Anisotropy in the GW background seen by Pulsar Timing Arrays

Transcription

1 Anisotropy in the GW background seen by Pulsar Timing Arrays Chiara M. F. Mingarelli TAPIR Seminar, Caltech 17 January 2014 animation: M. Kramer 1

2 Outline Are Gravitational Waves real? Pulsar Timing Arrays PTA Geometry and Signals The PTA overlap reduction function Estimate of Anisotropy Using PTAs to characterise anisotropy in the GW background Future Work 2

3 Gravitational Waves Ripples of space-time travelling at the speed of light. Emitted from any time-varying mass quadrupole. Like EM radiation, GWs are emitted by different sources at different frequencies. In general, sources include compact objects such as neutron stars and black holes. 3

4 Using pulsars to find GWs credit: Michael Kramer, 4 MPIfR Bonn

5 Millisecond Pulsars! WD? P<20ms P s/s Ultra-stable clocks many MSPs in binaries (companion spins up old pulsar) 80% of MSPs are ~1 kpc away not to scale 5

6 The Hulse-Taylor pulsar GWs have not yet been directly detected, although indirect evidence exists. Prediction: Hulse-Taylor (1975), binary pulsar: PSR Confirmation: Taylor & Weisberg (1982) 6

7 GWs are Real! 1993 Nobel Prize for Physics: Hulse and Taylor 7

8 Indirect to direct detection Ground based detectors such as LIGO: sources include compact objects, i.e. neutron star and stellar mass black hole mergers (khz) elisa (mhz) PTAs: supermassive black hole binaries, cosmic strings, relic GWs (nhz) 8

9 A Pulsar Timing Array Animation from John Rowe Animation/Australia Telescope National Facility, CSIRO 9

10 10 E. Hand, Nature 463, 147 (2010)

11 A Pulsar Timing Array Primary goal of PTAs is to detect a stochastic GW background. Currently 3 major PTAs: European PTA, Parkes PTA and NANOGrav. Together we form the IPTA: 39 MSPs (and counting) and 8 radio telescopes 11

12 GW Search: Strain Assuming circular binaries driven by GW emission only, can define a characteristic strain: Z Z! h 2 c f 4/3 dzdm d2 n 1 M5/3 dzdm (1 + z) 1/3 number of mergers per comoving volume, per unit redshift and chirp mass [(m 1 m 2 ) 3/5 M -1/5 ] (Phinney 2001, Sesana 2013).! h c = A! f yr 1 2/3 Shannon et al. (2013) recently reported the most stringent A=2.4 x yet. 12

13 GW Spectrum 13 P. Demorest et al., 2009

14 The GW signal Consider a plane wave expansion for the metric perturbation produced by a stochastic background: X Z Z h ij (t, x) = df dˆ h A (f, ˆ) e i2 f(t ˆ x) e A ij(ˆ) A S 2 where i,j are spatial indices, A is GW polarization, f is GW frequency, e A are pol n tensors and ha are the pol n amplitudes. Can decompose ˆ over ha and e A : h ij (t, ˆ) =e + ij (ˆ)h + (t, ˆ)+e ij (ˆ) h (t, ˆ) Pol n tensors uniquely defined once one specifies the wave principal axes described by the unit vectors m, n: e + ij (ˆ) = ˆm i ˆm j ˆn iˆn j e ij (ˆ) = ˆm iˆn j +ˆn i ˆm j 14

15 Typical PTA Geometry ˆ =(sin cos, sin sin, cos ) ˆm =(sin, cos, 0) ˆn = (cos cos, cos sin, sin ) Next: How are the pulsar signals modified by GWs? 15 image: S. Chamberlin, X. Siemens (2011)

16 Frequency Shift Consider a pulsar emitting at frequency ν 0 at position p.!! A GW source generates a metric perturbation which affects the actual frequency ν at which the radio pulses are received according to: where and 16

17 Frequency Shift The fractional frequency shift produced by a GW background is given by integrating z(t) over all directions: Z z(t) = = X A dˆ z(t, ˆ) Z Z df dˆf A (ˆ)h A (f, S 2 ˆ)e i2 ft h1 e where the F A are called the Antenna Beam Pattern : i2 fl(1+ˆ ˆp) i F A (ˆ) = apple 1 2 ˆp i ˆp j 1+ˆ ˆp ea ij(ˆ). 17

18 Overlap Reduction Function hr a(t j )r b (t k )i = Z tj dt 0 Z tk dt 00 z a(t 0 )z b (t 00 ) where and (ab) = = (f) ab(f, ˆ) Z tj dt 0 Z tk dt 00 Z +1 Z tj dt 0 Z tk dt 00 Z +1 Z 1 dˆ P (ˆ ) ab (f, ˆ ) 1 df 0 Z +1 1 df 00 z a(f 0 ) z b (f 00 )e i2 (f 0 t 0 f 00 t 00 ) df e i2 f(t0 t 00) H(f) (ab) (f). " X A F A a (ˆ )F A b (ˆ ) h 1 e i2 fl a(1+ˆ ˆp a ) ih 1 e i2 fl b(1+ˆ ˆp b ) i. # 18

19 Isotropic Background (?) Before now, background always considered to be isotropic. If background is detected, need a general way to analyze it: is its origin cosmological as predicted? Can we estimate the expected level of anisotropy? 19

20 Estimate level of anisotropy A background is anisotropic if by taking two conic sections of the sky centred around different directions the expected power is different. We call the fractional level of anisotropy ratio of the standard deviation of the energy density, and its expectation value, µ gw (f) an(f) gw(f) µ gw (f) gw(f) CMFM, Sidery, T., Mandel, I. and Vecchio, A. "Characterizing stochastic gravitational wave background anisotropy with Pulsar Timing Arrays", Phys. Rev. D 88, (2013). 20

21 How to estimate? Assume the universe is populated by identical sources with number density n. Compare the energy density in GWs centred on 2 different directions in the sky with solid angle d.!! ˆ ˆ 0!! d 21

22 Assumptions Consider sufficiently nearby sources such that we neglect effects of expansion/redshift. We will use the following geometry frequently:! dv = D 2 ddd Number of sources contributing to the background: dn = nd 2 ddd d d D + dd 22 D

23 Number of Sources Number of sources governed by Poisson statistics with mean µ = dn and variance 2 = dn.. Choose dv is sufficiently small such that dn<<1. Then the probability of finding one source is P (1) = dne dn dn 23

24 Number of Sources Total number of sources present in whole volume, within solid angle d btw Dm and DM, given by summing all the slices in the cone. The variance, 2 N from each conical slice: µ N = 2 N =, is the sum of the variances Z DM D m 24 nd 2 ddd " = n 3 d D3 M 1 µ N = dn Dm D M 3 #

25 GW Energy Density now want contribution to GW energy density per frequency interval and its variance know GW energy density scales as 1/D 2 assuming all sources are identical, contribution to the GW energy density per source is d GW dn = A D 2 25

26 GW Energy Density The expected GW energy density from sources in a small conical volume dv at distance D is dµ GW (D) P (1) d GW dn dn A D 2 D chosen so that it has a negligible probability of having more than one source. Recall n is source number density. = naddd 26

27 GW Energy Density The variance of the energy density from sources in the conical volume is d GW 2 (dµ GW(D)) 2 d 2 GW = P (1) dn n A2 D 2 ddd. (last obtained by using repeated applications of dn<<1) 27

28 Estimate Have now calculated and so can dµ GW d 2 GW proceed to calculate and. µ GW = Z DM D m µ GW 2 GW dµ GW = nad D M 1 D m D M and since the variance of a sum is the sum of the variance: 2 GW = Z DM D m GW = A µ GW d 2 GW = na 2 d p nd And D M DM D m D M D m DM D m D M D m 1/2 1 D m D M 28

29 Estimate Rewrite in terms of y=dm/dm GW µ GW = 1 d apple y nd 3 M (1 y)y2 1/2 GW µ GW = apple 4 d (y) N 0 1/2 where N 0 = 4 nd 3 3 M (1 y 3 ) is the total number of sources contributing to background and (y) (y 2 + y + 1)/(3y) 29

30 Estimate level of anisotropy Summary: gw (f) µ gw (f) = r (Dm,D M ) Nd where (D m,d M )=(y 2 + y + 1)/3y and y D m /D M NOW: Can try to estimate this quantity for supermassive black hole binaries.! Assume SMBHBs detectable up to z~1, and almost none after z~5 (e.g. Sesana Vecchio Colacino 2008, using Volonteri et al models) 30

31 Estimate level of anisotropy Using the results from Sesana, Vecchio, Colacino (2008) (Fig 4), can estimate total number of sources that contribute in a frequency interval of width Tobs: N f 10 8 Hz 11/3 5yr T obs Convert avg angular scale to multipole moment index l by using d =4 /2l gw (f) µ gw (f) 0.20 to obtain the fractional level of anisotropy: 11/6 1/2 1/2 f 5yr l 10 7 Hz T obs 2 1 1/2. 31

32 Characterizing Anisotropy We decompose the angular distribution function on the basis of the spherical harmonic functions! P (ˆ) X lm c m l Yl m (ˆ) Now write generalized correlation fns as: m l (f, ˆ ) Z dˆ P ( ) ab (f, ˆ ) 32 " X A F A a (ˆ )F A b (ˆ ) #

33 Recall PTA Geometry ˆ =(sin cos, sin sin, cos ) ˆm =(sin, cos, 0) ˆn = (cos cos, cos sin, sin ) This choice of geometry zeroes the F X antenna beam pattern. In anisotropic case, need to rotate. 33 image: S. Chamberlin, X. Siemens (2011)

34 Overlap Reduction Function Recall Z " X # (ab) (f) dˆ P (ˆ ) ab (f, ˆ ) F A a (ˆ )F A b (ˆ ) A and ab(f, ˆ) h 1 e i2 fl a(1+ˆ ˆp a ) ih 1 e i2 fl b(1+ˆ ˆp b ) i. In the short wavelength approximation, fl >>1 and La=Lb so apple ab 1+ ab more on this in Mingarelli & Sidery, in prep 34

35 1.0 Hellings and Downs, isotropy Overlap reduction function, Γ Γ Angular separation of pulsars, ζ 35

36 0.2 Overlap reduction function, Γ m Angular separation of pulsars, ζ Γ 1 1 Γ 0 1 Γ

37 Overlap reduction function, Γ m Γ 2 2 Γ 1 2 Γ 0 2 Γ 1 2 Γ Angular separation of pulsars, ζ 37

38 Overlap reduction function, Γ m Γ 3 3 Γ 2 3 Γ 1 3 Γ 0 3 Γ 1 3 Γ 2 3 Γ Angular separation of pulsars, ζ 38

39 State of Anisotropy work EPTA & NANOGrav embedding anisotropy in pipeline can currently inject and recover anisotropy up to quadrupole cross checking results across pipelines within EPTA before releasing anisotropic limits 39

40 Applications Each one of these new functions will be weighted by a clm factor which now becomes a search parameter. Only one parameter to replace in likelihood function evaluation! This method can be extended to search for single source GW sources at high l. 40

41 Conclusions Estimated 20% fractional level of anisotropy in stochastic GW background Now have a way of characterizing it on any angular scale General method: can confirm isotropic limits as well as impose new anisotropic ones. First anisotropic limits to be released shortly 41

42 Thank You! Slides available on 42