Spherical and nonspherical models of primordial black hole formation: exact solutions

# Spherical and nonspherical models of primordial black hole formation: exact solutions

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###### Abstract

We construct spacetimes which provide spherical and nonspherical models of black hole formation in the flat Friedmann–Lemaitre–Robertson–Walker (FLRW) universe with the Lemaitre–Tolman–Bondi solution and the Szekeres quasispherical solution, respectively. These dust solutions may contain both shell-crossing and shell-focusing naked singularities. These singularities can be physically regarded as the breakdown of dust description, where strong pressure gradient force plays a role. We adopt the simultaneous big bang condition to extract a growing mode of adiabatic perturbation in the flat FLRW universe. If the density perturbation has a sufficiently homogeneous central region and a sufficiently sharp transition to the background FLRW universe, its central shell-focusing singularity is globally covered. If the density concentration is sufficiently large, no shell-crossing singularity appears and a black hole is formed. If the density concentration is not sufficiently large, a shell-crossing singularity appears. In this case, a large dipole moment significantly advances shell-crossing singularities and they tend to appear before the black hole formation. In contrast, a shell-crossing singularity unavoidably appears in the spherical and nonspherical evolution of cosmological voids. The present analysis is general and applicable to cosmological nonlinear structure formation described by these dust solutions.

2,3]Sanjay Jhingan

## 1 Introduction

Black holes may have formed in the early Universe. These black holes are called primordial black holes (PBHs) [1, 2, 3]. Since PBHs can act as sources of Hawking radiation, strong gravitational fields and gravitational radiation, current observations can place a stringent upper limit on the abundance of PBHs [4]. Generally speaking, PBHs of mass were formed at an epoch when the total mass enclosed within the Hubble radius was of order , if we neglect critical behavior in gravitational collapse, mass loss due to Hawking evaporation, and mass gain due to accretion. The abundance of PBHs of mass depends primarily on the primordial fluctuations of mass and the state of the Universe at the formation epoch. Since the Hawking radiation and its final outcome depend on quantum gravity, PBHs can also provide observable phenomena of quantum gravity, even if no positive signal has been detected. Thus, PBHs can be seen as a probe into the early Universe, high energy physics, and quantum gravity. In particular, since PBHs can convey information about primordial fluctuations, the observational constraint on the abundance of PBHs is complementary to the cosmic microwave background anisotropy observation.

To convert primordial fluctuations in a given cosmological scenario to the abundance of PBHs, we need to understand the detailed process of PBH formation. The process of PBH formation entails big bang, gravitational collapse, apparent horizon, relativistic fluid, primordial fluctuations and cosmological expansion. We need dynamical and inhomogeneous solutions of the Einstein equation and the equations of motion for the matter fields. The threshold of PBH formation in spherical symmetry was analytically derived based on the Jeans criterion by Carr [5], and has recently been refined by Harada, Yoo and Kohri [6]. The threshold of PBH formation in spherical symmetry has also been studied by hydrodynamical simulations based on numerical relativity, which was pioneered by Nadezhin et al. [7]. For recent numerical studies, see Refs. [8, 10, 11, 9] and references therein.

In summary, these studies support efficient PBH production for a soft equation of state. The effective equation of state becomes very soft if particles are very massive [12] or if a scalar field oscillates in a quadratic potential, as in the ending phase of inflation [13, 14]. A fluid with a very soft equation of state can be well approximated by dust or pressureless fluid. Very recently, Torres et al. [15] numerically simulated cosmological nonlinear structure formation. They did a comparative study of an oscillating massive scalar field and dust in spherical symmetry.

However, nonspherical effects on PBH formation should be significant if the effective equation of state is very soft, as indicated by Carr [5]. This is because the Jeans scale is much smaller than the Hubble length and the deviation from spherical symmetry grows in Newtonian gravitational collapse of a uniform spheroid, which is known as the Lin–Mestel–Shu instability  [16]. Nonspherical effects on PBH formation have been studied by Khlopov and Polnarev [12] in the context of the phase transition of grand unification and their result has been applied to the ending phase of inflation [17, 18].

To study nonspherical effects on PBHs in a fully general relativistic, analytical, and exact manner, we focus on dynamical exact solutions with dust as an idealized model of a matter field with very soft effective equation of state. In the present context, we pay attention to the sequence of exact solutions, the Friedmann–Lemaitre–Robertson–Walker (FLRW) solution, the Lemaitre–Tolman–Bondi (LTB) solution [19, 20, 21], and Szekeres’s quasispherical solution [22]. The FLRW solution is a spatially homogeneous and isotropic solution with no arbitrary function. It can also describe the interior of a uniform density dust ball [23]. The LTB solution is a spherically symmetric and inhomogeneous solution with two arbitrary functions of one variable. This solution is particularly important because it is a general solution in spherical symmetry and includes the FLRW dust solution. The spherical model of PBH formation has been constructed with the LTB solution by Harada, Goymer, and Carr [24]. The Szekeres solution is a nonspherically symmetric and inhomogeneous solution with five arbitrary functions of one variable. It includes the LTB solution and it admits no Killing vector in general. However, the whole set of Szekeres solutions is a proper subset of the whole set of general dust solutions, of which the explicit expression is not known. The Szekeres solution has been intensively studied in the context of nonlinear perturbation of the Universe [25, 26, 27, 28].

A singularity which is not behind a black hole horizon, called a naked singularity, may appear in the course of gravitational collapse with regular initial data in the LTB solution [29, 30, 31, 32, 33, 34]. This is also the case in the Szekeres solution [35, 36, 37, 38]. Clearly this departure from spherical symmetry cannot avoid the formation of naked singularities. Goncalves [39] studied the cosmic censorship and the curvature strength for the shell-focusing singularities in the quasispherical Szekeres solutions with dust and a cosmological constant based on the radial null geodesics, which do not exist in general. Hellaby and Krasinski [40] studied the condition for the avoidance of shell-crossing singularities in the Szekeres solutions for the quasispherical, quasi-pseudospherical and quasiplanar cases in the very general formulations. The same authors [41] presented the geometrical interpretation of the Szekeres solutions for the quasi-pseudospherical and quasiplanar cases. Krasinski and Bolejko [42] defined an absolute apparent horizon in the quasispherical Szekeres solutions, discussed its difference from an apparent horizon, which is commonly used, and concluded that the apparent horizon can be regarded as the true horizon. Vrba and Svitek [43] rewrote the condition for the occurrence of shell-crossing singularities in terms of the maximum, minimum, and average density of the shell at the moment of occurrence. They also discussed the time evolution of the solutions, restricting themselves to the marginally bound case which is not relevant to the cosmological growing perturbation.

It is important to see the condition for the occurrence and non-occurrence of naked singularities and its physical interpretation in the context of the nonlinear evolution of cosmological primordial fluctuations. We adopt the simultaneous big bang condition to extract a growing mode of adiabatic perturbations. The formation and evolution of cosmological nonlinear perturbations have been analyzed on the same grounds [44, 45]. For these nonlinear cosmological perturbations, we investigate the formation of black holes and cosmological voids against the formation of shell-focusing and shell-crossing singularities and see how the deviation from spherical symmetry affects these physical situations for the first time. We also discuss the link between the current result and the physical interpretation by Khlopov and Polnarev [12] that shell-focusing naked singularities may be physically regarded as the onset of strong pressure gradient force.

This paper is organized as follows. In Sect. 2, we present and interpret the Szekeres solution. In Sect. 3, we describe the dynamics of the Szekeres solution. In Sect. 4, we analyze the shell-crossing and shell-focusing singularities under the simultaneous big bang condition. There, we find a necessary and sufficient condition for the Szekeres solution to describe the spherical and nonspherical formation of PBHs without suffering naked singularity formation. In Sect. 5, we propose several concrete models that describe the spherical and nonspherical formation of PBHs without naked singularities. Section 6 is devoted to our conclusions. In Appendix A, we briefly review the derivation of the Szekeres solution in order for the paper to be self-contained. We use geometrized units, where .

## 2 Szekeres’s quasispherical dust solution and its physical interpretation

### 2.1 Presentation of the metric

We present here a brief overview of Szekeres’s quasispherical dust solution. For completeness, the derivation of this solution is given in Appendix A. The line element is given by

 ds2=−dt2+e2λdr2+e2ω(dx2+dy2), (1)

where

 eω=ϕP, (2) eλ=P(eω),1W(r), (3) P=A(r)(x2+y2)+2B1(r)x+2B2(r)y+C(r), (4) A(r)C(r)−B21(r)−B22(r)=14, (5) (ϕ,0)2=W2(r)−1+S(r)ϕ, (6) ρ=PS(r)′−3S(r)P,18πϕ2[Pϕ,1−ϕP,1], (7)

, and . The prime denotes the ordinary derivative with respect to the argument and hence is only used for functions of one variable such as , , , , , and . For functions of more than one variable, the partial derivatives are denoted by a comma followed by the index of the differentiating variable. Equation (2.1) can be integrated to give

 ±∫ϕ0dφ√W2(r)−1+S(r)φ=t+H(r), (8)

where is an arbitrary function of . Thus, the solution contains seven arbitrary functions of . With one scaling freedom and one algebraic constraint taken into account, the solution contains five arbitrary functions.

Equation (2.1) implies that neither nor can vanish, while or can. We assume and without loss of generality. Equation (2.1) can be transformed into the following form:

 (9)

which implies and hence too is positive definite.

It is clear that the solution reduces to the LTB solution if and . See Refs. [47, 46] for the LTB solution, where , , and correspond to the energy, mass, and time functions, respectively, and gives the areal radius of the two-sphere of constant and . The coordinates on the two-sphere can be understood in terms of the stereographic projection. One of the three arbitrary functions corresponds to the gauge freedom. For example, can be fixed if we fix the radial coordinate at so that . The simplest choice to recover the Minkowski spacetime is , , and .

The energy density can be singular only if

 ϕ(t,r)=0 (10)

or

 ϕ(t,r),1P(r,x,y)−ϕ(t,r)P(r,x,y),1=0 (11)

is satisfied. The former and the latter are called shell-focusing and shell-crossing singularities, respectively.

Equation (2.1) describing the evolution of shells can be integrated explicitly. For , we have

 t+H(r)=±23√ϕ3S(r). (12)

For , we have

 ϕ=S1−W2(r)1−cosη2,  t+H(r)=S(r)(1−W2(r))3/2η−sinη2. (13)

For , we have

 ϕ=S(r)W2(r)−1coshη−12,  t+H(r)=S(r)(W2(r)−1)3/2sinhη−η2. (14)

The above solutions can be summarized into the following compact form:

 t+H(r)=√ϕ3S(r)G(1−W2(r)S(r)ϕ), (15)

where

 G(Y)=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩G+(Y)=arcsin√YY3/2−√1−YY(0

and where

 Y:=1−W2(r)S(r)ϕ. (17)

Note that for the expanding and collapsing branches are combined into a single complete solution with both big bang and big crunch. Since we are interested in the cosmological solutions, we focus on the branches which possess a big bang, so that for , while for . Figure 1 shows the relevant branches of for . We note that for admits the following expansion:

 G−(Y)=πY3/2−23−15Y−328Y2+O(Y3). (18)

### 2.2 Comoving coordinates and the dipole moment

Inspired by the functional form of Eq. (2.1), we make two successive coordinate transformations on the constant hypersurface. First from to new coordinates , where and are defined by

 p:=2A(r)(x+B1(r)A(r))  and  q:=2A(r)(y+B2(r)A(r)), (19)

and then from to based on the stereographic projection, where

 p=cotθ2cosφ  and  q=cotθ2sinφ. (20)

The function can now be expressed as

 P=14Asin2θ2, (21)

and the metric induced on the two-surface is given by the standard form of the two-sphere

 ds2|St,r=ϕ2(t,r)(dθ2+sin2θdφ2), (22)

whereas the line element in the spacetime contains off-diagonal terms and in general. Thus, we establish the following interesting relation:

 dM=ρeλ+2ωdrdxdy=~ρr2sinθdrdθdφ, (23)

where is the proper mass and is given by and is calculated to give

 ~ρ=S′8πr2W(1−3SS′P,1P). (24)

Therefore, the mass contained within the volume element is constant in time. This means the coordinates play the role of the comoving spherical coordinates and can be interpreted as the conserved mass density.

We have seen that gives the conserved mass density in terms of the comoving coordinates or . This suggests that with we can define a conserved dipole moment. In the comoving spherical coordinates, can be transformed to the standard form given in Eq. (2.2).

Thus, defining vectors and as

 →β:=−r(2(B′1−A′AB1),2(B′2−A′AB2),A′A), (25)

and

 →n:=(sinθcosφ,sinθsinφ,cosθ)=1r(~x,~y,~z), (26)

respectively, we find

 P,1P=−→n⋅→βr (27)

in the coordinate basis of the comoving Cartesian coordinates . Since the nonspherical dependence of the density distribution , as well as , appears only through this combination, the matter distribution has monopole and dipole moments only and the vector is proportional to the dipole moment. The absolute value of the vector is given by

 β2(r)=4r2[B′21(r)+B′22(r)−A′(r)C′(r)]. (28)

It can be easily shown that is positive definite. The definition of used here is the same as in Szekeres [35].

We can rewrite the expression for the energy density, Eq. (2.2), in the following form:

 ~ρ=~ρ0(1+→n⋅→d), (29)

where

 ~ρ0:=S′8πr2W  and  →d:=3SrS′→β. (30)

Thus, we can naturally define the nondimensional mass dipole moment localized in the shell labeled with .

On the other hand, the physical density can also be written in the following form:

 ρ=ρ0(1+χ), (31)

where

 ρ0:=S′8πϕ2ϕ,1  and%   χ:=(S′S−3ϕ,1ϕ)P,1P(ϕ,1ϕ−P,1P)S′S. (32)

We can interpret as the spherical part of the density field, which is identical to that of the reference LTB solution, and as the deviation from it. The nonsphericity does depend on time through . Although cannot be simply interpreted as the deviation due to the dipole moment, it is closely related to the vector . If , then vanishes identically. Conversely, if vanishes identically, then or .

### 2.3 Spherically symmetric and axially symmetric subclasses

As is seen in Eq. (2.1), the energy density depends only on and , if

 A′A=B′1B1=B′2B2=C′C (33)

or

 S′3S=ϕ,1ϕ (34)

is satisfied [48]. From Eqs. (2.1), Eq. (2.3) implies that all of , , , and are constant. Then, the spacetime becomes spherically symmetric and the solution reduces to the LTB solution. In this case, we can see . This supports our interpretation of as a dipole moment. Equation (2.3) holds if and only if is separable as we can show from Eq. (2.1). In this case, and therefore , i.e., the density is homogeneous. In this case, since the spatial component of the metric is written as , can be shown to be that of the constant curvature space in three dimensions and therefore this is the FLRW solution [22].

The above discussion also implies that the Szekeres solution can be matched to the Schwarzschild solution at any radius . In fact, we can always choose , , , , , , to be constant for . The mass parameter of the Schwarzschild black hole is given by . Then, the region for is spherically symmetric vacuum and hence the Schwarzschild solution by Birkhoff’s theorem.

Next, let us consider a subclass where identically but is still allowed to be a function of . Then, Eqs. (2.1) and (2.1) reduce to

 P(r,x,y)=A(r)(x2+y2)+14A(r). (35)

In the comoving spherical coordinates , we can easily find

 P,1P=A′Acosθ. (36)

Since the metric components in the comoving spherical coordinates do not depend on , the spacetime is axially symmetric. In the comoving Cartesian coordinates , we find

 →β=(0,0,−rA′A) (37)

Thus, is directed along the axis. This also supports our interpretation of as a dipole moment.

## 3 Dynamics of the Szekeres solution

### 3.1 Shell-crossing singularities

Following Szekeres [35], we quote two lemmas for quadratic forms.

###### Lemma 3.1.

A quadratic form (), where and are real and is complex, has no zeros in the complex plane if and only if its discriminant is positive, i.e., . Moreover, if and only if and , for any .

###### Lemma 3.2.
 Δ(θ′P−cθP,1)=14θ′2+c2θ2Δ(P,1), (38)

where is a constant and .

If we denote the time of the shell-crossing singularity by , where

 P(r,x,y)ϕ,1(tSC(r,x,y),r)−ϕ(tSC(r,x,y),r)P,1(r,x,y)=0, (39)

depends not only on but also on and in general. Thus, shell-crossing singularities are affected by nonsphericity. If we fix , a shell-crossing singularity occurs not on the two-surface but on a different two-surface in general. From Eq. (2.2), we can find that if and , then diverges to infinity at shell-crossing singularities. In other words, if the shell is nonspherical, it is the nonspherical rather than the spherical part of the density field that diverges at the shell-crossing singularities. Therefore, generic shell-crossing singularities in the Szekeres spacetime are essentially nonspherical.

Applying the lemmas presented above to the function , we can show that the two-surface possesses a shell-crossing singularity if and only if

 (ϕ,1ϕ)2≤4[(B′1)2+(B′2)2−A′C′], (40)

which can be rewritten as

 (ϕ,1ϕ)2≤β2r2, (41)

directly linking the shell-crossing condition with nonsphericity parameter . We define as the time of the earliest occurrence of shell-crossing singularity on so that the quadratic form with fixed begins to have a zero at . This implies

 (ϕ,1ϕ)2∣∣ ∣∣t=tSC(r)=β2(r)r2. (42)

### 3.2 Regularity condition

We assume the existence of a regular initial data surface (). Since the areal radius of the two-surface is , we also assume that is an increasing function of and scale so that . For the center to be locally Minkowski, Eq. (2.1) implies . For to be bounded, Eq. (2.1) implies . Then, for to be bounded as , Eq. (2.1) implies . This implies

 S(r)=O(r3)and1−W2(r)=O(r2). (43)

We further assume that the metric is in the comoving Cartesian coordinates. For to be in terms of , , and hence the nonspherical functions , , , and are continuous and differentiable at .

At , the initial data is shell-crossing free for all , , and if, and only if

 (ϕ0,1ϕ0)2>β2(r)r2. (44)

In turn, applying Lemma 3.2 to the function , we can see that is positive definite at if, and only if

 β2(r)4r2<(S′)236S2. (45)

Thus, regularity imposes the following condition on :

 β(r)

The above condition is automatically satisfied near , since .

### 3.3 Trapped surfaces, and apparent horizons

Following Szekeres [35], we consider a trapping condition for a spacelike two-surface . Since the tangent space of is spanned by , and , any normal vector to should satisfy . Thus, if we consider the congruence of null geodesics with tangent vector normal to , we find

 kμkμ = 0, (47) kμ ;νkν = 0 (48)

with on . Assuming , we can identify the null geodesics of () with outgoing (ingoing) ones. We can choose () on by choosing the scaling of the affine parameter. The sign of the expansion coincides with the sign of , which is calculated to give

 kμ ;μ=k0 ,0+k1 ,1+(λ,0+2ω,0)eλ+(λ,1+2ω,1)ϵ, (49)

where we put ,, and . Differentiating Eq. (3.3) with respect to , we find

 k0 ,0−ϵeλk1 ,0−eλλ ,0=0. (50)

In Eq. (3.3) with , we find

 eλk1 ,0+ϵ(k1 ,1+2λ ,0eλ)+λ,1=0. (51)

We eliminate from Eq. (3.3) by Eq. (3.3),, and eliminate from Eq. (3.3) by the resultant equation. Then, we find

 kμ ;μ=2ω,0eλ+2ϵω,1. (52)

Using Eqs. (2.1), (2.1), and (2.1), we can transform Eq. (3.3) to

 kμ ;μ=2(ϕ,1ϕ−P,1P)(ι√1+1W2(Sϕ−1)+ϵ), (53)

where . Equation (3.2) implies that the first factor is positive because it cannot change the sign without encountering a shell-crossing singularity. Thus, we establish that if the dust is collapsing (), the outgoing () null normal can have vanishing expansion if , while if the dust is expanding (), it is the ingoing () null normal that can have vanishing expansion if .

Here we identify a marginally trapped two-surface with an apparent horizon according to Krasinski and Bolejko [42]. Then we call an apparent horizon with vanishing outgoing (ingoing) null expansion a future (past) apparent horizon. Note that this is somewhat different from the notion of an apparent horizon defined by Hawking and Ellis [49]. Although a black hole horizon is usually identified with an event horizon in the asymptotically flat spacetime, such an identification is not so strongly motivated in the cosmological spacetime because of the teleological nature of an event horizon and because the asymptotic condition at null infinity is less physically meaningful in cosmology with a finite particle horizon. Although the local and dynamical definition of black hole horizon is ambiguous, we can identify a future apparent horizon with a local black hole horizon in this paper for not only its physical reasonableness but also its simplicity and usefulness for the analysis.

In the dust models, a noncentral shell-focusing singularity is always future trapped because and there. This means that noncentral singularity is causally disconnected from a distant observer.

### 3.4 Six critical events

As for the dynamics of the Szekeres solution, there are six important events: big bang, past apparent horizon, maximum expansion, future apparent horizon, big crunch and shell-crossing singularity. We denote the times of the occurrence of these six events at each shell with , , , , , and , respectively.

The order of these events is trivial except for shell-crossing singularity. This is because, except for shell-crossing singularity, the events are determined solely by each shell and its dynamics is identical to the FLRW spacetime.

For , we can find and there is no maximum expansion, no future apparent horizon and no big crunch. For the big bang time, Eq. (2.1) implies

 tBB(r)=−H(r). (54)

Since at apparent horizons, we find

 tPH(r)−tBB(r)=S(r)G+(1−W2(r)). (55)

For , we can find . The big bang time and the time of past apparent horizon are given by Eqs. (3.4) and (3.4). The time of future apparent horizon is given by

 tFH(r)−tBB(r)=S(r)G−(1−W2(r)). (56)

For the big crunch time, we find

 tBC(r)−tBB(r)=limϕ→0√ϕ3S(r)G−(1−W2(r)S(r)ϕ), (57)

and the result is

 tBC(r)−tBB(r)=πS(r)[1−W2(r)]3/2. (58)

The time of maximum expansion is given by , as seen from Eq. (2.1). Therefore, from Eqs. (2.1) and (2.1), we find

 tME(r)−tBB(r)=π2S(r)[1−W2(r)]3/2. (59)

The time of shell-crossing singularity is highly nontrivial because is determined not solely by the dynamics of the shell labeled , but that of the neighboring shells. We will study it in detail in the next section.

## 4 Szekeres solution as nonlinear cosmological perturbations

### 4.1 Simultaneous big bang condition

The simultaneous big bang condition is often used in the cosmological context and suitable for a nonlinear growing mode of adiabatic perturbation. This condition implies const. in Eq. (3.4). If we assume this condition, we find from Eqs. (2.1) and (2.1)

 ϕ→(9S4)1/3(t−tBB)2/3  and  ρ→16π(t−tBB)2 (60)

in the limit of , and hence the solution approaches the flat FLRW solution. This implies that the simultaneous big bang condition corresponds to extracting a purely growing mode of adiabatic perturbation from the flat FLRW universe. This condition in the LTB solution has been adopted for the construction of the primordial black hole formation model in the flat FLRW universe in Harada et al. [24]. The LTB solution as an exact model of black holes in the evolving Universe is also discussed from a very broad scope in Sect. 18.9 of Ref. [46].

### 4.2 Shell-crossing singularities

We should note that very general treatments of the occurrence of shell-crossing singularities for the LTB solution and the Szekeres solution are given in Sects. 18.10 and 19.7.4 of Ref. [46], respectively. Our aim in the next few subsections is to analyze the occurrence of shell-crossing singularities and get a physical insight into the nature of shell-crossing occurrence in the present cosmological setting.

#### 4.2.1 Expression for ψ=rϕ,1/ϕ

In order to see the occurrence of shell-crossing singularities, it is important to have the explicit form of . In the general case, this can be found by differentiating Eq. (2.1) with respect to as follows:

 ψ:=rϕ,1ϕ=1S(1−W2Y)3/2rH′+12r(lnS)′G−r(ln|1−W2|S)′dGdln|Y|32G+dGdln|Y|, (61)

where the derivative of is given by the following form:

 (62)

With the simultaneous big bang condition const., Eq. (4.2.1) is reduced to

 ψ=12r(lnS)′−r(ln|1−W2|S)′dlnGdln|Y|32+dlnGdln|Y|. (63)

Putting

 X(Y)=X±(Y):=2dlnG±dln|Y|2dlnG±dln|Y|+3,  a1(r):=−r[ln|1−W2|S2/3]′,  a0(r):=r(lnS1/3)′, (64)

we can rewrite in the following form:

 ψ(r,Y)=a1(r)X(Y)+a0(r). (65)

The function is plotted in Fig. 2.

For , we can see that the expression (4.2.1) is very singular. For this case, regularity is obvious in the following expression for :

 ψ=2dlnGdY[−r(1−W2)′ϕS+23Yr(lnS)′]2YdlnGdY+3+a0(r). (66)

In the limit where and are fixed, we find and . Therefore, for , we find

 ψ(r,~X)=~a1(r)~X+a0(r), (67)

where

 ~X:=ϕS,  ~a1(r):=−15r(1−W2)′, (68)

and is the same as in Eq. (4.2.1).

From Eq. (3.1), the two-surface possesses a shell-crossing if and only if According to Eq. (4.2.1) for and Eq. (4.2.1) for , we can now analyze rigorously the occurrence of shell-crossing singularity for given and under the simultaneous big bang condition.

We should note that Eq. (3.2) implies for and for . As for , Eq. (3.2) implies as . Since implies for sufficiently small , there is no shell-crossing singularity for sufficiently small . From Eqs. (4.2.1) and (4.2.1), since for and for both begin with 0 at the big bang, no shell-crossing singularity appears at least for a sufficiently short time interval after the big bang.

We now discuss in general the cases in terms of the shell labeled with , , and , separately. Hereafter, for simplicity, we choose the scaling of so that in accordance with Eq. (3.2), where is a positive constant. The result does not depend on the choice of the scaling. Recall that Eq. (3.2) holds for .

#### 4.2.2 Bound shell: 0<W(r)<1

For , as the shell labeled begins with a big bang, reaches maximum expansion, and ends in a big crunch, monotonically increases from 0, takes value 1 at maximum expansion, and then monotonically decreases to 0. Meanwhile, monotonically increases from to , which is its value at maximum expansion, and then switches to the branch, which monotonically increases from to as time proceeds. We should note that admits the following expansion:

 X−(Y)=3π2Y−3/2−3π4Y−1/2−3π16Y1/2+15Y+O(Y3/2). (69)

So, if , monotonically increases from to as time proceeds. Therefore, we can conclude that there appears no shell-crossing singularity. This is also the case if , where is constant in time. If , monotonically decreases from to . Since , this means that a shell-crossing singularity necessarily appears before the big crunch. In this case, it is important whether or not the shell-crossing singularity appears before the maximum expansion and the future apparent horizon. A maximum expansion is characterized by or . Since is a monotonically decreasing function of time, we can conclude that a shell-crossing singularity at appears before or coinciding with a maximum expansion if and only if . A future apparent horizon is characterized by in the collapsing branch. The value of on the future apparent horizon is given by . Since is a monotonically decreasing function of time, we can conclude that a shell-crossing singularity at appears before or coinciding with a future apparent horizon if and only if

 a1(r)X−(1−W2(r))+a0(r)≤β(r). (70)

If is a monotonically decreasing function of , and hence there is no shell-crossing singularity. Otherwise, there exists for which and a shell-crossing singularity appears. Then, Eq. (4.2.2) determines whether or not the shell-crossing singularity appears before or coinciding with a future apparent horizon. We can see that a large dipole moment can promote and advance the occurrence of shell-crossing singularity before the future apparent horizon.

#### 4.2.3 Unbound shell: W(r)>1

If , the shell labeled begins with a big bang and expands forever. In this case, monotonically decreases from to and monotonically decreases from to . So, if , monotonically increases from to as time proceeds. Therefore, we can conclude that no shell-crossing singularity appears. This is also the case if , where is constant in time. If , monotonically decreases from to . Since , this means that a shell-crossing singularity eventually appears in the course of expansion if and only if , i.e.,

 12r[ln|1−W2|]′≤β(r). (71)

This means that if has an extremum which is greater than 1, there necessarily appears a shell-crossing singularity, irrespective of the value of .

A cosmological void in the asymptotically flat FLRW spacetime is characterized by near the center and in the asymptotic region . Because of Eq. (3.2), must have a maximum which is greater than 1 in this case. Therefore, Eq. (4.2.3) implies that a shell-crossing singularity necessarily appears. This also applies to PBH formation which has a region where . Therefore, for PBH formation without shell-crossing singularities before a future apparent horizon, is necessary for all .

#### 4.2.4 Marginally bound shell: W(r)=1

For , the shell labeled begins with a big bang and expands forever. For this case, Eq. (4.2.1) is relevant. monotonically increases from 0 to as time proceeds. If , monotonically increases from to and hence there is no shell-crossing if Eq. (3.2) is satisfied. This is also the case if , where is constant in time. If , monotonically decreases from to . In this case, a shell-crossing singularity appears in the course of expansion.

Therefore, the condition for the occurrence of shell-crossing singularity is given by . If the region inside the shell with is unbound (), and the region outside is bound (), a shell-crossing singularity appears on this shell.

#### 4.2.5 Criterion in terms of the density distribution

We can define the following quantities:

 ¯ρ(t,r) := M,1(t,r)V,1(t,r), (72) ⟨ρ⟩(t,r) := ∫r0¯ρ(r′,t)W(r′)dV(r′,t)∫r0W(r′)dV(r′,t), (73) M(t,r) := ∫r0dr′∫∫dxdyρeλ+2β, (74) V(t,r) := ∫r0dr′∫∫dxdyeλ+2β. (75)

We can identify with the density averaged over the two-surface , and with the density averaged over the three-ball of which the surface is given by . The explicit expressions of the above quantities for the Szekeres solution are given by

 ¯ρ(t,r) = S′(r)ϕ2ϕ,1, (76) ⟨ρ⟩(t,r) = 3S(r)ϕ