post-newtonian theory and gravitational waves · deduced chirp mass would have a very large total...
TRANSCRIPT
Post-Newtonian theory and gravitational waves
Tanguy Marchand
1
Journée des thèses - APC - 10 Novembre 2016
David Langlois (APC) - Luc Blanchet (IAP)
The spacetime is a manifold of dimension 4 with a metric (which is a symmetric rank 2 tensor)
Matter follows geodesics in this curved spacetime
gab
General relativity in a nutshell
‣
‣
‣
3
5
Credit: P. S. Shawhan for the LIGO Scientific Collaboration and Virgo Collaboration (cf arxiv:1210.7173)
6PRL 116, 061102 (2016)
propagation time, the events have a combined signal-to-noise ratio (SNR) of 24 [45].Only the LIGO detectors were observing at the time of
GW150914. The Virgo detector was being upgraded,and GEO 600, though not sufficiently sensitive to detectthis event, was operating but not in observationalmode. With only two detectors the source position isprimarily determined by the relative arrival time andlocalized to an area of approximately 600 deg2 (90%credible region) [39,46].The basic features of GW150914 point to it being
produced by the coalescence of two black holes—i.e.,their orbital inspiral and merger, and subsequent final blackhole ringdown. Over 0.2 s, the signal increases in frequencyand amplitude in about 8 cycles from 35 to 150 Hz, wherethe amplitude reaches a maximum. The most plausibleexplanation for this evolution is the inspiral of two orbitingmasses, m1 and m2, due to gravitational-wave emission. Atthe lower frequencies, such evolution is characterized bythe chirp mass [11]
M ¼ ðm1m2Þ3=5
ðm1 þm2Þ1=5¼ c3
G
!5
96π−8=3f−11=3 _f
"3=5
;
where f and _f are the observed frequency and its timederivative and G and c are the gravitational constant andspeed of light. Estimating f and _f from the data in Fig. 1,we obtain a chirp mass of M≃ 30M⊙, implying that thetotal mass M ¼ m1 þm2 is ≳70M⊙ in the detector frame.This bounds the sum of the Schwarzschild radii of thebinary components to 2GM=c2 ≳ 210 km. To reach anorbital frequency of 75 Hz (half the gravitational-wavefrequency) the objects must have been very close and verycompact; equal Newtonian point masses orbiting at thisfrequency would be only ≃350 km apart. A pair ofneutron stars, while compact, would not have the requiredmass, while a black hole neutron star binary with thededuced chirp mass would have a very large total mass,and would thus merge at much lower frequency. Thisleaves black holes as the only known objects compactenough to reach an orbital frequency of 75 Hz withoutcontact. Furthermore, the decay of the waveform after itpeaks is consistent with the damped oscillations of a blackhole relaxing to a final stationary Kerr configuration.Below, we present a general-relativistic analysis ofGW150914; Fig. 2 shows the calculated waveform usingthe resulting source parameters.
III. DETECTORS
Gravitational-wave astronomy exploits multiple, widelyseparated detectors to distinguish gravitational waves fromlocal instrumental and environmental noise, to providesource sky localization, and to measure wave polarizations.The LIGO sites each operate a single Advanced LIGO
detector [33], a modified Michelson interferometer (seeFig. 3) that measures gravitational-wave strain as a differ-ence in length of its orthogonal arms. Each arm is formedby two mirrors, acting as test masses, separated byLx ¼ Ly ¼ L ¼ 4 km. A passing gravitational wave effec-tively alters the arm lengths such that the measureddifference is ΔLðtÞ ¼ δLx − δLy ¼ hðtÞL, where h is thegravitational-wave strain amplitude projected onto thedetector. This differential length variation alters the phasedifference between the two light fields returning to thebeam splitter, transmitting an optical signal proportional tothe gravitational-wave strain to the output photodetector.To achieve sufficient sensitivity to measure gravitational
waves, the detectors include several enhancements to thebasic Michelson interferometer. First, each arm contains aresonant optical cavity, formed by its two test mass mirrors,that multiplies the effect of a gravitational wave on the lightphase by a factor of 300 [48]. Second, a partially trans-missive power-recycling mirror at the input provides addi-tional resonant buildup of the laser light in the interferometeras a whole [49,50]: 20Wof laser input is increased to 700Wincident on the beam splitter, which is further increased to100 kW circulating in each arm cavity. Third, a partiallytransmissive signal-recycling mirror at the output optimizes
FIG. 2. Top: Estimated gravitational-wave strain amplitudefrom GW150914 projected onto H1. This shows the fullbandwidth of the waveforms, without the filtering used for Fig. 1.The inset images show numerical relativity models of the blackhole horizons as the black holes coalesce. Bottom: The Keplerianeffective black hole separation in units of Schwarzschild radii(RS ¼ 2GM=c2) and the effective relative velocity given by thepost-Newtonian parameter v=c ¼ ðGMπf=c3Þ1=3, where f is thegravitational-wave frequency calculated with numerical relativityand M is the total mass (value from Table I).
PRL 116, 061102 (2016) P HY S I CA L R EV I EW LE T T ER S week ending12 FEBRUARY 2016
061102-3
7PRL 116, 061102 (2016)
propagation time, the events have a combined signal-to-noise ratio (SNR) of 24 [45].Only the LIGO detectors were observing at the time of
GW150914. The Virgo detector was being upgraded,and GEO 600, though not sufficiently sensitive to detectthis event, was operating but not in observationalmode. With only two detectors the source position isprimarily determined by the relative arrival time andlocalized to an area of approximately 600 deg2 (90%credible region) [39,46].The basic features of GW150914 point to it being
produced by the coalescence of two black holes—i.e.,their orbital inspiral and merger, and subsequent final blackhole ringdown. Over 0.2 s, the signal increases in frequencyand amplitude in about 8 cycles from 35 to 150 Hz, wherethe amplitude reaches a maximum. The most plausibleexplanation for this evolution is the inspiral of two orbitingmasses, m1 and m2, due to gravitational-wave emission. Atthe lower frequencies, such evolution is characterized bythe chirp mass [11]
M ¼ ðm1m2Þ3=5
ðm1 þm2Þ1=5¼ c3
G
!5
96π−8=3f−11=3 _f
"3=5
;
where f and _f are the observed frequency and its timederivative and G and c are the gravitational constant andspeed of light. Estimating f and _f from the data in Fig. 1,we obtain a chirp mass of M≃ 30M⊙, implying that thetotal mass M ¼ m1 þm2 is ≳70M⊙ in the detector frame.This bounds the sum of the Schwarzschild radii of thebinary components to 2GM=c2 ≳ 210 km. To reach anorbital frequency of 75 Hz (half the gravitational-wavefrequency) the objects must have been very close and verycompact; equal Newtonian point masses orbiting at thisfrequency would be only ≃350 km apart. A pair ofneutron stars, while compact, would not have the requiredmass, while a black hole neutron star binary with thededuced chirp mass would have a very large total mass,and would thus merge at much lower frequency. Thisleaves black holes as the only known objects compactenough to reach an orbital frequency of 75 Hz withoutcontact. Furthermore, the decay of the waveform after itpeaks is consistent with the damped oscillations of a blackhole relaxing to a final stationary Kerr configuration.Below, we present a general-relativistic analysis ofGW150914; Fig. 2 shows the calculated waveform usingthe resulting source parameters.
III. DETECTORS
Gravitational-wave astronomy exploits multiple, widelyseparated detectors to distinguish gravitational waves fromlocal instrumental and environmental noise, to providesource sky localization, and to measure wave polarizations.The LIGO sites each operate a single Advanced LIGO
detector [33], a modified Michelson interferometer (seeFig. 3) that measures gravitational-wave strain as a differ-ence in length of its orthogonal arms. Each arm is formedby two mirrors, acting as test masses, separated byLx ¼ Ly ¼ L ¼ 4 km. A passing gravitational wave effec-tively alters the arm lengths such that the measureddifference is ΔLðtÞ ¼ δLx − δLy ¼ hðtÞL, where h is thegravitational-wave strain amplitude projected onto thedetector. This differential length variation alters the phasedifference between the two light fields returning to thebeam splitter, transmitting an optical signal proportional tothe gravitational-wave strain to the output photodetector.To achieve sufficient sensitivity to measure gravitational
waves, the detectors include several enhancements to thebasic Michelson interferometer. First, each arm contains aresonant optical cavity, formed by its two test mass mirrors,that multiplies the effect of a gravitational wave on the lightphase by a factor of 300 [48]. Second, a partially trans-missive power-recycling mirror at the input provides addi-tional resonant buildup of the laser light in the interferometeras a whole [49,50]: 20Wof laser input is increased to 700Wincident on the beam splitter, which is further increased to100 kW circulating in each arm cavity. Third, a partiallytransmissive signal-recycling mirror at the output optimizes
FIG. 2. Top: Estimated gravitational-wave strain amplitudefrom GW150914 projected onto H1. This shows the fullbandwidth of the waveforms, without the filtering used for Fig. 1.The inset images show numerical relativity models of the blackhole horizons as the black holes coalesce. Bottom: The Keplerianeffective black hole separation in units of Schwarzschild radii(RS ¼ 2GM=c2) and the effective relative velocity given by thepost-Newtonian parameter v=c ¼ ðGMπf=c3Þ1=3, where f is thegravitational-wave frequency calculated with numerical relativityand M is the total mass (value from Table I).
PRL 116, 061102 (2016) P HY S I CA L R EV I EW LE T T ER S week ending12 FEBRUARY 2016
061102-3
7PRL 116, 061102 (2016)
propagation time, the events have a combined signal-to-noise ratio (SNR) of 24 [45].Only the LIGO detectors were observing at the time of
GW150914. The Virgo detector was being upgraded,and GEO 600, though not sufficiently sensitive to detectthis event, was operating but not in observationalmode. With only two detectors the source position isprimarily determined by the relative arrival time andlocalized to an area of approximately 600 deg2 (90%credible region) [39,46].The basic features of GW150914 point to it being
produced by the coalescence of two black holes—i.e.,their orbital inspiral and merger, and subsequent final blackhole ringdown. Over 0.2 s, the signal increases in frequencyand amplitude in about 8 cycles from 35 to 150 Hz, wherethe amplitude reaches a maximum. The most plausibleexplanation for this evolution is the inspiral of two orbitingmasses, m1 and m2, due to gravitational-wave emission. Atthe lower frequencies, such evolution is characterized bythe chirp mass [11]
M ¼ ðm1m2Þ3=5
ðm1 þm2Þ1=5¼ c3
G
!5
96π−8=3f−11=3 _f
"3=5
;
where f and _f are the observed frequency and its timederivative and G and c are the gravitational constant andspeed of light. Estimating f and _f from the data in Fig. 1,we obtain a chirp mass of M≃ 30M⊙, implying that thetotal mass M ¼ m1 þm2 is ≳70M⊙ in the detector frame.This bounds the sum of the Schwarzschild radii of thebinary components to 2GM=c2 ≳ 210 km. To reach anorbital frequency of 75 Hz (half the gravitational-wavefrequency) the objects must have been very close and verycompact; equal Newtonian point masses orbiting at thisfrequency would be only ≃350 km apart. A pair ofneutron stars, while compact, would not have the requiredmass, while a black hole neutron star binary with thededuced chirp mass would have a very large total mass,and would thus merge at much lower frequency. Thisleaves black holes as the only known objects compactenough to reach an orbital frequency of 75 Hz withoutcontact. Furthermore, the decay of the waveform after itpeaks is consistent with the damped oscillations of a blackhole relaxing to a final stationary Kerr configuration.Below, we present a general-relativistic analysis ofGW150914; Fig. 2 shows the calculated waveform usingthe resulting source parameters.
III. DETECTORS
Gravitational-wave astronomy exploits multiple, widelyseparated detectors to distinguish gravitational waves fromlocal instrumental and environmental noise, to providesource sky localization, and to measure wave polarizations.The LIGO sites each operate a single Advanced LIGO
detector [33], a modified Michelson interferometer (seeFig. 3) that measures gravitational-wave strain as a differ-ence in length of its orthogonal arms. Each arm is formedby two mirrors, acting as test masses, separated byLx ¼ Ly ¼ L ¼ 4 km. A passing gravitational wave effec-tively alters the arm lengths such that the measureddifference is ΔLðtÞ ¼ δLx − δLy ¼ hðtÞL, where h is thegravitational-wave strain amplitude projected onto thedetector. This differential length variation alters the phasedifference between the two light fields returning to thebeam splitter, transmitting an optical signal proportional tothe gravitational-wave strain to the output photodetector.To achieve sufficient sensitivity to measure gravitational
waves, the detectors include several enhancements to thebasic Michelson interferometer. First, each arm contains aresonant optical cavity, formed by its two test mass mirrors,that multiplies the effect of a gravitational wave on the lightphase by a factor of 300 [48]. Second, a partially trans-missive power-recycling mirror at the input provides addi-tional resonant buildup of the laser light in the interferometeras a whole [49,50]: 20Wof laser input is increased to 700Wincident on the beam splitter, which is further increased to100 kW circulating in each arm cavity. Third, a partiallytransmissive signal-recycling mirror at the output optimizes
FIG. 2. Top: Estimated gravitational-wave strain amplitudefrom GW150914 projected onto H1. This shows the fullbandwidth of the waveforms, without the filtering used for Fig. 1.The inset images show numerical relativity models of the blackhole horizons as the black holes coalesce. Bottom: The Keplerianeffective black hole separation in units of Schwarzschild radii(RS ¼ 2GM=c2) and the effective relative velocity given by thepost-Newtonian parameter v=c ¼ ðGMπf=c3Þ1=3, where f is thegravitational-wave frequency calculated with numerical relativityand M is the total mass (value from Table I).
PRL 116, 061102 (2016) P HY S I CA L R EV I EW LE T T ER S week ending12 FEBRUARY 2016
061102-3
7
Post-Newtoniantheory
Numericalrelativity
BH perturbationsQNM
PRL 116, 061102 (2016)
Post-Newtonian theory‣ Perturbative expansion of relativistic effects
‣1 PN
‣More and more difficulties appear as we go to higher orders
Blanchet-Damour-Iyer formalism
8
⇣vc
⌘2
9
Blanchet-Damour-Iyer formalism
figure: www.virgo-gw.eu
Post-Minkowskian
Post-Newtonian
9
Blanchet-Damour-Iyer formalism
figure: www.virgo-gw.eu
Post-Minkowskian
Post-Newtonian
10
Once the equations of motions and the flux is known at n-PN:
10
Once the equations of motions and the flux is known at n-PN:
dEn
dt= Fn �n = fn(x)
10
Once the equations of motions and the flux is known at n-PN:
dEn
dt= Fn �n = fn(x)
11
0PN 0.5PN 1PN 1.5PN 2PN 2.5PN 3PN 3.5PNPN order
10�1
100
101
|��|
GW150914GW151226GW151226+GW150914
FIG. 7. The 90% credible upper bounds on deviations in the PNcoefficients, from GW150914 and GW151226. Also shown arejoint upper bounds from the two detections; the main contributoris GW151226, which had many more inspiral cycles in band thanGW150914. At 1 PN order and higher the joint bounds are slightlylooser than the ones from GW151226 alone; this is due to the largeoffsets in the posteriors for GW150914.
[41]. For convenience we list them again: (i) {d j0, . . . ,d j7}6
and {d j5l ,d j6l} for the PN coefficients (where the last twomultiply a term of the form f g log f ), (ii) intermediate-regimeparameters {d b2,d b3}, and (iii) merger-ringdown parameters{d a2,d a3,d a4}.7
In our analyses we let each one of the d pi in turn varyfreely while all others are fixed to their general relativity val-ues, d p j = 0 for j 6= i. These tests model general relativ-ity violations that would occur predominantly at a particu-lar PN order (or in the case of the intermediate and merger-ringdown parameters, a specific power of frequency in the rel-evant regime), although together they can capture deviationsthat are measurably present at more than one order.8
Given more than one detection of BBH mergers, posteriordistributions for the d pi can be combined to yield strongerconstraints. In Fig. 6 we show the posteriors from GW150914,generated with final instrumental calibration, and GW151226by themselves, as well as joint posteriors from the two eventstogether. We do not present similar results for the candidateLVT151012 since it is not as confident a detection as the oth-ers; furthermore, its smaller detection SNR means that its con-tribution to the overall posteriors is insignificant.
6 This includes a 0.5PN testing parameter d j1; since j1 is identically zero ingeneral relativity, we let d j1 be an absolute rather than a relative deviation.
7 We do not consider parameters that are degenerate with the reference timeor the reference phase, nor the late-inspiral parameters d si (for which theuncertainty on the calibration can be almost as large as the measurementuncertainty).
8 In [41], for completeness we had also shown results from analyses wherethe parameters in each of the regimes (i)-(iii) are allowed to vary simulta-neously; however, these tests return wide and uninformative posteriors.
For GW150914, the testing parameters for the PN coeffi-cients, d ji and d jil , showed moderately significant (2–2.5s )deviations from their general relativity values of zero [41]. Bycontrast, the posteriors of GW151226 tend to be centered onthe general relativity value. As a result, the offsets of the com-bined posteriors are smaller. Moreover, the joint posteriorsare considerably tighter, with a 1-s spread as small as 0.07for deviations in the 1.5PN parameter j3, which encapsulatesthe leading-order effects of the dynamical self-interaction ofspacetime geometry (the “tail” effect) [137] as well as spin-orbit interaction [66, 138, 139].
In Fig. 7, we show the 90% credible upper bounds onthe magnitude of the fractional deviations in PN coefficients,|d ji|, which are affected by both the offsets and widths ofthe posterior density functions for the d ji. We show boundsfor GW150914 and GW151226 individually, as well as thejoint upper bounds resulting from the combined posterior den-sity functions of the two events. Not surprisingly, the qualityof the joint bounds is mainly due to GW151226, because ofthe larger number of inspiral cycles in the detectors’ sensitivefrequency band. Note how at high PN order the combinedbounds are slightly looser than the ones from GW151226alone; this is because of the large offsets in the posteriors fromGW150914.
Next we consider the intermediate-regime coefficients d bi,which pertain to the transition between inspiral and merger–ringdown. For GW151226, this stage is well inside the sensi-tive part of the detectors’ frequency band. Returning to Fig. 6,we see that the measurements for GW151226 are of compa-rable quality to GW150914, and the combined posteriors im-prove on the ones from either detection by itself.
Last, we look at the merger-ringdown parameters d ai. ForGW150914, this regime corresponded to frequencies of f 2[130,300] Hz, while for GW151226 it occurred at f & 400 Hz.As expected, the posteriors from GW151226 are not very in-formative for these parameters, and the combined posteriorsare essentially determined by those of GW150914.
In summary, GW151226 makes its most important contri-bution to the combined posteriors in the PN inspiral regime,where both offsets and statistical uncertainties have signif-icantly decreased over the ones from GW150914, in somecases to the ⇠ 10% level.
An inspiral-merger-ringdown consistency test as performedon GW150914 in [41] is not meaningful for GW151226, sincevery little of the signal is observed in the post-merger phase.Likewise, the SNR of GW151226 is too low to allow for ananalysis of residuals after subtraction of the most probablewaveform. Finally, in [41], GW150914 was used to place alower bound on the graviton Compton wavelength of 1013 km.Combining information from the two signals does not signif-icantly improve on this; an updated bound must await furtherobservations.
VI. BINARY BLACK HOLE MERGER RATES
The observations reported here enable us to constrain therate of BBH coalescences in the local Universe more precisely
LIGO Scientific and Virgo collaboration arxiv:1606.04856
10
FIG. 6. Posterior density distributions and 90% credible intervals for relative deviations d pi in the PN parameters pi, as well as intermediateparameters bi and merger-ringdown parameters ai. The top panel is for GW150914 by itself and the middle one for GW151226 by itself,while the bottom panel shows combined posteriors from GW150914 and GW151226. While the posteriors for deviations in PN coefficientsfrom GW150914 show large offsets, the ones from GW151226 are well-centered on zero as well as being more tight, causing the combinedposteriors to similarly improve over those of GW150914 alone. For deviations in the bi, the combined posteriors improve over those of eitherevent individually. For the ai, the joint posteriors are mostly set by the posteriors from GW150914, whose merger-ringdown occurred atfrequencies where the detectors are the most sensitive.
up to 3.5PN. Since the source of GW151226 merged at⇠ 450 Hz, the signal provides the opportunity to probe thePN inspiral with many more waveform cycles, albeit at rel-atively low SNR. Especially in this regime, it allows us totighten further our bounds on violations of general relativity.
As in [41], to analyze GW151226 we start from the IMR-Phenom waveform model of [35–37] which is capable of de-scribing inspiral, merger, and ringdown, and partly accountsfor spin precession. The phase of this waveform is charac-terized by phenomenological coefficients {pi}, which includePN coefficients as well as coefficients describing merger andringdown. The latter were obtained by calibrating against nu-
merical waveforms and tend to multiply specific powers off , and they characterize the gravitational-wave amplitude andphase in different stages of the coalescence process. We thenallow for possible departures from general relativity, param-eterized by a set of testing coefficients d pi, which take theform of fractional deviations in the pi [135, 136]. Thus, wereplace pi ! (1+d pi) pi and let one or more of the d pi varyfreely in addition to the source parameters that also appearin pure general relativity waveforms, using the general rel-ativity expressions in terms of masses and spins for the pithemselves. Our testing coefficients are those in Table I of
11
The MPM algorithm
Gµ⌫(g↵� , @g↵� , @2g↵�) = 0
hµ⌫ ⌘p�ggµ⌫ � ⌘µ⌫ = Gh1µ⌫ + G2h2µ⌫ + . . .
⇤hiµ⌫ = ⇤(h1, . . . , hi�1)
@µhiµ⌫ = 0
11
n
The MPM algorithm
Gµ⌫(g↵� , @g↵� , @2g↵�) = 0
hµ⌫ ⌘p�ggµ⌫ � ⌘µ⌫ = Gh1µ⌫ + G2h2µ⌫ + . . .
Issue:
12
First issue: regularization
⇤�1⇤(x, t) =
Zd3x0⇤(x
0, t � | x� x
0 |)| x� x
0 |
⇤ ⇠r!01
rk, k � 3
⇤hiµ⌫ = ⇤(h1, . . . , hi�1)
@µhiµ⌫ = 0
n
Issue:
12
First issue: regularization
⇤�1⇤(x, t) =
Zd3x0⇤(x
0, t � | x� x
0 |)| x� x
0 |
⇤ ⇠r!01
rk, k � 3
FPB=0⇤�1
"✓r
r0
◆B
⇤
#
⇤hiµ⌫ = ⇤(h1, . . . , hi�1)
@µhiµ⌫ = 0
n
Issue:
12
First issue: regularization
⇤�1⇤(x, t) =
Zd3x0⇤(x
0, t � | x� x
0 |)| x� x
0 |
⇤ ⇠r!01
rk, k � 3
FPB=0⇤�1
"✓r
r0
◆B
⇤
#
⇤hiµ⌫ = ⇤(h1, . . . , hi�1)
@µhiµ⌫ = 0
n
FPB=0
2
4X
k��k0
gkBk
3
5 ⌘ g0
Second issue: tails
⇤hiµ⌫ = ⇤(h1, . . . , hi�1)
@µhiµ⌫ = 0
n
13
Second issue: tails
⇤hiµ⌫ = ⇤(h1, . . . , hi�1)
@µhiµ⌫ = 0
n
13
Second issue: tails
⇤hiµ⌫ = ⇤(h1, . . . , hi�1)
@µhiµ⌫ = 0
n
h2M⇥Mij
(t, r) ⇠ M
Z 1
0d⌧Mij(t� r � ⌧)Q(⌧)
13
14
⌫ =M1M2
(M1 +M2)2
x =
✓GM
tot
⌦
c
3
◆2/3
= O✓
1
c
2
◆
source image: virgo-gw.eu
14
124 Luc Blanchet
The orbital phase is defined as the angle �, oriented in the sense of the motion, between theseparation of the two bodies and the direction of the ascending node (called N in Section 9.4)within the plane of the sky. We have d�/dt = ⌦, which translates, with our notation, intod�/d⇥ = �5x3/2/⌫, from which we determine68
� = �1
⌫⇥5/8
⇢
1 +
✓
3715
8064+
55
96⌫
◆
⇥�1/4 � 3
4⇡⇥�3/8
+
✓
9275495
14450688+
284875
258048⌫ +
1855
2048⌫2
◆
⇥�1/2 +
✓
� 38645
172032+
65
2048⌫
◆
⇡⇥�5/8 ln
✓
⇥
⇥0
◆
+
831032450749357
57682522275840� 53
40⇡2 � 107
56�E +
107
448ln
✓
⇥
256
◆
+
✓
�126510089885
4161798144+
2255
2048⇡2
◆
⌫ +154565
1835008⌫2 � 1179625
1769472⌫3
�
⇥�3/4
+
✓
188516689
173408256x+
488825
516096⌫ � 141769
516096⌫2
◆
⇡⇥�7/8 +O✓
1
c8
◆�
, (317)
where ⇥0 is a constant of integration that can be fixed by the initial conditions when the wavefrequency enters the detector. Finally we want also to dispose of the important expression of thephase in terms of the frequency x. For this we get
� = �x�5/2
32⌫
⇢
1 +
✓
3715
1008+
55
12⌫
◆
x� 10⇡x3/2
+
✓
15293365
1016064+
27145
1008⌫ +
3085
144⌫2
◆
x2 +
✓
38645
1344� 65
16⌫
◆
⇡x5/2 ln
✓
x
x0
◆
+
12348611926451
18776862720� 160
3⇡2 � 1712
21�E � 856
21ln(16x)
+
✓
�15737765635
12192768+
2255
48⇡2
◆
⌫ +76055
6912⌫2 � 127825
5184⌫3
�
x3
+
✓
77096675
2032128+
378515
12096⌫ � 74045
6048⌫2
◆
⇡x7/2 +O✓
1
c8
◆�
, (318)
where x0 is another constant of integration. With the formula (318) the orbital phase is completeup to the 3.5PN order for non-spinning compact binaries. Note that the contributions of thequadrupole moments of compact objects which are induced by tidal e↵ects, are expected fromEq. (16) to come into play only at the 5PN order.
As a rough estimate of the relative importance of the various post-Newtonian terms, we givein Table 3 their contributions to the accumulated number of gravitational-wave cycles Ncycle inthe bandwidth of ground-based detectors. Note that such an estimate is only indicative, because afull treatment would require the knowledge of the detector’s power spectral density of noise, and acomplete simulation of the parameter estimation using matched filtering techniques [138, 350, 284].We define Ncycle as
Ncycle ⌘�ISCO � �seismic
⇡. (319)
The frequency of the signal at the entrance of the bandwidth is the seismic cut-o↵ frequencyfseismic of ground-based detectors; the terminal frequency is assumed for simplicity to be given
68 This procedure for computing analytically the orbital phase corresponds to what is called in the jargon the“Taylor T2 approximant”. We refer to Ref. [98] for discussions on the usefulness of defining several types ofapproximants for computing (in general numerically) the orbital phase.
Living Reviews in RelativityDOI 10.12942/lrr-2014-2
⌫ =M1M2
(M1 +M2)2
x =
✓GM
tot
⌦
c
3
◆2/3
= O✓
1
c
2
◆
source image: virgo-gw.eu
14
124 Luc Blanchet
The orbital phase is defined as the angle �, oriented in the sense of the motion, between theseparation of the two bodies and the direction of the ascending node (called N in Section 9.4)within the plane of the sky. We have d�/dt = ⌦, which translates, with our notation, intod�/d⇥ = �5x3/2/⌫, from which we determine68
� = �1
⌫⇥5/8
⇢
1 +
✓
3715
8064+
55
96⌫
◆
⇥�1/4 � 3
4⇡⇥�3/8
+
✓
9275495
14450688+
284875
258048⌫ +
1855
2048⌫2
◆
⇥�1/2 +
✓
� 38645
172032+
65
2048⌫
◆
⇡⇥�5/8 ln
✓
⇥
⇥0
◆
+
831032450749357
57682522275840� 53
40⇡2 � 107
56�E +
107
448ln
✓
⇥
256
◆
+
✓
�126510089885
4161798144+
2255
2048⇡2
◆
⌫ +154565
1835008⌫2 � 1179625
1769472⌫3
�
⇥�3/4
+
✓
188516689
173408256x+
488825
516096⌫ � 141769
516096⌫2
◆
⇡⇥�7/8 +O✓
1
c8
◆�
, (317)
where ⇥0 is a constant of integration that can be fixed by the initial conditions when the wavefrequency enters the detector. Finally we want also to dispose of the important expression of thephase in terms of the frequency x. For this we get
� = �x�5/2
32⌫
⇢
1 +
✓
3715
1008+
55
12⌫
◆
x� 10⇡x3/2
+
✓
15293365
1016064+
27145
1008⌫ +
3085
144⌫2
◆
x2 +
✓
38645
1344� 65
16⌫
◆
⇡x5/2 ln
✓
x
x0
◆
+
12348611926451
18776862720� 160
3⇡2 � 1712
21�E � 856
21ln(16x)
+
✓
�15737765635
12192768+
2255
48⇡2
◆
⌫ +76055
6912⌫2 � 127825
5184⌫3
�
x3
+
✓
77096675
2032128+
378515
12096⌫ � 74045
6048⌫2
◆
⇡x7/2 +O✓
1
c8
◆�
, (318)
where x0 is another constant of integration. With the formula (318) the orbital phase is completeup to the 3.5PN order for non-spinning compact binaries. Note that the contributions of thequadrupole moments of compact objects which are induced by tidal e↵ects, are expected fromEq. (16) to come into play only at the 5PN order.
As a rough estimate of the relative importance of the various post-Newtonian terms, we givein Table 3 their contributions to the accumulated number of gravitational-wave cycles Ncycle inthe bandwidth of ground-based detectors. Note that such an estimate is only indicative, because afull treatment would require the knowledge of the detector’s power spectral density of noise, and acomplete simulation of the parameter estimation using matched filtering techniques [138, 350, 284].We define Ncycle as
Ncycle ⌘�ISCO � �seismic
⇡. (319)
The frequency of the signal at the entrance of the bandwidth is the seismic cut-o↵ frequencyfseismic of ground-based detectors; the terminal frequency is assumed for simplicity to be given
68 This procedure for computing analytically the orbital phase corresponds to what is called in the jargon the“Taylor T2 approximant”. We refer to Ref. [98] for discussions on the usefulness of defining several types ofapproximants for computing (in general numerically) the orbital phase.
Living Reviews in RelativityDOI 10.12942/lrr-2014-2
⌫ =M1M2
(M1 +M2)2
x =
✓GM
tot
⌦
c
3
◆2/3
= O✓
1
c
2
◆
source image: virgo-gw.eu
4.5PN results: Marchand et al, 2016 (accepted to CQG)
Projects for 2nd and 3rd year
15
Projects for 2nd and 3rd year
‣ Working on the near-zone physics at 4PN
15
Projects for 2nd and 3rd year
‣ Working on the near-zone physics at 4PN
15
‣Studying the Vainshtein mechanism in some class of modified gravity theories.
Thank you