Baryonic Clumps and Dark halos

Globular Clusters and Dark Satellite Galaxies through the Stream Velocity

Smadar Naoz11affiliation: Institute for Theory and Computation, Harvard-Smithsonian Center for Astrophysics, 60 Garden St.; Cambridge, MA, USA 02138 22affiliation: Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA $\dagger$$\dagger$affiliation: Einstein Fellow , & Ramesh Narayan11affiliation: Institute for Theory and Computation, Harvard-Smithsonian Center for Astrophysics, 60 Garden St.; Cambridge, MA, USA 02138

The formation of purely baryonic globular clusters with no gravitationally bound dark matter is still a theoretical challenge. We show that these objects might form naturally whenever there is a relative stream velocity between baryons and dark matter. The stream velocity causes a phase shift between linear modes of baryonic and dark matter perturbations, which translates to a spatial offset between the two components when they collapse. For a () density fluctuation, baryonic clumps with masses in the range  M ( M) collapse outside the virial radii of their counterpart dark matter halos. These objects could survive as long-lived dark matter-free objects and might conceivably become globular clusters. In addition, their dark matter counterparts, which were deprived of gas, might become dark satellite galaxies.

1. Introduction

Observations indicate that globular clusters (GCs) contain practically no gravitationally bound dark matter (DM) (e.g., Heggie & Hut, 1996; Bradford et al., 2011; Conroy et al., 2011; Ibata et al., 2013). How did these objects form? Assuming a baryon-only universe, Peebles & Dicke (1968) suggested in early work that GCs formed via gravitational collapse of non-linear baryonic over-densities shortly after recombination (). Their model is no longer viable since we know now that DM dominates the matter content of the Universe.

Gunn (1980) suggested that GCs are formed in strong shocks when gas is compressed during galaxy mergers. The discovery of many massive young star clusters in the interacting Antennae system (e.g., Whitmore & Schweizer, 1995; Whitmore et al., 1999) supports this idea, and the scenario has been incorporated in cosmological hierarchical structure formation models (e.g. Harris & Pudritz, 1994; Ashman & Zepf, 1992; Kravtsov & Gnedin, 2005; Muratov & Gnedin, 2010).

Another currently popular paradigm is that GCs, like all structure, initially formed inside DM halos (Peebles, 1984), but these halos were later stripped by the tidal field of their host galaxies (e.g. Bromm & Clarke, 2002; Mashchenko & Sills, 2005; Saitoh et al., 2006; Bekki & Yong, 2012), leaving the central parts deficient in DM. However, some GCs are observed with stellar tidal tails, which is difficult to understand if the objects have extended DM halos (Grillmair et al., 1995; Moore, 1996; Odenkirchen et al., 2003; Mashchenko & Sills, 2005).

In the standard model of structure formation, because of baryon-radiation coupling, baryon over-densities at the time of recombination were about 5 orders of magnitude smaller than DM over-densities. Baryons and DM also had different velocities at recombination, with a relative speed that was coherent on comoving scales of a few Mpc (Tseliakhovich & Hirata, 2010). After recombination, the baryons decoupled from the photons and their subsequent evolution was dominated by the gravitational potential of the DM. They also cooled quickly, and their relative velocity with respect to the DM, called the “stream velocity”, became supersonic.

The stream velocity has important implications for the first structures (Stacy et al., 2011; Maio et al., 2011; Greif et al., 2011; Fialkov et al., 2012; Naoz et al., 2012, 2013; O’Leary & McQuinn, 2012; Bovy & Dvorkin, 2012; Richardson et al., 2013; Tanaka & Li, 2014), for the redshifted cosmological 21-cm signal (Dalal et al., 2010; Bittner & Loeb, 2011; Yoo et al., 2011; Visbal et al., 2012; McQuinn & O’Leary, 2012), and even for primordial magnetic fields (Naoz & Narayan, 2013).

Naoz et al. (2013) showed that the stream velocity can result in some halos becoming nearly baryon-free; the gas simply had too much relative velocity to fall into these DM halos. The question we ask here is: What happened to the baryons that failed to fall into these halos? We show that, in at least some cases, these baryons might have collapsed to form baryon-only bound objects that are physically separated from their parent DM halos. For an interesting range of masses, the spatial offset is larger than the virial radius of the DM halo, allowing the baryonic clumps to survive as independent DM-free objects. We suggest that these objects may have evolved into GCs. We also suggest that the corresponding gas-poor DM halos may be present day dark satellites or ultra-faint galaxies.

We begin in §2 by discussing the evolution of baryonic over-densities in the presence of a stream velocity. We then calculate in §3 the likelihood of forming spatially separated baryon-only objects. We analyze the survival of these objects in §4 and conclude with a discussion in §5. Throughout, we adopt the following cosmological parameters:  km sMpc).

2. Baryonic over-density and phase shift

We solve the coupled differential equations that govern the linear evolution of the dimensionless density fluctuations of the DM, , and the baryons, . Both quantities are complex numbers. In the baryon frame of reference, the evolution equations are:


where is the present day matter density as a fraction of the critical density, is the comoving wavenumber vector of the perturbation, is the relative velocity between baryons and DM in a local patch of the universe, is the scale factor of the universe, is the present day value of the Hubble parameter, is the mean molecular weight of the gas, is the mean temperature of the baryons, () is the cosmic baryon (DM) fraction, and is the dimensionless fluctuation in the baryon temperature. Derivatives are with respect to clock time. The above equations are a compact version of equations (5) in Tseliakhovich & Hirata (2010). We have used the fact that , and have included a pressure term appropriate to the equation of state of an ideal gas (Naoz & Barkana, 2005).

The density perturbation amplitudes and are complex numbers, with phases given by


The stream velocity introduces a phase shift, , between the baryons and DM (because of the third term in the left hand side of Eq. 1). This phase shift translates to a physical separation between the DM and baryon over-densities. The phase shift was discussed previously by Naoz et al. (2012), but the corresponding spatial shift was not resolved in their simulations. Here we use analytical linear theory.

Since the phase shift depends on , both the angle between and and the magnitude of are relevant. For concreteness, we present results corresponding to and , where is the (scale-independent) rms fluctuation of the stream velocity on small scales. In the top panel of Figure 1, we show as a function of redshift for four representative wavenumbers, and  Mpc, which correspond to baryon masses  M, respectively (Fig. 2). As seen in Figure 1, smaller scales (larger wave-numbers) develop a larger . The phase difference is related to the comoving distance between the baryon and DM fluctuation peaks, , by


This comoving separation , as well as the corresponding physical separation, , are shown in the lower two panels of Figure 1.

Figure 1.— Top panel: Phase shift between fluctuations in the baryons and DM as a function of redshift for modes with  Mpc, assuming a stream velocity and . Middle panel: Comoving spatial separation between baryons and DM (solid lines) and comoving virial radius of the DM halo (dashed lines). Bottom panel: Similar to the Middle panel, but shows the physical spatial separation.

To evaluate how significant the spatial displacement between the baryon and DM over-densities is, we compare it to the virial radius of the DM-only nonlinear object. The virial radius of an object that collapses at redshift is approximately (Bryan & Norman, 1998):


where, for simplicity, we have suppressed a weak dependence on the cosmological constant, which causes a slight decrease of the virial radius at low redshift (this effect is included in the numerical calculations). The comoving virial radius is given by .

Over a substantial range of , modes with  Mpc (or ) have spatial offsets between their baryonic and DM linear over-density peaks larger than the DM virial radius (Fig. 1). If baryons are able to collapse at these redshifts, they will form isolated baryonic clumps. In contrast, modes with smaller values of (larger ) have separations that lie within the DM virial radius.

Figure 2.— Top panel: The vertical axis measures the rarity of fluctuations (“number of ’s”) needed to produce a baryonic clump of mass (horizontal axis) at a given redshift (different solid lines); is the critical over-density for collapse and is the variance of (Eq. 3). Results are for a stream velocity and . Bottom panel: Corresponding results for a different measure of fluctuation rarity, .

3. Likelihood of baryon-only clump formation

When the baryon over-density amplitude approaches unity, the perturbation becomes non-linear and we expect the baryons to collapse. To estimate how rare such collapsed objects are we calculate the variance of as a function of baryon clump mass ,

where is the spectrum of baryon fluctuations calculated using Equations (1)–(2), , and the scale is the radius of a top-hat window function, with . The value of is normalized to of the total matter. If is the critical linear over-density for collapse, the rarity of clumps of mass at redshift is determined by the number of fluctuations needed: (e.g. Barkana & Loeb, 2001; Naoz & Barkana, 2007). For simplicity we have set to its standard value of (but note that may vary with time, Naoz et al., 2006; Fialkov et al., 2012).

We estimate the mass of a baryon-only clump that forms from a mode with comoving wavenumber by


In the top panel of Figure 2 we show the number of ’s by which an over-density of a given baryon mass must fluctuate in order for it to collapse at a given redshift. As an example, a baryon fluctuation with a mass  M collapses at .

The quantity is a little misleading since is computed as an integral over wavenumber . While the window function cuts off the contribution of all larger than about , there is no cutoff at low (large masses). Thus has a contribution from fluctuations with masses much larger than . This contamination is usually not important, but it is serious when lies below the Jeans scale so that the collapse of baryon over-densities is suppressed by gas pressure. In this situation, may indicate collapse at the of interest even though what is collapsing is actually some larger mass that is unaffected by gas pressure. To illustrate this point, we show in the bottom panel of Figure 2 a different measure of the rarity of fluctuations, , which focuses on the local power in baryon density fluctuations at wavenumber . Note the clear signature of the Jeans cutoff at low masses and high redshifts.

It is important to note that the collapse of baryonic clumps considered here is very different from collapse in the baryon-only universe considered by Peebles & Dicke (1968). In our case, baryons feel the gravitational potential of the DM; their collapse is driven primarily by the DM. However, because of the stream velocity, by the time the baryons collapse, they are spatially offset from the corresponding DM halo.

Figure 3.— Top Left panel: Spatial separation between baryon and DM clumps at the redshift of collapse, normalized by the virial radius of the DM halo, for (solid line) and (dashed line) fluctuations, plotted against the baryon clump mass . Collapse is defined by the condition , for and . Note that clumps with baryon mass larger than few are separated from their DM halos by less than the virial radius (they are “inside” the DM halo) and vice versa for clumps with smaller masses (“outside”). Calculations are for , . Bottom Left panel: Estimated survival time of baryon clumps against spiral-in and merger through dynamical friction (“inside” the DM halo) and loss of stars through evaporation (“outside” the halo). Baryon clumps with masses in the range are potentially long-lived, especially if tidal forces from other nearby clumps unbind them from their DM halos. Right panels: Similar to Left panels, except that collapse is defined by . The results are generally similar.

4. Survival of baryonic clumps

When a baryonic perturbation collapses, the spatial separation of the baryon clump from its DM halo is , where is obtained by evaluating Equation (4) at the redshift of collapse. The bottom panel in Figure 1 shows for some cases of interest. The upper panels in Figure 3 show the same information in a different format. For each , or equivalently each baryon clump mass , we have identified from Figure 2 the redshift at which a () fluctuation collapses111Since GCs account for only a small fraction of the baryon content of the universe, they are clearly rare, hence we focus on and fluctuations, rather than the more common fluctuations., and computed the ratio of at this to the virial radius (Eq. 5) of the just-collapsed DM halo. For example, a fluctuation with  M collapses with a spatial offset exactly equal to the virial radius of its DM halo. Clumps with smaller mass (larger wavenumber) form outside the virial radius, and vice versa. In the case of a fluctuation, the transition mass is  M.

Baryon clumps that form inside the virial radius of the DM host will spiral down to the center by dynamical friction. To estimate the time scale we adopt the fitting formula presented in Boylan-Kolchin et al. (2008):


where is the eccentricity of the baryon clump’s orbit, which we set to . We add to the epoch of collapse and show the sum in Figure 3 (lower panels).

Baryon clumps that form outside the virial radius are not affected by dynamical friction. However, these clumps generally have lower masses and are liable to lose any stars that they form via evaporation. Following Gnedin et al. (2014) and Gieles et al. (2011), we estimate the evaporation timescale as:


Again, we plot the sum .

Figure 3 shows that baryon clumps with  M may be able to survive destruction by either dynamical friction or evaporation, and may survive as independent long-lived clumps. These may be the objects we see today as GCs. Furthermore, their parent DM halos, which collapsed with a deficit of baryons, may today be ultra-faint galaxies and dark satellite galaxies.

While the above proposal seems attractive, could baryon clumps in the favorable mass range fall into their parent DM halos simply through the gravitational pull of the latter. The free-fall time from a separation is very short:

(Note: a fluctuation of has a physical separation of  kpc from its DM host.) The actual timescale is a little longer since the baryon clump will begin with an outward velocity (Hubble flow). However, this changes the result by less than a factor of two.

Newly-formed baryon and DM clumps will certainly free-fall and merge if they evolve in isolation. More often, however, we expect the two objects to participate in the hierarchical growth of structure in the universe. At least some baryon clumps that collapse outside the virial radius of their DM parent halos will experience strong tidal forces from neighboring objects and will become gravitationally unbound from their DM halos. It remains to be seen if a sufficient number can survive by this mechanism to explain the GCs we observe in the current universe.222Note that a previous study by O’Leary & McQuinn (2012) focused on fluctuations, which collapse at redshifts well below (according to our analysis), whereas their numerical simulation was limited to . Furthermore, because of low statistical sampling, they were unable to follow the evolution of and fluctuations. This may explain why they did not see any baryon-only clumps such as we predict.

5. Discussion

We have used linear theory to study the growth of baryonic and DM density fluctuations in the universe in the presence of a stream velocity between the two components. We focused on the fact that a non-zero stream velocity causes a phase shift between the complex amplitudes of the baryonic and DM density fluctuations, which results in a spatial separation between the two density peaks. When the perturbations go non-linear and collapse, the baryon clump forms at a different spatial location than its DM counterpart. For baryon clump masses less than about few , the separation is larger than the virial radius of the DM halo. Assuming tidal forces from other nearby objects are able to unbind the baryon clump from its DM halo, the clump could survive to the present day as a DM-free gravitationally self-bound object. We suggest that this may be how GCs formed in the universe. The corresponding baryon-deficient DM halos would similarly survive as dark satellite galaxies or ultra-faint galaxies, as suggested previously by Naoz et al. (2013).

Note that, in this picture, the collapse of a baryon clump is not driven purely by the self-gravity of the baryons. The primary driving agent is still the DM perturbation, whose effect on the baryons is nearly the same as in the standard () model of structure formation, so long as the phase shift is less than about a radian. For perturbations that satisfy this condition, baryonic collapse is almost as effective as in the standard model, and gravity is able to overcome gas pressure; the only difference is that the baryon and DM clumps form in spatially distinct locations.

In Figure 1 we showed phase and spatial offsets between the baryon and DM perturbations for linear modes with different wavenumbers . As an example, at , a mode with  Mpc (which corresponds to a baryonic mass  M), has a phase shift , which translates to a comoving distance between the baryon and DM density peaks of 7.3 kpc. A fluctuation with this collapses at (Fig. 2), and the spatial offset between the baryonic and DM clumps is about times the virial radius of the DM halo. For a fluctuation, collapse occurs at an earlier redshift and the separation is even larger.

Using the virial radius of the DM halo as a benchmark spatial separation to discriminate between baryon clumps that survive and those that merge through dynamical friction, we obtain a natural upper cutoff to the mass of baryon-only clumps of a few  (Fig. 3). There is similarly a natural lower cutoff at around , which arises from a combination of several effects: survival against evaporation of stars (Fig. 3), Jeans cutoff due to gas pressure, and too large a phase shift between baryons and DM (which is a serious effect for a few hundred Mpc). The resulting mass range of long-lived clumps, , agrees well with the masses of present-day GCs.333Even if star formation had a relatively low efficiency in these pristine gas clumps (say ), a  M baryon clump would still have produced enough stars to make it an attractive candidate progenitor for present-day GCs. The topic of star formation in our baryon-only clumps is beyond the scope of this paper. Note that all the numerical estimates given in this paper are for and . The range extends to larger masses for ; is equivalent to , but has no stream velocity effect (within linear theory).

A great deal of work is needed before one can be confident that the baryon-only clumps discussed here can indeed form naturally and that they are sufficiently long-lived to be interesting. Within an analytical approach, one has to consider higher-order non-linear techniques. Alternatively, a numerical approach is possible, but it will require much better mass resolution than one usually finds in galaxy formation simulations. Assuming one successfully demonstrates that the baryon clumps proposed here form and are long-lived, one must test whether or not stars will form inside the collapsed clumps, and whether a sufficient number of stars will survive down to with the correct spatial and metallicity distribution. These are challenging problems.

Finally we note that, even for baryonic (and DM) masses much larger than those discussed here (arising from modes with ), there is a significant spatial offset between the baryons and DM at the time of collapse ( kpc, depending on the redshift, Fig. 1). These baryon and DM clumps will quickly merge through dynamical friction. However, in the process, the baryons will likely cause substantial stirring of the DM fluid in the inner regions of the halo. This might well produce a core-like structure in the DM density distribution, potentially explaining a number of puzzling observations (for a review, see de Blok, 2010). Also, as Figure 1 indicates, the spatial offset evolves significantly with the redshift of formation. This means that galaxies that formed at higher redshifts are likely to have smaller dark matter cores, and their baryonic components will also be more compact, compared to galaxies that formed more recently (e.g., Daddi et al., 2005; van Dokkum et al., 2008). These topics are beyond the scope of this paper.


We thank Avi Loeb, Rennan Barkana, Steve Furlanetto, Mark Vogelsberger and the anonymous referee for useful comments. SN is supported by NASA through an Einstein Post–doctoral Fellowship awarded by the Chandra X-ray Center, operated by the Smithsonian Astrophysical Observatory for NASA, contract PF2-130096. RN was supported in part by NSF grant AST1312651.


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