Phase steps: “Nonparametric” extensions to inspiral template families GWDAW-9 Dec R. O’Shaughnessy, D. Jones Northwestern University

Context & Motivation Pipelines: –NS-NS PN –BH-BH BCV –BH-NS BCVspin …good (!)

Context & Motivation Pipelines: –NS-NS PN –BH-BH BCV –BH-NS BCVspin …less physical (detection only) …more templates, …higher threshold

Motivation Present pipelines: good ! –Will find what we’re looking for, if there SourceMatchCutoff (rho) N templates NS-NS (PN) >0.976~100 BH-BH (BCV) >0.97~8 number ~2000 BH-NS (BCVspin) …~ tba~large (100,000) )? [see owen latest]

Context & Motivation Potential improvements?: 1.Faster searching – Inspiral search is very CPU-intensive … – … but still need (eventually) to do real-time – Can we get comparable (or more) coverage w/ a fast search? (e.g., as w/ BCVspin -- fast search over extrinsic parameters)

Context & Motivation Potential improvements?: 1.Faster searching 2.Greater Coverage – No guarantees about physical signal FF… for example – BCV fits just what we’ve tried so far… (PN, EOB), but – Merger waves? – Non-GR - detecting signals from non-GR theories – Try models w/ more template parameters? – More coverage… – Only slow increase in threshold SNR with template number…  Problem : computational cost   w/ more templates / intrinsic parameters …but adding a few extrinsic parameters ok [ e.g. BCVspin ]

Context & Motivation Potential improvements?: 1.Faster searching 2.Greater Coverage Our proposal: Adds parameters to conventional families Many more ‘templates’ …. …but still fast search -- effectively extrinsic Result just good for detection (not estimation) Good coverage..requires careful tuning to get desired false rate very preliminary…

Analysis Assumptions Simple toy proposal & analysis Concreteness: Restrict study to –LIGO 1 noise …. gaussian only –BH-NS mass ranges –0 PN amplitude (i.e.  =0) –Extending nonspinning PN templates

Outline Context & Motivation [“why?”] Proposed form[“what?”] –Phase steps: why this form? –How much overlap improvement Complications[“works?”] –False alarms with steps Conclusions

Choosing a form Examine residuals?: –Expected residuals after template maximization? –…try to choose additions to compensate Concretely: Phase residuals only …

Choosing a form Examine phase residuals?: –Monotonic phase variation? well-fit by templates--> ~ irrelevant –Sinusoidal phase errors  except maximization over extrinsic [tc] automatically reshapes the phase error curve Example: Adjoin artificial sinusoidal phase error Maximize tc, phic

Choosing a form Examine phase residuals?: –Smooth phase variation --> irrelevant –Sinusoidal phase errors --> reshaped Example: Artificial sinusoidal phase error …before Hz 

Choosing a form Examine phase residuals?: –Smooth phase variation --> irrelevant –Sinusoidal phase errors --> reshaped Example: Artificial sinusoidal phase error Maximization requires phase constant near max sensitivity …after (FF=0.64)  Hz d  /df Hz

Choosing a form Examine phase residuals?: –Smooth phase variation --> irrelevant –Sinusoidal phase errors --> reshaped  residuals suggest use (smoothed) step functions in phase …after (FF=0.64)  Hz d  /df Hz

Phase steps Motivation: –Add to  smooth changes 0->2  Equations: …local effect on overlap d  /df Hz

Phase steps: Example Start –Sinusoidal phase Example (above) Optimize –Fix tc, phic –Add one by one –Optimize each spike independently Results –Initially:overlap = 0.64 –First step [Fc=50, w=8.2]: overlap = 0.74 –Second step [Fc=250, w=8.5]: overlap = 0.70 –Both:overlap = 0.84 Hz

Complications? Infinite-dimensional manifold …suggests slow search …but local character of steps --> fast (rough) search Easy to excite steps by noise … true, but manageable via search constraints (= rest of talk)

Controlling False Alarms Familiar Question –Improved overlap --> improved detection rate –More models --> higher false alarm rate e.g., want detection threshold for fixed false rate  goal: Tailor parameter ranges to minimize increase in false rate How to avoid accidents: 1.Steps disjoint 2.Steps aren’t too narrow or irrelevant 3.Each step makes a significant change in SNR 4.Look for steps only above a minimum SNR cutoff

Controlling False Alarms How to avoid accidents: 1.Steps disjoint Sanity constraint: Steps of opposite sign ~ cancel [ (+1) + (-1) = 0 in same place] Steps of same sign ~ “narrower” than w (I.e. d  /df larger) --> annoying  Hz Require: ( f1-f2 )> 6(w1+w2) for any pair of steps Side effect: z is ~ additive

Controlling False Alarms How to avoid accidents: 1.Steps disjoint 2.Steps aren’t too narrow or irrelevant Certain steps can’t change SNR much: Narrow steps (w << 1) Steps centered away from maximum …by definition, small-z steps However, easy to excite: –SNR in band of fraction z determines excitation –P(error) ~ Require: z>0.1 Interlude: SNR: band-limited issues Net coherent amplitude: z-band amplitude: Net noise power: z-band noise power: …relative bandwidth

Controlling False Alarms How to avoid accidents: 1.Steps disjoint 2.Steps aren’t too narrow or irrelevant Certain steps can’t change SNR much: Narrow steps (w << 1) Steps centered away from maximum …by definition, small-z steps However, easy to excite: –SNR in band of fraction z determines excitation –P(error) ~ Require: z>0.1 Z measures: 1) characteristic  /  due to step if indeed present 2) effective width of band (proportional to relative, power-weighted width) Interlude: SNR and steps: band-limited issues …relative bandwidth

Controlling False Alarms How to avoid accidents: 1.Steps disjoint 2.Steps aren’t too narrow or irrelevant Small-’z’ steps Narrow steps (w<<1), or Far from sensitivity maximum False positives with given step?: Probability of ‘false narrow step’ = probability N(0,1) > SNR-in-band = … can be large if z small ! Require: z>0.1 Models: Signal: SNR =  Two hypotheses (templates): Signal Signal + (one fixed z)

Controlling False Alarms How to avoid accidents: 1.Steps disjoint 2.Steps aren’t too narrow or irrelevant 3.Step makes a significant change in SNR Band-limited SNR: Total SNR change Band-limited SNR: higher because occurs only in smaller band  Accidents = unlikely, but still possible Require:

Controlling False Alarms How to avoid accidents: 1.Steps disjoint 2.Steps aren’t too narrow or irrelevant 3.Each step makes a significant change in SNR 4.Look for steps only above a minimum SNR cutoff Require: Don’t start looking unless SNR already high:  >  s (e.g, close to detection threshold:  s =  d --  ) Reason: Two moderately unlikely accidents required to find spike SNR >  s d(SNR) > 0.9

Aside: Nonparametric perspective Use more templates when SNR warrants … implicitly / automatically implemented by this constraint (INTERLUDE…)

Controlling False Alarms How to avoid accidents: 1.Steps disjoint 2.Steps aren’t too narrow or irrelevant 3.Each step makes a significant change in SNR 4.Look for steps only above a minimum SNR cutoff Detection strategy Conventional intrinsic search at each set of extrinsic params Trial step search Don’t try steps unless starting SNR > threshold  s Each step must improve SNR by at least  s Search over extrinsic parameters Detection if final SNR > threshold  d Toy #s:  s ~ 7  s ~ 1  d ~ 8

Summary & Conclusions Status –Generalizes any templates to “nonparametric” detection family Phase steps fix gross phase errors Family features match fine structure (e.g. spin) –“Pipeline” (one-detector) Probes into noise below detection threshold of “base” family Detects signals which would otherwise be rejected (because our templates only imperfectly model the “true” signal) and which should not have happened by accident. –Applications: merger waves, testing whether GR describes signals, etc Further tests: –Coverage of full “pipeline”, w/ constraints: vs SNR of input –Real noise

“Nonparametric” ? Templates used implicitly depend on SNR (!) –Broad coverage for high SNR –Low coverage for low SNR Pipeline as described : not nonparametric (yet)