1National Centre for Radio Astrophysics (TIFR), Post Bag 3, Ganeshkhind, Pune
- 411007, India.
2Tata Institute of Fundamental Research, Mumbai 400005, INDIA.
3Raman Research Institute, Bengaluru, Karnataka, India.
4The Institute of Mathetical Sciences, CIT Campus, Taramani, Chennai 600113,
Tamil Nadu, India
5 Homi Bhabha National Institute, Training School Complex, Anushakti Nagar,
Mumbai 400094, India.
6Dept of Physics, IIT Hyderabad, Kandi. Telangana 5022085 India.
7Department of Physics and Astrophysics, University of Delhi, Delhi.
8Amity Centre of Excellence in Astrobiology, Amity University Mumbai 410206,
Maharashtra, India.
9 Manipal Institute of Technology, Manipal 576104, Karnataka, India.
10Department of Physics, Indian Insitute of Technology Roorkee, Roorkee
247667, Uttarakhand, India.
11Jodrell Bank Centre for Astrophysics, University of Manchester, Manchester
M13 9PL, UK.
12The Indian Institute of Science Education and Research, Mohali, India.
13Kumamoto University, Graduate School of Science and Technology, Kumamoto,
860-8555, Japan.
14 Faculty of Advanced Science and Technology, Kumamoto University, Kumamoto,
8608555, Japan.
15Osaka City University Advanced Mathematical Institute, Osaka, 5588585,
Japan.
16Department of Physics, St. Xavier’s College (Autonomous), Mumbai 400001,
Maharashtra, India.
17Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn
(Germany).
18Universität Bielefeld, Fakultät für Physik, Universitätsstr. 25, D-33615
Bielefeld (Germany).
19Arecibo Observatory, University of Central Florida, Arecibo 00612, USA.
20University of Oxford, Department of Physics, Deny’s Wilkinson Building,
Keble Road, Oxford, UK OX1 3RH.
21Department of Physics, BITS Pilani Hyderabad Campus, Hyderabad 500078,
Telangana, India.
22Indian Institute of Space Science and Technology, Thiruvananthapuram, Kerala
695547, India.
23Inter-University Centre for Astronomy and Astrophysics (IUCAA), Pune, India
.
24School of Mathematics, National University of Ireland, Galway, University
Road, Galway H91TK33, Ireland.
25 National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100101, China.
26Faculty of Advanced Science and Technology, Kumamoto University, Japan.
27International Research Organization for Advanced Science and Technology,
Kumamoto University, Japan.

# Nanohertz Gravitational Wave Astronomy during the SKA Era: An InPTA
perspective

Bhal Chandra Joshi1,* Achamveedu Gopakumar2 Arul Pandian3 Thiagaraj Prabu3
Lankeswar Dey2 Manjari Bagchi4,5 Shantanu Desai6 Pratik Tarafdar4 Prerna Rana2
Yogesh Maan1 Neelam Dhanda Batra7 Raghav Girgaonkar8 Nikita Agarwal9
Paramasivan Arumugam10 Sarmistha Banik21 Avishek Basu11 Adarsh Bathula12
Subhajit Dandapat2 Yashwant Gupta1 Shinnosuke Hisano13 Ryo Kato14,15 Divyansh
Kharbanda6 Tomonosuke Kikunaga13 Neel Kolhe16 M. A. Krishnakumar17,18 P. K.
Manoharan19 Piyush Marmat10 Arun Naidu20 K. Nobleson21 Avinash Kumar Paladi22
Dhruv Pathak23 Jaikhomba Singha10 Aman Srivastava6 Mayuresh Surnis11 Sai
Chaitanya Susarla24 Abhimanyu Susobhanan25,1 Keitaro Takahashi26,27

###### Abstract

Decades long monitoring of millisecond pulsars, which exhibit highly stable
rotational periods, in pulsar timing array experiments is on the threshold of
discovering nanohertz stochastic gravitational wave background. This paper
describes the Indian Pulsar timing array (InPTA) experiment, which employs the
upgraded Giant Metrewave Radio Telescope (uGMRT) for timing an ensemble of
millisecond pulsars for this purpose. We highlight InPTA’s observation
strategies and analysis methods, which are relevant for a future PTA
experiment with the more sensitive Square Kilometer Array (SKA) telescope. We
show that the unique multi-sub-array multi-band wide-bandwidth frequency
coverage of the InPTA provides Dispersion Measure estimates with unprecedented
precision for PTA pulsars, e.g., $\sim 2\times$ 10-5 pc cm-3 for PSR
J1909$-$3744\. Configuring the SKA-low and SKA-mid as two and four sub-arrays
respectively, it is shown that comparable precision is achievable, using
observation strategies similar to those pursued by the InPTA, for a larger
sample of 62 pulsars requiring about 26 and 7 hours per epoch for the SKA-mid
and the SKA-low telescopes respectively. We also review the ongoing efforts to
develop PTA-relevant general relativistic constructs that will be required to
search for nanohertz gravitational waves from isolated super-massive black
hole binary systems like blazar OJ 287. These efforts should be relevant to
pursue persistent multi-messenger gravitational wave astronomy during the
forthcoming era of the SKA telescope, the Thirty Meter Telescope, and the
next-generation Event Horizon Telescope.

###### keywords:

gravitational waves—pulsars: general—stars: neutron—ISM: general.

<EMAIL_ADDRESS>
01 April 2022XXXXX

12.3456/s78910-011-012-3 #### 000 0000 1– 1

##  1 Introduction

Gravitational waves (GWs) are propagating ripples in the space-time curvature
as predicted by Einstein’s General Theory of Relativity (Einstein, 1918). The
routine detection of high-frequency GWs (30$-$800 Hz) by the LIGO-Virgo-KAGRA
collaboration111https://www.ligo.org/science/Publication-O3aCatalog/index.php
from around 100 merger events that involve black holes (BHs) and neutron stars
(NSs) in the last 6 years has opened a new window to the universe which is
complementary to the traditional electromagnetic window (Abbott $et~{}al$.,
2019, 2021a, 2021b). These observations are providing key insights into a
number of astrophysical puzzles that are difficult to tackle by traditional
electromagnetic astronomy (Arimoto $et~{}al$., 2021). This includes the
formation channels for stellar-mass BHs, Equation of State for NSs, measuring
the Hubble constant by employing standard and dark sirens, and testing general
relativity in ultra-strong regimes, as well as constraining alternative
theories which dispense with dark energy and dark matter (Arimoto $et~{}al$.,
2021; Mapelli, 2020; Banik & Bandyopadhyay, 2017; Abbott $et~{}al$., 2017a;
Boran $et~{}al$., 2018; Soares-Santos $et~{}al$., 2019; The LIGO Scientific
Collaboration $et~{}al$., 2021b). Therefore, it is reasonable to expect that
the other GW frequency windows should provide opportunities to explore many
aspects of physics, astrophysics and cosmology (Bailes $et~{}al$., 2021).
Specifically, nanohertz stochastic gravitational wave background (SGWB) formed
by the incoherent superposition of GWs from supermassive black hole binaries
(SMBHB) is expected to be detected in the coming years (Antoniadis $et~{}al$.,
2022). The discovery of such nanohertz (nHz) GWs will provide unique insights
into the formation and evolution of galaxies and their constituent SMBHs, a
precise description of solar system, cosmological standard sirens and
fundamental tests of gravitation (Burke-Spolaor $et~{}al$., 2019).

These nHz GWs can be detected by employing the precision timing of an ensemble
of radio millisecond pulsars (MSPs : Detweiler, 1979; Sazhin, 1978). A pulsar
timing array (PTA) is a dedicated experiment to time pulsars with the aim of
detecting low-frequency GWs (PTAs: Foster & Backer, 1990). At present, there
exist four such established efforts that employ the world’s best radio
telescopes along with three more emerging experiments222http://ipta4gw.org/.
PTAs require long-term high-cadence monitoring of a large ensemble of MSPs
using large highly sensitive radio telescopes. The high sensitivity of the
Square Kilometer Array (SKA) telescope with its multiple-beam design will not
only help to discover nHz GW sources, but also will strengthen the post-
discovery nHz GW science with sensitive high-cadence observations. This is one
of the main goals of the pulsar key science project of the SKA (Kramer &
Stappers, 2015; Janssen $et~{}al$., 2015).

While pulsar astronomy has traditionally been carried out with large single-
dish telescopes, SKA is designed to be an interferometer where the signal
collected over its large collecting area is recorded using a beam-former. Out
of the currently operating four PTAs, only the Indian Pulsar Timing Array
(InPTA) primarily uses an interferometer, the upgraded Giant Metre-wave Radio
Telescope (uGMRT), for its observations333It may be noted that Large European
Array for pulsars (LEAP: Bassa $et~{}al$., 2016) also uses a phased array of
multi-element telescopes, but these form a subset of European Pulsar Timing
Array experiment, which is largely based on single dish observations. Another
interferometer, which contributes data to European Pulsar Timing Array
experiment is the Westerbok Synthesis Radio Telescope. Recently, the MeerTime
experiment (Bailes $et~{}al$., 2016) has started collecting data with MeerKat,
which is also an interferometer. It may be noted that these telescopes have
not been used as sub-arrays unlike the uGMRT. The uGMRT is also a pathfinder
telescope for the SKA with features similar to the SKA. Hence, the observation
strategies uniquely adopted in the InPTA program are very much relevant for
initiating the SKA-era PTA program.

One of the unique features of InPTA is characterisation of propagation effects
with unprecedented precision, improving the overall precision of the pulsar
timing required for eventual nHz GW detection. The techniques developed for
the analysis of the low radio frequency data by the InPTA astronomers would
allow the Indian community to meaningfully contribute to PTA work with the
SKA. Additionally, InPTA researchers are developing approaches to model nHz GW
sources and associated detection techniques with a particular emphasis on
orbital eccentricities. These efforts should provide another pathway for us to
contribute to the SKA science efforts, particularly from the multi-messenger
perspectives.

We now present the relevance of the InPTA program for the discovery and
characterisation of nHz GW sources with the SKA telescope, motivated by the
above-mentioned considerations. The structure of the paper is as follows. A
brief review of the current state-of-art in the detection of nHz GWs and the
existing pulsar timing array experiments is provided in Section 2 followed by
a description of the InPTA experiment in Section 3. A comparison of the uGMRT
and the SKA from this perspective is discussed in Section 4 outlining features
of the InPTA experiment which are relevant to a future SKA program. A
discussion of analysis techniques and methods developed for the InPTA program,
which are likely to be carried forward to the SKA program, is presented in
Section 5. Possibilities for multi-messenger astronomy for individual sources
in the context of the SKA are outlined in Section 6. After a discussion on
contributions of the InPTA on development of the Indian and Japanese neutron
star community as future SKA ready users in Section 7, we conclude by listing
future directions in Section 8.

##  2 Pulsar Timing Arrays for Nanohertz GW Astronomy

SMBHBs with total masses in the $10^{8}-10^{10}\,M_{\odot}$ range are the
primary sources of nHz GWs (Detweiler, 1979). Such binaries emit nHz GWs
during their GW emission-dominated orbital evolution phases and are expected
to have orbital periods of years. This is because the natural (orbital)
frequency of such self-gravitating BH binary systems roughly follow
$\omega^{2}\,r^{3}=G\,M$, where $\omega,r$ and $M$ are the orbital angular
frequency, the associated separation and the total mass of the system
respectively (Sathyaprakash & Schutz, 2009). For a typical circular SMBHB
system, the frequency of the emitted GWs is given by (Detweiler, 1979)

$f_{\rm GW}\sim
20\,\text{nHz}\,\left(\frac{200\,M}{r}\right)^{3/2}\,\left(\frac{10^{10}\,M_{\odot}}{M}\right)\,,$
(1)

where we have used the fact that the frequency of GWs emitted from a circular
binary is twice its orbital frequency.

As noted earlier, the existing PTAs monitor an ensemble of MSPs to detect nHz
GWs emitted by SMBHBs (Foster & Backer, 1990; Hobbs & Dai, 2017). This is
because the times of arrival (TOAs) of MSP pulses can be modelled and
predicted with very high accuracy due to the extremely stable rotational
periods of MSPs (Taylor, 1992). It turns out that such predictions require the
use of an accurate timing model that incorporates various astrophysical and
instrumental delays (Edwards $et~{}al$., 2006). Typically, PTAs compare the
observed TOAs from MSPs with predictions for their arrival times based on a
pulsar timing model. The differences between the predictions and the actual
measurements are usually referred to as the pulsar ‘timing residuals’. When a
GW passes between the Earth and an MSP, it affects the geodesics along which
the pulses arrive at the Earth and therefore modulates the pulsar TOAs
(Estabrook & Wahlquist, 1975; Anholm $et~{}al$., 2009). The resulting GW-
induced timing residuals at a given epoch depends on the two GW polarization
states $h_{+,\times}$ such that (Susobhanan $et~{}al$., 2020)

$\displaystyle R(t)=$
$\displaystyle\begin{bmatrix}F_{+}&F_{\times}\end{bmatrix}\begin{bmatrix}\cos
2\psi&-\sin 2\psi\\\ \sin 2\psi&\cos 2\psi\end{bmatrix}$
$\displaystyle\qquad\int_{0}^{t}dt^{\prime}\,\begin{bmatrix}h_{+}(t^{\prime})-h_{+}(t^{\prime}+\Delta_{p})\\\
h_{\times}(t^{\prime})-h_{\times}(t^{\prime}+\Delta_{p})\end{bmatrix}\,,$ (2)

where $\Delta_{p}$ is a geometric delay, $F_{+,\times}$ are known as the
antenna pattern functions, $\psi$ is the GW polarization angle, and the
integration is over the coordinate time measured at the solar system
barycentre. The expressions for $\Delta_{p}$ and $F_{+,\times}$ may be found
in, e.g., Anholm $et~{}al$. (2009).

The existing PTAs are expected to detect first a stochastic GW background
(SGWB) created by the incoherent superposition of GWs from the whole cosmic
population of in-spiralling MBHBs (Pol $et~{}al$., 2021). Moreover, it is
customary to treat the resulting SGWB in the $10^{-9}-10^{-7}$ Hz (GW)
frequency window to be an isotropic Gaussian stationary signal which is
described by a characteristic amplitude $h_{c}$ that follows a power-law
spectrum, $h_{c}\propto f^{-2/3}$ (Phinney, 2001). Further, due to the
quadrupolar nature of GWs, the above SGWB is expected to induce the following
correlation between the timing residuals of each pulsar pair of PTAs (Hellings
& Downs, 1983):

$\displaystyle c(\theta)$
$\displaystyle=\frac{3}{2}x\,\ln{x}-\frac{x}{4}+\frac{1}{2}+\frac{1}{2}\delta(x)\,,$
(3)

where $x=(1-\cos{\theta})/2$ for an angle $\theta$ on the sky between two
pulsars, and $\delta(x)$ is the regular Dirac delta function (the resulting
curve is usually referred to as the Hellings-and-Downs curve). Therefore, the
actual detection of the nHz SGWB, characterised by certain $h_{c}$, involves
detailed investigations that determine how close the observed correlations
between the timing residuals of each pulsar pair of a given PTA follow the
Hellings-and-Downs curve. An incontrovertible detection of the SGWB due to
inspiralling SMBHBs will be made if the timing residuals of all pulsar-pairs
in a PTA are shown to follow the Hellings-and-Downs curve with a common red
spectrum. In order to achieve these goals, PTA experiments usually monitor
MSPs with a cadence of few weeks and the resulting timing data set now spans a
few decades for the three established PTAs (Perera $et~{}al$., 2019). This
ensures that the existing PTA data sets should be sensitive to GWs with
periods from weeks to years corresponding to a nHz GW frequency window
typically in the $10^{-8}-10^{-9}$ Hz range.

Currently, four such experiments are operational with a time-baseline of more
than a few years. While Parkes Pulsar Timing Array (PPTA : Manchester
$et~{}al$., 2013), North American Nano-hertz Observatory for Gravitational
Waves (NANOGrav : Arzoumanian $et~{}al$., 2018) and European Pulsar Timing
Array (EPTA : Desvignes $et~{}al$., 2016) have been gathering data for more
than a decade, an Indian effort that employs the uGMRT, called the Indian
pulsar timing array (InPTA), has been in operation since 2015 (Joshi
$et~{}al$., 2018). Additionally, PTA efforts are being initiated by employing
two more facilities, namely the Five hundred meter radio telescope (FAST) and
the MeerKat (Lee, 2016; Bailes $et~{}al$., 2016). The PTAs periodically pool
their data together to create the International Pulsar Timing Array (IPTA)
data releases. Two such data releases have happened during the last decade
(Verbiest $et~{}al$., 2016; Perera $et~{}al$., 2019). IPTA is a consortium of
consortia that is governed by a steering committee with members from all the
above mentioned PTAs.

All the well-established PTAs regularly make public their data sets along with
their GW astronomy-related analysis results (Alam $et~{}al$., 2020; Kerr
$et~{}al$., 2020; Desvignes $et~{}al$., 2016; Chen $et~{}al$., 2021). Within
the last two years, detailed and independent investigations of the most up-to-
date NANOGrav, PPTA and EPTA data sets reveal the presence of a common red
noise process in their pulsars with spectral signature consistent with an SGWB
due to in-spiralling SMBHBs (Arzoumanian $et~{}al$., 2020; Goncharov
$et~{}al$., 2021; Chen $et~{}al$., 2021). However, these data sets do not
strongly support the presence of the Hellings-and-Downs spatial correlation,
crucial to establish the presence of nHz SGWB that arises from merging SMBHBs.
This is being attributed to the presence of various statistical uncertainties
in the TOA measurements. Similar conclusions were drawn from a detailed
analysis of more sensitive IPTA DR2 data set formed from a combination of
older public data releases from the individual PTAs (Perera $et~{}al$., 2019;
Antoniadis $et~{}al$., 2022). At present, the constituent members of IPTA are
working in tandem by updating their data sets and employing multiple analysis
algorithms to increase the confidence in a possible detection of SGWB in the
coming years.

Unfortunately, the stochastic nature of our astrophysical GWB signal can be
mimicked by other random processes that occur in the PTA pulsars.
Specifically, the epoch to epoch variations in dispersion measure
(DM)444Dispersion measure is defined as the integral of the column density of
electrons over the line-of-sight to the pulsar and scattering variations need
to be accurately incorporated while obtaining TOA residuals from various PTA
data sets. These propagation effects are due to the relative motion of the
pulsar, the earth and the inhomogenous ionised inter-stellar and inter-
planetary medium (IISM and IPM). These are strong functions of frequency
thereby dominating low radio frequency observations of the PTA pulsars (Donner
$et~{}al$., 2020; Tiburzi $et~{}al$., 2021). The wide-band low-frequency
coverage in the InPTA experiment provides very high precision measurements of
these effects (Krishnakumar $et~{}al$., 2021; Nobleson $et~{}al$., 2022). A
comparable precision is expected from a similar future experiment with the
SKA. It was argued that one needs to model these variations appropriately to
ensure the correctness of the astrophysical interpretation of the expected
SGWB signals, extracted from the timing residuals (Lentati $et~{}al$., 2016).
Work is in progress to build such models for incorporating in a future IPTA
data release.

##  3 Indian Pulsar Timing Array experiment

Indian Pulsar Timing Array experiment (InPTA) was started in 2015 using the
Ooty Radio Telescope (ORT: Swarup $et~{}al$., 1971) and the legacy Giant
Metre-wave Radio Telescope (GMRT: Swarup $et~{}al$., 1991) as a pilot
experiment. The InPTA now utilises the unique capabilities of the upgraded
Giant Metre-wave Radio Telescope (uGMRT: Gupta $et~{}al$., 2017), which is
also a pathfinder telescope for the SKA. The salient features of this
experiment are described in this section.

Figure 1: Timing residuals for two PTA pulsars with the ORT during and after
the legacy phase of the proposal. The wander in the residuals before mid-2016
was traced to instability in the observatory time and frequency standards and
associated electronics and was subsequently rectified.

The experiment started as a pilot experiment in 2014, first using the ORT for
observing two pulsars, PSRs J1713+0747 and J1909-3744, to test the precision
timing capabilities of the Indian instruments. A newly commissioned pulsar
receiver called PONDER (Naidu $et~{}al$., 2015) was employed to coherently
dedisperse the data over 16 MHz band-pass at 326.5 MHz and generate time-
stamped integrated pulse profiles (IPPs) using topocentric pulse frequency
calculated from polynomial coefficients with the best known available timing
solutions. The folded time-series were analysed using TEMPO2 (Hobbs
$et~{}al$., 2006; Edwards $et~{}al$., 2006) pulsar timing software package to
obtain timing residuals. Early observations showed residuals varying over a
few milliseconds and with clock jitter (Figure 1). These were traced to
instability in the observatory time and frequency standard and associated
electronics. Based on these initial observations, the observatory
instrumentation was improved. Further observations showed stable timing
residuals demonstrating the feasibility of a PTA experiment and also helped to
obtain refined timing solutions for the next phase.

Encouraged by these results, a pilot program with a sample of 9 pulsars was
initiated using both the legacy GMRT and the ORT in 2015. The multi-frequency
data, acquired in this pilot phase, was used to test the feasibility of a PTA
with both instruments for the first time. The data were used (a) to plan
improvements in the observatory instrumentation, (b) to assess the timing
precision achievable with the legacy GMRT and the ORT, (c) to determine the
fixed pipeline delay between data acquisition systems (Susobhanan $et~{}al$.,
2021), (d) to build timing solutions for a larger sample of PTA pulsars and
(d) to train new members of the collaboration in observing and data reduction.
Different data reduction techniques using a variety of pulsar off-line
processing packages, such as PRESTO555https://www.cv.nrao.edu/s̃ransom/presto/
(Ransom, 2011), SIGPROC666http://sigproc.sourceforge.net/ (Lorimer, 2011),
PSRCHIVE777http://psrchive.sourceforge.net/ (Hotan $et~{}al$., 2004), and
DSPSR888http://dspsr.sourceforge.net/ (Van Straten & Bailes, 2011) were tried
on these data to decide the correct strategy for a bigger PTA experiment in
the future.

| Name | Period | DM | Median S/N | Median S/N | Obs time
---|---|---|---|---|---|---
|  | (s) | (pc cm-3) | 1460 | 500 | (min)
| J0645+5158 | 0.008853 | 18.25 | 7 | 52 | 30
| J1024-0719 | 0.005162 | 6.49 | 13 | 17 | 30
| J1455$-$3330 | 0.007987 | 13.57 | 9 | 19 | 30
| J1614-2230 | 0.003151 | 34.49 | 12 | 18 | 30
Exploratory InPTA | J1640+2224 | 0.003163 | 18.43 | 20 | 34 | 30
| J1730$-$2304 | 0.008123 | 9.62 | 61 | 56 | 30
| J1738+0333 | 0.005850 | 33.77 | 7 | 10 | 30
| J2317+1439 | 0.003445 | 21.91 | 8 | 21 | 30
| J2302+4442 | 0.005192 | 13.79 | 9 | 24 | 30
| J1643$-$1224 | 0.004622 | 62.41 | 129 | 423 | 50
| J1713+0747 | 0.004570 | 15.91 | 254 | 174 | 50
| J1857+0943 | 0.005362 | 13.31 | 74 | 134 | 30
Classic InPTA | J1909$-$3744 | 0.002947 | 10.39 | 77 | 289 | 50
| J1939+2134 | 0.001558 | 71.02 | 159 | 323 | 20
| J2145$-$0750 | 0.016052 | 9.00 | 103 | 946 | 50
| J2124$-$3358 | 0.004931 | 4.60 | 39 | 208 | 50
| J0437$-$4715 | 0.005757 | 2.64 | 958 | 2343 | 15
| J0613$-$0200 | 0.003061 | 38.77 | 61 | 119 | 50
| J0751+1807 | 0.003479 | 30.25 | 32 | 59 | 30
Expanded InPTA | J1012$+$5307 | 0.005255 | 9.02 | 259 | 359 | 30
| J1022$+$1001 | 0.016452 | 10.25 | 38 | 417 | 20
| J1600$-$3053 | 0.003597 | 52.32 | 71 | 67 | 50
| J1744$-$1134 | 0.004075 | 3.14 | 72 | 343 | 50
Table 1: The pulsar sample of the InPTA. The sample labelled ”Classic InPTA”
was observed from 2018$-$2022, whereas the sample labelled ”Exploratory InPTA”
was observed only in 2018$-$2019\. Likewise, the sample labelled ”Expanded
InPTA” was mostly observed only after 2021 with high sensitivity. The
”Exploratory InPTA” sample also included all pulsars listed in the ”Classic
InPTA” and ”Expanded InPTA” sample. The ”Classic InPTA” and the ”Expanded
InPTA” pulsars are being observed using Band 3 and 5 with 200 MHz bandwidth,
whereas ”Exploratory InPTA” sample was observed with Band 3, 4 and 5 with 100
MHz bandwidth.

The GMRT was upgraded between 2010 and 2017 (Gupta $et~{}al$., 2017) providing
a seamless frequency coverage from 50 to 1500 MHz with five new wide-band
feeds (Band 1 : 50$-$80 MHz; Band 2 : 120$-$250 MHz; Band 3: 250$-$500 MHz;
Band 4 : 550$-$850 MHz and Band 5 : 1050$-$1450 MHz). The wide-band data are
acquired with new wide-band receivers and back-end, capable of recording data
with 400 MHz band-pass, as beam-former time-series from 4 simultaneous beams.
After an initial exploratory experiment to assess the timing precision of the
uGMRT with 19 pulsars, a strategy to observe each pulsar in more than one
frequency band simultaneously with the uGMRT was adopted for the ongoing InPTA
experiment since 2018.

The uGMRT is a multi-element interferometer much like the SKA. The large
sensitivity of the uGMRT is achieved by compensating the geometric and
instrumental phase delays (including ionospheric delays) across the
interferometric array, consisting of 45-m diameter antennas, to form a phased
array. While a phased array of 20 antennas provides a collecting area of about
180-m diameter single dish, an eight antenna phased array synthesises a 125-m
single dish. In addition, the relatively smaller beam of the phased array
helps in both reducing the sky background as well as in filtering uncorrelated
noise as compared to a single dish telescope. The second advantage of an
interferometer like the uGMRT is a possibility of forming multiple phased
arrays using different groups of antennas, called sub-arrays. If different
sub-arrays are set up to observe at different frequencies, this allows
simultaneous multi-frequency observations covering a large bandwidth. On the
other hand, multiple sources can be covered at the same frequencies by
multiple sub-arrays. Thus, the uGMRT not only provided a sensitivity similar
to the large telescopes used in other PTA experiments, but an opportunity to
try out new observations strategies with an interferometer. The ongoing InPTA
experiment has explored this flexibility of an interferometer to optimise the
precision of the experiment.

In the beginning, the InPTA used three uGMRT bands (400$-$500, 650$-$750 and
1360$-$1460 MHz in Band 3, 4 and 5 of the uGMRT respectively) by grouping five
to 15 antennas in three sub-arrays using three out of four beams of the uGMRT.
As the uGMRT back-end (GMRT Wideband Backend - GWB : Reddy $et~{}al$., 2017)
provides a coherent dedispersion capability for a maximum of 200 MHz bandwidth
(De & Gupta, 2016), two of the Bands (Band 3 and 5) were coherently
dedispersed to remove dispersive effects of the IISM with 100 MHz band-pass
respectively, while Band 4 was observed with a 1024 channel digital filter-
bank. While this strategy sampled the 300$-$1500 MHz with three non-contiguous
bands, the smaller bandwidth as well as lesser number of antennas in each band
limited the achievable sensitivity of InPTA observations.

We addressed this shortcoming since 2019 by using only two sub-arrays with 10
and 15 antennas in Band 3 and 5. As most of PTA pulsars are relatively low DM
pulsars (2.6 to 30.0 pc cm-3), the dispersion smear for these pulsars can be
kept smaller than the sampling time employed (40.96 $\mu$s) by using a filter-
bank with 1024 sub-bands or more at Band 5. This permitted using the entire
200 MHz bandwidth for coherent dedispersion at Band 3 to eliminate the
significantly larger dispersive smear at frequencies near 300 MHz. Thus, the
data were observed in both bands with 200 MHz bandwidth with coherent
dedispersion used only in Band 3. The total integration time for all the
pulsars was also increased to one hour compared to 30 minute observations in
2018. This strategy provided 4 times larger sensitivity than earlier
observations significantly reducing ToA uncertainties as well as improving DM
precision as is shown later.

In this new strategy, the monitoring observations were started with 6 bright
pulsars and the sample is being gradually increased to 14 pulsars(Table 1).
New pulsars were selected from the pulsars which formed a part of the IPTA
Data Release I (Verbiest $et~{}al$., 2016). Apart from choosing only the
pulsars visible with the GMRT (Dec $>$ -50∘), the signal-to-noise ratio (S/N)
achieved in the exploratory phase of the experiment (labelled ”Exploratory
InPTA” in Table 1) with the selected configuration was used to make a
shortlist. As a good sampling of Hellings-Down overlap function (Hellings &
Downs, 1983) is required for detection of SGWB, new pulsars were added to
optimise a good coverage of this overlap function with pulsar pairs (Figure
2).

Figure 2: Histograms for the distribution of pairwise angular separations
between the InPTA pulsars. The sample was gradually increased to achieve a
more uniform sampling of spatial separations (Hellings-Downs overlap function
(Hellings & Downs, 1983)) over the course of the experiment. The red and green
histograms represent the sample over two 6 month observing semesters for the
uGMRT.

The current data set for InPTA, including the pilot, legacy and exploratory
uGMRT phases spans about 8 years. However, the span and cadence of individual
pulsars in our sample have varied over the years due to the considerations
mentioned above. While most of the recent dataset has been taken using the
upgraded GWB (Reddy $et~{}al$., 2017), the frequency bands and receivers
employed in observations prior to 2018 have varied. All the legacy GMRT and
the uGMRT data were acquired with two hands of polarisation, but the ORT data
were single polarisation data due to the nature of the instrument. Lastly, the
pipeline delays as well as other data acquisition features are well
characterised for the uGMRT receivers. As a consequence, the collaboration has
decided to limit its upcoming first public data release to observations with
the uGMRT alone to maintain the homogeneity in data properties.

Figure 3: Timing residuals obtained with data observed simultaneously over 300
to 1460 MHz for PSRs J1713+0747 and J1909$-$3744 are shown in this figure
after a DM fit over this range without using any frequency dependent FD
parameter or DMJUMP. Epoch by epoch fits provide very precise DMs in the InPTA
experiment.

The pulsar data generated by uGMRT beamformer observations are recorded in a
channelised time series binary format by the GWB, together with a timestamp
representing the start of the observation. A pipeline named
pinta999https://github.com/abhisrkckl/pinta (Susobhanan $et~{}al$., 2021) was
developed to process such datasets obtained using the uGMRT for the InPTA
experiment. This pipeline performs RFI mitigation using two different
softwares, RFIClean101010https://github.com/ymaan4/RFIClean (Maan $et~{}al$.,
2021) and gptool111111https://github.com/chowdhuryaditya/gptool, and produces
partially folded PSRFITS archives which can be used for further analysis using
standard pulsar software such as PSRCHIVE. After the data reduction offline
with this pipeline, the multi-band data were further analysed with post-
processing pipelines, DMCalc 121212https://github.com/kkma89/dmcalc.git and
PulsePotratiture (Pennucci $et~{}al$., 2014) to obtain precision DM
measurements. These pipelines were developed and/or tuned for the uGMRT
datasets and yield ToAs apart from DM measurements. These were analysed with
pulsar timing software, TEMPO2, to obtain the final timing solutions as well
as timing residuals, which form the input for GW analysis.

Figure 4: DM measurements using the data from 300 to 1460 MHz obtained for
PSRs J1713+0747 and J1909$-$3744 in the InPTA experiment.

The experiment has resulted in the highest precision measurements of DM
reported so far by any PTA, primarily due to a combination of simultaneous
Band 3 and Band 5 wide-band data covering 300$-$1460 MHz, which is unique to
our experiment. Such a combination is affected by misalignment due to profile
evolution and scatter broadening over this frequency range. As explained later
(Section 5), analysis of ToAs generated using frequency resolved templates
showed normally distributed residuals without using FD parameters or DMJUMP to
align Band 3 and Band 5 data (Figure 3). Consequently, we were able to achieve
high precision with the median uncertainty as small as 2 $\times$ 10-5 pc cm-3
for PSR J1909$-$3744\. The variation of DM for PSRs J1713+0747 and
J1909$-$3744 are shown in Figure 4 as an illustration. The SKA has a similar
frequency coverage and is likely to provide these measurements with similar or
better precision using a strategy similar to the InPTA. With these
measurements, we have also achieved $\mu$s post-fit residuals, comparable to
those obtained with the other PTA experiments and the preliminary results from
the upcoming public data release of the InPTA are shown in Figure 5.

Figure 5: Timing residuals for PSRs J1713+0747 and J1909$-$3744 with the data
obtained in the InPTA experiment are shown as a function of observation
epochs. The RMS post-fit residuals achieved are indicated in the legend

While we did not attempt observations of multiple sources using multiple
beams-sub-arrays, the experience from our strategy can be extrapolated to
evaluate such an observing strategy. The InPTA observations provide useful
experience to understand the phasing constraints from an ionospheric
turbulence point of view for different observations. This is important to
decide the configurations of interferometer antennas for the sub-arrays to be
used, particularly for the most distant antennas in the sub-array.

##  4 Lessons from InPTA for GW science with SKA

The multi-element interferometer nature of SKA makes it an instrument similar
to the uGMRT, albeit with a much larger sensitivity. Hence, our experience
with the uGMRT, which is also a recognised path-finder for the SKA, can inform
the observations and analysis strategy for a future PTA with the SKA. In this
section, some possibilities based on the InPTA experience are discussed.

The SKA consists of two telescopes, both of which will be constructed in two
phases. Here, we provide a top level description of the capabilities of the
phase-1 of the telescope, which will represent about 10 percent of the final
SKA collecting area. The SKA-mid will be located in Karoo, South Africa. It
will consist of 197 dishes spread over three spiral arms on an area of a
diameter of about 150-km. These include 133, 15-m dishes and 64, 13.5-m
MeerKat dishes. It is planned to have three frequency bands operational in the
first phase. While Band 2 covers 0.95 to 1.76 GHz and is likely to become
available first, Band 5 (4.6 $-$ 15.3 GHz) and Band 1 (0.35 $-$ 1.05 GHz) will
be commissioned subsequently. The planned back-end consists of a beam-former
providing 16 simultaneous beams with coherent dedispersion carried over 300 to
800 MHz band-pass. About 120 antennas, consisting of 46 MeerKat antennas, are
located in a cluster of dimensions 2 km by 2 km, whereas the rest of the 77
antennas are spread beyond these baselines. Further, about 20 antennas each
from the spiral arms (15 SKA dishes and 5 MeerKat dishes) can be used as two 3
km by 3 km clusters. While further arm antennas can also be used as phased
arrays, the baselines between these antennas are too long for a stable phase
solution at a frequency below 2 GHz. These baselines for the outer spiral arms
are probably useful as phased arrays at Band 5, where ionospheric propagation
effects are much smaller. However, the sensitivity of a Band 5 tied array is
smaller than Band 2 for the same number of antennas as the SKA-mid antennas
are less sensitive at Band 5. Moreover, pulsars are steep spectrum sources and
fainter at Band 5. Hence, only 120 to 140 antennas are considered here for
high sensitivity PTA observations.

GMRT | Single antenna | Subarray
---|---|---
Band 2 |  | 8 antennas
Aeff/Tsys ($m^{2}/K$) | 3.8 | 30
SEFD (Jy) | 727 | 91
Band 3 |  | 8 antennas
Aeff / Tsys ($m^{2}/K$) | 10.5 | 84
SEFD (Jy) | 263 | 33
Band 4 |  | 5 antennas
Aeff / Tsys ($m^{2}/K$) | 9.7 | 48
SEFD (Jy) | 286 | 57
Band 5 |  | 15 antennas
Aeff / Tsys ($m^{2}/K$) | 9.2 | 138
SEFD (Jy) | 300 | 20
Table 2:  The system equivalent flux density estimated for the uGMRT sub-array
configurations employed in the InPTA experiment

The SKA-low telescope consists of 131,072 dual polarised log periodic antenna-
based aperture arrays, located in Western Australia in Murchinson Radio-
astronomy Observatory. These dipoles are arranged in 512 stations in the
central core and three spiral arms located approximately 65 km apart with each
station consisting of 512 dipoles. The core consists of 224 stations with the
largest baseline of 1 km with 13 more clusters, each with 6 stations, located
within 2 km of the core. The central core can be used as two phased arrays,
each proving a physical aperture of 127020 m2 or as a single phased array of
two times that area.

SKA-mid uses a real-time digital back-end for pulsar timing, called Pulsar
Timing Engine (PST : Keane, 2018). The PST helps to record high time
resolution full-Stokes pulsar data. It receives 16 tied-array beams, with each
beam pointing on a pulsar to be timed concurrently. Observation durations can
be from 180 to 1800 seconds. Each beam can be coherently dedispersed over 300
to 800 MHz in real-time to provide high time resolution data, which can be
further processed to provide frequency-resolved sub-integrated time-stamped
IPP in the PSRFITS format (Hotan $et~{}al$., 2004). The multiple beams in both
SKA-mid and SKA-low allow multiple targets/bands to be observed concurrently
making both these instruments very flexible for different observing
strategies.

The uGMRT, which is a pathfinder telescope for the SKA, is similar to the SKA
due to its multi-band, multi-beam and multi-sub-array nature. The 30 antennas
of the GMRT can be grouped into 4 sub-arrays to operate over 5 frequency bands
using 4 beams provided by the GMRT back-end (See Section 3). Using our
experience for such a strategy, a PTA program with the SKA can utilise at
least 4 sub-arrays in the SKA-mid and two sub-arrays in the SKA-low for
simultaneous multi-frequency observations as well as concurrent observations
of multiple targets. Apart from optimal utilisation of telescope time for the
pulsar Key Science Project (KSP), such a strategy will also provide a wide-
frequency coverage for separating ISM noise from the precision timing in a way
similar to that in the InPTA. Some possibilities are explored below for a
suitable strategy for a PTA program with the SKA.

| Single | Single | All
---|---|---|---
SKA Low | station | subarray | subarrays
|  |  | within 1 km
Low Band | - | 100 | 200
(50-350 MHz) | - | stations | stations
Aeff / Tsys ($m^{2}/K$) | 1.10 | 110.00 | 220.00
SEFD (Jy) | 2,510.27 | 25.10 | 12.55
| Single | Single | All
SKA MID | antenna | subarray | subarrays
|  |  | within 2 km
Mid Band 1a |  |  |
(0.35-0.65 GHz) | - | 30 antennas | 120
Aeff/Tsys ($m^{2}/K$) | 2.10 | 63.00 | 252.00
SEFD (Jy) | 1,314.90 | 43.83 | 10.96
Mid Band 1b |  |  |
(0.65-1.05 GHz) | - | 30 antennas | 120
Aeff / Tsys ($m^{2}/K$) | 4.20 | 126.00 | 504.00
SEFD (Jy) | 657.45 | 21.92 | 5.48
Mid Band 2 |  |  |
(0.95-1.76 GHz) | - | 30 antennas | 120
Aeff / Tsys ($m^{2}/K$) | 11.00 | 330.0 | 1,320.00
SEFD (Jy) | 251.03 | 8.37 | 2.09
Table 3: The system equivalent flux density estimates for the proposed sub-
array configurations for a PTA experiment using the SKA Low and SKA Mid
telescopes

A comparison of the system equivalent flux density (SEFD) for the uGMRT arrays
employed in the InPTA experiment and those for the SKA-mid and SKA-low sub-
arrays, shown in Tables 2 and 3, suggests that a 30 antenna SKA sub-array is
equivalent to 6, 12 and 35 antenna uGMRT sub-array at band 3, 4 and 5 of the
uGMRT respectively. While such SKA sub-arrays provide marginally smaller
sensitivity than the uGMRT Band 3 sub-array used in the InPTA experiment, a
higher sensitivity is indicated for sub-arrays at other bands. With about 120
antennas in the central part of the SKA-mid, this suggests the use of a
maximum 4 sub-arrays using 4 beams with sensitivity similar to or greater than
the uGMRT. For weaker pulsars, one could instead use only two or even a single
SKA-mid sub-array to achieve almost four times the uGMRT sensitivity. We
recommend a possible configuration of 4 sub-arrays with the SKA-mid telescope
as shown in Figure 6. Given that the available bandwidth is 300/800 MHz
compared to 200 MHz used in the InPTA program, one would achieve much larger
sensitivity than the uGMRT sub-arrays in practice. While 16 sub-arrays are in
principle possible in the SKA, this comes at the cost of the sensitivity of
each sub-array and is not recommended based on the InPTA experience, where
only 2 sub-arrays are used instead of possible 4 in the current ongoing
observations. A similar comparison with the SKA-low configuration suggests
that two sub-arrays using each half of SKA-low core stations provide
significantly larger sensitivity than Band 2 of the uGMRT (See Tables 3 and
2). We recommend a possible configuration of 2 sub-arrays with the SKA-low
telescope as shown in Figure 7. Since SKA-mid and SKA-low are two different
telescopes with independent correlator and beam-former, coordinated
observations between the telescopes can provide the low-frequency support for
higher frequency observations with a SKA-PTA covering an unprecedented
frequency range of 50 $-$ 1800 MHz to provide the most sensitive PTA dataset
ever.

Figure 6:  The proposed sub-array configuration for a pulsar timing experiment
with the SKA-mid Phase-1 Telescope

The two to four sub-array configurations, discussed above, can be employed in
a variety of ways. One option would be to use the same observing strategy as
InPTA, where the different sub-arrays are used for simultaneous multi-
frequency observations of the same target pulsar (hereby referred to as OPTION
I). This provides a wide frequency coverage, useful for characterising the ISM
noise, the jitter noise and bias due to profile evolution across the
frequency. The other option would be to use each sub-array to observe a
different target pulsar concurrently (OPTION II). This option optimises the
use of telescope time to cover a much larger sample of PTA pulsars. A third
option is to use a hybrid strategy combining both multi-frequency and multi-
target strategies (OPTION III). Finally, fainter pulsars can be covered with
the full 120/200 antenna array, one source at a time, with three observations
employing a different frequency band (OPTION IV). It may be noted that the
brightness of each pulsar (or equivalently the achievable S/N) determines the
precision of ToAs. Consequently, the number of sub-arrays possible depends on
the distribution of flux density of the PTA pulsars, which also is a critical
factor in deciding the most useful observing strategy. Thus, given the pulsar
sample, all four of the observing strategies may be required.

Figure 7:  The proposed sub-array configuration for a pulsar timing experiment with the SKA-low Phase-1 Telescope  List of pulsars | Number of Pulsars | Observing time required
---|---|---
|  | Option I/II/III/IV
|  | (hrs) )
J0437-4715, J0621+1002, J0636+5128, J0900-3144, J1022+1001 |  |
J1045-4509, J1600-3053, J1603-7202, J1640+2224, J1643-1224 | 24 | 4/2/2
J1713+0747,J1730-2304, J1732-5049, J1747-4036, J1744-1134 |  |
J1832-0836, B1855+09, J1903+0327, J1909-3744, B1937+21 |  |
J1944+0907, J2124-3358, J2145-0750, J2234+0944 |  |
J0023+0923, J0030+0451, J0034-0534, J0218+4232, J0340+4130 |  |
J0610-2100, J0613-0200, J0711-6830, J0751+1807, J0931-1902 |  |
J1024-0719, J1453+1902, J1455-3330, J1614-2230, J1721-2457 |  |
J1738+0333, J1741+1351, J1802-2124, J1804-2717, B1821-24A | 38 | 22
J1853+1303, J1910+1256, J1911-1114, J1911+1347, J1918-0642 |  |
J1923+2515, J1946+3417, J1949+3106, B1953+29, J2010-1323 |  |
J2017+0603, J2033+1734, J2043+1711, J2129-5721, J2234+0611 |  |
J2214+3000, J2302+4442 J2317+1439 |  |
Table 4: Recommended observation strategy for a PTA program with the SKA
telescope. The pulsar sample was divided into two categories. The bright
pulsars could be observed for 10 minutes each using Option I (Single Target
simultaneous observations with three multi-band sub-arrays), Option II
(Multiple targets with 4 sub-arrays in three non-simultaneous multi-band
observations) or Option III(two targets with two band sub-arrays each). For
the fainter pulsars in the second category, option IV is recommended with the
full array utilised in three multi-band observations)

With the above considerations in mind, we classified the PTA sample into two
different categories. The first category consists of the brightest pulsars,
where we adopt OPTION I, II or III using the maximum number of sub-arrays
available for each telescope optimising the telescope time as well as
frequency coverage. The second category of pulsars consists of fainter
pulsars, where option IV is appropriate. Given the advertised parameters of
SKA telescopes, the required observation time for each pulsar in the
respective categories using these options was estimated from the radiometer
equation using the flux density estimates from the ATNF pulsar catalog
(Manchester $et~{}al$., 2005)
131313https://www.atnf.csiro.au/research/pulsar/psrcat/. These calculations
provide an estimate for the required telescope time. While details of some
parameters of SKA-telescopes are still being worked out in the construction
phase and a detailed optimisation is planned in future, some recommendations
are made below based on InPTA experience and available information.

A list of target pulsars to be observed using the suggested four options for
the SKA-mid telescope is provided in Table 4. The total observing time
required for each epoch of observations is also indicated in this table. We
find that a sample of 62 pulsars, visible with the SKA telescope, can be
observed with about 26 hours with the SKA-mid telescope. Likewise, seven hours
are needed for each epoch with the SKA-low telescope. Assuming a 10-day
cadence, the SKA key science program will require 936 hours for a PTA program
with the SKA. This estimate can vary depending on the number of available
antennas in the early construction phase of the SKA telescope (which increases
progressively from the current 64 MeerKat antennas during AA1 to AA* phase) as
well as on achievable SEFD and backend configuration and evolving radio
Frequency Interference environment. Finally, the observing proposed here can
be further optimised by reserving the SKA observations to weaker pulsars or
pulsars which are affected by significant inter-stellar propagation effects.
While a more detailed study is planned in future incorporating this
information, the recommendation in this section already provides a blue-print
for an SKA PTA, probably for the first time.

In conclusion, the experience gained with different configurations of the
uGMRT interferometer for frequencies between 300 to 1500 MHz in the InPTA
experiment is useful to propose observation configurations that can be used
once the SKA phase 1 telescope becomes available. Suggestions for the required
observing strategies are described in some detail in this section and can form
the basis of future SKA-PTA program of the SKA pulsar key science project.

##  5 Analysis methodology from InPTA relevant to SKA

The reduction and the analysis of the InPTA data, particularly that obtained
at the lower frequency band, required the development of new techniques or
tuning the techniques available in the literature. As mentioned in Section 3,
the uGMRT raw data were reduced by using an off-line pipeline, called pinta,
to obtain partially folded PSRFITS archives. While the SKA will employ a real-
time pipeline to process pulsar observations, the RFI mitigation and excision
techniques (RFIClean and gptool) will be relevant for the SKA pipeline as
well, particularly for Band 1 data from SKA-mid and data from SKA-low. The
final product of SKA real-time pipeline is also expected to be in the PSRFITS
format, which will allow the InPTA techniques for determination and
application of precision DMs to be incorporated in the SKA Science Data
processor easily. The experience relevant to the analysis of the data from the
SKA beam-former is described in this section.

###  5.1 Online RFI-excision before partially folding the data

Radio frequency interference (RFI) remains a growing concern to obtain quality
data and achieve the designed sensitivity. For most of the PTA experiments,
the data are processed in real-time to output data that are partially folded
over the pulsar spin period. While this real-time processing results in highly
reduced data rates as well as requirements of disk-space and computing
resources for off-line processing, it is much harder to identify and excise
RFI from the resultant partially integrated data. Moreover, a variety of RFI,
e.g., the periodic RFI, faint and short time-scale RFI, and temporally varying
RFI that mostly causes baseline variations, is not possible to mitigate from
the partially integrated data as it requires analysing contiguous temporal
sections of data to identify such RFI. The above understanding is reinforced
by our InPTA data processing experience, wherein mitigation of RFI in the
filterbank data using the excision tools RFIClean (Maan $et~{}al$., 2021) and
gptool greatly aid in the detection significance. In fact, in some particular
cases, the pulsars are detected only after RFI is excised using these tools.
The SKA will also employ a real-time pipeline to process pulsar observations.
To fully utilise the telescope’s sensitivity, it is desirable that the above
or similar RFI excision tools are employed in the SKA’s real-time data
reduction pipeline to mitigate RFI before the data are partially folded. The
computation time estimates based on the InPTA experiment (Susobhanan
$et~{}al$., 2021) suggest that these can run in real-time, particularly on
GPGPU based hardware as the source code is easily portable to GPGPU kernels.

###  5.2 Narrow-band technique for times-of-arrival and dispersion measure
estimation

Precise estimation of times-of-arrival (ToA) and DM at lower frequencies with
the help of traditional narrow-band technique is impeded by the following
factors: (1) the measured DM and frequency-dependent profile evolution are
intertwined, (2) uncorrected DM variations can introduce smear while
frequency-collapsing the data to degrade the S/N, and (3) systematic biases
can also get introduced in the estimated DM due to interstellar scatter
broadening. A frequency resolved narrow-band analysis method was developed for
the post-processing of the InPTA data. This method is called DMcalc and
utilises python interface of PSRCHIVE package (Krishnakumar $et~{}al$., 2021).

The ToAs are obtained in a timing experiment by cross-correlating the data
profiles with a noise-free template. Traditionally, a noise-free template is
obtained by averaging frequency-collapsed IPP from several epochs. Epoch to
epoch DM variations can introduce significant smear when multiple widely
separated epochs are averaged to obtain a noise-free template. Additionally,
an incorrect alignment of the pulse across the frequencies for pulsars showing
significant evolution of profiles with frequency leads to DM offsets. It may
be noted that both these effects are significantly larger at lower
frequencies. Indeed, our work shows that DM estimates using a single
frequency-collapsed template or an offset fiducial DM used to dedisperse the
template itself, can have constant offsets from the actual DMs (Krishnakumar
$et~{}al$., 2021). Traditionally, this is handled by using FD parameters or
DMJUMPS in the timing analysis.

In the InPTA narrow-band analysis, we use a frequency-resolved template and an
iterative procedure to estimate DM of each epoch. The number of sub-bands used
in the frequency resolved template is a trade-off between the profile
evolution and the achievable S/N in each sub-band and all DM measurements use
frequency resolved ToAs obtained by cross-correlating the data with frequency
resolved noise-free templates. The sub-band profiles in the templates for each
epoch are aligned by determining the epoch dependent DM in an iterative
procedure. Here, two or more sub-integrations Band 3 data of each epoch are
analysed using ToAs obtained by cross-correlating time-collapsed profiles of
the epoch aligned first with an assumed DM obtained from pdmp analysis
(Krishnakumar $et~{}al$., 2021). This analysis is repeated in subsequent
iterations progressively improving the DM estimate. A final frequency resolved
noise-free template is then formed by averaging the epochs aligned using the
DM obtained in the final iteration.

The main conclusion from the InPTA analysis with the procedure described here
is that FD parameters or DMJUMPS are not needed for any of the pulsars in the
InPTA sample. Considering that the sample includes pulsars which show
significant profile evolution from 300 to 1460 MHz as well as significant
scatter broadening at frequencies around 400 MHz (e.g. PSRs J1643$-$1224 and
J2145$-$0750), this suggests that using such frequency resolved analysis
appears to be more robust. Please note that our simultaneous multi-frequency
observations avoid any possibility of small variations in DM between near-
simultaneous high and low frequency observations adopted in other PTA
experiments. Additionally, we use higher precision DM measurements from 300 to
500 MHz. Thus, we recommend following an observing strategy and analysis
similar to the InPTA for a PTA experiment with the SKA. Thus, OPTION I and III
(See Section 4) are preferred options provided the required S/N is achievable
for the target pulsars.

The post processing pipeline DMcalc was developed for application on the
combined multi-band data with the noise-free template updated with the
fiducial DM obtained in the previous steps. It is straight-forward to
incorporate this in the SKA science data processor with a possibility of
providing real-time measurements. In turn, these will be useful for a quick
follow-up of DM events, such as those reported in PSR J1713+0747 (Lam
$et~{}al$., 2018), as well as updating DM for coherent dedispersion for the
subsequent observations.

###  5.3 Wideband technique for simultaneous measurement of times-of-arrival
and dispersion measure

Pulsar observations are increasingly using wideband receivers (defined as
receivers with a fractional band-pass of 100 percent or greater) to acquire
more precise TOAs (Gupta $et~{}al$., 2017; Johnston $et~{}al$., 2021). Pulse
profiles can evolve significantly over such wide bandwidths. This introduces a
bias in arrival times as well as DM measurements when such data are analysed
with a single band-collapsed 1-dimensional template. Pennucci $et~{}al$.
(2014) devised an algorithm for generating a 2-dimensional template and for
using these for simultaneous measurement of DMs and ToAs from wideband pulsar
data. They proposed pulsar timing with such wide-band ToAs(Pennucci, 2019).
This ”wideband timing technique” has been applied on various datasets, such as
the NANOGrav 12.5-year data (Alam $et~{}al$., 2021) and the CHIME and GBT-L
data in the 400$-$800 MHz frequency range (Fonseca $et~{}al$., 2021).
Recently, we applied this technique to the uGMRT data in the 300$-$500 MHz
frequency range (Nobleson $et~{}al$., 2022). As the profile evolution is much
more dominant for many pulsars below 400 MHz apart from a significant smearing
due to pulse broadening, this study demonstrated the effectiveness of this
technique below 400 MHz for the first time.

The wideband timing technique uses frequency-resolved high S/N IPP to
decompose this 2-dimensional data into principal component eigen-profiles
which capture the variation of the profile with the observing frequency. A
noise-free 2-dimensional PulsePortraiture is generated by fitting spline
functions to the coefficients of eigen-profiles required for modeling the
2-dimensional data 141414https://github.com/pennucci/PulsePortraiture
(Pennucci $et~{}al$., 2016). The phase offset of the wide-band data with the
pulse portrait is obtained by least-square minimization of the phase offset of
each sub-band pulse profile incorporating the cold-plasma dispersion law. This
yields both a high precision ToA and DM estimate. The timing solution and
residuals for a PTA pulsar are then obtained by maximising a likelihood
function, implemented in the software package TEMPO (Nice $et~{}al$., 2015),
with these ToAs and DM estimates as inputs.

Pulsars, such as PSRs J1643$-$1224 and J1939+2134, show significant profile
evolution due to scatter-broadening in the InPTA sample, whereas PSR
J2145$-$0750 shows an evolving profile. Analysis of Band 3 InPTA data needed
three eigen-profiles to model the pulse-portrait of the former two pulsars,
while two eigen-profiles were required for the latter (Nobleson $et~{}al$.,
2022). We also noticed that RFI contributes dominant eigenvalues and
corresponding noisy eigen-profiles, so it is important to remove RFI before
the principal component analysis for this technique to work well. The wavelet
and spline fits needed to be appropriately tuned for the technique to work at
our frequency. This experience can easily be adopted for SKA data.

We are currently developing the technique to include the likelihood function
used in the TEMPO2 (Hobbs $et~{}al$., 2006) software package which will use
the wideband DM measurements from the ToAs as priors to estimate the DM model
parameters. We also plan to develop a technique to combine multi-band data in
the regime of wideband analysis. The standardisation of techniques as part of
the InPTA analysis is directly relevant to the SKA due to the wide multi-band
frequency coverage of the SKA in view of the observations strategy proposed in
this paper.

##  6 Persistent Multi-Messenger GW Astronomy with the SKA

The LIGO-Virgo-KAGRA collaboration has inaugurated the era of GW astronomy due
to their routine detection of hecto-hertz GWs from around 100 stellar-mass BH
binaries since 2015 (Abbott $et~{}al$., 2020; The LIGO Scientific
Collaboration $et~{}al$., 2021a). Further, this collaboration heralded the era
of multi-messenger GW astronomy by the observations of transient hectohertz
GWs from a binary neutron star merger (GW170817) and its counterparts
(EM170817) across the electromagnetic spectrum (Abbott $et~{}al$., 2017b, c).
Strikingly, these multi-messenger observations of such a $100$ second-long
transient GW event are already providing fundamental contributions to the
areas of physics, astrophysics, and cosmology (Abbott $et~{}al$., 2017b).
Recall that multi-messenger astronomy usually involves combining data from
various messengers that arise from spectacular astrophysical events.

During the SKA era, we should be able to practice what we term the persistent
multi-messenger GW astronomy (Valtonen $et~{}al$., 2021). This is influenced
by the fact that the rapidly maturing PTAs should be detecting a stochastic
nHz GW background created by a population of merging SMBHBs in the coming
years (Antoniadis $et~{}al$., 2022). Thereafter, continuous nHz GWs from
bright individual SMBHBs that stand above this background should be routinely
detected with the help of the SKA-era PTA (Burke-Spolaor $et~{}al$., 2019).
The eventual detection of continuous nHz GWs from individual SMBHBs should
allow us to pursue multi-wavelength electromagnetic observations of such GW
sources as they are expected to reside in the active galactic nuclei (Burke-
Spolaor $et~{}al$., 2019; Xin $et~{}al$., 2021). In other words, persistent
multi-wavelength electromagnetic observations of such continuous GW sources
should allow us to pursue multi-messenger GW astronomy during the SKA era
(Valtonen $et~{}al$., 2021). It turns out that the resulting persistent multi-
messenger nHz GW astronomy can make profound contributions to the emerging
field of multi-messenger GW astronomy (Burke-Spolaor $et~{}al$., 2019).

The InPTA researchers are well-placed to contribute to the various aspects of
this emerging area of astronomy during the SKA era. This is partly due to our
involvement in the general relativity-based predictions of certain
Bremsstrahlung flares from the unique bright blazar OJ 287 and the successful
observational campaigns of these predicted flares (Dey $et~{}al$., 2018, 2019;
Laine $et~{}al$., 2020). These investigations and the associated campaigns
provided firm observational evidence for the presence of a nHz GW-emitting
SMBH binary in the blazar OJ 287. The underlying model, developed by Mauri
Valtonen and his collaborators, prescribes the observed double-peaked
Bremsstrahlung flares from OJ 287 to the repeated impacts of a secondary BH
with the accretion disk of the primary BH twice during its general
relativistic eccentric orbit. Notably, the 2019 Eddington flare from OJ 287,
predicted in Dey $et~{}al$. (2018), was successfully observed with the Spitzer
space telescope and this campaign even allowed us to provide a parametric
constraint on the celebrated BH no-hair theorem (Laine $et~{}al$., 2020).

These successful observational campaigns of a number of predicted impact
flares in optical wavelengths prompted us to explore the possibility of using
other electromagnetic wavelengths to probe the SMBH binary central engine
description for OJ 287 (Valtonen $et~{}al$., 2021). Fortunately, several high-
resolution Very Long Baseline Interferometry (VLBI) observational campaigns
have tried to image the radio jet of OJ 287 during the past three decades
(Hodgson $et~{}al$., 2017; Cohen, 2017). These observational campaigns reveal
that the position angle (PA) of the projected jet on the sky plane shows
systematic variations at both millimeter and centimeter wavelengths (Dey
$et~{}al$., 2021). Therefore, some of us pursued detailed investigations that
allowed us to explain the observed PA variations of OJ 287’s radio jet while
employing the now established SMBH binary central engine description (Dey
$et~{}al$., 2021). Further, we have provided specific predictions that should
be testable by the GMVA and EHT campaigns on OJ 287, planned for the coming
years. These efforts are ensuring that multi-band EM observations of OJ 287
will be pursued in the coming years to probe various aspects of its SMBH
binary central engine description (Valtonen $et~{}al$., 2021). It is expected
that such multi-wavelength and multidisciplinary efforts will be required to
pursue multi-messenger nHz GW astronomy with OJ 287 during the SKA era.

We are also developing general relativistic approaches that will be required
for extracting nHz GWs from isolated SMBH binaries from the SKA era PTA data
sets. Specifically, efforts are ongoing to develop ready-to-use
computationally efficient routines that model the expected pulsar timing
residuals induced by GWs from isolated SMBH binaries inspiraling along general
relativistic eccentric orbits (Susobhanan $et~{}al$., 2020). These approaches
employ post-Newtonian(PN) approximation for describing the general
relativistic SMBH binary dynamics and this approximation provides general
relativity based corrections to Newtonian dynamics in terms of a dimensionless
parameter that involves SMBH binary orbital velocity (Blanchet, 2014). In
Susobhanan $et~{}al$. (2020), we employed 3PN-accurate Keplerian parametric
solution to model general relativistic eccentric orbits of SMBH binaries while
an improved version of GW phasing approach was employed to incorporate the
effects of GW emission (Memmesheimer $et~{}al$., 2004; Damour $et~{}al$.,
2004). It may be recalled that these two elements, namely PN-accurate
Keplerian type parametric solution and the GW phasing approach, are employed
in the widely used Damour-Deruelle timing formula to time MSPs in general
relativistic eccentric orbits (Damour & Deruelle, 1986). Further, we are
incorporating the spin effects to model pulsar timing residuals induced by
inspiral GWs from spinning SMBH binaries in PN-accurate eccentric orbits with
the help of Königsdörffer & Gopakumar (2005)

In Figure 8, we show the amplitude of the expected GW-induced pulsar residuals
for pulsars across the sky for two SKA era nHz GW emitting SMBH binaries,
namely blazars OJ 287 and PKS 2131-021 while focusing only on the earth term
contributions (O’Neill $et~{}al$., 2022). The inference of a nHz GW emitting
SMBH binary in PKS 2131-021 is attributed to a very recent effort that
demonstrated two epochs of strong sinusoidal variation with essentially the
same period and phase in the 45.1-year radio light curve (O’Neill $et~{}al$.,
2022). A natural explanation is that this blazar is hosting an SMBH binary
with an orbital period of $\sim 4.75$ years (redshifted) in contrast to $\sim
12$ years for OJ 287. The heat maps in Figure 8 reveal that SKA-era PTA
efforts should be able to pursue persistent multi-messenger nHz GW astronomy
with these two sources that may involve other facilities like the Thirty Meter
Telescope and The Next Generation Event Horizon Telescope (Valtonen
$et~{}al$., 2021).

With the help of such general relativistic constructs, we are working to
constrain the presence of isolated eccentric SMBH binaries in PTA data sets,
influenced by Arzoumanian $et~{}al$. (2021). Additionally, we are modelling
general relativistic burst with memory PTA signals due to SMBH binaries in PN-
accurate hyperbolic orbits with the help of Cho $et~{}al$. (2018). These
efforts should allow us to contribute to the ongoing efforts to constrain the
presence of such isolated SMBH binaries in the existing IPTA data sets and
their updates. Clearly, such efforts will be crucial to detect routinely
isolated GW sources in the SKA era PTA data sets and therefore pursue
persistent multi-messenger GW astronomy during that era.

Figure 8: Distribution of the GW-induced Earth term amplitudes, namely $max-
min$ of the Earth term contributions to the timing residuals, for pulsars
around two different sky locations associated with two SMBHB candidates: OJ
287 and PKS2131-021. The binary parameters of OJ 287 are taken from Dey
$et~{}al$. (2018) while we employ O’Neill $et~{}al$. (2022) for PKS2131-021.
Influenced by O’Neill $et~{}al$. (2022), we let the total mass $M=2\times
10^{9}$ and mass ratio $q=1$ for the second candidate. The GW-induced
residuals are calculated by extending the method of Susobhanan $et~{}al$.
(2020) while incorporating PN-accurate GW emission effects for tracking the
evolution of the SMBHBs.

##  7 InPTA and the development of SKA user community in Japan and India

The InPTA consortium is playing an active role in building a scientific
community that will have expertise to use the SKA products to pursue its key
science projects. Taking a cue from the LIGO collaboration, which recruited
many high energy physicists into gravitational wave astronomy (after the
cancellation of the SSC), we have tried to build a community of researchers
from diverse areas such as Cosmology, General Relativity, Nuclear and High
Energy Physics, Astronomical Instrumentation, Neutron Star Physics, Solar
Physics to list a few examples. These researchers have nicely complemented the
original core group, which consisted of radio astronomers working on various
aspects of pulsar astronomy.

At present, InPTA hosts seven undergrad students, nine graduate students, and
ten postdoctoral researchers and we expect this collaboration to grow further
as we head into the SKA era. All these young researchers are now well
experienced in performing pulsar observations simultaneously at different
uGMRT bands in the sub-array mode, which will be a necessity in the SKA era.
There were some other undergraduate students, who were members of InPTA
earlier and then moved to different research fields for their Ph.D, as well as
some former postdocs who have moved on to non-academic jobs. We gather that
the training they received as members of InPTA remains invaluable to them.
Furthermore, several InPTA members who started with InPTA projects while being
based at Indian institutes have continued to remain active after received
their under-graduate and graduate degrees, in spite of taking on new positions
abroad.

To train new members, InPTA organises training schools and busy weeks (where
the whole collaboration members gather for in person interactions). Further,
young InPTA members are encouraged to present research results in national and
international levels and they participate in outreach activities. InPTA
members have organised a conference series titled “Asia SKA Initiative On NS”
twice, first time in 2016 in India and the second time in 2019 in Thailand.
Both of the meetings were attended by scientists from India, China, Japan,
Thailand, etc. The participants were from various stages of their careers.
InPTA members also played a key role to form a consortium called
‘Gravitational Radiation and Science with Pulsars’ (GRASP) that involve
researchers from South Africa and China in addition to InPTA members. Monthly
meetings of GRASP is playing a significant role in exchange of knowledge and
will help in future international collaborations in the SKA era. Finally,
InPTA hosted the annual meeting of the IPTA during June 2019 in Pune, India.
This meeting included a student training week followed by a conference week.
In addition to the 2019 meeting, InPTA members have acted as as members of the
organising committees of other IPTA meetings too, as well as delivered
lectures in student weeks and presented research works.

In what follows, we provide a brief timeline of our consortium. InPTA started
in 2015 as a pilot program consisting of only five members using the legacy
GMRT. It was accompanied by another pilot program using uGMRT during
2016$-$2017\. Thereafter, the two programs merged and InPTA program using
uGMRT emerged with 19 members during 2018. We were an associate member of the
IPTA consortium until 2021, when we were given the full membership. In 2021,
four researchers from Japan (based at Kumamoto University) joined the InPTA
consortium. Hence the word “InPTA” is now a moniker for an Indo-Japanese
pulsar timing array program using an Indian telescope, but is not confined to
only Indian researchers.

##  8 Conclusion

The observing strategy and analysis methods used in the InPTA are useful in
informing the design of a PTA experiment with the SKA telescope. A precision
better than 2 $\times$ 10-5 pc cm-3 for pulsars such as PSR J1909$-$3744,
already demonstrated in our analysis with the InPTA in this paper, is possible
with much larger sensitivity and frequency coverage provided by the SKA. We
recommend using the SKA-low and SKA-mid as one to four sub-arrays respectively
to cover frequencies from 50 to 1800 MHz concurrently based on the InPTA
experience. We show that such observing can be used for a 62 pulsar sample
three times a month requiring 936 hours similar to observing time required by
currently on-going PTA experiments. The development of general relativistic
constructs by the InPTA astronomers promise better tuned methods for search
for isolated SMBHB GW sources in the PTA data-sets in the SKA era as well as
constrain the astrophysics of these sources with multi-messenger observations.
Last, but not the least the InPTA experiment is developing a vibrant Indo-
Japanese user community for GW physics with the SKA telescope when this
telescope comes online later in the decade.

As the SKA-construction plan gets underway, it is possible to devise an
optimised observing plan complimentary to the four major PTA experiments (the
EPTA, the InPTA, the NANOgrav and the PPTA) for a higher cadence coverage of
an extended sample of PTA pulsars. This involves an optimisation in
sensitivity and frequency coverage using different combination of available
antennas as they become available in a phased manner from 2024 on-wards. A
design of such an optimal strategy is in progress and will be reported in
future.

A potential discovery of the nano-Hertz GW is likely in near future. This can
further inform the strategy best suited for post-discovery GW science in the
SKA-era. The discussion presented here will hopefully motivate a deeper
discussion amongst the major PTA experiments and the IPTA for future
directions of exciting GW astrophysics with the SKA, which is likely to
revolutionise the radio astronomy with its Square kilo-meter collecting area
in the next two decades.

## Acknowledgements

This work is carried out by InPTA, which is part of the International Pulsar
Timing Array consortium. We thank the staff of the GMRT who made our
observations possible. GMRT is run by the National Centre for Radio
Astrophysics of the Tata Institute of Fundamental Research. BCJ, PR, AS, SD,
LD, and YG acknowledge the support of the Department of Atomic Energy,
Government of India, under project identification # RTI4002. BCJ and YG
acknowledge support from the Department of Atomic Energy, Government of India,
under project # 12-R&D-TFR-5.02-0700. AS is supported in part by the National
Natural Science Foundation of China grant No. 11988101.

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