Gravitational Wave Astronomy - ICTP – SAIFR · Gravitational Wave Astronomy Riccardo Sturani...

Gravitational Wave Astronomy

Riccardo Sturani

International Institute of Physics - UFRN - Natal (Brazil)

Dark Universe ICTP-SAIFR, Oct 25th, 2019

Riccardo Sturani (IIP-UFRN) Gravitational Wave Astronomy DU - Oct 25th, 2019 1 / 29

GW basics in 1 slide

Gauged fixed metric perturbations hµν after discarding h0µ components,which are not radiative → transverse waves

hij =

h+ h× 0h× −h+ 00 0 0

2 pol. state like any mass-less particle

h+

h×


LIGO and Virgo: very precise rulers

Robert Hurt (Caltech)

Light intensity ∝ light travel difference in perpendicular armsEffective optical path increased by factor N ∼ 500 via Fabry-Perot cavitiesPhase shift ∆φ ∼ 10−8 can be measured ∼ 2πN∆L/λ→ ∆L ∼ 10−15/N m


Almost omnidirectional detectors

Detectors measure hdet : linear combination F+h+ + F×h×

-1 0 1F+

F×

L H

h+,× depend on sourcepattern functions F+,× depend on orientation source/detector


Almost omnidirectional detectors

Detectors measure hdet : linear combination F+h+ + F×h×

-1 0 1F+

F×

L V

h+,× depend on sourcepattern functions F+,× depend on orientation source/detector


Pattern functions:√

F 2+ + F2×


The LIGO and Virgo observatories

• Observation run O1 Sept ’15 - Jan ’16∼ 130 days, with 49.6 days of actual data, PRX(2016) 4, 041014, 2 detectors, 3BBH

• O2 Dec. ’16 – Jul’17 2 det’s + Aug ’17 3 det’s3(+4) BBH + 1BNS in double (triple) coinc.

• O3a: 3 detectors, Apr 2nd - Oct 1st 2019• O3b: Nov 1st – 6 monthsIn ∼ end of 2019 Japanese KAGRA and ∼ 2025Indian INDIGO will join the collaboration


Wave generation: localized sources

Einstein formula relates hij to the source quadrupole moment Qij

Qij =

∫d3xρ

(xixj −

1

3δijx

2

), v 2 ' GNM/r , η ≡ m1m2/M2

hij ∼ g(θLN )2GND

d2Qijdt2

' 2GNηMv2

Dcos(2φ(t))

f = 2kHz( r

30Km

)−3/2( M3M�

)1/2< fMax ' 12kHz

(M

3M�

)−1v = 0.3

(f

1kHz

)1/3(M

M�

)1/3<

1√6

Geometric factor g(θLN ) takes account of transversality projection(angular momentum L of the binary, observation direction N)

h+ ∼1 + cos2(θLN )

2ηMv 2

Dcosφ(ts/M, η, S

2i /m

4i , . . .)

h× ∼ cos(θLN ) ηMv 2

Dsinφ(ts/M, η, . . .)

Amplitudes of 2 polarizations modulated by θLN (h↗ for θLN ↘0), never both vanishingunlike dipolar motion for the electromagnetic case




Qij =

∫d3xρ

(xixj −

1

3δijx

2

), v 2 ' GNM/r , η ≡ m1m2/M2


d2Qijdt2

' 2GNηMv2

Dcos(2φ(t))

f = 2kHz( r

30Km

)−3/2( M3M�

)1/2< fMax ' 12kHz

(M

3M�

)−1v = 0.3

(f

1kHz

)1/3(M

M�

)1/3<

1√6


h+ ∼1 + cos2(θLN )

2ηM(1 + z)v 2

D(1 + z)cosφ(tO/(M(1 + z)), η, S

2i /m

4i , . . .)

h× ∼ cos(θLN ) ηM(1 + z)v 2

D(1 + z)sinφ(tO/(M(1 + z)), η, . . .)





Qij =

∫d3xρ

(xixj −

1

3δijx

2

), v 2 ' GNM/r , η ≡ m1m2/M2


d2Qijdt2

' 2GNηMv2

Dcos(2φ(t))

f = 2kHz( r

30Km

)−3/2( M3M�

)1/2< fMax ' 12kHz

(M

3M�

)−1v = 0.3

(f

1kHz

)1/3(M

M�

)1/3<

1√6


h+ ∼1 + cos2(θLN )

2ηMv 2

dLcosφ(tO/M, η, S2i /m4i , . . .)

h× ∼ cos(θLN ) ηMv 2

dLsinφ(tO/M, η, . . .)


h sensitive to red-shifted masses M → M(1 + z) ≡M


Binary systems and compact detected so far


https://www.gw-openscience.org/catalog/

See however Princeton’s groupresultsarXiv:1904.07214

20 40 60 80 100Msourcetot (M )

1.00

0.75

0.50

0.25

0.00

0.25

0.50

0.75

1.00

eff

Binary black hole events in O1 and O2

LVC-reportedThis workZackay et al. (2019)

0.0

0.2

0.4

0.6

0.8

1.0

p ast

ro


Sky localization of GW binary systems detected so far

0h 21h 18h 15h 12h 9h 6h 3h 0h0°

30°

60°60°

30°

-30°

-60° -60°

-30°GW

1708

17

GW17

0104

GW170104

GW170104

GW17

0823

GW170823

GW170608

GW17

0809

GW170814

0h 21h 18h 15h 12h 9h 6h 3h 0h0°

30°

60°60°

30°

-30°

-60° -60°

-30°

GW170

818

GW15

1012

GW151012

GW150914

GW15

1226

GW151226

GW17

0729

GW170729

Sky localization regions shrinks when Virgo in: 3rd detector matters!

LIGO/Virgo 1811.12907


Astro sources emitting transient GWs/EM/ν radiation

Coalescing binary systems of compact objects: η ≡ m1m2/M2

∆EGW ∼ 4× 10−2M�η(

M

3M�

)5/3(fmax

1 kHz

)2/3A good proxy of the fmax is the Innermost Stable Circular Orbit frequency

fISCO =1

6√

6

1

M' 2kHz

(M

M�

)−1For NS-NS/BH ejected material supposed to produce short (< 2 sec) GRBsEγ ∼keV-MeV, ejected sub-relativistic material may produce radio signalsν: baryon loaded jets may emit νs:

TeV-PeV ν @ γ emissionPeV-EeV ν @ optical afterglow

NS-NS/BH → energetic outflows at different timescale/wavelengthRapid neutron capture in the sub-relativistic ejecta can produce a kilonova ormacronova, with optical and near-infrared signal (hours - weeks)

Eventually radio blast from sub-relativistic outflow (months to years)

LIGO/Virgo + EM partners ApJ Let. 2016


Association with 20 short GRB during O1(before GW170817)

redshift

10−2 10−1 100

cumulativedistribution

10−2

10−1

100

BNS exclusion

NS-BH exclusion

BNS extrapolation

NS-BH extrapolation

EM observation

GW searched in a [-5,+1) sec window with GRBGW constraints far from EM exclusion distance, but extrapolation to design AdvancedLIGO sensitivity (2 years of data) will lead to detection/non-trivial constraints

LIGO/Virgo Astrophys.J. 841 (2017) no.2, 89Riccardo Sturani (IIP-UFRN) Gravitational Wave Astronomy DU - Oct 25th, 2019 12 / 29

GW170817

GW trigger on Aug 17th, 2017,ended at 12h 41’ 04.4” UTC, first inin LIGO Hanford, then confirmed asa triple coincidence → localized inan area of ∼ 28 deg2

GRB trigger from Fermi-GBM 1.7”after

first optical image 10.87 hrafterwards by One-Meter TwoHemisphere team with Swopetelescope at Las CampanasObservatory in Chile

X obtained by the X-Ray Telescopeon Swift after 14.9 h(NuSTAR 16.8 h)

radio (∼ 3, 6 GHz) by VLA 16 daysafter GW event

LIGO/Virgo & Partner Astronomy groups, Astrophys.J. 848 (2017) no.2, L12


An example of the optical image

Transient Optical Robotic Observatory of the South


GW170817 and ν detectors

Potential νs down-going for Antares and IceCube, grazing for Auger


GW170817 and ν detectors

Potential νs down-going for Antares and IceCube, grazing for AugerSource location at optical detection


ANTARES and ICECube look for ν

IceCube and ANTARES looked forµ − ν at ±500sec , observation for 2weeksno detectionPierre Auger for > 1017 eV, grazinggood for τ production at 1018 eV

10 3

10 2

10 1

100

101

102

103

E2 F

[Ge

Vcm

2 ]

±500 sec time-window

AugerIceCube

ANTARES

Kimura et al.EE optimistic

04

8

Kimura et al.EE moderate

04

Kimura et al.prompt0

GW170817 Neutrino limits (fluence per flavor: x + x)

102 103 104 105 106 107 108 109 1010 1011E/GeV

10 3

10 2

10 1

100

101

102

103

E2 F

[Ge

Vcm

2 ]

14 day time-window

Auger

IceCube

ANTARES

Fang &Metzger30 days

Fang &Metzger3 days

LIGO/Virgo, ANTARES, ICECUBE and Pierre Auger, Astrophys.J. 850 ’17 2, L35


BNS NSBH BBH

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

Rat

eE

stim

ates( M

pc−

3yr−

1)

Astrophysical predictions, upper limit from S5/6/O1 and observation from O1/O2

LIGO/Virgo CQG (’10) 27 173001, PRX (’16), APJ (2016), PRL 119 (’17)


Fundamental physics of Neutron stars

Tidal deformability Λ ' R−5Qab/ (∂a∂bUtidal ) depends on Equation ofState

Determinaton of Λ1,2(assuming same EoS for 1,2)

0 250 500 750 1000 1250Λ1

0

500

1000

1500

2000

Λ2

WFF1

APR4

SLy

MPA1

H4

MS1b

MS1

More Compact

Less Compact

Shading Common EoS for 2 bodies and constrain for mmax > 1.97M�Lines 50%, 90% countours, No mimimum mmax , independent EoSs

LIGO/Virgo collaboration, arXiv:1805.11581


Fundamental physics of Neutron stars II

Parametrizing EoS with Γ ≡ �+pp

dpd�

, Γ(x) = exp(∑

k γkxk), x ≡ log p

p0

γ0 ∈ [0.2, 2], γ1 ∈ [1.6, 1.7], γ2 ∈ [0.6, 0.6]γ3 ∈ [0.02, 0.02], Γ(p) ∈ [0.6, 4.5]

L. Lindblom PRD 82, 103011 (2010)

Constraints on Mass vs- radius and p vs. rho

8 10 12 14

R (km)

0.5

1.0

1.5

2.0

2.5

3.0

m(M�

)

WF

F1

AP

R4

SL

yM

PA

1

H4

BHlim

it

Buch

dahl

limit

0

1

0 1 1014 1015

ρ [g/cm3]

1032

1033

1034

1035

1036

1037

p[d

yn/c

m2 ]

ρnuc

2ρnuc

6ρnuc

lighter, heavier 90% prior, posterior

LIGO/Virgo collaboration, arXiv:1805.11581


Cosmology

From GWs only: luminosity distancedL ∼ 40+8−14Mpc, viewing angleθLN > 125

o

Fixing dL from info from EM location

+ Hubble relation dLH0 = z

1 Hubble constant fromSuperNovae (SHoES)148o < θLN < 166

o

Riess et al. ApJ 826, 56 ’16

2 from Planck157 < θLN < 177

o

Planck A&A, 594, A13 ’16

GW almost coincident with EM=⇒ ∆vGW /c ∼ 10−15

LIGO/Virgo, Fermi Gamma-Ray BurstMonitor, INTEGRAL Ap.J. 848 (2017)2, L13

50 60 70 80 90 100 110 120H0 (km s 1 Mpc 1)

1.0

0.9

0.8

0.7

0.6

0.5

0.4

cos

180170

160

150

140

130

120

(deg

)

GW170817Planck17

SHoES18

LIGO/Virgo Nature 551 ’17 7678, 85

h′′+× + 2

a′

ah′+,× + c

2s∇2h+,× = 0


Determining H0 without EM counterpart

Hubble law: z = H0dLDL can be measured, z degenerate with M, however if

the source in the sky has been localised (α, δ)

GW sources are in the galaxy catalogue with known red-shift

P(z ,DL|ci ) =∫

dM d~θ dα dδ P(DLM, ~θ, α, δ|ci )π(z , |α, δ)

Schutz, Nature (1986)323 310W. Del Pozzo,Phys.Rev.D86 (2012)043011


Galaxy catalogs and H0

20 40 60 80 100 120 140

H0 (km s−1 Mpc−1)

0.00

0.01

0.02

0.03

0.04

0.05

0.06

p(H

0)

(km−

1s

Mp

c)

Joint BBH+GW170817 Counterpart

Joint BBH

GW170817 Counterpart

Prior (Uniform)

Planck

SH0ES

sky localization from DES-Y1(GW170814)

GWENS (GW170818)

GLADE (GW150914, GW151226,

GW170608)

LIGO/Virgo, arXiv:1908.06060


Upper bounds on sub-solar mass compact binariescoalescence rate

Mass/M� Rate(Gpc−3yr−1) Horizon(Mpc)

1 < 106 1000.2 < 2× 104 30

SNR ∝ M5/6c , Mc ≡ η3/5MLIGO/Virgo arXiv:1808.04771, 1904.08976

Fraction f of MACHOs in binary∼ 10%× ΩMacho =⇒∼ 10−3ΩMacho in coalescing bina-ries coalescing within H−10

T. Nakamura et al. in Astrophys.J.L.

487 (1997) L139-L142


Ongoing O3: towards measuring merger distribution

At https://gracedb.ligo.org/superevents/public/O3Summary of candidate detections from O3a:33 candidate events, 7 of which possibly involving at least a NS

Expected merger distribution from Star Formation Rate (−) orSFR+Poissian distributed delay (−−)


Fundamental GR: inspiral analytic model

Inspiral h = A cos(φ(t)) ȦA � φ̇Virial relation:

v ≡ (GNMπfGW )1/3 η =m1m2

(m1 + m2)2

E (v) = −12ηMv2

(1 + #(η)v2 + #(η)v4 + . . .

)

P(v) ≡ −dEdt

=32

5GNv10(1 + #(η)v2 + #(η)v3 + . . .

)

E(v)(P(v)) known up to 3(3.5)PN

1

2πφ(T ) =

1

2π

∫ Tω(t)dt = −

∫ v(T ) ω(v)dE/dvP(v)

dv

∼∫ (

1 + #(η)v2 + . . .+ #(η)v6 + . . .) dvv6


Fundamental GR: inspiral analytic model

Inspiral h = A cos(φ(t)) ȦA � φ̇Virial relation:

v ≡ (GNMπfGW )1/3 η =m1m2

(m1 + m2)2

E (v) = −12ηMv2

(1 + #(η)v2 + #(η)v4 + . . .

)

P(v) ≡ −dEdt

=32

5GNv10(1 + #(η)v2 + #(η)v3 + . . .

)

E(v)(P(v)) known up to 3(3.5)PN

1

2πφ(T ) =

1

2π

∫ Tω(t)dt = −

∫ v(T ) ω(v)dE/dvP(v)

dv

∼∫ (

1 + #(η)v2 + . . .+ #(η)v6 + . . .) dvv6

PN Coefficients (tidal ∼ v10)Riccardo Sturani (IIP-UFRN) Gravitational Wave Astronomy DU - Oct 25th, 2019 25 / 29

Parameter estimation from φ(t)


Testing the PN series with O1/2

−1PN 0 P

N0.5

PN 1 PN

1.5PN 2 P

N

2.5PN

(l)

3 PN

3 PN(l)

3.5PN

10−3

10−2

10−1

100

101

|δϕ̂i|

GW150914GW151226GW170104GW170608GW170814CombinedCombined (SEOBNRv4)

Precision in measuring the PN coefficients ∼ 100%: one cannot both measure the astroparameters and test GR with same detection! Unique probe of high PN-orders

LIGO/Virgo arXiv:1903.04467


Precision gravity

Future 3rd generetaion detectors (Einstein Telescope, Cosmic Explore) / spacetelescope LISA will detect BBH binary signals with SNR 10− 102, with few goldenevents with SNR ∼ 103.Templates few % accurate OK for characterising a source with SNR O(10) (typical forLIGO/Virgo)for SNR ∼ 103 residual after extracting that source will have SNR ∼ O(10)

1 baising parameter estimation

2 contaminating the extraction of additional sources.

h(f ) ∝ f −7/6M5/6c

∆ti→m ∝ M−5/3c (f −8/3i − f−8/3

m )


Conclusions (Actually just the beginning!)

After the beginning of GW astronomy, we have witnessed the start ofmulti-messenger astronomy

New era for Cosmology: H0 from GWs! (and EM)

New input for fundamental physics: test of gravity in the strong fieldat short scale, radiative gravity sector tested at large distances andsoon precision gravity is going to be needed


Spare slides


How binary systems progenitors formed? I

Izzard et al. Proc. IAU 2012

A binary system of massive stars(Mbinary < 100Modot) can gothrough a common envelope phasethat shrinks considerably the orbit andalign the spins→ collapse to compactobjectsBNS requires complex evolution frommassive binaries + mass transfer and2 core-collapse SN explosions throughwhich the binary system survivesAsymmetries in collapse imparts na-tal kick to the (galactic pulsar propermotion)

LIGO/Virgo Ap.J. 850 (2017) L40

vs.


For BBH another mechanism is possible

Globular Cluster: medium with high density of black holes/stars, when 3black holes meet one is ejected and the binary shrinks

LIGO/VirgoAstrophys.J. 818 (2016) no.2, L22

Remnants of the first stars, produced at z ∼ 6 can give only a smallcontribution to the total rate

T. Hartwig et al. Mon.Not.Roy.Astron.Soc. 460 (2016) L74

Primordial black-holes? viable if 10−2 . MBH/M� . 1 (constraints fromevaporation, microlensing and CMB) but could grow later to 20-100 M�by mergerIf present in globular cluster may also explain their cosmologicalabundance as dark matter

Sasaki M. et al. PRL 2016, Bird S. et al. PRL 2016, Clesse S. and Garćıa-Bellido Phys.

Dark Univ. 2017


Metzer & Berger Astrophys.J. 746 (2012) 48


NS merger → short γRB → kilonova

Jet models for the ejecta (spherical fireball model)

↓BNS merger emitted γ rays, expected opening angles θj ' 3o − 10o :early time θj ∼ 1/γ (jet collimated and relativistic), prompt emission from energydissipation inside relativistic jet: X rays ∼ energy output and geometryShock wave from sGRB jet hitting surrounding medium, θj < 1/γlate

↓broadband synchrotron afterglow, with radio source ∼ months and UV, optical &IR emission from sub-relativistic ejectaνs expected only from near GRB (or in late emission by flares and plateaus)

Alexander et al., Astrophys. J. 848 (2017) no.2, L21

NS merger frees neutron-rich radioactive species, decaying in kilonova in ∼ daysObservation: Faint gamma ray emission: 1046 erg (isotropic), raising Xray flux 2.4- 5.4 day, continued up to 15 days (Chandra), delayed afterglow

↓short GRB viewed off-axis θobs & 20o

Berger E, 2014, ARA&A, 52, 43Riccardo Sturani (IIP-UFRN) Gravitational Wave Astronomy DU - Oct 25th, 2019 34 / 29

GW170817

BNS horizon (Mpc) was 218 for LL, 107 for LH, 58 for V

LIGO/Virgo Phys.Rev.Lett. 119 (2017) 161101


Detected waves (best fit)

0 6 12 18 24 30 36 42 48 54 60

time observable (seconds)

GW150914

LVT151012

GW151226

GW170104

GW170608

GW170814

GW170817

LIGO/University of Oregon/Ben Farr

Smaller objects can get closer =⇒ reach higher frequency before merger


Parameter estimation for GW170817

Low spin prior High spin prior

m1,2

1.4 1.5 1.6 1.7 1.8m1 (M�)

1.05

1.10

1.15

1.20

1.25

1.30

1.35

1.40m

2(M�

)TaylorF2

PhenomDNRT

PhenomPNRT

SEOBNRT

1.4 1.6 1.8 2.0 2.2 2.4m1 (M�)

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

m2

(M�

)

TaylorF2

PhenomDNRT

PhenomPNRT

SEOBNRT

Spin χeff = (m1S1z + m2S2z )/M - mass ratio q, distance - inclination

−0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12χeff

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

q

χ < 0.89

χ < 0.05

100◦ 120◦ 140◦ 160◦ 180◦

θJN

10

15

20

25

30

35

40

45

50

55

DL

(Mp

c)

volumetric prior

EM distance prior

LIGO/Virgo PRL 119, 161101 (2017), arXiv:1805.11579Riccardo Sturani (IIP-UFRN) Gravitational Wave Astronomy DU - Oct 25th, 2019 37 / 29

Templates need to be 2 orders of magnitude more accurate for use in LISA than inAdvanced LIGO/Virgo.

SNR ∝ GNM5/6c

D

(∫f −4/3

Sn(f )

df

f

)1/2=

(GNMc )5/6

DS−1/2n (f̄ )

(f−4/3

i − f−4/3

f

)1/2=

(GNMc )5/6

DS−1/2n (fi )

f−2/3

i√∆f

f7/3

i

' 200(

S1/2n Hz

1/2

10−24

)−1 (η

0.25

)1/2 ( M10M�

)5/6 (f

150Hz

)−2/3 ( D500Mpc

)−1' 40

(S

1/2n Hz

1/2

10−21

)−1 (η

0.25

)1/2 ( M70M�

)5/6 (f

10−4Hz

)−7/6 ( D100Mpc

)−1×(

∆f10−5Hz

)1/2


Long duration signals

Remember that:dt

df=

5

96πη(πGNMf )

−5/3 f −2

t = 10yrs

(M

70M�

)−5/3( f10−2Hz

)−8/3

∆f ' 96π5

(πGNMc f )5/3 f 2∆t ' 10−3Hz

( η0.25

)( M70M�

)5/3( f10−2Hz

)2

fISCO ' 1.5kHz(

M

3M�

)−1fmax ' 20kHz

(M

3M�

)−1

fgw ' 134Hz(

1.21M�Mc

)5/8(1secτ

)3/8

τ ' 2.2sec(

1.21M�Mc

)5/3(100Hzfgw

)8/3

Need for longer duration templates, as for the third-generation groundbased detectors, since the signals are present in the data stream for months

〈F 〉 ∼ 1− D − 12SNR2


Upper bounds on sub-solar mass compact binariescoalescence rate

Mass/M� Rate(Gpc−3yr−1) Horizon(Mpc)

1 < 106 1000.2 < 2× 104 30

SNR ∝ M5/6c , Mc ≡ η3/5MLIGO/Virgo arXiv:1808.04771

Fraction f of MACHOs in binary∼ 10%× ΩMacho =⇒∼ 10−3ΩMacho in coalescing bina-ries coalescing within H−10

T. Nakamura et al. in Astrophys.J. 487

(1997) L139-L142Riccardo Sturani (IIP-UFRN) Gravitational Wave Astronomy DU - Oct 25th, 2019 40 / 29

Formula for the N of cycles x ≡ v2 ≡ (Mωs)2/3 = (πMfGW )2/3

N = 2

∫ v

v0

ω(v ′)dE/dv ′(v ′)

F (v)dv ′ =

2ηGNM

η2GNM

∫1

v6(1 + #v2 + . . .

)∝ 1ηv−5

v = 0.12

(M

10M�

fGW10Hz

)1/3=⇒ v−5 = 4.7×104

(M

10M�

fGW10Hz

)−5/35×104


The EFT point of view on the 2-body problem

Different scales in EFT (borrowing ideas from NRQCD):

Very short distance. rs = GNMnegligible up to 5PN(effacement principle)

Short distance: potentialgravitons Hµν , kµ ∼ (v/r , 1/r)with r ∼ rs/v2Long distance: GW’skµ ∼ (v/r , v/r) GWs hGWµνcoupled to point particles withmoments

M. Beneke and V. A. Smirnov, NPB ’98; I. Stewart and W. Manohar, PRD ’07; W.Goldberger and I. Rothstein, PRD ’06


EFT for compact binary systems

Fundamental

Fundamental gravitationalfields

Fundamental coupling toparticle world line

Effective

Generic v -dependent potentialterms

Instantaneous couplings betweenworld-line particles

exp [iSeff (xa, hGW )] =

∫DH(x) exp [iSEH (H + hGW ) + iSpp(H + hGW )]

Spp = −m∫

dτ

(√GN(h00/2 + vih0i + v

iv jhij/2)

+ GNh200 . . .

)

SEH =1

32πGN

∫d4x

[(∂h)2 + h(∂h)2 + h2(∂h)2 . . .

]

Seff = m1

∫dt d3x

[1

2v 21 +

GNm22r

+1

8v 41 +

G 2Nm22r 2

(GNm1

2r+ v 21 + v1rv2r

)+ 1↔ 2

]3PN: Jaranowski, Schäfer, Damour; Blanchet, Faye; Itoh Futamase; Foffa & RS

4PN: Jaranowski, Schäfer, Damour; Blanchet et al, RS Foffa; RS Foffa, Mastrolia, Sturm



Fundamental



Effective





Spp = −m∫

dt d3x δ(~x − ~xa(t))(h00/2 + vih0i + v

iv jhij/2 + h200 . . .

)

(√GN(h00/2 + vih0i + v

iv jhij/2)

+ GNh200 . . .

)

SEH =1

32πGN

∫d4x

[(∂ih)

2 − (∂th)2 + h(∂h)2 + h2(∂h)2 . . .]

Seff = m1

∫dt d3x

[1

2v 21 +

GNm22r

+1

8v 41 +

G 2Nm22r 2

(GNm1

2r+ v 21 + v1rv2r

)+ 1↔ 2





Fundamental



Effective





Spp = −m∫

dt d3x δ(~x − ~xa(t))(√

GN(h00/2 + vih0i + v

iv jhij/2)

+ GNh200 . . .

)SEH =

1

2

∫d4x

[(∂ih)

2 − (∂th)2 +√

GNh(∂h)2 + GNh

2(∂h)2 . . .]

Seff = m1

∫dt d3x

[1

2v 21 +

GNm22r

+1

8v 41 +

G 2Nm22r 2

(GNm1

2r+ v 21 + v1rv2r

)+ 1↔ 2




How to compute effective potential?

Iteratively solve Einstein equations for point particles (J(x) ∼ δ3(x − x̄)):

−∫

dtV1−2(t, r) ∼∫

dt dt′ d3x d3x ′J(x)G(x − x ′)J(x ′)

= 32πGNm1m2

∫dt dt′

d4k

(2π)4e ik

µ(x1(t1)−x2(t2))µ

k2 − ω2

' GN∫

dt dt′δ(t − t′) m1m2|x1(t)− x2(t′)|

where k0 has been neglected. Considering effects of v

V ∝∫

dωd3ke−iωt12+ikx12

k2 − ω2 =∫

dωd3ke−iωt12+ikx12

k2

(1 +

ω2

k2+ . . .

)= δ(t12)

∫d3k

e ikx12

k2

(1 +

∂t1∂t2k2

+ . . .

)=

∫d3k

e ikx12

k2

(1− k · v1k · v2

k2+ . . .

)


Trading knowledge over the full trajectory with knowledge of all derivativesof the trajectory at equal time

time

→ +v̇ + v̈ + . . .


Newtonian potential:

Seff ∼∫dt GN m1m2r ∼ L

At 1PN:

Seff =∫dt

G 2N m21m2

r2∼ Lv2

v

v

v2

V1PN = −Gm1m2

2r

[1− GNm1

2r+

3

2(v21 )−

7

2v1v2 −

1

2v1r̂ v2r̂

]+

1↔ 2


The 4PN sector

G: : 3 diagrams

G2: 21 diagrams, RS and S. Foffa, PRD ’12

G3: 212 diagrams

G4: 317 diagrams RS and S. Foffa ArXiv:1902.xxxx

G5: 50 diagrams, RS et al. PRD ’18


4PN static G5N topologies

M0,1M1,1 M1,2

M1,3 M1,4 M2,2

M3,6


G5N master integrals

expressed via integration by parts reduction in terms of 7 master integrals:

M0,1M1,1 M1,2

M1,3 M1,4 M2,2

M3,6Mapping the problem into 4-loop 3-D self-enrgy diagrams

RS, S. Foffa, P. Mastrolia, C. Sturm PRD (2017) 104009


2 body dynamics expansions (spin-less)

Post-Minkowskian expansion parameter is GNM/r , vs PN expansion

L = −Mc2 + µv2

2+

GMµ

r+

1

c2[. . .] +

1

c4[. . .]

Terms known so far

N 1PN 2PN 3PN 4PN 5PN . . .0PM 1 v2 v4 v6 v8 v10 v12 . . .1PM 1/r v2/r v4/r v6/r v8/r v10/r . . .2PM 1/r2 v2/r2 v4/r2 v6/r2 v8/r2 . . .3PM 1/r3 v2/r3 v4/r3 v6/r3 . . .4PM 1/r4 v2/r4 v4/r4 . . .5PM 1/r5 v2/r5 . . .. . . . . . . . .

3PM recently computed (!) by Z. Bern, C. Cheung et al. arXiv:1901.04424What’s next? Amplitude program with modern methods: generalizedunitarity, gravity as a double copy of gauge theory. . .


Fundamental Gravity tests at cosmological scale

GWs travel at the speed of light, with deviations . 1015

Model of dark energy/modified gravity with a single scalar degree offreedom drastically constrained

↓

standard conformal coupling to the Ricci scalar (for Horndeski theories)

G. W. Horndeski, Int. J. of Th. Phys. (1974) 10, 6, 363384T. Baker, E. Bellini, P. G. Ferreira, M. Lagos, J. Noller, and I. Sawicki, Phys. Rev. Lett.

119, 251301 (2017)P. Creminelli & F. Vernizzi, Phys. Rev. Lett. 119, 251302 (2017)

J. Sakstein & J. Jain, Phys. Rev. Lett. 119, 251303 (2017)J. M. Ezquiaga & M. Zumalacrregui, Phys. Rev. Lett. 119, 251304 (2017)

L. Amendola, M. Kunz, I. D. Saltas and I. Sawicki arXiv:1711.04825


EM/ν from binary black hole coalescences?

BHs with accretion disks may lead to relativistic outflows with TeV - PeVν (hadrons accelerated by jets, if magnetic fields and long-lived debris fromBH progenitor, or dense gaseous environment) exotic but not impossible

e.g. R. Perna et al. Ap.J. 821 (2016) 1, L18, I. Bartos et al., Ap.J. 835 (2017) 2, 165

GW150914:γ,X optical, radio counterparts not found

Ap.J. 826 (2016) 1, L13

ν: E � O(100) GeV, time window ±500 sec, coincidences ∼ backg.Limit on ν spectral fluence (10% - 90%): < 1051 − 1054 erg

LIGO, ANTARES and ICECube Phys.Rev. D93 (2016) no.12, 122010

GW151226 (LVT151012): Eiso < 2× 1051 − 2× 1054 ergLIGO, ANTARES, IceCube, Phys.Rev. D96 (2017) 2, 022005

GW170104: No ν detection with a ±500 sec window, nor in ±3months

ANTARES Eur.Phys.J. C77 (2017) 12, 911

For reference: EGW 150914 ∼ 1054, long(short) GRB Eiso ∼ 1051, (1049 erg)Riccardo Sturani (IIP-UFRN) Gravitational Wave Astronomy DU - Oct 25th, 2019 52 / 29

EM/ν from binary black hole coalescences? Nope

BHs with accretion disks may lead to relativistic outflows with TeV - PeVν (hadrons accelerated by jets, if magnetic fields and long-lived debris fromBH progenitor, or dense gaseous environment) exotic but not impossible

e.g. R. Perna et al. Ap.J. 821 (2016) 1, L18, I. Bartos et al., Ap.J. 835 (2017) 2, 165

GW150914:γ,X optical, radio counterparts not found

Ap.J. 826 (2016) 1, L13

ν: E � O(100) GeV, time window ±500 sec, coincidences ∼ backg.Limit on ν spectral fluence (10% - 90%): < 1051 − 1054 erg

LIGO, ANTARES and ICECube Phys.Rev. D93 (2016) no.12, 122010

GW151226 (LVT151012): Eiso < 2× 1051 − 2× 1054 ergLIGO, ANTARES, IceCube, Phys.Rev. D96 (2017) 2, 022005

GW170104: No ν detection with a ±500 sec window, nor in ±3months

ANTARES Eur.Phys.J. C77 (2017) 12, 911

For reference: EGW 150914 ∼ 1054, long(short) GRB Eiso ∼ 1051, (1049 erg)Riccardo Sturani (IIP-UFRN) Gravitational Wave Astronomy DU - Oct 25th, 2019 52 / 29

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