Selection effects in pulsar searches

3.1 Selection effects in pulsar searches

3.1.1 The inverse square law and survey thresholds

The most prominent selection effect is the inverse square law, i.e. for a given intrinsic luminosity² , the observed flux density varies inversely with the distance squared. This results in the observed sample being dominated by nearby and/or high luminosity objects. Beyond distances of a few kpc from the Sun, the apparent flux density falls below the detection thresholds $Smin$ of most surveys. Following [101 ], we express this threshold as follows:

where $S∕Nmin$ is the threshold signal-to-noise ratio, $η$ is a generic fudge factor ( $≲ 1$ ) which accounts for losses in sensitivity (e.g., due to sampling and digitization noise), $npol$ is the number of polarizations recorded (either 1 or 2), $Trec$ and $Tsky$ are the receiver and sky noise temperatures, $G$ is the gain of the antenna, $Δ ν$ is the observing bandwidth, $tint$ is the integration time, $W$ is the detected pulse width and $P$ is the pulse period.

3.1.2 Interstellar pulse dispersion and multipath scattering

It follows from Equation (3 ) that the minimum flux density increases as $W ∕(P − W )$ and hence $W$ increases. Also note that if $W ≳ P$ , the pulsed signal is smeared into the background emission and is no longer detectable, regardless of how luminous the source may be. The detected pulse width $W$ may be broader than the intrinsic value largely as a result of pulse dispersion and multipath scattering by free electrons in the interstellar medium. The dispersive smearing scales as $3 Δ ν∕ν$ , where $ν$ is the observing frequency. This can largely be removed by dividing the pass-band into a number of channels and applying successively longer time delays to higher frequency channels before summing over all channels to produce a sharp profile. This process is known as incoherent dedispersion.

The smearing across the individual frequency channels, however, still remains and becomes significant at high dispersions when searching for short-period pulsars. Multipath scattering from electron density irregularities results in a one-sided broadening of the pulse profile due to the delay in arrival times. A simple scattering model is shown in Figure 12 in which the scattering electrons are assumed to lie in a thin screen between the pulsar and the observer [325 ]. The timescale of this effect varies roughly as $−4 ν$ , which can not currently be removed by instrumental means.

Figure 12: Left panel: Pulse scattering caused by irregularities in the interstellar medium. The different path lengths and travel times of the scattered rays result in a “scattering tail” in the observed pulse profile which lowers its signal-to-noise ratio. Right panel: A simulation showing the percentage of Galactic pulsars that are likely to be undetectable due to scattering as a function of observing frequency. Low-frequency ( $≲$ 1 GHz) surveys clearly miss a large percentage of the population due to this effect.

Dispersion and scattering are most severe for distant pulsars in the inner Galaxy where the number of free electrons along the line of sight becomes large. The strong frequency dependence of both effects means that they are considerably less of a problem for surveys at observing frequencies $≳$ 1.4 GHz [79 , 171 ] compared to the 400-MHz search frequency used in early surveys. An added bonus for such observations is the reduction in $Tsky$ which scales with frequency as approximately $−2.8 ν$ [210 ]. Pulsar intensities also have an inverse frequency dependence, with the average scaling being $ν− 1.6$ [237 ], so that flux densities are roughly an order of magnitude lower at 1.4 GHz compared to 400 MHz. Fortunately, this can be at least partially compensated for by the use of larger receiver bandwidths at higher radio frequencies. For example, the 1.4-GHz system at Parkes has a bandwidth of 288 MHz [243 ] compared to the 430-MHz system, where nominally 32 MHz is available [254 ].

3.1.3 Orbital acceleration

Standard pulsar searches use Fourier techniques [229 ] to search for a priori unknown periodic signals and usually assume that the apparent pulse period remains constant throughout the observation. For searches with integration times much greater than a few minutes, this assumption is only valid for solitary pulsars or binary systems with orbital periods longer than about a day. For shorter-period binary systems, the Doppler-shifting of the period results in a spreading of the signal power over a number of frequency bins in the Fourier domain, leading to a reduction in S/N [165 ]. An observer will perceive the frequency of a pulsar to shift by an amount $aT∕(P c)$ , where $a$ is the (assumed constant) line-of-sight acceleration during the observation of length $T$ , $P$ is the (constant) pulsar period in its rest frame and $c$ is the speed of light. Given that the width of a frequency bin in the Fourier domain is $1∕T$ , we see that the signal will drift into more than one spectral bin if $aT 2∕(P c) > 1$ . Survey sensitivities to rapidly-spinning pulsars in tight orbits are therefore significantly compromised when the integration times are large.

Figure 13: Left panel: A 22.5-min Arecibo observation of the binary pulsar B1913+16. The assumption that the pulsar has a constant period during this time is clearly inappropriate given the quadratic drifting in phase of the pulse during the observation (linear grey scale plot). Right panel: The same observation after applying an acceleration search. This shows the effective recovery of the pulse shape and a significant improvement in the signal-to-noise ratio.

As an example of this effect, as seen in the time domain, Figure 13 shows a 22.5-min search mode observation of the binary pulsar B1913+16 [154 , 359 , 360 ]. Although this observation covers only about 5% of the orbit (7.75 hr), the severe effects of the Doppler smearing on the pulse signal are very apparent. While the standard search code nominally detects the pulsar with S/N = 9.5 for this observation, it is clear that this value is significantly reduced due to the Doppler shifting of the pulse period seen in the individual sub-integrations.

It is clearly desirable to employ a technique to recover the loss in sensitivity due to Doppler smearing. One such technique, the so-called “acceleration search” [263 ], assumes the pulsar has a constant acceleration during the observation. Each time series can then be re-sampled to refer it to the frame of an inertial observer using the Doppler formula to relate a time interval $τ$ in the pulsar frame to that in the observed frame at time $t$ , as $τ(t) ∝ (1 + at∕c)$ . Searching over a range of accelerations is desirable to find the time series for which the trial acceleration most closely matches the true value. In the ideal case, a time series is produced with a signal of constant period for which full sensitivity is recovered (see right panel of Figure 13 ). This technique was first used to find PSR B2127+11C [6 ], a double neutron star binary in M15 which has parameters similar to B1913+16. Its application to 47 Tucanae [57 ] resulted in the discovery of nine binary millisecond pulsars, including one in a 96-min orbit around a low-mass ( $0.15 M ⊙$ ) companion. This is currently the shortest binary period for any known radio pulsar. The majority of binary millisecond pulsars with orbital periods less than a day found in recent globular cluster searches would not have been discovered without the use of acceleration searches.

For intermediate orbital periods, in the range 30 min – several hours, another promising technique is the dynamic power spectrum search shown in Figure 14 . Here the time series is split into a number of smaller contiguous segments which are Fourier-transformed separately. The individual spectra are displayed as a two-dimensional (frequency versus time) image. Orbitally modulated pulsar signals appear as sinusoidal signals in this plane as shown in Figure 14 .

Figure 14: Dynamic power spectra showing two recent pulsar discoveries in the globular cluster M62 showing fluctuation frequency as a function of time. Figure provided by Adam Chandler.

This technique has been used by various groups where spectra are inspected visually [248 ]. Much of the human intervention can be removed using a hierarchical scheme for selecting significant events [73 ]. This approach was recently applied to a search of the globular cluster M62 resulting in the discovery of three new pulsars. One of the new discoveries – M62F, a faint 2.3-ms pulsar in a 4.8-hr orbit – was detectable only using the dynamic power spectrum technique.

For the shortest orbital periods, the assumption of a constant acceleration during the observation clearly breaks down. In this case, a particularly efficient algorithm has been developed [82 , 304 , 175 , 307 ] which is optimised to finding binaries with periods so short that many orbits can take place during an observation. This “phase modulation” technique exploits the fact that the Fourier components are modulated by the orbit to create a family of periodic sidebands around the nominal spin frequency of the pulsar. While this technique has so far not resulted in any new discoveries, the existence of short period binaries in 47 Tucanae [57 ], Terzan 5 [309 ] and the 11-min X-ray binary X1820–303 in NGC 6624 [348 ], suggests that there are more ultra-compact radio binary pulsars that await discovery.