Constraints on massive graviton dark matter from precision pulsar timing and astrometry Konstantin POSTNOV (Sternberg Astronomical Institute) Collaborators:

Constraints on massive graviton dark matter from precision pulsar timing and astrometry

Konstantin POSTNOV (Sternberg Astronomical Institute)

Collaborators: Maxim Pshirkov (PRAO Lebedev Phyical Institute), Artyom Tuntsov (SAI), Aleksandr

Polnarev (QMC London UK), Deepak Baskaran (Cardiff UK)

QUARKS-2008

Plan

•Pulsars as GW detectors•Observational constraints on massive graviton CDM•“Surfing effect” of massive gravitons and limits on their propagation speed

Pulsars as GW detectors

Gravitational waves (1/2)

•General Relativity

• GW propagation velocity in empty space is с:

• Along axis z: )exp(ikzeh • Polarizatrion tensor have two non-zero components

• Monochromatic transverse GW has two polarizatios (GR)

e 11e 12e&

• GW energy density (monochromatic plane):

22222

~16

hhhG

c

• Stochastic isotropic background:

df

fdff g

cg

)()(

29 323 8 2 100c H G g cm Is the critical density

•Or:

cHkH /2 0

cfk /2

Pulsars as GW detectors

Gravitational waves (1/2)

Pulsars as monochromatic GW detectors

Monochromatic GW (1/3)

z

x

y

PSR

•GW changes the observed pulsar frequency (Sazhin (1978), Detweiler (1979))

dlt

hzz2

1

)exp()exp(1

sinRe~ 2 tiu

uDi

cos1u

•In GR interaction is independent of distance (if ) – no secular increase ~D.

cD 2

,)cos()(cos12cos2

1~

DDthth

),(,cos gwpgwp nnnnP

P

Is the GW polarization vector

Pulsars as GW detectorsMonochromatic GW (2/3)

•Variation of the observed frequency results in time residuals in time of arrival (TOA):

thdtttR

t

cos~/)()( 0

0

00

h

•Maximum sensitivity at frequencies ~ 1/Tobs

•Longer GWs also contribute to the observed Pulsar period and its derivativePP ,

1/Tobs 1/Tsamp 1/Tint

Tobs ~ 10 yearsTsamp~ 10 daysTint ~ 1 hour

h

In 2003 periodic motions in 3C66b were explained by binary SMBH (Sudou et al., 2003)-80 Mpc, 1.5x1010 M⊙

•Timing of PSR B1855+09 rejected this possibility (Jenet et al., 2004)

Pulsars as GW detectorsMonochromatic GW (3/3)

Pulsars as GW detectorsStochastic GWB (1/3)

• RMS of TOA residuals is (Detweiler, 1979):

•RMS of TOA residuals depend on GW energy density

•For flat GW spectrum of width Δf~f centered at f

dfdf

gc

f

G

f

GtR

43432

243

208

243

208)(

29 323 8 2 100c H G g cm - the critical density

yrsfhf

fhfS

yrgg 2524

54

2

11034.116

)()(

• For arbitrary GWB («red noise»):

01100

Hh

km s Mpc

Kaspi, Taylor, Ryba, 1994


Pair correlation of the TOA residuals for 20 pulsars (simulation, R,Manchester, 2007 )

•GW noise is the same for all pulsars

It is advantageous to observe ensemble of pulsars and correlate rms of TOA residuals between each pair of pulsars


Pulsars as GW detectorsPresent limits and prospects

(Manchester, 2007 – arXiv:0710.5026v2)

Tests in Solar systems

Doppler tracking (1/2)

•Estabrook & Wahlquist, 1975, principle similar to pulsar timing •Best current limits: Cassini mission, 10-3-10-6 Hz(Armstrong et al. 2003)

Solar system testsDoppler tracking (2/2)

Reynaud et al. 2008

•Future projects: Search for Anomalous Gravity using Atomic Sensors, SAGAS

Astrometric constraints

• A GW causes «drizzling» of visual position of a source on the sky (e.g, Kaiser&Jaffe, 1997):

h~

•The observed quantity is the arc length between two sources Ψ:

2sin~

h

• In the presence of a GW sources on the sky would oscillate w.r.t. to their true position with amplitude h. Modern ICRF precision (~100 μas) constrain low-frequency GWB: h<5x10-10

Theories with massive gravitons

• Massive gravity ( Rubakov 2004, Dubovsky 2004) with spontaneous Lorentz braking (Rubakov & Tinyakov arXiv:0802.4379 for a review)

• Healthy theory: no ghosts, no vDVZ discontinuity, no strong coupling at low scale

• Interesting phenomenology: DE-like term in Fridmann equations + possibility to produce massive gravitons in the early Universe copiously enough to explain all of CDM (Dubovsky, Tinyakov & Tkachev 2005)

• Taking graviton mass < (1015 cm)-1 (binary PSR constraints) and assuming all galactic CDM due to massive gravitons leads to a strong almost monochromatic (Δf/f~10-6) GW signal with amplitude

110

5~ 10

3 10

fh

Hz

Observational constraints: PTP08Pulsar timing (1/2)

• Isotropic GW background affects pulsar timing

• GW amplitude can be constrained from rms residuals of TOA of even one pulsar

• Strong monochromatic signal (e.g. if all of galactic DM is due to massive gravitons, as in Dubovsky et al 2005) will manifest itself at frequencies < 1/Tint (PSR integration time ~ 1-2 hrs) (PTP08):

)0(2Rr ,

• Limit on the GW amplitude from the existing rms residuals of TOA of pulsars:

2008arXiv0805.1519: Pshirkov, Tuntsov, Postnov

Constraints using existing rms TOA residuals (Manchester, 2007), PSR B1937+21

Observational constraints: PTP08Pulsar timing (2/2)

Observational constraints: BPPP08«Surfing effect» (1/4)

• Unlike in GR, massive gravitons propagate with velocity less than c :

• Mass of the graviton is expressed through phenomenological parameter ε :

• Pulsar frequency change by massive GW::

arXiv:0805.3103: Baskaran, Polnarev, Pshirkov, Postnov

http://xxx.lanl.gov/abs/0805.3103


• TOA residuals:

• Unlike GR, residuals seculary increase with distance to the source D !

• Above results for a monochromatic GW can be generalized to stochastic GWB:


• Response to any harmonics is known:

• The observed TOA residuals will be expressed through this «transfer function»:


• R(k) depends on ε (term )

I.

II.

• For example, power-law spectrum:

Observational constraints: BPPP08«Surfing effect»: limits (1/5)

• Depending on ε, PSR timing put bounds on energy density of GWB:

• Or some combination of GW energy density and ε :


• For known GW amplitude, the parameter ε can be constrained:

• For theoretically motivated GWB from SMBH:

or



• In terms of the graviton mass:

• From modern pulsar timing (Manchester 2007)

,

• which is by 3 orders of magnitude better than from Solar system bounds• can be increased by one order with increasing observational time• comparable to the future LISA constraints.

CONCLUSIONS

• Precise astronomical observations, especially pulsar timing, put strong bounds on massive graviton parameters:

• Cold massive gravitons cannot constitute all of the galactic dark matter

Спасибо за внимание!

Теории с массивным гравитономНаблюдаемые проявления (1/4)

(Тиняков 2007)



(Hi – параметр Хаббла в инфл. эпоху)



Принципы тайминга

Одиночные пульсары(1/4)

J 1022+ 10 J 1640+22

B1937+21 J2145- 07

Stairs, 2003



Радиотелескоп РТ-64 КРАО (ТНА-1500 ОКБ МЭИ)



•N-ый импульс от пульсара приходит на РТ в момент времени N. •Редукция в барицентр Солнечной системы. Момент прихода в барицентр СС:

62

32NN

N

tttN

•Считается, что пульсар вращается по известным законам. Момент прихода N-го импульса связан с его номером, частотой вращения и её производными и может быть предсказан.

•В действительности, между наблюдаемыми моментами прихода N-го импульса и предсказанными значениями всегда существует разница-остаточные уклонения:

)(

)( NobsN

tNNtR



00

2000

10 6

1)(

2

1)(

ttBttA

ttttttRR

•Уточнение параметров происходит по МНК. Минимизируются остаточные уклонения:

EEE crA sincoscos/

cossincossincos/ EEEE crB

p ,...,, -поправки к принятым значениям p,...,,


Остаточные уклонения

•После процедуры остаются остаточные уклонения моментов прихода импульсов

)(

)( NobsN

tNNtR

Остаточные уклонения пульсаров B1937+21 и B1855+09 (1985-1993, Аресибо),Kaspi, Taylor&Ryba(1994)


Двойные пульсары

•Движение в двойной системе описывается стандартными кеплеровскими параметрами:

•Период обращения: Pb

•Проекция большой полуоси:

•Эксцентриситет: e

•Долгота периастра: ω

•Эпоха периастра: T0

iax sin

•В сильных гравитационных полях появляются ПК-параметры ( и т.д. )

•Все эти параметры могут быть найдены из тайминга (аналогично, МНК-методом)

,bP


Алгоритм

1. Наблюдения, вычисление моментов прихода импульсов пульсаров (МПИ) в барицентре Солнечной системы.

2. Вычисление теоретических значений МПИ с использованием модели хронометрирования.

3. Определение отклонения значений теоретических МПИ от наблюдаемых (расчет остаточных уклонений – ОУ МПИ).

4. Уточнение параметров модели хронометрирования (далее к п.3 до сходимости модели).

Documents

Constraints on massive graviton dark matter from precision pulsar timing and astrometry Konstantin POSTNOV (Sternberg Astronomical Institute) Collaborators: