Relativistic virialization in the Spherical Collapse model for Einstein-de Sitter and $\mathbf{\Lambda}$CDM cosmologies

The question of how structures form in the Universe is a long-standing topic in theoretical cosmology and provides a lot of room for discussion. Since the fully non-linear regime cannot be accessed analytically, huge N-body simulations have been set up to describe structure formation by gravitationally interacting particles in an expanding background. However, these attempts are computationally costly and therefore perturbative approaches have been developed in order to keep the continuous character of general relativity and the FLRW model and make use of methods from fluid mechanics. A very simple semi-analytic model of this kind is the Spherical Collapse. A spherical, overdense patch evolves with the background expanding universe, slows down due to its self-gravity, turns-around and collapses. The object is stabilized by virialization which prevents it from collapsing into a singularity. Despite its simplicity and idealizations, this model gives a first insight into the formation of spherical halos at all mass scales. The underlying formalism dates back to Gunn and Gott in 1972 (see Gunn and Gott (1972)), but has been rediscovered and continuously extended in recent years (see Abramo et al. (2007); Bartelmann et al. (2006); Basilakos and Voglis (2007); Lee and Ng (2010); Manera and Mota (2006); Maor (2007); Maor and Lahav (2005); Mota and van de Bruck (2004); Nunes and Mota (2006); Pace et al. (2010); Wang and Steinhardt (1998); Wang (2006); Wintergerst and Pettorino (2010)).

In this work, we are going to use the results of Pace et al. (2010) (see Pace et al. (2010)) to investigate the process of virialization and try to find answers to some remaining questions in this context. The virial theorem provides a powerful tool to study systems in equilibrium, but in order to clarify its role in the framework of general relativity, a relativistic version is needed. After giving an overview of the classical concepts and the requirements of relativistic calculations in Sects. II and III, we will derive a relativistic version of the virial theorem based on the Tolman-Oppenheimer-Volkhoff equation (see Oppenheimer and Volkoff (1939); Tolman (1939) for the original references) in an Einstein-de Sitter and $\Lambda$CDM universe (see Sects. IV, V). In the following, this will be applied to the virialization equation in the spherical collapse model and a post-Newtonian expansion will be performed (see Sects. VII, VIII). The relativistically corrected results for the virial radius and virial overdensity will be discussed and leading order corrections are worked out in particular. (Sect. X). We will also dedicate a section to the problems occurring when this formalism is applied to general DE models and point out possible ideas to solve them (see Sect. IX). Throughout the paper, we will make use of natural units, i. e. $c=1$.

In the classical treatment of virialization, there are two major ingredients that have to be well-understood. First of all, structure formation in the present universe is highly non-linear on scales less than $10$ Mpc and an evolution equation for the spherical patch is needed that takes this non-linearity into account. Secondly, the virial theorem has to be combined with energy conservation to a virialization condition that allows determining the time when collapse stops and the system reaches an equilibrium. The key quantities, that are assigned to it, are the virial radius normalized to the turn-around one, $y_{\mathrm{vir}}=R_{\mathrm{vir}}/R_{\mathrm{ta}}$, and the virial overdensity with respect to the background, $\Delta_{V}=\rho(R_{\mathrm{vir}})/\rho_{\mathrm{b}}(a_{\mathrm{vir}})$. These are general functions of redshift and provide a characterization of the equilibrium state of the halo.

The non-linear evolution equation of a spherical overdensity of pressureless dark matter has already been treated in detail by many authors (see for example Abramo et al. (2007); Ohta et al. (2003); Pace et al. (2010)). The resulting equation

describes the non-linear evolution of a spherical, top-hat density contrast $\delta(a)=\frac{\rho-\rho_{b}}{\rho_{b}}$ with respect to the background dark matter density $\rho_{b}$. $E(a)$ contains all the dynamics of the background cosmological model and is related to the Hubble function via the expression $H(a)=H_{0}E(a)$. It will be important to express the virial overdensity $\Delta_{V}$ as a function of the turn-around one denoted by $\zeta$:

The virial radius, $y_{\mathrm{vir}}$, is obtained from the virialization equation in which the classical virial theorem $\overline{T}=\overline{\dfrac{R}{2}\dfrac{\partial U}{\partial R}}$ is combined with the assumption of energy conservation during collapse.¹¹1In the following, we will drop the bars over the time averaged quantities and implicitly assume time averaging. It should be mentioned that energy conservation is a very common assumption in the literature and it is not proven whether it can actually be applied. Maor & Lahav (2005), as well as Wang (2006), pointed out that a homogeneous DE component with $w\neq-1,\ -1/3$ clearly violates energy conservation between turn-around and collapse (see Maor (2007); Maor and Lahav (2005); Wang (2006))

In case of an Einstein-de Sitter universe, one simply obtains $y_{\mathrm{vir}}=\frac{1}{2}$ whereas the corresponding virial overdensity (evaluated at collapse scale factor) is given by $\Delta_{V}=18\pi^{2}\sim 178$ (see Wang (2006)). In case of $\Lambda$CDM and dynamical DE models, two major classical results have been proposed in the literature:

• •

  Wang & Steinhardt (1998) (WS afterwards) (see Wang and Steinhardt (1998))²²2Horellou & Berge (2005) (Horellou and Berge (2005)) have proposed a generalization due to dynamical DE models, but in the $\Lambda$CDM model both results agree. :

  $$y_{\mathrm{vir}}=\frac{1-\frac{\eta_{v}}{2}}{2+\eta_{t}-\frac{3}{2}\eta_{v}}$$

  (4)

• •

  Wang (2006) (PW afterwards) (see Wang (2006)):

  $$y_{\mathrm{vir}}=\frac{1-(1+3w)\frac{\eta_{t}}{2}}{2-(1+3w)\frac{\eta_{t}}{2}}\underset{(w=-1)}{=}\frac{1+\eta_{t}}{2+\eta_{t}}$$

  (5)

with the WS parameters $\eta_{v}$ and $\eta_{t}$, and the equation-of-state-parameter $w$ given by

The corresponding virial overdensities become functions of the collapse (virial) scale factor $a_{c}$ ($a_{\mathrm{vir}}$) and reach the EdS value asymptotically for small scale factors ($a_{c}<10^{-1}$) corresponding to the matter dominated era.³³3It has to be mentioned for completeness that this is only true for dynamical DE models that have negligible contribution in the matter dominated era. Counterexamples are early DE models (see results in Pace et al. (2010) and references therein).

Relativistic treatment of virialization in the same way as done in the classical case causes some trouble, because energy conservation is not global in general relativity. A second problem has been addressed by Komar (see Komar (1962, 1963)) stating that isolated bodies like a spherical halo can only be described exactly in asymptotically flat spacetimes which is generally not given in case of FLRW models. A promising way out of these problems is assuming that the scale of the halo is much smaller than the typical length scale of the background universe given by the Hubble radius. If the Killing vector field of the FLRW spacetime is considered with respect to this assumption, its time-like component greatly exceeds its spatial components, allowing to neglect the latter and at good approximation consider the Killing vector as time-like.

Fig. (1) illustrates that neighbouring Killing vectors are approximately parallel on the scale of the object, which means that the radial component of $K$ is extremely small on these scales and thus negligible. A detailed quantitative analysis of this issue is given in Appendix A.

From this argument, we can infer three major conclusions essential for the following considerations:

• •

  Since $r\ll R_{H}$, the solutions can be considered as nearly asymptotically flat such that isolated objects can be defined in GR and the halo mass $M$ is well-defined in the sense of a Komar integral.

• •

  Approximately, one can define the scale of the object as local and introduce a local coordinate frame which allows energy-momentum conservation, $\nabla_{\nu}T^{\mu\nu}=0$, to hold in that frame. Since the virialization equation, $T_{\mathrm{vir}}+U_{\mathrm{vir}}=U_{\mathrm{ta}}$, represents energy conservation, this condition is essential.

• •

  The approximately time-like Killing vector field $K$ of Eq. (83) allows static solutions of the field equations within an accuracy of $\sim 0.3\ \%$ (see Eq. (101)). In addition, the typical structure of the FLRW metric, especially the fact that a global time coordinate can be introduced and a space-like foliation with it, infers the orthogonality of $K$ to the underlying hypersurfaces such that the Frobenius condition⁴⁴4For $\omega$ being the corresponding dual vector to $K$, the Frobenius condition states that $\omega\wedge d\omega=0$ which turns out to be equivalent to $K$ being orthogonal to the spacelike sections spanned by suitably chosen spatial coordinates. (see Straumann (2004)) is naturally fulfilled. Thus, we can insert and compare static solutions at turn-around and virial redshift in the virialization equation $T_{\mathrm{vir}}+U_{\mathrm{vir}}=U_{\mathrm{ta}}$. Since the virial theorem will be applied to the final state which is static by definition (within the timescales we consider), this approximation holds in particular for the turn-around being a critical point, but not a static one in the exact treatment.

Relying on the assumptions of the previous section, we can set up a spherically symmetric, static spacetime with the metric:

In addition, we consider a system of two fluid components described by

in which

The equation of state of the cosmological constant fluid corresponds to $w=\frac{p_{\Lambda}}{\rho_{\Lambda}}=-1$.

Energy and momentum are locally conserved for the total system as well as for each component separately which means

Projecting the conservation equation of the (clustering) matter component onto the space perpendicular to the time direction leads to the relativistic Euler equation for the matter fluid

Working that out, one finds ⁵⁵5In the following sections, we define $\rho_{m}\equiv\rho$ and $p_{m}\equiv p$.

In case of a static configuration ($\nabla_{u}p=0$) and with the help of Eq. (6), Eq. (9) reduces to:

The field equations for the given metric are

Combining Eqs. (11) and (12) with (10) leads to

Eq. (15) is the TOV equation for the $\Lambda$CDM-model in case of homogeneous densities. For solving it, we assume the matter pressure to vanish at the boundary $p(r=R)=0$.⁶⁶6In the derivation of Eq. (1), the assumption of pressureless dark matter is a crucial argument. Nevertheless, for consistency with the TOV equation, we have to allow a pressure profile for the interior of the sphere. This issue will be discussed below.

This leads to

where

Inserting Eq. (15) and Eq. (16) into the hydrostatic equilibrium condition (Eq. (10)) and integrating with the boundary $a(R)=1/2\ln(1-AR^{2})$ (Schwarzschild-de Sitter solution) gives

and thus the full metric inside the sphere can be written as

which represents the metric of the interior Schwarzschild-de Sitter spacetime.

The well-known exterior Schwarzschild-de Sitter solution

matches continuously with Eq. (19) at $r=R$. In this particular case, it has to be mentioned that asymptotic flatness can only be reached approximately as discussed in Sect. (III). Since the scale of the halo is much smaller than the Hubble radius ($r/R_{H}\sim 10^{-3}$) we can still assume the object to be nearly isolated. We decided to embed the sphere into the Schwarzschild-de Sitter spacetime instead of an FLRW spacetime, because spacetime around the object can be assumed to be approximately static as well (due to the approximated time-like Killing vector field on these scales). In the ordinary Tolman-Oppenheimer-Volkhoff solution (see Oppenheimer and Volkoff (1939)), the perfect fluid sphere is embedded into vacuum described by the Schwarzschild solution. In order to be consistent with this approach, the generalization including a cosmological constant is embedded into the Schwarzschild-de Sitter spacetime. Nevertheless, it will turn out that the virial radius and overdensity can be predicted consistently with this approach, although a dark matter contribution outside the sphere is neglected (see Sect. (X) and the weak field limits in Sects. (VII) and (VIII)).

The pressure profile in Eq. (16) contains the radial dependence of the pressure in a sphere consisting of a cosmological constant fluid and collapsed dark matter embedded into a background Schwarzschild-de Sitter spacetime. When virialization starts, the system can be approximately assumed to be in equilibrium which means that it can really be described by Eq. (15)⁷⁷7The TOV-equation represents the equation of motion of the system in equilibrium.. In order to derive a virial theorem from that, one can take the first spatial moment which should usually lead to the virial theorem after time averaging. This means that small fluctuations around the equilibrium state are averaged out over time such that only time averaged quantities (energy expressions) are left in the virial theorem. Since the system is already in equilibrium and the TOV equation has no time-dependences, the time integral drops naturally and all quantities can be interpreted as time-averaged.

Eq. (15) is multiplied with $r$ and integrated (averaged) over the spacetime volume element (hence taking the spatial moment and time-averaging are performed in one step):

which becomes

Since all the quantities in the integral do not depend on time, the evaluation of the time integral cancels naturally and, while interpreting the given quantities as time-averaged, this becomes

Looking at the LHS of this equation in Euclidean space and performing a partial integration, we see that:

In consistency with the Euclidian case, we can propose that

Consequently, we define:

Inserting Eq. (16) and Eq. (6) into Eq. (28), we obtain

This is one version of a fully relativistic virial theorem for clustering dark matter in a $\Lambda$CDM-background model. Of course, other attempts exist in the literature to derive a relativistic virial theorem for several purposes. Chandrasekhar (Chandrasekhar (1965)) derived a post-Newtonian version of the tensor virial theorem by investigating the post-Newtonian hydrodynamic equations consistently with Einstein’s field equations. Bonazzola (1973) (Bonazzola (1973)) has proposed an integral identity consistent with general relativity in an asymptotically flat, stationary and axisymmetric spacetime. Vilain (1979) Vilain (1979) considers a scalar generalization of the virial theorem to general relativity which is valid for spherically symmetric, asymptotically flat spacetimes which has been successfully applied to stability studies of perfect fluid spheres. In addition, Vilain’s work allows to interprete the result of Bonnazzola (1973) geometrically in the spherical case. Gourgoulhon & Bonazzola (1994) (Bonazzola and Gourgoulhon (1994)) extended the work of 1973 to any stationary, asymptotically flat spacetime in general. Straumann (Straumann (2004)) proposes a virial expression in case of a spherically symmetric, static spacetime based on the Komar integral and asymptotic flatness. Except (Chandrasekhar (1965)), these remarkable results have in common that asymptotical flatness is a crucial assumption to the spacetime which is necessary in order to define isolated objects in the sense of a Komar integral (see Komar (1962, 1963)). We want to emphasize at this point that, strictly speaking, this condition has to be valid in our case as well. However, we make use of the fact that an isolated object can be approximately defined in the FLRW spacetime by assuming the scale of the halo to be much smaller than the corresponding Hubble radius.

The modified TOV solution can also be applied to find a relativistic expression for the gravitational potential energy of a spherical body. The derivation is inspired by the considerations of N. Straumann (see Straumann (2004)), but since it is quite technical, we refer to Appendix B and quote here only the final result:

with $\epsilon$ denoting the relativistic kinetic energy density which is defined in Eq. (112) (see Appendix B).

In case of small gravitational fields given for an object having a radius $r$ which is much larger than its Schwarzschild radius (this corresponds to $Ar^{2}\ll 1$), we can expand Eq. (30) and Eq. (31) to first order:

The kinetic energy will reduce to the specially-relativistic result if gravitational effects are neglected to zeroth order. The potential energy contains the Newtonian self-energy of a homogeneous sphere as a leading-order term. Thus, classical limits can be reproduced showing that Eq. (30) and Eq. (31) are consistently defined.

Assuming that energy conservation still holds during collapse, the virialization equation states

Let us now insert all derived energy expressions and perform a change of variable $r\rightarrow y\equiv r/R_{\mathrm{ta}}$. After simplifying the result, we end up with

with the definitions

Eq. (32) has to be solved numerically for $y_{\mathrm{vir}}$ at different redshifts (see Fig. 2 for the results). The turn-around radius, $R_{\mathrm{ta}}$, can be obtained by using the solution of Eq. (1) and Eq. (2):

Let us consider the classical limit with respect to two assumptions:

• •

  The sphere’s radius is much larger than its Schwarzschild radius $R_{S}=AR^{3}\ll R$, i.e. $AR^{2}\ll 1$.

• •

  The cosmological-constant density is much smaller than the dark matter density inside the sphere. Since $\rho_{\Lambda}$ is of the order of the critical density and $\rho=\Delta_{V}\rho_{cr}$ with $\Delta_{V}\sim 95-180$⁸⁸8See Pace et al. (2010) (Pace et al. (2010)) for their results in the $\Lambda$CDM case., this can be assumed safely in our case.

Expanding Eq. (33) to first order in $AR^{2}$ and $\rho_{\Lambda}/\rho$, and simplifying it, we end up with

Writing this in terms of the Wang-Steinhardt-parameters $\eta_{t}$ and $\eta_{v}$⁹⁹9See Wang & Steinhardt (1998) (Wang and Steinhardt (1998)) and Sect. II, this becomes

Eq. (38) can be solved approximately by

Thus, Eq. (33) reduces to the WS limit under the given assumptions.

The post-Newtonian expansion of the virialization equation can be done by simply performing a Taylor expansion of Eq. (33), however we choose a more elegant way including the equation of motion of the collapsing sphere.¹⁰¹⁰10This equation was first derived by Misner and Sharp (1964) (see Misner and Sharp (1964)).

We begin with a non-static, spherically symmetric spacetime described by

and use again the energy-momentum tensor of an ideal fluid with two components

The $\Lambda$-component satisfies an equation of state given by

Consider a comoving reference frame in which the four velocity $u$ has the components

The relativistic Euler equation for the (clustering) matter component $h_{\alpha\mu}\nabla_{\nu}T^{\mu\nu,(m)}=0$ states (in that frame)¹¹¹¹11In the following, we define again $\rho_{m}\equiv\rho$, $p_{m}\equiv p$.:

If we combine this relation with the field equations for the metric, we obtain the relativistic equation of motion for a spherically symmetric object (first derived by Misner and Sharp (1964) in Misner and Sharp (1964) and applied by Collins (1978) in Collins (1978))

In the presence of a cosmological constant $\Lambda$ with an equation of state like Eq. (42), this is slightly modified

In case of equilibrium, Eq. (44) and Eq. (45) reduce to the TOV equations with or without a cosmological constant:

If only small oscillations of the system around its equilibrium are considered, we can assume that $\dot{r}^{2}/c^{2}\ll 1$. Terms of this kind will be neglected in the following. After performing a Taylor-expansion up to the first post-Newtonian order $\mathcal{O}(\frac{1}{c^{2}})$ and inserting the zeroth-order expansion of the TOV-equation (Eq. (47)),

we arrive at

Since the derivation of the virial theorem requires an integration over the spacetime volume element, the metric has to be expanded as well. In our case, spacetime is described by the TOV metric given by

where

An expansion up to $\mathcal{O}(1/c^{2})$ leads to

The canonical volume form becomes

with $\mathrm{d}V_{T}=\mathrm{d}t\wedge\mathrm{d}V$ being the total volume element for a flat spacetime in spherical polar coordinates

In the following, we will also apply the definition of the canonical volume form of the spacelike 3-hypersurface $\Sigma$ described by the spatial coordinates

Taking the first spatial moment (multiplying with $r$ and integrating over the spatial volume) leads to the post-Newtonian version of Lagrange’s identity (see Collins (1978)):

In analogy to the classical case, we interpret¹²¹²12$I_{r}$ is defined to be the relativistic generalization of the classical moment of inertia. For a homogeneous sphere it is classically defined by $I=\frac{1}{2}\int_{V}{\rho r^{2}\mathrm{d}V}$. (see Collins (1978); Maor and Lahav (2005)

Using these definitions, Lagrange’s identity becomes the familiar expression

Dropping the corrections in $1/c^{2}$, the classical version of Lagrange’s identity is

Performing the time average will lead to the post-Newtonian virial theorem, because motions like oscillations around the equilibrium configuration are averaged out. Since we have to apply

the volume element of the averaged form changes $\mathrm{d}V_{\Sigma}=e^{b}\mathrm{d}V\rightarrow e^{a+b}\mathrm{d}V=\sqrt{-g}\mathrm{d}V$ while the time integration is performed.

Dropping all terms of $\mathcal{O}(1/c^{4})$, the post-Newtonian virial theorem is: