GW5Data analysis (II) and tests of GR Michele Vallisneri ICTP - PowerPoint PPT Presentation

GW5—Data analysis (II) 
 and tests of GR Michele Vallisneri ICTP Summer School on Cosmology 2016

2 GW detection in practice [see PRD 93, 122003 (2016)] condition and calibrate detector output filter detector output 
 with theoretical templates request coincidence and consistency among detectors apply data-quality cuts 
 and signal vetos estimate statistical significance (estimate background, using coincidence between time slides) follow up candidates 
 with detection checklist (estimate efficiency from injections, 
 number of galaxies within horizon) claim detection! get upper limit

“science” ~ signal(parameters) h ( t ; A , α , f ) = A e − t 2 2 α 2 sin(2 π ft ) 1.0 0.5 f 0.0 - 0.5 α - 1.0 - 5 0 5 1.0 1.0 1.0 0.5 0.5 0.5 0.0 0.0 0.0 - 0.5 - 0.5 - 0.5 - 1.0 - 1.0 - 1.0 - 5 0 5 - 5 0 5 - 5 0 5 1.0 0.5 0.0 - 0.5 - 1.0 - 5 0 5

noise = data – signal 3 3 2 2 1 1 – = 0 0 - 1 - 1 - 2 - 2 - 3 - 3 - 4 - 2 0 2 4 - 4 - 2 0 2 4 3 3 3 2 2 2 1 1 1 – = 0 0 0 - 1 - 1 - 1 - 2 - 2 - 2 - 3 - 3 - 3 - 4 - 2 0 2 4 - 4 - 2 0 2 4 - 4 - 2 0 2 4 3 3 2 2 1 1 – = 0 0 - 1 - 1 - 2 - 2 - 3 - 3 - 4 - 2 0 2 4 - 4 - 2 0 2 4

noise = data – signal hence p (signal parameters) = p (noise residual) 3 3 2 2 1 1 – = 0 0 - 1 - 1 - 2 - 2 - 3 - 3 - 4 - 2 0 2 4 - 4 - 2 0 2 4 3 3 3 2 2 2 1 1 1 – = 0 0 0 - 1 - 1 - 1 maximum likelihood - 2 - 2 estimate - 2 least noisy - 3 - 3 - 3 - 4 - 2 0 2 4 - 4 - 2 0 2 4 - 4 - 2 0 2 4 3 3 2 2 1 1 – = 0 0 - 1 - 1 - 2 - 2 - 3 - 3 - 4 - 2 0 2 4 - 4 - 2 0 2 4

noise = data – signal hence p (signal parameters) = p (noise residual) 3 3 2 2 1 1 – = 0 0 - 1 - 1 - 2 - 2 - 3 - 3 - 4 - 2 0 2 4 - 4 - 2 0 2 4 3 3 3 2 2 2 1 1 1 – = 0 0 0 - 1 - 1 - 1 maximum likelihood - 2 - 2 estimate - 2 least noisy - 3 - 3 - 3 - 4 - 2 0 2 4 - 4 - 2 0 2 4 - 4 - 2 0 2 4 3 3 2 2 1 1 – = 0 0 - 1 - 1 - 2 - 2 - 3 - 3 - 4 - 2 0 2 4 - 4 - 2 0 2 4

because of colored detector noise, detection and parameter estimation are sensitive to the frequency content of waveforms... acceleration noise: 
 position noise: 
 thermal, gravity, Sun optical sensing 10 17 eLISA 10 19 LISA S h f 10 21 adv LIGO 10 23 Einstein telescope 2 2 2 1 1 1 s ( t ) 0 0 0 10 25 - 1 - 1 - 1 10 4 10 4 - 2 - 2 - 2 0.01 1 100 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120 frequency Hz 2.0 2.0 2.0 1.5 1.5 1.5 s ( f ) 1.0 1.0 1.0 0.5 0.5 0.5 0.0 0.0 0.0 0 10 20 30 40 50 60 0 10 20 30 40 50 60 0 10 20 30 40 50 60

under the assumption of Gaussianity, the power spectral density yields the sampling distribution of noise R n ( f ) n ∗ ( f ) i / 2 σ 2 d f p ( n ) ∝ Π i e − n i n ∗ i = e − 2 S ( f )

See “Data analysis recipes: Fitting a model to data” Hogg, Bovy, and Lang 2010 http://arxiv.org/abs/1008.4686

Bayesian inference: we update our prior knowledge of physical parameters using the likelihood of observed data = p ( n = s − h ( θ i )) = N e − ( n,n ) / 2 p ( θ i | s ) = p ( θ i ) p ( s | θ i ) p ( s ) Z = p ( θ i ) p ( s | θ i ) d θ i 3 0.15 2 0.10 1 0.05 0.00 0 Ê - 0.05 - 1 - 0.10 f - 2 - 0.15 - 3 α - 4 - 2 0 2 4 - 0.6 - 0.4 - 0.2 0.0 0.2 0.4 0.6 deviates from high-SNR 
 � ✓ ∂ h ◆ ∂ h � covariance predicted F ij = � ∂θ i ∂θ j � with Fisher matrix

…but every noise realization will be different! 3 3 3 0.15 0.15 0.15 2 2 2 0.10 0.10 0.10 1 0.05 1 0.05 1 0.05 0.00 0.00 0.00 Ê 0 0 0 Ê Ê - 0.05 - 0.05 - 0.05 - 1 - 1 - 1 - 0.10 - 0.10 - 0.10 - 2 - 2 - 2 - 0.15 - 0.15 - 0.15 - 3 - 3 - 3 - 4 - 2 0 2 4 - 4 - 2 0 2 4 - 4 - 2 0 2 4 - 0.6 - 0.4 - 0.2 0.0 0.2 0.4 0.6 - 0.6 - 0.4 - 0.2 0.0 0.2 0.4 0.6 - 0.6 - 0.4 - 0.2 0.0 0.2 0.4 0.6 3 3 3 0.15 0.15 0.15 2 2 2 0.10 0.10 0.10 Ê Ê 1 0.05 1 0.05 1 0.05 0.00 0.00 0.00 0 0 0 Ê - 0.05 - 0.05 - 0.05 - 1 - 1 - 1 - 0.10 - 0.10 - 0.10 - 2 - 2 - 2 - 0.15 - 0.15 - 0.15 - 3 - 3 - 3 - 4 - 2 0 2 4 - 4 - 2 0 2 4 - 4 - 2 0 2 4 - 0.6 - 0.4 - 0.2 0.0 0.2 0.4 0.6 - 0.6 - 0.4 - 0.2 0.0 0.2 0.4 0.6 - 0.6 - 0.4 - 0.2 0.0 0.2 0.4 0.6 3 3 3 0.15 0.15 0.15 Ê 2 2 2 0.10 0.10 0.10 1 0.05 1 0.05 1 0.05 Ê Ê 0.00 0.00 0.00 0 0 0 - 0.05 - 0.05 - 0.05 - 1 - 1 - 1 - 0.10 - 0.10 - 0.10 - 2 - 2 - 2 - 0.15 - 0.15 - 0.15 - 3 - 3 - 3 - 4 - 2 0 2 4 - 4 - 2 0 2 4 - 4 - 2 0 2 4 - 0.6 - 0.4 - 0.2 0.0 0.2 0.4 0.6 - 0.6 - 0.4 - 0.2 0.0 0.2 0.4 0.6 - 0.6 - 0.4 - 0.2 0.0 0.2 0.4 0.6 3 3 3 0.15 0.15 0.15 2 2 2 0.10 0.10 0.10 1 0.05 1 0.05 1 0.05 Ê Ê 0.00 0.00 0.00 Ê 0 0 0 - 0.05 - 0.05 - 0.05 - 1 - 1 - 1 - 0.10 - 0.10 - 0.10 - 2 - 2 - 2 - 0.15 - 0.15 - 0.15 - 3 - 3 - 3 - 4 - 2 0 2 4 - 4 - 2 0 2 4 - 4 - 2 0 2 4 - 0.6 - 0.4 - 0.2 0.0 0.2 0.4 0.6 - 0.6 - 0.4 - 0.2 0.0 0.2 0.4 0.6 - 0.6 - 0.4 - 0.2 0.0 0.2 0.4 0.6

how to do this for many parameters?

Monte Carlo (Von Neumann and Ulam, 1946): 
 computational techniques that use random numbers

Monte Carlo (Von Neumann and Ulam, 1946): 
 computational techniques that use random numbers φ = 1 Z φ ( x ) dx → ˆ X φ ( x ( r ) ) R r

accuracy depends only on variance, 
 not on the number of dimensions φ = 1 Z φ ( x ) dx → ˆ X φ ( x ( r ) ) R r φ = var φ var ˆ R

unfortunately uniform sampling is extremely 
 inefficient in high-dimensional spaces (and so are importance sampling 
 and rejection sampling) π V box = (2 π ) d ( π ) d / 2 V ball = Γ ( n / 2 + 1) V box ∼ d d V ball − π − π π

Nicholas Metropolis and his 
 Mathematical Analyzer Numerical Integrator And Calculator

Marshall Rosenbluth and Edward Teller

Teller’s crucial suggestion: ensemble averaging... Z φ ( x ) p ( x ) dx , with p ( x ) ' e − E ( x ) / kT ⇓ φ ( x ) dp ( x ) ' 1 Z X φ ( x ( r ) ) with { x ( r ) } P R R

...with samples generated by the “Metropolis” algorithm • given x ( r ) , propose x ( r +1) by random walk • accept it if Δ E = E ( x ( r +1) ) – E ( x ( r ) ) < 0, 
 or with probability e – Δ E / kT if Δ E > 0 • if not accepted, set x ( r +1) = x ( r ) • the resulting detailed balance guarantees convergence to P

but why does it work?

• the Metropolis algorithm implements a initial condition Markov Chain { x ( r ) } with transition probability T ( x i ; x j ) = T ij • T is set by the proposal distribution Q and the transition rule (e.g., Metropolis) • if T ij satisfies certain properties, its repeated application to any initial probability distribution ρ 0 j eventually yields the equilibrium distribution ρ * i = P i (MacKay 2003) equilibrium distribution

(MacKay 2003) • the Metropolis algorithm is very general and very easy to implement but : • convergence, while guaranteed, is hard to assess • random-walk exploration is • need ( L / ε ) 2 ∼ ( σ max / σ min ) 2 steps 
 very inefficient to get independent sample try : • annealing, parallel tempering • Hamiltonian MCMC • affine-invariant samplers ( emcee ) • the evidence/partition function • thermodynamic integration is difficult to compute • reversible-jump MCMC Z • nested sampling ( MultiNest ) e − E ( x ) / kT dx , Z = Z p ( M ) = p (data | x ) p ( x ) dx

Testing GR: the standard hierarchy of theories of gravitation WEP Newton’s equivalence principle m I = m G 10 –13 EEP Einstein’s equivalence principle = WEP + local Lorentz invariance 
 10 –22 + local position invariance 10 –5 metric theories (what fields?) SEP EEP , but also for gravitational 
 experiments 10 –4 1981 → 2006 Dicke: test of EPs + PPN tests of metric theories

the PPN formalism: metric and potentials

the PPN formalism: parameters 10 –5 10 –4

Gravitational radiation... ...is predicted in virtually any metric theory of gravity that embodies Lorentz invariance, but it may differ from GR in: polarizations speed of waves (tested at low v 
 radiation reaction with binary pulsars) Unfortunately, no simple, principled framework like PPN exists for describing radiative systems or systems containing strong internal fields. So we must consider individual alternative theories, or perform 
 null tests of consistency.

Naïve and sentimental tests of GR consistency In order of difficulty and un-likelihood: • If we divide the waveform in segments, do individual SNRs pass a χ 2 test? • Is there a coherent residual? • What about the source parameters determined from each segment—are they consistent (within estimated errors) with the parameters determined from the entire waveform? • Is the shape of the likelihood surface consistent with what’s expected for this waveform family? • But before we suspect general relativity: 
 � ⌧ � ∂ h ∂ h instrument systematics, modeling, data 
 � ' 0 � ∂λ physical ∂λ non - GR � analysis, physical environments…

SNR in coherent burst analysis of data residual after subtracting best-fit GW150914 waveform

Polarization Tensor Scalar Vector