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\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Illustration of the definition of pitch angle. It is given as $\phi = \qopname  \relax o{tan}^{-1}\left (\frac  {\mathrm  {d}r}{\mathrm  {d}\theta }\ /\ r\right )$, or the angle between the spiral (red) and the tangent to a circle centred on the galaxy (blue).}}{3}{figure.2}}
\newlabel{fig:pitch-angle-example}{{2}{3}{Illustration of the definition of pitch angle. It is given as $\phi = \tan ^{-1}\left (\frac {\mathrm {d}r}{\mathrm {d}\theta }\ /\ r\right )$, or the angle between the spiral (red) and the tangent to a circle centred on the galaxy (blue)}{figure.2}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces A plot of redshift against stellar mass for the \textit  {stellar mass-complete sample} from \citet  {2017MNRAS.472.2263H}; the subset we use for analysis in Galaxy Builder are shown in red. At right we show a histogram of the stellar masses. This Figure is identical to Figure 4 from \citet  {2020arXiv200610450L} who use the same sample.}}{5}{figure.3}}
\newlabel{fig:stellarmass}{{3}{5}{A plot of redshift against stellar mass for the \textit {stellar mass-complete sample} from \citet {2017MNRAS.472.2263H}; the subset we use for analysis in Galaxy Builder are shown in red. At right we show a histogram of the stellar masses. This Figure is identical to Figure 4 from \citet {2020arXiv200610450L} who use the same sample}{figure.3}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Examples of spiral profiles fit using the hierarchical model described in Section \ref  {section:bhsm-model}. Deprojected points from \textit  {Galaxy Builder} clustered, cleaned spiral arms are shown in black; fit logarithmic spiral arms are shown in red, with the width of the line corresponding to the $2\sigma $ interval on predicted values of $\setbox \z@ \hbox {\frozen@everymath \@emptytoks \mathsurround \z@ $\textstyle r_\mathrm  {arm}$}\mathaccent "0365{r_\mathrm  {arm}}$. The two one-armed spirals in the top left panels are instances where the spiral clustering algorithm failed to identify all spiral arms present in the galaxy.}}{7}{figure.4}}
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\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Scatter plot showing how arm pitch angle compares to galaxy pitch angle for galaxies with different pitch angles and number of arms. The top panel shows a Gaussian KDE for $E[\phi _\mathrm  {gal}]$, and the right panel shows a Gaussian KDE for $E[\phi _\mathrm  {arm} - \phi _\mathrm  {gal}]$. The galaxy pitch angle is consistent with the mean of its arms, with large scatter and a slight bias against values near the lower bound of $0$ due to the lower limit applied.}}{8}{figure.5}}
\newlabel{fig:arm-pa-spread}{{5}{8}{Scatter plot showing how arm pitch angle compares to galaxy pitch angle for galaxies with different pitch angles and number of arms. The top panel shows a Gaussian KDE for $E[\phi _\mathrm {gal}]$, and the right panel shows a Gaussian KDE for $E[\phi _\mathrm {arm} - \phi _\mathrm {gal}]$. The galaxy pitch angle is consistent with the mean of its arms, with large scatter and a slight bias against values near the lower bound of $0$ due to the lower limit applied}{figure.5}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces The stellar mass distribution of the sample and the galaxy average pitch angle distribution shown as a 2D histogram (centre) and also projected along each axis. }}{8}{figure.6}}
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\@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces Density plot showing bulge strength ($B_\mathrm  {avg}$; left, orange) and bar strength ($p_\mathrm  {bar}$; right, green) against galaxy pitch angle ($\phi _\mathrm  {gal}$). Split points for the marginalized Anderson-Darling tests are labelled. There is no statistically significant relationship for either bulge or bar strength.}}{9}{figure.7}}
\newlabel{fig:bulge-bar-pa-hist}{{7}{9}{Density plot showing bulge strength ($B_\mathrm {avg}$; left, orange) and bar strength ($p_\mathrm {bar}$; right, green) against galaxy pitch angle ($\phi _\mathrm {gal}$). Split points for the marginalized Anderson-Darling tests are labelled. There is no statistically significant relationship for either bulge or bar strength}{figure.7}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {8}{\ignorespaces The results of marginalized two-sample Anderson-Darling tests examining whether pitch angles ($\phi _\mathrm  {gal}$ in blue and $\phi _\mathrm  {gal}$ in orange) for galaxies with $B_\mathrm  {avg} < 0.28$ and $B_\mathrm  {avg} \ge 0.28$ are drawn from the same distribution (top panel), and the results of marginalized three-sample Anderson-Darling tests for galaxies with no bar ($p_\mathrm  {bar} < 0.2$), a weak bar ($0.2 \le p_\mathrm  {bar} \le 0.5$) and a strong bar ($p_\mathrm  {bar} > 0.5$) (bottom panel). Confidence intervals are shown, with moving rightwards indicating more confidence in rejecting the null hypothesis that the compared values were drawn from the same parent distribution. We cannot reject the null hypothesis at the 1\% level for any of the tests conducted, meaning there is no evidence in this sample that bulge size or bar strength impacts pitch angle.}}{9}{figure.8}}
\newlabel{fig:ad-morphology-test}{{8}{9}{The results of marginalized two-sample Anderson-Darling tests examining whether pitch angles ($\phi _\mathrm {gal}$ in blue and $\phi _\mathrm {gal}$ in orange) for galaxies with $B_\mathrm {avg} < 0.28$ and $B_\mathrm {avg} \ge 0.28$ are drawn from the same distribution (top panel), and the results of marginalized three-sample Anderson-Darling tests for galaxies with no bar ($p_\mathrm {bar} < 0.2$), a weak bar ($0.2 \le p_\mathrm {bar} \le 0.5$) and a strong bar ($p_\mathrm {bar} > 0.5$) (bottom panel). Confidence intervals are shown, with moving rightwards indicating more confidence in rejecting the null hypothesis that the compared values were drawn from the same parent distribution. We cannot reject the null hypothesis at the 1\% level for any of the tests conducted, meaning there is no evidence in this sample that bulge size or bar strength impacts pitch angle}{figure.8}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {9}{\ignorespaces The distributions of pitch angles (orange and blue) relative to one uniform in $\qopname  \relax o{cot}\phi $ (black). Histograms have been normalised by the area between the limits such that they are comparable. The histogram was recalculated with identical bins for each posterior sample of $\phi _\mathrm  {gal}$ and $\phi _\mathrm  {arm}$, we plot the mean value of each bin, with the sample standard deviation shown as error bars. It is evident that the distributions are very similar between the chosen limits.}}{11}{figure.9}}
\newlabel{fig:pa-cot-distributions}{{9}{11}{The distributions of pitch angles (orange and blue) relative to one uniform in $\cot \phi $ (black). Histograms have been normalised by the area between the limits such that they are comparable. The histogram was recalculated with identical bins for each posterior sample of $\phi _\mathrm {gal}$ and $\phi _\mathrm {arm}$, we plot the mean value of each bin, with the sample standard deviation shown as error bars. It is evident that the distributions are very similar between the chosen limits}{figure.9}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {10}{\ignorespaces The results of a marginalized Anderson-Darling test for uniformity in $\qopname  \relax o{cot}$ for $\phi _\mathrm  {gal}$ (blue) and $\phi _\mathrm  {arm}$ (orange), with values corresponding to various confidence intervals shown. Moving rightwards on the x-axis implies greater confidence in rejecting the null hypothesis that the sample was drawn from a distribution uniform in $\qopname  \relax o{cot}$ between $15^\circ < \phi < 50.0^\circ $. In this instance, we would not be able to reject the null hypothesis at the 1\% level for either $\phi _\mathrm  {gal}$ or $\phi _\mathrm  {arm}$, meaning our sample is consistent with a cot uniform distribution. The larger error in $\phi _\mathrm  {gal}$ means that this result is more significant for $\phi _\mathrm  {arm}$, which is also physically motivated, as arms can wind independently.}}{11}{figure.10}}
\newlabel{fig:ad-cot-test}{{10}{11}{The results of a marginalized Anderson-Darling test for uniformity in $\cot $ for $\phi _\mathrm {gal}$ (blue) and $\phi _\mathrm {arm}$ (orange), with values corresponding to various confidence intervals shown. Moving rightwards on the x-axis implies greater confidence in rejecting the null hypothesis that the sample was drawn from a distribution uniform in $\cot $ between $15^\circ < \phi < 50.0^\circ $. In this instance, we would not be able to reject the null hypothesis at the 1\% level for either $\phi _\mathrm {gal}$ or $\phi _\mathrm {arm}$, meaning our sample is consistent with a cot uniform distribution. The larger error in $\phi _\mathrm {gal}$ means that this result is more significant for $\phi _\mathrm {arm}$, which is also physically motivated, as arms can wind independently}{figure.10}{}}
\@writefile{toc}{\contentsline {section}{\numberline {5}Acknowledgements}{11}{section.5}}
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\@writefile{toc}{\contentsline {section}{\numberline {6}Data Availability}{12}{section.6}}
\newlabel{lastpage}{{6}{12}{Data Availability}{section.6}{}}