T O M M A S O D E L O R E N Z O

TDELORENZO@PSU.EDU

C O N S T R A I N T S O N GW W AV E F O R M SIN COLLABORAT ION W I TH ABHAY ASHTEKAR AND NEEV KHERA

CATAN IA , 3RD FLAG MEET ING , JUNE 14TH 2019

INST I TUTE FOR GRAV I TAT ION AND THE COSMOS , PENN STATE UN IVERS I TY

1906.00913 [GR-QC & ASTR O-PH .HE]

R E S U LT

R E S U LT

W H Y ?

R E S U LT

W H Y ?

H O W ?

FULL GR PR OV IDES AN INF IN I TE AMOUNT OF CONSTRA INTS THAT ANY GW WAVEFORM PR ODUCED BY COMPACT B INARY COALESCENCES (CBC) MUST SAT I SFY !

W H Y ?

H O W ?

R E S U LT

FULL GR PR OV IDES AN INF IN I TE AMOUNT OF CONSTRA INTS THAT ANY GW WAVEFORM PR ODUCED BY COMPACT B INARY COALESCENCES (CBC) MUST SAT I SFY !

W H Y ?

H O W ?

LIGO/V IR GO : RAW DATA - NO I SE + MATCH-F I LTER ING VS THEORET I CAL MODEL

R E S U LT

FULL GR PR OV IDES AN INF IN I TE AMOUNT OF CONSTRA INTS THAT ANY GW WAVEFORM PR ODUCED BY COMPACT B INARY COALESCENCES (CBC) MUST SAT I SFY !

W H Y ?

H O W ?

LIGO/V IR GO : RAW DATA - NO I SE + MATCH-F I LTER ING VS THEORET I CAL MODEL

THEORET I CAL MODEL = WAVEFORMS = APPR OX IMAT IONS AND AMB IGU I T I ES

R E S U LT

FULL GR PR OV IDES AN INF IN I TE AMOUNT OF CONSTRA INTS THAT ANY GW WAVEFORM PR ODUCED BY COMPACT B INARY COALESCENCES (CBC) MUST SAT I SFY !

W H Y ?

H O W ?

LIGO/V IR GO : RAW DATA - NO I SE + MATCH-F I LTER ING VS THEORET I CAL MODEL

THEORET I CAL MODEL = WAVEFORMS = APPR OX IMAT IONS AND AMB IGU I T I ES

IMPR OVED SENS I T I V I T Y AND AB UNDANCE OF EVENTS ASKS FOR BETTER CONTR OL !

R E S U LT

FULL GR PR OV IDES AN INF IN I TE AMOUNT OF CONSTRA INTS THAT ANY GW WAVEFORM PR ODUCED BY COMPACT B INARY COALESCENCES (CBC) MUST SAT I SFY !

W H Y ?

H O W ?

LIGO/V IR GO : RAW DATA - NO I SE + MATCH-F I LTER ING VS THEORET I CAL MODEL

THEORET I CAL MODEL = WAVEFORMS = APPR OX IMAT IONS AND AMB IGU I T I ES

EXACT GR: ASYMPTOT I CALLY FLAT SPACET IMES

IMPR OVED SENS I T I V I T Y AND AB UNDANCE OF EVENTS ASKS FOR BETTER CONTR OL !

R E S U LT

FULL GR PR OV IDES AN INF IN I TE AMOUNT OF CONSTRA INTS THAT ANY GW WAVEFORM PR ODUCED BY COMPACT B INARY COALESCENCES (CBC) MUST SAT I SFY !

W H Y ?

H O W ?

LIGO/V IR GO : RAW DATA - NO I SE + MATCH-F I LTER ING VS THEORET I CAL MODEL

THEORET I CAL MODEL = WAVEFORMS = APPR OX IMAT IONS AND AMB IGU I T I ES

EXACT GR: ASYMPTOT I CALLY FLAT SPACET IMES

BMS FLUXES + BOUNDARY COND I T IONS DEF IN ING CBC

IMPR OVED SENS I T I V I T Y AND AB UNDANCE OF EVENTS ASKS FOR BETTER CONTR OL !

R E S U LT

T E M P L AT E B A N K S

T E M P L AT E B A N K Sh = h+ + ih× I N THE SENS I T I V I T Y BAND OF MY DETECTOR

INSP IRAL MER GER R INGDOWN

T E M P L AT E B A N K Sh = h+ + ih× I N THE SENS I T I V I T Y BAND OF MY DETECTOR

INSP IRAL MER GER R INGDOWN

BEST WAY : GENERAL RELAT IV I T Y ANALYT I CALLY IMPOSS IBLE

T E M P L AT E B A N K Sh = h+ + ih× I N THE SENS I T I V I T Y BAND OF MY DETECTOR

BEST WAY : GENERAL RELAT IV I T Y ANALYT I CALLY IMPOSS IBLE

2ND BEST WAY : NUMER I CAL RELAT IV I T Y (NR) YES…B UT

INSP IRAL MER GER R INGDOWN

T E M P L AT E B A N K Sh = h+ + ih× I N THE SENS I T I V I T Y BAND OF MY DETECTOR

2ND BEST WAY : NUMER I CAL RELAT IV I T Y (NR) YES…B UT

3RD BEST WAY : APPR OX IMAT IONS METHODS YES…B UT

INSP IRAL MER GER R INGDOWN

BEST WAY : GENERAL RELAT IV I T Y ANALYT I CALLY IMPOSS IBLE

T E M P L AT E B A N K Sh = h+ + ih× I N THE SENS I T I V I T Y BAND OF MY DETECTOR

2ND BEST WAY : NUMER I CAL RELAT IV I T Y (NR) YES…B UT

POST-NEWTON IAN NR

EOB(NR)/BOB

PHENOM

INSP IRAL MER GER R INGDOWN

PERTURBAT ION THEORY

BEST WAY : GENERAL RELAT IV I T Y ANALYT I CALLY IMPOSS IBLE

3RD BEST WAY : APPR OX IMAT IONS METHODS YES…B UT

T E M P L AT E B A N K Sh = h+ + ih× I N THE SENS I T I V I T Y BAND OF MY DETECTOR

2ND BEST WAY : NUMER I CAL RELAT IV I T Y (NR) YES…B UT

POST-NEWTON IAN NR

EOB(NR)/BOB

PHENOM

INSP IRAL MER GER R INGDOWN

PERTURBAT ION THEORY

BEST WAY : GENERAL RELAT IV I T Y ANALYT I CALLY IMPOSS IBLE

3RD BEST WAY : APPR OX IMAT IONS METHODS YES…B UT

T E M P L AT E B A N K Sh = h+ + ih× I N THE SENS I T I V I T Y BAND OF MY DETECTOR

2ND BEST WAY : NUMER I CAL RELAT IV I T Y (NR) YES…B UT

POST-NEWTON IAN NR

EOB(NR)/BOB

PHENOM

INSP IRAL MER GER R INGDOWN

PERTURBAT ION THEORY

BEST WAY : GENERAL RELAT IV I T Y ANALYT I CALLY IMPOSS IBLE

3RD BEST WAY : APPR OX IMAT IONS METHODS YES…B UT

A S Y M P T OT I C F L AT N E S S BOUNDARY COND I T IONS DEF IN ING I SOLATED GRAV I TAT IONAL SYSTEMS

A S Y M P T OT I C F L AT N E S S

i+

ℑ−

i−

uℑ+ ℑ+

u1

u2

naℓa

mai∘ i∘

ℑ−

qab

qabmana ℓa

BOUNDARY COND I T IONS DEF IN ING I SOLATED GRAV I TAT IONAL SYSTEMS

FUTURE NULL INF IN I TY : 3-SURFACE ℑ+ 𝕊2 × ℝ

A S Y M P T OT I C F L AT N E S S

i+

ℑ−

i−

uℑ+ ℑ+

u1

u2

naℓa

mai∘ i∘

ℑ−

qab

qabmana ℓa

BOUNDARY COND I T IONS DEF IN ING I SOLATED GRAV I TAT IONAL SYSTEMS

K INEMAT I CAL STRUCTURE ( ∘qab , ∘na)

FUTURE NULL INF IN I TY : 3-SURFACE ℑ+ 𝕊2 × ℝ

A S Y M P T OT I C F L AT N E S S

i+

ℑ−

i−

uℑ+ ℑ+

u1

u2

naℓa

mai∘ i∘

ℑ−

qab

qabmana ℓa

BOUNDARY COND I T IONS DEF IN ING I SOLATED GRAV I TAT IONAL SYSTEMS

3-PARAMETERS FAM I LY OF BOND I -FRAMES RELATE BY ASYMPTOT I C BOOSTS

(∘qab = ω2 ∘qab ,

∘na = ω−1na)

ω = γ (1 − v /c ⋅ x) γ = (1 − (v/c)2)−1

KINEMAT I CAL STRUCTURE ( ∘qab , ∘na)

FUTURE NULL INF IN I TY : 3-SURFACE ℑ+ 𝕊2 × ℝ

A S Y M P T OT I C F L AT N E S S

i+

ℑ−

i−

uℑ+ ℑ+

u1

u2

naℓa

mai∘ i∘

ℑ−

qab

qabmana ℓa

BOUNDARY COND I T IONS DEF IN ING I SOLATED GRAV I TAT IONAL SYSTEMS

3-PARAMETERS FAM I LY OF BOND I -FRAMES RELATE BY ASYMPTOT I C BOOSTS

(∘qab = ω2 ∘qab ,

∘na = ω−1na)

ω = γ (1 − v /c ⋅ x)

σ∘ = rhN = − ℒnσ∘ = − ·σ∘ NEWS = FREE DATA

γ = (1 − (v/c)2)−1

KINEMAT I CAL STRUCTURE ( ∘qab , ∘na)

DYNAM ICAL STRUCTURE

FUTURE NULL INF IN I TY : 3-SURFACE ℑ+ 𝕊2 × ℝ

SHEAR OF ℓa

A S Y M P T OT I C F L AT N E S S

i+

ℑ−

i−

uℑ+ ℑ+

u1

u2

naℓa

mai∘ i∘

ℑ−

qab

qabmana ℓa

BOUNDARY COND I T IONS DEF IN ING I SOLATED GRAV I TAT IONAL SYSTEMS

3-PARAMETERS FAM I LY OF BOND I -FRAMES RELATE BY ASYMPTOT I C BOOSTS

(∘qab = ω2 ∘qab ,

∘na = ω−1na)

ω = γ (1 − v /c ⋅ x)

σ∘ = rhN = − ℒnσ∘ = − ·σ∘ NEWS = FREE DATA

γ = (1 − (v/c)2)−1

LEAD ING ORDER OF WEYL : Ψ∘2 = lim

r→∞r3Cabcd manbℓcmd

Ψ∘1 = lim

r→∞r4Cabcd ℓambℓcnd

KINEMAT I CAL STRUCTURE ( ∘qab , ∘na)

DYNAM ICAL STRUCTURE

FUTURE NULL INF IN I TY : 3-SURFACE ℑ+ 𝕊2 × ℝ

SHEAR OF ℓa

BMS S Y M M E T R I E S

𝒫 = 𝒯 ⋊ L

ℑ−

i−

uℑ+ ℑ+

u1

u2

naℓa

mai∘ i∘

ℑ−

qab

qabmana ℓa

i+

BMS S Y M M E T R I E S

𝔅 = 𝒮 ⋊ L𝒫 = 𝒯 ⋊ L

ℑ−

i−

uℑ+ ℑ+

u1

u2

naℓa

mai∘ i∘

ℑ−

qab

qabmana ℓa

i+

BMS S Y M M E T R I E S

𝔅 = 𝒮 ⋊ L𝒫 = 𝒯 ⋊ L

SUPERTRANSLAT IONS 𝒮 ∈ ξa( f ) = f(θ, ϕ) ∘na

ℑ−

i−

uℑ+ ℑ+

u1

u2

naℓa

mai∘ i∘

ℑ−

qab

qabmana ℓa

i+

BMS S Y M M E T R I E S

𝔅 = 𝒮 ⋊ L𝒫 = 𝒯 ⋊ L

SUPERTRANSLAT IONS 𝒮 ∈

ℑ−

i−

uℑ+ ℑ+

u1

u2

naℓa

mai∘ i∘

ℑ−

qab

qabmana ℓa

B A L A N C E L A W

P( f )(u1) − P( f )(u2) = ℱ( f )(θ, ϕ)

i+

ξa( f ) = f(θ, ϕ) ∘na

BMS S Y M M E T R I E S

𝔅 = 𝒮 ⋊ L𝒫 = 𝒯 ⋊ L

SUPERTRANSLAT IONS 𝒮 ∈

B A L A N C E L A W

P( f )(u1) − P( f )(u2) = ℱ( f )(θ, ϕ)

P( f )(u) := −1

4πG ∮Cu

d2 ∘V f Re[Ψ∘

2 + ·σ∘σ∘]

ℱ( f )(u, θ, ϕ) :=1

4πG ∫u2

u1

du∮Cu

d2 ∘V f [ | ·σ∘ |2 − Re(ð2 ·σ∘)]

ℑ−

i−

uℑ+ ℑ+

u1

u2

naℓa

mai∘ i∘

ℑ−

qab

qabmana ℓa

i+

ξa( f ) = f(θ, ϕ) ∘na

∫u2

u1

du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = Re[Ψ∘2 + ·σ∘σ∘]

u2

u1

T H E R E S U LT

T H E R E S U LT

LET’S RESTR I CT TO CBCS : TWO ASSUMPT IONS

∫u2

u1

du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = Re[Ψ∘2 + ·σ∘σ∘]

u2

u1

T H E R E S U LT

LET’S RESTR I CT TO CBCS : TWO ASSUMPT IONS

1- ENER GY CONSERVAT ION :

u → ± ∞ 1/u1+ϵ ϵ > 0FOR SOME .AS THE NEWS GOES TO ZER O AS

∫u2

u1

du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = Re[Ψ∘2 + ·σ∘σ∘]

u2

u1

T H E R E S U LT

LET’S RESTR I CT TO CBCS : TWO ASSUMPT IONS

1- ENER GY CONSERVAT ION :

IN THE PAST REST-FRAME , AND

IN THE FUTURE REST-FRAME .

2- ASYMPTOT I C STAT IONAR I TY :

∂uΨ∘1 → 0 AS

AS

u → − ∞u → + ∞

∫u2

u1

du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = Re[Ψ∘2 + ·σ∘σ∘]

u2

u1

u → ± ∞ 1/u1+ϵ ϵ > 0FOR SOME .AS THE NEWS GOES TO ZER O AS

T H E R E S U LT

LET’S RESTR I CT TO CBCS : TWO ASSUMPT IONS

1- ENER GY CONSERVAT ION :

2- ASYMPTOT I C STAT IONAR I TY :

∫+∞

−∞du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = Re[Ψ∘

2 + ·σ∘σ∘]u=−∞

u=+∞

I N THE PAST REST-FRAME , AND

IN THE FUTURE REST-FRAME .∂uΨ∘

1 → 0 AS

AS

u → − ∞u → + ∞

u → ± ∞ 1/u1+ϵ ϵ > 0FOR SOME .AS THE NEWS GOES TO ZER O AS

T H E R E S U LT

LET’S RESTR I CT TO CBCS : TWO ASSUMPT IONS

∫+∞

−∞du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = Re[Ψ∘

2]u=−∞

u=+∞

1- ENER GY CONSERVAT ION :

2- ASYMPTOT I C STAT IONAR I TY :

IN THE PAST REST-FRAME , AND

IN THE FUTURE REST-FRAME .∂uΨ∘

1 → 0 AS

AS

u → − ∞u → + ∞

u → ± ∞ 1/u1+ϵ ϵ > 0FOR SOME .AS THE NEWS GOES TO ZER O AS

T H E R E S U LT

LET’S RESTR I CT TO CBCS : TWO ASSUMPT IONS

∫+∞

−∞du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = Re[Ψ∘

2]u=−∞

u=+∞

1- ENER GY CONSERVAT ION :

u → ± ∞ 1/u1+ϵ ϵ > 0FOR SOME .AS THE NEWS GOES TO ZER O AS

IN THE PAST REST-FRAME , AND

IN THE FUTURE REST-FRAME .

2- ASYMPTOT I C STAT IONAR I TY :

∂uΨ∘1 → 0 AS

AS

u → − ∞u → + ∞

Ψ∘2

SPHER I CALLY SYMMETR I C

T H E R E S U LT

LET’S RESTR I CT TO CBCS : TWO ASSUMPT IONS

1- ENER GY CONSERVAT ION :

u → ± ∞ 1/u1+ϵ ϵ > 0FOR SOME .AS THE NEWS GOES TO ZER O AS

IN THE PAST REST-FRAME , AND

IN THE FUTURE REST-FRAME .

2- ASYMPTOT I C STAT IONAR I TY :

∂uΨ∘1 → 0 AS

AS

u → − ∞u → + ∞

PAST REST-FRAME limu→−∞

Ψ∘2 = − GMi∘

FUTURE REST-FRAME limu→+∞

Ψ∘2 = − GMi+

Ψ∘2

SPHER I CALLY SYMMETR I C

∫+∞

−∞du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = Re[Ψ∘

2]u=−∞

u=+∞

T H E R E S U LT

LET’S RESTR I CT TO CBCS : TWO ASSUMPT IONS

1- ENER GY CONSERVAT ION :

u → ± ∞ 1/u1+ϵ ϵ > 0FOR SOME .AS THE NEWS GOES TO ZER O AS

IN THE PAST REST-FRAME , AND

IN THE FUTURE REST-FRAME .

2- ASYMPTOT I C STAT IONAR I TY :

∂uΨ∘1 → 0 AS

AS

u → − ∞u → + ∞

PAST REST-FRAME limu→−∞

Ψ∘2 = − GMi∘

FUTURE REST-FRAME limu→+∞

Ψ∘2 = − GMi+

limu→+∞

Ψ∘2 =

−GMi∘

γ3 (1 − vc ⋅ x)

3

Ψ∘2

SPHER I CALLY SYMMETR I C

∫+∞

−∞du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = Re[Ψ∘

2]u=−∞

u=+∞

T H E R E S U LT

LET’S RESTR I CT TO CBCS : TWO ASSUMPT IONS

1- ENER GY CONSERVAT ION :

u → ± ∞ 1/u1+ϵ ϵ > 0FOR SOME .AS THE NEWS GOES TO ZER O AS

IN THE PAST REST-FRAME , AND

IN THE FUTURE REST-FRAME .

2- ASYMPTOT I C STAT IONAR I TY :

∂uΨ∘1 → 0 AS

AS

u → − ∞u → + ∞

PAST REST-FRAME limu→−∞

Ψ∘2 = − GMi∘

FUTURE REST-FRAME limu→+∞

Ψ∘2 = − GMi+

limu→+∞

Ψ∘2 =

−GMi∘

γ3 (1 − vc ⋅ x)

3

Ψ∘2

SPHER I CALLY SYMMETR I C

∫+∞

−∞du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = GMi∘ −

GMi+

γ3 (1 − vc cos θ)

3

C O N C L U S I O N S

DEF IN I T ION OF A NEW MEASURE OF REL I AB I L I T Y OF WAVEFORMS FOR GW DATA ANALYS I S .

F IRST T IME WE CAN CHECK WAVEFORMS AGA INST EXACT GENERAL RELAT IV I T Y !

∫+∞

−∞du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = GMi∘ −

GMi+

γ3 (1 − vc cos θ)

3

C O N C L U S I O N S

F U T U R E W O R KTAKE THE WAVEFORMS IN THE BANKS AND TEST THEM!

F IRST T IME WE CAN CHECK WAVEFORMS AGA INST EXACT GENERAL RELAT IV I T Y !

∫+∞

−∞du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = GMi∘ −

GMi+

γ3 (1 − vc cos θ)

3

DEF IN I T ION OF A NEW MEASURE OF REL I AB I L I T Y OF WAVEFORMS FOR GW DATA ANALYS I S .

C O N C L U S I O N S

ANGULAR MOMENTUM AMB IGU I TY . IN PREPARAT ION .

F U T U R E W O R KTAKE THE WAVEFORMS IN THE BANKS AND TEST THEM!

F IRST T IME WE CAN CHECK WAVEFORMS AGA INST EXACT GENERAL RELAT IV I T Y !

∫+∞

−∞du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = GMi∘ −

GMi+

γ3 (1 − vc cos θ)

3

DEF IN I T ION OF A NEW MEASURE OF REL I AB I L I T Y OF WAVEFORMS FOR GW DATA ANALYS I S .

C O N C L U S I O N S

A NOVEL TEST OF GR? WORK IN PR OGRESS .

ANGULAR MOMENTUM AMB IGU I TY . IN PREPARAT ION .

F U T U R E W O R KTAKE THE WAVEFORMS IN THE BANKS AND TEST THEM!

F IRST T IME WE CAN CHECK WAVEFORMS AGA INST EXACT GENERAL RELAT IV I T Y !

∫+∞

−∞du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = GMi∘ −

GMi+

γ3 (1 − vc cos θ)

3

DEF IN I T ION OF A NEW MEASURE OF REL I AB I L I T Y OF WAVEFORMS FOR GW DATA ANALYS I S .

C O N C L U S I O N S

A NOVEL TEST OF GR? WORK IN PR OGRESS .

ANGULAR MOMENTUM AMB IGU I TY . IN PREPARAT ION .

F U T U R E W O R KTAKE THE WAVEFORMS IN THE BANKS AND TEST THEM!

F IRST T IME WE CAN CHECK WAVEFORMS AGA INST EXACT GENERAL RELAT IV I T Y !

E I THER SUPERTRANSLAT ION CONSERVAT ION OR SUPERR OTAT ION?

∫+∞

−∞du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = GMi∘ −

GMi+

γ3 (1 − vc cos θ)

3

DEF IN I T ION OF A NEW MEASURE OF REL I AB I L I T Y OF WAVEFORMS FOR GW DATA ANALYS I S .

C O N C L U S I O N S

A NOVEL TEST OF GR? WORK IN PR OGRESS .

ANGULAR MOMENTUM AMB IGU I TY . IN PREPARAT ION .

F U T U R E W O R KTAKE THE WAVEFORMS IN THE BANKS AND TEST THEM!

F IRST T IME WE CAN CHECK WAVEFORMS AGA INST EXACT GENERAL RELAT IV I T Y !

E I THER SUPERTRANSLAT ION CONSERVAT ION OR SUPERR OTAT ION?

∫+∞

−∞du [ | ·σ∘ |2 − Re(ð2 ·σ∘)] = GMi∘ −

GMi+

γ3 (1 − vc cos θ)

3

DEF IN I T ION OF A NEW MEASURE OF REL I AB I L I T Y OF WAVEFORMS FOR GW DATA ANALYS I S .

T H A N K Y O U !