Detecting Gravitational Waves with a pulsar timing array

Detecting Gravitational Waveswith a Pulsar Timing ArrayLindley LentatiCambridge University

Gravitational WavesFluctuations in curvature that propagate at the speed of light

Experiment in the lab:

Using an accurate clock, record when you measure ticks. Passing gravitational waves contract and expand space between observer and the clock.

Measure deviation from expected time -> Detected gravitational waves.

Expected length change:

width of an atomic nucleus in one meter baseline

Need *very* accurate clockOr huge distances

Segue into.

One problem..Using Pulsars to detect Gravitational WavesChange in path length from GWs: ~ few hundred meters = ~ 100 ns

far away ~ 1kpc = 3x1019 meters

Accurate Clocks

Almost There!Consider small perturbations to flat spaceLinearization of Einstein's equation leads to: The solution: A plane wave travelling at the speed of light

Gravitational Waves

Gravitational Waves from GRPlane Wave ExpansionWrite as a function of:Time Frequency of source Position of source on the sky Plane Wave ExpansionSum is over polarization states = [ + , x ]

+ x Plane Wave ExpansionEarthPulsarGW Source

Wavefront 1Wavefront 2

Perturbation at the pulsar at a time t_p

Perturbation at the Earth at a time t_e

Measure the difference between the two:Wavefront 1Wavefront 2Delay is purely geometric

L~1 kpc, can see frequency evolution over very large time scales.

RedshiftAntenna Beam PatternDefines sensitivity of pulsar to GW source on the sky

Total red shift obtained by integrating over all frequencies and all skyDont observe redshift:observe a fluctuation in the residuals left after subtracting a timing model from data

RedshiftSo far only one frequency, one source.

Redshift to ResidualsFrom a data challenge last year (Have a go)Real pulsars have possible sources of noise other than GWs: Intrinsic red noise Dispersion Measure variations JitterNeed some way to distinguish a gravitational wave background from this ->Redshift to Residuals

Signal due to Gravitational Waves is correlated between pulsarsother sources of noise (mostly) are not.

The Solution:For an isotropic background the angular correlation has an analytic solutionSmoking Gun of a real GW detection.

The Hellings-Downs Curve

Bayesianism and FrequentismStart at the heart of it..

Asks two different questions:

Frequentist: What is the probability of my data, given my model?Assumes model is fixed data random variable

Bayes:What is the probability of my model, given my data?Assumes data is fixed model is random variable

Start with a single pulsarFrom Earlier this week:

We want to build up a model signal s for the pulsar:Deterministic signal from timing modelStart with a single pulsarSignal function of timing model parameter and time:Barycentric Arrival TimeMatrix of Basis VectorsTiming modelSimple case might have:

Position (RA, DEC)Period/Spindown (F0, F1)Proper motion (PMRA, PMDEC)Parallax (PX)Others include Binary parameters etc

Timing model

= F0= F1= PMRA= PX= RATiming model

++++Start with a single pulsar

We Now Have sWhat about ?

Equation above assumes only uncorrelated white noise Almost certainly not trueStart with a single pulsarRewrite this equation:

Is a covariance matrix.Describes the correlations between the residuals . Diagonal = Uncorrelated (as before).Off Diagonal terms imply correlations between times .Start with a single pulsarSo to include correlated noise, this:

Becomes this:Start with a single pulsarSo to include correlated noise, this:

Becomes this:Is now dense much more computationally intensive to invert, not many other options.Marginalising over the timing modelTiming model parameters are nuisance parameters

Want to marginalise over them analytically so we dont have to sample

Consider 2d problem Probability density for parameters A and B.

Marginalising over the timing model

Can also marginalise analytically

Marginalisation

Volume MattersMarginalising over the timing modelNow proceed as in example(Uniform Prior)Marginalising over the timing modelEquivalent to a Projection orthogonal to the timing model

Decreased the dimensionality of the problemStill account for our uncertainty in the timing modelJust a Gaussian Can do this analyticallyNow lots of pulsars..Correlation between pulsars that defines the GWB signal

Pi = Noise covariance matrix for single pulsarContributions from GWB, intrinsic noise etcSij = cross correlated noise between pulsarsContributions from GWB, but also other thingsLarge!Inversion dominates computation timeThose other things.. Clock Errors

Same in every pulsar

Correlation = 1

Those other things.. Planet massesErrors in Planetary Ephemeris -> Error in location of barycentre

Orbits 0.2 -> 164 yearsIntroduces sine wavePeriod of orbital periodCorrelated between pulsars

Lets you measure mass of planets using pulsars

It all adds up..Total dimensionality ~ hundredsWeeks of compute time

Difficult problem!SamplingSaid we want to calculate P(X |D, M)

Non-trivial for non-trivial problems

Have to sample from posteriorMarkov-Chain Monte-CarloMarkov chain sequence of state changes that depends only on the most recent states, not the states that preceded them.

Simple example (from Wikipedia)

Probability of the weather.

Markov-Chain Monte-CarloP(Tomorrow is Sunny | Today is rainy) = 0.5P(Tomorrow is rainy | Today is rainy ) = 0.5

P(Tomorrow is rainy | Today is sunny) = 0.1P(Tomorrow is Sunny | Today is sunny) = 0.9

Markov-Chain Monte-CarloP(Tomorrow is Sunny | Today is rainy) = 0.5P(Tomorrow is rainy | Today is rainy ) = 0.5

P(Tomorrow is rainy | Today is sunny) = 0.1P(Tomorrow is Sunny | Today is sunny) = 0.9

P(Sun in 2 days| Sun) = P(S,S|S) + P(S,R | S) = 0.86P(Sun in 30 days | Sun) = ...= 0.833P(Sun in 100 days | Sun) = .= 0.833P(Sun in 100 days | rain) = .= 0.833

Markov-Chain Monte-CarloProbability of weather tomorrow depends only on the last few days.

Forgets about everything previous.

Important aspect of all samplers.

Random walk Metropolis HastingsRandom walk Metropolis HastingsHas its problems: Convergence rate depends on step size

Just rightToo smallToo bigStep Size:Random walk Metropolis HastingsBut will get there eventually

Just rightToo smallToo bigStep Size:

Random walk Metropolis HastingsVery poor for multi modal problems:

If step size allows jumps between modes,it will be too big within each mode.

If step size small enough to explore individual modes,it wont step between them.Nested Sampling (Skilling 2004)Solves a lot of these problems

Draw N points Uniformly from the priorLowest likelihood point = L0

Draw a new point with likelihood LiIf Li > L0 replace point with the new point

Otherwise try againNested Sampling (Skilling 2004)The Challenge:Draw new points from within the hard boundary L > L0Mukherjee (2005): Use ellipses to define the boundary

Still wasnt great for multi-modal problems.

MultiNest (Feroz & Hobson 2008)At each iteration:Construct optimal multi-ellipsoidal boundPick ellipse at random to sample new point

MultiNest (Feroz & Hobson 2008)Works great for multi-modal problems:

Polychord (Handley & Hobson 2015)Successor to MultiNest.

Still uses nested sampling.

Works in much higher dimensions (up to ~ 150)

Still a lot to doModellingSamplingMaking Datasets agree

Which is (maybe) why youre here!