Bayesian forward modelling of cosmic shear data

Natalia Porqueres, Alan Heavens, Daniel Mortlock, Guilhem Lavaux

► To cite this version:

Natalia Porqueres, Alan Heavens, Daniel Mortlock, Guilhem Lavaux. Bayesian forward modelling of cosmic shear data. Monthly Notices of the Royal Astronomical Society, 2021, 502 (2), pp.3035-3044. 10.1093/mnras/stab204. hal-03047660

HAL Id: hal-03047660

https://hal.science/hal-03047660

Submitted on 16 Aug 2022

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

# Bayesian forward modelling of cosmic shear data

Natalia Porqueres<sup>1,★</sup> Alan Heavens,<sup>1</sup> Daniel Mortlock<sup>1,2,3</sup> and Guilhem Lavaux<sup>4</sup>

Accepted 2021 January 21. Received 2021 January 20; in original form 2020 November 13

## ABSTRACT

We present a Bayesian hierarchical modelling approach to infer the cosmic matter density field, and the lensing and the matter power spectra, from cosmic shear data. This method uses a physical model of cosmic structure formation to infer physically plausible cosmic structures, which accounts for the non-Gaussian features of the gravitationally evolved matter distribution and light-cone effects. We test and validate our framework with realistic simulated shear data, demonstrating that the method recovers the unbiased matter distribution and the correct lensing and matter power spectrum. While the cosmology is fixed in this test, and the method employs a prior power spectrum, we demonstrate that the lensing results are sensitive to the true power spectrum when this differs from the prior. In this case, the density field samples are generated with a power spectrum that deviates from the prior, and the method recovers the true lensing power spectrum. The method also recovers the matter power spectrum across the sky, but as currently implemented, it cannot determine the radial power since isotropy is not imposed. In summary, our method provides physically plausible inference of the dark matter distribution from cosmic shear data, allowing us to extract information beyond the two-point statistics and exploiting the full information content of the cosmological fields.

**Key words:** gravitational lensing: weak—methods: data analysis—cosmology: large-scale structure of Universe.

## 1 INTRODUCTION

As light from distant galaxies propagates through the Universe, it is deflected by the gravitational field induced by the large-scale structures. This deflection results in a coherent distortion of observed galaxy images, inducing small changes in the ellipticity of observed galaxies, which is known as cosmic shear. The weak gravitational lensing effect is sensitive to the geometry of the Universe and the growth of cosmic structures, making it a powerful probe to study the matter distribution and the nature of dark matter and dark energy (see e.g. Kilbinger 2015, for a review).

The next-generation surveys like *Euclid* (Euclid Collaboration 2020), *Roman Space Telescope* (Spergel et al. 2015), and the Rubin Observatory (LSST Science Collaboration 2009) will provide unprecedented precision in cosmic shear measurements, performing wide-field cosmic shear surveys and measuring large and small scales. Harvesting the information from these datasets will present a challenge. Many of the current cosmic shear analyses focus on extracting information from the correlation function or the associated power spectrum (Kitching, Heavens & Miller 2011; Heymans et al. 2013; Kitching et al. 2014; Kitching, Heavens & Das 2015; Alsing et al. 2016; Kitching et al. 2016; Hildebrandt et al. 2017; Troxel et al. 2018; Hikage et al. 2019; Taylor et al. 2019). These analyses capture the two-point statistics, but they do not fully capture the non-Gaussian information encoded in the filamentary features of

the matter distribution (Bernardeau, van Wærbeke & Mellier 1997; Jain & Seljak 1997; van Wærbeke, Bernardeau & Mellier 1999; Schneider & Lombardi 2003; Takada & Jain 2003; Vafaee et al. 2010; Kayo, Takada & Jain 2013). While some approaches to access the non-Gaussian information are based on measuring high-order correlations (Bernardeau, van Wærbeke & Mellier 2003; Pen et al. 2003; Jarvis, Bernstein & Jain 2004; Semboloni et al. 2011; van Wærbeke et al. 2013; Fu et al. 2014), peak counts (Jain & van Wærbeke 2000; Dietrich & Hartlap 2010; Maturi, Fedeli & Moscardini 2011; Marian et al. 2012; Pires, Leonard & Starck 2012; Cardone et al. 2013; Lin & Kilbinger 2015; Liu et al. 2015b, a; Kacprzak et al. 2016; Petri et al. 2013; Peel et al. 2017) or using machine learning (Gupta et al. 2018), they rely on summary statistics that do not capture all the information and whose distributions are not well known.

Capturing the full information content of the large-scale structure requires a field-based approach to infer the matter distribution from observations. Böhm et al. (2017) presented a maximum likelihood estimator to reconstruct the matter density field from cosmic shear data, assuming a lognormal distribution for the density. The log-normal distribution reproduces the one- and two-point statistics but fails to reproduce higher-order statistics. Alsing et al. (2016), Alsing, Heavens & Jaffe (2017) presented a Bayesian hierarchical inference scheme to jointly infer shear maps and the corresponding power spectra, assuming Gaussian statistics of the shear field. From a Bayesian perspective, assuming a Gaussian distribution for the shear field is a well motivated prior since it constitutes the maximum

<footnote n="1"><sup>1</sup>Imperial Centre for Inference and Cosmology (ICIC) & Astrophysics group, Imperial College, Blackett Laboratory, Prince Consort Road, London SW7 2AZ, UK</footnote>

<footnote n="2"><sup>2</sup>Department of Mathematics, Imperial College London, London, SW7 2AZ, UK</footnote>

<footnote n="3"><sup>3</sup>Department of Astronomy, Stockholm University, Albanova, SE-10691 Stockholm, Sweden</footnote>

<footnote n="4"><sup>4</sup>CNRS & Sorbonne Université, UMR7095, Institut d'Astrophysique de Paris, F-75014, Paris, France</footnote>

<footnote>*E-mail: n.porqueres@imperial.ac.uk</footnote>
entropy prior once the mean and covariance are specified. However, more information coming from physics is available, and the Gaussian assumption is suboptimal. In this work, we address this limitation by including a gravity model in the Bayesian hierarchical model. For this, we build on the Bayesian Origin Reconstruction from Galaxies (BORG, Jasche & Kitaura 2010a; Jasche & Wandelt 2013a; Lavaux, Jasche & Leclercq 2019) framework, which employs a physical description of the dark matter dynamics and allows us to sample from the initial conditions, which are accurately described by Gaussian statistics. With this more complex data model, we get a better representation of the data, and we can extract information beyond the two-point statistics, exploiting the full information content of the shear fields.

One of the main challenges in the analysis of cosmic shear based on estimating the power spectrum is accounting for the masked regions within the survey area (see, e.g. Chon et al. 2004; Brown, Castro & Taylor 2005; Smith 2006). Our forward modelling approach circumvents these difficulties associated with the survey mask. Although the data do not provide information about the fields in the masked regions, the dynamical model still provides probabilistic information about the shear and density fields that are physically possible in those regions. In our method, the masked regions are treated as pixels with infinite noise, circumventing the need to treat unobserved areas as being cut from the analysis.

The paper is organised as follows. Section 2 describes the data model for the cosmic shear and the likelihood. Section 3 gives an overview of the Bayesian inference framework, BORG, as required for this work. In Section 4, we described the simulated data employed in testing and validating the method. The results are presented in Section 5, showing that the method provides unbiased matter density fields. In Section 7, we discuss the effect of the prior power spectrum in the results. Finally, Section 8 summarises the results.

## 2 THE DATA MODEL

The effect of weak gravitational lensing on a source can be described by two sky fields: the spin-2 shear, $\gamma$, which describes the distortion in the shape of the image, and the scalar convergence field, $\kappa$, which describes the change in angular size. These two fields are related to the lensing potential, $\phi$, by

$$ \kappa = \frac{1}{2} \partial \bar{\partial} \phi, \quad (1) $$

$$ \gamma = \gamma_1 + i\gamma_2 = \frac{1}{2} \partial \bar{\partial} \phi, \quad (2) $$

(see e.g. Kilbinger 2015), where $\gamma_1$ and $\gamma_2$ are the components of the shear distortion parallel and at $\pi/4$ to the coordinate axes, and $\partial = \partial_x + i\partial_y$ is the complex derivative on the sky, assuming the flat-sky approximation.

To connect the shear fields to the 3D dark matter distribution, we implemented a line-of-sight integration using the Born approximation, integrating along unperturbed paths. First, we generate convergence fields by integrating along the line-of-sight with the corresponding lensing weights as

$$ \kappa(\theta) = \frac{3H_0^2\Omega_m}{2c^2} \int_0^{r_{\text{lim}}} \frac{rdr}{a(r)} q(r) \delta^f(r\theta, r), \quad (3) $$

where $\theta$ is the position on the sky, $r$ is the comoving distance, $r_{\text{lim}}$ is the limiting comoving distance, $\delta^f$ is the final density field and

$$ q(r) = \int_{r}^{r_{\text{lim}}} dr' n(r') \frac{r' - r}{r'}, \quad (4) $$

where $n(r)$ is the source galaxy distribution. In our discrete implementation, this becomes

$$ \kappa_{mn}^{b} = \frac{3H_0^2\Omega_m}{2c^2} \sum_{j=0}^{N_2} \delta_{mnj}^{f} \left[ \sum_{s=j}^{N_2} \frac{(r_s - r_j)}{r_s} n^b(r_s) \Delta r_s \right] \frac{r_j \Delta r_j}{a_j}. \quad (5) $$

The index $b$ labels the tomographic bin and the subindices $mn$ indicate the 2D pixel on the sky, whose size is chosen to include typically many sources. The sum index $j$ indicates the slice in the radial direction, at a comoving distance $r_j$. $N_2$ is the total number of voxels along the radial axis. The voxels have a length of $\Delta r_j$. $\delta^f$ is the 3D dark matter overdensity at a scale factor $a$. The comoving radial distance $r_s$ indicates the distance to the source plane. The redshift distribution of sources for each tomographic bin is given by $n^b(z_s)$. In this initial proof-of-concept work, we focus on testing the inference and investigating the extent to which the 3D density field, and the 3D matter power spectrum, can be inferred from 2D shear maps. For these tests, we used a simplified scenario, assuming flat-sky and distant observer approximations. In future work, we will drop these approximations and consider the projection effects.

In the flat-sky approximation, we can obtain the shear values from the convergence field. On a flat-sky, the shear and the convergence are related in Fourier space. We, therefore, use a discrete Fourier transform (DFT) to obtain the shear values as

$$ \gamma_{mn}^{b} = \text{DFT}^{-1} \left[ \frac{(l_x + i l_y)^2}{l_x^2 + l_y^2} \text{DFT} \left( \kappa_{mn}^{b} \right) \right], \quad (6) $$

where $\bar{l} = (l_x, l_y)$ is the wave-vector written as a complex quantity. Since the convergence is also computed as part of the hierarchical model, this method has the advantage that it can analyse reduced shear,

$$ g = \frac{\gamma}{1 - \kappa}, \quad (7) $$

which, rather than the shear alone, controls the shape distortion.

To analyse cosmic shear observations in our Bayesian framework, we now built a likelihood based on this data model. We assume Gaussian pixel noise for the shear, corresponding to a negative log-likelihood, $\mathcal{L} = -\log P(\hat{\gamma} | \delta^f)$, that can then be written as

$$ \mathcal{L} = \sum_b \sum_{mn} \frac{[\hat{\gamma}_{1,mn}^b - \gamma_{1,mn}^b(\delta^f)]^2 + [\hat{\gamma}_{2,mn}^b - \gamma_{2,mn}^b(\delta^f)]^2}{2\sigma_b^2}, \quad (8) $$

where $\hat{\gamma} = \hat{\gamma}_1 + i\hat{\gamma}_2$ is the observed data. This is an estimate of the shear in the pixel, with a variance $\sigma_b^2$, which is determined from the shape noise and number of sources per pixel as $\sigma_c^2/N_{\text{sources}}$. We note that even if the ellipticity distribution is not Gaussian, provided many sources contribute to each pixel average the noise will become Gaussian according to the central limit theorem. An alternative would be to sample from the distribution in another level of the hierarchy, but this would be expensive, so we simplify this stage by using summary statistics of estimated shear and their variance.

This likelihood is then implemented into the large-scale structure sampler of the BORG framework. The corresponding physical forward modelling approach is illustrated in Fig. 1 and proceeds as follows. Using realisations of the three-dimensional field of primordial fluctuations, the dynamical structure formation model evaluates nonlinear realisations of the dark matter distribution, accounting for the light-cone effects inherent to deep observations. Using these 3D dark matter field realisations and the data model, BORG predicts shear fields that are compared to the observed data via the likelihood in equation (8).
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Hierarchical representation of the BORG inference framework for the analysis of cosmic shear data. Primordial fluctuations $\delta_{\text{ic}}$ encoded in a set of Fourier modes at $z \approx 1000$ are obtained from the prior $P(\delta_{\text{ic}}|\Omega)$, where $\Omega$ represents the cosmological parameters. These initial conditions are evolved using the gravity model $\mathcal{M}(\delta^{\text{ic}})$, which provides the evolved density $\delta_f$. The evolved density and the redshift distribution of sources $n^b(z)$ are then used to compute the convergence field for each tomographic bin $b$, $\kappa^b(\delta^l, n^b(z))$. From the convergence, we compute the cosmic shear $\gamma_1^b, \gamma_2^b$ in the flat-sky approximation. $\hat{\gamma}_1^b, \hat{\gamma}_2^b$ indicate the observational data. Purple boxes indicate deterministic transition while green boxes are probability distributions.</figcaption></figure>

## 3 THE BORG FRAMEWORK

This work extends the previously developed BORG algorithm to analyse the spatial matter distribution underlying cosmic shear observations. In this section, we provide a summary of the algorithm. A more detailed description of the BORG framework can be found in Jasche & Wandelt (2013a), Jasche, Leclercq & Wandelt (2015), Lavaux & Jasche (2016), Jasche & Lavaux (2019), Lavaux et al. (2019).

The BORG framework is a Bayesian inference method aiming at inferring the nonlinear spatial dark matter distribution and its dynamics from cosmological datasets. The underlying idea is to fit full dynamical gravitational and structure formation models to observations. By using nonlinear structure growth models, the BORG algorithm can exploit the full statistical power of high-order statistics of the matter distribution imprinted by gravitational clustering. This dynamical model links the primordial density fluctuations to the present large-scale structures. Therefore, the forward modelling approach allows the translation of the problem of inferring nonlinear matter density fields into the inference of the spatial distribution of the primordial density fluctuations, which are well described by Gaussian statistics (Planck Collaboration 2019). The BORG algorithm, therefore, infers the initial matter fluctuations, the dark matter distribution, and its dynamical properties from observations.

Motivated by inflation theory and observational data, the BORG algorithm employs a Gaussian prior for the initial density contrast at an initial cosmic scale factor of $a \simeq 10^{-3}$, time for which density perturbations are linearly growing. Initial and evolved density fields are linked by deterministic gravitational evolution mediated by various physics models of structure growth. Specifically, BORG incorporates several physical models based on Lagrangian Perturbation Theory

(LPT), fully nonlinear particle-mesh models (Jasche & Lavaux 2019), a model based on spatial COMoving Lagrangian Acceleration framework (Leclercq et al. 2020), and a semiclassical analogue to LPT (Porqueres et al. 2020). Any of these dynamical models can be straightforwardly employed within the flexible block sampling illustrated in Fig. 1. To test the inference method, in this work, we used LPT to approximately describe the gravitational clustering. However, in a future application of the method to real datasets, we will use the fully nonlinear particle-mesh (Jasche & Lavaux 2019) to have a better description of the matter density at small spatial scales, which undergo nonlinear dynamics. Though the particle mesh will be more costly, it will still be tractable. Tassev, Zaldarriaga & Eisenstein (2013) showed that the LPT begins to show significant deviations at $k > 0.2$ h/Mpc, but using the tCOLA modification of the equation of motion we can push the precision of an LPT-like simulation close to a full N-body simulation in a few time-steps, at the field level. Typically, it can be reached in at least as little as ten time-steps to reach 90 per cent correlation at $k = 1h\text{Mpc}^{-1}$ with a full N-body simulation such as one provided by Gadget-2 (Springel 2005).

At its core, the BORG framework employs MCMC techniques. This method allows inference of the full posterior distribution from which we can quantify the uncertainties in our results. However, the inference of the density field typically involves $\mathcal{O}(10^7)$ free parameters, corresponding to the discretised volume elements of the initial conditions. To explore this high-dimensional parameter-space efficiently, the BORG framework uses a Hamiltonian Monte Carlo (HMC) method, which exploits the information in the gradients and adapts to the geometry of the problem. We need, therefore, the adjoint gradient of the data model, which transforms the error vector from the likelihood space to the initial conditions. For the case of weak lensing, we derive this gradient in Appendix A. More details about the HMC and its implementation are described in Jasche & Kitaura (2010b) and Jasche & Wandelt (2013b).

## 4 THE MOCK DATA

To test the inference framework, we generated mock observations of cosmic shear with 30 sources per arcmin$^2$ as expected for the Euclid survey, with four tomographic bins and a nontrivial survey mask. In this section, we describe the properties of the synthetic data.

Mock data are constructed by first generating Gaussian initial conditions on a Cartesian grid of size $1h^{-1}$ Gpc $\times 1h^{-1}$ Gpc $\times 4h^{-1}$ Gpc with 128 $\times$ 128 $\times$ 256 voxels. To generate primordial Gaussian density fluctuations we used a cosmological matter power spectrum including the baryonic wiggles calculated according to the prescription provided by Eisenstein & Hu (1998, 1999). We further assumed a standard $\Lambda$CDM cosmology with the following set of parameters: $\Omega_m = 0.31$, $\Omega_\Lambda = 0.69$, $\Omega_b = 0.049$, $h = 0.6711$, $\sigma_8 = 0.8$, $n_s = 0.9624$. Here $H_0 = 100h\text{km s}^{-1}\text{Mpc}^{-1}$.

To generate realisations of the nonlinear density field, we evolve the Gaussian primordial fluctuations via LPT. This involves simulating displacements for $256^2 \times 512$ particles in the LPT simulation, accounting for light-cone effects inherent to deep observations. Final density fields are constructed by estimating densities via the cloud-in-cell scheme from simulated particles on the Cartesian grid. A cosmic shear field is generated by applying the data model described in Section 2, assuming the redshift distributions for tomographic bins shown in Fig. 2. Finally, we added Gaussian pixel-noise to the shear with variance corresponding to 30 sources per arcmin$^2$, as expected to be obtained from the Euclid survey (Laureijs et al. 2011), and with an error on intrinsic ellipticity given by $\sigma_\epsilon = 0.3$ (Kilbinger 2015). The total of 30 sources per arcmin$^2$ is then equally distributed
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Redshift distributions of sources for the four tomographic bins considered in this analysis.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 3.** Mask used to generate the mock data. The masked regions, with no contributing sources, are indicated in black. There are two different scales, corresponding to unobserved regions and bright stars.</figcaption></figure>

between the bins, corresponding to 7.5 galaxies per arcmin² for each tomographic bin. This corresponds to a signal-to-noise ratio of S/N = 0.5. We added a nontrivial survey mask, shown in Fig. 3. Since the data provides no direct information in the masked regions, these are treated as pixels with infinite noise.

# 5 RESULTS

Here, we present the results of applying our algorithm to simulated cosmic shear data. We show that our method infers unbiased density fields and corresponding power spectra at all scales considered in this work. We also perform a posterior predictive test for the shear, showing that the inferred densities can explain the data within the noise uncertainty.

<figure><img src="image_6.png" /><figcaption>**Figure 4.** Burn-in of the posterior initial matter power spectra. The colour scale shows the evolution of the matter power spectrum with the number of samples. The dashed lines indicate the underlying power spectrum and the 1- and 2-σ cosmic variance limits. The Markov chain is initialised with a Gaussian initial density field scaled by a factor 10⁻³ and the amplitudes of the power spectrum systematically drift towards the fiducial values, recovering the true matter power spectrum at the end of the warm-up phase.</figcaption></figure>

## 5.1 The warm-up phase of the sampler

In this first Bayesian approach, we keep the cosmology fixed, and specify a prior on the initial power spectrum. However, the power spectrum of the inferred matter distribution is conditioned by the data, and we can use the posterior $P(k)$ as a diagnostic for the effectiveness of the inference since the power spectrum of the posterior samples may differ from the prior. To monitor the initial warm-up phase of the Markov sampler, we follow a similar approach to our previous works (Jasche & Wandelt 2013a; Jasche & Lavaux 2017; Ramanah et al. 2019; Jasche & Lavaux 2019; Porqueres et al. 2019a, b): we initialised the Markov chain with an overdispersed state and traced the systematic drift of inferred quantities towards their preferred regions in the parameter space. Specifically, we initialised the Markov chain with a random Gaussian initial density field scaled by a factor 10⁻³ and monitored the drift of corresponding posterior power spectra during the warm-up phase. Fig. 4 presents the results of this exercise, showing successive measurements of the posterior power spectrum during the initial warm-up phase. The amplitudes of the posterior power spectrum show a systematic drift towards their fiducial values. By the end of the warm-up phase, the sampler has found an unbiased representation of the initial power spectrum at all Fourier modes considered in this work. Starting the sampler from an over-dispersed state, therefore, provides us with an important diagnostics to test the validity of the sampling algorithm.

## 5.2 Inferred density fields

As discussed above, our method uses a forward modelling approach, fitting a physical dynamical model to shear data and employs an
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Projections of the ground truth initial (left-hand upper panel), final density field (left-hand lower panel), inferred ensemble mean initial (middle upper panel) and ensemble mean final (middle-lower panel) density field computed from 500 MCMC samples. Since the information on the radial direction is not very informative, the density fields are projected on the sky, and the different slices of the 3D density field are weighted with the distribution of sources. Comparison between these panels shows that the method recovers the structure of the true projected density field with high accuracy. Right-hand panels show standard deviations of inferred amplitudes of the initial (upper right-hand panel) and final density fields (lower right-hand panel). The regions of high uncertainty correspond to the masked regions, where there are no contributing sources.</figcaption></figure>

MCMC sampler to explore the parameter space. This provides the full posterior distribution, from which we draw samples of the initial matter fluctuations.

Fig. 5 shows projections of the true fields, and the ensemble mean and variances of inferred three-dimensional fields. The mean and variance are estimated from 500 samples of the posterior distribution (the correlation length is ≈80 samples). To compare the ground truth to the inferred mean density field, we computed the projection of the density fields on the sky since the radial information is not very constraining. In this projection, the different slices of the 3D density field are weighted with the distribution of the lensing sources. A first visual comparison between ground truth and the inferred ensemble mean initial and final density fields shows that the algorithm correctly recovered the large-scale structures from cosmic shear data. As expected, the mean of the initial density samples exhibits a small degree of smoothing, a feature that is known from the Wiener filtering solution for Gaussian fields and Gaussian prior.

The right-hand panels of Fig. 5 show the corresponding standard deviations of the projected densities, which are estimated from the posterior samples. The high uncertainty regions correspond to the masked areas, where there are no contributing sources. While the data do not provide direct information about the density field in these masked regions, the dynamical model still provides probabilistic information about the density fields that are physically plausible in those regions. These results indicate that the method can deal with nontrivial survey masks, and account for the uncertainty in the unobserved areas. The standard deviation of the initial conditions is homogeneous, apart from mask effect, indicating that the dynamical model correctly propagates the information between the primordial matter fluctuations and the final density field.

## 5.3 Posterior predictive tests

Posterior predictions allow testing whether the inferred density fields provide accurate explanations of the data (see, e.g. Gelman et al. 2004). Generally, posterior predictive tests provide good diagnostics about the adequacy of data models in explaining observations and identifying possible systematic problems with the inference. In this section, we predicted the shear and convergence fields as the average computed from 500 posterior samples.

Fig. 6 presents the result of this test for one tomographic bin, showing that the posterior predicted shear and convergence recover the features of the true fields. The masked regions show higher standard deviation, indicating that the method can account for the uncertainties in the unobserved areas and provide probabilistic information of the physically plausible shear in those regions. While Fig. 6 shows a visual comparison, Fig. 7 shows the residuals between the true and the mean posterior-predicted shear fields. The green line in the plot indicates the noise distribution, showing that the distribution of the residuals is narrower than the noise distribution, and, therefore, the inferred quantities can explain the data at subnoise level, with the additional constraints coming from the cosmological prior.

Fig. 8 shows the power spectrum of the posterior-predicted shear, measured as the average of predicted shear fields from 500 randomly-drawn samples. The predicted power spectra match the true shear power spectrum at all scales. We note that we computed the true power spectrum from the noiseless data as the posterior-predicted shear does not contain noise. These shear power spectra are pure E-mode since they are obtained from the lensing equations under the Born approximation.
<figure><img src="image_8.png" /><figcaption>**Figure 6.** Posterior predicted shear and convergence for one tomographic bin. The left-hand column shows the shear data, including noise and masked regions; the second column shows the true shear and convergence fields, and third and fourth columns show the mean and standard deviation of the posterior-predicted shear and convergence, computed from 500 posterior samples. The method recovers the true cosmic shear correctly. The regions with higher standard deviation correspond to the masked regions.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 7.** Histogram of the residuals computed as the difference between the posterior predicted shear and the true shear. We note that the true shear does not include the noise. The residuals distribution is narrower than the distribution of pixel noise in the data, indicated in green, showing that the method recovers the true shear at subnoise level with additional constraints from the cosmological prior.</figcaption></figure>

Fig. 9 shows the correlation coefficient between the posterior-predicted shear maps and the true shear. This is readily understood, as it is similar to the Wiener solution for the posterior of a statistically homogeneous Gaussian field with signal power $S$ and Gaussian noise power $N$, (see e.g Jeffrey, Heavens & Fortio 2018). In this case the mean posterior has power suppressed by $(S^{-1} + N^{-1})N^{-1}$ and this is the correlation coefficient. As a result, we expect the tomographic bin centred at the lowest redshift (bin 0) to have a lower correlation coefficient because the lensing power at low redshift is smaller.

## 6 DISTRIBUTION OF THE CONVERGENCE FIELD

Previous Bayesian approaches (Alsing et al. 2016) rely on a Gaussian prior for the shear data. The Gaussian prior is well justified when only the mean and variance are known since it is the least informative prior. It is important to understand that the samples are not Gaussian fields, since they are conditioned on the data, so non-gaussianity in the field may be imposed by the data. However, we can make use of the fact that we have more information available from knowledge of gravitational physics and, for this reason, we include a gravity model in our Bayesian hierarchical model. The advantage here is that we sample from the initial field, which we know to be Gaussian, so rather than relying on an uninformative prior for the final shear fields, we use the correct Gaussian distribution for the initial conditions. What we do not yet do in this model is to vary the prior parameters of the power spectrum (as Alsing et al. (2016) do), and this will be the subject of future work. As described in Section 2, we obtained the
<figure><img src="image_10.png" /><figcaption>**Figure 8.** Posterior power spectrum of the shear field compared to the power spectrum of the true shear for each tomographic bin. The posterior power spectrum is the averaged of the power spectrum measured in 500 posterior samples. The orange line shows the noise power spectrum. The bottom plots show the ratio between the posterior and the true power spectrum, showing that the method recovers the true shear power spectrum at all scales.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 9.** Correlation coefficient between the posterior-predicted and true shear fields, from 500 posterior samples. This shows the expected Wiener-filter-like suppression of power where the noise is high.</figcaption></figure>

shear on a flat-sky from the convergence, which is computed from the nonlinear density field. Using our forward model, we have computed the convergence field in a wider range of tomographic bins, centred at different redshifts but with the same bin width ($\sigma = 0.1$), to illustrate more clearly how non-Gaussianity in the 1-point distribution evolves. Fig. 10 shows the distribution of these convergence fields. While the convergence shows a Gaussian distribution for tomographic bins centred at $z > 0.5$, it is skewed for tomographic bins at lower redshifts. This indicates that the Gaussian approximation is accurate at large redshift, but it is suboptimal for low-redshift bins.

<figure><img src="image_12.png" /><figcaption>**Figure 10.** Distribution of the convergence field for tomographic bins centred at different redshifts. For comparison, a Gaussian with the same mean and variance is plotted on top. The convergence distribution is skewed for tomographic bins centered at $z > 0.5$.</figcaption></figure>

# 7 PRIOR TEST

As discussed in Section 5.1, in this approach, we keep the cosmology fixed, and specify a prior on the initial power spectrum. Although our method does not sample the power spectrum, the power spectrum of the posterior samples is conditioned by the data. This means that, if the data require it, the posterior power spectrum deviates from the prior, and the density samples have a power spectrum that differs from the prior. To demonstrate that, we tested our method with mock data generated with $h = 0.6$, $\sigma_8 = 0.55$, $\Omega_m = 0.7$ and analyse these data with $h = 0.677$, $\sigma_8 = 0.8$, $\Omega_m = 0.3$, such that $\Omega_m h^2$ changes by a factor 2. These changes in the cosmological parameters give a different shape and amplitude of the power spectrum, as can be seen in Fig. 11. The resolution, total number of sources per
<figure><img src="image_13.png" /><figcaption>**Figure 11.** Burn-in phase of the dark matter power spectra for the prior test. We show the power spectrum for the modes parallel to the line-of-sight, $k_\parallel$ (upper panel), and the perpendicular modes, $k_\perp$ (lower panel), i.e. in the plane of the sky. The dotted line indicates the prior matter power spectrum, and the truth is indicated by the red dashed line. The colour scale indicates the sample number in the Markov chain. After the burn-in phase, the method recovers the correct matter power on the plane of the sky. However, the data is not constraining in the radial direction and the prior dominates. We note that as currently implemented, isotropy is not a requirement.</figcaption></figure>

square arcmin, the mask and tomographic bins remain as described in Section 4.

Fig. 11 presents the evolution of the power spectrum for this test, showing the burn-in phase of the Markov chain. In the plane of the sky, the posterior power spectrum agrees with the truth, demonstrating that the method is sensitive to the true power spectrum even when this differs from the prior. However, in the line-of-sight direction, the data is not constraining, and the prior dominates. These results are consistent with Simon, Taylor & Hartlap (2009). As currently implemented, isotropy is not required, and the method cannot determine the radial power accurately. The recovered matter power spectrum, however, suffices to explain the data, obtaining shear residuals below the noise level, as can be seen in Fig. 12. Fig. 13 shows the lensing power spectrum, indicating that the method recovers the correct lensing power at large scales where the noise is low. However, the small scales have lower S/N, and the posterior power spectrum drifts towards the prior. To avoid this prior sensitivity, we need to sample from the power spectrum as well as the field. This will be the subject of a future paper.

In future work, we will extend our Bayesian hierarchical model to jointly sample the cosmological parameters and the density field, using an approach compatible with the one presented in Ramanah et al. (2019). We expect that this future extension of the method will constrain the matter power spectrum, also in the

<figure><img src="image_14.png" /><figcaption>**Figure 12.** Histogram of the shear residuals for the prior test. The residuals are computed as the difference between the mean posterior-predicted shear and true shear. The distribution of shear residuals is narrower than the noise distribution, indicated in green.</figcaption></figure>

radial direction, through the assumption of isotropy, but we do not anticipate being able to recover the small-scale radial distribution at the field level, because of the width of the lensing kernel and the distance uncertainties. Meanwhile, the test presented here shows that the posterior power spectrum is conditioned by the data despite using a fixed cosmology.

## 8 SUMMARY AND DISCUSSION

We have developed a Bayesian physical forward model to infer the matter density field and primordial fluctuations from cosmic shear data. This framework consists of a Gaussian prior for the primordial fluctuations, a dynamical structure formation model that links the initial conditions and the evolved density field, and a likelihood based on a data model of the cosmic shear in the flat-sky approximation.

This forward modelling approach allows us to go beyond the common analyses of cosmic shear based on two-point statistics. While many studies of the cosmic shear focus on the power spectrum or the correlation function, the nonlinear dynamics of the large-scale structure encode significant information in higher-order statistics associated with the filamentary structure of the cosmic web. Our dynamical forward model reproduces the filamentary matter distribution and, in this way, allows using every data point, rather than relying on summary statistics that do not capture all the information and whose distributions are not well known. By employing a more accurate gravity model, our method also improves over previous Bayesian hierarchical approaches that assumed a Gaussian distribution of the shear field (Alsing et al. 2016).

We have tested our inference method with simulated data with four tomographic bins, a survey mask, and 30 sources per arcmin² as expected for the Euclid survey. These tests demonstrate that our method recovers the unbiased matter distribution and initial matter power spectrum from cosmic shear data. Posterior predictive tests showed that the inferred quantities are known to the sub-noise
<figure><img src="image_15.png" /><figcaption>**Figure 13.** Posterior power spectrum of the shear field compared to the power spectrum of the true shear for the prior test. The grey dotted line indicates the shear power spectrum computed using the prior cosmology, which differs from the truth. The orange line indicates the power spectrum of the noise. The bottom plots show the ratio between the posterior, prior, and noise and the true power spectrum, showing that the method recovers the true lensing power spectrum where the signal-to-noise is high, but is suppressed in the low S/N regions where the prior is low.</figcaption></figure>

level, with additional constraints coming from the cosmological prior.

Although our framework currently uses a fixed cosmology, we have shown that the method recovers the true power spectrum when this differs from the prior where the signal-to-noise is high. While we do not sample the power spectrum, the posterior power spectrum deviates from the prior if the data require it. To illustrate this, we performed a test using different values of $H_0$, $\sigma_8$ and $\Omega_m$ to generate the mock data and to analyse them. In this case, the prior power spectrum, therefore, differs from the true power spectrum. This test demonstrated that our method is sensitive to the underlying cosmology, and the power spectrum of the density samples is conditioned by the data, recovering the true matter power spectrum across the sky and the lensing power spectrum. However, in the radial direction, the data is not informative, and the prior dominates since we have not imposed isotropy. In future work, we will extend our Bayesian hierarchical approach to sample the cosmological parameters, through both the geometry and the power spectrum and its growth. We expect that this extension will also constrain the matter power spectrum in the radial direction through the imposition of isotropy, and remove the prior power spectrum sensitivity.

To summarise, this work demonstrates the feasibility of detailed and physically plausible inference of the large-scale structure from cosmic shear data. The proposed approach, therefore, improves the shear data model from previous methods by including a physical description of gravity, providing a better representation of the data and allowing us to extract information beyond the two-point statistics. In future work, we will explore the constraints on cosmology that this approach provides by jointly sampling the initial conditions and the cosmological parameters.

## ACKNOWLEDGEMENTS

This work was supported by STFC through Imperial College Astrophysics Consolidated Grant ST/5000372/1. GL acknowledges financial support from the ANR BIG4, under reference ANR-16-CE23-0002. This work was carried out within the Aquila Consortium.<fnref n="1" />

## DATA AVAILABILITY

The data underlying this article will be shared on reasonable request to the corresponding author.

## REFERENCES

* Alsing J., Heavens A., Jaffe A. H., Kiessling A., Wandelt B., Hoffmann T., 2016, MNRAS, 455, 4452

* Alsing J., Heavens A., Jaffe A. H., 2017, MNRAS, 466, 3272

* Bernardeau F., van Waerbeke L., Mellier Y., 1997, A&A, 322, 1

* Bernardeau F., van Waerbeke L., Mellier Y., 2003, A&A, 397, 405

* Böhm V., Hilbert S., Greiner M., Enßlin T. A., 2017, Phys. Rev. D, 96, 123510

* Brown M. L., Castro P. G., Taylor A. N., 2005, MNRAS, 360, 1262

* Cardone V. F., Camera S., Mainini R., Romano A., Diaferio A., Maoli R., Scaramella R., 2013, MNRAS, 430, 2896

* Chon G., Challinor A., Prunet S., Hivon E., Szapudi I., 2004, MNRAS, 350, 914

* Dietrich J. P., Hartlap J., 2010, MNRAS, 402, 1049

* Eisenstein D. J., Hu W., 1998, ApJ, 496, 605

* Eisenstein D. J., Hu W., 1999, ApJ, 511, 5

* Euclid Collaboration, 2020, A&A, 635, A139

<footnote n="1">¹https://aquila-consortium.org</footnote>
Fu L. et al., 2014, MNRAS, 441, 2725

Gelman A., Carlin J. B., Stern H. S., Rubin D. B., 2004, Bayesian Data Analysis, 2nd edn, Chapman and Hall/CRC

Gupta A., Matilla J. M. Z., Hsu D., Haiman Z., 2018, Phys. Rev. D, 97, 103515

Heymans C. et al., 2013, MNRAS, 432, 2433

Hikage C. et al., 2019, PASJ, 71, 43

Hildebrandt H. et al., 2017, MNRAS, 465, 1454

Jain B., Seljak U., 1997, ApJ, 484, 560

Jain B., van Waerbeke L., 2000, ApJ, 530, L1

Jarvis M., Bernstein G., Jain B., 2004, MNRAS, 352, 338

Jasche J., Kitaura F. S., 2010a, MNRAS, 407, 29

Jasche J., Kitaura F. S., 2010b, MNRAS, 407, 29

Jasche J., Lavaux G., 2017, A&A, 606, A37

Jasche J., Lavaux G., 2019, A&A, 625, A64

Jasche J., Wandelt B. D., 2013a, MNRAS, 432, 894

Jasche J., Wandelt B. D., 2013b, MNRAS, 432, 894

Jasche J., Leclercq F., Wandelt B. D., 2015, J. Cosmol. Astropart. Phys., 1, 036

Jeffrey N., Heavens A. F., Fortio P. D., 2018, Astron. Comput., 25, 230

Kacprzak T. et al., 2016, MNRAS, 463, 3653

Kayo I., Takada M., Jain B., 2013, MNRAS, 429, 344

Kilbinger M., 2015, Reports on Progress in Physics, 78, 086901

Kitching T. D. et al., 2014, MNRAS, 442, 1326

Kitching T. D., Heavens A. F., Miller L., 2011, MNRAS, 413, 2923

Kitching T. D., Heavens A. F., Das S., 2015, MNRAS, 449, 2205

Kitching T. D., Verde L., Heavens A. F., Jimenez R., 2016, MNRAS, 459, 971

Laureijs R. et al., 2011, preprint (arXiv:1110.3193)

Lavaux G., Jasche J., 2016, MNRAS, 455, 3169

Lavaux G., Jasche J., Leclercq F., 2019, preprint (arXiv:1909.06396)

Leclercq F., Faure B., Lavaux G., Wandelt B. D., Jaffe A. H., Heavens A. F., Percival W. J., 2020, A&A, 639, A91

Lin C.-A., Kilbinger M., 2015, A&A, 583, A70

Liu X. et al., 2015b, MNRAS, 450, 2888

Liu J., Petri A., Haiman Z., Hui L., Kratochvil J. M., May M., 2015a, Phys. Rev. D, 91, 063507

LSST Science Collaboration, 2009, preprint (arXiv:0912.0201)

Marian L., Smith R. E., Hilbert S., Schneider P., 2012, MNRAS, 423, 1711

Maturi M., Fedeli C., Moscardini L., 2011, MNRAS, 416, 2527

Peel A., Lin C.-A., Lanusse F., Leonard A., Starck J.-L., Kilbinger M., 2017, A&A, 599, A79

Pen U.-L., Zhang T., van Waerbeke L., Mellier Y., Zhang P., Dubinski J., 2003, ApJ, 592, 664

Petri A., Haiman Z., Hui L., May M., Kratochvil J. M., 2013, Phys. Rev. D, 88, 123002

Pires S., Leonard A., Starck J.-L., 2012, MNRAS, 423, 983

Planck Collaboration, 2020, A&A, 641, A9

Porqueres N., Kodi Ramanah D., Jasche J., Lavaux G., 2019a, A&A, 624, A115

Porqueres N., Jasche J., Lavaux G., Enßlin T., 2019b, A&A, 630, A151

Porqueres N., Hahn O., Jasche J., Lavaux G., 2020, A&A, 642, A139

Ramanah D. K., Lavaux G., Jasche J., Wandelt B. D., 2019, A&A, 621, A69

Schneider P., Lombardi M., 2003, A&A, 397, 809

Semboloni E., Schrabback T., van Waerbeke L., Vafaei S., Hartlap J., Hilbert S., 2011, MNRAS, 410, 143

Simon P., Taylor A. N., Hartlap J., 2009, MNRAS, 399, 48

Smith K. M., 2006, Phys. Rev. D, 74, 083002

Spergel D. et al., 2015, preprint (arXiv:1503.03757)

Springel V., 2005, MNRAS, 364, 1105

Takada M., Jain B., 2003, MNRAS, 344, 857

Tassev S., Zaldarriaga M., Eisenstein D. J., 2013, J. Cosmol. Astropart. Phys., 2013, 036

Taylor P. L., Kitching T. D., Alsing J., Wandelt B. D., Feeney S. M., McEwen J. D., 2019, Phys. Rev. D, 100, 023519

Troxel M. A. et al., 2018, Phys. Rev. D, 98, 043528

Vafaei S., Lu T., van Waerbeke L., Semboloni E., Heymans C., Pen U.-L., 2010, Astropart. Phys., 32, 340

van Waerbeke L. et al., 2013, MNRAS, 433, 3373

van Waerbeke L., Bernardeau F., Mellier Y., 1999, A&A, 342, 15

## APPENDIX A: ADJOINT GRADIENT OF THE DATA MODEL

The inference of the density field requires inferring the amplitudes of the primordial density at different volume elements of a regular grid, commonly between 128³ and 256³ volume elements. This implies 10⁶ to 10⁷ free parameters. To explore this high-dimensional parameter space efficiently, the BORG framework employs a Hamiltonian Monte Carlo (HMC) method, which adapts to the geometry of the problem by using the information in the gradients. Therefore, this algorithm requires the derivatives of the forward model. In this appendix, we derive the adjoint gradient of the shear model, which linearly transforms the error vector from the likelihood space to the parameter space of initial conditions.

More specifically, the HMC relies on the availability of a gradient of the posterior distribution. Therefore, we need to compute the gradient of the log-likelihood with respect to the initial density contrast, $\delta^{ic}$.

$$ \frac{\partial \log \mathcal{L}}{\partial \delta_p^{ic}} = \sum_b \sum_{mn} \frac{\partial \log \mathcal{L}}{\partial \gamma_{mn}^b} \frac{\partial \gamma_{mn}^b}{\partial \delta_p^{ic}} \quad (A1) $$

$$ \frac{\partial \log \mathcal{L}}{\partial \delta_p^{ic}} = \sum_b \sum_{mn} \frac{\hat{\gamma}_{1,mn}^b - \gamma_{1,mn}^b}{\sigma_{mn}^2} \frac{\partial \gamma_{1,mn}^b}{\partial \delta_p^{ic}} + \frac{\hat{\gamma}_{2,mn}^b - \gamma_{2,mn}^b}{\sigma_{mn}^2} \frac{\partial \gamma_{2,mn}^b}{\partial \delta_p^{ic}} \quad (A2) $$

$$ \frac{\partial \log \mathcal{L}}{\partial \delta_p^{ic}} = \sum_b \sum_{mn} \frac{\hat{\gamma}_{1,mn}^b - \gamma_{1,mn}^b}{\sigma_{mn}^2} \text{DFT}^{-1} \left[ \frac{l_x^2 - l_y^2}{l_x^2 + l_y^2} \text{DFT} \left( \frac{\partial \kappa_{mn}}{\partial \delta_p^{ic}} \right) \right] + \sum_b \sum_{mn} \frac{\hat{\gamma}_{2,mn}^b - \gamma_{2,mn}^b}{\sigma_{mn}^2} \text{DFT}^{-1} \left[ -\frac{2l_x l_y}{l_x^2 + l_y^2} \text{DFT} \left( \frac{\partial \kappa_{mn}}{\partial \delta_p^{ic}} \right) \right] $$

with

$$ \frac{\partial \kappa_{mn}}{\partial \delta_p^{ic}} = \frac{3H_0^2 \Omega_m}{2c^2} \left[ \sum_{s=p}^{N_2} \frac{(r_s - r_p) n^b(r_s) \Delta r_s}{r_s} \right] \frac{r_p \Delta r_p}{a} \frac{\partial \mathcal{M}(a, \delta^{ic})}{\partial \delta_p^{ic}} \quad (A3) $$

where $\mathcal{M}(a, \delta^{ic})$ is the dynamical forward model.

This paper has been typeset from a TeX/LA TeX file prepared by the author.
# List of astronomical key words (Updated on 2020 January)

This list is common to *Monthly Notices of the Royal Astronomical Society, Astronomy and Astrophysics*, and *The Astrophysical Journal*. In order to ease the search, the key words are subdivided into broad categories. No more than six subcategories altogether should be listed for a paper.

The subcategories in boldface containing the word 'individual' are intended for use with specific astronomical objects; these should never be used alone, but always in combination with the most common names for the astronomical objects in question. Note that each object counts as one subcategory within the allowed limit of six.

The parts of the key words in italics are for reference only and should be omitted when the keywords are entered on the manuscript.

## General

editorials, notices

errata, addenda

extraterrestrial intelligence

history and philosophy of astronomy

miscellaneous

obituaries, biographies

publications, bibliography

sociology of astronomy

standards

## Physical data and processes

acceleration of particles

accretion, accretion discs

asteroseismology

astrobiology

astrochemistry

astroparticle physics

atomic data

atomic processes

black hole physics

chaos

conduction

convection

dense matter

diffusion

dynamo

elementary particles

equation of state

gravitation

gravitational lensing: micro

gravitational lensing: strong

gravitational lensing: weak

gravitational waves

hydrodynamics

instabilities

line: formation

line: identification

line: profiles

magnetic fields

magnetic reconnection

(magnetohydrodynamics) MHD

masers

molecular data

molecular processes

neutrinos

nuclear reactions, nucleosynthesis, abundances

opacity

plasmas

polarization

radiation: dynamics

radiation mechanisms: general

radiation mechanisms: non-thermal

radiation mechanisms: thermal

radiative transfer

relativistic processes

scattering

shock waves

solid state: refractory

solid state: volatile

turbulence

waves

## Astronomical instrumentation, methods and techniques

atmospheric effects

balloons

instrumentation: adaptive optics

instrumentation: detectors

instrumentation: high angular resolution

instrumentation: interferometers

instrumentation: miscellaneous

instrumentation: photometers

instrumentation: polarimeters

instrumentation: spectrographs

light pollution

methods: analytical

methods: data analysis

methods: laboratory: atomic

methods: laboratory: molecular

methods: laboratory: solid state

methods: miscellaneous

methods: numerical

methods: observational

methods: statistical

site testing

space vehicles

space vehicles: instruments

techniques: high angular resolution

techniques: image processing

techniques: imaging spectroscopy

techniques: interferometric

techniques: miscellaneous

techniques: photometric

techniques: polarimetric

techniques: radar astronomy

techniques: radial velocities

techniques: spectroscopic

telescopes
## Astronomical data bases

astronomical data bases: miscellaneous
atlases
catalogues
surveys
virtual observatory tools

## Software

software: data analysis
software: development
software: documentation
software: public release
software: simulations

## Astrometry and celestial mechanics

astrometry
celestial mechanics
eclipses
ephemerides
occultations
parallaxes
proper motions
reference systems
time

## The Sun

Sun: abundances
Sun: activity
Sun: atmosphere
Sun: chromosphere
Sun: corona
Sun: coronal mass ejections (CMEs)
Sun: evolution
Sun: faculae, plages
Sun: filaments, prominences
Sun: flares
Sun: fundamental parameters
Sun: general
Sun: granulation
Sun: helioseismology
Sun: heliosphere
Sun: infrared
Sun: interior
Sun: magnetic fields
Sun: oscillations
Sun: particle emission
Sun: photosphere
Sun: radio radiation
Sun: rotation
*(Sun:) solar-terrestrial relations
*(Sun:) solar wind
*(Sun:) sunspots
Sun: transition region
Sun: UV radiation
Sun: X-rays, gamma-rays

## Planetary systems

comets: general

### comets: individual: ...

Earth
interplanetary medium
Kuiper belt: general

### Kuiper belt objects: individual: ...

meteorites, meteors, meteoroids

minor planets, asteroids: general

### minor planets, asteroids: individual: ...

Moon
Oort Cloud
planets and satellites: atmospheres
planets and satellites: aurorae
planets and satellites: composition
planets and satellites: detection
planets and satellites: dynamical evolution and stability
planets and satellites: formation
planets and satellites: fundamental parameters
planets and satellites: gaseous planets
planets and satellites: general

### planets and satellites: individual: ...

planets and satellites: interiors
planets and satellites: magnetic fields
planets and satellites: oceans
planets and satellites: physical evolution
planets and satellites: rings
planets and satellites: surfaces
planets and satellites: tectonics
planets and satellites: terrestrial planets
planet-disc interactions
planet-star interactions
protoplanetary discs
zodiacal dust

## Stars

stars: abundances
stars: activity
stars: AGB and post-AGB
stars: atmospheres
*(stars:) binaries (including multiple): close
*(stars:) binaries: eclipsing
*(stars:) binaries: general
*(stars:) binaries: spectroscopic
*(stars:) binaries: symbiotic
*(stars:) binaries: visual
stars: black holes
*(stars:) blue stragglers
*(stars:) brown dwarfs
stars: carbon
stars: chemically peculiar
stars: chromospheres
*(stars:) circumstellar matter
stars: coronae
stars: distances
stars: dwarf novae
stars: early-type
stars: emission-line, Be
stars: evolution
stars: flare
stars: formation
stars: fundamental parameters
*(stars:) gamma-ray burst: general
*(stars:) gamma-ray burst: individual: ...

stars: general
*(stars:) Hertzsprung-Russell and colour-magnitude diagrams
stars: horizontal branch
stars: imaging

### stars: individual: ...

stars: interiors
stars: jets

stars: kinematics and dynamics

stars: late-type

stars: low-mass

stars: luminosity function, mass function

stars: magnetars

stars: magnetic field

stars: massive

stars: mass-loss

stars: neutron

(stars:) novae, cataclysmic variables

stars: oscillations (including pulsations)

stars: peculiar (except chemically peculiar)

(stars:) planetary systems

stars: Population II

stars: Population III

stars: pre-main-sequence

stars: protostars

(stars:) pulsars: general

(stars:) pulsars: individual: ...

stars: rotation

stars: solar-type

(stars:) starspots

stars: statistics

(stars:) subdwarfs

(stars:) supergiants

(stars:) supernovae: general

(stars:) supernovae: individual: ...

stars: variables: Cepheids

stars: variables: Scuti

stars: variables: general

stars: variables: RR Lyrae

stars: variables: S Doradus

stars: variables: T Tauri, Herbig Ae/Be

(stars:) white dwarfs

stars: winds, outflows

stars: Wolf-Rayet

## Interstellar medium (ISM), nebulae

ISM: abundances

ISM: atoms

ISM: bubbles

ISM: clouds

(ISM:) cosmic rays

(ISM:) dust, extinction

ISM: evolution

ISM: general

(ISM:) HII regions

(ISM:) Herbig-Haro objects

## ISM: individual objects: ...

(except planetary nebulae)

ISM: jets and outflows

ISM: kinematics and dynamics

ISM: lines and bands

ISM: magnetic fields

ISM: molecules

(ISM:) photodissociation region (PDR)

(ISM:) planetary nebulae: general

(ISM:) planetary nebulae: individual: ...

ISM: structure

ISM: supernova remnants

## The Galaxy

Galaxy: abundances

Galaxy: bulge

Galaxy: centre

Galaxy: disc

Galaxy: evolution

Galaxy: formation

Galaxy: fundamental parameters

Galaxy: general

(Galaxy:) globular clusters: general

(Galaxy:) globular clusters: individual: ...

Galaxy: halo

Galaxy: kinematics and dynamics

(Galaxy:) local interstellar matter

Galaxy: nucleus

(Galaxy:) open clusters and associations: general

(Galaxy:) open clusters and associations: individual: ...

(Galaxy:) solar neighbourhood

Galaxy: stellar content

Galaxy: structure

## Galaxies

galaxies: abundances

galaxies: active

galaxies: bar

(galaxies:) BL Lacertae objects: general

(galaxies:) BL Lacertae objects: individual: ...

galaxies: bulges

galaxies: clusters: general

galaxies: clusters: individual: ...

galaxies: clusters: intracluster medium

galaxies: disc

galaxies: distances and redshifts

galaxies: dwarf

galaxies: elliptical and lenticular, cD

galaxies: evolution

galaxies: formation

galaxies: fundamental parameters

galaxies: general

galaxies: groups: general

galaxies: groups: individual: ...

galaxies: haloes

galaxies: high-redshift

galaxies: individual: ...

galaxies: interactions

(galaxies:) intergalactic medium

galaxies: irregular

galaxies: ISM

galaxies: jets

galaxies: kinematics and dynamics

(galaxies:) Local Group

galaxies: luminosity function, mass function

(galaxies:) Magellanic Clouds

galaxies: magnetic fields

galaxies: nuclei

galaxies: peculiar

galaxies: photometry

(galaxies:) quasars: absorption lines

(galaxies:) quasars: emission lines

(galaxies:) quasars: general
(galaxies:) quasars: individual: ...

(galaxies:) quasars: supermassive black holes

galaxies: Seyfert

galaxies: spiral

galaxies: starburst

galaxies: star clusters: general

galaxies: star clusters: individual: ...

galaxies: star formation

galaxies: statistics

galaxies: stellar content

galaxies: structure

Cosmology

(cosmology:) cosmic background radiation

(cosmology:) cosmological parameters

(cosmology:) dark ages, reionization, first stars

(cosmology:) dark energy

(cosmology:) dark matter

(cosmology:) diffuse radiation

(cosmology:) distance scale

(cosmology:) early Universe

(cosmology:) inflation

(cosmology:) large-scale structure of Universe

cosmology: miscellaneous

cosmology: observations

(cosmology:) primordial nucleosynthesis

cosmology: theory

Resolved and unresolved sources as a function of
wavelength

gamma-rays: diffuse background

gamma-rays: galaxies

gamma-rays: galaxies: clusters

gamma-rays: general

gamma-rays: ISM

gamma-rays: stars

infrared: diffuse background

infrared: galaxies

infrared: general

infrared: ISM

infrared: planetary systems

infrared: stars

radio continuum: galaxies

radio continuum: general

radio continuum: ISM

radio continuum: planetary systems

radio continuum: stars

radio continuum: transients

radio lines: galaxies

radio lines: general

radio lines: ISM

radio lines: planetary systems

radio lines: stars

submillimetre: diffuse background

submillimetre: galaxies

submillimetre: general

submillimetre: ISM

submillimetre: planetary systems

submillimetre: stars

ultraviolet: galaxies

ultraviolet: general

ultraviolet: ISM

ultraviolet: planetary systems

ultraviolet: stars

X-rays: binaries

X-rays: bursts

X-rays: diffuse background

X-rays: galaxies

X-rays: galaxies: clusters

X-rays: general

X-rays: individual: ...

X-rays: ISM

X-rays: stars

Transients

(transients:) black hole mergers

(transients:) black hole - neutron star mergers

(transients:) fast radio bursts

(transients:) gamma-ray bursts

(transients:) neutron star mergers

transients: novae

transients: supernovae

transients: tidal disruption events