Multipole moments as a tool to infer from gravitational waves the geometry around an axisymmetric body. Thomas P. Sotiriou SISSA, International School for Advanced Studies, Via Beirut, Trieste, Italy.

Motivation small body orbiting around a much more massive one † F D Ryan, PRD 52, 5707 (1995) ‡ T P Sotiriou and T A Apostolatos, PRD 71, (2005) geometry described by the mass and mass current multipole moments gravitational wave (GW) observables central object endowed with an electromagnetic field GWs carry all information needed to infer these moments as well geodesic motion tracer of the geometry of spacetime infer the moments information about the central object † two additional families of moments, the electric and the magnetic field moments possible in principle to get valuable information about the central object’s electromagnetic field ‡

The Model A test particle orbiting around a much more massive compact object Realistic description for a binary system composed of a 10^4 M ○ and a 10^6 M ○ BH for example Detection: LIGO, VIRGO etc.up to ~ 300 M ○ LISAfrom ~ 3×10 5 M ○ to ~3×10 7 M ○  Assumptions: 1.Spacetime is stationary, axisymmetric and reflection symmetric with respect to the equatorial plane. F μν respects the reflection symmetry 2.The test particle does not affect the geometry and its motion is equatorial 3.GW energy comes mainly from the quadrupole formula (emission quadrupole related to the system as a whole); no absorption by the central object

The quantities related to the test particle are:  Main frequency  Energy per unit mass  Orbital frequency We can relate these quantities to a number of observables

Observables GW Spectrum Periastron and Orbital Precession Frequency  Number of Cycles per Logarithmic Interval of Frequency where

The Multipole Moments We have expressed all quantities of interest with respect to the metric components. On the other hand the metric can be expressed in terms of the multipole moments. It has been shown by Ernst that this metric can be fully determined by two complex functions. One can use instead two complex functions that are more directly related to the moments.

These functions can be written as power series expansions at infinity where and can be evaluated from their value on the symmetry axis The coefficients m i and q i are related to the mass moments, M i, the mass current moments, S i, the electric field moments, E i, and the magnetic field moments, H i. LOM stands for “Lower Order Multipoles”

Algorithm Express the metric functions as power series in ρ and z, with coefficients depending only on the multipole moments The observable quantities discussed depend only on the metric functions. Thus they can also be expressed as power series in ρ, with coefficients depending only on the multipole moments Use the assumed symmetry to simplify the computation. M i is zero when i is odd and S i is zero when i is even. For e/m moments two distinct cases: E i is zero for odd i and H i is zero for even i (es), or vice versa (ms) Finally, Ω=Ω(ρ), so we can express ρ, and consequently the observables, as a series of Ω, or

Power Expansion Formulas Using this algorithm we get the observable quantities as power series of u where for example The results are similar for R n and Z n both in the electric symmetric case (es) and the magnetic symmetric case (ms)

The most interesting observable is the number of cycles since it is the most accurately measured where

Conclusions Each coefficient of the power expansions includes a number of moments. At every order, one extra moment is present compared to the lower order coefficient possible to evaluate the moments Accuracy of measuring the observed quantities affects the accuracy of the moment evaluation. Ryan: LIGO is not expected to provide sufficient accuracy but LISA is very promising and we expect to be able to measure a few lower moments Lower order moments can give valuable information: Mass, angular momentum, magnetic dipole, overall charge (e.g. Kerr- Newman BH). One can also check whether the moments are interrelated as in a Kerr-Newman metric. A negative outcome can indicate either that the central object is not a black hole or that the spacetime is seriously affected by the presence of matter in the vicinity of the black hole (e.g. an accretion disk)