aa r X i v : . [ a s t r o - ph . I M ] M a y Noise in pulsar timing arrays
Yan Wang
School of Physics, Huazhong University of Science and Technology, 1037 Luoyu Road, Wuhan,Hubei Province 430074, ChinaCenter for Advanced Radio Astronomy, University of Texas at Brownsville, 1 West UniversityBoulevard, Brownsville, Texas 78520, USAE-mail: [email protected]
Abstract.
To successfully detect gravitational waves with pulsar timing arrays, we need tohave a comprehensive understanding of the physical origins and statistical characteristics of thenoise in pulse arrival times and identify mitigation methods to reduce the noise. In this paperwe will review radiometer noise, phase jitter noise and timing noise in the noise budget of pulsartiming and show various efforts used to reduce them. We will briefly discuss the results of anoverall assessment of the components and physical causes of the timing residuals for millisecondpulsars in the North American Nanohertz Observatory for Gravitational Waves (NANOGrav).
1. Introduction
Pulsar timing arrays (PTAs) are striving to detect very low frequency (10 − − − Hz)gravitational waves (GWs) by observing a set of extremely stable millisecond pulsars (MSPs).Detection can be achieved by observing 20–40 pulsars over 5–10 years, assuming monthlyobservation cadence and 100 ns for root mean square (RMS) of timing residuals which ispresumably dictated by white Gaussian noise [1]. However, detection can be delayed by upto about 10 years depending on the level of red noise and the possible errors (e.g., ephemeris,polarization calibration, time transfer) in the highest timing precision [2]. So far, upper limitshave been reported for the amplitude of the stochastic GW background [3; 4; 5] and forcontinuous GW sources [6; 7] by three major PTAs (NANOGrav [4], PPTA [8] and EPTA[9]). Results from the International Pulsar Timing Array (IPTA) [10; 11; 12] which combine thedata sets from all PTAs will also become available in the near future.As in any gravitational wave detection experiment (e.g., LIGO [13], Virgo [14], eLISA[15]), noise characterization and mitigation are the central issues that need to be addressedin order to confidently detect and characterize the relatively weak GW signals, and carry outdetailed astrophysical interpretations of them. For PTAs, a comprehensive understanding of thenoise error budget of pulse arrival time is crucial to guide us to improve instruments, designexperiments, and develop algorithms.To achieve this goal, we need to build a complete measurement model that accounts forthe end to end errors in the highest timing precision. Here, one end is a rotating pulsar,which produces regular radio pulses. The spacially coherent radiation carrying these pulses willbe distorted by turbulent plasma in interstellar medium (ISM), causing dispersion, scattering,refraction, diffraction, etc. of the radio waves [16], along the line of propagation to the other end,a telescope. The faint EM signal is subsequently collected and focused by the large reflector,eceived by the radiometer and recorded by the backend system at the observatory. Fromemission to reception, there are various errors that can be introduced into the final estimationof pulse time of arrivals (TOAs).Here, we shall not discuss the propagation effects pertaining to the interstellar medium[17; 18], since it is covered in the account by L. Levin in this volume. Instead, we focuson the other errors such as radiometer noise, phase jitter noise and timing noise, which arecurrently under intensive scrutiny of the pulsar timing community [19; 20; 21]. The rest ofthis paper is organized as follows. In Sec. 2 we provide an overview of these noise sources andmethods of mitigation, and in Sec. 3 we discuss briefly the efforts and results in the NANOGravcollaboration on noise analysis. The paper is concluded in Sec. 4.
2. Noise
In timing analysis, it is a common practice to transform TOAs measured in the topocentricreference frame centered at a telescope to the inertial reference frame centered at the SolarSystem Barycenter [16; 22]. This transformation includes terms representing deterministic effectssuch as clock correction, time delay due to interstellar medium dispersion, geometric time delay(R¨ o mer delay), relativistic time delay (Shapiro delay, Einstein delay), etc. In addition, variouserror terms should be also added into the time transformation. A comprehensive list of timingerrors can be found in Table 1 of [23].In general, we can classify the errors into the ones pertinent to the time tagging of pulses(i.e. TOA measurement by template fitting) and the ones pertinent to the physical propertiesof pulsar or ISM. An example of the latter is the timing of a pulse with infinite SNR andknown profile which can be measured to an arbitrary precision (no time tagging error), but theirregularity of pulsar rotation or the stochastic fluctuation of dispersion measure may introduceadditional random components. The error can be achromatic which means its influence isindependent of the observation frequency (timing noise), weakly chromatic (radiometer noise,jitter noise), or strongly chromatic (effects rooted from interstellar medium).The power spectrum of the noise can be white or red. The TOA fluctuations caused bythe stochastic GW background have red spectra for individual pulsars, which are angularlycorrelated between pairs of pulsars [24]. Thus, for detecting this background, it is imperativeto assess and reduce, if possible, the confusing red noise components rooted from other sources,for example, pulsar and ISM. Radiometer noise is instrumental in origin (thermal electron fluctuation) combining thecontribution from the sky background (dominated by the synchrotron-radiating electrons in theplane of the Galaxy). Radiometer noise with a Gaussian probability density function is additiveto the pulse profile, of any of the Stokes parameters. Usually, the Stokes I (intensity) is used tomeasure the TOAs. Radiometer noise is weakly chromatic if the declination of pulse flux densitywith increasing of observation frequency cancels out much of the variation of the sky backgroundtemperature. The resulting TOA is estimated from template fitting of integrated pulse profilewith theoretical or integrated template. The minimum RMS error for TOA estimation due tofinite SNR and sampling rate is [23]: σ SNR = 1 µs (cid:18) W (cid:19) (cid:18) N (cid:19) − / (cid:18) (cid:19) (cid:18) ∆ W (cid:19) / , (1)= 0 . µs (cid:18) W (cid:19) / (cid:18) P (cid:19) − (cid:18) f . (cid:19) − α (cid:18) ∆ f (cid:19) − / (cid:18) N (cid:19) − / (cid:18) S sys S (cid:19) . (2) It could be strongly chromatic if the pulsar spectrum is flat in the observation band. ere W is the effective pulse width which equals to W FWHM / √ π ln 2 for a Gaussian pulseprofile, W FWHM is the full width at half maximum of the pulse, N is the number of pulseaveraged synchronously to yield an integrated profile, SNR is the single pulse SNR, ∆ is thesampling interval, and P is the spin period. Radio frequency f and bandwidth ∆ f are in GHz, S sys and S are the system equivalent flux density in Jy and mean flux density of the pulsarat 1 . T sys = T sys p n p t ∆ f , (3)and assuming that the flux density of pulsar follows a power law with spectral index α . T sys and ∆ T sys is the system temperature of the receiver and its RMS fluctuation, n p is the numberof polarization, and t is the integration time.Eq. 1-2 are obtained under the assumption that there is no variance for the integrated pulsetemplate over the frequency band, i.e. the profile has the same shape at high frequency as at lowfrequency. However, the profile evolution can arise from phenomena intrinsic to pulsar emissionbeam or ISM (DM change, scattering and scintillation). Ignoring it will degrade the timingprecision promised by the modern broad band receivers and backends (see Table 1). This socalled “the large-bandwidth problem” [26] is demonstrated in a recent 24-hour global observationfor the millisecond PSR 1713+0747 (c.f. Fig. 4 in [20]). Current treatment of this problem is togenerate templates for each frequency channel, add additional fitting parameters (JUMPs) tohandle the possible offsets between them, and allow the standard TOA analysis packages suchas tempo and tempo2 to find out the best fit value for them [4]. A more consistent and efficientmethod may be to create a two-dimensional pulse portrait (rather than one-dimensional profilesat different frequencies) which takes into account of differential profile evolution and time offsetas a smooth function of frequency. This strategy has been explored for broad band data in[27; 28]. Table 1.
Wide band receivers and backends currently used or planned for pulsar observation.Telescope D(m) Receiver f (GHz) T sys Backend ∆ f (MHz) Ref.GBT 100 L-band 1.15-1.73 20 GUPPI 800 [29]Arecibo 305 L-band 1.15-1.73 25 PUPPI 800 [30; 31]Parkes 64 Multibeam 1.23-1.53 28 APSR 1000 [8]Effelsberg 100 UBB 0.6-3 24 ASTERIX 512 [32; 33]FAST 500 L-band 1.15-1.72 25 – – [34] Increasing the SNR of the integrated profile will reduce the radiometer noise, however it is notthe only justification for using integrated pulse profile in TOA measurement. Individual pulseusually jitters in phase at the level of single pulse width, and its amplitude can change more than100% from pulse to pulse (c.f. Fig. 4 of [35] for PSR J1740+1000). Thus measuring individualpulses even with large SNR will result in a TOA uncertainty in the order of the pulse width.Phase jitter and amplitude modulation are related to the stability of the integrated pulse shapewhich is determined by the shape of individual pulse and the probability density function of thehase jitter. A stable pulse profile can be obtained by summing over at least several hundredsof individual pulses, and the associated TOA uncertainty due to jitter roughly scales inverselyas square root of the number of the individual pulses [36].Pulse phase jitter and amplitude modulation appear in all well studied pulsars. It is weaklychromatic. Jitter noise is not additive in nature as the radiometer noise. In fact, it changes theintegrated pulse profile in a statistical manner [21]. For a simple case in which we assume thepulse profile is Gaussian shape and the phase jitter follows a Gaussian distribution, the RMSerror caused by jitter noise can be written as [23] σ J = 0 . µs (cid:18) W i (cid:19) (cid:18) N (cid:19) − / (cid:18) f J / (cid:19) (cid:18) m (cid:19) / . (4)Here, m I is the amplitude modulation index defined as the ratio of the standard deviation ofthe amplitude to the mean at different pulse phases and m I ≈ f J is the dimensionless jitterparameter defined as the ratio of the standard deviation of the phase of single pulse to theintrinsic pulse width W i of the template and f J ≈ /
3. The RMS of total error σ t for estimatedTOA is the quadratic summation of radiometer noise and jitter noise, i.e. σ = σ + σ .By equating Eq. 1 with Eq. 4, we find that jitter noise will become more important thanradiometer noise when SNR exceeds only a few tenths. This sensitivity is accessible by thefuture radio telescopes such as FAST [37] and SKA [38] that have larger collecting areas andlower system temperatures. In this scenario, the noise will not be reduced by increasing theobservation bandwidth. Increasing the observation time will become an inevitable choice. As aresult, PTAs will need to request more observation time of radio telescopes. Timing noise, also known as spin noise, appears as the structures with temporal correlation intiming residuals that depart greatly from the measurement error alone. It may be caused bythe irregularity of rotational spin rate which could root from the changes in internal structureand/or magnetosphere of neutron star. It has been found in a number of canonical pulsars and afew MSPs (e.g., B1937+21, B1821-24) [39]. The RMS error of timing noise can be characterizedby a scaling law [19]: σ TN = Cν α | ˙ ν | β T γ , (5)where C , α , β and γ are the fitting parameters determined from the whole populations of pulsarswith measurable timing noise or upper limits, ν and ˙ ν is the pulsar spin frequency and frequencyderivative, T is the total span of observations. The best-fit values and ± σ confidence limitscalculated from canonical pulsar and MSP population are ln C = 1 . ± . α = − . ± . β = 1 . ± .
1, and γ = 2 . ± . γ ≈
2, it has a power spectral indexbetween − − h c of the stochastic GW background scales as h c = A ( f / yr − ) α , where α = − / T / associated with aspectral index of 2 α − − /
3. Noise assessment
As discussed above, the characterization of noise in PTA plays a central role in hunting forGWs. Due to its importance, the NANOGrav has formed a noise budget working group to usecomplementary methods to assess the constituents of timing residuals and their physical causes.The white noise and Gaussian statistics are two essential aspects of the assessments, especiallythe latter is a common assumption in forming the GW detection strategies [41; 42; 43; 44].Blindly applying these strategies without checking the presumption may lead to unreliableresults.Methods used include autocorrelation analyses, Bayesian inference, zero-crossing tests,Gaussianity tests, etc. A memorandum to consolidate the results from the overall assessment ofthe NANOGrav 5-yrs ASP/GASP data set for 17 MSPs [4] is in preparation. Further studiesextended to the NANOGrav 9-yrs (including 4-yrs PUPPI/GUPPI) data set for more than 30MSPs will be also carried out once the data are available.Initial results show that for the 5-yrs data most of the pulsars are consistent with the whitenoise assumption [45; 46], although it is possible that the red timing noise can appear in the9-yrs observations which have longer span and higher precision. Different levels of departurefrom Gaussian statistics are shown in most of the pulsars [46], it is suspected that the diffractiveinterstellar scintillation is the root cause. This suggests that the robust signal detection andcharacterization methods that are not sensitive to the non-Gaussianity should be implementedin GW data analysis.
4. Conclusions
Noise characterization and mitigation are the central issues in detecting GWs by pulsar timingarrays. In this article, we provide an overview of the features of radiometer noise, phase jitternoise and timing noise as well as the efforts to reduce them.The radiometer noise is dominant in the current timing precision, it will be continuouslymitigated with the developments of instrument and algorithm to a level smaller than the jitternoise. The jitter noise is intrinsic to a pulsar, it can only be mitigated by extending integrationtime, therefore affects the strategies on telescope time application and allocation.Timing noise is latent for most MSPs, it can potentially postpone the detection of GWs byPTAs. Finding more MSPs with excellent timing performance in the ongoing and future surveysis imperative in the competition between the red noise from GWs and pulsars themselves.
Acknowledgments
We wish to acknowledge the invitation from Prof. Andrea Lommen for the 10th LISA symposium,and the assistance from the local organizer at University of Florida, especially Prof. GuidoMueller for his help in conference registration. We would like to thank the NANOGrav membersfor helpful discussion and the Center for Gravitational Wave Astronomy at UTB for partialsupport under NASA grant NNX09AV06A. The NANOGrav project is supported by the NationalScience Foundation under PIRE award number 0968296. eferences [1] Jenet F A, Hobbs G B, Lee K J and Manchester R N 2005
Astrophysical Journal Letter
L123–L126 (
Preprint arXiv:astro-ph/0504458 )[2] Siemens X, Ellis J, Jenet F and Romano J D 2013
Classical and Quantum Gravity Preprint )[3] van Haasteren R, Levin Y, Janssen G H, Lazaridis K, Kramer M, Stappers B W, DesvignesG, Purver M B, Lyne A G, Ferdman R D, Jessner A, Cognard I, Theureau G, D’AmicoN, Possenti A, Burgay M, Corongiu A, Hessels J W T, Smits R and Verbiest J P W 2011
Monthly Notices of the Royal Astronomical Society
Preprint )[4] Demorest P B, Ferdman R D, Gonzalez M E, Nice D, Ransom S, Stairs I H, ArzoumanianZ, Brazier A, Burke-Spolaor S, Chamberlin S J, Cordes J M, Ellis J, Finn L S, Freire P,Giampanis S, Jenet F, Kaspi V M, Lazio J, Lommen A N, McLaughlin M, PalliyaguruN, Perrodin D, Shannon R M, Siemens X, Stinebring D, Swiggum J and Zhu W W 2013
Astrophysical Journal
94 (
Preprint )[5] Shannon R M, Ravi V, Coles W A, Hobbs G, Keith M J, Manchester R N, Wyithe J S B,Bailes M, Bhat N D R, Burke-Spolaor S, Khoo J, Levin Y, Oslowski S, Sarkissian J M, vanStraten W, Verbiest J P W and Want J B 2013
Science
Preprint )[6] Arzoumanian Z, Brazier A, Burke-Spolaor S, Chamberlin S J, Chatterjee S, Cordes J M,Demorest P B, Deng X, Dolch T, Ellis J A, Ferdman R D, Garver-Daniels N, Jenet F, JonesG, Kaspi V M, Koop M, Lam M T, Lazio T J W, Lommen A N, Lorimer D R, Luo J, LynchR S, Madison D R, McLaughlin M A, McWilliams S T, Nice D J, Palliyaguru N, PennucciT T, Ransom S M, Sesana A, Siemens X, Stairs I H, Stinebring D R, Stovall K, SwiggumJ, Vallisneri M, van Haasteren R, Wang Y, Zhu W W and NANOGrav Collaboration 2014
Astrophysical Journal
141 (
Preprint )[7] Zhu X J, Hobbs G, Wen L, Coles W A, Wang J B, Shannon R M, Manchester R N, Bailes M,Bhat N D R, Burke-Spolaor S, Dai S, Keith M J, Kerr M, Levin Y, Madison D R, Os lowskiS, Ravi V, Toomey L and van Straten W 2014
Monthly Notices of the Royal AstronomicalSociety
Preprint )[8] Manchester R N, Hobbs G, Bailes M, Coles W A, van Straten W, Keith M J, ShannonR M, Bhat N D R, Brown A, Burke-Spolaor S G, Champion D J, Chaudhary A, EdwardsR T, Hampson G, Hotan A W, Jameson A, Jenet F A, Kesteven M J, Khoo J, Kocz J,Maciesiak K, Oslowski S, Ravi V, Reynolds J R, Sarkissian J M, Verbiest J P W, Wen Z L,Wilson W E, Yardley D, Yan W M and You X P 2013
Publications of the AstronomicalSociety of Australia e017 ( Preprint )[9] Ferdman R D, van Haasteren R, Bassa C G, Burgay M, Cognard I, Corongiu A, D’AmicoN, Desvignes G, Hessels J W T, Janssen G H, Jessner A, Jordan C, Karuppusamy R, KeaneE F, Kramer M, Lazaridis K, Levin Y, Lyne A G, Pilia M, Possenti A, Purver M, StappersB, Sanidas S, Smits R and Theureau G 2010
Classical and Quantum Gravity Preprint )[10] Hobbs G, Archibald A, Arzoumanian Z, Backer D, Bailes M, Bhat N D R, Burgay M,Burke-Spolaor S, Champion D, Cognard I, Coles W, Cordes J, Demorest P, Desvignes G,Ferdman R D, Finn L, Freire P, Gonzalez M, Hessels J, Hotan A, Janssen G, Jenet F,Jessner A, Jordan C, Kaspi V, Kramer M, Kondratiev V, Lazio J, Lazaridis K, Lee K J,Levin Y, Lommen A, Lorimer D, Lynch R, Lyne A, Manchester R, McLaughlin M, Nice D,Oslowski S, Pilia M, Possenti A, Purver M, Ransom S, Reynolds J, Sanidas S, SarkissianJ, Sesana A, Shannon R, Siemens X, Stairs I, Stappers B, Stinebring D, Theureau G, vanHaasteren R, van Straten W, Verbiest J P W, Yardley D R B and You X P 2010
Classicaland Quantum Gravity Preprint )11] Manchester R N and IPTA 2013
Classical and Quantum Gravity Preprint )[12] McLaughlin M A 2014
ArXiv e-prints ( Preprint )[13] Abbott B P, Abbott R, Adhikari R, Ajith P, Allen B, Allen G, Amin R S, Anderson S B,Anderson W G, Arain M A and et al 2009
Reports on Progress in Physics Preprint )[14] Accadia T, Acernese F, Antonucci F and et al 2011
Classical and Quantum Gravity ArXiv e-prints ( Preprint )[16] Lorimer D R and Kramer M 2004
Handbook of Pulsar Astronomy [17] Rickett B J 1977
Annual review of astronomy and astrophysics Annual review of astronomy and astrophysics Astrophysical Journal
Preprint )[20] Dolch T, Lam M T, Cordes J, Chatterjee S, Bassa C, Bhattacharyya B, Champion D J,Cognard I, Crowter K, Demorest P B, Hessels J W T, Janssen G, Jenet F A, Jones G, JordanC, Karuppusamy R, Keith M, Kondratiev V, Kramer M, Lazarus P, Lazio T J W, Lee K J,McLaughlin M A, Roy J, Shannon R M, Stairs I, Stovall K, Verbiest J P W, MadisonD R, Palliyaguru N, Perrodin D, Ransom S, Stappers B, Zhu W W, Dai S, Desvignes G,Guillemot L, Liu K, Lyne A, Perera B B P, Petroff E, Rankin J M and Smits R 2014
Astrophysical Journal
21 (
Preprint )[21] Shannon R M, Os lowski S, Dai S, Bailes M, Hobbs G, Manchester R N, van Straten W,Raithel C A, Ravi V, Toomey L, Bhat N D R, Burke-Spolaor S, Coles W A, Keith M J,Kerr M, Levin Y, Sarkissian J M, Wang J B, Wen L and Zhu X J 2014
Monthly Notices ofthe Royal Astronomical Society
Preprint )[22] Edwards R T, Hobbs G B and Manchester R N 2006
Monthly Notices of the RoyalAstronomical Society
Preprint astro-ph/0607664 )[23] Cordes J M and Shannon R M 2010
ArXiv e-prints ( Preprint )[24] Hellings R W and Downs G S 1983
Astrophysical Journal Letter
L39–L42[25] Wilson T L, Rohlfs K and H¨uttemeister S 2009
Tools of Radio Astronomy (Springer-Verlag)[26] Lommen A N and Demorest P 2013
Classical and Quantum Gravity Preprint )[27] Pennucci T T, Demorest P B and Ransom S M 2014
Astrophysical Journal
93 (
Preprint )[28] Liu K, Desvignes G, Cognard I, Stappers B W, Verbiest J P W, Lee K J, ChampionD J, Kramer M, Freire P C C and Karuppusamy R 2014
Monthly Notices of the RoyalAstronomical Society
Preprint )[29] The Proposer’s Guide for the Green Bank Telescope 2014[30] [31] [32] [33] [34] Nan R, Li D, Jin C, Wang Q, Zhu L, Zhu W, Zhang H, Yue Y and Qian L 2011
InternationalJournal of Modern Physics D Preprint )35] McLaughlin M A, Arzoumanian Z, Cordes J M, Backer D C, Lommen A N, Lorimer D Rand Zepka A F 2002
Astrophysical Journal
Preprint astro-ph/0106371 )[36] Rathnasree N and Rankin J M 1995
Astrophysical Journal
ArXive-prints ( Preprint )[38] Smits R, Kramer M, Stappers B, Lorimer D R, Cordes J and Faulkner A 2009
Astronomyand Astrophysics
Preprint )[39] Hobbs G, Lyne A G and Kramer M 2010
Monthly Notices of the Royal Astronomical Society
Preprint )[40] Jenet F A, Hobbs G B, van Straten W, Manchester R N, Bailes M, Verbiest J P W, EdwardsR T, Hotan A W, Sarkissian J M and Ord S M 2006
Astrophysical Journal
Preprint astro-ph/0609013 )[41] Babak S and Sesana A 2012
Phys. Rev. D Preprint )[42] Ellis J A, Siemens X and Creighton J D E 2012
Astrophysical Journal
175 (
Preprint )[43] Ellis J A 2013
Classical and Quantum Gravity Preprint )[44] Wang Y, Mohanty S D and Jenet F A 2014
Astrophysical Journal
96 (
Preprint )[45] Perrodin D, Jenet F, Lommen A, Finn L, Demorest P, Ferdman R, Gonzalez M, Nice D,Ransom S and Stairs I 2013
ArXiv e-prints ( Preprint )[46] Wang Y and et al 2015)[46] Wang Y and et al 2015