ON THE BARYON FRACTIONS IN CLUSTERS AND GROUPS OF GALAXIES

There is a missing baryon problem in the nearby universe (see the review by Bregman 2007). We can see this by comparing the baryon fraction from direct measurements of galaxies and galaxy clusters to the value determined from the Wilkinson Microwave Anisotropy Probe (WMAP) three year data (Spergel et al. 2007), where the baryon fraction, the ratio of the baryonic mass to the total mass, is f_b = Ω_b/Ω_m = 0.175 ± 0.012⁶ independent of the Hubble constant. The baryon fraction of galaxies can also be measured by comparing the gravitating mass measured through gravitational lensing or the dynamics of satellites to the stellar and gas mass (e.g., Hoekstra et al. 2005; Heymans et al. 2006; Mandelbaum et al. 2006; Gavazzi et al. 2007; Jiang & Kochanek 2007). These studies find similar results. For example, Hoekstra et al. (2005) used isolated nearby galaxies to find f_b = 0.056 for spirals and f_b = 0.023 for ellipticals; on average, spirals have lost 2/3 of their initial baryons and ellipticals have lost 6/7 of their initial baryons. For galaxy clusters, where the gas mass dominates the baryon content, the missing baryon problem is less severe, particularly in the most massive clusters. Recent studies of relaxed massive clusters (e.g., Vikhlinin et al. 2006; Allen et al. 2008) show that after adding the stellar baryon component, the baryon fraction in most massive, relaxed clusters is close to the cosmological value. In the less massive galaxy groups, only a few are suitable for measuring the baryon fraction out to the outskirts of the groups, and they generally show significantly lower baryon fractions (e.g., Sun et al. 2009).

These observations indicate that the extent of baryon loss depends on the depth of the gravitational potential well: rich clusters retain their cosmological allotment of baryons, while galaxies are baryon-poor. The mass scale of the transition in the baryon fraction is not well constrained because there are so few measurements on the mass scales of groups. In groups, the baryon content should still be dominated by the gas mass, but the weakness of the X-ray emission from the outskirts of groups means that few groups have sufficiently deep X-ray observations to measure the baryon fraction out to a large enough radius to represent a complete inventory. Few are measured to r₅₀₀ (Sun et al. 2009), where the over-density inside the radius is 500 times the critical density of the universe ρ_c = 3H(z)²/8πG, and even fewer are observed to r₂₀₀, close to the virial radius. The alternative to individual measurements of the baryon fractions in clusters and groups is to stack the ROSAT All-Sky Survey (RASS; Voges et al. 1999) X-ray data to measure the average properties of groups and clusters selected by other means. In particular, Dai et al. (2007) were able to detect gas emission well beyond r₂₀₀ for galaxy groups with temperatures T>1.5 keV, significantly further out than the measurements of individual groups, using groups and clusters selected from the 2MASS (Skrutskie et al. 2006) catalogs using a matched filter algorithm (Kochanek et al. 2001). For other richness classes, the gas emission is detected beyond r₅₀₀, whereas X-ray observations of these individual poorest groups only have detections of their central regions. The stacking method has also been applied to the other large optically selected cluster catalogs (Rykoff et al. 2008; Shen et al. 2008).

In this paper, we combine the results from the Dai et al. (2007) stacking analysis with results for individual galaxies, groups, and clusters to summarize the dependence of the baryon fraction on potential well depth. This provides a more complete picture of baryon losses across a broad range of systems, which should enable us to better understand the origin of the losses. We calibrate all the measurement to the three year WMAP cosmology of H₀ = 73 km s^-1 Mpc⁻¹, Ω_m = 0.26, and Ω_Λ = 0.74 (Spergel et al. 2007).

The details of the RASS stacking analysis for the 2MASS groups and clusters are presented in Dai et al. (2007). Here, we briefly summarize the portions of analysis relevant to the calculation of the baryon fractions. We summarize the optical and X-ray results in Tables 1 and 2, where uncertainties are given with 1σ confidence. The 2MASS cluster catalog (Kochanek et al. 2001) is composed of more than 4000 clusters/groups that are selected from the 2MASS infrared survey (Skrutskie et al. 2006) using a matched filter algorithm that uses both the photometric catalogs and any available spectroscopic redshifts. Besides the cluster position, the algorithm also provides estimates of the optical richness, redshift, velocity dispersions, and the membership probabilities of individual galaxies. In Dai et al. (2007), we divided the catalogs into five richness classes, labeled 0–4 from richest to poorest, and then measured the stacked X-ray surface brightness profiles and spectra for each class using the RASS (Voges et al. 1999) X-ray images. Since RASS is an all-sky survey, we are able to subtract local backgrounds to each cluster. The stacked images have very well-defined backgrounds compared to typical pointed observations because they are averages over many images obtained in the scanning mode of the RASS observations.

Figure 1 shows the normalized surface brightness profiles from Dai et al. (2007) for the five richness classes, where we have multiplied the profiles by R² to reduce the dynamic range and then shifted them for clarity. The profiles are more heavily binned than in Dai et al. (2007) in order to show that we detect gas emission beyond r₅₀₀ (about 0.6r₂₀₀) and close to (or beyond) r₂₀₀ in all five richness classes. In particular, we detect gas emission beyond r₂₀₀ in richness classes 1 and 2. The X-ray data are stacked using the optical positions, so position errors smooth the inner profiles but have little effect at large radius (see Dai et al. 2007; Rykoff et al. 2008).

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We re-fit these surface brightness profiles with a β model, S(R) ∝ (1 + (R/R_c)²)^(−3β+1/2), which corresponds to a three-dimensional gas density profile of n(r) ∝ (1 + (r/r_c)²)^−3β/2, where R_c = r_c. Unlike our previous analysis, we do not include a constant component for subtracting any residual backgrounds. The fit results are generally consistent with those in Dai et al. (2007) given the uncertainties. The β model fits richness classes 3 and 4 best (Figure 1). For richness classes 0 and 1 the data drop below the best-fit β model at roughly r₂₀₀, as well as for the last radial bin of richness class 2. The deviations contribute to the χ² fit statistics by χ² = 13.8, 9.2, and 3.0 for the last 3, 6, and 1 data points of the richness class 0, 1, and 2 profiles. The significance of these deviations is 99.7% for richness class 0, 90% for richness class 1, and not significant for richness class 2, based on the F-test by comparing the β model fits to the fits for a modified β profile that falls more steeply at large radius. These deviations are not very significant due to large uncertainties and can be roughly modeled by allowing for a small residual in the background subtraction, as discussed in Dai et al. (2007), but they all have the same sense and could be real drops in cluster surface brightness profiles at large radius (r ∼ r₂₀₀). Such drops have been detected in previous observations (e.g., Neumann 2005) and seen in numerical models (e.g., Roncarelli et al. 2006). Based on these models we calculate the gas mass within r₅₀₀ and r₂₀₀ for each richness class using the β model density profiles normalized by the observed emission measures. Within r₅₀₀, the β model is adequate for all richness classes. When extending to r₂₀₀, only richness class 0 has a deviation of 2σ and we assume it is not significant and use the β model fit to calculate the masses within r₂₀₀.

For the spectral analysis, we use the same results as in Dai et al. (2007). We fit the stacked X-ray spectra with a Raymond–Smith model assuming a 0.33 solar metallicity and excluding clusters/groups with Galactic N_H column densities higher than 10²¹ cm^-2. Recent Chandra and XMM-Newton studies of hot clusters find declining metallicity from Z = 0.45 at the center to Z = 0.2 at 0.4 r₂₀₀ (e.g., De Grandi et al. 2004; Baldi et al. 2007; Leccardi & Molendi 2008). Compared with our metallicity assumption, we find that the difference affects little on the temperature estimates and the gas mass can be affected by from a few percent for the richness class 0 to 10% in the richness class 4. We find good fits to the spectra as in Dai et al. (2007), and our L–T relation based on stacked clusters matches well with those determined from individual clusters (Wu et al. 1999; Helsdon & Ponman 2000; Xue & Wu 2000; Rosati et al. 2002). This is in contrast to the stacking analysis of Rykoff et al. (2008) for the SDSS maxBCG cluster catalog (Koester et al. 2007), probably because the analysis of Rykoff et al. (2008) focused on massive clusters whose peak temperatures correspond to harder spectra peaking outside the ROSAT band.

We also measure the temperature profiles of the stacked clusters (Figure 2), except for richness class 4 where the signal-to-noise ratio of the spectra is too low after further binning the spectra by radius. We can compare our stacked temperature profiles to the empirical universal temperature profile of Vikhlinin et al. (2005) for relaxed clusters and those for 10 simulated clusters from Stanek et al. (2010). The Stanek et al. (2010) profiles with a temperature near 1 keV, comparable to our richness class 2, are from the Gas Millennium Simulation including a preheated plasma. In Figure 2, we show their "spectroscopic-like" temperature as defined in Mazzotta et al. (2004). We test the validity of this formula at low temperatures by comparing the simulated profiles with those derived from the X-ray photon event files produced by the X-ray Map Simulator (Gardini et al. 2004; Rasia et al. 2008). We find that our temperature profile is shallower than the empirical model of Vikhlinin et al. (2005), although we note that our cluster sample is optically selected and includes both relaxed and un-relaxed clusters. The models of Stanek et al. (2010) also predict shallower temperature profiles than Vikhlinin et al. (2005). Our temperature profiles are consistent with the flatter, large radius profiles of the Stanek et al. (2010) models. A recent temperature profile for galaxy groups has been measured in Sun et al. (2009), which is consistent with the temperature profile of Vikhlinin et al. to r₅₀₀, and then falls more rapidly than our profiles. These differences may be caused by the cluster selection method, where the optical sample shows a flatter temperature profile. This is partially supported by the simulation results (Stanek et al. 2010), which show flatter temperature profiles than the X-ray bright sample.

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We test if our flattened temperature profiles are caused by the ROSAT XMA-PSPC point-spread function (PSF) at large off-axis angles. Since we are dealing with scanning observations, the average off-axis angle is 2/3R_F = 38', where R_F is the radius of the PSPC field of view (57'). The FWHM of XMA-PSPC for this off-axis angle is 80'' ≃ 13 (Hasinger et al. 1993). For our richness classes 0–4, we measure r₂₀₀ = 1.57, 0.84, 0.70, 0.65, and 0.58 Mpc and 〈z〉 = 0.082, 0.068, 0.054, 0.039, and 0.027 (Table 1), thus the angles subtended by r₂₀₀ are 177, 112, 116, 146, and 186. Therefore, we conclude that the PSF cannot significantly flatten the temperature profiles at large radius from the center.

We fit the temperature profile of the four richness classes within r₂₀₀ with a single power law in radius to find that

$$\begin{equation} \log {T(r)} \propto -(0.10\pm 0.04) \log {r}. \end{equation} \tag{ 1 }$$

This profile is slightly shallower than an isothermal model in which the temperature is independent of radius. Given the temperature profile and our best-fit β model, we calculate the cluster mass using the equation of hydrostatic equilibrium to be

$$\begin{equation} M_{\rm grav}({<}r) = -\frac{kT(r)r}{G \mu m_p}\left(\frac{d\ln {\rho (r)}}{d\ln {r}}+\frac{d\ln {T(r)}}{d\ln {r}}\right). \end{equation} \tag{ 2 }$$

The resulting mass–temperature relation is consistent with previous estimates (Xu et al. 2001; Arnaud et al. 2005).

Since the stellar mass represents a non-negligible fraction of baryonic mass in groups and to a lesser extent clusters, we estimate its contribution to the total mass by converting the optical richness measurement into an estimate of the stellar mass. Kochanek et al. (2001) expressed the optical richness of the 2MASS clusters in terms of N_*666, defined as the number of L>L_* galaxies inside the (spherical) radius r_*666 where the galaxy overdensity is Δ_N = 200Ω⁻¹_M ≃ 666 times the mean density based on the K-band galaxy luminosity function of Kochanek et al. (2003). Here, we correct for the Poisson biases between the average richness at fixed temperature and the average temperature at fixed richness (see Section 3 of Dai et al. 2007) using a simple halo occupation model to obtain the intrinsic optical richness. Assuming the clusters have galaxy luminosity functions with the same shape as the average galaxy luminosity function, the total K-band luminosity of a cluster out to r_*666 is simply N_*666L_*Γ[2 + α, 1] where α = −1.09 and L_* = −23.39 + 5log h₁₀₀ are the faint-end slope and the characteristic luminosity of the Schechter function measured in the K band (Kochanek et al. 2003). Given this total luminosity, we convert to a stellar mass using a K-band mass-to-light ratio of 0.95 from Bell et al. (2003).

Figure 3 shows the baryon fraction f_b, gas fraction f_g, and stellar-to-gas mass ratio r_sg for our stacked 2MASS groups and clusters calculated within r₂₀₀, and the corresponding numerical values given in Table 2. We also show the measurements for individual clusters from the samples of Vikhlinin et al. (2006) and Sun et al. (2009), where f_g is measured to r₅₀₀ for a sample of groups and clusters. We correct the gas fraction measurements of individual clusters to r₂₀₀ upward by a factor of 1.24 to convert from f_g,500 to f_g,200 based on the ratio found for our stacked clusters in the first three richness classes 0, 1, and 2.

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3.1. Stellar-to-gas Ratio

We first examine the stellar-to-gas mass ratio r_sg of the stacked 2MASS groups and clusters shown in Figure 3. We fit the stacked data with a power law to find

$$\begin{equation} \log {r_{{\rm sg}, {\rm optical}}} = (-0.16\pm 0.03) - (1.17\pm 0.06) \log {T}. \end{equation} \tag{ 3 }$$

This trend is consistent with the other studies (e.g., David 1997; Giodini et al. 2009), who also found that the stellar-to-gas mass ratio increases at low temperatures and that stellar and gas masses are comparable at the low-temperature end of the distribution (∼0.9 keV).

In our calculation, we do not explicitly include the contribution from the intra-cluster star light (ICL) in the calculation of the stellar mass. The fraction of ICL mass in stellar mass (M_ICL/M_str) is difficult to measure with estimates ranging from 10% to 40% (Krick & Bernstein 2007; Gonzalez et al. 2005, 2007). In particular, Gonzalez et al. (2005, 2007) separate the stellar mass into three parts, the brightest cluster galaxy (BCG), the remaining cluster galaxies, and the ICL, and find that the BCG+ICL contribute to 40% of the stellar mass. In our approach, we model the cluster stellar mass using a model for the luminosity function of the cluster galaxies to correct for the magnitude limits of the galaxy catalogs. This obviously includes the BCG and observed cluster galaxies, but the extrapolation of the luminosity function also includes part of the emission that Gonzalez et al. treat as part of the ICL. Gonzalez et al. (2007) calculate that the stellar mass (BCG+galaxy+ICL) is comparable to the gas mass at M₅₀₀ = 5 × 10¹³ M_☉. This corresponds to our richness class 1 (Table 2); however, our stellar-to-gas ratio is 0.30 ± 0.03, a factor of 3 smaller than the estimate of Gonzalez et al. (2007). Considering that BCG+ICL contribute to 40% of the stellar mass in Gonzalez et al. (2007), our numbers are still off by a factor of 2 even excluding the ICL contribution. In our analysis, the stellar and gas mass are comparable at M₅₀₀ ∼ 2 × 10¹³ M_☉, which is consistent with the recent study of Giodini et al. (2009). Since the ICL fraction (∼11%) of Krick & Bernstein (2007) is consistent with the planetary nebula counting statistics and hostless supernova (SN) statistics, we proceed by assuming that the residual ICL missing from our calculation only contributes a small correction, less than 5%, to the total stellar mass after the extrapolation over the luminosity function.

We also show the r_sg values for individual X-ray bright clusters from Vikhlinin et al. (2006) and Sun et al. (2009). Wherever possible, we identify these clusters in the 2MASS cluster catalog to determine their richness and estimate their stellar masses in the same manner as for the stacked clusters. We find that individual, X-ray bright clusters tend to have lower stellar-to-gas ratios than the stacked clusters, with

$$\begin{equation} \log {r_{\rm sg,\ X\hbox{-}ray}} = (-0.65\pm 0.06) - (1.03\pm 0.14) \log {T}. \end{equation} \tag{ 4 }$$

This difference is consistent with recent findings that optically selected clusters tend to be X-ray under-luminous compared to the X-ray bright clusters due to the flux limits of the X-ray surveys (e.g., Stanek et al. 2006; Dai et al. 2007; Rykoff et al. 2008). Using the r_sg relation determined for the X-ray clusters in the 2MASS cluster catalogs, we can estimate the stellar mass and baryon fractions for the remaining individual X-ray clusters that lack 2MASS counterparts. Generally, the clusters without counterparts lie at redshifts beyond the completeness limits of the 2MASS cluster catalogs.

3.2. Baryon and Gas Fractions in Groups and Clusters of Galaxies

Next, we examine the baryon and gas mass fractions in groups and clusters of galaxies. Figure 3 shows that there is a break in these fractions at ∼1 keV. Below this temperature, both mass fractions drop significantly compared to the hotter systems above the break. For systems above 1 keV, we fit the baryon and gas fractions, f_b and f_g, with a power law, where we weight the stacked and individual data equally during the fit since they are reasonably consistent, to find that

$$\begin{equation} \log {f_g} = (-1.09\pm 0.02) + (0.30\pm 0.03) \log {T} \end{equation} \tag{ 5 }$$

and

$$\begin{equation} \log {f_b} = (-1.00\pm 0.02) + (0.20\pm 0.03) \log {T}. \end{equation} \tag{ 6 }$$

The gas mass fraction drops more steeply than the baryon mass fraction as we go from high temperature clusters to lower temperature groups. Below ∼1 keV, both the baryon and gas mass fractions drop significantly. For the gas fraction, the stack clusters in richness classes 3 and 4 deviate from the single power-law relation (Equation (5)) by Δχ² = 222.5, and for the baryon fraction the deviation from Equation (6) for the two richness classes is Δχ² = 41.2. We find that the breaks are at T_B = 1.2 ± 0.1 keV for gas fractions and T_B = 1.1 ± 0.2 keV for baryon fractions.

3.3. Baryon Losses from Galaxies to Clusters

Finally, we combine the baryon fractions in clusters with those measured for galaxies to probe baryon losses across a broad range of potential wells, as indicated by the virial velocities, from dwarf galaxies to rich clusters. For groups and clusters, we convert the temperatures to velocity dispersions through the scaling relation σ = 309 kms^-1(T/1 keV)^0.64 (Xue & Wu 2000), and then convert to the virial velocity assuming $V_c = \sqrt{2} V_{\sigma }$. Because of the broad range of virial velocities, we are not very sensitive to the effects of small offsets. For galaxies, we used the spiral galaxy sample of McGaugh (2005) and Stark et al. (2009) and the dwarf galaxy sample of Walker et al. (2007), where the rotation curve is used to determine the virial mass. For the dwarf galaxy sample of Walker et al. (2007), the baryon mass is assumed to be the stellar mass as the gas mass is negligible for these galaxies (Mateo 1998). McGaugh (2005) provide both stellar and gas masses, and the masses of Stark et al. (2009) sample are dominated by the gas. We also include the baryon fraction of the Milky Way (Sakamoto et al. 2003; Flynn et al. 2006) and that of an ensemble of early-type galaxies where the total mass is determined through gravitational lensing (Gavazzi et al. 2007). The results are shown in Figure 4.

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We can fit all these data with a broken power-law model to find that

$$\begin{equation} f_{b} = \frac{0.14 (V_c/440 \;{\rm km\;s^{-1}})^{a}}{{(1+(V_c/440 \;{\rm km\;s^{-1}})^{c})}^{b/c}}, \end{equation} \tag{ 7 }$$

where a = 1.6, b = 1.5, and c = 2. The baryon fraction, f_b, scales as f_b ∝ V^a−b=0.1_c above the break and f_b ∝ V^a=1.6_c below the break, and the parameter c in the equation is the smoothness of the broken power-law model. The model fits well for V_c ≳ 30 kms^-1, but most lower velocity dwarf galaxies lie above the curve, suggesting that the relation flattens again at very low masses. We test by fitting the archival data points without the stacking data points with the same broken power-law model and find consistent parameters.

We also calculate the dispersion of the data about our broken power-law fit in three ranges, systems with V_c ⩾ 440 kms^-1, 30 ⩽ V_c < 440 kms^-1, and V_c < 30 kms^-1 to find dispersions of 0.10, 0.29, and 1.2 dex, respectively. The dispersion decreases for more massive systems. This decrease is still significant if we remove the stacked data points where the scatter is reduced. For the systems with V_c ⩾ 30 kms^-1, the overall dispersion of 0.25 dex is relatively small considering the parameter range, although it is larger than that found by McGaugh (2005) using a more homogeneous sample.

In Figure 5, we show the stellar mass fraction for the McGaugh (2005) galaxies, our stacked groups and clusters, and individual clusters from Vikhlinin et al. (2006) and Sun et al. (2009). We exclude gas-rich galaxies and the dwarf galaxies in this plot. The stellar fraction of the stacked groups roughly joins onto the stellar fraction for galaxies. The stellar mass fraction increases from low mass to high mass galaxies, is roughly constant for massive galaxies and groups, and then declines as we move from groups to massive clusters.

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Baryon loss, compared to the global average, as a function of potential well depth over the range from rich clusters to dwarf galaxies should provide a good basis for understanding the causes of the losses. For deep potential wells, rich clusters with T ≳ 5 keV, baryon loss is not significant. The baryon fractions of these clusters, including both the gas and stellar components and corrected to r₂₀₀, are close to the cosmological value measured by WMAP. This is consistent with recent studies (e.g, Vikhlinin et al. 2006; Giodini et al. 2009; Stanek et al. 2010) of individual clusters. For clusters between 5 and 1 keV, baryon loss becomes increasingly important, a trend that can be modeled as a power law. Near ≈1 keV, there is a sharp decline in the baryon fraction of groups and this decline smoothly joins onto the observed baryon fractions in galaxies. Groups and galaxies of the same potential well depth appear to have the same baryon fractions. This suggests that baryons missing from galaxies do not reside in the potential wells of their parent groups.

The small scatter in the baryon fractions of galaxies, galaxy groups, and galaxy clusters with V_c ≳ 30 kms^-1 about this general trend indicates that the baryon loss mechanism of different systems must be fairly universal and primarily controlled by the depth of the system's potential well. Such a mechanism could be the pre-heating of baryons before they collapse. In this picture, the baryons never fall into galaxies and groups, but remain well beyond r₂₀₀. Alternatively, the gas falls into the potential wells and is subsequently removed by feedback provided from SNe and AGNs. It is hard to reconcile this scenario with similar baryon fractions in systems with similar potential wells but very different stellar-to-gas mass ratios. For example, if the feedback is dominated by SNe, whose rate is proportional to the stellar content, the gas-rich systems should have lower baryon losses than systems dominated by stellar mass at the same potential well depth. In some galaxies, Anderson & Bregman (2010) show that the energy from feedback is inadequate to expel the gas. It is possible that feedback processes are responsible for increasing the scatter of the mean relation, since the total feedback energy depends on a variety of aspects such as the AGN fraction in clusters or groups, the radio AGN fraction, the stellar-to-gas ratio, and the stellar metallicity. For massive clusters, to interpret the small scatter of baryon fractions, feedback must be dominated by one process or the feedback energy cannot be significant compared to the depth gravitational potential well so that it will have little effect on the baryon fraction of clusters. For less massive systems, such as galaxies, it is easier for the baryons to escape their potential wells and we do observe a larger scatter around the mean relation. For dwarf galaxies, one would expect the effect to be more pronounced and we observe a very large scatter.

It has been well established that the scaling relations in clusters and groups of galaxies deviate from the "self-similar" model (e.g., Kaiser 1986; Navarro et al. 1995). This includes the observed steeper slopes of L–T (e.g., White et al. 1997; Wu et al. 1999; Rosati et al. 2002) and M–T (e.g., Xu et al. 2001; Sanderson et al. 2003; Arnaud et al. 2005) relations, the break in the L–T relation (e.g., Helsdon & Ponman 2000; Xue & Wu 2000) at the group regime, and an entropy floor in the S–T relation (e.g., Helsdon & Ponman 2000). In particular, the breaks in these observed scaling relations occur at similar places as our break for baryon fractions (∼1 keV). This leads us to contemplate that the causes for these breaks are from the same origin. There are a number of models proposed to explain the deviations from the "self-similar" model (see Voit 2005 for a review), which involve either pre-heating (e.g., Evrard & Henry 1991; Kaiser 1991; Bialek et al. 2001; Kay et al. 2007; Gottlöber & Yepes 2007; Stanek et al. 2010) or combinations of cooling (e.g., Muanwong et al. 2002; Borgani et al. 2002; Davé et al. 2002; Kay et al. 2003; Valdarnini 2003) and heating from AGNs (McCarthy et al. 2008) or galactic winds (e.g., Davé et al. 2008). Some of these models only attempt to solve the problems in a particular mass range, such as in groups and clusters. We note that the baryon fraction measurement can extend all the way to low mass systems to dwarf galaxies, which provides a much longer baseline to constrain models. The small scatter about the broken power-law relation for baryon fractions from dwarf galaxies to clusters suggests a universal baryon loss mechanism. Therefore, it is unlikely that models with ingredients only involving physical processes in a limited mass range will explain such a relation.

The hierarchical structure growth of the CDM theory has been successful to explain many observations; however, it also encounters difficulties such as over-predicting low mass halos compared to observation (e.g., Kauffmann et al. 1993; Klypin et al. 1999; Moore et al. 1999). The discrepancy is reduced to be within a factor of ∼4 when counting the ultra-faint dwarf galaxies recently discovered from SDSS (e.g., Simon & Geha 2007). One hypothesis to solve this discrepancy is that re-ionization at high redshift (z ∼ 10) could suppress formation of dwarf galaxies (e.g., Bullock et al. 2000; Somerville 2002; Benson et al. 2002; Ricotti & Gnedin 2005; Moore et al. 2006; Simon & Geha 2007; Koposov et al. 2009). The entropy imposed by re-ionization make it more difficult for low mass halos to constrain gas and form stars. Therefore, a pre-heating model is capable of significantly affecting both the very low and high mass end of the systems. However, detailed modeling is needed to test whether such models can reproduce the baryon fractions in a wide range of systems.

In addition, our results also present another constraint to the hierarchical growth theory. We find that the baryon fractions are higher for more massive systems. If massive clusters are formed through merging from smaller structures, then the theory needs to explain why the baryon fraction increases after the mergers. Presumably, a large amount of cold or warm gas, which is outside the potential wells of smaller structures, is accreted during the merging process and falls in the potential well of the larger structure afterwards. Alternatively, it is possible that the smaller structures that we observe today are quite different from those that merge into larger systems.

The baryon fraction measurements of our stacked groups (T < 1 keV) have filled in the gap between measurements of individual galaxies and cluster/groups above 1 keV. The consistency between the baryon fraction measured in these poor groups and massive galaxies suggests a universal baryon loss mechanism that primarily depends on the depth of the system's potential well. More individual measurements of the baryon fractions for groups in this regime are needed to more accurately measure the average and begin to characterize its scatter. Optimally, the X-ray emission should be measured at quite large radius, beyond r₅₀₀, and the groups should be selected using a range of different methods, including both the X-ray and optically selected groups. For example, our stacked data for optically selected clusters show flatter surface brightness and temperature profiles at large radii than the X-ray bright clusters. The associated optical imaging data are also important to constrain the stellar content of poor groups, since our stacking analysis shows that in this regime the stellar mass is comparable to the gas mass, as also suggested by other studies (e.g., David 1997; Giodini et al. 2009).

We gratefully acknowledge valuable discussion and advice from G. Evrard, S. McGaugh, M. L. Mateo, J. A. Irwin, R. Dupke, E. Bell, E. Rykoff, and the anonymous referee. X.D. and J.N.B. acknowledge financial support for these activities through NASA grants NNX07AU2G, NNX08AB69G, and NNX08AT01G. X.D. and C.S.K. acknowledge support by NASA ADP grant NNX07AH41G. E.R. acknowledges support by NASA through the Chandra postdoctoral Fellowship grant No. PF5-70042.