# A uniform stellar origin for binary black holes revealed by lensing

Tom Broadhurst  Department of Theoretical Physics, University of the Basque
Country UPV-EHU, 48040 Bilbao, Spain  Donostia International Physics Center
(DIPC), 20018 Donostia, The Basque Country  IKERBASQUE, Basque Foundation for
Science, Alameda Urquijo, 36-5 48008 Bilbao, Spain  Jose M. Diego  Instituto
de Física de Cantabria, CSIC-Universidad de Cantabria, E-39005 Santander,
Spain  George F. Smoot  Donostia International Physics Center (DIPC), 20018
Donostia, The Basque Country  IAS TT & WF Chao Foundation Professor, IAS, Hong
Kong University of Science and Technology, Clear Water Bay, Kowloon, 999077
Hong Kong, China  Paris Centre for Cosmological Physics, Université de Paris,
CNRS, Astroparticule et Cosmologie, F-75013 Paris, France A, 10 rue Alice
Domon et Leonie Duquet, 75205 Paris CEDEX 13, France, Emeritus Physics
Department and Lawrence Berkeley National Laboratory, University of
California, Berkeley, 94720 CA, USA, Emeritus

###### Abstract

Although most gravitational wave events are claimed to be mergers of unusually
massive, $25-65M_{\odot}$, black holes, it is now clear that 20% of all
reported events comprise modest mass black holes, $5-15M_{\odot}$, like the
stellar black holes in the Milky Way. We show that such stellar mass black
hole binaries (BBH) if magnified by lensing galaxies can be detected at high
redshift, 1$<$z$<$5, with chirp masses increased by $1+z$, accounting for the
majority of apparently high mass BBH events. This simple lensing explanation
is manifested by the evident bimodality of BBH chirp masses now visible, with
80% of BBH events in a broad peak centered on $m_{chirp}\simeq 35M_{\odot}$,
and 20% of BBH events in a narrow, low mass peak at $m_{chirp}\simeq
8.5M_{\odot}$, matching well our prediction for lensed and unlensed events
respectively. This lensing interpretation is reinforced by the “graveyard
plot” when ranked by chirp mass, revealing a jump in chirp mass at
$m_{chirp}\simeq 10M_{\odot}$ that we show is caused by the large redshift
difference between unlensed events with $z<0.3$ and lensed events above $z>1$.
Furthermore, nearly all BBH events are seen to share a component mass ratio of
$m_{1}/m_{2}=1.45\pm 0.03$, indicating a common stellar origin for BBH events
across all chirp masses. This observed component mass uniformity implies most
binary black holes seldom pair up by random capture, instead we may conclude
that massive progenitor stars of BBH black holes typically formed in-situ, in
a well defined way over the full span of cosmic time accessed through
gravitational lensing.

Decades of EM-observations have established that black holes orbiting nearby
stars have a narrow range of masses, $5-15M_{\odot}$, peaking at $8M_{\odot}$,
with none known above $20M_{\odot}$, or below about 5$M_{\odot}$Remillard ;
El-Badry ; Stanway . Most local stellar black holes were identified by X-ray
gas emission but more recently new methods have uncovered a black hole of
$11.1\pm 2.2M_{\odot}$ from direct orbital motion Saracino2021 , and a micro-
lensing black hole of $7.1\pm 1.3M_{\odot}$ Sahu2022  has been determined from
stellar astrometry, thereby independently reinforcing the stellar black hole
mass mass range, $5-15M_{\odot}$. Here we show that this stellar black hole
mass range is shared by all 30 black holes that comprise the 15 lowest chirp
mass BBH detections reported to date, which also span the range,
$5-15M_{\odot}$, implying that these low chirp mass events comprise pairs of
conventional black holes where both members are remnants of massive stars.
This stellar mass population is clear in Figure 1 where we have ranked all BBH
events by chirp mass, showing these 15 lowest mass events are centered on a
chirp mass of $\simeq 8.5M_{\odot}$ and appear distinct form the majority of
events of higher chirp mass. This distinction is also apparent in the
component mass distribution of observed black hole masses shown in Figure 3,
where a sharp, low mass peak is distinct from the majority of black holes that
span a broad mass distribution centered on $\simeq 35M_{\odot}$, which we
discuss below in the context of lensing.

Figure 1: All reported binaries ranked by chirp mass This ranked version of
the “Graveyard” plot reveals that BBH events share a common component mass
ratio, forming a pair of remarkably parallel tracks when normalised by the
total binary mass (lower point set), defining a common mass ratio of $1.45\pm
0.03$ that spans the full range of chirp mass. We can also see here a step-
like transition between the “stellar mass” group of 15 lowest chirp mass BBH
events centered on a rest frame value of $8.5M_{\odot}$, with $z<0.3$, and all
the higher chirp mass events that follow a slowly rising curve, compatible
with being high redshift, lensed examples of the lower mass “stellar mass”
group. The commonality in mass ratio between high and low chirp mass BBH
events implies a common origin. Events marked as white lines with large mass
ratios are the reported NSBH events and the lower Mass-Gap and Mass-Asymmetric
events. The events marked in purple are discussed as possible second
generation BBH mergers.  Figure 2: Comparison with Lens Model. Predictions
from our lens model for one year of observation at O3 sensitivity in the
central and low panels, for comparison with the most trusted events reported
by the LVK team in the upper panel. The predictions show a clear distinction
between unlensed and lensed events marked with a vertical dashed line, arising
directly from the high redshift enhancement of the chirp masses by $1+z$ for
lensed events compared to the predicted unlensed events, for which $z<0.3$ \-
like the reported redshifts of the low chirp mass events. In the central panel
we allow the component masses to be drawn randomly from the input lognormal
mass distribution which has a wider spread of component mass ratios than in
the upper panel of the data, whereas, in the lower panel matches well the mass
ratios of the data when we restrict the component mass ratio to
$m_{1}/m_{2}=1.45$ with a small dispersion of $\simeq$ 10%.

It is also striking in Figure 1 that nearly all BBH events of high and low
mass appear to share a common characteristic mass ratio of $m_{1}/m_{2}\simeq
1.45$, resulting in two very uniform, parallel bands when normalising by the
total binary mass that span the full chirp mass range of Figure 1. To
interpret this uniformity in mass ratio, it should first be appreciated that
the cosmological expansion of GW waveform by $1+z$ exactly mimics a higher
chirp mass binary Wang ; Broadhurst2018  since gravitation is scale free, so
binaries with masses $(m_{1},m_{2})$ in the rest frame are indistinguishable
by frequency dependence from masses: $(\frac{m_{1}}{1+z},\frac{m_{2}}{1+z})$,
at redshift $z$, as the cosmological time dilation ensures the waveform is
shifted by precisely the necessary amount to preserve the chirp shape and
frequency by the same factor of $1+z$ for both the primary and secondary
masses. Hence, the uniformity of the component mass ratios is inherent to the
BBH black holes and independent of their redshifts.

We now draw attention to the striking transition visible in Figure 1, marked
by an upward step in chirp mass from $10.5M_{\odot}$ to $15.5M_{\odot}$, above
the range of stellar mass black holes, indicated in figure 1 between the 15
low mass events which we have argued are stellar mass BBH binaries at low
redshift, and all higher chirp mass events. In the context of lensing this
transition neatly distinguishes high-z lensed events from low-z unlensed
events because the chirp mass scales with redshift as $1+z$ via:

$M_{\rm Chirp}(z)=(1+z)M_{\rm Chirp}(0)$ (1)

Hence, this observed transition in chirp mass by $\simeq 5M_{\odot}$ is
equivalent to an increase in $1+z$ of $\simeq 50\%$ which translates into a
minimum redshift of $z>1.4$ for events above this transition relative to the
measured mean redshift of $\bar{z}\simeq 0.2$ reported for the stellar mass
BBH events lying below this transition.

Figure 3: Distribution of binary black hole masses Histogram of all 134
individual black hole masses for the 67 BBH events reported to date above an
SNR limit of 8 that we have modelled. The red curve shows our model prediction
for lensed events and the black curve is for unlensed events. Both predicted
curves have characteristic peaks because they derive from the same, relatively
narrow log-normal distribution of the known stellar black holes in our Galaxy
and together describe a bimodal distribution that matches the data well, with
a mean redshift of $z\simeq 0.2$ and $z\simeq 2.8$ for unlensed and lensed BBH
events respectively.

Regarding the origin of the high chirp mass events, we have argued that above
a chirp mass of $\simeq 15M_{\odot}$, all events are gravitationally lensed,
because BBH binaries with normal stellar mass black holes of $5-15M_{\odot}$
are detectable in the range $1<z<5$ at the current instrumental limits if
lensed, with chirp masses dominated by the $1+z$ expansion factor (eqn. 1). We
now apply our established lens model to generate distributions of lensed BBH
events, based on the simplest, empirically based a-priori assumptions, by
using the known mass function for stellar black holes and adopting the
universal form for the high magnification tail of fold caustics,
$\propto\mu^{-3}$ normalised by the known optical depth of galaxy lensing from
sky surveys. Lensing by galaxies or clusters does not of course alter the
frequencies of gravitational waves, but the detected strain is magnified, such
that the inferred distance will be substantially underestimated by
$\sqrt{\mu}$:

$D_{L}(z_{inferred})=\frac{{\rm M}_{chirp,z}^{5/6}}{h(t)}~{}F(t,{\rm
M}_{obs},\Theta)=\frac{D_{L}(z_{true})}{\sqrt{\mu}}$ (2)

For binary events the detection function $F(t,{\rm M}_{obs},\Theta)$ combines
the angular sky sensitivity, orbital inclination, spin and polarization of the
binary sourceFinn  and its distribution is numerically estimated with a
$\simeq 40\%$ dispersionNg ; Abbot_2016 . Our only major unknown is the
evolution of BBH events for which there is a wide range of ideasSig ; Ban ;
Ugo ; Farr , so we adopt an exponentially declining BBH event rate for $z<2$
as the simplest assumption that we find to reproduce the approximate 4-to-1
ratio of lensed to unlensed BBH events at the current depth of the data, as
shown in Figures 1 & 2, corresponding to an e-folding time of $\simeq 0.8Gyr$
(see appendix). We stress that the rate of evolution is effectively the only
degree of freedom we have in our lens model because the input mass function
and the lensing optical depth are set by real priors established before GW
events were detected, as outlined in detail in our first
calculationsBroadhurst2018  which we refer to as the BDS model.

Firstly, we show our predictions for unlensed BBH events with the BDS model,
which depends mainly on the input BBH component mass distribution and not on
lensing. Our predicted unlensed events span the range $z<0.3$ for the current
O3 sensitivity, with a peak in detections at a chirp mass of $8-10M_{\odot}$.
This is very similar to the inferred redshifts and chirp masses of the group
of 15 low mass events that are reported to span $0.07<z<0.32$, with a mean of
$z=0.17$, as listed in Abbott1+2  for periods 1&2 and in Abbott2021  for
period 3. The black hole masses comprising each binary are drawn in our model
from a log-normal mass distribution in two ways, firstly by drawing each one
randomly and secondly subject to the observed mass ratio of $1.45$ with a
small dispersion in the component mass ratio of $10\%$, shown for comparison
with the data in Figure 1 & 2\. We see this provides a very good reproduction
of observed transition above the stellar black hole mass events as the
transition from unlensed to lensed events and also matches the slowly rising
behaviour in the high mass region above this transition, with the majority of
high mass BBH events predicted to span $1<z<5$. This lens model also accounts
for the bimodal appearance of the distribution of black hole masses shown in
the histogram of Figure 3, with unlensed predicted events matching well the
sharp low mass peak, whereas lensed events match well the broad distribution
of high mass events peaking at $z\simeq 3$, corresponding to mean chirp mass
of $40M_{\odot}$. The relative numbers of events between these two peaks
allows us to determine that the intrinsic BBH event rate peaks at $z\simeq 3$,
at a level that is about $\simeq 3$ orders of magnitude higher than the rate
of unlensed detections at $z\simeq 0.2$, but is well below the rate of
formation of BHs implied by the rate of core SNe at $z\simeq 2$Diego_extreme
(see Appendix for rate evolution model details).

We also compare our lens model in the mass-distance plane shown in Figure 4,
for the reported values of chirp mass and distance, i.e. with no account for
lensing. This comparison demonstrates there is good consistency between the
unlensed input events with $M<20M_{\odot}$, in the stellar mass range, and
close agreement with the higher chirp BBH events that we predict are lensed in
the redshift range $1<z<5$, comprising 80% of all reported BBH events to date.

Figure 4: Distance-Mass plane comparison. This includes the most trusted
reported BBH events marked with their measurement errors. The open and
coloured circles are our unlensed and lensed event predictions respectively,
where the colour indicates magnification. Colored triangles indicate a
predicted second detection that is strong enough to be a repeat event. The
predictions are made as if lensing is unrecognised, i.e. for comparison with
the reported values. The simulated events correspond to 1 year of observation
at O3 sensitivity.

In addition to the dominant BBH population of uniform mass ratio identified
above, there are a few events with much higher mass ratios visible in Figure 1
(marked by white lines) centered on $m_{1}/m_{2}\simeq 5$. Of these, the two
with lowest chirp masses have been identified as low redshift NSBH
eventsAbbottNSBH  with neutron star masses of 1.5 & 1.9$M_{\odot}$ and with
black hole masses of 5.7 & 8.9$M_{\odot}$ reported, indicating a conventional
stellar origin for the black holes in these NSBH binaries. The next two events
with high mass ratios shown in Figure 1 have been classified as “mass-gap”
(MG), as their observed secondary component masses, 2.7 & 3.3$M_{\odot}$, are
more massive than neutron stars and less massive than stellar black
holesAbbottMG , which in the context of lensing we have simply concluded are
examples of lensed NSBH events at $z\simeq 1$, with conventional NS and BH
masses. This lensing interpretation can also be applied to the reported “mass-
asymmetric” event AbbottMA  because of its large mass ratio (Figure 1), which
in the context of lensing we have argued is simply a higher redshift NSBH
event, at $z\simeq 3.5$ so the NS mass then appears to exceed $5M_{\odot}$ and
thus misclassified as a black holeBDS:2020 .

Finally, we tentatively identify a possible population of 5 “second generation
events” marked in purple in Figure 1, with mass ratios centered on
$m_{1}/m_{2}\simeq 2.0$. In the context of our lensing interpretation such
events are anticipated, as the could comprise a typical remnant $\simeq
19M_{\odot}$ black hole (formed from typical first generation pairs of 8 & 11
$M_{\odot}$ seen in the “unlensed” region of Figure 1), paired with a typical
first generation black hole of $\simeq 10M_{\odot}$, thereby resulting in a
second generation event with $m1/m2\simeq 2$, like the observed events marked
in purple in Figure 1. Such second generation events should not be surprising
in our model given the high early rate of BBH events that we have concluded in
the context of lensing. Clearly, more data is required to determine whether
there is a such preferred mass ratio or whether these events are just the mass
ratio tail of the general BBH distribution.

##  I Discussion and Conclusions

Over 90 pairs of merging black holes have now been reported spanning a wide
range in chirp mass. These events now appear to be bimodal in terms of their
observed chirp masses (see Figure 3) and also most of these BBH events have
surprisingly similar component mass ratios across the full range of chirp mass
(see Figure 1&2), with a well defined mean value of $m_{1}/m_{2}=1.45\pm
0.03$. Here we have shown that a simple alteration in interpretation, by
invoking lensing, explains these features and accounts for the unusual events
from the two mass gap regions as conventional stellar remnants (NS and BH)
that are already observed in local group astronomy. Our conclusion that
lensing dominates current GW events contrasts with other estimates where the
input BBH mas distribution is based on the observed BBH events i.e. implicitly
assuming that lensing is not significant, and because the observed mass
distribution is so broad then lensed events are unrecognisable by mass because
they seldom exceed the chirp masses of intrinsically wide mass function
adopted Ng ; Otto  and are thus outnumbered by the predicted rate of unlensed
events. In our estimation the real BBH mass function is narrow, represented by
only the lowest 20% of chirp masses with $5-15M_{\odot}$, now clearly
recognisable as a population in the BBH data, and hence we come to the
opposite conclusion that lensing accounts for the majority of higher chirp
mass events as high redshift, $1<z<5$, stellar mass black holes.

Our conclusion that BBH events are predominantly lensed is analogous to the
brightest infrared galaxies detected in large sky surveys which are
predominantly lensed by intervening galaxies into clear Einstein rings with a
mean radius of $\simeq 0.85"$ Wardlow ; Negrello ; Bussmann , demonstrating
that lensing has magnified distant infrared galaxies (typically at $z\simeq
2$) to levels commonly exceeding the fluxes of the brightest unlensed
galaxies, such that they dominate FIR detections in the first decade of
fluxWardlow ; Negrello ; Bussmann . It is conceivable that these same lensed
galaxies are predominantly the source of lensed BBH events where massive
binary stars are abundant, so that a positive cross correlation may be
detected once positional triangulation is feasible for statistical samples of
BBH events.

Magnifications ranging over $10-500$ are implied by our lens model for the
majority of high chirp mass BBH detections, levels which we do know are
achievable for small sources when projected close to lensing caustics,
including the individually lensed stars recently discovered on the Einstein
radius of lensing clusters with magnifications of several thousand reported in
some casesKelly ; Venumadhav ; Diego ; Oguri . These highly magnified stars
limit the proportion of dark matter in primordial black holes insignificant
$<5$%, as only modest micro-lensing is seen, consistent with the projected
density of stars visible in the lensing clustersDiego ; Oguri , disfavouring
the primordial black hole hypothesis for BBH eventsBird ; Carr . Instead, this
absence of “LIGO-like” black holes may be regarded as support for our lensing
interpretation of the high chirp mass events as no new class of intrinsically
high mass black holes is required, just conventional stellar black holes that
are lensed at high redshift.

In addition to the case presented here, we have also raised a time delay
argument, finding that the time difference between pairs of BBH events with
indistinguishable waveforms and sky location LVT2021 , is consistent with the
time delays between lensed QSO images that range from days to weeks (Diego,
Broadhurst & Smoot 2021Diego2021 . Our lens model predicts that repeat events
should be detectable for a sizeable minority $\simeq 10\%-20\%$ of BBH
detections, set by the Earth-rotating angular sensitivity of
LIGO/VirgoBroadhurst2018  which limits detections to an overhead (and
underfoot) band of the sky and typically misses counter images in the current
configuration. It should also be appreciated that the relative fluxes of
close, highly magnified images of lensed QSO’s and SNeSNlens , typically
differ by a factor of 2 in brightness, implying substructure in lensing
galaxies is common on sub-arcsecond scalesTreu2019 ; Chan , in which case the
weaker counter image will typically fall below the GW detection threshold, as
most GW events are near the detection limit, SNR $\simeq 8$, with few
exceeding a level of SNR $>16$ required for detection of the weaker event to
overcome a factor $\simeq 2$ flux anomaly, with credible examples
proposedDai_pair .

Irrespective of lensing, the viability of the high masses inferred for BBH
events above $\simeq 50M_{\odot}$ is challenged theoretically by the physical
limit from pair instabilityWoosley  and also empirically there is a claimed
deficit of high mass stars $>40M_{\odot}$ that are metal poorSchoot  in the
SMC and similar shortages in the LMC and Milky Way, which if general would
disfavour larger stellar progenitors required at low redshift to account for
the reported high mass of BBH events. Instead, our conclusions appear
qualitatively aligned with simulations of high mass star formation, made well
before the GW detections, that predict a well defined process of fragmentation
and accretion for generating close multiples of higher mass starsKrum ,
qualitatively supporting the uniformity of the observed BBH component masses
and our lensing based conclusion that most BBH events comprise pairs of
conventional stellar mass black holes formed at early times.

An independent check on the viability of our lensing interpretation is
anticipated using the stochastic GW background that is integrated over all
redshift and binary mergers. This background is predicted to be higher in the
case of lensing, particularly at low GW frequencies due to the relatively high
rate of events at high redshift required by our lensing interpretationSuvodip
. Another complementary possibility is the direct detection of micro-lensing
modulated GW waveforms Diego_ML ; Otto  that would imply the presence of a
macro-lensing galaxy hosting the micro-lenses.

Finally, we emphasise that the broad black hole mass function derived from GW
events is not independently supported by local black hole masses, where there
is a complete absence of stellar black holes above $20M_{\odot}$. However,
there is excellent agreement between the narrow local stellar mass black hole
mass function and the sharp peak of BBH events seen at the low mass end, as we
have shown here. The main point to appreciate now is that lensing of these low
mass events does solve this seemingly contradictory situation by allowing the
detection of conventional stellar mass black holes at cosmological redshifts
where their chirp masses are high like the majority of BBH detections. Lensing
surely provides the simplest resolution of this apparent contradiction and
implies that 80% of the reported BBH events should be substantially revised
downwards in mass and upwards in distance.

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##  II  Appendix A: BBH event rate evolution

This appendix describes the model for the rate of mergers as a function of
redshift. This model is inferred from the observations and the assumption that
events with inferred chirp masses above 20 are all gravitationally lensed. Our
model consists of three basic ingredients; i) A low rate of mergers at low
redshift ($z<0.3$) which accounts for the events that are not being lensed.
ii) A large rate at redshift $z>2$, needed in order to compensate the low
probability of lensing.
iii) A transition between the low redshift and high redshift

We consider the simple scenario where the the low redshift rates can be
approximated by a constant, and the high redshift rate follows a functional
form similar to the cosmic star formation rate, beyond the point where star
formation peaks ($z\approx 2$). For the transition phase we adopt an
exponential form although other forms could be considered, such as a Gaussian
model. The current data does not allow to constrain the transition well, but
impose more clear limits on the low-z and high-z rates. For simplicity we
adopt the following functional form for the rate model;

$R(z)=N\times S(z)\times M(z)$ (3)

where $N\approx 6\times 10^{3}$ Gpc-3 yr-1 is the normalization constant,
$S(z)$ is proportional to the star formation rate given by the standard Madau
& Dickinson (2014) model (Eq. 15 in 2014, ARA&A, 52, 415),

$S(z)=\frac{(1+z)^{2.7}}{1+[(1+z)/2.9]^{5.6}}$ (4)

and $M(z)$ is a modulation function which is constant and equal to 1 for
$z>2$, i.e $M(z>2)=1$, and below $z=2$ is given by;

$M(z<2)=\frac{2\times 10^{-4}}{0.1+z^{1.9}}+e^{-\frac{|T(z)-T_{max}|}{T_{e}}}$
(5)

with $T(z)$ the lookback time at redshift $z$, $T_{max}$ the corresponding
lookback time at $z=2$, and $T_{e}$ the parameter controlling the speed at
which the transition between the low-z and high-z rates take place. The first
term in Equation 5 compensates the function $S(z)$ at low-z resulting in
$R(z)\approx Cte$ at low-z. Alternatively, one could simply replace the rate
$R(z)$ at $z<0.3$ by a constant, in order to reproduce the observed number and
distribution in the Mass-redshift plane of the not-lensed events.

With the above assumptions the rate implied by our lensing hypothesis is shown
in Figure 5 as a solid line. In order to reproduce the observed distribution
of mass and redshift for the low-z, low-Mass events, the local rate needs to
be below the implied rate by LIGO (dashed curve in Figure 5), since under our
lensing model hypothesis, some of the low-redshift events are in fact distant
but amplified by lensing events. At high redshifts, our model implies a rate
above $10^{4}$ events per year and Gpc3. This rate is considerably higher than
the one adopted by the LIGO collaboration under the no-lensing hypothesis.
Such a high rate is needed in order to compensate for the low probability of
lensing, but this rate is still well below the rate of SNe at these redshifts
(or BH production). At a rate of $R(z=2)=5\times 10^{4}$ Gpc-3 yr-1, ad
adopting a galaxy abundance of 0.08 gal Mpc3, consistent with UV the observed
luminosity function at this redshift (see Diego2019 ), we find that this rate
implies 1 merger per year and per 1500 galaxies at $z=2$. Finally, our lensing
interpretation favours a rapid decline in rate between $z\approx 2$ and
$z\approx 0.3$, perhaps tied to the relatively rapid formation of massive star
clusters.

Figure 5: Merger rate. The solid line shows the implied merger rate from our
lensing model, with $T_{e}=0.8$ Gyr. The dashed line shows a rate that traces
the cosmic star formation rate, and is similar to the models assumed by LIGO
under the assumption that no lensing is taking place.