© The Authors 2024

1. Introduction

In recent years, the long-awaited virial-shock (VS) signals around galaxy clusters and groups (for brevity, henceforth “clusters”) were finally detected in inverse-Compton (Keshet et al. 2017; Reiss et al. 2017; Reiss & Keshet 2018; Keshet & Reiss 2018), synchrotron (Keshet et al. 2017; Hou et al. 2023), and thermal Sunyaev–Zeldovich (Keshet et al. 2017, 2020; Hurier et al. 2019; Pratt et al. 2021; Anbajagane et al. 2022) signatures, both in stacking analyses and in individual clusters. The stacked leptonic signals indicate highly localized emission at normalized radii of 2.2 ≲ τ ≡ r/R₅₀₀ ≲ 2.5, where the subscript 500 refers to the radius around a cluster enclosing 500 times the critical mass density of the Universe at the respective redshift.

An unexpected signal was reported recently (Ilani et al. 2024, henceforth I24) in X-ray and radio catalog sources stacked around clusters from the Meta-catalogue of X-Ray detected Clusters of galaxies (MCXC; Piffaretti et al. 2011), with a highly localized excess at 2.25 < τ < 2.50 precisely coincident with the previous VS leptonic signals. These sources were found to be on average extended, hot (keV scales), magnetized, and radially polarized and so were tentatively identified as the shocked halos of infalling galaxies or galaxy aggregates, possibly including aging relativistic particles from previous galactic outflows (I24). However, the stacking analyses of leptonic emission or discrete sources relied on the same low-redshift MCXC clusters and their tabulated, X-ray-based, characteristic R₅₀₀ values.

We examined if the early eROSITA-DE Early Data Release (EDR) catalogs (described in Sect. 2) are sufficient to show an excess of VS X-ray sources within their 140 deg² field and, having identified a signal, characterize the properties of these sources. EDR catalog sources were thus stacked and radially binned around EDR catalog clusters (Sect. 3). The properties of the excess sources were then inferred statistically (Sect. 4) by comparing VS-region sources to their field counterparts. Finally, the results were analyzed and compared to the ROSAT–MCXC results (Sect. 5).

We generally followed the I24 methods and use their notations. A Λ cold dark matter model was adopted with an H₀ = 70 km s⁻¹ Mpc⁻¹ Hubble constant and an Ω_m = 0.3 mass fraction.

2. Catalog samples

We combined the EDR¹ catalogs of X-ray sources (Brunner et al. 2022), clusters (Liu et al. 2022), and cluster X-ray properties (Bahar et al. 2022). Better results are expected with the advent of the first eROSITA all-sky survey² and future all-sky catalogs³ when sufficiently accurate R₅₀₀ estimates become available.

Figure 1 presents the EDR and MCXC cluster catalogs in the phase space of M₅₀₀ mass versus the projected θ₅₀₀ = R₅₀₀/d_A angle, where d_A(z) is the angular diameter distance at redshift z. Thanks to the better resolution and sensitivity of eROSITA, EDR clusters, including massive ones, are available at higher redshifts and thus smaller θ₅₀₀ than in MCXC. However, due to the smaller field of view, the EDR catalog lacks the rare, highly extended or very massive clusters found in MCXC. We divided the 542 EDR clusters into four mass bins, each with about the same number of clusters, 135 or 136. Due to the ∼1′ resolution of the cluster catalog (Liu et al. 2022), we excluded the ten clusters with θ₅₀₀ < 1′. The mass bins and θ₅₀₀ cutoff are shown as lines in the figure.

Fig. 1. M500–θ500 phase space of eROSITA EDR clusters (blue disks), shown in comparison to MCXC clusters (red circles), with boundaries demarcating the four equal-sized mass bins (vertical dashed lines) and the ten excluded θ500 < 1′ clusters (horizontal dot-dashed lines).

Figure 2 presents the X-ray F_X flux histogram in the [0.2, 2.3] keV single-detection band (we used this band in the rest of the study unless stated otherwise) of all the 27 910 likelihood ℒ ≥ 6 EDR catalog sources and of the subset of the 541 extended sources. We focused on the regime between the minimal flux of the extended-source distribution, F_min ≃ 10^−13.57 erg s⁻¹ cm⁻², which approximately coincides with the mean flux (F_mean) of the full sample, and the upper cutoff, F_max ≃ 10⁻¹² erg s⁻¹ cm⁻², imposed to avoid the apparent bright outliers. This range comprises 5820 sources, 534 of which are extended. We further separated this subsample into faint and bright sources using the threshold F_th ≃ 10⁻¹³ erg s⁻¹ cm⁻², which approximately coincides with the extended-source median. The ∼10″ resolution of the source catalog (adopting the camera pixel size; Brunner et al. 2022) is sufficient for the analysis of θ₅₀₀ ≳ 1′ clusters.

Fig. 2. Nominal ([0.2, 2.3] keV) flux distributions of extended (lower, cyan histogram) and all (higher, pink histogram) EDR catalog sources, with vertical dashed lines showing (from left to right) the flux levels of the full-catalog median, the mean (Fmean ≃ Fmin), the threshold (Fth), and the upper cutoff (Fmax).

3. Binning and stacking

Denoting 𝒩(τ, c) as the number of sources found in a radial ring of normalized radius τ and width Δτ around cluster 1 ≤ c ≤ N_c, and ℱ(τ, c) as the number of combined foreground and background sources (henceforth field sources) expected in this ring, we adopted the field-only, 𝒩 = ℱ null hypothesis. A positive 𝒩 > ℱ excess is assigned a source-weighted (SW) significance

$$\begin{array}{c}\hfill S_{\mathrm{SW}}(\tau )=\frac{\mathcal{N}(\tau )-\mathcal{F}(\tau )}{\sqrt{\mathcal{F}(\tau )}}|\{\mathcal{N},\mathcal{F}\}(\tau )\equiv \sum _{\mathsf{c}=\mathsf{1}}^{N_c}\{\mathcal{N},\mathcal{F}\}(\tau ,\mathsf{c}),\end{array}$$(1)

or a cluster-weighted (CW) significance

$$\begin{array}{c}\hfill S_{\mathrm{CW}}(\tau )=\frac{1}{\sqrt{N_c}}\sum _{\mathsf{c}=\mathsf{1}}^{N_c}\frac{\mathcal{N}(\tau ,\mathsf{c})-\mathcal{F}(\tau ,\mathsf{c})}{\sqrt{\mathcal{F}(\tau ,\mathsf{c})}},\end{array}$$(2)

in the ℱ ≫ 1 normal-distribution limit. Modifications for a Poisson distribution, necessary for finite and especially small ℱ, are provided in Appendix A and incorporated henceforth.

For simplicity, given the EDR limited field of view, intermediate Galactic latitudes (20° ≲ b ≲ 40°), and large exposure variations, especially in the field periphery (Brunner et al. 2022), we approximated ℱ as a constant, measured in the {130° < RA < 142°, 0° < Dec < 4°} rectangle of fairly uniform exposure. The results are not sensitive to reasonable changes in the determination of ℱ(τ), including field estimates in the vicinity of each cluster and polynomial sky fits (see I24). We adopted the same nominal Δτ = 0.25 resolution used in previous stacking analyses. No substantial systematic errors are found when comparing SW to CW stacking, when varying the analysis parameters, or when analyzing, for each parameter set, > 10⁵ control samples, each analogous to a real sample but with randomized cluster sky coordinates. A discussion of systematic effects and deviations from circular symmetry is available in I24.

Away from the central τ ≲ 1 excess of sources associated with the intracluster medium (ICM), the extended (respectively, extended and bright) sources present a > 3σ excess (respectively, > 2σ) inside the anticipated VS radius, peaked at 1.75 < τ < 2.0 (respectively, 2.0 < τ < 2.25), as shown in Fig. 3 (top row). Also incorporating non-extended EDR sources yields a local 2.0 < τ < 2.25 excess in each mass bin (1–4), although the low-mass bin (bin 1) shows only a ∼0.5σ excess, as compared to S > 2 in each of the more massive bins (2–4); the co-added excess over these three bins, shown in Fig. 3 (bottom row), presents a ∼3σ (respectively, ∼4σ) excess for all sources (respectively, for bright sources only). Results for each mass bin and for all bins combined are shown in Appendix C. We note that while the VS signal emerges in extended or bright sources without any cuts, it vanishes if F_min is lowered, for example to the catalog median, which is sensitive to the ℒ cut.

Fig. 3. Top row: significance, S(τ), radial profiles of all (left panel) or only bright F > Fth (right) extended eROSITA sources, SW (blue diamonds, with solid lines to guide the eye) or CW (red circles, with dashed lines) stacked around massive clusters (bins 2–4). Also shown are the Poisson-statistics confidence levels (±{1σ, 2σ, 3σ, …}; dotted black curves), the control-sample median and the corresponding containment fractions (50%,16%,2.3%,… for SW; dot-dashed green lines), and the anticipated 2.2 < τ < 2.5 VS region (vertical yellow shading) based on previous stacked γ-ray (RK18) and radio (H23) continuum and discrete X-ray and radio source detections (I24). Bottom row: same as the top panels, but also including non-extended sources in the same Fmin < F < Fmax range.

The peripheral location of the excess and its proximity to the previously stacked signals tie it to the VS region. The narrowness of the signal indicates that the excess sources are directly associated with the shock, rather than being driven by the gradual changes in environment as infalling objects approach the cluster. The same findings in the ROSAT–MCXC analysis, along with the properties of the excess X-ray and radio sources, led us to the conclusion that these objects are likely shocked infalling gaseous halos, probably around galaxies or galaxy aggregates (I24). The VS excess typically peaks in the 2.0 < τ < 2.25 bin for EDR stacking, just inside the 2.25 < τ < 2.5 bin of maximal MCXC-stacking excess. This small offset might arise from differences in the catalog prescriptions for the R₅₀₀ determination, although some redshift dependence cannot be ruled out.

4. Excess source properties

Although the localized 2.0 < τ < 2.25 (henceforth the VS bin) excess is significant, the number of excess sources is small compared to the coincident field sources, so we are unable to tie individual sources to the VS. We can, however, characterize the excess sources statistically, despite the small sample size, by comparing catalog properties in the VS bin and in the field.

We can consider some source properties, ℙ, of interest, such as the source extent (SE) or luminosity (L). In the absence of source redshifts, we estimated ℙ by assigning sources the redshift of their tentative host cluster. We then compared the differential (ℕ(ℙ)≡dN/dℙ) and cumulative (N(< ℙ)) distributions among the 𝒩(2.0 < τ < 2.25, c) sources in the VS bin (subscript V) to the distributions of many random samples, each of ℱ(2.0 < τ < 2.25, c) sources, drawn from the field region (subscript F). The cumulative VS excess, N_V(< ℙ)−N_F(< ℙ), and its local significance, S(ℙ) = [ℕ_V − μ(ℕ_F)]/σ(ℕ_F), were then estimated. We also examined a reference ICM region, 1.0 < τ < 1.5, rescaling its 𝒩 to the solid angle of the VS bin.

Figure 4 presents the SE distribution in the VS bin as compared to the field and ICM regions. There is a significant (> 3σ) local excess of ∼7 sources with 60 kpc ≲ SE ≲ 300 kpc, broadly consistent⁴ with the results from I24. The ICM region shows a comparable or slightly smaller SE, also based on only a few sources. The results in this section are based on bright (F > F_th) sources around clusters in mass bins 2–4 (which show the most significant excess in Fig. 3), but similar results are obtained for the entire sample with all mass bins (see Appendix B).

Fig. 4. SE distribution for nominal cuts (F > Fth and mass bins 2–4). Differential distributions (ℕ(ℙ); top panel) and cumulative distributions (N(< ℙ); bottom panel, left axis) are shown for ℙ = log10SE in the VS bin (solid and dot-dashed green lines), ICM region (double dot-dashed purple), field sampling (dotted red; pink-shaded 1σ dispersion), and VS excess (dashed blue). The bottom panel also shows the local significance, S(ℙ), of the VS excess (solid green curve, right axis).

Low-redshift VS sources of L_X ∼ 10^{42 − 43} erg s⁻¹ luminosities were identified by I24 as thermal emission from shocked infalling halos, so one could naively expect L_X ∝ (1 + z)⁶ luminosities ∼10 times higher at typical EDR redshifts, in both nominal and higher-energy bands. Figure 5 presents the logarithmic distributions of the L_X, L_h, and L_u luminosities in the observed [0.2, 2.3], [2.3, 5], and [5, 8] keV bands, respectively. A significant VS excess of ∼15 sources in the 10^{43 − 44} erg s⁻¹ range is indeed found in L_X and L_h, whereas no significant excess can be identified in L_u. These sources are thus consistent with thermal emission from gaseous objects compressed and heated to keV-scale temperatures by the VS (see Appendix B).

Fig. 5. Luminosity ℙ = log10L distributions in the [0.2, 2.3] (top two panels), [2.3, 5] (middle panels), and [5, 8] keV (bottom panels) bands, using the same sample and notations of Fig. 4. We have added a ${\sigma }_{\text{smooth}}^2\equiv {(0.2L)}^2$ smoothing variance to L uncertainties for visibility.

5. Summary and discussion

By stacking EDR sources (Fig. 2) around 532 EDR clusters (Fig. 1), we see a significant excess of extended (3σ) or bright (3.5σ) sources narrowly localized at normalized 2.0 < τ < 2.25 cluster radii (Figs. 3 and C.1). The proximity to the ROSAT–MCXC (I24) and other stacked signals directly links these sources to the VS shock of the host cluster and renders look-elsewhere corrections negligible. The sources are on average extended (Figs. 4 and B.1), with L_X ≃ L_h ≃ 10^{43 − 44} erg s⁻¹, which is about ten times higher than in the I24 sources, and no L_u detection (Figs. 5 and B.2), as expected for keV-scale sources at EDR redshifts. The signal strengthens (4σ) without the low-mass quartile, M₅₀₀ < 10^14.3, suggesting a host-mass dependence not seen by I24.

The stacked signals emerge thanks to the omission of faint background sources, F < F_min ≃ F_mean (detected at lower likelihood levels), and the approximate similarity of clusters when scaled by R₅₀₀. The small offset of the 2.0 < τ < 2.25 EDR signal from previous 2.2 < τ < 2.5 MCXC-based VS signals hints at a possible dependence on redshift or F cut, but could also arise from differences in the catalog prescriptions for R₅₀₀ determination, which is based on X-rays in MCXC and weak lensing in EDR. Larger future studies could better resolve the offset and identify its origin, constrain the SE, L, and additional source properties, test the tentative M₅₀₀ dependence, and examine deviations from spherical symmetry, which were suggested by I24 and earlier MCXC-based studies but are too weak for the EDR.

Our results are consistent with the tentative identification of the X-ray sources as thermal emission from gaseous objects, likely galactic halos, compressed and heated to keV-scale temperatures by the VS, with a strong L_X(z) dependence. Their nonthermal counterpart, already seen as excess synchrotron radio sources, should also have nonthermal hard X-ray and possibly also γ-ray counterparts, so combining or cross-correlating broadband catalogs should uncover more VS-related phenomena (I24). We conclude that localized VS spikes in the distributions of discrete sources are a powerful probe of accretion from the cosmic web and of the underlying physical processes, ranging from the evolution of large-scale structure, to magnetization by the VS, to its dark matter splashback counterpart.

Acknowledgments

This research received funding from ISF grant No. 2126/22. May Gideon Ilani’s memory be a blessing.

References

Appendix A: Stacking equations

For a Poisson distribution of mean λ, where measurement k has probability p(k) = e^−λλ^k/k!, we can analytically sum the probabilities of equal or larger (respectively smaller) k values to quantify the significance of a positive excess (respectively a negative deficit) in units of Gaussian standard errors,

$$\begin{array}{c}\hfill S_p(k;\lambda )=\{\begin{array}{cc}\sqrt{2}{\mathrm{erfc}}^{-1}\left[\frac{\mathrm{\Gamma }(k)-\mathrm{\Gamma }(k,\lambda )}{\mathrm{\Gamma }(k)/2}\right]>0\hfill & \mathrm{if}k>\lambda ;\hfill \\ -\sqrt{2}{\mathrm{erfc}}^{-1}\left[\frac{\mathrm{\Gamma }(1+k,\lambda )}{\mathrm{\Gamma }(1+k)/2}\right]<0\hfill & \mathrm{if}k<\lambda ;\hfill \\ 0\hfill & \mathrm{if}k=\lambda ,\hfill \end{array}\end{array}$$(A.1)

where erfc(x) and Γ(x) are respectively the complementary error and the Euler Gamma functions. One can find S_SW for given S_p and ℱ by solving (cf. Eq. 1)

$$\begin{array}{c}\hfill S_p=S_p(\mathcal{N}=\mathcal{F}+S_{\mathrm{SW}}\sqrt{\mathcal{F}};\mathcal{F}),\end{array}$$(A.2)

which results in the S_SW(τ) profiles shown for integer S_p values in the S(τ) in Figs. 3 and C.1.

Appendix B: Source properties in the full sample

Figures B.1 and B.2 complement Figs. 4 and 5, respectively, showing the properties of all F_min < F < F_max sources around clusters in all mass bins. The signals are stronger here (i.e., they involve more excess sources than in Figs. 4 and 5) but are noisier. Regardless, our conclusions are qualitatively unchanged.

Fig. B.1. SE distribution of all sample sources around all clusters (same notations as in Fig. 4

Fig. B.2. Luminosity distributions of all sample sources around all clusters (same notations as in Fig. 5).

Appendix C: Mass dependence

Figure C.1 presents the stacked S(τ) profile in each of the four mass bins shown in Fig. 1, as well as the signal co-added over all four mass bins. While the higher-mass bins (2–4) each present a ∼2σ excess in the VS bin, the M ≲ 10^14.3 clusters in mass bin 1 show a negligible excess. This result differs from the comparable signals found by I24 in all their mass bins; however, due to the poor statistics of the present sample, we cannot conclude that there is a meaningful difference between the EDR and ROSAT–MCXC results in this regard.

Fig. C.1. Same as Fig. 3, but showing mass bins 1–4 separately (first four panels, top down) and combined (bottom panel).

All Figures

	Fig. 1. M500–θ500 phase space of eROSITA EDR clusters (blue disks), shown in comparison to MCXC clusters (red circles), with boundaries demarcating the four equal-sized mass bins (vertical dashed lines) and the ten excluded θ500 < 1′ clusters (horizontal dot-dashed lines).
In the text

	Fig. 2. Nominal ([0.2, 2.3] keV) flux distributions of extended (lower, cyan histogram) and all (higher, pink histogram) EDR catalog sources, with vertical dashed lines showing (from left to right) the flux levels of the full-catalog median, the mean (Fmean ≃ Fmin), the threshold (Fth), and the upper cutoff (Fmax).
In the text

Fig. 3. Top row: significance, S(τ), radial profiles of all (left panel) or only bright F > Fth (right) extended eROSITA sources, SW (blue diamonds, with solid lines to guide the eye) or CW (red circles, with dashed lines) stacked around massive clusters (bins 2–4). Also shown are the Poisson-statistics confidence levels (±{1σ, 2σ, 3σ, …}; dotted black curves), the control-sample median and the corresponding containment fractions (50%,16%,2.3%,… for SW; dot-dashed green lines), and the anticipated 2.2 < τ < 2.5 VS region (vertical yellow shading) based on previous stacked γ-ray (RK18) and radio (H23) continuum and discrete X-ray and radio source detections (I24). Bottom row: same as the top panels, but also including non-extended sources in the same Fmin < F < Fmax range.

In the text

Fig. 4. SE distribution for nominal cuts (F > Fth and mass bins 2–4). Differential distributions (ℕ(ℙ); top panel) and cumulative distributions (N(< ℙ); bottom panel, left axis) are shown for ℙ = log10SE in the VS bin (solid and dot-dashed green lines), ICM region (double dot-dashed purple), field sampling (dotted red; pink-shaded 1σ dispersion), and VS excess (dashed blue). The bottom panel also shows the local significance, S(ℙ), of the VS excess (solid green curve, right axis).

In the text

	Fig. 5. Luminosity ℙ = log10L distributions in the [0.2, 2.3] (top two panels), [2.3, 5] (middle panels), and [5, 8] keV (bottom panels) bands, using the same sample and notations of Fig. 4. We have added a ${\sigma }_{\text{smooth}}^2\equiv {(0.2L)}^2$ smoothing variance to L uncertainties for visibility.
In the text

	Fig. B.1. SE distribution of all sample sources around all clusters (same notations as in Fig. 4
In the text

	Fig. B.2. Luminosity distributions of all sample sources around all clusters (same notations as in Fig. 5).
In the text

	Fig. C.1. Same as Fig. 3, but showing mass bins 1–4 separately (first four panels, top down) and combined (bottom panel).
In the text