surface mass density; D L , D S and D LS are the angular diameter distances from the observer to the lens, to the source and from the lens to the source, respectively. We can get some simple but important results about the lensing efficiency for the DC14 model even before calculating the lensing probabilities. We notice that, from equation ( 2 ), NFW profile is a specific form that has (, , )(1, 3, 1), and SIS profile is similar to (, , )(2, 2, 2), where in the latter case, can be any non-zero number and we let 2 for definiteness. Therefore, it would be helpful to plot the parameters , and versus M halo from 10 11 to 10 14.9 M , as shown in Fig. 1 . We find that the inner slope has 1, an NFW like value, only in a narrow range of halo mass around 10 12 M , and it flattens for lower and higher mass ranges. This would result in a much lower lensing efficiency compared with that of NFW model and even further lower compared with SIS model. Figure 1. The parameters of , and as functions of M halo . Another parameter that helps us to understand the lensing efficiency is s in lensing equation ( 10 ). According to strong lensing theory, multiple images can occur only for sufficiently large values of s . Fig. 2 shows how s changes with M halo for both DC14 and NFW models, with a fixed typical value 0.45 of the lens redshift and the average value 1.56 of the source redshift (Inada etal. 2012 ). We find that the values of s for NFW model uniformly surpass that of DC14 in the whole range of halo mass of (10 11 , 10 14.9 M ), and the difference increases markedly for M halo 10 12 M . This would also mean an obviously lower lensing efficiency for the DC14 model compared with the NFW model. Figure 2. The parameter s as a function of M halo for NFW (dotted line) and DC14 (solid line). In both cases, the redshifts of source and lens are set to be 1.56 and 0.45, respectively. Figure 3. The relation between c and M 200 . Figure 4. The relation between i and M 200 . Figure 5. The relation between r s and M 200 . 2.2 <xref ref-type="bibr" rid="bib77">Schaller15</xref> model The Schaller15 model is derived from EAGLE simulations; the total (baryons plus DM) density profile consists of two terms (Schaller etal. 2015b ): where the first term is the NFW profile and the second term is NFW-like in that it shares the same asymptotic behaviour at small and large radii and has a slope of 2 at its scaleradius, r r i . We write the surface mass densities corresponding to the two terms as where and with For later calculations, we need to know the characteristic densities c and i , and characteristic radii r s and r i as functions of the total mass M 200 , which reads (Schaller etal. 2015b ) In practice, however, it is impossible to derive so many parameters from the only equation ( 16 ), we thus fit c etc. to M 200 with the data given by Schaller etal. ( 2015b ). The fitted results are displayed in Figs 3 5 . The fitting formulae are where X log10( M 200 ), and ca 0.06, cb

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation O B-Model O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O B-Model O O O O O B-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation O O O O B-Model O O O O O O O O O O O O O B-Model O O O B-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Tag I-Tag I-Tag O O B-Citation O O O O B-Software O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O B-Model O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O


we also define . We chose three different statistics: (). This statistic picks up the maximum deviation from the continuum spectrum for each simulated PDS. The observed value is then compared with the simulated distribution and the significance is evaluated directly. By construction, it implicitly accounts for the multitrial search performed all over the frequencies. is the Anderson–Darling (AD) statistic ( Anderson Darling 1952 ) obtained for the th set of compared with a distribution. Analogously, KS is the Kolmogorov–Smirnov (KS) statistic obtained for the th set of compared with a distribution. For each of the three statistics, comparing the values obtained from the observed PDS with the corresponding distribution of simulated values immediately yields the significance of possible deviations such as that of a QPO, or the goodness of the fit, as indicated by the AD and KS statistics. As in G12, in addition to the KS, we chose the AD statistic because it is sensitive to a few outliers from the expected distribution. For each GRB the choice between the two competing models was determined by the likelihood ratio test (LRT) in the Bayesian implementation described by V10. As for the aforementioned statistics, from the posterior predictive distribution we sampled the pdf of the statistic defined as where denotes the model obtained with the parameters at the mode of the posterior distribution of a generic model . The statistics in Eq. ( A.9 ) is then sampled using the simulated PDS () and compared with the observed value. For the LRT test, we performed simulations and accepted the bpl model when the probability of chance improvement was lower than 1% (see Sect. 2.3 ). As remarked in Sect. 2.3 , while this type of LRT accounts for the whole parameter space of the simpler model, the same does not hold for the alternative one, as is the case when we use the Bayes factor. Consequently, the rejection of pl does not necessarily imply that bpl is a better option. However, for the limited scope of this investigation, on independent grounds bpl represents a sensible alternative (Sect. A.1.1 ). Finally, after we determined the best–fitting model, we sampled through the MCMC simulation the joint posterior distribution of the model parameters and provided mean and standard deviation for each of them. As an example, Fig. A.1 shows the marginal posterior distributions of all the pairs of parameters of the bpl model for 050713A. Fig. A.1 Example of marginal posterior distributions for the pairs of parameters of the bpl model obtained from simulated posterior simulations for 050713A (only 2000 points are shown for clarity). Solid lines show the contour levels. The goodness of fit is established by the -values associated to the AD and KS statistics. In addition, we also compare the distribution of of the observed PDS against the expected distribution, as shown in Fig. A.2 for 050713A. Fig. A.2 Distribution of for 050713A (Fig. 1 ). The dashed line shows the

O O O O O O O O O B-TextGarbage O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O B-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-TextGarbage O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O B-Model O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O B-CelestialObject O O O O O O O O O O O O O O B-Model I-Model O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O B-CelestialObject O O O O O O O O


from 108 full-sky maps (Takahashi et al. 2017 ; Namikawa et al. 2019 ). The error bars are the standard deviations scaled by . The solid red, dashed orange, and dotted brown curves are the theoretical predictions based on BiHalofit, GM12 , and SC01 , respectively. The middle panels plot the relative deviations from the red curves: . In the bottom panels, these deviations are further divided by the relative standard deviation (). Although our fitting surely improved the prediction accuracy, noticeable discrepancies from the simulations were introduced by several sources. First, the finite thickness of the lens planes employed in the ray-tracing simulations may affect the simulations at l 200 (the same effect on convergence PS is demonstrated in Figure 10 of Takahashi et al. 2017 ). Second, the flat-sky formula in Equation ( 16 ) is inaccurate at large angular scales (the accuracy of the flat-sky approximation in the cosmic-shear PS is detailed in Kilbinger et al. 2017 ; Kitching et al. 2017 ). For example, in the squeezed limit, the minimum multipole is fixed as l 3 = 50, but a larger l 3 can mitigate the discrepancy (Namikawa et al. 2019 ). Since reducing the finite thickness of lens planes requires more numerically expensive simulations, we will leave the detailed study for future work. 6.2. Cosmic Shear Let us now consider the cosmic-shear signals in galaxy-shape measurements, which probe lower redshifts than CMB lensing. Sato et al. ( 2009 ) ran cosmological N -body simulations and subsequent ray-tracing simulations under the flat-sky approximation. Despite their small field of view (5 × 5 deg 2 ), they acquired sufficiently many weak-lensing maps (1000) for an accurate BS measurement. Their cosmological model was consistent with the WMAP 3 yr result (Spergel et al. 2007 ). Figure 13 plots the convergence BS at a source redshift of z s = 1, measured from the 1000 maps by Kayo et al. ( 2013 ). The theoretical and simulation results were consistently binned with . The simulation results were valid (within 5% error) up to l ≃ 4000, as confirmed by comparing the results with those from low- and high-resolution maps (Sato et al. 2009 ; Valageas et al. 2012 ). Overall, the plot shows a similar trend to the matter BS at z = 0.55 (see Figure 5 ). Our fitting formula well agreed with the simulation (within 10% level up to l = 4000). The deviation at small scales ( l ≳ 4000) was attributed to the limited resolution of the simulation. Figure 13. Convergence BSs measured from 1000 simulation maps at the source redshift (Sato et al. 2009 ; Kayo et al. 2013 ). Black circles are the measured averages, and the red curves were predicted by our model. The dashed orange and brown dotted curves were obtained by GM12 and SC01 , respectively. The field of view of each map is , and the error bars are the standard deviations scaled by . The bottom panels plot the relative deviations from the red curves. As the cosmological models in this and the previous

O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model B-Citation O O B-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-TextGarbage O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O B-URL I-URL I-URL I-URL O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O B-Model I-Model I-Model O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Dataset I-Dataset I-Dataset I-Dataset B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation O B-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


in previous studies (Kapranidis Liebert 1986 ; Dufour et al. 2007 ), atmosphere model codes used to analyze this object relied on constitutive physics that are not totally appropriate for the dense atmosphere of this cool white dwarf. In particular, the analysis of Kapranidis Liebert ( 1986 ) relied on models that assumed a Thomas–Fermi equation of state (Kapranidis 1983 ). Because of this assumption, they found that electron thermal conduction was an important energy transport mechanism, a conclusion that does not hold if we use more realistic equations of state (Bergeron et al. 1995 ; Kowalski et al. 2007 ). Also, Kapranidis Liebert ( 1986 ) computed their synthetic spectra from pure helium atmosphere structures, thus ignoring the impact of heavy elements on the pressure and temperature stratification. Moreover, a simple Lorentzian profile was assumed for the Ca i resonance line. Despite all those simplifying assumptions, they managed to obtain an excellent fit to the resonance line (with T eff = 4500 K and ) and were even able to find a good agreement to the unknown feature at 4500 Å. Unfortunately, no detail regarding this puzzling structure was given, so we cannot know how they were able to explain it. Dufour et al. ( 2007 ) revisited WD 2251−070 using more realistic atmosphere models, where heavy elements are included in the calculation of the atmosphere structures. That being said, a number of unjustified approximations remained, such as the ideal gas law, the ideal Saha ionization equilibrium, and Lorentzian profiles. Their best solution ( T eff = 4000 K and ) was in good agreement with the photometry and with the Ca i resonance line (but they were unable to explain the absorption feature near 4500 Å). Here, we revisit this star using our improved models (Blouin et al. 2018a ) and our new Ca i resonance line profiles. To do so, we use the spectroscopic observations of Dufour et al. ( 2007 ), the Gaia data release 2 (DR2) parallax measurement ( π = 117.15 ± 0.05 mas; Gaia Collaboration 2016 , 2018 ), photometry from Bergeron et al. ( 1997 ), the Panoramic Survey Telescope And Rapid Response System, Chambers et al. ( 2016 ), and the Two Micron All-Sky Survey. Our best fit to the spectroscopic and photometric observations is shown in Figure 9 , and the corresponding atmospheric parameters are given in Table 1 . Figure 9. Our best solution for WD 2251−070. The top panel shows our spectroscopic fit, and the bottom panel shows our fit to the BVRI , grizy, and JHK photometry. Table 1 WD 2251−070 Atmospheric Parameters To obtain this solution, we used the photometric technique (Bergeron et al. 1997 ) to determine T eff and . More precisely, the solid angle π ( R / D ) 2 and the effective temperature were found by adjusting the model fluxes to the photometric observations. From the solid angle, we computed the white dwarf radius R ( D is given by the Gaia parallax) and, from there, we found the mass and the surface gravity using the

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priors, the relative magnitudes of each component are able to vary freely. While fits performed without priors can produce biased results if the training sets cannot sufficiently describe the shapes of the components, the use of a prior derived from the training set applies constraints on both the shapes and the magnitudes of the components. Thus, caution should be taken to ensure that there is a high level of confidence that the given training set is sufficient to describe both the shape and magnitude of the component before a training set prior is applied. 12 12 In principle, if one is uncertain that the training set is fully representative of the data, it may be possible to apply a “weak” prior, which falls in between the highly constrained and uniform priors used in the examples here. A complicating factor in applying this method is that the posterior will no longer be Gaussian if the prior takes a non-Gaussian form. Note that when using a prior distribution, instead of minimizing the DIC in order to choose the number of SVD modes, we instead maximize the Bayesian evidence , given by where the parameters x are the coefficients of the SVD modes, is the likelihood, is the model composed of the SVD modes, and π ( x ) is the prior distribution. Note that the dimensionality N p of the parameter space will change when the number of SVD modes to be fit is varied. Maximizing the evidence ensures that the prior is properly taken into account. When using a training set prior, variations in modes unimportant to the training set are suppressed, but a larger number of modes may still be chosen by the evidence. Thus, the number of SVD modes may no longer provide as much information about the quality of the estimate of the signal. However, if the training sets do not encompass the true form of one or more of the components, the constraints provided by the prior will prevent the full data from being fit well. In this case, goodness-of-fit statistics such as or will be able to identify a poor fit. Further details regarding the use of these metrics for fits with priors are given in Section 3 . 2.4. Test Strategy In this section, we present a step-by-step outline for assessing the quality of the signal extraction in an observational setting when the true form of the signal is unknown. This test can be performed for fits with or without priors; however, we suggest that the test be performed without a prior distribution first owing to the fact that the use of an incorrect prior distribution will lead to significant bias in the extracted signal, as described in Section 2.3 . If the fit “passes” the test as described below, one may wish to include a training set prior to possibly improve the quality of the signal reconstruction further. 1. Fit many data realizations created from the training set curves. If no prior is used, create a distribution of the chosen numbers of SVD modes. This distribution will have the same dimensionality as the number of

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O B-Model O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O


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spread function of the system. Its energy corresponds to the incoming flux and is normalized to 1. At this point, it is important to note that the sensitivity is bounded. Since the mask transmission, | m |, cannot be greater than 1 (| m | ≤ 1), Eq. ( 7 ) implies that TF| f ≤ 2. Hence, given Eq. ( 8 ), we conclude that, in the frame of our normalizations, the sensitivity cannot be greater than 2: This is an important result as it defines the theoretical limit for the sensitivity of an FFWFS. 2.3. Application to the ZWFS The convolutional formalism introduced in the previous section was then applied to the ZWFS in order to find a simple formula of its sensitivity according to the mask parameters. As previously mentioned, the ZWFS mask is defined by two free parameters. The first is the depth (phase shift) of the dot δ ; for the classical ZWFS, δ = π /2. The second is its diameter p ; for the classical ZWFS, p = 1.06 λ / D . As described in N’Diaye et al. (2013) , the dot diameter value was chosen in order to get an equivalent flux inside and outside the focal plane dot. This configuration with p = 1.06 λ / D also allows a uniform reference intensity distribution to be obtained. The purpose of this section is to demonstrate that this choice of p = 1.06 λ / D is actually not optimal, and that the sensitivity can be improved with a larger dot. For the sake of clarity, we carried out this study in one dimension. As such, the spatial frequency vector f becomes the scalar frequency f . We further assumed that the weighting function ω , which corresponds to the PSF, can be described as a top-hat function of diameter radius a . This simplified geometry is summarized by Fig. 1 a. From this simplified geometry, we then computed TF| f and plot this quantity in Fig. 1 b. The full derivation can be found in Appendix B . Fig. 1. Simplified description of the ZWFS. ( a ): simplified one-dimensional framework for convolutional derivations; in black, the dot diameter is equal to p and the phase shift is equal to δ . In green, the PSF is approximated by a normalized top-hat function with a diameter a . ( b ): transfer function of the ZWFS for two cases; in red the dot size is smaller than the PSF, and in blue the dot is larger than the PSF. The optimal case appears for p = a and not p = a /2, as done for the classical ZWFS. From Fig. 1 b one can distinguish two cases. The first is p ≥ a (i.e., the dot diameter is larger than the PSF characteristic size). Frequencies above ( p + a )/2 reach the sensitivity 2sin( δ ). Frequencies below ( p − a )/2 are at 0. The second is p a : the dot diameter is smaller than the PSF size. Frequencies over p have a value of 2 p / a × sin( δ ), which is smaller than the theoretical limit of 2. From this simplified model one can first conclude that a phase shift of δ = π /2 will maximize the sensitivity, as expected. But surprisingly, it shows that for this phase shift a dot radius of p = a offers a sensitivity of the optimal

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O B-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O


) UVB model – scaled to the effective optical depth fitting function of Kim et al. ( 2007 ). The solid curves represent our fiducial model (purple), our model with α QSO = 1.3 (pink), with α QSO = 2.0 (black), and our model with an ionization fraction based on the quasar emissivity function of Kulkarni et al. ( 2019 , green). These curves have all been scaled by the same factor as our uniform UVB model. The effective optical depth varies up to tens of percent, with our α QSO = 1.3 model showing the largest discrepancy due to the largest variance in IGM temperature. Figure 6. The evolution of the effective optical depth of the H i Lyman α forest for three of our He ii reionization models compared with a model with a uniform UVB (dashed, teal curve). The solid purple curve shows our model with α QSO = 1.7, C He iii = 3, E max = 150 eV, and z reion,He ii = 2.8−4. The pink and black curves shows the same model, but with α QSO = 1.3 and 2.0, respectively. The light green curve shows our model based on the quasar emissivity function of Kulkarni et al. ( 2019 ). The error bars show measurements from Schaye et al. ( 2003 ), Kim et al. ( 2007 ), and Calura et al. ( 2012 ). The bottom panel shows the relative differences between our four He ii reionization models and the uniform UVB model. From our simulated H i Lyman α forest spectra, we calculate the flux PDF for our two simulations shown in Fig. 5 . Because the patchiness of reionization will result in multiple temperatures at the same gas density, the flux PDF will reflect more bimodality after He ii reionization is underway. The left-hand panel of Fig. 7 shows the flux PDF for a redshift at the onset of He ii reionization ( z = 4) and towards the end of reionization in the right-hand panel ( z = 3). The solid curve shows our model with inhomogeneous reionization and the dashed curve shows the effect of a homogeneous UVB. Indeed, in the midst of He ii reionization, we find a more bimodal distribution in comparison to our simulation without inhomogeneous He ii reionization. We compare these cases with observational measurements of the flux PDF from Kim et al. ( 2007 ) and Calura et al. ( 2012 ) at z ≈ 3. While these measurements show broad agreement with both simulations, and may slightly favour our inhomogeneous He ii reionization, the uncertainty in the measurements is nearly equal to, or greater than the difference between our model and the uniform UVB case. Figure 7. Flux PDF from a simulation with inhomogeneous He ii reionization model (solid) and from a simulation with a uniform UVB (dashed). The left-hand panel shows a redshift at the onset of He ii reionization ( z = 4) and the right-hand panel shows a redshift just before its completion ( z = 3). The error bars show measurements from Kim et al. ( 2007 ) and Calura et al. ( 2012 ). From these same simulations, we show the 1D flux power spectrum in Fig. 8 . The left-hand panel presents the flux power spectrum for a redshift at the onset of He

O B-Model I-Model O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O B-Model I-Model I-Model O O O O O O O O O O O O B-Model I-Model I-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength I-Wavelength O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula B-Formula I-Formula I-Formula I-Formula I-Formula B-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O B-Model I-Model I-Model O O O O O B-Wavelength I-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


study. Dashed lines show the previous (grey) and improved (blue) sensitivity limits. The cut at the bottom left corner seems to represent the increasing difficulty for detecting sources with the lowest synchrotron log() values. A possible explanation is that for low-synchrotron peak values, the IC peak may move to lower energies, hindering the detections; but also this could be connected to incompleteness of the 2WHSP sample close to the synchrotron peak cut, at ν ∼ 10 15 Hz. As reported in Chang et al. ( 2017 ) the incompleteness might be induced by the multifrequency selection criteria that was used to build the 2WHSP sample. 4.3 Notable sources The stacked γ-ray SEDs displayed in Fig. 2 present two distinct outliers, namely the sources 1BIGB J224910.6 − 130002 and 1BIGB J194356.2+211821. The source 1BIGB J224910.6 − 130002, also known as RBS 1899, is a blazar of unknown redshift classified as an EHSP due to its synchrotron peak position estimated at 10 17.5 Hz. Interestingly, the Fermi -LAT spectrum obtained with over 9.3 yr of data shows a steep decaying behaviour in log (ν) versus log (ν F ν ) space, with photon spectral index Γ ∼ 2.33 and relatively bright in γ-rays, as seen in Fig. 5(a) . However, this unusual behaviour is probably due to the averaging of the quiescent state with a series of γ-ray flares that happened along 2016 to 2017, as can be seen in the light curve reported in Fig. 5(b) . During these flaring episodes, the flux increased of ∼40 times with respect to the quiescent phase. This may also explain why the source was undetected in previous FGL and FHL catalogues. This is an important example of a γ-ray blazar that could represent a major target for the future CTA during periods of extreme activity (as detected during MJD 57662–57687, Fig. 5a in black). It also illustrates the strong impact of flaring activity in attempting a population study of extreme or faint blazars, since the most extreme γ-ray activity is smoothed over time when integrating over years of Fermi -LAT observations. Figure 5. Left plot: SED of 1BIGBJ224910.6 − 130002, showing the mean γ-ray SED obtained by integrating over 9.3yr in blue, and during the 25 d flare peak (MJD 57662.65–57687.65) in black. Right plot: Light curve of 1BIGBJ224910.6 − 130002 considering 1 month time bins, integrating over the broad-band of 900 MeV up to 500 GeV. Flux calculated only for > 3σ bins (red points), error bars, and upper limits (black cross) are calculated using Integral method considering 95 per cent of confidence level. The second peculiar source, 1BIGB J194356.2+211821 also known as HESS J1943+213, is the most extreme blazar of the 1BIGB catalogue, with log (ν p ) = 18.1. It was serendipitously discovered by the H.E.S.S. Collaboration (H.E.S.S. Collaboration et al. 2011 ) at VHE γ-rays during a galactic plane survey and later confirmed with VERITAS observations (VERITAS Collaboration et al. 2015 ). Due to the source’s position on the galactic plane and its

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Survey O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O B-Survey O O O O O O B-Wavelength O O O O O O O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject I-CelestialObject O B-CelestialObject I-CelestialObject O O B-CelestialObject I-CelestialObject I-CelestialObject I-CelestialObject O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength I-Wavelength O O B-Observatory B-Telescope O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Dataset I-Dataset I-Dataset I-Dataset O O O O O O O B-Wavelength O O O O O O O O O O B-Telescope O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O B-Observatory B-Telescope O O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject I-CelestialObject O O O O O O O O O O B-Wavelength I-Wavelength O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O B-CelestialObject I-CelestialObject O O O O O O O B-Dataset I-Dataset O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O B-Collaboration I-Collaboration B-Citation I-Citation I-Citation I-Citation I-Citation O O B-Wavelength I-Wavelength O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O B-Telescope O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O


the significance of the migration of the giant planets was recognized, JFCs and Oort cloud comets (OCCs) were thought to have formed in distinct regions, where JFCs formed beyond Neptune and OCCs formed between giant planets ( Dones et al. 2004 ). Except for P/SW3, we see little if any systematic difference between JFCs and OCCs, even when freshly exposed interior materials were measured. In terms of chemical composition, surveys of cometary volatiles have also shown no systematic difference between JFCs and OCCs, but that significant variations are present within each group ( Mumma Charnley 2011 ; A’Hearn et al. 2012 ). As growing observational evidence reveals that JFCs and OCCs formed within the same broad region, dynamical simulations, in the framework of planetary migration, also suggest that the Oort cloud and scattered disk are derived from a common parent population, i.e., the primordial trans-Neptunian disk ( Brasser Morbidelli 2013 ). Recent measurements of HDO/H 2 O in 67P ( Altwegg et al. 2015 ) by the ROSINA mass spectrometer on board the Rosetta spacecraft, also suggest that JFCs and OCCs formed in largely overlapping regions where the giant planets are today ( A’Hearn 2017 ). However, the proportions of comets formed at different heliocentric distances are not yet clear ( Dones et al. 2015 ). Füri Marty (2015) studied the distribution of N isotopes in the solar system and found that it follows a rough trend with increased 15 N abundances at larger radial distances from the Sun. Nitriles detected in cometary comae so far are consistently enriched in 15 N by a factor of ~3 relative to the protosolar nebula and a factor of 1.8 relative to the terrestrial planets ( Füri Marty 2015 ). The significant enrichment in 15 N is unlikely to be due to primordial nucleosynthetic heterogeneities, but rather a result of isotope fractionation processes during the formation of the solar system ( Füri Marty 2015 ). Possible processes include ion-molecule reactions in the interstellar medium under cold temperature with sufficient density ( Charnley Rodgers 2002 ) or photodissociation of N 2 by UV light from the proto-Sun or nearby stars ( Muskatel et al. 2011 ). Unlike nitrogen, the available carbon isotope ratios of comets, including the newly derived 12 C/ 13 C ratio of ER61, are consistent with the solar system abundance ratio of 90 ± 10 ( Wyckoff et al. 2000 ). The in situ measurements of the CO 2 isotopologs in the coma of 67P/Churyumov-Gerasimenko, taken by the Double Focusing Mass Spectrometer (DFMS) of the ROSINA, find that 12 C/ 13 C = 84 ± 4 ( Hässig et al. 2017 ). The DFMS measurements show a slight enrichment in 13 C in terms of the 12 C/ 13 C ratio compared to an average 12 C/ 13 C ratio of 91.0 ± 3.6 ( Manfroid et al. 2009 ; Bockelée-Morvan et al. 2015 ). However, given the large uncertainties of other ground-based observations, this difference seen in the DFMS measurements of 67P is not statistically significant. Our observations

O O O O O O O O O O O B-EntityOfFutureInterest O B-EntityOfFutureInterest I-EntityOfFutureInterest I-EntityOfFutureInterest B-EntityOfFutureInterest O O O O O O O O O B-EntityOfFutureInterest O O B-CelestialObject O B-EntityOfFutureInterest O O O O O B-Citation I-Citation I-Citation I-Citation O O O B-CelestialObject O O O O O O O O B-EntityOfFutureInterest O B-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest O B-EntityOfFutureInterest O O O O O O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O B-EntityOfFutureInterest O B-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O B-Citation I-Citation I-Citation O O O O O O O O B-CelestialObject O B-Citation I-Citation I-Citation I-Citation O O O B-Instrument O O O O O B-Mission O O O O B-EntityOfFutureInterest O B-EntityOfFutureInterest O O O O O O O O O O O O B-Citation I-Citation O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O B-Wavelength O O O B-EntityOfFutureInterest O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O B-CelestialObject I-CelestialObject O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O O O B-Instrument I-Instrument I-Instrument I-Instrument B-Instrument O O B-Instrument O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Citation I-Citation I-Citation I-Citation O O B-Instrument O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O B-Instrument O O B-CelestialObject O O O O O O


O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O B-CelestialObject O O O B-Survey I-Survey I-Survey I-Survey O O B-CelestialObject O O O O O O O O B-Instrument B-Citation I-Citation I-Citation I-Citation O B-Citation O O B-Software I-Software O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O B-Software I-Software I-Software I-Software O O O O O O B-Instrument O O O B-Wavelength O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software I-Software O B-Software I-Software I-Software O O O O B-Software I-Software O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation O O B-Telescope O O O O O O O O O O O O O O O O O O O O B-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Instrument O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software O O O O O O O O O O O O O O O O B-Instrument O O O B-Software O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation B-CelestialObject O O O O O O B-Software I-Software I-Software I-Software O O B-Software I-Software O O B-URL O O O O O O O O O O O B-Organization O O B-Instrument O O O B-URL O B-Formula O O O O O O O O O O O O O O O O O O O B-CelestialObject O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O B-Telescope O O O O O B-Mission O O O O O O O O B-Survey I-Survey I-Survey I-Survey B-Survey O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O B-CelestialObject B-Citation I-Citation I-Citation I-Citation O B-Survey I-Survey O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Mission O O O B-Mission O O O O


that discs are around half as common in binary systems closer than ∼20 au as they are around single stars or stars with wide stellar companions (Yelverton et al. 2019 ). There are competing theories for the formation of both giant planets and close binaries. The formation of close binaries may result from turbulent fragmentation of a molecular cloud, or may occur somewhat later via gravitational instability in a circumstellar disc (see e.g. Kratter 2011 for a review). Giant planets could either form via core accretion or, like close binaries, via the direct gravitational collapse of a circumstellar disc (see e.g. Mordasini et al. 2010 for a review). Thus, it is possible that giant planets and close binary companions form via essentially the same mechanism. If this were the case, then it may be expected that giant planets and close binaries have similar effects on debris discs, which is not what we observe. This could suggest that the two kinds of companion do not both form in the same way, though it could alternatively be the case that they do both form via gravitational instability but that the more massive stellar companions stir planetesimal discs more effectively and thus drive faster collisional decay. 4.4.2 Comparison with directly imaged planets Meshkat et al. ( 2017 ) found tentative evidence that giant planets at tens to hundreds of au – i.e. those probed by direct imaging – are more common in systems with detected debris discs than in those without, at the 88 per cent confidence level. While this result cannot be considered statistically significant, it is none the less suggestive that planets at separations comparable to typical debris discs have a stronger correlation with these discs than the much closer-in planets studied in this paper (recall from Table 6 that we found a difference at a lower confidence level of ∼70 per cent). The apparently contrasting results for planets on close and wide orbits could be understood in terms of a simple model in which the majority of planets form via core accretion regardless of separation, and in which RV planets like those in our study form within a few au (rather than migrating in from large semimajor axes). Within this model, detection of a planet through RV would suggest that planetesimals were able to form in the inner planetary system, while detection of a Kuiper belt analogue would indicate that planetesimals formed in the outer region of the system. However, it is not necessarily the case that the formation of planetesimals at close separations implies that the same should be possible at much larger separations (and vice versa), since, for example, the dust density in protoplanetary discs tends to decrease with stellocentric distance (e.g. Andrews et al. 2009 ), while ice particles that may aid planetesimal growth cannot exist within a few au of the star (e.g. Johansen et al. 2014 ). Additionally, the dynamical environment in the inner and outer systems may be different if undetected

O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation O O O O O O O O O B-Model I-Model O O O O O O O B-Model I-Model I-Model I-Model I-Model I-Model O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O


an ideal tool with which to calibrate other tracers of star formation. Finally, CO emission is the consequence of the interaction between CO molecules and molecular hydrogen (H 2 ) in regions where stars are formed – giant molecular clouds (GMCs). CO emission is correlated with the virial mass of GMCs observed in the Milky Way and in nearby spiral galaxies (Young Scoville 1991 ). Historically, the IR–RC–CO parameters have been studied using three empirical relationships, each one having different properties: first, the RC–IR correlation (e.g. Yun, Reddy Condon 2001 ; Ivison et al. 2010 ; Thomson et al. 2014 ; Magnelli et al. 2015 ; Dumas et al. 2011 ; Tabatabaei et al. 2013a , b ), valid up to z ∼ 2 and at a resolution of ≥ 500 pc. Second, the Schmidt–Kennicutt (SK) IR–CO law (Kennicutt 1998 ; Kennicutt Evans 2012 ), valid for bright galaxies with a high SFR as well as for low-surface-brightness galaxies with a low SFR. The form of the SK law is still debated because observations of disc and starburst galaxies have shown that this relationship can be fitted either by a single or by a bi-modal solution for the redshift range 0.05–2.5 (e.g. Genzel et al. 2010 ; Daddi et al. 2010 ; Ivison et al. 2011 ; Kennicutt Evans 2012 ; Tacconi et al. 2013 ; Santini et al. 2014 ; Freundlich et al. 2019 ). The slope of the SK law, which defines the depletion time of the gas in a galaxy (τ dep ≡ M gas /SFR), is central to our understanding of the mechanisms that govern star formation. According to models, a linear slope implies that star-formation activity is not driven only by the self-gravity of the galaxy (Semenov, Kravtsov Gnedin 2017 , 2019 ). Several studies dedicated to the spatially resolved KS relationship on subkiloparsec scales found a wide range in the value of the slope (indicated by the power-law index, N ∼ 0.6–3; ) of the KS relationship (Bigiel et al. 2008 ; Kennicutt Evans 2012 ; Rahman et al. 2011 ; Viaene et al. 2014 ; Casasola et al. 2015 ). The spread in the value of N may be intrinsic, suggesting that different star formation relationships exist; alternatively, it may result from the assumptions adopted in each study. Daddi et al. ( 2010 ) suggest that L IR (tracing SFR) and (tracing molecular gas mass) show no evolution in normal star-forming galaxies up to z = 1.5. However, Tacconi et al. ( 2018 ) suggest that τ dep ∝ (1 + z ) −0.62 , implying that τ dep decreases by a factor of ~2×, up to the redshift limit explored in our study, z = 1.5. Third, the RC–CO correlation (e.g. Adler et al. 1991 ; Murgia et al. 2002 , 2005 ; Leroy et al. 2005 ; Paladino et al. 2006 ; Schinnerer et al. 2013 ; Liu et al. 2015 ) is valid for a variety of galaxy types and also down to scales of a few hundred parsecs in some selected local galaxies. In this work, we derived equation ( 3 ) by combining RC–IR and IR–CO relationships from the literature to obtain the molecular gas mass density of the Universe, after deriving the integrated MGMF from measurements of

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O B-Citation I-Citation I-Citation O O O B-Wavelength O O O O O O O O O O O O O O O B-Model I-Model O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation O O O O B-Formula I-Formula I-Formula O O O O O O O O O O B-Model B-Model B-Model I-Model B-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O B-Model I-Model O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O B-Model I-Model B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O B-Wavelength O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O B-Model I-Model O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O B-Model I-Model O O O O O O O O O O O O O O O O B-Model I-Model O O O


them by 1000. For the labels, we normalize the total magnetic field strength by dividing it by 5000 and the inclination and azimuth angles by dividing them by π . The two numbers, 1000 and 5000, are used here because most of the Stokes measurements have values between −1000 and +1000, and their total magnetic field strengths range from −5000 to +5000 G. After obtaining the estimated vector magnetic field, which is inferred by our trained model, of a test data sample (pixel), we can derive the three Cartesian components of the magnetic field, namely, B x , B y , and B z , of the pixel as follows:where B total denotes the total magnetic field strength, φ is the inclination angle, and θ is the azimuth angle. 3. Methodology We use a CNN to infer vector magnetic fields from Stokes profiles of GST/NIRIS. Our CNN model helps in denoising inversions by exploiting the spatial information of the Stokes profiles. Figure 2 presents the architecture of our network. It contains an input layer, three convolutional blocks, two fully connected layers, and an output layer. The input layer receives a sequence of Stokes Q , U , and V components, each having 60 wavelength sampling points, with three channels. Each channel corresponds to a Stokes component. Figure 2. Architecture of our CNN. This network is comprised of an input layer, three convolutional blocks, two fully connected layers, and an output layer. The input of the CNN is a three-channel sequence of Stokes Q , U , and V components each having 60 wavelength sampling points. The intermediate outputs of the three convolutional blocks have 64, 128, and 256 channels, respectively. There are 1024 neurons activated by ReLUs in both of the two fully connected layers. The output layer has three neurons activated by the Tanh function, where each neuron produces a value in the range (−1, 1) representing the total magnetic field strength, inclination angle, and azimuth angle, respectively. After the input layer, there are three convolutional blocks with the following structures. The first convolutional block consists of two convolutional layers that take, as input, the output from the previous layer and filter it with 64 kernels of sizes 3 × 1 × 3 and 3 × 1 × 64, respectively, and a max-pooling layer with a pooling factor of 2. The second convolutional block consists of two convolutional layers with filters of 128 kernels of sizes 3 × 1 × 64 and 3 × 1 × 128, respectively, and a max-pooling layer with a pooling factor of 2. The third convolutional block consists of two convolutional layers with filters of 256 kernels of sizes 3 × 1 × 128 and 3 × 1 × 256, respectively. The third convolutional block does not contain a max-pooling layer. The activation functions used in both the convolutional and fully connected layers are rectified linear units (ReLUs; Goodfellow et al. 2016 ), defined asThe output of the three convolutional blocks is flattened into a sequence, which is then sent to the two fully connected layers

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O B-Telescope O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O


(Spergel et al. 2003 ). All the errors and upper limits are quoted at a 90% confidence level, unless otherwise specified. 2. Data and Catalogs The NuSTAR extragalactic survey program consists of tiered observations of well-known survey fields: (i) a set of deep tiled pointings covering the E-CDFS (Mullaney et al. 2015 ) and EGS (A. Del Moro et al. 2017, in preparation) fields, with an area of and arcmin 2 , respectively, and a total exposure of 1.49 Ms in each of the two fields, reaching a maximum exposure of ≈220 ks in E-CDFS and ≈280 ks in EGS (at 3–24 keV in each focal plane module (FPM), vignetting corrected); (ii) a medium-depth set of 121 tiled pointings covering ≈1.7 deg 2 of the COSMOS field (Civano et al. 2015 ) for a total exposure of 3.12 Ms, with a maximum depth of ≈90 ks in the central ∼1.2 deg 2 ; (iii) a serendipitous survey consisting of all serendipitous sources detected in any NuSTAR targeted observation (excluding all sources associated with the target; Alexander et al. 2013 ; Lansbury et al. 2017 ). This last tier of the survey spans a wide range of depths (see, e.g., Aird et al. 2015a ; Lansbury et al. 2017 ) and has the largest sky coverage, reaching deg 2 to date (still ongoing). Since in this work we require low-energy ( keV) Chandra data, together with the NuSTAR data, to produce broad X-ray band composite spectra, we limit our analyses to the sources detected in the E-CDFS, EGS, and COSMOS fields, which have good Chandra coverage and redshift completeness (spectroscopic and photometric), while we exclude the serendipitous survey sources owing to the heterogeneity of the available ancillary data. Although XMM-Newton observations are also available for the E-CDFS and COSMOS fields, we do not include these data in our analyses because combining the data from different X-ray instruments to produce the composite spectra can cause significant distortions in the resulting spectra (see Section 3.2.1 ). Our sample consists of 182 AGNs (see Table 1 ), 49 detected in E-CDFS (Mullaney et al. 2015 ), 42 detected in EGS (A. Del Moro et al. 2017, in preparation), and 91 detected in COSMOS (Civano et al. 2015 ). Of these sources, 64 (∼35%) are detected in the NuSTAR HB (8–24 keV). 25 25 Most of the sources that are formally HB undetected (∼65%) are actually detected also above 8 keV, but with lower significance than the false-probability threshold adopted by Civano et al. ( 2015 ), Mullaney et al. ( 2015 ), and A. Del Moro et al. (2017, in preparation). Table 1 NuSTAR Source Sample Summary Notes. a Number of HB (8–24 keV) detected sources. b Including both spectroscopic and photometric redshifts; the number of spectroscopic redshifts is reported in parentheses (also in the fifth and sixth columns). In Civano et al. ( 2015 ) and Mullaney et al. ( 2015 ), the NuSTAR sources have been matched to the Chandra and/or XMM-Newton point-source catalogs available in these fields (Lehmer et al. 2005 ; Brusa et al. 2010 ; Xue et al. 2011 ;

B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O B-Survey I-Survey I-Survey O O O O O O O O O O O O O O O O O O B-CelestialRegion B-Citation I-Citation I-Citation I-Citation O O B-CelestialRegion B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialRegion O O O O B-CelestialRegion O B-Wavelength I-Wavelength O O B-Instrument I-Instrument I-Instrument B-Instrument O O O O O O O O O O O O O O O O B-CelestialRegion I-CelestialRegion B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O B-Telescope O O O O B-Telescope O O O O B-Wavelength O O O O O O O O O O O O O B-CelestialRegion B-CelestialRegion O B-CelestialRegion I-CelestialRegion O O O B-Telescope O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope O O O O O O B-CelestialRegion O B-CelestialRegion I-CelestialRegion O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialRegion B-Citation I-Citation I-Citation I-Citation O O O O B-CelestialRegion B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O B-CelestialRegion B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O B-Telescope B-Wavelength B-Wavelength I-Wavelength O O O O O O O O O B-Wavelength O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O B-Telescope O O O O O O O B-Wavelength B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Telescope O O O O O O B-Telescope O B-Telescope B-Dataset I-Dataset O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O


Nonlinear regression is the best model, according to SIC. The break period varies between 2.35 log P 2.55 days at different wavelengths for all statistical methods. The F-test suggests that the variances of the quadratic and two-slope models are statistically similar. Further, the variances of a quadratic model and a linear relation for P 400 days are also similar in most cases. Therefore, we adopt a break period of 300 days () as the optimal period at which the change in the slope of PL and PLC relations occurs for O-rich Miras. The break period possibly shifts to longer periods (∼350 days, days) at NIR wavelengths. NIR PLRs are discussed in detail in Section 5 . Table 4 Results of the Statistical Tests to Find a Kink in the O-rich Mira PL and PLC Relations in the LMC Note. SIC: Schwarz Information Criterion used to estimate the break period in a two-slope model. RW: Random walk, TM: The Testimator, FT: F-test p ( f ): Probability of acceptance of the null hypothesis that the variance of a quadratic and a linear relation for P 400 days are equal. 4. Stability of Maximum Light for Miras 4.1. Photometric Variability at Max-light Optical photometry provides clear evidence that Miras display significantly smaller scatter in the PLRs at max-light than at mean-light. The pulsation properties and brightness variations of Miras are not strictly periodic and vary from cycle-to-cycle. Therefore, it is important to investigate the stability of max-light over different pulsation cycles. However, apart from OGLE I -band time-series observations, there is a lack of continuous photometry for Miras in the LMC, and even OGLE data do not necessarily cover the epochs of max-light during all pulsation cycles. High-precision photometry and subhour cadence from Kepler can be extremely useful to test the stability of max-light over different pulsation cycles. Bányai et al. ( 2013 ) analyzed variability in M giants from Kepler and identified a dozen long-period Mira-like variables. We estimate the scatter in the max-light magnitudes and median magnitudes for Miras over different pulsation cycles. For OGLE Miras, the sample is restricted to those stars for which GPR predicts max-light magnitudes with a precision better than three times their median uncertainty, and the time coverage is equivalent to the period of each Mira. The top panel of Figure 9 displays the comparison of the scatter in the median- and max-light magnitudes for OGLE and Kepler Mira candidates. When the scatter in both sets of magnitudes is smaller than 0.4 mag, max-light magnitudes seem to be more stable for most O-rich Miras. C-rich Miras often exhibit long-term trends and thus display large scatter both at median- and max-light. The bottom panel of Figure 9 presents the variation in min-light magnitudes versus the scatter in median magnitudes. For a majority of the stars, min-light magnitudes display significantly larger scatter than median magnitudes over multiple pulsation cycles. The

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Survey B-Wavelength I-Wavelength O O O O O O O O O O O O O B-CelestialObject O O B-Survey O O O O O O O O O O O O O O O O O O O B-Telescope O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O B-Telescope O O O O O O O O O O O O O O O O O O O O O O O O O B-Survey O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Survey O B-Telescope O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


analysis the Si q + ions are electrostatically deflected by 45°, then focused and collimated by a three-element einzel lens and a series of three apertures prior to entering the target gas cell. The ions undergo CE collisions with the target gas, lowering their charge states by (considered here) one and two exchanges. Currents for each CE step are measured in a retarding-potential analyzer placed at the end of the CE cell, followed by a deep Faraday beam trap. The target gas-cell pressure is monitored using a calibrated capacitance manometer (CM). Individual charge-state currents are measured through the application of a stepped retarding potential , the value of which depends on the incident charge-state q and ion energy through the relation = , where n is the number of exchanges. Thus, a series of retardations , , , will block the charge states q , q − 1, and q − 2, for a total of two exchanges, the maximum encountered in the present study. As in Machacek et al. ( 2014 ), the cross section σ q , q − n for the exchange of n electrons is given by Here, T is the target gas temperature, k is the Boltzmann constant, P is the gas pressure, L is the effective gas-cell length, and R q,q − n is the particles’ current ratio, given bywhere the currents , , correspond to measured current steps after application of the retarding potentials , , . Additional details on the 28 Si q + mass/charge spectrometry, retarding-potential stepping, measurement of background levels, presence of metastable ion states, tests for gas-cell pressure effects on CE cross sections, and data acquisition, together with sources and the levels of random and systematic errors, are given in Machacek et al. ( 2014 ). The He and H 2 used herein were ultrahigh purity-grade (99.999%) gases, and the 28 Si q + ions were produced by dissociative ionization of silane gas (SiH 4 , UHP-grade 99.999%) in the ECR plasma. 3. Results and Discussion A summary of the sources and magnitudes of the experimental errors is given in Table 1 . Shown in Figures 1 and 2 and listed in Table 2 are results of the present measurements of single and double total cross sections for 28 Si q + ions ( q = 7–10) colliding with He and H 2 at the indicated energies. Also given are the statistical error limits at the 2 σ level of confidence. In Table 2 note that the ratio of single-to-double CE cross sections for 28 Si q + + He is approximately 4.1 ± 1.2 (1 σ ) for all the charge states, with the exception of 28 Si 10+ at 2.25 keV/u, where the double-exchange cross section is small (0.104 × 10 −17 cm 2 ). The single-to-double ratio for 28 Si q + + H 2 shows larger variations, from 0.04 to 49.5. The reason for this is not understood. In order to provide some theoretical insight, calculations of the single-exchange CE cross sections were carried out in the static one-electron model (Mann et al. 1981 ). Here, the captured electrons mainly populate states with principal quantum number n that depend on the charge q of the

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-TextGarbage I-TextGarbage I-TextGarbage O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-TextGarbage I-TextGarbage O O O O O O O O O O O B-TextGarbage I-TextGarbage I-TextGarbage O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O


velocity, which was mostly determined by the NH 3 line observations (Table 1). Figure 5.50. 3775 The detected 22 GHz H 2 O maser spectra of the sources detected only at 22 GHz. Red and Black colors represent the first and second epochs, respectively. The source name is given in the top-left corner of the upper panel and “M” is attached for the source associated with 6.7 GHz CH 3 OH maser emission (see Section 3.1). The observing dates are shown in the top-right corner. The vertical dotted line indicates the systemic velocity, which was mostly determined by the NH 3 line observations (Table 1). Figure 5.51. 3816 The detected 22 GHz H 2 O maser spectra of the sources detected only at 22 GHz. Red and Black colors represent the first and second epochs, respectively. The source name is given in the top-left corner of the upper panel and “M” is attached for the source associated with 6.7 GHz CH 3 OH maser emission (see Section 3.1). The observing dates are shown in the top-right corner. The vertical dotted line indicates the systemic velocity, which was mostly determined by the NH 3 line observations (Table 1). Figure 5.52. 3866 The detected 22 GHz H 2 O maser spectra of the sources detected only at 22 GHz. Red and Black colors represent the first and second epochs, respectively. The source name is given in the top-left corner of the upper panel and “M” is attached for the source associated with 6.7 GHz CH 3 OH maser emission (see Section 3.1). The observing dates are shown in the top-right corner. The vertical dotted line indicates the systemic velocity, which was mostly determined by the NH 3 line observations (Table 1). Figure 5.53. 3878 The detected 22 GHz H 2 O maser spectra of the sources detected only at 22 GHz. Red and Black colors represent the first and second epochs, respectively. The source name is given in the top-left corner of the upper panel and “M” is attached for the source associated with 6.7 GHz CH 3 OH maser emission (see Section 3.1). The observing dates are shown in the top-right corner. The vertical dotted line indicates the systemic velocity, which was mostly determined by the NH 3 line observations (Table 1). Figure 5.54. 3880 The detected 22 GHz H 2 O maser spectra of the sources detected only at 22 GHz. Red and Black colors represent the first and second epochs, respectively. The source name is given in the top-left corner of the upper panel and “M” is attached for the source associated with 6.7 GHz CH 3 OH maser emission (see Section 3.1). The observing dates are shown in the top-right corner. The vertical dotted line indicates the systemic velocity, which was mostly determined by the NH 3 line observations (Table 1). Figure 5.55. 3914 The detected 22 GHz H 2 O maser spectra of the sources detected only at 22 GHz. Red and Black colors represent the first and second epochs, respectively. The source name is given in the top-left corner of the upper panel and “M” is attached for the source associated with 6.7 GHz CH 3 OH

O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O


can form a favourable overlap elsewhere on the backbone than at C2. Some electronic redistribution is required so it may not be as favourable for H to attach to C3 or N as it is at C2. This can account for the lower stability of the ketenimine product and the observed ∼9:1 ratio of the two products shown in Table 1 . We explored these two pathways in the related but simpler system H 2 CCN + H. Replacing the methyl group of CH 3 CHCN with H should not greatly affect the energetics of H-addition. Like CH 3 CHCN, the H 2 CCN radical shows resonance behaviour between two forms, as indicated by radical character on all three heavy atoms. For H 2 C1-C2-N, the Mulliken spin populations at the UMP2/aug-cc-pVTZ level on C1, C2, and N are 1.09, −0.64. and 0.65, respectively, which are quite similar to the spin densities on the analogous atoms in CH 3 -CH-CN above (there are also minor amounts of spin density on the two H atoms). We might therefore expect to form both CH 3 CN (methyl cyanide) and H 2 CCNH (ketenimine) by adding H to C1 and N of H 2 CCN, respectively. To test the likelihood of forming both products, we ran additional calculations. These calculations required a multireference treatment since closed-shell singlet molecules are being formed from open-shell doublet reactants. At the MRCI + Q/aug-cc-pVTZ level, we found that neither addition involves a reaction barrier. The issue thus becomes trying to understand why there is a preference for the cyanide product over the ketenimine product. With no barriers for either addition, the favourability of one product over the other must be due to the specific character of the full reaction surface. In addition of H to CH 3 CHCN, ethyl cyanide is more stable than the ketenimine product, and the same is true when adding H to H 2 CCN: methyl cyanide is more stable than ketenimine. The first channel has a ‘wider’ entrance on the potential energy surface; it is deeper and extends further out from C1. This is likely what favours cyanide formation over ketenimine formation in these reaction systems. To gain further insight into the ratio of the two products, we note that the study of delocalized radicals containing nitrogen atoms has been theoretically investigated in the case of the simplest of the cyanoalkyl radicals, CH 2 CN. Delbecq ( 1983 ) has shown using valence-bond analysis that cyanomethyl radical CH 2 CN has two different limit structures, C°H 2 CN and CH 2 = C = N° with relative weights of 83 per cent and 14 per cent, respectively. The H 2 CN π-delocalized structure was found with a low weight of the order of 3 per cent. Similarly, under our experimental conditions, the formation of CH 3 − C°H − C≡N through CH 2 = CH − C≡N + H reaction, would lead as shown in Fig. 8 , to the formation of a second localized form with ketenimine character CH 3 − CH = C = N° that explains the possibility of reaction at the nitrogen end to form CH 3 CH = C = NH. Figure 8. Two different limit structures and one

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O


of the relatively isolated star formation. External compression of the core (e.g., Frau et al. 2015 for the possibility of the cloud–cloud collision in the Pipe Nebula) may also initiate the star formation in the core. Rotation of the core does not affect its stability. The rotational kinetic energy of the core is only approximately 1% of the gravitational energy (Aguti et al. 2007 ). Uncertainty arising from the use of the plane-of-sky field component increases the importance of magnetic support. The magnetic field close to a pole-on geometry leads to the condition (subcritical). Therefore, 3D modeling of the polarization structure is desirable in order to determine the line-of-sight inclination angle of the core’s magnetic axis. The formation of a nearly critical core accompanied by a distorted magnetic field is an open problem. Consideration of the interplay among gravity, thermal pressure, and turbulence is necessary in order to understand the formation and evolution mechanism of FeSt 1–457. Quasi-static contraction under the support of a magnetic field (Shu 1977 ) requires a strong magnetic field, and this is not the case for FeSt 1–457 (, magnetically supercritical). In the other extreme, supersonic turbulence can produce cores that collapse dynamically and that are accompanied by highly supersonic infalling motion (e.g., Mac Low Klessen 2004 ). This is also not the case for FeSt 1–457 because nearly critical stability was shown in the present study and radio observations (Kandori et al. 2005 ; Aguti et al. 2007 ) showed quiescent kinematic gas motion in the core. Core formation between the two extreme models may account for the formation of FeSt 1–457 (e.g., Nakamura Li 2005 ; Basu et al. 2009a , 2009b ). Further theoretical studies on such a case are desirable. The relative importance of the magnetic field for the support of the core is investigated through the ratios of the thermal and turbulent energy to the magnetic energy, which are expressed as and , where , , and denote the isothermal sound speed at 9.5 K, the turbulent velocity dispersion, and the Alfvén velocity, respectively. The numerical coefficients differ in the equation depending on whether the velocity dispersion is in 1D or 3D. These ratios were found to be and . FeSt 1–457 is dominated by a static magnetic field and thermal support with a small contribution from turbulence. Turbulence could be an important parameter in the formation and development of the core, although it seems to have already mostly dissipated at the observed stage in the prestellar evolution. Therefore, a static magnetic field, the predominant component of anisotropic pressure in the core, should act as a key parameter controlling the star formation process in FeSt 1–457. Figure 8 shows the relationships among the orientations (position angles) of the axis of the elongation of the core (), the rotation axis (, Aguti et al. 2007 ), and the magnetic field axis (). Note that the elongation of the

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of SEGUE dwarfs. They measured, for the disc as a whole, a vertical metallicity gradient of dex kpc −1 , which is in agreement with our measurement. However, their low-α population has a gradient consistent with zero: d[M/H]/d dex kpc −1 . However, their intermediate α sub-population with 0.2 [α/M] 0.3 has d[M/H]/d dex kpc −1 , which agrees with our low-α metallicity gradient. We thus conclude that the discrepancy between our result and that of Schlesinger et al. ( 2014 ) is largely due to the chemical separation criteria (also see Ciucă et al. 2018 , who found an age-dependence for the thin disc vertically metallicity gradient, such that the youngest population has a flatter gradient). Mikolaitis et al. ( 2014 ) measured a slightly shallower gradient of d[M/H]/d z = −0.11 ± 0.01 for Gaia -ESO dwarfs and giants. The Gaia -ESO sample is more metal-poor overall, and Mikolaitis et al. ( 2014 ) separated thick disc stars by the location of underdensities in their [Mg i /M] histograms (their fig. 3). The dividing line is at different values of [Mg i /M] for different metallicity regimes. In fig. 10 of Mikolaitis et al. ( 2014 ), it is clear that their sample is biased against metal-rich stars, such that there are very few stars with [M/H] > 0 (also see Stonkutė et al. 2016 ). 5.2.2 The high-α population The vertical metallicity distribution of the thick-disc (high-α) stars is relatively flat compared to the low-α population, at −0.058 ± 0.003 dex kpc −1 . Several authors have measured the vertical gradient for the thick disc, using different methods to define this population. Earlier studies, such as Gilmore et al. ( 1995 ) and Prieto et al. ( 2006 ), reported no vertical metallicity gradient in the thick disc (Prieto et al. 2006 quoted an upper limit of d[M/H]/d z = 0.03 dex kpc −1 ). More recently, Boeche et al. ( 2014 ) concluded that the vertical metallicity gradient of the thick disc is consistent with zero, based on a sample of RAVE giants. However, other studies, using a combination of metallicity or kinematics to separate the thick disc, have reported a shallow metallicity gradient. Katz et al. ( 2011 ) observed sub-giants at two lines of sight: ( l, b ) = (51°, 80°) and (5°, 46°) at low resolution. Their MDFs show signs of bimodality, and the thick disc was defined as stars centred around [M/H] ≈ −0.5 dex. The vertical gradient measured by Katz et al. ( 2011 ) is −0.068 ± 0.009 dex kpc −1 , consistent with our value. Ruchti et al. ( 2011 ) observed a number of metal-poor thick disc candidates at high resolution using the MIKE, FEROS and UCLES spectrographs (λ/Δλ ≈ 35 000–45 000). They classified their stars based on a Monte Carlo simulation of space motion U, V, W , assuming Gaussian errors on the velocities and distances. By further restricting their α-enhanced sample with thick disc kinematics to metal-poor stars only ([M/H] ≤ −1.2), they avoid most thin disc contamination. The measured gradient is −0.09 ± 0.05 dex kpc −1 ,

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DR2 is designed to observe stars in the Milky Way, it does not always provide accurate measurements of sources that are extended or in very crowded regions. We urge caution and follow-up observations of the objects in Table 5 , especially those within 10′ of the center of NGC 5128. B.2. Conflicting Measurements Thirty-two of the identified GCs in NGC 5128 have conflicting classifications in the literature. A majority of these conflicting classifications occur because the target was measured to have a radial velocity within the standard acceptable range of ≈250–1000 km s −1 by one group but outside this range by another. Additionally, two objects were identified as GCs based on their radial velocities but later found to be stars based on HST/ACS imaging (see note 9 in Table 6 ). Another object was subsequently identified as a star because of its spectrum (see note 5 in Table 6 ). These objects with conflicting measurements have been matched between studies based on position and, in many cases, by their ID as well. When examining these objects with PISCeS data, several of them can be visually identified as background galaxies. We believe seven of these objects are indeed true GCs because a majority of their measurements classify them as such and they do not visually look like galaxies; thus, they are included in Table 4 . For each object with conflicting measurements, Table 6 first lists the R.A. and decl. in degrees (J2000). The position measurements are from the PISCeS data set to provide a uniform astrometric system. Next, we list the conflicting radial velocities and errors (kilometers per second) along with their sources, if applicable. We also include our classification of each object, as well as further information indicated by the “Notes” column. Many sources experience a discrepancy between Woodley et al. ( 2007 ) and Beasley et al. ( 2008 ). It appears that between sharing data with the Woodley group and publishing their own paper, the Beasley group updated measurements on several targets. We note that one of these objects has a mismatch in both its velocity measurement and coordinates. Woodley et al. ( 2007 ) gave an incorrect position for HH-048 as α = 20068900, δ = −43119111, with a radial velocity of 856 ± 56 km s −1 , citing Beasley et al. ( 2008 ) as the reference for this radial velocity measurement. Beasley et al. ( 2008 ), however, listed HH-048 as a background galaxy ( v r = 16,000 km s −1 ) with the same R.A. but a decl. of δ = −43011917 (note the single difference in the digit following the decimal point). Based upon visual inspection, the position listed in the Beasley catalog is correct and listed in Table 6 . Appendix C An Object with Too Many Names During our literature review, we found an instance where two confirmed GCs with different names shared the same position. The objects identified most recently as GC 0462/T17-1560 and GC 0200/T17-1561 have the same position in the Sinnott et al. ( 2010 ) catalog and positions

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