Cross-section measurements of top-quark
pair production in association with a hard photon
at 13 TeV with the ATLAS detector

25 years after the top quark's discovery, the Large Hadron Collider at CERN produces proton-proton collision data on unprecedented scales at unprecedented energies – and has heralded an era of top-quark precision measurements.
The production of a top-quark pair in association with a photon ($t\bar{t}\gamma$) gives access to the electromagnetic top-photon coupling, one of the fundamental properties of the top quark.
Various extensions of the Standard Model predict modifications of the coupling strength or structure, and deviations from the Standard Model prediction of the $t\bar{t}\gamma$ production cross-section would indicate new physics.
With enough statistics available from the Large Hadron Collider, the electron-muon channel has gained particular interest due to its high signal purity and precise available theory predictions.

This thesis presents results with the full Run 2 dataset collected with the ATLAS detector in proton-proton collisions at the Large Hadron Collider between 2015 and 2018 at 13 TeV centre-of-mass energy, corresponding to an integrated luminosity of 139 fb$^{-1}$.
In order to compare the results to fixed-order calculations that include non-doubly-resonant diagrams, a combined measurement of $t\bar{t}\gamma + tW\gamma$ is performed.
The focus is placed on a measurement of the fiducial inclusive cross-section in the electron-muon channel, where exactly one photon, one electron and one muon of opposite charge sign, and at least two jets, one of which must be $b$-tagged, are selected.
Furthermore, the ATLAS data is unfolded to parton level and measurements of differential cross-sections as functions of several observables are presented.
Both fiducial inclusive and differential results are compared to state-of-the-art fixed-order calculations at next-to-leading order in QCD.
An additional focus of the thesis is placed on studies to use machine-learning techniques, in particular deep neural networks, for the identification of prompt photons.

CHAPTER: ACKNOWLEDGEMENTS

A doctorate is not something to be achieved without help or guidance from others, most certainly not in experimental particle physics.
The list of people is long and I appreciate everyone I met and worked with over the course of my PhD studies – thank you!

I would like to take the opportunity to thank Arnulf Quadt, my PhD advisor, for his inspiration which had convinced me of particle physics long before even considering a PhD, for the possibility to join his group as a doctoral student and to continue our successful collaboration, for the numerous opportunities of new experiences at conferences, workshops, at , in Tokyo or in Cambridge, and for the constant support in particle-physics matters and beyond.
I am grateful to Stan Lai, the second member of my thesis committee,
to Johannes Erdmann as the second reviewer of this thesis who had many helpful comments in his convenor role,
and to Ivor Fleck as the additional reviewer of this thesis.
I would like to thank the German Academic Scholarship Foundation (Studienstiftung des Deutschen Volkes) for the PhD scholarship that supported me with my research.

I am thankful for the day-to-day supervision by Thomas Peiffer who always had sympathetic ears and helpful advice.
I would like to thank Elizaveta Shabalina for her supervision during my time at .
Her endless experience in top-quark physics, her positive attitude in spite of conference deadlines, and the discussions with her were most helpful.

An analysis is never a one-person job and I am grateful to everyone involved in the  project and the analysis.
Specifically, I would like to thank Julien Caudron, Yichen Li, Carmen Diez Pardos, Thomas Peiffer, Elizaveta Shabalina and Joshua Wyatt Smith for their guidance.
As my main contribution to the analysis was to the measurement of the fiducial inclusive , a special thanks goes to John Meshreki, my Siegen counterpart, who was the lead analyser for the differential measurements.

I am grateful to Osamu Jinnouchi who agreed to the Göttingen–Tokyo collaboration and hosted me within his working group at the Tokyo Institute of Technology.
I would like to extend my gratitude to all his group members and everyone else I met in Japan for welcoming me so kindly.
I am equally grateful to everyone I met during my time at , both at work and beyond (mostly in the mountains).
You made this a truly unique experience.
The same goes to all my colleagues and friends in Göttingen.

I would like to thank Boris, Josh, Lisa, Martin and Thomas for their attentive reading of this document and for providing many helpful comments.
Mein letzter Dank geht an meine Eltern für ihre bedingungslose Unterstützung in allen Zeiten.

CHAPTER: CONTRIBUTIONS BY THE AUTHOR

An experiment as large and complex as the detector needs a collaboration of thousands of physicists to develop and operate the experiment successfully, and to conduct measurements using its recorded collision data.
Since many different steps contribute to publications, these papers are always published in the name of the entire collaboration and are purposely not attributed to individuals.
The author's work documented in this thesis entered two publications: a measurement of the process in the and dilepton channels using data corresponding to 36 of integrated luminosity <cit.>, and a measurement in the channel using 139 of integrated luminosity <cit.>.
The following paragraphs are meant to provide a (non-exhaustive) list of the author's contributions to these measurements.

The 36 analysis <cit.> uses the Prompt Photon Tagger () tool in the channels, a project initiated as a collaborative effort of the Göttingen group.
The tool was designed by B. Völkel and by as part of their MSc thesis and PhD thesis projects, respectively <cit.>.
After the architecture and the design of the had been established and finalised, the author took over from B. Völkel  and contributed to the implementation of the into the analysis, in particular to the treatment of systematic uncertainties associated with the (<ref>).
In addition to the contributions, the author performed studies related to the implementation of other systematic uncertainties in Ref. <cit.>.

The author made central contributions to the measurement <cit.> using data corresponding to 139 and was the lead analyser for the fiducial inclusive measurement:
he wrote the ntuple production software and produced and processed them, studied sample overlap-removal strategies and the categorisation of photons (<ref>), conducted event-selection studies (<ref>) and established the treatment of systematic uncertainties and their evaluation ( <ref>).
These steps are prerequisites for both the fiducial inclusive and differential measurements performed in Ref. <cit.>.
The latter are only summarised briefly in this thesis (<ref>) as they were not the focus of the author's work.
The lead analyser for these differential measurements was J. Meshreki and more details about them are to appear in his PhD thesis.
The author of this thesis conducted the fiducial inclusive measurement and all related studies and established its strategy (<ref>).

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

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titleMeasurements of inclusive and differential cross-sections of combined $t\bar{t}\gamma$ and $tW\gamma$ production in the $e\mkern-2mu\mu$ channel at 13 TeV with the ATLAS detector

University of Göttingen

titleStudies of the discrimination between prompt photons and hadron fakes using neural networks
typeMSc thesis

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# Dark Matter Constraints from the Eccentric Supermassive Black Hole Binary OJ
287

Ahmad Alachkar  King’s College London, Strand, London, WC2R 2LS, United
Kingdom  John Ellis  King’s College London, Strand, London, WC2R 2LS, United
Kingdom  Theoretical Physics Department, CERN, Geneva, Switzerland  National
Institute of Chemical Physics & Biophysics, Rävala 10, 10143 Tallinn, Estonia
Malcolm Fairbairn  King’s College London, Strand, London, WC2R 2LS, United
Kingdom

###### Abstract

OJ 287 is a blazar thought to be a binary system containing a $\simeq 18$
billion solar mass primary black hole accompanied by a $\simeq 150$ million
solar mass secondary black hole in an eccentric orbit, which triggers
electromagnetic flares twice in every $\simeq 12$ year orbital period when it
traverses the accretion disk of the primary. The times of these emissions are
consistent with the predictions of general relativity calculated to the 4.5th
post-Newtonian order. The orbit of the secondary black hole samples the
gravitational field at distances between ${\cal O}(10)$ and ${\cal O}(50)$
Schwarzschild radii around the primary, and hence is sensitive to the possible
presence of a dark matter spike around it. We find that the agreement of
general-relativistic calculations with the measured timings of flares from OJ
287 constrains the mass of such a spike to $\lesssim 3$% of the primary mass.

KCL-PH-TH/2022-40, CERN-TH-2022-115

OJ 287 is an active galactic nucleus (AGN) situated near the ecliptic in the
constellation of Cancer at a cosmological redshift $z=0.3056$ Sitko and
Junkkarinen (1985); Nilsson et al. (2010). It is categorised as a BL Lacertae
(BL Lac) object with a relativistic jet aligned very close to our line of
sight Stein et al. (1976). Due to its proximity to the ecliptic, OJ 287 was
often unintentionally photographed in the past, with an optical database
dating back to 1888 Sillanpaa et al. (1988); Hudec et al. (2001); Visvanathan
and Elliot (1973); Abdo et al. (2010); Hudec et al. (2013), in addition to
dedicated observing programmes Takalo (1996); Goyal et al. (2018); Hudec et
al. (2013). This huge data set contains quasi-periodic pairs of
electromagnetic flares every $\sim 12$ years Valtonen et al. (2006), which are
explained by Lehto and Valtonen Lehto and Valtonen (1996); Sundelius et al.
(1997) as impacts of a secondary black hole (BH) with the accretion disk of
the primary BH as it describes an eccentric orbit. These impacts occur during
periapsis approach and retreat, and the intervals between the flares are
modulated by the precession of the orbit and the emission of gravitational
waves Valtonen et al. (2008); Dey et al. (2018); Laine et al. (2020a).

In this scenario, the secondary BH punches through the geometrically thin,
optically thick and radiation-dominated accretion disk of the primary BH with
hypersonic velocity, shocking gas and generating hot bubbles of plasma on each
side of the disk that expand, cool down adiabatically and eventually radiate
by thermal bremsstrahlung after becoming optically thin Lehto and Valtonen
(1996); Sundelius et al. (1997), producing the observed flares Lehto and
Valtonen (1996). The time delay between the emergence of a bubble at the
impact site and the epoch when it becomes transparent is an important aspect
of the binary BH model. While this can be estimated via detailed astrophysical
modelling of the disk impact shock and outflow evolution Pihajoki (2016), it
necessarily introduces some uncertainty into the calculations.

The observations constrain the primary BH to have a mass $\sim 18.35\times
10^{9}$ $M_{\odot}$ Dey et al. (2018), with the secondary BH having a mass
$\sim 150\times 10^{6}$ $M_{\odot}$. The orbit of the secondary BH has an
eccentricity of $\sim 0.65$, with a periapsis $\sim 9$ and an apoapsis $\sim
48$ times the primary’s Schwarzschild radius, which is $\sim 360$ AU. These
properties make OJ 287 a very powerful laboratory for probing general
relativity (GR) and other aspects of fundamental physics. The GR predictions
for the binary system have been calculated to 4.5th post-Newtonian order,
including the dissipative effects of the emission of gravitational waves, and
these calculations have been used to predict successfully the time of the 2019
burst of electromagnetic emission, which arrived within a few hours of the
predicted time Dey et al. (2018).

In this paper we use the successful comparison of GR predictions with the data
on this unique astrophysical system to provide for the first time a constraint
on physics beyond the Standard Model of particle physics, specifically on
models of dark matter. It was argued in Gondolo and Silk (1999) that if cold
dark matter is present at the centre of a galaxy, a massive central BH
distributes it adiabatically into a ‘spike’ with a radial profile of the form

$\rho(r)\;\propto\;F(r)\left(\frac{R_{\rm sp}}{r}\right)^{\gamma_{\rm sp}}\,,$
(1)

with $F(r)=(1-2R_{S}/r)^{3}$, where $R_{S}$ is the Schwarzschild radius of the
BH, and $R_{\rm sp}$ scales the size of the spike. (See Ullio et al. (2001);
Merritt et al. (2002) for a critique of the possible existence of a spike.)
The density slope parameter, $\gamma_{\rm sp}$, is expected to lie between
2.25 and 2.5. The total mass of cold dark matter in the spike is obtained by
integrating (1) over $r$ until the density sinks below the normal galactic
dark matter density, and is unknown a priori because of the unknown
normalisation of $F(r)$ that depends on the BH being considered and its
environment.

Evidence for the existence of such a spike has been sought in the centre of
the Milky Way. In particular, measurements of the orbit of the star S2, whose
orbit samples the gravitational field of Sgr A* down to radii $\sim 10^{-2}$
pc, were found in Lacroix (2018) to constrain the spike mass to $\sim 1$ % of
the mass of Sgr A*, assuming $\gamma_{\rm sp}=7/3$ and $R_{\rm sp}=100$ pc,
the upper limit obtained from VLT measurements of the orbit of S2. (A weaker
constraint is provided by the concordance between the estimates of the mass of
Sgr A* based on these orbital measurements and the radius of the innermost
stable circular orbit measured by the Event Horizon Telescope,
$r_{\text{ISCO}}\simeq 50$ $\mu$as Akiyama et al. (2022), which requires that
the spike mass be $\lesssim 10$ % of the mass of Sgr A*.)

The eccentricity of the orbit of the secondary BH in the OJ 287 system implies
that it is sensitive to the density of any such spike around the primary BH at
radii between the binary separations at periaspsis and apoapsis, potentially
causing deviations from the GR predictions. We use simulations to model these
deviations and fit the OJ 287 data with modified parameters that include dark
matter spikes of different magnitudes. We treat the gravitational effects of
the spike in the Newtonian approximation, assuming that it is centred on the
primary black hole and spherical, that shells with radii less than the
separation between the primary and secondary have the same effect as a
pointlike mass co-located with the primary, and that the effects of shells
with radii larger than the separation can be neglected. We follow Lacroix
(2018) in assuming $\gamma_{\rm sp}=7/3$, while we fix $R_{\rm sp}$ at
$5R_{S}$ (of the primary BH) throughout the analysis, and allow $\rho_{\rm
sp}$ to vary such that the total DM mass enclosed within a region
$2R_{S}\lesssim r\lesssim 50R_{S}$ (encompassing the trajectory traversed by
the secondary BH around the primary from periapsis to apoapsis) equals the
studied $m_{\rm sp}/m_{1}$ ratio (ranging from 0 to 0.03). This is well within
the gravitational radius of influence of the primary BH (defined as the radius
at which the enclosed DM mass is twice the mass of the primary BH). We neglect
dynamical friction and post-Newtonian effects due to the spike, which is
justified a posteriori by the upper limit on its mass that we establish.

We use the Post-Newtonian (PN) scheme to calculate the orbital dynamics up to
4.5PN order. We work in the centre-of-mass (CoM) frame, converting the two-
body problem to an effective one-body problem, and use harmonic coordinates.
The harmonic condition, (de Donder gauge) is imposed, breaking general
covariance. The relative two-body acceleration and the precessional dynamics
of the spin of the primary, written in terms of the unit vector
$\mathbf{s}_{1}$, can be estimated using

$\displaystyle\ddot{\mathbf{x}}\equiv\frac{d^{2}\mathbf{x}}{dt^{2}}$
$\displaystyle=$
$\displaystyle\ddot{\mathbf{x}}_{0}+\ddot{\mathbf{x}}_{1\mathrm{PN}}+\ddot{\mathbf{x}}_{2\mathrm{PN}}+\ddot{\mathbf{x}}_{3\mathrm{PN}}$
$\displaystyle+$
$\displaystyle\ddot{\mathbf{x}}_{2.5\mathrm{PN}}+\ddot{\mathbf{x}}_{3.5\mathrm{PN}}+\ddot{\mathbf{x}}_{4\mathrm{PN}\mathrm{(tail)}}+\ddot{\mathbf{x}}_{4.5\mathrm{PN}}$
$\displaystyle+$
$\displaystyle\ddot{\mathbf{x}}_{\mathrm{SO}}+\ddot{\mathbf{x}}_{\mathrm{SS}}+\ddot{\mathbf{x}}_{\mathrm{Q}}+\ddot{\mathbf{x}}_{\mathrm{4PN(SO-
RR)}}$ $\displaystyle\frac{d\mathbf{s}_{1}}{dt}$ $\displaystyle=$
$\displaystyle(\mathbf{\Omega}_{\mathrm{SO}}+\mathbf{\Omega}_{\mathrm{SS}}+\mathbf{\Omega}_{\mathrm{Q}})\times\mathbf{s}_{1},$

where $\mathrm{\mathbf{x}}=\mathrm{\mathbf{x_{1}}}-\mathrm{\mathbf{x_{2}}}$ is
the CoM relative separation vector between the BHs with masses $m_{1}$ and
$m_{2}$, $\ddot{\mathbf{x}}_{0}=-\frac{Gm}{r^{3}}\mathrm{\mathbf{x}}$ is the
zeroth-order Newtonian acceleration, where $m$ is the total mass of the binary
and $r=|\mathrm{\mathbf{x}}|$, and
$\mathbf{S}_{1}=\frac{Gm_{1}^{2}\chi_{1}}{c}\mathbf{s}_{1}$ is the spin of the
primary with the Kerr parameter $\chi$ $\in[0,1]$ in GR. The terms
$\ddot{\mathbf{x}}_{\mathrm{SO}}$ and $\ddot{\mathbf{x}}_{\mathrm{SS}}$ denote
the spin-orbit (SO) and spin-spin (SS) couplings in GR, entering at 1.5PN and
2PN at leading orders, while $\ddot{\mathbf{x}}_{\mathrm{Q}}$ is the classical
spin-orbit coupling (Q) arising from the quadrupole deformation of a Kerr BH,
at 2PN order, and the $\mathbf{\Omega}_{i}$ are the SO, SS and Q contributions
to the precession of the primary BH spin vector. Our calculations of the
various terms in (Dark Matter Constraints from the Eccentric Supermassive
Black Hole Binary OJ 287) are based on an extensive body of work Iyer and Will
(1993, 1995); Jaranowski and Schäfer (1997); Königsdörffer et al. (2003); Itoh
(2009); Gopakumar et al. (1997) and will be set out in detail in an upcoming
publication.

The baseline binary BH model has 9 relevant parameters: the two BH masses
$m_{1,2}$, the primary BH Kerr parameter $\chi_{1}$, the initial apocentre
eccentricity $e_{0}$, the initial semimajor axis $a$ and its angle of
orientation $\theta_{0}$, an ambiguity parameter $\gamma$ for the leading-
order hereditary contributions to GW emission in the BBH dynamics implemented
following Equation (5) in Dey et al. (2018), an azimuthal angle $\theta_{S1}$
and a polar angle $\psi_{S1}$ parametrising the orientation of the primary
spin vector.

When making fits, the plane-crossing epochs must be corrected to incorporate
astrophysical processes between an impact and an observed optical outburst,
with a time delay $t_{\mathrm{del}}$ added to account for the time lag between
the generation of the plasma bubbles and the epoch at which they become
optically thin. An additional timing correction $t_{\mathrm{adv}}$ is applied
to model the tidal force of the approaching secondary BH that warps the disk
and advances the impacts. We adopt empirical models of the time delay and
advance based on the best-fit orbit of Dey et al. (2018), which our good fits
resemble. In our orbital fitting procedure, the observational uncertainties
are all assumed to be Gaussian and uncorrelated, and the optimisation
algorithm adopted is the Nelder-Mead algorithm Nelder and Mead (1965), which
we implement using the constrNMPy package.

Fig. 1 displays the evolution of the OJ 287 system in our best no-DM fit over
a period of 120 y, corresponding to 10 orbits of the secondary BH (represented
by the smaller black spot), in a coordinate system centred on the primary BH
(represented by the larger black spot). We see clearly the large orbital
precession and the passages of the secondary BH through the accretion disc
(represented as a shaded plane) as it approaches and retreats from the
periapsis.

Figure 1: The precessing orbit of the secondary BH (smaller black spot) in a
coordinate system centred on the primary BH (larger black spot). Note the
passages of the secondary BH through the accretion disc (represented as a
shaded plane).

Table 1 lists in the first column the starting times of flares of OJ 287
starting with that at the end of 1912, with the corresponding uncertainties
shown in the second column. The timings of the flares not written in bold were
not measured accurately and are not used in our fits. Ref. Dey et al. (2018)
used 10 flares, corresponding to 9 time differences that they fitted using the
9 parameters listed above. In our analysis we have one additional parameter
characterizing the spike, and our fits include the well-measured 2019 flare as
well as the less accurately measured flares in 1959 and 1994. The contribution
of the spike to the dynamics is included at zeroth order and follows the mass
profile obtained from (1). The third column in the Table lists the predictions
for the starting times in our best-fit no-DM spike model, the fourth column is
our best fit for a spike mass ratio $m_{\rm sp}/m_{1}=0.02$, and the fifth
column is that for a spike mass ratio of 0.03.

Flare times, uncertainties and model estimates
Julian year uncertainty No spike 2% spike 3% spike 1912.980 ±0.020 1912.982
1912.968 1912.958 1922.529 - 1922.536 1922.537 1922.537 1923.725 - 1923.730
1923.725 1923.722 1934.335 - 1934.337 1934.340 1934.340 1935.398 - 1935.403
1935.402 1935.400 1945.818 - 1945.819 1945.822 1945.822 1947.283 ±0.002
1947.285 1947.284 1947.283 1957.095 ±0.025 1957.087 1957.078 1957.073 1959.25
±0.05 1959.216 1959.213 1959.213 1964.231 - 1964.242 1964.216 1964.204
1971.126 - 1971.129 1971.127 1971.127 1972.935 ±0.012 1972.934 1972.921
1972.916 1982.964 ±0.0005 1982.964 1982.965 1982.965 1984.125 ±0.01 1984.120
1984.116 1984.114 1994.77 ±0.1 1994.595 1994.599 1994.599 1995.841 ±0.002
1995.839 1995.838 1995.838 2005.745 ±0.015 2005.747 2005.754 2005.754
2007.6915 ±0.0015 2007.693 2007.691 2007.692 2015.875 ±0.025 2015.882 2015.882
2015.872 2019.569 ±0.0005 2019.569 2019.568 2019.569

Table 1:  Starting epochs (in Julian years) of the observed optical flares of
OJ 287 from 1912 onwards. First column: The data points prior to 1970 were
obtained from archival photographic plates while the historical 1912/3 flare
time is from Hudec et al. (2013). We only use the flares listed in bold, the
starting times of the other flares are not known accurately and are not used
in the fits. Second column: The uncertainties in the starting times. Third,
fourth and fifth columns: Our predictions for the starting times in the best-
fit no-DM spike model and in the best-fit models with spike mass ratios of 2
and 3%.

Fig. 2 shows the minimum values of $\chi^{2}$ that we found for the indicated
choices of the spike mass ratio, $m_{\rm sp}/m_{1}$. The best-fit no-DM model
has $\chi^{2}=6.05$, and the best fits for values of $m_{\rm sp}/m_{1}<0.03$
all have $\chi^{2}\lesssim 10$, corresponding to $p$-values $\gtrsim 20$%.
However, the best fit for $m_{\rm sp}/m_{1}=0.03$ has $\chi^{2}\simeq 13.6$,
corresponding to a $p$-value $\simeq 3$%, and the values of $\chi^{2}$ rise
sharply as $m_{\rm sp}/m_{1}$ increases beyond $0.03$. For example,
$\chi^{2}\simeq 99$ already for $m_{\rm sp}/m_{1}=0.035$, corresponding to a
$p$-value $\simeq 10^{-18}$. On the basis of these results we conclude that
the current OJ 287 data set an upper limit $m_{\rm sp}/m_{1}\lesssim 0.03$.

Figure 2: The dependence of the $\chi^{2}$ likelihood function on the ratio
$m_{\rm sp}/m_{1}$, as obtained from the sampling of model parameters
described in the text.

The upper panel of Fig. 3 compares the predictions of our best-fit models with
no DM spike (shown as red spots), $m_{\rm sp}/m_{1}=0.02$ (shown as blue
crosses) and $m_{\rm sp}/m_{1}=0.03$ (shown as purple triangles) with the
observations (shown in black). We see a high level of consistency for the no-
DM model, the most noticeable deviation being that for the 1994 flare.
However, the observations of this flare were the least precise among the
flares fitted, and its timing is the most uncertain, so we do not regard this
deviation as being significant. There is also a good level of consistency for
the best fit with $m_{\rm sp}/m_{1}=0.02$. However, the best fit with $m_{\rm
sp}/m_{1}=0.03$ clearly does not fit the data as well, enabling the eye to
confirm the numerical analysis presented in the previous paragraph.

Figure 3: Upper panel: The best-fit predictions for the flare timings
calculated in a model without a dark matter spike (red spots), as obtained
from the sampling of model parameters described in the text, compared with the
measured timings (black). Also shown are the timings for the best fits with
$m_{\rm sp}/m_{1}=0.02$ (blue crosses) and $m_{\rm sp}/m_{1}=0.03$ (purple
triangles). Lower panel: A scatter plot of the predictions for the interval
between the 2022 and 2019 flares (vertical axis) for our best fits with the
indicated values of $m_{\rm sp}/m_{1}$ (horizontal axis). We also show the
prediction of Valtonen et al. (2022) for the the interval between the 2022 and
2019 flares (horizontal red dashed line).

Our no-DM spike prediction for the very well measured 2019 flare shown in
Table 1 agrees with the measured timing to within a few hours, within the
uncertainty in the starting time of the 2019 flare. The timing of the 1982
flare is as well known as that in 2019, and our no-DM spike prediction for it
again agrees with the measurement within a few hours.

As further support for our conclusion that $m_{\rm sp}/m_{1}\lesssim 0.03$, we
have considered two possible side-effects of interactions between the
secondary black hole and the spike. One is the dissipative effect of back-
reaction on the orbit of the secondary black hole, and the other is the
possible disruption of the spike by the secondary black hole. Following the
approach of Kavanagh et al. (2020), we find that both effects can be neglected
in a first approximation.

According to the model of Valtonen et al. (2008); Dey et al. (2018); Laine et
al. (2020a) another flare of OJ 287 is expected shortly, though there are
considerable astrophysical uncertainties in its timing, as discussed in
Valtonen et al. (2022). We now discuss how the timing of this flare might
differ from the no-spike pure GR case discussed in Valtonen et al. (2022). As
seen in the lower panel of Fig. 3, our estimates for $m_{sp}/m_{1}\lesssim
0.03$ suggest that the next flare might be advanced by $\lesssim 5$ days
relative to the no-spike prediction. On the other hand, we estimate that it
would be delayed by $>10$ days for $m_{sp}/m_{1}\geq 0.035$. In view of the
uncertainty in the arrival time estimated in Valtonen et al. (2022), our
results suggest that observations of the next flare of OJ 287 may not be able
to strengthen our constraint $m_{sp}/m_{1}\leq 0.03$ based on the timings of
previous flares.

GR has already been tested by observations of OJ 287 to the 4.5PN order,
including the effects of GW emission Laine et al. (2020b), and will be tested
again by observations of future flares. We have shown in the paper, for the
first time, how observations of OJ 287 can be used to constrain models of new
physics beyond the Standard Model of particle physics. Specifically,
observations of OJ 287 can constrain the possible existence of a cold dark
matter spike surrounding its primary BH. They already constrain the possible
mass of such a spike to $m_{\rm sp}/m_{1}<0.03$ at the 99.9% CL, and
measurements of the timing of the expected future flares and more detailed
modelling have the potential to constrain further the spike mass.

There are certainly ways in which the modelling of dark matter effects on the
evolution of OJ 287 could be improved. For example, we have considered the
possible static gravitational effects of a cold dark matter spike but have not
considered other possible interactions between the secondary black hole and
the spike, such as tidal forces or non-gravitational interactions. Another
important area for future research will be to refine the modelling of the
dynamical interactions between the secondary black hole and the accretion
disk, and the time delay between the nominal impact on the accretion disk and
electromagnetic emissions from the cooling of the plasma bubble that it
generates, which is the dominant uncertainty in the prediction of the timing
of any future flare.

OJ 287 has barely begun to demonstrate its potential for probing not just
astrophysics and GR, but also other aspects of fundamental physics.

## Acknowledgements

We are grateful to Mauri Valtonen and Pauli Pihajoki for fruitful discussions
and valuable comments. The work of AA was supported by a UK STFC studentship,
that of JE and MF was supported in part by STFC Grants ST/P000258/1 and
ST/T000759/1, and that of JE also in part by the Estonian Research Council via
a Mobilitas Pluss grant.

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