Prospects of detecting soft X-ray emission from typical WHIM filaments around massive clusters and the coma cluster soft excess

ABSTRACT

While hot ICM in galaxy clusters makes these objects powerful X-ray sources, the cluster’s outskirts and overdense gaseous filaments might give rise to much fainter sub-keV emission. Cosmological simulations show a prominent ‘focusing’ effect of rich clusters on the space density of the warm-hot intergalactic medium (WHIM) filaments up to a distance of |$\sim 10\, {\rm Mpc}$| (∼ turnaround radius, r_ta) and beyond. Here, we use Magneticum simulations to characterize their properties in terms of integrated emission measure for a given temperature and overdensity cut and the level of contamination by the more dense gas. We suggest that the annuli |$(\sim 0.5-1)\times \, r_{ta}$| around massive clusters might be the most promising sites for the search of the gas with overdensity ≲ 50. We model spectral signatures of the WHIM in the X-ray band and identify two distinct regimes for the gas at temperatures below and above |$\sim 10^6\, {\rm K}$|⁠. Using this model, we estimate the sensitivity of X-ray telescopes to the WHIM spectral signatures. We found that the WHIM structures are within reach of future high spectral resolution missions, provided that the low-density gas is not extremely metal-poor. We then consider the Coma cluster observed by SRG/eROSITA during the CalPV phase as an example of a nearby massive object. We found that beyond the central r ∼ 40 arcmin (⁠|$\sim 1100\, {\rm kpc}$|⁠) circle, where calibration uncertainties preclude clean separation of the extremely bright cluster emission from a possible softer component, the conservative upper limits are about an order of magnitude larger than the levels expected from simulations.

1 INTRODUCTION

Warm-hot intergalactic medium (WHIM) with temperature from 10⁵ K to a few times 10⁶ K and overdensity (relative to the mean baryonic density of the Universe) from a few to a few tens is believed to contain almost a half of all baryons in the modern Universe (Cen & Ostriker 1999; Davé et al. 2001; Bertone et al. 2010; Martizzi et al. 2019; Tuominen et al. 2021). On the low temperature and density end, it is located in sheets of matter collapsed in only one direction, while on the high temperature and density end, it corresponds to the hot and tenuous outskirts of virialized objects, i.e. galaxies, groups, and clusters.

The relatively high temperature and low density of this gas make its direct observation very hard. Indeed, on the one hand, thermally-excited emission is very faint and falls into UV and soft X-ray bands, suffering from high line-of-sight absorption and foreground emission of our own Galaxy. On the other hand, the degree of hydrogen and helium ionization in this medium is extremely high (due to both collisional and photoionization in the radiation field of cosmic X-ray and UV background) prohibiting any detectable absorption signal akin to Lyman forest absorption readily observed in the spectra of quasars at redshifts above two.

This picture changes dramatically if WHIM is allowed to be significantly enriched with metals, which are capable of staying only partially ionized under WHIM-relevant physical conditions. Both emission (Paerels et al. 2008) and absorption (Richter, Paerels & Kaastra 2008) in lines of the metals have long been considered as the most promising venue for the direct observation of this medium, in particular, thanks to the exquisite redshift-sensitivity of the signal enabling the possibility of cross-correlation of the X-ray signal with galaxies overdensities inferred from the optical surveys (Bregman et al. 2019; Nicastro, Fang & Mathur 2022 for recent reviews).

An alternative approach is to focus on the overdense regions which are naturally present in the vicinity of massive galaxy clusters. Namely, the expected overdensity of all structures near a turn-around radius of the cluster is ∼2–3 (see e.g. fig. 2 in Diemer & Kravtsov 2014 or fig. 5 in O’Neil et al. 2021), and one can expect WHIM gas to be concentrated there as well, in particular in the form of the matter filaments with an overdensity of few tens. As a result, the excess X-ray emission from such regions would naturally contain the contribution of WHIM, and this can be used even without the selection of individual filaments traced by the overdensity of galaxies at the redshift of the central cluster.

Given that the surface brightness of the WHIM emission does not depend on the distance to the object (for small redshifts), it is primarily the sky area covered by the object that determines the total integrated X-ray signal. In this regard, deep observations of nearby massive clusters are well suited for the purposes of WHIM detection with X-ray telescopes featuring large Field of View (FoV).

For future X-ray missions with microcalorimetric spectral resolution in the soft X-ray band (≲ a few eV near energies of the most prominent oxygen lines at 0.5–0.7 keV), the situation is different thanks to the possibility of selecting redshift windows which are less affected by the Galactic foreground emission, effectively reducing the astrophysical background level from the bright foreground emission lines by two orders of magnitude. In this case, deep observations of ∼5–10 Mpc regions around massive clusters at redshift ∼0.1 offer best opportunities.

The Coma cluster (Abell 1656) is particularly interesting in the context of the WHIM detection in X-rays since evidence for such emission has been reported based on EUVE and ROSAT data (e.g. Lieu et al. 1996, 1999; Bonamente et al. 2022). This object is one of the few X-ray brightest nearby (z = 0.0231) massive (⁠|$M_{500}\sim 6\times 10^{14}\, M_\odot$|⁠; Planck Collaboration X 2013) clusters, which subtends several square degrees on the sky. Here we revisit the question of the so-called ‘soft-excess’, i.e. a distinct spectral component at sub-keV energies, which might be associated with |$\sim 10^6\, {\rm K}$| gas in the vicinity of Coma.

The paper is organized as follows. In Section 2 we introduce quantities that we use to characterize the amount of gas (hydrogen and helium) associated with the WHIM. In Section 3 we use the data of numerical simulations in order to extract these quantities from the data of numerical simulations. We then analyze the excess of the WHIM gas within a turn-around radius near massive clusters (Section 4). The level of the WHIM signal contamination by denser gas present in the simulation is estimated in Section 5. The expected spectral signatures of the WHIM gas are calculated in Section 6. The sensitivity of X-ray telescopes to this signal is estimated in the same section. In Section 7 the observations of the Coma cluster with SRG/eROSITA are discussed and compared with the predictions of numerical simulations. In Section 8 we discuss the implications of our results in the context of WHIM search strategies and mention a possible boost of the WHIM signatures by the ‘side illuminations’. Finally, Section 9 summarizes our findings.

Throughout the paper, we assume a flat Lambda cold dark matter (ΛCDM) cosmology with Ω_m = 0.3, Ω_Λ = 0.7, H₀ = 70 km s⁻¹ Mpc⁻¹. For Ω_b we use Planck-2015 value 0.0486 (Planck Collaboration XIII 2016).

2 MEASURES OF LINE-OF-SIGHT EMISSION AND THOMSON OPTICAL DEPTH

As we discuss later in the text, the detectable X-ray signal associated with the WHIM largely comes from spectral signatures of metals (and their amplitude scales essentially linearly with the total gas metallicity), except for a pure Thomson scattering and the part of the bremsstrahlung emission due to hydrogen and helium. The latter two components do not have sharp spectral features and it is not easy to differentiate them from other backgrounds and foregrounds. On the other hand, parameters like gas density and temperature of the WHIM appear to be more robustly predicted by numerical simulations than metal abundances. It therefore makes sense to use densities and temperatures from simulations and treat the metal abundance as an additional parameter. In this section, we discuss only the density-related quantities, which determine the amplitude of the X-ray signals, either in absorption or in emission.

A convenient and commonly used way of characterizing the density-related dependence of the intensity of an optically thin plasma emission (when collisions are the dominant excitation mechanism) is via the line-of-sight emission measure

which we have reformulated in terms of the critical.¹ density of the Universe |$\rho _c\approx 9.21\times 10^{-30}\, {\rm g\, cm^{-3}}$| and the baryon fraction Ω_b = 0.0486 at z = 0. μ ≈ 0.608 is the mean atomic weight for fully ionized plasma with the abundance of heavy elements relative to the Solar photosphere Z/Z_⊙ = 0.2, m_p is the proton mass, |$\left(\frac{n_e}{n_t}\right)\approx 0.52$| and |$\left(\frac{n_H}{n_t}\right)\approx 0.44$| are the fractions of electrons and protons relative to the total number of particles (including helium and heavier elements), (1 + δ) is the gas density relative to the mean baryon density and L is the length of the region occupied by the gas along the line of sight.

For z = 0, we can define a factor

and |$\langle n_e\rangle =2.3 \times 10^{-7}\, {\rm cm^{-3}}$|⁠, |$\langle n_H\rangle =1.9\times 10^{-7}\, {\rm cm^{-3}}$|⁠. As a function of z, η obviously changes as |$\eta =\eta _{\, 0}(1+z)^6$|⁠.

When dealing with objects, which size is small compared to cosmological scales, it is convenient to move η outside of the integral and characterize the line-of-sight emission measure by the quantity

where l is in units of Mpc. We use this approach below and often quote EM_l in units of Mpc. With this definition, for a 10 Mpc size object with overdensity δ = 9, |${\rm EM_{\it l}}=(9+1)^2\times 10=10^3 \, {\rm Mpc}$|⁠.

For comparison with the emission measure normalization convention in a widely used xspec package (Arnaud 1996), it is convenient to provide conversion coefficients from EM_l to |${\rm K_{XSPEC}}=\frac{10^{-14}}{4\pi \left[D_A (1+z)\right]^2}\int n_e n_H dV$|⁠. Since were are dealing with the diffuse emission, we give the conversion factors per unit of solid angle: for |${\rm EM_{\it l}=1\, Mpc}$|⁠, K_XSPEC = 1.08 × 10⁻⁴(1 + z),  3.30 × 10⁻⁸(1 + z), and 9.16 × 10⁻¹²(1 + z) per steradian, square degree, and square arcminute, respectively.

Another, similarly useful, quantity is the Thomson optical depth of a gas lump, which for a uniform density at z ∼ 0 is

Below, we use both quantities to characterize the expected signal from the low-density gas. It turns out that for the WHIM gas in the relevant range of overdensities, EM_l is a more suitable measure for temperatures |$T\gtrsim 10^6\, {\rm K}$|⁠, while at low temperature τ_T is more appropriate, because of photoionization effect of the CXB photons.

3 EXPECTED LEVELS OF THE WHIM EMISSION MEASURE FROM COSMOLOGICAL SIMULATIONS

A number of cosmological hydro simulations have been used to predict density, temperature, and spatial distribution of the WHIM, either using Eulerian (Cen & Ostriker 1999; Smith et al. 2011; Shull, Smith & Danforth 2012; Planelles et al. 2018; Vazza et al. 2019; Zhu, Zhang & Feng 2021), or Lagrangian (Davé et al. 2001; Springel & Hernquist 2003; Yoshikawa et al. 2003; Bertone et al. 2010; Rahmati et al. 2016; Khabibullin & Churazov 2019; Martizzi et al. 2019; Galárraga-Espinosa et al. 2021; Tuominen et al. 2021) approaches.

To estimate the X-ray signal that can be associated with the WHIM gas, we used two cubical cut-outs from the Magneticum² suite of cosmological simulations (Dolag, Komatsu & Sunyaev 2016).

Specifically, we extracted two boxes of 57.5 Mpc on each side from the full simulation Box2b/hr at redshift z = 0.252 (the final redshift for this simulation). The Box2b/hr simulation has the total comoving volume of (640 h⁻¹ cMpc)³ with dark matter m|$_\mathrm{DM}=6.9\times 10^8 \ \rm {M_\odot }$| and gas m|$_\mathrm{gas}=1.4\times 10^8 \ \rm {M_\odot }$| mass resolutions across it, simulated with 2880³ particles by an improved version (Beck et al. 2016) of the N-body code gadget 3, which is an updated version of the code gadget 2 (Springel 2005) involving a Lagrangian method for solving smoothed particle hydrodynamics (SPH).

In addition to solving for the cosmological hydrodynamical evolution of the gas, the simulation follows the evolution of various metal species and their relative composition in the interstellar, circumgalactic, intracluster, and intergalactic media via computing continuous enrichment by supernovae of type Ia and type II, and asymptotic giant branch star winds in the fully cosmological context. The enrichment is computed self-consistently with the underlying evolution of the stellar populations (for details, see Tornatore et al. 2007) and matter outflows from the galaxies initiated by star formation and active galactic nuclei (AGN) feedback (Hirschmann et al. 2014). The simulations were capable of reproducing detailed metal distributions within galaxies and galaxy clusters at redshift z ∼ 0 (Dolag, Mevius & Remus 2017), while achieving significant enrichment of the intergalactic medium already at z ∼ 2–3 (Biffi et al. 2017).

The first cut-out box was centred on the second most massive cluster in the simulated volume, while the second one was taken at a random position to serve as a reference box. Namely, the cluster has the total mass of M_{500, c} ≈ 7.5 × 10¹⁴ h⁻¹M_⊙ ≈ 10¹⁵ M_⊙ and R_{500, c} = 1.3h⁻¹ Mpc≈1.8 Mpc.

For a massive galaxy cluster at z ∼ 0, the turn-around radius (the position of the non-expanding shell which is furthest away from the cluster centre) is equal to ∼3R_200m ∼ 10 Mpc, where R_200m is the radius for which mean enclosed matter density exceeds the mean matter density ρ_m of the Universe by a factor of 200. This relation can be obtained from the self-similar spherical collapse model (Fillmore & Goldreich 1984; Bertschinger 1985) specialized for ΛCDM (e.g. Shi 2016) or derived from numerical simulations (e.g. Hansen et al. 2020; Korkidis et al. 2020). The mean enclosed density within the turn-around radius is ρ(< R_ta)/ρ_m ∼ 10 (e.g. Shi 2016; Tanoglidis, Pavlidou & Tomaras 2016). At the turn-around radius, numerical simulations predict local density |$\displaystyle \rho (R_{ta})/\rho _m \simeq 2-3$| (see e.g. fig. 2 in Diemer & Kravtsov 2014 or fig. 5 in O’Neil et al. 2021). Clearly, our cut-outs from the simulations are few times bigger than the turn-around radius of the cluster, and we expect convergence to the ‘mean’ properties in the outer parts of the cluster box.

The particle-based data of the SPH simulation were remapped onto a uniform cartesian grid with the voxel size of (100 kpc)³, where we used a so-called gather approach to compute the SPH smoothed quantities at the centre of the grid cells, resulting in 575³ boxes with physical parameters of the gas density, temperature, and metallicity within them. Although this size of the grid cell is significantly bigger than the smoothing length size of the particles within strongly overdense structures, e.g. in the inner parts of galaxies, groups, and clusters, the primary focus of the current study is on the diffuse intergalactic medium, where such gridding is fully adequate. The consistency of the grid-based and particle-based predictions for emission and absorption by the intergalactic medium has been confirmed by both statistical and point-by-point direct comparisons between them.

In Fig. 1, we show maps of EM_l projected along one of the axes of the box. The upper and lower rows show such maps for the ‘cluster’ and ‘reference’ boxes, respectively. The maps shown also differ by the selection of cells contributing to EM_l. In particular, the left column (A) shows the total emission measure including all cells with overdensity δ < 10⁴. On the contrary, the right column (D) corresponds to the most stringent selection criteria: only the low overdensity (δ < 50) cells with the gas temperature in the range |$0.5\times 10^6\, {\rm K}\lt T\lt 2\times 10^6 \, {\rm K}$|⁠. The middle two panels illustrate the importance of selecting only the gas with the temperature |$\sim 10^6\, {\rm K}$| and excluding dense regions, which likely correspond to cores of virialized halos. In particular, columns B and C use the same temperature range, but different cuts in overdensity: δ < 10⁴ and δ < 200 for B and C, respectively. We note in passing, that (1 + δ) refers to the local density, rather than to the enclosed density, which for virialized halos is always larger. Therefore, for halos, the above overdensity cuts effectively remove cores that correspond to a factor of 1.5–2 larger enclosed overdensity.

The selection cuts used above are further illustrated in Fig. 2, which show the mass-weighted gas distribution in the overdensity-temperature plane for the ‘cluster’ and ‘reference’ cubes. The white dashed line delineates the selection criteria corresponding to column B in Fig. 1, namely |$0.5\times 10^6\, {\rm K}\lt T\lt 2\times 10^6\, {\rm K}$| and δ < 10⁴. In the simulated data, these cuts account for ∼20 per cent of all baryons in the box with ∼43 per cent and ∼33 per cent in colder and hotter phases, respectively. As we show below, low-density and cool gas is still able to produce observable signatures in the X-ray spectra. We therefore defined as ‘WHIM’ the gas with the density (1 + δ) < 50 and the temperature in the range |$10^4\, {\rm K}\lt T\lt 2\times 10^6\, {\rm K}$|⁠. To indicate the extended range of temperatures and the overdensity cut, we will call this phase ‘eWHIM50’. As a further justification of the density and temperature cuts used here, we note that for a massive halo, a local matter overdensity ∼40–50 is found near the radius r_200m where the enclosed matter overdensity if a factor of 200 larger than the mean matter density of the Universe, (e.g. Diemer & Kravtsov 2014; O’Neil et al. 2021). In simulations, the radial density profile is the steepest in the vicinity of r_200m, suggesting that this radius can be used as one of the definitions of a boundary separating the cluster gas from a more diffuse WHIM. Our (1 + δ) < 50 cut therefore broadly corresponds to this definition. We have also completely ignored the gas with |$T\lesssim 10^4\, {\rm K}$| when considering the WHIM. Such gas might be present either as cold and dense clumps in galaxies or a low-density gas outside halos. In the former case, no X-ray emission is expected and the absorption signatures are confined to galaxies. In the latter case, the density and the mass of such gas are so small that it is safe to ignore this phase.

The abundance of metals predicted by simulation for different gas phases is shown in Fig. 2. The abundance is relative to the Solar photospheric abundance and is estimated as the ratio of the total mass of elements heavier than helium to the hydrogen mass and scaled by the Solar value of ≈2.6 × 10⁻² (e.g. Anders & Grevesse 1989). We note here, that the scatter of abundances inside each overdensity/temperature bin can be substantial (see Appendix  B) and the mean value does not necessarily characterize the bulk of the gas with a given density and temperature. However, it is this value that one would get if the components with different abundances are physically co-spatial and mixing between them is allowed to take place (the enrichment model implemented in Magneticum does not allow metals exchange between the particles).

As expected, EM_l maps (Fig. 1) show a network of filaments converging on the massive cluster. These filaments are often a few tens of Mpc long and ∼4-6 Mpc thick. Once the selection based on the gas temperature (or temperature and overdensity) is applied, the inner parts of the filaments (inside the hot and dense cluster region) become suppressed. In the reference box maps, similar filaments are present, but their number is significantly smaller, and in the absence of a prominent cluster at the centre of this box, the distribution of filaments does not show clear convergence of filaments on a single centre. Instead, there is one very prominent and largely one-dimensional filament that is traced by a chain of lower-mass halos. At the edges of the ‘cluster’ and ‘reference’ boxes their appearance is similar, suggesting that at these distances (⁠|$\gtrsim 20\, {\rm Mpc}$|⁠) the influence of the overdensity around the main cluster on the distribution of matter is not very strong. This, in particular, reflects the higher density of all types of structures near the turn-around radius of a massive cluster.

4 RADIAL PROFILES

A more quantitative characterization of EM_l maps is sheown in Fig. 3 left-hand panel, where the radial profiles for concentric rings around the cluster are shown. The horizontal lines show the mean levels of EM_l in the reference box with the same temperature and overdensity cuts. For a homogeneous box having the mean baryonic density of the Universe, the expected value of EM_l is, obviously, the size of the box, i.e. 57.5 Mpc.

These radial profiles illustrate several characteristic properties:

• The level of EM_l is a strong function of the applied gas phase selection criteria.

• For all temperature and overdensity cuts, the cluster box profiles converge to the mean level of the reference box beyond ∼20 Mpc, i.e. ∼5–7R_200m. This further strengthens the suggestion that the chosen size of the box is large enough and the impact of the cluster on the matter density at larger distances is subdominant, at least for the purpose of this study.

• Inside ∼10–20 Mpc, the projected emission measure due to ‘focused’ filaments is significantly higher than for the reference box.

• In the inner region (⁠|$\lt 1\, {\rm Mpc}$|⁠), the scatter in the EM_l radial profiles for the warm |$10^4T\lt 2\times 10^6\, {\rm K}$| gas is large, as illustrated by error bars shown in the right-hand panel of Fig. 3. These error bars were crudely evaluated as the root-mean-square variations among radial profiles calculated for three different projections (along the X, Y, and Z axes of the simulated box). The level of the projected emission measure can be higher or lower than the corresponding values for the reference box. This partly reflects the fact, that not much of the 10⁶ K gas is present inside the cluster and, more importantly, the stochastic fluctuations due to a highly inhomogeneous gas density distribution in cluster outskirts are dominated by a rather small number of the most prominent filaments.

• There is a range of projected radii (⁠|$\sim 1 - 10 \, {\rm Mpc}$|⁠), where the scatter is moderate.

Given that both the cluster and reference boxes have similar levels of EM_l beyond ∼20 Mpc, it makes sense to calculate the ‘excess’ emission associated with the focusing effect of the cluster on the distribution of filaments. We do this by subtracting the mean levels obtained for the reference box from the radial profiles. The results are shown in the right-hand panel of Fig. 3. There we average three projections to reduce the scatter and we assign an error bar to each radial bin, which characterizes this scatter.³ Unlike the left-hand panel of Fig. 3 the excess emission should be independent of the size of the box (once it is larger than the correlation length in the density distribution). Fig. 3 (right-hand panel) also corroborates the conjecture that the range of radii of ∼5–10 Mpc is well suited for the search of |$T\sim 10^6~K\, {\rm gas}$| in the vicinity of a massive cluster (apart from possible observational limitations).

Instead of calculating the excess emission of the WHIM in radial bins, one can search for bright localized spots associated with the WHIM within 5–10 Mpc of the cluster. The probability of finding EM_l above a given value in a projected image of the simulated ‘cluster’ box can be characterized by the probability distribution function PDF(EM_l), which is shown in Fig. 4 for the entire box and the 5–10 Mpc ring. This plot shows that the probability of finding excess values of |${\rm EM_{\it l}}\sim 10^3\, {\rm Mpc}$| associated with eWHIM50 is ∼10 per cent in the 5–10 Mpc ring. We repeated the same procedure for two other massive clusters from the Magneticum and found very similar probability distribution functions of the EM_l.

5 CONTAMINATION OF THE WHIM SIGNAL BY HIGH-DENSITY REGIONS

One of the issues that might hamper the detection of the WHIM signal is its contamination by higher-density gaseous lumps seen in projection. In Figs 1, 3, and 4, the regions of high-overdensity have been removed from the original (3D) data. Of course, this can only be done for the simulated data. The discussion on what approach should be used in application to real data (see e.g. Hickox & Markevitch 2007) is beyond the scope of this paper. However, we can simply estimate the probability of such contamination by using the same simulation data. The size of the cube, corresponding to the redshift bin of Δz ∼ 0.01 is well suited for this purpose. The chances of a random superposition of two structures can already be easily estimated from Fig. 4.

However, the largest signal is coming from collapsed halos and from the regions adjacent to them. It is therefore likely that a random superposition approach severely underestimates the level of the contamination. To characterize this effect we have calculated the ratio of the two EM_l maps

where the map in the numerator accounts for the WHIM only, while the denominator includes all gas phases. The larger the value of R, the higher the contribution of the WHIM-like gas to the total signal. Therefore, masking the regions with low R retains a cleaner WHIM signal. The impact of the masking is illustrated in Fig. 5 for R > 0.2 and 0.5. In principle, the low value of R does not mean that it is not possible to distinguish the contributions of the low and high-density components, provided that their spectra are sufficiently different. On the other hand, R > 0.5 ensures that the signal is indeed dominated by the low-density gas. All in all, this exercise suggests that if real WHIM properties are reproduced by simulations used here, then (i) about half of the area must be excluded (for the 5–10 Mpc annulus around the most massive clusters) and (ii) the maximum amplitude of the WHIM signal in the ‘uncontaminated’ regions reduces by a factor of ∼2 compared to the unmasked case. This is further illustrated by Fig. 6 that shows the probability of finding EM_l above a given value for the projected images without contaminated regions.

6 EXPECTED SPECTRA AND DETECTION SENSITIVITY

Given the density and temperature of the WHIM gas, we calculate the ionization balance of heavy elements by taking into account photoionization by cosmic UV and X-ray background radiation fields at z ∼ 0 (as we have done in Churazov et al. 2001; Khabibullin & Churazov 2019). For the thermal emission, MEKAL model (Mewe, Gronenschild & van den Oord 1985; Liedahl, Osterheld & Goldstein 1995) as implemented in the xspec package (Arnaud 1996) was used. A comparison of the predicted spectra with the results from Khabibullin & Churazov (2019) that uses cloudy (see Ferland et al. 2017) shows good consistency that is fully sufficient for the purposes of this study. The role of photoionization is particularly important for low temperatures and small overdensities. Indeed, as illustrated in Fig. 7, photoionization by CXB photons leads to the appearance of O vii and O viii ions over a broad range of (low) temperatures and overdensities, which are characteristic of the WHIM component. We note that while the time needed to establish ionization equilibrium in the low-density WHIM phase can be long (see e.g. fig. 3 in Yoshikawa & Sasaki 2006), this mostly concerns the case of recombining hot plasma. Here, we are more focused on the case of photoionized plasma, when the CXB photons provide relatively fast channels of reaching approximate ionization equilibrium for warm gas. We therefore assume that the ionization equilibrium has been reached.

The following processes can contribute to the potentially observable X-ray spectral signatures (either in emission or absorption) of a given gas lump:

• Photoelectric absorption, Thomson, and resonant scattering of the background X-ray emission, dominated by the CXB. For the WHIM case, the role of the Thomson scattering is not particularly important. These processes remove CXB photons from the line of sight going through the WHIM, resulting in a decrement of the CXB emission in direction of the gas lump compared to an ‘empty’ field.

• Thermal emission of the gas, where due to the strong contribution of photoionization, the role of Radiative Recombination Continuum (RRC) is especially prominent. We also note here, that the photoelectric absorption and RRC can partly cancel each other – these are two reciprocal processes that affect the spectrum above the ionization edges of corresponding ions.

• Fluorescent and resonantly scattered photons. Fluorescent photons are not important for the WHIM emission unlike the resonantly scattered photons at the energies of the most prominent lines, principally of O vii and O viii. As discussed in Churazov et al. (2001) and Khabibullin & Churazov (2019), this component is visible only if a significant fraction of the X-ray background is resolved or there is an additional bright source that illuminates the gas from a side (an AGN or a galaxy cluster).

All these spectral components are shown in the left-hand panel of Fig. 8. In this plot, the absorbed and emitted spectra are shown separately. In reality, only the sum of all components can be seen in the direction of a filament, with the additional possibility to resolve some fraction of compact sources that compose CXB.

The presence or absence of a filament can be tested by comparing the spectrum toward the filament and the spectrum toward an ‘empty’ field. Here we assume that the spectrum of the ‘empty’ field consists of a pure CXB and that the fraction of the resolved CXB is the same as in the filament data. Of course, there are other foreground/background components contributing to the observed spectra, but those will cancel once the difference ‘filament’ minus ‘empty’ is calculated. This difference is essentially the signal that we are looking for.

The corresponding spectrum is shown in the right-hand panel of Fig. 8 for some fiducial parameters of a WHIM filament. One of the spectra corresponds to the case when the CXB is not resolved (blue line). In this case, a negative signal at low energies is due to photoelectric absorption, while the positive signal is due to fluorescence, recombination, and collisionally excited lines. For comparison, the red line shows the difference spectrum (hereafter on-off spectrum) when the CXB is fully resolved (hypothetical case). In this limit, the photoelectric absorption signal is gone, while the resonant lines become very prominent.

Apart from the difference spectrum, the spectrum of resolved CXB sources collected from the filament region will feature absorption lines that collectively account for the same number of photons that are seen as resonantly scattered lines in the difference spectrum. This is of course true only on average. As discussed in Khabibullin & Churazov (2019) comparing the absorbed and emitted flux one can determine the properties of the gas that constitutes the filament.

Here, we focus more on the detection of the WHIM via the on-off spectra. Fig. 9 shows how the on-off spectra look like when convolved with the spectral response (a combination of the effective area and the energy redistribution matrix) of different instruments, namely, ROSAT/PSPC (Truemper 1982), SRG/eROSITA (Predehl et al. 2021), and proposed soft X-ray microcalorimeter LEM (Kraft et al. 2022). For all plots, the gas metallicity is set to 0.1, overdensity δ = 30, the Thomson optical depth τ_T = 10⁻⁴, and foreground absorbing column density |$N_H=10^{20}\, {\rm cm^{-2}}$|⁠. The resolved CXB fraction is set to zero for ROSAT/PSPC and to 0.5 for eROSITA and LEM. It is assumed that the filament fills the entire FoV, which is a strong assumption, especially for PSPC which has the largest FoV.

The spectra, shown in Fig. 9 have been convolved with the standard spectral responses of these instruments. At low temperatures, the on-off spectrum can be negative. This is especially clear in the PSPC and LEM spectra. As discussed above, the negative features are caused by the photo-electric absorption of the CXB signal in the direction of the filament. This negative signal would diminish if most of the CXB signal is resolved, although this is not feasible with these three instruments. At higher temperatures, negative parts go away and the spectra become more close to the thermal emission plus resonantly scattered emission in the lines of elements like oxygen or neon. The latter signal depends linearly on the CXB resolved fraction.

One can use the on-off model spectra to estimate the sensitivity of a given instrument to the presence of the filament. Since the spectra shown in Fig. 9 represent the difference between the on-filament and off-filament spectra, one has to take into account the inevitable Poissonian noise of the signal that comes from the full spectrum. To this end, we model the total spectrum in the filament direction as a combination of five components: the Local Hot Bubble, the Milky Way diffuse emission, the unresolved part of the CXB, the instrumental background, and the filament emission itself. We assume that the ‘off-filament’ spectrum is known and it does not introduce additional errors. Furthermore, we assume that the abundance, temperature, and overdensity of the gas inside the filament are known. With these assumptions, the only unknown quantity is the normalization of the filament signal. We further assume that the exposure is long enough so that the χ² statistic is applicable. With these assumptions, the sensitivity was calculated for several values of overdensity, temperature, and abundance.

In Fig. 10 we show the 3σ values expressed in terms of EM_l (blue curves) and the Thomson optical depth (red curves). All curves with different metalicities are scaled to Z/Z_⊙ = 0.2. Clearly, the error on the filament strength scales inversely with Z. This result is a natural consequence of the fact that much of the signal is associated with spectral signatures of metals. It is also clear that there are two different regimes and low and high temperatures, respectively.

At low temperatures (⁠|$T\lesssim 10^6\, {\rm K}$|⁠), the production of photons via excitation of important transition by electron collisions is subdominant and the expected signal is dominated by CXB photons which are ‘transformed’ by interactions with the WHIM, namely absorbed photons are re-emitted as the recombination radiation and fluorescent lines, while the resonant scattering changes the direction of photons in the lines of heavy elements, principally, oxygen. In this regime, the strength of the signal scales approximately linearly with τ_T. For instance, for |$T=10^5\, {\rm K}$| and (1 + δ) ≳ 10 the fraction of O vii ions decreases with density (see e.g. Churazov et al. 2001; Yoshikawa et al. 2003; Khabibullin & Churazov 2019; Wijers et al. 2019) and the column density of these ions remains weakly sensitive to the actual gas density (see also Appendix  A).

In the opposite limit (⁠|$T\gtrsim 10^6\, {\rm K}$|⁠), the collisional excitation/ionization can be much more efficient than the photoionization, provided that the gas density is not too low. As a result, even for moderately high densities, the dominant signal is largely due to thermal emission that scales as the density squared. The same scaling remains valid in the low-density limit when photoionization strongly dominates. As a result, the fraction of O vii and O viii ions becomes a linear function of density, and therefore the total signal scales with the density squared. Fig. 10 clearly illustrates this behavior: in terms of EM_l, all sensitivity curves overlap with each other (once scaled with the abundance).

From Fig. 10 it follows that for eROSITA (with the exposure time accumulated in the CalPV phase towards the Coma cluster) the levels of the detectable signal corresponds to |${\rm EM_{\it l}} \approx 10^{4}(Z/\mathrm{ Z}_\odot /0.2)\, {\rm Mpc}$| for |$T\lesssim 2\times 10^6\, {\rm K}$|⁠. This value exceeds expectations by ∼ order of magnitude. For LEM, exposure times ∼1 Ms would bring the sensitivity down to |$\approx 10^{3}(Z/Z_\odot /0.2)\, {\rm Mpc}$|⁠, i.e. close to expectations. From Fig. 10 it is also clear that detecting |$T\gtrsim 3\times 10^6\, {\rm K}$| gas is much easier to detect than the cooler gas (see also Fig. A5 in the Appendix). Given the linear scaling of the signal with Z, the abundance of metals at the level of a few per cent of the Solar values would make the detection of this gas extremely difficult.

7 COMA CLUSTER

We now proceed with the comparison of the aforementioned spectral model and the estimates of EM_l from the SRG/eROSITA observations of the Coma cluster.

7.1 Observational data

The SRGX-ray observatory (Sunyaev et al. 2021) was launched on 2019 July 13 from the Baikonur cosmodrome. It carries two wide-angle grazing-incidence X-ray telescopes, eROSITA (Predehl et al. 2021) and the Mikhail Pavlinsky ART-XC telescope (Pavlinsky et al. 2021), which operate in the overlapping energy bands of 0.2–8 and 4–30 keV, respectively.

The main data set used here is similar to the one described in (Churazov et al. 2021, hereafter Paper I). Briefly, dedicated SRG observations of the Coma cluster were performed in two parts, on 2019 December 4–6 and 2020 June 16–17, in the course of Calibration and Performance Verification phases (hereafter CPV). The observatory was operating in a scanning mode when a rectangular region of the sky is scanned multiple times to ensure uniform exposure of the region at the level of |$\sim 20\, {\rm ks}$| per point. For the spectral analysis, we use the data of five eROSITA telescopes equipped with the on-chip filter. These telescopes have a very similar response and are not prone to optical light leakage that might complicate the analysis of the soft part of the spectrum (see Predehl et al. 2021). In addition, we used the data from the SRG/eROSITA All-sky Survey in order to determine the typical sky background level around the cluster.

7.2 Results

The spectra were extracted from a set of concentric annuli centred on the Coma cluster (see Fig. 11). The width of each annulus is 20 arcmin (⁠|$\sim 560\, {\rm kpc}$|⁠), so that the outermost annulus goes up to |$\sim 3.4\, {\rm Mpc}$|⁠.

Since we are looking for signatures of faint emission from the |$T\sim 10^6\, {\rm K}$| gas, the magnitude of Galactic interstellar absorption is very important in order to properly estimate the flux. Shown in Fig. 12 is the total effective column density (NH_t) of the Milky Way in the region ∼16 × 16 square degrees in size that includes the Coma cluster. The value of NH_t was evaluated by combining the HI data from the HI4PI survey (HI4PI Collaboration 2016) and E(B − V) map (Meisner & Finkbeiner 2015) following the recipe described in Willingale et al. (2013).

The eROSITA background can be schematically decomposed into three components: (1) intrinsic detector background, (2) X-ray background due to distant sources well beyond the boundaries of the Milky Way, e.g. AGN, clusters, galaxies, etc., and (3) X-ray background associated with the compact and diffuse sources in the Milky Way. The first component (detector background) is, of course, not affected by extinction. For the second component (distant objects) the total NH_t determines the absorption. The third component is the foreground (for Coma) and is mostly associated with the Milky Way soft diffuse emission, which itself consists of several components. This makes the background (foreground for Coma) model somewhat uncertain.

In the direction of Coma, the mean value of NH_t is |$1.0\times 10^{21}\, {\rm cm^{-2}}$|⁠. We have selected a large region (radius  = 5°) close to the Coma field (see Fig. 12), where the mean NH_t is essentially identical to the Coma field. Of course, the agreement in the total column density between two regions does not guarantee that the distribution of the absorbing material is the same, and, consequently, the impact of the diffuse Milky Way emission might be different. However, selecting the background region in the vicinity of the source region should minimize this effect.

This choice is also not without a flaw; however, according to Fig. 3, the excess surface brightness is rather flat between 1 and 10 Mpc from the cluster centre. This means, that the background region is expected to contain some ‘excess’ related to the Coma cluster too. Therefore, the signal in the difference between the source and background regions might be attenuated. We decided that for the purpose of this paper, the advantages of using the background region close to Coma outweigh the disadvantages. The size of the background region is large enough so that the cumulative exposure (of the all-sky survey data) is comparable with that of the Coma rings and therefore the noise introduced by the background does not increase the uncertainties in measured net Coma fluxes significantly.

Apart from the problem of selecting the background field, it is desirable to remove bright X-ray sources unrelated to the soft diffuse emission. The main issue here is not resolving a significant fraction of the background, but rather suppressing the level of fluctuations caused by fluctuations of the number of objects (i.e. the shot noise and even their intrinsic variability) in the Coma and background regions. Since the background region is covered in the survey, where the exposure time is much smaller than in the CalPV data in the Coma field, we selected a rather high threshold for excising compact sources, namely |$F_X\gt 10^{-13}\, {\rm erg\, cm^{-2}\, s^{-1}}$| in the 0.5–2 keV band to ensure a uniform level of the background resolved fraction.

From the LogN-LogS curve of AGN obtained by Chandra (see e.g. Luo et al. 2017), we estimate that about 16 per cent of the flux, associated with AGN, will be resolved, and the level of fluctuations of the remaining AGN flux is ∼5 per cent on scales of order 1 deg². This estimate corresponds to pure Poissonian fluctuations of the number of sources and does not include other terms. In addition, we have removed four regions manually. Namely, a bright and variable star 41 Com, two highly variable AGNs, and a 20 arcmin circle (radius) covering a compact (in projection) group of AGNs. None of these excised regions have a strong impact on the results.

The spectra were extracted for each of the annuli shown in Fig. 11 and corrected for the particle and sky background. The errors were calculated following the procedure outlined in Churazov et al. (1996). Namely, to avoid the bias due to the low number of counts per bin when minimizing χ² for unbinned spectra (provided that the total number of counts associated with every independent spectral component is large), the errors were calculated from the smoothed spectra. An example of various components (including detector intrinsic background and sky background) contributing to the total observed spectrum is shown in Fig. 13. The spectra for the annulus between 60 and 80 arcmin (∼1.7–2.2 Mpc) were used.

Clearly, for this annulus, both particle and sky backgrounds make very large contributions. The ‘cleaned’ spectrum, i.e. corrected for all backgrounds, is shown with blue crosses. For comparison, the histogram (blue) shows the best-fitting single-temperature APEC model. In this model, the interstellar absorption, redshift, and abundance of heavy elements were fixed at |$NH=1.0\times 10^{20}\, {\rm cm^{-2}}$|⁠, z = 0.0231 and Z/Z_⊙ = 0.2, respectively. The best-fitting temperature is |$\sim 3.3\, {\rm keV}$|⁠. Clearly, at the level of accuracy provided by these data, the one-temperature model appears to be sufficient to describe the data in this annulus.

For comparison, the black and magenta dashed lines show the APEC model with |$T=10^6\, {\rm K}$|⁠, Z/Z_⊙ = 0.2 and normalizations corresponding to EM_l ≈ 10⁶ and 10³ Mpc, respectively. The former corresponds to the normalization suggested in (Bonamente et al. 2022), i.e. δ ∼ 300 and L ∼ 10 Mpc, while the latter with EM_l = 10³ Mpc is motivated by the Magneticum simulations.

Fig. 13 illustrates the difficulties of measuring the emission of the filaments in X-rays. In the inner regions (exemplified by the orange data points), where the Coma emission is bright, the expected signal is very small compared to the flux from the hot and dense Coma core. The response of the instrument (in particular, its off-diagonal part) has to be extremely well-calibrated to place reliable constraints on the WHIM signal. At larger radii, the Coma contribution becomes subdominant, but the sky background dominates (Fig. 13). In this case, the uncertainties in response are less important, but now the background level has to be calibrated to high accuracy. These requirements make the whole exercise very challenging. As we see below, both issues are important.

To estimate a possible contribution of the WHIM signal to the observed Coma spectra, we used a model consisting of two components. The first component is the APEC model which represents the emission associated with the cluster itself. The second component is the ‘on-off’ WHIM spectrum considered in Section 6. We fixed all major parameters of the WHIM model: metal abundance Z = 0.2 (abundance table of Lodders 2003), temperature |$T=1.5\times 10^6\, {\rm K}$|⁠, gas density (1 + δ) = 30, so that the temperature and density follow the correlation of the most widespread WHIM phases (see the diagonal line in Fig. 2). This choice of T and (1 + δ) is rather arbitrary, but it serves the purpose of illustrating the characteristic sensitivity of the present data set to the potential WHIM emission. The same metal abundance Z = 0.2 was assumed for the APEC model. The effective column density of the Galactic absorption is fixed at |${\rm NH}=1.0\times 10^{20}\, {\rm cm^{-2}}$|⁠.

The resulting model has three free parameters: the normalizations of both components and the temperature of the cluster emission. The resulting constraints on the emission measure associated with the WHIM component are shown as 3σ upper limits (red crosses) in Fig. 14. For one radial bin, the best-fitting normalization of the WHIM component is just above the 3σ level, but it could well be the result of a too simplistic model rather than the evidence for the WHIM signal. For comparison, the coloured dashed lines show levels of the excess emission measure associated with the WHIM predicted by simulations (see Section 6 and Appendix). It is clear that even for a very optimistic value of the WHIM metallicity Z = 0.2, the upper limits are more than one order of magnitude above the expectations.

The thick grey line in Fig. 14 shows the (crudely estimated) level of systematic uncertainties associated with existing calibration uncertainties of the low-energy tail of the spectral response. These systematic uncertainties scale with the brightness of the cluster emission and therefore increase dramatically in the inner region. Other sources of systematic errors include the uncertainties in the stray light contribution (see e.g. Churazov et al. 2023) and time variable Solar Wind Charge Exchange emission. However, given that the pure statistical noise is much larger than the anticipated signal, the additional noise introduced by these uncertainties is not crucially important.

We reiterate here that given the discussion of the contaminating emission from the cluster itself (and its immediate outskirts) in Section 5, the SRG/eROSITA CalPV observations of the Coma cluster may not be the most promising data for the WHIM search since they cover only the region within ∼3 Mpc from the cluster centre. In reality parts of the region that we use here as the ‘background’ field (see Fig. 12) might be better suited for this purpose. Nevertheless, typical filaments are clearly too faint for a secure detection with eROSITA as illustrated by Fig. 14.

8 DISCUSSION

The detection of WHIM signatures is particularly interesting on two grounds. First of all, a significant fraction of baryons can be ‘hidden’ in a low-density warm gas (e.g. Cen & Ostriker 1999). Secondly, the enrichment of this gas with metals is sensitive to the physical processes that control the escape of metals from halos where most of the star formation takes place (e.g. Biffi et al. 2017). Since the WHIM signal in X-rays is primarily sensitive to the spectral features induced by metals, the detection of WHIM would help answer these questions.

The metal distribution predicted in the Magneticum simulations is rather complex (see Appendix  B). In particular, all metal-rich particles (i.e. gas mass elements) are confined to regions within a few virial radii around halos, while it is ‘zero-metallicity’ gas that forms a network of filaments. This does not necessarily mean that the filaments are made of pure hydrogen and helium. Rather it shows that the feedback and enrichment models implemented in Magneticum and shown to be able to reproduce observed properties of galaxies, groups, and clusters (including their stellar masses and metal contents) might produce close-to-zero metallicity of the diffuse intergalactic medium with eWHIM properties. Therefore, the detection of the WHIM metal signatures with a sensitive X-ray instrument, providing a lower limit on the gas metallicity, can be used as a diagnostic of the actual enrichment process. Similarly, upper limits on the gas metallicity can be used to constrain models characterized by stronger feedback spreading the metals over larger Intergalactic Medium (IGM) volumes.

From the observational point of view, traditionally, two approaches for the search for the WHIM signal in X-rays are considered: either via absorption lines in the spectra of a bright distant AGN (see e.g. Nicastro et al. 2022 for the recent review) or via diffuse emission (Churazov et al. 2001; Paerels et al. 2008; Bertone et al. 2010; Takei et al. 2011; Planelles et al. 2018; Khabibullin & Churazov 2019; Vazza et al. 2019; Parimbelli et al. 2022), including stacking over large sky areas (Tanimura et al. 2020, 2022). In the former case, a serendipitous intersection of a WHIM structure by the line of sight towards the bright source is used to probe the WHIM. In this case, the brightness of the background objects is the main criterion. In the latter case, the region selected by the overdensity of galaxies (or of other tracer objects) is used as a proxy for a possible WHIM patch. In particular, the identification of a long and massive filament oriented approximately along the line of sight is a viable strategy (e.g. Markevitch 1999).

In this study we mostly advocate for another approach, namely using regions enclosed by the turn-around radius of massive clusters, where, by definition, the density of WHIM gas should be higher than in other locations. This approach has a few attractive features: (i) the redshift of the expected signal is known and (ii) the size of the affected region is large enough so that the emission from the cluster itself does not contaminate the WHIM signal, and (iii) any cluster can be used as a target and (iv) typical rather exceptional WHIM patches are probed.

Along the same line, the analysis of regions around massive clusters can be used to generate two spectra: the combined spectrum of all resolved AGN more distant than the cluster and the remaining ‘unresolved’ spectrum of the diffuse emission (see Appendix  A). While the former spectrum contains absorption signatures Joint analysis of the two spectra not only increases the significance of the WHIM signal detection but also allows one to better constrain the temperature and density of the gas constituting the filament. As in other cases, removing the contaminating signal from smaller halos is a serious issue.

This approach implies specific requirements on the instrument characteristics. Since the aim is to detect faint diffuse emission spread over substantial areas of the sky at low redshifts, the best-suited future X-ray mission would need to have a large grasp (i.e. the product of the effective area and size of the FoV), high spectral resolution, and stable instrumental background (at least in terms of spurious spectral features). While the large grasp allows obtaining sensitive maps of large areas in a reasonable amount of time, the latter two requirements ensure good control over systematic uncertainties in the estimation of the astrophysical and instrumental background signals.

Examples of such missions are ATHENA (Nandra et al. 2013) and the wide-field X-ray microcalorimeter LEM (Kraft et al. 2022). For nearby objects that subtend large solid angles in the sky, the grasp, i.e. the product of the effective area and the size of the FoV matters most, while for distant objects, the effective area is more important. Therefore, LEM with a factor ∼10 larger grasp is more promising for z ∼ 0 objects, while ATHENA with ∼6 times larger effective area is more suitable for the more distant ones (e.g. Vazza et al. 2019; Parimbelli et al. 2022).

In principle, yet another effect that might boost the WHIM signal in the vicinity of a cluster is the illumination of the gas by an additional X-ray flux (on top of the CXB) that might come from an AGN (e.g. Churazov et al. 2001) or the cluster itself (e.g. Štofanová et al. 2022). This affects the ionization fractions, but more importantly, the extra illumination boosts the resonant scattering signal. The most advantageous configuration is when the illumination is coming from the side rather than from behind the WHIM patch so that the illuminating flux does not contaminate the WHIM signal. One can consider two flavors of this boost factor. In the first case, the illuminating flux contains a narrow resonant emission line scattered in the WHIM region. This is a standard case of the resonant scattering that can boost the signal from cluster outskirts (e.g. Gilfanov, Sunyaev & Churazov 1987; Zhuravleva et al. 2013). In the second case, the illuminating spectrum is featureless like the CXB. The only difference between these two cases is that a fraction of the total illuminating flux that can be scattered is higher in the former case.

The obvious condition for the importance of the resonant scattering effects with the ‘side illumination’ is that the flux density of the illuminating source across the width of the line is comparable (or exceeds) that of the CXB. In the case of clusters like Coma that lack prominent emission lines below a few keV, it is sufficient to compare the cluster surface brightness within a given radius I_cl(< R_cl) with that of the CXB (I_CXB) at the same energy, i.e.

Clearly, this is still a close vicinity of the cluster, but it might be important when studying the outskirts. In this case, resonant lines will be present in emission even if CXB sources are not resolved.

If the illumination is provided by an AGN, a more convenient estimate is via its luminosity at (νL_ν) at the line energy (Churazov et al. 2001)

A further variation of the same type is a beamed source, e.g. a blazar that is pointing away from the observer and could illuminate a conical region in the WHIM.

9 CONCLUSIONS

We describe in full detail spectral signatures associated with the WHIM in the soft X-ray band and assess the prospects of its detection and diagnostic.

We factorize the strength of the WHIM signal into several components. The amount of the gas is quantified via the quantities (i) EM_l = ∫(1 + δ)²dl (in units of length), where (1 + δ) is the ratio of baryon density to the cosmic mean value and/or (ii) the Thomson optical depth τ_T = ∫(1 + δ)〈n_e〉σ_Tdl. The density and temperature of the gas determine the shape of the spectral distortions, while the net signal amplitude is proportional to the abundance of metals in the WHIM (Section 2), except for a small contribution from a pure H + He bremsstrahlung and the Thomson scattering.

We further extend our previous models (Churazov et al. 2001; Khabibullin & Churazov 2019) of spectral distortions associated with WHIM. The model takes into account collisional and photo-ionisation (by the CXB photons), photoelectric absorption, resonant scattering, fluorescence, and local production of X-ray photons. The model predicts the difference of the spectrum in the direction of a ‘WHIM patch’ and from the ‘empty’ region. It takes into account the fraction of the CXB resolved into individual sources and can simultaneously describe the absorption signal in the resolved spectrum and the total signal in the diffuse part (Section 6 and Appendix  A).

At low temperatures (⁠|$T\lesssim 10^6\, {\rm K}$|⁠), the production of photons via excitation of important transition by electron collisions is subdominant and the expected signal is dominated by CXB photons which are ‘transformed’ by interactions with the WHIM, namely absorbed photons are re-emitted as the recombination radiation and fluorescent lines, while the resonant scatterings change the direction of photons in the lines of heavy elements, principally, oxygen. In this regime, the strength of the signal scales linearly with τ_T and gas metallicity.

At higher temperatures and moderate overdensities (∼10), the excitation of major transitions by electrons becomes significant and the strength of the signal scales with EM_l. As before, the scaling with metallicity is linear as well.

Motivated by the results of the Magneticum simulations and the analysis of expected spectral distortions, we define a specific region in the density-temperature diagram, which is the focus of our studies. Namely, we selected the gas with an extended (compared to other definitions of WHIM) range of temperatures |$10^4\, {\rm K}\lt T\lt 2\times 10^6 \, {\rm K}$| and the density (1 + δ) ≲ 50. To distinguish it from more restrictive WHIM definitions, we call this gas eWHIM50 (Section 3).

One promising strategy for finding WHIM (and, in particular, eWHIM50) is by focusing on the region within the turnaround radius |$r_{ta}\sim 10\, {\rm Mpc}$| of a massive cluster and avoiding the central region within ∼0.5r_ta. We found that in the resulting ∼5–10 Mpc annulus, the probability of finding localized patches with |${\rm EM_{\it l}}\gtrsim 10^3\, {\rm Mpc}$| is ∼10 per cent. In terms of τ_T, the excess associated with these patches is ∼2 × 10⁻⁵. The annulus-averaged value of the EM_l is a factor of a few lower. These levels are within reach of future high spectral resolution missions like LEM (Kraft et al. 2022), although this critically depends on the metallicity of the actual metallicity of the WHIM gas, which might not be tightly constrained in the simulations (Section 4).

A similar strategy of probing the volume inside r_ta but still far from the virial or splashback radii of a massive cluster, could be used to study other types of objects in ‘representative’ volumes of the Universe at well-defined redshifts (= redshift of the cluster). At least the objects for which the overdensity of matter of order 2–3 does not strongly affect their properties, but simply increases their space density compared to the field.

The level of contamination of the eWHIM50 signal by regions with larger overdensities has been estimated by comparing in projected images the contribution of the (1 + δ) ≲ 50 gas to the EM_l with the total EM_l provided by the gas at any overdensity. The condition that the low-density gas contributes more than half of the total EM_l excludes much of the region within 5 Mpc from the cluster centre, but less than ∼50 per cent of the area in the 5–10 Mpc annulus (and ∼20 per cent of the projected image of the entire cluster box; see Fig. 5). This suggests that it should be possible to find uncontaminated regions in real data, although a practical recipe for doing this selection has to be identified (Section 5).

If the abundance of metals in the eWHIM50 gas (located well beyond the virial radii of halos) is at the level of Z/Z_⊙ ∼ 10⁻², the prospects of detecting this gas with X-ray spectrometers are rather pessimistic. The abundance of metals does not affect the dispersion measure and/or SZ signals as possible proxies for the presence of the gas, although in this case, the lack of a precise redshift ‘tag’ complicates the detection of individual structures.

Considering the Coma cluster that was observed by SRG/eROSITA during the CalPV phase, combining them with shallower all-sky survey data for background estimation, we concluded that in the central region (≲ 40 arcmin) the current level of the calibration uncertainties precludes the robust characterization of the soft emission potentially associated with the eWHIM50 signal on top of the bright cluster emission. In the outer regions (40–120 arcmin), the impact of the systematic uncertainties is less prominent. There, 3σ upper limits are about one order of magnitude higher than the typically expected levels for the eWHIM50 gas (Section 7).

Finally, we note that here we are specifically targeting typical filaments/overdense regions. This does not exclude the possibility of finding rare objects, e.g. exceptionally massive filaments oriented along the line of sight, that have much stronger integral WHIM signal. On contrary, the approach proposed here can be applied to the surroundings of any massive cluster, in particular, those located at the redshifts optimizing detection prospects versus bright Galactic foreground emission. We show that e.g. proposed soft X-ray microcalorimetric mission LEM might be able to detect WHIM signal from a turnaround radius of a massive cluster in the redshift window around z ≈ 0.08, given that of the WHIM gas is enriched to ∼0.1 of the solar value.

ACKNOWLEDGEMENTS

We are grateful to our reviewer for useful comments and suggestions. This work is partly based on observations with the eROSITA telescope onboard SRG space observatory. The SRG observatory was built by Roskosmos in the interests of the Russian Academy of Sciences represented by its Space Research Institute (IKI) in the framework of the Russian Federal Space Program, with the participation of the Deutsches Zentrum für Luft-und Raumfahrt (DLR). The eROSITA X-ray telescope was built by a consortium of German Institutes led by MPE, and supported by DLR. The SRG spacecraft was designed, built, launched, and is operated by the Lavochkin Association and its subcontractors. The science data are downlinked via the Deep Space Network Antennae in Bear Lakes, Ussurijsk, and Baikonur, funded by Roskosmos. The eROSITA data used in this work were converted to calibrated event lists using the eSASS software system developed by the German eROSITA Consortium and analysed using proprietary data reduction software developed by the Russian eROSITA Consortium.

IK and KD acknowledge support by the COMPLEX project from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program grant agreement ERC-2019-AdG 882679. KD acknowledges support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC-2094 - 390783311. The calculations for the hydrodynamical simulations were carried out at the Leibniz Supercomputer Center (LRZ) under the project pr83li and we are especially grateful for the support through the Computational Center for Particle and Astrophysics (C2PAP).

DATA AVAILABILITY

X-ray data analysed in this article were used by permission of the Russian SRG/eROSITA consortium. The data will become publicly available as a part of the corresponding SRG/eROSITA data release along with the appropriate calibration information. All other data are publicly available and can be accessed at the corresponding public archive servers.

Footnotes

References

APPENDIX A: WHIM SPECTRAL MODEL

Fig. A1 illustrates the WHIM spectrum formation adopted in this study (see also fig. 1 in Khabibullin & Churazov 2019 for another version of the sketch).The ‘ON’ and ‘OFF’ regions correspond to the directions toward the WHIM filament and away from it, respectively. Corresponding model spectra are shown in Fig. A2. For display purposes, the spectra were convolved with a Gaussian, FWHM = 2 eV. As in Fig. A1, the foreground emission is ignored for clarity. Accordingly, the ‘OFF’ spectrum (middle panel) is a pure CXB (broken power law approximation), and the normalizations of the total, resolved, and unresolved spectra are related as 1: f_res: (1 − f_res), reflecting another simplifying assumption that shape of the resolved and unresolved CXB spectra are the same.

For the ‘ON’ direction (left-hand panel in Fig. A2), the total spectrum contains shows the photons produced by electron excitation in the WHIM gas, some signatures of photo-electric absorption, fluorescent photons, and recombination continuum and lines. Conservative scattering processes (Compton and resonant) do not show up in the total spectrum (Churazov et al. 2001; Khabibullin & Churazov 2019). The resolved spectrum on the contrary shows the absorption lines (resonant scattering from the line of sight) and photoelectric absorption. The attenuation by Compton scattering reduces the intensity of the resolved spectrum, but this effect is too small to see it in Fig. A2. For the sake of clarity, we have also assumed that the solid angle associated with resolved sources is very small and therefore the contribution of the diffuse components to the resolved spectrum can be neglected. In the unresolved spectrum the thermal emission, resonantly-scattered and fluorescent lines are present (red lines).

Finally, the right-hand panel of Fig. A2 shows the difference between the ON and OFF spectra, i.e. pure signatures of the presence of the WHIM filament. For the total spectrum the ‘locally produced photons’ are the most prominent for |$T=10^6\, {\rm K}$| gas. In the resolved spectrum, the absorption lines are prominent, while in the unresolved spectrum, the resonantly scattered lines add to local emission. The key process contributing to the excess emission is indicated on the top-right side, while the processes leading to negative/absorption structures are listed on the bottom-right side.

We now proceed with a slightly more elaborate model that accounts for the contribution of the Milky Way foreground and takes into account the effective area and the resolution of the LEM mission.

As discussed in Section 6, the WHIM ‘on-off’ signal can be conveniently factorized into three components

where A characterize the amount of hydrogen in a patch of the WHIM, Z/Z_⊙ is the metallicity and S(E, δ, T) describes the shape of the spectrum for a given δ and T. This prescription ignores the contributions of two continuum components that do not scale with the abundance of heavy elements, namely, the Thomson scattered continuum and a pure H + He thermal bremsstrahlung. In the above equation, A is expressed via |$\tau _T\, \times \, {\rm arcmin^2}$| (an equivalent formulation can be done in terms of |${\rm EM_{\it l}} \, \times \, {\rm arcmin^2}$|⁠; see equation 4). The arcmin² appears in the normalization factor since we are considering the diffuse source and mostly consider nearby objects so that the strength of the WHIM signal increased with the solid angle subtended by the source. When calculating the sensitivity (Section 6) we assumed that the source fills the entire FoV of a given instrument. Z_⊙ stays for the Solar abundance (e.g. Anders & Grevesse 1989) and S(E, δ, T) is the shape of the WHIM signal.

In reality, observational data (spatially resolved spectra of a sky region where the WHIM signal is present) can be split into two parts: S_r(E) and S_u(E). The former is the combined spectrum of bright distant compact sources (primarily AGN) that constitute a large part of the CXB that can be resolved in the data. This spectrum is simply a sum of spectra of small circles around these sources. We assume that a fraction f_CXB ∼ 0.5 can be resolved this way.

Accordingly

where S_CXB(E) is the CXB spectrum⁴ and s_dif(E) is an additional (small) contribution of the diffuse emission to the circles around resolved sources. The s_dif(E) is small as long the solid angle subtended by the circles is small. Finally, B(E) is the sum of the instrumental background and the Milky Way foreground.

Here τ_a(E) = τ_T + τ_ph(E) + τ_rs(E) is the total optical depth due to the Thomson scattering, photoelectric absorption, and resonant scattering. Here and below we assume that τ_a(E) ≪ 1 at any energy and any second-order terms in τ can be neglected. In general, this spectrum will contain a number of spectral features including absorption edges and absorption lines. Studying such spectra is similar to the normal absorption studies (see Nicastro et al. 2022 for the recent review), albeit instead of a single bright source, the spectrum combined from a large number of individual sources is used.

The second component (S_u(E)) is the spectrum of the remaining (unresolved) spectrum of the WHIM region without excised regions

where S_fl(E) is the fluorescent emission and S_th(E) is the ‘thermal’ emission of the WHIM, consisting of recombination emission, and emission caused by excitation of various transitions by electrons and the thermal bremsstrahlung (see Section 6). The examples of S_r(E) and S_u(E) are shown in Fig. A3 with light-blue and light-red lines, respectively. There we show the spectra convolved with the LEM spectral response. Since the solid angle for S_r(E) is much smaller⁵ than for S_u(E), all diffuse components are subdominant in S_r(E), while the CXB makes equal contributions to the spectra (for f_CXB = 0.5).

In reality, when searching for the WHIM signatures, one has to compare the signal in the direction of the WHIM patch and the signal from the ‘blank’ field, i.e. where the WHIM signal is absent (at least at the same redshift). Therefore, a more revealing is the difference between the on-WHIM and off-WHIM spectra, in which many contributions, e.g. the instrumental background and the Milky Way foreground as well as parts of the CXB signal are absent. We denote these difference spectra as S_r, d(E) and S_u, d(E) for resolved and unresolved spectra, respectively.

The resolved difference spectrum is

i.e. it simply shows the missing flux in the spectrum of resolved CXB sources. A small term related to the absorption of the diffuse background behind the WHIM patch is neglected.

Similarly, the unresolved difference spectrum is

Both S_r, d(E) and S_r, d(E) are shown in Fig. A3 as the bright blue and red lines. To illustrate the dependence of the difference spectra on the density and temperature, Fig. A4 shows a set of difference spectra for T = 3 × 10⁵ and 2 × 10⁶ K and densities (1 + δ) = 2, 20, 200 for a fixed τ_T (left plot) and fixed EM_l (right plot). It is clear that at low temperatures and/or small overdensities, both spectra are dominated by the effects of the resonant scattering, i.e. the resolved spectrum shows absorption lines, while the unresolved spectrum shows the same lines in emission (provided that a significant part of the CXB is resolved). For high temperatures and substantial overdensities, the signal is dominated instead by the ‘locally’ produced photons. This means that the joint analysis of the resolved and unresolved spectra can boost the sensitivity by a factor |$\sim \sqrt{2}\,$| for low temperatures and/or overdensities, while in the opposite limit, the unresolved spectra bear most of the information on the presence of the WHIM patch. This is further illustrated by Fig. A5, where the sensitivity is calculated for the WHIM subsample when the gas temperature and density are correlated as |$T=4\times 10^5\, {\rm K} \times (1+\delta)$| (see black dashed line in Fig. 2). For this plot we assumed that the CXB resolved fraction is ∼0.5, the CXB dominates the continuum, and the WHIM lines are shifted from the brightest lines of the Milky Way foreground. Should the angular resolution of an instrument be high enough so that a large fraction of the CXB can be excised from the data without much impact on the remaining diffuse emission, the sensitivity of the unresolved spectra to the WHIM emission could be higher. This is not the case for eROSITA or LEM discussed here.

A1 Limitations of the model

The spectral model used here has a number of limitations/assumptions that stem from attempts to keep the model simple and target the regime when the expected WHIM signal is very small. Here we list the most important of these limitations that one has to keep in mind.

• The model applies to z ∼ 0.

• Ionization fractions correspond to the equilibrium state (including photoionization).

• The electron temperature is not affected by the photoionization of metals by CXB photons.

• The abundance ratios among metals are fixed to the Solar photospheric values.

• Resonant lines are not saturated and treated by the model as Dirac delta functions of energy. In general, the optical depth τ(E) is assumed to be small at any energy, i.e. the Taylor expansion e^−τ(E) ≈ 1 − τ(E) is valid for any E.

• Any second-order terms in τ are neglected.

• A reduced list of transitions is used when the resonantly scattered emission is evaluated.

APPENDIX B: DISTRIBUTION OF METALS IN THE INTERGALACTIC MEDIUM (IN MAGNETICUM SIMULATIONS)

Although the total amount of metals accumulated over the lifetime of the Universe can be predicted rather robustly, since it is closely connected with the overall star formation history, their split between different phases and corresponding spatial distribution is rather uncertain (e.g. Péroux & Howk 2020). In particular, this is determined by the efficiency of the metal transport from the sites of star formation (i.e. dark matter halos of various sizes) and their spread across intergalactic volumes (including mixing and diffusion).

Similarly to the situation with the magnetic field in the IGM (e.g. Kronberg, Lesch & Hopp 1999), the former part is especially sensitive to the kinetic energy associated with the stellar and AGN-driven winds in the first galaxies at high redshift, z > 2 (e.g. Biffi et al. 2017). Various simulation sets differ in this regard substantially, resulting in different predictions for the efficiency and spatial distribution of this so-called early enrichment (e.g. Tuominen et al. 2021).

The Magneticum model of the feedback is rather gentle, lacking extreme energetic outbursts and kinetic energy injections from supermassive black holes (Dolag et al. 2016). As a result, the spread of metals is limited to a few viral radii of the halos. Consequently, the diffuse intergalactic medium, and WHIM in particular, stays largely not enriched, although the average metallicity of all gas components stays constant at 0.2 (Angelinelli et al. 2022). We illustrate this by showing the projected emission measure distribution of the SPH particles in the ‘cluster’ box with zero and non-zero metallicity in the left- and right-hand panels of Fig. A6, respectively.

Taken at the face value, this prediction would preclude the detection of this gas in X-rays (either in emission or in absorption) with any reasonable parameters of the future X-ray facilities. Namely, even a high effective area of the detector would not solve the problem given the inability to single out the signal corresponding to a particular redshift range of the ‘focusing’ galaxy cluster or galaxy overdensity. The same is true for the thermal and kinematic SZ effects and other redshift-insensitive probes. Detection of the dispersion measure jump for a line-of-sight distributed sources (e.g. FRBs) might be used for that, requiring though the rather high density of the sources. Perhaps, more promising could be the detection of Thomson reflection of the light from extremely bright and moderately collimated lighthouses, namely that their illumination cones can be resolved spatially and cross an intergalactic filament almost perpendicularly (e.g. Churazov et al. 2001).