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Paul D. Lasky, Andrew Melatos, Vikram Ravi, George Hobbs, Pulsar timing noise and the minimum observation time to detect gravitational waves with pulsar timing arrays, Monthly Notices of the Royal Astronomical Society, Volume 449, Issue 3, 21 May 2015, Pages 3293–3300, https://doi.org/10.1093/mnras/stv540
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Abstract
The sensitivity of pulsar timing arrays to gravitational waves is, at some level, limited by timing noise. Red timing noise – the stochastic wandering of pulse arrival times with a red spectrum – is prevalent in slow-spinning pulsars and has been identified in many millisecond pulsars. Phenomenological models of timing noise, such as from superfluid turbulence, suggest that the timing noise spectrum plateaus below some critical frequency, fc, potentially aiding the hunt for gravitational waves. We examine this effect for individual pulsars by calculating minimum observation times, Tmin(fc), over which the gravitational wave signal becomes larger than the timing noise plateau. We do this in two ways: (1) in a model-independent manner, and (2) by using the superfluid turbulence model for timing noise as an example to illustrate how neutron star parameters can be constrained. We show that the superfluid turbulence model can reproduce the data qualitatively from a number of pulsars observed as part of the Parkes Pulsar Timing Array. We further show how a value of fc, derived either through observations or theory, can be related to Tmin. This provides a diagnostic whereby the usefulness of timing array pulsars for gravitational-wave detection can be quantified.
1 INTRODUCTION
Pulsar timing arrays (PTAs; e.g. Kramer & Champion 2013; McLaughlin 2013; Manchester et al. 2013) seek to detect nanohertz gravitational waves from cosmological and extragalactic sources by looking for correlations between contemporaneously measured pulse arrival times from multiple radio pulsars (Hellings & Downs 1983). The sensitivity of a PTA is limited by pulsar timing noise, i.e. stochastic wandering of pulse arrival times. External noise sources include interstellar plasma turbulence, jitter noise and errors in terrestrial time standards; see Cordes (2013) for a description of all dominant noise sources and an estimate of their magnitudes. Intrinsic noise sources have been attributed to microglitches (Cordes & Downs 1985; D'Alessandro et al. 1995; Melatos, Peralta & Wyithe 2008), post-glitch recovery (Johnston & Galloway 1999), magnetospheric state switching (e.g. Kramer et al. 2006; Lyne et al. 2010), fluctuations in the spin-down torque (Cheng 1987a,b; Urama, Link & Weisberg 2006), variable coupling between the crust and core or pinned and corotating regions (Alpar, Nandkumar & Pines 1986; Jones 1990), asteroid belts (Shannon et al. 2013b) and superfluid turbulence (Greenstein 1970; Link 2012; Melatos & Link 2014). Analyses of long-term millisecond pulsar timing data indicate that timing noise power spectra are typically white above some frequency and red below it (Kaspi, Taylor & Ryba 1994; Shannon & Cordes 2010; van Haasteren et al. 2011; Shannon et al. 2013a).
Red timing noise power spectra cannot extend to arbitrarily low frequencies, as the infinite integrated noise-power implies divergent phase residuals and hence (if phase residuals arise from torque fluctuations) unphysical pulsar angular velocities. One therefore expects the spectrum to plateau, or even become blue, below some turn-over frequency fc. A number of physical models naturally predict low-frequency plateaus, including superfluid turbulence (Melatos & Link 2014) and asteroid belts (Shannon et al. 2013b). We discuss the former in detail below. A low-frequency plateau enhances prospects for the detection of a stochastic gravitational wave background. As the gravitational wave spectrum is a steep power law for most cosmological sources (e.g. Maggiore 2000; Phinney 2001; Grishchuk 2005), it rises above the plateau below some frequency as long as it too does not have a low-frequency cut-off (e.g. Wyithe & Loeb 2003; Sesana 2013a; Ravi et al. 2014; McWilliams, Ostriker & Pretorius 2014, and discussion below).
In this paper, we quantify how a low-frequency timing noise plateau affects the direct detection of gravitational waves with PTAs. Specifically, we calculate the minimum observation time for any individual pulsar to become sensitive to gravitational wave stochastic backgrounds from binary supermassive black holes (SMBHs) and cosmic strings. We note this minimum observation time is only an indicative quantity for determining when a gravitational wave signal will dominate the timing residuals for an individual pulsar; it does not account for algorithms that correlate noise properties between pulsars, a point we discuss in more detail throughout. We do this in two ways, first by parametrizing the timing noise in a model independent way, and secondly by applying the superfluid turbulence model of Melatos & Link (2014). In the first approach, we express this minimum time in terms of three pulsar observables; the amplitude and spectral index of the timing noise power spectral density and the turn-over frequency. The second approach is included as an example of how to relate PTA observables to neutron star internal properties in the context of one particular physical model with only two free parameters. It does not imply any theoretical preference for the superfluid turbulence model, and will be extended to other physical models in the future.
The paper is set out as follows. In Section 2, we define a phenomenological model for timing noise, and review predictions for the power spectral density of the phase residuals induced by gravitational waves from SMBHs and cosmic strings. In Section 3, we calculate minimum observation times for hypothetical pulsars as a function of their timing noise spectral index, normalization, and turn-over frequency. In Section 4, we apply the superfluid turbulence model to data and extract ‘by-eye’ parameter estimates for various pulsars in the Parkes Pulsar Timing Array (PPTA). We then determine criteria for selecting ‘optimal’ pulsars in Section 5 and conclude in Section 6.
2 POWER SPECTRUM OF THE PHASE RESIDUALS
2.1 Timing noise
Equation (5) provides a quantitative method for determining when the gravitational wave signal will dominate the timing residual power spectrum. We emphasize that this is only an indicative threshold for detection; it is not a substitute for a careful signal-to-noise estimate given desired false alarm and false dismissal rates. Cross-correlation search algorithms look simultaneously at a range in f (e.g. Hellings & Downs 1983; Jenet et al. 2005; Anholm et al. 2009; van Haasteren et al. 2009). For example, our definition (5) is equivalent to the boundary between the ‘weak signal limit’ and the ‘intermediate regime’ as defined in Siemens et al. (2013). While Siemens et al. (2013) calculate a scaling of gravitational wave detection significance with time assuming only white timing noise, they also perform simulations with red noise assuming q = −3. A future research project is therefore to introduce red noise with and without a low-frequency turn-over into the analytic calculations of Siemens et al. (2013).
It is likely that the near future will see an increasing number of PTA pulsars satisfy the condition ΦGW(f) > ΦTN(f), and that this will occur before a statistically significant detection is announced. Equation (5) and the analysis presented in this paper therefore provide an important input into the time-scale on which this condition will be met by individual pulsars, as a prelude to a cross-correlation detection strategy.
2.2 Cosmological gravitational wave background
PTAs are sensitive to gravitational wave backgrounds generated by two cosmological sources1: binary SMBHs and vibrations from cosmic strings.
2.2.1 Supermassive binary black holes
Source . | |$A_{\rm GW}^{\rm min}$| . | |$A_{\rm GW}^{\rm max}$| . | α . |
---|---|---|---|
SMBHs (Sesana 2013b, 68 per cent) | 3.5 × 10−16 | 1.5 × 10−15 | −2/3 |
SMBHs (Ravi et al. 2015, 95 per cent) | 5.1 × 10−16 | 2.4 × 10−15 | −2/3 |
Cosmic strings | 10−16 | 10−15 | −1 ≲ α ≲ −0.8 |
Source . | |$A_{\rm GW}^{\rm min}$| . | |$A_{\rm GW}^{\rm max}$| . | α . |
---|---|---|---|
SMBHs (Sesana 2013b, 68 per cent) | 3.5 × 10−16 | 1.5 × 10−15 | −2/3 |
SMBHs (Ravi et al. 2015, 95 per cent) | 5.1 × 10−16 | 2.4 × 10−15 | −2/3 |
Cosmic strings | 10−16 | 10−15 | −1 ≲ α ≲ −0.8 |
2.2.2 Cosmic strings
Cosmic strings are topological defects that may form in phase transitions in the early Universe and produce strong bursts of gravitational radiation, which may be detectable in PTAs (Damour & Vilenkin 2000, 2001, 2005). A cosmic string-induced stochastic background of gravitational waves is characterized by three-dimensionless parameters: the string tension, Gμ, the reconnection probability, p, and a parameter, ϵ, related to the size of loops. The best quoted limit of Gμ ≲ 1.2 × 10−8 is derived from PTA limits of the stochastic gravitational wave background (van Haasteren et al. 2011, 2012), although a more stringent constraint (still to be computed), is possible with existing data sets [see Sanidas, Battye & Stappers (2013) for projected constraints in the near future]. Combined observations using the ground-based Laser Interferometer Gravitational Wave Observatory (LIGO) and Virgo constrain the ϵ–Gμ plane to be 7 × 10−9 < Gμ < 1.5 × 10−7 and ϵ < 8 × 10−11 (Abbott et al. 2009; Aasi et al. 2014). Limits on Gμ are model dependent; the reconnection probability is inversely proportional to ΦGW(f), and smaller values of ϵ increase the minimum gravitational wave frequency emitted. This can take the maximum of the stochastic background out of the sensitivity band for PTAs (e.g. Siemens, Mandic & Creighton 2007; Ölmez, Mandic & Siemens 2010).
Despite the above caveats, a power-law model for the characteristic strain spectrum from cosmic strings given by equation (6) with −1 ≲ α ≲ 0.8 is a good approximation for the PTA frequency band (Maggiore 2000). The predicted range for AGW is quoted in Table 1.
3 MODEL INDEPENDENT MINIMUM OBSERVATION TIME
To attain adequate sensitivity to gravitational waves at a frequency, f, in the phase residuals of an individual pulsar, we must have ΦGW(f) > ΦTN(f) for that pulsar, subject to the caveats regarding specific data analysis algorithms expressed in the text following equation (5). If equation (6) applies across all relevant frequencies, and ΦTN(f) turns over below fc, then ΦGW(f) > ΦTN(f) is always satisfied for some |$f=T_{\rm min}^{-1}$|, as in equation (5).
In Fig. 1, we plot ΦTN and ΦGW as functions of f. The coloured shaded regions and the region enclosed by the black dotted curves in the left-hand plot contain all the ΦGW curves in the parameter range in Table 1 for SMBHs. The blue shaded region is the 95 per cent confidence interval from Ravi et al. (2015) as described in Table 1. The black dotted curves enclose the 68 per cent confidence interval from Sesana (2013b). The shaded grey region is the predicted range from Ravi et al. (2014) that includes low-frequency-whitening of ΦGW(f) due to non-circular binaries. In the right-hand plot, the green shaded region is the parameter space enclosed by the cosmic string predictions from Table 1 with α = −1. The dotted black curves are specific, representative calculations of the cosmic string background with p = ϵ = 1 and Gμ = 1.2 × 10−8 (top curve) and 1.0 × 10−10 (bottom curve)2. The black, red and purple curves in each panel are indicative examples of pulsar timing noise as described by equation (3), with values of fc, ATN, q and AW given in the caption to Fig. 1. The correspondingly coloured dashed curves extrapolate backwards the power-law scaling (equivalently assuming fc → 0). Finally, the dot labelled ‘PPTA’ in both panels marks the lowest limit on the stochastic gravitational wave background from Shannon et al. (2013a).
Fig. 1 illustrates the principal idea of this paper. If ΦTN is a simple power law without a low-frequency turn-over, and for moderate values of q, timing noise masks the gravitational wave background down to low frequencies, and Tmin is correspondingly long (we quantify this below). A turn-over in ΦGW(f) at some fc is therefore critical for practical PTA experiments with any millisecond pulsar that exhibits a steep timing noise spectrum with q ≳ 2. The low-frequency plateau in ΦGW from elliptical binary SMBHs (the grey shaded region in the left-hand panel of Fig. 1) makes the need for a turn-over in ΦTN(f) even more acute.
In Fig. 2, we plot the minimum observation time, Tmin, defined by equation (5), as a function of the asymptotic (high-f) timing noise spectral index, q, for a hypothetical pulsar with P = 10 ms and various values of fc and ATN in each panel. The shaded regions and dotted black curves delineate the ranges of Tmin(q) for binary SMBHs and cosmic strings, following the same colour scheme as in Fig. 1 and as detailed in the caption of Fig. 2. The coloured dashed curves give the limits on Tmin(q) if ΦTN does not turn-over (i.e. fc → 0). The horizontal dashed black line marks the PPTA observing time of 11.3 yr used for the lowest limit on the stochastic background published to date (Shannon et al. 2013a).
To help interpret Figs 1 and 2, consider a hypothetical pulsar with ATN = 10−8 yr (i.e. the two left-hand panels) and q = 2. If the timing noise spectral density turns over at fc = 1/5 yr−1 or 1/20 yr−1, the minimum observation time given the most optimistic scenario from Ravi et al. (2015) is Tmin = 20.5 yr or 37.2 yr, respectively. On the other hand, if ΦTN(f) does not turn over, then the dashed blue curves show that the pulsar is insensitive to a gravitational wave signal until Tmin = 70.7 yr. The effect of a plateau in ΦTN(f) is therefore quite striking. Pulsars without a plateau and q ≳ 3 (depending less sensitively on ATN) are relatively inferior as a tool for detecting gravitational waves.
4 TIMING NOISE FROM SUPERFLUID TURBULENCE: A WORKED EXAMPLE
In Section 3, the description of timing noise is model independent, in the sense that ΦTN is parametrized phenomenologically by equation (3), without reference to a specific underlying, physical model. In this section, we repeat the analysis in Section 3 for the timing noise model of Melatos & Link (2014) and Melatos, Link & Lasky (in preparation), which attributes the fluctuating phase residuals to shear-driven turbulence in the interior of the neutron star. We emphasize that we do not express any theoretical preference for this model ahead of other models in the literature (see Section 1). We focus on it here only because (i) it is predictive, (ii) its results can be expressed in compact, analytic form and, (iii) the theoretical formula for ΦTN(f) depends on just three internal neutron star parameters, so it is easy to infer constraints on these parameters by combining the model with data.
The value of the exponent, q, in equation (8) depends on the form of the superfluid velocity two-point decorrelation function. Melatos & Peralta (2010) executed a first attempt to calculate the velocity correlation function numerically on the basis of Hall–Vinen–Bekarevich–Khalatnikov superfluid simulations (Peralta et al. 2008), but it is not well understood for terrestrial turbulence experiments, let alone for a neutron star interior, especially when stratification plays a role (e.g. Lasky, Bennett & Melatos 2013, and references therein). An empirical choice is therefore made that reproduces the asymptotic power-law dependence from timing noise data, i.e. ΦTN ∝ f−q as f → ∞ [for details see Melatos & Link (2014); Melatos et al., (in preparation)]. We emphasize equation (8) is not a unique choice, nor can it be inverted uniquely to infer the underlying velocity correlation function (Melatos et al., in preparation).
In Fig. 3, we show four examples of millisecond pulsar phase residual power spectra measured by the PPTA (Manchester et al. 2013). Overplotted on the data are reasonable ‘by-eye’ fits generated by the superfluid turbulence model for q = 2, 4 and 6. The fits are neither unique nor optimal (e.g. in a least-squares sense), but they are representative. It is outside the scope of this paper to extract detailed fits and values for λ, η(R−1), and q for each pulsar3. We simply note that a broad range of parameters fit the phase residuals for any given pulsar. The pulsars shown in Fig. 3 have been chosen as they appear to have moderate to high levels of timing noise, cf. other PPTA pulsars. All exhibit a relatively red spectrum. In the context of superfluid turbulence, they imply fc ≳ 10−2 yr−1, so that the plateau is potentially observable in the not-too-distant future4.
In Fig. 4, we plot two further examples of millisecond pulsar phase residuals. These objects exhibit the lowest level of timing noise in the PPTA sample. For the superfluid turbulence model to remain consistent with these data, the objects must have long decorrelation time-scales, i.e. fc ≲ 102 yr−1. The data show the white noise component, AW, and the turbulence-driven red-component sits below AW. Under these circumstances, the turnover in ΦTN(f) occurs too low in frequency to be observed, and the main factor limiting PTA detection is AW.
5 OPTIMAL PULSARS
What pulsars are best placed to detect a gravitational wave background, given the longest time one is prepared to wait? In Fig. 5, we plot 1/fc against ATN, for different values of q and Tmin in each panel. The left-hand vertical axis displays the results for the model-independent form of ΦTN in equation (3). The right-hand vertical axis registers the decorrelation time 1/η(R−1), in the superfluid turbulence model in Section 4. The dashed grey curves are curves of constant λ. Overplotted are the superfluid turbulence model ‘fits’ to the PPTA pulsar data in Fig. 3, where the open circles, filled circles, open squares and filled squares are PSRs J1024−0719, J1643−1224, J1824−2452A and J1939−2134, respectively.
Fig. 5 allows us to ask whether, for example, 20 yr of timing a specific pulsar will allow for sensitivity to the most optimistic SMBH gravitational wave strain of AGW = 2.4 × 10−15. In the middle set of panels, the latter strain limit appears as the rightmost boundary of the blue shaded region. A pulsar with timing noise below this curve is sensitive to a gravitational wave signal in Tobs ≤ 20 yr. Sensitivity depends on q as illustrated in the three different panels running vertically. It also depends on fc. For example, a hypothetical pulsar with q = 4 and ATN ≈ 10−11 yr is only sensitive to a gravitational wave background for 1/fc ≲ 12 yr. This is an interesting constraint: a pulsar in a PTA that tolerates Tmin ≤ 20 yr is sensitive to a gravitational wave background if ΦTN(f) exhibits a plateau after ≲ 12 yr of timing.
The superfluid turbulence model fits from Fig. 3 give an indication as to the usefulness of individual pulsars from the PPTA data set. For example, consider PSR J1939−2134 (filled squares). If one again tolerates Tmin ≤ 20 yr, the fits imply a pulsar is sensitive to a conservative prediction for the gravitational wave background for q ≲ 4, although for q ≈ 4 this requires the timing noise spectrum to plateau after approximately 15 yr of timing. We emphasize again that the model fits should only be taken as indicative; careful and detailed analysis is required to extract the true timing noise signal parameters from the data.
6 CONCLUSION
PTA limits on the cosmological gravitational wave background are continually dropping to the point where they usefully constrain galaxy formation models (Shannon et al. 2013a). Positive detections, on the other hand, require a cross-correlation algorithm to simultaneously analyse timing residuals from multiple pulsars. Such a detection will likely occur when the gravitational wave background is the largest component in the unmodelled portion of many individual pulsar's timing residuals (Siemens et al. 2013). If the timing noise spectrum is steeper asymptotically (at high f) than the gravitational wave spectrum, this is only possible if the timing noise spectrum flattens below some frequency, fc. In this paper, we calculate the minimum observation time required, given fc, before the gravitational wave background rises above the timing noise plateau in any specific pulsar. We calculate this minimum observation time both in a model-independent way, and for timing noise arising from superfluid turbulence. The latter model is selected not because it is necessarily preferred physically, but because it is simple, predictive and analytically tractable and therefore provides a test-bed for repeating the calculation with other physical models in the future.
Our results rely on the timing noise spectrum whitening below some threshold frequency, fc. This provides an observational diagnostic that can be used to infer the effectiveness of an individual pulsar in a PTA. If, upon observing a pulsar for some T > 1/fc, one finds that ΦTN(f) has not whitened below fc, that pulsar's capacity for assisting usefully in the detection of a gravitational wave background is severely diminished. The fc for a given pulsar is a function of the rotational parameters of the pulsar, and the gravitational wave amplitude and spectral index. Therefore, using the prescription outlined in this paper, one can predict fc for a given pulsar and a given gravitational wave background.
In reality, measuring fc in a single pulsar is difficult. First, the noise in a given pulsar timing power spectrum is large, and secondly, the power in the lowest frequency bin is generally dominated by the fact that a quadratic polynomial is fit to the timing residuals [see van Haasteren & Levin (2013)]. These two effects potentially mimic a low-frequency turn-over, implying multiple low-frequency bins are required to confirm the existence of a low-frequency cut-off.
Many data analysis algorithms simultaneously fit the timing model and the unknown noise contributions for any individual pulsar. In this sense, one can include a low-frequency plateau into gravitational-wave detection algorithms, e.g. by way of a Bayesian prior on the form of the power spectral density. Physically motivated models for timing noise, such as the superfluid turbulence model discussed herein, could be used to guide such priors.
We are grateful to the anonymous reviewer for the thoughtful and thorough review of the manuscript. PDL and AM are supported by Australian Research Council (ARC) Discovery Project DP110103347. PDL is also supported by ARC DP140102578. VR is a recipient of a John Stocker Postgraduate Scholarship from the Science and Industry Endowment Fund. We thank Yuri Levin for comments on the manuscript and Ryan Shannon for comments on an earlier version. Calculations of the cosmic string stochastic background used the GWPlotter website: http://homepages.spa.umn.edu/∼gwplotter.
Calculations used the GWPlotter website: http://homepages.spa.umn.edu/∼gwplotter.
The amplitude and spectral index of red-noise in pulsar timing residuals are highly covariant, especially when only the lowest few frequency bins show evidence for red noise (e.g. van Haasteren et al. 2009; van Haasteren & Levin 2013). Finding best-fitting parameters for the superfluid turbulence model is therefore a non-trivial task that will be the subject of future work.
We note that PSR J1824−2452A resides in a globular cluster (Lyne et al. 1987), implying most of the timing noise is likely a result of motions within that cluster rather than superfluid turbulence. The curves shown in Fig. 3 therefore represent an upper limit on the contribution from superfluid turbulence.