The local nanohertz gravitational-wave landscape from supermassive black hole binaries

Articles
https://doi.org/10.1038/s41550-017-0299-6
© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
1
Max Planck Institute for Radio Astronomy, Bonn, Germany. 2
TAPIR, MC 350-17, California Institute of Technology, Pasadena, CA, USA. 3
Jet Propulsion
Laboratory, California Institute of Technology, Pasadena, CA, USA. 4
School of Physics and Astronomy and Institute of Gravitational Wave Astronomy,
University of Birmingham, Birmingham, UK. 5
Department of Astrophysical Sciences, Princeton University, Princeton, NJ, USA. 6
Department of Astronomy,
University of California Berkeley, Berkeley, CA, USA. 7
Department of Physics and Astronomy, West Virginia University, Morgantown, WV, USA. 8
Center for
Gravitational Waves and Cosmology, West Virginia University, Morgantown, WV, USA. Present address: 9
Center for Computational Astrophysics, Flatiron
Institute, New York, NY, USA. *e-mail: cmingarelli@flatironinstitute.org
S
upermassive black holes (SMBHs) are widely held to exist at the
heart of massive galaxies1
. Galaxy mergers should form super-
massive black hole binary (SMBHB) systems, which eventually
emit gravitational waves and merge2
. Galaxy mergers are a funda-
mental part of hierarchical assembly scenarios, forming the back-
bone of current structure formation models. Thus, the detection of
gravitational waves from merging SMBHs would be of far-reaching
importance in cosmology, galaxy evolution and fundamental phys-
ics, providing information not accessible by any other means.
Pulsar timing arrays (PTAs) can detect nanohertz gravitational
waves by monitoring radio pulses between millisecond pulsars,
which are highly stable clocks. Gravitational waves change the
proper distance between the pulsars and the Earth, thus inducing a
delay or advance of the pulse arrival times. The difference between
the expected and actual arrival times of the pulses—the timing
residuals—carries information about the gravitational waves that is
extracted by cross-correlating the pulsar residuals3–6
. Current PTAs
include European PTA7
(EPTA), the North American Nanohertz
Observatory for Gravitational Waves (NANOGrav)8
, the Parkes
PTA9
and the International PTA (IPTA)10
, the latter being the union
of the former three.
Here we introduce a bottom-up approach to constructing both
realistic gravitational-wave skies and future IPTA projections: we
use IPTA pulsars with their real noise properties, and galaxies
from the 2 Micron All Sky Survey (2MASS)11
, together with galaxy
merger rates from the Illustris cosmological simulation project12,13
,
to form multiple probabilistic realizations of the local gravitational-
wave Universe. In each realization, we search for SMBHB systems
that emit continuous gravitational waves (CGWs) in the PTA band,
and also compute their contribution to the nanohertz gravitational-
wave background (GWB) and its anisotropy14,15
. We report on the
physical properties of the most frequently selected SMBHBs and
their host galaxies, and estimate their time to detection.
Galaxy selection
SMBHB merger timescales can be of the order of 109
years after the
galaxy merger, and therefore morphological merger signatures can be
difficulttoidentify.Ourfocushereisonmassiveearly-typegalaxies,as
these are likely to have formed from major mergers and would there-
fore host SMBHBs with approximate mass ratios, q, of 0.25 ≤ q ≤ 1.
Toapproximateamassselection,weselectintheK-bandusingthe
2MASS11
Extended Source Catalog16
, following the procedure out-
lined in detail elsewhere17
, but to a distance of 225 Mpc and over the
fullsky.Wedonotexcisethegalacticplane,buttherearefewerreliable
sources there. We convert from the 2MASS K-band luminosity (MK)
to the stellar mass (M*) through = . − . +M Mlog ( ) 10 58 0 44( 23)
*10 K ,
appropriateforearly-typegalaxies18
,andapplyaK-bandcutMK ≤ −25
to select galaxies with stellar mass of > ~ ⊙M M10
*
11
, as these are
likely to host SMBHBs19
. At distances >225 Mpc, 2MASS itself
begins to become incomplete at ≈ −M 25K . Although our sample is
not formally volume-complete within our chosen mass and distance
cuts, it includes the majority of massive, nearby galaxies, which are
expected to dominate the signal for PTAs.
This process creates a galaxy catalogue with 5,110 early-type gal-
axies. In earlier work15,17
, it was found that 33 of these galaxies contain
dynamically measured SMBHs (Fig. 1a). Moreover, we manually
add a further nine galaxies from 2MASS: NGC 4889, NGC 4486a,
NGC 1277, NGC 1332, NGC 3115, NGC 1550, NGC 1600, NGC
7436 and A1836 BCG, which did not make the luminosity cut, but
which also host dynamically measured SMBHs.
The SMBHB total mass, M = M1 + M2, is estimated by taking
the stellar mass to be the bulge mass, and applying the M•−Mbulge
The local nanohertz gravitational-wave landscape
from supermassive black hole binaries
Chiara M. F. Mingarelli1,2,3,9
*, T. Joseph W. Lazio3
, Alberto Sesana4
, Jenny E. Greene5
, Justin A. Ellis3
,
Chung-Pei Ma6
, Steve Croft6
, Sarah Burke-Spolaor7,8
and Stephen R. Taylor2,3
Supermassive black hole binary systems form in galaxy mergers and reside in galactic nuclei with large and poorly constrained
concentrations of gas and stars. These systems emit nanohertz gravitational waves that will be detectable by pulsar timing
arrays. Here we estimate the properties of the local nanohertz gravitational-wave landscape that includes individual supermas-
sive black hole binaries emitting continuous gravitational waves and the gravitational-wave background that they generate.
Using the 2 Micron All-Sky Survey, together with galaxy merger rates from the Illustris simulation project, we find that there are
on average 91 ± 7 continuous nanohertz gravitational-wave sources, and 7 ± 2 binaries that will never merge, within 225 Mpc.
These local unresolved gravitational-wave sources can generate a departure from an isotropic gravitational-wave background at
a level of about 20 per cent, and if the cosmic gravitational-wave background can be successfully isolated, gravitational waves
from at least one local supermassive black hole binary could be detected in 10 years with pulsar timing arrays.
Nature Astronomy | www.nature.com/natureastronomy

Articles Nature Astronomy
empirical scaling relation20
with scatter ε0 between M• and Mbulge
(where M• is the black hole mass), as described in the Methods.
Dynamically measured SMBH masses are fixed.
Probability of a galaxy hosting a SMBHB in the PTA band. We
attack this problem by estimating two quantities: the probability
that a SMBHB is in the PTA band, and the probability that a given
galaxy has a SMBHB. The former is estimated from quantities that
are derived from 2MASS, such as each galaxy’s stellar mass, M*,
and the SMBHB mass M as estimated through the M•−Mbulge rela-
tionship. The latter is obtained from the cumulative galaxy–galaxy
merger rates from the Illustris cosmological simulation project12,13
.
A realization of the local nanohertz gravitational-wave sky is
created as follows. The first quantity is the probability that a SMBHB
is in the PTA band, tc/Tz, where π= ∕ − ∕ − ∕
t f(5 256)( )c
8 3
c
5 3
is the
time to coalescence of the binary and = ∕ +∕ ∕
 q q M[ (1 ) ]c
5 3 2 5 3
is
the chirp mass.
In each realization, all galaxies in the sample are assigned
an SMBH with total mass M, and with mass ratio q drawn from
a log-normal distribution in [0.25,1]. The time to coalescence is
computed from 1 nHz, which is the beginning of the PTA band.
Tz is the effective lifetime of the binary: this is the sum of the
dynamical friction21
(tdf) and stellar hardening22
(tsh) timescales
(details in Methods).
The second quantity is the probability that a galaxy hosts a
SMBHB. This is computed from the Illustris cumulative galaxy–
galaxy merger rate, μN t M zd / d ( , , )
* *
where μ* is the stellar mass ratio
of the galaxies, taken at the beginning of the binary evolution. This
75°
60°
45°
22 h 20 h 18 h 16 h 14 h 12 h 10 h 8 h 6 h 4 h 2 h
22 h 20 h 18 h 16 h 14 h 12 h 10 h 8 h 6 h 4 h 2 h
22 h 20 h 18 h 16 h 14 h 12 h 10 h 8 h 6 h 4 h 2 h
30°
15°
–15°
–30°
–45°
–60°
–75°
2MASS galaxy distribution
a b
dc
log10(h), GW sky at f = 3.79 × 109
Hz
log10(h), strain of EPTA-detected SMBHBs in 2MASS
–14.75 –14.50 –14.25 –14.00 –13.75 –13.50 –13.25
–14.4
10–11
10–12
10–13
10–14
10–15
10–9
10–8
Gravitational-wave frequency (Hz)
EPTA 2016
Detected with sky maps
10–7
10–6
Strain
–14.3 –14.2 –14.1 –14.0 –13.9 –13.8
0°
75°
60°
45°
30°
15°
–15°
–30°
–45°
–60°
–75°
0°
75°
60°
45°
30°
15°
–15°
–30°
–45°
–60°
–75°
0°
Fig. 1 | The best pulsars boost the number of continuous gravitational-wave detections by a factor of 4. a, Distribution of galaxies in the 2MASS galaxy
catalogue. b, Example of an all-sky gravitational-wave (GW) strain sensitivity map (where h is the strain), shown here at f = 3.79 nHz, with our own
reprocessing of data from previous work25
. The strain upper limit on the sky near to the best six pulsars (orange stars) is better constrained by a factor of
four, with respect to the average strain, compared to that near the 35 other pulsars (white stars). c, All detected SMBHB host galaxies over multiple Monte
Carlo realizations. We find that 131 gravitational-wave skies out of 75,000 contain SMBHB emitting detectable continuous gravitational waves in the PTA
band, using the all-sky gravitational-wave sensitivity maps, as in panel b. Orange stars as in panel b. The size and colour of the circles indicate the relative
strain of the source. d, The sky-averaged strain upper limit of CGW sources25
. Only 34 of the 131 detected sources (red dots) from panel c lie above the
upper limit curve, demonstrating that sky-averaged strain underestimates the number of detectible gravitational-wave sources by a factor of about 4.
For IPTA projections, see Table 1.
Table 1 | Probability of detecting nearby CGWs with IPTA
Sky-averaged strain All-sky gravitational-wave strain sensitivity map
FAP 15 years 20 years 25 years 15 years 20 years 25 years
5 × 10−2
2% (0.1%) 24% (0.3%) 100% (0.8%) 8% (0.4%) 96% (1%) 100% (3%)
3 × 10−3
0.5% (0.03%) 9% (0.2%) 48% (0.3%) 2% (0.1%) 36% (0.8%) 100% (1%)
1 × 10−4
0.3% (0.01%) 4% (0.08%) 27%(0.2%) 1% (0.04%) 16% (0.3%) 100% (0.8%)
The predictions are reported for different false alarm probabilities (FAP), for total length of IPTA dataset, and if the pulsars are dominated by white (red) noise. This probability is the total number of
sources from all realizations lying above a given detection curve, divided by the total number of realizations, with a maximum of 100% (even though multiple sources may be detected; Supplementary
Fig. 1). For Gaussian distributions, 1-FAPs are 2σ, 3σ and 3.9σ. PTA datasets are now between 10 years and 15 years long; hence, 25-year projections are likely 10 years from now. Using an all-sky
gravitational-wave strain sensitivity map may add an additional factor of 4 in gravitational-wave strain sensitivity, as seen in Fig. 1, which highlights the importance of pulsar positions on the sky with
respect to the gravitational-wave source.

ArticlesNature Astronomy
redshift z is calculated at Tz with Planck cosmological parameters23
.
The merger rate is then multiplied by time elapsed since then, Tz. Thus,
the probability, pi, of galaxy i hosting a SMBHB in the PTA band is
∫ μ μ=
.
p
t
T
N
t
M z Td
d
d
( , , ) (1)
* * *i
i
z
z
c,
1
0 25
Calculating Tz is essentially rewinding the SMBHB evolution:
starting in the gravitational-wave emission phase, we calculate how
long the binary spent in a stellar hardening phase, and then in a
dynamical friction phase, for binary separations out to the effective
radius of the galaxy. We can only rewind by 12.5 Gyr (z = 4), as this
is the maximum z from ref. 12
. If Tz > 12.5 Gyr (equivalently z > 4),
the evolution of the binary has probably stalled, and we set pi = 0.
The total number of SMBHBs emitting gravitational waves in the
PTA band, for each Monte Carlo realization, is the sum of all these
probabilities: = ∑N pi iSMBHB . This number varies from realization
to realization. We draw NSMBHB galaxies from the galaxy catalogue
according to the probability distribution P = pi/NSMBHB.
For each of these selected galaxies, we compute the strain h, aver-
aged over inclination and polarization,
= π +
∕
∕
h
D
f z
32
5
[ (1 )] (2)c
5 3
L
2 3
assuming circular binary orbits, where f is the gravitational-wave
frequency and DL is luminosity distance. Each of the probabilis-
tically selected galaxies hosts an SMBHB in the PTA band with
f > 1 nHz, where the evolution is assumed to be dominated by
gravitational-waveemission.WeassigneachSMBHBagravitational-
wave frequency24
:
= π −− − ∕
− ∕








f t t
256
5
( ) (3)1
c
5 8
c
3 8
by drawing tc from a uniform distribution in [26 Myr, 100 yr] (for-
mally, we set tc = 0 and sample in −t), which is the time to coales-
cence of an SMBHB with = = ⊙M M M101 2
9
from 1 nHz and 100 nHz,
respectively. We note that the upper bound of 26 Myr for tc does not
limit the detectability of lower mass binaries: if we consider a gal-
axy with MK = −25 and therefore = . × ⊙M M2 88 10
*
11
, then, via the
relationship M•−Mbulge, = . × ⊙M M8 76 108
. As the maximum tc is
26 Myr, this binary has a minimum gravitational-wave frequency of
1.7 nHz, but one can see in Supplementary Fig. 3e that this is one of
the likeliest gravitational-wave frequencies for detectable SMBHBs.
Galaxy distances are estimated by means of techniques outlined in
the Methods section. We use the approximation z = 0 only in equa-
tion (2), since gravitational-wave sources are all <225 Mpc away.
Projections for the IPTA. Continuous nanohertz gravitational-
wave upper limits come in two types: (1) a limit as a function of
sky location of the gravitational-wave source, gravitational-wave
frequency and pulsar sky location (Fig. 1b), and (2) a sky-averaged
strain upper limit as a function of gravitational-wave frequency,
which averages over all pulsar sky locations (Fig. 1d). Because
upper limit curves, and not detection curves, were reported in ref. 25
,
we use the upper limit as a proxy for detection to underline the
importance of the pulsar and SMBHB sky location. Indeed, the PTA
response to CGWs is maximal when the gravitational-wave source
lies very close to the pulsar5,26
.
For IPTA projections, we are more rigorous, constructing an
IPTA-like array to estimate the time to detection of CGW sources in
2MASS. We start with the 49 IPTA pulsars10
, and from 2016 onward
we add four pulsars per year from sky locations accessible with IPTA
telescopes, using the median white noise uncertainty of 300 ns.
Detection probability curves are computed using thep statistic27
and false alarm probabilities28
(FAPs) of 5 × 10−2
, 3 × 10−3
and
1 × 10−4
. For Gaussian distributions, these 1-FAPs are 2σ, 3σ and
3.9σ, respectively.
Gravitational-wave backgrounds. The search for the GWB is ongo-
ing7–10,29,30
, with detection expected31
in 5 to 7 years. Although most
PTAs publish limits on isotropic GWBs, searching for and charac-
terizing GWB anisotropy is gaining momentum32
. This is done by
decomposing the sky on a basis of spherical harmonics14,33,34
, Ωℓ
Y ( )m ,
and with pixel bases35
.
In an effort to understand the contribution of these CGW
sources to the GWB, we transform a gravitational-wave sky real-
ization to a GWB. This is done by casting each individual binary’s
strain contribution to the closest pixel in a HEALPix sky map, and
computing the characteristic strain Δ= ∑ ∕h h f fk k kc
2 2
, where hk is the
strain of source k, f is its gravitational-wave frequency, and Δf is the
inverse observation time25
.
The total power on the sky is 4π, and is decomposed as
Ω Ω= ∑ℓ ℓ ℓ
 P c Y( ) ( )m m m,
, where Ω is the direction of gravita-
tional-wave propagation. Anisotropy is described in terms of
= ∑ ∕ ℓ +ℓ =−ℓ
+ℓ
ℓC c (2 1)m m
2
, normalized to the isotropic compo-
nent, C0. We calculate the angular power spectrum of a gravita-
tional-wave sky and take the monopole value as the contribution to
the overall isotropic GWB.
Results
Galaxies hosting SMBHBs. We compute the probability of each
galaxy in the catalogue containing an SMBHB emitting gravi-
tational waves in the PTA band, f ≥ 1 nHz. We carry out multiple
realizations of the galaxy catalogue, sampling over black hole mass,
mass ratio and time to coalescence. We find that, on average, there
are 91 ± 7 galaxies hosting SMBHBs in the PTA band and 7 ± 2
stalled SMBHBs, despite the inclusion of a stellar hardening phase36
to overcome the ‘final parsec problem’ (Supplementary Fig. 2).
Over multiple realizations, fewer than 1% of gravitational-wave
skies hosted a currently detectable SMBHB. We also note that for
25-year datasets, binaries with c approximately equal to ⊙M109
are the likeliest to be detected, with more-massive binaries being
disfavoured (Supplementary Fig. 3).
Six pulsars dominate the EPTA’s sensitivity to CGWs, leading to
an upper-limit map of the sky that is approximately dipolar, with the
strain sensitivity behind the best pulsars being a factor of 4 higher
(Fig. 1b). Although we use the EPTA as an example, results are simi-
lar for NANOGrav, PPTA and the IPTA.
Anisotropy and contributions to the gravitational-wave back-
ground. In Fig. 2a, we show an example realization of the local
nanohertz gravitational-wave sky with a loud SMBHB, chosen at
random. In this realization, NGC 4472 hosted a PTA-detectable
SMBHB. Of course, NGC 4472 was only one of 87 galaxies in this
realization, but it was the only one that contained a binary that
was loud enough to be detected. Assuming an isotropic GWB with
an amplitude of a few times 10−16
and a 25-year dataset, we find
that such a single CGW source contributes less than 1% to the
overall strain budget. This is to be expected, because a CGW is
the ultimate anisotropy, and therefore contributes very little to an
isotropic GWB. The strong CGW source does, however, dominate
the angular power spectrum of the sky (Fig. 2d), where we mea-
sure GWB anisotropy. When the strong CGW source is removed
(Fig. 2c), we find anisotropy from the superposition of gravita-
tional waves from undetected CGW sources at the level of about
20%, dominating the angular power spectrum of the GWB up to
ℓ≈ 10 (Fig. 2d).

Time to detection. Evidence for nanohertz gravitational waves will
increase slowly and continuously. We estimate the time to detec-
tion of nearby gravitational-wave sources with the IPTA, requiring
a 95% detection probability under different FAPs: 5 × 10−2
, 3 × 10−3
,
1 × 10−4
(2σ, 3σ and 3.9σ for Gaussian distributions). This is done
for 15-year, 20-year and 25-year datasets, noting that current datas-
ets are 10–15 years. Hence, 25-year predictions are for 10 years from
now. Results for detections with 3 × 10−3
FAP are in Supplementary
Fig. 1, and summarized in Table 1.
We find that strong red noise in pulsar residuals greatly dimin-
ishes the chance of detecting CGW sources in the next 10 years.
If, however, the pulsar noise is white or if CGW sources can be
extracted from an unresolvable GWB, then there is a 50% chance
of detecting a local CGW source with a 3 × 10−3
FAP in the next
decade. Even more encouragingly, when one uses the all-sky gravi-
tational-wave strain sensitivity map for detections, a detection with
10−4
FAP is possible in 10 years (Table 1).
Discussion
Using the sky location and noise properties of IPTA pulsars, mas-
sive galaxies in 2MASS and galaxy merger rates from Illustris,
we estimate when and where PTAs are likely to detect CGW
sources in the local Universe. Over multiple realizations of the
local gravitational-wave sky, we find that ≪1% of gravitational-
wave sources would have been detected with current PTA data30
(Fig. 1c), supporting the conclusions that current non-detection is
unsurprising25
.
In making IPTA predictions, we did not include new telescopes
that will come on line in the next 10 years, such as MeerKAT37
and
possibly SKA Phase 138
. Both telescopes will greatly increase PTA
sensitivity in the Southern Hemisphere.
Future IPTA detections depend on how successfully the GWB
can be subtracted. The red noise in the pulsars is meant to emulate an
unresolved GWB with A = 4 × 10−16
(where A is the amplitude of the
GWB), consistent with ref. 39
. Although some pulsars exhibit intrin-
sic red noise, the pulsars that offer the highest-precision timing—
which contribute most to CGW sensitivity—are broadly consistent
with being white-noise dominated.
An overview of the detected CGW parameters is given in
Supplementary Fig. 3. Interestingly, massive galaxies such as M87 have
a lower probability of being selected, as this depends on ∝ − ∕
tc c
5 3
.
Therefore, binaries in, for example, M104 are more likely to host
SMBHBs in the PTA band.
We performed a brief literature search on the likeliest galaxies
to host SMBHBs (Supplementary Fig. 3a, b) to assess whether they
showed signs of merger or a current candidate binary. We find that
NGC 3115 is the only object currently under investigation as a can-
didate binary or recoiling black hole40
. Although many of the other
candidates in the list of top SMBH red-noise candidates show signs
of recent or ongoing merging activity, many galaxies in this mass
range are involved in merging, and a more complete comparison
between the merger properties of this sample and those of the gen-
eral population is beyond the scope of this work.
Methods
Galaxy selection from 2MASS. We select our initial sample from the Two Mass
Redshift Survey (2MRS41
). We first make a very broad selection of objects with
K < −22 and radial velocity distance D < 250 Mpc. We then cross-correlate with the
2 h4 h6 h8 h10 h12 h14 h16 h18 h20 h22 h
–75°
–60°
–45°
–30°
–15°
0°
15°
30°
45°
60°
75°
1 2 3 4 5 6
× 10–15
× 10–17
× 10–16
Strain, h
a
0.0 0.5 1.0 1.5 2.0 2.5
Characteristic strain, hc
b
0 1 2 3 4 5 6
Characteristic strain, hc
c
0 10 20 30
ℓ
40 50
0.0
0.2
0.4
0.6
0.8
1.0
Cℓ/C0
All sky
Sky without NGC 4472
PTA angular resolution
d
2 h4 h6 h8 h10 h12 h14 h16 h18 h20 h22 h
2 h4 h6 h8 h10 h12 h14 h16 h18 h20 h22 h
–75°
–60°
–45°
–30°
–15°
0°
15°
30°
45°
60°
75°
–75°
–60°
–45°
–30°
–15°
0°
15°
30°
45°
60°
75°
Fig. 2 | The GWB from nearby CGW sources. a, Example realization in which an SMBHB was detected in NGC 4472 (yellow star). There were 86 other
galaxies hosting SMBHBs with f > 1 nHz (purple circles), but these were too faint to be detected. b, The 87 CGW sources from panel a turned into a GWB,
assuming a 25-year IPTA dataset. This background is clearly dominated by the strong CGW source in NGC 4472. c, The GWB formed by sources in panel
a without the CGW source in NGC 4472. d, The angular power spectrum of the GWBs both with and without NGC 4472 (solid blue line and dashed-dot
orange line respectively), assuming a 25-year dataset. The angular resolution of current PTAs is 𝓁 ≈ 4 (vertical dashed green line). Even without the CGW
source in NGC 4472, the other 86 SMBHBs produce anisotropy at the level ≥20% up to 𝓁 = 15.

Crook group catalogue42
to correct all radial velocities for galaxies in our sample to
the radial velocity of the group. In practice, this has a negligible impact on most of
the galaxies in the sample, which reside at the centres of their groups. Finally, with
group distances and magnitudes in hand, our sample has 5,110 massive early-type
galaxies with K < −25 and D < 225 Mpc.
To make our sample cleaner, and enable the clean conversion of stellar mass
to an inferred black hole mass, we select only early-type galaxies (elliptical or S0),
using morphologies from HyperLeda43
. We visually inspected approximately 1,000
of the galaxies and found the sample to be very clean.
The galaxy catalogue is augmented by adding nearby galaxies that host
SMBHs with dynamical mass measurements15,17
. We added 33 such galaxies, nine
of which were not previously in the 2MASS sample (for example because of low
MK luminosity). For these galaxies, we use the measured SMBH mass instead of
inferring it from an empirical scaling relation.
Supermassive black hole masses. We estimate the SMBHB mass by converting
the 2MASS K-band luminosity to the stellar mass, take this stellar mass to
be the bulge mass, Mbulge, and further apply the scaling relation20
between
M• and Mbulge to obtain the total BH mass, M•. When computing the total
SMBHB mass this way, we incorporate an intrinsic scatter in the relation, ε0,
as follows: each logarithmic realization of M•−Mbulge is a random draw from a
normal distribution with mean μ from M• −Mbulge, and σ = 0.34. The answer is
exponentiated. As the exponent is drawn from a normal distribution, it follows
that the total black hole mass follows a log-normal distribution over many
realizations, thus favouring smaller masses. We therefore consider the binary
mass to be conservative.
IPTA predictions. We add four pulsars per year to the existing 49 IPTA pulsars,
chosen from sky locations in the field of view of the current IPTA telescopes.
The white noise level is modelled as a combination of radiometer noise and jitter
noise. The radiometer noise is estimated as the harmonic mean of the measured
error bars (for each backend and observing frequency) to avoid overestimation
due to times of arrival (TOAs) with low signal-to-noise ratio, which would have
large error bars. Jitter noise is obtained for each observing frequency44
. We then
compute an infinite-frequency TOA uncertainty from the low-frequency and
high-frequency noise estimates in order to simulate dispersion-measure fitting.
From 2016 onward, we add four pulsars per year using the median white noise
uncertainty of the existing pulsars in the array, typically around 300 ns. Further,
we assume a new wide-band timing back-end installed in 2018 at Arecibo and the
Green Bank Telescope, which reduces the root-mean-square (r.m.s.) white noise by
a factor of about 1.7.
This back-end upgrade is the dominant factor in the improved white-noise
detection curves (Supplementary Figs. 1 and 4) owing to the fact that the signal-
to-noise ratio, ρ, of a CGW detection is roughly ρ ∝ < 〈NTc/σ2
〉1/2
, assuming that
the N pulsars have identical intrinsic properties, T is the length of the dataset, c
is the cadence of the pulsar observation and σ is the white noise r.m.s.28
. For red
noise with spectral index γ, ρ ∝ 〈NTc/f−γ
〉1/2
= 〈NT1−γ
c〉 1/2
at f = 1/T. There are,
of course, other factors that motivate a large and expanding PTA, including the
geometric term from the antenna beam pattern Ω Ω∝ + ⋅+ × −
 F p( ) (1 )
, 1
, where
Ω is the direction of propagation of the gravitational wave, p is the direction to
the pulsar5,26,28
, and +, × is the gravitational-wave polarization. Therefore, when
Ω ⋅ ≈ − p 1(when the direction to the source, Ω− , is aligned with the pulsar) the
response is maximal (Fig. 1b).
Projections are made for 15-year, 20-year and 25-year datasets with various
false alarm probabilities, FAP. One can convert from the FAP, x, to multiples of
the standard deviation σ via xσ = erf ∕( )x (2) . For example, a FAP of 10−4
= 3.9σ,
assuming a Gaussian distribution.
Currently, a full Bayesian analysis is computationally intractable when
performing these kinds of detection sensitivity analyses. Because we are
performing these analyses as a function of CGW frequency and PTA
configuration, we must compute the detection statistic (Bayes factor for Bayesian
analysis, FAP for frequentist analysis) millions of times. In comparison to the
frequentist FAP statistic, which takes fractions of a second to compute, the Bayes
factor computation requires several hours to compute. Thus, for the number of
simulations required in this work, a full Bayesian analysis for all simulations is
intractable at this time, and we have instead used a frequentist proxy assuming
only white noise in order to emulate the possible resolving capability of the
Bayesian analysis.
Note that for pulsars with strong red noise, approximately 100 times fewer
sources are detected than with the white-noise-dominated pulsars (Table 1). The
presence of unresolved red noise alters the position of the minimum frequency
and achievable strain, shifting the lowest gravitational-wave frequency accessible
by PTAs to higher frequencies. Hence, the sources must be much closer (we find
that most of these are within 20 Mpc) in order to be detected. We also find that,
on average, the galaxies in our catalogue underwent a major merger at z = 0.3
(Supplementary Fig. 2c).
Generating gravitational-wave sky maps. Gravitational-wave sky maps are created
by interpolating a set of 128 original data points at 87 gravitational-wave
frequencies. We interpolate between the points using a bivariate spline
approximation over a rectangular mesh on a sphere, and project the resulting
sky on a hierarchical equal-area isolatitude (HEALpix) map. This is done for 87
gravitational-wave frequencies, resulting in 87 HEALpix maps (saved as FITS
files which are freely available). It is possible that more sources could be detected
if future data points are extended to a greater range in declination. Interpolation
errors close to the poles required us to make a hard cut at declinations of ±70°,
eliminating potential galaxies as CGW sources.
Galaxies hosting SMBHBs in the PTA band. In a preliminary study of local
potential nanohertz CGW sources45
, the authors assembled a 90% complete
galaxy catalogue out to 150 Mpc, including galaxies with SMHBs with
≥ ⊙M M107
. In that study, however, it was assumed that there was an equal
probability for all galaxies to host a SMBHB, and that this was an equal-mass
binary. Moreover, the authors of ref.19
used a top-down approach to predict
gravitational-wave skies by creating a simulated galaxy catalogue and matching
SMBHB merger rates from the Millennium Simulation46
to the Sloan Digital Sky
Survey47
(SDSS), in an effort to identify the characteristics of potential SMBHB
host galaxies. However, SDSS has a limited sky coverage, and does not allow a
full-sky investigation of the loudest potential gravitational-wave sources in the
nearby Universe. The work presented here marks the use of a galaxy survey to
identify local massive galaxies, to assign each galaxy a probability that it hosts a
SMBHB based on Illustris galaxy–galaxy merger rates, and to estimate the time to
detection of these sources with PTAs.
We now give an example of how probabilities are assigned to galaxies in
the catalogue derived from 2MASS, in our bottom-up approach. Consider,
for example, galaxy NGC 4594, which has MK = −25.88 (although NGC 4594
has a dynamically measured SMBH mass, we illustrate how its mass would be
assigned by means of the empirical M•−Mbulge relation). Through the MK−M*
empirical scaling relation18
, the stellar mass is = . × ⊙M M
*
7 03 1011
, and via
M•−Mbulge with scatter ε0, we calculate that there is an SMBHB with total mass
= . × ⊙M M1 92 109
. The mass ratio of the binary, q, is drawn from a log-normal
distribution in [0.25, 1.0], from which we randomly draw q = 0.47. The chirp mass
is therefore = . × ⊙ M7 69 10c
8
.
The dynamical friction timescale21
is computed assuming that the Coulomb
logarithm is log(Λ) = 10:
= −
⊙


















t
a v M
M
264 Myr
2kpc 250 kms
10
(4)df
2
c
1
8
2
where M2 = qM is the mass of the secondary black hole, σ=v 2c with
σ = . + . ∕ ⊙M M Mlog (
*
) 2 3 0 3log( *
10 )
11
(ref. 48
) and a is the galaxy’s effective radius,
Reff, taken from Eq. 4 of ref. 49
(Supplementary Fig. 5).
We scale the stellar mass of the galaxy, M*, by a factor of 0.7 (refs 50,51
)
(see Supplementary Fig. 2), to estimate the mass of the descendant galaxy when
we begin the SMBHB evolution, in the dynamical friction phase. This scaled M* is
also used to estimate the velocity dispersion σ. The parameters drawn are therefore
= . × ⊙M M6 14 102
8
, σ = 319 km s–1
, Reff = 7.3 kpc, which, when input in equation
(4), yield a dynamical friction timescale of tdf = 1.03 Gyr.
The stellar hardening timescale36
is computed as in equations (6) and (7) of ref. 22
:
σ
ρ
σ
ρ
= =
∕






t
H a
a
M M M
H
*
where
*
64
5
(5)sh
inf
inf ,GW
,GW
inf 1 2
inf
1 5
where σ is computed via M*−σ as above48
, and ρinf is the density profile evaluated
at the influence radius, rinf, with γ = 1 (corresponding to a Hernquist profile52
);
more details are given elsewhere22
. Here we implicitly assume circular binaries
and that the hardening constant H = 15, and find that the hardening timescale is
tsh = 2.54 Gyr.
The sum of the dynamical friction and hardening timescales is
1.03 + 2.54 = 3.57 Gyr, or z = 0.3 using Planck23
cosmological parameters.
The cumulative galaxy–galaxy merger rate (Table 1 of ref. 12
) requires only
M* and z as inputs, because the dependence on μ* is removed by integrating
it between 0.25 ≤ μ* ≤ 1; see Equation (1). We scale M* by 0.7, and find that is
dNmerg(M*,z)/dt = 0.11.
Finally, we compute the time that this binary will spend in the PTA band.
The time to coalescence of the binary, tc, is taken from the lower limit of the PTA
band: fmin = 10−9
Hz, resulting in tc = 31.8 Myr. From equation (1), the probability
of NGC 4594 hosting a SMBHB in the PTA band in this particular realization is
p = (31.8 Myr) × (0.11 mergers Gyr–1
) = 3.5 × 10−3
.
Data availability. The authors use open-source scientific tools for Python53–58
and L. Singer’s open-source plot.py59
. We are pleased to provide the reader with
a series of Jupyter Notebooks which reproduce our figures and results via
C.M.F.M.’s git repository60
, https://github.com/ChiaraMingarelli/nanohertz_GWs.
Here, we also share the underlying detection curves and gravitational-wave
sensitivity sky maps.

Received: 9 May 2017; Accepted: 4 October 2017;
Published: xx xx xxxx
References
1. Ferrarese, L. & Ford, H. Supermassive black holes in galactic nuclei: past,
present and future research. Space Sci. Rev. 116, 523–624 (2005).
2. Begelman, M. C., Blandford, R. D. & Rees, M. J. Massive black hole binaries
in active galactic nuclei. Nature 287, 307–309 (1980).
3. Estabrook, F. B. & Wahlquist, H. D. Response of Doppler spacecraft tracking
to gravitational radiation. Gen. Relat. Gravit. 6, 439–447 (1975).
4. Sazhin, M. V. Opportunities for detecting ultralong gravitational waves.
Sov. Astron. 22, 36–38 (1978).
5. Detweiler, S. Pulsar timing measurements and the search for gravitational
waves. Astrophys. J. 234, 1100–1104 (1979).
6. Hellings, R. W. & Downs, G. S. Upper limits on the isotropic gravitational
radiation background from pulsar timing analysis. Astrophys. J. Lett. 265,
L39–L42 (1983).
7. Lentati, L. et al. European pulsar timing array limits on an isotropic
stochastic gravitational-wave background. Mon. Not. R. Astron. Soc. 453,
2576–2598 (2015).
8. Arzoumanian, Z. et al. The NANOGrav Nine-year Data Set: limits on
the isotropic stochastic gravitational wave background. Astrophys. J.
821, 13 (2016).
9. Shannon, R. M. et al. Gravitational waves from binary supermassive black
holes missing in pulsar observations. Science 349, 1522–1525 (2015).
10. Verbiest, J. P. W. et al. The International Pulsar Timing Array: first data
release. Mon. Not. R. Astron. Soc. 458, 1267–1288 (2016).
11. Skrutskie, M. F. et al. The Two Micron All Sky Survey (2MASS). Astron. J.
131, 1163–1183 (2006).
12. Rodriguez-Gomez, V. et al. The merger rate of galaxies in the Illustris
simulation: a comparison with observations and semi-empirical models.
Mon. Not. R. Astron. Soc. 449, 49–64 (2015).
13. Genel, S. et al. Introducing the Illustris project: the evolution of galaxy
populations across cosmic time. Mon. Not. R. Astron. Soc. 445,
175–200 (2014).
14. Mingarelli, C. M. F., Sidery, T., Mandel, I. & Vecchio, A. Characterizing
gravitational wave stochastic background anisotropy with pulsar timing
arrays. Phys. Rev. D 88, 062005 (2013).
15. Schutz, K. & Ma, C.-P. Constraints on individual supermassive black hole
binaries from pulsar timing array limits on continuous gravitational waves.
16. Jarrett, T. H. et al. 2MASS Extended Source Catalog: overview and
algorithms. Astron. J. 119, 2498–2531 (2000).
17. Ma, C.-P. et al. The MASSIVE Survey. I. A volume-limited integral-field
spectroscopic study of the most massive early-type galaxies within 108 Mpc.
Astrophys. J. 795, 158 (2014).
18. Cappellari, M. Effect of environment on galaxies’ mass-size distribution:
unveiling the transition from outside-in to inside-out evolution. Astrophys. J.
Lett. 778, L2 (2013).
19. Rosado, P. A. & Sesana, A. Targeting supermassive black hole binaries and
gravitational wave sources for the pulsar timing array. Mon. Not. R. Astron.
Soc. 439, 3986–4010 (2014).
20. McConnell, N. J. & Ma, C.-P. Revisiting the scaling relations of black hole
masses and host galaxy properties. Astrophys. J. 764, 184 (2013).
21. Binney, J. & Tremaine, S. Galactic Dynamics 2nd edn. (Princeton Univ. Press,
Princeton, 2008).
22. Sesana, A. & Khan, F. M. Scattering experiments meet N-body. I. A practical
recipe for the evolution of massive black hole binaries in stellar environments.
Mon. Not. R. Astron. Soc. 454, L66–L70 (2015).
23. Planck Collaboration et al. Planck 2015 results. XIII. Cosmological
parameters. Astron. Astrophys. 594, A13 (2016).
24. Peters, P. C. Gravitational radiation and the motion of two point masses.
Phys. Rev. 136, 1224–1232 (1964).
25. Babak, S. et al. European Pulsar Timing Array limits on continuous
gravitational waves from individual supermassive slack hole binaries.
26. Mingarelli, C. M. F. & Sidery, T. Effect of small interpulsar distances in
stochastic gravitational wave background searches with pulsar timing arrays.
Phys. Rev. D 90, 062011 (2014).
27. Ellis, J. A., Siemens, X. & Creighton, J. D. E. Optimal strategies for
continuous gravitational wave detection in pulsar timing arrays. Astrophys. J.
756, 175 (2012).
28. Arzoumanian, Z. et al. Gravitational waves from individual supermassive
black hole binaries in circular orbits: limits from the North American
Nanohertz Observatory for Gravitational Waves. Astrophys. J.
794, 141 (2014).
29. The NANOGrav Collaboration. et al. The NANOGrav nine-year data set:
observations, arrival time measurements, and analysis of 37 millisecond
pulsars. Astrophys. J. 813, 65 (2015).
30. Desvignes, G. et al. High-precision timing of 42 millisecond pulsars
with the European Pulsar Timing Array. Mon. Not. R. Astron. Soc.
458, 3341–3380 (2016).
31. Taylor, S. R. et al. Are we there yet? Time to detection of nanohertz
gravitational waves based on pulsar-timing array limits. Astrophys. J. Lett.
819, L6 (2016).
32. Taylor, S. R. et al. Limits on anisotropy in the nanohertz stochastic
gravitational wave background. Phys. Rev. Lett. 115, 041101 (2015).
33. Taylor, S. R. & Gair, J. R. Searching for anisotropic gravitational-wave
backgrounds using pulsar timing arrays. Phys. Rev. D 88, 084001 (2013).
34. Gair, J., Romano, J. D., Taylor, S. & Mingarelli, C. M. F. Mapping
gravitational-wave backgrounds using methods from CMB analysis:
application to pulsar timing arrays. Phys. Rev. D 90, 082001 (2014).
35. Cornish, N. J. & van Haasteren, R. Mapping the nano-hertz gravitational
wave sky. Preprint at https://arxiv.org/abs/1406.4511 (2014).
36. Quinlan, G. D. The dynamical evolution of massive black hole binaries I.
Hardening in a fixed stellar background. New Astron. 1, 35–56 (1996).
37. Booth, R. S., de Blok, W. J. G., Jonas, J. L. & Fanaroff, B. MeerKAT key
project science, specifications, and proposals. Preprint at https://arxiv.org/
abs/0910.2935 (2009).
38. Janssen, G. et al. Gravitational wave astronomy with the SKA. Advancing
Astrophysics with the Square Kilometre Array (AASKA14) 037 (2015).
39. Sesana, A., Shankar, F., Bernardi, M. & Sheth, R. K. Selection bias in
dynamically measured supermassive black hole samples: consequences for
pulsar timing arrays. Mon. Not. R. Astron. Soc. 463, L6–L11 (2016).
40. Wrobel, J. M. & Nyland, K. Discovery of a flat-spectrum radio nucleus in
NGC 3115. Astron. J. 144, 160 (2012).
41. Huchra, J. P. et al. The 2MASS Redshift Survey—description and data release.
Astrophys. J. Suppl. S 199, 26 (2012).
42. Crook, A. C. et al. Groups of galaxies in the Two Micron All Sky Redshift
Survey. Astrophys. J. 655, 790–813 (2007).
43. Paturel, G. et al. HyperLeda. I. Identification and designation of galaxies.
Astron. Astrophys. 412, 45–55 (2003).
44. Lam, M. T., Cordes, J. M., Chatterjee, S. & Dolch, T. Pulsar timing errors
from asynchronous multi-frequency sampling of dispersion measure
variations. Astrophys. J. 801, 130 (2015).
45. Simon, J. et al. Gravitational wave hotspots: ranking potential locations of
single-source gravitational wave emission. Astrophys. J. 784, 60 (2014).
46. Springel, V. et al. Simulations of the formation, evolution and clustering of
galaxies and quasars. Nature 435, 629–636 (2005).
47. Abazajian, K. N. et al. The seventh data release of the Sloan Digital Sky
Survey. Astrophys. J. Suppl. Ser. 182, 543–558 (2009).
48. Zahid, H. J., Geller, M. J., Fabricant, D. G. & Hwang, H. S. The scaling
of stellar mass and central stellar velocity dispersion for quiescent galaxies at
z < 0.7. Astrophys. J. 832, 203 (2016).
49. Dabringhausen, J., Hilker, M. & Kroupa, P. From star clusters to dwarf
galaxies: the properties of dynamically hot stellar systems. Mon. Not. R.
Astron. Soc. 386, 864–886 (2008).
50. De Lucia, G. & Blaizot, J. The hierarchical formation of the brightest cluster
galaxies. Mon. Not. R. Astron. Soc. 375, 2–14 (2007).
51. McBride, J., Fakhouri, O. & Ma, C.-P. Mass accretion rates and histories of
dark matter haloes. Mon. Not. R. Astron. Soc. 398, 1858–1868 (2009).
52. Dehnen, W. A family of potential-density pairs for spherical galaxies and
bulges. Mon. Not. R. Astron. Soc. 265, 250–256 (1993).
53. Jones, E., Oliphant, T., Peterson, P. et al. SciPy: Open source scientific
tools for Python (2001); http://www.scipy.org/ (SciPy, accessed
24 August 2016).
54. The Astropy Collaboration. et al. Astropy: a community Python package for
astronomy. Astron. Astrophys. 558, A33 (2013).
55. van der Walt, S., Colbert, S. C. & Varoquaux, G. The numpy array:
a structure for efficient numerical computation. Comp. Sci. Eng.
13, 22–30 (2011).
56. Barrett, P., Hunter, J., Miller, J. T., Hsu, J.-C. & Greenfield, P. In Astronomical
Data Analysis Software and Systems XIV (eds Shopbell, P., Britton, M. &
Ebert, R.) 91–95 (Astronomy Society of the Pacific, 2005).
57. Górski, K. M. et al. HEALPix: a framework for high-resolution discretization
and fast analysis of data distributed on the sphere. Astrophys. J. 622, 759–771
(2005).
58. Perez, F. & Granger, B. E. Ipython: a system for interactive scientific
computing. Comput. Sci. Eng. 9, 21–29 (2007).
59. Singer, L. P. et al. The first two years of electromagnetic follow-up with
advanced LIGO and Virgo. Astrophys. J. 795, 105 (2014).
60. Mingarelli, C. Chiaramingarelli/nanohertz_gws: first release (2017);
https://doi.org/10.5281/zenodo.838712
Acknowledgements
We thank S. Babak, J. Verbiest, D. Kaplan, E. Barr, K. Górski and E. Sheldon for
discussions. This publication makes use of data products from the Two Micron All
Sky Survey, which is a joint project of the University of Massachusetts and the Infrared

Processing and Analysis Center/California Institute of Technology, funded by NASA
and the National Science Foundation (NSF). C.M.F.M. was supported by a Marie Curie
International Outgoing Fellowship within the European Union Seventh Framework
Programme. S.R.T was partly supported by appointment to the NASA Postdoctoral
Program at the Jet Propulsion Laboratory, administered by Oak Ridge Associated
Universities and the Universities Space Research Association through a contract with
NASA. A.S. is supported by a University Research Fellowship of the Royal Society.
Parts of these computations were performed on the Zwicky cluster at Caltech, which
is supported by the Sherman Fairchild Foundation and NSF award PHY-0960291. Part
of this research was carried out at the Jet Propulsion Laboratory, California Institute
of Technology, under a contract with NASA. This work has also been supported by
NSF award 1458952, NSF AST-1411945 and 1411642. The NANOGrav project receives
support from NSF Physics Frontier Center award number 1430284. The Flatiron Institute
is supported by the Simons Foundation.
Author contributions
C.M.F.M. modelled the supermassive black hole evolution, developed and ran the Monte
Carlo simulations used here to explore their evolution, analysed the resulting data,
produced the figures and table, and was the primary author of this paper. C.M.F.M.,
T.J.W.L. and S.B.S. developed the concept of this work. A.S., C.P.M., S.C. and T.J.W.L.
advised on supermassive black hole astrophysics and helped to interpret the results.
J.E.G. and S.C. assembled and inspected the galaxy catalogue. J.A.E. developed the time
to detection methods for the IPTA. S.R.T. helped to develop the methods used to turn
CGW sources into a GWB, and explore its angular power spectrum.
Competing interests
The authors declare no competing financial interests.
Additional information
Supplementary information is available for this paper at https://doi.org/10.1038/
s41550-017-0299-6.
Reprints and permissions information is available at www.nature.com/reprints.
Correspondence and requests for materials should be addressed to C.M.F.M.
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in
published maps and institutional affiliations.

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