Detection of the Missing Baryons toward the Sightline of H1821+643

Based on high-resolution UV data taken with the Hubble Space Telescope, Tripp et al. (1998) studied the Lyα absorbers in the spectrum of H 1821+643. These authors also measured spectroscopic redshifts of 154 galaxies in the ∼1° field centered on the AGN. Therefore, the redshifts of the UV absorption line systems and the intervening galaxies could be cross-correlated. Based on this, Tripp et al. (1998) identified 35 absorption line systems that may be associated with foreground galaxies. Since the redshifts of several Lyα absorbers are similar, in some cases the same foreground galaxy is assigned to multiple absorption line systems. For the foreground galaxies with multiple absorption line systems, we retained one Lyα absorber, whose redshift was the closest to that of the foreground galaxy. This reduced our sample to 21 absorption line systems.

The mass distribution of galaxies is different in voids and in filaments. Specifically, almost all massive galaxies (with halo mass of ${M}_{\mathrm{halo}}\gtrsim 3\times {10}^{11}\ {M}_{\odot }$) reside in filaments, while walls and voids are dominated by low-mass (${M}_{\mathrm{halo}}\lesssim 3\times {10}^{11}\ {M}_{\odot }$) galaxies (Aragón-Calvo et al. 2007; Hahn et al. 2007b; Cautun et al. 2014). To ensure that the Lyα absorbers are associated with filaments, we further filtered the foreground galaxies based on their mass. To derive the stellar mass of the foreground galaxies, we used the FAST (Fitting and Assessment of Synthetic Templates) code that fits stellar population synthesis templates to broadband photometry data (Kriek et al. 2009). We collected SDSS photometric data in the u, g, r, i, z bands for each foreground galaxy and we used the spectroscopic redshift of the galaxies as an input. For our model, we assumed solar metallicity, a Salpeter initial mass function (Salpeter 1955), and a Milky Way dust law for the exctinction (Cardelli et al. 1989). To this end, we only retained galaxies whose stellar mass exceeds ${M}_{\star }\gtrsim {10}^{10}\ {M}_{\odot }$, which approximately corresponds to halo mass of ${M}_{\mathrm{halo}}\gtrsim 3\times {10}^{11}\ {M}_{\odot }$ according to the stellar mass–halo mass relation of Behroozi et al. (2010).

Based on the mass selection, we find that our sample consists of 17 absorption line systems, which are likely associated with massive (${M}_{\star }\gtrsim {10}^{10}\ {M}_{\odot }$) foreground galaxies. In Table 2 we show the redshift of the absorption line systems and the characteristics of the associated galaxies, including impact parameter (b) and virial radius (${R}_{\mathrm{vir}}$). We emphasize that the impact parameter of almost all foreground galaxies exceeds their virial radius, which implies that the present set of absorption line systems probe WHIM filaments rather than the halos of galaxies. We also note that the average Lyα equivalent width for our sample is ∼174.4 ${\rm{m}}\mathring{\rm A}$. Thus, our stacking mostly concentrates on relatively strong Lyα absorbers.

The resolving power of the Chandra LETG is significantly lower than that of modern UV spectrographs. As such, the detected UV absorption lines may be spaced so close together in redshift space that they cannot be resolved by the LETG. This would result in stacking the same X-ray spectrum multiple times, which could distort our results. Therefore, a stacking study must avoid any overlaps by filtering duplicate redshifts. We define overlapping UV absorption lines as those where the wavelengths corresponding to their redshifts are within δλ = 0.0125 Å, which is equivalent with one resolution element of Chandra ACIS-LETG. For these systems, we would compute the mean redshifts of the overlapping systems and only include this mean redshift in the stacks. However, in our sample, we did not find overlapping redshifts. Therefore, we carry out our stacking analysis using the redshift of 17 Lyα absorber. The positions of the foreground galaxies, relative to H 1821+643, are shown in Figure 1.

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In this work, we analyze the Chandra ACIS-LETG observations of H 1821+643. This instrument/grating configuration is ideal to search for low-energy absorption lines. The AGN was observed in four pointings and the total exposure time of the data is t_exp = 470.2 ks. Details about the observations are provided in Table 1. To prepare the data for the analysis, we rely on standard CIAO (version 4.9 and CALDB 4.7.6) software package tools (Fruscione et al. 2006).

The main steps of the analysis are as follows. First, we reprocessed the data using the chandra_repro CIAO tool to ensure that the most up-to-date calibration is applied. In addition, this tool creates the region masks, extracts the spectra, and builds the corresponding response files. In this work, we only considered the first order dispersed spectra and we combined the ±1 orders of the grating spectra. Since H 1821+643 was observed four times, we combined the individual observations using the combine_grating_spectra CIAO tool. This resulted in a single spectrum and the corresponding response files.

Because we stack the X-ray spectrum of H 1821+643 multiple times corresponding to the redshifts of the UV absorption line systems, the spectra and the corresponding response matrix file and ancillary response file must be shifted. Specifically, we blueshifted the spectrum and the response files using ${\lambda }_{\mathrm{rest}}={\lambda }_{\mathrm{obs}}{(1+z)}^{-1}$, where z is the redshift of the individual absorption lines. This, in turn, shifts the lines to their rest-frame wavelengths, allowing us to stack the spectra associated with different UV absorption line systems.

To stack the spectra and response files that are shifted to the rest-frame wavelength, it is necessary to have them on the same wavelength grid. However, the LETG wavelength grid is non-uniform, and hence the blueshifted spectra and response files cannot be directly co-added. To overcome this issue, we applied two operations on the spectra and response files: rebinning and cropping.

To rebin the data, we first defined a universal wavelength grid, which consists of bins with uniform widths of 0.0125 Å. The particular choice was motivated by the fact that this bin size corresponds to a factor of four oversampling of the 0.05 Å resolution of LETG and represents the default bin size of LETG. Therefore, the rebinned spectra have approximately the same number of bins as the original spectra, which means that rebinning does not distort any of the observed spectral features. To further confirm this, we rebinned the spectra and the response files using different binning factors, such as 0.025 Å or 0.05 Å. We conclude that the results presented in this work are not affected in any statistically significant way by the particular rebinning factor.

Since the relevant metal lines are in the λ = 10–35 Å wavelength range, we cropped the spectra and retained only this wavelength range. Following these steps, we stacked the 17 blueshifted spectra and response files using combine_grating_spectra task.

The end result of our stacking procedure is a single co-added blueshifted spectrum of H 1821+643. This spectrum was used to search for absorption lines originating from various metal lines. Given that we stacked the spectrum N_abs = 17 times, the total exposure time of our stacked spectrum is ${t}_{\mathrm{stack}}\,={t}_{\exp }\times {N}_{\mathrm{abs}}=8.0$ Ms.

In this work, we aim to probe whether the Chandra LETG spectrum of H 1821+643 exhibits X-ray absorption lines originating from the most abundant ions. Specifically, we investigate the N vi, N vii, O vii, O viii, Ne ix, and Ne x ions, which are the most abundant ions in the WHIM and the LETG bandpass. Although we may also expect absorption lines at the wavelengths of C v and C vi, at the (redshifted) wavelength of these lines the effective area of ACIS-LETG is about an order of magnitude lower than that at the wavelength of the (redshifted) O vii line. Therefore, it is virtually impossible to probe the existence of C v and C vi absorption lines in the stacked spectrum of H 1821+643 using ACIS-LETG observations.

Before we investigate the stacked dataset, we probe whether the blueshifted but unstacked X-ray spectra exhibit absorption lines. We find that none of the individual spectra show statistically significant absorption lines. This implies that our stacked spectrum is not influenced by a single absorbing system. In the next step, we study the X-ray spectrum that was stacked based on the redshift of UV absorption line systems (Section 2.1).

In Figure 2, we present the stacked spectrum of H 1821+643 in the 21.3–21.9 Å wavelength range. Visual inspection of the spectrum reveals an absorption line feature at λ ≈ 21.60 Å, which is consistent with the rest-frame wavelength of the O vii ion (λ_{O vii} = 21.602 Å). Specifically, we detect a feature with four bins below the continuum level, where the central and deepest bin is located at λ ≈ 21.60 Å. In addition, the feature exhibits a symmetric, Gaussian profile, which is consistent with that of absorption lines. These arguments imply that we have detected an O vii absorption line in the stacked spectrum of H 1821+643.

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To characterize the absorption line at the wavelength of the O vii ion and to probe the statistical significance of this feature, we carry out spectral fitting with XSpec (Arnaud 1996). We utilize a two component model that consists of an absorbed power law model that represents the continuum of the background AGN and a Gaussian absorption line profile, which is an appropriate model for weak lines. During the fitting procedure, the only fixed parameter is the wavelength of the line, which was fixed at λ = 21.602 Å. The intrinsic line width was set to 0 and, along with other model parameters (line normalization, power law normalization, and photon index of power law), it was a free parameter of the fit. To determine the detection significance, we use the normalization parameter of the Gaussian line profile and the corresponding uncertainties.

The spectral fitting confirms the existence of an absorption line at λ = 21.602 Å. Based on the XSpec fitting, we establish that the statistical significance of the absorption line is 3.3σ. We emphasize that this detection is not the result of a blind search. Indeed, in this work—unlike in many previous studies— we did not employ numerous redshift trials to identify a tentative absorption line. Instead, the redshifts of the Lyα absorption lines, which were used to perform the stacking analysis, were known a priori from UV studies. Therefore, the derived 3.3σ statistical significance represents the actual confidence level of the detection. To confirm that the centroid of the line agrees with the rest-frame wavelength of the O vii ion, we repeated the fitting procedure, but allowed the line centroid to vary. We obtained the best-fit line centroid of ${\lambda }_{\mathrm{obs}}={21.593}_{-0.006}^{+0.013}$ Å, which is consistent with the wavelength of the O vii ion. Thus, for the first time, we detect a definite O vii absorption line in the stacked spectrum of H 1821+643.

For the O vii detection toward the sightline of H 1821+643, we calculated the equivalent width of the line and obtained ${{EW}}_{{\rm{O}}{\rm{VII}}}=(4.1\pm 1.3)$ mÅ. Given that the source is optically thin, and thus the detected absorption line is unsaturated, we can convert ${{EW}}_{{\rm{O}}{\rm{VII}}}$ to the column density of the O vii ion using the linear formula:

$$\begin{eqnarray}\begin{array}{rcl}N({\rm{O}}\,{\rm{VII}}) & = & 3.48\times {10}^{15}\ {\mathrm{cm}}^{-2}\left(\displaystyle \frac{{{EW}}_{{\rm{O}}{\rm{VII}}}}{10\,{\rm{m}}\mathring{\rm A} }\right)\\ & = & (1.4\pm 0.4)\times {10}^{15}\ {\mathrm{cm}}^{-2}\end{array}\end{eqnarray} \tag{ 1 }$$

where ${{EW}}_{{\rm{O}}{\rm{VII}}}$ is the rest-frame equivalent width. We emphasize that the stacked O vii equivalent width, and thus the column density, represent average values according to the definition of equivalent width (${EW}\equiv {{\rm{F}}}_{\mathrm{line}}/{{\rm{F}}}_{\mathrm{continuum}}$).

In Figure 3, we present the stacked spectra at the rest-frame wavelength of Ne ix, Ne x, N vi, N vii, and O viii, where we highlight 0.6 Å regions around the expected position of these lines. The data do not reveal the existence of prominent absorption lines at the wavelengths of these ions. We also carry out spectral fitting, which confirms the non-detection of these lines in the stacked spectrum of H 1821+643. In the absence of detections, we computed 1σ upper limits on the equivalent widths and column densities of the other major metal lines. The results are summarized in Table 3.

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Based on the XSpec analysis of the stacked spectrum of H 1821+643, we obtained a 3.3σ detection of an absorption line at the rest-frame wavelength of the O vii ion (Section 3). To further confirm the statistical significance of this detection, we utilize a complementary and independent technique. Specifically, we perform Monte Carlo simulations and assess the possibility that the detection is the result of a chance coincidence.

To carry out the simulations, we mimic the observed data by generating a set of 17 random redshifts in the range of z = 0–0.297, where the upper limit corresponds to the redshift of H 1821+643. Using the random set of redshifts, we analyze the spectrum of H 1821+643 following the steps described in Section 2.2. Specifically, we corrected for overlapping redshifts, we blueshifted the spectra and response files to the rest-frame wavelength, and we stacked the spectra. We fit the stacked spectrum with XSpec following Section 3.1 and we determined the detection significance of a potential absorption line feature at 21.602 Å. To obtain statistically meaningful results, we generated 10⁴ random redshift sets (each containing 17 redshifts) and carried out this procedure for every randomly generated set of spectra.

In Figure 4, we present the distribution of the statistical significances of a possible line for the 10⁴ random redshift sets. The observed distribution is well described with a standard normal distribution. After fitting the randomly stacked spectra, we find that three random redshift sets show an absorption feature with ≥3.3σ significance and four random redshift sets exhibit an emission feature with ≥3.3σ significance. Hence, the chance coincidence of detecting a ≥3.3σ absorption line is 3 × 10⁻⁴. This result is fully consistent with the detection significance obtained from the XSpec fitting. Thus, our Monte Carlo simulations independently demonstrate that the detection of the O vii absorption line in the stacked spectrum of H 1821+643 is at the 3.3σ confidence level.

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To probe whether the randomly stacked spectra exhibit the characteristics of typical absorption or emission lines, we visually inspected the 7 spectra that exhibit a ≥3.3σ detection. Typically, a prototypical Gaussian absorption/emission line has a symmetric profile and the lowest/highest data points are coincident with the rest-frame wavelength of the O vii ion. As discussed in Section 3.1, the O vii absorption line detected in the stacked spectrum of H 1821+643 exhibits these characteristic (Figure 2). However, only one out of the seven random spectra with ≥3.3σ significance, shows these characteristics. Hence, the random occurrence rate of an absorption/emission line with ≥3.3σ statistical significance with a Gaussian profile is 10⁻⁴. Assuming standard normal distribution, this chance coincidence corresponds to a statistical significance of 3.9σ.

It is necessary to verify our stacking analysis because individual (i.e., unstacked) spectral features are not detectable, hence visual confirmation of the stacking is not possible. To this end, we used two approaches to probe the accuracy of our technique, while assuming an equal contribution from individual spectral lines.

First, we confronted the single and co-added spectra and we then examined them from bin to bin. This demonstrates that the stacked spectrum is the sum of the individual spectra, thereby confirming the applicability of our analysis.

Second, in a more extensive approach, we repeated our stacking analysis on simulated spectra containing undetectable (i.e., low statistical significance) absorption features, similar to the real data, and we then examined the line detection significance of the simulated, stacked spectra.

To simulate the observations, we used the Xspec command fakeit, which allows us to create spectra using the model and instrumental response files of the observed data. The two important parameters of the applied model, containing a power law and a Gaussian line profile (Section 3.1), are the normalization and the wavelength of the absorption line. We adjusted the line normalization in correspondence with the detected absorption feature. According to the number of foreground systems in the sightline of H 1821+643, we defined 17 evenly spaced redshifts within the redshift of the AGN. We then used the corresponding wavelengths (with the O vii rest-frame wavelength as zero-point) as the location of the simulated absorption features. For the remaining model parameters, we adopted the values obtained from the fitting of the real unstacked data and we also used the unstacked value of 470 ks for the exposure time. This way, we reproduced the spectrum of H 1821+643 17 times, each with a low statistical significance absorption feature at one of the 17 fake wavelengths.

Because of the random nature of fakeit simulations, the statistical significance of the stacked spectral feature is expected to follow a normal distribution. If our stacking analysis is valid, then the distribution should peak at ∼3σ. To have a sufficient sample size, we repeated the fakeit simulation 1000 times, resulting in 17,000 unstacked spectra (and 1000 stacked spectra) using the same redshift set for each run. These spectra were then stacked and fitted following the method given in Sections 2.2 and 3.2. The detection significances of the stacked spectra resulted in a normal distribution with a mean value of −2.8σ and standard deviation of 1.1, which implies that our stacking analysis is valid. In addition, we also fit the unstacked spectra at the simulated wavelengths, which also produced a normal distribution for the detection significances with mean value of −0.8σ and standard deviation of 1.0, corresponding to individually undetectable absorption lines. This verifies the suitability of the simulated spectra.

The detection of the O vii ion in the stacked spectrum of H 1821+643 allows us to derive the cosmological mass density of the O vii absorbers. Following the calculations carried out for UV absorption lines (e.g., Tripp et al. 2000), we express the O vii baryon density as

$$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Omega }}}_{{\rm{b}}}({\rm{O}}\,{\rm{VII}}) & = & \displaystyle \frac{\mu {m}_{{\rm{p}}}{H}_{0}}{{\rho }_{{\rm{c}}}c}{\left[\left(\displaystyle \frac{{\rm{O}}}{{\rm{H}}}\right){f}_{{\rm{O}}{\rm{VII}}}Z/{Z}_{\odot }\right]}^{-1}\cdot \\ & & \cdot \,\displaystyle \frac{{\displaystyle \sum }_{i}{N}_{i}({\rm{O}}\,{\rm{VII}})}{{\rm{\Delta }}X}\end{array}\end{eqnarray} \tag{ 2 }$$

where μ = 1.3 is the mean atomic weight, m_p is the proton mass, ρ_c is the critical density of the universe, $(O/H)\,=4.94\,\times \,{10}^{-4}$ is the solar abundance of oxygen relative to hydrogen (Asplund et al. 2009), ${f}_{{\rm{O}}{\rm{VII}}}$ is the O vii ionization fraction, $Z/{Z}_{\odot }$ is the metallicity of the absorber, ${\sum }_{i}{N}_{i}({\rm{O}}\,{\rm{VII}})$ is the measured O vii column density summed up for i = 17 absorption line systems, and ΔX is the absorption distance interval defined by Bahcall & Peebles (1969). Assuming that all 17 absorption line systems contribute equally, the O vii baryon density is:

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{{\rm{b}}}({\rm{O}}\,{\rm{VII}})=(0.0023\pm 0.0007){[Z/{Z}_{\odot }{f}_{{\rm{O}}\mathrm{VII}}]}^{-1}\end{eqnarray} \tag{ 3 }$$

However, if fewer than 17 systems contribute, then this value corresponds to an upper limit. Hence the O vii baryon density can be expressed as:

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{{\rm{b}}}({\rm{O}}\,{\rm{VII}})=\displaystyle \frac{i}{17}(0.0023\pm 0.0007){[Z/{Z}_{\odot }{f}_{{\rm{O}}\mathrm{VII}}]}^{-1},\end{eqnarray} \tag{ 4 }$$

where i is the number of absorbers contributing to the signal and also where i must be more than a few because none of the individual systems are detectable (see Section 3.1). In this formula, the major sources of uncertainty are the unconstrained metallicity of the WHIM and the ionization fraction of O vii.

Cosmological simulations show that prominent filaments with higher column densities have higher metallicities (Cen 2012). Specifically, for the column density of ${N}_{{\rm{O}}{\rm{VII}}}=1.4\,\times {10}^{15}\ {\mathrm{cm}}^{-2}$, the expected metallicity is $Z={0.18}_{-0.09}^{+0.17}\,{Z}_{\odot }$.

Specifying the ionization fraction of different species in the WHIM is not straightforward. Generally, ionization comes from two distinct processes: collisional ionization and photoionization. Assuming only collisional ionization, O vii dominates the widest temperature range of $T\approx {10}^{5.5}\mbox{--}{10}^{6.3}$ K, and at lower temperatures, O v has the highest ionization fraction, overwhelming the O vi ion. However, in the low-density WHIM, collisional ionization plays a minor role in the formation of O v–O viii ions at temperatures between ∼10⁴–10⁷ K but may be relevant at higher temperatures ($T\gt {10}^{7}$ K) or hydrogen densities above 10⁻³ cm⁻³ (Hellsten et al. 1998). Therefore, at temperatures relevant for WHIM filaments, photoionization is the dominant process. Specifically, UV radiation is responsible for the ionization of lithium-like species (e.g., O vi), while X-ray radiation plays a role in the ionization of helium-like and hydrogen-like ions (e.g., O vii and O viii). The effect of photoionization depends strongly on the intensity and the shape of the external radiation field, and the hydrogen number density of the WHIM (e.g., Chen et al. 2003; Cen & Fang 2006; Kawahara et al. 2006). However, collisional and photoionization equilibrium is still an oversimplification. For the O vii–O ix ions at densities of the WHIM, the timescales required to reach ionization equilibrium may be longer than the Hubble time (Yoshikawa & Sasaki 2006). While general ionization processes (i.e., collisional ionization and photoionization) can be modeled by photoionization codes, the importance of the latter effect requires hydrodynamical simulations. Other hydrodynamical effects, such as supernova feedback, gas shocks produced by AGN or star formation activity, merging or accretion processes, and the anisotropic temperatures occurring at sites of nonlinear structure formation, may also be significant sources of ionization and cannot be neglected. Based on hydrodynamical simulations of Cen (2012), we assume a temperature of ∼10⁶ K for a single WHIM filament in the sightline of H 1821+643. Considering the large parameter space involved by the different ionizing mechanisms, we rely on non-equilibrium simulations from Ji et al. (2005) and results of Mathur et al. (2003) to estimate the ionization fraction of O vii, for which we adopt f_{O vii} = 0.75.

Using $Z\approx 0.18\,{Z}_{\odot }$ for metallicity and f_{O vii} = 0.75 for the O vii ionization fraction with ${H}_{0}=70\ \mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$, and assuming that all 17 systems contribute equally to the detected signal, we estimate that the O vii baryon mass density is ${{\rm{\Omega }}}_{{\rm{b}}}({\rm{O}}\,{\rm{VII}})=0.017\pm 0.005$. Given that the cosmic baryon mass density is ${{\rm{\Omega }}}_{{\rm{b}}}=0.045$, we conclude that O vii absorbers contribute (37.5 ± 10.5)% of the total baryon content of the Universe. This is in good agreement with recent results from IllustrisTNG simulations, where the WHIM constitutes ∼47 % of the baryons at z = 0 (Martizzi et al. 2018).

The galaxies in the vicinity of H 1821+643 exhibit a wide range in impact parameters (b = 0.1–4.4 Mpc), which covers regions from the outer halos of galaxies (∼0.1 Mpc) to the scales of the WHIM (∼4.4 Mpc). To study the O vii absorption line as a function of impact parameter, we binned the absorption line systems as a function of their impact parameter. Specifically, we divided the absorption line systems into two subsamples. The first subsample includes eight galaxies with $0.1\,\mathrm{Mpc}\lt b\lt 1$ Mpc, while the second subsample includes nine galaxies with $1\,\mathrm{Mpc}\lt b\lt 4.4$ Mpc. We emphasize that the impact parameter exceeds the virial radius for almost all foreground galaxies (Table 2), except for two galaxies whose virial radius exceeds the impact parameter. However, both of these galaxies have relatively low halo mass (${M}_{\mathrm{halo}}\lesssim 7\times {10}^{11}\ {M}_{\odot }$), implying that these galaxies are not likely to host extended X-ray halos around them. Thus, both the inner and outer subsample primarily probe the large-scale WHIM filaments.

Following the procedure outlined in Section 2.2, we stack the absorption line systems in each subsample and fit the O vii absorption line with a Gaussian absorption model as described in Section 3.1. Given the fewer absorption line systems in the subsamples, we obtain weak detections in the inner and outer subsamples at the 2.9σ and 1.7σ level, respectively. We present the spectra and the best-fit models in Figure 5. Taking these signals as detections, we derive the O vii equivalent width and column density, and obtain ${{EW}}_{\mathrm{in}}=(5.2\pm 1.6)\ {\rm{m}}\mathring{\rm A}$ and ${{EW}}_{\mathrm{out}}=(3.1\pm 1.8)\ {\rm{m}}\mathring{\rm A}$ for the subsamples with smaller and larger impact parameters, respectively. These values correspond to O vii column densities of $N{({\rm{O}}{\rm{VII}})}_{\mathrm{in}}=(1.8\pm 0.6)\times {10}^{15}$ cm⁻² and $N{({\rm{O}}{\rm{VII}})}_{\mathrm{out}}=(1.1\pm 0.6)\times {10}^{15}$ cm⁻².

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In the left-hand panel of Figure 6, we display the column density as a function of impact parameter. Within statistical uncertainties, we find that the column density is invariant as a function of impact parameter. While a decreasing trend may exist, the present data is insufficient to better constrain the dependence of column density as a function of impact parameter.

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In the right-hand panel of Figure 6, we depict the hydrogen gas density distribution as a function of impact parameter. We deduced the n_H hydrogen density for the inner and outer subsamples using:

$$\begin{eqnarray}&&{n}_{{\rm{H}}}=\displaystyle \frac{N({\rm{O}}\,{\rm{VII}})}{{f}_{{\rm{O}}{\rm{VII}}}\ Z/{Z}_{\odot }\ (O/H)\ r}\end{eqnarray} \tag{ 5 }$$

where N(O vii) is the average O vii column density of a single WHIM filament, $Z/{Z}_{\odot }\approx 0.18$ is the metallicity of the gas (Section 4.1), $(O/H)=4.94\,\times \,{10}^{-4}$ is the solar abundance of oxygen relative to hydrogen (Asplund et al. 2009), ${f}_{{\rm{O}}{\rm{VII}}}=0.75$ is the ionization fraction of O vii (Section 4.1), and r is the path length. Given that the observed signal represents the large-scale WHIM, we assumed r = 5 Mpc, which is the typical width of WHIM filaments (Cautun et al. 2014). We note that filaments have complex shape and structure, and they also vary in length and width, depending on their environment (Dolag et al. 2006). We find that the gas density of a WHIM filament is ${n}_{{\rm{H}}}\simeq (1-2)\times {10}^{-6}$ cm⁻³, which corresponds to overdensities of $\delta \simeq 5-9$. These values are typically expected for WHIM filaments (Hahn et al. 2007a; Aragón-Calvo et al. 2010; Cautun et al. 2014). Thus, our detections are consistent with a picture in which the observed absorption lines originate from large-scale WHIM filaments.

In the right-hand panel of Figure 6, we also show the gas density profile for the massive spiral galaxy, NGC 1961. Emission studies have explored the gas density profile to about 60 kpc radius around the galaxy (the CGM), which corresponds to about ∼15% of the virial radius (Bogdán et al. 2015; Anderson et al. 2016). Combining these data with our absorption line study allows us to construct, for the first time, a gas density profile that simultaneously explores the outer scales of the CGM and the WHIM. We also adopted and display three model curves from Smith et al. (2017).⁶ The steepest curve (${n}_{{\rm{H}}}\sim {r}^{-2.8}$) represents a Navarro–Frenk–White halo, assuming collisionless particles. The curve with the lowest slope (${n}_{{\rm{H}}}\sim {r}^{-3/4}$) models a mass distribution where most of the baryons reside within r₂₀₀. Finally, the ${n}_{{\rm{H}}}\sim {r}^{-3/2}$ curve is fitted to a simulated galactic halo distribution. When comparing the observations, we find that the average density of the WHIM filaments is about two orders of magnitude lower and is also significantly lower than predicted from the extrapolated density profile established for NGC 1961. A better agreement is obtained with the ${n}_{{\rm{H}}}\sim {r}^{-2.8}$ or ${n}_{{\rm{H}}}\sim {r}^{-3/2}$ curves, within statistical uncertainties. Although this measurement is insufficient to determine an exact model for circumgalactic regions, it excludes predictions with higher densities.