Testing General Relativity
with Relativistic Binary
Pulsars
Ingrid Stairs
UBC
GWPAW
Milwaukee
Jan. 29, 2011
Jodrell Bank
Green Bank
Telescope
Parkes
Arecibo
Outline
• Intro to pulsar timing
• Equivalence principle tests
- “Nordtvedt”-type tests
- Orbital decay tests
• Relativistic systems
- Timing
- Geodetic precession
- Preferred-frame effects
• Looking to the future
Pulsars and other
compact objects probe
theories near objects
with strong gravitational
binding energy. These
tests are qualitatively
different from Solar-system
tests!
D. Page
WD: 0.01% of mass
NS: 10--20% of mass
BH: 50% of mass
is in binding energy.
Pulsar Timing in a Nutshell
Standard
profile
Measure
offset
Observed
profile
Record the start time of the observation with the
observatory clock (typically a maser).
The offset gives us a Time of Arrival (TOA) for
the observed pulse.
Then transform to Solar System Barycentre,
enumerate pulses and fit the timing model.
Pulsars available for GR tests
Young
pulsars
Recycled
pulsars
MSPs are often very stable, nearly as good as
atomic clocks on timescales of years.
This should allow the use of an array of them
to detect gravitational waves.
This is the goal of the International Pulsar
Timing Array – European, Australian and
North American collaborations.
Equivalence Principle Violations
Pulsar timing can:
set limits on the Parametrized
Post-Newtonian (PPN) parameters α1, α3 (, ζ2)
test for violations of the Strong Equivalence
Principle (SEP) through
- the Nordtvedt Effect
- dipolar gravitational radiation
- variations of Newton's constant
(Actually, parameters modified to account for
compactness of neutron stars.)
(Damour & Esposito-Farèse 1992, CQG, 9, 2093; 1996, PRD, 53, 5541).
SEP: Nordtvedt (Gravitational Stark) Effect
Lunar Laser Ranging: Moon's orbit is not polarized toward Sun.
Constraint:   (4.4  4.5) 10
4
  4    3 
WD
10
2
2
1
  1   2  1   2
3
3
3
3
(Williams et al. 2009, IJMPD, 18, 1129)
 pulsars: NS and WD fall
Binary
differently in gravitational
field of Galaxy.
NS
 m grav 
 inertial  1  i
m

E grav  E grav 2
 1 
 
  ...
m
m
 i   i 
Constrain Δnet = ΔNS -ΔWD
Result is a polarized orbit.

(Damour & Schäfer 1991, PRL, 66, 2549.)
Deriving a Constraint on Δnet
After Wex 1997, A&A, 317, 976.
Use pulsar—white-dwarf binaries
with low eccentricities ( <10-3).
Eccentricity would contain a “forced”
component along projection of
Galactic gravitational force onto
the orbit. This may partially
cancel “natural” eccentricity.
Constraint ∝ Pb2/e. Need to estimate orbital inclination and masses.
Formerly: assume binary orbit is randomly oriented on sky.
Use all similar systems to counter selection effects (Wex).
Ensemble of pulsars: Δnet < 9x10-3 (Wex 1997, A&A, 317, 976; 2000, ASP Conf. Ser.).
Now: use information about
longitude of
periastron (previously
unused) and measured
eccentricity and a Bayesian
formulation to construct
pdfs for Δnet for each
appropriate pulsar,
representing the full
population of similar objects.
Gonzalez et al., in prep.
Result: |Δnet| < 0.0045 at 95% confidence (Gonzalez
et al, nearly submitted, based on method used in
Stairs et al. 2005).
Constraints on α1 and α3
α1: Implies existence of preferred frames.
Expect orbit to be polarized along projection of velocity (wrt CMB)
onto orbital plane. Constraint ∝ Pb1/3/e.
Ensemble of pulsars: α1 < 1.4x10-4 (Wex 2000, ASP Conf. Ser.).
Comparable to LLR tests (Müller et al. 1996, PRD, 54, R5927).
This test now needs updating with Bayesian approach...
α3: Violates local Lorentz invariance and conservation of
momentum. Expect orbit to be polarized, depending on crossproduct of system velocity and pulsar spin. Constraint ∝ Pb2/(eP),
same pulsars used as for Δ test.
Ensemble of pulsars: α3 < 5.5x10-20 (Gonzalez et al. in prep.;
slightly worse limit than in Stairs et al. 2005, ApJ, 632, 1060, but
more information used).
(Cf. Perihelion shifts of Earth and Mercury: ~2x10-7 (Will 1993,
“Theory & Expt. In Grav. Physics,” CUP))
Orbital Decay Tests
These rely on measurement of or
constraint on orbital period derivative, PÝb .
This is complicated by systematic biases:
PÝb 
PÝb 
PÝb  PÝb  PÝb 
PÝb 
           
  
 
Pb Observed Pb Accel Pb GÝ Pb mÝ Pb Quadrupolar Pb Dipolar
PÝb 
PÝb 
PÝb 
  
    
Pb Accel Pb gravitational field Pb Shklovskii
PÝ 
2 d
b

 
c
Pb Shklovskii


Dipolar Gravitational Radiation
Difference in gravitational binding energies of NS and WD implies
dipolar gravitational radiation possible in, e.g., tensor-scalar theories.
2
2
4

G* m1m2
Ý
Pb Dipole  3
 c1   c 2 

c Pb m1  m2
Damour & Esposito-Farèse
1996, PRD, 54, 1474.
Test using pulsar—WD systems in short-period orbits. Examples:
PSR B0655+64, 24.7-hour orbit:
2
 cp   0  < 2.7x10-4 (Arzoumanian 2003, ASP Conf. Ser. 302, 69).
PSR J1012+5307, 14.5-hour orbit:
2
 cp   0  < 6x10-5 (Lazaridis et al. 2009, MNRAS 400, 805).

PSR J0751+1807, 6.3-hour orbit: PÝb measurement in Nice et al.
2005 ApJ 634, 1242 has major revision with more data.


PSR J1141-6545
Young pulsar with a white-dwarf companion, eccentric, 4.45-hour orbit
ω, γ and PÝb measured
through timing. Sin i
measured by scintillation.

Because of eccentricity, dipolar
radiation predictions can be large.
Bhat et al PRD 77, 124017 (2008)

Agreement with GR here sets
limits of  02  2.1105 for
weakly nonlinear coupling and
 02  3.4 106 for strongly
nonlinear
coupling (Bhat et al

2008).
Orbital decay tests rely on measurement of or
constraint on orbital period derivative, PÝb .
This is complicated by systematic biases:
PÝb 
PÝb 
PÝb  PÝb  PÝb 
PÝb 

           
  
 
Pb Observed Pb Accel Pb GÝ Pb mÝ Pb Quadrupolar Pb Dipolar
PÝb 
PÝb 
PÝb 
  
    
Pb Accel Pb gravitational field Pb Shklovskii
PÝ 
2 d
b

 
c
Pb Shklovskii
Variation of Newton's Constant
Spin: Variable G changes moment of inertia of NS.
Expect PÝ GÝ depending on equation of state, PM correction...
P

G
GÝ
Various millisecond pulsars, roughly:  2 1011 yr 1
G
PÝb GÝ

Orbitaldecay: Expect
, test with circular NS-WD binaries.
Pb G
GÝ

 (1.3  2.7) 1011 yr 1
PSR B1855+09, 12.3-day orbit:
G
(Kaspi, Taylor & Ryba 1994, ApJ, 428, 713; Arzoumanian 1995, PhD thesis, Princeton).

PSR J1713+0747, 67.8-dayorbit:
GÝ
 (1.5  3.8) 1012 yr 1
G
(Splaver et al. 2005, ApJ, 620, 405, Nice et al. 2005, ApJ, 634, 1242).

PSR J0437-4715, 5.7-day orbit:
GÝ
 (0.5  2.6) 1012 yr 1
G
(Verbiest et al 2008, ApJ 679, 675, 95% confidence, using slightly different assumptions).
GÝ
 (4  9) 1013 yr 1
Not as constraining as LLR (Williams et al. 2004, PRL 93, 361101):

G
Combined Limit on GÝ and
Dipolar Gravitational Radiation

Lazaridis et al. 2009 (MNRAS 400, 805) combine thePÝb limits
from J1012+5307 and J0437-4715 to form a combined limit
on these two quantities:
GÝ
12
1
 (0.7  3.3) 10 yr
G
( cp   0 )
3
D 
 (0.3  2.5) 10
2
S
2
and


Relativistic Binaries
Binary pulsars, especially double-neutron-star systems:
measure post-Keplerian timing parameters in a
theory-independent way (Damour & Deruelle 1986, AIHP, 44, 263).
These predict the stellar masses in any theory of gravity.
In GR:
Pb 5 / 3
1
2/3
Ý 3  T0 M  1  e 2 

2 
Pb 1/ 3 2 / 3 4 / 3
  e  T0 M m2 m1  2m2 
2 
5 / 3


 73
7 / 2
192
P
37 
5/3
PÝb 
 b  1 e 2  e 4 1  e 2  T0 m1m2 M 1/ 3
5 2   24
96 
r  T0 m 2
Pb 2 / 3 1/ 3 2 / 3 1
s  x  T0 M m2
2 
M  m1  m2
T0  4.925490947 s
The Original System: PSR B1913+16
Highly eccentric double-NS
system, 8-hour orbit.
Ý and γ parameters
The 
predict the pulsar and
companion masses.

The PÝb parameter is in
good agreement, to ~0.2%.
Galactic acceleration
 modeling now limits this test.
Weisberg & Taylor 2003, ASP Conf. Ser. 302, 93
(Courtesy Joel Weisberg)
Orbital Decay of PSR B1913+16
The accumulated shift of
periastron passage time,
caused by the decay of
the orbit.
A good match to the
predictions of GR!
Weisberg, Nice & Taylor, 2010
(Courtesy Joel Weisberg)
Mass-mass diagram for
the double pulsar
J0737-3039A and B.
In addition to 5 PK
parameters (for A), we
measure the
mass ratio R. This
is independent of
gravitational theory
⇒ whole new
constraint on gravity
compared to other
double-NS systems.
As of July 2010; Kramer et al. in prep.
Numbers reported in 2006 (Kramer et al, Science):
We can use the measured values of R = 0.9336±0.0005
and 
Ý = 16.8995±0.0007 o/year to get the masses and
then compute the other parameters as expected in GR:

Expected in GR:
Observed:
γ = 0.38418(22) ms
γ = 0.3856 ± 0.0026ms
PÝ = -1.24787(13)×10-12 PÝ = (-1.252 ± 0.017)×10-12
r = 6.153(26) ms
r = 6.21 ± 0.33 ms
0.00016
0.00013
s

0.99974
s  0.999870.00048 
0.00039
b
b
PÝb 
4

10
 
Pb Accel
In particular:

sobs
 0.99987  0.00050
pred

s
Deller et al 2008, Science
This was a 0.05% test of
strong-field GR – now down to
a ~0.02% test!
Using Multiple Pulsars
Strong constraints on
parameters in
alternate theories can be
achieved by combining
information from multiple
pulsars plus solar-system
tests (Damour &
Esposito-Farese).
Damour and Esposito-Farese
Geodetic Precession
Precession of either NS's spin axis about
the total (~ orbital) angular momentum.
Why would we expect this?
Before the second
supernova:
all AM vectors
aligned.
Before the second
supernova:
all AM vectors
aligned.
After second supernova:
orbit tilted,
misalignment angle δ
(shown for recycled A);
B spin pointed
elsewhere (defined
by kick?).
Geodetic Precession
Precession period: 300 years for B1913+16, 700 years for B1534+12,
265 years for J1141-6545 and only ~70 years for the J0737-3039 pulsars.
Observed in B1534+12
(Arzoumanian 1995)
and measured by
comparison to
orbital aberration:

spin
1
Observed in PSR B1913+16,
(Weisberg et al 1989, Kramer 1998)
which will disappear ~2025.
Kramer 1998

 0.44
0.48 o/yr
0.16
(68% confidence)
cf GR prediction:
0.51 o/yr
(Stairs et al 2004)
What about the double pulsar?
There appear to be no changes
in A's profile (Manchester
et al 2005, Ferdman et al 2008).
This starts to put strong
constraints on the
misalignment angle:
<37o for one-pole model,
<14o for two-pole model
(95.4% confidence limits,
Ferdman PhD thesis, UBC 2008;
plot updated to 2009).
But B has changed a lot....
and has now disappeared completely!
Dec. 2003
Jan. 2009
Perera et al., 2010
B's effects on A
provide another
avenue to precession:
Eclipses of A by B
occur every orbit
(Lyne et al 2004,
Kaspi et al 2004).
A's flux is
modulated by the
dipolar emission
from B.
McLaughlin et al. 2004a
See Lyutikov &
Thompson 2005
for a simple
geometrical model,
illustrated here for
April 2007 data
by Breton et al. 2008.
René Breton
See Lyutikov &
Thompson 2005
for a simple
geometrical model,
illustrated here for
April 2007 data
by Breton et al. 2008.
This model makes
concrete
predictions if B
precesses...
Eclipse geometry, Breton et al 2008.
If B precesses, ϕ will change with
time, and the structure of the
eclipse modulation should also
change... and it does!
Dec. 2003
Nov. 2007
Courtesy René Breton
The angles α and θ stay
constant, but ϕ
changes at a rate of
4.77+0.66-0.65o yr-1. This is
nicely consistent with the
rate predicted by GR:
5.0734(7) o yr-1.
And because we measure
both orbital semi-major
axes, this actually constrains
a generalized set of
gravitational strong-field
theories for the first time!
Breton et al 2008
Preferred-frame effects with double-NS
systems
Wex & Kramer 2007
Wex & Kramer showed that pulsars similar to the double pulsar
are sensitive to preferred-frame effects (within semi-conservative
theories) via their orbital parameters. Here they show that the
double pulsar (left), a similar system at the Galactic centre
(middle) and their combination can lead to strong constraints on
the directionality of the preferred frame (PFE antenna).
Future Telescopes
The Galactic Census with the
Square Kilometre Array
should provide:
~30000 pulsars
~1000 millisecond pulsars
~100 relativistic binaries
1—10 pulsar – black-hole
systems.
With suitable PSR—BH systems, we may be able
to
4
c Q
c spin
S
measure the BH
and
quadrupole
moment
q


G2 M 3
G M2
, testing
2
q



the Cosmic
Censorship conjecture and

the No-hair theorem
.
Pulsar with 100 μs timing precision in 0.1-year, eccentricity 0.4
orbit around Sgr A*: weekly observations over 3 years lead to a
measurement of the spin (via frame-dragging) and quadrupole
moment (below) to high precision, assuming Sgr A* is an extreme
Kerr BH.
Wex et al., in prep.
ΔS/S ~ 10-4...10-3
 ΔQ/Q = 0.008
The challenge
will be finding
such pulsars!!
Future Prospects
Long-term timing of pulsar – white dwarf systems
⇒ better limits on GÝ/G and dipolar gravitational radiation, as
well as PPN-type parameters
⇒ better limits on gravitational-wave background
Long-term
timing of relativistic systems
⇒ improved tests of strong-field GR.
Ý in 0737A: we may be
Potential to measure higher-order terms in
able to measure the neutron-star moment of inertia!
Profile changes and eclipses in relativistic binaries
⇒ better tests of precession rates, geometry determinations.
Large-scale surveys, large newtelescopes...
⇒ more systems of all types... and maybe some new “holy grails”
such as a pulsar—black hole system... stay tuned!
⋅
Constraints on α3 (and ζ2)
α3 can also be tested by isolated pulsars.
Self-acceleration and Shklovskii effect contribute to observed
period derivatives:
P
n a self
c
d
Ṗ pm = P 2
c
Ṗ =
3
Young pulsars: α3 < 2x10-10 (Will 1993, “Theory & Expt. In Grav.
Physics,”CUP).
Millisecond pulsars: α3 < ~10-15 (Bell 1996, ApJ, 462, 287; Bell & Damour
1996, CQG, 13, 3121).
.
α3+ζ2 also accelerate the CM of a binary system ⇒ variable P
in eccentric PSR B1913+16: (α3+ζ2) < 4x10-5 (Will 1992, ApJ, 393, L59).
But geodetic precession and timing noise can mimic this effect,
so not really a good test.
PSR B1534+12
Measure same parameters as
for B1913+16, plus Shapiro
delay.
The parameters ˙ , s and γ
form a complementary test
of GR.
After Stairs 2005, ASP Conf. Ser. 328, 3.
The measuredṖ b contains
a large Shklovskii v2/d
contribution. If GR is correct,
the distance to the pulsar is
1.05 ± 0.05 kpc.
What about the double pulsar?
Geodetic precession
appears not
to be happening
in 0737A, despite
short 75-year period.
No pulse shape
changes over 14.5
months, plus back to
discovery observation
(Manchester et al 2005).
But B has shown evidence of
orbital (likely due to A)
changes and long-term (likely
precession) changes from the
start.
Burgay et al. 2005
See Lyutikov &
Thompson 2005
for a simple
geometrical model,
illustrated here for
April 2007 data
by Breton et al. 2008.