Searching for the missing in clusters

Bilhuda Rasheed, Neta Bahcall1, and Paul Bode

Department of Astrophysical Sciences, 4 Ivy Lane, Peyton Hall, Princeton University, Princeton, NJ 08544

Edited by Marc Davis, University of California, Berkeley, CA, and approved January 10, 2011 (received for review July 8, 2010)

Observations of clusters of galaxies suggest that they contain few- fraction in the richest clusters, it is still systematically below er baryons (gas plus stars) than the cosmic fraction. This the cosmic value. This baryon discrepancy, especially the gas frac- “missing baryon” puzzle is especially surprising for the most mas- tion, is observed to increase with decreasing cluster mass (14, 15). sive clusters, which are expected to be representative of the cosmic This raises the questions: Where are the missing baryons? Why matter content of the universe (baryons and ). Here we are they “missing”? show that the baryons may not actually be missing from clusters, Attempted explanations for the missing baryons in clusters but rather are extended to larger radii than typically observed. The range from preheating or other energy inputs that expel gas from baryon deficiency is typically observed in the central regions of the system (16–22, and references therein), to the suggestion of clusters (∼0.5 the virial radius). However, the observed gas-density additional baryonic components not yet detected [e.g., cool gas, profile is significantly shallower than the mass-density profile, faint stars (10, 23)]. Simulations, which do suggest a depletion of implying that the gas is more extended than the mass and that cluster gas in the inner regions of clusters, do not yet contain all the gas fraction increases with radius. We use the observed density the required physics [stellar and Active Galactic Nuclei (AGN) profiles of gas and mass in clusters to extrapolate the measured feedback, cosmic ray heating, magnetic fields, etc.] for providing baryon fraction as a function of radius and as a function of cluster accurate comparisons to the data. Therefore, we use only obser- mass. We find that the baryon fraction reaches the cosmic vations in this work; the collective data should help shed light on value near the virial radius for all groups and clusters above which physical processes are most essential. ∼ × 13h−1M In this paper we investigate the possibility that the “missing 5 10 72 ⊙. This suggests that the baryons are not missing, they are simply located in cluster outskirts. Heating processes baryons” are not missing at all, but are rather located in the out- (such as shock-heating of the intracluster gas, supernovae, and skirts of clusters where few detailed observations have yet been Active Galactic Nuclei feedback) likely contribute to this expanded made. The missing baryons problem is typically observed within distribution. Upcoming observations should be able to detect these the central regions of clusters, generally within a radius of R500 baryons. (where the enclosed mass-density is 500 times the critical den- sity). This radius is ∼0.5 of the virial radius of the cluster [where cosmology ∣ hot intracluster gas the enclosed density is ∼100 times the critical density for the current Lambda Cold Dark Matter (LCDM) cosmology (24, 25)]. Thus for a virial radius of ∼1.5 Mpc, the typical missing baryon lusters of galaxies, the largest virialized systems in the uni- ∼0 75 Cverse, are powerful tools in constraining cosmology and tra- problem is observed only at . Mpc from the cluster center. ASTRONOMY cing the large-scale structure of the universe (1–4, and references Observations show that the gas density profile in the outer 14 15 −1 parts of clusters decreases with radius slower than the mass therein). The large mass of clusters (∼10 to 10 h72 M⊙) implies that their contents—dark and baryonic matter—have been profile in these regions. Using gravitational lensing, the latter accreted from very large regions of ∼10 comoving Mpc, and has been observed out to large radii (11, 26, 27) and is consistent therefore should be representative of the mean matter content with the expected Navarro, Frank, White (NFW) profile (28). of the universe; on these large scales there are no clear mechan- Whereas the cluster mass density declines with radius approxi- mately as r−2.6 in these outer regions, the gas density declines only isms to separate dark and baryonic matter (e.g., refs. 5 and 6). r−2.2 The strong gravitational potential of clusters also implies that as . This implies that the gas is more extended than the total mass, and therefore the gas fraction increases with radius beyond baryons cannot easily escape from these systems. Therefore, clus- R ters are expected to retain the cosmic baryon fraction, the relative the observed radius of 500. A shallow slope of the gas profile fraction of baryons to total matter on large scales. This basic (as compared with the mass profile) is indeed expected if gas heating occurs in the clusters (e.g., from shock-heating of the expectation of a representative baryon fraction in clusters was gas, supernovae, and AGNs). The heating makes the gas less used in 1993 (6) to suggest that the mass-density of the universe bound relative to the dark matter potential, and spreads it out must be low, since the observed baryon fraction in clusters was to larger radii. considerably larger than expected for a critical density universe. Here we use the observed slopes of the gas-density and mass- Most of the baryons in clusters reside in the X-ray emitting hot density profiles in the outer regions of clusters to extrapolate the intracluster gas, which approximately traces the cluster gravita- observed gas fraction from R500 to larger radii, up to the virial tional potential dominated by dark matter. The rest of the bar- radius [R ¼ R100 (24, 25)]. We add the observed stellar fraction yons are in the luminous galaxies and in isolated stars that vir to the extrapolated gas fraction to find the baryon fraction at comprise the small amount of faint diffuse intracluster light large radii. We perform this extrapolation as a function of cluster (ICL). mass from groups to rich clusters, and as a function of radius from A puzzle has been raised, however, over the last few years: R500 to R . Note that this analysis is based entirely on obser- Detailed X-ray observations from Chandra, XMM-Newton, vir vations. and ROSAT suggest that the cluster baryon fraction (gas plus We find that the baryon fraction increases systematically with stars relative to total mass) is considerably lower than the cosmic radius, and show that there is no missing baryon problem in rich value. The cosmic baryon fraction is well determined both from Big-Bang nucleosynthesis (7, 8) and from observations of the cos- mic microwave background to be f b ¼ 0.1675 0.006 (WMAP7: Author contributions: B.R., N.A.B., and P.B. designed research, performed research, 9). The cluster gas fraction has been reported by observations contributed new reagents/analytic tools, analyzed data, and wrote the paper. (10–15) to be only 60–80% of the cosmic value, with stars con- The authors declare no conflict of interest. tributing only a small (∼10%) additional amount of baryons. This article is a PNAS Direct Submission. Whereas the baryon fraction appears to approach the cosmic 1To whom correspondence should be addressed. E-mail: [email protected].

www.pnas.org/cgi/doi/10.1073/pnas.1009878108 PNAS ∣ March 1, 2011 ∣ vol. 108 ∣ no. 9 ∣ 3487–3492 Downloaded by guest on September 30, 2021 clusters when the data is extrapolated to near the virial radius, bins, from groups to rich clusters (15). (We do not include 13 −1 where the baryon fraction becomes consistent with the cosmic the lowest mass bin at 10 h72 M⊙ that contains only two groups value. Most of the missing baryons are therefore expected to with large error bars.) Our cluster sample has a mass range of R R 3 × 1013h−1M M 1015h−1M be in the outskirts of clusters, between 500 and vir. This result 72 ⊙ < 500 < 72 ⊙ and a redshift range of can be tested with upcoming observations of the Sunyaev–Zeldo- 0.012 < z < 0.23. The mean observed gas fraction for each mass vich (SZ) effect in clusters [e.g., South Pole Telescope (SPT) (29); bin is listed in Table 1 and shown in Fig. 1.p Theffiffiffiffiffiffiffiffiffiffiffiffi error on the mean Atacama Cosmology Telescope (ACT) (30)] as well as with more is the rms standard deviation divided by N − 1. The horizontal sensitive X-ray observations. bars are the mass ranges for the bins. Also presented in Fig. 1 is Observations have shown that the missing baryon problem at the cosmic baryon fraction observed by the WMAP7 microwave R500 becomes more severe for lower mass clusters and groups of background measurements. One can see that the cluster gas galaxies than for rich clusters; the observed gas fraction decreases fraction at R500 is significantly lower than the cosmic baryon frac- considerably with decreasing cluster mass. This too would be tion. The gas fraction decreases significantly from rich to poor expected if the heating processes expand the gas: the lower grav- clusters; whereas rich clusters contain about 12% gas within itational potential of the smaller systems will not be able to hold R500, the gas fraction in poor clusters and groups is only ∼6–7%. on to their gas as well as the higher mass clusters. The gas-density profile in small groups is indeed observed to be shallower than Stellar Fraction. The galactic stellar fraction has been measured in the gas-density profile in massive clusters, suggesting that the nearby clusters using multiband optical and infrared surveys com- gas in low-mass systems is more spread out. We extrapolate bined with stellar population models. We use the results obtained the observed baryon fraction as described above for clusters as from the COSMOS survey (15) and 2MASS survey (31) for near- a function of their mass—from groups to rich clusters. We find by clusters. The combined COSMOS and 2MASS sample covers R 3 × 1013–1015h−1M that for this entire mass range the baryon fraction within vir our entire cluster mass range, 72 ⊙. We bin the is flat and is consistent with the cosmic value. observed stellar fraction into the same four logarithmic mass bins In the next section we describe the observations and analysis. A as the gas fraction sample. The mean stellar fraction declines with −0 370 04 discussion of the results and our conclusions follow. We use a cluster mass as M . . (see ref. 15 and Fig. 5), exhibiting an LCDM cosmology with h ¼ 0.72 and Ωm ¼ 0.258. opposite trend to that of the gas fraction. To determine the contribution of the diffuse intracluster light Observations, Analysis, and Results to the stellar fraction, we use observations by Zibetti (32), who The gas fraction in clusters of galaxies has been measured for stacked hundreds of clusters from the Sloan Digital Sky Survey a relatively large sample of groups and clusters within R500. to reach unprecedented depth and cluster-centric distances. They The total baryon fraction is obtained by adding the stellar mass find the ICL is centrally concentrated, and that on average the fraction observed for these systems within the same radius. This ICL contributes ∼10% of the stellar light within the central baryon fraction is systematically lower than the cosmic value mea- 500 kpc for all cluster masses. We add this 10% contribution sured by WMAP7; the discrepancy increases, especially for the to the galactic stellar fraction discussed above for all clusters. gas fraction, with decreasing cluster mass (refs. 14 and 15 and We note that the ICL contribution may decline to less than references therein). Here we investigate the possibility that the 10% when extending to larger cluster radii; but since the ICL missing baryons are spread out to larger radii, beyond R500. is a very small contribution to the total baryon fraction, this effect We investigate this possibility by extrapolating the observed has negligible consequences (see Discussion). The total stellar gas fraction to larger radii, from R500 up to the virial radius, using fraction for the four mass bins is summarized in Table 1. It is the mean observed gas and mass density profiles in these outer added to the gas fraction to obtain the total average baryon frac- regions; these density profiles have been measured up to the virial tion for each mass range. The baryon fraction within R500 is listed radius (and occasionally beyond). The observed stellar mass frac- in Table 1 and plotted as a function of mass in Fig. 1. The defi- tion, including the small contribution from the ICL, is added to ciency of baryons within R500 relative to the cosmic value is clearly the gas fraction to yield the total baryon fraction. The baryon seen in Fig. 1; the deficiency becomes more severe for lower mass fraction is then investigated as a function of radius, from R500 clusters. R to vir, and as a function of mass, from groups to rich clusters. Gas and Mass-Density Profiles. Extrapolating the observed gas frac- Gas Fraction at R500. Although some observations extend to R200, tion to larger radii beyond R500 requires the knowledge of the the gas fraction has been accurately measured for a sufficiently observed gas and mass-density profiles in these regions. The large sample of nearby clusters only out to R500. We use X-ray gas-density profile has been measured well in the outer parts R − R observations of the gas fractions for 39 nearby clusters observed of clusters ( 500 vir) using X-ray observations of nearby clus- with Chandra and XMM-Newton (12–14). These authors use ters with ROSAT, Chandra, XMM-Newton, and Suzaku. The similar methods of data reduction and analysis. We use the com- observed gas profile in the outer regions fits well to a beta-model −α 3β ¼ α ρ ∝ r gas pilation by Giodini et al. (15) of these groups and clusters above with a density slope of gas gas (where gas ). We use 13 −1 the mass of M500 ¼ 3 × 10 h72 M⊙ (where M500 is the mass with- observations that have small uncertainties on the gas-density in R500). After conversion to a common cosmology, the three slope at large radii and that cover our entire cluster mass range samples have been binned into four logarithmically spaced mass (12, 33–36).

Table 1. Cluster gas, stellar, and total baryon fractions within R500

−1 gas starsþICL b Bin hM500i (h72 M⊙) f 500 f 500 f 500 1 5.1 × 1013 0.068 ± 0.005 0.050 ± 0.002 0.118 ± 0.005 2 1.2 × 1014 0.080 ± 0.003 0.040 ± 0.004 0.120 ± 0.005 3 3.0 × 1014 0.103 ± 0.008 0.023 ± 0.002 0.126 ± 0.009 4 7.1 × 1014 0.123 ± 0.007 0.021 ± 0.002 0.143 ± 0.007

Gas, stellar (including ICL), and baryon fraction of clusters within R500 for four cluster mass bins: Bin 1 14 −1 14 −1 14 −1 (0.3–0.7 × 10 h72 M⊙), Bin 2 (0.7–1.7 × 10 h72 M⊙), Bin 3 (1.7–4.2 × 10 h72 M⊙), and Bin 4 (4.2–10× 14 −1 gas stars 10 h72 M⊙). f 500 are averages from Chandra and XMM observations (12–15). f and the 10% ICL contribution are from (15, 31, 32). The error bars are 1-σ errors on the mean.

3488 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1009878108 Rasheed et al. Downloaded by guest on September 30, 2021 β Fig. 2. Observed average weighted gas-density slope gas (where −3β ρ ∝ r gas ≥R gas )at 500 from ROSAT, Chandra, XMM-Newton, and Suzaku (12, 33–36) as a function of cluster temperature. Filled circles are weighted bin-averages of 39 ROSAT clusters (33); filled diamonds are ROSAT averages of hundreds of stacked optical clusters (34; their richness classes 0 and 1); empty circles are bin-averages of 10 Chandra clusters (12); empty diamond is Suzaku observation for Abell 1795 (35); empty square is XMM observation for A1983 β T (36). The dashed line illustrates the average trend of gas with . Error bars are 1-σ errors on the mean. Horizontal bars represent the temperature bin size.

refs. 37 and 38. For our four mass bins (Table 1 and Fig. 1) we find α ¼ 3 × β ¼ 1 8 0 2 the following mean gas-density slopes: gas gas . . for Bin 1, 1.9 0.07 for Bin 2, 2.1 0.02 for Bin 3, and 2.3 0.02 for Bin 4. The error bars are thep standardffiffiffiffiffiffiffiffiffiffiffiffi deviation of the average β-values in each bin divided by N − 1. When comparing a slope at a given temperature bin in Fig. 2 to a given mass bin in Fig. 1, we use the observed M500 − T relation (12); the results are not sensitive to the exact conversion because of the slowly varying β ðTÞ gas relation. The final piece required for extrapolating the gas (and baryon) fraction to large radii is the total mass-density profile. This has

been observed to follow the NFW profile (28) to large radii, ASTRONOMY by using weak lensing observations with the Sloan Digital Sky Sur- vey and other observations (11, 26, 27, and references therein). Fig. 1. The observed cluster gas and baryon fraction within R500 as a func- We use the average observed value of the concentration para- tion of cluster mass, M500. The observed cosmic baryon fraction is shown by meter c200 ¼ 5 for our mass range (26). The NFW profile has the shaded band (1-σ; ref. 9). (A) Observed cluster gas fraction from Chandra a mass-density slope of αm ¼ 2.6 in the radius range from R500 and XMM-Newton within R500 (12–15). The average for the four mass bins is R α ¼ 2 7 R R ρ ∝ r−αm shown by the filled circles. (B) Observed cluster gas and baryon fraction with- to 200, and m . from 200 to vir (where m ). These in R500 for the averages of the four mass bins. Error bars are 1-σ errors on the slopes are considerably steeper than the corresponding gas- mean. Horizontal bars represent the mass range of each bin. density slopes, thus yielding an increasing gas fraction with radius in cluster outskirts. The weighted average observed β-values of the gas-density slope are presented as a function of cluster temperature in Fig. 2. Extrapolation and Results. Using the observed gas-density and The data include measurements for 51 clusters as well as average mass-density slopes at large radii, we extrapolate the observed slopes for stacked samples of hundreds of optical clusters; the gas fraction from R500 as a function of radius up to the virial slopes are binned in temperature. The error bars on the bin-aver-pffiffiffiffiffiffiffiffiffiffiffiffi radius. The gas-fraction increases with radius as: N − 1 aged gas slopes are the standard deviation divided by . −α z 0 25 ρ ðRÞ ∝ R gas The ROSAT observations of 39 < . clusters by Vikhlinin gas gas αm−α f ð< RÞ ∝ −α ∝ R gas ;r>R500: et al. (33) cover the full outer regions of clusters, from 0.3R500 ρmðRÞ ∝ R m 1 5R ∼R to . 180 ( vir). The best-fit slope to these outer regions is R R R determined for a wide range of cluster temperature, from The extrapolated gas fraction from 500 to 200 and to vir is 2 keV to over 10 keV. The Chandra observations of 10 massive presented as a function of cluster mass in Fig. 3A. Because the clusters by Vikhlinin et al. (12) provide β-fits to gas-density slopes gas-density slope is shallower in groups than in rich clusters, the near R500; their weighted average values are consistent with the increasing trend of gas fraction with mass becomes weaker at the trend shown by the slopes of the previous sample (33). For the outer radii. 13 −1 lowest mass bin (M500 ≈ 5 × 10 h72 M⊙) we use the observed The baryon fraction is presented for the different radii—R500, R R — B density slope from ROSAT by Dai et al. (34; their richness class 200, and vir as a function of mass in Fig. 3 ; the baryon frac- 1), who obtain integrated X-ray gas profile for stacked samples of tion within these radii is the sum of the gas fraction (Fig. 3A) and R hundreds of low-mass optical clusters out to vir. We also include the stellar mass fraction discussed above, including the 10% ICL. Bautz et al. (35) who measure the gas profile of Abell 1745 to R200 We assume that this fraction remains constant with increasing using Suzaku observations. radius; the main results do not change significantly if this assump- The mean observed beta slopes presented in Fig. 2 are consis- tion is changed because of the relatively small contribution of the tent with each other, and show a shallower slope for lower mass stellar component (see Discussion). The error bar on the extra- systems than for massive clusters. Similar results were noted by polated gas fraction is the propagated errors of the gas fraction at

Rasheed et al. PNAS ∣ March 1, 2011 ∣ vol. 108 ∣ no. 9 ∣ 3489 Downloaded by guest on September 30, 2021 Fig. 4. The dependence of the baryon fraction on radius, from R500 to R ð¼ R Þ vir 100 , for two representative mass bins (bins 2 and 4; the others show a similar trend). The radius is presented in units of R200. The slow but steady increase of the baryon fraction with radius is apparent, reaching the cosmic R 1 σ value near vir for all cluster masses. The error-bars are - errors on the mean.

baryon fraction increases with radius for clusters of all masses; it reaches the cosmic baryon fraction near the virial radius (R100). This suggests that baryons are not missing in clusters, they are simply located in cluster outskirts. Recent observations of the SZ effect in 15 massive clusters using the South Pole Telescope(29) measure the gas-density pres- sure profile in these clusters as a function of radius up to the virial radius (and beyond); they are well fit with a beta-model in the outer regions. The detection of the gas to these large radii, and their observed beta-model slope (when corrected for the R R Fig. 3. Cluster gas (A) and baryon fraction (B) within 500 (observed), 200 temperature profile), are nicely consistent with our results and and R ð¼ R100Þ (extrapolated using observational data as discussed in vir the conclusion that the baryons are out between R500 and R , the paper). The gas and baryon fractions are presented as a function of vir cluster mass for the four averaged binned samples. The error-bars are the although the gas-density slope cannot yet be accurately deter- 1-σ error on the mean for each bin; horizontal bars represent the mass range mined from the SZ measurements. Similarly, George et al. of each bin. The observed cosmic baryon fraction is presented by the shaded (39) use Suzaku X-ray observations to trace the gas-density pro- band (1-σ). file in the massive cluster PKS0745-191 up to the virial radius, observing the baryons at the cluster outskirts and measuring a R500 and the gas slope used for extrapolation. The error bar on shallow gas-density profile in these outer regions (with lower the baryon fraction is the combined errors of the gas and stellar resolution). fractions. The gas-density profile is observed to be even shallower in low- The results in Fig. 3B show that the baryon fraction flattens mass clusters than in massive clusters. This is qualitatively con- considerably as a function of cluster mass when extrapolated sistent with the gas being heated via shocks and feedback (e.g., to larger radii; this is due to the combined effect of the shallower from supernovae and AGN). This feedback will be more signifi- gas-density profile in groups, which results in more gas in their cant in low-mass systems when compared to the binding energy of outskirts, plus the larger observed stellar mass fraction in groups, the gas. If star formation is more efficient in groups than in clus- which adds more baryons in the smaller systems. In fact, at the ters (e.g., ref. 31), this will further increase the gas entropy in virial radius we find that the baryon fraction is essentially flat these systems, because the star formation removes the lowest-en- from groups to rich clusters, at a level consistent with the cosmic tropy gas from the , leaving behind gas with baryon fraction. This suggests that there are no missing baryons: higher average entropy (40). A higher stellar fraction also implies Most of the missing baryons are likely located in the outskirts of more supernovae/AGN activity per unit mass. This explains why, clusters, extending to nearly the virial radius. at R500, groups have a lower baryon fraction than clusters. How- The extrapolated baryon fraction is presented as a function of ever, by the virial radius, we are able to account for all the baryons radius for two of our mass bins (Bins 2 and 4) in Fig. 4; the results expected from the cosmic value. show the slow but steady increase in the baryon fraction with The stellar fraction has been observed for a large number radius. of clusters by Gonzalez et al. (41), who report a somewhat higher stellar fraction for low-mass clusters and a slightly lower Discussion stellar fraction for high-mass clusters than the observations by The gas fraction at R500 is often cited to claim that clusters con- Giodini et al. (15) and Lin et al. (31). The higher stellar fraction tain fewer baryons than the universal baryon fraction and there- in groups (41) is likely due to a selection bias toward systems fore exhibit a missing baryon problem. Here we show, based with dominant Brightest Cluster Galaxies in the small groups; purely on observational results, that the gas (and baryon) fraction the large, centrally located galaxy in such systems dominates increases substantially with radius beyond R500. Using the ob- the stellar fraction. Their observed trend of stellar fraction with served gas and mass-density profiles, we extrapolate the observed mass is therefore somewhat steeper, log f stars ¼ð7.57 0.08Þ− baryon fraction (gas and stars) as a function of radius from R500 to ð0.64 0.13Þ log M500. Using this stellar fraction in our analysis the virial radius. Since the gas density is observed to decline more does not change our result significantly; the baryon fraction at slowly with radius than the total mass, we find that the average the virial radius decreases by ≈5%, to 0.164, for the most massive

3490 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1009878108 Rasheed et al. Downloaded by guest on September 30, 2021 bin, and increases by ≈7%, to 0.179, for the lowest mass bin. Conclusions These are within our 1-σ error bars. We investigate the missing baryon problem in clusters of galaxies. We use the observed 10% contribution of the ICL to the stellar Observations show that the baryon fraction (gas plus stars) mea- fraction (32). If the ICL fraction changes at the outer radii, it will sured within a radius of R500 in groups and clusters is significantly not affect our results since the ICL contribution is only below the cosmic value; this baryon discrepancy increases with ∼0.2–0.4% of total mass. Similarly, we assume in our analysis that decreasing cluster mass. This gives rise to the puzzle: Where the stellar fraction does not change when extrapolated from R500 are the missing baryons? Why are they missing? We show that R to vir, since no observations of the stellar fraction have been the baryons may not be missing at all, but rather are spread R made beyond 500. If the stellar mass fraction decreases some- out to larger radii, beyond R500, and can be found in the cluster what with radius, then the baryon fraction will slightly decrease. outskirts. Based entirely on observations, we investigate the If we assume, for example, a 20% decrease in stellar fraction from dependence of the baryon fraction on radius for clusters of dif- R R 500 to vir, we find that the baryon fraction is lowered by only 2% ferent masses, from groups to rich clusters. We use the mean R for the massive clusters and 8% for the low-mass groups (at vir); observed gas and mass-density profiles in clusters to extrapolate 1 σ these values are within - of our baryon fraction results even for the observed gas fraction as a function of radius from R500 to the the lowest mass groups. virial radius. Since the gas-density profile is significantly shal- For the mass-density profile, we use the mean observed value lower than the mass-density profile, the gas-fraction increases c ¼ 5 of the concentration parameter of the NFW profile, 200 ,as with radius; it increases more rapidly for groups than for rich clus- observed from weak lensing (26) for our mass range. As Mandel- ters because of shallower gas-density slope in groups. We add the baum et al. (26) discuss, this value is slightly lower than results observed stellar fraction and the diffuse intracluster light to c from simulations and some previous studies. If 200 is larger than the gas fraction to obtain the total baryon fraction. We find that 5, the mass profile will be more concentrated; i.e., fall off even the average baryon fraction for all groups and clusters with steeper with respect to the gas profile at the outer radii. This 13 −1 M500 ≥ 5 × 10 h72 M⊙ increases steadily with radius, reaching would cause the gas fraction to increase even more with radius. the cosmic value and becoming flat as a function of mass when The effect is small, however, and a change of c200 to 7 induces a 5% measured within the virial radius (for the LCDM cosmology). change in baryon fraction at the virial radius of < . This suggests that baryons are not missing in clusters, but are sim- The observations presented above are qualitatively consistent ply located in cluster outskirts. This picture is qualitatively con- with an energy input in clusters that heats the intracluster gas and sistent with heating processes (such as shock-heating of the makes it less bound than the dark matter. Hydrodynamic simula- intracluster gas, as well as supernovae and AGN feedback) caus- tions of cluster formation that include some of the relevant phy- ing the gas to expand to the cluster outskirts. Upcoming observa- sics are in qualitative good agreement (within ∼10%) with the tions in the X-rays and SZ should be able to detect these baryons. picture presented here. Using simulations without cooling, star formation, or feedback, the baryon fraction is roughly constant, ACKNOWLEDGMENTS. B.R. thanks her research adviser, collaborator, and friend at 90% of the cosmic value, from R500 to R200 (42). When these Neta Bahcall for her judicious and delightfully rewarding stewardhip processes are included (with feedback coming from AGN), then of her senior thesis, which culminated in this paper. B.R. also thanks the baryon fraction is found to be lower at R500, but instead of Princeton University for enabling this experience. The authors thank Rachel

being flat it increases with radius (43, 44). Similar results, based Mandelbaum, Greg Novak, David Spergel, Michael Strauss and Alexey ASTRONOMY on cluster energetics, are found in the models of Bode et al. (22). Vikhlinin for their helpful comments. Computational work was performed at the TIGRESS high performance computer center at Princeton University, However, simulations do not yet contain all the physics needed which is jointly supported by the Princeton Institute for Computational for accurate comparison with the observations (including sources Science and Engineering and the Princeton University Office of Information of nonthermal pressure and possible nonequipartition effects). Technology.

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