the ‘resultant’ Kerr BH. It is being developed by Damour and his collaborators since 1999. Main idea of EOB approach: To extend the domain of validity of PN and BH perturbation theories so as to approximately cover non-perturbative features of BBH evolution.

Utility (partially expected) of EOB formalism It provides accurate templates needed for detection of GW signals (and estimation of their parameters) originating from coalescences of BBHs, it gives a physical understanding of dynamics and radiation of BBHs, it can be extended to BH-NS or NS-NS systems (after incorporating tidal interactions).

Black-hole binary: EOB vs numerical relativity

T.Damour, A.Nagar, Phys.Rev.D 79, 081503(R) (2009)

0.3 Numerical Relativity (Caltech-Cornell) EOB (a5 = 0; a6=-20)

0.2

0.1 /ν ]

22 0 [Ψ

ℜ −0.1

−0.2 1:1 mass ratio

−0.3

500 1000 1500 2000 2500 3000 3500 4000 t

0.3

0.2

0.1

−0.1

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−0.3 Merger time

3800 3820 3840 3860 3880 3900 3920 3940 3960 3980 4000 t

1 Post-Newtonian gravity and gravitational-wave astronomy

2 Polarization waveforms in the SSB reference frame

4 Effective one-body formalism EOB-improved 3PN-accurate Hamiltonian Usage of Padéapproximants EOB flexibility parameters

i In the coordinate system (t, xs ) centered on a gravitational-wave source the wave polarization functions can be computed from the equations

G d2J R d2J R h (t, x i ) = xx t − − yy t − , (1a) + s c4 R dt2 c dt2 c

2G d2J R h (t, x i ) = xy t − , (1b) × s c4 R dt2 c for the wave propagating in the +z direction of the coordinate system.

Let us introduce coordinate system (t, x i ) about which we assume that some solar-system related observer measures the gravitational-wave ﬁeld TT hµν in the neighborhood of its spatial origin. In the discussion of the detection of gravitational waves by solar-system-based detectors, the origin of these coordinates will be located at the solar system barycenter (SSB). Let us denote by x∗ the constant 3-vector joining the origin of the coordinates (x i ) (i.e. the SSB) i with the origin of the (xs ) coordinates (i.e. the ‘center’ of the source). We also assume that the spatial axes in both coordinate systems are i i parallel to each other, i.e. the coordinates x and xs diﬀer just by constant shifts determined by the components of the 3-vector x∗,

i ∗i i x = x + xs . (2)

TT It means that in both coordinate systems the components hµν of the gravitational-wave ﬁeld are numerically the same.

Equations (1) are valid only when the z axis of the TT coordinate system is parallel to the 3-vector x∗ − x joining the observer located at x (x is the 3-vector joining the SSB with the observer’s location) and the gravitational-wave source at the position x∗. If one changes the location x of the observer, one has to rotate the spatial axes to ensure that Eqs. (1) are still valid. Let us now ﬁx, in the whole region of interest, the direction of the +z axis of both coordinate systems considered here, by choosing it to be antiparallel to the 3-vector x∗ (so for the observer located at the SSB gravitational wave propagates along the +z direction). To a very good accuracy one can assume that the size of the region where the observer can be located is very small compared to the distance from the SSB to the gravitational-wave source, which is equal to

r ∗ := |x∗|.

Our assumption thus means that r r ∗, where r := |x|.

Then x∗ = (0, 0, −r ∗) and the Taylor expansion of R = |x − x∗| and n := (x − x∗)/R around x∗ reads

z x 2 + y 2 |x − x∗| = r ∗ 1 + + + O(x l /r ∗)3 , (3a) r ∗ 2(r ∗)2

x xz y yz x 2 + y 2 n = − + , − + , 1 − + O(x l /r ∗)3. r ∗ 2(r ∗)2 r ∗ 2(r ∗)2 2(r ∗)2 (3b)

One can compute the TT projection of the reduced mass quadrupole moment at the point x in the direction of the unit vector n (which in general is not parallel to the +z axis). One gets 1 J TT = −J TT = (J − J )+ Ox l /r ∗, (4a) xx yy 2 xx yy

TT TT l ∗ Jxy = Jyx = Jxy + O x /r , (4b)

TT TT l ∗ Jzi = Jiz =0+ O x /r for i = x, y, z. (4c)

TT i To obtain the gravitational-wave ﬁeld hµν in the coordinate system (t, x ) one should plug Eqs. (4) into the quadrupole formula. If one neglects in Eqs. (4) the terms of the order of x l /r ∗, then in the whole region of interest covered by the single TT coordinate system, TT the gravitational-wave ﬁeld hµν can be written in the form

TT + × l ∗ hµν (t, x) = h+(t, x) eµν + h×(t, x) eµν + O x /r , (5)

+ × where e and e are the polarization tensors and the functions h+ and h× are of the form given in Eqs. (1).

Dependence of the 1/R factor in the amplitude of the wave polarizations (1) on the observer’s position x (with respect to the SSB) is usually negligible, so1 /R in the amplitudes can be replaced by 1/r ∗ [this is consistent with the neglection of the O(x l /r ∗) terms we have just made in Eqs. (4)].

This is not the case for the 2nd time derivative of Jij in (1), which determines the time evolution of the wave polarizations phases and which is evaluated at the retarded time t − R/c. Here it is usually enough to take into account the ﬁrst correction to r ∗ [given by Eq. (3)].

After taking all this into account the wave polarizations (1) take the form

G d2J z + r ∗ d2J z + r ∗ h (t, x i ) = xx t − − yy t − , (6a) + c4 r ∗ dt2 c dt2 c

2G d2J z + r ∗ h (t, x i ) = xy t − . (6b) × c4 r ∗ dt2 c

The wave polarization functions (6) represent a plane gravitational wave propagating in the +z direction.

P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw Leading-order waveforms (Newtonian binary dynamics) Post-Newtonian gravity and gravitational-wave astronomy Leading-order waveforms without radiation-reaction effects Polarization waveforms in the SSB reference frame Leading-order waveforms with radiation-reaction effects Relativistic binary systems Post-Newtonian corrections Effective one-body formalism Post-Newtonian spin-dependent effects

1 Post-Newtonian gravity and gravitational-wave astronomy

2 Polarization waveforms in the SSB reference frame

4 Effective one-body formalism EOB-improved 3PN-accurate Hamiltonian Usage of Padéapproximants EOB flexibility parameters

1 Post-Newtonian gravity and gravitational-wave astronomy

2 Polarization waveforms in the SSB reference frame

4 Effective one-body formalism EOB-improved 3PN-accurate Hamiltonian Usage of Padéapproximants EOB flexibility parameters

We consider a binary system made of two bodies with masses m1 and m2 (we will always assume m1 ≥ m2). We introduce m m M := m + m , µ := 1 2 , (7) 1 2 M so M is the total mass of the system and µ is its reduced mass. It is also useful to introduce the dimensionless symmetric mass ratio m m µ ν := 1 2 = . (8) M2 M The quantity ν satisﬁes 0 ≤ ν ≤ 1/4, the case ν = 0 corresponds to the test-mass limit and ν = 1/4 describes equal-mass binary.

We start from deriving the wave polarization functions h+ and h× for waves emitted by a binary system in the case when the dynamics of the binary can reasonably be described within the Newtonian theory of gravitation.

Let r1 and r2 denote the position vectors of the bodies, i.e. the 3-vectors connecting the origin of some reference frame with the bodies. We introduce the relative position vector,

r12 := r1 − r2. (9)

The center-of-mass reference frame is deﬁned by the requirement that

m1 r1 + m2 r2 = 0. (10)

Solving Eqs. (9) and (10) with respect to r1 and r2 one gets m m r = 2 r , r = − 1 r . (11) 1 M 12 2 M 12

In the center-of-mass reference frame we introduce the spatial coordinates (xc, yc, zc) such that the total orbital angular momentum vector J of the binary is directed along the +zc axis.

Then the trajectories of both bodies lie in the (xc, yc) plane, so the position vector ra of the ath body (a = 1, 2) has components

ra = (xca, yca, 0),

and the relative position vector components are

r12 = (xc12, yc12, 0),

where xc12 := xc1 − xc2 and yc12 := yc1 − yc2.

It is convenient to introduce in the coordinate (xc12, yc12) plane the usual polar coordinates (r, φ):

xc12 = r cos φ, yc12 = r sin φ. (12)

Within the Newtonian gravity the orbit of the relative motion is an ellipse (we consider here only gravitationally bound binaries). We place the focus of the ellipse at the origin of the (xc12, yc12) coordinates. In polar coordinates the ellipse is described by the equation

a(1 − e2) r(φ) = , (13) 1 + e cos(φ − φ0) where a is the semimajor axis, e is the eccentricity, and φ0 is the angle of the orbital periapsis. The time dependence of the relative motion is determined by the Kepler’s equation, 2π φ˙ = (1 − e2)−3/21 + e cos(φ − φ )2, (14) P 0 where P is the orbital period of the binary, r a3 P = 2π . (15) GM

The binary’s binding energy E and the modulus J of its total angular orbital momentum are related to the parameters of the relative orbit through the equations GMµ E = − , J = µpGMa(1 − e2). (16) 2a

Let us now introduce the TT ‘wave’ coordinates (xw, yw, zw) in which the gravitational wave is traveling in the +zw direction and with the origin at the SSB. The line along which the plane tangent to the celestial sphere at the location of the binary’s center-of-mass [this plane is parallel to the (xw, yw) plane] intersects the orbital (xc, yc) plane is called the line of nodes. Let us adjust the center-of-mass and the wave coordinates in such a way, that the x axes of both coordinate systems are parallel to each other and to the line of nodes.

Then the relation between these coordinates is determined by the rotation matrix S,

   ∗      xw xw xc 1 0 0 ∗ yw = yw + S yc , S := 0 cos ι sin ι , (17) ∗ zw zw zc 0 − sin ι cos ι

∗ ∗ ∗ ∗ where (xw, yw, zw) are the components of the vector x joining the SSB and the binary’s center-of-mass, and ι ( 0 ≤ ι ≤ π) is the angle between the orbital angular momentum vector J of the binary and the line of sight (i.e. the +zw axis). We assume that the center of mass of the binary is at rest with respect to the SSB.

In the center-of-mass reference frame the binary’s moment of inertia tensor has changing in time components which are equal

ij i j i j Ic (t) = m1 xc1(t) xc1(t) + m2 xc2(t) xc2(t). (18)

ij Making use of Eqs. (11) one ﬁnds the matrix Ic built from the Ic components of the inertia tensor,

 2  (xc12(t)) xc12(t) yc12(t) 0 2 Ic(t) = µ xc12(t) yc12(t)(yc12(t)) 0 . (19) 0 0 0

The inertia tensor components in wave coordinates are related with those in center-of-mass coordinates through the relation

3 3 i j ij X X ∂x ∂x k` I (t) = w w I (t), (20) w ∂x k ∂x ` c k=1 `=1 c c what in matrix notation reads

T Iw(t) = S ·Ic(t) · S . (21)

To obtain the wave polarization functions h+ and h× we plug the components of the binary’s inertia tensor into general equations [note that in these equations the components Jij of the reduced quadrupole moment can be replaced by the components Iij of the inertia tensor]. We get [here R = |x∗| is the distance to the binary’s center-of-mass and tr = t − (zw + R)/c is the retarded time] Gµ h (t, x) = sin2 ιr˙(t )2 + r(t )¨r(t ) + c4R r r r 2 2 2 2 + (1 + cos ι) r˙(tr) + r(tr)¨r(tr) − 2r(tr) φ˙(tr) cos 2φ(tr) 2 2 − (1 + cos ι) 4r(tr)˙r(tr)φ˙(tr) + r(tr) φ¨(tr) sin 2φ(tr) , (22a) 2Gµ h (t, x) = cos ι 4r(t )˙r(t )φ˙(t ) + r(t )2φ¨(t ) cos 2φ(t ) × c4R r r r r r r 2 2 2 + r˙(tr) + r(tr)¨r(tr) − 2r(tr) φ˙(tr) sin 2φ(tr) . (22b)

1 Post-Newtonian gravity and gravitational-wave astronomy

2 Polarization waveforms in the SSB reference frame

4 Effective one-body formalism EOB-improved 3PN-accurate Hamiltonian Usage of Padéapproximants EOB flexibility parameters

Gravitational waves emitted by the binary diminish the binary’s binding energy and total orbital angular momentum. This makes the orbital parameters changing in time. Let us ﬁrst assume that these changes are so slow that they can be neglected during the time interval in which the observations are performed. Making use of Eqs. (14) and (13) one can then eliminate from the formulae (22) the ﬁrst and the second time derivatives of r and φ, treating the parameters of the orbit, a and e, as constants.

a(1 − e2) r(φ) = (13) 1 + e cos(φ − φ0) 2π φ˙ = (1 − e2)−3/21 + e cos(φ − φ )2 (14) P 0

The result is the following:

4G 2Mµ h (t, x) = A (t ) + A (t ) e + A (t ) e2 , (23a) + c4Ra(1 − e2) 0 r 1 r 2 r

4G 2Mµ h (t, x) = cos ι B (t ) + B (t ) e + B (t ) e2 , (23b) × c4Ra(1 − e2) 0 r 1 r 2 r

where the functions Ai and Bi , i = 0, 1, 2, are deﬁned as follows

1 2 A0(t) = − (1 + cos ι) cos 2φ(t), 2

1 2 1 2 A1(t) = sin ι cos(φ(t) − φ0) − (1 + cos ι) 5 cos(φ(t) + φ0) + cos(3φ(t) − φ0) , 4 8

1 2 1 2 A2(t) = sin ι − (1 + cos ι) cos 2φ0, 4 4

B0(t) = − sin 2φ(t), 1 B1(t) = − sin(3φ(t) − φ0) + 5 sin(φ(t) + φ0) , 4 1 B2(t) = − sin 2φ0. 2

In the case of circular orbits the eccentricity e vanishes and the wave polarization functions (23) simplify to

2 2G Mµ 2 h+(t, x) = (1 + cos ι) cos 2φ(tr), (25a) − c4Ra 4G 2Mµ h×(t, x) = cos ι sin 2φ(tr), (25b) − c4Ra where a is the radius of the relative circular orbit and 2π φ(t) = φ0 + (t t0). P −

1 Post-Newtonian gravity and gravitational-wave astronomy

2 Polarization waveforms in the SSB reference frame

4 Effective one-body formalism EOB-improved 3PN-accurate Hamiltonian Usage of Padéapproximants EOB flexibility parameters

Let us now consider observational intervals short enough so it is necessary to take into account eﬀects of radiation reaction changing the parameters of the bodies’ orbits. In the leading order the rates of emission of energy and angular momentum carried by gravitational waves are given by the formulae

3 3 * 3 2+ 3 3 3 2 3 gw G X X d Jij gw 2G X X X d Jj` d Jk` L = , L = ijk . E 5c5 dt3 Ji 5c5 dt2 dt3 i=1 j=1 j=1 k=1 `=1

We plug into them the Newtonian trajectories of the bodies and perform time averaging over one orbital period of the motion. We get

32 G 4µ2M3 73 37 Lgw = 1 + e2 + e4 , (26a) E 5 c5a5(1 − e2)7/2 24 96

32 G 7/2µ2M5/2 7 Lgw = 1 + e2 . (26b) J 5 c5a7/2(1 − e2)2 8

It is easy to obtain equations describing the leading-order time evolution of the relative orbit parameters a and e. We ﬁrst diﬀerentiate both sides of equations GMµ E = − , J = µpGMa(1 − e2), 2a with respect to time treating a and e as functions of time. Then the rates of changing the binary’s binding energy and angular momentum, dE/dt and dJ/dt, one replaces by minus the rates of gw gw emission of respectively energy LE and angular momentum LJ carried by gravitational waves, given in Eqs. (26).

The two equations such obtained can be solved with respect to the derivatives da/dt and de/dt:

da 64 G 3µM2 73 37 = − 1 + e2 + e4 , (27a) dt 5 c5a3(1 − e2)7/2 24 96

de 304 G 3µ2M2 121 = − e 1 + e2 . (27b) dt 15 c5a4(1 − e2)5/2 304

Let us note two interesting features of the evolution described by the above equations: Eq. (27b) implies that initially circular orbits (for which e = 0) remain circular during the evolution; the eccentricity e of the relative orbit decreases with time much more rapidly than the semimajor axis a, so gravitational-wave reaction induces the circularization of the binary orbits (roughly, when the semimajor axis is halved, the eccentricity goes down by a factor of three).

Evolution of the orbital parameters caused by the radiation reaction inﬂuences gravitational waveforms. To see this inﬂuence we restrict our considerations to binaries along circular orbits. The decay of the radius a of the relative orbit is given by the equation [this is Eq. (27a) with e = 0]

da 64 G 3µM2 = − . (28) dt 5 c5a3 Integration of this equation leads to

1/4 tc − t a(t) = a0 , (29) tc − t0

where a0 := a(t = t0) is the initial value of the orbital radius and tc − t0 is the formal ‘lifetime’ or ‘coalescence time’ of the binary, i.e. the time after which the distance between the bodies is zero.

It means that tc is the moment of the coalescence, a(tc) = 0; it is equal to

5 c5a4 t = t + 0 . (30) c 0 256 G 3µM2

If one neglects radiation reaction eﬀects, then along the Newtonian circular orbit the time derivative of the orbital phase, i.e. the orbital angular frequency ω, is constant and given by [see Eqs. (14) and (15)] r 2π GM ω = φ˙ = = . (31) P a3

We now assume that the inspiral of the binary system caused by the radiation reaction is adiabatic, i.e. it can be thought as a sequence of circular orbits with radii a(t) which slowly diminish in time according to Eq. (29). Adiabacity thus means that Eq. (31) is valid during the inspiral, but now angular frequency ω is a function of time, because of time dependence of the orbital radius a.

Making use of Eqs. (29)–(31) we get the following equation for the time evolution of the instantaneous orbital angular frequency during the adiabatic inspiral:

256(GM)5/3 −3/8 ω(t) = ω 1 − ω8/3(t − t ) , (32) 0 5c5 0 0

where ω0 := ω(t = t0) is the initial value of the orbital angular frequency and where we have introduced the new parameter M (with dimension of a mass) called a chirp mass of the system:

M := µ3/5M2/5. (33)

To get the polarization waveforms h+ and h× which describes gravitational radiation emitted during the adiabatic inspiral of the binary system, we use the general formulae (22). In these formulae we neglect all the terms proportional tor ˙ ≡ a˙ or ¨r ≡ ¨a (the radial coordinate r is identical with the radius a of the relative circular orbit, r ≡ a). By virtue of Eq. (31) φ¨ ∝ a˙, therefore we also neglect terms ∝ φ¨. All these simpliﬁcations lead to the following formulae:

4(GM)5/3 1 + cos2 ι h (t, x) = − ω(t )2/3 cos 2φ(t ), (34a) + c4R 2 r r

4(GM)5/3 h (t, x) = − cos ι ω(t )2/3 sin 2φ(t ). (34b) × c4R r r

Making use of Eq. (31) one can express the time dependence of the waveforms (34) in terms of the single function a(t):

4G 2µM 1 + cos2 ι 1 √ Z tr h (t, x) = − cos 2 GM a(t0)−3/2dt0 + φ , + c4R 2 a(t ) 0 r t0 (35a)

4G 2µM 1 √ Z tr h (t, x) = − cos ι sin 2 GM a(t0)−3/2dt0 + φ , × c4R a(t ) 0 r t0 (35b)

where φ0 := φ(t = t0) is the initial value of the orbital phase of the binary.

It is sometimes useful to analyze the chirp waveforms h+ and h× in terms of the instantaneous frequency fgw of the gravitational waves emitted by a coalescing binary system.

The frequency fgw is deﬁned as

1 dφ(t) f (t) := 2 × , (36) gw 2π dt where the extra factor 2 is due to the fact that [see Eqs. (34)] the instantaneous phase of the gravitational waveforms h+ and h× is 2 times the instantaneous phase φ(t) of the orbital motion (so the gravitational-wave frequency fgw is twice the orbital frequency).

Making use of Eqs. (29)–(31) one gets that for the binary with circular orbits the gravitational-wave frequency fgw reads

53/8c15/8 1 f (t) = (t − t)−3/8. (37) gw 8πG 5/8 M5/8 c

The chirp waveforms given in Eqs. (35) can be rewritten in terms of the gravitational-wave frequency fgw as follows

1 + cos2 ι Z tr h (t, x) = −h (t ) cos 2π f (t0)dt0 + 2φ , (38a) + 0 r 2 gw 0 t0 Z tr 0 0 h×(t, x) = −h0(tr) cos ι sin 2π fgw(t )dt + 2φ0 , (38b) t0

where h0(t) is the changing in time amplitude of the waveforms,

4π2/3G 5/3 M5/3 h (t) := f (t)2/3. (39) 0 c4 R gw

1 Post-Newtonian gravity and gravitational-wave astronomy

2 Polarization waveforms in the SSB reference frame

4 Effective one-body formalism EOB-improved 3PN-accurate Hamiltonian Usage of Padéapproximants EOB flexibility parameters

P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw PN corrections for EOM/luminosities without spin-dependent eﬀects

EOM N 1PN 2PN 2.5PN 3PN 3.5PN 4PN 4.5PN 5PN 5.5PN 6PN 6.5PN

Energy/ angular momentum luminosity — — — N — 1PN 1.5PN 2PN 2.5PN 3PN 3.5PN 4PN

Leading-order waveforms (Newtonian binary dynamics) Post-Newtonian gravity and gravitational-wave astronomy Leading-order waveforms without radiation-reaction effects Polarization waveforms in the SSB reference frame Leading-order waveforms with radiation-reaction effects Relativistic binary systems Post-Newtonian corrections Effective one-body formalism Post-Newtonian spin-dependent effects

There are two problems, usually analyzed separately: problem of ﬁnding equations of motion (EOM), problem of computing gravitational-wave luminosities (of energy, angular momentum, and linear momentum).

PN corrections for EOM/luminosities without spin-dependent eﬀects

EOM N 1PN 2PN 2.5PN 3PN 3.5PN 4PN 4.5PN 5PN 5.5PN 6PN 6.5PN

Energy/ angular momentum luminosity — — — N — 1PN 1.5PN 2PN 2.5PN 3PN 3.5PN 4PN

The response function of the laser-interferometric detector to gravitational waves from coalescing compact binary (made of nonspinning bodies) in circular orbits:

C h(t) = φ˙(t)2/3 sin 2φ(t) + α, D D is the distance of the binary to the Earth, C and α are some constants, and φ(t) is the orbital phase of the binary (so φ˙(t) ≡ dφ(t)/dt is the angular frequency).

Dimensionless post-Newtonian parameter for circular orbits: 1 x ≡ (GMφ˙)2/3. c2

The orbital phase φ(t) evolution is computed from the balance equation dE = −L, dt which has the following PN expansion: d 1 1 1 1 E + E + E + E + E + O(v/c)9 dt N c2 1PN c4 2PN c6 3PN c8 4PN 1 1 1 1 = − L + L + L + L + L N c2 1PN c3 1.5PN c4 2PN c5 2.5PN 1 1 1 + L + L + L + O(v/c)9 . c6 3PN c7 3.5PN c8 4PN

Bounding energy in the center-of-mass frame for circular orbits up to the 4PN order

µc2x E(x; ν) = − 1 + e (ν) x + e (ν) x 2 + e (ν) x 3 2 1PN 2PN 3PN 448 + e (ν) + ν ln x x 4 + O(v/c)10 , 4PN 15

3 1 27 19 1 2 e1PN(ν) = − − ν, e2PN(ν) = − + ν − ν , 4 12 8 8 24

675 34445 205 2 155 2 35 3 e3PN(ν) = − + − π ν − ν − ν , 64 576 96 96 5184

3969 498449 3157 2 2 301 3 77 4 e4PN(ν) = − + c1ν + − + π ν + ν + ν 128 3456 576 1728 31104

(Jaranowski & Sch¨afer2012–2013, Foﬀa & Sturani 2013),

123671 9037 2 896 c1 = − + π + (2 ln 2 + γ)) (γ is the Euler’s constant) 5760 1536 15

(Le Tiec, Blanchet, & Whiting 2012, Bini & Damour 2013).

Gravitational-wave luminosity for circular orbits up to the 3.5PN order

32c5 L(x; ν) = ν2x 5 1 + ` (ν) x + 4π x 3/2 + ` (ν) x 2 5G 1PN 2PN 856 + ` (ν) x 5/2 + ` (ν) − ln(16x) x 3 2.5PN 3PN 105 7/2 8 + `3.5PN(ν) x + O (v/c) ,

1247 35 44711 9271 65 2 `1PN(ν) = − − ν, `2PN(ν) = − + ν + ν , 336 12 9072 504 18 8191 535 `2.5PN(ν) = − − ν π, 672 24

6643739519 16 2 1712 134543 41 2 94403 2 775 3 `3PN(ν) = + π − γ + − + π ν − ν − ν , 69854400 3 105 7776 48 3024 324

16285 214745 193385 2 `3.5PN(ν) = − + ν + ν π. 504 1728 3024

Post-Newtonian results Gravitational-wave luminosities for energy and angular momentum in the case of quasi-elliptical orbits were computed up to the 3PN order beyond the leading-order formula. PN eﬀects modify not only the orbital phase evolution of the binary, but also the amplitudes of the wave polarizations h+ and h×: —for quasi-circular orbits the 3PN-accurate corrections were computed; —for quasi-elliptical orbits the 2PN-accurate corrections were computed.

It is not easy to obtain the wave polarizations h+ and h× as explicit functions of time taking into account all known higher-order PN corrections. In the case of quasi-elliptical orbits one has to deal with three diﬀerent time scales: orbital period, periastron precession, and radiation reaction time scale.

1 Post-Newtonian gravity and gravitational-wave astronomy

2 Polarization waveforms in the SSB reference frame

4 Effective one-body formalism EOB-improved 3PN-accurate Hamiltonian Usage of Padéapproximants EOB flexibility parameters

The spin of a rotating body is of the order

S ∼ mavspin,

where m and a denote the mass and typical size of the body, respectively, and vspin represents the velocity of the body’s surface. We are interested in compact bodies, so Gm v a ∼ , and then S ∼ Gm2 spin . c2 c2

Nomenclature on PN spin-dependent effects Gm2 Maximally rotating bodies: v ∼ c =⇒ S ∼ . spin c Spin-orbit (i.e. linear in S) effects in EOM: 1.5PN + 2.5PN + ··· ; spin-spin effects in EOM: 2PN + 3PN + ··· . Gm2v Slowly rotating bodies: v c =⇒ S ∼ spin . spin c2 Spin-orbit (i.e. linear in S) effects in EOM: 2PN + 3PN + ··· ; spin-spin effects in EOM: 3PN + 3PN + ··· .

More nomenclature on PN spin-dependent eﬀects Just words, no 1/c counting: leading-order (LO) + next-to-leading-order (NLO) + next-to-next-to-leading-order (NNLO) + ··· . Formal counting: PN orders are counted in terms of 1/c originally present in the Einstein equations, i.e. the spin variables do not contribute to counting of 1/c. Then, e.g., spin-orbit eﬀects in EOM start as follows: 1PN + 2PN + ··· .

P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw Post-Newtonian gravity and gravitational-wave astronomy EOB-improved 3PN-accurate Hamiltonian Polarization waveforms in the SSB reference frame Usage of Padéapproximants Relativistic binary systems EOB flexibility parameters Effective one-body formalism

1 Post-Newtonian gravity and gravitational-wave astronomy

2 Polarization waveforms in the SSB reference frame

4 Effective one-body formalism EOB-improved 3PN-accurate Hamiltonian Usage of Padéapproximants EOB flexibility parameters

Structure of EOB formalism

1 Computation of inspiral + plunge waveform hinsplunge(t): PN conservative dynamics −→ EOB-improved Hamiltonian; PN radiation losses −→ EOB radiation-reaction force; PN waveform. 2 Computation of ringdown waveform hringdown(t): BH perturbation theory.

EOB insplunge ringdown EOB-waveform: h (t) = θ(tm t) h (t) + θ(t tm) h (t), − − where θ(t) denotes Heaviside’s step function and tm is the time at which the two waveforms hinsplunge, hringdown are matched.

1 Post-Newtonian gravity and gravitational-wave astronomy

2 Polarization waveforms in the SSB reference frame

4 Effective one-body formalism EOB-improved 3PN-accurate Hamiltonian Usage of Padéapproximants EOB flexibility parameters

Within the ADM approach the 3PN-accurate 2-point-mass conservative Hamiltonian H x1, x2, p1, p2 was derived. We need its reduction to the center-of-mass frame.

Relative motion in the center-of-mass frame (p1 + p2 = 0): x1, x2, p1, p2 (q, p); → reduced variables in the center-of-mass frame:

x1 x2 p1 p2 t q − , p = , bt . ≡ GM ≡ µ − µ ≡ GM

‘Non-relativistic’ Hamiltonian of the 2-point-mass system:

2 H (m1 + m2)c Hb NR − . ≡ µ Conservative 3PN-accurate Hamiltonian describing relative motion

NR 1 Hb (q, p) = HbN (q, p) + Hb1PN (q, p) c2 1 1 + Hb2PN (q, p) + Hb3PN (q, p) , c4 c6

1 2 1 HbN (q, p) = p ,... 2 − q

(the functional form of Hb NR (q, p) is gauge dependent).

Units: c = G = 1.

Real two-body problem (two masses m1, m2 orbiting around each other)

l Eﬀective one-body problem (one test particle of mass m0 eﬀective moving in some background metric gαβ )

The mapping rules between the two problems (motivated by quantum considerations): H the adiabatic invariants (the action variables) Ii = pi dqi are identiﬁed in the two problems; the energies are mapped through a function f :

= f ( ), Eeﬀective Ereal f is determined in the process of matching.

eﬀective One looks for a metric gαβ such that the energies of the bound states eﬀective of a particle moving in gαβ are in one-to-one correspondence with the energies of the two-body bound states:

(Ii ) = f ( (Ii )) . Eeﬀective Ereal

Static and spherically symmetric eﬀective metric (in Schwarzschild-like coordinates):

D(q0) ds2 = A(q0) dt02 + dq02 + q02 (dθ02 + sin2 θ0 dϕ02), eﬀ − A(q0)

2 3 4 0 M0 M0 M0 M0 A(q ) = 1 + a1 + a2 + a3 + a4 + , q0 q0 q0 q0 ··· 2 3 0 M0 M0 M0 D(q ) = 1 + d1 + d2 + d3 + . q0 q0 q0 ···

The energy-map for the “non-relativistic” energies NR R NR R m0 and M: Eeﬀ ≡ Eeﬀ − Ereal ≡ Ereal −

NR NR NR NR 2 NR 3 ! eﬀ real real real real E = E 1 + α1 E + α2 E + α3 E + . m0 µ µ µ µ ···

It is natural to require that

m0 µ = m1 m2/(m1 + m2), ≡ M0 M = m1 + m2, ≡

with these choices the Newtonian limit tells us that a1 = 2. −

At each PN level one has only three arbitrary coefficients: the 1PN level: a2, d1, and α1; the 2PN level: a3, d2, and α2; the 3PN level: a4, d3, and α3. How many independent equations has to be satisfied when mapping the real problem onto the effective one?

The structure of the nPN conservative relative-motion real Hamiltonian in the center-of-mass frame:

1 Hb NR (q, p) = p2(n+1) + p2n + p2(n−1)(np)2 + + (np)2n nPN q ···

1 1 + p2(n−1) + + (np)2(n−1) + + ; q2 ··· ··· qn+1

NR (n + 1)(n + 2) Hb (q, p) has CH (n) = + 1 coeﬃcients. nPN 2

The identiﬁcation of the action variables guarantees that the two problems are mapped by a canonical transformation, with generating function

˜ 0 i 0 0 G(q, p ) = q pi + G(q, p ),

i so the relation between the real phase-space coordinates (q , pi ) 0i 0 and the eﬀective phase-space coordinates (q , pi ) reads

0 0 0i i ∂ G(q, p ) 0 ∂ G(q, p ) q = q + 0 , pi = pi + i . ∂ pi ∂ q

The structure of the nPN generating function:

0 0 02n 1 02(n−1) 0 2(n−1) Gn (q, p ) = (q p ) p + p + + (np ) PN · q ··· 1 + + ; ··· qn

n(n + 1) G (q, p0) has C (n) = + 1 coeﬃcients. nPN G 2

The diﬀerence ∆(n) between the number of equations to satisfy and the number of unknowns at the nPN level:

∆(n) = CH (n) (CG (n) + 3) = n 2 ; − −

∆(1) = 1 , − ∆(2) = 0 ,

∆(3) = +1 .

2PN-accurate eﬀective (geodesic) Hamiltonian

2 αβ 0 0 0 0 = µ + geﬀ (x ) pα pβ

⇓ v u ! u A(q0) Hb R (q0, p0) = tA(q0) 1 + p02 + 1 (n0 p0)2 . eﬀ D(q0) − ·

P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw Matching results at the 2PN level

No degrees of freedom left.

Energy map coeﬃcient: α2 = 0.

Post-Newtonian gravity and gravitational-wave astronomy EOB-improved 3PN-accurate Hamiltonian Polarization waveforms in the SSB reference frame Usage of Padéapproximants Relativistic binary systems EOB flexibility parameters Effective one-body formalism

Matching results at the 1PN level

One unrestricted degree of freedom is used to impose the condition d1 = 0, i.e. that the linearized eﬀective metric coincides with the linearized Schwarzschild metric. 1 Energy map coeﬃcient: α = ν. 1 2

Matching results at the 1PN level

Matching results at the 2PN level

No degrees of freedom left.

Energy map coeﬃcient: α2 = 0.

3PN-accurate eﬀective (nongeodesic) Hamiltonian

2 αβ 0 0 0 αβγδ 0 0 0 0 0 0 = µ + geﬀ (x ) pα pβ + A (x ) pα pβ pγ pδ

⇓ v u ! u A(q0) Hb R (q0, p0) = tA(q0) 1 + p02 + 1 (n0 p0)2 + Z(q0, p0) , eﬀ D(q0) − ·

0 0 1 02 2 02 0 0 2 0 0 4 where Z(q , p ) z1(p ) + z2 p (n p ) + z3(n p ) . ≡ q02 · ·

Matching results at the 3PN level

The parameters z1, z2, and z3 satisfy the linear constraint:

8z1 + 4z2 + 3z3 = 6(4 3ν)ν . − Energy map coeﬃcient: α3 = 0 .

To simplify the 3PN eﬀective dynamics of circular orbits, one chooses

z1 = 0 = z2 , z3 = 2(4 3ν)ν . −

1 The values α1 = 2 ν at 1PN, α2 = 0 at 2PN, and α3 = 0 at 3PN correspond exactly to

R ( R )2 m2 m2 Eeﬀ = Ereal − 1 − 2 . µ 2m1m2

R Solving above equation with respect to real one obtains the real EOB-improved 3PN-accurate Hamiltonian:E

s HR q0(q, p), p0(q, p) µ HR (q, p) = M 1 + 2ν eﬀ − . real µ

1 Post-Newtonian gravity and gravitational-wave astronomy

2 Polarization waveforms in the SSB reference frame

4 Effective one-body formalism EOB-improved 3PN-accurate Hamiltonian Usage of Padéapproximants EOB flexibility parameters

Pad´eapproximant of (k, l)-type (with k + l = n) for series n σ(x) = c0 + c1 x + + cn x (with c0 = 0): ··· 6

k Nk (x) Pl (σ(x)) , ≡ Dl (x) where the polynomials Nk (of degree k) and Dl (of degree l) k are such that the Taylor expansion of Pl (σ(x)) coincides with σ(x) up to (x n) terms. O

Let us consider the reduced angular momentum of the system in the center-of-mass reference frame:

j J . ≡ Gm1m2 In the test-mass limit and along circular orbits: 1 j(x; ν = 0) = , px(1 3x) − the pole x = 1/3 corresponds to ‘light ring’ (or the last unstable circular orbit).

As a result of the 3PN-accurate PN computations one gets

1 1 1 16 j 2(x; ν) = 1 + (9 + ν) x + (36 − ν)(9 − 4ν) x 2 + w (ν) x 3 , x 3 36 3 5

where 81 1 7321 23 1 w (ν) ≡ + 41π2 − ν + ν2 + ν3. 5 16 64 6 64 216

One can construct the following sequence of near-diagonal Pad´esof j 2(x):

1 1 j 2 (x; ν) ≡ P0 1 + (9 + ν) x , P1 x 1 3

1 1 1 j 2 (x; ν) ≡ P1 1 + (9 + ν) x + (36 − ν)(9 − 4ν) x 2 , P2 x 1 3 36

1 1 1 16 j 2 (x; ν) ≡ P2 1 + (9 + ν) x + (36 − ν)(9 − 4ν) x 2 + w (ν) x 3 . P3 x 1 3 36 3 5

The results of computation read: 1 j 2 (x; ν) = , P1 1 x 1 − 3 + 3 ν x

1 + 1 ν + 25 νx j 2 (x; ν) = 9 12 , P2 1 17 1 2 x 1 + 9 ν − 3 − 12 ν + 27 ν x

1 17 1 2 1 2 1 + 3 + ν − w6(ν) x + 9 − ν + ν − 3 + ν w6(ν) x j 2 (x; ν) = 3 4 9 3 , P3 x 1 − w6(ν)x

where 192 w (ν) ≡ w (ν). 6 (36 − ν)(9 − 4ν) 5 The test-mass limit is recovered exactly,

2 2 2 1 lim jP (x; ν) = lim jP (x; ν) = lim jP (x; ν) = . ν→0 1 ν→0 2 ν→0 3 x(1 − 3x)

eﬀ 0 The EOB metric coeﬃcient g00 (u) = A(u) (with u 1/q ) has the following 3PN-accurate− truncated Taylor expansion:≡

3 4 5 A(u; ν) = 1 2 u + 2ν u + a4(ν) u + u , − O

where the 3PN coeﬃcient a4(ν) reads 94 41 2 a4(ν) = π ν. 3 − 32

We want now to factor a zero of A(u; ν) (and no longer a pole), which will be smoothly connected with the test-mass-limit (i.e.

ν 0) value u = 1/2, therefore the Pad´e-improved APn (u; ν) of A(→u; ν) we deﬁne at the nPN level as

1 AP (u; ν) P [Tn+1[A(u; ν)]] . n ≡ n

1 Post-Newtonian gravity and gravitational-wave astronomy

2 Polarization waveforms in the SSB reference frame

4 Effective one-body formalism EOB-improved 3PN-accurate Hamiltonian Usage of Padéapproximants EOB flexibility parameters

EOB formalism synergy Numerical Relativity ←→ ←→

Select sample of NR results

Calibrating EOB ﬂexibility↓ parameters

NR-improved↓ EOB (accurate ‘analytical’ waveform templates; ‘analytical’ predictions)

Flexibility parameters in the metric coeﬃcient g eﬀ(u) = A(u) − 00

Instead of using (maybe in Pad´e-improved form)

3PN 3 4 A (u; ν) = 1 2 u + 2ν u + a4(ν) u , − one considers two-parameter class of extensions of A3PN(u; ν) deﬁned by

1 3PN 5 6 A(u; ν, a5, a6) P A (u; ν) + ν a5 u + ν a6 u . ≡ 5

“Contrary to the hint of Mroué,Kidder, and Teukolsky (2008), that EOB is ‘only ... a very good fitting model’, we think that EOB incorporates a lot of the real physics of coalescing BBH, and that its parametric flexibility is a way to complete it with the ‘missing’ non-perturbative physics provided by NR simulations.”

(T.Damour, 2008)

P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw