Decrease in the Brightness of the Cosmic X-ray and Soft Gamma-ray Background toward Clusters of Galaxies
aa r X i v : . [ a s t r o - ph . H E ] A p r to be published in Astronomy Letters, 2019, Vol. 45, No. 12, pp. 791–820(in Russian: Pis’ma v Astronomicheskii Zhurnal, 2019, Vol. 45, No. 12, pp. 835–865). DECREASE IN THE BRIGHTNESS OF THE COSMICX-RAY AND SOFT GAMMA-RAY BACKGROUNDTOWARD CLUSTERS OF GALAXIES
S. A. Grebenev and R. A. Sunyaev Received February 25, 2019; revised September 18, 2019; accepted October 23, 2019
Abstract —
We show that Compton scattering by electrons of the hot intergalactic gas in galaxy clustersshould lead to peculiar distortions of the cosmic background X-ray and soft gamma-ray radiation — anincrease in its brightness at hν ∼ < hν ∼ hν ∼ hν ∼ > z ∼ <
1) clusters theline at hν ∼ > ∼ K) plasma of their peripheral( ∼ < hν ∼ . ∼ ∼
9, and ∼
500 keV) to lower energies. Thus, in contrast to the microwavebackground radiation scattering effect, this effect depends on the cluster redshift z , but in a very peculiarway. When observing clusters at z ∼ >
1, the effect allows one to determine how the X-ray backgroundevolved and how it was “gathered” with z . To detect the effect, the accuracy of measurements should reach ∼ . DOI:
Keywords: cosmic background radiation, galaxy clusters, hot and warm-hot intergalactic plasma, Comptonscattering, recoil effect, Doppler effect, photoionization, bremsstrahlung and recombination radiation.
INTRODUCTIONIn recent years the effect of a decrease in the brightnessof the cosmic microwave background radiation towardgalaxy clusters has turned from an elegant theoreticalidea (Sunyaev and Zel’dovich 1970, 1972, 1980, 1981;Sunyaev 1980; Zel’dovich and Sunyaev 1982) into one ofthe most important tools for studies in the field of ob-servational cosmology and astrophysics of the early Uni-verse. The energy redistribution of photons in the back-ground radiation spectrum after their Thomson scatter-ing by electrons of the hot ( kT e ∼ −
15 keV) intergalac-tic cluster gas underlies this effect. In this case, a deficitof photons is formed in the low-frequency part of thespectrum (at energies hν ∼ < . kT r , where T r ≃ . * e-mail: [email protected] source with an unusual spectrum appears in the highfrequency part. This effect is unique in that its ac-tion is determined by the optical depth of the clustergas for scattering by electrons along the line of sight τ T = σ T R N e ( l ) d l , i.e., it is proportional to the gasdensity and not to the density squared, as the bright-ness of the intrinsic thermal radiation from the hot gas.Here, σ T is the Thomson scattering cross section. Sur-prisingly, the amplitude of the effect does not decreasewith cluster distance (redshift z ); the spectral shape ofthe background distortions does not depend on z either.Owing to these properties, the effect is widely used todetermine the parameters of clusters and to effectivelysearch for them. The effect is successfully observed withthe specially constructed SPT (South Pole Telescope,Carlstrom et al. 2002; Williamson et al. 2011; Bleemet al. 2015) and ACT (Atacama Cosmology Telescope,Hasselfield et al. 2013) telescopes and a number of othertelescopes (Birkinshaw 1999); the Planck satellite (Ade79192 GREBENEV, SUNYAEVet al. (Planck Collaboration) 2014, 2015, 2016a) hasmade an enormous contribution to the investigation ofthe effect.In this paper we consider a similar effect — the dis-tortions arising due to scattering by electrons of the hotgas of clusters in the cosmic X-ray background. Theexistence of such an effect has already been mentionedin Sunyaev and Zel’dovich (1981), Zel’dovich and Sun-yaev (1982), and Khatri and Sunyaev (2019). Based onsimple nonrelativistic ( hν ≪ m e c ) estimates, these au-thors concluded that it is impossible to directly observethe effect against the intrinsic thermal X-ray backgroundof the cluster gas. At the same time, they noted that itis potentially important to take into account this effectwhen considering the thermal balance of the gas. Inthis paper we performed relativistically accurate MonteCarlo computations of the effect and investigated anddiscussed the prospects for its observation in a harder( hν ∼ >
60 keV) energy range.The X-ray background (diffuse) radiation differs fromthe cosmic microwave background radiation by its origin(it is a superposition of the radiation spectra of a largenumber of AGNs — active galactic nuclei and quasars),but, at the same time, it is also characterized by a highdegree of isotropy and homogeneity. In the standard( hν ∼ <
10 keV) X-ray range from 60 to 80% of the back-ground radiation has already been resolved into separate(point) sources by telescopes with grazing-incidence mir-rors (see, e.g., Hasinger et al. 1998; Miyaji et al. 2000;Giacconi et al. 2001); another ∼
10% of the backgroundhas been explained by the thermal radiation of the gas ingalaxy clusters. An extrapolation of the X-ray spectrafor detected AGNs to the harder ( hν ∼ >
10 keV) energyrange, which takes into account the evolution of theirnumber and the composition of their population with z ,shows that here they should also make a dominant con-tribution to the background spectrum (Sazonov et al.2008; Ueda et al. 2014; Miyaji et al. 2015). It followsfrom the simulations by Ueda et al. (2014) that the ob-served background spectrum is formed mainly at z ∼ z , while the intensity (in the rest frame) decreasesslowly (in the range z ∼ . − z ∼ < z ∼ > hν > ∼ S ( E ) ≃ . E − . e − E/ . , at E < . E/ − . + (1)0 .
504 ( E/ − . + , at E > . . E/ − . Here, S ( E ) is the energy flux expressed in keVcm − s − keV − sr − and E is the photon energy hν in keV. This fit agrees well with the gamma-ray back-ground measurements at energies 1 MeV – 100 GeV bythe COMPTEL and EGRET telescopes of the CGROobservatory. The undistorted background spectrum cor-responding to this fit is indicated in Fig. 1a by the thicksolid (blue) line. The dotted line indicates the spectrumcorresponding to the extension of the soft component inEq. (1) to the hard range.As we will see below, the distortions in this spectrumthat arise when the background radiation interacts withelectrons of the hot gas in a galaxy cluster are fairly smalland do not exceed the current accuracy of our knowledgeof the spectrum shape and parameters. In this regardthe effect being discussed seems more difficult to mea-sure than the distortion of the microwave backgroundradiation whose spectrum has an almost ideal Planckianshape. However, if the accuracy of X-ray and gamma-raymeasurements will increase to the required level in fu-ture, then the undistorted spectrum will be remeasuredsimultaneously with the distorted one, which will allowthe deviations to be revealed. The estimates of the rel-ative deviations of the background spectrum presentedin this paper remain valid.MONTE CARLO COMPUTATIONSAs the initial approximation we will assume the hot gasin a cluster to be distributed spherically symmetricallywith uniform electron density N e and temperature kT e within its radius R c . The optical depth of such a gascloud for scattering calculated along the line of sight ASTRONOMY LETTERS Vol. 45 No. 12 2019
ECREASE IN THE BRIGHTNESS OF THE COSMIC X- AND GAMMA-RAY BACKGROUND 793
Fig. 1. (a) Spectra of the cosmic X-ray background (solid thick blue line) and the thermal (bremsstrahlung and recombina-tion) X-ray radiation from the hot intergalactic gas (thin green line, 5-eV resolution) of a cluster with a core radius R c = 350kpc and uniform electron temperature and density distributions inside it, kT e = 5 keV and N e = 1 . × − cm − (so that τ T = 2 × − ), the gas metallicity is Z = 0 . Z ⊙ . The dashed line indicates the exponential extension of the background fitat low ( hν ∼ <
60 keV) energies. (b) The relative background distortions due to scattering by cluster gas electrons with thesame temperature kT e = 5 keV, but different optical depths for Thomson scattering, τ T = 2 , , ,
12 and 16 × − (alongthe line of sight passing through the cluster center). (c) Attenuation of the distortions when they are observed at differentimpact parameters ρ = 0 . , . , . , . , . . R c relative to the center for a cluster with a β density distributionwith N c = 7 . × − cm − and the same core radius and temperature as those on other panels. The thin (green) linesindicate the distortion profile at the center. The thermal gas radiation is included in the distortions. The dotted straightlines correspond to the unperturbed background for each case. passing through its center will be τ T = 2 σ T N e R c = 2 τ c .Our computations of the Compton scattering of thebackground radiation in such a cloud were performedby the Monte Carlo method in accordance with the al-gorithms developed by Pozdnyakov et al. (1983). Thebackground radiation was assumed to be incident on thecloud isotropically. The angle-averaged radiation leav-ing the cloud was considered as the emergent one. Inthis sense, characterizing below the amplitude of thedistortions in the spectrum by the optical depth alongthe line of sight passing through the cloud center, by τ T we mean the characteristic of the cloud itself. Thegas cloud with an optical depth τ T = 1 × − has amass M g ≃ . × ( τ T / . r c /
350 kpc) M ⊙ . Thetotal mass M of the corresponding cluster, includingthe dark matter, should be greater at least by an order of magnitude. This is a moderate-mass cluster like theComa cluster. Below we will also consider more massiveclusters (see Table 4).The hydrogen and helium in the cluster gas were as-sumed to have normal cosmic abundances, X ≃ . Y ≃ .
24 by mass (Allen et al. 1973), respectively, N e ≃ ( X + 0 . Y ) ρ/m p ≃ . ρ/m p . As a rule, theabundance of the iron-group elements was taken to be Z = 0 . Z ⊙ , but it could change. At a gas temper-ature typical for clusters the atoms of most elementsare ionized fully, iron is ionized to the hydrogen- andhelium-like states, and nickel is ionized to the lithium-like state. Photoabsorption by Fe XXVI and Fe XXVions introduces distortions into the spectrum compara-ble in relative amplitude to the distortions due to scat-tering by electrons. Therefore, this process should be ASTRONOMY LETTERS Vol. 45 No. 12 2019
94 GREBENEV, SUNYAEVtaken into account in the computations.The degree of iron ionization for a plasma of the re-quired temperature was obtained from the code by Ray-mond and Smith (1977), which is used to compute theionization balance of an optically thin plasma. Apartfrom the Fe XXV and Fe XXVI ions, the absorption byFe XXII – Fe XXIV ions was taken into account whencomputing the distortions of the background spectrum.The absorption by Ni XXIII – Ni XXVIII was taken intoaccount in the same way. The ionization by backgroundphotons in the ionization balance was neglected. Weused the fits of the cross sections for photoabsorptionby various ions from Verner and Yakovlev (1995) andVerner et al. (1996). The bremsstrahlung and recom-bination radiation spectrum of the intergalactic clusterplasma was also computed using the Raymond-Smithcode. The thin solid (green) line in Fig. 1a indicatesthis spectrum for a cluster with a plasma temperature kT e = 5 keV. On the whole, the recombination onto theiron atoms ionized as a result of cosmic background pho-toabsorption is analogous to the recombination onto theions formed by collisional processes. The correspond-ing recombination radiation gives only a small ∼ < The Model with a Uniform Density Distribution
Figure 1b presents the distortions in the background ra-diation spectrum (Eq. (1)) computed for various op-tical depths of the cloud . The plasma temperaturewas assumed for all computations to be the same andequal to kT e = 5 keV, the heavy element abundance was Z = 0 . Z ⊙ . It can be seen from the spectrum corre-sponding to τ T = 1 . × − , for which the thin (green)line indicates the result of our computation without ab-sorption by iron ions, that Compton scattering slightly(by ∼ . hν ∼ <
40 keV. However, photoabsorption reduces this risebeyond the iron ionization threshold hν ∼ > ∼ .
2% near the threshold is observed in the backgroundspectrum due to absorption at these energies. The max-imal increase and decrease in intensity in the spectrumare ∼ < .
005 and ∼ . − .
04% for the spectrum cor-responding to the smallest optical depth τ T = 2 × − of those considered and reach ∼ .
03 and ∼ . τ T = 1 . × − .The background spectrum also exhibits a weak absorp-tion line on the L shell of iron ions with the threshold at hν ∼ > ∼ .
04% (for a cluster In our computation of the photoabsorption in a warm-hot in-tergalactic medium with kT e = 0 . Or various electron densities N e , because at a fixed clustercore radius the optical depth and density are related uniquely, τ T ∼ N e . Table 1.
Parameters of the MeV dip in the backgroundspectrum toward a cluster with a uniform gas densitya τ T , hν th b hν γ c h ∆ ν γ d W γ e × − keV keV MeV keV2 29 584 2.3 1.14 33 584 2.7 3.48 35 584 2.9 10.012 33 584 3.0 18.116 35 604 3.0 27.1a kT e = 5 keV.b The energy of the beginning of the dip.c The energy of the deepest point of the dip.d The FWHM of the dip.e The equivalent width of the dip. with τ T = 1 . × − ).At energies hν ∼ >
60 keV scattering by electrons leadsto a “dip” in the background spectrum due to the recoileffect. As a result of this effect, the photons lose a certainfraction of their energy and are shifted downward alongthe frequency axis. These shifted photons make a certaincontribution to the intensity excess in the spectrum atenergies hν ∼ <
60 keV (to be more precise, at hν ∼ < kT e ,see the low-frequency asymptotics to Eq. (10) below andthe curve corresponding to kT e = 0 keV in Fig. 7). Thesoft photons shifted upward along the frequency axis (to hν ∼ kT e ) due to the Doppler effect make a majorcontribution to the excess.In the range hν ∼ −
600 keV, where the depthof the dip in the background spectrum due to the re-coil effect is maximal, the drop in background radiationbrightness reaches 0 . − .
2% for realistic optical depthsof the cluster, τ T ∼ (4 − × − . At τ T = 1 . × − the dip deepens to ∼ . W γ for various Thom-son optical depths τ T of the gas. Given that the lineis broad and the background intensity along its profilecan change greatly, we defined the equivalent width asfollows: W γ = R ∞ ν th ( F − F ν ) /F d hν , where F ( ν ) is thephoton spectrum of the background [ S ( ν ) = hν F ( ν )].We considered the same model cluster with a uniformdensity, R c = 350 kpc, and kT e = 5 keV. The table alsogives the energies of the beginning, hν th , and the center, hν γ , of the dip in the spectrum and its full width at halfmaximum (FWHM) h ∆ ν γ . Whereas the hν th and hν γ variations in Table 1 can be attributed to the error ofour calculation, the changes in h ∆ ν γ and W γ are real,they reflect a drop in the dip amplitude as τ T decreases.Note that the dips in the background spectrum relatedto the recoil effect and photoabsorption are formed in theradiation going through the cluster from its back side.To a first approximation, their depth is proportional tothe mean optical depth of the cluster gas in this direc- ASTRONOMY LETTERS Vol. 45 No. 12 2019
ECREASE IN THE BRIGHTNESS OF THE COSMIC X- AND GAMMA-RAY BACKGROUND 795
Fig. 2.
Comparison of the background distortions arisingin the hot cluster gas (relative to its initial spectrum) inthe direct escape radiation (black curve) and scattered ra-diation (green curve). The blue (thick) curve indicates thesum of these distortions. The cluster gas is assumed to bedistributed uniformly within the radius R c = 350 kpc and tohave an optical depth τ T = 1 . × − , temperature kT e = 5keV, and metallicity Z = 0 . Z ⊙ . tion: < τ T > = 2 R Z R c τ T ( ρ ) ρ dρ = 23 τ T , (2)where τ T ( ρ ) is the Thomson optical depth of the clusteralong the line of sight passing at an impact parameter ρ from the center. It is τ T ( ρ ) = τ T (cid:0) − ρ / R (cid:1) / , at ρ < R c , at ρ ∼ > R c . (3)The amplitude of these dips in the background spectrumshould be very sensitive to its spatial fluctuations. Toreliably detect the effect of a decrease in backgroundbrightness at the corresponding energies, it is necessaryto use extended (nearby) clusters.In contrast, the increase in the background intensitybelow hν <
60 keV is associated with the scattered pho-tons. These are the photons of the radiation incidenton the cluster from all sides, least of all from its backside (the photons coming from the back side are scat-tered at small angles and do not contribute noticeablyto the spectral distortion). Figure 2 shows separatelythe spectra of the distortions arising in the direct es-cape radiation and scattered radiation in a cluster witha temperature kT e = 5 keV and a Thomson optical dept τ T = 1 . × − . It is clearly seen that all “negative”features (due to photoabsorption and the recoil effect)are contained only in the direct escape spectrum; thescattered photon spectrum is smooth and has no dis-tinct features. Formally, the scattered photons are also subject to photoabsorption and are shifted to lower ener-gies after recoil (in secondary interactions with the gas),but these effects are negligible due to the smallness ofits optical depth.The contribution of various sky regions to the scat-tered radiation spectrum is defined by the phase func-tion, which in the nonrelativistic limit hν ≪ m e c hasa simple form, dσ T ( θ ) = (3 / σ T (1 + cos θ ) sin θ d θ ,where θ is the scattering angle (and, in view of the sym-metry of the phase function, the photon arrival anglerelative to the line of sight toward the cluster). This isa smooth function. Clearly, the photons responsible forthe increase in background intensity in the cluster dueto the Doppler effect are collected from the entire skyand, hence, these distortions should not be sensitive tospatial background fluctuations. Dependence on the Gas Temperature
Figure 1a shows that the detection of distortions in thebackground spectrum in the X-ray range hν ∼ <
60 keVwill be greatly complicated due to the intrinsic thermalradiation of the intergalactic gas.The chances to detect the distortions increase for re-laxed clusters with a lower temperature. This is illus-trated by Fig. 3, in which the changes in the ampli-tude and shape of the distortion spectrum are shownas a function of the gas temperature. We again con-sider the model cluster with a uniform density distri-bution within the radius R c = 350 kpc and an opti-cal depth along the line of sight passing through thecenter τ T = 1 . × − . In Fig. 3a the gas tempera-ture in the cluster was assumed to be 2 keV. The figuregives a general idea of the relationship between the back-ground spectrum and the thermal radiation spectrum ofthe cluster gas (just as Fig. 1a, which gives such an ideafor a cluster with kT e = 5 keV). Figure 3b shows the rel-ative distortions (in percent) arising in the backgroundspectrum after scattering and absorption in the gas ofsuch a cluster. We considered various gas temperatures kT e = 2 , , , , , , and 20 keV.The depth of the MeV dip in the background spectrumdue to the recoil effect after scattering by electrons isalmost independent of the gas temperature. As wouldbe expected, the amplitude of other “negative” changesin the spectrum (the dips due to photoabsorption) reachits maximum for a cold gas with kT e = 2 keV. Thedepth of the dip beyond the threshold hν ∼ ∼ . ∼ . Z = 0 . Z ⊙ . The amplitude ofthe absorption lines changes as the metallicity increasesand decreases. In the figure this change is indicated bythe thin (green) lines for kT e = 2 keV ( Z = Z ⊙ ) and kT e = 20 keV ( Z = 0).In contrast to the “negative” changes in the back-ground spectrum, the amplitude of the “positive” de-viation (the excess of background radiation due to the ASTRONOMY LETTERS Vol. 45 No. 12 2019
96 GREBENEV, SUNYAEV
Fig. 3.
Same as Fig. 1, but for τ T = 1 . × − , R c = 350 kpc ( N e = 8 . × − cm − ), and various temperatures: (a) kT e = 2keV, (b, c) kT e = 2 , , , , , , and 20 keV. The metallicity is Z = 0 . Z ⊙ . The green line on panel (b) indicatesour computation without photoabsorption ( Z = 0) for kT e = 20 keV and with photoabsorption at metallicity Z = Z ⊙ for kT e = 2 keV. The background distortions on panel (c) include the thermal plasma radiation (the thin green lines — withoutthis radiation). Doppler effect) is maximal for the hottest gas with kT e = 20 keV. At energies hν ∼ −
80 keV a broademission feature (line) whose relative amplitude reaches ∼ . − .
15% is formed in the corresponding spectrumof the background distortions partly due to the Comp-ton processes and partly due to the properties of thebackground spectrum itself.In Fig. 3c the thermal radiation of the intergalactic gaswas added to the background distortions (the measure-ments are assumed to be performed toward the clustercenter). We see that the energy range that allows theCompton distortions to be directly observed in the back-ground spectrum, without any illumination by the ther-mal gas radiation, turns out to be sufficiently wide onlyfor relaxed clusters with a gas temperature kT e ∼ < hν ∼
400 keV.Note that the iron and nickel ions were assumed to beat rest when computing the photoabsorption line pro- files. This is admissible, because the Doppler broaden-ing and smearing related to the thermal motion of ions,for example, for the profile of the absorption jump at hν th = 9 keV at typical gas temperatures for clustersis only ∼ hν th (2 kT e / m p c ) / ∼ kT e / / eV. The resolution of the computations presented inFigs. 1 and 3 and most of the succeeding figures, h ∆ ν ∼ . hν th ≃
300 eV, is much coarser. Even in Fig. 4,which shows a detailed profile of this line (with a resolu-tion that is better by several times), to demonstrate itscomplexity and multicomponent structure, the thermalmotion of iron and nickel ions could smooth only slightlythe sharpest features of the fine line structure.Figure 4 shows that the edge of the absorption linenear the threshold is strongly distorted even without anythermal broadening. It has the shape of a more or lessregular “step” only in the case of kT e = 3 keV. At lowertemperatures the beginning of both lines is shifted left-ward, the threshold turns into a semblance of a “flight ofstairs” consisting of several successive steps. At highertemperatures additional steps appear on the right, shift-ing the threshold of the lines by 300–400 eV to greater ASTRONOMY LETTERS Vol. 45 No. 12 2019
ECREASE IN THE BRIGHTNESS OF THE COSMIC X- AND GAMMA-RAY BACKGROUND 797
Fig. 4.
Formation of absorption lines in the X-ray back-ground spectrum due to photoionization of iron and nickelions in the hot gas of a galaxy cluster. A cluster with auniform density, optical depth τ T = 1 . × − , gas temper-atures kT e = 1 . , , , , , and 12 keV, and metallicity Z = 0 . Z ⊙ . energies. On reaching kT e = 8 keV, the absorption linewith the threshold at ∼ hν ∼ ∼ W X .Since this line is formed in a region of enhanced back-ground flux (through the Compton continuum formingin the cluster due to the Doppler effect after scatter-ing by electrons), its equivalent width W X = R ∞ ν th ( F c − F ν ) /F c d hν was determined relative to this enhancedlevel F c ( ν ) (the background photon spectrum distortedin the cluster obtained in the limit Z → hν th and the en-ergy of the deepest point of its profile hν X are shiftedby 400–500 eV to higher energies. The cause can beunderstood from Fig. 4 — a new step related to the ab-sorption of photons by more strongly ionized iron andnickel ions appears in the structure of the left line edge.For the same reason, the line width h ∆ ν X (FWHM ),which slightly increases with temperature probably dueto a change in the shape of the Compton continuum, de-creases abruptly by ∼
500 eV on reaching kT e = 12 keV.At this time the step height reaches half the absorptionline depth. The Model with a β Density Distribution
Peripheral observations of the gas in clusters with a real(decreasing with radius) density distribution could also
Table 2.
Parameters of the photoabsorption line with thethreshold at ∼ kT e , hν th b hν X c h ∆ ν X d W X ekeV keV keV keV eV Z = 0 . Z ⊙ . Z = 1 . Z ⊙ . τ T = 1 . × − .b The threshold energy.c The energy of the deepest point of the line.d FWHM.e The equivalent width.f The value reflects the resolution of our computation. have a certain advantage in combatting the thermal ra-diation. Indeed, the thermal radiation intensity is pro-portional to N , while the distortions due to scatter-ing by electrons are proportional to N e . Accordingly,the contribution of the thermal radiation should dropto the cluster edge faster than that of the scattered one(Zel’dovich and Sunyaev 1982).The action of this effect is demonstrated by Fig. 1c, inwhich the distortions of the background spectrum (in-cluding the thermal plasma radiation) are representedby the thick (blue) lines for various impact parameters ρ from the cluster center. For comparison, the thin(green) line indicates the spectrum of the distortionsthat should be observed toward the center. In the case oflarge ρ the lines intersect, suggesting that the contribu-tion of the thermal radiation vanishes at lower energiesthan those for the observations toward the cluster cen-ter. The spectra shown in this figure were computedfor a β gas density distribution (Cavaliere and Fusco-Femiano 1976), which agrees satisfactorily with the ob-served X-ray brightness distribution of many clusters(Arnaud 2009), N e = N c (cid:18) R R (cid:19) − β/ . (4)It follows from observations (Jones and Forman 1984)that for most clusters β ≃ /
3. At such β the cluster sur- ASTRONOMY LETTERS Vol. 45 No. 12 2019
98 GREBENEV, SUNYAEVface emission measure EM ( ρ ) = 2 R ∞ N ( ρ ) d l , definingthe thermal radiation intensity, and the Thomson opticaldepth along the line of sight τ T ( ρ ) = 2 σ T R ∞ N e ( ρ ) d l ,defining the amplitude of the spectral distortions due toscattering and absorption, at an impact parameter ρ are,respectively, EM ( ρ ) = π (cid:18) ρ R (cid:19) − / N R c (5)and τ T ( ρ ) = π (cid:18) ρ R (cid:19) − / σ T N c R c . (6)The gas emission measure EM in a real cluster is seento drop with increasing ρ much faster than the opticaldepth τ T . Therefore, at large ρ the thermal radiationshould cease to hinder the background observations atlower energies than in the observations toward the clus-ter center. This is illustrated by Fig. 1c, based on Eqs.(5) and (6). Unfortunately, the extension of the rangefavorable for the observation of background distortionsafter scattering turns out to be moderately large, whilethe amplitude of the effect proper, in turn, decreasesquite rapidly with increasing ρ .In the computation whose result is used here, as be-fore, the intergalactic gas temperature was taken tobe kT e = 5 keV, the cluster core radius is R c = 350kpc, and the Thomson optical depth of the gas alongthe line of sight passing through the cluster center is τ T = 1 . × − . The distortions in the backgroundspectrum toward the center (at ρ = 0) were computedby the Monte Carlo method by assuming the densityprofile to break at the “outer” radius R b = 2 R c .The dependence of the results of our computations on R b is investigated in Fig. 5. The spectra of the back-ground distortions arising in clusters with a β gas den-sity distribution, the same temperatures kT e and op-tical depths along the line of sight toward the center τ T , but different break radii of the β density profile R b = 1 , , , and 10 R c are shown here. For com-parison, the thin lines indicate our computations of thebackground distortions in a cluster with a uniform den-sity distribution (with the same kT e , R c , and τ T ).A change in R b slightly changes the optical depthof the gas τ T ( ρ = 0 , R b ) in a real cluster relative to τ T ( ρ = 0) = πσ T N c R c following from Eq. (6) (derivedin the limit R b → ∞ ). Nevertheless, it is appropriate tocompare the clusters of equal optical depths for the pureeffect of different cluster geometries to be seen. For thispurpose, in the computations in Fig. 5 the central den-sity N c of the cluster profile in Eq. (4) was multipliedby 0 . π/ arctan( R b /R c ). With or without this correc-tion, the model clusters considered, of course, cannot bedeemed identical, if only because the cluster gas massincreases noticeably with R b from M g = 3 . × M ⊙ for R b = 1 R c to M g = 6 . × M ⊙ for R b = 10 R c . Fig. 5.
Comparison of the relative background distortionsarising in a “real” galaxy cluster (thick blue lines) with a β gas density distribution (Eq. (4)) and a cluster with auniform density distribution (thin green line). The gas in theclusters has the same Thomson optical depths τ T = 1 . × − (along the line of sight through the cluster center), core radii R c = 350 kpc, electron temperatures kT e = 5 keV, and Z =0 . Z ⊙ . We considered the cases with various break radii R b of the β profile. Figure 5 shows that the cluster with a real density dis-tribution with R b = 1 R c leads to virtually the samebackground distortions in amplitude and shape of theenergy dependence as does the cluster with a uniformdensity distribution. It can also be seen from the figurethat even despite the increase in the mass of the clus-ter with a real density distribution with increasing R b ,the amplitude of the background distortions decreasesrapidly in this case. This behavior can be explained bytaking into account the fact that the Thomson opticaldepth of the intergalactic gas averaged over the visiblearea of the cluster with a β profile, < τ T > , decreaseswith increasing R b . Indeed, by integrating the opticaldepth from Eq. (6) over the area 2 π R R b τ T ( ρ ) ρ d ρ andnormalizing to πR , we find < τ T ( R b ) > = 2 τ T R R "(cid:18) R R (cid:19) / − (7) ≃ . τ T at R b = 1 R c and ≃ . τ T at R b = 10 R c . ASTRONOMY LETTERS Vol. 45 No. 12 2019
ECREASE IN THE BRIGHTNESS OF THE COSMIC X- AND GAMMA-RAY BACKGROUND 799
Fig. 6.
Formation of absorption lines in the thermal radia-tion spectrum of the cluster gas due to the ionization of ironions: relative to (a) the thermal radiation itself and (b) thecosmic background radiation. A cluster with a uniform den-sity, radius R c = 350 kpc, τ T = 1 . × − , Z = 0 . Z ⊙ , andvarious gas temperatures kT e . Note that we compute the average spectrum of thebackground distortions in the cluster by the Monte Carlomethod. To obtain the background distortions towardits center ( ρ = 0) presented in Fig. 1c, their amplitudewas properly corrected for the above decrease in opti-cal depth when averaged over the visible cluster area π (2 R c ) .¿From the viewpoint of effective detection of the back-ground distortions due to the interaction with the clus-ter gas, Fig. 5 clearly indicates that the cluster obser-vations by a telescope with an aperture (angular reso-lution) covering the central part of the cluster with aradius ∼ < R c . A similar conclusion can be drawn froman analysis of the background distortions arising in acluster with the density distribution predicted by the Navarro-Frenk-White model (hereafter NFW, Navarroet al. 1997). Such a cluster is analyzed in the Appendix. Spectral Distortions of the Intrinsic Gas Radiation
The detection of the background distortions associatedwith its interaction with the hot gas of galaxy clustersis complicated not just by the presence of intense intrin-sic gas radiation. In turn, distortions whose amplitudeexceeds noticeably the relative amplitude of the back-ground distortions appear in the spectrum of this ther-mal radiation.Indeed, the Raymond-Smith code, along with othercodes used to compute the bremsstrahlung and recom-bination radiation spectrum of an optically thin plasma,suggests that the optical depth of the plasma τ T → hν ∼ > τ T = 1 . × − , kT e = 2 , , , or 12 keV and Z = 0 . Z ⊙ . We used the same code as that for the back-ground distortion computations, but the source of pho-tons was assumed to be uniformly distributed through-out the cluster and its radiation spectrum was takenfrom our computations of the optically thin plasma spec-trum by the Raymond-Smith code for a cluster tempera-ture kT e (only the continuum bremsstrahlung was takeninto account).We see that the distortions arising in the spectrum ofthe thermal gas radiation resemble those in the back-ground spectrum. However, there is also a difference.The bremsstrahlung spectrum abruptly breaks at ener-gies hν ∼ > kT e . At the same time, Comptonization shiftsthe photons upward along the frequency axis, tendingto form a Wien spectrum with a characteristic break en-ergy ∼ kT e . This process explains the sharp increasein radiation intensity at energies hν ∼ > kT e . The pro-cess is well known in X-ray astronomy and is success-fully used to explain the observed spectra of accretingblack holes (see, e.g., Shapiro et al. 1976; Sunyaev andTitarchuk 1980). The upward shift of the photons alongthe frequency axis is determined by the Doppler effect;after each scattering the average change in photon fre-quency is ∆ ν/ν ∼ ( kT e /m e c ) / . A competing processis the recoil effect, which in the nonrelativistic limit low-ers the photon frequency after scattering, on average, by∆ ν/ν ∼ − hν/m e c (these estimates can be easily ob- ASTRONOMY LETTERS Vol. 45 No. 12 2019
00 GREBENEV, SUNYAEV
Table 3.
Parameters of the total (including the thermalradiation distortions) photoabsorption line with the thresh-old at ∼ kT e , hν th b hν X c h ∆ ν X d W X ekeV keV keV keV keV Z = 0 . Z ⊙ Z = 1 . Z ⊙ R c = 350 kpc, τ T = 1 . × − . b The threshold energy.c The energy of the deepest point of the line.d FWHM.e The equivalent width. tained from the Kompaneets equation, see below). Theprocesses are balanced at hν ∗ ∼ ( kT e m e c ) / ∼
32 ( kT e / / keV . (8)Accordingly, for clusters with a low gas temperature kT e ∼ < − ∼ −
50 keV is formed in the distortion spec-trum (Fig. 6); for hotter clusters the distortions growwith energy until the complete cutoff of the thermalspectrum. Formula (8) to some extent also explains theenergy of the broad emission feature appearing in thedistortion spectrum of the background in hot galaxyclusters (Fig. 3b).The amplitude of the distortions in the thermalplasma radiation spectrum is ∼ τ T and accounts for frac-tions of percent of the intensity of the spectrum itself(just as the amplitude of the cosmic background distor-tions). However, since the thermal radiation intensityin the X-ray range exceeds the background intensity bytwo or three orders of magnitude, these distortions arecomparable to the background intensity. When the ob-served cluster radiation spectrum is fitted by the thermalradiation model of an optically thin plasma, these distor-tions, including the iron ion absorption lines, will not besubtracted and will lead to a noticeable enhancement ofthe distortions formed directly in the background spec-trum. This is clearly seen from Fig. 6b, in which thebackground distortions and the thermal gas radiationdistortions are added together and are given in percentrelative to the background spectrum. The background level is shown by dotted lines. The amplitude of thedistortions in the iron and nickel photoabsorption linesreaches 100% or more. Note that the absorption linewith the 9-keV threshold greatly weakens in the distor-tion spectra of cold clusters with kT e = 2 − ∼ hν ∼ <
100 keV exceed noticeablythe background distortions. They are very difficult totake into account.Below we will assume that the question about the sub-traction of the thermal gas radiation spectrum, given itsdistortions due to the finite optical depth, from the ob-served cluster spectrum has somehow been solved. Wewill consider only the distortions arising directly in thecosmic background. The distortions of the intrinsic ther-mal gas radiation will be considered in detail in a sepa-rate paper (Grebenev and Sunyaev 2020).ANALYTICAL ESTIMATESWhen investigating the Compton scattering of the cos-mic microwave background radiation in the hot gas of agalaxy cluster, important analytical estimates (Sunyaevand Zel’dovich 1980; Zel’dovich and Sunyaev 1982) wereobtained by solving the Kompaneets (1957) equation,which describes the photon energy redistribution energyin the diffusion approximation. The validity of applyingthis equation to an optically thin gas typical for clusterswas tested and confirmed by Sunyaev (1980). Similarquite interesting estimates can also be obtained for theproblem under consideration.The Kompaneets equation with relativistic corrections(Cooper 1971; see also Arons 1971; Illarionov and Sun-yaev 1972; Grebenev and Sunyaev 1987), but at a mod-erately high electron temperature ( kT e ≪ m e c ) can berepresented as ∂F ν ∂τ c = hm e c ∂∂ν (cid:20) ν ξ ( T e )1 + βν + γν (cid:18) F ν ν + kT e h ∂∂ν F ν ν (cid:19)(cid:21) , where F ν is the intensity of the photon spectrum, τ c is the Thomson radial optical depth of the cloud, β =9 × − keV − , γ = 4 . × − keV − ,ξ ( T e ) = 1 + 52 kT e m e c . We neglected the term responsible for the induced scat-tering and retained only the term of the first order in kT e /m e c in ξ ( T e ) (see Cooper 1971). Substituting the ASTRONOMY LETTERS Vol. 45 No. 12 2019
ECREASE IN THE BRIGHTNESS OF THE COSMIC X- AND GAMMA-RAY BACKGROUND 801
Fig. 7.
Comparison of the relative hard X-ray backgrounddistortions due to scattering in the hot cluster gas calculatedfrom the analytical formulas (9) and (10) (solid thick bluelines) and by the Monte Carlo method (thin green lines).The photoabsorption by iron ions was disregarded. We con-sidered a cluster with a uniform density distribution, coreradius R c = 350 kpc, optical depth τ T = 1 . × − , andvarious electron temperatures kT e = 2 , , , , , and 20keV. For kT e = 20 keV the dashed line indicates the calcula-tion including only the recoil effect after scattering. background intensity F ν ( ν ) = S ν ( ν ) /hν in the form F ν = Aν − α exp ( − ν/ν ) (the part of the spectrum fromEq. (1) corresponding to low energies hν ∼ <
60 keV)into the right-hand side of this equation, for the rela-tive changes in the background we find∆ F ν F ν = hνm e c τ c ξ ( T e )1 + βν + γν (cid:20) − kT e hν + (9)+ (cid:18) − α − νν − νν (cid:19) (cid:18) − kT e hν − kT e hν (2 + α ) (cid:19)(cid:21) . Here, we introduce the notation ν for the function ν ( ν ) = 1 + βν + γν β + 2 γν . The distortion spectrum for the harder, hν ∼ >
60 keV,(power-law) part of the spectrum can be found fromEq. (9) by passing to the limit ν → ∞ :∆ F ν F ν = hνm e c τ c ξ ( T e )1 + βν + γν × (10) × (cid:20) (cid:18) − α − νν (cid:19) (cid:18) − kT e hν (2 + α ) (cid:19)(cid:21) . In the limit βν ≪ ν ≪ ν Eq. (10) gives∆ F ν F ν = (2 + α ) τ c kT e ξ ( T e ) m e c (cid:20) α − − α α hνkT e (cid:21) . Accordingly, to a first approximation, for hν ∼ < kT e , theamplitude of the effect is proportional to the Comptonparameter y C = τ c ξ ( T e ) kT e /m e c . In the opposite limit hν ≫ kT e the formula for the distortions takes the form∆ F ν F ν = − τ c ξ ( T e ) hνm e c α − ν/ν βν + γν , i.e., the amplitude of the effect is proportional to τ c . The thick solid (blue) lines in Fig. 7 indicate the re-sults of applying Eqs. (9) and (10) to a cluster with aradial optical depth τ c = 6 × − (which corresponds tothe optical depth along the line of sight passing throughthe center τ T = 1 . × − ) and electron temperatures kT e = 2 , , , , and 20 keV. At hν >
60 keV weused a superposition of the solutions (10) for the spec-tra with different photon indices in accordance with themodel of the background spectrum (Eq. (1)). Since thebackground spectrum was fitted by different functions atlow and high energies, its derivative can have a discon-tinuity at 60 keV. In the distortion spectra calculatedfrom the approximate formulas (9) and (10), a jump orbreak is observed near this energy for many tempera-tures. Therefore, the analytical solution in the narrowregion ±
10 keV near 60 keV is not shown in the figurefor clarity. For clarity, we also disregard the photoab-sorption of background photons by strongly ionized ironions. This process can be easily included in the analyti-cal solution (see Grebenev and Sunyaev 1987).The thin (green) lines in Fig. 7 indicate the resultsof our Monte Carlo computations of the distortions inthe background spectrum. The computations were per-formed for the same cluster parameters as those forthe analytical solution and allow them to be compared.However, they differ not only by the method of solu-tion, but also by the boundary conditions: for the Kom-paneets equation the isotropic background source is lo-cated at the gas cloud center, while for the Monte Carlomethod the background radiation penetrates the cloudfrom outside. Note that an indistinct feature, which, ob-viously, is associated with the piecewise continuous fit ofthe background spectrum (Eq. (1)), is also observed near60 keV in many of the spectra computed by the MonteCarlo method.
ASTRONOMY LETTERS Vol. 45 No. 12 2019
02 GREBENEV, SUNYAEV
Fig. 8.
Expected cosmic background (its intensity) distortions after scattering and absorption in the hot cluster gas versusits redshift z (thin green lines). Clusters with a uniform density, optical depth τ T = 1 . × − , and core radius R c = 350 kpc.The gas temperature is kT e = (a) 2 and (b) 5 keV, Z = 0 . Z ⊙ . The thick (blue) lines indicate the background distortionsincluding the thermal plasma radiation (shown incompletely on panel (a)). On the whole, it can be said that the calculation basedon Eqs. (9) and (10) correctly reproduces the spectralshape of the numerically computed distortions, althoughit smoothes the deepest part of the absorption featureat energies hν ∼ −
600 keV arising due to the recoileffect. It is possible that a better coincidence could beachieved here by numerically integrating the relativistickernel of the kinetic equation for Compton scattering(Sazonov and Sunyaev 2000), but in this case an expres-sion for the spectrum could not be obtained explicitly.The presented formulas give quite reasonable estimatesof the distortions in the background spectrum. The an-alytical solution also allows the nature of the variouscomponents in the distortion spectrum to be easily clar-ified. For example, the dashed line in Fig. 7 indicatesour calculation of the distortions in the limit T e → hν ∼ <
150 keV and the shape of the left edge of the ab-sorption feature at hν ∼ >
150 keV are associated with theMaxwellian velocities of electrons and are formed due tothe Doppler effect. DEPENDENCE ON Z One of the remarkable properties of the effect of mi-crowave background radiation scattering by the hot gasof galaxy clusters is its independence of the redshift z .Indeed, no matter how far the cluster is, the distortionsare observed in its present-day well measured spectrumcharacterized by a temperature T r = 2 . T r was a factor of (1 + z ) higher atthe time of its interaction with the cluster, this doesnot manifest itself in any way, because the equation de-scribing the Doppler spectral distortions depends onlyon x = hν/kT r (Zel’dovich and Sunyaev 1982), i.e., it isinvariant in z .The situation with the X-ray background scatteringis different. Here all distortions are also observed in thepresent-day spectrum, although they are formed in thebackground spectrum at the cluster redshift z . However,apart from the Doppler ones, among them there are thedistortions that arise due to photoabsorption and therecoil effect at quite specific energies — the absorptionthresholds hν ∼ hν ∼
500 keV. In thespectrum observed at z >
ASTRONOMY LETTERS Vol. 45 No. 12 2019
ECREASE IN THE BRIGHTNESS OF THE COSMIC X- AND GAMMA-RAY BACKGROUND 803shifted to low energies. Figure 8 presents the spectra ofthe background distortions that should be recorded fromclusters with the same parameters ( τ T = 1 . × − , R c = 350 kpc, and kT e = 2 keV in Fig. 8a or 5 keVin Fig. 8b), but located at different redshifts. The spec-tra were obtained by recalculating the present-day back-ground spectrum (Eq. (1)) to the corresponding z us-ing standard formulas (see, e.g., Zel’dovich and Novikov1975), computing the distortions there, and then recal-culating the spectrum back to z = 0 . Both (photoab-sorption and recoil effect) lines are seen to be actuallygreatly shifted leftward; at z = 2 the absorption line isat ∼ ∼
200 keV. However, the case of z = 2 is, in a sense, extreme — the clusters are subjectto strong evolution and at z ∼ z , like themicrowave background distortions.As z increases, the spectrum of the thermal plasmaradiation in the cluster is also shifted to low energiesand its intensity decreases. This is indicated by the thick(blue) lines in Fig. 8. This shift (and attenuation) allowsthe effects of Compton scattering and photoabsorptionof the background in distant clusters to be investigatedat lower energies. The fundamental difference in theshape of the z dependence of the amplitude of Comp-ton X-ray background distortions compared to the fluxof bremsstrahlung and recombination radiation is illus-trated by Fig. 9. The solid (dark green) lines in thisfigure indicate the variation in the spectral flux of ther-mal radiation wth z for a cluster with a uniform den-sity distribution, τ T = 1 . × − , R c = 350 kpc, and kT e = 2 keV expected during its observation by a tele-scope with an aperture radius of 5 ′ (FWHM). The curvesfrom top to bottom correspond to the fluxes at energies hν = 2 , ,
12 and 20 keV. The dashed line indicates the z dependence of the integrated flux. The initial flux riseup to z ≃ .
06 is related to the increase in the volumeemission measure of the cluster gas visible within theaperture: EM = 4 π R " − (cid:18) − ρ R (cid:19) / , where ρ z is the impact parameter in the cluster framecorresponding to the specified aperture width. At high z z, when the cluster is already completely within thefield of view, the integrated flux of its thermal radia-tion drops with z as a power law; the spectral fluxes athigh energies drop more rapidly due to the cutoff of thebremsstrahlung and recombination radiation spectrumat the corresponding energies. The dotted line indicates In our calculations we adopted the standard ΛCDM cosmolog-ical model with Ω M = 0 . , Ω Λ = 0 . , and H = 70 km s − Mpc − . Fig. 9.
Variations in the spectral flux (recorded within thetelescope aperture with a 5 ′ radius) of thermal X-ray radia-tion from the hot gas in a galaxy cluster (dark green lines)and the absolute amplitude of the background distortionsdue to Compton scattering and photoabsorption in this gas(lines with dots) with z . The energy hν for which the flux isgiven is specified near each curve. The dashed line indicatesthe change in the integrated flux of thermal radiation. Thedotted line indicates the X-ray background flux at 5 keV.The cluster gas has an optical depth for Thomson scattering τ T = 1 . × − , core radius R c = 350 kpc, and temperature kT e = 2 keV. the X-ray background flux at 5 keV falling into this aper-ture.Because of the shift of the photoabsorption lines andthe absorption feature related to the recoil effect to-ward low energies, the z dependence of the amplitudeof the Compton distortions and photoabsorption takesa fairly complex shape (Fig. 9, especially 6 and 8 keV).The abrupt jumps on these curves are associated withthe passage of the threshold of the absorption line at ∼ z changes, given its fine structure. Suchjumps should also be observed at low energies — left-ward of the absorption threshold at ∼ ASTRONOMY LETTERS Vol. 45 No. 12 2019
04 GREBENEV, SUNYAEV
Table 4.
Parameters of the individual clusters selected to estimate the effect of scattering and absorption of the X-raybackground in their hot intergalactic gas ∗ Cluster namea z R , b R c , θ c , kT e , N c c M g M b Y SZ d Ze, τ T ,f Referencegmain alternative kpc kpc ′ keV 10 M ⊙ Z ⊙ − M of a cluster with a mean density equal to 500 ρ cr ( z ) of the Universe.c The gas density at the cluster center, 10 − cm − .d The microwave background scattering “efficiency” (Kravtsov et al. 2006), 10 M ⊙ keV.e The abundance of the iron-group elements compared to the normal cosmic abundance.f The Thomson optical depth along the line of sight passing through the cluster center.g 1 – Menanteau et al. (2012); 2 – Bulbul et al. (2019); 3 – Bleem et al. (2015); 4 – White et al. (1997);5 – Jones and Forman (1984); 6 – David et al. (1993); 7 – Williamson et al. (2011); 8 – Aghanim et al.(Planck Collaboration) (2011); 9 – Markevitch et al. (2002); 10 – Ade et al. (Planck Collaboration) (2013);11 – Herbig et al. (1995); 12 – Forman and Jones (1982); 13 – Ade et al. (Planck Collaboration) (2016b);14 – Gavazzi et al. (2009); 15 – Vikhlinin et al. (2006); 16 – McDonald et al. (2015). ∗ The underlined estimates were obtained using the formulas and dependences from Navarro et al. (1997) andKravtsov et al. (2006). previous figures presented the relative distortion ampli-tude (in %). Here (just as above when considering thethermal cluster radiation) we took into account the ef-fect of observation of only the part of the cluster at low z (due to the excess of its size above the aperture size) andthe effect of observation of a noticeable fraction of theundistorted background at high z (because the clusterbegins to occupy only part of the aperture). In clus-ters with a smooth density distribution (described bythe β or NFW models — see the Appendix) the curvesin Fig. 9, modified by these effects will be smoother.As follows from the figure, the amplitude of the back-ground distortions due to scattering and absorptionchanges with z much more weakly than the flux of ther-mal radiation. In this case, the probability of detectingthe distortions in the background spectrum from the in-teraction with the gas of distant clusters may turn outto be even higher than that from the interaction withthe gas of nearby clusters. In any case, the detection ofsuch distortions remains a very challenging problem.INDIVIDUAL CLUSTERSTable 4 gives basic characteristics (the temperature andcentral density of the intergalactic gas, other parameters of the β model density distribution) for several knownrich clusters that exhibit strong microwave backgroundradiation distortions. In particular, this can be seen fromthe high values of the parameter Y SZ characterizing theamplitude of the background distortions (Kravtsov etal. 2006) given in column 11 of Table 4. Such clustersas Phoenex, SPT-CL J0615-5746, SPT-CL J2106-5844,and El Gordo were even discovered owing to this effect— by SPT (Williamson et al. 2011), ACT (Menanteauet al. 2012), or the Planck satellite (Aghanim et al.(Planck Collaboration), 2011). These are mostly verymassive hot clusters with kT e ∼ >
10 keV, but the coldnearby Virgo, A 1367, and A 1991 clusters were also in-cluded in the list. The detectability of distortions in theX-ray background from a cluster, and this list was com-piled precisely for its estimation, is determined by manyfactors, and a reliable detection of the effect in the mi-crowave background by no means implies that it can bedetected in X-rays.For a number of clusters we failed to find the measuredvalues of some density distribution parameters. Theseparameters were then estimated from their dependenceon M ( Y SZ ) determined by Vikhlinin et al. (2006)and Kravtsov et al. (2006). In Table 4 their values areunderlined for clarity. For all clusters we calculated the ASTRONOMY LETTERS Vol. 45 No. 12 2019
ECREASE IN THE BRIGHTNESS OF THE COSMIC X- AND GAMMA-RAY BACKGROUND 805
Fig. 10.
Expected distortions in the spectrum of the X-raybackground intensity due to Compton scattering and pho-toabsorption in the hot gas of several known galaxy clusters(thin green lines). The thick (blue) lines also take into ac-count the thermal gas radiation. optical depth for Thomson scattering along the line ofsight passing through their center (column 13 in the ta-ble). Note that for such nearby and extended clusters asVirgo, Coma, and Perseus it turns out to be only a factorof 2–3 smaller than the optical depth of distant super-massive clusters like El Gordo, Bullet, SPT-CL J2106-5844, and SPT-CL J0615-574. For convenience, column6 gives the angular sizes of the clusters correspondingto the values of R c specified in Table 4. For all clustersin the table we computed the relative distortions pro-duced by them (the gas kept by their gravity) in the X-ray background radiation spectrum by the Monte Carlomethod under the assumption of a β density distributionof the intergalactic gas. The computations were carriedout for the same exponent of the distribution β = 2 / β values for separate clusters did not exceed the errors of their determination. The abun-dance of the iron-group elements Z in the gas relativeto the cosmic abundance (column 12 in the table) wastaken from the literature. The cluster redshifts weretaken into account.The results of our computations of the relative back-ground radiation distortions (in percent to the initialspectrum) are presented in Fig. 10. They correspond tothe observations toward the cluster center by a telescopewith a narrow aperture corresponding to the cluster coreradius ( ∼ ′ for most clusters, see Table 4). As one re-cedes from the cluster center or when observing by atelescope with a wide aperture, the distortion amplitudeshould drop. The thin (green) and thick (blue) lines in-dicate the background distortions proper and the distor-tions including the thermal gas radiation, respectively.We see that the rich hot clusters in the lower part of thefigure lead to large deviations of the background overthe entire spectrum — positive at hν ∼ <
100 keV (theexcess radiation due to the Doppler effect) and negativeat energies above hν ∼ >
100 keV (the dip in the spectrumdue to the recoil effect). The feature at ∼ ∼ <
200 keV.For the cold clusters in the upper part of the figure,such as Virgo, A 1991, Coma, and Perseus, the ampli-tude of the positive background distortions at low ener-gies is negligible, while the amplitude of the MeV dip re-mains fairly large; it differs from the amplitude of the dipfor the SPT-CL J2106-5844, SPT-CL J0615-5746, and ElGordo clusters, which were believed to be most massivein the Universe, only by a factor of 2 or 3. The featuresdue to photoabsorption in the spectrum of the back-ground distortions by these clusters are expectedly largerthan those for the hot clusters due to the low gas tem-perature — the feature at 9 keV is even comparable inamplitude to the dip at high energies. The thermal radi-ation from the clusters, also expectedly, begins to hinderthe detection of the effect at appreciably lower ( hν ∼ < The observations of suchextended clusters even at a slightly smaller amplitude ofthe background distortions than that for rich, but distantclusters may turn out to be much more significant andfruitful.
ASTRONOMY LETTERS Vol. 45 No. 12 2019
06 GREBENEV, SUNYAEVX-RAY BACKGROUND FLUCTUATIONSThe detection of hard X-ray background deviations ata level of fractions of percent is not something abso-lutely unattainable per se. Such measurements aimed atsearching for background fluctuations have already beencarried out by both HEAO-1 (Boldt 1987; Treyer et al.1998) and RXTE (Gruber et al. 1999b; MacDonald etal. 2001) observatories.In particular, the HEXTE/RXTE instrument per-formed almost simultaneous observations of two sky re-gions spaced 3 ◦ apart. The differences in the fluxes mea-sured in these regions within the field of view of the in-strument with an area of 1.1 sq. deg allowed backgroundfluctuations much weaker than was possible in individ-ual observations to be searched for. Such fluctuationswere actually detected at a flux level of (0 . ± . , (0 . ± . , and (0 . ± . L X = 1 × erg s − and apower-law spectrum with an exponential cutoff at highenergies typical for AGNs at the center of our stan-dard model cluster with a uniform density, optical depth τ T = 1 . × − , core radius R c = 350 kpc, electron tem-perature kT e = 5 keV, and metallicity Z = 0 . Z ⊙ . Wewill take the photon index of the spectrum to be α = 1 . E c = 300 keV (Sazonov et al.2008; Ueda et al. 2014). The radiation spectrum of suchan AGN is indicated in Fig. 11a by the dotted red line forcomparison with the spectra of the hot gas in the galaxycluster and the cosmic background radiation. Note thatthe X-ray luminosity of the thermal radiation from thecluster gas is L c = 6 . × erg s − . The green lineat the bottom in Fig. 11b indicates the spectrum of therelative distortions arising in the AGN radiation whenit passes through the hot cluster gas. It is very simi-lar to the spectrum of the relative distortions arising inthe background radiation (indicated in the same figureby the black line). Note only the shape of the MeV dipslightly skewed to low energies, which, obviously, reflectsthe difference in shape between the initial AGN radia-tion spectrum and the background spectrum (primarilythe exponential cutoff at high energies in the AGN spec-trum).In reality, however, we are interested not in the distor-tions in the AGN radiation spectrum, but in its radiationthat was scattered in the cluster and became diffuse, be-cause the direct escape radiation will be perceived as theradiation of a compact source (AGN) when analyzing thedata and, naturally, should be subtracted.The blue line in Fig. 11b indicates the spectrum ofscattered AGN photons (relative to its initial spectrum),while the red dotted line indicates the same spectrum,but relative to the initial background radiation spec-trum. Remarkably, the scattered photon spectrum issmooth and does not contain any negative features re-lated to photoabsorption or the recoil effect after scat-tering. Its amplitude relative to the initial AGN radia-tion spectrum is more than 0.6%, i.e., it is much greaterthan the amplitude of the final AGN radiation distor-tions. We noted these properties of the scattered ra-diation (only that of the background) previously when ASTRONOMY LETTERS Vol. 45 No. 12 2019
ECREASE IN THE BRIGHTNESS OF THE COSMIC X- AND GAMMA-RAY BACKGROUND 807
Fig. 11.
Influence of the AGN radiation on the observa-tions of background distortions in the hot cluster gas: (a)the AGN radiation spectrum (red dotted line) in comparisonwith the background and cluster gas spectra; (b) the dis-tortions of the AGN (green) and background (black curves)spectra, the scattered AGN radiation relative to its initialspectrum (blue curve) and relative to the initial backgroundspectrum (red dotted line); (c) the background distortionsincluding the scattered AGN radiation. The optical depth ofthe gas toward the center is τ T = 1 . × − , the cluster radiusis R c = 350 kpc, kT e = 5 keV, and Z = 0 . Z ⊙ . The AGNX-ray luminosity is L X = 1 × erg s − , the green line onthe lower panel indicates the case of L X = 1 × erg s − . discussing Fig. 2. The amplitude of the scattered AGNradiation relative to the background spectrum turns outto be small everywhere, except the standard X-ray band hν ∼ <
10 keV. The enhancement of the distortions in thesoft X-ray band is related to the differences in the AGNand background spectra: the AGN intensity here beginsto approach and even exceed the background intensity.However, many AGNs exhibit a low-energy cutoff in theradiation spectrum related to the absorption of their ra-diation in the gas-dust torus surrounding the supermas-sive black hole in the galactic nucleus. Clearly, whenthe absorption is taken into account, the rise of the rel-ative background distortions due to the soft X-ray AGNradiation should be less distinct or even disappear alto-gether. The important thing is that the spectrum of the scat-tered AGN radiation contains no absorption featuresand, therefore, it cannot reduce or smear such features inthe spectrum of the background distortions arising whenit passes through the hot cluster gas. In the case of verybright AGN flares occurring on a time scale of hundredsof thousands of years, narrow lines associated with res-onance scattering of the AGN X-ray emission by Fe, S,and Si ions can appear in the background spectrum atthese energies (Sazonov et al. 2002).Indeed, as Fig. 11c shows, adding the AGN radia-tion scattered in the cluster gas to the spectrum of theemerged background distortions does not smear the fea-tures related to background photoabsorption on the K and L shells of strongly ionized iron and nickel ions inthe cluster gas. What is especially remarkable, the scat-tered AGN radiation does not reduce and does not smearthe MeV dip in the background spectrum related to therecoil effect when its photons are scattered by electronsin the cluster gas.At the same time, the AGN radiation can lead to a risein background intensity at low energies hν ∼ < t ∼ R c /c ≃ . c is the speed oflight) years after the decay of the X-ray activity of theAGN itself. Similar radiation of the former activity ofthe black hole at the center of our Galaxy was observedfrom molecular clouds (Markevitch et al. 1993; Sunyaevet al. 1993). On the other hand, at such a long time ofthe reaction to AGN activity ( ≃ z ∼ < ∼ / Ly α hydrogen lines at z ∼ ∼ B = 0 .
044 (Rauch 1998). It isbelieved that ∼ / ASTRONOMY LETTERS Vol. 45 No. 12 2019
08 GREBENEV, SUNYAEV
Fig. 12.
Expected distortions in the X-ray backgroundspectrum due to scattering and absorption in the envelopeof the warm-hot intergalactic medium (WHIM) surroundingthe galaxy cluster toward its center: (a) without the back-ground distortions in the hot cluster gas and (b) with thedistortions (the green lines indicate the background spec-tra before the distortion in WHIM). The hot gas (HIM) inthe cluster has a Thomson optical depth toward the center τ T = 1 . × − , core radius R c = 350 kpc, electron tem-perature kT e = 2 , , Z = 0 . Z ⊙ ;the WHIM envelope has a factor of 4 larger mass, radius R w = 2 or 3 Mpc, electron temperature kT w = 0 . Z w = 0 . , . , or 0 . Z ⊙ . servation are contained in the moderately hot plasmawith a temperature ∼ K located in filaments andother similar structures on the far periphery of galaxyclusters (Cen and Ostriker 1999). This gas phase iscalled the Warm-Hot Intergalactic Medium (WHIM).The existence of WHIM was confirmed by the detec-tion of a soft X-ray excess from a plasma with kT ∼ . Ly α line at 0.65 keV in the spectraof several galaxy clusters (see, e.g., Finoguenov et al.2003; Kaastra 2004a, 2004b).Can the interaction of the X-ray background withWHIM give rise to additional distortions in its spec-trum? Suppose, for simplicity, that each galaxy clusteris surrounded by a thick spherical WHIM layer (shell)with a quasi-uniform density distribution ρ w = N w m p .The outer radius of the layer is R w ≃ − M g within the radius R c ) contains ∼ M w ≃ (2 / B ) / (0 . × / B ) M g ≃ M g . The mean electron density in this layer is then N w = 4 N e ( R c /R w ) ≃ . × − × (cid:18) τ T . × − (cid:19) (cid:18) R c
350 kpc (cid:19) (cid:18) R w (cid:19) − cm − . This estimate agrees well with the WHIM density mea-surements from the O VII and O VIII lines in the X-rayspectra of galaxy clusters (Kaastra 2004a). The electrondensity in the hot cluster gas N e expressed here via itsThomson optical depth along the line of sight passingthrough the cluster center, τ T = 2 σ T N e R c = 1 . × − . The Thomson optical depth of the WHIM envelope alongthe same line of sight is τ w ≃ σ T N w ( R w − R c ) ≃ τ T ( R c /R w ) ≃ . × − × (cid:18) τ T . × − (cid:19) (cid:18) R c
350 kpc (cid:19) (cid:18) R w (cid:19) − . Obviously, the distortions related to purely Comptonscattering in WHIM will be the same as those after scat-tering in the hot cluster gas (Hot Integrgalactic Mediumor HIM), but smaller in absolute value, because the op-tical depth of the envelope is small. In this case, the ironand nickel atoms in WHIM will be ionized much moreweakly than those in the hot cluster gas (at kT w ≃ . ρ ≫ R c . We considered theenvelopes with various outer radii R c (and, accordingly,various densities N w and optical depths τ w ) and metal-licities Z w . The WHIM temperature was taken to be kT e = 0 . K shellof iron and nickel is seen to be shifted leftward alongthe energy axis compared to the spectrum of the distor-tions in HIM (from ≃ . ≃ . L shell is shifted even more strongly —from ≃ .
03 to ≃ .
27 keV. Whereas the hard line has anegligible depth, the depth of the absorption line at 1.27keV is unexpectedly large even for the lowest metallicityconsidered Z w = 0 . Z ⊙ . ASTRONOMY LETTERS Vol. 45 No. 12 2019
ECREASE IN THE BRIGHTNESS OF THE COSMIC X- AND GAMMA-RAY BACKGROUND 809
Fig. 13.
Expected distortions in the thermal radiation spec-trum of the hot cluster gas (HIM) in a galaxy cluster dueto scattering and absorption in the envelope of the warm-hot intergalactic medium (WHIM) surrounding the galaxycluster toward its center (blue lines). The hot gas (HIM) inthe cluster has a Thomson optical depth toward the center τ T = 1 . × − , radius R c = 350 kpc, electron tempera-ture kT e = 2 , , Z = 0 . Z ⊙ ; theWHIM envelope has a factor of 4 larger mass, radius R w = 3Mpc, optical depth τ w ≃ . × − , electron temperature kT w = 0 . Z w = 0 . Z ⊙ . The greenlines indicate the cluster HIM spectra before the interactionwith WHIM. Absorption lines are also formed at energies below ∼ ∼ . , . , . − . , and 1 .
87 keV in the WHIM ra-diation spectrum. Note that iron and nickel ions withvacancies on the lower electronic shells are formed asa result of background photoabsorption in the WHIMplasma, which could not be formed in it due to colli-sional processes. Although these ions recombine mainlythrough the Auger effect, the formation of photons of flu-orescent lines (similar to the K α line with energy 6.4 keVemitted with a 34% probability when neutral iron atomsare ionized) is also possible. There are no such emissionlines in the intrinsic thermal recombination spectrum ofWHIM. These lines are narrow and much harder to de-tect than the photoabsorption lines; therefore, they aredisregarded in this paper.The total spectra including the distortions in both WHIM and HIM are presented at the bottom (inFig. 12b). The radius of the WHIM envelope was takento be R c = 3 Mpc, the temperature is again kT w = 0 . Z w = 0 . Z ⊙ . The opticaldepth of the hot cluster gas is τ T = 1 . × − , thetemperature is kT e = 2 , , or 5 keV, and its metal-licity is Z = 0 . Z ⊙ . The spectra of the backgrounddistortions formed in HIM were previously presented inFig. 4. Substantial changes are seen to occur in the back-ground spectrum when passing through the WHIM en-velope: (1) a new intense broad absorption line appearsat ∼ . ∼ ∼ . ∼ . ∼ . − . ∼ ∼ ∼ . z ∼ <
1) clusters. The formationof the absorption line at ∼ . ASTRONOMY LETTERS Vol. 45 No. 12 2019
10 GREBENEV, SUNYAEVit passes through the hot intergalactic gas in galaxy clus-ters. We investigated the dependence of the distortionamplitude and shape on the parameters of the cluster gas— its temperature, optical depth, and density distribu-tion. The analogous distortions arising in the microwavebackground radiation are well known, are actively stud-ied, and are widely used in observational cosmology. Weshowed the following:1. Compton scattering by electrons of the intergalacticgas in clusters leads to peculiar distortions of thebackground radiation — a rise in its brightness at hν ∼ < y C = τ T kT e /m e c , averaged over the visible (withinthe telescope aperture) part of the cluster (or theparameter Y SZ in the case of a distant cluster), forthe hottest clusters with kT e ∼
15 keV it reaches ∼ .
1% at energies ∼
100 keV;3. the decrease in background brightness due to therecoil effect has a maximum at energies ∼ − τ T averaged over the visible (in the aperture)part of the cluster, it has an amplitude of ∼ . .
3% for the optically thickest clusters and does notdepend on the temperature;4. photoabsorption by strongly ionized ions of the iron-group elements also leads to a decrease in the back-ground with the formation of two absorption lines ofcharacteristic shape with threshold energies hν ∼ kT ∼ K) plasma located on the distant ( ∼ < z ∼ <
1) galaxy clustersnoticeably enhances the absorption line at hν ∼ hν ∼ . ∼ z (in contrast to thespectral shape of the microwave background distor-tions), although the distortion amplitude does notdepend on z ;8. the detection of background distortions at energies hν ∼ < kT e is complicated by the presence of in-trinsic thermal radiation from the intergalactic gas,which makes it very difficult to measure the effect ofa rise in the background in the X-ray range; the ob-servations of the MeV dip in the background spec-trum are free from this noise for most clusters;9. the detection of X-ray background distortions isalso complicated by the presence of AGNs in (ornear) the cluster with luminosities L X ∼ − erg s − at present or even in the distant (up to ∼ hν ∼ <
150 keV; the scattered radia-tion does not contain any photoabsorption lines andeven the MeV dip in its spectrum and, therefore,does not change their amplitude in the spectrum ofthe background distortions.We considered a number of real galaxy clusters andpredicted the shape and amplitude of the spectra of thebackground deviations expected for them based on theavailable parameters. Although the background inten-sity distortions toward the cluster center are maximalfor the most massive hot clusters, cold nearby clusterslike Virgo and Coma have real chances for detecting theeffect in the hard X-ray and gamma-ray range alreadynow. It seems possible that the hard X-ray backgroundfluctuations detected by the HEAO-1 and RXTE obser-vatories are associated in part with the effect being dis-cussed in the paper — with the background distortionsin distant galaxy clusters.CONCLUSIONSIt is clear from the foregoing that the main obstacle toobserving the distortions in the cosmic background spec-trum due to scattering by electrons in the X-ray range hν ∼ < −
100 keV (in the cluster rest frame) is the in-trinsic plasma thermal radiation . If it were not for theillumination by this radiation, modern X-ray telescopeswith grazing-incidence optics like NuSTAR (Harrison et Besides, although the background distortions due to scatteringand photoabsorption grow with increasing τ T of the cluster gas,the thermal radiation intensity, proportional to N , grows muchfaster. ASTRONOMY LETTERS Vol. 45 No. 12 2019
ECREASE IN THE BRIGHTNESS OF THE COSMIC X- AND GAMMA-RAY BACKGROUND 811al. 2013) might well observe these distortions (or at leastthey have fallen just short of this). The X-ray calorime-ters with the resolution and sensitivity required to mea-sure the absorption line profile in the background beingdesigned at present would be able to determine the gastemperature in a cluster and its composition only fromthe absorption line profile in the background.In December 2019 the SRG orbital astrophysicalobservatory with the highly sensitive ART-XC andeROSITA telescopes onboard is going to begin to scanthe sky in X-rays for four years. The ART-XC telescope,operating in the range 5–30 keV (Pavlinsky et al. 2018),will be able to observe the distortions in the backgroundspectrum due to scattering by electrons in the hot gasof clusters and photoabsorption by iron and nickel ions;the eROSITA telescope, operating in the range 0.3–8keV (Predehl et al. 2018), will be able to investigate theabsorption lines in the background spectrum at lowerenergies. Even before the survey, within the Russianpart of the Performance Verification Program, the SRGobservatory is going to scan the nearby bright Comacluster.We could attempt to subtract the thermal radiationspectrum of the cluster gas itself from the measurementsusing theoretical models and knowing the gas tempera-ture, density, and metallicity. However, (1) in real clus-ters the gas temperature usually changes with radiusand in a rather complicated way, which greatly com-plicates highly accurate simulations; (2) because of thefinite optical depth of the cluster gas, the distortions dueto scattering and absorption should appear in the spec-trum of its thermal radiation, which are much greater inabsolute value than the background distortions that areadded to them when subtracting the ideal spectrum ofan optically thin plasma; (3) the intense thermal radia-tion of the gas should introduce a noticeable statisticalerror into the background measurements, reducing thesignal-to-noise ratio
S/N . Therefore, to detect the ef-fect, it is more preferable to investigate cold, massive,nearby (extended) clusters using the peripheral clusterregions, along with the central ones, for observations.The presence of AGNs in a galaxy cluster can ad-ditionally complicate the X-ray observations of the ef-fect, because their radiation scattered in the cluster gaswill be perceived as a “positive” background distortion.Scattered radiation can exist in a cluster even hundredsof thousands of years after the decay of the galactic nu-cleus — due to the “X-ray echo” effect. At the sametime, it contains no absorption lines and MeV dip in thespectrum and does not distort these features appearingin the X-ray background spectrum when interacting withthe hot cluster gas.It may well be that it would be more preferable tosearch for the Compton cosmic background distortionsin the hard X-ray and gamma-ray spectral ranges, wheretheir relative amplitude is maximal and the contribu-tion of the thermal radiation from the hot cluster gas falls exponentially. However, it should be borne in mindthat the absolute amplitude of the distortions in theseranges decreases due to the peculiar shape of the X-raybackground spectrum (see Fig. 9 above). Besides, the in-struments operating in these ranges are still noticeablyinferior in their capabilities to modern X-ray telescopes,despite the abundance of problems and the general un-derstanding of the importance of studies in this field.The incessant efforts (see, e.g., Fryer et al. 2019) todesign more sensitive MeV gamma-ray telescopes like e-ASTROGAM (Tatischeff et al. 2016; De Angelis et al.2017) or ASTENA/LAUE (Virgilli et al. 2017) allowone to look with optimism at the prospects for detectingthe effect in this range in the immediate future.Concluding this paper, we emphasize once again thatall of the expected background distortions in the hotcluster gas have a very small ( ∼ < ∼ . APPENDIX
THE NAVARRO–FRENK–WHITE MODELThe dark matter density distribution in galaxy clustersis successfully described by the Navarro-Frenk-White(hereafter NFW) profile (Navarro et al. 1997) found byN-body simulations of equilibrium configurations in thetheory of hierarchical clustering of cold dark matter: ρ D ( R ) = ρ s R s R (cid:18) RR s (cid:19) − . Here, ρ s is the dark matter density parameter and R s is the scale parameter (cluster core radius). For such ASTRONOMY LETTERS Vol. 45 No. 12 2019
12 GREBENEV, SUNYAEV
Fig. 14.
Equilibrium density distributions of an isothermal gas in the gravitational field of a cluster with dark matter havingthe NFW profile (Navarro et al. 1997) and their dependence on cluster mass M . The thin solid (blue) lines indicate thedistributions with kT e = 5 keV. The central gas density is everywhere assumed to be the same (see the text). The dashed(black) line for M = 5 × M ⊙ indicates the distribution with kT e = 6 keV. The concentration parameter is c = 2 . β gas distribution with β = 2 / R c = 0 . R s (the normalization ensures that the β distribution near R ∼ R s coincides with the gas distribution in the NFW model for M = 5 × M ⊙ ; the dotted lineindicates the 10% level of the dark matter density distribution ρ D /m p for this case). a density profile the cluster dark matter mass withinradius R is M D ( < R ) = 4 πρ s R (cid:20) ln (cid:18) RR s + 1 (cid:19) − RR + R s (cid:21) . (A1)Given the dark matter distribution, the cluster gas dis-tribution law can be found (refined). In particular, inseveral papers it was proposed to modify the β densitydistribution by including a central cusp and an addi-tional wider component (see, e.g., Vikhlinin et al. 2006;Arnaud et al. 2010). Because of the larger number ofparameters, the modified distribution acquired greaterfreedom in changing the shape and successfully fittedthe observed brightness profiles of clusters. A differentapproach was realized by Shi and Komatsu (2014) andShi et al. (2016), who used the hydrostatic equilibriumequation for the gas in the gravitational field of darkmatter to find the cluster gas distribution. However,they obtained the gas density (pressure) profiles in anexplicit form only for specific temperature distributions.As will be shown below, this approach itself can alsobe extended to the clusters with a constant temperatureinvestigated in this paper. Neglecting the gas self-gravity, the hydrostatic equi-librium equation has a simple form: M D ( < R ) = − kT e Gm p d ln( N e ) d ln( R ) R, (A2)where G is the gravitational constant and m p is the pro-ton mass. Substituting (A1) into this equation and in-tegrating the resulting equation, we find the cluster gasdensity distribution N e ( R ) = N s e − ǫ (cid:18) RR s (cid:19) ǫ R s /R . (A3)Here, N s is the central gas electron density; generallyspeaking, it is not directly related to the central darkmatter density ρ s . The shape of the distribution is de-fined by only one dimensionless parameter ǫ = 2 πρ s R ( Gm p /kT e ) . (A4)Using the characteristic values of the cluster radius R ,mass M = 500 ρ cr ( z )(4 / πR , and the concentra-tion parameter c proposed by Navarro et al. (1997) to ASTRONOMY LETTERS Vol. 45 No. 12 2019
ECREASE IN THE BRIGHTNESS OF THE COSMIC X- AND GAMMA-RAY BACKGROUND 813describe the cluster properties: R = c R s M = 4 πρ s R E ( c ) − , where E ( c ) = [ln(1 + c ) − c/ (1 + c )] − , and ρ cr ( z ) = 3 / (8 πG ) H ( z ) is the mean (critical) den-sity of the Universe at cluster redshift z , we will trans-form (A4) to ǫ = GM m p kT e R s E ( c ) ≃ (A5)9 . cE ( c ) (cid:18) M M ⊙ (cid:19) (cid:18) R
500 kpc (cid:19) − (cid:18) kT e (cid:19) − . Figure 14 shows how the equilibrium gas density dis-tributions in a cluster with the NFW dark matter profilecalculated from Eqs. (A3) and (A4) depend on the clus-ter mass M and gas temperature kT e . The concen-tration parameter c was taken to be 2 .
5. For the clusterwith M = 5 × M ⊙ the central gas density N s wasdetermined by assuming the mass of the gas containedwithin the radius R to be M g ( R < R ) ≃ . M (Kravtsov et al. 2006). The central gas density in theremaining clusters was fixed at this value for the conve-nience of tracing the change in profile shape (obviously,the condition M g ≃ . M no longer must be fulfilledin this case). The gas distribution depends on kT e asstrongly as it does on M (compare the distributionfor the cluster with M = 5 × M ⊙ and a gas tem-perature kT e = 5 keV indicated by the solid blue line andthe distribution for the same cluster, but with kT e = 6keV, indicated by the dashed black line). This would beexpected, because these quantities enter into (A5) in thecombination M /kT e .The thick (green) line in the figure indicates the β gas distribution with β = 2 / r c = 0 . R s . Thecharacteristic values of R c and R are marked in thefigure by the vertical dotted (red) lines. The normaliza-tion of the gas density was chosen in such a way thatthis distribution roughly coincided with the gas distri-bution in the NFW model near R ∼ R s . The β distri-bution is seen to have a much gentler (flatter) profilein the central ( R ∼ < R s ) part of the cluster than theNFW profile. For comparison, the dotted (black) curvein the figure indicates the NFW dark matter distribu-tion proper, its normalization was set equal to 10% ofthe density ρ D ( R ) /m p for M = 5 × M ⊙ . The ra-dial Thomson optical depth for such a cluster computedby integrating the distribution (A3) is τ r ≃ . × − , which accounts for ∼ / τ c = σ T N s R c , and is evensmaller for more massive clusters. For the β distributionwith a flat top the radial optical depth is 0 . π τ c (see Eq. This is how the cluster gas would be distributed if the shapeof its radial density profile closely coincided with the shape of thedark matter profile.
Fig. 15.
The relative background distortions (thick bluelines) arising in a cluster where the dark matter is distributedaccording to the NFW law, while the gas — according toEq. (A3). The cluster mass is M = 1 × M ⊙ (the coreradius is R c ≃
300 kpc), the radial optical depth of the gasis τ r ≃ . × − . The cases with various break radii R b areconsidered. The background distortions in a cluster with auniform density distribution (thin green lines) with a radialoptical depth τ c = 6 × − are shown for comparison. Thegas temperature and metallicity in both cases are the same, kT e = 5 keV and Z = 0 . Z ⊙ . (6)). This difference is related to the more rapid drop ingas density with radius in the central part of the NFWmodel cluster.The results of our computations of the cosmic back-ground distortions for a cluster with the NFW dark mat-ter profile are indicated in Fig. 15 by the thick (blue)lines. Here we consider a more massive cluster with M = 1 × M ⊙ , a radial optical depth τ r =1 . × − ( τ c = 3 . × − ), R s = 600 kpc ( R c = 300kpc), a gas temperature kT e = 5 keV, and metallicity Z = 0 . Z ⊙ . For comparison, the thin (green) linesindicate the results of our computations of the back-ground distortions for a cluster with a uniform gas den-sity with a radial optical depth τ c = σ T N e R c = 6 × − and the same temperature kT e = 5 keV. We consid- ASTRONOMY LETTERS Vol. 45 No. 12 2019
14 GREBENEV, SUNYAEVered the cases with various density break radii R b =0 . , , , , and 10 R s . Although the real Thomson op-tical depth of the gas in the NFW cluster exceeds thatin the homogeneous model by a factor of ∼
2, the dis-tortions arising in the NFW model due to the stronggas concentration to the center are comparable to thedistortions in a cluster with a uniform density (for thecomputations with R b ≤ R s ). For larger R b the distor-tion amplitude drops in the same way as in the case of acluster with a β density distribution (Fig. 5). Obviously,the distortion depth is determined by the optical depthaveraged over the visible part of the cluster < τ T > .Just as in the case of a β distribution, it is clear thatthe efficiency of background distortion observations bya telescope with an aperture (angular resolution) with aradius more than ∼ − R c (at given z ) should droprapidly. REFERENCES
1. P. A. R. Ade, N. Aghanim, M. Arnaud, M. Ashdown, F.Atrio-Barandela, J. Aumont, C. Baccigalupi, A. Balbi,et al. (Planck Collaboration), Astron. Astrophys. ,A140 (2013).2. P. A. R. Ade, N. Aghanim, C. Armitage-Caplan, M. Ar-naud, M. Ashdown, F. Atrio-Barandela, J. Aumont, H.Aussel, et al. (Planck Collaboration), Astron. Astro-phys. , A29 (2014).3. P. A. R. Ade, N. Aghanim, C. Armitage-Caplan, M. Ar-naud, M. Ashdown, F. Atrio-Barandela, J. Aumont, H.Aussel, et al. (Planck Collaboration), Astron. Astro-phys. , A14 (2015).4. P. A. R. Ade, N. Aghanim, M. Arnaud, M. Ashdown, J.Aumont, C. Baccigalupi, A. J. Banday, R.B. Barreiro,et al. (Planck Collaboration), Astron. Astrophys. ,A27 (2016a).5. P. A. R. Ade, N. Aghanim, M. Arnaud, M. Ashdown, J.Aumont, C. Baccigalupi, A. J. Banday, R.B. Barreiro,et al. (Planck Collaboration), Astron. Astrophys. ,A101 (2016b).6. N. Aghanim, M. Arnaud, M. Ashdown, F. Atrio-Barandela, J. Aumont, C. Baccigalupi, A. Balbi, A. J.Banday, et al. (Planck Collaboration), Astron. Astro-phys. , A26 (2011).7. C.W. Allen,
Astrophysical Quantities , 3d ed. (Athlone,London, 1973).8. A. de Angelis, V. Tatischeff, M. Tavani, U. Oberlack, I.Grenier, L. Hanlon, R. Walter, A. Argan, P. von Ball-moos, et al., Exp. Astron. , 25 (2017).9. M. Arnaud, Astron. Astrophys. , 103 (2009).10. M. Arnaud, G. W. Pratt, R. Piffaretti, H. B¨ohringer, J.H. Croston, and E. Pointecouteau, Astron. Astrophys. , A92 (2010).11. J. Arons, Astrophys. J. , 437 (1971).12. M. Birkinshaw, Phys. Rep. , 97 (1999).13. L. E. Bleem, B. Stalder, T. de Haan, K. A. Aird, S. W.Allen, D. E. Applegate, M. L. N. Ashby, M. Bautz, etal., Astrophys. J. Suppl. Ser. , 27 (2015). 14. E. Boldt, Phys. Rep. , 215 (1987).15. S. Boughn and R. Crittenden, Nature , 45 (2004).16. S. P. Boughn and R. G. Crittenden, Mon. Not. Roy.Astron. Soc. , 1013 (2005).17. E. Bulbul, I.-N. Chiu, J. J. Mohr, M. McDonald, B. Ben-son, M. W. Bautz, M. Bayliss, and L. Bleem, Astrophys.J. , 50 (2019).18. J. E. Carlstrom, G. P. Reese, and D. Erik, Ann. Rev.Astron. Astrophys. , 643 (2002).19. A. Cavaliere and R. Fusco-Femiano, Astron. Astrophys. , 137 (1976).20. R. Cen and J. P. Ostriker, Astrophys. J. , 1 (1999).21. E. Churazov, M. Haehnelt, O. Kotov, and R. Sunyaev,Mon. Not. Roy. Astron. Soc. , 93 (2001).22. E. Churazov, R. Sunyaev, M. Revnivtsev, S. Sazonov,S. Molkov, S. Grebenev, C. Winkler, A. Parmar, et al.,Astron. Astrophys. , 529 (2007).23. G. Cooper, Phys. Rev. D , 2312 (1971).24. L. P. David, A. Slyz, C. Jones, W. Forman, S. D. Vrtilek,and K. A. Arnaud, Astrophys. J. , 479 (1993).25. A. C. Fabian, Nature , 672 (1977).26. A. Finoguenov, U.G. Briel, and J.P. Henry, Astron. As-trophys. , 777 (2003).27. W. Forman and C. Jones, Ann. Rev. Astron. Astro-phys. , 547 (1982).28. C. L. Fryer, F. Timmes, A.L. Hungerford, A. Couture, F.Adams, M. Avila, W. A. Aoki, A. Arcones, D. Arnett,et al., “A White Paper for the 2020 Decadal Survey” ,astro-ph:1902.02915 (2019).29. M. Fukugita, C. Hogan, and P. J. E. Peebles, Astrophys.J. , 518 (1998).30. R. Gavazzi , C. Adami, F. Durret, J.-C. Cuillandre, O.Ilbert, A. Mazure, R. Pell´o, and M. P. Ulmer, Astron.Astrophys. , L33 (2009).31. R. Giacconi, P. Rosati, P. Tozzi, M. Nonino, G.Hasinger, C. Norman, J. Bergeron, S. Borgani, et al.,Astrophys. J. , 624 (2001).32. S. A. Grebenev and R. A. Sunyaev, Sov. Astron. Lett. , 438 (1987).33. S. A. Grebenev and R. A. Sunyaev, Mon. Not. Roy.Astron. Soc., in preparation (2020).34. D. E. Gruber, J. L. Matteson, L. E. Peterson, and G. V.Jung, Astrophys. J. , 124 (1999a).35. D. E. Gruber, D. MacDonald, R. E. Rothschild, E.Boldt, R. F. Mushotzky, and A. C. Fabian, Nucl. Phys.B – Proc. Sup. , 625 (1999b).36. F. A. Harrison, W. W. Craig, F. E. Christensen, C. J.Hailey, W. W. Zhang, S. E. Boggs, D. Stern, W. R.Cook, et al., Astrophys. J. , 103 (2013).37. G. Hasinger, R. Burg, R. Giacconi, M. Schmidt, J.Trumper, and G. Zamorani, Astron. Astrophys. ,482 (1998).38. M. Hasselfield, M. Hilton, T. A. Marriage, G. E. Ad-dison, L. F. Barrientos, N. Battaglia, E. S. Battistelli,J.R. Bond, et al., J. Cosmol. Astropart. Phys. , 008(2013).ASTRONOMY LETTERS Vol. 45 No. 12 2019 ECREASE IN THE BRIGHTNESS OF THE COSMIC X- AND GAMMA-RAY BACKGROUND 815
39. T. Herbig, C. R. Lawrence, A. C. S. Readhead, and S.Gulkis, Astrophys. J. , L5 (1995).40. A. F. Illarionov and R. A. Syunyaev, Sov. Astron. ,45 (1972).41. C. Jones and W. Forman, Astrophys. J. , 38 (1984).42. J. S. Kaastra, Proc. IAU Colloquium “Outskirts ofGalaxy Clusters: Intense Life in the Suburbs” (ed. A.Diaferio) , 105 (2004a).43. J. S. Kaastra, J. Korean Astron. Soc. , 375 (2004b).44. R. Khatri and R. A. Sunyaev, in preparation (2019).45. A. S. Kompaneets, Sov. Phys. JETP , 730 (1957).46. A. V. Kravtsov, A. Vikhlinin, and D. Nagai, Astrophys.J. , 128 (2006).47. M. S. Longair and R. A. Sunyaev, Nature , 719(1969).48. D. R. MacDonald, D. E. Gruber, and E. A. Boldt, AIPConf. Proc , 734 (2001).49. P. Madau and G. Efstathiou, Astrophys. J. , L9(1999).50. M. Markevitch, G. R. Blumenthal, W. Forman, C. Jones,and R. A. Sunyaev, Astrophys. J. , 326 (1992).51. M. Markevitch, R. A. Sunyaev, and M. N. Pavlinsky,Nature , 40 (1993).52. M. Markevitch, A. H. Gonzalez, L. David, A. Vikhlinin,S. Murray, W. Forman, C. Jones, and W. Tucker, As-trophys. J. , L27 (2002).53. M. McDonald, B. R. McNamara, R. J. van Weeren, D.E. Applegate, M. Bayliss, M. W. Bautz, B. A. Benson,J. E. Carlstrom, et al., Astrophys. J. , 111 (2015).54. F. Menanteau, J. P. Hughes, C. Sif´on, M. Hilton, J.Gonz´alez, L. Infante, L. F. Barrientos, A. J. Baker, etal., Astrophys. J. , 7 (2012).55. T. Miyaji, G. Hasinger, and M. Schmidt, Astron. As-trophys. , 25 (2000).56. T. Miyaji, G. Hasinger, M. Salvato, M. Brusa, N. Cap-pelluti, F. Civano, S. Puccetti, M. Elvis, et al., Astro-phys. J. , 104 (2015).57. J. F. Navarro, C. S. Frenk, and S. D. M. White, Astro-phys. J. , 493 (1997).58. M. Pavlinsky, V. Levin, V. Akimov, A. Krivchenko, A.Rotin, M. Kuznetsova, I. Lapshov, A. Tkachenko, et al.,Proc. SPIE , 106991Y (2018).59. L. A. Pozdnyakov, I. M. Sobol’, and R. A. Syunyaev,Sov. Sci. Rev., Sec. E: Astrophys. Space Phys. Rev. ,189 (1983).60. P. Predehl, W. Bornemann, H. Br¨auninger, H. Brunner,V. Burwitz, D. Coutinho, K. Dennerl, J. Eder, et al.,Proc. SPIE , 106995H (2018).61. M. Rauch Ann. Rev. Astron. Astrophys. , 267(1998).62. J. C. Raymond and B. W. Smith, Astrophys. J. Suppl.Ser. , 419 (1977).63. S. Yu. Sazonov and R. A. Sunyaev, Astrophys. J. ,28 (2000).64. S. Yu. Sazonov, R. A. Sunyaev, and C. K. Cramphorn,Astron. Astrophys. , 793 (2002). 65. S. Sazonov, R. Krivonos, M. Revnivtsev, E. Churazov,and R. Sunyaev, Astron. Astrophys. , 517 (2008).66. S. L. Shapiro, A. P. Lightman, and D. M. Eardley, As-trophys. J. , 187 (1976).67. X. Shi and E. Komatsu, Mon. Not. Roy. Astron. Soc. , 521 (2014).68. X. Shi, E. Komatsu, D. Nagai, and E. T. Lau, Mon.Not. Roy. Astron. Soc. , 2936 (2016).69. R. A. Sunyaev, Sov. Astron. Lett. , 213 (1980).70. R. A. Sunyaev, M. Markevitch, and M. Pavlinsky, As-trophys. J. , 606 (1993).71. R. A. Sunyaev and L. G. Titarchuk, Astron. Astrophys. , 121 (1980).72. R. A. Sunyaev and Ya. B. Zeldovich, Ap&SS , 3 (1970).73. R. A. Sunyaev and Ya. B. Zeldovich, Comm. Astrophys.Space Phys. , 173 (1972).74. R. A. Sunyaev and Ya. B. Zeldovich, Ann. Rev. Astron.Astrophys. , 537 (1980).75. R. A. Sunyaev and Ya. B. Zeldovich, Sov. Sci. Rev.,Sec. E: Astrophys. Space Phys. Rev. , 1 (1981).76. V. Tatischeff, M. Tavani, P. von Ballmoos, L. Hanlon,U. Oberlack, A. Aboudan, A. Argan, D. Bernard, et al.,Proc. SPIE , 99052N (2016).77. M. Treyer, C. Scharf, O. Lahav, K. Jahoda, E. Boldt,and T. Piran, Astrophys. J. , 531 (1998).78. Y. Ueda, M. Akiyama, G. Hasinger, T. Miyaji, and M.G. Watson, Astrophys. J. , 104 (2014).79. L. A. Vainshtein and R. A. Sunyaev, Sov. Astron. Lett. , 353 (1980).80. D. A. Verner and D. G. Yakovlev, Astron. Astrophys.Suppl. Ser. , 125 (1995).81. D. A. Verner, G. J. Ferland, K. T. Korista, and D. G.Yakovlev, Astrophys. J. , 487 (1996).82. A. Vikhlinin, A. Kravtsov, W. Forman, C. Jones, M.Markevitch, S. S. Murray, and L. van Speybroeck, As-trophys. J. , 691 (2006).83. E. Virgilli, V. Valsan, F. Frontera, E. Caroli, V. Lic-cardo, and J. B. Stephen, JATIS , 044001 (2017);arXiv:1711.03475.84. D. A. White, C. Jones, and W. Forman, Mon. Not. Roy.Astron. Soc. , 419 (1997).85. R. Williamson, B. A. Benson, F. W. High, K. Vander-linde, P. A. R. Ade, K. A. Aird, K. Andersson, R. Arm-strong, et al., Astrophys. J. , 139 (2011).86. Ya. B. Zel’dovich and I. D. Novikov, The Structure andEvolution of the Universe (M.: Nauka, 1975).87. Ya. B. Zel’dovich and R. A. Sunyaev, in
Astrophysicsand Space Physics (ed. by R. A. Sunyaev, Nauka-Fiz-matlit, Moscow, 1982), p. 9 [in Russian].