text:
Belews Lake loses protections under Clean Water Act overhaul, lawyer says | Local
BELEWS CREEK — The Trump administration's overhaul of the Clean Water Act earlier this year removes Belews Lake from its protection and environmentalists are at odds with federal and Duke Energy officials over how this potentially affects that and at least a dozen other lakes.
Environmentalists are suing the federal government, claiming changes made to the landmark environmental act put Belews Lake and many other water bodies at risk, including some that are sources for drinking water.
Duke Energy owns the 3,864-acre lake, a popular recreational site for boating and fishing located about 20 miles northwest of Greensboro.
President Donald Trump campaigned on overhauling the act, saying that it squelched private property rights. Trump signed an executive order shortly after taking office, directing federal agencies to begin the work of repealing and replacing it.
The Southern Environmental Law Center, on behalf of several environmental groups, filed for an injunction in April to throw out the rule changes issued by the Environmental Protection Agency and the U.S. Army Corps of Engineers. The U.S. District Court in Charleston, S.C., is scheduled to hear summary motions in SELC's lawsuit on Dec. 1, but a ruling won't likely come until later, according to Frank Holleman, senior attorney for the SELC.
Holleman said the change not only prevents provisions of the act from being applied to the lake, which was created as a steam station cooling pond, but also takes away a vital avenue for the community to hold Duke Energy accountable if it is polluted.
The change revises the definition of "waters of the United States" to exclude cooling ponds. The 1972 law regulates and sets standards for pollutant discharges into these waters, which includes lakes, rivers, wetlands and coastal areas.
Cooling ponds
And that's where Belews Lake and similar lakes factor in. Under the new definition, Holleman said the entire lake now is considered part of the steam station's wastewater treatment facility and not subject to the federal environmental law.
However, Nathan Craig, a principal with Duke Energy who specializes in water and natural resource issues, said the lake always has been treated as and operated under regulatory standards set for wastewater treatment systems involving cooling lakes.
"No other wastewater or pollutant is treated in Belews Lake nor is allowed to be treated in Belews Lake," he said. "Except for temperature, the lake and any discharge into the lake must meet strict Clean Water Act and state water quality requirements."
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As for Duke Energy's interest in the rule, Craig said it was to ensure the "waste treatment exclusion" in the act applies to ash basins, cooling ponds and other components of the waste treatment system. "The company wanted clarity to ensure these systems can continue to operate under the same regulatory requirements as they have since 1979," when wastewater treatment systems were excluded from the act.
Officials with the EPA and the N.C. Department of Environmental Quality, which issues permits for the lake, did not respond to messages and emails seeking comment for this story. Neither did the Edison Electric Institute, the national association of shareholder-owned electric utilities, which lobbied for the changes.
Belews Lake was created by Duke Power Co., now known as Duke Energy, in the early 1970s by damming Belews Creek to provide water for the Belews Creek Steam Station.
The station is one the Southeast's largest coal-burning power plants, although recent changes give operators the option to also run on natural gas. It generates up to 2,240 megawatts, enough to bring power to nearly 1.8 million average homes at full load.
Lake water is used in the process, whether the plant runs on coal or natural gas. While lake water doesn't come into contact with the station's steam, it is returned to the lake at a higher temperature.
"Duke Energy discharges hot water into Belews Lake and if that heat is not adequately controlled, that affects the health of the lake and the fish in the lake," Holleman said. The heat is the primary pollutant for the lake, he said.
That discharge was covered under the Clean Water Act before the revised definition of "cooling ponds" and putting the entire lake out of the act's purview.
"This is an unprecedented change, to remove these lakes from coverage of the Clean Water Act, Holleman said.
But Duke Energy officials say changes have no real impact on how the company monitors the lake and that the state permitting process assures safety.
"We still have very, very strict state requirements to meet, so that doesn't change, the lake remains protected," Duke spokesman Bill Norton said.
Craig said the discharge into Belews Lake is primarily cooling water and possibly some stormwater.
Pollutants such as bromides, mercury, arsenic, selenium and chromium generated by the plant are diverted to the Dan River, which still remains under the protection of the Clean Water Act. Temperature limits also are placed on this wastewater.
"This change isn't really going to change how this station is currently operated and how the station has been operated since coming online back in the '70s," Craig said.
Past pollution
But Holleman said the Clean Water Act has played a vital part in ensuring Duke Energy cleans up when something goes awry.
"What worries us is will these assurances stand up when the rule is not written to protect the lake," he said. Holleman cited the utility's pollution problems involving selenium leaking into Belews Lake in the 1970s and early '80s, and the 2014 coal ash spill into the Dan River at the site of the utility's retired steam station in Eden, which was torn down in 2017.
The steam stacks from Duke Energy's Belews Creek Steam Station are seen at Belews Lake from Humphrey's Ridge in Stokesdale, Wednesday, Sept. 30, 2020. The Trump administration's rewrite of the Clean Water Act means Belews Lake will no longer be covered by the environmental protection law.
Walt Unks, Winston-Salem Journal
Shortly after the Belews Creek Steam Station was built, a coal ash basin — a type of waste pond — leaked, sending toxic amounts of the wastewater byproduct selenium into Belews Lake. Within years, 17 of the lake's 20 fish species were wiped out.
Holleman said the N.C. Department of Environmental Quality, using provisions in the Clean Water Act, required Duke to change how it managed the contaminants.
"(By) applying the Clean Water Act, the coal ash pollution of the lake was greatly reduced over time by routing the discharge from the coal ash site out of the lake," Holleman said. "So that's an example of how the Clean Water Act can protect the lake."
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The act also was vital in forcing Duke Energy to move 12 million tons of coal ash submerged for years in a storage basin to a lined, on-site landfill, Holleman said.
The power company agreed to the plan in a settlement after the SELC filed a lawsuit on behalf of the Stokes County and state branches of the NAACP and Appalachian Voices North Carolina environmental group. The lawsuit sought to force Duke Energy to move ash submerged for years in a storage basin to a lined, on-site landfill and clean up groundwater at the site.
The civil rights groups were involved because the coal ash site is near a predominantly African American community in Walnut Cove.
Duke has upgraded technology at the Stokes County plant so that all coal ash is handled in dry form — much of it recycled — and the basin is no longer in use.
Lawsuits such as this, where residents can directly sue, aren't allowed under North Carolina law, Holleman said. "So the local community would not have the ability to enforce permit requirements like they do under the Clean Water Act," he said.
The removal also opens the door for operations other than Duke to use the lake for wastewater purposes, Holleman said.
Norton said this was unlikely because Duke owns all the land surrounding the lake and would have to grant an easement from the power company to access the lake.
"We do not anticipate such a request, and any such entity would have to apply for state and federal permits, such as with the Army Corps of Engineers," he said.
Political sway
When he announced the rule changes in January, EPA Administrator Andrew Wheeler told reporters that states were still free to step in with state protections of newly vulnerable waterways if they chose.
And Duke spokesman Norton said even without the Clean Water Act provisions, the company's activities are monitored and well-regulated by the state.
"There's not just one tier of protection," he said. "There were also state laws there that protected, that drove the decisions that we ultimately made (surrounding the coal ash disposal)."
Norton said that under the state regulations and permits, Duke has monitored Belews Lake since the mid-1970s, with surface water testing since 1977 and fish studies since 1976.
"We're held accountable for what we say," Norton said. "We use third-party labs that are certified, independently of us, on those test results. So when you have got that level of protection. You have the state watching out, you have independent labs doing the results, and then you've got federal rules as well."
But Holleman said such rules can be influenced by the political winds — especially at the state level.
"Would North Carolina require something of businesses and industry and polluters that the federal government does not," Holleman asked. "That's always the hard question because of the political pressure that's brought to bear on state agencies.
"The state agencies often are … reluctant to put strong requirements on polluters, particularly major business," Holleman said. "It is really the backstop of the national Clean Water Act and the national EPA requirements that the state agencies rely upon in order to resist local political pressure."
And he pointed out former Gov. Pat McCrory's ties to the company.
"You may remember the prior governor used to work for Duke Energy and he fought against requiring them to clean up their coal ash," Holleman said. "The new governor's administration is requiring them to clean up their coal ash. Well, who's going to be the governor eight years from now? And who's going to be the secretary of the environment?"
Locals' reaction
Josh Snow, owner of Reel-Livin Outdoor Adventures in King, fishes on Belews Lake while waiting for clients at the Pine Hall public boat launch, Wednesday, Sept. 30, 2020. The Trump administration's rewrite of the Clean Water Act means Belews Lake will no longer be covered by the environmental protection law.
Phillip and Nancy Ridge, who've lived beside Belews Lake since the 1970s, said they trust Duke Energy to protect it. The couple lives in a small mobile home park on West Ridge Circle, just yards away from its shoreline.
The couple said that the company has been a good steward of the lake and it doesn't bother them that it's no longer covered by the Clean Water Act.
Phillip Ridge, 79, said it's in Duke Energy's interest to use the best practices available in operating the facility.
"They're down there now working on the dam," he said. "They're trying to keep everything up-to-date.
"Why would they mess up something they need," he said. "I mean, that's where they're making their money."
Nancy Ridge said she goes in the lake every summer, "all summer long," noting that it's safe. "That's my exercise in the summer."
Still, others who use the lake said they would rather see the additional protection offered by the Clean Waters Act.
"I think it's a shame, people swim in this lake," said Josh Snow, who recently formed Reel-Fishin Outdoor Adventures and runs fishing tours at the lake. "I think the people who use the lake ought to have a voice for it."
Lewis Pritchard of Walnut Cove, who was out kayaking on the lake last week, said he'd rather see the act applied to the lake.
"I guess the Clean Water Act would help motivate them more," Pritchard said. "I like to keep the lake pretty. I kayak a lot out here … and I eat the fish, too."
Those activities are why, Holleman said, keeping the lake under the Clean Water Act is so important.
"It gives the local community power to protect their water resources when the state and federal government are not taking those actions," Holleman said.
Belews Lake and Steam Station
High-voltage power lines extend eastward from Duke Energy's Belews Creek Steam Station as seen from Oak Level Church Road in Stokesdale, Wednesday, Sept. 30, 2020. The Trump administration's rewrite of the Clean Water Act means Belews Lake will no longer be covered by the environmental protection law.
Pontoon boats are moored at Humphrey's Ridge on Belews Lake, Wednesday, Sept. 30, 2020. Duke Energy's Steam Station is seen in the background. The Trump administration's rewrite of the Clean Water Act means Belews Lake will no longer be covered by the environmental protection law.
A couple of fishermen fish in one of the fingers of Belews Lake, Wednesday, Sept. 30, 2020. The Trump administration's rewrite of the Clean Water Act means Belews Lake will no longer be covered by the environmental protection law.
Josh Snow, owner of Reel-Livin Outdoor Adventures in King prepares his boat for clients at the Pine Hall public boat launch, Wednesday, Sept. 30, 2020. The Trump administration's rewrite of the Clean Water Act means Belews Lake will no longer be covered by the environmental protection law.
Duke Energy's Belews Creek Steam Station sits on the edge of Belews Lake as seen from the Pine Hall Public Boat Launch, Wednesday, Sept. 30, 2020. The Trump administration's rewrite of the Clean Water Act means Belews Lake will no longer be covered by the environmental protection law.
Lindy Ritz, Winston-Salem Journal
Belews Lake is seen from the Pine Hall Public Boat Launch, Wednesday, Sept. 30, 2020. The Trump administration's rewrite of the Clean Water Act means Belews Lake will no longer be covered by the environmental protection law.
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\section{Introduction}
The second generation \ac{GW} detectors — Advanced LIGO~\cite{TheLIGOScientific:2014jea}
and Virgo~\cite{TheVirgo:2014hva} — have so far published observations of
11 compact binary mergers from the
first two observing runs~\cite{LIGOScientific:2018mvr}, including
one binary neutron star merger that was also observed across the electromagnetic
spectrum~\cite{GBM:2017lvd}.
The third observing run is currently underway, with further sensitivity improvements planned in the coming
years~\cite{Aasi:2013wya}.
\ac{GW} observations have already begun to constrain models of the formation and rates of stellar mass
compact binary mergers~\cite{LIGOScientific:2018jsj}, and to make strong-field tests of the general theory of
relativity~\cite{LIGOScientific:2019fpa}.

Models for the \ac{GW} signal, parametrized in terms of the properties of the system (such as masses, spins, and orientation), are compared with detector data to
infer the source properties of \ac{GW} events.
The \ac{GW} signal is commonly expressed in a multipole expansion where we
denote terms beyond the leading order quadrupole contribution as ``higher order multipoles''.
These higher order multipoles are typically much weaker than
the dominant quadrupolar multipole, but grow in relative strength
for systems that are more asymmetric in mass.
Past studies have shown that for events where the signal contains measurable
power in the higher multipoles,
parameter estimates can be biassed when using only a dominant-multipole model.
Conversely, it is also true that for some systems we are able to measure
the source parameters more accurately using a higher multipole model
~\cite{PhysRevD.92.022002,PhysRevLett.120.161102,Chatziioannou:2019dsz,Kalaghatgi:2019log,PhysRevLett.121.191102,2019arXiv191102693S}.

Another important physical effect is spin precession, where
couplings
between the orbital and spin angular
momenta can cause the orbital plane to precess
and thus cause modulations of the observed \ac{GW}~\cite{PhysRevD.49.6274,PhysRevD.52.821}.
In terms of the \ac{GW} multipoles, precession
mixes together different orders ($m$-multipoles) of the same degree ($\ell$-multipoles), complicating
a simple description of the
waveform~\cite{Schmidt:2010it,PhysRevD.84.124011,PhysRevD.85.084003,PhysRevD.86.104063,PhysRevD.88.024040,Boyle:2014ioa,PhysRevD.84.124002,Fairhurst:2019vut}.
By not taking into account precession and higher order multipoles
in our waveform models we may not be able to confidently
detect and accurately characterize signals where
these effects are important~\cite{Capano:2013raa,PhysRevD.96.124024,CalderonBustillo:2017skv,PhysRevD.94.024012,Harry:2017weg,PhysRevD.93.084019}.
These events are also likely to be very interesting astrophysically,
providing valuable information about \ac{BBH} formation mechanisms
and hence are events with high
scientific gain that we wish to model and measure accurately.

The field of waveform modeling has seen sustained development over almost two
decades and is currently thriving,
with improvements to current and development of novel methods
allowing for more accurate and efficient
models to be applied in data analysis pipelines
~\cite{PhysRevD.59.084006,Pan:2013rra,PhysRevD.62.064015,Pan:2011gk,
Taracchini:2013rva,Blanchet2014,Hannam:2013oca,Ajith:2007qp,
PhysRevD.82.064016,Khan:2015jqa,PhysRevLett.120.161102,
PhysRevD.98.084028,PhysRevD.95.044028,PhysRevD.95.024010,
PhysRevD.96.024058, PhysRevD.98.104052,Mehta:2019wxm,
Varma:2018mmi, Williams:2019vub,Setyawati:2018tqp,
PhysRevD.100.024059,PhysRevD.89.044021,Doctor:2017csx,Blackman:2017dfb,Varma:2019csw}.
In this work we take a step towards including as many important physical
effects as possible in waveform models, by constructing the most physically complete
phenomenological model to date.
We present a frequency-domain model for the \ac{GW} signal
from the inspiral, merger and ringdown of a \ac{BBH} system.
The \acp{BH} are allowed to precess and we also model the contribution
to the \ac{GW} signal from higher order multipoles.
This combines the progress made in two earlier models: a precessing-binary model
that includes accurate two-spin precession effects during the
inspiral~\cite{Chatziioannou:2017tdw,Chatziioannou:2016ezg}
({\tt PhenomPv3}~\cite{PhysRevD.100.024059}), and an approximate higher-multipole
aligned-spin model ({\tt PhenomHM}~\cite{PhysRevLett.120.161102}).

\begin{figure}[t!]
\includegraphics[width=0.98\columnwidth]{figs/vertical-td-wEOB.pdf}
 \caption{
 Comparison of the detector response strain $h(t)$ viewed at an inclination angle of $\pi/2$.
 Solid grey: \ac{NR} simulation (SXS:BBH:0058).
 A mass-ratio $q \equiv m_1/m_2 = 5$, precessing \ac{BBH}
 simulation with a dimensionless-spin magnitude of $\chi_1 = 0.5$ generated with a total mass of $80M_\odot$.
 The \ac{NR} signal contains all the $\ell,m$ modes up to and including $\ell=4$.
 Top panel, blue: Precessing model {\tt SEOBNRv3}~\cite{PhysRevD.89.084006}, with the $((2,\pm 2), (2,\pm 1))$
 modes in the co-precessing frame.
 Middle panel, orange: Precessing model {\tt PhenomPv3}~\cite{PhysRevD.100.024059} with only the $2,\pm 2$
 modes in the co-precessing frame.
 Bottom panel, green: Precessing model presented here, {\tt PhenomPv3HM}, with the
 $(\ell, |m|) = ((2,2), (2,1), (3,3), (3,2), (4,4), (4,3))$ modes in the co-precessing frame.
 The orientation-averaged mismatch $1-\overline{\mathcal{M}}$ (see Sec.~\ref{sec:matches})
 is $8\,\%$ for the top panel,
 $7\,\%$ for the middle and $1\,\%$ for the bottom.
 We only plot the last $\sim$7 \ac{GW} cycles for clarity but the behaviour is qualitatively
 the same throughout the 29-orbit inspiral ($\sim$60 \ac{GW} cycles).
 }
 \label{fig:TDComparison}
\end{figure}

Figure~\ref{fig:TDComparison} demonstrates the improved accuracy that is
achievable by our new model, {\tt PhenomPv3HM}, compared to other existing models
that include the effect of spin precession, but not higher order multipoles.
We compare the observed \ac{GW} signal
predicted by our new model against a high-mass-ratio, precessing
\ac{NR} simulation\footnote{The \ac{NR} waveform is SXS:BBH:0058 from the SXS
public catalogue~\cite{sxsonline}. It has a
mass-ratio of $q=5$ with spin on the larger \ac{BH}
directed in the orbital plane with a dimensionless spin magnitude of $\chi_1 = 0.5$.} (thick grey line).
We plot the \ac{GW} signal observed at an inclination
angle\footnote{Here we define the inclination as the angle between the orbital
angular momentum and the line of sight at the beginning of the waveform.}
of $\pi/2$ rad to emphasise the effect of precession.
We use all multipoles in the range $2 \leq \ell \leq 4$ when computing
the \ac{NR} \ac{GW} polarisations.

We compute the mismatch (defined in Section~\ref{sec:matches}) between
three different precessing waveform models and the \ac{NR} waveform, and average
over all possible orientations.
The top panel shows the optimal
waveform (in blue) when we use
{\tt SEOBNRv3}~\cite{PhysRevD.95.024010} and
the middle panel shows (in orange) the result when
we use {\tt PhenomPv3}.
In this context the optimal waveform maximises the overlap over
coalescence time, template phase and polarisation angle and the intrinsic
parameters are fixed to the values from the \ac{NR} simulation.
As shown in~\cite{PhysRevD.100.024059} {\tt SEOBNRv3} and {\tt PhenomPv3}
have overlaps of $\sim99\,\%$ and $\sim 98\,\%$ respectively to this \ac{NR}
waveform when only the $\ell=2$ multipoles are considered.
When we include higher order multipoles in the \ac{NR} waveform
we find the overlap drops to only $\sim 92\,\%$ and $\sim 93\,\%$ respectively.
This is an example where the exclusion of higher multipoles in template
models can lead to unacceptable losses in \ac{SNR}.
The bottom panel shows, in green, the best fitting {\tt PhenomPv3HM}
template.
We find remarkable agreement, even through the inspiral, merger
and ringdown stages. The overlap is now $99\,\%$ and the subtle modulation
visible is accurately captured by our model.
It is useful to point out here that, in {\tt PhenomPv3HM}, the higher multipole
and the precession elements of the model have not been
calibrated to \ac{NR} simulations, but when this is done
we expect the accuracy to improve further.

The rest of the paper is organized as follows. In Sec.~\ref{sec:method}
 we describe how our model is constructed.
 In Sec.~\ref{sec:matches} we present results where we have compared our
model against precessing \ac{NR} simulations including higher order mutlipoles
up to and including $\ell = 4$ to demonstrate its accuracy across
the parameter space where we have \ac{NR} simulations.
We have also performed a parameter estimation study to quantify the impact
on parameter recovery when using a model that includes both higher multipoles
and precession, the results of which are presented in section~\ref{sec:pe}.

Finally, in section~\ref{sec:gw170729} we have analysed data for
the GW170729 event, publicly available
at the Gravitational Wave Open Science Center
(GWOSC)~\cite{Vallisneri:2014vxa},
which has evidence for non-zero \ac{BH} spin~\cite{LIGOScientific:2018mvr}
and unequal masses~\cite{Chatziioannou:2019dsz}.

\section{Method}
\label{sec:method}

Our method to build a model for the \ac{GW} signal from precessing
\acp{BBH} is based upon the novel ideas of
Refs.~\cite{PhysRevD.84.024046, PhysRevD.84.124002, PhysRevD.84.124011},
where the \ac{GW} from precessing binaries
can be modelled as a dynamic rotation of non-precessing systems.
In  Refs.~\cite{Hannam:2013oca,Taracchini:2013rva,PhysRevD.89.084006} the authors
used these ideas to build the first precessing \ac{IMR} models.

Our goal is to derive frequency-domain expressions for the
\ac{GW} polarisations $\tilde{h}_{+/\times}(f)$ in terms of the
 multipoles $\tilde{h}_{\ell m}(f)$.
We start from the complex \ac{GW} quantity, $h = h_{+} - i h_{\times}$,
in the time-domain and decompose
this into spin weight $-2$ spherical harmonics,

\begin{equation}
\label{equ:htd}
  h(t,\vec{\lambda},\theta,\phi)  = \sum_{\ell \geqslant 2} \sum_{-\ell\leqslant m \leqslant \ell} h_{\ell,m}(t,\vec{\lambda}) {}_{-2}Y_{\ell,m}(\theta,\phi) \, .
\end{equation}

This is a function of the time $t$, the intrinsic source parameters (masses and spin angular momenta of the bodies)
 denoted by $\vec{\lambda}$, and the polar angles $\theta$ and $\phi$ of a coordinate system
whose $z-$axis is aligned with the total angular momentum $\vec{J}$ of the binary
at some reference frequency.
To approximate the precessing multipoles $h^{prec}_{\ell,m}(t)$ we
perform a dynamic rotation of the non-precessing multipoles
$h_{\ell,m}^{ \substack{non- \\ prec}}(t)$,

\begin{equation}
\label{equ:rot}
h^{prec}_{\ell,m}(t) = \sum_{-\ell\leqslant m' \leqslant \ell} h^{ \substack{non- \\ prec}  }_{\ell,m'}(t) \, D^{\ell}_{m',m}(\alpha(t), \beta(t), \epsilon(t)) \, .
\end{equation}

We define the Wigner D-matrix as
$D^{\ell}_{m',m}(\alpha, \beta, \epsilon) = e^{i m \alpha} d^{\ell}_{m',m}(-\beta) e^{-i m' \epsilon}$
and the Wigner \emph{d}-matrix is given in Ref.~\cite{Ajith:2007jx}.

Next we transform to the frequency domain using the stationary phase approximation~\cite{Droz:1999qx}
under the assumption that the precession angles modify the signal via a slowly
varying amplitude, giving us an expression for the frequency-domain
multipoles in terms of the co-precessing frame multipoles,

\begin{equation}
\label{equ:hlmprec}
\tilde{h}^{prec}_{\ell,m}(f) = \sum \limits_{-\leqslant \ell< m' \leqslant \ell} \tilde{h}^{ \substack{non- \\ prec}  }_{\ell,m'}(f) \, D^{\ell}_{m',m}(\alpha, \beta, \epsilon) \, .
\end{equation}

For brevity we omit the explicit dependence on frequency for the
the precession angles $(\alpha, \beta, \epsilon)$ but they
are evaluated at the stationary points $t(f) = 2\pi f/m'$
~\cite{PhysRevD.82.064016}.

The frequency-domain \ac{GW} polarisations $\tilde{h}_{+/\times}(f)$ are defined as the
Fourier transform (FT) of the real-valued \ac{GW} polarisations $h_{+/\times}(t)$,
which we write as,

\begin{eqnarray}
  \label{equ:hp_1}
  \tilde{h}_{+}(f) &=& {\rm{FT}}\left[ {\rm{Re}}(h(t)) \right] =  \frac{1}{2}\left(\tilde{h}(f) + \tilde{h}^{*}(-f)\right) \, , \\
  \label{equ:hc_1}
  \tilde{h}_{\times}(f) &=& {\rm{FT}}\left[ {\rm{Im}}(h(t)) \right] =  \frac{i}{2}\left(\tilde{h}(f) - \tilde{h}^{*}(-f)\right) \, .
\end{eqnarray}

To arrive at the final expression for the frequency-domain \ac{GW} polarisations
we substitute Eq.~(\ref{equ:hlmprec}) into Eqs.~(\ref{equ:hp_1}) and (\ref{equ:hc_1}),
assuming $f>0$ and symmetry through the orbital plane in the co-precessing
frame\footnote{This leads to the simplification $\tilde{h}_{\ell, m'}(f) = (-1)^{\ell} \tilde{h}^{*}_{\ell, -m'}(-f)$.}, leading to,

  \begin{align}
    \tilde{h}^{prec}_{+}(f) &= \frac{1}{2} \sum_{\ell \geqslant 2} \sum_{m'>0}
 \tilde{h}_{\ell,m'}^{ \substack{non- \\ prec}}(f) \sum_{m=-\ell}^{\ell} \left( A^{\ell}_{m',m} + (-1)^{\ell} A^{*\ell}_{-m',m} \right) \, ,
 \label{eq:hp}
  \end{align}
  \begin{align}
\tilde{h}^{prec}_{\times}(f) &= -\frac{i}{2} \sum_{\ell \geqslant 2} \sum_{m'>0}
 \tilde{h}_{\ell,m'}^{ \substack{non- \\ prec}}(f) \sum_{m=-\ell}^{\ell} \left( A^{\ell}_{m',m} - (-1)^{\ell} A^{*\ell}_{-m',m} \right) \, .
  \label{eq:hc}
  \end{align}

To shorten the expression we define the auxillary matrix
$A^{\ell}_{m',m} \equiv {}_{-2}Y_{\ell,m} D^{\ell}_{m',m}$
and omit the explicit angular dependence of ${}_{-2}Y_{\ell,m}$
and the precession angles in $D^{\ell}_{m',m}$.
The summation over $\ell$ and $m'$ are over the modes included in the
co-precessing frame. Here we use the {\tt PhenomHM} model~\cite{PhysRevLett.120.161102},
which contains the $(\ell, |m'|) = ((2,2), (2,1), (3,3), (3,2), (4,4), (4,3))$ modes.

Due to precession the properties of the remnant \ac{BH} in the precessing system
are different to those in the equivalent non-precessing system.
We use the same prescription as described in Sec.III.C of
Ref.~\cite{PhysRevD.100.024059}
to include the in-plane-spin contribution to the spin of the remnant \ac{BH}.
This modified final spin vector changes the ringdown spectrum of the
aligned-spin multipoles.

Lastly, we note that the models for the three ingredients
(the non-precessing model, the precession angles, and the \ac{BH} remnant model)
are independent in our construction, and can therefore
each be updated when any of them are improved.

\section{Waveform assessment}

\subsection{Mismatch Computation}
\label{sec:matches}

\begin{figure}[t!]
  \includegraphics[width=0.98\columnwidth]{figs/{mismatches-mass-ratio-single-inc}.pdf}
 \caption{The results of the comparison between {\tt PhenomPv3} (top),
 {\tt PhenomPv3HM} (bottom) and the precessing \ac{NR} simulations from the public SXS catalogue.
 The figure shows the mismatch average over a reference phase and polarisation
 angle ($1 - \overline{\mathcal{M}}$) as a function of the total mass
 for an inclination of $\iota = \pi/3$.
 The worst case is SXS:BBH:0165, a short ($\sim$6 orbits) signal with mass ratio 1:6 and high precession.
 }
 \label{fig:matches}
\end{figure}

\begin{table*}[t]
\centering
\begin{tabular}{lllllll}
\hline \hline
Waveform Model                    & \multicolumn{3}{l}{{\tt PhenomPv3}} & \multicolumn{3}{l}{{\tt PhenomPv3HM}} \\ \hline
Mass-Ratio ($\#$)                          & $0$           & $\pi/3$       & $\pi/2$      & $0$     & $\pi/3$     & $\pi/2$    \\ \hline
$1 \leqslant q \leqslant 2$ (72) & $0.998^{ 0.999 }_{ 0.993 }$ & $0.989^{ 0.996 }_{ 0.977 }$ & $0.982^{ 0.993 }_{ 0.967 }$ &  $0.997^{ 0.999 }_{ 0.993 }$ & $0.993^{ 0.996 }_{ 0.986 }$ & $0.987^{ 0.992 }_{ 0.972 }$ \\
$q=3$ (15) & $0.989^{ 0.999 }_{ 0.974 }$ & $0.959^{ 0.967 }_{ 0.950 }$ & $0.941^{ 0.946 }_{ 0.933 }$ & $0.993^{ 0.997 }_{ 0.985 }$ & $0.987^{ 0.993 }_{ 0.975 }$ & $0.984^{ 0.989 }_{ 0.974 }$ \\
$q=5$ (2)  & $0.973^{ 0.978 }_{ 0.968 }$ & $0.941^{ 0.951 }_{ 0.931 }$ & $0.911^{ 0.925 }_{ 0.897 }$ & $0.990^{ 0.990 }_{ 0.989 }$ & $0.971^{ 0.975 }_{ 0.966 }$ & $0.978^{ 0.989 }_{ 0.968 }$  \\
$q=6$ (1)  & $0.863$ & $0.919$ & $0.939$ & $0.950$ & $0.914$ & $0.898$  \\ \hline \hline
\end{tabular}
\caption{Match results from Sec.~\ref{sec:matches}.
We quote the mean value of the match for each inclination
angle considered ($\iota \in [0, \pi/3, \pi/2]$ rad) and averaged
over all cases in the mass-ratio
category for the $M_{tot}=100M_\odot$ case.
The subscript and superscript are the minimum and maximum values of the
match for the mass-ratio category considered.
}
\label{table:matches}
\end{table*}

The standard metric to assess the accuracy of \ac{GW} signal models is
to calculate the noise-weighted inner product between the
\emph{template} model and an accurate \emph{signal} waveform.
As our signal we use \ac{NR} waveforms from the publicly available
SXS catalogue~\cite{PhysRevLett.111.241104,sxsonline,Boyle:2019kee} generated
using the
\ac{NR} injection infrastructure in {\tt LALSuite}~\cite{Schmidt:2017btt}.
From this catalogue we select the precessing
configurations with the highest numerical resolution. This set contains
90 systems with $q \in [1,6]$, however, the majority of cases
have $q \leqslant 3$. We have 2 cases at $q=5$ and one case at $q=6$.
There are six cases that have at least one \ac{BH} with a dimensionless
spin magnitude $|\chi| >0.5$ whereas the majority of cases have
$|\chi| \leqslant 0.5$
\footnote{During the concluding stages of this project
the SXS collaboration updated their catalgoue
to include $\sim 2000$ new simulations~\cite{Boyle:2019kee}.
We defer comparison to this catalogue to a future date.}.
For the exact list of \ac{NR}
configurations and specific details on how the mismatch calculations
were performed we refer the reader to Ref.~\cite{PhysRevD.100.024059}
where we presented an identical analysis but restricted the signals
to contain the $\ell = 2$ multipoles.

Since {\tt PhenomPv3} is constructed from {\tt PhenomD} it only
has the $\ell=|m|=2$ modes in the co-precessing frame and therefore
we expect this model to perform poorly when the contribution to the
signal due to higher modes is not negligible.
As {\tt PhenomPv3HM} is constructed from {\tt PhenomHM}
and contains the $(\ell, |m|) = ((2,2), (2,1), (3,3), (3,2), (4,4), (4,3))$
modes in the co-precessing frame we expect it to outperform {\tt PhenomPv3}.
In the \ac{NR} signal we include multipoles with $\ell \in [2,4]$ to be
consistent with the highest modeled $\ell$ mode in {\tt PhenomPv3HM}.

We use the expected noise curve for Advanced LIGO operating at design
sensitivity~\cite{aligozerodethp}
\footnote{See~\cite{aligozerodethpnew} for a more recent reference.}
with a low-frequency cutoff of $10{\rm{Hz}}$.
Due to the presence of higher modes the orbital phase of the binary is
no longer degenerate with the phase of the observed waveform, which
means the standard method to analytically maximise over the template
phase is not applicable. It is possible, however, to
analytically maximise over the template polarisation using
the skymax-SNR derived in Ref.~\cite{PhysRevD.94.024012}.
In our match calculation we analytically maximise over the template polarisation
and relative time shift and numerically optimise over the template orbital
reference phase and frequency. Finally we average the match, weighted
by the optimal SNR, over the signal orbital reference phase and polarisation
angle. See Sec.~III A in Ref.~\cite{PhysRevD.100.024059} for details.

Figure~\ref{fig:matches} shows the orientation-averaged-mismatch~\cite{PhysRevD.100.024059}
as a function of the total mass of the binary for an inclination angle $\iota = \pi/3$.
Here $\iota$ is the angle between the Newtonian orbital
angular momentum and the line of sight at the start frequency of the \ac{NR}
waveform. The first row uses the dominant multipole-only model, {\tt PhenomPv3},
and the second row uses the new higher multipole model, {\tt PhenomPv3HM},
presented here. We clearly see that for $q \geqslant 3$ that it is important
to include higher modes in the template model.

In Table~\ref{table:matches} we summarise the results of our validation study
by tabulating the results as the match (as opposed to mismatch as in
Fig.~\ref{fig:matches}) for each model according to mass ratio and inclination
angle. Next to the mass-ratio range in parentheses is the number of \ac{NR}
cases in that mass-ratio category.
Each entry in the table is calculated as follows: for the
$M_{tot} = 100 M_\odot$ case we average the match over all cases in the
mass-ratio category and write the minimum and maximum match as
subscript and superscript respectively.
\paragraph*{$1 \leqslant q \leqslant 2$:}

In this mass-ratio range both {\tt PhenomPv3} and {\tt PhenomPv3HM}
perform comparably, most likely due to the strength of
higher multipoles scaling with mass ratio.

\paragraph*{$q=3$:}

Here we start to see the importance of the higher multipoles to accurately
describe the \ac{NR} signal.
For $\iota = 0$ {\tt PhenomPv3}
has an average match of $0.989$. However, as the inclination angle increases,
thus emphasising more of the higher multipole content of the signal,
the average match drops to $0.941$ and can be as low as $0.933$.
On the other hand, {\tt PhenomPv3HM} is able to describe the \ac{NR} data
to an average accuracy of $0.984$ with a minimum value of $0.974$ for
inclined systems.

\paragraph*{$q=5$:}

At this mass-ratio the loss in performance for {\tt PhenomPv3} is
noticable even for low inclination values. At $\iota = 0$
the average match is $0.973$ dropping to $0.911$ at $\iota = \pi/2$.
The match for {\tt PhenomPv3HM} at $\iota = 0$ remains high
at $0.99$, but reduces to $0.978$ at $\iota = \pi/2$.
Note that we only have two \ac{NR} simulations at $q=5$ and are thus unable to
rigorously test the model at this and similar mass ratios.

\paragraph*{$q=6$:}

When comparing to this \ac{NR} simulation we find both models perform substantially
worse than the $q=5$ cases with even {\tt PhenomPv3} outperforming
{\tt PhenomPv3HM} with matches as low as $0.898$.
We have verified that we obtain matches of $\sim 0.97$ when restricting the
\ac{NR} waveform to just the $\ell = 2$ multipoles, consistent with
our previous study~\cite{PhysRevD.100.024059}.
We conclude that either our
model is outside its range of validity or that this \ac{NR} simulation
is inaccurate for the higher multipoles, however, our results are robust
against \ac{NR} simulations of this configuration at multiple resolutions.
This \ac{NR} simulation, SXS:BBH:0165 is exceptional for a few
reasons. First, it is a high mass-ratio system where higher multipoles
are more important. Second, it is a strongly precessing system
with primary  $\chi_1 = (0.74, 0.19, -0.5)$ and secondary
$\chi_2 = (-0.19, 0., -0.23)$ spin vectors. Finally, it is also
very short, only containing $\sim 6.5$ orbits. We encourage more
\ac{NR} simulations in this region by different \ac{NR} codes to (i) cross
check the results and (ii) populate this region with more data with which
to test and refine future models.

We conclude from our study that {\tt PhenomPv3HM} greatly improves the
accuracy towards precessing \acp{BBH} for systems with mass-ratio up to 5:1.
We expect to be able to greatly improve the accuracy and extend towards
higher mass-ratio by further calibrating the higher order multipoles
and precession effects to \ac{NR} simulations.

\subsection{Parameter Uncertainty}
\label{sec:pe}

One of the main purposes of a waveform model is to estimate the source
parameters of \ac{GW} events.
With models we can quantify the expected parameter uncertainty as a function
of the parameter space
~\cite{
  PhysRevD.95.064053,PhysRevD.95.064052,Kumar:2018hml,
  Ghosh:2015jra,
  Pankow:2016udj,
  Vitale:2014mka,Farr:2015lna,
  PhysRevD.93.084042,Stevenson:2017dlk,PhysRevD.95.064053}.
Instead of a computationally intense systematic parameter estimation campaign
we have chosen to focus on one configuration and study in detail the
dependency of parameter recovery on \ac{SNR}.
We wish to study a system where both precession and higher modes are important
and guided by previous studies~\cite{Capano:2013raa,PhysRevD.96.124024,PhysRevD.95.104038}
we chose to study a
double precessing spin, mass-ratio 3 \ac{BBH}
signal with a total mass of $150 M_\odot$ in the detector frame.
Starting at a frequency of $10$ Hz this system produces a waveform with about
20 \ac{GW} cycles and merges at a frequency of about $120$ Hz.
See Table~\ref{table:injection-pars} for specific injection values,
where $(\theta_{JN}, \phi)$ define the direction of propagation in the source
frame, $(\alpha, \delta)$ are the right-ascension and declination of the source
and $\psi$ is the polarisation angle.

We simulate this fiducial signal with {\tt PhenomPv3HM} and recover its
parameters using the parallel tempered MCMC algorithm implemented as
{\tt LALInferenceMCMC} in the publicly available LALInference
software~\cite{PhysRevD.91.042003} with {\tt PhenomPv3HM} as the template
model.
We perform three separate, zero-noise, injections to investigate how our results
depend on the injected \ac{SNR}. Specifically we inject the signal at
luminosity distances of
3000 Mpc, 1500 Mpc and 300 Mpc corresponding to a three-detector network \ac{SNR}
of 17, 35 and 176 respectively.
We use the design sensitivity noise curves for
the LIGO Hanford, LIGO Livingston and Virgo detectors~\cite{aligozerodethp}.

We present our results by tabulating the median and 90\% credible
interval on binary parameters in Tab.~\ref{table:injection-pars}
and source frame parameters in Tab.~\ref{tab:source_frame}.
We also plot the 90\% credible interval as a function of the injected \ac{SNR}
for a few chosen parameters in
Figs.~\ref{fig:90width-mass}, \ref{fig:90width-spin}
and \ref{fig:90width-spin-angle}.
In the high \ac{SNR} limit the uncertainty on the parameters
should decrease linearly wtih SNR i.e., as $1/\rho$~\cite{Vallisneri:2007ev},
which is shown as a dashed black line in these figures.

In the following discussion we
change our convention for the mass-ratio to $q \equiv m_2/m_1 \in [0, 1]$
and will abbreviate
the width of the 90\% credible interval of
parameter $X$ at an \ac{SNR} of $\rho$ as $C_{90\%}^{\rho}(X)$.

{\renewcommand{\arraystretch}{1.5
\begin{table}
  \centering

    \begin{tabular}{p{1.7cm}p{1.7cm}p{1.7cm}p{1.7cm}l}
    \hline \hline
    Parameter                         &   \shortstack{Injection \\ Value} & \shortstack{$\rho = 17.6$ \\ $D_L = 3000$}   & \shortstack{$\rho = 35.2$ \\ $D_L = 1500$}   & \shortstack{$\rho = 176$ \\ $D_L = 300$}   \\
    \hline
    $m^{\rm{det}}_{1}/M_\odot$              & 112.500                           & $102.98^{+13.38}_{-12.26}$                   & $107.71^{+7.96}_{-7.43}$                     & $112.38^{+1.73}_{-1.75}$                   \\
    $m^{\rm{det}}_{2}/M_\odot$              & 37.500                            & $40.62^{+5.92}_{-5.29}$                      & $39.00^{+2.88}_{-2.66}$                      & $37.55^{+0.54}_{-0.52}$                    \\
    $M_{\rm{total}}^{\rm{det}} / M_\odot $  & 150.000                           & $143.64^{+11.28}_{-9.74}$                    & $146.78^{+6.48}_{-5.94}$                     & $149.93^{+1.49}_{-1.46}$                   \\
    $\mathcal{M}_{c}^{\rm{det}} / M_\odot $ & 54.940                            & $55.08^{+3.37}_{-3.20}$                      & $55.05^{+1.69}_{-1.69}$                      & $54.95^{+0.36}_{-0.34}$                    \\
    $q$                                     & 0.333                             & $0.39^{+0.11}_{-0.08}$                       & $0.36^{+0.05}_{-0.04}$                       & $0.33^{+0.01}_{-0.01}$                     \\
    $\theta_1$ / rad                              & 1.052                             & $1.14^{+0.36}_{-0.37}$                       & $1.10^{+0.27}_{-0.19}$                       & $1.05^{+0.04}_{-0.04}$                     \\
    $\theta_2$ / rad                              & 2.090                             & $1.73^{+1.01}_{-1.21}$                       & $2.04^{+0.72}_{-1.09}$                       & $2.09^{+0.14}_{-0.12}$                     \\
    $\Delta \phi_{12}$ / rad                      & 1.571                             & $2.82^{+3.11}_{-2.49}$                       & $1.79^{+3.42}_{-1.35}$                       & $1.58^{+0.24}_{-0.24}$                     \\
    $\theta_{JN}$ / rad                           & 1.050                             & $1.62^{+0.60}_{-0.73}$                       & $1.21^{+0.92}_{-0.23}$                       & $1.05^{+0.03}_{-0.03}$                     \\
    $\cos(\phi)$                            & 1.000                             & $-0.04^{+1.03}_{-0.96}$                      & $0.46^{+0.54}_{-1.46}$                       & $1.00^{+0.00}_{-0.01}$                     \\
    $\alpha$ / rad                                & 1.047                             & $4.235^{+0.124}_{-3.213}$                    & $1.070^{+3.277}_{-0.036}$                    & $1.047^{+0.004}_{-0.004}$                  \\
    $\delta$ / rad                                & 1.047                             & $-1.020^{+2.098}_{-0.125}$                   & $1.025^{+0.037}_{-2.155}$                    & $1.047^{+0.004}_{-0.004}$                  \\
    $\psi$ / rad                                  & 1.047                             & $1.52^{+0.66}_{-0.76}$                       & $1.28^{+0.80}_{-0.39}$                       & $1.05^{+0.04}_{-0.04}$                     \\
    $\chi_{\rm{eff}}$                            & 0.200                             & $0.204^{+0.129}_{-0.136}$                    & $0.201^{+0.070}_{-0.074}$                    & $0.200^{+0.016}_{-0.017}$                  \\
    $\chi_{p}$                              & 0.700                             & $0.681^{+0.186}_{-0.285}$                    & $0.705^{+0.098}_{-0.105}$                    & $0.699^{+0.020}_{-0.024}$                  \\
    $|\chi_1|$                              & 0.806                             & $0.77^{+0.15}_{-0.27}$                       & $0.80^{+0.07}_{-0.09}$                       & $0.81^{+0.02}_{-0.02}$                     \\
    $|\chi_2|$                              & 0.806                             & $0.45^{+0.47}_{-0.40}$                       & $0.59^{+0.35}_{-0.42}$                       & $0.80^{+0.13}_{-0.11}$                     \\
    $D_L$ / Mpc                                   & see heading                       & $3086.50^{+739.44}_{-571.98}$                & $1465.76^{+177.84}_{-157.87}$                & $300.12^{+7.14}_{-6.94}$                   \\
     \hline \hline
    \end{tabular}
  \caption{
    Injection parameters and results from parameter estimation of
    simulated signals.
    We quote the median and 90\% credible interval.}
  \label{table:injection-pars}
\end{table}

{\renewcommand{\arraystretch}{1.5
\begin{table*}
  \begin{tabular}{p{1.8cm}p{1.8cm}p{1.8cm}p{1.8cm}p{1.8cm}p{1.8cm}p{1.8cm}}
    \toprule
    & \multicolumn{2}{l}{\quad \quad \quad \quad $\rho$=17}          & \multicolumn{2}{l}{\quad \quad \quad \quad$\rho$=34}          & \multicolumn{2}{l}{\quad \quad \quad \quad$\rho$=176}         \\\hline
Parameter   & Inj.       & Rec. & Inj.       & Rec. & Inj.       & Rec. \\\hline
$m^{\rm{src}}_{1}/M_\odot$             &   74.56  & $67.60^{+8.22}_{-8.35}$       &   87.789 & $84.28^{+5.36}_{-5.24}$       & 105.724 & $105.54^{+1.55}_{-1.58}$  \\
$m^{\rm{src}}_{2}/M_\odot$             &   24.853 & $26.60^{+3.95}_{-3.55}$       &   29.263 & $30.48^{+2.47}_{-2.21}$       &  35.241 & $35.27^{+0.51}_{-0.50}$   \\
$M_{\rm{total}}^{\rm{src}} / M_\odot $ &   99.413 & $94.35^{+6.64}_{-7.16}$       &  117.052 & $114.82^{+3.97}_{-3.96}$      & 140.965 & $140.80^{+1.28}_{-1.27}$  \\
$M_{c}^{\rm{src}} / M_\odot $          &   36.412 & $36.11^{+2.29}_{-2.34}$       &   42.872 & $43.04^{+1.31}_{-1.28}$       &  51.631 & $51.61^{+0.31}_{-0.30}$   \\
$D_L / \rm{Mpc}$                       & 3000     & $3086.50^{+739.44}_{-571.98}$ & 1500     & $1465.76^{+177.84}_{-157.87}$ & 300     & $300.12^{+7.14}_{-6.94}$  \\
$z$                                    &    0.509 & $0.52^{+0.10}_{-0.08}$        &    0.281 & $0.28^{+0.03}_{-0.03}$        &   0.064 & $0.065^{+0.001}_{-0.001}$ \\
\hline \hline
\end{tabular}
  \caption{
    Source frame injection parameters and
    results from parameter estimation of
    simulated signals.
    We quote the median and 90\% credible interval.}
  \label{tab:source_frame}
\end{table*}

\subsubsection{Masses}

\begin{figure}[t]
  \includegraphics[width=1\columnwidth]{figs/masses-source-width.pdf}

\caption{90\% credible intervals for the source frame mass parameters as a
function of injected SNR}
\label{fig:90width-mass}
\end{figure}

Figure~\ref{fig:90width-mass} shows the source frame mass parameters;
primary mass $m^{\rm{src}}_{1}$, secondary mass $m^{\rm{src}}_{2}$,
chirp mass $\mathcal{M}_{c}^{\rm{src}}$, total mass $M_{\rm{total}}^{\rm{src}}$ and mass-ratio $q$.
We find good scaling with respect to $1/\rho$ for all source frame mass parameters.

Table~\ref{tab:source_frame} shows the injection and recovered values.
Even at the high total masses we consider here we find that the chirp mass
is still the best measured parameter with
$C_{90\%}^{17}(\mathcal{M}_{c}^{\rm{src}}) = 4.63 M_\odot$ and $C_{90\%}^{176}(\mathcal{M}_{c}^{\rm{src}}) = 0.61 M_\odot$.
The total mass is the next best measured mass parameter with low and high SNR accuracies of
$C_{90\%}^{17}(M_{\rm{total}}^{\rm{src}}) = 13.8 M_\odot$ and $C_{90\%}^{176}(M_{\rm{total}}^{\rm{src}}) = 2.55 M_\odot$ respectively.
We find the primary mass can be measured to an accuracy of
$C_{90\%}^{17}(m^{\rm{src}}_{1}) = 16.57 M_\odot$ for low SNR
and $C_{90\%}^{176}(m^{\rm{src}}_{1}) = 3.13 M_\odot$ for high SNR.
And for the secondary mass we find
$C_{90\%}^{17}(m^{\rm{src}}_{2}) = 7.5 M_\odot$
and $C_{90\%}^{176}(m^{\rm{src}}_{2}) = 1.01 M_\odot$ for low and high SNR
respectively.
Finally, we are able to constrain the mass-ratio to $C_{90\%}^{17}(q) = 0.19$
and $C_{90\%}^{176}(q) = 0.02$.

\subsubsection{Spins}

\begin{figure}[t]
  \includegraphics[width=1\columnwidth]{figs/spins-width.pdf}

\caption{90\% credible intervals for the \ac{BH} spin magnitudes
and effective spin parameters as a function of injected SNR.}
\label{fig:90width-spin}
\end{figure}

Figure~\ref{fig:90width-spin} shows the primary and secondary spin magnitude $|\chi_1|, |\chi_2|$,
the effective aligned-spin $\chi_{\rm{eff}}$ and effective precessing-spin $\chi_{\rm{p}}$ parameters.

With the exception of $|\chi_2|$ we find good agreement with the $1/\rho$ scaling.
This suggests that for $|\chi_2|$ the two weaker injections do not have high enough \ac{SNR}
for the posterior distribution function for this parameter to be approximated by a Gaussian~\cite{Vallisneri:2007ev}.
That being said, we do observe the 90\% width decrease with \ac{SNR} albeit at a slower rate.
At \ac{SNR} of $17$ and $34$ we find we are not able to place strong constraints on $|\chi_2|$
with $C_{90\%}^{17}(|\chi_2|) = 0.87$ and $C_{90\%}^{34}(|\chi_2|) = 0.77$.
However, at the high SNR of $176$ we begin to constrain the spin magnitude at the level of
$C_{90\%}^{176}(|\chi_2|) = 0.24$, approximately the same level of uncertainty
as $\chi_{\rm{eff}}$ at a \ac{SNR} of 17. This is consistent with the study of non-precessing binaries in
Ref.~\cite{Purrer:2015nkh}, which concluded that the secondary spin will not be measurable for SNRs
below $\sim$100, but our results suggest that this carries over to precessing systems.

The primary spin magnitude is measured with much higher precision than the
secondary spin magnitude. However, constraining this parameter to a 90\% width
of less than 0.2 requires an SNR of $\sim 30$. This parameter
does follow the $1/\rho$ scaling very well and for high SNR cases we
estimate the statistical uncertainty to be $C_{90\%}^{176}(|\chi_1|) = 0.04$.

Of the effective spin parameters the effective aligned parameter $\chi_{\rm{eff}}$ is the best
measured quantity.
This is closely related to the leading order spin effect in \ac{PN} theory~\cite{PhysRevD.52.848,PhysRevD.84.084037}
appearing at 1.5 PN order.
For all three SNRs the median value is always within $10^{-3}$
of the true value with the uncertainties ranging from
$C_{90\%}^{17}(\chi_{\rm{eff}}) = 0.265$ to
$C_{90\%}^{176}(\chi_{\rm{eff}}) = 0.033$.

Turning towards the effective precession spin parameter, $\chi_p$,
at the lowest \ac{SNR} we find the marginalised posterior
for $\chi_p$ has a median value of $0.681$, close to the true value
but with a wide uncertainty of $C_{90\%}^{17}(\chi_p) = 0.47$, spanning
almost half of the full range.
The evolution of the median value does not change significantly with increasing
\ac{SNR} however,
our measurement uncertainty does decrease with increasing \ac{SNR} as expected
and we find $C_{90\%}^{35}(\chi_p) = 0.203$ for the medium \ac{SNR} and
$C_{90\%}^{176}(\chi_p) = 0.044$ for the high \ac{SNR} case.

\begin{figure}[t]
  \includegraphics[width=1\columnwidth]{figs/spin-angles-width-deg.pdf}
\caption{90\% credible intervals for the \ac{BH} spin orientation
parameters as a function of injected SNR.}
\label{fig:90width-spin-angle}
\end{figure}

Figure~\ref{fig:90width-spin-angle} shows the spin orientation parameters.
$\theta_1$ and $\theta_2$ are the polar angles of the primary and secondary spin vectors
with respect to the orbital angular momentum at the reference frequency.
The angle $\Delta \phi_{12}$ is the angle between the primary and secondary spin vectors
projected into the instantaneous orbital plane at the reference frequency.
This angle is particularly useful when characterising precessing binaries
as $\Delta \phi_{12} = 0 $ or $\Delta \phi_{12} = \pi $ are resonant spin configurations
(if other conditions on the mass-ratio and spin magnitudes are met)~\cite{Schnittman:2004vq}.

We find $\theta_1$ has good \ac{SNR} scaling with
$C_{90\%}^{17}(\theta_1) = 0.73$ rad ($\sim 42$ deg)
and $C_{90\%}^{176}(\theta_1) = 0.08$ rad ($\sim 5$ deg).
Furthermore, $\theta_2$ and $\Delta \phi_{12}$ are measured much less accurately and
require SNRs of $\sim 60$ and $\sim 100$ to achieve statistical uncertainties
of $\sim 1$ rad ($\sim 60$ deg), respectively.
However, in the event of a high SNR signal we find we are able to constrain
$\theta_2$ to $C_{90\%}^{176}(\theta_2) = 0.26$ rad ($\sim 15$ deg)
and $\Delta \phi_{12}$ to $C_{90\%}^{176}(\Delta \phi_{12}) = 0.48$ rad ($\sim 28$ deg).

In summary, we find that the primary spin magnitude $|\chi_1|$ and polar angle
$\theta_1$ can be constrained at an \ac{SNR} of $\sim 30$,
while the seconday spin magnitude $|\chi_2|$ and polar angle
$\theta_2$, as well as the the information about the relative orientation of
the spin vectors $\Delta \phi_{12}$ are not constrained until we reach an
\ac{SNR} of $\sim 200$.

\subsubsection{Waveform Systematics}
\label{sec:wfsys}

Parameter estimation on a \ac{GW} event with a waveform model
that does not include relevant physics effects could result in biased results.
To quantify the size of the bias due to neglecting
higher modes and/or precession for this signal we repeat our
parameter estimation analsis with four additional models.

The waveform models we use are listed in Tab.~\ref{tab:models},
where we mark whether or not each model contains precession and/or higher modes.
{\tt PhenomD} is the baseline model upon which the other {\tt Phenom} models
used in this work are built. We include two different precessing models
{\tt PhenomPv2} and {\tt PhenomPv3} to gauge systematics on precession.
{\tt PhenomHM} includes higher modes but is a non-precessing model and finally
the precessing and higher mode model {\tt PhenomPv3HM} presented in this article.

Our results are presented in Fig.~\ref{fig:injres}. From left to
right the columns show the one dimensional marginalised posterior
distribution for the; $m_1^{det}$, $m_2^{det}$, $\chi_{\rm{eff}}$
and $\chi_{\rm{p}}$.
The rows from top to bottom show the results for the low $(\rho=17)$,
medium $(\rho=34)$ and high $(\rho=176)$ SNR injections.
The true value is shown as a vertical dashed black line.
For all SNRs we find biases in the recovered masses for all models other than
PhenomPv3HM i.e., the model that was used to produce the synthetic signal.
This suggests that for real GW signals that are similar to this injection
require analysis with models that contain \emph{both} the effects
of precession and higher modes.
For the high SNR case multi-mode posteriors are found for the {\tt PhenomD} case.
For $\chi_{\rm{eff}}$ we find that for the low SNR injection
the true value is within the $90 \%$ credible interval (CI) and therefore not considered biased
however, as the SNR of the injection is increased we find that $\chi_{\rm{eff}}$
can become heavily biased for the two precessing models
but remains unbiased for the non-precessing models.
For $\chi_{\rm{p}}$ we find that the precessing and non-higher mode models
({\tt PhenomPv2} and {\tt PhenomPv3}) consistently favour larger values of $\chi_{\rm{p}}$
as the SNR increases.
Interestingly we also start to find large differences between
{\tt PhenomPv2} and {\tt PhenomPv3} at the high SNR cases.

\begin{table}
\begin{tabular}{l l l}
\toprule
\textbf{Model} & Precession & Higher Modes \\
{\tt PhenomD} \cite{PhysRevD.93.044006,Khan:2015jqa} &\xmark &\xmark \\
{\tt PhenomPv2} \cite{Hannam:2013oca,PhysRevD.91.024043} &\cmark &\xmark \\
{\tt PhenomPv3} \cite{PhysRevD.100.024059} &\cmark &\xmark \\
{\tt PhenomHM} \cite{PhysRevLett.120.161102} &\xmark &\cmark \\
{\tt PhenomPv3HM}  & \cmark & \cmark  \\ \hline \hline
\end{tabular}
\caption{Waveform models that we use to analyse GW170729 and highlighting
which physical effects are included each model.}
\label{tab:models}
\end{table}

\begin{figure*}
\includegraphics[width=0.5\columnwidth]{figs/m1.pdf}
\includegraphics[width=0.5\columnwidth]{figs/m2.pdf}
\includegraphics[width=0.5\columnwidth]{figs/chi_eff.pdf}
\includegraphics[width=0.48\columnwidth]{figs/chi_p.pdf}
\caption{One dimensional marginal posterior probability distributions
for detector-frame primary and secondary masses (1st and 2nd columns respectively),
effective aligned-spin $\chi_{\rm{eff}}$ and effective precession spin $\chi_{\rm{p}}$
parameters (3rd and 4th columns respectively).
Each row, from top to bottom, shows results for the low $(\rho=17)$,
medium $(\rho=34)$ and high $(\rho=176)$ SNR injections.
The true value is marked as a vertical black dashed line.
The prior is shown as a black historgram.
We show results for {\tt PhenomD} (red), {\tt PhenomHM} (purple), {\tt PhenomPv2} (green),
{\tt PhenomPv3} (blue) and {\tt PhenomPv3HM} (orange).
Note the results for $\chi_{\rm{p}}$ do not show {\tt PhenomD} or {\tt PhenomHM}
as they are aligned-spin models only.
}
\label{fig:injres}
\end{figure*}

\subsection{GW170729 Analysis}
\label{sec:gw170729}

Of the ten binary-black-hole observations reported by the LIGO-Virgo collaborations~\cite{LIGOScientific:2018mvr},
GW170729 shows the strongest evidence for unequal masses, making it the most
likely signal for which higher modes could impact parameter measurements.
This motivated the study in Ref.~\cite{Chatziioannou:2019dsz}, where the authors analysed GW170729 with
two new aligned-spin and higher mode models ({\tt SEOBNRv4HM}~\cite{PhysRevD.98.084028}
and {\tt PhenomHM}~\cite{PhysRevLett.120.161102}). They found that the
models preferred to interpret the data as the \ac{GW} signal coming from
a higher mass-ratio system with estimates for the mass-ratio changing from
$0.62^{+0.36}_{-0.23}$ for {\tt PhenomPv2} to $0.52^{+0.26}_{-0.21}$ for {\tt PhenomHM}
($90 \%$ credible interval). This event also has evidence for a
positive $\chi_{\rm{eff}}$, although when analysed with higher modes
the $90 \%$ credible interval for $\chi_{\rm{eff}}$ extended to include zero.
This event has also been analysed in~\cite{2019arXiv190505477P}
with the aligned-spin and higher mode model {\tt NRHybSur3dq8}~\cite{Varma:2018mmi} where the the authors draw similar conclusions.
Motivated by this we prioritise GW170729 to
analyse first with {\tt PhenomPv3HM} and compare to existing results.
We use the posterior samples for {\tt PhenomHM} from~\cite{Chatziioannou:2019dsz},
and for {\tt PhenomPv2} from~\cite{gwtc1samples}.
Results for {\tt PhenomD}, {\tt PhenomPv3} and {\tt PhenomPv3HM}
were computed for this work using the {\tt LALInferenceMCMC} code~\cite{Veitch:2014wba}.

In Fig.~\ref{fig:gw170729results} we show the joint posterior for
the source frame component masses ($m^{\rm{src}}_{1}$, $m^{\rm{src}}_{2}$)
in the upper left;
the aligned effective spin and mass-ratio ($\chi_{\rm{eff}}$, $q$) in the upper right
and finally the luminosity distance and inclination angle ($D_L$, $\iota$)
in the bottom plot.
The quantitative parameter estimates for the source properties are provided
in Tab.~\ref{table:gw170729table}.
Our posterior on the effective precession parameter $\chi_p$ is consistent
with previous results and shows no significant differences due to
different choice of precession model (between {\tt PhenomPv2} and {\tt PhenomPv3})
or including both precession and higher modes as in {\tt PhenomPv3HM}.
We find that the marginal posterior effective aligned-spin parameter $\chi_{\rm{eff}}$
and luminosity distance $D_L$ are
remarkably similar to the results from {\tt PhenomHM}.
The posterior for the inclination angle $\iota$ for {\tt IMPhenomPv3HM}
has more support for more inclined viewing angles; however, the
change is minor.

Interestingly, not only do we find remarkably consistent results between
{\tt PhenomD} and {\tt PhenomPv2} as discussed in~\cite{Chatziioannou:2019dsz}
but also with {\tt PhenomPv3}. This indicates that precession alone
does not influence our inference for this event.
However, including precession in addition to higher modes in the analysis
does noticeably shift the posterior, albeit not very significantly
in terms of the 90\% CIs, which mostly overlap.

We find that the one-dimensional marginal posterior for the mass-ratio is
pushed further
towards lower mass-ratio values (more asymmetric) when using {\tt PhenomPv3HM}
where we find $q=0.47^{+0.28}_{-0.16}$ ($90\%$ level),
implying that including precession \emph{and} higher modes reinforces the
findings of ~\cite{Chatziioannou:2019dsz}.
As more asymmetric masses are favoured the estimate for
the primary mass (source frame) is shifted towards higher values
and the secondary mass is shifted towards lower values
where we find $m_1^{src} = 58.25^{+11.73}_{-12.53} M_\odot$
and $m_2^{src} = 28.18^{+9.83}_{-7.65} M_\odot$.

By favouring larger mass estimates for the primary \ac{BH} we challenge
formation models to describe this event through
standard stellar evolution mechanisms.
In particular our results inform the
pulsational pair-instability supernova (PPISN)
mechanism~\cite{10.1093/mnras/stv3002, Woosley_2017}.
The population synthesis analysis in~\cite{2019arXiv190402821S}
investigated the resulting distribution of \ac{BH} masses subject to
different PPISN models.
They find that in three out of the four models that they explore, the
maximum \ac{BH} mass is $\sim 40 M_\odot$
~\cite{refId0,2018arXiv181013412M,2019ApJ...878...49W},
and in one of the models the maximum \ac{BH} mass is $\sim 58 M_\odot$
~\cite{Woosley_2017}.
In Fig.~\ref{fig:gw170729m1} we show the one-dimensional marginal posterior
for the source-frame primary mass resulting from the analysis using
{\tt PhenomPv2} (blue), {\tt PhenomHM} (orange) and {\tt PhenomPv3HM} (purple).
The 90\% credible interval of each result is shown as the shaded area under
their respective curves.
The vertical black dashed lines denote the maximum
\ac{BH} mass from the four different PPISN models that were investigated
in~\cite{2019arXiv190402821S}.
We do not show the posterior for {\tt PhenomD} or {\tt PhenomPv3}
as these are consistent with the {\tt PhenomPv2} posterior.

When using {\tt PhenomPv2} to analyse the data we find that the
maximum \ac{BH} mass for all PPISN models are consistent with the
posterior. When we include non-precessing, higher modes ({\tt PhenomHM})
the PPISN models that predict maximum \ac{BH} masses of $\sim 40 M_\odot$
~\cite{refId0,2018arXiv181013412M,2019ApJ...878...49W} are excluded at the
following level.
In the posterior $1.3 \%$ of samples have a mass of $\leqslant 40 M_\odot$.
As noted previously when we include both precession and higher modes
the primary mass shifts slightly higher resulting in
$0.6 \%$ of samples having a mass of $\leqslant 40 M_\odot$.
If we assume that the primary \ac{BH} in the GW170729 binary underwent
a PPISN then the following PPISN
models~\cite{refId0,2018arXiv181013412M,2019ApJ...878...49W}
are disfavoured at greater than 90\% credibility
and the maximum \ac{BH} mass as predicted by~\cite{Woosley_2017}
is consistent with our results.
There are some caveats to these results however.
In~\cite{2019arXiv190402821S} the authors uses a linear fit to the PPISN
model of~\cite{Woosley_2017} that systematically predicts
larger remnant \ac{BH} masses for pre-supernova helium (core) masses
$M_{\rm{He}} > 60 M_\odot$ than the model of~\cite{Woosley_2017} predicts.
This in turn leads to larger maximum \ac{BH} masses for this particular model.
However, the size of this systematic uncertainty is unknown.
Another caveat in the analysis of~\cite{2019arXiv190402821S}
is that the models \cite{2018arXiv181013412M, 2019ApJ...878...49W} have
an uncertianty of $\sim 5 M_\odot$ to account for the difference
between the gravitational and baryonic mass~\cite{2012ApJ...749...91F}.

\begin{figure*}
\includegraphics[width=0.88\columnwidth]{figs/gw170729-m1-m2.pdf}
\includegraphics[width=0.92\columnwidth]{figs/gw170729-q-c.pdf}
\includegraphics[width=0.92\columnwidth]{figs/gw170729-iota-d.pdf}
\caption{GW170729 parameter estimation results.
Top left: component source-frame masses ($m^{\rm{src}}_{1}$, $m^{\rm{src}}_{2}$).
Top right: effective spin and mass-ratio ($\chi_{\rm{eff}}$, $q$).
Bottom: inclination angle and luminosity distance ($\iota$, $D_L$).
The contour lines correspond to a credible level of $90\%$.
}
\label{fig:gw170729results}
\end{figure*}

\begin{table*}
\centering
\begin{tabular}{llllll}
\hline
Parameter                                                 & {\tt PhenomD}                         & {\tt PhenomHM }                      & {\tt PhenomPv2  }                     & {\tt PhenomPv3 }                      & {\tt PhenomPv3HM }                   \\
\hline
Primary Source Mass: $m^{\rm{src}}_{1}/M_\odot$           & $50.55^{+14.02}_{-10.64}$ & $56.36^{+11.08}_{-12.41}$ & $51.22^{+16.19}_{-10.99}$ & $51.39^{+16.35}_{-11.58}$ & $58.25^{+11.73}_{-12.53}$ \\
Secondary Source Mass: $m^{\rm{src}}_{2}/M_\odot$         & $32.18^{+10.18}_{-8.84}$  & $29.45^{+9.72}_{-8.36}$   & $32.43^{+9.75}_{-9.46}$   & $31.67^{+10.43}_{-9.27}$  & $28.18^{+9.83}_{-7.65}$   \\
Total Source Mass: $M_{\rm{total}}^{\rm{src}} / M_\odot $ & $82.80^{+15.29}_{-10.82}$ & $85.16^{+14.00}_{-10.53}$ & $83.93^{+14.74}_{-10.91}$ & $83.52^{+14.94}_{-11.09}$ & $86.18^{+13.42}_{-10.77}$ \\
Mass-Ratio: $q$                                           & $0.64^{+0.31}_{-0.24}$    & $0.52^{+0.31}_{-0.18}$    & $0.63^{+0.32}_{-0.26}$    & $0.62^{+0.33}_{-0.26}$    & $0.48^{+0.28}_{-0.16}$    \\
Effective Aligned Spin: $\chi_{\rm{eff}}$                 & $0.34^{+0.19}_{-0.26}$    & $0.28^{+0.22}_{-0.28}$    & $0.36^{+0.19}_{-0.28}$    & $0.34^{+0.19}_{-0.27}$    & $0.27^{+0.21}_{-0.28}$    \\
Effective Precession Spin: $\chi_{\rm{p}}$                & N/A                       & N/A                       & $0.44^{+0.35}_{-0.29}$    & $0.44^{+0.36}_{-0.30}$    & $0.42^{+0.39}_{-0.29}$    \\
Luminosity Distance: $D_L$ / Mpc                          & $2749^{+1353}_{-1359}$    & $2241^{+1391}_{-1065}$    & $2831^{+1371}_{-1340}$    & $2797^{+1386}_{-1318}$    & $2270^{+1307}_{-974}$     \\
redshift: $z$                                             & $0.48^{+0.19}_{-0.21}$    & $0.40^{+0.20}_{-0.17}$    & $0.49^{+0.19}_{-0.21}$    & $0.48^{+0.19}_{-0.20}$    & $0.41^{+0.19}_{-0.16}$    \\
\hline
\end{tabular}
\caption{Parameter estimation results for GW170729. Masses are quoted in the source frame.
We quote the median and the 90\% symmetric credible interval of the
one-dimensional marginalised posterior distributions.
}
\label{table:gw170729table}
\end{table*}

\begin{figure}
\includegraphics[width=\columnwidth]{figs/m1_PPISN.pdf}
\caption{one-dimensional marginal posterior distribution for the primary
source-frame mass.
The posteriors for three waveform models are shown: {\tt PhenomPv2} (blue),
{\tt PhenomHM} (orange) and {\tt PhenomPv3HM} (purple).
The 90\% credible interval of each result is shown as the shaded area under
their respective curves.
We also plot as vertical black dashed lines the maximum
\ac{BH} mass from four different PPISN models, which were investigated
in Ref.~\cite{2019arXiv190402821S}.
Ref.~\cite{Woosley_2017} predict a maximum mass of $58.4 M_\odot$,
Ref.~\cite{refId0} predicts $40.5 M_\odot$ and
Refs.~\cite{2018arXiv181013412M,2019ApJ...878...49W} both predict
$39.5 M_\odot$.
}
\label{fig:gw170729m1}
\end{figure}

\section{Discussion and Future}

In this work we have presented
the first, frequency-domain, phenomenological \ac{IMR} model for
spin-precessing \acp{BBH} that also includes the effects of subdominant
multipoles --- beyond the quadrupole --- in the co-precessing frame.
By comparing to a large set of precessing \ac{NR} simulations we find that
our simple model is able to accurately reproduce the expected \ac{GW} signal
with an accuracy of $99\%$ ($97\%$) for small (high) inclinations, a significant
improvement over models that do not include subdominant multipoles, which
have accuracies of $97\%$ ($91\%$) for small (high) inclinations.

Precise measurements of \ac{BH} spins from \ac{GW} observations
requires high \ac{SNR} events in part due to
the relatively high \ac{PN} order
that spin effects appear at.
We performed an idealized parameter estimation analysis
to quantify the precision to which the \ac{BH} spin magnitude and orientation
can be measured, ignoring any effects of systematic error
on the waveform.
We find, for this particular system (See Table~\ref{table:injection-pars}),
that the primary spin
parameters are more tightly constrained than the secondary spin,
as expected for an unequal-mass system such as this.
In the following discussion we remind the reader that the low, medium and
high SNR cases have corresponding values of $17$, $34$ and $176$ respectively.
The primary spin magnitude can be constrained to a 90\% CI of
$0.42$ for the low SNR case (about half the width of the physical range)
and to a 90\% CI of $0.04$ for the high SNR case.
The secondary spin magnitude cannot be meaningfully constrained until the
high SNR case with a 90\% CI of $0.24$.
The primary spin polar angle shows reasonably good agreement with the
expected SNR scaling and can be constrained to $\sim 42$ deg (low SNR)
and $\sim 5$ deg (high SNR) at 90\% CI.
The secondary spin polar angle shows poor agreement with the
expected SNR scaling and we find can only be meaningfully constrained
($\sim 15$ deg) for the high SNR case.
The azimuthal angle between the spins ($\Delta \phi_{12}$)
shows poor scaling with \ac{SNR}. We find that only the highest \ac{SNR}
case was able to constrain $\Delta \phi_{12} \lesssim 28$ deg.
Our parameter estimation study is only a point estimate for the
size of the uncertainty on binary properties and
a systematic study that explores the parameter space of precessing binaries
is required to draw more general
conclusions~\cite{PhysRevD.95.064053,PhysRevD.95.064052}.
However, recent work in understanding precession better
may help make such a study tractable by focusing on regions where
we expect precession to
be measurable~\cite{2019arXiv190800555F, 2019arXiv190805707F}.

We have analysed the \ac{GW} event GW170729 with the new
precessing and higher mode model.
We have shown that while the general interpretation of this event
is unchanged we find that even small shifts in the posteriors due
to using different waveform models, with different physical effects
incorporated, can be enough to inform astrophysical
models such as the PPISN mechanism as we considered in this paper.
If we assume that the primary \ac{BH} in the GW170729 binary underwent
a PPISN then we disfavour the PPISN
models from \cite{refId0,2018arXiv181013412M,2019ApJ...878...49W}
at greater than 90\% credibility and our results are consistent with~\cite{Woosley_2017}.
See~\cite{Farmer:2019jed} for a recent investigation into the location
of the PPISN model mass-gap.

Our model is analytic and natively in the frequency-domain, and as such it can
be readily used in likelihood acceleration methods such as \ac{ROQ}~\cite{PhysRevD.94.044031}
or ``multibanding'' techniques~\cite{Vinciguerra:2017ngf}.
This model can be used to determine the impact on \ac{GW} searches,
event parameter estimation and
population inference due to the effects of precession and higher modes.

We expect to be able to greatly improve {\tt PhenomPv3HM}, and similar models, by using
models for the underlying higher multipole aligned-spin model that have
been calibrated to \ac{NR} waveforms~\cite{cecilioplus}. Likewise, a model for the precession dynamics
tuned to precessing \ac{NR} simulations will improve its performance~\cite{hamiltonplus}.
Although our model is a function of the 7 dimensional intrinsic parameter space of
non-eccentric \ac{BBH} mergers it is not 7 dimensional across the entire
coalescence. It is true during the inspiral, but during the pre-merger
and merger we use an effective aligned-spin parametrization.
Work is underway to develop an \ac{NR} calibrated aligned-spin model with the
effects
of two independent aligned-spins~\cite{geraintplus}. In addition, promising
attempts to dynamically enhance incomplete models via
singular-value-decomposition have recently been
presented~\cite{Setyawati:2018tqp}, and the model introduced here can easily
be employed by such an automated tuning process.

With regards to higher modes we only include a subset of the complete
list of modes, specifically $(\ell, |m|) = ((2,2), (2,1), (3,3), (3,2), (4,4), (4,3))$.
We also ignore mode mixing~\cite{Berti:2014fga}
and the asymmetry between the $+m$ and $-m$ modes, which are
responsible for out of plane recoils~\cite{Brugmann:2007zj}.

We plan to extend this model to include tidal effects
as introduced in~\cite{Dietrich:2018uni,Dietrich:2019kaq} as well as implement
a model for the \ac{GW} suitable for neutron star-black hole binaries
where the effects of spin-precession, subdominant multipoles and tidal effects
could all become important. The model presented here could be
used as a baseline for such a model.
Finally this model is implemented in the
LSC Algorithm Library (LAL)~\cite{lalsuite}.

\begin{acknowledgments}

We thank Geoffrey Lovelace, Lawrence Kidder and Michael Boyle for support
with the SXS catalogue.
We thank Carl J. Haster and Yoshinta Setyawati for useful discussions.
S.K. and F.O. acknowledge support by the
Max Planck Society's Independent Research Group Grant.
The Flatiron Institute is supported by the Simons Foundation.
M.H. was supported by Science and Technology Facilities Council (STFC) grant ST/L000962/1
and European Research Council Consolidator Grant 647839, and thanks the Amaldi Research
Center for hospitality.
We thank the Atlas cluster computing team at AEI Hannover.
The authors are grateful for computational resources provided by
the LIGO Laboratory and supported by National Science Foundation Grants
PHY-0757058 and PHY-0823459.
This work made use of numerous open source computational packages
such as python~\cite{python}, NumPy, SciPy~\cite{scipy}, Matplotlib~\cite{Hunter:2007},
the \ac{GW} data analysis software library {\tt pycbc}~\cite{pycbc-software}
and the LSC Algorithm Library (LAL)~\cite{lalsuite}.
This research has made use of data, software and/or web tools obtained from the Gravitational Wave Open Science Center (https://www.gw-openscience.org), a service of LIGO Laboratory, the LIGO Scientific Collaboration and the Virgo Collaboration. LIGO is funded by the U.S. National Science Foundation. Virgo is funded by the French Centre National de Recherche Scientifique (CNRS), the Italian Istituto Nazionale della Fisica Nucleare (INFN) and the Dutch Nikhef, with contributions by Polish and Hungarian institutes.

\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


text:
Rori Harmon
Relentless No. 25 Texas Women's Basketball beats Baylor at home 68-55
January 23, 2023 January 23, 2023 Deyona Gaston, Rori Harmon, Shaylee Gonzales, Sonya Morris, Taylor Jones, Texas Women's Basketball
It has been a few years coming, five to be exact, since Texas Women's Basketball has beaten Baylor Bears at home in Waco. The relentless double digit scoring from five Texas players sealed the win 68-55.
Texas Women's Basketball advance to 2020 NCAA Sweet 16
March 21, 2022 March 21, 2022 Aaliyah Moore, Jody Conradt, MarchMadness, Retha Swindell, Rori Harmon, sweet 16
In the last game ever at the Frank Erwin Center, Texas Women's Basketball crushes Utah 78-56
No. 7/9 Texas Women's Basketball Win 2022 Big 12 Tournament Championship
March 14, 2022 March 15, 2022 2022 Big 12 Champions, Baylor, Lauren Ebo, Rori Harmon, TexasWBB
KANSAS CITY, Mo. — For the first time since 2003, the No. 7/9 Texas Women's Basketball have won the Phillips 66 2022 Big 12 Tournament Championship over No. 4 Baylor 67-58.
[VIDEO] Powerful message from Vic Schaefer after win over Iowa State 82-73
March 12, 2022 March 13, 2022 Joanne Allen-Taylor, Lauren Ebo, Post game press conference, Rori Harmon, TexasWBB, Vic Schaefer
KANSAS CITY, Mo. – In a battle to determine who gets to continue and play for the Big 12 Championship, Texas and Iowa State needed an overtime to settle it. No. 3 Texas beat No. 2 Iowa State 82-73.
Texas Women's Basketball advance to Phillips 66 Big 12 Championship Semifinals
March 12, 2022 March 12, 2022 Aliyah Matharu, Audrey Warren, Big 12 Championship, Joanne Allen-Taylor, Lauren Ebo, Rori Harmon
KANSAS CITY, Mo. — No. 7/9 Texas Women's Basketball win over Kansas State 72-65 moves them up to the semifinals in the Phillips 66 Big 12 Championship.
Texas Women's Basketball Coach Schaefer talks behind the scenes
February 24, 2022 February 24, 2022 Audrey Warren, Joanne Allen-Taylor, Rori Harmon, Texas Women's Basketball, TexasWBB, Vic Schaefer
AUSTIN, Texas – With their 9th straight 20+ win season, head coach Vic Schaefer shares what happens behind the scenes that make the players and team successful.


Prickwillow Museum, formerly known as the Prickwillow Drainage Engine Museum, tells the story of the changing face of the Fens and its network of drainage systems and pumping stations. The museum is housed in the old pumping station in Prickwillow, 4 miles east of the city of Ely in Cambridgeshire. The Fens are very low-lying compared with the chalk and limestone uplands that surround them – in most places no more than 10 m above sea level. Indeed, as a result of drainage and the subsequent shrinkage of the peat fens, many parts of the Fens now lie below mean sea level. Although one writer in the 17th century described the Fenland as entirely above sea level (in contrast to the Netherlands), the area now includes the lowest land in the United Kingdom, Holme Fen in Cambridgeshire. Within the Fens there are a few hills, which have historically been called "islands" as they remained dry when the low-lying fens around them were flooded. The largest of the fen-islands is the Isle of Ely, on which the cathedral city of Ely was built: its highest point is 39 m above mean sea level.
Though some signs of Roman hydraulics survive, and there were also some medieval drainage works, land drainage was begun in earnest during the 1630s by the various investors who had contracts with King Charles I to do so. The leader of one of these syndicates was the Earl of Bedford, who employed Cornelius Vermuyden as engineer. The scheme was imposed despite huge opposition from locals who were losing their livelihoods based on fishing and wildfowling. Fenmen known as the Fen Tigers tried to sabotage the drainage efforts. Two cuts were made in the Cambridgeshire Fens to join the River Great Ouse to the sea at King's Lynn – the Old Bedford River and the New Bedford River, the latter being known also as the Hundred Foot Drain. Both cuts were named after the Fourth Earl of Bedford who, along with some gentlemen adventurers (venture capitalists), funded the construction and were rewarded with large grants of the resulting farmland. The work was directed by engineers from the Low Countries. Following this initial drainage, the Fens were still extremely susceptible to flooding, so windpumps were used to pump water away from affected areas. However, their success was short-lived. Once drained of water, the peat shrank, and the fields lowered further. The more effectively they were drained, the worse the problem became, and soon the fields were lower than the surrounding rivers. By the end of the 17th century, the land was under water once again.
The major part of the draining of the Fens was effected in the late 18th and early 19th century, again involving fierce local rioting and sabotage of the works. The final success came in the 1820s when windpumps were replaced with powerful coal-powered steam engines, such as Stretham Old Engine, which were themselves replaced with diesel-powered pumps, such as those at Prickwillow Museum and, following World War II, the small electric stations that are still used today. The dead vegetation of the peat remained undecayed because it was deprived of air (the peat being anaerobic). When it was drained, the oxygen of the air reached it, since then the peat has been slowly oxidizing. This, together with the shrinkage on its initial drying and the removal of soil by the wind, has meant that much of the Fens lies below high tide level. As the highest parts of the drained fen are now only a few metres above mean sea level, only sizeable embankments of the rivers, and general flood defences, stop the land from being inundated. Nonetheless, these works are now much more effective than they were.
The collection includes six large diesel pumping engines, all British-built and of early to mid twentieth century design, rescued from Fenland pumping stations. The Mirrlees engine is the original engine installed at Prickwillow pumping station in 1924. The large engines are complemented by a collection of small, portable, engines by Lister, Petter and other manufacturers. All the engines in the collection have been restored and are maintained by a team of volunteers. There is wheelchair access to the majority of the Museum. Assistance dogs are welcome. Tuesdays are usually work-days for the volunteer engineers so why not visit then to watch and talk to them!
Transport: Ely (National Rail) then bus. Bus Routes : 129 stops very close by.


A Complete Teacher's Manual for Darker The Night- For use as a comparison/contrast to Holocaust stories.
Darker The Night* examines life in World War II Germany from the civilian perspective. Teachers in middle and upper grades are using the novel as a comparison and contrast to the more familiar Holocaust stories-The Boy in The Striped Pajamas by John Boyne, The Diary of Anne Frank, and Night by Elie Wiesel.
Sabrina Beasley, MA, NBCT EA/ELA, has designed this teaching tool to give educators engaging activities, studies and projects. Her tools offer many options for differentiation.
and digital resources including Quizlets and interviews with the woman who inspired the novel.
Your students can read Darker the Night alongside any of the Holocaust stories for a more complete understanding and analysis of life during the war.
* Darker The Night has a Lexile score of HL 640L.


Like or not app google.
As of fallthe overwhelming majority of Android apps have been installed by only a small number of users. So far so good, Google. To gain more insight into the nature of the app universe as a whole and the permissions that apps require to run, Pew Research Center collected information about over 1 million apps in the Google Play Store. Clean of unknown text. Compatible with iPhone, iPad, and iPod touch. Surprise party for Kristen! How to find permissions There are several places users can find the permissions an app is requesting. Complicating the matter even further for users, app developers cannot edit the description of each permission and therefore cannot include information about why each permission is needed. Stay up to date on your favorite topics with Discover. This version of Android, however, will not be available to most users at the time this report is released.
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Video: Like or not app google 8 Dangerous Apps! Do Not Install These Popular Android Apps !
This puts apps at the center of debates about privacy in the digital age.
To gain more insight into the nature of the app universe as a whole and the permissions that apps require to run, Pew Research Center collected information about over 1 million apps in the Google Play Store. Keep every thought Try Google Keep.
When and where you need it Need to remember to pick up some groceries? In addition, permissions will be displayed not at the moment of download, but when an app requires the particular permission.
All of this information can be crucial to the functioning of mobile apps. Set a location-based reminder to pull up your grocery list right when you get to the store.
Like or not app google Find what you're looking for even faster, and let Keep do the remembering for you.
Nov 8, Version No need for text messages back and forth. How to find permissions There are several places users can find the permissions an app is requesting. At other times, such access can be a helpful convenience that allows the app to function more broadly, but it is not critical to the core mechanism of how the app functions.
Need to finish a to-do?
Third-party sites and apps are created by companies or developers that ar.
you' d prefer not to share, like pictures of your family or copies of official documents. Note: Some of these steps work only on Android and up. Manage apps connected to your Google Account, like those with Google Sign-In permissions. You can get apps, games, and digital content for your device using the Google Play Store app. The Play Store app comes pre-installed on Android devices that.
We fixed user-reported issues and added new features to provide a better search experience.
Fresh look for Discover: Keep works on your phone, tablet and computer. Using this distinction, 70 permissions could allow an app to access user information, while allow an app to control some hardware function of the device, such as allowing the app to control the vibration function of the device or control the camera flash.
By category, communication and business apps require the largest number of permissions. Keep every thought Try Google Keep.
Sandra bullock dated bob schneider Pew Research Center does not take policy positions.
Every tree shall be known by its fruit.
Syncing across your devices requires internet connection. The average mean app in this dataset required five permissions before a user could install it. Need to remember to pick up some groceries?


Students are encouraged to TAKE THEIR TIME WITH THESE ASSESSMENTS. They can pause an assessment, do one at a time, and retest as needed. WHS Guidance charges $2 per test. Free/Reduced lunch test fees are waived.