# Non-comoving baryons and cold dark matter in cosmic voids

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## Abstract

We examine the fully relativistic evolution of cosmic voids constituted by baryons and cold dark matter (CDM), represented by two non-comoving dust sources in a \(\varLambda \)CDM background. For this purpose, we consider numerical solutions of Einstein’s field equations in a fluid-flow representation adapted to spherical symmetry and multiple components. We present a simple example that explores the frame-dependence of the local expansion and the Hubble flow for this mixture of two dusts, revealing that the relative velocity between the sources yields a significantly different evolution in comparison with that of the two sources in a common 4-velocity (which reduces to a Lemaître–Tolman–Bondi model). In particular, significant modifications arise for the density contrast depth and void size, as well as in the amplitude of the surrounding over-densities. We show that an adequate model of a frame-dependent evolution that incorporates initial conditions from peculiar velocities and large-scale density contrast observations may contribute to understand the discrepancy between the local value of \(H_0\) and that inferred from the CMB.

## 1 Introduction

The generic term “Cosmic Voids” denotes \(\sim \) 10–120 Mpc sized round shaped low density regions surrounded by overdense filaments and walls, all of which conform the “cosmic web” of large-scale structure (baryons and CDM) revealed by observations and N-body simulations [1]. There is a large body of literature on cosmic voids, from seminal early work [2, 3] to recent extensive reviews [4, 5, 6, 7] and detailed catalogues (see a summary in [8]). As revealed by these reviews and references therein: (i) cosmic voids enclose only 15% of cosmic matter-energy (within the \(\varLambda \)CDM paradigm) but constitute about 77% of cosmic volume; (ii) they form from early negative density contrast perturbations; (iii) they roughly keep their rounded shape and (iv) their dynamics is relatively insensitive to considerations from baryon physics. This relatively simple and pristine dynamics renders them as ideal structure systems to improve the theoretical modeling of generic cosmological observations [9, 10, 11], and to assess several open problems in cosmology: the nature of dark matter and dark energy [12, 13, 14, 15, 16, 17, 18, 19, 20, 21], redshift space distortions [22, 23, 24, 25], Cosmic Microwave Background (CMB) properties [26, 27, 28, 29, 30], Baryonic Acoustic Oscillations (BAO) [31], alternative gravity theories [32, 33, 34, 35, 36, 37], local group kinematics and peculiar velocity fields [38, 39, 40, 41, 42, 43, 44], as well as theoretical issues such as gravitational entropy [45, 46].

The usual approach to study the nature and dynamics of cosmic voids is through Newtonian gravity [47] (see reviews [4, 5, 6, 7] and references cited therein for examples of studies based on analytic work, perturbations and N-body simulations). These studies have put forward various forms of “universal” density profiles [48, 49] that fit observations and catalogues. However, given the fact that cosmic voids are approximately spherical structures that tend to become more spherical as they evolve (see [50] for the first proof of this fact known as the “*bubble theorem*”^{1} and also [4, 5, 6, 7, 51, 52, 53] for further discussions and comparison with N-body simulations), it is also feasible to study them by means of spherically symmetric, exact and numerical solutions of Einstein’s equations. As examples of analytic and semi-analytic general relativistic studies, there are many based on Lemaître–Tolman–Bondi (LTB) dust models [54, 55, 56, 57], or the more general non-spherical (but quasi-spherical) Szekeres models [58, 59, 60, 61]. While numerical relativity techniques have already been applied in a cosmological context beyond spherical symmetry [62, 63, 64, 65, 66, 67], most relativistic numerical studies on cosmic voids still rely on metric-based techniques involving the Misner-Sharp mass function, and thus their validity is restricted to spherical symmetry [68, 69, 70, 71]. In particular, four relevant studies that specifically examine general relativistic void dynamics for spherical symmetry preceding our study are [72, 73, 74, 75].

In the present article we examine the fully general relativistic evolution of a spherically symmetric cosmic void, assuming as matter source a mixture of non-interactive baryons and CDM species, each evolving along a different 4-velocity. Specifically, we consider the evolution of a generic cosmic void suitably embedded in a \(\varLambda \)CDM background. Since CDM is the dominant clustering source, we assume a frame in which its 4-velocity is comoving, whereas the baryons evolve along a non-comoving 4-velocity that defines a non-trivial field of spacelike relative velocities with respect to the CDM frame. Consequently the 4-velocities of the two dust sources are related by a boost, and the energy-momentum tensor of the mixture (as described in the comoving frame) no longer has the form of a perfect fluid, but that of a complicated “fluid” energy-momentum tensor that contains effective pressures and energy flux terms associated with the relative velocity field.

In order to deal with the general relativistic dynamics for this energy-momentum tensor, we do not resort to the traditional metric based methods of [72, 73, 74, 75], but consider the system of first order partial differential equations (PDEs) provided by the “fluid-flow” (or “1 + 3”) representation of Einstein’s equations [76, 77, 78] in terms of evolution equations and constraints for the covariant quantities associated with the CDM comoving 4-velocity. These covariant variables are (i) kinematic: expansion scalar, 4-acceleration, shear tensor and relative velocity, all computed from the CDM frame; (ii) source terms: the total energy density, pressure and energy flux that arise from the relative velocity and (iii) the electric Weyl tensor. These equations (evolutions and constraints) must be supplemented by spacelike constraints.

The plan of the paper is as follows. In Sect. 2, we introduce a model for the evolution of a generic mixture of fluids in a spherically symmetric spacetime. This model is specialized to the case of two non-interacting dust-like fluids, namely CDM and baryons, in Sect. 3. We examine the numerical solutions of the resulting system of partial differential equations (PDEs) in a void formation scenario. Through representative numerical examples, we look at the influence of the relative baryon-CDM velocity on the evolution and present-day final structure. Our results are summarized and discussed in Sect. 4. Finally, we have included three appendices that complement the main text. “Appendix A” provides the general tensorial evolution equations of the \(1+3\) description; while in “Appendix B” and “Appendix C”, we present the dimensionless baryon-CDM system of PDEs and show its (single-fluid) LTB limit.

## 2 A mixture of multiple fluids in spherical symmetry

*N*,

*B*and

*Y*are functions of the radial and time coordinates. Notice that this metric contains as a particular case the Robertson–Walker line element, a solution recovered in our examples at scales larger than \(\sim 100\) Mpc at \(z=0\).

^{2}Hence the family of fundamental observers will evolve with comoving 4-velocity,

*p*and anisotropic pressure \(\pi _{\mu \nu }\) and energy flux \(q_\mu \) of the source given by the energy-momentum tensor, as well as the electric Weyl tensor \(E_{\mu \nu }=C_{\mu \alpha \nu \beta } \, u^{\alpha }u^{\beta }\) (the magnetic Weyl tensor identically vanishes). This system (displayed in “Appendix A”) is ideal for a numerical treatment in which all variables have a clear physical and geometric meaning. Since it is based on a 4-velocity flow it is fully covariant, and thus it is readily applicable to spacetimes that are not spherically symmetric.

*W*and \(\varSigma \) are scalar functions,

*i*” labels the components and \(v_{(i)}^2=g_{\mu \nu }v_{(i)}^\mu v_{(i)}^\nu \).

*p*, \(\pi ^{\mu \nu }\) and \(q^{\mu }\) are the energy density, isotropic and anisotropic pressures,

^{3}and the energy flow measured by the fundamental observers along \(u^\mu \). These components are determined by projecting the energy-momentum tensor parallel and orthogonal to \(u^\mu \) [78, 79]:

*i*as measured in the \(u^\mu \)-frame. In absence of non-gravitational interaction these energy-momentum tensors are separately conserved: \( J^\mu _{(i)}=0\) for all

*i*.

### 2.1 A mixture of non-interacting perfect fluids

*A*is defined in (5), \(Q=\sum \limits _i Q_{(i)}\) and \(\varPi =\sum \limits _i \varPi _{(i)}\). This system must be complemented by the following constraints:

*M*is the Misner-Sharp function

*W*through the constraint (17a) that allows us to eliminate the evolution equation (16f) for \(\dot{W}\) (and

*W*in (16e)), though, since

*M*is fully expressible through (19) in terms of the variables of (16), we do not need to use this function explicitly to integrate this system (we just eliminate

*M*with (19)).

^{4}Besides these constraints, we also need to supplement the system with the conservation equation for each fluid:

## 3 Void evolution from a mixture of two decoupled dusts

### 3.1 A numerical example of the two-component mixture

### 3.2 Evolution of density profiles

To look at the effect of the relative velocity on voids and wall formation we develop a code capable of handling test cases with given initial densities for each species, a given curvature profile, and a series of profiles for the relative velocity (\(V_{\mathrm{peak}}\)).

*V*, so that smaller voids result from initially negative values for the relative velocity. We also illustrate the evolution of the density contrast profiles for the specific case of \(V_{\mathrm{peak}}\sim 7\times 10^{-3}\) (corresponding to the red curves in Fig. 2), snapshots for different values of

*z*are displayed in Fig. 3.

Notice that as the evolution proceeds the density contrast at the surrounding wall increases, reaching probably a shell-crossing singularity. We interpret this as the onset of an intricate virialization process, a stage of structure formation that marks the limit of validity of the dust model, and that lies beyond the scope of this work (discussed elsewhere in the literature [83, 84]). Since our purpose is to look at the simultaneous evolution of the matter-energy components (CDM and baryons) within the void before the onset of virialization, we have chosen \(z=23\) as the initial time slice, simply because it is easier to set the initial conditions at this time than at, say, the linear regime of the last scattering time \(z\simeq 1100\), well before gravitational clustering becomes dynamically significant. However, these initial conditions are idealized but not fine-tuned or unrealistic, they simply correspond to a spherically symmetric realization of the generic spectrum of random CDM and baryon perturbations, characteristic of the linear regime at the last scattering surface \(z \simeq 1100\), which evolve to produce a void of the desired size.

### 3.3 Local expansion of the components

Figure 4 shows the difference between the two expansions \(H_{\hbox {\tiny {B}}}\) and \(H_{\mathrm{DM}}\) at \(z=0\) for the solutions whose density contrast are depicted with red and blue lines in Fig. 2, corresponding to \(V_{\mathrm{peak}}\) equal to \(7\times 10^{-3}\) (red lines) and \(5 \times 10^{-3}\) (blue lines). Note that this difference can be of the order of \( \mathrm {km/ (s \, Mpc)}\), around the maximum of the baryonic matter density (even larger differences are expected at times close to virialization). This estimation is roughly that of the discrepancy between the values of \(H_0\) reported by CMB and SNe observations [81, 85, 86], thus suggesting that considering a relative velocity between baryons and CDM may provide interesting clues to understand this issue (though this task lies beyond the scope of the present work).

### 3.4 The baryon-CDM relative velocity

*B*that generalizes the background scale factor. In a quasi-homogeneous perturbative regime \(B\sim a(t,r)\,r\) and thus \(B_{,r}>0\) should hold, while \(V_{,r}>0\) should also hold because relative velocities increase from the centre onwards as

*r*increases. Therefore the derivative of \((V^2)/(2B^2)\) should be positive and thus the right-hand side of the equation above negative. As a consequence, \(\dot{V}<0\) holds and relative velocities dilute asymptotically during cosmic expansion. However, this is not applicable to a non-perturbative regime where large gradients of the involved variables may occur and/or change signs, so that the relative velocity can be amplified by a local inhomogeneity.

*V*by looking at a definition of peculiar velocity often used in a perturbative approach: the difference between the local Hubble flow relative to the Hubble flow of the background, which can be estimated as \(v_{pec}=(H_{\tiny \mathrm{{local}}}-H_{\tiny \mathrm{{FLRW}}}) Y\) [87]. Then, once again neglecting the highest power of

*V*in Eq. (26), we get

## 4 Discussion and final remarks

We have considered the fully relativistic evolution of spherically symmetric cosmic voids made up of a mixture of two non-comoving dust components, identified as CDM and baryonic matter. Specifically, we looked at the effects of the baryon-CDM relative velocity on the void properties. We found that for baryons converging to the centre of the void, as seen from the CDM frame, the final density profile shows an effective reduction on the size of the void (see Fig. 2). On the other hand, if the baryon component is receding from the centre, the void presents a deeper (baryonic) underdensity, and the walls manifest a larger density contrast as illustrated in Figs. 2 and 3.

*background*(common) expansion, reached at large radii. Such differences could be interpreted as the velocity bias field, here evolved to non-linear stages. Figure 5 shows that such bias manifests most prominently at the peak of the density contrast (walls of the void).

The spherical void model we work with is qualitatively analogous to earlier models [72, 73, 74, 75]. As in these models, we obtain qualitatively analogous results that depict the expected streaming of baryons determined by the rapid void expansion, a characteristic also present in Newtonian models. However, the dynamical equations employed in the past are based on a numerical scheme constructed from the Misner-Sharp mass that is completely tied to spherical symmetry. As a contrast, the system of evolution equations and constraints here considered is based on covariant fluid-flow scalars that can be computed for any spacetime, regardless of the symmetry considerations.

Our approach to void dynamics could also represent an important improvement over the “silent models” of [67] that try to address this issue through “emergent” spatial curvature. Silent models (characterized by a non-rotating dust source with purely electric Weyl tensor) are theoretically handicapped by the conjecture stating that Einstein’s equations may not be integrable in general under the “silence” assumption [90, 91] (the models in [67] also neglect matching conditions among the different silent cells). By assuming dust sources with different (non-comoving) 4-velocities, the resulting models are based on similar physical assumptions but are no longer silent because of the non-trivial energy and momentum flux among the dust sources.

In conclusion, the fully relativistic evolution of baryons and CDM along different 4-velocity frames can provide important clues in understanding the observational tension in the estimation of the value of \(H_0\) from local observations and through interpretation of the Planck data. A concrete example is furnished by the study of the Hubble flow in the non-spherical models examined in [64], which tries to understand this tension, but did not consider different 4-velocities for the baryon and CDM components. This work could be improved by allowing for a non-comoving baryon 4-velocity that would provide more degrees of freedom as we have done in this paper. Likewise the multiple fluid approach can provide important corrections to the usual study of the process of formation and growing of large-scale structure in the universe. Finally, we emphasize the fact that the system of evolution equations and constraints used in our numerical modeling has been constructed with covariant fluid flow variables, and thus it is readily applicable (under certain restrictions) to examine self-gravitating systems that are much less idealized that those under the assumption of spherical symmetry.

## Footnotes

- 1.
The “Bubble Theorem” states that an isolated underdensity (void) tends to evolve into a spherical shape, explicitly: “

*as the void becomes bigger, its asphericities will tend to disappear*” [50]. - 2.
This choice, however, is arbitrary and one could also choose the fundamental observers moving with the baryon component or even a frame in which the total momentum flux density vanishes \(q^\mu =0\).

- 3.
In this setup, the cosmological constant is implicitly considered by the substitution \(\rho \rightarrow \rho + \varLambda \) and \({\mathrm {p}}\rightarrow {\mathrm {p}}-\varLambda \).

- 4.
To work beyond spherical symmetry we can easily do away with the usage of the Misner-Sharp mass function and work with the electric and/or magnetic Weyl tensor.

## Notes

### Acknowledgements

The authors acknowledge support from research Grants SEP-CONACYT 282569 and 239639. I.D.G. also acknowledges valuable discussions with S. Fromenteau.

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