Spherical collapse in quintessence models

with zero speed of sound

[0.5cm]

Paolo Creminelli, Guido D’Amico, Jorge Noreña,

[.1cm] Leonardo Senatore and Filippo Vernizzi

[0.5cm]

Abdus Salam International Centre for Theoretical Physics

Strada Costiera 11, 34151, Trieste, Italy

SISSA, via Beirut 2-4, 34151, Trieste, Italy

INFN - Sezione di Trieste, 34151 Trieste, Italy

School of Sciences, Institute for Advanced Study,

Olden Lane, Princeton, NJ 08540, USA

CEA, IPhT, 91191 Gif-sur-Yvette cédex, France

CNRS, URA-2306, 91191 Gif-sur-Yvette cédex, France

Abstract

We study the spherical collapse model in the presence of quintessence with negligible speed of sound. This case is particularly motivated for as it is required by stability. As pressure gradients are negligible, quintessence follows dark matter during the collapse. The spherical overdensity behaves as a separate closed
FLRW universe, so that its evolution can be studied exactly. We derive the critical overdensity for collapse and we use the extended Press-Schechter theory to study how the clustering of quintessence affects the dark matter mass function. The effect is dominated by the modification of the linear dark matter growth function. A larger effect occurs on the total mass function, which includes the quintessence overdensities. Indeed, here quintessence constitutes a third component of virialized objects, together with baryons and dark matter, and contributes to the total halo mass by a fraction . This gives a distinctive modification of the total mass function at low redshift.

## 1 Introduction

One of the most important open questions for cosmology is whether dark energy is a dynamical component of the universe or a cosmological constant. A plethora of experiments and future observations are currently planned with the aim of improving our understanding of this question (see for instance [1] for a review). One of the most popular models of dynamical dark energy is quintessence, where the acceleration of the universe is driven by a scalar field with negative pressure. Standard quintessence is described by a minimally coupled canonical scalar field [2]. In this case, scalar fluctuations propagate at the speed of light and sound waves maintain quintessence homogeneous on scales smaller than the horizon scale [3]. Quintessence clustering takes place only on scales of order the Hubble radius, so that its effect is strongly limited by cosmic variance. However, quintessence modifies the growth evolution of dark matter through its different expansion history. Thus, future galaxy catalogs and weak lensing surveys will have a great potential in constraining the dark energy properties, in particular its equation of state, through a detailed study of the evolution of dark matter.

The study of quintessence perturbations – where by quintessence we indicate a dark energy sector described by a single scalar degree of freedom – has been recently revived in [4]. Here the most general theory of quintessence perturbations around a given background was derived using the tools developed in [5, 6], formulated in the context of an effective field theory. An important conclusion is that quintessence with an equation of state can be free from ghosts and gradient instabilities [5, 4]. In this regime, stability can be guaranteed by the presence of higher derivative operators [7, 5] with the requirement that the speed of sound of propagation is extremely small, . On cosmological scales, higher derivative operators are phenomenologically irrelevant and quintessence simply behaves as a perfect fluid with negative pressure but practically zero speed of sound [4]. Interestingly, this description applies also in the non-linear regime, i.e. when perturbations in the quintessence energy density become non-linear, as long as the effective theory remains valid.

This study motivates the possibility that quintessence has a practically zero speed of sound. Apart from these theoretical considerations, the fact that the speed of sound of quintessence may vanish opens up new observational consequences. Indeed, the absence of quintessence pressure gradients allows instabilities to develop on all scales, also on scales where dark matter perturbations become non-linear. Thus, we expect quintessence to modify the growth history of dark matter not only through its different background evolution but also by actively participating to the structure formation mechanism, in the linear and non-linear regime, and by contributing to the total mass of virialized halos.

In the linear regime, a series of articles have investigated the observational consequences of a clustering quintessence. In particular, they have studied the different impact of quintessence with or on the cosmic microwave background [8, 9, 10, 11], galaxy redshift surveys [12], large neutral hydrogen surveys [13], or on the cross-correlation of the integrated Sachs-Wolfe effect in the cosmic microwave background with large scale structures [14, 15]. On non-linear scales, the dependence of the dark matter clustering on the equation of state of a homogeneous quintessence, i.e. with , has been investigated using N-body simulations in a number of articles (see for instance [16] and references therein for a recent account).

A popular analytical approach to study non-linear clustering of dark matter without recurring to N-body simulations is the spherical collapse model [17]. In this approach, one studies the collapse of a spherical overdensity and determines its critical overdensity for collapse as a function of redshift. Combining this information with the extended Press-Schechter theory [18, 19] one can provide a statistical model for the formation of structures which allows to predict the abundance of virialized objects as a function of their mass. Although it fails to match the details of N-body simulations, this simple model works surprisingly well and can give useful insigths into the physics of structure formation. Improved models accounting for the complexity of the collapse exist in the literature and offer a better fit to numerical simulations. For instance, it was shown in [20] that a significant improvement can be obtained by considering an ellipsoidal collapse model. See also [21, 22] for recent theoretical developments and new improvements in the excursion set theory.

The spherical collapse can be generalized to include a cosmological constant (see for instance [23]) and quintessence with [24] (see also [25, 26] for subsequent applications). If quintessence propagates at the speed of light it does not cluster with dark matter but remains homogeneous. Indeed, pressure gradients contribute to maintain the same energy density of quintessence between the inner and outer part of the spherical overdensity. A study of the spherical collapse model with different quintessence potentials was performed in [27]. For a nice review on structure formation with homogeneous dark energy see also [28].

In this paper we study the spherical collapse model in the case of quintessence with zero speed of sound. This represents the natural counterpart of the opposite case .
Indeed, in both cases there are no characteristic length scales associated to the quintessence clustering^{1}^{1}1The characteristic length scale associated to the quintessence clustering is the sound horizon scale, i.e., . As mentioned above, this vanishes for so that clustering takes place on all scales. For we have , which is much larger than the scales associated to the spherical collapse. and the spherical collapse remains independent of the size of the object.
For this study we describe quintessence using the model developed in [4]. As explained, this description remains valid also when perturbations become non-linear and can thus be applied to the spherical collapse model.

As the spherical collapse occurs on length scales much smaller than the Hubble radius, we will describe it using a convenient coordinate system where the effect of the Hubble expansion can be treated as a small perturbation to the flat spacetime. In these “local” coordinates the description of the spherical collapse becomes extremely simple and the well-known cases can be easily extended to quintessence with . In this case pressure gradients are absent and quintessence follows dark matter during the collapse. Thus, in contrast with the non-clustering case where quintessence and dark matter are not comoving, for the collapsing region is described by an exact Friedmann-Lemaître-Robertson-Walker (FLRW) universe. Note that even though the energy density of quintessence develops inhomogeneities as long as the collapse proceeds, the pressure inside and outside the overdense region remains the same. Thus, as explained below, our model does not give the same description of clustering quintessence as that proposed by [27] and studied, for instance, in [29, 30, 31].

We will see that, besides quantitative differences with respect to the case – a different threshold for collapse and a different dark matter growth function – the case has a remarkable qualitatively new feature. Quintessence clusters together with dark matter and participates in the total mass of the virialized object, contributing to their gravitational potential.

The plan of the paper is the following. In section 2 we describe quintessence models with . In section 3 we study spherical collapse solutions first in known cases (dark matter only, CDM and quintessence) and then in the case of a clustering dark energy, . It turns out to be much simpler to describe these solutions in coordinates for which the metric is close to Minkowski around a point in space. The equation for the evolution of the spherical collapse are solved in section 4 and the threshold for collapse is calculated in the various cases. This leads to the calculation of the dark matter mass function in section 5. In section 6 we study the accretion of quintessence to the dark matter haloes and its effect on the total mass function. The contribution of quintessence to the mass may be distinguished from the dark matter and baryon component in cluster measurements, as we briefly discuss in section 7. Conclusions and future directions are discussed in section 8.

## 2 The model: quintessence with

Let us consider a -essence field described by the action [32, 33]

(1) |

The evolution equation of derived from this equation is

(2) |

where . The energy-momentum tensor of this field can be derived using

(3) |

and can be written in the perfect fluid form as [34]

(4) |

once we identify

(5) |

Let us initially neglect perturbations of the metric and assume a flat FLRW universe with metric . The energy-momentum tensor of the field can be perturbed around a given background solution corresponding to a background energy density and pressure,

(6) |

where . To describe perturbations it is useful to write the scalar field as [5, 4]

(7) |

where describes the difference between the uniform time and scalar field hypersurfaces.^{2}^{2}2We assume that is a monotonous function of .
Then, eq. (5) can be expanded linearly in using
and . This yields, for the perturbations of the energy density, pressure and velocity,

(8) |

where we have used eq. (6) and defined , where has the dimension of a mass.

To describe the evolution of perturbations we can expand the action (1) up to second order in as done in [4],

(9) |

The second term proportional to can be integrated by parts so that the part of the action linear in can be written using eq. (6) as , where is the Hubble rate. This part cancels due to the background equation of motion. Furthermore, we can manipulate the last two terms of the action integrating by parts the last term, proportional to , and making use of the background equation of motion, to rewrite them as . Finally, it is convenient to rewrite the coefficients left in this expansion in terms of the background energy density and pressure using eq. (6). This yields

(10) |

The coefficients of this quadratic action are completely specified by the background
quantities and . The latter is a function of time which we expect to vary slowly with a rate of order Hubble.^{3}^{3}3The time variation of is expected to be even slower than Hubble, i.e. of order , which is the typical time variation of . As shown in [6, 4], eq. (10) is the most general action describing quintessence in absence of operators with higher-order spatial derivatives. Note that this action is even more general than the starting Lagrangian (1) as it can be generically derived using only symmetry arguments [6].
An advantage of eq. (10) is that its coefficients are written in terms of observable quantities. Indeed, is proportional to , where is the equation of state of quintessence, which we will take here and in the following to be constant.
The parameter is related to the speed of sound of quintessence, given by

(11) |

As can be seen from this equation, absence of ghost – i.e., positiveness of the time kinetic-term in eq. (10) – implies that has the same sign as [35, 5, 4]. In particular, for one has , which signals the presence of gradient instabilities. As shown in [5, 4] stability can be guaranteed by the presence of higher derivative operators but requires that the speed of sound is extremely small, practically zero [4].

Regardless of the motivations expressed above on the stability of single field quintessence for , in the following we will be interested in considering the limit , which is obtained when . We will see that what turns out to be physically relevant are the density and pressure perturbations on surfaces of constant , i.e. of constant . These are the perturbations in the so-called velocity orthogonal gauge, and using eq. (8) they are given by

(12) |

Indeed, defined in eq. (10) can be written as [4]

(13) |

Thus, the pressure perturbation is suppressed with respect to the energy density perturbation by the smallness of the speed of sound. As we will see, in the limit this implies that pressure forces are negligible and quintessence follows geodesics, remaining comoving with the dark matter.

In the limit , the energy density perturbation on velocity orthogonal slicing becomes

(14) |

Note that since , the difference between and is negigible for small speed of sound, . All these conclusions hold independently of the value of , provided that the effective theory described by action (10) remains valid, i.e. for [36, 4]. In particular, they hold also when perturbations in the energy density of quintessence become non-linear, i.e., for .

Gravitational perturbations can be straightforwardly included in the action (10) as in [4]. As a warm-up exercise we will here, instead, study the evolution of quintessence in the spherical collapse solution. According to the spherical collapse model, the overdensity can be described as a closed FLRW universe with a scale factor which is different from the one of the background . This remains true also when we take into account quintessence with negligible speed of sound. Indeed, eq. (13) shows that there is no pressure difference between the inside and the outside of the overdense region. Therefore, inside the overdensity we can describe quintessence using eq. (9), where the time evolution of the metric is described by the scale factor and we thus replace by .

With this new action, the second term proportional to can be integrated by parts and the coefficients of the linear part of the action rewritten in terms of and using eq. (6). However, now the linear part of the action does not cancel but can be written, using the background equation of motion, as , where we have defined , with . We are thus left with a linear term in the action, due to the difference between the rates of expansion inside and outside the overdensity. After manipulations of the last two terms in eq. (9), similarly to what was done to derive eq. (10), the action inside the overdensity becomes^{4}^{4}4We will not include in the action the contribution to coming from the curvature of the closed FLRW universe. Indeed, as it is time independent, it does not affect our discussion.

(15) |

Using that , neglecting time variations of and discarding terms suppressed when (i.e., in the limit ) the equation of motion of derived from this action reads

(16) |

As expected, the quintessence perturbation induced by the overdensity is proportional to , i.e. it vanishes in the limit of the cosmological constant. Note that the source term on the right-hand side of this equation can be written as and is suppressed by the smallness of . This implies that, even for large overdensities, i.e. , variations of due to the gravitational potential well are extremely small, , inside the regime of validity of the effective theory. Furthermore, this also implies that the difference in between the homogeneous and closed FLRW solutions is also tiny, . Thus, the quintessential scalar field practically lies on the same point of its potential.

Equation (16) can be written, using eq. (14) (and ), as

(17) |

Note that, as is always negative, the sign of is the same as that of . Remarkably, this implies that for dark matter halos accrete negative energy from quintessence, as was noticed at linear level in [9]. Combining eq. (17) with the background continuity equation, , we obtain

(18) |

This equation describes the evolution of the energy density of quintessence with inside a spherical overdensity dominated by dark matter. Note that the pressure perturbation is absent, as it is suppressed by . Indeed, this equation differs from the description currently given in the literature for clustering dark energy. In particular, the analogue of this equation given in [27, 29] includes the pressure perturbation . Including the pressure perturbation leads to an incorrect description even in the linear regime, in contrast with eq. (18) which does match the linear theory for small overdensities.

In the following two sections we will make this analysis more complete and derive all the equations necessary to describe the spherical collapse with quintessence.

## 3 Spherical collapse in local coordinates

As we did in the former section, the spherical collapse is usually treated using FLRW coordinates, as in the simplest cases the overdensity evolves as an independent closed universe. This somewhat obscures a crucial simplification of the problem, i.e. that the collapse of dark matter haloes occurs on scales much smaller than
the Hubble radius. In this limit one can treat gravity as a small perturbation of Minkowski space.^{5}^{5}5For a recent use of this approximation in cosmology see [37]. As the dynamics of quintessence is not completely intuitive, we want to make use of a coordinate system where this simplification is explicit; this will also make the dynamics of the other cases of spherical collapse clearer. We thus choose a coordinate system
around a given point, such that the deviation of the metric from Minkowski is suppressed by
, where is the distance from the point, for any time.^{6}^{6}6In the spherically symmetric case, the range of validity of this approximation goes to zero close to the collapse singularity. However, this is not relevant because the
singularity is anyway an artifact of the spherical symmetry. In the
real case the curvature of space remains small and the halo reaches
virialization. Notice we do not want to limit the validity of
our approximation to times shorter than because this is also the typical
time-scale of the evolution of a dark matter halo. These requirements define the so called Fermi
coordinates.
Note also that we are not taking any Newtonian limit: as we are interested in quintessence we cannot neglect pressure as source of gravity.

A particular choice of Fermi coordinates are the so-called Fermi normal coordinates [38] where the deviation of the metric from Minkowski can be written as a Taylor expansion around the origin whose leading coefficients are components of the Riemann tensor. These are (with the convention of [39])

(19) | |||||

(20) | |||||

(21) |

Here we will be interested only in spherically symmetric solutions. As vanishes because of rotational symmetry, must be of order higher than . Thus we can neglect it in the following discussion. Furthermore, rotational symmetry implies that the corrections to will be proportional to while those to will be proportional either to or to . It is possible to make a redefinition of the radial coordinate such as to get rid of the term without affecting and at . Note that in such a way we are using Fermi coordinates which are not of the normal form. In this case the metric can be written in the Newtonian gauge (not to be confused with the cosmological perturbation theory Newtonian gauge) form as

(22) |

where and are proportional to . In this gauge the component of the Einstein equation gives

(23) |

The part of the Einstein equation proportional to the identity gives

(24) |

As the typical time scale is of order Hubble, the term is suppressed with respect to by and can therefore be neglected. Thus, using eq. (23) we obtain

(25) |

As a first step, let us show how one can use these coordinates to describe an unperturbed FLRW solution with non-vanishing curvature. In isotropic comoving coordinates this metric is written as

(26) |

where is the curvature parameter. With the change of coordinates and [37], with and evaluated at rather than at , one gets at first order in ,

(27) |

which is indeed of the Fermi form (22), where the corrections from flat spacetime are given by

(28) |

Let us now show that the metric (27) is a solution of the Einstein equations (23) and (25). In the coordinates , and are not space independent; however, their space dependence is suppressed by so that it can be neglected. With spherical symmetry, assuming regularity at the origin the two Einstein equations (23) and (25) are then solved by

(29) |

and

(30) |

Comparing these expressions with (28) we recover the two Friedmann equations, respectively,

(31) |

and

(32) |

Note also that the traceless part of the Einstein equation, , is trivially satisfied by the expressions above. Matter stays at fixed in the original FLRW coordinates; therefore it moves in the new coordinates as , i.e. with velocity . Finally, using this equality one can check that also the component of the Einstein equation is satisfied.

Let us now look at the dynamical equations for the fluid. The time component of the conservation of the energy-momentum tensor gives (see for example [40]) the continuity equation,

(33) |

This equation is the same as in Minkowski spacetime as the gravitational corrections only induce terms suppressed by . When the velocity is simply given by an unperturbed Hubble flow we obtain the standard conservation equation in expanding space, .

The spatial component of the conservation of the energy-momentum tensor gives the Euler equation,

(34) |

where we have assumed that . At leading order in gravitational perturbations enter only through the last term on the right-hand side of this equation. In the particular case of an isotropic and homogenous solution the first term on the right-hand side exactly cancels: as , the gradient of the pressure cancels with the term coming from its time dependence. This is not surprising as what matters in the Euler equation is the 4-dimensional gradient of pressure perpendicular to the fluid 4-velocity. In this case eq. (34) reduces to

(35) |

This equation is verified by the Hubble flow since we get

(36) |

which is clearly satisfied by the explicit expression for , eq. (28).

We can now use these local coordinates to describe the spherical collapse in various cases, starting from the simplest.

### Dark matter only

Let us take a spherically symmetric distribution of matter around the origin. As both the gravitational potentials and satisfy the Poisson equation, we do not need to know how the mass is radially distributed to solve for the gravitational background outside a given radius . We just need the total mass inside the radius . In particular (see figure 1) if inside a given radius , a distribution contains as much matter as the unperturbed cosmological solution, from the outside it will look exactly as the unperturbed background. This implies that we can smoothly glue this solution at to the cosmological background, and that the latter will not be affected by the gravitational collapse inside. This is of course a linearized version of Birkhoff’s theorem in General Relativity.

Conversely, the solution inside a given radius is not affected by what happens outside. In particular, if we assume a homogeneous initial condition inside a radius (with ), this central region will evolve as if these homogeneous initial conditions were extended outside, i.e. as a complete FLRW solution [17]. The central overdense region will remain exactly homogeneous, reaching maximum expansion and then collapsing.

Without further assumptions, the evolution of the layer does not enjoy particular simplifications and its evolution must be computed as a function of the initial profile. In any case, this is usually irrelevant as we are only interested in the fate of the region. If we assume that the layer is empty, the Poisson equations (23) and (25) give solutions for the potentials, like for a source localized at the origin. This is the linearization of the exact Schwarzschild solution.

### Dark matter and a cosmological constant

The considerations above also hold when we include a cosmological constant. Although and now solve different equations (because ), they are both of the Poisson form. Thus, we still have an unperturbed evolution outside if the total matter inside matches the background value. Assuming initial homogeneity, the central region will evolve like a complete FLRW universe [23]. Although now pressure does not vanish, it just comes from the cosmological constant which does not define a preferred frame and is therefore comoving with dark matter both inside and outside the overdensity.

### Non-clustering quintessence:

When quintessence has a speed of sound , it does not effectively cluster but it keeps on following the cosmological background solution, irrespective of the dark matter clustering [24].

As before, outside there is an unperturbed cosmological background. What is new now is that the central region does not behave as a complete FLRW solution, even if we start with a homogeneous overdensity. Indeed, quintessence and dark matter do not have a common velocity: while dark matter slows down and eventually starts collapsing, quintessence keeps following the external Hubble flow . Note that, on the other hand, in the cosmological constant case one cannot define a dark energy 4-velocity as its energy-momentum tensor is proportional to the metric. To study the evolution of the dark matter overdensity one must use the Euler equation (36). Here, what defines the velocity flow of dark matter is the effective “scale factor” , . This yields

(37) |

Using the explicit solution (30) for , this equation becomes [24]

(38) |

where we have separated the contribution of dark matter and quintessence to . Notice that, although this equation looks like one of the Friedmann equations, the dynamics of is not the same as for a FLRW universe. Indeed, evolves following the scale factor , while the quintessence follows the external scale factor . In a FLRW universe, from eq. (38) together with the continuity equation one can derive the first Friedmann equation, . Here, as the different components follow different scale factors, this is not longer possible and the first Friedmann equation does not hold.

### Clustering quintessence:

Let us now move to the subject of this paper. We want to show that in the limit of vanishing speed of sound quintessence remains comoving everywhere with dark matter. In particular, this implies that in the region quintessence follows dark matter in the collapse and the overdensity behaves as an exact FLRW solution so that, contrary to the case, also the first Friedmann equation holds. The fact that quintessence remains comoving with dark matter can be understood both by using the fluid equations or directly from the scalar field equation of motion.

In the fluid language the dynamics is described by the Euler equation (34). In general, in the presence of sizable pressure gradients a fluid does not remain comoving with dark matter, i.e. it does not follow geodesics. Since quintessence has a sizable pressure, the fact that it moves following geodesics may be unexpected but it is obtained in the limit . This can be easily seen by rewriting the Euler equation for quintessence in covariant form as

(39) |

where is the quintessence 4-velocity. When the right-hand side of this equation vanishes, the 4-velocity solves the geodesic equation. Notice that the pressure gradient is multiplied by the projector perpendicular to the fluid 4-velocity. This is the same as projecting on surfaces of constant and it is equivalent to a gradient of the velocity-orthogonal pressure perturbation that appears in equation (12), which involves only , and not . By eq. (13) this is negligible in the limit and thus the right-hand side of (39) vanishes.

This result is even clearer in the scalar field language. Taking the derivative of the equation defining the quantity in (1),

(40) |

and writing it in terms of the 4-velocity we have

(41) |

and therefore

(42) |

Equation (39) is recovered using eq. (5) and taking into account that and that the first term vanishes when multiplied by the projector orthogonal to . From this we clearly see that what matters is only the gradient of the pressure on const hypersufaces. This vanishes in the limit and thus we have geodesic motion. We stress that, although quintessence with follows geodesics, its dynamics is quite different from dark matter. Pressure does not accelerate the quintessence 4-velocity but it does affect the energy conservation equation (33). Moreover, quintessence does not enjoy a conserved current, while dark matter particle number is conserved; this is related to the absence of the shift symmetry in the scalar field Lagrangian (see for example [41]).

As discussed in section 2, the different dark matter evolution inside and outside the overdensity changes the quintessence solution by a very tiny amount : the quintessence field sits at the same position along its potential, , apart from negligible corrections. Notice that this was derived using two different Friedmann coordinate systems, one following dark matter inside the overdensity and one following the unperturbed Hubble flow outside. Thus, in reality we have two solutions and respectively. Once these two solutions are written in the same local coordinates (27), the solution for becomes for and for . This implies that in these coordinates has to jump in the layer between the two regions, by an amount , and that this jump is not suppressed by . One may expect that the scalar field would “react” to this gradient between the inside and the outside. However, this does not happen in the limit as the spatial kinetic term is very suppressed. Let us see this explicitly.

To study the scalar field equations in the local coordinates, one can start by writing the equations in Minkowski space and then check a posteriori that the deviation of the metric from flat space only gives relative corrections . The evolution equation for , eq. (2), reads in Minkowski space

(43) |

If we try a comoving solution of the form we end up with the standard FLRW equation for , inclusive of the friction term,

(44) |

Metric fluctuations give only a correction to this equation of order . As we discussed, the two homogeneous solutions for and are different so that we expect gradient terms to smooth out the initial top-hat profile. To estimate the thickness of the layer over which the smoothing takes place, we can study perturbations around a top-hat profile and require the spatial and time kinetic term of the perturbation in eq. (10) to be comparable,

(45) |

Using that , from this comparison we obtain . This makes perfect sense: our top-hat profiles are smoothed out over a distance comparable to the sound horizon.^{7}^{7}7It is straightforward to check that this estimate is not altered by the higher derivative operators that are required for stability when [5].

In conclusion, the solutions outside and inside the overdensity are exact FLRW with quintessence comoving with dark matter. Gradient terms will smooth out this solutions on scales of order of the sound horizon, which vanishes for . This discussion also tells us that taking will be correct only for objects which are much bigger than the sound horizon . In the opposite limit of an object which is much smaller than the sound horizon, one can treat quintessence as unperturbed as discussed above in the case. For example, if one is interested in objects larger than Mpc, one can neglect the speed of sound as long as .

## 4 Solving the spherical collapse

In this section we derive the equations for the spherical collapse of dark matter in the presence of quintessence with vanishing speed of sound and we compute their solutions numerically.

### The background universe

The background is described by a flat FLRW metric with scale factor satisfying the Friedmann equation,

(46) |

where and are the background energy density of dark matter and quintessence, respectively. For later purposes, we express and in terms of the fractional abundance of dark matter ,

(47) |

Dark matter redshifts with the expansion as the physical volume, , while the energy density of quintessence scales as . The dark matter contribution to the critical density can be written as a function of its value today, , and , the scale factor normalized to unity today (at ),

(48) |

This yields

(49) |

Equation (47) can be then rewritten as

(50) |

where the second equation follows from (49). Furthermore, rescaling the time variable by defining

(51) |

one can rewrite the Friedmann equation as

(52) |

The initial condition for can be imposed at some small initial time during matter dominance, . Then, eqs. (49) and (52) completely describe the background evolution of the metric and energy-momentum tensors.

### The linear evolution

Before studying the collapsing spherical overdensity we derive the evolution equations of perturbations of dark matter and quintessence in the linear regime. As we consider scales much smaller than the Hubble radius, the gauge dependence is not important. We will thus perturb the continuity and Euler equations in local coordinates, eqs. (33) and (35), adding small inhomogeneous perturbations and to the homogeneous energy density and Hubble flow velocity,

(53) |

Let us start from the dark matter. Perturbing at linear order eq. (33) with yields

(54) |

where we have specified that we are describing perturbations using local spatial coordinates . On the other hand, on the left-hand side of this equation one recognizes the time derivative at fixed comoving coordinates , i.e.,

(55) |

Indeed, here we are interested in describing the evolution of an overdensity of dark matter contained in a comoving volume. Thus, we describe and as a function of the comoving coordinates, which simply gives

(56) |

To close this equation we need the evolution of the dark matter peculiar velocity . This can be obtained by perturbing at linear order the Euler equation (35). Using comoving coordinates the perturbed Euler equation becomes

(57) |

where is the perturbation of the Newtonian potential,

(58) |

Equations (56)–(58) have been derived for instance in [42] in the context of Newtonian mechanics described with expanding coordinates, for a pressureless fluid in the presence of vacuum energy. Here the Poisson equation for is sourced by both dark matter and quintessence perturbations,

(59) |

where we have used that . The final step is to eliminate the peculiar velocity by subtracting the divergence of eq. (57) from the time derivative of eq. (56). With the Poisson equation (59) we obtain

(60) |

For quintessence we perturb the continuity equation (33) which gives, in comoving coordinates, using ,

(61) |

To eliminate the divergence of the peculiar velocity we can use eq. (56) taking quintessence to be comoving with dark matter. Indeed, as explained above, both the dark matter and quintessence follow geodesics and are dragged by the same potential well and the growing mode of their velocities is the same. Thus

(62) |

In matter dominance, when , the solution of this equation is [4]

(63) |

Note that the denominator on the right-hand side further suppresses the perturbation of quintessence with respect to the naive estimate.

In terms of the dimensionless variables and , respectively defined in eqs. (48) and (51), equations (60) and (62) rewritten as

(64) |

where we have used eq. (50), and

(65) |

The initial conditions are set in terms of the initial dark matter density contrast . In matter dominance i.e., , while the value of is fixed by through equation (63).

### The spherical overdensity

We now study the evolution of a spherical homogeneous overdensity of radius in a FLRW background that satisfies the Friedmann equation (46). We denote the energy densities of dark matter and quintessence inside the collapsing ball by and , respectively. Since dark matter is pressureless and since quintessence pressure perturbation is negligible, , we can take quintessence pressure to be the unperturbed one .

In local coordinates, the evolution of the scale factor is described by the Euler equation (36). Using the appropriate scale factor – i.e., instead of – and replacing the potential using eq. (30), the divergence of this equation can be written as

(66) |

(Note that for a non-clustering quintessence the equation for is the same with replaced by [24].)

For the evolution equations for and we use the continuity equation (33). Inside the ball this reads, for dark matter,

(67) |

whose solution is simply

(68) |

For dark energy eq. (33) becomes

(69) |

which can be rewritten in terms of the nonlinear density contrast as

(70) |

To solve eqs. (66) and (70) numerically it is convenient to use , the radius of the ball normalized to unity at the initial time,

(71) |

and change and to the dimensionless variables , . Using eq. (50), eq. (66) can be rewritten as

(72) |

where we have used eq. (68) and that in the linear regime, where the initial conditions are set, . Equation (70) yields

(73) |

As initial conditions we have by definition; the expansion rate of a collapsing sphere with dark matter only and in the linear regime can be written as [43]

(74) |

which fixes the first derivative of , . For the dark energy perturbation we use that is linear at early times, , and thus is fixed in terms of by eq. (63).

By solving numerically eq. (52) for the background evolution described by and plugging the result into the coupled eqs. (72) and (73), one can compute the evolution of as a function of time , from the initial time to the time of collapse . The evolution of is shown in figure 2 for four different models: CDM only, CDM, quintessence and quintessence in the cases