Coalescenza quadrupolo  Coalescing binary systems: main target of ground based interferometers quadrupole approach: point masses on a circular orbit + radiation reaction If the two stars have different masses reduced mass The frequency increasesThe orbital radius evolves as CHIRP

For binary systems far from coalescence the quadrupole formalism works. could these signals be detectable by LISA? Flusso sistema binario

CATACLISMIC VARIABLES semi-detached with small orbital period Primary star: White dwarf Secondary: star filling its Roche-lobe and accreting matter on the companion Remember that we are computing the radiation emitted because of the Orbital motion ONLY There exist other sources which may be interesting for LISA PSR

could these signals be detectable by LISA? Flusso sistema binario THERE IS HOPE ! PSR Cat. Var.

the quadrupole formalism assumes that  BINARY PULSAR PSR OK  PULSATING NEUTRON STARS  WHEN THE SYSTEM IS CLOSE TO COALESCENCE the condition is no longer satisfied: STRONG FIELD EFFECTS

Sistemi planetari extrasolari 1 GW’s emitted by a binary systems carry information not only on the features of the orbital motion, but also on processes that may occur inside the stars EXTRASOLAR PLANETARY SYSTEMS (Wolsczan & Frail) Discovery: 1992 (Wolsczan & Frail) Since then ~ 60 have been discovered in our neighbourhood Solar type star + one or more planets 46 with mass [ ] Juppiter mass 12 with bigger masses (brown dwarfs) PECULIAR FEATURES: smaller than that of Mercury from the Sun More than 1/3 orbit at a distance smaller than that of Mercury from the Sun Some of them have an orbital period of the order of hours (Mercury: P=88 days) Mass and e radius of the central star + mass and orbital parameters of the planets can be inferred from observations THEY ARE VERY CLOSE TO US!!! D  10 pc Could a planet be so close to the central star as to excite Its proper modes of oscillation?

-Could a planet be so close to the central star as to excite its proper modes of oscillation? - How much energy would be emitted in GWs by a system in this resonant condition with respect to the energy due to orbital motion (quadrupole formula+point particle approximation)? - for how long can a planet stay in this resonant situation? A more appropriate formalism to describe these phenomena is based on a PERTURBATIVE APPROACH: Sistemi planetari extrasolari 2 I Is an exact solution of Einstein + Hydro eqs. (TOV-equations) which describes the central sun-like star We assume that the star is perturbed by the planet which moves on a circular or eccentric orbit. This is a reasonable assumption because Mp << M*

We obtain a set of linear equations, in r and t, which couple the perturbations of the metric with the perturbations of the thermodynamical variables We perturb Einstein’sequations + Eqs. of Hydrodynamics We expand in tensor spherical harmonics and separate the equations On the right hand side of the equations there is a forcing term: the stress-energy tensor of the planet moving on a circular or elliptic orbit; the planet is assumed to be a point mass Mp << M*. The perturbed equations are solved numerically to find the GW signal As a first thing we find the frequencies of the quasi-normal modes: they are solutions of the perturbed equations, which satisfy the condition of being regular at r=0, and that behave like a pure outgoing wave at radial infinity. They belong to complex eigenfrequencies: the real part is the pulsation frequency, the imaginary part is the damping time, due to the emission of Gravitational Waves

A mode of the star can be excited if the mode frequency and the orbital frequency (circular orbit) are related by the constraint We find which non-radial mode, i.e. which quasi-normal mode, can be excited quasi-normal modes of stars The quasi-normal modes of stars are classified depending on the restoring force which is prevailing g - g - modes f – f – mode p - p - modes w w ‘pure spacetime oscillations’ We put the planet on a circular orbit at a given radius and check, by a Roche-lobe analysis, if it can stay on that orbit without being disrupted by the tidal interaction, i.e. without accreting matter from the star (and viceversa)

quasi-normal modes of stars: The quasi-normal modes of stars: are classified depending on the restoring force which is prevailing g - g - modes f – f – mode p - p - modes w w ‘pure spacetime oscillations’ How much time can a planet stay close to a resonance? A planet like the Earth can stay on an orbit such as to excite a mode g4 or higher, whithout melting or being disrupted by tidal forces A Juppiter like planet can excite the mode g10 or higher The orbital energy is a known function of R 0 (geodesic equations) Modi quasi-normali The grav. Luminosity is found by Numerical integration

A Brown Dwarf : can stay, for instance, on an orbit resonant with the mode g 4 emitting waves with an amplitude > 2x for 3 years Juppiter : g 10 mode – with amplitude > 3x for 2 years LISA V. Ferrari, M. D'Andrea, E. Berti Gravitational waves emitted by extrasolar planetary systems Int. J. Mod. Phys. D9 n.5, (2000) E. Berti,V. Ferrari Excitation of g-modes of solar type stars by an orbiting companion Phys. Rev. D63, (2001)

Can we obtain better estimates of the radiated GW for binary systems close to coalescence? Post-Newtonian formalism: The equations of motion and Einstein’s eqs are expandend in powers of V/c to compute energy flux and waveforms. In this manner the treatment of the radiation due to the orbital motion is refined Quadrupole formalism + Post-Newtonian corrections Describe with extreme accuracy the coalescence of BLACK HOLES (point masses) PN -formalism NON-ROTATING BODIES - test-particle (m 1 << m 2 ) : everything is known up to (V/c) 11 - equal masses : -orbital motion up to (V/c) 6 (3PN) beyong Newtonian acceleration GW- emission up to (V/c) 7 (3.5PN) beyond the quadrupole formula

In conclusion : For coalescing, non rotanting BLACK HOLES we know how to describe the signal up to the ISCO (Innermost Stable Circ. Orbit) 1) What happens after the ISCO is reached? 2) What do we know about GW emitted by rotating black holes? Few events per year detectable by LIGO and VIRGO for systems with 20 M  < M tot < 40 M  Much work to do : post-newtonian+perturbative: the signal must be modeled as a function of (a 2, a 2, m 1, m 2 ), and of the orbital parameters. fully non-linear numerical simulations to describe the merging (Grand-Challenge, Potsdam) + perturbative approaches for the quasi-normal mode ringing The detection of this part of the signal using these templates will allow to determine the total mass of the system Conclusioni buchi neri

Pert. Stelle di neutroni1 WHAT DO WE KNOW ABOUT THE COALESCENCE OF NEUTRON STARS? When they are far apart, the signal is correctly reproduced by the Quadrupole formalism : point masses in circolar orbit + radiation reaction When they reach distances of the order of 3-4 stellar radii the orbital part of the emitted energy can be refined by computing the post-newtonian corrections (same as for BH) At these distances, the tidal interaction may excite the quasi-normal modes of oscillation of one, or both stars This process can be studied by a perturbative approach

picch i Perturbative approach: True star + point mass We perturb Einstein’s eqs. + Hydrodynamical eqs. We solve them numerically We find that differences with respect to black holes due to the internal structure appear when v/c > 0.2 Last cycles before Coalescence! We compute the orbital evolution, the waveform and the emitted energy for different EOS’ Gualtieri, Pons, Berti, Miniutti, V.F. Phys. Rev D, 2001, 2002 P(v)= E GW / E ORB

discussione Why are we interested in effects that are so small? Our knowledge of nuclear interactions at supranuclear densities is very limited: we do not know what is the internal structure of a NS Observations allow to estimate the mass of NS’ (in some cases) but not the RADIUS : we are unable to set stringent constraints on the EOS of nuclear matter at such high densities. If we could detect a ‘clean’ GW signal coming from a NS oscillating in a quasi-normal mode, we could have direct information on its internal structure and consequently on the EOS of matter in extreme conditions of density and pressure unaccessible from experiments in a laboratory

Phase transitions from ordinary nuclear matter to quark matter, or to Kaon-Pion condensation, occurring in the inner core of NS’ at supranuclear densities, would produce a density discontinuity. A g-mode of oscillation would appear as a consequence Miniutti, Gualtieri, Pons, Berti, V.F. Non radial oscillations as a probe of density discontinuity in NS

EURO EURO - Third Generation GW Antenna I n May 1999 the funding agencies in Britain, France, Germany and Italy commissioned scientists involved in the construction and operation of interferometric gravitational wave detectors in Europe (GEO and VIRGO) to prepare a vision document to envisage the construction of a third generation interferometric gravitational wave detector in Europe on the time scale of 2010.

In conclusion: gravitational radiation can be studied by using different approaches -Quadrupole formalism -Perturbations about exact solutions -Numerical simulations in full GR 1)To study the coalescence of BH-BH binaries post-newtonian calculations have to be extended to the rotating case (already started) 3) The merging phase has to be studied through fully non linear numerical simulations 4) About the excitation of quasi-normal modes, we need to understand how the energy is distributed among them in astrophysical situations: known sources need to be studied in much more detail 2) To study the coalescence of NS-NS or NS-BH binaries, the perturbative approach has to be generalised to the case of equal masses and to rotating stars