The Search for Gravitational Waves from Spinning Neutron Stars

Part I: Continuous Gravitational Waves Part II: Data-analysis methods Part III: Transient Continuous Waves

The Search for Gravitational Waves from Spinning Neutron Stars Reinhard Prix Albert-Einstein-Institut Hannover

RIT Seminar, Jan 2011

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GWs from spinning NS


Outline

1

Part I: Continuous Gravitational Waves

2

Part II: Data-analysis methods

3

Part III: Transient Continuous Waves

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Detection of gravitational waves y Ly Lx

x

h(t)

t

Measure scalar Strain h(t) ≡

Lx (t)−Ly (t) 2L

+ “listening” to the Universe

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Worldwide Network of Detectors

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Detector Sensitivity 10−18

S1: Aug 2002, 17 days S2: Feb 2003, 2 months S3: Nov 2003, 2 months S4: Feb 2005, 1 month S5: Nov 2005, 2 years S6: Jul 2009, 1.2 years

10−19

10−21

√

Sn [Hz−1/2 ]

10−20

10−22

10−23

10−24

10

100

1000

Freq [Hz]

h0 =

∆L L

∼ 3 × 10−23 =⇒ ∆L ∼ 10−19 m = 10−4 fm!! Reinhard Prix



LSC Data Analysis

LIGO (H1, H2, L1), Virgo and GEO600 data analyzed within the LIGO Virgo Collaboration (LVC): ∼ 80 institutions, ∼ 700 authors 4 major search groups (different targets and methods): Binary inspirals (CBC): short modeled signals Bursts: short unmodeled signals (supernovae, merger) Stochastic background: cosmological background GWs Continuous Waves: spinning deformed neutron-stars (NSs)

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GWs from spinning NS Quadrupole formula (Einstein 1916). GW luminosity (axial ellipticity ): LGW ∼

c 5 2 R s 2 V 6 G R c |{z} |{z} |{z}

∼1059 erg/s ∼0.4

∼0.1

(Optimistic) amplitude h0 : Izz ν 2 100 pc −25 h0 ∼ 4 × 10 100 Hz d 10−6 1045 g cm2 √ Best S5 LIGO sensitivity: Sn (150Hz) ∼ 3 × 10−23 Hz−1/2

+ CW signals buried in the noise =⇒ need “matched filtering” √ SNR ∝ √hS0 T observation time T ∼ O(days − months) n

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What do we know about Neutron Stars?

M ∼ 1.4 M

Open questions:

R ∼ 10 − 20 km =⇒ RRs B∼

=

108

2GM c2R

−

glitches, pulse emission

∼ 0.4

equation of state, strange matter

G

mountains, unstable oscillations

1014

ν ∈ [0.1 Hz, 716 Hz]

+ Gravitational waves? Reinhard Prix



Continuous GWs from Spinning Neutron Stars Continuous waves: quasi-sinusoidal fGW ∼ O (ν) quasi-infinite & Tobs Emission mechanisms: “Mountains” (fGW = 2ν) Oscillations (r-modes: fGW ∼ 4ν/3)

Free precession (fGW ∼ ν, 2ν)

Accretion (driver) Reinhard Prix



Types of CW searches ˆ and fGW = 2ν Targeting known pulsars: skyposition n + computationally cheap O (laptop) ˆ but fGW ∈ ±∆f Directed searches: known skyposition n + computationally challenging O (cluster)

Wide-parameter searches for unknown NSs: cover large ˆ , frequency f , spindown f˙ , . . . parameter regions in sky n + coherent matched filtering impossible √ SNR ∝ √hS0 T BUT computing cost Cp ∝ Np T ∝ T 7 n

Semi-coherent methods: break data into Nseg shorter “segments” of duration ∆T , and combine incoherently =⇒ better sensitivity per computing cost maximize available computing power by using Einstein@Home Reinhard Prix



What is Einstein@Home? Maximize available computing power Cut parameter-space λ in small pieces ∆λ • Send workunits ∆λ to participating hosts • Hosts return finished work and request next Public distributed computing project, launched Feb. 2005 runs on GNU/Linux, Mac OSX, Windows,.. Currently ∼100,000 active participants, ∼300Tflops Search for isolated neutron stars f ∈ [50, 1500] Hz Analyzed LIGO data from S3, S4, S5, now S5 Hierarchical Since 2009 ∼ 30% spent on short-period binary search in Arecibo Data: First E@H pulsar discovery [Science, Aug 2010] Reinhard Prix



Spindown upper-limit for CWs from known pulsars Rotational energy lost: E˙ rot ∝ Izz |{z} ν ν˙

observed

2 2 Energy emitted in GWs: E˙ GW ∝ ν 6 Izz

Spindown upper limit: Spindown fully due to GW emission E˙ GW = E˙ rot Assumed Izz (from EOS) and known distance d: r 1 5G |ν| ˙ =⇒ upper limit: h0sd = Izz 3 d 2c ν Reinhard Prix



CW Sensitivity Progression -21

10

ScoX1[S2] Targeted[S2] Targeted [S4] Targeted [9Months S5] Directed [9Months S5]

10-22

Fstat[S2] ] Hough[S2

Vela

10-23

3] H [S E@ [S4] E@H [S5] E@H Statistical UL

Crab

r]

5 Hie

[S E@H

h0

10-24

-I LIGO B1951+32

O eLIG

Virgo

10-25

J1913+1011

J0537-69

10-26

ScoX1 J0437-4715

(S5)

)

9 (200

14)

O (20

AdvLIG

B1937+21 10-27

10

100 GW Frequency [Hz]

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1000



Continuous Wave Signal Model o NS frame: quasi-sinusoidal wave, slowly varying frequency 1˙ 2 Phase φ(τ ) = 2π f τ + 2 f τ + ... GW frequency for triaxial NS: f = 2 ν , r-modes: f = 4/3 ν, precession: f ≈ ν

2 polarization amplitudes: A+ , A× =⇒ Components in wave frame (ψ): h× (τ ) = A+ cos (φ(τ ) + φ0 ) h+ (τ ) = A× sin (φ(τ ) + φ0 ) ˆ ) dependent modulations: o Detector frame t: sky-position (n ˆ) Phase: Doppler-effect due to earth’s motion τ = τ (t; n ˆ) Amplitude: rotating Antenna-pattern F+,× (t, ψ; n Reinhard Prix



Parametrized CW signal family: Amplitude parameters: A = {A+ , A× , ψ, φ0 } ˆ , f , f˙ , . . .} Doppler parameters: λ = {n ˆ ) A+ cos φ(t; λ, φ0 ) s(t; A, λ) = F+ (t; ψ, n ˆ ) A× sin φ(t; λ, φ0 ) + F× (t; ψ, n Jaranowski, Królak, Schutz 1998: factorize signal model use new coordinates Aµ = Aµ (A+ , A× , ψ φ0 ) 4 P s(t; A; λ) = Aµ hµ (t; λ) µ=1

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Two ways of doing “Statistics” Frequentist probability: limiting frequency of “random variables”: p(x) ≡ limN→∞ NNx

+ “Mind projection fallacy” (Jaynes) Needs imagined “repeated experiments” to define p(x)

Bayesian probability: quantified uncertainty about a statement H assuming I: P (H|I) ∈ [0, 1] + inductive extension of logic: P (H|I) = 1 ⇐⇒ H true given I

What does observed data x say about hypothesis H? Limited to likelihood p(x|H) Trial and error on good “tests”

+ directly express P (H|x I) Axiomatic and straightforward

Same rules for manipulating probabilities. Bayesian probability more powerful (optimal information use) and flexible.

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Hypothesis Testing I Two competing hypotheses: No signal: HN : x(t) = n(t) CW signal: HS : x(t) = n(t) + s(t; A, λs )

[assume λs known]

Given x(t), how to “optimally” decide between HN and HS ? If A = As known: Neyman-Pearson lemma

+ Likelihood ratio L(x) is the “most powerful” test: L(x) ≡

P (x|HS (As )) > Lth =⇒ accept HS , else HN P (x|HN )

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Hypothesis Testing. II. Testing power false-alarm probability: pfA ≡ P(pick HS |HN ) detection probability: pdet ≡ P(pick HS |HS ) Receiver-Operator-Characteristic (ROC): pdet (fA ): SNR = 4.0, cos ι = 0.00, ψ = 0.00 detection probability η = 1 − fD

1 0.8 0.6 0.4 0.2 0 0.001

L(As ) 0.01

0.1

false-alarm probability fA Reinhard Prix


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Hypothesis Testing. III. F-statistic Generally amplitude parameters A = {h0 , cos ι, ψ, φ0 } unknown =⇒HS ≡ {HS (A) for any A} is a composite hypothesis: + Likelihood ratio is function L(x; A) + Neyman-Pearson lemma does not apply

Frequentist ad-hoc approach: maximize L over A: Λ(x) ≡ max L(x; A) = max A

A

P (x|HS (A)) P (x|HN )

F-statistic: Jaranowski, Królak, Schutz (1998) Analytic maximization over {Aµ }:

Λ(x) ≡ max L(x; Aµ ) = eF (x) {Aµ }

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Bayesian Hypothesis testing Directly compute the posterior probability for any hypothesis Hi : P (Hi |x I) ∝ P (x|Hi I) P (Hi |I) | {z } | {z } | {z } posterior

likelihood

(“Bayes’ theorem”)

prior

Posterior odds ratio:

OSN (x) ≡

P (HS |x) P (x|HS ) P (HS ) = , P (HN |x) P (x|HN ) P (HN ) | {z } | {z } Bayes factor prior odds

Bayes factor BSN generalizes likelihood ratio L(x): Z P (x|HS ) BSN (x) ≡ = L(x; A) P (A|HS ) d 4A | {z } P (x|HN ) A−prior

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Bayesian CW analysis: going beyond the F-statistic natural prior P (A|Hs ): isotropic NS spin + P (h0 ) × P (cos ι, ψ, φ0 ) ∼ uniform R =⇒ B(x) ≡ L(x; A) dh0 dcos ι dψ dφ0 B(x) is more powerful than F(x)

unphysical uniform prior P (Aµ ) + recover F-statistic: R BF (x) = L(x; A) d 4 A ∝ eF (x)

R Prix, B Krishnan, CQG 26 (2009)

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Transient CWs from Spinning Neutron Stars New class of GW signals: transient CWs O (hours − weeks) bridge gap between O (ms) Bursts and O (∞) CWs

strong GWs more likely related to transient events in NSs Glitches:

∆ν ν

. 10−6

seen in ∼ 25 “young” pulsars

Relaxation τ ∼ hours − weeks Data-analysis: new search parameters {t 0 , τ } statistic, confidence, cost, ...

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Pulsar glitches and the 2-fluid model Neutron Stars believed to contain a superfluid of neutrons, spinning at νs > ν, with moment of inertia Is ∼ 10−2 Ic νs

ν

=⇒

Two-fluid glitch model: at critical lag ∆ν = νs − ν =⇒ sudden transfer of angular momentum Is νs → Ic ν spin-up Glitch δν, followed by relaxation over timescale τ Reinhard Prix



Astrophysical estimates for transient CWs most “traditional” CW mechanism have poorly understood stability/timescales: precession, r-modes, accretion non-axisymmetric Ekman flow in pulsar glitch relaxation VanEysden, Melatos, CQG25 (2008), Bennett, VanEysden, Melatos, MNRAS409 (2010) + predicts transient CWs detectable by AdvLIGO to ET energy estimate from superfluid “excess” rotation energy: Es = 12 Is (νs2 − ν 2 ) → EGW + SNR2 ∝ SnEνGW 2 d2 Assuming 1yr between glitches, S6-sens. (3LIGO + Virgo) Pulsar B0531+21 (Crab) B0833-45 (Vela) J1813-1749

ν [Hz] 30 11 22

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d [kpc] 2 0.29 4.7

SNR 12 4 2



Transient CW detection method CW rect exp

HN : x(t) = n(t) , HS : x(t) = n(t) + h(t; A, λs , t 0 , τ )

h(t)

Prix, Giampanis, Messenger, to be submitted (2011)

Use uniform priors on t 0 and τ Use “F-statistic priors” on A

R

t0

t0 + τ

t0 + 3τ

0 eF (x;t ,τ ) dt 0 dτ

+ Bayes factor: BF (x) ∝ R 0 + Parameter estimation: P t 0 |xHS ∝ eF (x;t ,τ ) dτ R 0 P (τ |xHS ) ∝ eF (x;t ,τ ) dt 0

Maximum likelihood (ML): Fmax (x) ≡ max{t 0 ,τ } F(x; t 0 , τ ) maximum-likelihood estimators {t 0 ML , τ ML } Reinhard Prix



Transient CW detection power 0.6

pdet − pfA

r BF e BF

̟s = r

0.5

r Fmax e Fmax

0.4 0.3 0.2 0.1

fixed sky-position

0 0

0.4

0.6

0.8

1

pf A

0.6

τ ∈ [0.5, 2.5] days

r Fmax e Fmax

0.4 0.3

SNR = 3 t 0 ∈ [0, 6] days

r BF e BF

̟s = e

0.5 pdet − pfA

0.2

0.2 0.1 0 0

0.2

0.4

0.6

0.8

1

pf A Reinhard Prix



Transient CW parameter estimation Example posteriors (SNR=5): 0.8

0.3

pdf inj ML

0.6

0.2

0.4 0.1

0.2 0

0 0

5

10

15

20

25

30

2

4

6

0

t [days]

8

10

12

14

τ [days]

Parameter-estimation quality Monte-Carlo (N = 104 ): 14

t0 = rand

12

rms [τ − τs ]

rms t0 − t0s

10

8 6 4

ML MP

2 0 0

2

4

6

8

10

12

9 8 7 6 5 4 3 2 1 0

τ = τmin τ = rand

ML MP 0

2

4

SNR ρ

6 SNR ρ

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8

10

12