Multiple nonspherical structures from the extrema of Szekeres scalars.
We examine the spatial extrema (local maxima, minima and saddle points) of the covariant scalars (density, Hubble expansion, spatial curvature and eigenvalues of the shear and electric Weyl tensors) of the quasi–spherical Szekeres dust models. Sufficient conditions are obtained for the existence of distributions of multiple extrema in spatial comoving locations that can be prescribed through initial conditions. These distributions evolve without shell crossing singularities at least for ever expanding models (with or without cosmological constant) in the full evolution range where the models are valid. By considering the local maxima and minima of the density, our results allow for setting up elaborated networks of “pancake” shaped evolving cold dark matter over–densities and density voids whose spatial distribution and amplitudes can be controlled from initial data compatible with standard early Universe initial conditions. We believe that these results have an enormous range of potential application by providing a fully relativistic non–perturbative coarse grained modelling of cosmic structure at all scales.
pacs:04.20.Jb, 98.80.Jk, 98.65.Dx, 04.40.-b
The current era of “Precision Cosmology” has produced a large amount of high quality observational data at all astrophysical and cosmic scales whose theoretical interpretation requires a robust modelling of self–gravitating systems. Conventionally, the large cosmic scale dynamics of these sources is examined through linear perturbations on a CDM background Liddle (); malikwand (); EMM (), while Newtonian gravity (perturbative and non–perturbative Padma (), as well as numerical simulations simu1 (); simu2 ()) is often employed for self–gravitational systems at smaller galactic and galactic cluster scales .
A non–perturbative approach by means of analytic or numerical solutions of Einstein’s equations EMM (); kras1 (); kras2 (); BKHC2009 () is less favoured to analyse observations because (unless powerful numerical methods are employed) the high non–linear complexity of Einstein’s equations renders realistic models mathematically untractable. As a consequence, extremely idealised toy models are used in most cosmological applications based on fully relativistic and non–perturbative methods. The most prominent example are the spherically symmetric Lemaître–Tolman (LT) models Lema1933 (); Tolm1934 (); Bond1947 () (see extensive reviews in EMM (); kras1 (); kras2 (); BKHC2009 ()), which have been used to describe fully relativistic non–perturbative evolution of cold dark matter (CDM) sources at all scales: from local collapsing structures of galactic scale embedded in an FLRW background (spherical collapse and “top hat” models Padma (); BKHC2009 ()) to Gpc sized cosmic voids in the recent effort to explore the possibility of fitting large scale cosmological observations without the assumption of a dark energy source or a cosmological constant BCK2011 (); marranot (); bisnotval ().
Even if admitting that LT void models have failed to fit large scale observations redlich (); zibin1 (); zibin2 (); bull (); finns (), non–perturbative general relativistic models are still needed and can be useful to probe structure formation scenarios and to provide theoretical support to cosmological observations within the current CDM paradigm. However, less idealised models that are not restricted by spherical symmetry are required for this purpose, as the CDM structures we observe in all scales (from galactic surveys cosmography1 (); cosmography2 ()) is far from spherically symmetric. In this context, an improvement on the limitations of LT models is furnished by the well known Szekeres solutions S75 (); Sz75 (); GoWa1982 () (see their derivation and classification in kras1 (); kras2 (); BKHC2009 (); BCK2011 ()), which are still restrictive but are endowed with more degrees of freedom. While an extensive literature already exists on the usage of Szekeres models to address various problems in cosmic structure modelling and observations fitting Bsz1 (); Bsz2 (); IRGW2008 (); Bole2009-cmb (); KrBo2010 (); Bole2010-sn (); BoCe2010 (); NIT2011 (); MPT (); PMT (); BoSu2011 (); sussbol (); WH2012 (); Buckley (); Vrba (); kokhan1 (); kokhan2 (), their full potential for theoretical and empirical applications to Cosmology is still open for future development.
A specific issue in which Szekeres solutions (specially quasi–spherical models of class I kras1 (); kras2 (); BKHC2009 ()) are particularly helpful is the modelling of cosmic structure. Previous work on this issue Bsz1 (); Bsz2 (); PMT (); BoSu2011 (); sussbol (); WH2012 (); Buckley (); kokhan1 (); kokhan2 () (see review in BKHC2009 ()) has addressed the study of various aspects of non–spherical sources, allowing for a coarse grained description of CDM structures currently observed (see specially Bsz1 (); Bsz2 (); BoSu2011 ()), which typically consist of spatial distributions of spheroidal under–dense regions (voids) of typical 30–50 Mpc size surrounded by a web of elongated filamentary over–dense regions struct ().
Since a distribution of over–densities and voids can be identified with a distribution of spatial maxima and minima of the density, we aim in this work (a complementary related work is found in nuevo ()) at extending and improving this literature by undertaking a comprehensive study of the spatial extrema (local maxima, minima and saddle points) of all Szekeres covariant scalars (not just the density).
By working in spatial spherical coordinates, we show how the conditions for the existence and location of these extrema can be split into interdependent angular and radial conditions, the latter requiring to be handled numerically. However, by considering Szekeres models compatible with a time preserved Periodic Local Homogeneity property based on the existence of a sequence of local homogeneity 2–spheres (analogous to localised FLRW backgrounds), we are able to find rigorous sufficient conditions for the existence of distributions of arbitrary numbers of these spatial extrema.
We also examine how spatial extrema can arise by means of a “simulated shell crossing” mechanism for generic models that do not admit this property. We discuss extensively the classification of the extrema, the conditions for their avoidance of shell crossings and present a procedure to specify their spatial location at all times from selected initial conditions.
These results allow us to set up elaborated networks of an arbitrary number of evolving pancake–like over–densities and density voids, in which the spatial density maxima or minima can be placed in prescribed comoving spatial locations at all times by realistic initial data. We believe that such evolving networks can provide a non–trivial coarse grained description of observed structures in a wide range of astrophysical and cosmological scales (specially in the supercluster scale superstruct ()), and as such can be a very useful tool in current cosmological research.
The section by section contents of the paper are described as follows: we introduce in section II the Szekeres models in spatial spherical coordinates and in terms of the q–scalars and their exact fluctuations sussbol (). The Szekeres dipole and its angular extrema are discussed in section III (particular cases of Szekeres models from special orientation of the dipole are displayed in Table 1). The notion of the “LT seed model” is introduced in section IV. The conditions that define the location of the spatial extrema of all Szekeres scalars are listed and discussed in section V. In section VI we introduce Periodic Local Homogeneity, showing that this property is a sufficient condition for the existence of an arbitrary number of spatial extrema of Szekeres scalars. In section VII we show how spatial extrema can also arise in generic models not compatible with this property by means of a “simulated shell crossings” mechanism. The classification of all spatial extrema as maxima, minima and saddle points is found by qualitative arguments in section VIII. The possibility of avoiding shell crossing singularities in the time evolution of these extrema and possible concavity “inversions” of the extrema (a local maximum evolving into a local minimum) are discussed in sections IX and X. The design of Szekeres models that allow for the description of multiple evolving cosmic structures (defined by the spatial maxima and minima of the density) is discussed in section XI and a numerical example is examined in section XII. A summary and our conclusions are given in section XIII. The article contains four appendices: Appendix A provides the relation between the metric variables and the spherical coordinates we have used and the standard ones, Appendix B lists the regularity conditions at the origin and to avoid shell crossings, Appendix C provides a rigorous classification of the spatial extrema and Appendix D the proof of a necessary condition for the existence of spatial extrema of the scalars.
Ii Szekeres models in spatial spherical coordinates.
The metric of Szekeres models in spherical coordinates is given by Bsz2 (): 111By “Szekeres models” we will refer henceforth to quasi–spherical models of class I with a dust source (see details on the obtention and classification of these models in kras1 (); kras2 (); BKHC2009 ()). The spherical coordinates that we use are those defined as a “stereographic” projection in kras2 (). The standard diagonal representation of the Szekeres metric and the transformations relating it to (1)–(3) are given in Appendix A.
where is defined in (11), the time dependence is contained in the scale factors:
while the Szekeres dipole and its magnitude are given by
where are arbitrary functions satisfying the regularity conditions and .
The main covariant scalars associated with Szekeres models are: the density ; the Hubble scalar with ; the spatial curvature scalar with the Ricci scalar of the hypersurfaces arbitrary constant ; the eigenvalues of the shear and electric Weyl tensors ( is the nonzero conformal invariant of Petrov type D spacetimes). These scalars are expressible in the following concise and elegant form:
where we have introduced the “q–scalars” and their exact fluctuations (for ) defined as sussbol (): 222The q–scalars and their fluctuations have been widely used in various applications of LT models sussmodes (); part1 (); part2 (); perts (); RadProfs (). Their properties are extensively discussed in these references. In references WH2012 (); Vrba () the density q–scalar is denoted by “”.
where and is the determinant of the spatial part of the metric (1)–(3). The q–scalars are related to proper volume averages of their corresponding scalars with weight factor BoSu2011 (); sussbol (). At each 2–sphere of constant the are exact fluctuations around this average, which defines an FLRW background as . The q–scalars and their fluctuations are covariant objects sussbol () and reduce in their linear limit to standard variables of cosmological perturbations in the isochronous comoving gauge part2 (); perts ().
Evaluating the integral in (10) for each scalar yields the following scaling laws and constraints
where the subindex denotes (and will denote henceforth) evaluation at an arbitrary fixed , thus allowing for the study of the dynamics of the models within an initial value framework. Every Szekeres model becomes fully specified by initial conditions furnished by six free parameters: , two of plus the dipole parameters . The models become determined either by solving the Friedman equation (12) or by integrating the evolution equations for the (see sussbol (); nuevo ()).
It is very important to emphasise that the spherical coordinates we are using lack a covariant meaning. Hence, “angular directions” indicate locations along curves with constant (radial rays), which are not spacelike geodesics as in spherically symmetric spacetimes RadAs (). The origin worldline 333We assume henceforth that the LT seed model (section IV) admits a symmetry centre at , and thus its associated Szekeres models admit an origin worldline whose regularity conditions are given in Appendix B.1. The symmetry centre or origin worldline is a physically motivated assumption, but it is not absolutely necessary, as regular LT and Szekeres and models exist that admit either two such worldlines or none (for time slices with the topology of a 3–sphere or of a “wormhole” szwh ()). The results we obtain in the present article can be easily extended to this type of models (see Appendix D of sussbol ()). is not a symmetry centre. However, as opposed to the standard dipole coordinates ) (see Appendix A), the angular coordinates are bounded and thus are very useful to specify coordinate locations and for an intuitive understanding of the properties of the models, for example: it is evident from the metric (1)–(3) that the surfaces of constant and constant are 2–spheres, as setting yields the metric of a 2–sphere with surface areas . These surfaces constitute a smooth foliation of any time slice by non–concentric 2–spheres (see figure 1). This follows from the fact that depends on the angular coordinates, hence the proper radial length along radial rays from to points in any 2–sphere smoothly depends on the angular coordinates of the points.
Iii Angular orientation: the Szekeres dipole.
It follows directly from (1)–(3), (7), (8) and (10) that the angular coordinate dependence of the metric and of all covariant scalars is entirely contained in the Szekeres dipole kras2 (). The spherical coordinates allow us to obtain a very precise specification of a unique angular coordinate location of the dipole at each 2–sphere of constant from the extrema of each 2–sphere: the “angular extrema” that follow from the condition:
which yields two antipodal positions in the coordinates:
which cross the origin. The magnitude of at the curves is
where the subindex denotes (and will denote henceforth) evaluation along the curves and is the dipole magnitude defined in (6). Therefore, the angular extrema at each surface of constant necessarily lie in their intersection with the curves (see figure 2). Since along and along we have at each 2–sphere of constant
while allowing for the radial dependence of the full spatial 3–dimensional extrema of are located in the curves at some such that .
Restrictions on the orientation of the dipole in the coordinates correspond to particular cases of Szekeres models that follow from restrictions on the parameters . These particular cases are listed in Table 1.
|Dipole oriented along a radial ray: one free parameter.|
|Parameter restrictions||Spherical coordinates||Rectangular coordinates|
|two of nonzero,||(general ray)|
Dipole contained in a plane crossing the origin: two free parameters.
|Parameter restrictions||Spherical coordinates||Rectangular coordinates|
Iv The LT seed model.
We shall use the term “LT seed model” to denote the unique spherically symmetric LT model that follows by setting in the metric (1)–(3) and in the covariant scalars (7)–(8). Evidently, every Szekeres model can be constructed from its LT seed model by specifying the dipole parameters that define . Therefore, Szekeres models inherit some of the properties of their LT seed model: the scale factors and the q–scalars in (11) are common to both. In fact, the volume integral (10) evaluated for LT scalars yields the same q–scalars and Friedman equation (11) and (12) sussmodes (); part1 (); perts (). Hence, the LT scalars satisfy the same relations (7)–(8) 444Notice that the density LT fluctuation is exactly the term “” that frequently appears in WH2012 (); Vrba (). :
with the exact LT fluctuations related to Szekeres fluctuations by:
Since the LT fluctuations only depend on , it is useful to express the Szekeres scalars and exclusively in terms of LT objects and , either as:
or, alternatively, in terms of and
with the resulting advantage in both decompositions that all the angular dependence is contained in the dipole term . However, the decomposition (25) is more useful, and is thus preferable, because the q–scalars satisfy simple scaling laws like (11) and thus are easier to manipulate than the standard LT scalars . We assume henceforth that regularity conditions to avoid shell crossings (LABEL:noshx1)–(86) hold everywhere, hence
hold everywhere and thus we can consider the zeroes and signs of both fluctuations interchangeably.
V Location of the spatial extrema of Szekeres scalars.
The conditions for the existence of a spatial 3–dimensional local extremum of at any time slice are given by
with similar conditions for and (unless stated otherwise all extrema defined by (28) will be local). We examine separately below the angular and radial derivatives.
v.1 Angular extrema.
We obtain from (25) the conditions for the angular extrema of at any fixed :
with analogous expressions for and . Evidently we have , but the converse is false. However, the condition defines for all a trivial or degenerate angular extremum of 555The condition may hold in specific values for all (see section VI). It implies (from (10) and (24)), and thus becomes independent of the angular coordinates. The determinants of the Hessian matrix and all its minors (see C) vanish at points that fulfil , and thus this condition marks the same degenerate angular extrema of as a constant function for which all derivatives are trivially zero. The equality also holds at coordinate values such that , but the derivatives of do not vanish at these values. . As we explain in section VI, this condition leads to the Comoving Homogeneity Spheres. Therefore, we have the following important result:
The (non–degenerate) angular extrema of all Szekeres scalars , as well as and , coincide at all with the angular extrema of the Szekeres dipole . Since is a necessary (not sufficient) condition for (28), the spatial extrema of all Szekeres scalars are necessarily located along the curves .
v.2 Radial location of the spatial extrema: extrema of the radial profiles.
The missing condition in (28) for a spatial extremum of is the radial equation for from (25) (and similar equations and from (26)). However, the spatial extrema are necessarily located in the curves , hence these radial conditions must be evaluated for the “radial profiles” of the scalars evaluated in these curves (which for fixed depend only on ). As a consequence, finding the location of the spatial extrema reduces to finding the extrema of the “radial profiles” of the scalars given explicitly by:
and for and
where we used (24) to eliminate and in terms of and . The solutions of (32)–(LABEL:condr2) or (LABEL:condrSP) yield the extrema of the radial profiles (30)–(31), which together with the angular extrema satisfying (LABEL:AEab) lead to the full 3–dimensional spatial extrema complying with (28). Therefore, the location of all spatial extrema of Szekeres scalars in the spherical coordinates is given by
where are given by (16)–(17) and are solutions of the radial conditions (32)–(LABEL:condr2) (for extrema of ) or (LABEL:condrSP) (for extrema of and ). We can ascertain the following points before finding (or discussing existence) of solutions of the radial conditions:
The spatial extrema of all scalars are not comoving. The constraints (32)–(LABEL:condr2) and (LABEL:condrSP) depend on time, hence their solutions are (in general) different for different values of and define constraints of the form . However, to simplify the notation we will denote these radial coordinates simply by .
Spatial extrema of different scalars. Evidently, the radial conditions (32)–(LABEL:condr2) have different solutions for different scalars.
However, the location and classification of these extrema cannot be achieved as long as the main technical difficulty remains: finding solution of the radial conditions (32)–(LABEL:condr2) and (LABEL:condrSP) for . Evidently, solving these radial conditions, for and for generic models and without further assumptions, is practically impossible without numerical methods. However, it is possible to obtain sufficient conditions for the existence of such solutions in specified comoving radial ranges without actually having to solve these constraints.
Vi Sufficient conditions for the existence of spatial extrema.
Sufficient conditions for the existence of spatial extrema of Szekeres scalars follows from imposing “local homogeneity” defined (in a covariant manner) by demanding that the shear and Weyl tensors vanish (which implies ) for all , but only for selected worldlines or surfaces (as opposed to global homogeneity leading to the FLRW limit if this condition holds everywhere kras1 (); kras2 (); sussbol ()). In fact, such local homogeneity holds at the origin worldline if standard regularity conditions are satisfied (see B.1). This type of local homogeneity can also be imposed on a discrete set of comoving 2–spheres marked by fixed radial comoving coordinate values :
Periodic Local Homogeneity (PLH). Consider a sequence of arbitrary nonzero increasing radial comoving coordinate values and open intervals between them
A Szekeres model is compatible with PLH in the spacetime region bounded by the spherical world–tube (see figure 3) if at each 2–sphere the scalars comply at all with
where the subindex denotes evaluation at the fixed comoving radii (LABEL:rstar). We will use the term Comoving Homogeneity Spheres to denote the non–concentric 2–spheres where local homogeneity holds. The origin worldline can be regarded as the Comoving Homogeneity Spheres of zero area.
Szekeres models admitting PLH are characterised by the following properties:
They can be specified by initial conditions in which any two of the tree initial value functions comply with
Notice that these initial conditions are sufficient to define PLH at all (pending shell crossings), since (8) and the constraints (13)–(14) and (24) that relate the and the fluctuations and are preserved for the whole time evolution 666If condition holds only for one of the scalars , it will not hold for the remaining ones (see examples for generic LT models in RadProfs ()). Since does not hold (though one of these scalars may vanish because of (8)), this case does not lead to PLH. Also, for a single scalar does not occur in comoving surfaces and (in general) it only holds for restricted ranges of the time evolution. We discuss this case in section X. . We only need two of the three scalars to comply with (LABEL:localhom) and (LABEL:localhom1ab) because the fluctuations are related by these constraints.
They become partitioned by inhomogeneity shell regions (see figure 3) bounded by Comoving Homogeneity Spheres. Since condition (LABEL:localhom) that defines these 2–spheres holds in an asymptotic FLRW background perts (), each pair of these contiguous spheres plays the role of time preserved localised FLRW backgrounds that surround these inhomogeneity shells where the extrema of the scalars are located.
Proposition 1. PLH defined by (LABEL:localhom) and (LABEL:localhom1ab) constitutes a sufficient condition for the existence of spatial extrema (not located in the origin) of the Szekeres scalars and of for all .
the radial coordinates of each pair are (for each scalar) solutions of the radial conditions (32)–(LABEL:condr2) or (LABEL:condrSP) in the intervals , with the given by (LABEL:rstar).
spatial extrema at are located in the curve and those at in the curve .
Proof. As we prove in Appendix D, PLH defined by (LABEL:localhom) is a sufficient condition for the existence, at all , of extrema of each of the radial profiles in the radial ranges . Therefore, the radial coordinates of the pairs of spatial extrema with correspond to the radial coordinates of the extrema with of and given in (30)–(31), as these are the radial profiles of the scalars and along the curves , and thus satisfy (LABEL:AEab). Since they are solutions of the radial constraints (32)–(LABEL:condr2) or (LABEL:condrSP) for all , they also satisfy (28). Since PLH is preserved by the time evolution, the existence and location of the extrema at the intervals in the curves are also preserved in time (pending shell crossings, see section IX).
Proposition 1 can also be proven by qualitative arguments assisted by figure 4. The following inequalities that follow directly from (30)–(31) hold at all for any Szekeres model complying with standard regularity conditions (see Appendixes B.1 and B.2)
These inequalities, which also hold for Szekeres models compatible with PLH, lead to a qualitative but robust inference of the time preserved concavity of the radial profiles (30)–(31) showing various possible sequences of maxima and minima along the radial intervals . This is illustrated in figure 4:
Panels (a) and (b): the fluctuations keep the same sign in all . There are saddle points of at the , the maxima and minima of and lie inside the intervals and at the .
Panels (c) and (d): the fluctuations alternate signs along the sequence of . Maxima and minima of occur at the , while maxima and minima of lie inside the intervals .
As the figures reveal, the order of the maxima and minima depends on the type of extrema at the origin (or the sign of in the first interval ). Evidently, any combination of the four patterns displayed can also be defined by PLH initial conditions (LABEL:localhom1ab).
PLH allows for the existence at all of an arbitrary number of extrema in the specified intervals and in specified angles , which follow from (16)–(17) through a choice of dipole parameters . In principle, even models with an infinite number of extrema can be considered for an infinite sequence of . However, simple dipole configurations (as those found in most of the literature Bsz2 (); NIT2011 (); BoCe2010 (); MPT (); PMT (); WH2012 ()) can also be constructed by PLH initial conditions (LABEL:localhom1ab) that admit a single Comoving Homogeneity Sphere at some , leading to three extrema: one at the origin and the other two located in coordinates with .
Vii Extrema from “simulated shell crossings” induced by the dipole parameters.
PLH only provides sufficient (not necessary) conditions for the existence of spatial extrema of all Szekeres scalars. As discussed in Appendix D.1, extrema for monotonic radial profiles may occur in LT and Szekeres models under special assumptions on the profiles. However, as shown in the numerical examples of Bsz2 (); BoSu2011 (); Buckley (), spatial extrema of all Szekeres scalars may occur, even without the special conditions discussed in Appendix D.1, in models defined by generic initial conditions, and thus admitting completely arbitrary radial profiles of and of the scalars (LABEL:LTscals) of the LT seed model.
Generating spatial extrema in generic Szekeres models is specially straightforward for extrema along the curve , since (see figure 5) the profiles of grow significantly in a given radial range if the term (with given by (6)) in the denominator of (30)–(31) becomes sufficiently small for specific choices of dipole parameters , which may occur regardless of the choice of initial value functions . Since marks a shell crossing singularity at the curve (see B.2), and thus the radial profiles diverge (which implies that the scalars themselves diverge), then a choice of dipole parameters in which is positive but close to zero produces a maximum or minimum via a large growth (or decay) of these profiles (shown in figure 5) by creating (in specific radial ranges) conditions close to a shell crossing (i.e. regular conditions that simulate or approximate the behaviour near a shell crossing singularity).
In order to illustrate that a maximum of (and thus a spatial maximum along ) is easy to achieve from a simulated shell crossing, we consider a choice of dipole parameters such that has a minimum at some for a fixed . Let (for sufficiently small ) be the minimal value of , hence necessarily holds. Under these simple generic assumptions the radial conditions (32) and (LABEL:condrSP) for an extremum of either one of the radial profiles become for
and are easily satisfied for a very wide range of generic initial conditions and radial profiles of in which holds but and have opposite signs, as this combination of signs is very common: grows () with the concavity of a maximum () or decays () with the concavity of a minimum (). The type of extremum obtained by a minimum of at depends on the concavity of the profile of , which is determined by the sign of and follows from the type of extremum at the origin that is common to , but does not depend on the choice of dipole parameters because . Hence, as illustrated in figure 5, if there is a central minimum of and (LABEL:noPLHcond1ac) hold, the extremum at is necessarily a maximum (and vice versa).
which for clearly conveys a large growth or decay of these profiles depending on the sign of (or, equivalently, the sign of ).
An extremum of (and thus a spatial extremum along ) can also occur without assuming PLH, but the conditions for this are far more stringent even if we attempt a simulated shell crossing, since following (