Precision Timing of PSR J0437–4715 with the IAR Observatory and Implications for Low-frequency Gravitational Wave Source Sensitivity

The North American Nanohertz Observatory for Gravitational Waves (NANOGrav; McLaughlin 2013; Ransom et al. 2019) collaboration is working toward the detection and characterization of low-frequency GWs by means of monitoring an array of nearly 80 precisely timed MSPs using three telescopes: the 305 m William E. Gordon Telescope of the Arecibo Observatory (Arecibo), the 100 m Robert C. Byrd Green Bank Telescope of the Green Bank Observatory (GBO), and the 27 element 25 m antennas of the National Radio Astronomy Observatory's (NRAO) Karl G. Jansky Very Large Array (VLA). NANOGrav has already placed upper limits on the amount of GWs in the universe, which have provided strong astrophysical constraints on merging galaxies and cosmological models (Arzoumanian et al. 2018a, 2020; Aggarwal et al. 2019, 2020). NANOGrav is poised to detect and begin characterization of the stochastic GW background from an ensemble of unresolved supermassive black hole binaries (SMBHBs) within the next 3–5 yr (Rosado et al. 2015; Taylor et al. 2016; Arzoumanian et al. 2018a; Arzoumanian et al. 2021, in preparation). The first detection of a continuous-wave (CW) source of GWs from a single SMBHB is expected by the late 2020s (Rosado et al. 2015; Mingarelli et al. 2017; Kelley et al. 2018; Aggarwal et al. 2019). In addition, NANOGrav is sensitive to the direct signature of SMBHB mergers, known as a "burst with memory" (BWM; Aggarwal et al. 2020).

The Argentine Institute of Radio Astronomy (IAR; Instituto Argentino de Radioastronomía) has begun upgrades of two 30 m antennas primarily for the purpose of pulsar observations (Gancio et al. 2020). Located at latitude ≈−37°, the observatory covers declinations δ < −10°. The two antennas, A1 and A2, are 30 m equatorial mount telescopes separated by 120 m. Both are capable of observations around 1.4 GHz, each with current specifications listed in Table 1.

While 30 m antennas fall below the typical size of radio telescopes used in precision timing experiments (e.g., Perera et al. 2019), the IAR Observatory has several advantages for high-precision timing, specifically in the area of GW detection and characterization. With access to the southern sky below δ < −10° as well as many galactic disk pulsars, in addition to the ability to track objects for nearly four hours, IAR has demonstrated the capability of high-cadence timing on a number of those pulsars, particularly of the bright millisecond pulsar J0437−4715 (Gancio et al. 2020; Sosa Fiscella et al. 2021). Their preliminary observations suggest that future measurements will be a significant contributor to PTA sensitivity.

Discovered in the Parkes Southern Pulsar Survey (Johnston et al. 1993), PSR J0437−4715 is the brightest known MSP and has the lowest dispersion measure (DM; the integrated line-of-sight electron density) of any MSP; it is the second-lowest of any pulsar (Manchester et al. 2005). Its bright pulses can be precisely timed with high S/N, allowing for this pulsar to be used in tests of general relativity (van Straten et al. 2001; Verbiest et al. 2008) and pulsar emission physics (Osłowski et al. 2014), as a source in the development of new high-precision timing calibration and methodology (van Straten 2006; Liu et al. 2011; Osłowski et al. 2013), and as a sensitive component to pulsar timing arrays (Perera et al. 2019; Kerr et al. 2020). Some pulsar properties are listed in Table 2.

Table 2. Parameters for PSR J0437−4715

Parameter	Value
Spin Period a (P)	5.76 ms
Dispersion Measure a (DM)	2.64 pc cm−3
Effective Width b (Weff)	667.6 μs
Flux Density at 400 MHz a (S400, )	550.0 mJy
Flux Density at 1000 MHz c (S400)	223 mJy
Flux Density at 1400 MHz a (S1400)	160.0 mJy
Spectral Indexc (α)	−0.99
Jitter rms for 1 hr at 730 MHz (σJ,730)	61 ± 9 ns
Jitter rms for 1 hr at 1400 MHz (σJ,1400)	48.0 ± 0.6 ns
Jitter rms for 1 hr at 3100 MHz (σJ,3100)	41 ± 2 ns
Phase Bins (Nϕ , assumed)	512
Median σS/N for Antenna 1 (σS/N, A1)	200 ns
Median σS/N for Antenna 2 (σS/N, A2)	330 ns
Median S/N for Antenna 1c (${\tilde{S}}_{{\rm{A}}1}$)	148.8
Median S/N for Antenna 2c (${\tilde{S}}_{{\rm{A}}2}$)	90.5

Notes.

^a Values taken from psrcat (Manchester et al. 2005). ^b Effective width estimated from profile in Kerr et al. (2020). ^c Derived parameters.

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In addition to the ability to measure high S/N pulses, given its low DM, we expect that unmodeled chromatic (i.e., radio-frequency-dependent) interstellar propagation effects are reduced for this pulsar (Cordes & Shannon 2010), thereby limiting one of the significant sources of timing noise found in many other pulsars. PSR J0437−4715, however, is not timed without complications. The pulsar has measurable achromatic noise consistent with rotational spin fluctuations (Shannon & Cordes 2010; Lam et al. 2017), quasi-periodic intensity and polarization variations are seen in subpulse components (De et al. 2016), and a systematic variation in the pulse shape has also been observed (Kerr et al. 2020). While these effects complicate the pulsar's use in tests of fundamental physics with high-precision timing, its demonstrated stability nonetheless suggests that it can be used in these tests (e.g., Verbiest et al. 2009), including in searches for stochastic and deterministic GW signals.

In this paper, we demonstrate the specific capabilities of the IAR to observe PSR J0437−4715 and provide unparalleled sensitivity to the pulsar for the purpose of single-source CW detection and future characterization. In Section 2, we provide the framework for our TOA sensitivity estimates. We consider the current IAR observations of PSR J0437−4715 in Section 3, and then extrapolate to future possible operating modes in Section 4, comparing the modes to observations of the pulsar by other international observatories as well. In Section 5, we demonstrate how the IAR will improve sensitivity to single-source CWs when the data are combined with that of NANOGrav. As the collaboration observing the most pulsars (Ransom et al. 2019) and with the most recent and stringent CW limits (Aggarwal et al. 2019; Arzoumanian et al. 2020), we will focus our analyses primarily on improving upon a projected NANOGrav data set with observations of PSR J0437−4715, noting that upcoming analyses (e.g., Alam et al. (2021a, 2021b), which already include more time baseline and more pulsars) imply greater sensitivity in the future. We describe other possible pulsar targets in Section 6, and conclude in Section 7.

We use the framework of Lam et al. (2018) to estimate the noise contributions to pulsar TOA uncertainties—or equivalently, how precisely we can measure the TOAs. Gancio et al. (2020) show the first set of timing residuals (the observed TOAs minus a model for the expected arrival times) for PSR J0437−4715 with the IAR, and they demonstrate excess noise beyond the uncertainties derived from the common practice of matching a pulse template to the observed pulsar profile to estimate their TOAs (Taylor 1992).

2.1.1. Short-timescale Noise

Lam et al. (2016a) describe a model for white noise in timing residuals on short timescales, ≲1 hr, i.e., the time of typical observations. In that analysis, they assume perfect polarization calibration and radio frequency interference (RFI) removal; the former is currently impossible in the case of A1, with only one measured polarization channel. However, assuming both are true, the three white noise components described are the template fitting, jitter, and scintillation noise terms.

Template fitting errors rely on the matched-filtering assumption that the observed pulse profiles are an exact match to a template shape with some additive noise. These are the minimum possible errors for TOA estimates, and are given by

$$\begin{eqnarray}&&{\sigma }_{{\rm{S}}/{\rm{N}}}(S)=\displaystyle \frac{{W}_{\mathrm{eff}}}{S\sqrt{{N}_{\phi }}},\end{eqnarray} \tag{ 1 }$$

where W_eff is the effective width of the pulse, N_ϕ is the number of samples (bins) across the profiles, and S is the signal-to-noise ratio (written this way in equations for clarity), taken to be the peak to off-pulse rms. The effective width is related to the spin period of the pulsar P and the template profile normalized to unit height, U(ϕ), by

$$\begin{eqnarray}&&{W}_{\mathrm{eff}}=\displaystyle \frac{P}{{N}_{\phi }^{1/2}{\left[{\displaystyle \sum }_{i=1}^{{N}_{\phi }-1}{\left[{U}_{\mathrm{obs}}({\phi }_{i})-{U}_{\mathrm{obs}}({\phi }_{i-1})\right]}^{2}\right]}^{1/2}}.\end{eqnarray} \tag{ 2 }$$

Pulse profiles are frequency-dependent, and so this error will apply to each profile measured at a given frequency. However, given the low bandwidths currently available at the IAR Observatory, it is sufficient, given the S/N regime of the observations, to average the data across not only time but frequency as well, to obtain a single TOA per epoch per antenna. Similarly, many other frequency-dependent variations within the band will be small and we will ignore those here.

While W_eff and N_ϕ are constants, the pulse S/N will vary on short timescales, due to diffractive scintillation, with a known probability density function (PDF) written as (Cordes & Chernoff 1997; Lam et al. 2016a)

$$\begin{eqnarray}&&{f}_{S}(S| {S}_{0},{n}_{\mathrm{ISS}})=\displaystyle \frac{{\left({{Sn}}_{\mathrm{ISS}}/{S}_{0}\right)}^{{n}_{\mathrm{ISS}}}}{S{\rm{\Gamma }}({n}_{\mathrm{ISS}})}{e}^{-{{Sn}}_{\mathrm{ISS}}/{S}_{0}}{\rm{\Theta }}(S),\end{eqnarray} \tag{ 3 }$$

where S₀ is the mean S/N, Γ is the Gamma function, Θ is the Heaviside step function, and n_ISS is the number of scintles in the observation of length T and bandwidth B, given by

$$\begin{eqnarray}&&{n}_{\mathrm{ISS}}\approx \left(1+{\eta }_{t}\displaystyle \frac{T}{{\rm{\Delta }}{t}_{{\rm{d}}}}\right)\left(1+{\eta }_{\nu }\displaystyle \frac{B}{{\rm{\Delta }}{\nu }_{{\rm{d}}}}\right).\end{eqnarray} \tag{ 4 }$$

The scintillation parameters Δt_d and Δν_d describe the characteristic timescale and bandwidth of intensity maxima, or scintles, in a dynamic spectrum. The η_t and η_ν parameters are the scintle filling factors ≈0.2 (Cordes & Shannon 2010; Levin et al. 2016).

Gwinn et al. (2006) found two diffractive scintillation scales for PSR J0437−4715 made at an observing frequency of 328 MHz. The first has Δt_d = 1000 s and Δν_d = 16 MHz, while the second has Δt_d = 90 s and Δν_d = 0.5 MHz. For a Kolmogorov medium, we can scale these quantities as Δt_d ∝ ν^6/5 and Δν_d ∝ ν^22/5 (Cordes & Rickett 1998; Lam et al. 2018). Therefore, when scaled to an observing frequency of 1.4 GHz, we have Δt_d = 5700 s and Δν_d = 9.5 GHz for the first scale, and Δt_d = 510 s and Δν_d = 300 MHz for the second scale. Keith et al. (2013) measured the scintillation parameters at 1.5 GHz for PSR J0437−4715; when scaled to 1.4 GHz, they found Δt_d = 2290 s and Δν_d = 740 MHz, in between the two scales observed by Gwinn et al. (2006).

Using these measurements of the scintillation parameters, we will estimate n_ISS and thus the predicted impact on the PDF of S. Given the current small bandwidths of the receiver, since B < Δν_d in all cases, the second of the two components in Equation (4) will be ≈1 and so we have n_ISS ≈ 1 + η_t T/Δt_d. For T = 13200 s, this quantity will range between 1.5 and 5.2. The higher n_ISS is, the more that the PDF will tend toward the mean S/N value S₀, whereas when n_ISS is close to 1, the distribution becomes exponential. Therefore, this value will heavily dictate the timing uncertainties achievable, which we will briefly discuss when applying to the real data.

The two other short-timescale contributions to the white noise discussed in Lam et al. (2016a) are the jitter and scintillation-noise terms; the latter is separate from the S/N change due to scintillation. Both cause stochastic deviations to the observed pulse shapes such that the matched-filtering assumption of template fitting no longer applies. Jitter, due to single-pulse stochasticity, becomes a significant noise contribution for well-timed, high-S/N MSPs (Lam et al. 2019). Shannon et al. (2014) measured the timing uncertainty due to jitter for an hour-long observation to be σ_J = 48.0 ± 0.6 ns, significantly less than the template fitting errors shown in Gancio et al. (2020). In addition, since the rms jitter scales inversely with the square root of the number of pulses, this contribution to the TOA uncertainty for a 3^h 40^m observation will be even smaller in IAR data, and therefore can be ignored.

Scintillation noise is caused by the finite-scintle effect and results in pulse-shape stochasticity due to an imperfectly known pulse broadening function (Cordes et al. 1990; Cordes & Shannon 2010; Coles et al. 2010). Its maximum value is approximately the scattering timescale τ_d, which is inversely proportional to Δν_d (Cordes & Rickett 1998; Cordes & Shannon 2010):

$$\begin{eqnarray}&&{\tau }_{{\rm{d}}}=\displaystyle \frac{{C}_{1}}{2\pi {\rm{\Delta }}{\nu }_{{\rm{d}}}},\end{eqnarray} \tag{ 5 }$$

where C₁ ≈ 1 is a coefficient that depends on the geometry of the intervening medium. Since scattering tends to be more significant at higher DMs (Bhat et al. 2004), we already expect τ_d to be small for PSR J0437−4715. For the minimum value of Δν_d above, we find a maximum value of τ_d ≲ 1 ns, and therefore scintillation noise can also be ignored for this pulsar.

2.1.2. Dispersion Measure Estimation

Estimation of DM and the subsequent correction of dispersion are critical for proper precision timing. Since the scattering timescale is small, the dominant component to DM estimation errors is that due to the white noise described above (Lam et al. 2018). For measurements taken at two frequencies ν₁ and ν₂, with the frequency ratio defined as r ≡ ν₂/ν₁ ≥ 1, the timing uncertainty due to DM estimation is (see Appendix A of Lam et al. 2020)

$$\begin{eqnarray}&&{\sigma }_{\widehat{\mathrm{DM}}}=\displaystyle \frac{\sqrt{{\sigma }_{{\nu }_{1}}^{2}+{r}^{4}{\sigma }_{{\nu }_{2}}^{2}}}{{r}^{2}-1},\end{eqnarray} \tag{ 6 }$$

where the σ_ν values are the timing uncertainties at each frequency. Both Cordes & Shannon (2010) and Lam et al. (2018) describe the matrix formalism for calculating this uncertainty when multichannel measurements are available. Estimating the DM with measurements taken at multiple frequencies results in reduced uncertainties even if the covered range r is the same.

If DM variations are not accounted for in a timing model, as in Gancio et al. (2020), then additional uncertainties arise. Many types of effects give rise to both stochastic and systematic variations in DM (Lam et al. 2016b). For example, an Earth–pulsar line of sight passing close to the Sun will cause an increase in the DM from the increased electron density of the solar wind (You et al. 2007; Tiburzi et al. 2019; Madison et al. 2019); while the motion through this electron density profile gives rise to systematic variations, any fluctuations in the solar wind will cause variations that are random in nature. The turbulent interstellar medium will also give rise to DM fluctuations that are random though correlated over time (Foster & Cordes 1990; Phillips & Wolszczan 1991).

The turbulent medium is typically parameterized by a power-law electron density wavenumber spectrum over many orders of magnitude (Armstrong et al. 1995). An equivalent formulation that one can derive from this spectrum is the DM structure function (e.g., Lam et al. 2016b), given by

$$\begin{eqnarray}&&{D}_{\mathrm{DM}}(\tau )\equiv \langle {\left[\mathrm{DM}(t+\tau )-\mathrm{DM}(t)\right]}^{2}\rangle ,\end{eqnarray} \tag{ 7 }$$

where τ is the time lag and the brackets denote the ensemble average. The structure function of the timing perturbations δ t is related to the DM structure function by

$$\begin{eqnarray}&&{D}_{\delta t}(\tau )={K}^{2}{\nu }^{-4}{D}_{\mathrm{DM}}(\tau ),\end{eqnarray} \tag{ 8 }$$

where K ≈ 4.149 × 10³ μ s GHz² pc⁻¹ cm³ is the dispersion constant in observationally convenient units, i.e., for a DM of 1 pc cm⁻³ and radio emission at 1 GHz, we expect 4149 μs of delay. This is a significantly larger delay than the achievable timing precision, because DM estimates are known much more precisely and therefore the delay can be corrected for to high accuracy.

We can directly relate the rms timing fluctuations for a given time lag to the structure function by

$$\begin{eqnarray}&&{\sigma }_{\delta t}(\tau )={\left[\displaystyle \frac{1}{2}{D}_{\delta t}(\tau )\right]}^{1/2}.\end{eqnarray} \tag{ 9 }$$

For a Kolmogorov turbulent spectrum, we can write the structure function in the power-law form (Lam et al. 2016b):

$$\begin{eqnarray}&&{D}_{\delta t}(\tau )=\displaystyle \frac{1}{{\left(2\pi \nu \right)}^{2}}{\left(\displaystyle \frac{\tau }{{\rm{\Delta }}{t}_{{\rm{d}}}}\right)}^{5/3}=0.0253\,{\mathrm{ns}}^{2}\,{\nu }_{\mathrm{GHz}}^{-2}\,{\left(\displaystyle \frac{\tau }{{\rm{\Delta }}}{t}_{{\rm{d}}}\right)}^{5/3}.\end{eqnarray} \tag{ 10 }$$

For PSR J0437−4715, we will first consider the middle value of Δt_d provided by Keith et al. (2013), with Δt_d = 2290 s. On the timescale of the τ ≈ 100 days as shown in Gancio et al. (2020), the rms timing perturbations for unaccounted for DM variations would be approximately 110 ns at 1.4 GHz, and so should be a negligible part of their error budget shown. Nonetheless, if we consider this error for longer timespans, as for 1000 and 10,000 days, the rms perturbations are 740 ns and 5.0 μs, respectively. ⁴ Note that the predicted structure function extrapolated from Δt_d as given above, using the value of Δt_d estimated from scintillation measurements, is actually lower (by a factor of 5) than what is empirically estimated from the DM time series by Keith et al. (2013). This could potentially arise from additional structures in the interstellar medium, unrelated to general turbulence (Lam et al. 2016b).

2.1.3. Additional Sources of Uncertainty

GW detection relies on measuring the signature of TOA perturbations among pulsars. In the case of a stochastic background—for instance, from the unresolvable sum of signals from SMBHBs—one relies on cross-correlations between these perturbations. For individual SMBHBs and other single-source signals, one uses a deterministic model. In both cases, the sky location and sensitivity of individual pulsars affect the ability of the full PTA to detect gravitational waves. We use the framework discussed in Hazboun et al. (2019) and Lam (2018) to estimate the sensitivity to single sources, which we describe here briefly.

While the ability to claim a detection of GWs from a single SMBHB requires more than one pulsar, the sensitivity to a single source depends on the signal strength and noise in each individual pulsar. These signals are then added up across the network to build a robust detection. There are a number of statistics defined in the literature to search for single-source GWs (Babak & Sesana 2012; Ellis et al. 2012a, 2012b; Taylor et al. 2016). Here, we focus on assessing and optimizing the sensitivity of PTAs (Lam 2018; Hazboun et al. 2019) given different pulsars and changes in the observing strategies. The framework of Hazboun et al. (2019) uses a matched-filter statistic tailored to studying sky-dependent sensitivity. The S/N, $\rho (\hat{k})$, is dependent on the response of an individual pulsar's sky position and noise characteristics, but has been averaged over inclination angle and GW polarization:

$$\begin{eqnarray}&&\langle {\rho }^{2}(\hat{k}){\rangle }_{\mathrm{inc}}=2{T}_{\mathrm{obs}}{\int }_{{f}_{{\rm{L}}}}^{{f}_{{\rm{H}}}}{df}\,\displaystyle \sum _{i}\displaystyle \frac{4}{5}\displaystyle \frac{{T}_{i}}{{T}_{\mathrm{obs}}}\displaystyle \frac{{S}_{h}(f)}{{S}_{i}(f,\hat{k})}.\end{eqnarray} \tag{ 11 }$$

Here, S_h (f) is the strain power spectral density (PSD) of a monochromatic SMBHB signal, T_i /T_obs is the fraction of total observation time for the PTA covered by a particular pulsar labeled by i, and ${S}_{i}(f,\hat{k})$ is the strain-noise PSD for a particular pulsar. The dependence on sky location, $\hat{k}$, comes from the quadrupolar response function of pulsars to GWs. This S/N shows that the signals from individual pulsars add independently, but this really only tells part of the story. While the quantity $\rho (\hat{k})$ captures the sensitivity of a PTA to individual sources, one would need a significant signal in a few pulsars in order to claim a detection.

The PSD ${S}_{i}(f,\hat{k})$ is related to the usual noise spectra, written in Lam (2018) as the sum of white noise, red noise, and the stochastic GW background,

$$\begin{eqnarray}&&{P}_{i}(f)={P}_{{\rm{W}},i}(f)+{P}_{{\rm{R}},i}(f)+{P}_{\mathrm{SB}}(f),\end{eqnarray} \tag{ 12 }$$

where the stochastic background term takes a power-law form of P_SB(f) ∝ f^−β with β = 13/3 for an ensemble of SMBHBs (Jenet et al. 2006). The PSD ${S}_{i}(f,\hat{k})$ also takes into account the timing model fit of the pulsar, through the inverse noise-weighted transmission function and the response function, which maps the strain from the GWs to the induced residuals in the pulsar's TOAs. See Hazboun et al. (2019) for more details. For uniform white noise given by σ_W,i between observations with cadence c_i , the white noise term is simply ${P}_{{\rm{W}},i}(f)=2{\sigma }_{{\rm{W}},i}^{2}/{c}_{i}$. This form demonstrates the strength of the IAR Observatory in observing PSR J0437−4715: the cadence can be a factor of ∼30 larger than for other telescopes, thus significantly reducing the white noise term.

The strain power spectrum for a single CW source with GW frequency f₀ and amplitude h₀ is (Thrane & Romano 2013; Lam 2018; Hazboun et al. 2019)

$$\begin{eqnarray}&&{S}_{h}(f)\equiv \displaystyle \frac{1}{2}\,{h}_{0}^{2}\left[{\delta }_{T}(f-{f}_{0})+{\delta }_{T}(f+{f}_{0})\right].\end{eqnarray} \tag{ 13 }$$

where δ_T (f) is the finite-time approximating function of a Dirac delta function,

$$\begin{eqnarray}&&{\delta }_{T}(f)\equiv \displaystyle \frac{\sin (\pi {fT})}{\pi f},\quad \mathop{\mathrm{lim}}\limits_{T\to \infty }{\delta }_{T}(f)=\delta (f).\end{eqnarray} \tag{ 14 }$$

That is, given the simple CW approximation often used in PTA analyses of an unchanging orbital frequency, over an infinite time, the power spectrum would be a delta function at f₀. While more realistic signals, such as those that include a phase shift from the delayed pulsar term and frequency evolution, are usually used in real PTA analyses (e.g., Aggarwal et al. 2019), the sensitivity is accurately estimated with a circular model. Additionally, it should be noted that this framework includes the signal from the time-delayed pulsar term. An analysis using the pulsar term requires very accurate pulsar distances in order to gain an appreciable signal; perfect knowledge of the distance boosts the squared S/N by a factor of 2. While standard analyses (Aggarwal et al. 2019) use the pulsar term, it is unclear how much is gained by its inclusion. This has no effect on the ratios of detection thresholds examined here, because the factor (of $\sqrt{2}$) for thresholds would cancel.

To estimate the TOA uncertainties for each system, we combined the various noise components in the following manner. We scaled the A1 and A2 median S/Ns as estimated in Section 3, and scale by the ratio of the flux densities from 1400 MHz to the new center frequency, the ratio of the square root of the bandwidths and number of polarization channels, and inversely by the ratio of the receiver temperatures. With the new S/N, we calculated a new template fitting error for the receiver. With the two center frequencies, we were able to calculate the TOA uncertainty due to DM estimation (Equation (6)); we assumed that, in all future timing analyses, DM will be estimated on a time-varying basis such that the rms timing fluctuations are negligible (e.g., from Equation (9)), otherwise the unmodeled DM variations will make any high-precision timing experiment impossible. Last, we added in 100 ns of uncertainty due to polarization miscalibration, comparable to the value calculated by van Straten (2006).

Even though we approximated the observations across each band as single measurements, ignoring the varying spectral index with the band, changes in pulse width, etc., for simplicity we calculated the TOA uncertainties as if there are two "spot" measurements at each center frequency. While this is a simplification, it provided us with a conservative estimate, since a full least-squares fit formalism will lead to reduced TOA uncertainties.

In summary, our TOA uncertainties consisted of template fitting errors, DM estimation errors, and polarization miscalibration errors, added together in quadrature.

We considered several different configurations: both the current and future setups as described in Gancio et al. (2020), as well as setups where additional receivers are built. We restricted ourselves to discussing the configurations broadly, assuming that the receiver frontend is matched with a backend that will be able to adequately sample and coherently dedisperse the data. Gancio et al. (2020) describe the future planned upgrades to the antennas, which include receivers reaching system temperatures T_sys < 50 K, a 500 MHz bandwidth in a range between 1 and 2 GHz, and a new FPGA-based CASPER board (Hickish et al. 2016) backend capable of processing the increased bandwidth. For other configurations we list, we assumed similar matching, as we primarily wanted to focus on estimating the TOA uncertainties given the assumed development of a specific system. We estimated these uncertainties under two scenarios with the current equipment: three in which one or more of these higher-bandwidth systems is deployed, and one optimistic scenario in which ultrawideband receivers are deployed on both antennas. The full list of assumptions is provided in Table 3, along with the TOA uncertainties we estimate. The gains of the antennas in our analysis did not vary, though some dish or efficiency improvements may be possible in the future, e.g., resurfacing the dishes in the future may improve the aperture efficiencies slightly as compared with their current materials (Testori et al. 2001; Gancio et al. 2020).

We describe the rationales for considering each configuration below.

• C1:  
  Current. This configuration is as described in Gancio et al. (2020); see their Table 1.
• C2:  
  Optimized. Since the center frequencies are tunable, we allowed for the maximum separation in frequencies, to minimize TOA uncertainties from DM misestimation. We also halved A1's bandwidth coverage in favor of dual-polarization measurements (see their Section 2.1 regarding these two modes for A1), a requirement for high-precision timing.
• C3:  
  Wideband. With a modest upgrade of two receivers and backends covering 500 MHz each, as discussed in Gancio et al. (2020), the two antennas can cover the 1–2 GHz range. The target system temperature is T_sys < 50 K. While some parts of the band will be lost due to RFI, no significant segments of the band are currently lost across this frequency range (Gancio et al. 2020).
• C4:  
  Low Frequency. Instead of two receivers covering the full L-band range, we instead selected one 500 MHz receiver to cover the top end of L band (1.5–2.0 GHz) for A1, and then for A2 considered a receiver from 400–450 MHz. This lower frequency range is used by NANOGrav at Arecibo for some pulsars. Its primary allocation in Argentina ⁵ is for maritime communication and radionavigation (see ITU-R 2016 for Region 2), which helps to limit fixed sources of RFI. In addition, scattering will minimally affect PSR J0437−4715, given its low DM, and so the increased flux density (see Table 2) can lead to high-S/N TOAs, also providing a significant frequency difference to estimate DM. For the L-band receiver, we used the higher end of the possible bandwidth range in order to minimize TOA uncertainties due to DM misestimation.
• C5:  
  High Frequency. As a parallel to C4, we considered instead using a second receiver at higher frequencies, in the 2.5 GHz (S band) range, which is also used by NANOGrav at Arecibo for some pulsars. This region of the frequency spectrum often contains significant RFI due to overlap with wireless communications. Nonetheless, we considered the potential for such a system. For the L-band receiver, we used the lower end of the possible bandwidth range in order to minimize TOA uncertainties due to DM misestimation.
• C6:  
  Dual Ultrawideband. As an optimistic setup, we considered the receiver systems as described in Hallinan et al. (2019) for the DSA-2000. This observatory will employ low-cost receiver systems from 0.4–2.0 GHz for each of its 2000 antennas. While their target system temperature is 25 K, we kept 50 K for use in this considered configuration. Since the receiver setup is identical for both antennas, for simplicity we assumed two bands centered at 700 and 1500 MHz with 600 and 1000 MHz of bandwidth, respectively, to cover the 400–2000 MHz range, and included a factor of $\sqrt{2}$ in our calculation.

As discussed in Section 2.1.1, one astrophysical cause of the S/N changing is diffractive scintillation. When considering wider bandwidths, the number of scintles will grow, thus causing the pulse S/N to tend more toward a mean value rather than covering an exponentially distributed range. In practice, this means that, for low-bandwidth setups such as the current configurations, there are epochs in which the template fitting error in Equation (1) breaks down (Arzoumanian et al. 2015) and TOAs cannot be reliably estimated, resulting in a loss of usable data. In addition to the mean S/N S₀ increasing due to wider bandwidths, the change in the diffractive scintillation PDF means that the TOAs will tend toward higher mean/median values, without a loss of data. Thus, we considered the extrapolation from the median S/N in Section 3 to be more robust.

For our analysis, we assumed that configurations C3 through C6 are operational by 2023, a rapid timeline, and assumed that the observing conditions are static into the future. As with all facilities, future upgrades may yield additional sensitivity. In addition, growing RFI will impact timing sensitivity as well. Gancio et al. (2020) show that ∼10% of observing times are significantly affected due to RFI, with significantly less time at night. RFI affects both antennas differently, given their differing proximities to the local IAR offices. The RFI tends to be narrowband, and thus can be excised while retaining the majority of the band. In our work, we ignored the role of RFI, noting that it will play an important but small role in the assumptions we are making, regardless.

Using the single-pulsar-dependent terms in Equation (11) and the instrumental/telescope-dependent components of the TOA uncertainty, defined below, we constructed a metric that describes the overall timing precision of a pulsar in an array (with the subscript i being dropped):

$$\begin{eqnarray}\begin{array}{rcl}M\equiv {\left(\displaystyle \frac{{Tc}}{{\sigma }_{\mathrm{eff}}^{2}}\right)}^{1/2} & = & 34.6\,\mu {{\rm{s}}}^{-1}\,{\left(\displaystyle \frac{T}{10\mathrm{yr}}\right)}^{1/2}{\left(\displaystyle \frac{c}{12{\mathrm{yr}}^{-1}}\right)}^{1/2}\\ & & \times {\left(\displaystyle \frac{{\sigma }_{\mathrm{eff}}}{0.1\mu s}\right)}^{-1}.\end{array}\end{eqnarray} \tag{ 15 }$$

When T is the total observing baseline in years, c is the observing cadence in years⁻¹, and σ_eff is the effective telescope-dependent TOA uncertainty in μs, then M_i has units of μs⁻¹. We define σ_eff here as the combination (quadrature sum) of the short-timescale white noise, the DM estimation uncertainty due to the white noise (Equation (6)), and polarization calibration uncertainty (taken to be 100 ns; see Section 4.1. The short-timescale white noise includes template-fitting, jitter, and scintillation-noise uncertainties. As discussed in Section 2, jitter becomes negligible for the duration of observations conducted by the IAR, but not necessarily for other observatories. Scintillation noise is negligible, given the small scattering timescale.

The quantity M acts like a signal-to-noise ratio, where larger M implies a higher sensitivity, though we note that it is not a direct relationship. Alternatively, this quantity can be viewed as proportional to the (square) inverse of the error on the mean of the residuals.

Figure 1 shows curves of M as a function of time for a number of different telescopes and configurations. The three blue curves denote three different IAR configurations as listed in Table 3, where we take the start time for C2 to be mid-2020, and we assume an aggressive program start time of 2023 January for all other upgraded configurations. Delays will shift the curves to the right. Nonetheless, it is quite clear how drastically instrumentation improvements can improve M.

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Comparisons to other observatories are described in the following subsections.

4.2.1. The Parkes Telescope

4.2.2. The Very Large Array

4.2.3. MeerKAT

4.2.4. Comparing M for Different Observatories