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Power spectra used in this analysis, from both BOSS DR12 data, and MultiDark-Patchy mocks. For each separate chunk (at a different sky location and redshift bin), we show both the monopole ($\ell=0$, upper two spectra) and quadrupole ($\ell=2$, lower two spectra), before (darker colors) and after (lighter colors) density field reconstruction. Colored lines and shaded regions indicate the mean and $1\sigma$ variations between 999 mock catalogs in each chunk, with the data shown as black points. Errorbars indicate the square root of the covariance diagonal, estimated from the same set of mocks. We note that reconstruction sharpens the Fourier-space BAO wiggles, whilst slightly reducing the overall amplitude and removing most of the large-scale quadrupole power. Further note that the NGC data appears much smoother, due to the larger effective volumes of these regions.

Posterior distribution of the Alcock-Paczynski (AP) parameters $\alpha_\parallel$, $\alpha_\perp$ (defined in Eq.\,\ref{eq: AP-params}) obtained from analysis of the high-z NGC BOSS DR12 power spectrum, after density-field reconstruction. Posterior samples are obtained by minimizing a likelihood consisting of a linear model with an additional theoretical error to account for the poorly-understood post-linear shape of the spectrum, as described in Sec.\,\ref{subsec: recon-analysis}. We show results from three choices of hyperparameters in the analysis; the fiducial choice (red), adopting a much narrower prior on the non-linear damping scale $\Sigma_\mathrm{NL}$ (blue) and inflating the theoretical error kernel (Eq.\,\ref{eq: theoretical-err-kernel}) by a factor of five. These contours were generated from $\sim 10^4$ posterior samples obtained from running 16 MCMC chains in parallel. We note negligible difference in the AP parameters from imposing a tight prior on $\Sigma_\mathrm{NL}$, with a slight bias obtained by inflating the theoretical error.

Distribution of the \resub{best-fit} AP parameters obtained from applying the BAO analysis method of Sec.\,\ref{subsec: recon-analysis} to 999 MultiDark-Patchy mock galaxy samples. For each mock, we plot the best-fit value obtained from an MCMC analysis which fits the corresponding power spectrum against a theoretical model and outputs a posterior distribution for $\{\alpha_\parallel, \alpha_\perp\}$. The dotted lines indicate the expected value for the mock cosmology (Sec.\,\ref{subsec: mocks}), with the black cross showing the average and $1\sigma$ deviation across all mocks. Note that the error bar is not normalized by the number of mocks, thus it represents the expected variation from a single mock. The red cross and dashed lines show the best-fit obtained (and its 68\% and 95\% confidence levels (CLs)) from analyzing the true BOSS data-set in this chunk (as tabulated in Tab.\,\ref{tab: AP-results}). \resub{A comparison of the posterior contour shapes for mocks and data is shown in Fig.\,\ref{fig: ap-param-shape}.}

\resub{Posterior distribution shapes for the AP parameter estimates shown in Fig.\,\ref{fig: all-AP-params}. We overplot the 68\% and 95\% CL contours from the BAO analysis of 999 mock galaxy samples in red, shifting the distribution to have zero mean. The corresponding posterior for the data in each patch is shown in blue. Note that the data contours are consistent with a random draw from the set of mock contours.}

Covariance of the unreconstructed monopole power and the AP parameters for the low-z NGC chunk, using data from 999 Patchy mocks. Black crosses (red circles) show the covariance of the parallel (perpendicular) parameter and we normalize by the unreconstructed power measurements and AP variance in each case. The blue line shows a rough estimate based on a simple model of the post-reconstruction wiggly power spectrum (Eq.\,\ref{eq: cov-alpha-pk-estimate}), and we note that this is capture the functional form well.

CMB-independent cosmological constraints obtained from this work for the baseline $\nu\Lambda$CDM model, as tabulated in Tab.\,\ref{table0}. The `FS+BAO' data-set refers to the combination of full-shape (FS) modelling of unreconstructed power spectra via a one-loop full-shape model and BAO-modelling of reconstructed power spectra to compute Alcock-Paczynski parameters, incorporating the theoretical error methodology of Ref.\,\citep{2016arXiv160200674B}, with a joint sample covariance used to unite the two approaches. The `FS' data-set (equivalent to the full-shape analysis of Sec.\,\ref{subsec: unrecon-analysis}) was presented in Ref.\,\cite{2019arXiv190905277I} and `Planck 2018' refers to Ref.\,\cite{2018arXiv180706209P}. This plot shows the cosmological constraints obtained from combination of four BOSS DR12 data chunks, which are displayed separately in Fig.\,\ref{fig:wns-separate}. $H_0$ is quoted in $\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1}$ units.

Cosmological constraints obtained from the joint FS and BAO analysis of four disjunct BOSS DR12 data chunks, compared to the constraints from the latest Planck analysis \citep{2018arXiv180706209P}. The joint analyses of all four chunks yields the contours of Fig.\,\ref{fig:wns}.

Cosmological constraints obtained from this work, using the CMB-independent $\nu\Lambda$CDM model, but imposing Planck priors on the spectral slope $n_s$. The FS+BAO constraints obtained from analyzing the four data chunks in combination and separately are shown in the left and right plots respectively, which have the same forms as Figs.\,\ref{fig:wns}\,\&\,\ref{fig:wns-separate}, where $n_s$ was left unconstrained.

Cosmological constraints obtained from this work, using the CMB-independent $\nu\Lambda$CDM model, but imposing Planck priors on the spectral slope $n_s$. The FS+BAO constraints obtained from analyzing the four data chunks in combination and separately are shown in the left and right plots respectively, which have the same forms as Figs.\,\ref{fig:wns}\,\&\,\ref{fig:wns-separate}, where $n_s$ was left unconstrained.

Marginalized one-dimensional posterior distribution and two-dimensional probability contours (at the 68\% and 95\% CL) for the parameters of the $\nu\Lambda$CDM model, including varied neutrino masses, as obtained from analyses of the Planck likelihood \resub{separately} and in combination with BAO and FS information from BOSS. $N_{\text{eff}}$ is fixed to the standard model value $3.046$ and we quote $H_0$ in $\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1}$, with $M_{\text{tot}}$ given in eV.

As Fig.\,\ref{fig:mnuPl}, but for the cosmological parameters of the $\nu\Lambda$CDM+$N_\mathrm{eff}$ model, additionally varying the number of relativistic degrees of freedom $N_\mathrm{eff}$.

Parameter constraints obtained from analyzing the mean of the Patchy mocks corresponding to the low-z NGC BOSS galaxy sample, with the FS+BAO or FS-only likelihoods. Dashed lines represent the true values used in the simulations, which are $1\sigma$ consistent in all cases. This uses a slightly simplified cosmological model compared to the main analysis, as described in the text.

Parameter constraints obtained from analysis of the low-z (left panel) and high-z (right panel) NGC BOSS data using the $\nu\Lambda$CDM model assuming a fixed spectral tilt $n_s$. As discussed in the text, we observe significant improvements in $H_0$ from the inclusion of BAO data only for the high-z sample.

Parameter constraints obtained from analysis of the low-z (left panel) and high-z (right panel) NGC BOSS data using the $\nu\Lambda$CDM model assuming a fixed spectral tilt $n_s$. As discussed in the text, we observe significant improvements in $H_0$ from the inclusion of BAO data only for the high-z sample.