ONDE GRAVITAZIONALIteoria e sorgenti

Gianluca Gemme - INFN Genova

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LIGO Livingston ObservatoryLouisiana, USA

LIGO Hanford ObservatoryWashington, USA

Virgo, Cascina, Italy

This is the 100th anniversary of a revolution in physics

Space and time are more interesting than Newton had thoughtSpace isn’t just a “stage”,

but has its own dynamicsOrbits aren’t stableMatter can “disappear”,

leaving behind only its gravity

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First direct observation of gw14 Sep 2015

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229,000 paper downloads from APS in the first 24 hours

Phys. Rev. Lett. 116, 061102 (2016)

http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.061102

Contents

Part I: Theory of gravitational wavesPropertiesInteraction with test massesWave generation: the quadrupole formulaBasic estimates

Part II: Gravitational wave sourcesBinary systems: the Hulse-Taylor pulsarCompact binary systems

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Part ITheory of gravitational waves

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The Einstein field equations“matter tells spacetime how to curve”

The Einstein field equations describe how mass and energy (as represented in the stress–energy tensor) are related to the curvature of space-time (as represented in the Einstein tensor):

where Gab is the Einstein tensor, c is the speed of light in a vacuum and G is the gravitational constant, which comes from Newton's law of universal gravitation

𝐺𝐺𝑎𝑎𝑎𝑎 =8𝜋𝜋𝐺𝐺𝑐𝑐4

𝑇𝑇𝑎𝑎𝑎𝑎

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𝑑𝑑𝑑𝑑2 = 𝑔𝑔𝑎𝑎𝑎𝑎𝑑𝑑𝑑𝑑𝑎𝑎𝑑𝑑𝑑𝑑𝑎𝑎

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Zoology

Inverse metric gij

Christoffel symbols

Riemann curvature tensor

Ricci and scalar curvatures

Einstein tensor

Energy-momentum tensor

The energy–momentum is a tensor quantity that describes the density and flux of energyand momentum in spacetime, generalizing the stress tensor of Newtonian physics

It is an attribute of matter, radiation, and non-gravitational force fields

The stress–energy tensor is the source of the gravitational field in the Einstein field equations of general relativity, just as mass density is the source of such a field in Newtonian gravity

Fluid in equilibrium

Electromagnetic field

Scalar field (Klein-Gordon equation)

𝐺𝐺𝜇𝜇𝜇𝜇 ≡ 𝑅𝑅𝜇𝜇𝜇𝜇 −12𝑔𝑔𝜇𝜇𝜇𝜇𝑅𝑅

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13

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If we choose k along the z axis we have

And

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Geodesics

The trajectory in the spacetime of a free test particle is called geodesicIn special relativity these are straight lines in

Minkowski space

In general relativity geodesics describe the motion of inertial (free) test particles, i.e. particles not subject to external, non-gravitational forces (free-fallingparticles)

𝑑𝑑2𝑑𝑑𝛼𝛼

𝑑𝑑𝜏𝜏2= −Γ𝜇𝜇𝜇𝜇𝛼𝛼

𝑑𝑑𝑑𝑑𝜇𝜇

𝑑𝑑𝜏𝜏𝑑𝑑𝑑𝑑𝜇𝜇

𝑑𝑑𝜏𝜏

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Generation of gws

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Interaction of gws with test masses

TT frame

Let’s look at geodesic equation in the TT frame assuming that a test mass is at rest at τ = 0

In the TT frame, particles which were at rest before the arrival of the wave remain at rest even after the arrival of the wave

In general relativity the physical effects are not expressed by what happens to the coordinatesDoes this mean that GWs have no physical effect at all?Let’s give a look at proper distances or proper timesConsider two events (t, x1, 0, 0) and (t, x2, 0, 0). In the TT

gauge the coordinate distance x2 - x1 = L does not change even in presence of a GW propagating along the z axisThe proper distance is given by

More generally if the spatial separation between the two events is given by L, the proper distance is given by

and to linear order in h

If the two test masses are mirrors between which a light beam travels back and forth, it is the proper distance that determines the time taken by the light to make a round-trip

So the fact that the GWs affect the proper distance means that they can be detected measuring the round-trip time

Effect of a GW on test masses

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PART IISources of gravitational waves

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GW Sources‘Gravitational waves are generated whenever a dynamical system is seen to have a changing silhouette, i.e. a pencil rotating about its axis would not produce gravitational waves, but if it were tumbling it would’

A strong source of GWs: • masses of the bodies must be large• motion must be fast• gravitational fields must be large

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Where do gravitational waves come from?

From stars living in galaxies…

Supernova explosionsRotating stars (pulsars)

Binary systems(black holes, neutron stars)

Credit: NASA/COBE

..and from the beginning of the Universe!

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Sources And MethodsLong

durationShort

duration

Known Signal (matched filter

search) Pulsars Compact Binary Inspirals

Unknown Signal (template-less

methods)

Stochastic Background Bursts

Compact Binary Coalescences

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Neutron Stars - Extremely dense object which remains after the collapse of a massive star. ~1.4 times the mass of the Sun compressed into a ball the size of Manhattan.

Black hole - A region of space-time caused by an extremely compact mass where the gravity is so intense it prevents anything, including light, from leaving

Compact binary - A system of two remnants of collapsed stars, for example a neutron star and a black hole, orbiting around each other very closely

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((( )))

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The Fate of B1913+16

Gravitational waves carry away energy and angular momentum

Orbit will continue to decay (inspiral) over the next ~300 million years, until…

The neutron stars will merge !And possibly collapse to form a black holeFinal few minutes will be in audio frequency band

Gravitational wave detectors can listen for signals like these

h(t)

CBC Injection

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((( )))Information from the InspiralTime evolution of GW amplitude and frequency depend on the masses, spins and orbit orientation of the binary system

Compact objects: white dwarfs, neutron stars, black holes

First-order effect: “chirp rate” when not too close to merger

Characteristic time scale: 𝜏𝜏 ∝ 𝑚𝑚1+𝑚𝑚21/3

𝑚𝑚1𝑚𝑚2So higher mass chirps more quickly

Inspiral ends at innermost stable circular orbit (ISCO)Depends on masses and spins; 𝑓𝑓ISCO ∝

1𝑚𝑚1+𝑚𝑚2

So higher mass signal cuts off at a lower frequency

Relative amplitude and phase of polarization components (ℎ+, ℎ×) indicate the orientation of the orbit

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((( )))

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Relativistic Corrections

Orbital phase vs. time orbital phase vs. frequency during chirp

“Post-Newtonian expansion” if spins are negligible:

where

So phase evolution near merger gives individual masses

( ) ( )

( )

( )

( )

+

+++

−

++

+=Ψ

−

−

−

−

3/12

3/2

1

3/5

144617

10085429

10160643058673

6415

83

411

336743

965

12832

fm

fm

fm

fmtff c

πηηη

πηπ

πηη

πη

π

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21 ,)( mmmmmm =+= η

1PN

1.5PN

2PN

Newtonian

Relativistic effects

((( )))Into the MergerMerger dynamics are driven by strong-field gravity

Post-Newtonian expansion loses accuracyNeutron star tidal deformation can affect final part of inspiralBlack hole spins can cause orbital plane to precess and strongly influence final “plunge”

Numerical relativity to the rescue !

48Mroue & Pfeiffer

Baker et al., PRL 99, 181101 (2007)

Precessing binary:

PN

NR

((( )))Expansion History of the UniverseGR predicts the absolute luminosity of a binary inspiral+merger detection of a signal measures the luminosity distance directly

“Standard siren” – neutron star binaries out to z~1, BH binaries anywherePrecision depends on SNR, ability to disentangle orbit orientation

GW signal alone does not determine redshiftGW signal is redshifted, but that looks just like a change in masses

Identifying an optical counterpart provides redshiftHost galaxy redshift can be measuredKnowing exact sky position of the source helps analysis

With a sample of events, can trace out distance-redshift relatione.g. measure cosmological w parameter to within a few percentOne systematic: weak lensing

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What can an observation teach us?

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• Is general relativity the correct theory of gravity?

• How many black holes are there in the universe?

• How do the first generations of stars live and die?

• How does a core collapse power a supernova?

• What is the nuclear equation of state of a neutron star?

• What is the engine that powers a short gamma ray burst?

• What new physics lies beyond the CMB?• What happened at the earliest moments of

creation?

Multimessenger Astronomy

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GW data EM data Neutrino data

The complete picture

ONDE GRAVITAZIONALI�teoria e sorgentiThis is the 100th anniversary of a revolution in physicsFirst direct observation of gw�14 Sep 2015ContentsPart IThe Einstein field equations�“matter tells spacetime how to curve”Slide Number 7Slide Number 8ZoologySlide Number 10Energy-momentum tensorSlide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16GeodesicsSlide Number 18Slide Number 19Generation of gwsSlide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Interaction of gws with test massesTT frameSlide Number 29Slide Number 30Effect of a GW on test massesSlide Number 32PART IIGW SourcesWhere do gravitational waves come from?Sources And MethodsCompact Binary CoalescencesSlide Number 38Slide Number 39Slide Number 40Slide Number 41Slide Number 42Slide Number 43The Fate of B1913+16CBC InjectionInformation from the InspiralRelativistic CorrectionsInto the MergerExpansion History of the UniverseWhat can an observation teach us?Multimessenger Astronomy