How empty are the voids?
Anton N. Baushev




bau]A.N. Baushev
baushev@gmail.com
[bau]Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research,
141980 Dubna, Moscow Region, Russia
 The department of Physics, Helsinki University, Helsinki, Finland




§ ABSTRACT 
We find an analytical solution for the minimal matter density of a void, its central density.  It turns out that the voids are not so empty: most of the voids have the central underdensity Δ_c ∼ -50% (which means that the matter density in their centers is only two times lower than in the Universe on average). For small voids (of radius R_0≃ 5-10 Mpc), the underdensity can be significantly greater, but the number of voids decreases rapidly with increasing of |Δ_c| over 50%, and voids with Δ_c < -80% are practically absent. The large voids (R_0≥ 40 Mpc) always have |Δ_c| < 50%.



voidcosmologylarge scale structure of the Universedark matter
 98.65.Dx98.65.-r98.80.-k98.62.Py


How empty are the voids?
    [
    September 9, 2024
========================






§ INTRODUCTION

Voids are vast (∼ 10-100 Mpc) areas of the large-scale structure of the Universe that contain no galaxy clusters and almost no bright galaxies. The voids are elliptical and occupy a significant part of the Universe volume. It is still not quite clear what is the ratio between the matter density inside voids and of the Universe on average. On the one hand, the density of bright galaxies is tens times lower in the void than in the Universe on average, and it implies that the voids are really quite empty. On the other hand, one may easily observe only the stellar component of sufficiently bright galaxies in the voids. There are reasons to believe that the structure formation in the voids is suppressed <cit.>, and a significant quantity of matter forms there a massive diffuse component. The voids may contain almost arbitrary quantity of dark matter (hereafter DM),   both as the homogeneous component and as dark halos. The baryon matter in form of dwarf galaxies, hot gas, ultradiffuse galaxies (UDGs) etc. is also hardly detectable in the voids. Therefore, the real emptiness of the voids is hard to be measured. 

N-body simulations support the opinion that the voids are very underdense: they suggest that the matter density in the void center is almost ten times lower than of the Universe on average (see the Discussion for details). However, the simulations may suffer from numerical effects, and it would be very useful to check them by analytical calculation. The analytical investigation of void formation have a long history; here we mention only a few remarkable works from the very extensive literature: <cit.>. As a rule, calculations were carried out under the assumption of spherical symmetry of the system; in addition, in early works the cosmological constant was ignored for obvious reasons. The spherical symmetry allows to find a solution for any initial conditions. However, the result depends on the shape of the initial perturbation, which is unknown. In addition, the spherical symmetry is obviously violated when the voids enter the nonlinear regime and begin to "collide" with each other, forming a flat wall between them.

We solve a much simpler problem in this paper. We want to estimate the minimal density in a void, the density at its center. As we show, the density profile of voids is very flat at the center, and thus the region with the density close to the minimal one may occupy a significant part of the void volume. Contrary to the full density profile, the minimal void density can be found analytically, and it depends only on the amplitude of the initial perturbation, which can be found assuming that the primary perturbations were Gaussian. We use the analytical method offered in <cit.>, correcting a minor calculation error that was made there and generalizing the result to the case of a non-spherical void.

The structure of the paper is the following: in the second section we calculate the central density of a void in the spherically-symmetric case, in the third — we consider the case of a non-spherical void and show that the central density should be the same as in the spherically-symmetric case, in the Discussion section we estimate the underdensities of real voids and discuss them.



§ CALCULATIONS

Now we find the central density of a spherically-symmetric void. First of all, we need to set the initial conditions. We choose a redshift z_1 deeply on the matter-dominated stage of the Universe, so that we may already neglect the radiation term, but the perturbation (which later transforms into the void) is still small, say, z_1=10. We will denote the values of variables at this moment by subscript '1'. Let the characteristic size of the perturbation at this moment be λ_1.

The future void contains a lot of substructures, and therefore the real density profile of the underdensity at z=z_1 is quite complex. However, the substructures do not affect much the void formation, and we may get rid of them by 'averaging' the velocity and density profiles (for instance, with the help of a Gaussian filter). Hereafter we will consider this 'softened' profile instead of the real one.

We denote the scale factor of the Universe by a(t), and the Hubble constant - by H≡daa dt. The redshift is bound with a by trivial equation z+1=a_0/a. The subscript '0' corresponds to the present-date values of the quantities. We choose the point of the density minimum in the center of the void as the origin of coordinates and surround it by a sphere of radius R_1≪λ_1, which expands with the same rate as the Universe: R=R_1 aa_1. Since the size of the perturbation λ changes alike, ratio R/λ remains constant. We need to underline that the sphere is not quite comoving with the substance: the void center expands slightly faster, than the Universe on average, and therefore the matter crosses the sphere outwards.

 The outer regions of the void do not influence the matter inside R, since a spherically symmetric layer does not create gravitational field inside it (the Birkhoff's theorem). Thus, though a void is always surrounded by huge masses of matter, it hardly can create a strong gravitational field in the void center. As we show below, this conclusion is very probably valid even for a non-spherical void. The only significant perturbations inside R may be created by other nearby elements of the cosmic web, outside of the void and its walls, for instance, by the walls of a nearby void, since these structures are distributed asymmetrically with respect to the void that we consider. However, this tidal perturbations are small on the scales of R (they are ∼ (R/λ)^3), and we neglect them.

 The matter density inside R, q_m, does not depend on coordinates (since r=0 is an extremum of ρ(r)), and the matter velocity v⃗∝r⃗ (because of the spherical symmetry). Moreover, the only component that has cosmologically significant (p∼ρ) pressure after z_1 is the dark energy, but its pressure is everywhere constant. So the system has no pressure gradients. 
 
 We conclude that the evolution of the Universe inside R may be described by usual Friedman's equation <cit.>, though its parameters should slightly differ from those of the undisturbed Universe. We denote the critical, matter, dark energy, and curvature densities of the undisturbed Universe by ρ_c, ρ_m, ρ_Λ, ρ_a, of the 'universe' inside R — by q_c, q_m, q_Λ, q_a, respectively. We'd like to remind that the curvature density ρ_a≡3c^28π Gya^2, where y=-1, 1, 0 for a closed, open, and flat Friedmann's universe, respectively.  We denote the scale of the 'universe' inside R by b(t), and the Hubble constant — by Q≡dbb dt. The critical densities are 

    ρ_c=3H^2/8π G;       q_c=3Q^2/8π G

We assume the standard ΛCDM cosmological model, i.e., the dark energy is a non-zero cosmological constant (ρ_Λ=q_Λ=  const), the Universe is flat (ρ_a (but not q_a!) is equal to zero). Since we consider the Universe at z≤ 10, we neglect the radiation term, and therefore Ω_m+Ω_Λ≡ρ_mρ_c+ρ_Λρ_c=1. The present-day ratios Ω_m,0=0.315, Ω_Λ,0=0.685, H_0=73 (km/s) Mpc^-1 <cit.>. It is convenient to introduce h so that H_0=h· 100 (km/s) Mpc^-1. Thus h=0.73.  

Since the Universe is flat, a(t) is defined up to an arbitrary factor (b(t) is the radius of the 3-space of the 'universe' inside R, and so it is defined uniquely). Thus we always may choose a_1=b_1. The Friedman's equations for the Universe and the central region of the void may be written as

    H^2=8π G/3[ρ_Λ+ρ_m,1(a_1a)^3],
    
    Q^2=8π G/3[q_Λ+q_m,1(b_1b)^3+q_a,1(b_1b)^2].

The values of ρ_Λ, q_Λ, and ρ_m,1 can be easily expressed through Ω_m,0 and ρ_c,0=3H_0^2/(8π G):

    ρ_m,1=(a_0/a_1)^3 ρ_c,0Ω_m,0;   ρ_Λ = q_Λ=ρ_c,0 (1-Ω_m,0).

At z=z_1 the future void is only a shallow underdensity |δ_1|=|Δρ_m/ρ_m,1|≪ 1. The density contrast on the matter-dominated stage is proportional to a(t) <cit.>, and it is more convenient to characterize the underdensity by 

    ℵ≡ (z+1) δ=(a_0/a) δ;  ℵ= (a_0/a_1) δ_1,

since ℵ is constant on the matter-dominated stage, and it does not depend on the choice of z_1. Thus 

    q_m,1=(1-δ_1)(a_0/a_1)^3 ρ_c,0Ω_m,0,
    
    q_m=(1-δ(t))(a_0/a)^3 ρ_c,0Ω_m,0

As we have already mentioned, the sphere R is not quite comoving with the substance: the void center expands slightly faster, than the Universe on average, and therefore the matter crosses the sphere outwards with some speed v(t). The speed of sphere R with respect to the origin of coordinates at z=z_1 is equal to H_1 R_1, while the matter speed is equal to Q_1 R_1. Thus 

    Q_1 R_1= H_1 R_1 +v_1.

The speed can be easily calculated from the continuity consideration. The flux of matter through sphere R at z=z_1 is (we substitute eqn. (<ref>) for q_m,1)

    dM/dt=-4π R^2_1 v_1 q_m,1= -4π R^2_1 v_1 (1-δ_1)(a_0/a_1)^3 ρ_c,0Ω_m,0.

On the other hand,

    M(t)=4/3π R^3_1 (a/a_1)^3 q_m=
    
    =4/3π R^3_1 (1-δ(t))(a_0/a_1)^3 ρ_c,0Ω_m,0.

Here we substitute eqn. (<ref>) for q_m. Only multiplier (1-δ(t)) depends on time in the obtained equation. We may calculate the time derivative of it with the help of eqn. (<ref>): since ℵ is constant on the matter-dominated stage, dℵ=0 and from (<ref>) it follows that  

    a(t) dδ(t)=δ(t) d a(t);  dδ(t)/dt=δ(t)da/adt=δ(t) H

Then we obtain from (<ref>):

    dM/dt
    =δ(t) H(t) 4/3π R^3_1 (a_0/a_1)^3 ρ_c,0Ω_m,0.

Equating equations (<ref>) and (<ref>) at z=z_1, we obtain

    v_1=1/3δ_1 R_1 H_1

As we can see, v_1/R_1≪ H_1, the velocity perturbation is small with respect to the Hubble expansion. Then we may rewrite (<ref>) as Q_1 = (1+δ_1/3) H_1. Squaring it and neglecting the small δ^2_1 term, we obtain Q^2_1 = (1+2δ_1/3) H^2_1. Substituting here equations (<ref>) and (<ref>) at z=z_1, we get:

    (1+2/3δ_1)(ρ_Λ+ρ_m,1)=q_Λ+q_m,1+q_a,1.

In accordance with (<ref>) and (<ref>), ρ_Λ=q_Λ, q_m,1=(1-δ_1)ρ_m,1, and we may convert (<ref>) into

    q_a,1=2/3δ_1ρ_Λ+5/3δ_1 ρ_m,1.

However, ρ_Λ is ∼ 3 orders of magnitude smaller, than ρ_m,1, and we may neglect the first term. Thus

    q_a,1=5/3δ_1 ρ_m,1=5/3δ_1 (a_0/a_1)^3 ρ_c,0Ω_m,0=
    
    =5/3ℵ(a_0/a_1)^2 ρ_c,0Ω_m,0.

We used (<ref>) in the last transformation. Now we substitute equations (<ref>), (<ref>), and (<ref>) for q_Λ, q_m,1, and q_a,1 into (<ref>). We reduce b_1 and use equation (<ref>) in the form of 8π Gρ_c,0=3H_0^2:

    Q^2=H^2_0[Ω_Λ,0+Ω_m,0(a_0b)^3
    +5/3ℵ Ω_m,0(a_0b)^2].

We also neglected small term δ_1 ((1-δ_1)≃ 1) in equation (<ref>) for q_m,1. This equation[In work <cit.>, a minor mathematical error was made: the numerical coefficient before the last term under the root in expression (<ref>) for Q was 3 instead of the correct value 5/3, as in this work.] for Q(t) defines the evolution of the region inside R. In order to finish our calculation, we use a well-known trick <cit.>: the age of the Universe, as well as the age of the 'universe' inside R, may be found from the Hubble constants:  

    t_U=∫ dt=∫_0^a_0da/aH,    t_R=∫_0^b_0db/bQ,

where H(a) and Q(b) are given by equations (<ref>) and (<ref>), respectively. But, of course, t_U should be equal to t_R. 

It is essential that b_0 a_0: the void center expands much stronger than the Universe on average. Let us denote k=b_0/a_0. It is apparent that the central density of a void is q_m,0=ρ_m,0/k^3. As is customary, we will characterize voids by their present-day central underdensity

    Δ_c≡q_m,0-ρ_m,0/ρ_m,0=1/k^3-1

For voids Δ_c is always negative. Thus, our problem is reduced to the determination of k. We need to calculate both the integrals in (<ref>) and equate them. The first integral may be calculated analytically

    t_U=2/3H_0(1/√(Ω_m,0))√(1-Ω_m,0)

The second one may be transformed into (we use equation (<ref>) for H_0)

    t_R=1/H_0∫^k_0 √(x) dx√(x^3 Ω_Λ,0+Ω_m,0(1+5/3ℵ x)),

Equating the age of the Universe calculated in two different ways, t_U=t_R, we obtain the final equation that bounds 'the factor of additional expansion' of the void, k, with the amplitude of initial perturbation, ℵ, where Ω_m,0 is the only parameter (we take into account that Ω_m,0+Ω_Λ,0=1):

    2/3(1/√(Ω_m,0))√(1-Ω_m,0)=∫^k_0 √(x) dx√(x^3+Ω_m,0(1+5/3ℵ x - x^3)).

This equation (together with equation (<ref>)) solves the problem of the void central density in the spherically-symmetric case. It does not imply that the final void is in linear regime, being correct for arbitrarily strong final perturbations. As it should be, equation (<ref>) does not depend on the choice of the initial moment, which we denoted by '1': the central density is totally defined by the amplitude of initial perturbation, ℵ, and Ω_m,0.





§ THE CASE OF AN ELLIPSOIDAL VOID


Thus, the central density of a void may be found analytically in the spherically-symmetric case. However, this case is degenerate, and it would be more reasonable to consider the general case of a small initial perturbation having an ellipsoidal shape. Will the central density (<ref>) be a reasonable estimation in this case?

First of all, the statement that the outer regions of the void and huge masses of matter surrounding it hardly can create a strong gravitational field in the void center is very probably correct for a general ellipsoidal void as well. Indeed, the density profile of an ellipsoidal void looks like (we take into account that the average density of the Universe is equal to the critical one):

    ρ=ρ_c - ϱ(x^2/e^2_1+y^2/e^2_2+z^2/e^2_3).

This profile may be divided into a set of thin homoeoids of constant densities (a homoeoid is a region bounded by two concentric, similar ellipsoids). A homoeoid homogeneously filled with matter does not create gravitational field inside itself, and so it does not differ from spherical layers in this sense. Thus, if we could neglect the influence of outer layers in the case of a spherical void, we may do the same in the case of an ellipsoidal one.

Second, let us consider how the asymmetry of distribution (<ref>) evolves with time. In the case of an overdensity the answer is well-known and can be based on a simple qualitative consideration <cit.>. Until the perturbation is linear, its form changes little <cit.>. Apparently, this result is valid for an underdensity as well. As an ellipsoidal overdensity reaches the nonlinear regime, its asymmetry rapidly grows. Indeed, in this case the second term in (<ref>) is positive. Let us suppose that e_1 is the shortest length in (<ref>), i.e., the ellipsoid is compressed the most along the x axis. Then the gravitational acceleration towards the center is also the largest along this axis. All the velocity field on the nonlinear stage of the structure is determined by the gravitational field of the forming structure. Thus, not only the gravitational attraction is the strongest along the shortest axis of the ellipsoid, but the velocity and acceleration of the contraction are the highest. As a result, the asymmetry rapidly grows with time, and finally the ellipsoid transforms into a Zeldovich pancake <cit.>.



It is easy to understand by the same qualitative consideration that the situation is opposite in the case of a void: the asymmetry in the center decreases with time. Indeed, in this case the second term in (<ref>) is negative: the underdensity creates a sort of 'gravitational repulsion' with respect to the case of undisturbed Universe. The repulsion is the strongest along the shortest axis of the ellipsoid: as a result, the ellipsoid expands faster in this direction, and the asymmetry decreases with time. Thus, even in the case of an ellipsoidal void the central part of it tends to spherical symmetry (e_1=e_2=e_3). The process of spherization is relatively slow when the density contrast of the void is small, but gets very fast on the nonlinear stage of the void formation.

Third, let us consider the central density evolution in the case of an ellipsoidal void. The evolution is determined by the continuity equation

    ρ̇/ρ= - v⃗.

Thus the density evolution is totally determined by v⃗, and we need to find how it depends on the ellipsoid parameters. The density (<ref>) in the void center depends only on time, since this is an extremum point. The gravitational potential in the void center looks like <cit.> 

    ϕ=ϕ_u + 1/2(ϕ_xx x^2+ϕ_yy y^2+ϕ_zz z^2),

where ϕ_u is the gravitational potential of the undisturbed Universe. Of course, v⃗ created by ϕ_u coincides with that in the spherically-symmetric case. Now we need to find v⃗ created by the second term in (<ref>), which is actually the perturbation of the velocity field created by the void. Coefficients ϕ_xx, ϕ_yy, ϕ_zz are determined by the central density and the ratios between e_1, e_2, and e_3 <cit.> and depend only on time. It follows from Δϕ = 4π G ρ that 

    ϕ_xx +ϕ_yy +ϕ_zz = 4π G ϱ(0)= 4π G (ρ-ρ_c).

Thus, though each coefficient ϕ_ii depends on the void asymmetry, their sum depends only on the density. 

On the stage when the perturbation is linear we may easily calculate the velocity field created by it. For instance, if we consider the x component, dv_x/dt=xϕ_xx(t). At some previous moment of time, τ,

    dv_x/dτ=xa(τ)/a(t)ϕ_xx(τ).

Here we neglect the difference between the scale factor inside and outside the void, which is a reasonable approximation on the linear stage: the particle gets some additional velocity in the perturbation, but the time is too short to shift it significantly with respect to the undisturbed Universe. This approximation fails only when the perturbation gets nonlinear <cit.>. From (<ref>) we have

    v_i=i∫_0^t a(τ)/a(t)ϕ_ii(τ) dτ,

where i=x,y,z.  Now we may calculate the velocity divergence:

    dv_x/dx+dv_y/dy+dv_z/dz=∫_0^t a(τ)/a(t)(ϕ_xx+ϕ_yy+ϕ_zz)dτ,

and with the help of equation (<ref>) we finally obtain

    v⃗=4π G∫_0^t a(τ)/a(t)(ρ(τ)-ρ_c(τ))dτ.

As we can see from equation (<ref>), the velocity distribution is anisotropic in the case of an ellipsoidal perturbation: coefficients ϕ_xx, ϕ_yy, ϕ_zz are essentially different. The velocity components are not completely independent, however: they are implicitly bound by (<ref>), and as a result the velocity divergence (<ref>) does not depend on the void ellipticity at all. Thus, equation (<ref>) shows that the central density evolution of an ellipsoidal void is exactly the same as in the spherically-symmetric case, at least, on the linear stage of the void formation.

To summarize this section: the statement that the outer regions of the void hardly can create a strong gravitational field in the void center is very probably correct for a general ellipsoidal void as well. At the linear stage the central density evolution of an ellipsoidal void coincides with that of the spherically-symmetric one with the same initial central density. The nonlinear stage requires further consideration, but the ellipticity itself rapidly decreases on this stage, making the density profile spherically symmetric. Thus, equations (<ref>, <ref>) for the central density in the spherically-symmetric case give, at least, a very good estimation of the central density even for an elliptical void. 



§ DISCUSSION

Figure <ref> presents relationship (<ref>) between perturbation amplitude ℵ and the 'factor of additional expansion' of the void, k; figure <ref> — the dependence of the  central underdensity, Δ_c, from ℵ. The solid lines represent the relationships corresponding to the real Universe with Ω_m,0=0.315. The dashed lines correspond to the case, when the universe has the same H_0 and no curvature term, as the real Universe, but contains no dark energy (i.e., Ω_m,0=1). As we can see, the presence of the dark energy suppresses the void formation: the voids in the hypothetical universe without dark energy expand stronger and have lower central density.

This is not surprising: the presence of dark energy suppresses the growth of cosmological perturbations, both overdensities and underdensities. This effect may be taken into account by multiplying the perturbation amplitudes by corrective multiplier g(z) (see, for instance, <cit.> for details). At the matter-dominated stage g(z)=1, but even now, when the dark energy dominates (Ω_Λ,0=0.685), g(0)≡ g_0≃ 0.7879, i.e., the influence of the dark energy on the structures is moderate.

As equation (<ref>) shows, the deeper the initial perturbation is, the larger k is, i.e., the deepest regions of the initial underdensity corresponding to the future void expand the most. This leads to the characteristic void density profile: it is very flat in the center (i.e., the matter density is almost constant in the central part of the void), but grows rapidly towards the edge of the void. For instance, N-body simulations suggest the following profile <cit.>:

    ρ(r)/ρ_M,0=A_0+A_3(r/R_V)^3.
 
The authors obtained the best fit A_0=0.13± 0.01, A_3=0.70± 0.03 for voids of R_V=8 h^-1 Mpc. In reality the central part may be even flatter <cit.>. Equations (<ref>, <ref>) define a one-to-one correspondence between the amplitude ℵ of the perturbation that later forms the void and the void central underdensity Δ_c. One should distinguish Δ_c from the average void underdensity Δ_av, which is also used in literature. Of course, |Δ_c| ≥ |Δ_av|. Moreover, Δ_av is much worse defined: it strongly depends on how the void boundary is determined. The density profile is very steep there, and if we slightly shift the border (by changing the void criterion), Δ_av changes significantly, while Δ_c remains the same.



In order to compare our results with observations, we need to estimate the values of ℵ corresponding to voids. Following observational results <cit.>, we assume Gaussianity of the primordial perturbations. In particular, it means that on the linear stage perturbations with the same |ℵ| and size are equally probable, regardless of whether they are under or overdensities. Let us consider an overdensity and an underdensity of the same size and |ℵ| at the linear stage. The underdensity now forms a void of radius R_0 and central underdensity Δ_c, which is bound with |ℵ| by equations (<ref>, <ref>). The overdensity collapses into a galaxy cluster of mass

    M_0≃4/3π R^3_0 ρ_m,0

at z=z_col, when its amplitude Δρ_m/ρ_m≃ 1, i.e., 

    g(z_col)|ℵ|z_col+1=1.

For instance, a galaxy cluster of mass 2· 10^14 M_⊙ approximately corresponds to a void of radius R_0≃ 11 Mpc. There is one-to-one correspondence between the central underdensity of the void, Δ_c, and amplitude |ℵ| of the initial perturbation. Equation (<ref>) binds |ℵ| and z_col of the cluster. Since we suppose that the void and the cluster had the same amplitude of initial perturbation |ℵ|, there is also a one-to-one correspondence between Δ_c and z_col, which is represented in figure <ref>.

Thus, if we consider voids and galaxy clusters formed from primordial perturbations of the same size (i.e., the void radius and the cluster mass are bound by (<ref>)) and the same |ℵ| (i.e., their Δ_c and z_col depend as it is shown in Fig. <ref>), their number densities in the Universe should be equal because of the perturbation Gaussianity. Therefore, instead of measuring Δ_c of the voids of some given radius R_0 (which is difficult) we may observe z_col of the corresponding galaxy clusters. 

Observations <cit.> show no clusters of the present-day mass exceeding 3· 10^14 M_⊙ form before z=1.75. It means that there are no (or extremely rare) voids of radius R_0≥ 12.5 Mpc with |Δ_c| > 75%. The first clusters of lower mass may occur at z<2.5 <cit.>, which corresponds to Δ_c = -80%. Large voids (R_0≥ 40 Mpc) correspond to clusters of mass > 10^16 M_⊙, which do not exist, because the perturbations of this mass are still linear. Fig. <ref> shows that the central underdensities of the supervoids are Δ_c > -55%: they are also still linear.

Let us emphasize that we estimated the maximum possible densities of voids in the previous paragraph. Indeed, only the first clusters with the present-day mass 3· 10^14 M_⊙ appear at z=1.75, while most of the clusters of this mass collapse much later. By analogy, there may be only rare (one of a hundred) voids of radius R_0≥ 12.5 Mpc having Δ_c ≃ -75%, but most of the voids of this radius have a significantly higher central density. 

In order to estimate not the maximum possible , but the average |Δ_c| of the voids of a given radius, we use the Gaussianity of the primordial perturbations again. Let us randomly choose a comoving sphere of radius L/((z+1)h) (so L/h is the present-day radius of the sphere) deeply on the matter-dominated stage, when the perturbations corresponding to voids were linear.  The Gaussianity of the perturbations means that the probability f that the average density contrast Δρ_m/ρ_m inside L is equal to δ_L has the Gaussian form

    p(δ_L)=1/σ_L √(2π)exp(-δ^2_L/2σ^2_L).

The value of σ_L is proportional to g(z)/(z+1). In view of the foregoing, we assume that the distribution of ℵ looks like

    f(ℵ)=1/σ_8 √(2π)exp(-ℵ^2/2σ^2_8),

where σ_8=0.811 <cit.>. Thus, we assume that the distribution of ℵ coincides with that of relatively small perturbations (R_0=L_0/h=8h^-1 Mpc ≃ 11 Mpc). This size roughly corresponds to small voids, while the largest voids exceed R_0=70 Mpc <cit.>. Actually, we suppose that σ_L= const=σ_8. In fact the value of σ decreases with the object size, and thus our approach somewhat overestimates ℵ and the emptiness of large voids.

When the perturbations reach nonlinear stage, the voids expand significantly faster than the Universe on average. As we have discussed, a void has two regions: a large density plateau in the center, where the matter density is almost constant and equal to q_m,0, and the void walls with very steep density profile. The plateau part expands k^3(ℵ) times more than the Universe on average. Combining it with (<ref>), we find that the volume V(Δ_c) occupied by the regions with matter density less than Δ_c is proportional to

    V(Δ_c(ℵ))∝k^3(ℵ)/σ_8 √(2π)exp(-ℵ^2/2σ^2_8).

By expressing ℵ in terms of Δ_c using expressions (<ref>) and (<ref>), we can integrate (<ref>) and obtain the fraction of the Universe volume occupied by regions emptier than Δ_c, as a function of Δ_c. This dependence is presented in Fig. <ref>.



It is reasonable to expect that the plateau regions of voids occupy at least 1/3 of the Universe volume, which correspond to Δ_c ≃ -52%. This is probably the most reasonable estimate of the average central underdensity Δ_c for the voids. According to figure <ref>, regions with Δ_c ≃ -75% and Δ_c ≃ -80% occupy ∼ 1.5% and ∼ 0.1%  of the Universe volume, respectively. These fractions are quite consistent with the above considerations that only one of a hundred small voids may have Δ_c ≃ -(75-80)%, but this is a very rare exception, not a rule.

Let us sum it up. Most of the voids have Δ_c ∼ -50%, there are no voids with Δ_c < -80%. For small voids (R_0≃ 5-10 Mpc), the underdensity can be significantly greater than |Δ_c| = 50%, but the number of voids decreases rapidly with increasing of |Δ_c|, and voids with |Δ_c| > 80% are practically absent. The largest voids (R_0> 40 Mpc) always have |Δ_c| < 50%: they correspond to overdensities of mass 2· 10^16 M_⊙, which exceeds the mass of the richest galaxy clusters. Overdensities of this length are still linear in the modern Universe, so the underdensities also could not yet enter the nonlinear regime. Finally, the voids with R_0= 15-50 Mpc have average values of their parameters: their average Δ_c ∼ -50%, the number of voids decreases rapidly with increasing of |Δ_c|, and the upper limit of |Δ_c| depends on the void radius (larger voids are less underdense) and is always smaller than |Δ_c| =75%.

Our estimates seriously contradict the estimates of void density obtained in N-body simulations. For instance, equation (<ref>) is obtained by fitting N-body simulation and suggests average Δ_c = -87% (A_0=0.13) even for small voids with (R_0=8h^-1 Mpc ≃ 11 Mpc). Recent simulations also suggest very underdense voids. <cit.> find that voids occupy ∼ 83% of the Universe volume, and their average underdensity is Δ_av = -65%. The fact that voids occupy 83% of the Universe in the simulations makes it clear that we are talking about the average, not the central density of voids. The relationship between Δ_av and Δ_c depends on the model, but if we use the generally accepted (<ref>), then Δ_av = -65% corresponds to Δ_c = -90%. Thus, simulations consistently give an average of Δ_c = -(85-90)% for voids, while this analytical calculation gave an estimate Δ_c = -50% for this value.

This contradiction is difficult to attribute to the simplifying assumptions we made in our analytical calculation. Section <ref> shows that the void ellipticity very probably doesn't affect the central void density at all. We also show that the tidal influence on the void center is moderate. Moreover, tidal perturbations created by an external body (at least, to the first approximation) lead to a deformation of a dust cloud on which they act, but not to a change in its volume. Thus, there is no reason to believe that the simplifying assumptions made in our calculations may be responsible for the discrepancy. The N-body simulations are known to suffer from some significant numerical effects <cit.>, and this could be the reason why the simulations significantly overestimate the emptiness of the voids. 

The overestimation of the void emptiness by the N-body simulations is evident from the result of  simulations <cit.> that the voids with average underdensity Δ_av = -65% occupy ∼ 83% of the Universe volume. Comparing it with equation (<ref>), we may see that this situation is only possible if σ_L≫ 1: all the underdense region are already deeply nonlinear (Δ_av = -65%), they have already expand much more than the Universe on average and occupy 83% of the Universe instead of 50% in the linear regime. However, the voids are large, and they correspond to L ≳ 8 Mpc. Since σ_L decreases with L, for voids σ_L ≲σ_8≃ 0.811<1, and the regions with Δ_av = -65% may occupy only ∼ 10% of the Universe volume, not 83% (see figure <ref>).

As equation (<ref>) shows, the "Hubble constant" inside a void Q differs from that of the Universe H and depends on |ℵ|. It is curious to check if we may determine the void underdensity by measuring Q. Substituting k=b_0/a_0 into (<ref>), we obtain the relationship binding the present-day ratio Q_0/H_0 and ℵ.  With the help of equations (<ref>, <ref>) we may express Q_0/H_0 through Δ_c, the dependence is reproduced in figure <ref>. As we can see, the "Hubble constant" inside voids indeed always exceeds H. However, the prospects for measuring Δ_c by observing Q_0 seem bleak: as Δ_c changes from 0 to 100%, Q_0 increases only by ∼ 25%. Thus, Q_0 is not very sensitive to Δ_c, and it is difficult to draw the value of Δ_c from observations of Q_0 inside the void.



Finally, a question arises: if the voids contain so much matter, why do they look so empty? Indeed, the density of bright galaxies in the voids is tens of times lower than in the Universe on average, while the density of matter in most voids, as we have seen, is only two to three times lower than the average one. The reason for this discrepancy is probably quite simple. We saw that the interior of the void can be considered as a part of a Friedmann space, but its parameters differ from those of our Universe. Firstly, the moment of equality of the matter and dark energy densities  occurs somewhat earlier, and so the value of g_0 is noticeably higher in it. For this reason alone, the growth of cosmological perturbations in the void is suppressed significantly stronger than in the Universe on average. Secondly, the "universe" inside the void corresponds to the open Friedmann model: the three-dimensional space there has Lobachevsky geometry. The open universe expands faster than the flat one, and this also suppresses the perturbation growth.

The problem of the smaller perturbation growth inside voids certainly requires additional study. Note, however, that both of the above-mentioned suppression mechanisms are activated comparatively late, when the amplitude of the void-forming underdensity becomes noticeable (z<5). Thus, large structures in the voids, like large galaxies and galaxy clusters, are strongly suppressed. But low-mass structures, such as dwarf galaxies or dark matter clamps, collapse at z>10 and therefore form in the void in almost the same way as in the rest of the Universe: their present-day numerical density there is only k^3∼ 2-3 times lower than the average in the Universe, it follows the matter density. Thus, voids contain a significant amount of dark and baryonic matter, but it forms only low-mass, faintly luminous objects and a massive diffuse component, being therefore not visible.

We would like to thank the Academy of Finland mobility grant 1341541, for the financial support of this work.




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