ARECIBO PULSAR SURVEY USING ALFA. III. PRECURSOR SURVEY AND POPULATION SYNTHESIS

Our current knowledge of the non-recycled (hereafter normal) pulsar and millisecond pulsar (MSP) Galactic populations²¹—their spatial, period, and luminosity distributions—primarily comes from the results of the Parkes Multibeam Pulsar Survey (PMPS; Manchester et al. 2001; Morris et al. 2002; Kramer et al. 2003; Hobbs et al. 2004; Faulkner et al. 2004; Lorimer et al. 2006). Analyses of these results have shown that the Galactic normal pulsar population is made up of 30,000 ± 1100 sources beaming toward Earth with luminosities above 0.1 mJy kpc²; their radial density profile is best described by a gamma function and their distance from the Galactic plane, by an exponential function with a scale height of 0.33 kpc (Lorimer et al. 2006). A more physically realistic treatment of pulsar luminosities involves using a log-normal luminosity function, which is demonstrated from pulsar population syntheses (e.g., Faucher-Giguère & Kaspi 2006). The advantage of this approach is that it allows predictions of the total normal pulsar population size—not just the number above a certain luminosity cutoff; Faucher-Giguère & Kaspi (2006) predict that there are 120,000 ± 20,000 detectable, normal pulsars in the Galaxy.

Since there are only ∼10% as many known MSPs as normal pulsars (Manchester et al. 2005), we do not have the same level of knowledge about recycled pulsars' population parameters. The High Time Resolution Universe (HTRU) Survey (Keith et al. 2010) has added more normal pulsar discoveries to the PMPS haul and many MSPs as well (e.g., Bates et al. 2011; Burgay et al. 2013; Morris et al. 2002; Hobbs et al. 2004; Mickaliger et al. 2012). Recent analysis of the intermediate latitude portion of HTRU MSP detections by Levin et al. (2013) uses a scale factor method (Vivekanand & Narayan 1981; Lorimer et al. 1993) and 50 detected MSPs to place a lower limit of 30,000 ± 7000 on the Galactic MSP population size (considering sources whose luminosities exceed 0.2 mJy kpc²). The scale height of the MSP population is fairly well established to be 500 pc (Lorimer 2005; Cordes & Chernoff 1997), but the spatial, period and luminosity functions are currently less well understood. Although many models can be ruled out, plausible MSP populations with a variety of underlying distributions are consistent with the observed sample (Lorimer 2010).

Despite the fact that Arecibo's latitude does not permit observations close to Galactic center like those at Parkes, the unique combination of Arecibo's sensitivity, paired with the high spectral resolution of its back-ends, provides a much deeper view through the Galaxy's dispersive medium, which often smears out signals from distant sources. Although the PMPS and HTRU surveys have sampled much of the sky surrounding the Galactic center—an area of the sky with high pulsar density—and have discovered over 1000 pulsars, Pulsar Arecibo L-band Feed Array (PALFA) provides a glimpse of the population density at larger Galactic radii (R > 5 kpc), which will help improve the spatial features of future pulsar population models. Arecibo's ability to reach competitive sensitivity limits with short integration times (one to five minutes) makes acceleration searches for binaries unnecessary for all but the most exotic systems. Finally, Arecibo's unrivaled sensitivity allows PALFA to probe the low-luminosity end of the Galactic pulsar population, leading to a better understanding of the underlying luminosity distribution.

With Arecibo's unique capabilities, PALFA has great potential to discover many normal pulsars as well as MSPs, thus improving our statistical picture of each population's characteristics. Given the number of discoveries by PMPS, it has historically been used to refine pulsar population modeling assumptions for normal pulsars. Recent efforts have been made to discover additional MSPs in archival PMPS data (Mickaliger et al. 2012) with motivation to improve MSP population models. With higher sensitivity to dispersed sources and MSPs, the PALFA survey's influence on normal and millisecond pulsar population studies will complement those of the PMPS and HTRU surveys. MSPs are essential for the direct detection of gravitational waves by pulsar timing array projects (e.g., Demorest et al. 2013). The best way to increase our sensitivity to the stochastic background is to add new MSPs to the array (Siemens et al. 2013).

In this paper, we present the detections and discoveries from the initial phase of the PALFA survey, hereafter referred to as the "precursor survey." In Section 2, we describe the PALFA precursor survey parameters and sky coverage and introduce two pipelines used to process the raw data. We present the 45 detections made by the precursor survey in Section 3 and include an evaluation of the survey's efficacy based on measured and theoretically calculated signal-to-noise (S/N) ratios. In Section 4 we discuss the portion of sky in the precursor survey that overlapped with the PMPS and show preliminary evidence that PALFA will indeed be probing more distant, fainter sources. Comparing population simulations to precursor survey detection statistics, we generate probability density functions (PDFs) for normal and millisecond pulsar populations in Section 5. These PDFs inform the predictions we make about the total number of pulsars (normal and MSP) we expect to have detected when the full PALFA survey is complete. We conclude in Section 6, stating the most probable normal and millisecond pulsar population sizes according to the precursor survey results.

The PALFA precursor survey covered portions of two Galactic sectors—an inner Galaxy region, 36° ≲ ℓ ≲ 75°, tiled with 865 pointings, and an outer Galaxy region, 170° ≲ ℓ ≲ 210°, covered by 919 pointings. All pointings were within one degree of the Galactic plane (|b| < 1°) and had dwell times of 134 and 67 s for inner- and outer-Galaxy regions respectively. The precursor survey used the Arecibo L-band Feed Array (ALFA) 7-beam receiver in conjunction with the Wideband Arecibo Pulsar Processor (WAPP) back-end (Dowd et al. 2000), which was set up to record 256 channels covering a 100 MHz bandwidth, centered at 1.42 GHz, every 64 μs. Each ALFA pointing includes seven distinct beam positions in a hexagonal pattern. As PALFA continues, the sky coverage will increase slightly in Galactic longitude (32° ≲ ℓ ≲ 77° and 168° ≲ ℓ ≲ 214°) and will extend to Galactic latitude ±5°. For the remainder of the paper, we will refer to this extended spatial coverage (accompanied by a three-fold increase in bandwidth) as the full PALFA survey. The precursor survey, optimized for maximum efficiency and sensitivity, used a "sparse sampling" technique described in detail in Cordes et al. (2006); gaps left by the precursor survey will be covered in multiple passes by the full PALFA survey. PMPS overlaps with the southernmost regions covered by Arecibo in the PALFA precursor survey, corresponding to 36° ≲ ℓ ≲ 50°. In Section 3, we will compare the performance of the two surveys in this overlap region to make a statement about the efficacy of the PALFA precursor survey.

Data from the PALFA precursor were previously analyzed in Cordes et al. (2006). That analysis used a quasi-real-time Quicklook pulsar search pipeline in which the data were decimated in time and frequency by factors of 8 and 16, respectively, yielding 32 spectral channels and 1024 μs time resolution. Using the decimated data, 11 pulsars were discovered and 29 previously known pulsars were detected. Timing and spectral characteristics from follow-up observations of the newly discovered pulsars are given in Nice et al. (2013).

We have analyzed these same data files at native full time- and frequency-resolution using the PALFA survey's Presto 1 pipeline. The full resolution search of the precursor survey data did not yield any pulsar discoveries (and in fact missed some sources flagged by the Quicklook pipeline), but revealed two more previously known normal pulsars (J1946+2611, B1924+16) and the bright MSP B1937+21. The Presto 1 zaplist, a list of frequencies and their harmonics related to known sources of radio frequency interference (RFI), may be responsible for this scant improvement over Quicklook results since it was fairly restrictive, "zapping" ∼8% of the spectral region between 0 and 10 Hz (∼84% of known pulsars have spin frequencies in this range). At least one previously known source, B1925+188, fell inside a zapped portion of the spectrum, but its fourth harmonic was still detectable in Presto 1 results. Four other sources that were detected by Quicklook (J1913+1000, B1919+14, J2002+30 and J2009+3326) were not detectable in Presto 1 results. Of the 12,488 PALFA precursor beams, 183 (1.5%) were not processed by the Presto 1 pipeline, including beams where J1913+1000 and B1919+14 should have been detected. PSRs J2002+30 and J2009+3326 were processed by Presto 1 and their spin frequencies were outside zapped portions of the spectrum; why these two sources were not detectable remains unknown, although it is plausible that harmonics of their true spin frequencies could have been "zapped," causing these sources to fall below a detectable threshold.

After the precursor survey was complete, raw data products were decimated to 4-bit resolution and saved in that form. In the process, some files were lost or corrupted (i.e., detection data files for J1913+1000, B1919+14 and B1924+16), so results from Cordes et al. (2006) were used when necessary. We used a complete list of precursor beam positions to determine minimum offset angles from each known source in the survey region, then refolded corresponding 4-bit data files, yielding two additional detections (J1906+0649 and J1924+1631). Table 1 outlines the means by which all sources in the PALFA precursor survey were detected.

2.1. Presto 1 Pipeline

2.2. Detection S/N Measurements

For all sources detected by the Quicklook and Presto 1 processing pipelines, we refolded data files from beam positions nearest those sources using known pulsar parameters and calculated measured signal-to-noise (S/N)_meas values. For each pulse profile, we used a simple algorithm to determine on- and off-pulse bins, then summed on-pulse intensities and divided by the maximum profile intensity to get an equivalent top-hat pulse width W_eq (in bins). Finally, (S/N)_meas is computed with

$$\begin{equation} ({\rm S/N})_{\rm meas} =\frac{1}{\sigma _{p}\sqrt{W_{\rm eq}}}\sum _{i=1}^{n_{\rm bins}} (p_i-\bar{p}), \end{equation} \tag{ 1 }$$

as in Lorimer & Kramer (2005), where $\bar{p}$ and σ_p are the mean and standard deviation of off-pulse intensities, respectively, p_i is the intensity of an individual profile bin and each profile had n_bins = 128. We divided W_eq by the number of bins in a profile n_bins to convert to duty cycle δ for each detection. Computed δ and (S/N)_meas values are listed in Table 1.

To measure the effectiveness of a pulsar survey, we look at the known sources that fall inside the survey region and compare the number of detections to the number of expected detections. Effectiveness will then be evaluated by whether the survey meets/exceeds expectations for detecting individual sources.

3.1. Defining Detectability

The PALFA multibeam receiver is composed of seven beams, each with an average FWHM of ∼335; adjacent beams are separated by ∼55, or ∼1.6 half-power beamwidths. Outer beams and the central beam have gains of 8.2 and 10.4 K Jy⁻¹ respectively (Cordes et al. 2006). Although previous population studies have modeled gain patterns using Gaussian functions (e.g., Lorimer et al. 2006), we use an Airy disk function to better model the additional gain from the side lobes of individual beams. Although this is not a perfect representation of the PALFA survey's true gain pattern—in fact, the side lobes of the outer ALFA beams are highly asymmetric (see Spitler et al. 2014, for a more precise model)—the Airy disk captures Arecibo's off-axis gain better than the Gaussian model and still provides the simplicity required to run population simulations quickly.

The theoretical signal-to-noise ratio (S/N)_th for a given pulsar with flux density (S₁₄₀₀) measured in mJy at 1400 MHz, spin period P, and pulse width W is given by

$$\begin{equation} ({\rm S/N})_{\rm th} =\frac{S_{1400} G\sqrt{n_p t_{\rm obs} \Delta f}}{\beta T_{\rm sys}}\sqrt{\frac{1-\delta }{\delta }}, \end{equation} \tag{ 2 }$$

where δ = W/P is the pulse duty cycle; G is the gain in K Jy⁻¹ of a specific beam, n_p = 2 is the number of summed polarizations, t_obs is the integration time (134 s and 67 s for inner- and outer-Galaxy observations, respectively), Δf = 100 MHz is the bandwidth, β = 1.16 is a correction factor that accounts for losses in the digitization process and T_sys is the system temperature measured in K (Dewey et al. 1985). Flux densities S₁₄₀₀ were obtained from the ATNF Pulsar Catalog (Manchester et al. 2005) for known pulsars and Nice et al. (2013) for pulsars discovered by the PALFA precursor survey. Equation (2) is an approximation since this treatment assumes top-hat pulse profiles and ignores the considerable variability in pulse shape. The majority of pulsars have Gaussian-shaped profiles, however, so this approximation works well in most cases.

Hereafter (S/N)_th will refer to theoretical signal-to-noise ratios, computed using Equation (2), while (S/N)_meas refers to signal-to-noise ratios measured from PALFA detections as described in Section 2.2 and specifically Equation (1).

Since gain is a function of a source's angular offset from the beam center, we model it as an Airy disk so that the gain

$$\begin{equation} G=G_0 \left (\frac{2 {\rm J_1}(k a \sin (\theta))}{k a \sin (\theta)}\right)^2, \end{equation} \tag{ 3 }$$

where J₁ is a Bessel function of the first kind, G₀ is the maximum on-axis gain of the beam, k = 2π/λ is the wavenumber (λ, the observation wavelength), a is the effective aperture radius (∼220 m), and θ is the angular offset of a source from the beam center, measured in radians. In predicting S/N for a given pulsar, the pulsed nature of its emission must be taken into account, as shown by the final term in Equation (2). For all pulsars that were detected in the precursor survey, we computed W_eq, then δ as described in Section 2.2. For sources that were not detected, we divide the pulse width at half maximum (W50), from the ATNF Pulsar Catalog (Manchester et al. 2005), by the period to compute δ, then (S/N)_th. Finally, T_sys includes the receiver temperature (T_rec = 24 K) and sky temperature (T_sky), which varies as a function of position and frequency as shown by Haslam et al. (1982). Since this sky temperature map describes T_sky at 408 MHz, we convert these values into 1.4 GHz sky temperatures using an assumed spectral index of α = 2.6, that is T_sky∝ν^−α.

Although there are many factors involved, we assume a 1:1 relationship between (S/N)_meas and (S/N)_th in order to use S/N as a prediction tool for the detectability of known sources. The true relationship between (S/N)_meas and (S/N)_th can be seen in Figure 1.

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Using a complete list of beam positions, we found the survey observations carried out closest to known pulsars in the precursor region (i.e., minimizing angular offset, θ). For each of these positions, we found the maximum expected gain for a given pulsar using Equation (3). Previously measured parameters for known pulsars allowed us to compute a theoretical signal-to-noise, (S/N)_th, as shown in Equation (2). We define a known pulsar to be detectable if we find (S/N)_th > 9 for that pulsar. A full list of pulsars detected by the precursor survey as well as those considered detectable due to their (S/N)_th values can be found in Table 1. Before PALFA began, there were 84 known pulsars positioned inside the target precursor survey region, although this sky area was not covered uniformly; 31 of 84 were deemed detectable, while 33 were actually detected, and seven had no previous flux measurements. Of the 51 non-detections, most can be attributed simply to the sources not being close to a PALFA precursor survey beam pointing, as the survey had only limited coverage in this region. Figure 2 shows the portion of the precursor survey that overlaps with the PMPS), an example of this limited coverage. Three of the 51 non-detections (B1910+10, J1913+1145, and B1911+11) were unexpected, since (S/N)_th > 9 for these sources; one of the 33 detections (B1929+20) was also unexpected, given its low (S/N)_th value. The non-detections could be due to a variety of factors—most likely RFI. Scintillation could have also suppressed the expected signal during precursor survey observations or boosted the signal during initial flux measurements. It is unlikely scintillation affected the detectability of J1913+1145, however, because of this source's high DM (637 pc cm⁻³). Given the short integration time near each of these sources (134 s), the pulse-to-pulse variability may have strongly affected (S/N)_meas since relatively few pulses were recorded. Also, because of the large error bars on (S/N)_th (∼30% fractional error) due to uncertainties in flux measurements, the sources may simply be weaker than expected.

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Although most sources with high (S/N)_th values were detected by the precursor survey's processing pipelines, five such sources were not. For each of these cases, we employed the same procedure as introduced in Section 2.2, using known periods and dispersion measures to dedisperse and fold the data from the closest pointing to each source. For the three sources mentioned earlier (B1910+10, B1911+11, and J1913+1145), no pulsations were detected; for the other two, J1906+0649 and J1924+1631, pulsations are evident, but relatively weak. PSR J1906+0649 was likely missed because of the RFI environment at Arecibo.

In addition to the 33 re-detected pulsars in the region, PSR J1924+1631 was discovered shortly after the precursor survey was completed, when the PALFA survey underwent an upgrade to a new backend with three times more bandwidth. This source was then retroactively found in precursor survey data with (S/N)_meas just above the detection threshold and has therefore been included in analysis that follows. Strong RFI present in the refolded precursor data explains why this source was not discovered earlier. P. Lazarus (2014, in preparation) will describe the most recent processing pipeline in detail, address the RFI environment and its effect on the PALFA survey's "true" sensitivity.

The PALFA precursor survey region overlaps the region covered by the PMPS in Galactic longitude, 36° ≲ ℓ ≲ 50°. Although there were 58 previously known pulsars in this longitude range and within ∼1° of the Galactic plane when the precursor survey took place (see Figure 2), we compare the PMPS and precursor survey detections only based on sources deemed detectable by the precursor survey. We justify this criterion based on the fact that, due to patchy coverage, only ∼10% of the overlap region lies within an angular offset Δθ ∼ 12 of a precursor beam center. We choose 12 since this is the average angular offset $\langle \Delta \theta \rangle ={\rm FWHM}/2\sqrt{2}$ for the precursor survey. Half of all sources that fall within a radius R = FWHM/2 of the nearest beam center will also be within the average angular offset 〈Δθ〉.

The PMPS discovered or detected all 24 of the previously known pulsars in this region considered detectable by the PALFA precursor survey. The precursor survey detected 21 of these, and discovered an additional four sources in this region. The PMPS retroactively detected two of these four precursor discoveries in archival data (e.g., Lorimer et al. 2006).

One of the three detectable known pulsars that the precursor survey missed, B1910+10, had a (S/N)_th value of ∼11 (see Table 1), just above the detectability threshold of (S/N)_th = 9; the other two, J1913+1145 and B1911+11, were expected to be detected with (S/N)_th = 23 and 36, respectively. Error in (S/N)_th is ∼30%, which reflects the error in flux measurements and can easily explain the first non-detection. It is much harder to explain why J1913+1145 and B1911+11 were not detected, given their high (S/N)_th values, but other systematics such as RFI and scintillation may explain these discrepancies.

The four precursor survey discoveries—J1901+0621, J1904+0738, J1905+0902 and J1906+0746—have relatively high dispersion measures and were all detected near the signal-to-noise threshold with 15 < (S/N)_meas < 22, so it is not surprising that they were not detected by previous surveys. PSR J1906+0746 is a 144 ms pulsar in a relativistic, 3.98 hr orbit and was initially missed during manual inspection of PMPS candidate plots due to RFI with a period similar to that of the pulsar (Lorimer et al. 2006). Both J1906+0746 and J1901+0621 were found retroactively in Parkes data, which was expected, given that both are moderately bright sources with flux densities at 1400 MHz of about 0.5 mJy. The other two discoveries, J1904+0738 and J1905+0902, are much fainter—0.23 and 0.097 mJy, respectively (Nice et al. 2013). These discoveries show preliminary evidence that with Arecibo's high sensitivity, the PALFA precursor survey probed a deeper and lower-luminosity pulsar population than previous surveys. However, the three unexpected non-detections suggest that the PALFA precursor survey did not realize its full sensitivity and more work is required to better understand Arecibo's RFI environment and develop mitigation techniques.

The relative sensitivity limits as a function of period and DM for the PMPS and precursor surveys are compared in Figure 3. To generate these curves, we used an average T_sky value for each survey region, assumed a constant pulse duty cycle of δ = 0.05, and applied the empirical pulse broadening function from Bhat et al. (2004) to account for multipath scattering in the interstellar medium. For the three objects that were detected at Parkes, but not in the PALFA precursor survey (B1910+10, J1913+1145 and B1911+11), all have periods between 300 and 600 ms, a regime where the PMPS nominal sensitivity limit in Figure 3 is about twice as high as the precursor survey's. However, the angular offsets to these sources (66, 47 and 25, respectively, for PMPS and precursor values can be found in Table 1) imply that both surveys were equally sensitive to them since the PALFA precursor beam (FWHM ∼ 335) is much narrower than that of the PMPS (FWHM ∼ 144) and its sensitivity therefore drops off more quickly as a function of Δθ. Taking angular offsets into account, B1910+10 (S₁₄₀₀ = 0.22 mJy) falls below the adjusted minimum sensitivity limit (∼0.26 mJy for both surveys), but B1911+11 and J1913+1145 do not, so angular offsets alone do not explain why these sources went undetected. Since other sources with lower flux densities and similar angular offsets were detected (i.e., J0628+0909, J1906+0649, J1906+0912, J1907+0740, J1907+0918, J2011+3331), we conclude that transient effects such as RFI decreased the signal-to-noise ratios of B1910+10, B1911+11 and J1913+1145 and possibly scintillation for the former two.

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The analysis presented here uses PsrPopPy—a package that models the Galactic population and evolution of pulsars. With this software, we populated a synthetic galaxy with pulsars whose attributes like cylindrical spatial coordinates, period, DM, luminosity, etc. were chosen from pre-determined PDFs (Lorimer et al. 2006). PsrPopPy²⁴ is a Python implementation of PSRPOP,²⁵ which was written in Fortran (Lorimer et al. 2011); it shares much of the same functionality, but the object-oriented nature of Python and improved modularity of the code make it more readable and easier to write plug-ins for specific modeling purposes. Further details on the PsrPopPy software package are forthcoming (Bates et al. 2014).

5.1. Generating Pulsar Population PDFs

In order to deduce the sizes of the underlying Galactic normal and millisecond pulsar populations, we compared the results of PsrPopPy simulations to the PALFA precursor survey's detection statistics for each of these two classes of pulsar. In each case, we made a set of assumptions about the underlying population (see Table 2) and drew spatial and intrinsic pulsar parameters from assumed distributions to form a synthetic Galactic population. We simulated a survey of this synthetic population by computing (S/N)_th as was discussed in Section 3. Again, detections were then defined as sources with (S/N)_th > 9. The assumptions that went into our simulations, outlined in Table 2, were largely drawn from the work by Lorimer et al. (2006) for the normal pulsar population. In that paper, however, the luminosity distribution for normal pulsars was assumed to behave as a power law with a low-luminosity cutoff of 0.1 mJy kpc². Since the PALFA precursor survey's sensitivity dips below this cutoff value in some cases, we instead adopt a log-normal luminosity distribution, introduced by Faucher-Giguère & Kaspi (2006).

Table 2. Parameters Used in Population Simulations

Prior Distribution Parameter	Normal PSR Simulations	MSP Simulations
Luminosity	Log Normal: μ = −1.1; σ = 0.9	Log Normal: μ = −1.1; σ = 0.9
Period	Log Normal: μ = 2.7,σ = −0.34	(see Figure 4)
Radial	Gamma Function: (see Lorimer et al. 2006)	Gaussian: σ = 6.5 kpc
Scale height	0.33 kpc	0.5 kpc
Duty Cycle	(explained in Section 5.1)	(explained in Section 5.1)
Electron Model	NE2001	NE2001

Notes. Assumed parameter values/distributions for normal and millisecond pulsar populations respectively. These parameters are used as input values to the appropriate PsrPopPy functions, which generate an underlying, synthetic population. Changing input parameters directly affects the number of detections expected from a given simulated survey.

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Since far fewer MSPs are known, we have very little information about the population's spatial and intrinsic parameter distributions, so some assumptions are simply adopted from the normal pulsar population (luminosity and radial distributions), while others are grounded in some preliminary experimental results (scale height, period, and duty cycle distributions). In this case, we used a Gaussian radial distribution with a standard deviation of 6.5 kpc and an exponential scale height larger than that of normal pulsars to reflect the fact that MSPs are distributed more uniformly across the sky. The Gaussian radial model for MSPs in the Galaxy is similar to that of normal pulsars, but makes no assumption about a deficiency of sources toward the Galactic center, an effect observed from full normal pulsar population synthesis and modeled with a Gamma function (Lorimer et al. 2006).

We adopted the period distribution shown in Figure 4 from Lorimer (2013), where it was initially realized by adjusting the weights of various bins from a flat distribution (in log P) until preliminary simulations matched the sample of observed MSPs from PMPS. Unlike normal pulsar duty cycles, which show inverse proportionality to the square root of spin period (i.e., shorter-period pulsars have wider pulses), MSPs tend to exhibit relatively constant duty cycle across period, with larger scatter about some mean value than the normal pulsar population (Kramer et al. 1998; Smits et al. 2009). Therefore, our simulations assumed MSP duty cycles to be independent of period.

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To make the simulated detections as realistic as possible, we used precursor survey parameters in signal-to-noise ratio calculations, and modified PsrPopPy to accept the survey's true pointing positions, as well as corresponding integration times and specific beam gain values. For each population class, we performed simulated precursor surveys across a range of trial population sizes (85,000–130,000 for normal pulsars and 5000–50,000 for MSPs). For each trial, we performed 2000 simulated realizations of independent Galactic populations for MSPs and normal pulsars respectively. To form a likelihood function describing pulsar population size, we compared the results of these simulations to the true number of detections for each population class in the precursor survey. The precursor survey only detected a single MSP (B1937+21), so the likelihood was computed by dividing the number of simulations that resulted in a single detection by the total number of simulations at that population size.

Of the 45 detections listed in Table 1, we exclude B1937+21 (MSP) from our normal pulsar analysis. Although J1906+0746 is in a binary system, it is a young pulsar with a characteristic age of 112 kyr and has likely not undergone recycling from its companion, so we include it in our analysis. The likelihood function was formed by dividing the number of simulations that detected 44 pulsars by the total number of simulations at a given trial population size. We fit binomial distributions to simulated likelihood functions for normal and MSP populations (shown in Figure 5) in order to smooth simulation results and provide integrable functions to determine confidence intervals. For an underlying population of size N, a given simulation has n successes (detections) and N − n failures (non-detections); these kinds of binary outcomes are nicely modeled by binomial distributions.

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The binomial distributions provide the functional form

$$\begin{equation} p(n|N,\theta)=\frac{N!}{n!(N-n)!}\theta ^n(1-\theta)^{N-n}, \end{equation} \tag{ 4 }$$

which describes the probability of drawing n pulsars from a total population of N given some detection probability θ. To select the θ value that produces posterior PDFs that best match the simulated data, we chose the one that minimized χ², computed by comparing simulated to expected population distributions. Finally, the posterior population size PDFs are normalized so that they could be used to quote confidence intervals. With some number n_success of successful realizations (simulations in which the target number of detections is reached), the Poissonian error is given by $\sqrt{n_{\rm success}}$. Data points that reflect the probability of detecting exactly the target number of pulsars at a trial population size and their error bars are multiplied by the same constant required to normalize the best fit PDF. After looking at multiple realizations of the simulated data presented in Figure 5 and comparing the standard deviation of data points at each population size to assumed Poissonian error bar magnitudes, we determined that the Poisson model accurately reflects the uncertainties in population sizes.

By integrating the PDFs shown in Figure 5, we find the mode and 95% confidence interval for the normal pulsar population size to be 107,000$^{+36,000}_{-25{,}000}$. We find a lower mode for the MSP population size, 15,000$^{+85{,}000}_{-6000}$ and the high uncertainty in the corresponding 95% confidence interval reflects the fact that our prediction depends on a single MSP detection in the precursor survey. These results describe the respective Galactic pulsar populations that are beaming toward Earth and errors on most likely population sizes account only for statistical uncertainties due to the limited number of detections in the PALFA precursor survey, not for other sources (e.g., uncertainties in scale height, luminosity distribution, electron density model, etc.).

The confidence interval that the precursor survey places on the normal pulsar population is consistent with earlier results; Faucher-Giguère & Kaspi (2006) predict 120,000 ± 20,000 detectable normal pulsars, also using a log-normal distribution to model the pulsar luminosity function. The predicted MSP population size is also consistent with previous estimates; the upper limit we find easily encompasses the population size prediction made by Levin et al. (2013), although the lower limit quoted in that paper, 30,000 ± 7000, is more constraining. Neither of these 95% confidence intervals is tight enough to put strict constraints on normal or millisecond pulsar population sizes, but the consistency is encouraging and we expect the full PALFA survey to place much more stringent constraints on these populations when complete.

Using input parameters from Table 2 to generate a synthetic, Galactic normal pulsar population, we found that the PALFA precursor survey should be expected to detect ∼40 sources. Through periodicity searches, 43 were found, which indicates that current population parameters, initially determined using PMPS results, are already quite accurate and applicable to a variety of situations. As we mentioned in Section 3.1, three sources that we expected to detect were not detected, but it is common for (S/N)_th and (S/N)_meas values to not match perfectly. Due to uncertainties in initial flux measurements, there can be as much as ∼30% fractional error in (S/N)_th. Referring again to Figure 1, we show a general trend toward a slope of unity when plotting theoretical versus measured S/N for the detections made by the precursor survey, but there is significant scatter in these comparisons. Scatter like this can be caused by scintillation, RFI, poor prior flux measurements or some combination of all of these.

The precursor survey discovered 11 pulsars, 4 of which fell inside the region overlapping PMPS, allowing us to directly compare their respective sensitivities. While PMPS detected almost three times as many sources in this region, this discrepancy was largely due to the differences in sky coverage—PMPS covered this area uniformly, while the precursor survey had large blocks of coverage missing and slight gaps between pointings due to a "sparse sampling" technique. In fact, only ∼25% of the overlap region was covered by the precursor survey to a sensitivity greater than or equal to that of PMPS. Even so, the PALFA precursor survey discovered four pulsars that PMPS missed; two of these four were retroactively found by reanalyzing archival data but the others(J1904+0738 and J1905+0902) have high dispersion measures and very low fluxes—an encouraging, albeit small, piece of evidence that Arecibo's sensitivity gives PALFA a glimpse at fainter and more distant pulsars. Figure 3 in Nice et al. (2013) uses more recent PALFA discoveries to show further evidence of PALFA probing deeper than previous surveys as do recent discoveries mentioned in Crawford et al. (2012).

We simulated a range of Galactic pulsar populations—both non-recycled and recycled—of various sizes and used the PALFA precursor survey's detection statistics to place limits on normal and millisecond pulsar population sizes respectively. By comparing experimental results to simulations, we formed PDFs for normal and MSP population sizes, then integrated these PDFs to define confidence intervals.

Assuming the most probable normal and millisecond population sizes according to the simulations described in Section 5.1 are correct, we ran 1000 trials with the same distribution parameter assumptions for each population to determine the most likely number of detections by the beginning of 2014 and after PALFA is complete. Averaging the results of these 1000 trials in each case, we determine a predicted number of detections, then quote errors that are directly proportional to the 95% confidence limits from normal and millisecond pulsar population PDFs. Following this procedure, we expect the full PALFA survey to detect $1000^{+330}_{-230}$ normal pulsars (this includes previously known sources that are re-detected) and $30^{+200}_{-20}$ MSPs. Identical estimation techniques predict that $490^{+160}_{-115}$ normal pulsars and $12^{+70}_{-5}$ MSPs should have been detected by the beginning of 2014, but at the time, PALFA had detected 283 normal pulsars and 31 MSPs, respectively.²⁶

The discrepancy between observed and predicted detection rates is notable for the normal pulsar population. Given the numbers quoted here, PALFA has currently detected just over 50% of the expected number of normal pulsars, according to simulations. These simulations do not yet take into account the local RFI environment of the PALFA survey, which certainly plays a role in the perceived dearth of pulsar detections as of early 2014. Two pulsars that went undetected by both Quicklook and Presto 1 pipelines in the precursor survey, J1906+0649 and J1924+1631, provide evidence that initial processing techniques were not optimal and improvements are necessary. In repeated simulations of precursor detections in the inner Galaxy region, we find 30–50% of simulated, detectable sources had S/N values between 9 and 15 (just above the detection threshold). In the precursor survey, only about 10% of detections had (S/N)_meas values in this regime. Although the precursor survey discovered mostly low flux density sources, the fact that only a small fraction of detections were near the S/N threshold suggests that some sources were missed or assumptions that determine our sensitivity curves are not entirely correct.

A potential factor of two lower sensitivity to normal pulsars because of RFI would bring the survey yield and simulated population into agreement. The most recent PALFA survey pipeline will be described in depth by P. Lazarus (2014, in preparation) and that paper will also construct PALFA's "true" sensitivity curve, taking into account the RFI environment by injecting artificial signals of varying strength into real data. In future work, we will reprocess precursor survey data with the current pipeline to see if it improves the shortcomings of earlier versions (e.g., inconsistent detection statistics, noted in Table 1).

The assumed radial distribution of pulsars in the Galaxy (see Table 2) could also contribute to the discrepancy between expected (simulated) and true pulsar yields. Since the distribution is based on extrapolated results from the PMPS, which surveyed higher-populated regions of the sky, population density estimates for longitudes farther from Galactic center may be inaccurate. Overestimated pulsar population densities in the Galactic longitude range surveyed by the PALFA precursor survey could be a factor in the discrepancies we find between expected and actual pulsar detections there. Future refinement of pulsar population models using PALFA results will provide consistency checks for existing population model parameters.

We note that the current number of MSPs detected by PALFA is consistent with predictions, but this is not surprising, given the high uncertainties in our model due to the precursor survey only detecting one MSP. As the number of detections increases, future predictions will be far more constraining so that we can re-examine initial assumptions about the MSP population characteristics.

Future population studies with the complete PALFA survey will contribute substantially to current population models because of the Galactic longitude ranges covered and Arecibo's unrivaled sensitivity (especially in the millisecond pulse period regime). As the number of normal and millisecond pulsar detections increases, our ability to refine specific, simulated model parameters that describe each underlying population will improve significantly.

The Arecibo Observatory is operated by SRI International under a cooperative agreement with the National Science Foundation (AST–1100968), and in alliance with Ana G. Méndez-Universidad Metropolitana, and the Universities Space Research Association. M.A.M. and J.K.S. are supported through NSF PIRE award 0968296. D.J.N. is supported through NSF grant 0647820. V.M.K. was supported by an NSERC Discovery and Accelerator Grant, the Canadian Institute for Advanced Research, a Canada Research Chair, Fonds de Recherche Nature et Technologies, and the Lorne Trottier Chair in Astrophysics. J.W.T.H. acknowledges funding from NWO and ERC. Work at Cornell was supported by NSF Grants 0507747 and 1104617 and made use of the Cornell Center for Advanced Computing. Pulsar research at UBC is supported by an NSERC Discovery Grant and Discovery Accelerator Supplement and by the Canada Foundation for Innovation. P.L. acknowledges support of IMPRS Bonn/Cologne and NSERC PGS–D.