the$best$laboratories:$pulsars$with$companions · the$best$laboratories:$pulsars$with$companions$...

The$best$laboratories:$Pulsars$with$companions$Kramer & Stairs (2008)

~"2000""radio"pulsars"

1.40$ms$$(PSR$J1748?2446ad)$8.50$s$$$$$(PSR$J2144?3933)$$

~"140""binary"pulsars"

Orbital(period(range(

95$min$$(PSR$J0024?7204R)$5.3$yr$$$(PSR$J1638?4725)$

Companions)

MSS,$WD,$NS,$planets$

"and"1"Double"Pulsar!"

PSR$J0737?3039A/B$$22.7$ms$/$2.77$s$Porb$=$147$min$e($$$$=$0.088"

RelaQvisQc$effects$observed$and/or$precisely$measured:$

$GravitaQonal$physics$tested$by$pulsars$

•  Precession$of$periastron$•  GravitaQonal$redshiW$•  Shapiro$delay$due$to$curved$space?Qme$•  GravitaQonal$wave$emission$•  GeodeQc$precession,$relaQvisQc$spin?orbit$coupling$$•  Speed$of$gravity$•  …$

Concepts/principles$deeply$imbedded$in$theoraQcal$framework:$•  Strong$Equivalence$Principle$(grav.$Stark$effect)$•  Lorentz$invariance$of$gravitaQonal$interacQon$•  Non?existence$of$preferred$frames$•  ConservaQon$of$total$momentum$•  Non?variaQon$of$gravitaQonal$constant$•  …$

Limits$on$alternaQve$theories,$e.g.$tensor?scalar$theories,$TeVeS$…$

$GravitaQonal$physics$tested$by$pulsars$

Class. Quantum Grav. 26 (2009) 073001 Topical Review

10-8 10

-710

-6

| !" |

10-32

10-31

10-30

10-29

10-28

10-27

10-26

10-25

10-24

10-23

10-22

10-21

|d! "

/dt|

[s-1

]

J0737-3039

J1906+0746

B2127+11C

B1913+16

J1756-2241

B1534+12

J1829+2456

J1811-1736

J1518+4904

B1820-11

J1753-2240

Figure 1. Orbital energy–energy loss diagram for known and likely double neutron star binaries.!O denotes the orbital energy of a binary system divided by the reduced mass energy (µc2). It isa direct measure for the strength of post-Newtonian effects in the orbital dynamics. Its derivatived!O/dt is the reduced gravitational wave luminosity of the system according to general relativity. Itis an indicator for the strength of radiative effects that cause secular changes to the orbital elementsdue to gravitational radiation damping. This figure illustrates that the double pulsar is the mostrelativistic binary pulsar known to date.

quantity immediately accessible to the observer. The conversion is possible since we knowthe inclination angle of the orbit, i, which we can determine with the help of the relativisticShapiro effect. We will discuss this effect and a number of other manifestations of relativisticgravity in the following section (section 3), where we will see that the small but nonzeroorbital eccentricity of e = 0.088 is important.

The combination of these system parameters means that the double pulsar is currently themost interesting source for tests of theories of gravity. This is even true without the uniquefeature of having both neutron stars visible as radio pulsars. This is illustrated in figure 1where we attempt to compare the known double neutron star systems by plotting the orbitalenergy of each binary system divided by the reduced mass energy (as a direct measure forthe strength of post-Newtonian effects) versus its derivative (as as an indicator of the strengthof radiative effects that cause secular changes to the orbital elements due to gravitationalradiation damping). The double pulsar clearly stands out at the top, marking its importancefor experiments discussed in this work.

3


10-8 10

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10-32

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|d! "

/dt|

[s-1

]

J0737-3039

J1906+0746

B2127+11C

B1913+16

J1756-2241

B1534+12

J1829+2456

J1811-1736

J1518+4904

B1820-11

J1753-2240




3


10-8 10

-710

-6

| !" |

10-32

10-31

10-30

10-29

10-28

10-27

10-26

10-25

10-24

10-23

10-22

10-21

|d! "

/dt|

[s-1

]

J0737-3039

J1906+0746

B2127+11C

B1913+16

J1756-2241

B1534+12

J1829+2456

J1811-1736

J1518+4904

B1820-11

J1753-2240




3


10-8 10

-710

-6

| !" |

10-32

10-31

10-30

10-29

10-28

10-27

10-26

10-25

10-24

10-23

10-22

10-21

|d! "

/dt|

[s-1

]

J0737-3039

J1906+0746

B2127+11C

B1913+16

J1756-2241

B1534+12

J1829+2456

J1811-1736

J1518+4904

B1820-11

J1753-2240




3

Redu

ced$gravita

Qonal$w

ave$luminosity

$

Orbital$energy$(μc2)$

Hulse?Taylor$

Kramer$&$Wex$(2009)$

Double$Pulsar$

Here:$Double$Neutron$Stars$(more$on$NS?WD$later)$

Latest$update$on$Hulse?Taylor$Pulsar$$

Observed$value:$$

$$$$dPb/dt=$?2.423$±$0.001$×$10?12$

• $First$binary$pulsar$• $Decay$of$orbit$as$first$evidence$for$existence$of$gravitaQonal$waves$• $Data$in$agreement$with$GR’s$$gravitaQonal$quadrupole$emission:$$

[ Weisberg et al. 2010 ]

Test$limited$by$systemaQc$$effects$$due$to$relaQve$$acceleraQon$in$the$GalacQc$gravitaQonal$potenQal$and$our$(limited)$knowledge$of$it.$

The$Double$Pulsar$(Burgay$et$al.$2003,$Lyne$et$al.$2004)$• $Old$22?ms$pulsar$in$$a$147?min$orbit$with$young$2.77?s$pulsar$• $Orbital$velociQes$of$1$Mill.$km/h$• $Eclipsing$binary$in$compact,$slightly$eccentric$and$$edge?on$orbit$• $Ideal$laboratory$for$gravitaQonal$and$fundamental$physics$• $In$parQcular,$exploitaQon$for$GR$(Kramer$et$al.$2006,$Breton$et$al.$2008,$Kramer$&$Wex$2009)$• $Recent$very$significant$improvement$(Kramer$et$al.$in$prep.)$

Orb

ital

pha

se →

Comparison$Hulse?Taylor$vs$Double$Pulsar$

PSR$B1913+16$ PSR$J0737?3039A/B$

Sun$

PSR$B1913+16$

PSR$J073773039A/B$

More$compact…$

…$and$much$closer!$

Double$Pulsar:$most$relaQvisQc$system$

4$x$larger$than$Hulse?Taylor!$

?$Measured$within$a$few$days$of$observaQons!$?$One$full$revoluQon$in$about$20$years!$(cf.$to$3$Million$years$for$Mercury)$

•  Huge$orbital$precession$of$16.8991$±$0.0001$$deg/yr!$$


•  Huge$orbital$precession$of$16.8991$±$0.0001$$deg/yr!$•  Clock$variaQon$due$to$gravitaQonal$redshiW:$383.9$±$0.6$microseconds!$

?$As$other$clocks,$pulsars$run$slower$in$deep$gravitaQonal$potenQals$

?$Changing$distance$to$companion$$(and$felt$grav.$potenQal)$during$ellipQcal$orbit$


•  Huge$orbital$precession$of$16.8991$±$0.0001$$deg/yr!$•  Clock$variaQon$due$to$gravitaQonal$redshiW:$383.9$±$0.6$microseconds!$•  Shapiro$delay$in$edge?on$orbit:$$$s$=$sin(i)=0.99975$±$0.00009$$

?$At$superior$conjuncQon,$pulses$from$pulsar$A$pass$B$in$20,000km$distance$?$Space?Qme$near$companion$is$curved$!AddiQonal$path$length$$$$$$$$$$!$Delay$in$arrival$Qme$–$depending$on$geometry$and$companion$mass$

20,000km$i=88.7±0.2$deg$


•  Huge$orbital$precession$of$16.8991$±$0.0001$$deg/yr!$•  Clock$variaQon$due$to$gravitaQonal$redshiW:$383.9$±$0.6$microseconds!$•  Shapiro$delay$in$edge?on$orbit:$$$s$=$sin(i)=0.99975$±$0.00009$•  Shrinkage$of$orbit$due$to$GW$emission:$$ΔPb=107.79$±$0.11$ns/day!$$

?$Pulsars$approach$each$other$by$$$$$$$$$$$$$$$$$$$$$7.152$±$0.008$mm/day$$$$?$Merger$in$85$Million$years$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$AnimaQon$by$Rezolla/AEI$

Are$these$numbers$in$agreement$with$GR$predicQons?$$$$

TesQng$theories$of$gravity$

RelaQvisQc$effects$measured$as$correcQons$to$Keplerian$orbit:$

•  Binary$period,$Pb$$•  Projected$semi?major$axis,$$$$$x$=$ap$sin(i)$/$c$•  Eccentricity,$e$•  Longitude$of$periastron,$ω$•  Epoch$periastron,$T0$

Post?Keplerian$Parameters$$=$theory$independent$$$$$correcQons$to$describe$$$$$pulse$arrival$Qmes$

Among$others:$•  Shapiro$delay,$r$and$s$•  GravitaQonal$redshiW,$γ$$•  Decay$of$orbit,$dPb/dt$•  Precession$of$orbit,$dω/dt$

Idea:$Compare$measured$magnitude$of$PK$parameters$with$theory$predicQon!$

Keplerian$parameter:$ Post?Keplerian$(PK)$parameters:$

Strong?field$tests$with$binary$pulsars$

Elegant$method$to$test$(falsify!)$any$theory$of$gravity$$(Damour$&$Taylor$‘92)$

All$PK$parameter$can$be$wriven$as$funcQon$of$only$observed$Keplerian$and$the$masses$of$pulsar$$and$companion,$e.g.$in$GR$we$can$write$orbital$$precession$rate$as:$

( )2

3/23/53/2

123/

emmPTdtd cpb

−

+"#

$%&

'=−

πω !

Periastron$advance$

For$every$post?Keplerian$parameter,$we$can$write:$$$$$Mass$companion$=$FuncQon$Theory$(Mass$pulsar$|$Keplerian,$PK$paramteters)$


All$lines$given$by$PK$measurements$need$to$meet$in$a$single$point$for$theory$to$pass$test!$$We$need$2$PK$parameters$to$define$intersecQon$point.$$Every$addiQonal$PK?line$can$potenQally$miss$this$intersecQon$point$and$hence$tests$the$theory!$


NPK$–$2$tests$possible$





PK3$

Fail!$

PK1$

PK2$





PK1$

PK3$Pass!$

PK2$

GR$test$with$the$Double$Pulsar$

Five(!)$unique$strong?field$tests,$represented$in$a$single$mass?mass$plot:$

MB=(1.2489±0.0007)M"$

MA=(1.3381±0.0007)$M"$

Kramer et al (2006), Breton et al. (2008), Kramer & Wex (2009)

GR$test$with$the$Double$Pulsar$

Five(!)$unique$strong?field$tests,$represented$in$a$single$mass?mass$plot:$

Kramer et al (2006), Breton et al. (2008), Kramer & Wex (2009)

Periastron$advance$

Mass$raQo$(two$orbits!)$

Shapiro$delay$

Shapiro delay

GravitaQonal$redshiW$

Spin?orbit$coupling$

B

A

A

B

mm

xx

R =≡

Mass$raQo$&$6$PK$parameters$⇔ 7?2$=$5$tests$of$GR!$

More$than$in$any$system!$

R$is$independent$of$strong$(self?)field$effects$to$1PN!$

$QualitaQvely$$

different$&$unique$constraint!$

$

Grav.$wave$emission$

Best$strong?field$test$of$General$RelaQvity$

Recent$big$improvements:$Kramer et al (2006) Kramer et al (in prep.)

Precision$measurements,$e.g.$$P$(ms)$$=$22.6993785996213$±$0.0000000000002$(measured$to$0.2$picoseconds!)$$Pb$(d)$=$$$0.102251562452$±$0.000000000008$$$$$(i.e.$2.45h$measured$to$691$ns!)$dPb/dt$$=$(?1.248±0.001)$x$10?12$$?$agreement$with$GR$at$0.1%$?$best$radiaQon$test!$$$$

• $Shapiro$delay$measured$to$9x10?5$!$–$More$precise$than$we$can$predict!$• $Best$radiaQve$test$ever:$0.1%$$• $Precision$will$improve$further$with$Qme!$$$$$$$$$$$$$$$$•  Note:$for$$5th$test$see$talk$by$Vicky!$

Double$Pulsar:$Latest$tests$of$GR$

Expected$in$GR:$ Observed:$

γ$=$0.3840(4)$ms$ 0.3839(6)$$ms$

dPb/dt=?1.248(1)x10?12$ ?1.248(1)x10?12$

r$=6.151(3)$µs$ 6.26(14)$µs$

s=0.99988(50)$ 0.999750(90)$

Based$on:$R$=$1.0714±0.0011$&$ώ=16.8991±0.0001$deg/yr$

1.000(2)$

1.000(1)$

0.98(2)$

1.0000(5)$

RaQo:$

shape s of Shapiro delay

[ Kramer et al. 2006 ] R and sin i

Proj. semi-major axis: xA = 1.415032(1) xB = 1.5161(16)

R ⌘ xA/xB = mB/mA +O �c

�4�

[ Kramer et al. 2006 ]

[ Damour & Deruelle 1986, Damour 2005 ]

Generic$tests$with$the$Double$Pulsar$Modified$Einstein?Infeld?Hoffmann$(mEIH)$formalism:$$$$L(x,$v;$mA,$mB,$GAB,$eAB,$xAB)$?$developed$by$Eardley,$Will$and$Damour$&$Taylor$?$provides$a$generic$descripQon$for$the$moQon$of$a$compact$bodies$

21

!

[ Kramer & Wex 2009 ]

!

GABmA = 1.339± 0.003 GM�

GABmB = 1.250± 0.002 GM�

Double Pulsar measurements

Mass ratio

R ! xA/xB = mB/mA + O!

c!4"

From R and s = sin i

GABmA = 1.339 ± 0.003 GM"

GABmB = 1.250 ± 0.002 GM"

Limits from the double pulsar

! " 2"AB # #AB = 4.998 ± 0.008 [0.2%]

$ " (G0B + %A&BA)/GAB = 1.005 ± 0.010 [1.0%]

rSh " (G0B/GAB)("0B + 1) = 4.04 ± 0.22 [5.4%]

!B " "AB/GAB = 1.88 ± 0.26 [13%]

If "AB/GAB = 1

2 ("AB + 1)

"AB = 2.8 ± 0.5 [18%]

"AB/GAB = 1.88 ± 0.26 [13%]

Strong field PPN parameters

Orbital precession: 2$AB # 'AB = 0.999 ± 0.004 [0.4%]

Pulsar rotation: (G0B + %A&BA)/GAB = 1.00 ± 0.01 [1%]

Signal propagation: (G0B/2GAB)($0B + 1) = 1.01 ± 0.05 [5%]

Spin precession: "AB/2GAB = 0.89 ± 0.13 [18%]

Alternative theories of gravity

JFBD theory

Field equations

R#µ! =

8(G#

c4

!

T #µ! # 1

2T #g#µ!

"

+ 2)µ*)!*

gµ!# $#

µ$#!* = #

4(G#

c4+0 T#

Physical metric

gµ! = g#µ! exp (2+0*)

7

E

22

In$strong$fields,$effecQve$coupling$constants$become$object?dependent:$e.g.$for$α0$=$3×10?5$

|α|

10-4

10-3

10-2

10-1

100

m [MSun]1 1.2 1.4 1.6 1.8

β0 = -5.5

β0 = -4.5

β0 = -5.0

β0 = -4.0

Solar System|α|

10-4

10-3

10-2

10-1

100

ε = mb/m - 10.04 0.06 0.08 0.1 0.12

β0 = -5.5

β0 = -5.0

β0 = -4.5

β0 = -4.0

Solar System

Note

!20 !

1

2"BD + 3

# = 1 "2!2

0

1 + !20

# 1 " 2!20

GAB/G = 1 " 2!20 (sA + sB " 2sAsB) + O(!4

0)

$AB = 3 " 4!20 (1 " sA)(1 " sB) + O(!4

0)

%AB = 1 " 8!20 sAsB + O(!4

0)

Quadratic model

Field equations

R!µ! =

8&G!

c4

!

T !µ! " 1

2T !g!µ!

"

+ 2'µ('!(

gµ!! $!

µ$!!( = "

4&G!

c4(!0 + )0()T!

and

!0 ! )0(0

Physical metric

gµ! = g!µ! exp!

2!0( + )0(2"

Various actions and action terms

Sgravity =c3

16&G!

#

d4x%"g! [R! " 2f(gµ!

! 'µ('!()]

Sgravity =c3

16&G!

#

d4x%"g! [R! " 2f(gµ!

! (,µ(,!)] + Svector

$

U[µ;!], U(µ;!), Uµ;µ, U!Uµ;! , UµUµ

%

Svector[U[µ;!]U[µ;!], U

(µ;!)U(µ;!), U"Uµ;"U#Uµ;# , Uµ;µ, UµUµ]

Svector[Uµ; g!µ! ]

Smatter[*; gµ! ]

Smatter[*; gµ! ! A2(()g!µ! + B(()UµU! ]

8

Physical$metric:$ [$Damour(&(Esposito9Farèse(1996(]$

Tensor?Scalar$theories$

gµ⌫ = g⇤µ⌫ exp(2↵0'+ �0'2)

Best$studied:$„quadraQc$model“$by$$Damour$&$Esposito?Farese$$

23

Parameters$as$funcQon$of$Egrav/mc2$of$A$and$B$

solar system

Example:$Strong$Field$Effects$in$the$b0=$?6$``QuadraQc$Model´´$

(GAB / G) - 1 ωQM / ωGR A

A

B

B

Solar$system$tests$are$not$sufficient$to$test$theories.$Even$the$Double$Pulsar$alone$is$not$sufficient!$

Dipolar$GravitaQonal$RadiaQon$in$binary$systems$

But PSR – WD system also effective lab – in particular if PSR is massive! See talk by Vicky Kaspi!

PSR-BH system would be best as BH would not have no “scalar hair” (see e.g. Wex & Koepikin 1998, in particular Liu et al. 2012a,b)

P dipole

b

= �4⇡2

Pb

GmA

mB

c3(mA

+mB

)

1 + e2/2

(1� e2)5/2(↵

A

� ↵B

)2

A B

Unlike$GR,$most$alternaQve$theories$of$gravity$predict$dipole$radiaQon$that$dominates$the$energy$loss$of$the$orbital$dynamcis$(1.5$pN):$

Next$best$thing:$PSR$J1738+0333$–$a$PSR?WD$system$

Precision$Qming$observaQons$of$PSR$ OpQcal$observaQons$of$the$WD$companion$[$Freire(et(al.,(2012]$ [$Antoniadis(et(al.,(2012$]$

From$spectrum$(surface$gravity$&$T):$((((((mc$=$0.181$±0.008$Msun$

From$radial$velociQes:$(((((((q(=(mp/mc$$=$8.1$±$0.2$

Hence:$$$$$$mp(=(1.47(±(0.07(Msun(

(((((((((((((((

From$pulsar$Qming:$(

(((((((((((Dp$=$1.4$±$0.1$kpc(((((((((((

(((((((((((Pb/dt$=$(?27.7$±$1.5)$!$10?15$$

P$$$=$5.85$ms,$Pb$$=$8.51$h$

Gemini+ Keck observations

Limits$on$dipole$radiaQon$and$change$in$G$

Observed$orbital$decay$agreement$with$GR‘s$quadrupole$emission:$

GR$predicQon$

Hence,$limit$on$contribuQon$from$dipole$rad$or$change$in$G:(((((((((((((((

Pbexc = Pb

obs − PbGR = Pb

dipole + PbG

22

242

112 pDb

Sunp

c

b

excb s

PT

sMm

GG

PP

κµ

π−$%

&'(

)*+

,-.

/ +−−=

Freire$et$al.$(2012)(((((((((((((((

LLR(((((((((((((((

"�

Limits$bever$than$solar$system$limits$for$most$of$the$parameter$space:$

Best$limits$on$Tensor?scalar$theories$

€

a(ϕ) = α0(ϕ −ϕ0) +12β0(ϕ −ϕ0)

2 + ...

Note:$• $In$GR,$$a(φ)$=$0$• $Jordan?Fierz?Brans?Dicke:$$$$β0=0$and$a(φ)$=$α0(φ?φ0)$

Coupling$of$maver$to$scalar$field:$PSR J1738+0333. II. The most stringent test of scalar-tensor gravity. 31

LLR

LLR

SEP

J1141–6545

B1534+12

B1913+16

J0737–3039

J1738+0333

!6 !4 !2 2 4 60

0

0

0|

10

10

10

10

10

Cassini

Figure 7. Solar-system and binary-pulsar constraints on the matter-scalar coupling constants !0 and "0. Note that a loga-rithmic scale is used for the vertical axis |!0|, i.e., that general relativity (!0 = "0 = 0) is sent at an infinite distance down thisaxis. LLR stands for lunar laser ranging, Cassini for the measurement of a Shapiro time-delay variation in the Solar System,and SEP for tests of the strong equivalence principle using a set of neutron star-white dwarf low-eccentricity binaries (seetext). The allowed region is shaded, and it includes general relativity. PSR J1738+0333 is the most constraining binary pulsar,although the Cassini bound is still better for a finite range of quadratic coupling "0.

LLR

SEP

J1141–6545

B1534+12

B1913+16

J0737–3039

J1738+0333

10

!6 !4 !2 0 2 4 60

0|

100

Tuned

TeVeS

Inconsistent

TeVeS

10

10

10

Figure 8. Similar theory plane as in Fig. 7, but now for the (non-conformal) matter-scalar coupling described in the text,generalizing the TeVeS model. Above the upper horizontal dashed line, the nonlinear kinetic term of the scalar field may be anatural function; between the two dashed lines, this function needs to be tuned; and below the lower dashed line, it cannot existany longer. The allowed region is shaded. It excludes general relativity (!0 = "0 = 0) because such models are built to predictmodified Newtonian dynamics (MOND) at large distances. Note that binary pulsars are more constraining than solar-systemtests for this class of models (and that the Cassini bound of Fig. 7 does not exist any longer here). For a generic nonzero "0,PSR J1738+0333 is again the most constraining binary pulsar, while for "0 ! 0, the magnitude of |!0| is bounded by theJ0737"3039 system.

c# 0000 RAS, MNRAS 000, 000–000

Freire et al. (2012)

GR (α0=β0=0)

MOND?like$tensor?vector?scalar$(TeVeS)$theories$

dark matter

[ Milgrom 1983 ]

Modified Newtonian dynamics (MOND)

To reproduce MOND:

TeVeS with aquadratic kinetic term:

[ Bekenstein & Milgrom,Bekenstein ]

,

“Natural“$MOND$and$binary$pulsars$

MOND potential appears at a distance larger than:

~ 7000 AU ! α02

If α02 < 0.003, MOND effects would be apparent in the solar system dynamics,

hence:

Such value would lead to a significant emission of dipolar gravitational radiation...

[ Bruneton & Esposito-Farese 2007 ]

|↵0| & 0.05

Limits$from$PSR$J1738+0333$for$general$class$of$TeVeS$

Double$Pulsar$and$PSR?WD$systems$complement$each$other$perfectly$MOND?like$TeVeS$theories$NEED$TO$BE$TUNED$and$deviate$from$its$original$form$Depending$on$future$$improvement,$TeVeS$might$eventually$become$inconsistent$theory$$$$

PSR J1738+0333. II. The most stringent test of scalar-tensor gravity. 31

LLR

LLR

SEP

J1141–6545

B1534+12

B1913+16

J0737–3039

J1738+0333

!6 !4 !2 2 4 60

0

0

0|

10

10

10

10

10

Cassini

Figure 7. Solar-system and binary-pulsar constraints on the matter-scalar coupling constants !0 and "0. Note that a loga-rithmic scale is used for the vertical axis |!0|, i.e., that general relativity (!0 = "0 = 0) is sent at an infinite distance down thisaxis. LLR stands for lunar laser ranging, Cassini for the measurement of a Shapiro time-delay variation in the Solar System,and SEP for tests of the strong equivalence principle using a set of neutron star-white dwarf low-eccentricity binaries (seetext). The allowed region is shaded, and it includes general relativity. PSR J1738+0333 is the most constraining binary pulsar,although the Cassini bound is still better for a finite range of quadratic coupling "0.

LLR

SEP

J1141–6545

B1534+12

B1913+16

J0737–3039

J1738+0333

10

!6 !4 !2 0 2 4 60

0|

100

Tuned

TeVeS

Inconsistent

TeVeS

10

10

10

Figure 8. Similar theory plane as in Fig. 7, but now for the (non-conformal) matter-scalar coupling described in the text,generalizing the TeVeS model. Above the upper horizontal dashed line, the nonlinear kinetic term of the scalar field may be anatural function; between the two dashed lines, this function needs to be tuned; and below the lower dashed line, it cannot existany longer. The allowed region is shaded. It excludes general relativity (!0 = "0 = 0) because such models are built to predictmodified Newtonian dynamics (MOND) at large distances. Note that binary pulsars are more constraining than solar-systemtests for this class of models (and that the Cassini bound of Fig. 7 does not exist any longer here). For a generic nonzero "0,PSR J1738+0333 is again the most constraining binary pulsar, while for "0 ! 0, the magnitude of |!0| is bounded by theJ0737"3039 system.

c# 0000 RAS, MNRAS 000, 000–000

Beckenstein‘s TeVeS (=rel. MOND)

New unpublished 0737 limit

Freire et al. 2012

TeVeS inconsistent

[$Kramer,(Stairs,(Freire,(Esposito9Farèse,(Wex,…$(in$prep.)$]$

ity up to!!!!!!!!!!!!!!!!!!!GM!=a0

p" 7000 AU, and the MOND dynam-

ics beyond. A reasonable value "# 1=10 would also suf-fice for the refined model (2.18). However, this theorywould be inconsistent by 5 orders of magnitude withpost-Newtonian tests in the solar system, because thescalar field would be much too strongly coupled to matter.

A possible solution to the above problem would be tofine-tune the function f$s% even further. In order to get theMOND regime for r *

!!!!!!!!!!!!!!!!GM=a0

p, as required by galaxy

rotation curves, one would need f0$s% "!!!!sp

for !s & !4 <10&10. On the other hand, in order to obtain the Newtonianregime within the solar system, say for r & rmax # 20 or30 AU, one would need f0$s% " 1 for !s *$!4GM!=a0r2

max%2 # 10&10 (this second occurrence of10&10 is a numerical coincidence). Therefore, there wouldexist a brutal transition between the MOND andNewtonian regimes around !s# 10&10. Not only wouldthe introduction of such a small dimensionless number bequite unnatural, but this model would also predict that theanomalous acceleration caused by the scalar field remainsapproximately equal to the constant11 a0 between 30 and7000 AU. As illustrated in Fig. 3, this would be a way toreconcile the MOND acceleration

!!!!!!!!!!!!!!GMa0p

=r at large dis-tances with the experimentally small Newtonian contribu-tion !2GM=r2 of the scalar field at small distances.Although this is not yet excluded experimentally, it wouldhowever suffice to improve by 1 order of magnitude thepost-Newtonian constraint on !2 to rule out such a fine-tuned model (the planned astrometric experiment GAIA[114] should reach the 10&6 level for !2, and the proposedLATOR mission [115,116] should even reach the 10&8

level). Therefore, one should not consider it seriously.An a priori better solution to the above problem would

be to recall that the Newtonian limit f0$s% " 1 is actuallyunnecessary. Since the metric g'"# already generates aNewtonian potential &GM=r, it suffices that the scalar-field contribution remain negligible (even at the post-Newtonian level) in the solar system. One may try afunction f0$s% whose shape looks like the one displayedin Fig. 4, for instance

f0$s% ( ")!!!!!!j !sj

p

$1) j !sj%1)1=n; (2.20)

where n is a positive constant and as before !s * !6c4s=a20.

Then the transition occurs again at !s# 1, i.e., aroundrtrans ( !2

!!!!!!!!!!!!!!!!GM=a0

p, but Eq. (2.15) shows that the force

mediated by the scalar field reads !@r’c2 ( $a0=!2%+$r=rtrans%n in the large-s limit [assuming that the " ofEq. (2.20) has a negligible influence]. Therefore, even if

! ( 1, so that the Newton-MOND transition occurs atrtrans (

!!!!!!!!!!!!!!!!GM=a0

pas expected, the anomalous scalar force

would be negligible with respect to post-Newtonian rela-tivistic effects for r, rtrans and n large enough. However,the above scalar field @r’ / rn happens not to be a solutionof Eqs. (2.15) and (2.20) for r < rtrans, where " mustactually dominate. In such a case, the scalar force takesthe Newtonian form !@r’c2 ( !2GM=r2" and is evenincreased by a factor 1=" with respect to model (2.17).This suffices to rule out the class of models (2.20). Onemay also notice that for "! 0, they do not satisfy thehyperbolicity condition (b) in the large-s limit unless n (1; but even for this limiting case n ( 1, the scalar force isnot negligible at small distances.

In conclusion, the above discussion illustrates thatRAQUAL models are severely constrained, although theyinvolve a free function f$s% defining the kinetic term of thescalar field. Contrary to some fears in the literature, thepossible superluminal propagations do not threaten cau-sality, and the two conditions (a) and (b) are the only oneswhich must be imposed to guarantee the field theory’sconsistency. For instance, monomials f$s% ( sn are al-lowed if n is positive and odd [except on the possiblehypersurfaces where s vanishes, which would violate the

FIG. 3. Fine-tuned function f0$s% such that Newtonian andpost-Newtonian predictions are not spoiled in the solar system,although the MOND dynamics is predicted at large distances.The right panel displays the quite unnatural contribution of thescalar field to the acceleration of a test mass, as a function of itsdistance with respect to the Sun.

FIG. 4. Typical shape of a function f0$s% such that scalar-fielddeviations from GR are / rn, n > 0, in the large-s limit.

11Note that the MOND constant a0 is too small by a factor 7 toaccount for the anomalous acceleration of the two Pioneerspacecrafts. Actually, we will also see in Sec. V C below thatthere exists a crucial difference between the MOND dynamicsand the Pioneer anomaly.

FIELD-THEORETICAL FORMULATIONS OF MOND-LIKE . . . PHYSICAL REVIEW D 76, 124012 (2007)

124012-13

[$Bruneton(&(Esposito9Farèse(2007$]$

31

TeVeS$becomes$unnatural$

a02$~$10?4$requires$an$extremely$unnatural$behaviour$of$f(x):$ϕ)

Summary$&$Conclusions$

• $Pulsars$are$unique$tools$for$the$study$of$fundamental$physics,$esp$gravitaQon$

• $They$provide$clean$and$simple$experiments$as$test$masses$with$clocks$

• $The$best$example,$the$Double$Pulsar,$just$gets$bever$and$bever$

• $Best$strong?field$tests$(incl.$radiaQon$test)$provided$by$Double$Pulsar$

• $No$Qme$to$talk$about:$SEP$tests$$?$See$review$by$Freire,$Kramer$&$Wex$(in$press)$

• $Best$limits$for$alternaQve$theories$of$gravity$(general,$tensor?scalar,$TeVeS)$

$$(!$you$may$need$to$invoke$Chameleon/Galileon$screening)$

• $MOND?like$TeVeS$theories$may$be$ruled$out$eventually$

• $Best$test$would$be$a$PSR?BH$system$–$we’re$looking$for$it$$

• $New$systems$are$being$discovered$–$some$are$very$exciQng…!!$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$to$be$conQnued$–$with$Vicky’s$talk!$

the$best$laboratories:$pulsars$with$companions · the$best$laboratories:$pulsars$with$companions$...

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