Emanuele BertiEmanuele Berti

EB, A. Buonanno, C. M. Will,EB, A. Buonanno, C. M. Will, gr-qc/0411129+gr-qc/0504017gr-qc/0411129+gr-qc/0504017

EB, A. Buonanno, Y. Chen, EB, A. Buonanno, Y. Chen, in preparationin preparationEB, V. Cardoso, C. M. Will, EB, V. Cardoso, C. M. Will, in preparationin preparation

LISA images and animations from:LISA images and animations from:

http://lisa.jpl.nasa.govhttp://lisa.jpl.nasa.gov

Estimating spinning binary Estimating spinning binary parameters parameters

and testing alternative theories of and testing alternative theories of gravity gravity

with LISAwith LISA

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 ProgressEffect of systematic errors on parameter

estimationInspiral vs. ringdown for SMBHs

OutlineOutline

Joint ESA-NASA spaceborne GW observatorysensitive to 10-5 Hz<f<1 Hz

3 spacecraft in heliocentric trajectorytrailing 20° behind the Earth

Equilateral triangle, armlength=5x106 km

60° inclination with respect to the ecliptic

LISA orbitLISA orbit

A simple model of LISA responseA simple model of LISA response

LISA orbit - continuedLISA orbit - continued

Unbarred coords: LISA frame

Barred coords: Solar System frame

Taking into accountLISA’s orbital motion

we can estimate* distance to the

binary* angular resolution

Cutler 97

LISA noise and sourcesLISA noise and sources

* SMBH binaries: M~105-107 Msun

IMBH binaries: M~102-104 Msun

* Neutron star-IMBH binaries: MIMBH~102-104 Msun

LISA reachLISA reachfor NS-BH and SMBH binariesfor NS-BH and SMBH binaries

Luminosity distance DL

for NS-BH binariesdetected with SNR=10

Event rate is uncertain !!Gair et al. 04, Will 04

SNR for equal massBH-BH binaries

as a function of total massat DL=3 Gpc (z~0.5)

SMBH binariesSMBH binariesRelativity:

* 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 ratesare uncertain

SMBH binaries and galaxy SMBH binaries and galaxy mergersmergers

* SMBHs are present in bulges of nearly all local galaxies

* Galaxy mergers produce SMBH binariesGalaxy 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 friction2) 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)

SMBH binary event ratesSMBH 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<69 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

GW templates for GW templates for circularcircular orbits 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

Parameter estimationParameter estimation

Monte Carlo distributions for a SMBH binary – 1

Monte Carlo distributions for a SMBH binary Monte Carlo distributions for a SMBH binary - 2- 2

Bound on the Bound on the graviton Compton wavelengthgraviton Compton wavelength

We only include spin-orbit(spin-spin: Fisher matrix becomes non-invertible)

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

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

Correlations Correlations and angular resolutionand angular resolution

Maximum increaseswhen we include

spin-orbit

Insensitive to inclusion

ofspin-orbit

Bound on Brans-Dicke parameterBound on Brans-Dicke parameter

We only include spin-orbit(spin-spin: Fisher matrix becomes non-invertible)

Average errors for a SMBH binary as a Average errors for a SMBH binary as a function of Mfunction of M

Graviton Compton wavelengthGraviton Compton wavelengthWill 98 (LIGO and LISA),

Will & Yunes 04 (LISA SCG, angle-average, no spins)

Binary atDL=3 Gpc (z~0.5)

Solar system bound(Yukawa-term

deviations from Kepler’s third law):

bound

g~3x1012 km

Alternatives: Sutton & Finn 02 (binary pulsars); Cutler, Hiscock & Larson 03 (GW phase vs. orbital phase); Jones 04

(eccentricity)

Brans-Dicke parameter (SNR=10)Brans-Dicke parameter (SNR=10)Will 94, Damour & Esposito-Farése 98 (LIGO)

Scharre & Will 02, Will & Yunes 04 (LISA, angle-average, no spins)

Average errors as functionsAverage errors as functionsof cosmological redshiftof 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 (106+106) (107+107)

Percentage of Percentage of “useful” SMBH binaries“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)

(106+106) (107+107)

Parameter estimationParameter estimationand the LISA low-frequency noiseand the LISA low-frequency noiseDegradation 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

Systematic errors Systematic errors in parameter estimationin parameter estimation

Spin effects known only up to 2PN: we work at 2PNOmitting 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)

From inspiral to ringdownFrom inspiral to ringdown

Ringdown: damped exponential oscillations withcomplex (QNM) frequencies lm,n=R+iI

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=rdM

Standard lore: l=m=2 (“bar” mode) should dominate

Inspiral vs. ringdownInspiral vs. ringdown

Angular momentum:small effect

a=0.98a=0.80

a=0rd is crucial:

neednumericalrelativity

How is energy distributed with (l,m,n)??Brandt & Seidel 96: distorted BHs

Sperhake et al. 05: head-on

ConclusionsConclusionsBinary 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), ~20-100 (reduced mass) including the SO term;

additional factor ~3-5 including also the SS term

Brans-Dicke bound reduced by ~30-80 by SOMassive graviton bound reduced by ~4-5 by SO

------------------------------------------------------------------------------------

Future workFuture 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

(Very partial) reference list(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/0212218+CQG 20, S65 (2003):

Cosmology with SMBHs; EM counterparts, lensing* Hughes & Menou, astro-ph/0410148: “golden binaries”* Vecchio, PRD 70, 042001 (2004): precessing binaries

* EB, Buonanno & Will, gr-qc/0411129+0504017: parameter estimation

* Seto, PRD 66, 122001 (2002): finite armlength* Damour, Iyer & Gopakumar, PRD 70, 064028 (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/0410364: SMBH binary formation

Tidal deformations in the pre-merger phase Tidal deformations in the pre-merger phase for binary neutron starsfor 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

1) Quasiequilibrium2) Synchronized (corotating) or better, irrotational flow

3) Conformal flatness4) Barotropic EOS (usually a polytropic with =2)

Post-Newtonian diagnostic for binary Post-Newtonian diagnostic for binary neutron starsneutron stars

Mora & Will 03; EB, Iyer & Will, work in progress

Point massesk2=k3=0

Uniform densityk2=3/4,k3=3/8

“Half uniform density”k2=3/4,k3=3/8

=2 polytropek2=0.260,k3=0.106

=2 polytrope+eccentricity