astrosismologie relativiste et ondes gravitationnelles

Astrosismologie relativiste et ondes gravitationnelles :ce que pourrait nous dire la danse des etoiles a neutrons

Loıc Villain

DFA, Universidad de AlicanteE-03690 Alicante, Espana

etLUTH, Observatoire Paris-Meudon

F-92195 Meudon, [email protected]

base sur des collaborations avecSilvano Bonazzola (LUTH), Pawe l Haensel (CAMK, Warszawa)

Annecy, Seminaire LAPTH, Avril 2006

Loıc Villain (DFA/LUTH) Astrosismologie et OGs Seminaire LAPTH, Annecy, Avril 2006 1 / 61

mailto:[email protected]

Outline

1 General relativity and gravitational waves

2 Neutron stars

3 Instabilities and oscillations of compact objects

4 Inertial mode instability and spectral methods

5 Summary, conclusions and perspectives


General relativity and gravitational waves

1

Gravitational waves in general relativity



1

Gravitational waves in general relativityA : brief history and main features

By. A. NAGAR



Prediction of gravitational waves (GWs)

First mention

1906, Poincare : any theory of gravitation with Lorentz invariance should lead tosome “ondes gravifiques” (but no valid theory found)

Oscillations in general relativity

1916, Einstein : “gravitationalwaves” exist in general relativity(GR) : progressive wave

1918, Einstein : “quadrupoleformula” (→ GW emissivity) forslow motions and weak gravitationalfield found in a given system ofcoordinates → coordinate wavesor physical waves ? ? ?



Theoretical proof

GWs and characteristic surface

Finzi (1949) : covariante proof of the deplacement at c of characteristicsurfaces associated to Einstein equations → quite unknown work (privatecommunication by S. Bonazzola)

GWs and energy

Pirani (1956) : “What would happen to my detector if a GW would gothrough my lab ?” → energy deposite → physical detection possible

Bondi, Schoen, Yau, Witten (and others) (1962-1982) : “What wouldhappen to an objects emitting GWs ?” → mass loss (≡ energy) until apositive value is reached → physical emission possible



GWs observation

Observational proof

Hulse & Taylor (1982) : discoveryand precise observation of a binarypulsar PSR B1913+16 → Nobelprize in 1993

First try of direct detection

Weber (1965) : first bar, no actualdetection



Hulse & Taylor : indirect observation

indirect experimental verification ofGWs existence but also very precisetest of GR : theoretical prediction

versus observational data



1

Gravitational waves in general relativityB : multipoles and emission



GWs emission and mass multipoles

Electromagnetism

at least time variations of electric dipole to emit electromagnetic waves

breaking of spherical symmetry (∼ massless spin 1 boson)

Gravitation

at least time variations of mass quadrupole to emit GWs in GR

∼ breaking of spherical and axial symmetries (∼ massless spin 2 boson)

Notes

axial symmetry with time variation of radial distribution also leads to GWs

in others relativistic theories of gravitation : scalar waves also



Current multipoles

Quadrupolar flux (l = m = 2) leading toa current quadrupole and GWs

Electromagnetism

other possibility : magnetic dipole

∼ breaking of electric flux sphericalsymmetry

Gravitation

in GR “current multipoles” lead toGWs

basic example : spherical mass wtihconstant density but non-symmetricinternal motions



Criteria for relevant emissivity

Emissivity

dE

dt∼ GN

c5s2ω6M2R4

s breaking of symmetry, M mass, R typical radius, ω typical frequency

problem : GN /c5 ∼ 3 × 10−53 I.S. → very weak emission

no GWs in lab (current quadrupole : another 1/c factor)

Astrophysical emissivity

solution : rewrite the formula with Schwarzschild radius, Rs = 2MGN /c2,typical velocity v, and ω ∼ v/R

dE

dt∼ c5

GNs2

(Rs

R

)2 (v

c

)6

with astrophysical units the 10−53 factor has become a 10+53 factor...



Astrophysical GWs

Typical amplitude in astrophysical units

h ∼ 2× 10−19

(M

M

)1/2 (1 Mpc

d

) (1 kHz

f

) (1 ms

τ

)1/2

ε1/2 ,

f frequency, d distance to the source, M solar mass, Mpc megaparsec(∼ 3× 1022 m) : Local group of galaxies, τ duration of the emission, εefficacity (= ratio between emitted and mass energies)

N.B. : detection of h and not h2 → 1/d signal but weak...

relevant sources

relativistic (v ∼ c) compact (high M/R) object with coherent internalmotions (to avoid destructive interferences)

compact binaries (black holes, neutron stars, strange stars) + isolatedcompact object : oscillating, rotating (without axial symmetry) and/oraccreting



1

Gravitational waves in general relativityC : direct detection and gravitational astronomy



Result of a GW

Recipies to detect a GW

several test masses (6= electromagnetism)

continuous (bar) or discrete (mirrors of an interferometer) mass distribution

narrow (bar) or wide (interferometers) band



GW detection in 2006

Detectors

Bars (and sphere) : numerous, more sophisticated than Weber’s ; looking forcorrelations ; drawbacks : narrow band...

Interferometers : various taking data (LIGO, TAMA, GEO), VIRGO in“commissionning”, LISA Pathfinder for 2007 ( ?) ; drawbacks : moreexpansive, more complex...



Historical parenthesis : Meudon Observatory’s detector

From left to right : Jean Thierry-Mieg, Georges Herpe, Silvano Bonazzola andMichel Chevreton (around 1965).



Worldwide network of bars and interferometers



European network



VIRGO sensibility curve

Noise

at low frequency : mainly sismic noise

at high frequency : quantum fluctuations of laser beam, thermal noise ofmirrors



All sensibility curves

Detectors

at low frequency : LISA (no sismic noise in space)

at high frequency : ground detectors (here LIGO)


Neutron stars

2

Compact astrophysical objects (material ones)


Neutron stars

2

Compact astrophysical objects (material ones)A : (pre-)history


Neutron stars

Prehistorical footsteps

Theoretical birth

1932, Chadwick : discovery of neutron

1932, Landau (following a legend) : idea of neutron stars on the same dayof the announcement ; → reality : “prediction” before the discovery(Haensel, et al.)

1934, Baade & Zwicky : very cautious proposal

With all reserve we advance the view that supernovæ represent thetransitions from ordinary stars into neutrons stars, which in their final stagesconsist of extremely closely packed neutrons

1939, Tolman, Oppenheimer et Volkoff : first structure calculation of aneutron star modelized as a relativistic Fermi gas of neutrons → modernmodels more complex : degeneracy pressure ∼ 1

3 of the resistance (stronginteraction crucial)


Neutron stars

Discovery

History

1967, Bell & Hewish : firstobservation of a pulsar (lighthousemodel Gold and Pacini)

1996, Walter et al. : first opticalobservation of an isolated neutron star

2002, Cottam et al. : observation ofgravitational redshift of a NS’satmosphere spectrum

2004, Lyne et al. : first double pulsar(PSR J0737-3039A and PSRJ0737-3039B)

2006 : more than 1500 known pulsars


Neutron stars

A parallel history : strange stars

X-ray observation of 3C58 byChandra

History

1971, Bodmer : true fundamental stateof matter = deconfined quark plasmawith possible hypercharge

1984, Witten : same idea + proposalof “strange stars”

1986, Haensel et al., Alcock et al. :first numerical models (MIT bag model)

2002, NASA : announcement ofrevolutional discoveries : 2 isolatedneutron stars said to be strange stars →nothing more than a perfect illustrationof what a scientist should not do...


Neutron stars

2

Compact astrophysical objects (material ones)B : birth and main features


Neutron stars

Birth of compact objects

Gravitational collapse of massive stars iron cores

iron core mass > Chandrasekhar mass ∼ 1.5 Solar mass

collapse (in a few ms) → density increases → electronic captures +photo-dissociations

fission of nuclei, free neutrons appear

bounce (at saturation density) and possible ejection of outer layers (fallingslower : 1 s)

initial mass of the star > 45M : no supernova, black hole formation

central remnant = warm soup (T ∼ 6.1011 K) of neutrons, protons, electronsand neutrinos


Neutron stars

Gravitational waves from the collapse

Main mechanisms

during collapse, fast variation of mass quadrupole (Q) → GWs

strong convective motions in the remnant → GWs

bounce also gives fast variation of Q and GWs

if formation of BH : Q increases and then decreases

excited BH : loss of “gravitational hair” to give a Kerr BH (damping ofquasi-normal modes in a few ms)


Neutron stars

Characteristics at birth

After collapse

(Proto)Neutron Star (PEN) with mass 1.2 to 2 M (∼ iron core mass)

T ∼ 6.1011 K : npeν matter opaque to ν but fast cooling

possible fall back of surrounding matter → possible black hole and GWs(typically : Mi > 20M → BH)

N.B. : true story depends on metallicity, angular momentum, existence of acompanion, magnetic field, etc. (stellar wind in very massive stars :Mi ∼ 60M can lead to a NS)

Birth of a neutron star (8− 10 < Mi/M < 20)

cooling through diffusion/emission of ν → ν-transparency in t < 1 minute(T < 3.1011 K)

anisotropic emission of ν → GWs

npe plasma becomes a degenerated Fermi liquide (TF ∼ 1011 K) of n and p(with ultrarelativistic e gas) at supranuclear density (ρ ∼ 10−14 g.cm−3)


Neutron stars

More exotical features

Physics in very dense matter

strong interaction → possible formation of nucleon Cooper paires → BCSsuperfluidity (for some densities since effective interaction) : TF ∼ 1011 K,Ts ∼ 1010 K

at very high densities → exotical particles and states (superfluid hyperons,deconfined quarks, etc.)

→ during collapse, possible phase-transition leading to GWs

Collapse and compact object

radius ∼ 102 km to 10 km→ compactness parameter (M/R)= 0.1 after 10 s→ relativistic object (Sun : ∼ 10−6 ; Schw. black hole : R = 2 M)

magnetic flux conservation → very high magnetic fields : 108−9 G to 1013 G

(almost) conservation of angular momentum during collapse → very highangular velocity (period P ∼ 1 ms) → electromagnetic radiation (lighthousemodel) and possible instabilities leading to GWs


Neutron stars

Structure of an old compact star (Source F. Weber)


Neutron stars

Density profile

Typical energy density profiles of a neutron and strange stars [Glendenning (1997)]

(rough) approximation : constant density (better for strange stars and moremassive ones)

N.B. : surface density of strange star 6= 0 (self-bound object due to QCD)


Instabilities and oscillations of compact objects

3

Instabilities and oscillations of rotating compactobjects



3

Instabilities and oscillationsA : link between modes and instabilities



Modes of neutron stars

Modes of isotropic stars

a spherical star without“anisotropic physics” (stress inthe rigid crust, magnetic field,etc.) → all modes with the sameazimuthal number m have thesame frequency : wlm ≡ wl

Pattern speed (“apparentvelocity”) :positive m →σ+ = w

m = dφdt > 0 → prograde

negative m →σ− = w

m = dφdt < 0 →

retrograde

no axial part for the velocity field[parity (−1)l for polar, (−1)l+1

for axial]

NSs main modes for various equationsof state (from Kokkotas et al.)



Modes of rotating stars

Rotating star

degeneracy breaking (splitting wlm 6= wl(m−1)) + modification offrequencies

wi = wr −m Ω +Clm(Ω2)

axial modes enter into the game

progrades and retrogrades modes are not affected in the same way : “sign” ofthe pattern speed may change

↔ trace of an instability



Non-axisymmetric instabilities of rotating stars

Main classes

Dynamical : hydrodynamical timescale (here millisecond)

Secular : related to some “dissipative process” → longer timescale [ex. :viscosity (τ > 103 s), angular momentum loss, etc.]

Dissipative processes triggers of secular instabilities in NSs

viscosity : conservation of angular momentum, dissipation of vorticity[physical basis : n-n diffusion (low T) or beta reactions n ↔ p + e + ν(high T)]

gravitational radiation : dissipation of angular momentum, conservation ofvorticity

Summary of the principle

viscosity : evolution from (E0, L, C0) → (E < E0, L, C < C0)

GWs : evolution from (E0, L0, C) → (E < E0, L < L0, C)→ both made possible by spontaneous symmetry breaking but competition



Secular instabilities

Some (almost) detail on the secular instabilities

viscosity driven instability tries to make the star non-axisymmetric and rigidlyrotating → GW emission

GW driven instability tries to make the star keep a given non-axisymmetricfixed shape → inner fluxes → differential rotation → damping due to viscosity

possibility to define some “canonical energies” (in the inertial frame or in therotating frame) such that for a given eigenmode, a negative value of theenergy means that the mode is unstable



CFS criterion

Chandrasekhar-Friedman-Schutz (1970-1978)

in a rotating fluid, modes that are progrades in the rotating frame butretrogrades in the inertial frame are driven to instability by any mechanismthat can evacuate angular momentum away from the fluid

Principle ∼ such a mode gives a negative contribution to the total angularmomentum of the star, but, through GWs (for instance), carries away apositive amount → total angular momentum of the star decreases → GWsemission makes the angular momentum of the mode more and more negative→ instability...

verified when wr wi < 0 ↔ wr (wr ± m Ω) < 0 with azimuthal number mand star angular velocity Ω



Direct consequences of the CFS criterion

Generic instability of rotating relativistic fluids

any rotating relativistic perfect fluid is unstable → for every mode, there is aminimal angular velocity of the star above which an instability appears : Ωl,m

for increasing values of m :

- (Ωl,m ) → (instability )

- (viscous timescale ) → damping of the instability easier

- [growing time of the instability (related to multipoles) ] → (instability )

relevance in actual relativistic stars (neutron stars) obtained through awindow of instability for each mode in the plane (Temperature, AngularVelocity)

but in actual stars, various phenomena make the game more tricky :compressibility, equation of states, magnetic field, differential rotation etc.

→ need to proceed to realistic and detailed numerical studies



f mode window of instability (l = m = 2)

f mode window of instability (l = m = 2) : angular velocity of the star in Keplervelocity units (= maximal velocity for which the star is not losing matter at the

equator). N.B. : minimal value > 90% ΩK .



3

Instabilities and oscillationsC : inertial modes instability



Inertial modes inertiels (in Newtonian perfect fluids)

Main features

inertial modes ↔ generated byCoriolis force : only in rotatingfluids

bidimensional : Rossby modes(cf. atmospheric physics)

mainly fluxes → very smalldensity perturbations → almostno mass quadrupole...

CFS instability

but in 1998, Andersson and Friedman & Morsink : r modes verify CFScriterion ∀Ω → large window of instability

explanation : wr = 2 m Ωl(l+1) → wi = −Ω m (l+2)(l−1)

l+1 → wi wr < 0



R-mode window of instability (l = m = 2)

R-mode window of instability (l = m = 2) : much larger than for all othermodes...



Possible implications of the instability

Baby NSs

explanation of a limit for young NSs period ? (Andersson et al., Lindblom etal.)

a way to identify strange stars among compact astrophysical objects ?(Andersson et al.)

Oldest NSs

NSs in low mass X-ray binaries

continuous transfert of angularmomentum (and matter) byaccretion → the reason why LowMass X-ray Binaries have verysimilar frequencies ? (Bildsten,Wagoner)



Relevance of the r-mode instability ?

Open questions

influence of frame-dragging ? (∼ Lense-Thirring) → ∼ differential rotationΩ = Ω(r, θ)

exact characteristic of the modes in fast rotating relativistic stars ?

growing of the instability in young and hot NSs (PNSs) ?

influence of the crust ?of a magnetic fields ?

coupling to other modes ?

exotical composition :→ viscosity ? (hyperons superfluidity ?)→ fast cooling → time spent in the instability window ?

r-modes in superfluid stars ?

non-linear saturation amplitude ?

in binary systems, influence of accretion ?


Inertial mode instability and spectral methods

4

Inertial mode instability and spectral methodscf. Villain, Bonazzola & Haensel (2005)



Villain, Bonazzola & Haensel 2005

Goals

improve the study of inertial modes (typical time millisecond) in GR (mostof studies done with Newtonian gravitation and post-newtonian corrections)

investigate the influence of realistic equations of state (not only polytropsP ∼ nγ or barotrops P = P (n) as in most of past studies)

take into account stratification (inhomogeneous composition) and its effectthrough the existence of various microscopical timescales



Methodology

Spectral numerical simulation

time evolution (linear here) through resolving of relativistic Euler equations(coming from ∇T = 0 for a perfect fluid)

tridimensional spectral code based on spherical coordinates → highnumerical stability → long simulations → very precise spectra

initial version of the code (analytical tests in simplest Newtonian andpost-Newtonian situations) Villain & Bonazzola (2002) upgraded to take intoaccount stratification



Numerical and technical parenthesis

Usual solving of partial differential equations

functions known on a discrete lattive, derivative calculated by finitedifferences

Main advantage : fast implementation

Main drawback : no way to use spherical coordinates (singular operators) →badly defined surface of stars

Spectral solving

generalisation of Fourier transforms (cf. spherical harmonics for Poissonequation) → functions known by their coefficients in a reciprocal space,derivatives calculated by linear algebra ↔ (semi-analytic method)

Main advantage : very high precision (no numerical viscosity, less numericalpoints needed), spectral basis chosen to fit with the physical situationgeometry, etc.

Main drawback : implementation takes more time (elementary operations toimplement one by one), need to change everything for new topology, etc.



Spectral methods in general relativity

Numerical relativity and relativistic astrophysics at Meudon Observatory (LUTH)

since the early 90ies DARC later LUTH : Silvano Bonazzola and Jean-AlainMarck, later E. Gourgoulhon, J. Novak, P. Grandclement, ...

relativistic astrophysic : compact objects and gravitational waves sources

up to recently : stationnary situations (in dynamical situations no easytemporal spectral basis)

hydrodynamics : finite differences for time coordinate



Hypothesis

Main features and timescales

npe degenerated matter (T TF ) without superfluidity

perfect fluids (viscous time >> 1 ms) in comotion (strong interactioneffect)

electric neutrality (plasma frequency ∼ 10−20 s)

breaking of beta equilibrium (n ↔ p + e) due to millisecond oscillations(relaxation time through weak interaction : between some hours and somemonths)

influence of the modes on the background metric neglected : Cowlingapproximation, GW signal from multipoles



Chosen problem

Questions

no relaxation → frozen composition for any lump of matter (nulleLagrangian perturbation of proton fraction x)

equations of state = effective barotrops at equilibrium Peq (nb, xeq(nb))(baryonic number density and proton fraction) but non-barotropic EOSs fordynamical situation

coupling between composition g modes and r-modes (l = m = 2)



Illustration of numerical results

Power spectra of density (m = 2) perturbation time evolution for various angularvelocity. Splitting of the two components m = ± 2 can be seen for g-modes ;

r-mode increases with Ω



Energy fluxes (without stratification)



Energy fluxes (with stratification)



Summary, conclusions and perspectives



NSs oscillations and high frequencies GWs ?

NSs are natural laboratories to study nuclear matter at very high density andgravitation in the strong field regim, but current observations giveinformations only on global quantities (M,R,M/R) or on what happens atthe surface

to better understand the outside (magnetosphere, bursts, etc.) knowledge ofinner structure needed : modes are witnesses of what happens inside

moreover NSs are relativistic objects → GWs are direct witnesses of the innerstructure

during gravitational collapses, many mechanisms can emit GWs, among themoscillations

but not all modes emit GWs in a relevant way : instabilities are probablymore interesting

CFS criterion is a very useful tool to identify instabilities, but microphysicsmakes the game more tricky

r-modes are very promising, but probably other instabilities are still to bediscovered...

plenty of reasons to study the dynamics of compact objects and to improveGW detectors in the high frequency band



Works in progress and perspectives

coupling to Maxwell equations → MHD → study of soft gamma repeaters(cf. 17/12/2004 magnetar)

non-linear study → saturation amplitude of the instability and consequentlyof the expected gravitational wave signal

proto-neutron stars : warm fast rotating objects [collaboration with Alicante,cf. Villain et al. (2004)]

prediction of the gravitational signal by simulations with less approximations(collaboration with Meudon, Bonazzola, Gourgoulhon, Grandclement, Novak)



References

Villain, Bonazzola & Haensel, Inertial modes in stratified rotating neutronstars : An evolutionary description, Phys.Rev. D71 (2005) 083001

Villain & Bonazzola, Inertial modes in slowly rotating stars : an evolutionarydescription, Phys.Rev. D66 (2002) 123001

Bonazzola, Gourgoulhon & Marck, Spectral methods in general relativisticastrophysics, J.Comput.Appl.Math. 109 (1999) 433

Kokkotas & Schmidt, Quasi-Normal Modes of Stars and Black Holes, LivingRev. Relativity 2, (1999), 2. [Online article] :http ://www.livingreviews.org/lrr-1999-2

Stergioulas, Rotating Stars in Relativity, Living Rev. Relativity 6, (2003), 3.[Online article] : http ://www.livingreviews.org/lrr-2003-3/


astrosismologie relativiste et ondes gravitationnelles

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