Astrometric Properties of a Stochastic Gravitational Wave Background

Astrometric Properties of a Stochastic Gravitational Wave Background Éanna Flanagan, Cornell Conference on Cosmology since Einstein Hong Kong University of Science and Technology 31 May 2011 Laura Book, EF, PRD 83, 024024 (2011) 1

Outline of Talk Review of stochastic gravitational wave backgrounds: sources and detection methods High precision astrometry Constraining gravitational waves with astrometry: history, existing upper limits, order of magnitude estimates Deflection pattern produced by a stochastic background, correlation function and spectrum Optimal data analysis method and implications 2

Stochastic Gravitational Wave Backgrounds Assume: Gaussian Stationary Homogeneous Isotropic Spectrum: Ω gw (f) = 1 ρ c dρ gw d ln f f 2 H 2 0 h rms (f) 2 3

Gravitational Waves Probe the Early Universe 4

(LIGO, Nature, 2009) 5

QSO Astrometry 6

High Precision Astrometry Radio VLBI Optical Satellites HIPPARCOS (1989-93) N 10 6 stars δθ few mas δθ δt D λ D δφ 1 λ D δφ 100 µas 1 cm 10 4 km 1 SKA N 10 6 quasars δθ 10 µas GAIA (2013) N 10 6 quasars N 10 9 stars δθ 10 µas SIM?? 7

Astrometry and Gravitational Waves: History Grav. waves cause angular deflection, δθ h Initial idea (Fakir, 1993) δθ h 1 b Does not work (Damour, Kopekin) δθ h 1 b 3 Second idea (Braginsky, 1991): search for stochastic background δθ(d) d Find δθ(d) h, not random walk, Gravitational waves cause apparent angular velocities, correlated across the sky Distant sources have small proper motions Search in data for expected statistical deflection pattern 8

Order of Magnitude Estimates Signal: δθ rms (f) h rms (f) H 0 f Ω gw (f) δω rms (f) fδθ rms (f) H 0 Ω gw (f) 10 2 µ as yr 1 Ωgw 10 6 1/2 Monitor N sources, time T, accuracy θ = ω = Ω gw θ T N θ 2 NT 2 H 2 0 10 6 2 1 2 θ N T 10 µ as 10 6 1yr Existing VLBI upper limit Limit applies to N 10 3, Ω gw 10 3 ln(1/t ) ln H 0 d ln f Ω gw (f) 9

Order of Magnitude Estimates (cont.) Source Proper motions: Stars in Milky Way: ω v D 10 4 µ as yr 1 v 100 km s 1 D 1kpc 1 Quasars: ω 0.1 µ as yr 1 v 1 D 10 3 km s 1 3Gpc In fact jets produce apparent proper motion of quasars of order ω 10 µ as yr 1 10

Outline of Talk Review of stochastic gravitational wave backgrounds: sources and detection methods High precision astrometry Constraining gravitational waves with astrometry: history, existing upper limits, order of magnitude estimates Deflection pattern produced by a stochastic background, correlation function and spectrum Optimal data analysis method and implications 11

Angular Deflection Computation Metric: ds 2 = a(η) 2 dη 2 +(δ ij + h ij )dx i dx j Find perturbed geodesic for which Detection event unchanged Null Emission frequency unchanged Intersects source worldline ko = ω O 1+δz uo + n i + δn i (n,t) eî O S Result: x α (λ) = (η 0 + λ, n i λ), P ij = δ ij n i n j δn i = 1 2 P ik n j h jk (0) + 1 λ S P ik n j λs 0 dλ h jk (λ) + 1 2 (λ S λ)n l h jl,k (λ) 12

Approximations Four lengthscales: Three approximations: 1) Subhorizon modes only, For simplicity No superhorizon contribution expected 2) Distant source limit, λ L H λ D Not a restriction for quasars Could be removed Also invoked in pulsar timing, different justification 3) Slowly varying modes only, λ ct For simplicity only. Removed in present work Not invoked in pulsar timing 13

Simplified Deflection Formula Specialize to plane wave mode: q +2 a a q + Ω 2 q =0, q(η) 1 a(η) e iωη In distant source and subhorizon limits, find δn i (n) = 1 2 n jh ij (0) + ni + p i 2(1 + n p) [n jn k h jk (0)] h ij (η, x) = Re H ij e iωp x q(η) (WKB near observer) For stochastic background, find deflection correlation function δn i (n,t)δn j (n,t ) = 0 df e 2πif(t t ) H2 0 Ω gw (f) 16π 2 f 3 H ij (n, n ) where H ij = α(θ)[a i A j B i C j ], cos Θ = n n, A = n n, B = n A, C = n A, α(θ) = 1 2 sin 2 Θ (7 cos Θ 5) 48 sin6 (Θ/2) sin 4 Θ ln[sin(θ/2)] 14

Angular Deflection Power Spectrum We compute power spectrum as a function of frequency and angular scale Expand angular deflection as δn(n,t)= Q=E,B l 2 l m= l δn Qlm (t) Y Q lm (n) Find δn Qlm (t)δn Q l m (t ) = δ QQ δ ll δ mm with S Ql (f) = H2 0 Ω gw (f) α l 2πf 3 2l +1, 5 α l = 6, 7 60, 3 100, 11 1050,..., α l =1 l=2 0 df cos[2πf(t t )]S Ql (f) 15

Typical Deflection Pattern 16

Typical Deflection Pattern 16

Outline of Talk Review of stochastic gravitational wave backgrounds: sources and detection methods High precision astrometry Constraining gravitational waves with astrometry: history, existing upper limits, order of magnitude estimates Deflection pattern produced by a stochastic background, correlation function and spectrum Optimal data analysis method and implications 17

Optimal Data Analysis Method For slowly varying modes δn(n,t) δn(n,t 0 )+ δn(n,t 0 )(t t 0 )+... Data: average positions, velocities, accelerations n A, ω A, α A, 1 A N Model: ω A = Qlm Y Q lm (n A) δn Qlm (t 0 ) + δω A, or ω = M s + δω, with s s T = Σ s, δω δω T = Σ w Noise δω due to measurement error and source proper motion Detection statistic: Q =tr ω T A ω for some A with tr [A Σ w ]=1 Choose A to maximize the signal to noise ratio. Result is S 2 N 2 Q 2 Q 2 = 1 no signal 2 tr D 2 TF, D =(Σ w ) 1 M Σ s M T 18

Prediction for Sensitivity of Astrometry S 2 N 2 = l 2 N 2 H 4 0 α 2 l 2l +1 d ln f Ωgw (f) 2 σ 4 ω + d ln f (2πf) 2 Ω gw (f) 2 σ 4 α +... Parameters: Measurement error: Stellar motion: Quasar motion: σ ω 10µ as yr 1, σ ω 10 4 µ as yr 1, σ ω 10µ as yr 1, σ α 0 σ α 10µ as yr 2 σ α 10 3 µ as yr 2 Pristine sources: Dominated by measurement error Quasars today Velocity data best Non-pristine sources: Dominated by proper motion errors Stars today, quasars in future? Acceleration data also useful 19

Prediction for Sensitivity of Astrometry (cont.) Generalizing analysis to include time dependence of deflections gives S 2 N 2 = l 2 N 2 H 4 0 α 2 l (2l + 1)σ 4 ω ln(1/t ) d ln f Ω gw(f) 2 + 9 2π 4 ln(1/t ) d ln f Ω gw(f) 2 f 5 T 5 20

Conclusions Future optical (GAIA) and radio (SKA) astrometry data can provide interesting constraints on stochastic gravitational waves, at the level of Ω gw 10 6 over a broad band. The limit is scale invariant at frequencies low compared to the observation time, but rises as Ω gw f 5/2 at high frequencies For GAIA, using the acceleration data from the may be competitive with the velocity data from the quasars at certain frequencies 10 9 10 6 stars 21

Conclusions (cont.) The future: As technology improves, how far can this technique be pushed? Achieving the inflation-signal regime of Ω gw 10 14 would require θ 10 4 10µ as n as. Even if this were achieved there are several other challenges: Background from Scalar Modes: Scalar perturbations give a l=2, electric signal equivalent to Ω gw,eff d ln k δ rms (k) 2 (H 0 /k) 4 10 9, but do not contribute to the l=2, magnetic signal at linear order. Narrow Band: Acceleration rather than velocity data of quasars would have to be used, giving a narrowband limit. Solar System Modeling: Extremely accurate models of the Solar System would be required to subtract from the data kinematic effects (acceleration and rotation of our reference frame) and light bending by the Sun and planets. Black Hole Background: A stochastic background of gravitational waves from coalescing supermassive black holes might be present at frequencies of f yr 1 and swamp any early Universe signal. 22