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\title{Calibrating semi-analytic VT's to injections in O3a}
\author{Daniel Wysocki}
\date{\today \\ LIGO-T2000432-v1}


\begin{document}

\maketitle

In the O2 populations paper, we took SNR-based $VT$ grids in mass $(m_1, m_2)$ and mass--aligned spin $(m_1, m_2, \chi_{1z}, \chi_{2z})$ space, and calibrated them against pipeline-based injection $VT$'s in mass-space only, which are only usable when averaged over the entire parameter space.
We assumed that the injection $VT$'s had some underlying functional form, such that
%
\begin{equation}
  VT_{\mathrm{inj}}(m_1, m_2) =
  f(m_1, m_2) \, VT_{\mathrm{grid}}(m_1, m_2),
\end{equation}
%
where $f$ is some unknown function.
We took $f$ to be a $K$th order polynomial, $\sum_a^K \sum_b^K \lambda_{ab} m_1^a \, m_2^b$, such that
%
\begin{equation}
  \langle VT_{\mathrm{inj}}(m_1, m_2) \rangle =
  \sum_{a=0}^K \sum_{b=0}^K
    \lambda_{ab} \, \langle m_1^a \, m_2^b \, VT_{\mathrm{grid}}(m_1, m_2) \rangle
\end{equation}
%
allowing us to estimate the coefficients $\lambda_{ab}$ via least squares approximation, by taking the averages over a number of different population models.
Due to decisions made early on with the injection-based $VT$'s, these population models all followed the Model A powerlaw from the O2 populations paper, with fixed $m_{\mathrm{min}} = 5 \, M_\odot$, $\beta_q = 0$, and variable $\alpha_m$ and $m_{\mathrm{max}}$ in order to cover a wide range of parameter space.

Now in O3a, we have injections in mass--aligned spin space, and want to perform calibration again.
Extending the calibration na\"{i}vely to the 4D mass--aligned spin space will result in a very high order basis with $K^4$ terms.
Not much sensitivity information is present in $(m_1, m_2, \chi_{1z}, \chi_{2z})$ that isn't present in $(m_1, m_2, \chi_{\mathrm{eff}})$, so a simple dimensionality reduction can be made here, taking our calibrating function to be $\sum_a^K \sum_b^K \sum_c^K \lambda_{abc} m_1^a \, m_2^b \, \chi_{\mathrm{eff}}^c$.

All that's left to be decided is a population model.
It may first be wise to extend the mass calibration from Model A to Model B, still fixing $m_{\mathrm{min}}$ (but now lowered to our injection campaign's new minimum of $3 \, M_\odot$), but allowing $\beta_q$ to vary.
For the spin {d.o.f.}, the population models being used in our analysis are no longer viable, so instead we propose using a rescaled beta distribution in $\chi_{\mathrm{eff}}$, and varying $\mathbb{E}[\chi_{\mathrm{eff}}]$ and $\mathrm{Var}[\chi_{\mathrm{eff}}]$, restricting them to values which produce non-singular distributions (i.e., $\alpha_{\chi_{\mathrm{eff}}}, \beta_{\chi_{\mathrm{eff}}} \geq 1$).


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