The role of pulsars’ The role of pulsars’ timing in GW timing in GW detection detection ANDREA POSSENTI ANDREA POSSENTI INAF INAF – – Osservatorio Astronomico di Cagliari Osservatorio Astronomico di Cagliari VESF SCHOOL on VESF SCHOOL on GRAVITATIONAL WAVES GRAVITATIONAL WAVES V Edition V Edition 26 26 - - 30 Jul 2010, SCfA 30 Jul 2010, SCfA – – Sesto Pusteria (Italy) Sesto Pusteria (Italy) 26 JULY 2010 26 JULY 2010
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The role of pulsars’ The role of pulsars’ timing in GW timing in GW
detectiondetection
ANDREA POSSENTIANDREA POSSENTIINAF INAF –– Osservatorio Astronomico di CagliariOsservatorio Astronomico di Cagliari
VESF SCHOOL on VESF SCHOOL on GRAVITATIONAL WAVESGRAVITATIONAL WAVES
2.2. Double neutron star merger rateDouble neutron star merger rate
3.3. Pulsar Timing conceptsPulsar Timing concepts
4. Gravitational Waves emission 4. Gravitational Waves emission constraints from binary pulsarsconstraints from binary pulsars
5. Gravitational Waves detection using pulsar 5. Gravitational Waves detection using pulsar timing arrays: the idea and the sensitivity timing arrays: the idea and the sensitivity
6. Gravitational Waves detection using pulsar 6. Gravitational Waves detection using pulsar timing arrays: ongoing experimentstiming arrays: ongoing experiments
•• Manchester & Taylor 1977 “Manchester & Taylor 1977 “PulsarsPulsars””•• Lyne Lyne & Smith 2005 “& Smith 2005 “Pulsar AstronomyPulsar Astronomy””•• Lorimer Lorimer & Kramer 2005& Kramer 2005 ““ Handbook of Pulsar Handbook of Pulsar AstronomyAstronomy””•• AA.VV. 2009 “AA.VV. 2009 “Physics of relativistic objects in compact binaries: fromPhysics of relativistic objects in compact binaries: from
birth to coalescencebirth to coalescence”, Springer”, Springer
•• Stairs 2003: Stairs 2003: Testing General RelativityTesting General Relativitywith with pulsar timingpulsar timing•• WillWill , 2006: , 2006: The confrontation btw General Relativity and experimentThe confrontation btw General Relativity and experiment•• Lorimer 2008: Lorimer 2008: Binary and millisecond pulsarsBinary and millisecond pulsars
VESF SCHOOL on VESF SCHOOL on GRAVITATIONAL WAVESGRAVITATIONAL WAVES
ELECTROMAGNETIC OBSERVATIONS OF ELECTROMAGNETIC OBSERVATIONS OF PULSARS AND BINARIESPULSARS AND BINARIES
VESF SCHOOL on VESF SCHOOL on GRAVITATIONAL WAVESGRAVITATIONAL WAVES
V EditionV Edition2626--30 July 2010 30 July 2010 –– SCfA SCfA –– Sesto Pusteria (Italy)Sesto Pusteria (Italy)
What is a Pulsar?WhatWhat is is a Pulsar?a Pulsar?AA PULSAR PULSAR is is a a rapidlyrapidly rotatingrotating andand highlyhighly
magnetizedmagnetizedneutronneutron starstar, , emittingemitting a a pulsedpulsedradio radio signalsignal as aas aconsequence of consequence of aa lightlight --house effecthouse effect
@K
ram
er
The rotating magnetized NS in vacuumThe rotating magnetized NS in vacuumThe rotating magnetized NS in vacuum
Assuming that the rotational energy lossAssuming that the rotational energy loss
LL sdsd = d/dt (E= d/dt (Erotrot ) = d/dt (I) = d/dt (IΩΩ22/2) = I /2) = I ΩΩ ΩΩmatches the emitted powermatches the emitted power(derived (derived
from the basic electrodynamics formula):from the basic electrodynamics formula):
LL dipole dipole = [ 2/3c= [ 2/3c33] | ] | µµ | | 22one can inferone can infer……
··
····
µµ = ½ B= ½ Bpp RR33
is the magnetic is the magnetic momentmomentR = NS radiusR = NS radiusBBpp = polar magn. field= polar magn. field
Derived parameters: age & magnetic fieldDerived Derived parameters: age & magnetic fieldparameters: age & magnetic field
• Actual age of pulsar is function of initial periodand braking index n=(n=(νν νν )/ )/ νν 22 (assumed constant)
• For PP00 << P<< P, n = n = 33 , have “characteristic age”“characteristic age”
• If true age known,one can compute initial period
• From braking equation, one can derive B0 at NS equator with R = NS radius. Valueat at polepole is 2B2B00
• Typically assumedR=10R=10 km, km, I=10I=104545 gm cmgm cm22, , n=3n=3
(from Manchester & Taylor)
·· ·
Radio pulsar basic parametersRadio pulsar basic parametersRadio pulsar basic parameters
Radio pulsars are powered byRadio pulsars are powered byrotational energyrotational energy
SpinSpin--down age:down age:
SpinSpin--down power:down power:
Surface magnetic field: Surface magnetic field:
P P ……and allows one to place a pulsar on the basic and allows one to place a pulsar on the basic PP vs vs [ or[ or P vs BP vs Bsurfsurf ] diagram…] diagram…
BBsurfsurf = 1.0 · 10= 1.0 · 101212 [ P P[ P P--1515 ]]½½ GaussGauss
ττc c = 1.6 · 10= 1.6 · 1077 P / PP / P--1515 yryr
The observation of the spin periodThe observation of the spin periodPP and of its derivativeand of its derivativeallows one to give an estimate of various physical quantities:allows one to give an estimate of various physical quantities:
P P
The Bs vs P diagram
The BThe Bss vs P vs P diagramdiagram
A pulsar is put on it A pulsar is put on it once both P and dP/dt once both P and dP/dt
are measured, from are measured, from which which
BBs s = 3.2= 3.2··101019 19 [ P P ]½ G[ P P ]½ G..
How to explain How to explain this group of this group of
pulsars ?pulsars ?
A dichotomy in the
population
A dichotomy A dichotomy in the in the
populationpopulation
ATNF Pulsar Catalogue
Young pulsar Line
P = 1.557 P = 1.557 msms
Extreme physical conditions Extreme physical conditions occur in millisecond pulsarsoccur in millisecond pulsars
VVtangtang = 0.13 = 0.13 cc !!!!
tangential velocity
Discovery of the first millisecond pulsar B1937+21 (1982)
DiscoveryDiscovery of the of the first millisecondfirst millisecond pulsar pulsar B1937+21 (1982)B1937+21 (1982) [Backer et al. 1982][Backer et al. 1982]
First promise of putting First promise of putting constraints to the Equation of constraints to the Equation of
State for nuclear matter !State for nuclear matter !
Short spin periods: 1.39 ms < P < 200 ms (conventional)
Much higher fraction of binarityMuch higher fraction of binarity : f: f binbin> 70%> 70%
Slower mean 3D velocity: v Slower mean 3D velocity: v ~ 130 km/s ~ 130 km/s [Toscano et al 1999][Toscano et al 1999]
Half of the objects moving towards the Galactic plane Half of the objects moving towards the Galactic plane [Toscano et al 1999][Toscano et al 1999]
A tendency to wider duty cycles: W A tendency to wider duty cycles: W ~ 0.1~ 0.1--0.4 P 0.4 P [Kramer et al 1998][Kramer et al 1998]
Similar mean spectral index: Similar mean spectral index: αα ~ ~ -- 1.7 1.7 [Kramer et al. 1998] [Kramer et al. 1998]
Slightly less average radio luminosity Slightly less average radio luminosity [Kramer et al. 1998 ][Kramer et al. 1998 ]
Higher degree of polarization Higher degree of polarization [Xilouris et al. 1998][Xilouris et al. 1998]
The MSP vs ordinary pulsar features The MSP vs ordinary pulsar features The MSP vs ordinary pulsar features
Recycling scenarioRecycling scenario: Millisecond pulsars are old neutron stars spun up by accretion of matter and angular momentum from a companion star in a
multiple system [Bisnovati[Bisnovati--Kogan & Kronberg 1974, Alpar et al. 1982]Kogan & Kronberg 1974, Alpar et al. 1982]
The MSP formation paradigm The MSP formation paradigm The MSP formation paradigm
A died pulsar could be spun up and rejuvenated by an A died pulsar could be spun up and rejuvenated by an evolving binary companionevolving binary companion
1000 yr
deat
hlin
e
Hubble time
A died pulsar could be spun up and rejuvenated by an A died pulsar could be spun up and rejuvenated by an evolving binary companionevolving binary companion
Dan
a B
erry
@ N
AS
AD
ana
Ber
ry @
NA
SA
A newly born fast spinning pulsar
1000 yr
Hubble time
deat
hlin
e
A recycled A recycled pulsar spins down again due to pulsar spins down again due to magnetodipole brakingmagnetodipole braking
Bin
ary
Evo
lutio
nB
inar
y E
volu
tion
Bin
ary
Evo
lutio
n
[ Sta
irs 2
004
] [ S
tairs
200
4 ]
the current sample !the current samplethe current sample !!More & more pulsars:More & more pulsars:More & more pulsars:
Until 1997: ~ 750~ 750
Now in the Atnf Catalog: ~ ~ 18801880
~~ 2020 Extragalactic (LMC/SMC)
~~ 150150 Binary (140140somehow recycled)
~~ 8080 Young (age<10000 yr)
~~ 2626 Vela-like (i.e. very young)
33 Radio emitting “magnetars”
99 Double Neutron star binaries
11 Double pulsar
140140in 2626 Globular Clusters
++ Rrats, Intermittent PSRs, …
TOTALTOTAL known sampleknown sample
GC search > 70
Drift scan search > 30
GBTGBT discoveriesdiscoveries
Parkes PM = 725
Parkes SWI+SWII = 69+25
Parkes PH = 18
Parkes PA > 10
Parkes HTRU > 40
Total using multibeam >887Parkes GC search > 34
ParkesParkesdiscoveriesdiscoveries
Galactic Plane search > 50
AreciboArecibo discoveriesdiscoveries
2.2.Double neutron star Double neutron star
merger ratemerger rate
ELECTROMAGNETIC OBSERVATIONS OF ELECTROMAGNETIC OBSERVATIONS OF PULSARS AND BINARIESPULSARS AND BINARIES
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Coalescence rate R (empirical approach)Coalescence rate Coalescence rate R R (empirical approach)(empirical approach)
Lifetime of a system = current age + merging time of a pulsar of a system
Correction factor : correction for pulsar beaming
Lifetime of a systemNumber of sources × correction factorR R =
[ fro
m C
hung
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Kim
][ f
rom
Chu
ngle
e K
im ]
Properties of pulsars in DNSsProperties of pulsars in Properties of pulsars in DNSsDNSs
B1913+16 59.03 8.6x10-18 7.8 0.617 2.8 (1.39)
B1534+12 37.90 2.4x10 -18 10.0 0.274 2.7 (1.35)
Galactic disk Galactic disk pulsars merging in less than an Hubble timepulsars merging in less than an Hubble time
Properties of pulsars in DNSs (cont)Properties of pulsars in Properties of pulsars in DNSsDNSs (cont)(cont)
B1913+16 110 320 4º.23
B1534+12 250 2500 1º.75
τc (Myr) τmerg (Myr) dω/dt (deg/yr)
J0737-3039 200 86 16º.90
Galactic disk Galactic disk pulsars merging in less than an Hubble timepulsars merging in less than an Hubble time
J1756-2251 400 15900 2°.59
J1906+0746J1906+0746 0.11 320 7°.57
Coalescence rate R (empirical approach)Coalescence rate Coalescence rate R R (empirical approach)(empirical approach)
Lifetime of a system = current age + merging time of a pulsar of a system
Correction factor : correction for pulsar beaming
Number of sources : number of pulsars in coalescing DNSs in the galaxy of a given type
Lifetime of a systemNumber of sources × correction factorR R =
How many pulsars How many pulsars ““ similarsimilar ”” to each of the known to each of the known DNSsDNSsexist in our exist in our Galaxy? It needs estimating the SCALE factorGalaxy? It needs estimating the SCALE factor
[ fro
m C
hung
lee
Kim
][ f
rom
Chu
ngle
e K
im ]
Results for Double Neutron StarsResults for Double Neutron StarsResults for Double Neutron Stars
[ Chunglee Kim 2008 ][ Chunglee Kim 2008 ]
The Galactic coalescence rate R for Double Neutron Star BinariesThe The GalacticGalactic coalescence rate coalescence rate R R
for Double Neutron Star Binariesfor Double Neutron Star Binaries
118+174-79 27+80
-23
RR (current)(current) (Myr(Myr --11) R (previous) (Myr) R (previous) (Myr--11) ) Coalescence Coalescence raterate
ForFor the reference model (at 95% CL):the reference model (at 95% CL):
Detection rate of Double Neutron Star inspiralsDetection rate of Detection rate of Double Neutron StarDouble Neutron Star inspiralsinspirals
Rdet (advanced) =
Rdet (initial) =
TheThe most probable most probable inspiralinspiral detection rates for detection rates for LIGO/VIRGOLIGO/VIRGO
~ 1 event per 8 yr (95% CL, most optimistic)
~ 600 events per yr (95% CL, most optimistic)
Rates may be significantly higher if a substancial populationRates may be significantly higher if a substancial populationof highly eccentric binary systems exists. It could escape of highly eccentric binary systems exists. It could escape detection due the short lifetime before GW inspiraldetection due the short lifetime before GW inspiral
Many uncertainties in the modelMany uncertainties in the model
One year of observation with LISA of the Double Pulsar wouldOne year of observation with LISA of the Double Pulsar woulddetect the continuous emission at freq=0.2 mHz with a S/N detect the continuous emission at freq=0.2 mHz with a S/N ≈≈ 22
[ Kalogera 2004 ][ Kalogera 2004 ]
[ Lor
imer
200
8 ]
[ Lor
imer
200
8 ]
Results for NS-WD binariesResults for NSResults for NS--WD binariesWD binaries
[ Chu
ngle
e K
im 2
008
][ C
hung
lee
Kim
200
8 ]
Only 3 known coalescing systems known to dateOnly 3 known coalescing systems known to date
The Galactic coalescence rate R for Neutron Star-White Dwarf Binaries
The The GalacticGalactic coalescence rate coalescence rate R R for Neutron Starfor Neutron Star--White Dwarf BinariesWhite Dwarf Binaries
ForFor the reference model (at the reference model (at 68%68% CL), not corrected for beaming:CL), not corrected for beaming:
J0751+J1757+J1141J0751+J1757+J1141[ Chunglee Kim et al 2004 ][ Chunglee Kim et al 2004 ]
Coalescing frequencies are in the LISA band: Coalescing frequencies are in the LISA band: but expected event rate not very encouragingbut expected event rate not very encouraging
[ Chunglee Kim et al 2004 ][ Chunglee Kim et al 2004 ]
Combination of various observational constraintsCombination of various observational constraintsresulting from binary population synthesis coderesulting from binary population synthesis codeare very promising are very promising
[ O’Shaughnessy et al 2008 ][ O’Shaughnessy et al 2008 ]
The presented approach is an empirical oneThe presented approach is an empirical oneThe alternate option is to run extended Monte CarloThe alternate option is to run extended Monte Carlosimulations of the most likely evolutionary scenario starting simulations of the most likely evolutionary scenario starting from a population of primordial binariesfrom a population of primordial binariesThe uncertainties in the assumption for the initial state of theThe uncertainties in the assumption for the initial state of thebinaries make the range of the predicted merging rates binaries make the range of the predicted merging rates larger than with the empirical approach larger than with the empirical approach
[ Dewey & Cordes 1987 ][ Dewey & Cordes 1987 ][ Lipunov et al 1996 ][ Lipunov et al 1996 ][ Belczynski, Kalogera, Bulik 2002 ] [ Belczynski, Kalogera, Bulik 2002 ] [ Belczynski et al 2008 ][ Belczynski et al 2008 ]
3.3.Pulsar Timing Pulsar Timing
ConceptsConcepts
ELECTROMAGNETIC OBSERVATIONS OF ELECTROMAGNETIC OBSERVATIONS OF PULSARS AND BINARIESPULSARS AND BINARIES
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Timing idea: observationsTiming idea: observationsTiming idea: observationsPerforming Performing repeated observations of the Times of Arrivalrepeated observations of the Times of Arrival
(ToAs)(ToAs) at the telescope of the pulsations from at the telescope of the pulsations from a given pulsara given pulsar
++searching the ToAs for systematic trendssearching the ToAs for systematic trendson many different on many different
timescales, from minutes to decadestimescales, from minutes to decades
Timing of a radio pulsar: operations for getting a ToA
Timing Timing of a radio pulsar: operations of a radio pulsar: operations for getting a ToA for getting a ToA
@ Lorimer
The dedispersionTheThe dedispersiondedispersion
Single pulse profilevs
integrated profile
Single pulse profileSingle pulse profilevsvs
integrated profileintegrated profile
Determination of the TOPOCENTRIC Times of Arrival (ToAs)
Determination of the TOPOCENTRIC Determination of the TOPOCENTRIC TimesTimes of Arrival (ToAs)of Arrival (ToAs)
ToA uncertainty (ToA uncertainty (ωω = width of the pulse, P=pulsar period)= width of the pulse, P=pulsar period)::
Timing idea: modelingTiming idea: modelingTiming idea: modelingif a physical model adequately describes the systematic trends iif a physical model adequately describes the systematic trends in n the ToAs, it is applied with the smallest number of parametersthe ToAs, it is applied with the smallest number of parameters
when a model finally describes accurately the observed ToAs, when a model finally describes accurately the observed ToAs, the values of the the values of the model’s parameters shed light onto the physical model’s parameters shed light onto the physical
propertiesproperties of the pulsar and/or of its environment of the pulsar and/or of its environment
otherwiseotherwise
if a physical model is not adequate, if a physical model is not adequate, it is extended (adding parameters) or rejected in favour of it is extended (adding parameters) or rejected in favour of
another model another model
The TOPOCENTRIC ToAs must be corrected, calculating them to to infiniteinfinite frequency at Solar System frequency at Solar System BarycentreBarycentre (SSB) thus
obtaining the BARYCENTERED ToAs: the time scaleis (Tempo2) the Barycentric Coordinate Time (TCB), i.e. the proper time of an observer at SSB were the gravity field of Sun and Planets absent
ttSSBSSB : Calculated BARYCENTERED ToA at INFINITE frequencyttobs obs : Observed TOPOCENTRIC ToAttclk clk : Observatory clock correction to TAI (= UTC + leap sec), via GPSD/fD/f22 : Dispersion term ∆∆RR : Roemer delay (propagation delay) to SSB (need SS ephemeris, e.g. DE405) ∆∆SS : Shapiro delay in Solar-System ∆∆EE : Einstein delay at Earth
Timing key quantity: the residualsTiming key quantity: the residualsTiming key quantity: the residualsGiven the full set of parameters (aGiven the full set of parameters (a11, a, a22, …, a, …, ann) of a model, the i) of a model, the i--th th residual rresidual rii is the difference in rotational phase is the difference in rotational phase ΦΦ (with (with --0.5<r0.5<rii<+0.5)<+0.5)
between the observed phase of arrival of the ibetween the observed phase of arrival of the i--th pulse and the th pulse and the phase of arrival of that pulse as predicted by the modelphase of arrival of that pulse as predicted by the model
rr ii = = ΦΦobserved observed ((ii --th pulseth pulse) ) –– ΦΦmodel(amodel(a11, a, a22, , ……, a, ann))((ii --th pulseth pulse))
In an iterative procedure, In an iterative procedure, one leastone least--square fitssquare fits on suitable on suitable subsets of the possible parameters (asubsets of the possible parameters (a11, a, a22, …, a, …, ann) of the model, ) of the model, in the aimin the aim to remove apparent trends and thus eventually to remove apparent trends and thus eventually toto
approach rapproach rii << 1<< 1
Thanks to the leastThanks to the least--square fit procedure, one square fit procedure, one can iterativelly solve for can iterativelly solve for
The quality of the timing solution is usually given in term The quality of the timing solution is usually given in term
of the root mean square of the root mean square rmsrms of the residuals:of the residuals:
the smaller rms is, the smaller physical effects the smaller rms is, the smaller physical effects can be measuredcan be measured
Timing model: isolated pulsarsTiming model: isolated pulsarsTiming model: isolated pulsarsFrom timing of an isolated pulsar over a long enough
time span, one can in principle get
RA & DEC : Celestian coordinatesPMRA & PMDEC : Proper Motionπ : Trigonometric Parallax (i.e. Distance) DM : Accurate Dispersion MeasureDM1 : Time Derivative of Dispersion Measure P0: Rotational PeriodP1: Time derivative of P0P2: Second time derivative of P0P3: Third time derivative of P0…
Since 1974 pulsars in binary systems are known
Since 1974 pulsars in binary Since 1974 pulsars in binary systems are knownsystems are known
The PULSARCENTRIC ToAs (i.e. ToAs expressed in pulsar proper time) must be corrected, calculating them
atat the Pulsar Systemthe Pulsar SystemBarycenterBarycenter (PSB)
Correcting ToAs to the binary barycenter Correcting ToAs to the binary barycenter Correcting ToAs to the binary barycenter
ttPSRPSR--BARYBARY : Time at pulsar system barycenterTTpsr psr : Time in pulsar proper time (measured as at pulsar surface)∆∆R,bR,b : Roemer delay (propagation delay) from pulsar to PSB ∆∆S,bS,b : Shapiro delay in pulsar binary ∆∆E,bE,b : Einstein delay in pulsar binary∆∆AA : Aberration delay due to pulsar rotation
tPSR-BARY = Tpsr + ∆R,b + ∆E,b + ∆S,b + ∆A
Those terms contain various parameters of the binary systemThose terms contain various parameters of the binary systemand thus a least-square fit to the residuals of a model
including those parameters can allow to measure them…
( ) ( )( )2
3
2
32 sinsin4),(
cp
c
orb
pcp
mm
im
P
ia
Gmmf
+== π
Mass function:
forfor ii = 90= 90o o MinimumMinimum companion masscompanion mass
forfor ii = 60= 60o o MedianMedian companion masscompanion mass
For most binaries, 5 For most binaries, 5 kepleriankeplerian parameters are measured parameters are measured and they are (well) enough to satisfactorily describe all and they are (well) enough to satisfactorily describe all
the datathe dataPb : Orbital periodx = ap sin i : Projected semi-major axisω : Longitude of periastrone : Eccentricity of orbitT0 : Time of periastron
PulsarPulsar periods periods cancan sometimes besometimes bemeasured measured with unrivalled with unrivalled precisionprecision
e.ge.g. on Jan 16, 1999, PSR J0437. on Jan 16, 1999, PSR J0437--4715 had a period of 4715 had a period of
16 significant digits!
5.757451831072007± 0.000000000000008 ms
Millisecond pulsars (MSPs) as clocks
Millisecond pulsars (MSPs) Millisecond pulsars (MSPs) as clocks as clocks
Atomic clocks vs pulsar timing Atomic clocks vs pulsar timing Atomic clocks vs pulsar timing
Unfortunately only a subsample of the recycled pulsars is able to achieve such a rotational stability
The majority of the ordinary pu lsars undergo timing irregularitiThe majority of the ordinary pu lsars undergo timing irregulariti eses
[ Har
tnet
t & L
uite
n 2
010
][ H
artn
ett &
Lui
ten
201
0 ]
ATNF Pulsar Catalogue
Young pulsar
Line
log
Recycled Recycled pulsars: pulsars: ~ 140~ 140known objects; known objects;
NSNSageage > 10> 1088--101099 yryr
The most rapidly rotating The most rapidly rotating are known as are known as millisecond millisecond
pulsarspulsars
High precision pulsar timing: which targets?High precision pulsar timing: which targets?High precision pulsar timing: which targets?OrdinaryOrdinary pulsars: pulsars:
~ 1650~ 1650known objects; known objects; NSNSageage< few 10< few 1077 yryr
4.4. Gravitational Waves emission Gravitational Waves emission constraints from binary pulsarsconstraints from binary pulsars
5.5. Gravitational Waves detection using pulsar Gravitational Waves detection using pulsar timing arrays: the idea and the sensitivitytiming arrays: the idea and the sensitivity
6.6. Gravitational Waves detection using pulsar Gravitational Waves detection using pulsar timing arrays: ongoing experimentstiming arrays: ongoing experiments
Binary systems: the classic lawsBinary systems: the classic lawsBinary systems: the classic laws
……for some binary pulsars, the for some binary pulsars, the accuracy of the ToA data is so high accuracy of the ToA data is so high that that -- by using by using only the keplerianonly the keplerian
descriptiondescription -- one can obtain one can obtain no acceptable timing solutionno acceptable timing solution!!
Additional physics is needed! Additional physics is needed!
Tests of Gravity theories in the weakTests of Gravity theories in the weak--field limitfield limit
The Parametrized Post Newtonian formalism is well tailored for The Parametrized Post Newtonian formalism is well tailored for describing the outcomes of these tests describing the outcomes of these tests [e.g Will 2006 ][e.g Will 2006 ]
102
10−≅==cR
GM
E
E
Earth
Earth
rest
gravEarthε 6
210−≅==
cR
GM
E
E
Sun
Sun
rest
gravSunε
WeakWeak in which sensein which sense??
In term of the compactness parameter In term of the compactness parameter εε
All the Solar System tests fall in this category… since the All the Solar System tests fall in this category… since the experiment about the light deflection by Sun experiment about the light deflection by Sun [Eddington 1919][Eddington 1919]
and the observation of the Mercury advance of perihelionand the observation of the Mercury advance of perihelion
So far, So far, GR GR has passed all these tests has passed all these tests with with full marks and cum laudefull marks and cum laude
But But is GR still the bestis GR still the bestavailable theory for describing Nature available theory for describing Nature also under also under extremeextremephysical conditionsphysical conditions? ?
This is NOT an ACADEMIC question:This is NOT an ACADEMIC question:
e.g. e.g. extreme conditionsextreme conditionsare certainly those at which any long are certainly those at which any long sought sought unified modelunified model for interactions appliesfor interactions applies[ e.g. Antoniadis 2005 ][ e.g. Antoniadis 2005 ]
There exist alternative metric gravity theories (e.g. a subclassThere exist alternative metric gravity theories (e.g. a subclassamong the tensoramong the tensor--scalar theories) which would pass ALL Solar scalar theories) which would pass ALL Solar System (weakSystem (weak--field limit) tests, but would be violated as soon as field limit) tests, but would be violated as soon as
extreme conditions (strongextreme conditions (strong--field limit) are reached field limit) are reached [Damour & Esposito[Damour & Esposito--Farese 1996]Farese 1996]
Moreover, is enough to Moreover, is enough to test alternative theories only test alternative theories only in thein theweakweak--field limitfield limit ? ?
Not on Earth or on Solar System…Not on Earth or on Solar System…but in the Cosmo...very interesting targets are but in the Cosmo...very interesting targets are
“relativistic objects in compact binaries”“relativistic objects in compact binaries”
Where Where to find a laboratory for testing GR in to find a laboratory for testing GR in extreme conditionsextreme conditions??
““ compact” binariescompact” binariesGravitational radiation inspiral affects binary evolution within an Hubble time
NSs and BHs are NSs and BHs are “relativistic” objects“relativistic” objects
0.2εNS ≅==2cR
GM
E
E
NS
NS
rest
grav 0.5εBH ≅==2cR
GM
E
E
BH
BH
rest
grav
Tests of Gravity theories in the strongTests of Gravity theories in the strong--field limitfield limitStrongStrong in which sensein which sense??
In term of the compactness parameter In term of the compactness parameter εε the source should satisfythe source should satisfy
Astrophysical contextsAstrophysical contexts::•• during late stages of coalescence of a binary hosting relativistduring late stages of coalescence of a binary hosting relativistic ic object(s), the orbital velocity approaches object(s), the orbital velocity approaches cc and the orbital separation and the orbital separation approaches the size of the star(s), whence physical processes ocapproaches the size of the star(s), whence physical processes occur in cur in strongstrong-- field conditions:field conditions: according to the according to the BH massBH mass, they are wonderful , they are wonderful targets for targets for LIGO, VIRGOLIGO, VIRGO , for the Pulsar Timing Arrays (, for the Pulsar Timing Arrays (PTAsPTAs) and, ) and, in future, in future, LISALISA
•• emission processes occurring in relativistic objects close to themission processes occurring in relativistic objects close to the event e event horizon: e.g. spectral and timing features in the electromagnetihorizon: e.g. spectral and timing features in the electromagnetic c emission (often Xemission (often X-- ray) from the neighbourhood of the last stable orbit ray) from the neighbourhood of the last stable orbit of accretion disks surrounding NS or BH hosted in a binary systeof accretion disks surrounding NS or BH hosted in a binary system: m: some hints from some hints from XMMXMM-- NewtonNewton and and RossiRossi-- XTEXTE but targets for future but targets for future high energy (most Xhigh energy (most X-- ray) observatories: ray) observatories: LOFT, XEUSLOFT, XEUS… …
•• compact relativistic binary pulsars: compact relativistic binary pulsars: targets for targets for currentcurrent TIMING TIMING observations in the observations in the RADIORADIO bandband
Tests of Gravity theories in the strongTests of Gravity theories in the strong--field limitfield limit
Tests of Gravity theories in the strongTests of Gravity theories in the strong--field limitfield limitWait a minute! Wait a minute! Orbits of known binary pulsars are never Orbits of known binary pulsars are never
entering the strongentering the strong--field limit...field limit...
But in most alternative theories of gravity (e.g. in the tensorBut in most alternative theories of gravity (e.g. in the tensor--scalar ones) the scalar ones) the orbital motion and the gravitational radiation orbital motion and the gravitational radiation damping depend on the gravitational binding energydamping depend on the gravitational binding energy (i.e. self (i.e. self
gravity, e.g. gravity, e.g. εεNSNS≈≈0.2, 0.2, εεBHBH≈≈0.5) of the involved bodies 0.5) of the involved bodies [e.g. Esposito[e.g. Esposito--Farese 2005, Will 2006]Farese 2005, Will 2006]
352
1010 −−
−
−− −≅==
ca
GM
E
E
psrbin
psrbin
rest
gravpsrbinε 35 1010 −−− −≅
c
V psrbin
If enough accuracy in the measurements is provided, If enough accuracy in the measurements is provided, significant significant effectseffectsare expected to be detectable even are expected to be detectable even in the weakin the weak--field limit field limit
Tests of Gravity theories in the strongTests of Gravity theories in the strong--field limitfield limit
A suitable and successful framework for testing and constrainingA suitable and successful framework for testing and constraininga very large class of gravity theories is that of the a very large class of gravity theories is that of the PostPost--Keplerian Keplerian
22ndnd →→ In ANY specific gravity theory (picked in a large range of In ANY specific gravity theory (picked in a large range of metric theories), and for negligible spin contributions, the metric theories), and for negligible spin contributions, the PK PK
parametersparameterscan be written only as a can be written only as a function of the masses of the function of the masses of the two stars and of the keplerian parameterstwo stars and of the keplerian parametersof the binary system of the binary system
[Damour & Deruelle 1986][Damour & Deruelle 1986]
11stst →→ PK parameters are operationally definedPK parameters are operationally defined: :
i.e. they are phenomenological quantities, which there is a i.e. they are phenomenological quantities, which there is a prescription to measure forprescription to measure for
What do we learn What do we learn when observing also thewhen observing also thePostPost--kepleriankeplerian parameters ?parameters ?
PeriastronPeriastron precessionprecession
Time Time dilationdilation & & gravitationalgravitational redshiftredshift
ShapiroShapiro delaydelay ((amplitudeamplitude))
ShapiroShapiro delaydelay ((shapeshape))
OrbitalOrbital periodperiod decaydecay
……where…where…
• e e orbitalorbital eccentricityeccentricity
•• PPbb orbitalorbital periodperiod
•• x x projected semimajor axisprojected semimajor axis
•• mmpp pulsar masspulsar mass
•• mmcc companioncompanion star massstar mass
•• MM == mmpp + + mmcc total system total system lagrangianlagrangian massmass
ObservingObserving the the valuesvalues of of onlyonly 2 2 PK PK parametersparameters
One can One can measuremeasure the pulsar and the pulsar and companion star companion star massesmasses withwithunrivalledunrivalled precisionprecision
Once more Once more thanthan 2 2 relativisticrelativistic PK PK parametersparametersare known, one derives the masses ofare known, one derives the masses ofthe 2 bodies and hence predicts the further the 2 bodies and hence predicts the further PK par on the basis of a given Gravity PK par on the basis of a given Gravity TheoryTheory
AA test test forfor GravityGravity TheoriesTheories
ω&
The pulsar and companion star masses are The pulsar and companion star masses are unconstrainedunconstrained
Mass Function Mass Function constraintconstraint
NOT ALLOWED
( ) ( )( )2
3
2
32 sinsin4),(
cp
c
orb
pcp
mm
im
P
ia
Gmmf
+=π=
sin sin i = 1i = 1
One PKOne PK--parameter: constraining massparameter: constraining mass
γ
Two PK parameters: mass determined Two PK parameters: mass determined withinwithin a theorya theory
Three PK parameters: in Three PK parameters: in correct theory lines meetcorrect theory lines meet!!
γ
But But not in a wrongnot in a wrong theory !!!theory !!!
γ
Now the catalog contains 8/9Double Neutron Star BinariesNow theNow the catalogcatalog contains contains 8/98/9Double Neutron Star BinariesDouble Neutron Star Binaries
Now the catalog contains 8/9Double Neutron Star BinariesNow theNow the catalogcatalog contains contains 8/98/9Double Neutron Star BinariesDouble Neutron Star Binaries
Most precise NS mass determination to date:Most precise NS mass determination to date:1.4414(2) M1.4414(2) Msunsun + 1.3867(2) M+ 1.3867(2) Msun sun [ Weisberg & Taylor 2004][ Weisberg & Taylor 2004]
[ Wei
sber
g 20
07 ]
[ Wei
sber
g 20
07 ]
The (in?)direct proof of GW existence:PSR B1913+16
The (in?)direct proof of The (in?)direct proof of GW existence:GW existence:PSRPSR B1913+16B1913+16
GR provides an accurate GR provides an accurate description of the system as description of the system as orbiting POINT MASSES: orbiting POINT MASSES: i.e. NS structure does not i.e. NS structure does not
affect orbital motionaffect orbital motion
NOBEL PRIZENOBEL PRIZE19931993
TaylorTaylor & & HulseHulse
The measurementsThe measurementsof Russell Hulseof Russell Hulseandand of Joe Taylorof Joe Taylor……The prediction of theThe prediction of theEinstein’s equationsEinstein’s equations……
Pulsar Pulsar + Pulsar+ PulsarSpin period = 22.7 ms + 2.77 sSpin period = 22.7 ms + 2.77 sOrbital period = 2.5 hrsOrbital period = 2.5 hrsEccentricity = 0.09Eccentricity = 0.09
PSR J0737-3039A/B (orb params)PSR PSR J0737J0737--3039A/B 3039A/B (orb params)(orb params)
Discovered on 2003 Discovered on 2003 [ Burgay et al 2003, Lyne et al 2004 ][ Burgay et al 2003, Lyne et al 2004 ]
Measured 5 PK pars: Measured 5 PK pars: ωω γγ PPb b s rs r····+ mass ratio + mass ratio R R
++ geodetic precession rate geodetic precession rate ΩΩprec prec [ Breton et al 2008][ Breton et al 2008]
The The origin of the doubleorigin of the double pulsarpulsar
The double pulsar PSR J0737-3039A/BThe doubleThe double pulsar PSR J0737pulsar PSR J0737--3039A/B3039A/B
MassMass--mass diagram for J0737mass diagram for J0737--3039A&B3039A&B
at J
an 2
004
at J
an 2
004
Mass function A
MassMass--mass diagram for J0737mass diagram for J0737--3039A&B3039A&B
at J
an 2
004
at J
an 2
004
Mass function B
MassMass--mass diagram for J0737mass diagram for J0737--3039A&B3039A&B
at J
an 2
004
at J
an 2
004
Mass ratio
MassMass--mass diagram for J0737mass diagram for J0737--3039A&B3039A&B
at J
an 2
004
at J
an 2
004
Periastronadvance
MassMass--mass diagram for J0737mass diagram for J0737--3039A&B3039A&B
at J
an 2
004
at J
an 2
004
Grav. Redshift+ 2nd order Doppler
MassMass--mass diagram for J0737mass diagram for J0737--3039A&B3039A&B
at J
an 2
004
at J
an 2
004
Shapiro s
MassMass--mass diagram for J0737mass diagram for J0737--3039A&B3039A&B
at J
an 2
004
at J
an 2
004
MassMass--mass diagram for J0737mass diagram for J0737--3039A&B3039A&B
Shapiro rat J
an 2
004
at J
an 2
004
The very last mass-mass diagram for J0737-3039A/B
The very last massThe very last mass--massmass diagram for diagram for J0737J0737--3039A/B3039A/B
jul 2008jul 2008
[Bre
ton
et a
l [B
reto
n et
al
2008
]20
08]
5 independent tests of GR!MB=1.249(1)M
MA=1.338(1)M
%05.0exp
obs
≈s
s
Radiative GR tests for J0737-3039 system may reach 0.01% level in a decade [ Deller et al 2009 ]
Radiative GR tests for J0737Radiative GR tests for J0737--3039 system may 3039 system may reach 0.01% level in a decade reach 0.01% level in a decade [ Deller et al 2009 ][ Deller et al 2009 ]
Current radiative GR tests for Current radiative GR tests for J0737J0737--30393039system system are at are at ~~1%1% level level [ Kramer et al 2006 ][ Kramer et al 2006 ]
Prospects for timing are Prospects for timing are excellent:excellent:
•• precision precision ωω ≈≈ timetime 1.5 1.5 PPb b
•• precision precision γγ ≈≈ time time 1.5 1.5 PPb b 1.31.3
Radiative predictions of GR tested at Radiative predictions of GR tested at better than better than ~~6%6% levellevel [ Bhat et al 2008][ Bhat et al 2008]
The case of PSR J1141-6541The case of PSR J1141The case of PSR J1141--65416541
Masses of the two components are similarMasses of the two components are similarMM NSNS = ( 1.27 = ( 1.27 ±± 0.01 ) M0.01 ) Msunsun
MM WDWD = ( 1.02 = ( 1.02 ±± 0.01 ) M0.01 ) Msunsun……but the radii are certainly very different, leading to a signifbut the radii are certainly very different, leading to a significant difference inicant difference in
the degree of compactnessthe degree of compactnessεε (i.e. in the(i.e. in theselfself--gravity gravity ) of the two bodies:) of the two bodies:
2.02
≅==cR
GM
E
E
NS
NS
rest
gravNSε 4
210−≅==
cR
GM
E
E
WD
WD
rest
gravWDε
TensorTensor--scalar scalar theories predicts the emission of atheories predicts the emission of alarge large amount of DIPOLAR scalar wavesamount of DIPOLAR scalar waves(as opposed to the (as opposed to the
dominant QUADRUPOLAR radiation predicted by GRdominant QUADRUPOLAR radiation predicted by GR) ) inin such asuch avery asymmetric systemvery asymmetric system
By the 2012 By the 2012 δδrel rel PPbb ≈≈ 2% at which galactic & kin 2% at which galactic & kin corrections become dominant, but likely a 1% corrections become dominant, but likely a 1%
determination will be achievable for PSR J1141determination will be achievable for PSR J1141--6541 6541
Pulsars as GW detectorsPulsars as GW detectorsPulsars as GW detectorsThe PulsarThe Pulsar--Earth path can be used as the armEarth path can be used as the armof of
a a huge cosmichuge cosmicgravitational wavegravitational wave detectordetector
PerturbationPerturbation in spacein space--timetime can becan bedetecteddetectedin in timing residuals over a timing residuals over a suitable long observation time spansuitable long observation time span
Sensitivity (rule of thumb):Sensitivity Sensitivity (rule of thumb):(rule of thumb):
T
σ(f)~h TOA
c
Radio Radio PulsarPulsar
wherehc(f) is the dimensionless strain at freq f
σTOA is the rms uncertainty in Time of Arrival
T is the duration of the dataspan
wherewhere
hhcc(f) (f) is the dimensionless strain at freq is the dimensionless strain at freq ff
σσTOA TOA is the rms uncertainty in Time of Arrival is the rms uncertainty in Time of Arrival
T T is the duration of the dataspan is the duration of the dataspan
Source Source of GWsof GWs
EarthEarth
An instructive applicationAn instructive applicationAn instructive application
[ Jen
et e
t al 2
004
][ J
enet
et a
l 200
4 ]
[ Jen
et e
t al 2
004
]
The radio galaxy 3C66 (at The radio galaxy 3C66 (at z z = 0.02) was claimed to harbour a = 0.02) was claimed to harbour a double SMBH with a total mass of 5.4 double SMBH with a total mass of 5.4 ·· 10101010 MM sunsun and an and an
orbital period of order orbital period of order ~~yr yr [ Sudou et al 2003][ Sudou et al 2003]
Timing residuals from PSR B1855+09 Timing residuals from PSR B1855+09 exclude exclude such a massive double BH such a massive double BH at 95 c.l.at 95 c.l.
The GW background from Massive BH binaries
The GW background from Massive The GW background from Massive BH binariesBH binaries
The current paradigm is that The current paradigm is that [e.g. Ferrarese & Merrit 2000][e.g. Ferrarese & Merrit 2000]
•• mergers aremergers arean an essentialessentialpart in galaxy formation and evolutionpart in galaxy formation and evolution•• nucleinuclei of most (all?) large galaxies of most (all?) large galaxies host Massive BH(s)host Massive BH(s)(MBH:(MBH:i.e. mass larger than 10i.e. mass larger than 1066 MM sunsun))
There should There should be plenty of SMBH binariesbe plenty of SMBH binariesin the early universe, in the early universe, sinking to the their galaxy center (due to dynamical friction?) sinking to the their galaxy center (due to dynamical friction?)
The The frequency of GWfrequency of GW emitted by these systems is typicallyemitted by these systems is typically
When reaching orbital separation When reaching orbital separation less than about 1 pc, GW emission less than about 1 pc, GW emission become the dominantbecome the dominantterm in energy loss, making the MBH binary term in energy loss, making the MBH binary
to shrink faster and faster to shrink faster and faster
2/32/1
9 pc01.010 nHz3
−
a
M
Mf~
sun
[ Ses
ana,
Vec
chio
et a
l 200
8][ S
esan
a,V
ecch
io e
t al 2
008]
The GW background from Massive BH binaries
The GW background from Massive The GW background from Massive BH binariesBH binaries
The expected amplitude spectrum form the ensemble of these The expected amplitude spectrum form the ensemble of these MBH binaries is MBH binaries is [ e.g. Phinney 2001; Jaffe & Backer 2003][ e.g. Phinney 2001; Jaffe & Backer 2003]
with a strain amplitude with a strain amplitude theoretically expected in the range theoretically expected in the range
[ e.g. Jaffe & Backer 2003, [ e.g. Jaffe & Backer 2003, Sesana,Vecchio et al 2008]Sesana,Vecchio et al 2008]
hhcc ≈ ≈ 1010--1616 →→ 1010--1515
3/2; =− αα(f)~fhc8 nHz
100 nHz
z~1around frequencyaround frequency ffGWBGWB = 1 = 1 yryr --11
Max contribution from BH Max contribution from BH
binaries at binaries at z z ≈ ≈ 11
The GW background from Massive BH binaries
The GW background from Massive The GW background from Massive BH binariesBH binaries
An alternative representation of the results/limitsAn alternative representation of the results/limits (better for (better for cosmological backgrounds) involves cosmological backgrounds) involves
32,0
2920 g/cm102
8
3Hc h
G
H −•≅=π
ρkm/s)/Mpc(100 ,00 HhH ⋅=
critical energy density for closing the critical energy density for closing the Universe, with the Hubble constant Universe, with the Hubble constant expressed as expressed as
GWρ energy density of the GW background energy density of the GW background
fGW
cGW logd
logd1 ρρ
=Ω energy density of the GW background per energy density of the GW background per logarithmic frequency interval relative to logarithmic frequency interval relative to ρρcc
3/22,0 )( ffh GWH ∝Ω
then, the expected spectrum of the GW background goes as then, the expected spectrum of the GW background goes as
The GW background from Relic GWsThe GW background from Relic GWsThe GW background from Relic GWs
Hogan 2006
••During Phase TransitionsDuring Phase Transitions•• Bubble collisionsBubble collisions•• Topological defectsTopological defects•• Primordial turbulencePrimordial turbulence•• Magnetic fieldMagnetic field
[ e.g. Grishchuck 2005; Boyle & Buonanno 2008][ e.g. Grishchuck 2005; Boyle & Buonanno 2008][ e.g. Grishchuck 2005; Boyle & Buonanno 2008]
The GW background from Cosmic StringsThe GW background from Cosmic StringsThe GW background from Cosmic Strings
Loops Loops are formed from strings, they are formed from strings, they oscillate and oscillate and emit GWsemit GWs++
There is a whole range of loop sizesThere is a whole range of loop sizes
↓↓this leads to a stochastic background of GWsthis leads to a stochastic background of GWs
Spa
ce.c
om
δβ −−− ≈≈Ω )3/1(222,0 )( fffh GWH
;β−(f)~Afhc 267;1@
1010 1416
→≈≈
−≤ −−
βyrf
A
[ e.g. Caldwell et al 96; Maggiore 2000; Damour & Vilenkin 2005][ e.g. Caldwell et al 96; Maggiore 2000; Damour & Vilenkin 2005][ e.g. Caldwell et al 96; Maggiore 2000; Damour & Vilenkin 2005]
The “best” case using a single pulsarThe “best” case using a single pulsarThe “best” case using a single pulsarRemembering the approx formula Remembering the approx formula
T
σ~(f)h TOA
c
one can estimate that for detecting the expected GW background fone can estimate that for detecting the expected GW background from merging rom merging of SMBHs (strain amplitude hof SMBHs (strain amplitude hcc ~ 10~ 10--1616--1010--1515) would require ) would require at least a timing at least a timing stabilitystability σσTOATOA < < 1010--100 ns over few years 100 ns over few years
The best result so far using a single source is from 8The best result so far using a single source is from 8--yr timing of PSR yr timing of PSR B1855+09 at Arecibo implying limit B1855+09 at Arecibo implying limit for for f~7 nHzf~7 nHz [ Kaspi et al 94][ Kaspi et al 94]
Extended dataset led to ΩGW h0,H2 (1/17 yr)<~ 2 10-9 [Lommen et al 2002][Lommen et al 2002]but not
confirmed yet by independent analyses[Jenet et al 2006][Jenet et al 2006]
Subject to uncontrollable timing noise effects!
A pulsar timing array (PTA)A pulsar timing array (PTA)A pulsar timing array (PTA)Using a Using a number of pulsarsnumber of pulsarsdistributed across the sky it is possible distributed across the sky it is possible
to separate the timing noise contribution from each pulsar from to separate the timing noise contribution from each pulsar from the the signature of the signature of the GW backgroundGW background, which , which manifests as a localmanifests as a local(at (at Earth) Earth) distortion in the times of arrivaldistortion in the times of arrival of the pulses which is of the pulses which is
common to the signal from all pulsarscommon to the signal from all pulsars
@ K
ram
er
The pulsar timing array conceptThe pulsar timing array conceptThe pulsar timing array conceptA(t)A(t) dimensionless amplitude of the GW at time tdimensionless amplitude of the GW at time tNNii(t)(t) intrinsic timing noise of the iintrinsic timing noise of the i--th pulsar at time tth pulsar at time tααii geometric term dependent on pulsar sky coord and GW prop&polar vgeometric term dependent on pulsar sky coord and GW prop&polar vectors ectors ννii rotation frequency of the irotation frequency of the i--th pulsarth pulsarδνδνii fractional frequency shift detected in the ifractional frequency shift detected in the i--th pulsarth pulsar
)()( tNtA iii
i +=αυδυ
By crossBy cross--correlating correlating ‹‹brackets…brackets…›› the observations of ithe observations of i--th and jth and j--th pulsars, one getsth pulsars, one gets
)()()()()()()(2 tNtNtNtAtNtAtA jiijjiji +++ αααα
Since GW amplitude and intrinsic timing noise are uncorrelated Since GW amplitude and intrinsic timing noise are uncorrelated the the latter 3 terms tend to become negligiblelatter 3 terms tend to become negligiblewhile the while the dataspandataspan(i.e. number (i.e. number
of observations) and the of observations) and the number of pulsars become large enoughnumber of pulsars become large enough
Clock errorsClock errorsAll pulsars have the same TOA All pulsars have the same TOA variations: variations: MonopoleMonopole signaturesignature
Can separate these effects provided there is a Can separate these effects provided there is a sufficient number of widely distributed pulsarssufficient number of widely distributed pulsars
[ adapted from Manchester ]
Idea first discussed by Idea first discussed by Romani Romani [1989][1989] and and Foster & Backer [1990]Foster & Backer [1990]
A pulsar timing array (PTA) for detecting a stocastic Background of GW (GWB)
A pulsar timing array (PTA) for detecting A pulsar timing array (PTA) for detecting a stocastic Background of GW (GWB) a stocastic Background of GW (GWB)
abababab
ab δϑϑϑθζ2
1
2
1)
2
cos1(
4
1)
2
cos1log()
2
cos1(
2
3)( ++−−−−=
θab
Pulsar Pulsar aaPulsar Pulsar bb
Hellings & Downs [1983]Hellings & Downs [1983]: correlation that an : correlation that an isotropic and stocastic GWBisotropic and stocastic GWBleaves on the leaves on the timing residuals of 2 pulsars timing residuals of 2 pulsars a andand b separeted by an anglesepareted by an angleθab in skyin sky
A too simple A too simple (interpretation of the) (interpretation of the) sensitivity curve…sensitivity curve…
Pulsar timing arrays for stocastic GWB: a typical sensitivity curve
Pulsar timing arrays for stocastic GWB: Pulsar timing arrays for stocastic GWB: a typical sensitivity curvea typical sensitivity curve
Limited by total Tspan≈ few yrs ≈ few 108 sec
Limited by interval btw observations: days→weeks ≈ 106-107 sec
For pulsar with white timing noise, best sensitivity for f ≈1/Tobs
White timing noise contribution
for the GWB due to SMBH for the GWB due to SMBH N = number of epochsN = number of epochsM = number of pulsarsM = number of pulsars)1(3/5, −•
∝MMNT
hspan
TOAGWBc
σ
Detailed simulations Detailed simulations are required for are required for more realistic more realistic sensitivity curves…sensitivity curves…
Spherical harmonic decompositionSpherical harmonic decomposition[Burke 1975, Dettweiler 1979, Jaffe & Backer 2003, Demorest et a[Burke 1975, Dettweiler 1979, Jaffe & Backer 2003, Demorest et al 2005]l 2005]
Data analysis for a stocastic GWBData analysis for a stocastic GWBData analysis for a stocastic GWB
Two point correlationTwo point correlation
Correlating the time derivative of the Correlating the time derivative of the residualsresiduals [Hellings & Downs 1983][Hellings & Downs 1983]
Directly correlating the time residualsDirectly correlating the time residuals[Jenet et al 2005][Jenet et al 2005]
Bayesian analysisBayesian analysis[van Haasteren, Levin, McDonald, Lu 2008][van Haasteren, Levin, McDonald, Lu 2008]RobustRobust: deals easily with unevenly sampled data, variable number of tracked pulsars, etc.MarginalisationMarginalisation: deals easily with all systematics of knownfunctional form, including the timing modelCapable to simultaneously measure the Capable to simultaneously measure the amplitude and the shapeamplitude and the shapeof the GWBof the GWB
Data analysis methodologiesData analysis methodologiesData analysis methodologiesBayesian analysis of the timing residuals of an ensemble of pulsBayesian analysis of the timing residuals of an ensemble of pulsars ars
[van Haasteren, Levin, McDonald, Lu 2008][van Haasteren, Levin, McDonald, Lu 2008]
Sanity check tests: Sanity check tests:
[@ v
an H
aast
eran
n 20
08]
Duration of the experiment Duration of the experiment
Typical rms of the timing data Typical rms of the timing data
Number of pulsars Number of pulsars
Rate of data taking Rate of data taking
Useful for optimizing PTA(s) experimental setupUseful for optimizing PTA(s) experimental setup [@ van Haasteren 2008]
S/N>2S/N>2
rms < 200 nsrms < 200 ns
≈≈ 2020--2525>≈ >≈ 55--10 yr10 yr
GW from discrete sources: a spiral-in binaryGW from discrete sources: a spiralGW from discrete sources: a spiral--in binaryin binaryFor a coalescing BH binaryFor a coalescing BH binary[ e.g Thorne 87 ][ e.g Thorne 87 ]
3/24
)]1([5
24 zf
Dc
GMh c
s += π5/1
213/5
21 )()( −+= MMMMMc
f = freq of GWf = freq of GWD = comoving distance of the sourceD = comoving distance of the sourcez = redshift of the sourcez = redshift of the sourceMM cc = =
The expected signature is a periodic GW signal with period twicThe expected signature is a periodic GW signal with period twice e the orbital period of the binary: well away from the last stablethe orbital period of the binary: well away from the last stableorbit it is expected a orbit it is expected a sinusoidal effectsinusoidal effecton the pulsar timing residualson the pulsar timing residuals
To give an order of magnitude estimate, at the last stable orbitTo give an order of magnitude estimate, at the last stable orbit(i.e. immediately (i.e. immediately before merging), the expected strain is before merging), the expected strain is [[Sathyaprakash & Schutz 2009Sathyaprakash & Schutz 2009]]
≈ −
DM
Mh
sun
BHLSOs
Gpc1
1010 10
13
,at a frequency at a frequency
≈
BH
sunLSO M
Mf
1010
nHz440
[ Yar
dley
et a
l 201
0 ]
[ Yar
dley
et a
l 201
0 ]
A spiral-in binary: a typical PTA sensitivity curve
A spiralA spiral--in binary: a typical in binary: a typical PTA sensitivity curve PTA sensitivity curve
Limited by total Tspan≈ few 108 sec and need to fit for spin derivative and jumps btw dataset
Limited by interval btw observations: weeks ≈ 106-107 sec
1 yr-period fitted out in fitting procedures
Ares ~ hs / freq
for inspiraling BHsfor inspiraling BHshhss = strain = strain ≈ ≈ f f 2/32/3
DDpulpul = psr distance= psr distanceΘΘ = source to psr angle= source to psr angleΦΦ = = GW polariz angleGW polariz angleffobsobs = GW freq obs= GW freq obs
At least one SMBH+SMBH will induce timing At least one SMBH+SMBH will induce timing residual of order 5residual of order 5--50 nanosec 50 nanosec [ Sesana et al 2009 ][ Sesana et al 2009 ]
A spiral-in binary: some interesting cases A spiralA spiral--in binary: some interesting cases in binary: some interesting cases
GWs from discrete sources: GW bursts with memory
GWs from discrete sources: GWs from discrete sources: GW bursts with memoryGW bursts with memory
2
,,
≈
c
V
D
Rh asphejSchw
memb
hhb.,memb.,mem = burst memory strain = burst memory strain D = source distanceD = source distanceVVej,asphej,asph = vel of aspherically ejected particles = vel of aspherically ejected particles RRSchwSchw = Schwarzschild radius of source= Schwarzschild radius of source[ e.g. Braginsky & Thorne 87][ e.g. Braginsky & Thorne 87]
In generalIn general, produced when there is a net change in the time derivatives of, produced when there is a net change in the time derivatives ofmultipole multipole moments characterizing the system moments characterizing the system [e.g. Zeldovich & Polnarev 74][e.g. Zeldovich & Polnarev 74]
In astrophysicsIn astrophysics, typically occur in events which are accompanied by large amoun, typically occur in events which are accompanied by large amount of t of mass or radiation ejected in an asymmetric fashionmass or radiation ejected in an asymmetric fashion[e.g. Braginsky & Thorne 87][e.g. Braginsky & Thorne 87]
If If ejected particles are ejected particles are gravitonsgravitons, it is dubbed Christodoulou effect , it is dubbed Christodoulou effect [Payne 83; [Payne 83; Christodoulou 91; Blanchet & Damour 92] Christodoulou 91; Blanchet & Damour 92]
GW burst events characterized by a final nonGW burst events characterized by a final non--zero change in the gravitational zero change in the gravitational wave field wave field ∆∆hhijij
TTTT (burst memory)(burst memory) after a characteristic time after a characteristic time δδt (burst duration)t (burst duration)
GW bursts with memory: merging of a Super Massive BH binary
GW bursts with memory: merging GW bursts with memory: merging of a Super Massive BH binaryof a Super Massive BH binary
For equal mass BHFor equal mass BH--binary at the binary at the most favourable anglemost favourable angle
This corresponds to:This corresponds to:
⋅≈ −
DM
Mh
sun
BHmemb
Gpc1
10102
8
16
,
Detecting GW bursts with memory with a PTA
Detecting GW bursts with Detecting GW bursts with memory with a PTAmemory with a PTA
The jumps in the metric is permanent and thus it produces a The jumps in the metric is permanent and thus it produces a linear increasing of pulsar timing residuals with time linear increasing of pulsar timing residuals with time [Pshirkov et al 2010 ][Pshirkov et al 2010 ]
Undistinguishable from a period glitch in a single pulsar, but Undistinguishable from a period glitch in a single pulsar, but distinguishable in a pulsar timing array. distinguishable in a pulsar timing array.
DetectableDetectablewith current facilities for SMBH binary of 10with current facilities for SMBH binary of 10 88 MM sun sun
up to ≈ 1 Gpc, or for 10up to ≈ 1 Gpc, or for 101010 MM sunsun everywhere in the Universeeverywhere in the Universe[van Haasteren and Levin 2010 ][van Haasteren and Levin 2010 ]
HoweverHowever 0.1 − 0.01 detected mergers during the current PTAs 0.1 − 0.01 detected mergers during the current PTAs lifetime of about 10 yearslifetime of about 10 years[[Sesana et al 2007 ]Sesana et al 2007 ]
LISAAdv LIGO/VIRGO
CMB-POL
Pulsar Timing array(s): the frequency spacePulsar Timing array(s): the frequency spacePulsar Timing array(s): the frequency space
Note the Note the complementarity in explored frequencies complementarity in explored frequencies with respect with respect to the current and the future GW observatories, like LIGO, to the current and the future GW observatories, like LIGO,
advLIGO, advVIRGO and LISA advLIGO, advVIRGO and LISA
I. Current projects: PPTAI. Current projects: PPTAI. Current projects: PPTAParkes Pulsar Timing Array: PPTAParkes Pulsar Timing Array: PPTAAustralian based, using Parkes 64m dishAustralian based, using Parkes 64m dish
Running since Running since ~ ~ 2003 and currently achieving the 2003 and currently achieving the best results so farbest results so far
@ M
.Bu
rgay
The currently used set of observed millisecond The currently used set of observed millisecond pulsars in the PPTA australian projectpulsars in the PPTA australian project
P < 20 ms and not in globular clustersP < 20 ms and not in globular clusters
[ @D
.Man
ches
ter
]
[ Hob
bs e
t al.
Dec
200
8 ]
[ Hob
bs e
t al.
Dec
200
8 ]
For full PPTA (rms of 100 ns For full PPTA (rms of 100 ns over 5 yr for many MSPs) over 5 yr for many MSPs)
Factor >10 improvement on Factor >10 improvement on hhcc and on and on ΩΩgwgw limitslimits
hhc c [1/(1 yr)]< 1.1 [1/(1 yr)]< 1.1 ×× 1010--1414
With ~ 2 yr of useful data and 7 MSPs used (5 with a rms < 300 ns)
With ~ 2 yr of useful data and 7 MSPs used (5 with a rms < 300 ns)
North American Nanohertz Observatory for North American Nanohertz Observatory for Gravitational Waves: NANOGravGravitational Waves: NANOGrav
USA & Canada based, using the excellent Arecibo 300m USA & Canada based, using the excellent Arecibo 300m dish and GBT 101m dish and statedish and GBT 101m dish and state--ofof--art backendsart backends
Running only since Running only since ~ ~ 20082008 @ NRAO
@ Cornell
II. Current projects: NANOGravII. Current projects: NANOGravII. Current projects: NANOGrav
European Pulsar Timing Array European Pulsar Timing Array
++
Large European Array for PulsarLarge European Array for Pulsar
III. Current projects: EPTA-LEAPIII. Current projects: EPTAIII. Current projects: EPTA--LEAPLEAP
European basedEuropean based
Running since Running since ~ ~ 20062006
University of Manchester, JBO, GB ASTRON,Un.Leiden,Un.AmsterdamNL
Max-Planck Institut fur Radioastronomie, GERINAF Osservatorio Astronomico di Cagliari, ITA Nancay Observatory, FR
Andrea PossentiMarta BurgayNichi D’AmicoMaura Pilia
Michael KramerAxel JessnerKosmas Lazaridis
Jason HesselsYuri LevinRutger van Haasteren
Current limits from EPTA dataCurrent limits from EPTA dataCurrent limits from EPTA data
…and applying a Bayesian analysis [e.g. van Haasteren 2009][e.g. van Haasteren 2009]
@ v
an H
aast
eren
Using the data from 6 pulsars:J1640+22 (dataspan 12 yr ; rms=1.6µs) J1855+09 (dataspan 23 yr ; rms=1.70 µs)J1713+0747 (dataspan 11 yr ; rms=0.73 µs) J1744-1134 (dataspan 10 yr ; rms=0.55 µs)J1909-1134 (dataspan 4 yr ; rms=0.11 µs) J1918-0642 (dataspan 7 yr ; rms=2.24 µs)
hhc c [1/(1 yr)]< 1.9 [1/(1 yr)]< 1.9 ×× 1010--1414 …only a factor ≈ 1.7 worse than the current published PPTA limit
Careful analysis of the Careful analysis of the ““ redred”” component of the timing noise was performed while component of the timing noise was performed while calculating the current upper limit for a GWB signal in the Eptacalculating the current upper limit for a GWB signal in the Epta data data [van Haasteren 2009] [van Haasteren 2009]
Wes
terb
ork
Effe
lsbe
rg
@ v
an H
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Long term advantages of EPTALong term advantages of EPTALong term advantages of EPTA
Larger total number of TOAsCommensurate scheduling will allow for improved binary
and yearly phase coverageA wide range of frequencies can be sampled and then
compared in quasi-simultaneous sessionsSimultaneous same frequency observations can be used to
check polarisation calibration and overall timing offsetsTelescope, Instrumentation, or Observatory clock based
errors can be quickly identified and corrected
++The data will be combined with those provided by
LEAP…The data will be combined with those provided by
LEAP…
Phased array of the 5 major European telescopes
Funded by the EU Research Council: 2.5 M
Phased array of the 5 major European telescopes
Funded by the EU Research Council: 2.5 M
Large European Array for Pulsars: LEAPLarge European Array for Pulsars: LEAPLarge European Array for Pulsars: LEAP
Sensitivity equivalent to illuminated AreciboSensitivity equivalent to illuminated Arecibo
But able to see much more or the skyBut able to see much more or the sky
People involved: 2 staff, 2 senior postDoc and 2 junior postDocDuration: 5 years since mid 2009People involved: 2 staff, 2 senior postDoc and 2 junior postDocDuration: 5 years since mid 2009
Expected sensitivity of EPTA+LEAP after 5 yrs of Expected sensitivity of EPTA+LEAP after 5 yrs of observations will largely improve the current best limits for observations will largely improve the current best limits for
the GW Background Amplitudethe GW Background Amplitude
Ada
pted
from
Ver
bies
t et a
l [20
09]
Ada
pted
from
Ver
bies
t et a
l [20
09]
~ a factor 10~ a factor 10~ a factor 10
[ Verbiest et al 2009 ][ Verbiest et al 2009 ]
Pulsar Timing array(s): exploring the futurePulsar Timing array(s): exploring the futurePulsar Timing array(s): exploring the futurefor the GWB due to SMBH for the GWB due to SMBH N = number of epochsN = number of epochsM = number of pulsarsM = number of pulsars)1(3/5, −•
∝MMNT
hspan
TOAGWBc
σ
Hardware Hardware and and data analysisdata analysis points required points required for for obtaining imobtaining improvements on provements on σσ
•• RealReal--time mitigation of interferencestime mitigation of interferences
•• Knowledge of the interplanetary weather (i.e. the solar wind cKnowledge of the interplanetary weather (i.e. the solar wind component) and of the omponent) and of the interstellar weatherinterstellar weather
•• Full polarimetric calibrationFull polarimetric calibration
•• Frequency dependent template pulse profilesFrequency dependent template pulse profiles
•• TimeTime--scale for stabilization of the template pulse profilescale for stabilization of the template pulse profile
•• A timing software capable to handle all the non GWA timing software capable to handle all the non GW--induced effects down to ~1 nsinduced effects down to ~1 ns
•• Improved solar system ephemerisImproved solar system ephemeris
[ Verbiest et al 2010 ][ Verbiest et al 2010 ]
Pulsar Timing array(s): exploring the futurePulsar Timing array(s): exploring the futurePulsar Timing array(s): exploring the futurefor the GWB due to SMBH for the GWB due to SMBH N = number of epochsN = number of epochsM = number of pulsarsM = number of pulsars)1(3/5, −•
∝MMNT
hspan
TOAGWBc
σ
Limits to the imLimits to the improvements onprovements on σσ/T/T5/35/3 are related are related toto two intrinsictwo intrinsic pulsar propertiespulsar properties
σσminmin : : the lowest achievable RMS residualsthe lowest achievable RMS residualstiming stabilitytiming stability :: the potential of the timing data of a given pulsarpulsarto keep constant (low) RMS residuals at all time-scales up to the time-span of a PTA project,
Pulsar Timing array(s): exploring the futurePulsar Timing array(s): exploring the futurePulsar Timing array(s): exploring the future
Nominally, σTOA should decrease linearly with 1/(S/N)1/(S/N)
these data taken on the pulsar J0437these data taken on the pulsar J0437--47154715demonstrate the potential demonstrate the potential to achieve to achieve TOA precisionsTOA precisions down to down to σσmin min ~~ 20 ns 20 ns !!
Systematic worsening of the TOA
uncertainties at high S/N, showing a
scaling with 1/sqrt(S/N) 1/sqrt(S/N)
[ Hobbs et al 2009; [ Hobbs et al 2009; Verbiest et al 2010 ]Verbiest et al 2010 ]
Pulsar Timing array(s): exploring the futurePulsar Timing array(s): exploring the futurePulsar Timing array(s): exploring the future
it therefore appears likely that the it therefore appears likely that the GWB signal will dominateGWB signal will dominate our timing our timing on on timetime--scales between 5 and 10 years,scales between 5 and 10 years,provided rms timing residuals are provided rms timing residuals are
decreased enoughdecreased enough
[ Ver
bies
t et a
l 200
9 ]
[ Ver
bies
t et a
l 200
9 ]
Using the stability parameter
σz of Matsakis et al [1997]Matsakis et al [1997]
τ = time-scalec3 = 3rd polynomial fitted to a subset of the residual of length τ
effect of a hypothetical GWB
1 µs rms
0.1 µs rms
Timing array(s): the future for GWBs detection
Timing array(s): the future for Timing array(s): the future for GWBs detectionGWBs detection
@ S
tapp
ers
Current projects are evolving in pace with predictions. Then at Current projects are evolving in pace with predictions. Then at least least very significant limits on GWB (and hopefully a detection) will very significant limits on GWB (and hopefully a detection) will be be
achieved within 5achieved within 5-- 10 years 10 years
A detailed scientific investigation of the GWBackground A detailed scientific investigation of the GWBackground is warranted with SKAis warranted with SKA
Timing array(s): the future for GWs detection of discrete sourcesTiming array(s): the future for Timing array(s): the future for
GWs detection of discrete sourcesGWs detection of discrete sources
SKA will lead to discover them and doing science (e.g. testing GSKA will lead to discover them and doing science (e.g. testing GR vs R vs alternate theories)alternate theories)