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A&A
Volume 695, March 2025
Article Number A128
Number of page(s) 12
Section Extragalactic astronomy
DOI https://doi.org/10.1051/0004-6361/202451282
Published online 12 March 2025

© The Authors 2025

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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1. Introduction

Active galactic nuclei (AGN) are among the most luminous sources in the Universe. They are powered by accretion onto supermassive black holes (SMBHs) in their centres (Lynden-Bell 1969). As X-ray emission is ubiquitous in AGN, the XMM-Newton and Chandra X-ray missions have probed with unprecedented accuracy the AGN demographics, and hence they have mapped in detail the accretion history in the Universe (e.g. Ueda et al. 2014; Aird et al. 2015b; Miyaji et al. 2015; Ranalli et al. 2016; Fotopoulou et al. 2016; Georgakakis et al. 2017; Pouliasis et al. 2024). The majority of AGN are obscured by large amounts of dust and gas. This obscuring screen is believed to have the form of a torus, although direct imaging at mid-IR and sub-millimetre wavelengths have revealed a more complicated structure (e.g. Hönig et al. 2012; García-Burillo et al. 2016).

Additional constraints on the demographics of the AGN population come from the diffuse X-ray radiation that permeates the whole Universe, the X-ray background (Giacconi et al. 1962). This radiation comes from the superposition of all discrete point sources in their vast majority AGN (Mushotzky et al. 2000). The energy density of the X-ray background peaks at around 30 keV (Frontera et al. 2007; Churazov et al. 2007; Revnivtsev et al. 2003). Based on population synthesis models that attempt to reproduce the shape of the X-ray background, and especially its 30 keV hump, authors have found that a significant fraction of the sources that constitute this radiation must be associated with Compton-thick sources (Comastri et al. 1995; Gilli et al. 2007; Treister et al. 2009; Akylas et al. 2012; Ananna et al. 2019). These heavily obscured sources present column densities in excess of ≈1024 cm−2, and hence the obscuration is caused by Compton scattering on electrons rather than photoelectric absorption (e.g. Hickox et al. 2017). Nevertheless, the X-ray background synthesis models appear to produce divergent results regarding the number of Compton-thick sources. On the low end, the models of Akylas et al. (2012) and Treister et al. (2009) yielded Compton-thick fractions of less than 20%. On the other end, the models of Ananna et al. (2019) predict Compton-thick fractions of the order of 50%. Recently, Carroll et al. (2023) attempted to constrain the fraction of Compton-thick sources by forward modelling the column density distribution of mid-IR selected AGN up to a redshift of z = 0.8. They estimate a Compton-thick fraction of 55% (but see Georgakakis et al. 2020).

A number of works have attempted to detect directly these highly obscured sources using the superb sensitivity of the XMM-Newton and Chandra missions (Georgantopoulos et al. 2009, 2013; Lanzuisi et al. 2015, 2017; Brightman et al. 2014; Koulouridis et al. 2016; Buchner et al. 2014; Georgakakis et al. 2017; Corral et al. 2019; Laloux et al. 2023). The identification of the Compton-thick sources is based on either the direct detection of the absorption turnover or the detection of a large equivalent width FeKα line, which is the smoking gun of heavy obscuration. However, as these missions operate in the relatively soft 0.5–10 keV band, they may miss a significant number of these heavily obscured sources.

The Burst Alert Telescope (BAT) instrument (Barthelmy et al. 2005) on board the Neil Gehrels Swift mission, (Gehrels et al. 2004), all-sky survey was very prolific in detecting a large number of candidate Compton-thick AGN due to its energy passband that extends to very hard energies, 14 − 195 keV. The 70-month BAT survey detected 728 non-blazar AGN. Ricci et al. (2015) identified 55 Compton-thick AGN among this sample up to a redshift of z ∼ 0.1. Also using the 70-month BAT sample, Akylas et al. (2016) applied a Bayesian approach and identified a few tens of candidate Compton-thick AGN and assigned a column density probability distribution to each object. Both Ricci et al. (2015) and Akylas et al. (2016) found an observed Compton-thick fraction of less than 10% of the total AGN population. However, when Torres-Albà et al. (2021) used a volume-limited sample of Compton-thick AGN with NuSTAR spectra up to z = 0.01, they estimated an intrinsic Compton-thick fraction of about 20%. Their results refer to the column density range NH = 1024 − 25 cm−2, the so-called transmission-dominated Compton-thick sources. This is the same column density range routinely probed by the Ricci et al. (2015) and Akylas et al. (2016) works. The work of Torres-Albà et al. (2021) suggests that a number of Compton-thick sources are so heavily obscured that their detection is arduous even in the BAT high energy band (see also Burlon et al. 2011). Some X-ray background synthesis models (e.g. Ananna et al. 2019) predict much higher Compton-thick fractions of up to 50%. This is because they assume an equally numerous fraction of Compton-thick AGN with extreme column densities in the range 1025 − 26 cm−2, which have been dubbed reflection-dominated sources.

Based on their BAT sample, Akylas et al. (2016) derived the luminosity function of Compton-thick AGN. Ananna et al. (2022) derived the luminosity function using the Ricci et al. (2015) Compton-thick BAT sample, finding reasonable agreement with the work of Akylas et al. (2016). However, it soon became evident that the BAT’s moderate sensitivity and spectral resolution may hamper the reliable identification of a number of faint sources as bona fide Compton-thick AGN. For this reason NuSTAR observations of the BAT-selected AGN have been systematically employed in order to accurately pinpoint the absorbing column density (e.g. Tanimoto et al. 2022; Torres-Albà et al. 2021; Marchesi et al. 2017, 2018; Georgantopoulos & Akylas 2019; Zhao et al. 2021; Silver et al. 2022).

2. Data

2.1. Neil Gehrels Swift

The Swift Gamma-Ray Burst (GRB) observatory (Gehrels et al. 2004) was launched in November 2004 and has been continually observing the hard X-ray (14 − 195 keV) sky with the BAT. The BAT is a large coded-mask telescope optimised to detect transient GRBs, and it was designed with a very wide field of view of ∼60 × 100 degrees.

The data presented in this paper stem from the analysis of the sources detected during the 70 months of observation by the BAT hard X-ray detector on the Neil Gehrels Swift GRB observatory (Baumgartner et al. 2013). The 70-month BAT survey is an almost uniform hard X-ray all-sky survey, with a sensitivity of 1.34 × 10−11 ergs s−1 cm−2 over 90% of the sky and 1.03 × 10−11 ergs s−1 cm−2 over 50% of the sky in the 14–195 keV band.

2.2. NuSTAR

The Nuclear Spectroscopic Telescope Array (NuSTAR, Harrison et al. 2013), launched in June 2012, is the first orbiting X-ray observatory that focuses light at high energies (E > 10 keV). It consists of two co-aligned focal plane modules (FPMs) that are identical in design. Each FPM covers the same 12 × 12 arcmin portion of the sky and comprises of four Cadmium-Zinc-Tellurium detectors. NuSTAR operates between 3 and 79 keV, and it provides an improvement of at least two orders of magnitude in sensitivity compared to previous hard X-ray observatories operating at energies E > 10 keV. This is because of its high spatial resolution 58 arcsec half-power diameter.

2.3. The Compton-thick active galactic nuclei in the BAT survey

In the 70-month BAT survey, 1171 hard X-ray sources have been detected down to 4.8σ and associated with 1210 counterparts. The majority of these sources are associated with AGN. In particular, there are 752 non-blazar AGN among these sources (Koss et al. 2022). Ricci et al. (2015) derived the X-ray spectra for the 70-month BAT survey sources, combining the Neil Gehrels Swift spectra with available XMM-Newton, ChandraSUZAKU, and ASCA data. The X-ray spectra showed that 55 sources are associated with Compton-thick obscuration. In another study, Akylas et al. (2016), again using the BAT 70-month survey, identify 42 sources with a probability above 30% of being Compton-thick. Although there is significant overlap between the two samples, there are a number of sources that have been characterised as Compton-thick in one sample but not in the other. Specifically, 35 sources are common between the two samples. Apart from the different methodology, it has to be noted that Akylas et al. (2016) uses only the Neil Gehrels Swift – that is, the BAT and the X-ray telescope (XRT) – data in their spectral analysis.

2.4. NuSTAR confirmation of BAT-selected Compton-thick active galactic nuclei

It was soon realised that the limited energy resolution of BAT may compromise the secure classification of Compton-thick AGN. In particular, it could be unclear whether a source was a bona fide Compton-thick AGN or just a heavily obscured source with a column density bordering the Compton-thick regime. Tanimoto et al. (2022) systematically analysed the NuSTAR observations of 52 candidate Compton-thick AGN in the sample of Ricci et al. (2017). They find that 28 objects are Compton-thick AGN at the 90% confidence interval. Their analysis was based on the XCLUMPY model of Tanimoto et al. (2019). The Clemson University group has performed another search for Compton-thick AGN in the BAT catalogues (e.g. Marchesi et al. 2017, 2018, 2019; Torres-Albà et al. 2021; Zhao et al. 2020, 2021; Silver et al. 2022; Sengupta et al. 2023). Their candidate Compton-thick AGN were originally selected either because their properties (either their Seyfert 2 optical type or the absence of a bright ROSAT counterpart) suggest the presence of substantial intrinsic absorption or because they lack soft 0.3–7 keV coverage. The authors primarily used the BORUS02 model of Baloković et al. (2019), but they also employed the MYTORUS model of Murphy & Yaqoob (2009). Despite the use of different models, the derived line-of-sight column densities agree reasonably well with Tanimoto et al. (2022), within the uncertainties. Georgantopoulos & Akylas (2019) analysed the available NuSTAR observations of 19 sources from the Akylas et al. (2016) sample. The vast majority of these sources are common with the Ricci et al. (2015) sample. Georgantopoulos & Akylas (2019) used the absorption model of Murphy & Yaqoob (2009) together with a physically motivated thermal Comptonisation model (Titarchuk 1994) to represent the coronal emission. They find that all but two sources are associated with Compton-thick obscuration. Additional NuSTAR observations of the six Compton-thick candidates of Akylas et al. (2016) that are not common with Ricci et al. (2015) show that three sources are indeed Compton-thick: 2MASX J00253292+6821442, ESO426-G002, and NGC4941. The other three (NGC 3081, NGC 3588NED01, and ESO234-IG063) present lower obscuring column densities (Traina et al. 2021; Silver et al. 2022).

2.5. Models for Compton-thick absorption

In the following, we present a very concise overview of the models most often used to quantify the absorption and reflection in the NuSTAR observations of BAT-selected Compton-thick AGN. In Compton-thick objects, obscuration is primarily caused by scattering of X-ray photons on electrons and subsequent photoelectric absorption. There are a number of spectral codes available that utilise Monte Carlo simulations to model the absorption and reflection on the obscuring screen.

The MYTORUS Murphy & Yaqoob (2009) model assumes a tube-like azimuthally symmetric torus. The half-opening of the torus is assumed to be 60°, corresponding to a scenario where there are equal numbers of obscured and unobscured AGN. The model can decouple the line-of-sight and equatorial column densities. This is important for taking into account continuum variability and time delays between the direct (zeroth order or unscattered) continuum and the Compton-scattered continuum.

However, it has become apparent that the AGN torus is composed of many individual clouds. This is primarily because the 10 μm silicate feature appears as both an emission and absorption feature in Seyfert-2 galaxies, while the smooth torus models predict this feature only in absorption. In the infrared band, Nenkova et al. (2008) constructed spectral models for clumpy tori. They assumed a power law in the radial direction and a normal distribution in the elevation direction for the configuration of the clouds. Following this model, Tanimoto et al. (2019) constructed a new X-ray clumpy torus model designated as XCLUMPY by adopting the same geometry of clump distribution. The clumpy torus models predict a higher fraction of unabsorbed reflection components as observed in many obscured AGN. The line-of-sight column densities do not appear to differ significantly from the smooth tori model (see discussion in Torres-Albà et al. 2021).

One of the most widely used models is BORUS02 (Baloković et al. 2019). The reprocessing medium is assumed to be a sphere of uniform density with conical cutouts at both poles, approximating a torus with a variable covering factor. The half-opening angle of the polar cutouts, θtor, is measured from the symmetry axis towards the equator, and it ranges from zero (full covering) to 83°, corresponding to a disc-like 10% covering. The line-of-sight component, NHlos, can have a different column density than the column density of the torus (NHtor) in order to account for clouds passing in front of the line of sight. In practice, this feature mimics the properties of a patchy torus.

Finally, Buchner et al. (2021) developed a clumpy torus named UXCLUMPY. Naturally, the patchy geometry results in strong Compton scattering, causing soft photons to escape also along Compton-thick sight lines. This model introduces an additional Compton-thick reflector near the corona, a necessary feature in order to achieve acceptable spectral fits to the NuSTAR spectra. This additional component can be interpreted as being part of the dust-free broad-line region, an inner wall or rim, or a warped disc.

3. The sample

3.1. Sample selection

We compiled our sample of Compton-thick AGN in the local Universe based on the Neil Gehrels Swift BAT 70-month survey. We started by using the Compton-thick AGN samples selected by Ricci et al. (2015) and Akylas et al. (2016). Finally, we complemented our sample with a few additional Compton-thick sources reported by the Clemson group (see Table A.1). The condition for inclusion of these sources in our sample is that they are detected in the 70-month survey, and thus they are included in the sample of Baumgartner et al. (2013). These sources evaded classification by both Ricci et al. (2015) and Akylas et al. (2016). As NuSTAR observations exist for all of these sources, there is an accurate determination of the column density available. We included all sources with column densities consistent with NH = 1024 cm−2 at the 90% confidence level. We note that the selected threshold value is somewhat below the column density that corresponds to τ = 1 for Compton scattering. We included only sources with redshift z < 0.05, corresponding roughly to 200 Mpc. Our sample contains 45 sources and is presented in Table A.1. The first reference in column (8) of Table A.1 denotes the origin of the column density. Nearly all of our sources have a column density much lower than 1025 cm−2, with the exception being NGC 1068, which has a borderline column density of N H = 10 − 0.80 + 100 × 10 25 cm − 2 $ \mathrm{N_H}= 10^{+100}_{-0.80}\times 10^{25}~\mathrm{cm}^{-2} $. We chose to exclude this source and restrict our analysis only to the 1024 − 1025 cm−2 sources. This is because most current models for Compton-thick absorption do not produce reliable results at such extreme column densities.

The majority of the column densities comes from the work of either Zhao et al. (2021), Marchesi et al. (2018), or Tanimoto et al. (2022). For a vast majority of the sources, the differences between the column densities are consistent within the uncertainties. An exception is ESO 005-G004, where Tanimoto et al. (2022) using XCLUMPY find that the source is heavily obscured, while Marchesi et al. (2018), using MYTORUS, find that the source is marginally Compton-thick. Zhao et al. (2021), using BORUS02, find a column density of NH ≈ 5 × 1024 cm−2. The differences are probably attributable to the fact that the different models prefer a significantly different value for the photon-index that counteracts the derived column density. Finally, we note that the luminosities are estimated in a consistent manner, as described in the section below.

3.2. Estimation of absorption-corrected luminosities

The selected sample of local Compton-thick AGN has been studied with a variety of spectral models in the literature. While estimations of the line-of-sight column density, NH, are reasonably robust and model independent (e.g. Saha et al. 2022) if X-ray data above 10 keV are included, the remaining parameters can be quite uncertain. This can cause significant discrepancies, particularly when estimating the absorption-corrected luminosity. For consistency, we assumed a single X-ray spectral model for all the sources. Given this model, the observed BAT flux, and the NH estimates from the literature, we calculated the corresponding intrinsic luminosity for each AGN in our sample. We used the spectral model assumed by Ananna et al. (2022) in their study of the X-ray luminosity function of NuSTAR sources. It is a modification of the model assumed by Ueda et al. (2014) where the torus absorption and reflection are instead modelled using BORUS02. For Compton-thick sources, a photon index of Γ = 1.8 was used with an energy cut-off of 200 keV, an inclination angle of 72deg, and a half-opening angle of 60deg. To check the validity of our derived luminosities, we also used the XCLUMPY model. We found no significant differences in our derived luminosities.

For a rigorous treatment of the NH and flux uncertainties, for each source we generated 2000 random samples of NH flux pairs and calculated the corresponding luminosity. The sampling was done assuming a generalised extreme distribution parameterised in such a way that its median corresponds to the quoted value of the NH, flux in Table A.1 and the fifth and 95th percentiles correspond to the respective error interval. This distribution is recommended to reproduce strong asymmetric confidence intervals Possolo et al. (2019).

3.3. Detection probability and selection biases

The sensitivity curve (flux limit versus area covered) of the BAT survey is not uniform across the sky (Baumgartner et al. 2013). Towards the flux limit’s bright end (low sensitivity), the area covered is the largest, but a considerable number of sources are less likely to be detected. This is because they will present a faint flux due to having a low luminosity or a higher column density or simply because they are found at a higher redshift. This introduces some biases that have to be addressed and quantified in order to accurately estimate the X-ray luminosity function of the intrinsic population of the Compton-thick sources. We then needed to estimate the probability that a Compton-thick source with a column density NH, redshift z, and intrinsic luminosity LX will be detected in the BAT survey. To this end, we used the BORUS02 model as above to build the sensitivity maps. We assumed a photon index of Γ = 1.8, an inclination angle i = 72deg, and a torus half-opening angle σ = 60deg. Then we calculated the expected flux at a given set of z, LX, and NH using a grid of 50 bins in each parameter. The probability of detecting a source was derived by convolving the expected flux with the area curve of BAT (Baumgartner et al. 2013). The upper and lower panels of Fig. 1 present the detection probability maps as a function of redshift and intrinsic X-ray luminosity for a source with NH ∼ 1024 cm−2 and NH ∼ 1025 cm−2, respectively.

thumbnail Fig. 1.

Detection probability maps as a function of the intrinsic X-ray luminosity and redshift for a source with an intrinsic column density of NH ∼ 1024 cm−2 (upper) and NH ∼ 1025 cm−2 (lower). The red lines indicate the probability of detection at 1% and 99%.

3.4. Column density, redshift, and luminosity distributions

The redshift distribution of our sample is given in Fig. 2. The redshift distribution peaks at z ≈ 0.015. In the same figure we present the distribution of the unabsorbed luminosity in the 10–50 keV band as well as the hydrogen column density, NH, distribution. The column density distribution is dominated by sources just above the threshold of NH = 1024 cm−2, while there is a significant deficit of sources with log(NH)[cm−2] > 24.5. This can most probably be attributed to a selection effect. The widest possible range of column densities can be detected only at the lowest redshifts, owing to the limited sensitivity of the BAT survey. This is evident in Fig. 3, where we plot the NH values versus the redshift of each source. It is apparent that there is a strong correlation between NH and redshift in the sense that at the highest redshifts, we can detect only Compton-thick AGN with relatively low column densities. In the previous section, we presented the detection probability as a function of intrinsic X-ray luminosity and redshift for a given column density. From Fig. 1, we observed that sources such as NGC1068 with an intrinsic luminosity LX ≈ 1044 erg s−1 and column density NH ∼ 1025 cm−2 can be detected (detection probability Pdetection > 0.5) up to a redshift of z = 0.03. A heavily obscured Compton-thick source with NH = 1025 cm−2 must have an intrinsic luminosity of LX > 3 × 1044 erg s−1 in order to be detected at the redshift limit of our survey z = 0.05.

thumbnail Fig. 2.

Distributions of the hydrogen column density (top panel), the 10 − 50 keV absorption-corrected luminosity (middle panel), and redshift (bottom panel) for our sample.

thumbnail Fig. 3.

Column density versus redshift (top panel); column density versus X-ray absorption-corrected luminosity (middle panel); X-ray absorption-corrected luminosity versus redshift (bottom panel).

4. Analysis

4.1. X-ray luminosity function

We defined the differential X-ray luminosity function, ϕ, as the number of sources N per co-moving volume V and per logarithmic interval logLX as a function of redshift, z, and luminosity, LX,

ϕ ( L X , z ) = d Φ ( L X , z ) d log L X = d 2 N ( L X , z ) d V d log L X . $$ \begin{aligned} \phi (L_X,z) = {\mathrm{d}\Phi (L_X,z)}{\mathrm{d} \log L_X} = \dfrac{d^2{N}(L_X,z)}{\mathrm{d} V \mathrm{d} \log L_X}. \end{aligned} $$(1)

Following Akylas et al. (2016) and Ananna et al. (2022), we modelled the differential luminosity function with a broken power law (Maccacaro et al. 1984; Barger et al. 2005), defined as

d Φ ( L X ) d log L X = A × [ ( L X L ∗ ) γ 1 + ( L X L ∗ ) γ 2 ] − 1 , $$ \begin{aligned} {\mathrm{d}\Phi (L_X)}{\mathrm{d} \log L_X} = A \times \left[\left(\dfrac{L_X}{L_*}\right)^{\gamma _1} + \left(\dfrac{L_X}{L_*}\right)^{\gamma _2}\right]^{-1}, \end{aligned} $$(2)

where A is a normalisation factor and L* is the break luminosity, while γ1 and γ2 are the slopes of the power law at the faint-end and the bright-end slopes, respectively.

Furthermore, we calculated the binned luminosity function for visualisation purposes following the Page & Carrera (2000) method that is based on the 1/Vmax method (Schmidt 1968; Avni & Bahcall 1980). After dividing the sample of Compton-thick sources into redshift, luminosity, and hydrogen column density bins, the binned luminosity function could be estimated as

ϕ ( L X , z , N H ) = ⟨ N ⟩ ∫ ∫ ∫ Ω ( L X , z , N H ) d V d z d log L X d log N H d z , $$ \begin{aligned} \phi (L_X,z,N_H) = \dfrac{\langle N \rangle }{\int \int \int \Omega (L_X,z,N_H) {\mathrm{d}V}{\mathrm{d}z}\, \mathrm{d} \log L_X\,\mathrm{d} \log N_H~\mathrm{d} z} , \end{aligned} $$(3)

where ⟨N⟩ is the number of sources in each bin, dVdz is the differential co-moving volume, and Ω represents the survey area (Sect. 3.3).

4.2. Fit and parameter estimation

We used Bayesian inference to estimate the parametric form of the X-ray luminosity function following the approach of Loredo (2004). The full description of the method is given in Pouliasis et al. (2024). Here, we just outline the basic concepts. Given a data set of n observations, D = {di; i = 1, …, n}, and a model for the X-ray luminosity function defined by a set of parameters Θ, according to Bayes’ theorem, the posterior probability (i.e. the probability of obtaining the selected model given the observational data) is

P ( Θ | D ) = P ( D | Θ ) P ( Θ ) P ( D ) . $$ \begin{aligned} P(\boldsymbol{\Theta } | D) = \dfrac{P(D|\boldsymbol{\Theta }) P(\boldsymbol{\Theta })}{P(D)}. \end{aligned} $$(4)

The likelihood, ℒ = P(D|Θ), is the probability of obtaining the observational data given the model. The prior, P(Θ), is the a priori probability for the parameters of the model. The term P(D) is the evidence of the model: P(D) = ∫P(Θ|D)dΘ. We derived the posterior probability distribution of the model parameters using the nested sampling Monte Carlo algorithm MLFriends (Buchner & Bauer 2017), implemented in the UltraNest package. Nested sampling algorithms allow for the tracing of the posterior distribution of the model, given a data set, while at the same time calculating the Bayesian evidence. The Bayesian approach allows for a rigorous treatment of the uncertainties in the X-ray properties of the sources. During the inference process, we assumed flat priors for the model parameters, either uniform or log-uniform, that span a reasonably broad range of the parameter space according to previous studies in the literature (Akylas et al. 2016; Ananna et al. 2022). The range of our priors is consistent with the 1/Vmax non-parametric luminosity function. In Table 1, we provide the minimum and maximum values allowed in the flat priors we used for the parameters of our X-ray luminosity function model.

Table 1.

Best-fit values and 2σ errors for the parameters of the luminosity function.

The log-likelihood of this process can be written as

ln L ( { d i } | Θ ) = − λ + ∑ i ln ∫ ∫ ∫ P i ( L X , z , N H | Θ ) d V d z dlog N H dlog L X d z . $$ \begin{aligned}&\ln {\mathcal{L} }(\{d_i\} | \boldsymbol{\Theta }) = \nonumber \\&-\lambda + \sum _i \ln \int \int \int P_i({L_{\rm X}}, z, {N_{\rm H}} | \boldsymbol{\Theta })~{\mathrm{d}V}{\mathrm{d}z}{\mathrm{dlog} }{N_{\rm H}}~{\mathrm{dlog} }{L_{\rm X}}~\mathrm{d}z. \end{aligned} $$(5)

The parameter λ is the expected number of observed sources for a Poisson process, given an X-ray luminosity function model with parameters Θ:

λ = ∫ ∫ ∫ ϕ ( L X , z , N H | Θ ) Ω ( L X , z , N H ) d V d z dlog N H dlog L X d z , $$ \begin{aligned} \lambda = \int \int \int \phi ({L_{\rm X}}, z, {N_{\rm H}} | \boldsymbol{\Theta }) \Omega ({L_{\rm X}}, z, {N_{\rm H}})~{\mathrm{d}V}{\mathrm{d}z}{\mathrm{dlog} }{N_{\rm H}}~{\mathrm{dlog} }{L_{\rm X}}~d z, \end{aligned} $$(6)

where Ω is the survey sensitivity function and ϕ is the luminosity function. The parameter Pi in Eq. 5 is given by

P i ( L X , z , N H | Θ ) = p ( d i | L X , z , N H ) ϕ ( L X , z , N H | Θ ) Ω ( L X , z , N H ) , $$ \begin{aligned} P_i({L_{\rm X}}, z, {N_{\rm H}} | \boldsymbol{\Theta }) = p(d_i | {L_{\rm X}}, z, {N_{\rm H}})~\phi ({L_{\rm X}}, z, {N_{\rm H}} | \boldsymbol{\Theta })~\Omega ({L_{\rm X}}, z, {N_{\rm H}}), \end{aligned} $$(7)

where p(di|LX, z, NH) is the probability of the source i being at redshift z with column density NH and luminosity LX. This probability is given by the posterior probability distributions we obtained during the X-ray spectral analysis. We have included in this term the sensitivity function of the survey Ω. The integral involving Pi can be calculated using an importance sampling Monte Carlo integration technique (Press et al. 1992). The integration limits used in Eq. (5) are [0,0.05], [41.0,46.0], and [23.5,26.0] for the parameters z, log LX, and log NH, respectively.

5. Results

5.1. The 10–50 keV X-ray luminosity function

Here we present the best-fit results and the confidence intervals for the parameters of the luminosity function (Eq. (2)). These parameters are the normalisation, A; the break, L*; and the faint-, γ1, and bright-end, γ2, slopes. The values together with the 68% confidence intervals are given in Table 1. In Fig. 4, we present the corner plots. The relationships between the various parameters are depicted through scatter plots of all possible sets of parameters (e.g. L⋆ versus γ1). The diagonal boxes illustrate the marginalised posterior distribution of each parameter. The break luminosity is log L⋆[erg s−1]≈43, while the faint-end of the luminosity function is quite flat, having a slope of γ 1 = 0 . 01 − 0.74 + 0.51 $ \gamma_1=0.01^{+0.51}_{-0.74} $. This suggests that there may not be ample room for a faint population of Compton-thick AGN.

thumbnail Fig. 4.

One-dimensional (diagonal panels) and two-dimensional marginalised posterior distributions for the double power-law model parameters. The shaded areas in the 2D posterior distributions correspond to 1σ and 2σ confidence levels (2D values; i.e. 39% and 86% respectively). The shaded areas for the 1D posteriors correspond to the 1σ confidence level.

In Fig. 5, we present the X-ray luminosity function in the 10–50 keV band. In the left panel, we compare the parametric luminosity function with the non-parametric one derived using the 1/Vmax method. In the right panel of Fig. 5, we compare our luminosity function with those of Akylas et al. (2016) and Ananna et al. (2022). These luminosity functions have been derived in the 20–40 keV and 14–195 keV band, respectively. We homogenised the different luminosity functions to the 10–50 keV band using the corresponding spectral models assumed by each work.

thumbnail Fig. 5.

The X-ray luminosity function in the 10–50 keV band. Left panel: Compton-thick (logNH[cm−2] = 24 − 25) X-ray luminosity function in the redshift range 0.0 ≤ z ≤ 0.05. The shaded regions represent the 68% and 95% confidence intervals. The points show the binned 1/Vmax luminosity function with the corresponding 68% uncertainties. Right panel: Our luminosity function compared with those of (Akylas et al. 2016; Ananna et al. 2022) and also by the population synthesis models of Ananna et al. (2019). The best-fitting model of the latter is evaluated at the mean redshift of our analysis (z = 0.025).

The luminosity function of Ananna et al. (2022) lies somewhat above our estimate, especially around L⋆. It has to be noted that the two luminosity functions contain different sets of objects despite the fact that they both come from the 70-month BAT survey. Moreover, it has to be taken into account that the luminosity function of Akylas et al. (2016) is not corrected for obscuration. In Fig. 5 (right panel), we also compare our luminosity function with the X-ray background synthesis models of Ananna et al. (2022). These refer to both the Compton-thick population with column densities NH = 1024 − 25 cm−2 and NH = 1025 − 26 cm−2, respectively. We can compare our luminosity function only with the former, as they refer to the same column density range. It appears the model predicts a very high number of Compton-thick AGN at low luminosities, a few times 1042 erg s−1. This is rather at odds with our findings and even with the luminosity function of Ananna et al. (2022).

5.2. Comparison with the 2–10 keV luminosity function

Next, we compare our luminosity function with other Compton-thick luminosity functions derived in the softer 2–10 keV band. Ueda et al. (2014) constructed the luminosity function in the 2–10 keV band by compiling data from various missions operating in these wavelengths. They also used the BAT nine-month survey, which contains a number of known Compton-thick AGN, to determine their fraction in the local Universe. Their luminosity function has been used to model the spectrum of the X-ray background in the energy range 1–1000 keV, resulting in very good agreement with the data. In Fig. 6, we compare the Ueda et al. (2014) with our luminosity function in the 2–10 keV band. We plot the luminosity function of Ueda et al. (2014) that refers to the 1024 − 25 cm−2 column density range. For the conversion of our luminosity function in the 2–10 keV band, we used the same spectral model used in Sections 3.2 and 3.3. Their luminosity function is in good agreement with ours at luminosities brighter than LX ≈ 1042 erg s−1. However, the former appears to present an upturn at faint luminosities.

thumbnail Fig. 6.

X-ray luminosity function (logNH[cm−2] = 24 − 25) in the 2–10 keV band. The shaded regions represent the 68% and 95% confidence intervals. The green dash line corresponds to the Compton-thick luminosity function of Ueda et al. (2014). The dotted line denotes the luminosity function of Aird et al. (2015a). Finally, the red fish-shaped diagram corresponds to the luminosity function of Buchner et al. (2014) (see text for details).

In Fig. 6 we also plot the luminosity function derived by Buchner et al. (2015) as a fish-shaped diagram. This region gives the 10%–90% quantiles as a range for the uncertainty of the luminosity function. We note that the luminosity function of Buchner et al. (2015) typically refers to the NH = 1024 − 26 cm−2 column density range. However, the reflection-dominated sources with NH = 1025 − 26 cm−2 are very sparsely sampled because of the extreme obscuration (e.g. Buchner et al. 2015). It is therefore reasonable to assume that the luminosity functions of Buchner et al. (2015) quoted in the NH = 1024 − 26 cm−2 column range provide very good approximations of the luminosity function in the NH = 1024 − 25 cm−2 column density range. Finally, we plot the luminosity function derived by Aird et al. (2015b) for the NH = 1024 − 25 cm−2 column density range.

5.3. The observed fraction of Compton-thick active galactic nuclei

We estimated the observed cumulative fraction of Compton-thick AGN in different redshift bins. For a given redshift, zi, this is defined as the ratio of the Compton-thick AGN up to zi over the total number of (non-blazar) AGN in the same redshift bin. The fraction is given in Fig. 7. The fraction is the highest in the first redshift bin z < 0.01, with f ≈ 0.17 ± 0.06. According to Fig. 1, an obscured source with NH ≈ 1024 cm−2 and LX < 3 × 1042 erg s−1 cannot be easily detected even in our first redshift bin, that is, up to z = 0.01 (≈43 Mpc). This is exacerbated at higher column densities. An AGN that is obscured by a column density of 1025 cm−2 can be detected at z = 0.01 only if it has a luminosity above LX > 1043 erg s−1. This means that our estimated fraction f = 0.17 can only be considered as a lower limit to the fraction of Compton-thick AGN.

thumbnail Fig. 7.

Observed fraction of Compton-thick AGN as a function of redshift.

5.4. The intrinsic fraction of Compton-thick active galactic nuclei based on the luminosity function

Our obscuration-corrected luminosity function provides us with the opportunity to estimate the intrinsic fraction of Compton-thick AGN. Towards this end, we estimated the intrinsic number of Compton-thick AGN in the 2–10 keV band using the following expression:

∫ z = 0 z = 0.05 ∫ logL X = 42 log L X = 45 Φ ( L ) d V d z dlog L X d z . $$ \begin{aligned} \int _{\rm z=0}^\mathrm{z=0.05} \int _{\rm logL_X=42}^{\log {L_X}=45} \Phi (L) ~{\mathrm{d}V}{\mathrm{d}z}~{\mathrm{dlog} }{L_{\rm X}}~\mathrm{d} z. \end{aligned} $$(8)

Our luminosity function predicts 112 Compton-thick AGN with a luminosity of LX > 1042 erg s−1 (10–50 keV) over the whole sky up to a redshift of z = 0.05. Then we compared with the number of Compton-thin AGN (NH = 1020 − 24 cm−2) derived using the above expression and the Ueda et al. (2014) Compton-thin luminosity function. The number of Compton-thin AGN is 359. The fraction of Compton-thick sources is defined as

f 24 − 25 = N 24 − 25 / ( N 20 − 24 + N 24 − 25 ) , $$ \begin{aligned} f_{24-25} = N_{24-25}/(N_{20-24}+N_{24-25}), \end{aligned} $$(9)

where N24 − 25 and N20 − 24 are the number densities of objects in the column density range NH = 1024 − 25 cm−2 and NH = 1020 − 24 cm−2; The above number densities are derived from this current work and Ueda et al. (2014), respectively. The fraction of Compton-thick AGN versus the total number of AGN is 24 ± 5%. The error is derived by sampling the uncertainty space of our Compton-thick luminosity function parameters at the 68% confidence level.

Next, we compare our findings with the Compton-thick fractions derived in the literature. Ricci et al. (2015) derived a fraction of 27 ± 4% by modelling the absorption distribution of the NH = 1024 − 25 cm−2 Compton-thick AGN from the BAT 70-month survey. This figure is entirely compatible with our estimates. Ueda et al. (2014) derived the absorption function in the local Universe using data from the BAT nine-month survey. They found similar fractions for Compton-thick AGN in the same column density range (NH = 1024 − 25 cm−2). In general, it appears that a consensus has been reached at least regarding the number of Compton-thick AGN with NH = 1024 − 25 cm−2 in the local Universe (see also Burlon et al. 2011; Georgantopoulos & Akylas 2019; Torres-Albà et al. 2021). All the above results have been derived on the basis of BAT detections. This is not surprising since the large pass-bands of the Neil Gehrels Swift mission facilitates the detection of Compton-thick AGN.

Recently, Boorman et al. (2024) constrained the Compton-thick fraction in the local Universe (z < 0.044) using a sample of 122 AGN primarily selected to have warm IRAS colours. By fitting the available X-ray spectra, they estimated a Compton-thick fraction of 35 ± 6%. In a similar work, Akylas et al. (2024) used a sample of WISE-selected AGN up to redshifts of z = 0.02. The estimated Compton-thick fraction is 0.25 ± 0.05 %.

Beyond the local Universe, the results based primarily on the modelling of Chandra and XMM-Newton spectra present considerable scatter. A summary of the estimated fraction of Compton-thick AGN in the NH = 1024 − 25 cm−2 column density range is given in Table 2. We note that some of these works (Buchner et al. 2015; Laloux et al. 2023) quote the fraction of Compton-thick AGN in the column density range NH = 1024 − 26 cm−2 despite the fact that the number of detected sources with > 1025 cm−2 is extremely small (see discussion in Buchner et al. 2015). For this reason, it can be safely assumed that the observed fraction of Compton-thick AGN in the NH = 1024 − 25 cm−2 column density range is approximately equal to the fraction of Compton-thick sources in the NH = 1024 − 26 cm−2 range.

Table 2.

Compton-thick AGN fractions.

It is unclear whether the fraction of Compton-thick AGN increases with redshift. Buchner et al. (2015) find that the fraction is consistent with being constant, while Lanzuisi et al. (2017) find a steep increase at high redshifts. If indeed the fraction increases with redshift, this may mark a genuine evolution of the obscuring medium in Compton-thick AGN with cosmic time. For example, it has been proposed that the excess obscuration at higher redshifts is associated with the host galaxy of the AGN (e.g. Gilli et al. 2022). Alternatively, the excess number of Compton-thick AGN could be an artefact of the moderate photon statistics at higher redshifts combined with the limited pass-band of the Chandra and XMM-Newton missions. Interestingly, Laloux et al. (2023) analysed the X-ray spectra of AGN in the COSMOS field using a novel method. They derived the 6 μm luminosity of the AGN component to use it as a prior for the determination of the X-ray obscuration. At redshifts of z < 0.5, their method yields results that are compatible with the BAT results in the local Universe. At higher redshifts, Laloux et al. (2023) could only derive upper limits for the Compton-thick fraction.

5.5. NuSTAR number counts constraints

In this section, we explore the possible constraints that can be posed by the serendipitously selected NuSTAR Compton-thick AGN. Lansbury et al. (2017b) present the sources that have been detected in the 40-month serendipitous source catalogue covering 13 deg2. Making use of this catalogue, Lansbury et al. (2017a) selected the candidate Compton-thick AGN by applying a hardness ratio criterion (i.e. a hard-to-soft band ratio BRNu > 1.7) and using the 3–8 keV and 8–24 keV bands. This hardness ratio was chosen based on the spectrum of a Compton-thick AGN when assuming the model of Baloković et al. (2014). They found four candidate sources at small redshifts within z < 0.07. Subsequent NuSTAR spectral analysis (Lansbury et al. 2017a) appears to be consistent with Compton-thick absorption. Recently, Greenwell et al. (2024) presented the new serendipitous NuSTAR catalogue covering 40 deg2. They found three additional Compton-thick AGN by applying the same hardness ratio criterion, BRNu > 1.7, within the same redshift range. Out of these seven sources, only four are within the range of our derived luminosity function. We used these candidate Compton-thick sources to derive the number count distribution in the 8–24 keV band in order to compare it with our Compton-thick luminosity function. In Fig. 8 (left), we compare the NuSTAR and BAT number counts within z = 0.05 with the predictions of our luminosity function. In the same figure (right), we compare the NuSTAR number counts within z = 0.07 with the predictions of our luminosity function extrapolated to z = 0.07 in order to exploit the better number statistics. We assigned weights for each source in the NuSTAR number counts. For sources with X-ray spectral fits in Lansbury et al. (2017a), the weight was calculated taking the full NH distribution into account. For the remaining sources, we used the error on the BRNu hardness ratio as the weight. The errors on the BAT and NuSTAR number counts were estimated following Mateos et al. (2008).

thumbnail Fig. 8.

Observed and predicted number counts in the 8–24 keV band. (a) Left: Redshift range z < 0.05. The BAT 70-month survey number counts are plotted (red shaded diagram) together with the predictions of our luminosity function (shaded curve). The NuSTAR number counts are shown with the green diagram. (b) Right: Redshift range z < 0.07. The green shaded diagram corresponds to the NuSTAR number counts. The predictions of our luminosity function are depicted with the shaded curve. The red dotted line denotes the predictions of the X-ray background synthesis model of Ananna et al. (2019) in the 1024 − 1026 cm−2 column density range.

In the same plot we give the predictions of the Ananna et al. (2019) X-ray background synthesis model. It appears that our luminosity function lies well below the NuSTAR number counts in the z < 0.07 redshift range. The luminosity function of Ananna et al. (2019) derived from their X-ray background synthesis model, which includes the constraints of the NuSTAR Compton-thick number counts, is actually much closer. Their model includes a number of reflection-dominated Compton-thick NH = 1025 − 26 cm−2 AGN equal to the number of the transmission-dominated Compton-thick AGN. Taken at face value, this result could imply that the vast majority of the Compton-thick population are associated with reflection-dominated AGN that remain undetected by BAT due to its limited sensitivity. However, caution has to be exercised in the interpretation of the NuSTAR number counts. Akylas & Georgantopoulos (2019) demonstrated that the NuSTAR number counts of the full X-ray source population in the 8–24 keV and the 3–8 keV bands are incompatible with the BAT and Chandra number counts, respectively. In particular, there is an upturn in the 8–24 keV NuSTAR number counts at fluxes of a few 10−14 erg cm−2 s−1. This could be attributed to a very strong Eddington bias at these faint fluxes Civano et al. (2015).

5.6. Possibility of a missing population of Compton-thick active galactic nuclei

Additional constraints on the number of Compton-thick AGN can be provided by the X-ray background synthesis models. As we have explained in the previous section, many of these models assume the existence of large numbers of extremely obscured Compton-thick sources with NH > 1025 cm−2. However, given the uncertainties of the X-ray background spectrum at high energies, around 30 keV, it is by no means certain that the addition of these reflection-dominated sources is required to fit the background (e.g. Akylas et al. 2012). Actually, Comastri et al. (2015) have argued that a large number of heavily buried Compton-thick AGN could be present without violating the X-ray and IR background constraints. NGC 4418 is usually considered the prototype for this population (e.g. Sakamoto et al. 2010). One way to compile large samples of these extreme sources would be to study in X-rays the volume-limited samples of nearby AGN detected in either the optical or the IR bands. Akylas & Georgantopoulos (2009) examined the XMM-Newton spectra of the Seyfert-2 galaxies in the Ho et al. (1997) sample of nearby galaxies. They found a Compton-thick AGN fraction compatible with the findings in this work. More recently, Asmus et al. (2020) compiled the most comprehensive sample of candidate AGN in the nearby Universe, z< 100 Mpc. Detailed studies of these sources with NuSTAR have provided a significant advance in the study of the less luminous local Compton-thick AGN (Akylas et al. 2024).

6. Summary

We have compiled a new sample of bona fide Compton-thick AGN based on the initial selection of candidates from the 70-month BAT survey. Then, we confirmed that these are Compton-thick sources using the column densities derived by NuSTAR spectral analysis. Our final sample consists of 44 sources up to a redshift of z = 0.05 with intrinsic luminosities as faint as LX ≈ 3 × 1041 erg s−1 in the 10–50 keV band. All of these sources have column densities in the range NH = 1024 − 25 cm−2. Our primary goal in this work is to derive a robust X-ray luminosity function for Compton-thick AGN in the local Universe and, based on this, to securely estimate the fraction of Compton-thick AGN. The derivation of the luminosity function follows a Bayesian methodology where the errors on the column densities and the luminosities are fully taken into account. Our results can be summarised as follows:

  • The luminosity function is described with a double power law where the faint- and bright-end slopes are γ 1 = 0 . 01 − 0.74 + 0.51 $ \gamma_1=0.01^{+0.51}_{-0.74} $ and γ 2 = 1 . 72 − 0.37 + 0.40 $ \gamma_2=1.72^{+0.40}_{-0.37} $, respectively. The break of the luminosity function is logL ∗ [ erg s − 1 ] = 42 . 9 − 0.43 + 0.44 $ \mathrm{logL}_*[\mathrm{erg~s}^{-1}]=42.9^{+0.44}_{-0.43} $. The flat slope of the faint end of the luminosity function rather argues against a numerous population of faint Compton-thick AGN.

  • The fraction of Compton-thick AGN relative to the total AGN population is 24 ± 5%. This was estimated using our Compton-thick luminosity function and the Compton-thin luminosity function of Ueda et al. (2014).

In conclusion, there appears to be a consensus on the number of Compton-thick AGN in the local Universe, at least when these are derived from the BAT data. At higher redshifts, a significant scatter is observed in the estimated fraction of Compton-thick AGN. If a higher fraction is indeed confirmed by future studies, such as those that will be performed with the ATHENA mission, this would suggest a strong evolution of the AGN Compton-thick population with cosmic time.

Acknowledgments

We thank the anonymous referee for very constructive comments. We acknowledge financial support by the European Union’s Horizon 2020 programme “XMM2ATHENA” under grant agreement No 101004168. The research leading to these results has also received funding from the European Union’s Horizon 2020 Programme under the AHEAD2020 project (grant agreement n. 871158).

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Appendix A: The Compton-thick active galactic nuclei

Table A.1.

The Compton-thick AGN sample.

All Tables

Table 1.

Best-fit values and 2σ errors for the parameters of the luminosity function.

In the text
Table 2.

Compton-thick AGN fractions.

In the text
Table A.1.

The Compton-thick AGN sample.

In the text

All Figures

thumbnail Fig. 1.

Detection probability maps as a function of the intrinsic X-ray luminosity and redshift for a source with an intrinsic column density of NH ∼ 1024 cm−2 (upper) and NH ∼ 1025 cm−2 (lower). The red lines indicate the probability of detection at 1% and 99%.
In the text
thumbnail Fig. 2.

Distributions of the hydrogen column density (top panel), the 10 − 50 keV absorption-corrected luminosity (middle panel), and redshift (bottom panel) for our sample.
In the text
thumbnail Fig. 3.

Column density versus redshift (top panel); column density versus X-ray absorption-corrected luminosity (middle panel); X-ray absorption-corrected luminosity versus redshift (bottom panel).
In the text
thumbnail Fig. 4.

One-dimensional (diagonal panels) and two-dimensional marginalised posterior distributions for the double power-law model parameters. The shaded areas in the 2D posterior distributions correspond to 1σ and 2σ confidence levels (2D values; i.e. 39% and 86% respectively). The shaded areas for the 1D posteriors correspond to the 1σ confidence level.
In the text
thumbnail Fig. 5.

The X-ray luminosity function in the 10–50 keV band. Left panel: Compton-thick (logNH[cm−2] = 24 − 25) X-ray luminosity function in the redshift range 0.0 ≤ z ≤ 0.05. The shaded regions represent the 68% and 95% confidence intervals. The points show the binned 1/Vmax luminosity function with the corresponding 68% uncertainties. Right panel: Our luminosity function compared with those of (Akylas et al. 2016; Ananna et al. 2022) and also by the population synthesis models of Ananna et al. (2019). The best-fitting model of the latter is evaluated at the mean redshift of our analysis (z = 0.025).
In the text
thumbnail Fig. 6.

X-ray luminosity function (logNH[cm−2] = 24 − 25) in the 2–10 keV band. The shaded regions represent the 68% and 95% confidence intervals. The green dash line corresponds to the Compton-thick luminosity function of Ueda et al. (2014). The dotted line denotes the luminosity function of Aird et al. (2015a). Finally, the red fish-shaped diagram corresponds to the luminosity function of Buchner et al. (2014) (see text for details).
In the text
thumbnail Fig. 7.

Observed fraction of Compton-thick AGN as a function of redshift.
In the text
thumbnail Fig. 8.

Observed and predicted number counts in the 8–24 keV band. (a) Left: Redshift range z < 0.05. The BAT 70-month survey number counts are plotted (red shaded diagram) together with the predictions of our luminosity function (shaded curve). The NuSTAR number counts are shown with the green diagram. (b) Right: Redshift range z < 0.07. The green shaded diagram corresponds to the NuSTAR number counts. The predictions of our luminosity function are depicted with the shaded curve. The red dotted line denotes the predictions of the X-ray background synthesis model of Ananna et al. (2019) in the 1024 − 1026 cm−2 column density range.
In the text

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