Download Tests of Alternative Theories of Gravity

Tests of Alternative Theories of Gravity
Gilles Esposito-Farese (GReCO, IAP, France)
General relativity passes all tests with flying colors
⇒ WHY considering alternative theories?
∃ theoretical motivations for alternatives to G.R.:
Quantization of gravity & unification with other forces
[strings] predict the existence of PARTNERS to graviton
Useful to contrast their predictions with G.R.:
– What theoretical information can we extract from experimental data?
– What can be further tested?
∃ some puzzling exprimental issues:
Dark energy (72%), dark matter (24%), Pioneer anomaly
GENERAL RELATIVITY
c3
S=
16πG
action
Z
√
matter, g ]
d x − gR + Sstandard [allfields
µν
4
dynamics of gravity
SPIN 2 FIELD
model
coupling of matter to gravity
MINIMAL COUPLING TO gµν
MATTER–GRAVITY COUPLING
Smatter [ matter , gmn ]
Metric coupling chosen to satisfy the (weak) equivalence principle
Impossible to determine
from a local experiment
if there is acceleration
or gravitation
(Einstein 1907)
acceleration
€
gravitation
MATTER–GRAVITY COUPLING
Smatter [ matter , gmn ]
Metric coupling chosen to satisfy the (weak) equivalence principle
freely falling
elevator
€
(special relativity)
Earth
MATTER–GRAVITY COUPLING
Metric coupling: Smatter [ matter , gµν ]
⇒
Freely falling elevator
(= Fermi coordinate system)
1
gµν =
1
1
1
λ
µν
=0
1 Constancy of the constants
2 Local Lorentz invariance
Space & time independence of coupling constants
and mass scales of the Standard Model
Local non-gravitational experiments are
Lorentz invariant
Oklo natural fission reactor
.
|α/α| < 7×10–17 yr–1 << 10–10 yr–1 (cosmo)
Isotropy of space verified at the 10–27 level
[Shlyakhter 76, Damour & Dyson 96]
[Prestage et al. 85, Lamoreaux et al. 86,
Chupp et al. 89]
3 Universality of free fall
4 Universality of gravitational redshift
Non self-gravitating bodies fall with the same
acceleration in an external gravitational field
In a static Newtonian potential
g00 = –1 + 2 U(x)/c2 + O(1/c4)
the time measured by two clocks is
τ1/τ2 = 1 + [U(x1)–U(x2)]/c2 + O(1/c4)
Flying hydrogen maser clock: 2×10–4 level
[Vessot et al. 79–80, Pharao/Aces will give 5×10–6]
Laboratory: 4×10–13 level [Baessler et al. 99]
: 2×10–13 level [Williams et al. 04]
4 Universality of gravitational redshift (time dilation)
acceleration
Doppler effect
(cf. fire-truck siren)
gravitation
⇒ Whatever their composition,
lower clocks are slower
(⇒ impossible to
synchronize
even static clocks)
Theoretical motivations for non-metric coupling?
∃ dilaton ϕ, partner of graviton in 10 dimensions
• SUPERstrings:
• Dimensional reduction: ∃ moduli ϕ
•…
⇒ ∃ dilatonic coupling of a scalar field to gauge fields
⇒ Effective coupling constant k eff = k 0 e−ϕ/2 depends on x
⇒ Masses m(ϕ) depend also on x
⇒ Violations of universality of free fall:
Newton force =
spin 2
mA
+
spin 0
Z
SEM =
gmn =
( )
gµν Aµ
Aν ϕ
eϕ
2 √
4
−
d
x
F
µν
g
2
4 k0
ϕ
conformal invariant ⇒ e cannot be
eliminated by redefining g~µν = f(ϕ) gµν
δ a = –∇ ln m( x )
depends on composition of mA and mB
mB ∂ϕmA ∂ϕmB
Theoretical motivations for non-metric coupling?
∃ dilaton ϕ, partner of graviton in 10 dimensions
• SUPERstrings:
• Dimensional reduction: ∃ moduli ϕ
•…
⇒ ∃ dilatonic coupling of a scalar field to gauge fields
⇒ Effective coupling constant k eff = k 0 e−ϕ/2 depends on x
⇒ Masses m(ϕ) depend also on x
⇒ Violations of universality of free fall:
Newton force =
Newton
Einstein
Strings
spin 2
mA
geometry
coupling
constants
rigid
soft
soft
rigid
rigid
soft
+
spin 0
Z
SEM =
gmn =
( )
gµν Aµ
Aν ϕ
eϕ
2 √
4
−
d
x
F
µν
g
2
4 k0
ϕ
conformal invariant ⇒ e cannot be
eliminated by redefining g~µν = f(ϕ) gµν
δ a = –∇ ln m( x )
depends on composition of mA and mB
mB ∂ϕmA ∂ϕmB
Theoretical motivations for non-metric coupling?
∃ dilaton ϕ, partner of graviton in 10 dimensions
• SUPERstrings:
• Dimensional reduction: ∃ moduli ϕ
•…
Z
⇒ ∃ dilatonic coupling of a scalar field to gauge fields
SEM =
⇒ Effective coupling constant k eff = k 0 e−ϕ/2 depends on x
⇒ Masses m(ϕ) depend also on x
⇒ Violations of universality of free fall:
Newton force =
Newton
Einstein
Strings
spin 2
mA
geometry
coupling
constants
rigid
soft
soft
rigid
rigid
soft
+
spin 0
gmn =
( )
gµν Aµ
Aν ϕ
eϕ
2 √
4
−
d
x
F
µν
g
2
4 k0
ϕ
conformal invariant ⇒ e cannot be
eliminated by redefining g~µν = f(ϕ) gµν
δ a = –∇ ln m( x )
depends on composition of mA and mB
mB ∂ϕmA ∂ϕmB
Tree-level predictions of strings
Experiment
∆a/a ~ 10–5
.
α/α ~ H0 ~ 10–10 yr–1
>>
>>
γPPN–1 ~ O(1)
βPPN–1 ~ O(40)
>>
5×10–13
7×10–17 yr–1
10–5
How can strings be saved?
• Add a mass to dilaton?
BUT – no natural mechanism to generate masses for all scalar fields in the theory
– difficult cosmological problems
[e.g. Polonyi: too much energy stored in cosmological oscillations of ϕ(t)]
• String loops!
a(ϕ)
Transform eϕ into ea(ϕ)
Large
slope
Cosmological
evolution
Small
:
slope
ϕmin
ϕln
m(ϕmin) ≈ 0
ϕ
If ≈ same ϕmin for all elementary particles [cf. S-duality, i.e., symmetry under gstring = eϕ → 1/gstring = e−ϕ],
then expected deviations from general relativity are
[ ϕln m(ϕnow)]2 ∼ 10–10 → 10–19
[Damour & Polyakov 94, Damour & Vilenkin 95–96, Damour, Piazza, Veneziano 02]
Experimental data in the context of this string model
Composition
independent
tests
Coupling strength
to a dilaton
∼ [ ϕln m(ϕ)]2
10–3.5
Gravity Probe B (orbiting gyroscope)
SORT
(heliocentric clocks time delay)
LATOR
(light deflection by Sun)
10–5
10–5.5
10–6
10–7
10–8
10–9
possible
10–11
most probable
10–14
Composition
dependent
tests
.
Oklo reactor: |α/α| < 7×10–17 yr–1
ground clocks
geocentric clocks: redshifts at 10–4 level
equivalence principle tests: |∆a/a| < 2×10–13
heliocentric clocks (PHARAO, ASTROD, …)
MICROSCOPE:
10–15 accuracy in ∆a/a
[Damour, Piazza, Veneziano]
EXPECTED
DILATONIC
EFFECTS
[Damour & Polyakov]
(satellite test of the equivalence principle):
10–18 accuracy in ∆a/a
STEP
⇒ Within this string-inspired framework,
free-fall experiments are the most precise
DYNAMICS OF GRAVITY
Now, assume metric coupling of matter to gravity: S = Sgravity + Smatter [ matter , gµν ]
?
Phenomenological approach: PPN formalism
• Do not assume anything about Sgravity
• Write the most general form that gµν can take in presence of matter,
at the first post-Newtonian order [Newton × 1/c2]
Basic idea
[Eddington 1923]:
– g00 = 1 – 2
gij = δij
"
Gm
rc2
PPN
+ 2β
Gm
rc2
PPN
1 + 2 γ
Gm
rc2
2
+ …
+ …
#
Generalization [Will & Nordtvedt 1972]: 10 parameters including βPPN and γPPN
Conclusion of experimental tests in the Parametrized Post-Newtonian formalism
γPPN
Lunar Laser Ranging
2
Mercury perihelion shift
General
Relativity
1.5
Mars radar ranging
&
Very Long Baseline Interferometry
&
Time delay for Cassini spacecraft
1
LLR
1.004
0.5
PPN
0
0.5
1
1.5
2
β
1.002
1
ξ
α1,2,3
ζ1,2,3,4
γPPN
Cassini
VLBI
GENERAL RELATIVITY
is essentially the only
theory consistent with
weak-field experiments
0.998
general
relativity
0.996
0.996
0.998
1
1.002
1.004
β
PPN
[Table from C.M. Will gr-qc/0504086]
space curvature created by mass
nonlinearity in superposition law
preferred-location effects
10–4
preferred-frame effects
combination of other parameters
violation of conservation
of total momentum
5¥10–4
Tests of the “strong equivalence principle” and of preferred-frame effects
• The different accelerations (due to a third
C.M.
body or to their absolute velocity with respect
to a preferred frame) induce a polarization
of the periastron towards a precise direction
A
aA
.
wR
e
eF
• $ several binary
pulsars with e ª 0
.
wR
eR
fixed direction
|eFixed|
|aA – aB|
eR
eF
fi statistical argument to constrain PPN parameters
[Damour, Schäfer, GEF, Bell, Camilo, Wex, …]
B
fixed direction
e
aB aA
Tests of the “strong equivalence principle” and of preferred-frame effects
B
• The different accelerations (due to a third
C.M.
body or to their absolute velocity with respect
to a preferred frame) induce a polarization
of the periastron towards a precise direction
eF
aA
eR
fixed direction
|eFixed|
|aA – aB|
• Earth-Moon-Sun system [Nordtvedt]
G
• $ several binary
pulsars with e ª 0
.
wR
e
A
.
wR
e
aB aA
G = G (1 + d + d )
eR
eF
fi statistical argument to constrain PPN parameters
[Damour, Schäfer, GEF, Bell, Camilo, Wex, …]
= G (1 + d + d )
dA =
grav.
^
dA + (4b–g–3) EA
equivalence due to
principle
dilaton
violation coupling
–13
|Da/a| < 2¥10
experiment
PPN
contribution
fi
/mAc2
~ 10–10
dilaton coupling < 10–8
PPN constraint |4b–g–3| < 10–3
DYNAMICS OF GRAVITY (continued)
Brane models imply (long and) short-distance modifications of Newton’s law
GM
V =
( 1 + α e − r/λ )
r
5th dimension
gravitation
our 4-dimensional
space-time
(maybe other
parallel spaces)
[C.D. Hoyle et al., Phys. Rev. D70 (2004) 042004, hep-ph/0405262]
Constraining the graviton mass?
!
∃ no clean theory of massive graviton
(“vDVZ” discontinuity, ghosts, or predictions not yet worked out)
⇒ phenomenological point of view…
• Solar system [C. Talmadge et al., Phys. Rev. Lett. 61 (1988) 1159]
Gm –r/λg
e
r
⇒ mg < 4×10–22 eV/c2 ⇔ λg = h/(mgc) > 3×1012 km
Yukawa-type Newtonian potential VN =
• Binary pulsars
[L.S. Finn and P.J. Sutton, Phys. Rev. D 65 (2002) 044022; Class. Quantum Grav. 19 (2002) 1355]
⇒ mg < 10–19 eV/c2 ⇔ λg = h/(mgc) > 1010 km
• LISA correlated with optical observations
photons
E = γ mc2 ⇔ vg2/c2 = 1 – mg2c4/E2 (dispersion relation)
⇒ mg < 6×10–24 eV/c2 ⇔ λg = h/(mgc) > 2×1014 km
gravitons
[S.L. Larson, W.A. Hiscock, Phys. Rev. D 61 (2000) 104008;
C. Cutler, W.A. Hiscock, S.L. Larson, Phys. Rev. D 67 (2003) 024015;
A. Cooray, N. Seto, Phys. Rev. D 69 (2004) 103502]
• GW interferometers alone
vg≈ c
[C.M. Will, Phys. Rev. D 57 (1998) 2061;
E. Berti, A. Buonanno, C.M. Will, Phys. Rev. D 71 (2005) 084025]
LIGO/VIRGO ⇒ mg < 2×10–22 eV/c2 ⇔ λg = h/(mgc) > 6×1012 km
LISA (BH-BH) ⇒ mg < 2×10–26 eV/c2 ⇔ λg = h/(mgc) > 6×1016 km
vg< c
high frequency gravitational waves
low frequency gravitational waves
DYNAMICS OF GRAVITY (continued)
S = Sgravity + Smatter [ matter , gµν ]
?
Field-theoretical approach
• Now, gµν
is assumed to be a combination of fields which propagate in a consistent way:
* + a1 BµBν + a2 gµν
* BρB*ρ + a3 BµρBν* ρ + …]
gµν = A2(ϕ)[gµν
spin 0 spin 2
antisymmetric
tensor field
vector fields (spin 1)
* ) lead in general to many flaws:
• However, vector (Bµ) or tensor (Bµν) partners to graviton (gµν
– discontinuities in the field degrees of freedom
– negative-energy modes
– causality violations
– ill-posedness of the Cauchy problem
– no theoretical motivations for being coupled to matter in the “metric” way Smatter [ matter , gµν ]
–…
⇒ best-motivated and consistent alternatives to General Relativity:
S=
1
16 π G
∫
Tensor−scalar theories
−g*
{
R*− 2
spin 2
( µϕ)2 } + S
spin 0
[matter , g
matter
µν
*
A2(ϕ) gµν
physical metric
[N.B.: more than one scalar and massive scalars are also possible]
]
Simplest example:
Nordström’s purely scalar theory (1913)
1
S=
8πG
Z
d4 x η µν ∂µ ϕ∂ν ϕ + Smatter [matter, gµν ≡ ϕ2 ηµν ]
spin 0
conformally flat
physical metric
– Correct Newtonian limit
– Satisfies weak and strong equivalence principles (⇒ no Nordtvedt effect on Moon’s orbit)
– But post-Newtonian predictions inconsistent with experiment:
ds2 = gµν dxµ dxν = 0 ⇒ ηµν dxµ dxν = 0
Light rays follow geodesics of flat spacetime ⇒ no light deflection!
Light deflection and the equivalence principle
⇒ Modification
of the stars’
apparent position
acceleration
gravitation
Sun
Earth
Light deflection and the equivalence principle
⇒ Modification
of the stars’
apparent position
acceleration
gravitation
Sun
Earth
Light deflection and the equivalence principle
⇒ Modification
of the stars’
apparent position
acceleration
gravitation
Sun
Earth
In 1914, Einstein predicts
half the correct value
[Eddington 1919]
This is because ∃ also a deformation of space:
Nordström’s
theory 1913
Einstein’s
general relativity 1915
Earth
Sun
Sun
Earth
For general tensor−scalar theories
1
16 π G
∫
−g*
{
R*− 2
( µϕ)2 } + S
spin 2
• Matter−scalar interaction:
Effective Newton’s constant:
1
β
2 0
µν
matter
spin 0
ln A(ϕ) = α0 (ϕ–ϕ0) +
matter
[matter , g
]
ln A(ϕ)
2
(ϕ–ϕ0) + …
ϕ
ϕ
ϕ
ϕ
*
A2(ϕ) gµν
physical metric
...
S=
Geff = G ( 1 + α02 )
graviton
scalar
α0
α0
ϕ
ϕ
curvature
β0
slope α0
ϕ0
ϕ
For general tensor−scalar theories
∫
−g*
{
R*− 2
( µϕ)2 } + S
spin 2
ln A(ϕ) = α0 (ϕ–ϕ0) +
Effective Newton’s constant:
1
β
2 0
ϕ
ϕ
ϕ
SEM =
⇒ no photon−scalar vertex !
⇒ light deflection
ϕ
ϕ
curvature
β0
Geff = G ( 1 + α02 )
scalar
α0
conformal invariance
]
ln A(ϕ)
2
(ϕ–ϕ0) + …
ϕ
*
A2(ϕ) gµν
physical metric
graviton
• Photon−scalar interaction:
µν
matter
spin 0
• Matter−scalar interaction:
matter
[matter , g
...
1
16 π G
Z
√
µρ νσ
− g g g Fµν Fρσ =
ϕ
,
ϕ
ϕ
slope α0
α0
,
GM
Geff M
=
< G.R. result
rc2
(1 + α02 )rc2
Z
ϕ0
√
− g∗ g∗µρ g∗νσ Fµν Fρσ
ϕ
...
S=
ϕ
ϕ
, … =0
ϕ
Other post-Newtonian predictions
ln A(ϕ) = α0 (ϕ–ϕ0) + 12 β0 (ϕ–ϕ0)2 + …
ϕ
ϕ
ϕ
ϕ
...
matter
ϕ
ϕ
matter
ln A(ϕ)
|α0|
perihelion
shift
curvature
β0
0.035
0.030
slope α0
0.025
LLR
ϕ
ϕ0
0.020
0.015
VLBI
0.010
Geff = G ( 1 + α02 )
graviton
PPN
γ
–1
βPPN– 1
α02
α02 β0
ϕ
scalar
α0
LLR
0.005
Cassini
α0
β0
α0
α0
−6
−4
−2
0
2
4
6
matter
β0
ϕ
ϕ
General Relativity
Vertical axis (β0 = 0) : Jordan–Fierz–Brans–Dicke theory α02 =
Horizontal axis (α0 = 0) : perturbatively equivalent to G.R.
1
2 ωBD + 3
Higher-order deviations from G.R.
• At any order in
1 , the deviations involve at least two α factors:
0
cn
graviton
α0
α0
α0
…
α0
= small deviations!
scalar
α0
α0
• But nonperturbative strong-field effects may occur:
"
deviations = α20 × a0 + a1
< 10−5
Gm
Rc2
+ a2
LARGE for
Gm
Rc2
Gm
Rc2
2
+ …
≈ 0.2 ?
#
matter-scalar
No deviation from
coupling function
General Relativity
ln A(ϕ)
in weak-field conditions
α0 = 0 ϕ
0
β0 < 0
neutron star
ϕc
scalar charge
|αA|
ϕ
0.6
large slope ~ αA
⇒ large deviations from
0.4
General Relativity
for neutron stars
all
sm
Energy
E≈
∫[
0.2
0
1
2
—
(∇ϕ)
2
0.5
β0ϕ2/ 2
+ρe
]
R(
m/
n)
Su
/R
al m
tic
cri
R
m/ r)
ge sta
lar tron
u
(ne
1 R
—
2
“spontaneous scalarization”
2
ϕc2 + m eβ0ϕc / 2
parabola
ϕ0
ϕ
Gaussian
if β0< 0
ϕc
(at the center of the star)
[T. Damour & G.E-F 1993]
1
1.5
critical
mass
2
2.5
maximum
mass in GR
3
maximum
mass
baryonic mass
—
mA/m
Weak-field experiments
– g00 = 1 – 2
gij = dij
"
Gm
rc2
PPN
+ 2b
1 + 2 g
Gm
rc2
PPN
Gm
rc2
2
+ …
+ …
#
Strong-field tests ?
solar system
–10
0 7¥10
–6
2¥10
neutron
star
black
hole
0.2
0.5
⇒
binary pulsars
companion
moving clock
giving information
about this stronggravity region
pulsar
radio waves
observer
self-gravity
Gm
"
Rc2
deviation from
flat space
#
Binary-pulsar tests
pulses
pulsar = (very stable) clock
binary
=
pulsar
t
pulses
moving
clock
t
P
• Time of flight across orbit
– orbital period
– eccentricity
– periastron angular position
–…
• Redshift
size of orbit
c
P
e
w
G mB
rAB c2 + second order Doppler effect
– parameter
“Keplerian” parameters
vA 2
2 c2 (“Einstein time delay”)
gTiming
• Time evolution of Keplerian parameters
– periastron advance
(“Roemer time delay”)
.
1
w (order c2 )
.
1
– gravitational radiation damping P (order c5 )
“post-Keplerian” observables
[PSR B1913+16 • Hulse & Taylor]
3
observables
–
2
unknown
masses mA, mB
=
1
test
Plot the three curves [strips]
theory
observed
gTiming(mA, mB) = gTiming
. theory
. observed
w
(mA, mB) = w
. theory
.observed
P
(mA, mB) = P
. .
“ g - w - P test ”
PSR B1913+16
in general relativity
companion
mB/m
. th
. exp
P (mA,mB) = P
2.5
2
intersection
1.5
γthT (mA,mB) = γexp
T
1
.
.
ωth(mA,mB) = ωexp
.
ω = 4.22661°/yr
GR
γT = 4.294 ms
.
P = –2.421 × 10–12
⇒
0.5
s≤1
0
0.5
1
1.5
2
Discovered by R. Hulse and J. Taylor in 1974
2.5
mA/m
pulsar
mA = 1.4408 m
mB = 1.3873 m
PSR B1534+12
in general relativity
companion
mB/m
.
w
2.5
.
P
2
1.5
1
intersection
.
w
r, s
gT
s
5 observables - 2 masses = 3 tests
.
“Galactic” contribution to P
r
[Damour–Taylor 1991]
g
Doppler n.v
0.5
0
.
P
0.5
1
1.5
Discovered by A. Wolszczan in 1991
2
2.5
mA/m
pulsar
fi
d Doppler
v2^
n.a +
d PSR
dt
PSR J1141–6545
in general relativity
companion
mB/m
2.5
Asymmetrical system
neutron star – white dwarf
.
P
.
ω
Neutron star born after white dwarf
⇒ eccentricity e = 0.17 large
and nonrecycled pulsar
2
intersection
1.5
s≤1
.
P = –4 × 10–13
Mass function
1
3
(mB sin i )
γ
2
(mA+ mB)
0.5
0
0
0.5
1
1.5
2
2.5
Discovery Kaspi et al. 1999, Timing Bailes et al. 2003
mA/m
pulsar
=
2
3
2 π (x c)
P
G
PSR J0737–3039
in general relativity
2nd pulsar
mB/m
pulsar A
pulsar B
.
ω
2.5
.
P
xA/xB
observer
2
s
P = 2 h 27 min 14.5350 s
1.5
.
ω = 16.90°/yr
r
1
γ
xB mA
xA = mB = 1.07
intersection
0.5
6 observables − 2 masses = 4 tests
0
0
0.5
1
1.5
2
2.5
Timing Burgay et al. 2003, Double pulsar Lyne et al. 2004
mA/m
1st pulsar
neutron star
ϕ
scalar charge
|αA|
0.6
Strong-field effects
0.4
0.2
0
eff
GAB
= G ( 1 + αA αB )
A
B
graviton
A
B
αA
αB
scalar
depends on internal
structure of bodies A & B
Similarly for (γPPN– 1) and (βPPN– 1)
A
B
αA
αB
B
A
βB
αA
⇒
all post-Newtonian effects
A
αA
Quadrupole
+O
c5
2 Dipole
Monopole
Quadrupole
1
+
0+ 2 +
+
+O
c
c
c3
c5
Energy flux =
(αA–αB)2
1
c7
spin 2
1
c7
spin 0
0.5
1
1.5
critical
mass
2
2.5
maximum
mass in GR
3
maximum
mass
baryonic mass
—
mA/m
mB/m
PSR B1913+16
in scalar-tensor theories
.
ω
γ
1.5
1
mB/m
0
.
P
γ
General relativity
passes the test
0.5
0.5
1
1.5
mA/m
1
mA/m
1.5
mB/m
2.5
0.5
0
.
P
0.5
.
ω
1.5
1
A tensor–scalar theory
which passes the test
(β0 = –4.5, α0 small enough)
γ
γ
2
1.5
.
ω
A tensor–scalar theory
which does not pass the test
(β0 = –6, any α0)
1
0.5
0
.
P
0.5
1
1.5
2
2.5
mA/m
Solar-system & PSR B1913+16 constraints
on scalar-tensor theories of gravity
matter
matter-scalar
coupling function
|α0|
ln A(ϕ)
ϕ
PSR
B1913+16
α0
β0 < 0
0.040
β0 > 0
0.035
α0
ϕ
0.025
0.020
0.015
−6
VLBI
0.010
Cassini
0.005
−4
−2
binary pulsars
impose β0 > −4.5
i.e.
βPPN– 1
< 1.1
PPN
γ –1
0
2
4
6
β0
matter
ϕ
ϕ
general relativity
[T. Damour & G.E-F 1998]
Vertical axis (β0 = 0) : Jordan–Fierz–Brans–Dicke theory
Horizontal axis (α0 = 0) : perturbatively equivalent to G.R.
α02 =
1
2 ωBD + 3
The four accurately timed
binary pulsars in general relativity
PSR B1913+16
mB
.
P
.
w
2.5
.
w
2.5
2
s£1
0
0
0.5
1
2
1.5
2
2.5
mA
0
0
0.5
1
1.5
2
2.5
.
w
.
P
intersection
s£1
mA
xA/xB
2
1.5
2.5
PSR J0737-3039
mB
.
P
.
w
g
0.5
PSR J1141-6545
mB
2.5
r
1
g
s
intersection
1.5
1.5
0.5
.
P
2
intersection
1
PSR B1534+12
mB
s
1.5
r
1
1
g
g
0.5
0
intersection
0.5
0
0.5
1
1.5
2
2.5
mA
0
0
0.5
1
1.5
2
2.5
mA
Solar-system & best binary-pulsar constraints
on scalar-tensor theories of gravity
matter-scalar
coupling function
matter
ln A(ϕ)
|α0|
α0
β0 < 0
ϕ
0.050
β0 > 0
PSR
B1913+16
0.045
α0
ϕ
0.040
0.035
0.025
0.020
0.015
VLBI
Cassini
−6
0.005
−4
−2
binary pulsars
impose β0 > −4.5
i.e.
PSR
J1141–6545
0.010
βPPN– 1
γPPN– 1
< 1.1
0
2
4
6
β0
matter
general relativity
ϕ
ϕ
[T. Damour & G.E-F 2005]
Vertical axis (β0 = 0) : Jordan–Fierz–Brans–Dicke theory
Horizontal axis (α0 = 0) : perturbatively equivalent to G.R.
α02 =
1
2 ωBD + 3
Solar-system and best binary-pulsar
constraints on tensor–scalar theories
(updated July 2005)
matter
matter
Logarithmic
scale for a0
j
|a0|
100
j
|a0|
0.2
SEP
B1534+12
SEP
10-1
J0737–3039
0.175
B1913+16
B1534+12
0.15
All pulsars
10-2
0.125
J1141–6545
LLR
Cassini
0.1
0.075
J0737–3039
0.05
B1913+16
J1141–6545
0.025
-6
-4
-2
0
10-3
2
4
All pulsars general relativity
6
matter
matter
b0
10-4
-6
-4
-2
0
2
j
j
general relativity
(a0 = b0 = 0)
4
6
j
j
b0
|Φ0|
10−1
J1141–6545
J0737–3039
Experimental constraints on another
natural class of scalar-tensor theories:
S=
1
16 π G
∫
[matter , g ]
+ Smatter
−g
{(
)−(
R 1+ξΦ2
LLR
B1534+12
10−2
)2 }
All pulsars
µΦ
SEP
Cassini
µν
10−3
B1913+16
10−4
0
1
N.B.: ∃ other classes of scalar-tensor theories
(e.g., some where the SEP tests are the most constraining,
whereas PSR J1141–6545 does not tell us much!)
general relativity
2
3
4
ξ
Gravitational wave antennas
LIGO/VIRGO/LISA
signal
Signal emitted by an
inspiralling binary system:
time
•••
Effect on a detector:
LIGO
(depends on hundreds
of post-Newtonian
coefficients)
VIRGO
Gravitational wave antennas
LIGO/VIRGO/LISA
One needs accurate (3.5 PN)
templates to extract the
signal from the noise
Gravitational waves
in scalar-tensor gravity
Quadrupole
1
+
O
c5
c7
Monopole . 1 2 Dipole Quadrupole
1
+
σ+ 2 +
+
+
O
c
c
c3
c5
c7
Energy flux =
Collapsing star
Earth
Factor α0 =
Energy flux
= (strong field)2
= Monopole/c
>> usual Quadrupole/c5
1
1−γPPN
≈
< 0.003
2ωBD+3
2
Detection
= (strong field) × (weak field)
= too small for LIGO/VIRGO
[J. Novak's thesis, PRD 57, 4789; 58, 064019 (1998)]
and not in LISA's frequency band
spin 2
spin 0
Gravitational waves
in scalar-tensor gravity
Quadrupole
1
+
O
c5
c7
Monopole . 1 2 Dipole Quadrupole
1
+
s+ 2 +
+
+
O
c
c
c3
c5
c7
Energy flux =
Collapsing star
Earth
Factor a0 =
Energy flux
= (strong field)2
= Monopole/c
>> usual Quadrupole/c5
1
1-gPPN
ª
< 0.003
2wBD+3
2
Detection
= (strong field) ¥ (weak field)
= too small for LIGO/VIRGO
[J. Novak's thesis, PRD 57, 4789; 58, 064019 (1998)]
and not in LISA's frequency band
Inspiralling binary
Even if no helicity-0 wave is detected, the time-evolution of
the (helicity-2) chirp depends on the Energy flux = (strong field)2
fi A priori possible to detect indirectly the presence of j:
If binary inspiral detected with GR templates
fi bound on matter–scalar coupling strength
[matched-filter analysis: C.M. Will, Phys.Rev. D 50 (1994) 6058]
spin 2
spin 0
Chirp evolution in general relativity
signal
time
For a given binary system
signal
time
Chirp evolution in a tensor–scalar theory
Chirp evolution in general relativity
signal
time
For an unknown mass of the system
signal
in phase
out of phase
time
Chirp evolution in a tensor–scalar theory
matter
Solar-system and possible LIGO/VIRGO
constraints on scalar-tensor gravity
|α0|
ϕ
[Damour & GEF 1998]
0.050
LIGO/VIRGO
0.045
NS-BH
0.040
0.035
0.030
0.025
0.020
0.015
VLBI
0.010
Cassini
−6
−4
−2
0.005
0
2
4
6
β0
matter
ϕ
ϕ
general relativity
Vertical axis (β0 = 0) : Jordan–Fierz–Brans–Dicke theory
Horizontal axis (α0 = 0) : perturbatively equivalent to G.R.
α02 =
1
2 ωBD + 3
Solar-system, possible LIGO/VIRGO, and binary-pulsar
constraints on scalar–tensor theories of gravity
matter
ϕ
|α0|
LIGO/VIRGO
0.2
NS-BH
0.175
B1534+12
0.15
LIGO/VIRGO
NS-NS
0.125
0.1
0.075
J0737–3039
0.05
B1913+16
J1141–6545
0.025
−6
−4
Bad news: LIGO/VIRGO will not probe scalar effects
Good news! ⇒ GR templates can be used securely
−2
0
2
4
All pulsars general relativity
6
matter
β0
ϕ
ϕ
Possible LISA constraints
on scalar-tensor theories of gravity
matter
|α0|
PSR
J1141-6545
LISA will probe |α0| ~ 1.5 × 10–3 if 1.4 m NS – 1000 m BH
observed with S/N = 10
0.050
0.040
0.035
-15
10
0.030
-16
No scalar-field effect
10
compar
able ma
)
-1/2
1/2
PSR
B1913+16
0.045
[Scharre & Will 2002; Will & Yunes 2004; Berti, Buonanno & Will 2005]
ss BH-B
-17
Sn (f) (Hz
ϕ
10
0.025
H binar
0.020
ies
Strong scalar-field effects
NS-IM
BH bin
aries
-18
10
-19
10
LISA
with spin-orbit and
spin-spin effects
LISA
with spin-orbit effects
0.015
VLBI
0.010
Cassini
-20
10
0.005
LISA
-21
10
-5
10
-4
10
-3
10
-2
10
f (Hz)
-1
10
0
10
−6
−4
−2
⇒ Tight constraints if detection of binary inspirals with GR templates
But if no detection, what would we conclude?
0
2
4
general relativity
6
β0
matter
ϕ
ϕ
matter
|α0|
Future binary-pulsar constraints
on scalar-tensor theories of gravity
ϕ
0.050
0.045
0.040
0.035
PSR
J1141−6545.
0.030
LISA
with spin-orbit and
spin-spin effects
0.025
+ 1% accurate P
0.020
LISA
with spin-orbit effects
0.015
0.010
Cassini
0.005
LISA
−6
Binary pulsars will probably probe such scalar-tensor
theories before LISA is launched
Good news: GR templates can be used securely
−4
−2
0
2
4
general relativity
6
β0
matter
ϕ
ϕ
Logarithmic
scale for a0
|a0|
100
10-1
LISA NS-BH
All pulsars
spin-orbit & spin-spin
spin-orbit
10-2
Cassini
10-3
J1141–6545
+ 1% Pdot
10-4
-6
-4
-2
0
2
4
6
b0
Logarithmic
scale for a0
|a0|
Future binary-pulsar constraints
on scalar-tensor theories of gravity
100
10-1
LISA NS-BH
All pulsars
spin-orbit & spin-spin
spin-orbit
10-2
PSR-BH
1% Pdot
Cassini
Pulsar-white dwarf
and
Pulsar-black hole
are the most constraining systems
for alternative theories
(large dipolar radiation)
10-3
J1141–6545
+ 1% Pdot
10-4
-6
-4
-2
0
2
4
6
b0
Puzzling issues
• Pioneer 10 & 11 anomaly in solar system (~70 AU):
extra acceleration ~ 8.5 ¥10–10 m s–2
• Cosmological observations:
72% of “dark energy”
24% of “dark matter”
4% of baryonic matter
“photograph” of Universe at 380 000 years (now 13.7 billion years)
Dark energy
– Why ΩΛ = 0.72 ~ Ωm = 0.28 today? (type Ia supernovae combined with CMB)
∃ hints from some models
but no clean answer yet
−122
– Why is Λ ≈ 3 × 10
c3
so small? (type Ia supernovae notably)
/hG
V(ϕ)
Possible explanation
via “quintessence”
t
today
Λ
ϕ
ϕ0
• New qualitative difference between cosmological observations and solar-system/binary-pulsar ones
ϕ
...
matter
ϕ
ϕ
matter
ϕ
matter
ϕ
ϕ
Cosmological observations give access
to the full shape of matter-scalar coupling A(ϕ)
and/or scalar-field potential V(ϕ)
• Usual cosmology:
– Assume particular forms of V(ϕ) [and A(ϕ)] for theoretical reasons
– Predict all observable quantities
– Compare them to experimental data
• Phenomenological approach:
Reconstruct A(ϕ) & V(ϕ) from observational data.
Result:
If luminosity distance DL(z) and
δρ
density fluctuations δm(z) = ρ are both
known as functions of the redshift z,
then A(ϕ) & V(ϕ) can be reconstructed.
[Boisseau, GEF, Polarski & Starobinsky 2000]
N.B.: A priori obvious, since one “fits” two observed functions
[DL(z) & δm(z)] with two unknown ones [A(ϕ) & V(ϕ)] !
• Semi-phenomenological approach:
[δm(z) not yet well measured]
– Theoretical hypotheses on V(ϕ) or A(ϕ)
– Reconstruct the other one from DL(z)
N.B.: A priori obvious too, since one fits one observed
function [DL(z)] with one unknown function [A(ϕ) or V(ϕ)].
However, this naive reasoning works only locally (small interval).
Result:
∃ tight constraints if DL(z) measured
on a wide interval z ∈ [0, ∼2],
even with large error bars!
[GEF & Polarski 2001]
Constraints come mainly from positivity of energy :
Egraviton ≥ 0 ⇔ A2 > 0 ⇔ ΦBD > 0
Eϕ ≥ 0 ⇔ − ( µϕ)2 ⇔ ωBD > −3/2
Dark matter
= pressureless and noninteracting component of matter
• Imposed notably by rotation curves of galaxies and clusters:
fi $ really some dark matter (many theoretical candidates notably from SUperSYmmetry),
or modification of Newton’s law at large distances?
• Milgrom’s phenomenological “MOND” proposal:
= GM/r 2
√
√
a = a0 aN = GMa0 /r
a=
aN
(OK for galaxies, not for clusters)
if a > a0 ≈ 1.2 × 10- 10 m.s- 2
if a < a0
Relativistic formulations of MOND?
$ various attempts but
– some change the field-theory action depending
on galaxy !
√
– some predict e.g. a = kM 2 /r instead of M/r ,
and assume then k = M - 3/2 !
– many contain tachyons or ghosts (fi unstable) !
Relativistic formulations of MOND?
Best present candidate: Bekenstein-Sanders model
– Many years of research fi seems very fine tuned
but works (even for Pioneer anomaly if one wishes).
• 1 usual (spin-2) graviton
• 1 (spin-1) vector field
• 2 (spin-0) scalar fields
Idea: use one of the scalar fields as
a “Lagrange parameter” to create a
nonlinear F[( f)2] √
for the other one
fi obtain the right M dependence
Relativistic formulations of MOND?
Best present candidate: Bekenstein-Sanders model
– Many years of research fi seems very fine tuned
but works (even for Pioneer anomaly if one wishes).
• 1 usual (spin-2) graviton
• 1 (spin-1) vector field
• 2 (spin-0) scalar fields
Idea: use one of the scalar fields as
a “Lagrange parameter” to create a
nonlinear F[( f)2] √
for the other one
fi obtain the right M dependence
– Stability not yet fully clear (tachyons & ghosts?).
F
local
effects
cosmology
1
-2
-4
-6
-8
2
3
4
Relativistic formulations of MOND?
Best present candidate: Bekenstein-Sanders model
– Many years of research fi seems very fine tuned
but works (even for Pioneer anomaly if one wishes).
• 1 usual (spin-2) graviton
• 1 (spin-1) vector field
• 2 (spin-0) scalar fields
Idea: use one of the scalar fields as
a “Lagrange parameter” to create a
nonlinear F[( f)2] √
for the other one
fi obtain the right M dependence
– Stability not yet fully clear (tachyons & ghosts?).
– Matter coupled to a physical metric involving
the vector field fi $ preferred frame (ether).
Idea: couple differently the scalar
field f to “g00” and “gij” in order
to get the right light deflection
(cf. conformal coupling fi no extra deflection)
F
local
effects
cosmology
1
-2
-4
-6
-8
2
3
4
Relativistic formulations of MOND?
Best present candidate: Bekenstein-Sanders model
– Many years of research fi seems very fine tuned
but works (even for Pioneer anomaly if one wishes).
• 1 usual (spin-2) graviton
• 1 (spin-1) vector field
• 2 (spin-0) scalar fields
Idea: use one of the scalar fields as
a “Lagrange parameter” to create a
nonlinear F[( f)2] √
for the other one
fi obtain the right M dependence
– Stability not yet fully clear (tachyons & ghosts?).
F
local
effects
cosmology
1
2
3
-2
-4
-6
-8
– Matter coupled to a physical metric involving
the vector field fi $ preferred frame (ether).
Idea: couple differently the scalar
field f to “g00” and “gij” in order
to get the right light deflection
Bekenstein
+ large Wn
(cf. conformal coupling fi no extra deflection)
– CMB predictions [Skordis et al., astro-ph/0505519]
need Wn = 0.17 (not far from WDM = 0.24)
fi $ dark matter!
Bekenstein
without Wn
Standard
LCDM
4
Conclusions
• ∃ theoretical motivations for considering alternatives to General Relativity.
• Contrasting G.R. with alternatives ⇒ understand which features have been tested
⇒ compare probing power of different observations
solar-system,
–5
[weak field regime tested at the 10 level]
binary-pulsar,
[strong field regime]
and cosmological observations.
[time evolution]
first derivative of ln A(ϕ)
matter
nonperturbative effects
matter
ϕ
ϕ
ϕ
⇒ second derivative of ln A(ϕ)
a priori full shape of ln A(ϕ)
but much more noisy
• General Relativity passes all tests with flying colors.
• ∃ still some puzzling experimental facts ⇒ understand them either theoretically
or experimentally
matter
ϕ
...
Qualitative difference between
ϕ
ϕ