1. Estimating spinning binary parameters and testing alternative theories of gravity with LISA
Emanuele Berti
EB, A. Buonanno, C. M. Will, gr-qc/ gr-qc/
EB, A. Buonanno, Y. Chen, in preparation
EB, V. Cardoso, C. M. Will, in preparation
LISA images and animations from:

2. Outline 1) Brief overview of the LISA mission and noise
2) Typical target binaries for LISA
* High mass-ratio inspirals
* Supermassive black hole (SMBH) binaries
(observational evidence and event rates)
3) Gravitational-wave templates for nonprecessing binaries
* Monte Carlo simulations for parameter estimation
* Bounds on alternative theories of gravity
(Brans-Dicke, massive-graviton theories)
4) LISA and the formation history of SMBH binaries
(effect of LISA low-frequency noise)
5) Work In Progress
Effect of systematic errors on parameter estimation
Inspiral vs. ringdown for SMBHs

3. LISA orbit Joint ESA-NASA spaceborne GW observatory
sensitive to 10-5 Hz<f<1 Hz
3 spacecraft in heliocentric trajectory
trailing 20° behind the Earth
Equilateral triangle, armlength=5x106 km
60° inclination with respect to the ecliptic

4. A simple model of LISA response

5. * distance to the binary
LISA orbit - continued
Unbarred coords:
LISA frame
Barred coords:
Solar System frame
Taking into account
LISA’s orbital motion
we can estimate
* distance to the binary
* angular resolution
Cutler 97

6. LISA noise and sources * SMBH binaries: M~105-107 Msun
IMBH binaries: M~ Msun
* Neutron star-IMBH binaries: MIMBH~ Msun

7. LISA reach for NS-BH and SMBH binaries
Luminosity distance DL
for NS-BH binaries
detected with SNR=10
Event rate is uncertain !!
Gair et al. 04, Will 04
SNR for equal mass
BH-BH binaries
as a function of total mass
at DL=3 Gpc (z~0.5)

8. Orbital dynamics following galaxy merger
SMBH binaries
Relativity:
* Inspiral PN waveforms -> High-precision tests of GR
* Merger waveforms -> Tests of nonlinear gravity
* Ringdown waveforms -> Tests of the no-hair theorem
Astrophysics:
* Cosmic history of SMBHs and IMBH/SMBH formation from high z
to the present; indirectly, hints on structure formation
* “Final parsec problem”: how does a SMBH binary become
“hard” enough for GWs to drive merger?
LISA measures M=(1+z)Msource
High SNR, good sensitivity only in a certain SMBH mass/redshift range
Orbital dynamics following galaxy merger
and event rates
are uncertain

9. SMBH binaries and galaxy mergers
* SMBHs are present in bulges of nearly all local galaxies
* Galaxy mergers produce SMBH binaries
Some circumstantial observational evidence
for “hard” SMBH binaries:
Blazar OJ 287 (optical variability with P=11.86yrs)
Radio galaxy 3C 66B (elliptical motion of radio core with P=1 yr)
SMBH binary formation (Merritt & Milosavljevic 04)
1) Galaxies merge, BHs sink to center by dynamical friction
2) Gravitational slingshot interactions with stars inside the “loss cone” (or gas accretion onto the binary) increase the binary’s binding energy
3) Gravitational radiation dominates (“final parsec problem”: need mechanism to refill the loss cone)
Some observational evidence that coalescence is the norm (Haenhelt 03)

10. SMBH binary event rates
* Haenhelt 94: rate can be high if BHs reside in DM halos
* Menou et al. 01: ubiquity of BHs in galaxies today
consistent with small occupation number at high z
Total of about 10 events/yr out to z=5
* Rhook & Wyithe 05: revise Wyithe & Loeb 03 (optimistic)
and find consistent rates of about 15 events/yr;
most events originate at z=3-4
* Sesana et al. 04: binaries can be ejected from core,
a 3 year LISA mission could resolve 35 mergers at 2<z<6
9 events/yr with one binary more massive than 105 Msun
* Enoki et al. 04: events visible
out to z=1 (M=108 Msun) or out to z=3 (M=106 Msun)
* Islam et al., Koushiappas: more optimistic rates,
different mechanism to “seed” galaxies with BHs at large z
“Average” prediction: expect ~10 events/year at 2<z<6

11. GW templates for circular orbits
Spin-orbit, 1.5PN Spin-spin, 2PN
Brans-Dicke: dipole radiation; best bounds from NS-IMBH
Massive graviton: D-dependent delay in wave propagation
Best bounds from SMBH binaries at large distance

12. Parameter estimation

13. Monte Carlo distributions for a SMBH binary – 1

14. Monte Carlo distributions for a SMBH binary - 2

15. Bound on the graviton Compton wavelength
We only include spin-orbit
(spin-spin: Fisher matrix becomes non-invertible)

16. Monte Carlo distributions for a NS-IMBH binary - 1

17. Monte Carlo distributions for a NS-IMBH binary - 2

18. Correlations and angular resolution
Insensitive to inclusion of
spin-orbit
Maximum increases
when we include
spin-orbit

19. Bound on Brans-Dicke parameter
We only include spin-orbit
(spin-spin: Fisher matrix becomes non-invertible)

20. Average errors for a SMBH binary as a function of M

21. Graviton Compton wavelength
Will 98 (LIGO and LISA),
Will & Yunes 04 (LISA SCG, angle-average, no spins)
Binary at
DL=3 Gpc (z~0.5)
Solar system bound
(Yukawa-term deviations from Kepler’s third law):
bound
lg~3x1012 km
Alternatives:
Sutton & Finn 02 (binary pulsars); Cutler, Hiscock & Larson 03 (GW phase vs. orbital phase); Jones 04 (eccentricity)

22. Brans-Dicke parameter (SNR=10)
Will 94, Damour & Esposito-Farése 98 (LIGO)
Scharre & Will 02, Will & Yunes 04
(LISA, angle-average, no spins)

23. Average errors as functions of cosmological redshift
Hughes 02: LISA only measures redshifted combinations of the intrinsic binary parameters (masses, spin) of the form
M=(1+z)Msource, J=(1+z)2Jsource
Measuring (luminosity) distance DL(z,cosmology) and assuming
cosmology is known, find z(DL) and remove degeneracy
( ) ( )

24. Percentage of “useful” SMBH binaries
How many (nonspinning) binaries can we observe with:
* Error on the chirp mass <0.1%?
* Error on the reduced mass <1%? (“Golden” binaries)
*Error on distance <10%? (~uncertainty on cosmology)
( ) ( )

25. Parameter estimation and the LISA low-frequency noise
Degradation in parameter estimation for SMBH binaries
if we can only trust the noise curve down to f=10-4 Hz (“Standard” value in BBW: f=10-5 Hz)
Results for nonspinning binaries at DL=3 Gpc

26. Systematic errors in parameter estimation
Spin effects known only up to 2PN: we work at 2PN
Omitting spin effects, the waveform is now known to 3.5PN
Blanchet, Damour, Esposito-Farése & Iyer 04
Effect on parameter estimation for LIGO-Virgo:
EB & Buonanno (unpublished)
Arun, Iyer, Sathyaprakash & Sundararajan 05
Estimates of chirp mass (reduced mass) improve by 19% (52%)
respectively from 2PN to 3.5PN, but improvement is non-monotonic
LISA: the effect of high-PN orders can be dramatic!
Number of cycles contributed by high PN orders is usually large
EB, Buonanno & Chen, work in progress:
* Study convergence of the PN series for LISA
* Truncate signal 1 day/week/month before merger?
(idea: get rid of region where PN convergence is poor)

27. From inspiral to ringdown
Ringdown: damped exponential oscillations with
complex (QNM) frequencies wlm,n=wR+iwI
The SNR for detection of each QNM depends on:
* The black hole’s angular momentum, a=J/M
* The energy radiated in the mode, Erad=erdM
Standard lore: l=m=2 (“bar” mode) should dominate

28. Inspiral vs. ringdown erd is crucial: a=0.98 a=0.80 a=0
Angular momentum:
small effect
a=0.98
a=0.80
a=0
erd is crucial:
need
numerical
relativity
How is energy distributed with (l,m,n)??
Brandt & Seidel 96: distorted BHs
Sperhake et al. 05: head-on

29. Conclusions Future work
Binary parameters in the gravitational-wave phase are
highly correlated: adding parameters dilutes available information
Errors on parameters increase when we include spins aligned/antialigned with orbital angular momentum
For SMBH binary mass measurements are degraded by factors:
~10 (chirp mass), ~ (reduced mass) including the SO term;
additional factor ~3-5 including also the SS term
Brans-Dicke bound reduced by ~30-80 by SO
Massive graviton bound reduced by ~4-5 by SO
Future work
* Spin precession could help decorrelate parameters (Vecchio)
* Eccentricity could remove degeneracies (Hellings & Moore)
* Systematics might be comparable to statistical errors for large SNR
* Inspiral vs. ringdown: multi-mode formalism for parameter estimation

30. (Very partial) reference list
* Flanagan & Hughes, PRD 57, 4535 (1998): inspiral, merger, ringdown
* Cutler, PRD 57, 7089 (1997): parameter estimation formalism
* Schutz, Nature 323, 310 (1986): binaries as standard candles
* Markovic, PRD 48, 4738 (1993): cosmology with binaries
* Hughes, MNRAS 331, 805 (2002): cosmology, inspiral & ringdown
* Holz & Hughes, astro-ph/ CQG 20, S65 (2003):
Cosmology with SMBHs; EM counterparts, lensing
* Hughes & Menou, astro-ph/ : “golden binaries”
* Vecchio, PRD 70, (2004): precessing binaries
* EB, Buonanno & Will, gr-qc/ : parameter estimation
* Seto, PRD 66, (2002): finite armlength
* Damour, Iyer & Gopakumar, PRD 70, (2004): PN with eccentricity
* Hellings & Moore, CQG 20, S181 (2003): higher harmonics
* Rogan & Bose, CQG 21, S1607 (2004): dependence on sky position
* Miller & Colbert, IJMPD 13, 1 (2004): IMBHs
* Merritt & Milosavljevic, astro-ph/ : SMBH binary formation

31. Tidal deformations in the pre-merger phase for binary neutron stars
In Newtonian theory (see eg. Kopal) rotational deformations and tidally-induced deformations in a binary (equilibrium tides) are described by the so-called apsidal constants kl. For homogeneous stars, kl=3/[4(l-1)]
The l-th apsidal constant is related to the l-th multipole moment
(eg., for a Kerr black hole k2 is the same
as for a homogeneous Newtonian star)
In Newtonian theory, to linear order,
k2 for tidal and rotational deformations is the same
State of the art simulations of the pre-merger phase
(eg. the LORENE code) assume
Quasiequilibrium
Synchronized (corotating) or better, irrotational flow
Conformal flatness
Barotropic EOS (usually a polytropic with G=2)

32. Post-Newtonian diagnostic for binary neutron stars
Mora & Will 03; EB, Iyer & Will, work in progress
Point masses
k2=k3=0
Uniform density
k2=3/4,k3=3/8
“Half uniform density”
G=2 polytrope
k2=0.260,k3=0.106
G=2 polytrope+
eccentricity