pulsar timing for gwb - samsis.n. lahiri (ncsu) lect2 3 / 11. block boostrap caliration of the test...

PulsarTimingforGWB:Sta5s5calMethods

G. Jogesh BabuPennStateCollaborators:

Soumen Lahiri (NCSU),

Justin Ellis (JPL/Caltech) & Joseph Lazio (JPL)

Gravita5onalwavesareripplesinspace-5me(greengrid)producedbyaccelera5ngbodiessuchasinterac5ngsupermassiveblackholes.Thesewavesaffectthe5meittakesforradiosignalsfrompulsarstoarriveatEarth.(Credits:DavidChampion)

Apulsarwithastrongmagne5cfield(linesinblue)producesabeamoflightalongthemagne5caxis.Astheneutronstarspins,themagne5cfieldspinswithit,sweepingthatbeamthroughspace.IfthatbeamsweepsoverEarth,weseeitasaregularpulseoflight.(Credit:NASA/GoddardSpaceFlightCenterConceptualImageLab)

Pulsarsarehighlymagne5zedneutronstars,therapidlyrota5ngcoresofstarsleSbehindwhenamassivestarexplodesasasupernova.

PulsarTiming•  Thebestsciencebasedonpulsarobserva5onshascomefromtheiruseas

toolsviapulsar5ming.

•  Pulsar5mingistheregularmonitoringoftherota5onoftheneutronstarbytrackingthe5mesofarrival(TOA)oftheradiopulses.

•  Thepulsar5mingaccountsforeverysinglerota5onoftheneutronstaroverlongperiods(yearstodecades)of5me.

•  Thisveryprecisetrackingofrota5onalphaseallowsustoprobetheinteriorphysicsofneutronstarsandtestgravita5onaltheories.

•  Forpulsar5ming,radiodatais“folded”modulotheinstantaneouspulseperiodPorpulsefrequencyf=1/P.Averagingovermanypulsesyieldsahighsignal-to-noiseaveragepulseprofile.

•  Althoughindividualpulseshapesvaryconsiderably,theshapeoftheaverageprofileisquitestable.

GWfromPulsarTiming•  Therecentdetec5onofgravita5onalwaves(GW)bytheLaser

InterferometerGravita5onal-WaveObservatory(LIGO)camefromtwoblackholes,eachabout305mesthemassofoursun,mergingintoone.

•  Gravita5onalwavesspanawiderangeoffrequenciesthatrequiredifferenttechnologiestodetect.

•  AnewstudyfromtheNorthAmericanNanohertzObservatoryforGravita5onalWaves(NANOGrav)hasshownthatlow-frequencygravita5onalwavescouldsoonbedetectablebyexis5ngradiotelescopes.

•  Nanohertzgravita5onalwavesareemi\edfrompairsofsupermassiveblackholesindistantgalaxiesorbi5ngeachother,eachofwhichcontainmillionsorabillion5mesmoremassthanthosedetectedbyLIGO.

•  Detec5ngthissignalispossibleifweareabletomonitorasufficiently

largenumberofpulsarsspreadacrosstheskyandlikelyseeingthesamepa\ernofdevia5onsinallofthem.

PulsarTimingArray(PTA)•  APTAisasetofradiopulsarsthatcanbeusedintandemtosearch

forgravita5onalwaves.

•  ThedifferencebetweenthemeasuredandpredictedTOAswillresultinastreamof5mingresiduals,whichencodetheinfluenceofgravita5onalwavesaswellasanyotherrandomnoiseinthemeasurement.

•  ByhavingaPTA,onecancorrelatetheresidualsacrosspairsofpulsars,leveragingthecommoninfluenceofagravita5onal-wavebackground(GWB)againstunwanted,uncorrelatednoise.

•  ThekeypropertyofaPTAisthatthesignalfromaGWBwillbecorrelatedacrosspulsars,whilethatfromtheothernoiseprocesseswillnot.

•  ThismakesaPTAfunc5onasagalac5c-scale,gravita5onal-wavedetector.

APTAnoisemodelAGWtraversingtheGalaxywillaffecttheperiodPofanemi\edpulsetrainsuchthatthearrival5meofapar5cularpulsewillbeperturbedfromthearrival5meexpectedinwave-freespace.Thefrac5onalchangeinthepulsefrequencyνi=1/Pifori-thpulsarmaybemodeledasΔνi/νi =αi h(t)+ni(t),whereh(acon5nuousrandomfunc5onal)isGWsignalcommontoallpulsars,αiistheanglefactorforthei-thpulsarandnirepresentsallnoisesourcesuniquetothei-thpulsar.

Time-DomainImplementa5on•  GWsinducearedshiSinthesignalfromthepulsarthatdependsonthegeometryofthepulsar-Earthsystem.

•  Inpulsar5ming,theobservablequan5tyisnottheredshiS,butthe!mingresidual,whichisjusttheintegraloftheredshiS.

•  ASersomederiva5ons,the5mingresidualscanbewri\en/approximatedasδt = Mβ+ nM is a design matrix (δt need not be linear, but iteration of the base model is a good approximation)

•  Byleastsquares,wegetanes5matorofβ

•  By assuming Gaussian noise n and evaluating log likelihood ratio (GW present vs. noise), we get optimal cross correlation statistic for a PTA which is an estimate of E(GW amplitude)2):

for pulsar pairs ij. Sij are estimates of cross covariances E(ri rj

T)Piisautocovariancematrixofpulsari,riare5mingresiduals.

Statistical Tests

• Our goal is to test the Hypotheses:

H0 : A2 = 0 vs H1 : A2 6= 0.

• To this end, we need the null distribution of the statistic A2.

• One way to accomplish this is to apply a version of the BlockBootstrap method!

S.N. Lahiri (NCSU) Lect2 1 / 11

Statistical Tests

• (Re-)Write the timing residual series for the jth pulsar as

Y(j) = M(j)β(j) + n(j)

• Note that under H0, the noise processes n(j) areuncorrelated!

• We need to make sure that the Bootstrap constructionreproduces this structural restriction!


Block Bootstrap for the null distribution

• Next, consider the regression residuals

n(j) = Y(j) −M(j)β(j)

and their centered version n(j).

• Resample BLOCKS of values in n(j), independently of theother residual series, to reconstruct the Bootstrap versionn∗(j) of n(j).

• Define the Bootstrap version of the Y(j) as

Y∗(j) = M(j)β(j)

+ n∗(j)

• Note that the random vectors Y∗(j) are independent (andhence uncorrelated) for different j-s.


Block Boostrap caliration of the test

• Let A∗2 denote the Bootstrap version of A2, obtained byreplacing the original Y(j) values by Y∗(j).

• Then, a test of H0 : A2 = 0 vs H1 : A2 6= 0 at level α ∈ (0, 1)is given by

A2 > aα

where aα denotes the (1− α)-quantile of the conditionaldistribution of A∗2.


A frequency domain test

• Let dj(ω) denote the discrete Fourier transform (DFT) ofn(j), the jth residual series, j = 1, . . . ,m.

• Let d(ω) denote the vector of the m DFTs.

• Then, the matrix valued periodogram is

I(ω) = d(ω)d(ω)∗.

• The off-diagonal elements of I(ω) give (raw) estimates ofpairwise cross-spectral densities.


A frequency domain test

• Under H0 : A2 = 0, all of these components must beidentically equal to zero at ALL frequencies ω.

• A potential test statistic is

T1 ≡∑i<j

∫ω∈J

‖di(ω)dj(ω)‖1|di(ω)||dj(ω)|

dω

where ‖(a+ ιb)‖1 = |a|+ |b|, and |(a+ ιb)| =√a2 + b2,

a, b ∈ R and ι =√−1.

• An alternative test statistic is:

T∞ ≡ supω∈J

∑i<j

‖di(ω)dj(ω)‖1|di(ω)||dj(ω)|

.


Calibration of the test statistics in FD

• In practice, we will use a finte set J of frequencies in thedefinitions of T1 and T∞.

• Under some regularity conditions, if the frequencies in J arewell-separated,

the real- and the imaginary parts of the DFTs areapproximately Gaussian, and independent!!

• We shall use this fact to find a calibration to T1 and T∞.



• Let Cj(ω) and Sj(ω) are respectively the cosine and the sinetransforms of the jth series!

• Under suitable regularity conditions and under H0:([Cj(ω)/σj(ω)]mj=1; [Sj(ω)/σj(ω)]mj=1

)ω∈J

≈(

[Zj(ω)]mj=1; [Wj(ω)]mj=1

)ω∈J

,

where Zj(ω)] and Wj(ω) are independent and identcallydistributed (iid) N(0,1) random variables.

• Here, the scaling factors σj(ω) depend on the locations tjk’sand the spectral density of the jth series ONLY!!!



• Define the ”limit” analogs of the scaled dj(ω) as

Dj(ω) = Zj(ω) + ιWj(ω), ω ∈ J, j = 1, . . . ,m.

• Then, it follows that

T1 =∑i<j

∑ω∈J

∥∥∥di(ω)dj(ω)∥∥∥1/[σi(ω)σj(ω)]∣∣∣di(ω)

∣∣∣∣∣∣dj(ω)∣∣∣/[σi(ω)σj(ω)]

≈d∑i<j

∑ω∈J

∥∥∥Di(ω)Dj(ω)∥∥∥1∣∣∣Di(ω)

∣∣∣∣∣∣Dj(ω)∣∣∣



• Thus, the test statistic T1 can be calibrated by simplygenerating a set of iid N(0,1) random variables!!

• This is expected to be more accurate than the limitdistribution (given by a linear combination of independentChi-squared rando variables), as m is not very large!!

• A similar approximation holds for the statistic T∞.


GWB Testing

Thank you!!


pulsar timing for gwb - samsis.n. lahiri (ncsu) lect2 3 / 11. block boostrap caliration of the test...

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