Morgan Le Delliou
email: mor-gan.ledelliou@uam.es

Filipe C Mena
email: fmena@math.uminho.pt

José P Mimoso
email: jpmimoso@cii.fc.ul.pt

The role of shell crossing on the existence and stability of trapped matter shells in spherical inhomogeneous Λ-CDM models

Keywords: numbers: 98.80.-k, 98.80.Cq, 98.80.Jk, 95.30.Sf, 04.40.Nr, 04.20.Jb

We analyse the dynamics of trapped matter shells in spherically symmetric inhomogeneous Λ-CDM models. The investigation uses a Generalised Lemaître-Tolman-Bondi description with initial conditions subject to the constraints of having spatially asymptotic cosmological expansion, initial Hubble-type flow and a regular initial density distribution. We discuss the effects of shell crossing and use a qualitative description of the local trapped matter shells to explore global properties of the models. Once shell crossing occurs, we find a splitting of the global shells separating expansion from collapse into, at most, two global shells: an inner and an outer limit trapped matter shell. In the case of expanding models, the outer limit trapped matter shell necessarily exists. We also study the role of shear in this process, compare our analysis with the Newtonian framework and give concrete examples using density profile models of structure formation in cosmology.

I. INTRODUCTION

Studies of non-linear structure formation in cosmology, namely spherical top hat collapse models, often assume that there is no influence of the cosmological background on a finite domain which has disconnected from the background dynamics (see e.g. [1][2][3][4]).

We have looked at this problem in more detail in Ref. [5] and found local conditions under which such separation could be justified for inhomogeneous cosmological models. In particular, we have studied the possibility for perfect fluid solutions to exhibit locally defined separating shells between collapsing and the expanding (cosmological) regions.

The simplest examples given in [5] were set in an inhomogeneous universe of dust with a positive cosmological constant and the nature of the dust spherical shells allowed the system to be entirely determined from its initial conditions, at least, until the eventual occurrence of shell crossing.

However, shell crossings or caustics are expected to happen in these settings with more general initial conditions than in [5] and an interesting question is whether our previous results are robust with respect to the occurrence of shell crossing. This is the main concern in this paper, which can be regarded a natural follow up of our previous work [5].

Shell crossing in spherical symmetry has already been studied in several past works, although in contexts different from the one of present paper. For Lemaître-Tolman-Bondi (hereafter LTB) spacetimes shell crossing conditions were established by Hellaby and Lake [6] in terms of the metric data and more recently re-written by Sussman in terms of quasi-local scalars [8,[START_REF] Sussman | A dynamical systems study of the inhomogeneous Lambda-CDM model[END_REF]. Goncalves [START_REF] Goncalves | [END_REF], has shown that shell crossing exists for Λ-LTB spacetimes with charge. In [11], it has been shown that shell-crossing occurs for a large class of initial conditions in models of formation of voids and some cases of fluids with pressure gradients.

There were also several works about the strength of shell crossing singularities, with the general conclusion that it is a weak singularity in the sense of Tipler [12]. This then raised the question of the continuity of the metric across these singularities and, very interestingly, solutions of dynamical extension through shell crossing singularities of LTB have been proved to exist, by Nolan [13], while the case including a cosmological constant and electric charges has been discussed by Gonçalves [START_REF] Goncalves | [END_REF].

A complementary treatment was given by Nunez et al. [14] for metric extensions through shell crossing based on the interactions between shells, which translate in a conservation relation between mass and momenta, for timelike massive shells. Physically, this conservation relation summarises the microphysics of the fluid, however for dust, only purely gravitational interaction occurs between crossing shells, hence the rest mass of each shell is conserved [15].

Here, we shall not deal with the problem of metric extensions after shell crossing and, motivated by the above results, we shall assume the validity of the field equations in between shell crossing events and the continuity of the radial coordinates. Our main concern here will be to study the effects of shell crossing on the existence and stability of separating shells in spherical symmetry. In this paper, we shall also discuss the role of shear in the formation of shells which separate expanding from collapsing regions, we shall compare our results with Newtonian cases and give a concrete example of initial data which develops shell crossing and exhibits separating shells in a Λ-dust model.

The models considered in this paper obey the following properties: (a) spherically symmetric dust (the rest mass of infinitesimal pressureless shells is conserved under shell crossing) with a cosmological constant in Generalised LTB (GLTB) system ; (b) Lagrangian treatment of the radial coordinates (assume there are metric extensions through shell crossings); (c) asymptotic spatial cosmological behaviour (Friedmann-Lemaître-Robertson-Walker, hereafter FLRW, at spatial infinity); (d) initial Hubbletype flow (outgoing initial velocities); (e) regular initial density distribution (no finite mass for infinitely thin shell, and no singularity or zero density at the centre).

The paper is organized as follows: in a first part (II) we recall the conditions for the existence of matter trapped shells and study the role of shear on the existence of those shells in Λ-LTB models. Section (III) is devoted to the study of the effect of shell crossing in Λ-LTB models. In particular, we perform a dynamical analysis and separate this study into a local and global effects. We give concrete examples in section (IV) before presenting the final conclusions.

II. TRAPPED MATTER SHELLS IN Λ-CDM

A. Conditions for the existence of trapped matter shells

In this section, we briefly recall some results of our previous paper [5] which did not consider shell-crossings.

The GLTB system proposed in Refs. [5,16], has the following simple form for the case of a Λ-dust model where P ′ = 0 and P = P dust = 0 (here we set G = 1 = c, Λ > 0, α is the lapse function, r (T, R) the areal radius and E the energy of spatial hypersurfaces 1 )

ds 2 = -α(t, R) 2 dt 2 + (∂ R r) 2 1 + E(t, R) dR 2 + r 2 dΩ 2 . (1)
The Bianchi identities projected along and orthogonal to the timelike flow n = ∂ t yield (P is the pressure, ρ the density, the prime ′ denotes ∂ R , a dot ˙stands for ∂ t and Θ is the expansion along the flow)

ρ = -(ρ + P ) 3 Θ, (2)
- P ′ ρ + P = α ′ α = 0 ⇒ αdt = dt * ⇒ α = 1, (3)
and the Einstein Field Equations (M is the Misner-Sharp mass [17], defined as M = ´R 0 4πρr 2 r ′ dR)

Ėr ′ = -2 ṙ 1 + E ρ + P P ′ = ∓2 1 + E ρ + P P ′ α 2 M r + 1 3 Λr 2 + E (4)
⇒ Ė =0, E = E(R), unless there is shell crossing, ( 5)

Ṁ = -ṙ4πP r 2 = ∓4πP r 2 α 2 M r + 1 3 Λr 2 + E = 0 (6)
⇒ M =M (R), unless there is shell crossing,

ṙ2 =2 M r + 1 3 Λr 2 + E. (7)
2 Time derivation of Eq. ( 8) gives a Raychaudhuri equation related to the gTOV function of Ref. [5]:

gTOV = M r 2 - Λ 3 r = -r. (9)
The dynamical analysis detailed in Ref. [5] yields the motion of separated non-crossing shells in their respective effective potential

E = V (r) ≡ - 2M r - Λ 3 r 2 , (10)
where the unstable saddle point, for which gTOV = 0, gives a local separating shell (see [5], figures 1 and 2, and repeated in Fig. 1), in the case when the shell's energy reaches its critical value. This separating (or "cracking", by analogy with Herrera et al. [18]) shell is characterised by

r lim = 3 3M Λ , (11)
E lim = -(3M ) 2 3 Λ 1 3 = -Λr 2 lim , (12)
1 Actually, 3 R = -2 (Er) ′ r 2
so E is related to the 3-curvature. 2 In case of shell crossing, Ė can be nonzero as r ′ = 0 and M gets changed by the loss or gain of the mass from infinitesimal shell crossings, so E = E(t, R) and M = M (t, R), in that case.

r 3 3M Λ E > E lim -(3M ) 2 3 3 √ Λ E < -2M r -Λ 3 r 2 r ṙ Figure 1.
Effective potential kinematic analysis (left) and phase space analysis (right) from [5]. The kinematic analysis for a given shell of constant M and E depict the fate of the shell, depending on E relative to E lim . It either remains bound (E< < E lim ) or escapes and cosmologically expands (E> > E lim ). There exists a critical behavior where the shell will forever expand, but within a finite, bound radius (E = E lim , r ≤ r lim ). The maximum occurs at r lim = 3 3M/Λ. The corresponding phase space behaviour follows, the scales are set by the value of r lim = 3 3M/Λ while the actual kinematic of the shell is given by E.

while the energy follows

E = ṙ2 + V (r). ( 13
)
Definition 1. Local trapped matter shells in Λ-LTB are defined in GLTB coordinates as the locus R ⋆ such that

Θ 3 + a ≡ ṙ r = 0 and L n Θ 3 + a ≡ ṙ r = 0. (14)
This definition follows from Eqs. (3.11) and (3.16) of [5] applied to dust with Λ.

In Λ-LTB, conditions ( 14) are reached by shells at time-infinity which are characterised by Eqs. (8) and (12) so that (see footnote 2)

E (t = ∞, R ⋆ ) = E lim (t = ∞, R ⋆ ) (defining R ⋆ ) i.e. 3 Θ 3 + a 2 =2 M (R ⋆ ) r 3 (T, R ⋆ ) + 1 3 Λ - (3M (R ⋆ )) 2 3 Λ 1 3 r 2 (T, R ⋆ ) . (15)
So, since here the Misner-Sharp mass M and energy E of each shell is conserved in time (without shell crossing), and E is thus set by initial M (R) and ṙi (R) profiles, one can characterise local trapped matter, or limit, shell by the intersections E = E lim (see [5], for details). Global shells emerge from the neighbourhood behaviour around those intersections which local study we give in Secs. III A and III C. Before studying the occurrence of shell-crossing we will now examine more carefully the role of shear in these settings.

B. The role of shear in the existence of trapped matter shells

In Ref. [5], we derived the relation between expansion and shear (see Eq. III.10) and found that, in the presently studied model, the shear could be put in the form

a = ∓ 1 6 E + 2 M r + Λ 3 r 2 × × E ′ r ′ - 2E r + 2 r M ′ r ′ - 3M r . (16)
In the latter equation the terms within the brackets measure the departures from the profiles E = Ē(t) r 2 and M = M (t) r 3 that one would expect from a homogeneous, uniformly curved models . Indeed, in FLRW models E = kr 2 and M ∝ ρ(t)r 3 . Moreover, we should stress that Eq. ( 16) yields the shear in terms of non-local (integral) quantities (E and M ). We can now evaluate the expansion and shear at the limit shell defined by setting Eqs. ( 8) and ( 9) to zero at time infinity in Ref. [5], which, with the conservation of E and M , is simply defined by Eqs. ( 8) and (12). Combining those equations yields

E ′ = - 2M ′ r lim . ( 17
)
First on the limit shell we can write, setting

E = E lim , a = ∓ 2 M ′ r ′ 1 r -1 r lim + 2Λ r 2 lim r 1 -r lim r 6 Λ 3 2 r 3 lim r + r 2 -3r 2 lim . (18)
With the definition of mass issued from Ref. [5]'s Eq. II.27 in GLTB coordinates so

M ′ =4πρr 2 r ′ , (19)
we then express the shear of the limit shell as

a lim = ∓ Λ 3 r r lim 3 1 -4πρ(r) Λ r r lim 3 2 + r r lim , (20)
which in the limit of time infinity simplifies into

a lim∞ = ∓ Λ -4πρ(r lim ) 3 √ Λ . (21)
This quantity does not vanish in general. Since at that locus we have Θ = 3 ṙ r -a , the expansion then reads

Θ lim = ± 3Λ r r lim 3    2 -3 r r lim + r r lim 3 + 1 -4πρ(r) Λ r r lim 3 2 + r r lim    , (22)
which in the limit of time infinity simplifies into

Θ lim∞ = ± Λ -4πρ(r lim ) √ Λ . ( 23
)
We shall now use a particular form of initial data in order to study in more detail the role of shear in the appearance of the diving shell. In the examples below we shall assume M > 0, ρ > 0, Λ > 0 and E < 0 around the origin. So, consider analytic initial data for Λ-LTB as in [19,20] 4 :

M (R) = R 3 ∞ i=0 m i R i , m 0 > 0 (24)
E(R) = R 2 ∞ i=0 E i R i , E 0 < 0 4
This data ensures that the solution approaches FLRW at the origin which is therefore regular.

then, from the expressions above, we derive

a lim (R) = ±Λ 1/2 1 3 - 2 3 2/3 m 1 3 0 + 2m 1 m 2/3 0 R + ( 3m 2 m 2/3 0 - m 2 1 m 5/3 0 )R 2 + O(R 3 ) r lim (R) = ( 3 Λ ) 1 3 m 1 3 0 R + m 1 3m 2/3 0 R 2 + O(R 3 ) E lim (R) = -3 2 3 Λ 1 3 m 2 3 0 R 2 + 2m 1 3m 1/3 0 R 3 + O(R 4 ) (25)
also, for the re-scaling r(t 0 , R) = R, we get an expression for the initial shear distribution as (see also [21]):

a(t 0 , R) = ± E 1 + 2m 1 6A R ± 1 6 2E 2 + 4m 2 A - (E 1 + 2m 1 ) 2 2A 3 R 2 + O(R 3 ) with A(R) = E 0 + Λ/3 + 2m 0 .
It is interesting to see that for a fixed shell R near the centre, bigger M (i.e. bigger m 3 ) means smaller initial shear but bigger |E lim | and r lim for that shell. On the other hand, since bigger initial shear implies smaller |E lim | (i.e. smaller departures from E lim = 0) and smaller r lim , one can argue that, at least around the origin (and for the above initial data), shear contributes to the appearance of "cracking" limit shells. This is in agreement with the results of Herrera et al. [18]. We summarize this result as:

Result 1. Consider a neighbourhood U of the origin where the Λ-LTB initial data can be written as (24). Then, bigger values of the initial shear |a(t 0 , R)| in U , imply smaller |E lim | and favour the occurrence of trapped matter shells in U .

For data which is asymptotically Friedmann at infinity we take functions which, at infinity, can be expanded in the form 5 :

M (R) = +∞ i=1 m i R 3 i , E(R) = +∞ i=1 E i R 2 i
with m 1 = 0 and E 1 = 0.By taking asymptotic expan-

R E lim R ⋆i E = E < (R < R ⋆i ) E = E > (R > R ⋆i ) E E = E lim (R ⋆i )
Bound shells

Unbound shells

Figure 2. Overcoming local configuration of E intersecting E lim . Phase space and effective potential trajectories from dynamical analysis of [5] give the local qualitative behaviour, emphasised on the radial axis. Inner shells on bound trajectories and outer shells on unbound paths forecast no shell crossing locally. Considering E lim as corresponding to the Newtonian zero radial velocity axis in [24,25], this configuration is analogous to, e.g., figure 1 of [26].

sions we find :

r lim (R) = 3 Λ 1 3 m 1 3 1 R + m 2 3m 2 3 1 1 R 1 2 + O( 1 R ) E lim (R) = -3 2 3 Λ 1 3 m 2 3 1 R 2 + 2 3 m 2 m 1 3 1 R 1 2 + 2 3 m 3 m 1 3 1 + O( 1 R 1 4
)

while the initial shear is:

a(t 0 , R) = ∓ E 2 2 √ 3 √ 3E 1 + Λ + 6m 1 1 R + O( 1 R 4 3
) So, again, bigger values of the initial shear |a(t 0 , R)| near infinity, imply smaller |E lim | and favour the occurrence of trapped matter shells. We shall return to this issue in section IV where we study other examples in more detail.

III. SHELL-CROSSING AND TRAPPED MATTER SHELLS

A. Sufficient conditions for shell-crossing

In terms of the comoving coordinates of metric (1), shell-crossing is defined as a surface for which ∂ R r = 0 and the density diverges 6 . In geometrical terms, at shellcrossing there is a discontinuity both in the extrinsic curvature K ij and in the spacetime metric. For the spacetimes considered here, those discontinuities are finite and the magnitude of the jump in K ij can be read from the expressions derived in [15,23].

Hellaby and Lake [6] (see also [7]) have derived necessary and sufficient conditions for the occurrence of shell-crossing in LTB, in terms of the free initial data. Other works have used other type of conditions which are sufficient to avoid shell crossing 7 and therefore imply ∂ R r = 0. For example, in the case of LTB, Landau and Lifshitz [START_REF] Landau | The Classical Theory of Fields[END_REF] simply assume ∂ R r > 0 and, in [START_REF] Hellaby | [END_REF], Hellaby and Lake impose the condition for a simultaneous big bang in their local analysis around the initial singularity.

Here, we shall take a different point of view and write sufficient conditions for the occurrence of shell-crossing in terms of the local behaviour of M and E in the neighbourhood of some intersection, when it exists, of the energy E with the critical energy E lim . In order to do that we first observe that two local configurations are possible in the neighbourhood of the intersection: either

E ′ > E ′ lim or E ′ < E ′ lim . In the case E ′ > E ′
lim , shells just inside the intersection radius will have a lower E than their respective E lim and will therefore be trapped in closed trajectories, following the dynamical analysis presented in Fig. 1. In that case, shells just outside the intersection will display higher E than their respective E lim and will accordingly be free to escape to infinity on unbound trajectories. That shell distribution will lead to the separation of neighbouring shells, those inside the intersection being bound to a finite region while those outside will escape to infinity. This case doesn't entail neighbouring shell crossings and is presented on Fig. 2.

On the contrary, in the case E ′ < E ′ lim , shells just inside the intersection will have a higher E than their respective E lim and will accordingly be free to escape to infinity on unbound trajectories whereas shells just outside the intersection will display a lower E than their respective E lim and will therefore be trapped in closed trajectories. Because of the configuration of that shell distribution, shell crossings of neighbouring shells occur: those inside the intersection escaping to infinity will have to cross those outside which are bound to a finite region. This case is presented on Fig. 3. We summarize this result as: 6 There can exist cases where ∂ R r = 0 and the density does not diverge. At those regular extrema, the extrinsic curvature is discontinuous while the metric is continuous and finite [6,22]. 7 A comment on the occurrence of caustics when using a synchronous reference frame can be found in Ref. [START_REF] Landau | The Classical Theory of Fields[END_REF] ( §97). We must notice though that the latter assumes that the strong energy condition holds, whereas in our present case the cosmological constant evades that assumption.

E E lim E = E > (R < R ⋆j ) E = E < (R > R ⋆j ) R ⋆j E = E lim (R ⋆j ) R Unbound shells Bound shells Figure 3. Undercoming local configuration of E intersecting E lim .
Phase space and effective potential trajectories from dynamical analysis of [5] give the local qualitative behaviour, emphasised on the radial axis. Outer shells on bound trajectories and inner shells on unbound paths will lead to local shell crossing. Considering E lim as corresponding to the Newtonian zero radial velocity axis in [24,25], this configuration is similar to, e.g., figure 2b of [24].

Result 2. Let ∆ = E -E lim and consider a Λ-LTB spacetime where there is R ⋆ such that ∆| R⋆ = 0. Then, a sufficient condition for the existence of shell crossing is

∆ ′ | R⋆ < 0.
We point out that, for Λ = 0, our condition leads to E ′ < 0, which is the condition implicitly considered in [6,7] 8 . In that case, we simply obtain E ′ lim = -8πρr 2 r ′ Λ 3M 1 3 = 0. In this sense, our sufficient condition generalises, for Λ = 0, the result of Ref. [6,7] 9 . We also note that their condition on bang times t b (R) is t ′ b (R) ≤ 0, while we can also allow for t ′ b (R) > 0 as long as t b (R) is less than the initial time t 0 considered here.

There is an interesting analogy between our shellcrossing condition and a similar condition in Newtonian theory. In fact, the Newtonian approach used in [24,25] considers kinematic configurations in velocity-radius two dimensional phase space which lead to one (and three) dimensional Zel'Dovich pancakes (see Refs. [25,29,30] 8 For LTB with Λ = 0, we recall that the necessary and sufficient conditions for no-shell crossing in [6] are:

T ′ B ≤ 0, E ′ ≥ 0, M ′ ≥
0 where T B is the bang time, while the necessary and sufficient conditons for no-shell crossing in [7] are:

T ′ B ≤ 0, E ′ > 0, M ′ ≥ 0.
Therefore, if one of these conditions fails then there will be shell crossing. 9 Although our interest is in the neighbourhood of radius where ∆ = 0, our analysis can be extended to other locations.

for the classical cosmological spherical context). Their behaviour is similar to the local evolutions of the dynamical configurations in Figs. 2 and3. While in [24,25] the authors take the radial axis to separate collapsing and expanding kinematics, here we take E lim locally as a deformed radial axis.

B. Hypotheses and dynamical analysis

Since part of our analysis is based on the E -E lim diagram, it is useful to clarify the constraints introduced by the set of hypotheses we propose.

Regular density distribution

A regular density distribution is motivated by standard cosmological models [11, 29, e.g.]. In the weak energy condition, the density remains positive so the mass profile is initially always monotonously increasing, thus E lim , from Eq. 12, is initially always monotonously decreasing,

∂M ∂R ≥ 0 ⇒ ∂E lim ∂R ≤ 0. ( 27
)
The regularity implies finiteness of the mass and nonzero values for the density at the centre. This constraints their logarithmic slope in the following manner: suppose a value -ǫ for the slope of the density in the centre (ρ ∝ r -ǫ ), then the mass shall behave accordingly as r 3-ǫ . Finiteness of the mass implies then ǫ ≤ 3 and no vacuum in the centre implies ǫ ≥ 0, from the density.

Initial Hubble-type flow

This simplifies the initial velocity profile into one that only admits outgoing radial velocities (positive ṙ), in the fashion of expanding initial conditions in a Hubble flow, although less restrictive. As a consequence of this and the previous condition, the profiles in the centre always respect, in initial conditions, the hierarchy E < E lim , which is crucial for the emergence of a bound core. In this case

E lim = -(3M ) 2 3 Λ 1 3 ∼ R → 0 R 2-2 3 ǫ → 0 as ǫ ≤ 3, E = ṙ2 - 2M R - Λ 3 R 2 ∼ R → 0 -R 2-ǫ → 0, since ṙ ∼ R → 0 R → 0 so the ṙ2 and Λ 3 R 2 both tend to
zero as R 2 and are thus dominated by the -2M R term for ǫ > 0. Thus around the centre,

E E lim ∼ R → 0 = 2 3 2 3 R M Λ 1 3 ∼ R → 0 2 3 2 3 Λ 1 3 R -ǫ 3 > 1 ⇒ E < E lim , for ǫ > 0, E lim < 0.
In the peculiar case of a constant central density (ǫ = 0),

we have M ∼ R → 0 4π 3 ρ 0 R 3 , ṙ ∼ R → 0 ∂ R ṙ0 R = H c R so E = H 2 c -8π 3 ρ 0 -Λ 3 R 2 = H 2 c -8π 3 (ρ 0 + ρ Λ ) R 2 .
In that case, the Hubble-type flow needs to remain moderate in the centre to respect the constraint

∂ R ṙ0 < 4π ρ 2 3 0 (2ρ Λ ) 1 3 + 2 3 (ρ 0 + ρ Λ ) .
In the rest of the paper, we assume the conditions for E < E lim in the centre are met.

Asymptotic spatial cosmological behaviour

If we restrict our explorations to asymptotically cosmological (FLRW) solutions, this implies that at radial infinity the mass and velocity initial profiles, constraining the energy and E lim profiles for all time, shall obey

M -→ R → ∞ 4π 3 ρ b R 3 with 3M 4πR 3 -→ R → ∞ ρ b = ρ b (t) ⇒ E lim -→ R → ∞ -(4πρ b ) 2 3 Λ 1 3 R 2 , ṙi (R) -→ R → ∞ H i R ⇒ E -→ R → ∞ -K R 2 . ( 28
)
We note that the value of the curvature K of the asymptotic FLRW solution compared with the equivalent (4πρ b )

2 3 Λ 1 3
FLRW critical curvature will determine, together with the central constraint E < E lim , the occurence of, at least, one intersection of E and E lim of the E ′ > E ′ lim kind, not inducive of shell crossing (see Sec. III A). Definition 2. Supposing there exists n ∈ N shells verifying equation (15), we order them by initial radius and

denote them R ⋆i , i ∈ [1, n], • R ⋆out ≡ R ⋆n the outermost intersections E = E lim of the initial profiles • R ⋆in ≡ R ⋆1 the innermost initial intersections E = E lim of the initial profiles.

Local mass conservation and Lagrangian frame

Since in our system, the cosmological constant is inert by definition and dust purely interacts gravitationally, we assume, as in [15], that the rest mass of each crossing infinitesimal shell is conserved. The shell crossing event can thus be viewed as an infinitesimal exchange of the relative positions and integrated masses while each shell conserves its own velocity.

δM crossing For r × < r lim δR⋆i R ⋆i δ[E -Elim](R) E +δ E lim+δ E lim E R -2δM r × (R) -2δM r lim R ⋆i+δ
As shell masses M and energies E are conserved between shell crossing events, Eq. ( 8) will govern the motion of individual shells. Keeping initial R = r(0, R) as Lagrangian labels, we can follow the dynamics of the shells using a the simple prescription obtained above without the need to reorder the radial labels as would require a metric extension. Instead, we keep the initial labels all throughout and follow each shell's evolution using Eq. ( 8) and the shell crossing prescription of Sec. III B 4, as e.g. in Ref [24,29].

C. Local effects of shell crossing on trapped matter shells

In this section, we will detail how a test crossing shell affects locally the values of E and E lim around trapped matter shells.

Since each shell conserves its infinitesimal mass, the local effect of an elementary crossing of a system's shell by a test, neighbouring, shell will just exchange their non-local mass in the exchange of their positions10 . As a consequence, their values of E and E lim will also change. The change of E, in Eq. ( 4), is allowed by the shell crossing event.

R δM crossing 2δM r × (R) For r × < r lim E lim+δ E lim δR⋆j δ[E -Elim](R) E 2δM r lim E +δ R ⋆j+δ R ⋆j
A shell crossed at some r × by an infinitesimal mass δM (δM > 0 for inward crossing, < 0 for outward crossing) will see its values shifted as follow (the reciprocal is true for the crossing shell with-δM )

E +δ =E - 2δM r × , (29)
E lim+δ ≃E lim + 2 3 δM M E lim . (30)
Thus, for an inward (resp. outward) crossing, both E and E lim will decrease (resp. increase). Their relative separation, crucial around intersections, will follow

δ∆ ≃2δM 1 r lim - 1 r × , (31)
which generalises the conditions from [6,7] (see Result 2). The sign of this shift is determined by the initial position r(t 0 , R) = R = r i of shells with respect to their r lim . Bound shells can never cross their respective r lim and shells with E = E lim reach their r lim at infinity in time. Thus crossing events involving one bound shell, satisfy 1 r lim -1 r× < 0. However, once escaping shells go beyond their respective r lim , they experience the opposite relative effect on their δ∆. Thus, it is possible to have a crossing of two escaping shells beyond their respective r lim that produce shifts in the opposite direction. However, once beyond their r lim , even drastic changes cannot put shells on closed orbits linked with the centre as they would correspond to points on the outer side of the effective potential (Fig. 1a). Since intersections E = E lim take place in the neighbourhood of bound shells (those with E under their E lim ) we can restrict ourselves to consider local shell crossing in r × < r lim .

E E lim E lim+δ δ[E -Elim](R) δM crossing 2δM r × (R) 2δM r lim δR⋆i For r × < r lim R E +δ R ⋆i+δ R ⋆i
To first order, for inward-going crossing shells, we have δ∆ < 0, as illustrated on Figs. 4 and7, while outwardgoing shells have δ∆ > 0, see Figs. 6 and5. As a consequence, the limit shell defined by the intersection shifts forward (resp. backward) for the two cases of local configurations. The resulting cases are overcoming inward crossings and undercoming outward crossings (resp. overcoming outward crossings and undercoming inward crossings) and are illustrated on Figs. 4 and5 (resp. 6 and 7).

To simplify the qualitative study of the system, we will first consider a prescription where both M and E are conserved, in Secs. III D 1 and III D 2. We will then drop this assumption and include the evolution of trapped matter shells' neighbourhoods, building from infinitesimal shell crossing as described below in Sec. III D 3 to ascertain the qualitative evolution of the system, in Sec. III D 4.

D. Global effect of shell crossing on limit trapped matter shells

Simplest model with shell crossing

In order to study the simplest set of initial conditions where shell crossing occurs, given the constraints of sec. III B from Result 2, we shall consider a model with a single undercoming configuration. The topological constraints coming from the two dimensional E vs. R dia- grams 11 , together with the choice of an open background at infinity 12 leads to initial conditions for E and E lim with three intersections (see Fig. 8), the middle one verifying Result 2. We thus have a model with R ⋆1 = R ⋆in , R ⋆2 and R ⋆3 = R ⋆out defined in its initial conditions. We can now consider the inner system, also called the system, to be circumscribed by R ⋆out . Unbound shells inside this system are in position to escape it and hence define a remarkable shell outside the system:

E lim E -2δM r lim E lim+δ δR⋆j δ[E -Elim](R) ∀R, r × < r lim R inward δM crossing E +δ -2δM r × (R) R ⋆j R ⋆j+δ

Remark 1. The inner or non-bound shells of initial conditions in E and E lim induce a few remarkable features defined as follows:

• We will consider all shells inside R ⋆out as the initial inner system. • We will denote by E max the maximum value of non-bound E in the set of shells inside R ⋆out or outside of it but with horizontal tangent, i.e.

E max = max {E : ((E ′ = 0) ∨ (0 < R ≤ R ⋆out )) ∧ (E ≥ E lim )} • R max is the largest value for which E = E max , i.e. R max = max {R : E (R) = E max } • R f ree is the furthest shell outside R ⋆out with in- creasing E = E max , when it exists, i.e. R f ree = max {R : (R ≥ R ⋆out ) ∧ (E = E max ) ∧ (E ′ (R) > 0)} • We will note E f ree the value of E, when it exists, as E f ree = E (t = t 0 , R f ree ).
With the above definitions, we will now examine the effects of shell crossing on trapped matter shells. 11 For example in the centre (E < E lim ). See Fig. 8. 12 This means E

-→ R → ∞ -k F LRW .R 2 with k F LRW < 0 and E lim -→ R → ∞ -(4πρ b ) 2 3 Λ 1 3 .R 2 E lim E innermost R ⋆ = R ⋆in outermost R ⋆ = R ⋆out R < R ⋆out never crossed by outer shells R E → -k F LRW .R 2 R f ree by inner shells never crossed R > R f ree E f ree E lim → -(4πρ b ) 2 3 Λ 1 3 R 2 R ⋆2 E max Figure 8.
Open background with arbitrary central mass distribution and a single local undercoming intersection. It always gives protected inner shells as well as unmodified cosmological expansion, when keeping integrability despite shell-crossing. Shell crossing entails no fundamental modification.

Limit trapped matter shells in the integrable dynamical system

In a model where both M and E are conserved through shell crossings, we can extend the analysis of [5], as each shell's dynamics remains integrable and is governed by the Lagrangian Eq. ( 8) 13 . In this case, the qualitative dynamical behaviour of the system is entirely determined from the shape of its initial conditions in a E and E lim vs. R diagram.

As we will see in Sec. III D 4, when including the full effects of shell crossing on E and E lim , the key properties of trapped matter shells will be obtained in the limit t → ∞. Since in this section, M and E are assumed to be conserved with time, all the properties deduced here will remain unchanged in that limit. We will therefore express our results in the limit t → ∞, using definitions which evolve from Rem. 1 and are detailed in Appendix A.

From Fig. 8, we can see that all the bound shells will remain under r lim⋆out = r lim (R ⋆out ), while all unbound shell of the inner system will escape it 14 . Thus, considering that bound shells will eventually turnaround and orbit back and forth between the centre and their turnaround radius, we find that all shells inside r (t, R) = r lim⋆out will be crossed from both sides (interior and exterior) 15 . Only the shell R ⋆out will remain uncrossed from outside shells. This leads to the following definition: Definition 3. The outer limit trapped matter shell R t⋆out∞ verifies Def. 1 in the limit t → ∞, in addition to being the outermost such shell which locally is not shell-crossing inducive 16 i.e.

R t⋆out∞ = {R : max {R ⋆∞ } ∧ (E ′ (t → ∞) > E ′ lim (t → ∞))} Note, from Def. 2, that R ⋆out : verifies Def. 3; defines R t⋆out , if E ′ > E ′
lim ; and verifies R t⋆out = R t⋆out∞ in the limit t → ∞.

Remark 2. In Λ-LTB with asymptotic cosmological evolution (FLRW at radial infinity) and initial Hubblelike flow (outwards going) for which shell crossing occurs, the outer limit trapped matter shell is a surface S with the following properties:

• The matter exterior to S follows trapped geodesics, remaining in that exterior.

• The matter inside S can expand and collapse protected from the crossing of outside shells.

• S is the shell with largest R for which the energy E intersects the critical energy E lim , from bound to unbound shells.

The condition for existence of R t⋆out∞ , follows from the properties of R t⋆out , so we obtain the following result: Result 3. Sufficient conditions for the existence of an outer limit trapped matter shell are:

• The FLRW curvature of the background k F LRW < (4πρ b ) 2 3 Λ 1 3 , or • R t⋆out exists, or • The local configuration around R ⋆out is such that E ′ ⋆out > E ′ lim⋆out . Proof. k F LRW < (4πρ b ) 2 3 Λ 1 3 ⇒ E (R → ∞) > E lim (R → ∞) so the last intersection E (R) = E lim (R) is such that E ′ > E ′
lim from a corollary to Bolzano-Weierstrass theorem and the Definition 3 is verified.

We show, in Fig. 8, a diagram with data such that the inner limit trapped matter shell R t⋆out is at R ⋆out . The exterior of the system will include all the unbound shells escaping to infinity. However, the dynamics from Eq. ( 8) allows us to study under what conditions unbound system's shells will never cross shells located in the exterior of the system 17 . Take two different shells R 1 < R 2 , eventually crossing each other at a given radius 18 r × , and with the outer shell more open than the inner shell (i.e. 16 Recall that R⋆∞is defined by solutions in initial R of Eq. 15 taken at t → ∞. 17 Their escape velocity at infinity should never exceed that of exterior shells. 18 This radius is allowed to tend to radial infinity.

E 1 < E 2 for M 1 < M 2 ): E 1 =v 2 1 - Λ 3 r 2 × - 2M 1 r × , with E 1 <E 2 , (32)
E 2 =v 2 2 - Λ 3 r 2 × - 2M 2 r × and M 1 <M 2 , (33)
⇒ ∆v 2 =∆E + 2∆M r × > 0 and (34)

∆v 2 ∼ r × → ∞ ∆E > 0 ⇒ ∀t, v 2 2 >v 2 1 .
(

) 35
Thence shells with E 1 < E 2 and M 1 < M 2 will always remain in the same radial order and the shell with E = E max will then escape all other system's shells. It then appears that, when R f ree exists, all shells with E > E f ree will never be crossed by any shell inside R f ree .

The counterpart to Def. 3 can thus be formulated by defining first E max . In turn, the condition for the existence of E max is that E ≥ E lim , in the limit t → ∞. Thus, Rem. 1 can be adapted here as 19 Definition 4. Suppose E max∞ exists and is defined, in the initial conditions, as

E max∞ = max E| (t→∞) : ((E ′ = 0) ∨ (0 < R ≤ R ⋆out∞ )) ∧ (E ≥ E lim ) (t→∞) . ( 36
)
Then, inner limit trapped matter shells are defined as the locus R f ree∞ such that

R f ree∞ = max {R : (R ≥ R ⋆out∞ ) ∧ (E = E max∞ ) t→∞ ∧ (E ′ (t → ∞, R) > 0)} . ( 37
)
Remark 3. In Λ-LTB with asymptotic cosmological evolution (FLRW at radial infinity) and initial Hubblelike flow (outwards going) for which shell crossing occurs (and E max∞ is defined), the inner limit trapped matter shell is a surface S with the following properties:

• The matter interior to S follows trapped geodesics which remain in that interior.

• The matter exterior to S expands, protected from the crossing of inside shells.

• S is the shell outside the system (defined with R ⋆out∞ ) with energy equal to that of the highest E of non-bound shells, and starting inside of the system, or outside it but with horizontal tangent.

The conditions for existence of R f ree∞ combine the existence of E max∞ with constraints on the background: 19 The following can be formulated also in terms of gauge invariant Lie derivatives, expansion and shear, as seen in appendix B.

Result 4. Sufficient conditions for the existence of an inner limit trapped matter shell are (a) the existence of E max∞ and (b) the existence of E f ree∞ : (a) • There exist initially a non-bound, system shell, or a non-bound shell with horizontal tangent: ∃R :

(0 < R ≤ R ⋆out ∨ E ′ = 0) ∧ E (R) ≥ E lim (R), or • R ⋆out∞ exist s, or • There exist at least one R ⋆i (b) • E max∞ < E (R → ∞), or • ∃R : R ≥ R ⋆out∞ ∧ E (t → ∞, R) = E max∞ ∧ E ′ (R) > 0 Proof. (a) 1/ If we have R such that (0 < R ≤ R ⋆out ∨ E ′ = 0) ∧ E (R) > E lim (R), then,
either it is a maximum so E max exists and, by time evolution of its neighbourhood, E max∞ exists, or, by continuity, in the case when it is not a shell with E ′ = 0 (local maximum), there is a shell with larger E which satisfies Rem. 1 for E max and thus one in its neighbourhood satisfying Def. 5 for E max∞ . 2/ If R ⋆out∞ exists, it is not bound at time infinity and is inside the system, therefore, even if it is the only unbound system shell, it can at least define E max∞ . 3/ If there is only one R * , then it is R ⋆out by Def. 2. We are then in the case 2/ above as this guarantees the existence of R ⋆out∞ .

(b) Since: (i) E max∞ is, by definition, the largest value of E reached at time infinity by inner or outer local maxima shells, (ii) uncrossed outer shells have their E conserved, (iii) asymptotic cosmological conditions render E monotonous near infinity, (iv) evolution of the inner shell R max∞ follows Eqs. 35, (v) the energy profile is continuous, therefore, E ⋆out∞ ≤ E max∞ and by continuity, since

E max∞ < E (R → ∞), exterior shells will obey E ∈ [E ⋆out∞ , E (t → ∞, R → ∞)[ [E max∞ , E (t → ∞, R → ∞)[,
thus there exist at least one shell at time infinity with E = E max∞ . Moreover, for the outermost exterior shell

R xmax∞ = max {R : R ≥ R * out∞ , E (R) = E max∞ } with E = E max∞ , since E max∞ < E (R → ∞), by continuity, all shells outside of it will verify E > E max∞ . Therefore E ′ (R xmax∞ ) > 0 and R xmax∞ = R f ree∞ is fulfilling Def. 4.
We show a diagram in Fig. 8 where we indicate the outer limit trapped matter shell for which R f ree = R f ree∞ , in the model where both E and M are conserved between shell crossings. We thus have found, for that model, that extending the analysis of [5] in the context of shell crossing leads to the emergence of two remarkable shells: an inner limit trapped matter shell and an outer limit trapped matter shell. From their definitions 3 and 4, we can deduce other properties depending on the background cosmological model, namely:

From Result 3, any background with E > E lim will admit an outer limit trapped matter shell. This includes some closed models and all flat and open models.

From Result 4, and under the assumptions of this section, any closed background in our models cannot foster an inner limit trapped matter shell as the finite value of E max∞ is always larger than its energy at radial infinity. Conversely, open models always have an inner limit trapped matter shell (see the example of section IV) and only flat models with moderate enough energy fluctuations (i.e. for which E max < 0 = E (R → ∞)) can allow the existence of an inner limit trapped matter shell. In summary:

Summary 1. Consider a Λ-LTB spacetime with asymptotic cosmological evolution (FLRW at radial infinity) and initial Hubble-like flow (outwards going) for which shell crossing occurs. Then:

1. The global limit trapped matter shell found in the no-shell crossing Λ-LTB examples of [5] is split, if shell crossing occurs, into at most, two global shells, namely an inner limit trapped matter shell and an outer limit trapped matter shell.

For open or flat expanding spacetimes,

(a) there exists always an outer limit trapped matter shell at R t⋆out∞ .

(b) The inner limit trapped matter shell exists in flat backgrounds for sufficiently small initial velocities inside the system limited by R ⋆out∞ .

3. For closed spacetimes, outer limit trapped matter shells are present if k F LRW < (4πρ b ) 2 3 Λ 1 3 and inner limit trapped matter shells cannot be defined if shell crossing occurs, with our definitions.

In the Λ-CDM examples of global limit trapped mat-

ter shell found in [5], inner and outer limit trapped matter shells reduce to one single surface.

E f ree = E ⋆out , so R f ree∞ = R ⋆out = R t⋆out∞ .
In this section, we have assumed that E and M were conserved through shell crossings. In the next section, we drop this assumption and investigate whether our previous results remain true.

Global effect of shell crossing

Since the sign of ∆ = E -E lim determines the binding property of the system, it is useful to give the final values of E and E lim for each shell, labeled i, in terms of the initial R i and M i , reaching areal radius r after crossing shells, with

M (r(R, t), t) = M i + ˆdM in -ˆdM out = M i + ∆M i
where the index in refers to inward crossing, out to outward crossing, j to the shells crossing shell i.

Using definition (12) and integrating Eq. ( 29) over all crossing shells, we get

E lim (r) =E lim (R i ) - 1 + ∆M i M i 2 3 -1 3M i r lim (R i ) , (38)
E(r) =E(R i ) -8π ˆdr j,in -ˆdr j,out ρ (r j ) r 2 j r ×i (r j ) , (39)
where r ×i is a crossing radius. Because of their qualitatively simple shell crossing histories, we can look at the changes for three peculiar shells, singled out on Fig. 8: the innermost limit shell, the outermost limit shell and the maximum E shell initially lying in the interior of the outermost limit shell. The innermost limit shell will only be crossed by more bound shells exterior to it, so ∆M 1 > 0 and

E(r(R ⋆1 )) =E(R ⋆1 ) -8π ˆdr j,in ρ (r j ) r 2 j r ×1 (r j ) . ( 40
)
Since

1 3 ∆M 1 M 1 2 + 2 3 ∆M 1 M 1 3 > 0 (41) ⇔ 1 + ∆M 1 M 1 2 3 -1 < 2 3 ∆M 1 M 1 (42)
and

- 1 r ×1 (r j ) < - 1 max [r ×1 (r j )] < - 1 r lim (R ⋆1 ) , (43)
as the innermost limit shell becomes a bound shell, we get that

∆ [E -E lim ] 1 <2∆M 1 1 r lim (R ⋆1 ) - 1 max [r ×1 (r j )] < 0.
(44)

Thus, the innermost limit shell will globally shift outwards, following the qualitative analysis of Fig. 4.

innermost R ⋆ = R ⋆in R, r.a i /a ∞ E lim → -(4πρ b ) 2 3 Λ 1 3R 2
E max by outer shells

R ⋆2 E → -k F LRW .R 2 E lim E∞ E E lim∞ outermost R ⋆∞ = R ⋆out ∞ R < R ⋆out∞ never crossed outermost R ⋆ = R ⋆out Figure 9
. Illustration, on an open background with arbitrary central mass distribution, of the effect of shell crossing on the inner global limit shell previously defined as the outermost local limit shell. The time variation of the locus of the outermost local limit shell leads to defining it as the time infinity outermost limit shell: this is shown on the extrapolated time infinity energy profiles and linked to the initial energy profile by a connecting curve. The global inner limit shell is then just shifted inwards, compared with the integrable analysis.

In turn, the outermost limit shell will be only crossed by all the unbound shells interior to it, so

E(r(R ⋆out )) =E(R ⋆out ) + 8π ˆdr j,out ρ (r j ) r 2 j r ×out (r j ) ≡E(R ⋆out ) + 2 ∆M out r ×out (R ⋆out ) , (45)
where ∆M out is the positive mass loss of the outermost limit shell and r × is a reduced crossing radius. Note that, by construction, M out > M (R ⋆out , t → ∞) = M out -∆M out > 0. Now, supposing the density distribution remains finite we can decompose the crossing of the outermost limit shell by all escaping inner shells into a series of infinitesimal shell crossings. Thus following Eq. ( 31) we get

d [E -E lim ] = -8πdr j,out ρ (r j,out ) r 2 j,out × 1 r lim (M ⋆out (t ×j )) - 1 r ×out (r j,out ) . (46)
As all shells cross outwards 20 and

1 r ×out (R ⋆out ) > 1 r lim ( r ×out ) ≥ 1 r lim (R ⋆out )
, (47) 20 Note that R⋆out starts as a marginally bound shell well inside its limit radius.

E lim E E lim → -(4πρ b ) 2 3 Λ 1 3 R 2 innermost R ⋆ = R ⋆in R E → -k F LRW .R 2
never crossed by inner shells The time variations of the inner shells' maximum energy function from shell crossings lead to defining it as the outer shell with same energy as the time infinity inner shells' maximum energy function E: this is shown with the highest of extrapolated time infinity Es of inner shells peaks. The global outer limit shell is then just shifted inwards, compared with the integrable analysis.

E f ree∞ R f ree∞ R ⋆2 R ⋆out∞ R > R f ree∞ outermost R ⋆ = R ⋆out E max∞ E max
then, in this case, we have

∆ [E -E lim ] out = 2∆M out 1 r ×out (R ⋆out ) - 1 r lim (R ⋆out ) > 0. (48)
Thus, the outermost limit shell will shift relatively inwards, following the qualitative analysis of Fig. 6. Finally, the maximum E shell initially inside R ⋆out , or with horizontal tangent (its initial radius is R max ), will be only crossed inwards by all the shells starting with radii above it and having an E below E(R max , t → ∞), at the moment of crossing. This shell will then follow

∆ [E -E lim ] max < 2∆M max × 1 r lim (R max ) - 1 max [r ×max (r j )] < 0, (49)
similarly as for the innermost limit shell. We summarize the main result of this section as:

Result 5. Consider a Λ-LTB spacetime where shell crossing exists. Then the metric and extrinsic curvature are discontinuous and the discontinuity in E is given by (29). Furthermore, at R ⋆out , ∆ [E -E lim ] out > 0 and, at R max , ∆ [E -E lim ] max < 0.

Qualitative analysis of limit trapped matter shells

In this section, we argue that the results contained in Summary 1 remain true for the case where M and E are not conserved through shell crossing. We discussed the behaviour of the outermost limit shell R ⋆out and of the outward escaping highest energy inner shell R max in Sec. III D 3. As those determine the two separating shells R t⋆out∞ and R f ree∞ studied above, their modifications by shell crossing will indicate that the effective limit shells are just displaced but obey the same general properties. We illustrate this on the open background example (Fig. 8), for which we separated the study of each limit shell.

In Fig. 9, we represent the construction of the outer trapper matter shell, using the qualitative evolution of R ⋆out and its neighbouring shells from Eq. 48.

In Fig. 10, using the qualitative evolution of E max and its neighbouring shells from Eq. 49, we represent the construction of the inner trapper matter shell for open initial conditions. The subsequent modifications proceed from those qualitative changes and do not modify the formulations of the results from their counterparts in the model where both E and M are conserved between shell crossings.

In the case where E and M are not conserved, the effect of shell crossing on R ⋆out given by Eq. ( 48) implies only that R t⋆out∞ < R t⋆out without qualitative changes and the Definition 3 is verified. In turn, the effect of shell crossing on R f ree depends on the effect on E f ree from Eq. (49) and by the monotonous increase of E near infinity and only implies that R f ree∞ < R f ree .

Therefore, the findings of Sec. III D 2, extending the analysis of [5] in the context of shell crossing, are only quantitatively modified as full shell crossing effects only displace inwards both the inner and outer limit trapped matter shells: the initial outermost intersection of E and E lim gets unbound and the system at infinity gets con- sequently reduced in Lagrangian initial radius. In turn, the maximum energy of the inner regions gets lowered, so the inner limit trapped matter surface is also drawn inwards. This displacement modifies only marginally the conclusions obtained in Sec. III D 2, namely: (i) the splitting of the local trapped matter shell is maintained when those shells exist. (ii) Open, flat and closed models with existing R t⋆out (with k F LRW < (4πρ b )

2 3 Λ 1
3 ) all retain an R t⋆out∞ and (iii) the modification of the maximum energy of the inner regions allows just more asymptotic cosmological flat models to keep their inner limit trapped matter shell, if the shift from E max tends to E max∞ < 0. Therefore, from the sufficient conditions for inner and outer limit trapped matter shells (Res. 4 and 3), the results contained in Summary 1 remain true in the case where M and E are not conserved through shell crossing.

IV. EXAMPLES: NFW PROFILES WITH ONE UNDERCOMING INTERSECTION

In [5], we studied examples of trapped matter shells using cosmological models with a Navarro, Frenk and White (NFW) density profile [31] and a simple parabolic E profile. Here we adapt those profiles in order to present one intersection with E lim of the undercoming type as in the local configuration of Fig. 3, and thus ensure, at the local level, the appearance of shell crossing.

To do so, we use a fourth order polynomial in the canonical Lagrange form, to provide for the behaviour in the intersecting region, that we cut off with an exponential so that an open FLRW term dominates at infinity. We chose the profile to be 0 at the origin and at a characteristic radius R 0 set near the last possible intersection point with the NFW E lim given by Eq. (4.21) of [5],

R 0 ≤ R -1 ,E lim (R -1 ) = -1,
so as to secure the crossings in the physical region. The remaining three points of interpolation are chosen to be set alternately below and above the E lim curve inside the region set by 0 and R 0 . The form is set by

E(R) = m 1 x x 1 x -x 2 x 1 -x 2 x -x 3 x 1 -x 3 x -1 x 1 -1 + m 2 x x 2 x -x 1 x 2 -x 1 x -x 3 x 2 -x 3 x -1 x 2 -1 + m 3 x x 3 x -x 1 x 3 -x 1 x -x 2 x 3 -x 2 x -1 x 3 -1 + ǫ 1 x e -x -k ∞ R 2 ǫ 0 x 2 (x 2 -1) ǫ 0 + 1 , (50)
where x = R/R 0 , we have denoted the three intermediate points as x 1 , x 2 and x 3 , the values of the polynomial at those points by m 1 , m 2 and m 3 , ǫ 1 is a small constant making sure we have the freedom to fit E(R 0 ) = E(0) = 0 where the polynomial itself is built to vanish, ǫ 0 is a small constant making sure the polynomial dominates in the interesting range but allowing the curvature at radial infinity to be set by a Friedmann-type k ∞ . The form (50) automatically vanishes at 0. We chose the polynomial values such that at those points, E is alternately below, above and again below E lim , the last one making sure it remains above -1:

E(R 0 ) =0 = ǫ 1 e -1 -k ∞ R 2 0 ǫ 0 , ⇒ ǫ 0 = ǫ 1 k ∞ R 2 0 e , (51)
E(R 1 ) =gE lim (R 1 ) = (m 1 + ǫ 1 x 1 ) e -x1 -k ∞ R 2 0 ǫ 0 x 4 1 (x 2 1 -1) ǫ 0 + 1 = (m 1 + ǫ 1 x 1 ) e -x1 - ǫ 1 x 4 1 e -(1 -x 2
We illustrate this in Figs. (11) and (12) after choosing the values of the density profile identical to those in [5].

The cut off leaves the region under R 0 almost unaffected by the Friedmann term, so the value of k ∞ is not very relevant there but we set it to -1. We summarize the result of this section as: Result 6. For NFW density and initial data given by (50-54), the Λ-LTB spacetime has three shells R ⋆ such that E| R⋆ = E lim | R⋆ . Furthermore, for this data, there is shell crossing and R ⋆out -R ⋆out∞ < 0 and R f ree -R f ree∞ < 0.

V. CONCLUSIONS

We have studied the effects of shell crossing on the existence of trapped matter shells in Λ-LTB spacetimes. In particular, we have considered initial conditions such that: (i) our models approach a FLRW solution at radial infinity and have an initial outgoing Hubble-type flow (ii) the shell crossing of dust remains pressureless and the mass of infinitely thin shells remains finite.

We have shown that the local trapped matter shells discussed in Ref. [5] split in two shells: one outer limit trapped matter shell and one inner limit trapped matter shell.

We have established sufficient conditions for the existence of such shells in Λ-LTB spacetimes, in terms of initial data for which shell crossing occurs. Furthermore, we have derived a number of properties for those shells using a qualitative approach inspired in newtonian-like frameworks of cosmological kinematical models, as in [24,25].

We have also studied the role of shear in these settings and concluded, as in [18], that shear favours the emergence of trapped matter shells.

Finally, we have given concrete examples where shell crossing occurs and the inner and outer limit trapped matter shells emerge, using NFW data.

As potential applications of our models we note that (i) Due to mass conservation and integrability in the absence of shell crossing at the boundary, the background asymptotic conditions remain FLRW over all time. Therefore, this gives an interesting setting to study the extendability of Birkhoff's theorem to cosmological expanding backgrounds; (ii) Extensions of this work to unsmooth distri-butions of mass should be possible and might give support to current structure formation analyses using the spherical top hat collapse model, in the case of ΛCDM.

L n Θ 3 + a ≡ ṙ r = 1 r 2 r lim r E lim -E ⇔ E(R) = r lim R E lim -R 2 L n Θ 3 + a = r 2 E lim r lim r 3 -L n Θ 3 + a ,
so the condition of existence for E max , that E ≥ E lim , translates into initial condition with the inequality

L n Θ 3 + a = r lim R -1 E lim R 2 + 1 R 2 (E lim -E) ≤ r lim R -1 E lim R 2 < 0,
or at time infinity into

L n Θ 3 + a (t→∞) ≤ r lim r -1 E lim r 2 (t→∞)
< 0.

We get then R max and E max from

R max = max R, E R 2 = -min L n Θ 3 + a - E lim r lim R 3 ∧((E ′ = 0) ∨ (0 < R ≤ R * out ))∧ L n Θ 3 + a ≤ r lim R -1 E lim R 2 < 0 ⇒ E max = -R 2 max min L n Θ 3 + a - E lim r lim R 3 = max R 2 E lim r lim R 3 -L n Θ 3 + a , ((E ′ = 0) ∨ (0 < R ≤ R * out )) ∧ L n Θ 3 + a ≤ r lim R -1 E lim R 2 < 0 .
Taken at t → ∞, this translates into exists. Then, inner limit trapped matter shells are defined, in the models considered with GLTB coordinates, as the locus R f ree∞ such that

R max∞ = max R, E = max -r 2 L n Θ 3 + a - E lim r lim R 3 (t→∞) ∧ ((E ′ = 0) ∨ (0 < R ≤ R * out∞ )) ∧ L n Θ 3 + a ≤
R f ree∞ = max R, (R ≥ R * out∞ ) ∧ Θ 3 + a = E max∞ r 2 + 2 M r 3 + 1 3 Λ t→∞ ∧ (E ′ (t → ∞, R) > 0) . (B2)

Figure 4 .

 4 Figure 4. Effect of an ingoing, infinitesimal test shell-crossing on the energy and critical energy profiles, around the local initial configuration for the overcoming of E lim by E. The initial intersection shell becomes bound on such perturbations and the local intersection shell shifts outwards in radius.

Figure 5 .

 5 Figure 5. Effect of an outgoing, infinitesimal shell-crossing on the energy and critical energy profiles, around the local initial configuration for the undercoming of E lim by E. The initial intersection shell becomes unbound on such perturbations and the local intersection shell shifts outwards in radius.

Figure 6 .

 6 Figure 6. Effect of an outgoing, infinitesimal shell-crossing on the energy and critical energy profiles, around the local initial configuration for the overcoming of E lim by E. The initial intersection shell becomes unbound on such perturbations and the local intersection shell shifts inwards in radius.

Figure 7 .

 7 Figure 7. Effect of an ingoing, infinitesimal test shell-crossing on the energy and critical energy profiles, around the local initial configuration for the undercoming of E lim by E. The initial intersection shell becomes bound on such perturbations and the intersection shell shifts inwards in radius.

Proof. 1 : 4 :

 14 Direct from Definitions 3 and 4 and Result 2 which leads to shell crossings at some R ⋆ . 2(a): From Result 3. 2(b): Direct from Result 4, as open and flat expanding spacetimes admit E (R → ∞) ≥ 0. Some flat spacetimes can exhibit E f ree∞ > 0 while their E -→ R → ∞ 0. For those cases, Definition 4 is never verified. 3: Using Result 3, for closed spacetimes, E -→ R → ∞ -∞ ≪ E f ree∞ , so from Result 4 , Definition 4 is never verified. Applying Definitions 3 and 4 to configurations where there is only one intersection R ⋆ = R ⋆1 = R ⋆out verifying E ′ > E ′ lim , no shell crossing occurs. Thus all E values are constant over time so R ⋆out = R t⋆out∞ , and given the open background,

Figure 10 .

 10 Figure 10. Illustration, on an open background with arbitrary central mass distribution, of the effect of shell crossing on the outer global limit shell previously defined as the outer shell with same energy function as the inner shells' maximum. The time variations of the inner shells' maximum energy function from shell crossings lead to defining it as the outer shell with same energy as the time infinity inner shells' maximum energy function E: this is shown with the highest of extrapolated time infinity Es of inner shells peaks. The global outer limit shell is then just shifted inwards, compared with the integrable analysis.

Figure 11 .

 11 Figure 11. NFW with background E lim and an example of E profile given by Eq. (50-54), setting R0 = 3000, arbitrarily setting x1 = 1 4 , x2 = 1 2 , x3 = 3 4 , ǫ1 = 10 -2 , g = 2 and ǫ = e -1 so that the figure is proportioned.

Figure 12 .

 12 Figure 12. NFW with background E lim and an example of E profile given by Eq. (50-54) in log (-E)-log (-R) scale.

1 ) ǫ1 k∞R 2 0⇒ m 1 + ǫ 1 x 1 x 3 - 1 , 2 0ǫ 0 x 4 2 (x 2 2 - 1 ) ǫ 0 + 1 = (m 2 + ǫ 1 x 2 ) e -x2 + ǫ 1 x 4 2 e- 1 , 2 0ǫ 0 x 4 3 (x 2 3 - 1 ) ǫ 0 + 1 = (m 3 + ǫ 1 x 3 ) e -x3 + ǫ 1 x 4 3 e -( 1 -x 2 3 ) ǫ1 k∞R 2 0⇒ m 3 + ǫ 1 x 3 x 3

 12131222112221233113331233 =gE lim (R 1 )e x1 (52)E(R 2 ) = E lim (R 2 ) g = (m 2 + ǫ 1 x 2 ) e -x2 -k ∞ R (53) E(R 3 ) =E lim (R 3 ) -(E lim (R 3 ) + 1) (1 -ǫ) = (E lim (R 3 ) + 1) ǫ -1 = (m 3 + ǫ 1 x 3 ) e -x3 -k ∞ R = [(E lim (R 3 ) + 1) ǫ -1] e x3

⇒ E max∞ = max r 2 E lim r lim r 3 Definition 6 .

 236 ′ = 0) ∨ (0 < R ≤ R * out∞ ))Suppose that E max∞ defined asE max∞ = max r 2 E lim r lim r 3 ′ = 0) ∨ (0 < R ≤ R * out∞ ))

Note that we only assume this data form at infinity and not around the origin. Otherwise, we would have a non-regular origin.

In this process the other shells of the system, not involved, will remain unaffected and conserve their masses.

The final fate of each shell will always remain on horizontal lines and their gravitational nature, whether bound or unbound will also remain the same throughout their history.

This is indicated on Fig. 8 by horizontal arrows.

This includes the R ⋆in and R ⋆2 shells, locally considered trapped matter shells.

ACKNOWLEDGMENTS

The work of MLeD is supported by CSIC (Spain) under the contract JAEDoc072, with partial support from CICYT project FPA2006-05807, at the IFT, Universidad Autonoma de Madrid, Spain. Financial support from the Portuguese Foundation for Science and Technology (FCT) under contract PTDC/FIS/102742/2008 and contract CERN/FP/109381/2009 is gratefully acknowledged by J. P. M.. F. C. M. is supported by CMAT, Univ. Minho, through FCT plurianual funding, Project No. PTDC/MAT/108921/2008 and CERN/FP/116377/2010 and Grant No. SFRH/BSAB/967/2010.

Appendix A: Time infinity definitions Definition 5. The inner or non-bound shells of initial conditions in E and E lim , in the limit t → ∞, induce a few remarkable features defined as follows:

taken at time infinity but singled out by its radius in the initial profile of E; in particular we note R ⋆out∞ ≡ R ⋆n (t → ∞) for the outermost intersection and R t⋆out∞ ≡ R ⋆out∞ when we add the condition

• We will note E max∞ the maximum value taken at time infinity, but singled out in the initial profile, of non-bound E in the set of shells inside R ⋆out∞ or outside but with initial horizontal tangent, i.e.

Appendix B: Gauge invariant definitions for inner limit trapped matter shells

We can rewrite E in terms of gauge invariant quantities with Eqs. (9, 8, 14, 12 and 11)


Mirja Reichel
email: mirja.reichel@bode-chemie.de

Anke Heisig

Peter Heisig

Günter Kampf

Bacterial response to human β-Defensin-3 after chlorhexidine exposure 1 Skin bacteria after chlorhexidine exposure -Is there a difference in response to human ß-Defensin-3?

Keywords: time-kill study, chlorhexidine, human beta-Defensin, Staphylococcus spp., sequence homology, gyr

Purpose:

We investigated whether exposure to sub-lethal concentrations of chlorhexidine digluconate (CH) changed the response of 5 Staphylococcus spp. to human β-Defensin-3.

Methods:

The change in response for each strain was determined in-vitro with time-kill-experiments in suspension by comparing the mean log 10 -reduction caused by hBD-3 at 1.5 and 3 h in exposed and non-exposed bacteria. Identity of staphylococcal species was verified by DNA

sequence homology in the gyrA genes in comparison with reference strains.

Results:

Baseline sub-lethal concentrations allowing visible bacterial growth were between 0.0625 and 0.25 μg/ml. Sub-lethal CH concentrations increased within 3 days in two isolates. For Staphylococcus capitis 19/2, CH-exposed cells were less susceptible to 0.5 μg/ml hBD-3 (log 10 -reduction 0.78 versus 2.06 at 1.5 h; p <0.001; t-test). For Staphylococcus aureus, however, CH-exposed cells were more susceptible to 1 μg/ml hBD-3. The observed changes between CH-exposed and non-exposed cells did not indicate a general trend in response to hBD-3.

Conclusions:

Overall, we found no consistent evidence that 3 days of exposure to CHG changed the response of 5 Staphylococcus spp. to hBD-3. The use of CHG for skin antisepsis is, based on our data, unlikely to change the natural defence activity of hBD-3.

Background 1

One of the main functions of the skin is to protect the host from microbial infections; however, 2 the skin itself is a potential source of endogenous infections. The skin has established differ-3 ent physiological mechanisms to prevent infections, which can be caused by resident and 4 transient skin microorganisms. Firstly, it acts as a physical barrier. In addition, the skin shows 5 a diversity of immune-relevant processes to control the microbial ecology present on human 6 skin. 7 8 One important mechanism for controlling colonization of the skin is the secretion of constitu-9 tively or inducibly expressed antimicrobial peptides (AMP) which are part of the innate im-10 mune system. These peptides are effective against pathogenic microorganisms and some of 11 them e.g., defensins, are multifunctional signaling molecules that trigger adaptive immune 12 responses [START_REF] Funderburg | Human -defensin-3 activates professional antigen-presenting cells via Tolllike receptors 1 and 2[END_REF]. [START_REF] Reichel | Alcohols for skin antisepsis at clinically relevant skin sites[END_REF] 14 Human β-Defensin-3, a cationic, broad spectrum AMP of the defensin family, is one of the 15 AMPs with multiple biological functions [START_REF] Gallo | Biology and clinical relevance of naturally occurring antimicrobial peptides[END_REF][START_REF] Dhople | The human beta-defensin-3, an antibacterial peptide with multiple biological functions[END_REF]. In contrast to human beta-Defensin-1 and -2, 16 hBD-3 shows potent antimicrobial activity against Gram-positive bacteria. These include S. 17 aureus [START_REF] Harder | Isolation and characterization of human beta -defensin-3, a novel human inducible peptide antibiotic[END_REF][START_REF] Kisich | The constitutive capacity of human keratinocytes to kill Staphylococcus aureus is dependent on beta-defensin 3[END_REF], which is one of the most common causes of nosocomial infections, such as 18 catheter-related bloodstream infections [START_REF] Mangram | Guideline for prevention of surgical site infection, 1999. Hospital Infection Control Practices Advisory Committee[END_REF][START_REF] O'grady | Guidelines for the prevention of intravascular catheter-related infections[END_REF], but also multidrug-resistant nosocomial patho-19 gens, e.g. methicillin-resistant S. aureus (MRSA) and vancomycin-resistant enterococci [4, 20 8]. The hBD-3 peptide is inducibly expressed by keratinocytes of the human skin [START_REF] Sorensen | Wound healing and expression of antimicrobial peptides/polypeptides in human keratinocytes, a consequence of common growth factors[END_REF]. Various 21 stimuli which influence hBD-3 expression have been identified, e.g., the upregulation of hBD-22 3 expression is associated with inflammatory skin diseases like psoriasis. Enhanced amount 23 of hBD-3 has been found after contact of the skin with bacteria [START_REF] Dhople | The human beta-defensin-3, an antibacterial peptide with multiple biological functions[END_REF][START_REF] Harder | Isolation and characterization of human beta -defensin-3, a novel human inducible peptide antibiotic[END_REF][START_REF] Schittek | The role of antimicrobial peptides in human skin and in skin infectious diseases[END_REF]. In addition, current 24 research findings have shown that sterile wounding of the skin may also result in increased 25 hBD-3 expression [START_REF] Sorensen | Injury-induced innate immune response in human skin mediated by transactivation of the epidermal growth factor receptor[END_REF]. [START_REF] Zorko | Alexidine and chlorhexidine bind to lipopolysaccharide and lipoteichoic acid and prevent cell activation by antibiotics[END_REF] Maisetta et al. investigated synergisms in antimicrobial activity between hBD-3 and other 28 antibiotics against oral bacteria. They showed that a combination of chlorhexidine with hBD-3 29 had a beneficial antibacterial effect [START_REF] Maisetta | Activity of human beta-defensin 3 alone or combined with other antimicrobial agents against oral bacteria[END_REF]. [START_REF] Li | Gram-positive threecomponent antimicrobial peptide-sensing system[END_REF]

31

CH is used in skin antisepsis during the placement of central venous catheters (CVC); This is 32 in accordance with recommendations of the Centers for Disease Control (CDC) to use alco-33 hol or a 2% chlorhexidine-based preparation for this procedure [START_REF] O'grady | Guidelines for the prevention of intravascular catheter-related infections[END_REF]. In a previous study, we [START_REF] Brissette | Mechanisms of decreased susceptibility to betadefensins by Treponema denticola[END_REF] were able to show that chlorhexidine digluconate (CHG) in propan-1-ol slows down recoloni-35 zation of the skin for at least 72 h [START_REF] Reichel | Alcohols for skin antisepsis at clinically relevant skin sites[END_REF]. We therefore assumed that CHG persists on the skin 36 for at least 3 days after application when the skin is covered with an adhesive dressing after 37 evaporation of the alcohol. 38 39 CVC may remain in place for an average of 6-28 days and dressings are changed several 40 times during this period [START_REF] Walder | Prevention of bloodstream infections with central venous catheters treated with anti-infective agents depends on catheter type and insertion time: evidence from a meta-analysis[END_REF]. Normally the skin around the catheter site is treated with a CH-41 containing antiseptic repeatedly every time the catheter dressing is changed. Thus, the skin 42 bacteria at the catheter-site are permanently exposed to CH and the physical skin barrier, 43 which is essential for protection of the bloodstream against microorganisms, is permanently 44 disrupted for a long period of time. The results of Sørensen et al. showed that the concentra-45 tion of hBD-3 increases within 4 days after sterile wounding [START_REF] Sorensen | Injury-induced innate immune response in human skin mediated by transactivation of the epidermal growth factor receptor[END_REF]; therefore we assumed that 46 the bacteria are first exposed to CH for at least 3 days before the hBD-3 concentration on 47 skin increases. 48 49 It has been reported that different bacterial species are able to adapt to sub-lethal concentra-50 tions of CH [START_REF] Thomas | Development of resistance to chlorhexidine diacetate in Pseudomonas aeruginosa and the effect of a "residual" concentration[END_REF][START_REF] Suller | Antibiotic and biocide resistance in methicillin-resistant Staphylococcus aureus and vancomycin-resistant enterococcus[END_REF][START_REF] Westergren | In vitro development of chlorhexidine resistance in Streptococcus sanguis and its transmissibility by genetic transformation[END_REF][START_REF] Mcbain | Effects of a chlorhexidine gluconate-containing mouthwash on the vitality and antimicrobial susceptibility of in vitro oral bacterial ecosystems[END_REF][START_REF] Block | Association between intensity of chlorhexidine use and microorganisms of reduced susceptibility in a hospital environment[END_REF][START_REF] Braoudaki | Adaptive resistance to biocides in Salmonella enterica and Escherichia coli O157 and cross-resistance to antimicrobial agents[END_REF] and that this adaptation affects the susceptibility to different biocides [START_REF] Tattawasart | Development of resistance to chlorhexidine diacetate and cetylpyridinium chloride in Pseudomonas stutzeri and changes in antibiotic susceptibility[END_REF] 51 and antibiotics in Pseudomonas stutzeri [START_REF] Suller | Antibiotic and biocide resistance in methicillin-resistant Staphylococcus aureus and vancomycin-resistant enterococcus[END_REF][START_REF] Tattawasart | Development of resistance to chlorhexidine diacetate and cetylpyridinium chloride in Pseudomonas stutzeri and changes in antibiotic susceptibility[END_REF][START_REF] Russell | Possible link between bacterial resistance and use of antibiotics and biocides[END_REF]. Moore et al. showed that different skin 52 bacteria were able to adapt to CH after repeated exposure [START_REF] Moore | In vitro study of the effect of cationic biocides on bacterial population dynamics and susceptibility[END_REF]. 53

To the best of our knowledge, there are no studies on whether CHG exposure changes the 55 response of Gram-positive bacteria to antimicrobial peptides, such as hBD-3. 56

57

The aims of this study were to analyze, using in-vitro time-kill-studies, if exposure to CHG at 58 sub-lethal concentrations for 3 days influences the response of different Staphylococcus spp. 59 to hBD-3, and if this treatment reduces the potency of the natural defense.

Methods 61

Collection and growth of the bacterial strains 62

Six different isolates of skin bacteria were taken from different skin sites of two healthy, male 63 volunteers (volunteer number 16 and 19). From volunteer 16, we obtained two isolates from 64 the upper back after application of 89.5% (v/v) propan-1-ol and one skin isolate from the ab-65 domen. In addition, two isolates were taken from the forehead of volunteer 19 after applica-66 tion of 70% (v/v) propan-2-ol or 89.5% (v/v) propan-1-ol and one isolate was collected from 67 the abdomen. The skin bacteria were collected using a swab method which has previously 68 been described [START_REF] Reichel | Alcohols for skin antisepsis at clinically relevant skin sites[END_REF]. The skin bacteria were grown aerobically on casein-peptone soymeal-69 peptone agar (CASO agar) at 37°C for 48 h; thereafter, six single colonies were isolated, 70 plated separately on CASO agar and aerobically incubated at 37°C for 24 h. For susceptibili-71 ty testing, isolate 19/3, 16/2 and 16/3 were excluded from the study due to clonal-identity 72 (see below; isolate); the chosen isolates are listed in Table 1. The bacteria were stored at -73 74°C using the CRYOBANK-system (MAST Diagnostica Laboratoriumspräparate GmbH, 74 Reinfeld, Germany). 75 76

Identification and characterization of skin isolates 77

All isolates were Gram-positive; they were characterized by macroscopic and microscopic 78 analyses. They were plated on CASO agar, sheep blood agar and selective agar plates (ka-79 namycin and mannitol-salt). The presence of clumping factor, protein A and capsular poly-80 saccharide was analyzed with Staphytect™ plus test (Oxoid Deutschland GmbH; Wesel, 81 Germany). For genotypic characterization of the clonal relationship of the skin isolates 82 PFGE-analysis using the criteria of Tenover et al., 1995 [24] was used (data not shown). Template DNA was isolated from 3 to 5 individual colonies of a type strain of the respective 102 staphylococcal species and from skin isolates of volunteers using the QIAGEN Tissue Purifi-103 cation Kit  according to the manufacturer's protocol. Using 2.5 µl of template DNA in a total 104 volume of 50 µl respective intragenic fragments of gene gyrA were amplified using the follow-105 ing reaction conditions: 1.25 mM MgCl 2 , 30 (40) pMol of each primer of gyrA 50 µmol 106 dNTPs, 2.5 U Taq DNA polymerase (NE Biolabs, Germany) in reaction buffer supplied by the 107 manufacturer. After an initial denaturation at 94°C for 5 min. primer annealing followed for 45 108 s at 50°C , DNA synthesis at 72° for 1 min. and denaturation at 94°C for 30 s over 35 cycles. 109 PCR fragments were purified using the Macherey-Nagel NucleoSpin Extract II Kit  according 110 to the manufacturer's protocol. Samples were eluted with 50 µl of a 1:10 diluted EB-Puffer  111 (QIAGEN, Hilden, Germany).

Between 40 and 80 ng of purified PCR product were used for the subsequent cycle sequenc-113 ing reactions, which were run for 30 cycles (denaturation at 96°C for 20 s; annealing at 50° 114 for 5 s and polymerisation at 60°C for 4 min, in a total volume of 20 µl using 10 pMol of the 115 respective 5'-primer, Quick Start Kit (Beckman Coulter Krefeld, Germany) according to the 116 manufacturer's instructions. 117 118 After completing the cycle sequencing reactions DNA fragments were precipitated by adding 119 5 µl of precipitation solution (2 µl 3M sodium acetate solution pH 5,2; 2 µl 100 mM EDTA pH 120 8,0 and 1 µl Glycogen) and mixed with 60 µl of absolute ethanol. Precipitated DNA was col-121 lected by subsequent centrifugation (15 min, 15,000 rpm, room temperature) washed with 122 ethanol (70%). Samples were air-dried and resuspended in 30 µl SLS ® (Beckman-Coulter , 123 Krefeld, Germany), and immediately cooled to 4°C before applying the DNA samples to the 124 capillary electrophoresis column. To assign an isolate to a staphylococcal species the DNA 125 sequence determined for the respective gyrA gene fragment was aligned to a database of 126 staphylococcal reference strains yielding homologies equal to or higher than 90%, whereas 127 homologies below 90% were found for gyrA gene fragments from different species. 128 129 To assign skin isolate 16/1 to a bacterial species without doubt, an API ID 32 Staph-test 130 (bioMérieux Deutschland GmbH, Nürtingen, Germany) was carried out for biochemical identi-131 fication. 132 133

Determination of oxacillin and chlorhexidine susceptibility (MIC) 134

The MIC determinations were carried out by microdilution [START_REF]European Committee on Antimicrobial Susceptibility Testing (2003) Determination of minimum inhibitory concentrations (MICs) of antibacterial agents by broth microdilution[END_REF] After overnight incubation, each tube was shaken and screened for visible growth. 100 μl of 159 the control sample were added to 9.9 ml of TSB without any additional substances (second 160 and third passage of control in broth). From the CHG-containing samples, the sample with 161 the highest CHG concentration and visible growth was chosen. For inoculation of 9.9 ml TSB 162 containing CHG (as described above), 100 μl of the suspension were taken for each sample. 163

The control and the CHG-dilution series were again incubated for 24 h. 164 165 Day 4: 100 μl of the control sample and the sample with the highest CHG concentration with visible 167 growth were plated homogenously on CASO agar containing 0.1% L-histidine, 0.3% lecithin, 168 and 3% polysorbate 80 and incubated for 24 h. 169 170 Day 5: 171

The lyophilized recombinant hBD-3 (RELIATech, Braunschweig, Germany) was solubilized 172 and diluted in aqua bidest (pH 6.5-6.8) as recommended by the manufacturer of the peptide, 173 to obtain a 10 μl solution containing the defined concentration of hBD-3 (Table 2). To ensure 174 identical conditions for each test, the reconstituted hBD-3 was stored in cooled metal blocks 175 until the test was performed. The tests were carried out within several minutes after prepar-176 ing the diluted peptide solutions. Before the tests started, preliminary tests were carried out 177 to find suitable hBD-3 concentrations for each isolate, to enable us to carry out time-kill-178 studies at 1.5 and 3 h. For each strain two specific hBD-3-concentrations were chosen, and it 179 was predefined that the high concentration must contain fourfold more hBD-3 than the low 180 concentration. 181 182 Colonies on each CASO agar plate were rinsed off with 10 ml phosphate buffer (10 mM), the 183 suspensions were transferred into sterile glass cylinders with glass pearls and homogenized 184 by mixing for at least 10 min. Each suspension was diluted with phosphate buffer (10 mM) to 185 an optical density of 0.125-0.175 as described above. A dilution series was plated in dupli-186 cate to determine the baseline bacterial density. 187 188 All tests were carried out with the CHG-exposed and the non-exposed bacteria. Tests in so-189 lution were carried out in triplicate as follows: 190 191 100 μl of the 1:10 dilution of the adjusted suspension of bacteria were added to each of the 192 batch so that a total volume of 1 ml was obtained in all cases. None of the skin isolates showed hemolysis on blood agar. All skin isolates were susceptible 233 to oxacillin (MIC < 8 μg/ml). The MICs for chlorhexidine were comparable for all tested 234 strains (MIC 0.5 -2 μg/ml). 235

236

The comparison of the resulting DNA sequences with those determined for the staphylococ-237 cal reference strains revealed that isolates 19/1 and 19/2 should be assigned to Staphylo-238 coccus capitis subsp. capitis (homology > 98 %). DNA sequencing results of an internal 284 239 bp PCR fragment of gene gyrA obtained from isolate 16/1 revealed sequence homologies of 240 87%, 88%, 90%, and 89% compared to the reference strains of S. aureus, S. epidermidis, 241 S. warneri and S. capitis subsp. capitis, respectively. The API 32 ID Staph (bioMérieux 242 Deutschland GmbH, Nürtingen, Germany) confirmed the gyrA gene sequence-based identifi-243 cation of isolate 16/1 as Staphylococcus warneri with %id = 99.9% and t = 0.95; 0 biochemi-244 cal results were found against the assignment to this species. 245

246

CHG exposure followed by hBD-3 time-kill-test 247

The concentrations of the CHG exposition as well as the results for the hBD-3-time-kill-tests 248 are presented in Table 2.

250

A twofold decrease in CHG susceptibility was detected for two isolates (19/2 and 16/1) after 251 3 days exposure to CHG. No change in susceptibility was detected for 19/1 and S. epider-252 midis ATCC 12228. However, we found only low turbidity after the first incubation of 253 S. aureus ATCC 6538 with 1 μg/ml CHG and an increasing susceptibility in the second pas-254 sage. 255

256

The results for hBD-3-time-kill testing were inconsistent (Table 2). In 13 out of 20 experi-257 ments we found no statistically significant difference in response to hBD-3 with and without 258 previous CHG exposure. In four cases hBD-3 killed significantly fewer bacteria which had 259 been previously exposed to CHG, whereas in one of these the difference was less than 0.1 260 log-scale. We found a significant decrease in hBD-3 susceptibility for both exposure times 261 only for S. capitis 19/2 and after 3h exposure to hBD-3 for S. warneri 16/1.In three cases 262 more CHG-exposed bacteria were killed by hBD-3 compared to non-exposed bacteria, al-263 though in one case the difference was less than 0.4 log-scales (Figures 12345).

Discussion 265

We found no clear evidence that exposure to sub-lethal concentrations of CHG for 3 days 266 influenced the response of Staphylococcus spp. to hBD-3. Our results show variable 267 changes in time-kill-studies for different strains. In strains that showed a decrease in CHG 268 susceptibility within the 3 day exposure period (S. capitis 19/2 and S. warneri 16/1) we found 269 a significant decrease in response to hBD-3. This may indicate a link between CHG adapta-270 tion and a decrease in response to hBD-3. Further investigation is needed to verify this hy-271 pothesis. 272 CHG concentrations were much lower than the concentrations used in clinical practice. On 320 the other hand, a beneficial effect of CHG has been shown in the prevention of catheter-321 related bloodstream infections, and a synergism with hBD-3 has been demonstrated. If a 322 significant influence of CHG exposure on hBD-3-susceptibility can not be confirmed, the 323 presence of both substances on the skin in combination may contribute to the positive effect 324 seen in the prevention of catheter-related bloodstream infections due to a potentially syner-325 gistic effect against skin bacteria. 326 327 However, the power of this study is limited due to the small number of strains tested. Moreo-328 ver, different results may be obtained with a longer CHG exposure time prior to hBD-3-time-329 kill-tests. It is possible that a time period of 3 days is not long enough to select bacteria which 330 show changes in response to hBD-3. Further investigation with more test strains and other 331 bacterial species is needed to clarify whether CHG-adapted bacteria are significantly less 332 susceptible to hBD-3. Although we carried out all tests in triplicate and found comparable 333 results in the tests for the strains which showed a decrease in susceptibility after CHG expo-334 sure, it would be interesting to see if the same results can be found with another sample of 335 the same peptide. Our data, however, do not allow to give an answer because all tests were 336 carried out with the same batch of hBD-3. If these findings can be verified by other studies, it 337 would be interesting to determine if comparable or even identical resistance mechanisms are 338 the cause. 339 340 Conclusions 341

Apparent differences in the response to hBD-3 were found in bacteria which showed a de-342 crease in susceptibility after exposure to sub-lethal concentrations of CHG for 3 days. How-343 ever, we found no consistent evidence that 3 days of exposure to CHG changed the re-344 sponse of 5 Staphylococcus spp. to hBD-3. The use of CHG for skin antisepsis is, based on 345 our data, unlikely to change the natural defence of hBD-3. 346

Figures

Figure 1 A-E

  type strains were used to amplify the respective intragenic gyrA 86 fragment for sequence-based identification of skin isolates: Staphylococcus aureus ATCC 87 6538, Staphylococcus, warneri ATCC 27836, Staphylococcus warneri ATCC 155, Staphylo-88 coccus capitis subsp. capitis ATCC 27841, Staphylococcus epidermidis ATCC 14990. 89 90 Primers 91Based on a multiple DNA sequence alignment of genes gyrA from different staphylococcal 92 species, regions of high homology were chosen for the design of a primer pair to be used for 93 PCR amplification and for cycle sequencing of the quinolone resistance-determining region 94 (QRDR).The primer pair includes 5'-primer gyrA_[A70V]_5'_:+73_Sau 5'-95 TATGCGATG(A/T)(G/C)(A/C/T)GTTAT(A/C/T)GT -3' corresponding to nts. 73 to 82) and 3'-96 primer gyrA_[A640R]_3'_+661_Sau 5'-CCGTTGG(A/G)AA(A/G)TC(A/T)GG(A/C/T)CC-3' 97 complementary to nts. 661 to 680) giving rise to a 608 bp intragenic PCR fragment. For cycle 98 sequencing reactions primer gyrA-A70V was used.

  . The tests were carried out 135 with the skin isolates, S. aureus ATCC6538 and S. epidermidis ATCC12228. S. aureus 136 ATCC29213 was used as the control strain and oxacillin (InfectoStaph, Infectopharm) was 137 used as the reference antibiotic. The oxacillin-containing Mueller-Hinton broth (DIFCO, 138 Augsburg, Germany) was supplemented with 3% NaCl. 139Preparation of test bacteria and time-kill-analyses with hBD-3 141All tubes and plates were incubated aerobically at 37°C. The following procedure was carried 142 out separately for each isolate. 143 144 Day 0: 145The second passage of each test strain was generated by incubation on CASO agar for 24 htest suspension: A small number of single colonies taken from CASO agar 149 were suspended in physiological saline to adjust the suspension to 10 6 to 10 7 bacteria (opti-150 cal density of 0.125-0.175 at 550 nm; measured with Eppendorf ECOM 6122 (Eppendorf AG, 151 Hamburg, Germany)). 152 153 9.9 ml of tryptic soy broth (TSB) without any additional substances (control without CHG ex-154 posure) as well as 9.9 ml TSB containing different concentrations of CHG were inoculated 155 with 100 μl of bacterial suspension. Both were incubated for 24 h



  susceptibility test with low hBD-3-concentration: 10 μl low concentration of 195 hBD-3-solution in 890 μl phosphate buffer (10mM) 196  susceptibility test with high hBD-3-concentration: 10 μl high concentration of 197 hBD-3-solution in 890 μl phosphate buffer (10mM) 198 199 After the exposure time of 1.5 and 3 h, dilution series were prepared with TSB supplemented 200 with 3% polysorbate 80, 0.3% lecithin, 0.1% L-histidine, and 0.1% L-cysteine; 100 μl-aliquots 201 were plated in duplicate on CASO agar containing 0.1% L-histidine, 0.3% lecithin, and 3% 202 polysorbate 80 and incubated for 48-72 h. The CFU were counted after incubation and trans-203 ferred into log 10 -values. 204 205 Statistical analysis 206 All computer-based statistical calculations were performed with SPSS 13.0 or 15.0 (SPSS 207 Inc., Chicago, IL, USA). 208 209 All plates, with CFU between 15 and 300 were evaluated. The results were evaluated as 210 arithmetic means of the log 10 -values of each duplicate analysis. When the results of two dif-211 ferent dilution steps were within the defined CFU range, the weighted means were calcu-212 lated. 213 214 The log 10 -reduction for both exposure times was calculated by subtracting the means of the 215 triplicate analyses of log 10 CFU count for hBD-3 treated samples from the mean log 10 CFU 216 count of the phosphate buffer control. The change in response to hBD-3 was determined by 217 comparing the mean log 10 -reduction at 1.5 h as well as 3 h in exposed and non-exposed 218 (control) bacteria. 219 220 A t-test was carried out to determine the differences between the log 10 -reduction of the CHG-221 exposed and non-exposed samples after 1.5 h and 3 h for both test concentrations of hBD-3. Results 223 Characterization of isolates and strains 224 Two of the isolates obtained from the forehead of volunteer 19 (19/2 and 19/3), and all three 225 isolates from two different body sites of volunteer 16 (16/1 to 16/3) showed identical PFGE-226 patterns (data not shown) although they were obtained from different body sites respectively 227 different test areas at the forehead. 228 229 The chosen isolates varied in their ability to obtain agglutination with the Staphytect plus™ 230 test (Oxoid Deutschland GmbH; Wesel, Germany). Only isolate 16/1 showed a positive ag-231 glutination reaction. All skin isolates grew on mannitol-salt agar and fermented mannitol. 232

-D. Isolate 16 / 1 ( 4 - 1

 16141 Figure 1 A-E -Reduction of CFU counts of five staphylococcal strains by hBD-3; previously exposed (closed symbols) or not-exposed (open symbols) for 3 days to sub-lethal concentrations of CHG

Table 2 -Comparison of log 10 -reduction of five Staphylococcus spp. by hBD-3 with and without previous CHG-exposure for 3 days

 2

		Turbidity after 24 h with		exposure time hBD-3; 1.5 h	exposure time hBD-3; 3 h
	Isolate	CHG in broth (μg/ml) day 1 day 2 day 3	concentration hBD-3 (μg/ml)	CHG exposed	non CHG exposed	P-value	CHG exposed	non CHG exposed	P-value
						RF±SD	RF±SD		RF±SD	RF±SD
	S. capitis 19/1	0.25	0.25	0.25	0.5 2	2.48 ± 0.36 4.02 ± 0.06	2.53 ± 0.30 3.74 ± 0.32	0.88 0.28	3.29 ± 0.23 5.13 ± 0.58	3.01 ± 0.26 5.13 ± 0	0.24 0.99
	S. capitis 19/2	0.0625	0.125	0.25	0.125 0.5	0.12 ± 0.03 0.78 ± 0.04	0.20 ± 0.02 2.06 ± 0.11 <0.001 0.01	0.23 ± 0.03 1.16 ± 0.09	0.24 ± 0.06 2.58 ± 0.20	0.93 <0.001
	S.warneri 16/1	0.25	0.5	1	1 4	1.06 ± 0.51 1.83 ± 0.27	1.22 ± 0.11 2.38 ± 0.28	0.62 0.07	1.16 ± 0.40 3.63 ± 0.63	2.20 ± 0.34 3.89 ± 1.18	0.03 0.75
	S. epidermidis ATCC 12228	0.25	0.25	0.25	0.125 0.5	0.30 ± 0.11 -0.01 ± 0.11 1.73 ± 0.16 1.49 ± 0.04	0.03 0.06	0.25 ± 0.08 1.90 ± 0.34	-0.26 ± 0.28 1.36 ± 0.04	0.04 0.05
	S. aureus ATCC6538	1 (low turbidity)	0.5	0.5	0.25 1	0.26 ± 0.26 1.65 ± 0.37	0.15 ± 0.04 1.32 ± 0.15	0.50 0.22	0.60 ± 0.14 3.18 ± 0.46	0.28 ± 0.20 1.87 ± 0.35	0.09 0.02

RF = log 10 -reduction

On the one hand, further discussion may be needed if studies verify that CHG-adapted bac-

teria show a reduced response to hBD-3, however, it should be kept in mind that the tested

Acknowledgements 348

The study was funded by Bode Chemie GmbH, Hamburg, Germany. GK and MR are paid 349 employees of Bode Chemie GmbH, Hamburg, Germany. We thank A: Schnasse and T. 350

Claußen for excellent technical assistance with DNA sequence analysis and MIC determina-351 tion, respectively. Furthermore we thank P.-M. Kaulfers for the PFGE-analysis.