1.
Effective Field Theory
Approach to Testing GR
AIMS, Cape Town
September 1st 2010
Hillary Sanctuary
Friday, September 17, 2010
2.
Outline
I. Brief introduction to GWs
II. Motivation : probe non-linearities of GR
III. Tools : Effective Field Theory Method (NRGR)
IV. Measuring deviations from GR. Physics beyond GR?
Friday, September 17, 2010
3.
Results
• Measurement of radiation loss from the Hulse-
Taylor Binary provides an important conﬁrmation
of the gauge structure of GR to 0.1%.
• Careful: provides experimental bounds unknown
physics IF General Relativity is the correct low-
energy theory of gravity.
Friday, September 17, 2010
4.
I. Introduction to GW
• Gravitational waves are oscillations of the
“fabric” of spacetime.
• To see this (weak ﬁeld limit) of Einstein’s eqns:
gµν ηµν hµν
|hµν |
Friday, September 17, 2010
6.
Binary Pulsar : important test
A pulsar is a highly magnetized, rotating neutron star that emits EM radiation.
A binary pulsar is a pulsar + companion.
Friday, September 17, 2010
7.
Hulse-Taylor Binary (1974)
The pulsar rotates on its axis17 times per second, pulse period is 59ms.
The orbital period is 7.75 hours.
Separation at periastron: 1,000,000 km.
Inspiral for 300,000,000 years.
Gravitational waves?
Friday, September 17, 2010
8.
Inferred detection.
Orbital decay due to energy loss via gravitational radiation.
The data points indicate the observed
change in the epoch of periastron with
date while the parabola illustrates the
theoretically expected change in epoch
according to GR. astro-ph/0407149
Friday, September 17, 2010
9.
Quest for direct detection
Network of GW detectors
(LIGO,Virgo, LISA,etc)
Friday, September 17, 2010
10.
A closer look
Animation: h ∼ 0.5
f ∼ 2 seconds
BH coalescence:
(10M⊙ , r = 10RS , R = 100 Mpc)
h ∼ 10−21
f ∼ 100 seconds−1
Friday, September 17, 2010
11.
Estimates at interferometers
Physical effect of a passing GW, perturb relative positions of freely-falling masses:
δL
∼h
L
h L
δL ∼ 10 −16
cm
10 −21 km
Bohr radius (atom): a0 ∼ 5 × 10−9 cm
Nucleus: 1 f m = 10−13 cm
Need to measure miniscule perturbations.
Friday, September 17, 2010
12.
Sources Sensitivity Curves
SB
SB
h
th
ed
SB
rs
nc
h
Fi
Best candidates for
ed
ha
nc
En
25BH/25BH
va
10-20 Inspiral @ 150Mpc
Rin
earth-based detectors:
Ad
Collision gd
SN hang
own
NS/NS @ up @ 10
30Mpc o 0km, 15M
r 10BH/1
BH and NS
0BH @ 1 pc
hc = h!n 50Mpc
gup c
han 15Mp
10-21
@2
SN km ,
0
coalescences.
Rin
Inspiral
gdo
Collision
NS/NS @
300Mpc
wn
or 10BH
/10BH @
150
30B 0Mpc
SN Boiling @ z H/30B
10-22 at 20 kpc =1 H
10 100 1000 10000
frequency, Hz
igure 3: LIGO’s projected broad-band noise sensitivity to bursts hSB (Refs. [12, 15]) compared
ith the characteristic amplitudes hc of the waves from several hypothesized sources. The signal
√
o noise ratios are 2 higher than in Ref. [12] because of a factor 2 error in Eq. (29) of Ref. [11].
Friday, September 17, 2010
13.
Example: NS-NS coalescence 3.1 Inspiral of binary stars 137
!
Fig. 3.2 The time evolution of the GW amplitude in the inspiral phase of a binary
system.
1.45M⊙
Using the explicit expression (3.16) we ﬁnd (recall that ∂/∂τ = −∂/∂t)
Φ(τ ) = −2 45 kmτ
5GM
c3
c
−5/8
5/8
+ Φ0 , (3.29)
where Φ0 = Φ(τ = 0) is an arXiv:0804.0594
integration constant, equal to the value of Φ
at coalescence. Since both fgw , given in eq. (3.16) and Φ in eq. (3.29)
depend only on the combination τ , the GW amplitude can be expressed
directly in terms of the time t of the observer, and the time of coalescence
of the observer, tcoal ,
5/4 1/4
1 GMc 5 1 + cos2 ι
h+ (t) = cos [Φ(τ )] ,
r c2 cτ 2
5/4 1/4
1 GMc 5
h× (t) = cos ι sin [Φ(τ )] , (3.30)
r c2 cτ
where
Friday, September 17, 2010
14.
Example: NS-NS coalescence 3.1 Inspiral of binary stars 137
!
Fig. 3.2 The time evolution of the GW amplitude in the inspiral phase of a binary
system.
1.45M⊙
Using the explicit expression (3.16) we ﬁnd (recall that ∂/∂τ = −∂/∂t)
Φ(τ ) = −2 45 kmτ
5GM
c3
c
−5/8
5/8
+ Φ0 , (3.29)
where Φ0 = Φ(τ = 0) is an arXiv:0804.0594
integration constant, equal to the value of Φ
at coalescence. Since both fgw , given in eq. (3.16) and Φ in eq. (3.29)
depend only on the combination τ , the GW amplitude can be expressed
directly in terms of the time t of the observer, and the time of coalescence
of the observer, tcoal ,
5/4 1/4
1 GMc 5 1 + cos2 ι
h+ (t) = cos [Φ(τ )] ,
r c2 cτ 2
5/4 1/4
1 GMc 5
h× (t) = cos ι sin [Φ(τ )] , (3.30)
r c2 cτ
where
Friday, September 17, 2010
15.
II. Motivation
• GR is non-linear
• Two labs where non-linearities are at work
(binary pulsars, coalescences)
• Non-linearities (non-abelian vertices):
• Can we measure these vertices? YES
(Tevatron, LEP for SM, deviations from GR?)
Friday, September 17, 2010
16.
Post-Newtonian Expansion
• What is it? Post-Newtonian terms in GR are
corrections to Newtonian Gravity in powers of
v/c 1 (c = 1)
• Why? GW experiments such as VIRGO/LIGO
need high order corrections for detection (phase).
• Several techniques available to calculate PN
corrections, UV divergences.
Friday, September 17, 2010
17.
EFT method
• UV divergences due to ﬁeld theory coupled to
point-particles.
• Reformulation of PN corrections in terms of
Feynman diagrams hep-th/0409156
• NRGR : binary inspiral as at EFT calculation
Track this contribution in observables.
Friday, September 17, 2010
18.
III. Tools: NRGR Tutorial
• “Non relativistic General Relativity”
• Translates PN corrections of N-body (binary)
systems into Feynman diagrams.
• Powerful toolbox of EFT techniques (UV
divergences), manifest power-counting.
Friday, September 17, 2010
19.
Main Ingredients of NRGR
• Identify relevant dof at the scale of interest.
gµν , x (λ)
µ
• Construct effective Lagrangian allowed by
µ
symmetries. GCI :x → x (x)
µ
Worldline RPI :λ → λ (λ)
• Separate into conservative gravitons H
and radiative gravitons h
• Identify how terms scale in powers of v
• Collect Feynman diagrams to desired powers in v
Friday, September 17, 2010
20.
Relevant Scales of a Binary System
We want two EFTs,
one at the orbital scale of the binary,
the other at the scale of radiation.
Friday, September 17, 2010
21.
Starting point
Theory of relativistic point particles coupled to gravity:
S = SEH + Spp
√
SEH =−2m2 l
P
4
d x gR(x)
Spp =− ma dτa + ... O(v 10 ) Finite size effect
Neglect spin
a
dτ 2 = gµν dxµ dxν
Friday, September 17, 2010
22.
Potential radiation gravitons
Friday, September 17, 2010
23.
H and h scale differently
k µ ∼ (v/r, 1/r)
k µ ∼ (v/r, v/r)
Friday, September 17, 2010
25.
Feynman Building Blocks:
Graviton Vertices
1 ¯
gµν = ηµν + (Hµν + hµν )
MP l
Friday, September 17, 2010
26.
Feynman Building Blocks:
Matter Vertices
1 ¯
gµν = ηµν + (Hµν + hµν )
MP l
Friday, September 17, 2010
27.
Which diagrams are allowed?
Friday, September 17, 2010
28.
Orbital Scale EFT
Newton’s Potential (0 PN)
Friday, September 17, 2010
29.
Orbital Scale EFT
EIH Lagrangian (1 PN)
Friday, September 17, 2010
30.
Tools
Basic Results in NRGR
The EIH Lagrangian
The potential Lagrangian (orbital scale)
1
L = L0 + L2
c2
1 2 1 2 Gm1 m2
L0 = m1 v1 + m2 v2 +
2 2 r
1 2 1 4 Gm1 m2
L2 = m1 v2 + m2 v2 +
8 8 2r
2 2 G(m1 + m2 )
× 3(v1 + v2 ) − 7v1 · v2 − (ˆ · v1 )(ˆ · v2 ) −
r r
r
The effective Lagrangian (conservative dynamics) valid to 1PN.
Friday, September 17, 2010
31.
The 2PN Radiation Lagrangian
The 1PN radiation Lagrangian is identically zero in the CM. The
Radiation Scale EFT
2PN radiation Lagrangian is the next-to-leading correction :
v2
h00 1 2 Gm1 m2
Lrad = − ma va −
2MP l 2 a
r
1 1 O(v 5/2 )
− ijk Lk ∂j hi0 + ma rai raj R0i0j
2MP l 2MP l a
Friday, September 17, 2010
32.
Tools
Basic Results in NRGR
Quadrupole Radiation Forumla
Extracting the Three-Graviton and Four-Graviton Vertices from Binary Pulsars and Coalescing Binaries
Tools
Quadrupole Radiation (2.5PN)
Basic Results in NRGR
Quadrupole Radiation Forumla
2
1 1 1
Im −i 2MP l a,b dt1 dt2 Qij (t1 )Qkl (t2 ) R0i0j (t1 , Xcm )R0k0l (t2 , Xcm )
2
1 1 1
Im −i 2MP l a,b dt1 dt2 Qij (t1 )Qkl (t2 ) R0i0j (t1 , Xcm )R0k0l (t2 , Xcm )
GN ... ...
P = Q... Q
G5 ... ij ij
N
P = Qij Qij
5
where the quadrupole moment is deﬁned by:
where the quadrupole moment is deﬁned by:
1 12 2
Qij ij =
Q= ma rraiajaj −δij rδij ra
ma ai r r −
3 3a
aa
Friday, September 17, 2010
33.
v2
The radiation Lagrangian (2.5PN)
h00 1 2 Gm1 m2
Lrad = − ma va −
2MP l 2 a
r
1 1
− ijk Lk ∂j hi0 + ma rai raj R0i0j
2MP l 2MP l a
The effective Lagrangian responsible for radiation to leading order.
Friday, September 17, 2010
34.
Extracting the Three-Graviton and Four-Graviton Vertices from Binary Pulsars and Coalescing Binaries
Tools
Basic Results in NRGR
NRGR in a nutshell
NRGR Summary
We saw:
The diﬀerent length scales at hand (rs , r ∼ H, h)
How to build the NRGR EFT
How to get Lconservative with PN corrections
(Newton + EIH + ...)
How to get Lradiation with PN corrections.
More importantly, we have seen exactly how the 3- and 4- graviton
More importantly, wePN corrections. how the
vertices contribute to these have seen
three graviton vertex contributes to the
-1PN correction (EIH)
- quadrupole radiation
Friday, September 17, 2010
35.
Extracting the Three-Graviton and Four-Graviton Vertices from Binary Pulsars and Coalescing Binaries
Results
Modiﬁcation of the 3-graviton vertex
IV. Measuring deviations (GR)
Modifying the 3g Vertex
arxiv:0907.2186v2
→ (1 + β3 )×
This factor will aﬀect in particular the following diagrams HHH
and HHh:
(1 + β3 ) (1 + β3 )
Friday, September 17, 2010
36.
Extracting the Three-Graviton and Four-Graviton Vertices from Binary Pulsars and Coalescing Binaries
Results
Comparing with Classical Tests of GR
Estimating β3 :Estimating β3
Classical Tests (orbit)
By modifying the three-graviton vertex, we have modiﬁed the
conservative part of the Lagrangian:
Modiﬁcation to the 1PN Lagrangian
G2 m1 m2 (m1 +m2 )
∆Lcons = −β3 N r2
In terms of PPN parameters, we can identify β = 1 + β3 and γ = 1.
Perihelion of Mercury
|β − 1| 3 · 10−3 −→ |β3 | 3 · 10−3
Lunar Laser Ranging
|4β − γ − 3| 9 · 10−4 −→ |β3 | 2 · 10−4
Friday, September 17, 2010
37.
N
possible vertices, and we must Qij Qij the framework of multiscalar-tensor theories, the e
(17)
ction of
PQQ = G compute, the the PPN formalism introduced in Ref [11] a
in imaginary(22)
part of PQQ = 5 N Q Q , of
tension
··· ···
(22)
ij lows for a consistent treatment of both the conservati
ij
(18) 5
ith our as already found in [34]. Computingdynamics (including the eﬀects. strongly self-gravitatin
4 the otherradiative case of
contribu-
−i bodies)band of
(18) a b a
tions 2 ﬁnd thatdt2 I[34]. )Ikl (t2 ) Sij (t1 Pqq (t2 ) interesting to see what bounds on β3 ca
as already found in ijterms PQq andIt)Skl vanish(21)
we dt1 the (t1 Computing the other contribu-
is clearly , identi-
8MPl
Estimating β (radiation)
cally. In fact, the Qq and qQ graphs vanish becauseidenti- probe the radiati
tions we ﬁnd that the terms PQq and Pqq vanishtheQij thatbinary pulsars or th
be obtained from experiments
(19) a,b=1
j
h(19)
3 sector of because δij δkl of of
GR, such as
cally. In fact, the qδ , and δqQβgraphs vanish on the Q
Qq β V , Z )observation because
timing
is traceless, while the qq graph vanishes of the coalescence ij compact binaries at i
ij )] , a
where Iij = (Qij , ij 3 ij 3 ij terferometers. The eﬀective Lagrangian describing th
depends
is the gives zero when contracted δik δjl + vanishes2 δijh kl , on δij δkl
is traceless, while S a = graph δil δ , δ− 3 of, δij ) which
matter variables andthe qq (R0i0j , R0i0jjk ij h002 the binary system with radiation gravito
because
(16) interaction
is the tensor thatﬁeld. ijWhenδboth the il δjk −bythefunctionthree graphs in Fig. 2 (co
the gravitational contracted from +is obtained 3 δij δdia-which
gives zero when comes out ik δmore complicated.
δ two-point kl ,
0i0i ,and
is is
rce the The R0k0l . The formula and V Z graphs Fig. 6 of ref.for and the introduction
R the
radiation QV , qZ is responding to vanish [34]),
jl vertices of computing the
0i0i ,=
β3 is gram in Fig. 3that proportionalfrom the two-pointone in Fig. 2c.
is 0i0j tensor are comes out to the 3quadrupole, vertex
β aﬀects the HHh
function
similar R0k0l . The QV , qZ and V contributions come in ref. [34], but with o
obtains reasons, so the only relevant Z graphs vanish as
R0i0j the usual GR result Computing these graphs for
r orbit
β3 =
reorbit
the
(17) from thereasons, so the G ···and we ﬁnd three-graviton vertex, we ﬁnd
similar QZ and qV graphs, relevant contributions come
only ··· modiﬁed
N
cation
re the from the QZ andQQ =graphs, and ,we ﬁnd1 [Qij R0i0j + qR0i0i + β3 (3V h00 + Z ij hij )]
P qV
5
Qij Qij
··· Lrad =
˙
(22)
PQZ = −2β3 GN Qij Zij , 2MPl (23)
cation(18) in ···
˙
as already found P [34]. ComputingQ where Qij is the quadrupole moment of the source an
theZ , contribu- (23)
other
(1
upled and QZ = −2β3 GN ij ij
tions we ﬁnd that the terms PQq and weqq vanish identi-
P deﬁne
upled
oupled cally. In fact, the Qq and qQ graphs vanish because Qij 1
(19) and ··· ˙
PqV = −6β3 Gvanishes. because δ q = 3 ma x2 ,
terms
oupled is traceless, while the qq graph N qV (24) (1
ij δkl
a
a
··· ˙ 2 δ δ , which
)terms
is the
e can As for thewhen and ZZ= −6β3+they V 3 ij contributionm1 m2 ,
gives zero
V V contracted δik δjl Gδil δq give a kl V (r) = GN r
PqV graphs, N jk − . (24) (1
R0i0i , is is the tensor that comes out from the two-point function
that, from theVpoint of view of the multipole expansion,m1 m2 ri rj
As0i0j R0k0l . The QV , qZ and V Z graphs vanish(r) = GN 3
R for the V and ZZ graphs, they give a contribution
no same order .
ation
or β3 =
ce can Z ij for , (1
Note:thedipole radiationas the quadrupole radiation but pro- r
is of from the so the only relevant contributions expansion,
e orbit similar reasons, point of view of the multipole come
that,
uation
(20)
2
portional QZ β3 , and graphs, and we ﬁnd r = x1 − x2 . The term Qij R0i0j in eq. (16) is th
to and qV can be neglected. where
from the same order as the quadrupole radiation but pro-
are the is of the
We can now use these results to perform the compari-The second term, qR0i0i ,
2
usual quadrupole interaction.
ﬁcation
(20) portional to β3 , and can be neglected.
···
˙ ijnon-radiating when β3 =radiated powersee thatthe orb
0, but we will for β3
he or- son with binary QZ = −2β3 GN Qij Z 0 it contributes to(23)
P pulsars and with interferometers. the
, when
We can now use these results to isperform the compari- in eq. (16) are th
non-circular. The last two terms
on for
coupled 17,and
Friday, September 2010 explicit β3 -dependent terms induced by the modiﬁcatio
38.
Extracting the Three-Graviton and Four-Graviton Vertices from Binary Pulsars and Coalescing Binaries
Results
Comparing with the Hulse-Taylor Binary
The Hulse-Taylor Binary Pulsar
Estimating inβ3
Extracting the Three-Graviton and Four-Graviton Vertices from Binary Pulsars and Coalescing Binaries
The Hulse-Binary Pulsar was ﬁrst detected 1974.
Results
Modiﬁcation of the 3-graviton vertex
The classical GR results for the period slow-down is:
Calculating the Power
Period Slow-Down
Plugging in the Newtonian equations of motion for elliptic orbits,
˙
P GR GR −8/3
Pb
96 5/3
we have: 5/3
b
GR
Pb
= − 5 GN ν M 2π [f (e)]
32G4 µ2 M 3 1 73 37
PQQ = N
1 + e2 + e4 ,
5a5 (1 − e2 )7/2 24 96
32G4 µ2 M 3 1 5 175 85
PQZ = β3period5 as calculated with the2 modiﬁed,
The slow-down
N
+ e + e4
5a (1 − e2 )7/2 2 24 96
three-graviton vertex:4 µ2 M 3
32GN 1 5 2 5
˙β PqV = −β3 β
β3−8/3 2 7/2 e + e4 .
Pb 96 5/3 5a5 Pb
5/3 (1 − e ) 16 64
β =− 5 GN ν M 2π [f (e) + β3 g(e)]
Pb
What M =can + m2extract m1 m2 /M 2
information m1 we ν = from this? M1 M2
˙β ˙ ν=
0≤e Pb = 0 P GR .
Compare≤ 1, ewithfor bcicular orbits. M
M = M1 + M2
Friday, September 17, 2010
39.
Extracting the Three-Graviton and Four-Graviton Vertices from Binary Pu
acting the Three-Graviton and Four-Graviton Vertices from Binary Pulsars and Coalescing Binaries
esults Results
Extracting the Three-Graviton and Four-Graviton Vertices from Binary Pulsars and Coalescing Binaries
Comparing with the Hulse-Taylor Binary Comparing with the Hulse-Taylor Binary
Results
Comparing with the Hulse-Taylor Binary
stimating β3 Estimating β33
Estimating β
Estimating β3
The total mass M
(Tentative(Tentative are
andcalculations.) Replacing M → Mβ and ν M ν→
the ratio ν calculations.) Replacing → β
determined by the
˙1β / β GR
˙˜
PbβωP GR = Pb + P3 g (e) = 1 +
periastron shift˙ /˙
and the Einstein
where g (e) 2.70gfor the Hulse-Binary Pulsar.
˜ where ˜(e)
time delay γ. The 2.70 for the Hulse-Binary
Compared to experiment: ωβ3 = (1
point is that these experiment:− β3 /3) ωGR
Compared to
ng the Three-Graviton and Four-Graviton Vertices from Binary Pulsars and Coalescing Binaries ˙
Pbobs /PbGR γβ3 = 1
two parameters = 1.0013(21) γGR + 2.70β3
˙
ts
˙
˙ obs M 3 = = 1.0013(21)
are in principle Pb /PβGR (1 + β3 /2) MGR
mparing with the Hulse-Taylor Binary b
we get: modiﬁed by β3 . νβ3 = (1 + Cβ3 )νGR
mating β3 we get:
Hulse-Taylor → β3
(Tentative calculations.) Replacing M → Mβ and ν → νβ Hulse-Taylor
we get, → β3
Pbβ /P GR = 1 + β3 g (e)
˙ ˙ ˜
β3 = (4.0 ± 6.4) · 10−4
where g (e) 2.70 for the Hulse-Binary Pulsar.
˜
Compared to experiment: β3 = (4.0 ± 6.4) · 10
˙ ˙
Pbobs /PbGR = 1.0013(21) 1 + 2.70β3
we get:
Friday, September 17, 2010
40.
Interpretation
• Radiation from Hulse-Taylor binary pulsar provides
measurement of the gauge structure of GR (0.1%).
• Assumes that GR is the low-energy limit of a high-
energy theory gravity, then experimental accuracy
provides bounds on possible deviations generated
by new physics.
Friday, September 17, 2010
41.
Not the end of the story...
• If GR is not the correct low-energy EFT,
could have different bounds.
Friday, September 17, 2010
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