On spherical dust fluctuations: the exact vs. the perturbative approach
Abstract
We examine the relation between the dynamics of Lemaître–Tolman–Bondi (LTB) dust models (with and without ) and the dynamics of dust perturbations in two of the more familiar formalisms used in cosmology: the metric based Cosmological Perturbation Theory (CPT) and the Covariant Gauge Invariant (GIC) perturbations. For this purpose we recast the evolution of LTB models in terms of a covariant and gauge invariant formalism of local and non–local “exact fluctuations” on a Friedmann–Lemaître–Robertson–Walker (FLRW) background defined by suitable averages of covariant scalars. We examine the properties of these fluctuations, which can be defined for a confined comoving domain or for an asymptotic domain extending to whole time slices. In particular, the non–local density fluctuation provides a covariant and precise definition for the notion of the “density contrast”. We show that in their linear regime these LTB exact fluctuations (local and non–local) are fully equivalent to the conventional cosmological perturbations in the synchronouscomoving gauge of CPT and to GIC perturbations. As an immediate consequence, we show the timeinvariance of the spatial curvature perturbation in a simple form. The present work may provide important theoretical connections between the exact and perturbative (linear or no–linear) approach to the dynamics of dust sources in General Relativity.
pacs:
98.80.k, 04.20.q, 95.36.+x, 95.35.+dI Introduction.
Galaxy surveys represent a key probe of the fundamental properties of our universe. Inhomogeneities in the distribution of galaxies can be related to the underlying inhomogeneous distribution of dark matter. Consequently, by observing fluctuations in the galaxy distribution at different redshifts, one can both study the growth of dark matter perturbations and probe the nature of the gravitational action. An essential tool for describing and understanding cosmic dynamics on different scales is the study of perturbations on a FLRW background. The most favored approach to perturbations is the framework generically known as Cosmological Perturbation Theory (CPT) which relies on the smallness of quantities that describe fluctuations from the homogeneous and isotropic FLRW spacetime (see e.g. Zel:70 (); tomita (); bardeen () for pioneering work). This approach is based on suitably defined Gauge Invariant quantities whose definition and evolution equations can be found in the essential reviews (e.g. Mukhanov (); Bernardeau (); malik:wands ()). While CPT is based on metric perturbations, there is an alternative and equivalent “Gauge Invariant Covariant” (GIC) formalism based on covariant tensorial quantities defined by a 4–velocity field BDE (); 1plus3 (); TCM ().
Perturbations based on CPT are adequate (and widely employed) in the study of cosmic sources during the early stages of evolution of the Universe, where it is safe to assume near homogeneous conditions. This approximation is supported by the nearly isotropic (to within one part in ) Cosmic Microwave Background Radiation, which together with the Almost Geren and Sachs Theorem almost:geren () provides a strong motivation for using a spacetime close to a FLRW model.
At late times, however, relativistic linear perturbations based on CPT are only adequate for scales comparable to the Hubble radius . On scales much smaller than , the formation of cosmic structure is the dominant gravitational process. This is highly nonlinear but assumed to take place in non–relativistic Newtonian conditions, so it is usually studied by means a wide range of Newtonian gravity models ranging from simple toy models (“Top hat” or “spherical collapse” Padma ()) to more sophisticated numerical Nbody simulations Bernardeau ().
The study of gravitational collapse through CPT has improved by extending the scope to the nonlinear regime (see e.g. BruMaMol (); tomita:nonlinear (); bruni:2013 (); bruni:2014 (); rampf:rigo ()). Within the perturbative approach, however, only the mildly nonlinear regime can be modelled, a far from complete analysis of the collapse process up to the virilarisation stage where the density contrast is of order .
This leaves an important area unexplored, namely, how nonlinear relativistic corrections impact on the formation of large scale structure, see for example nonlinear () for an extensive review. Indeed, some of these effects have begun to be taken into account in Nbody methods, which make use of relativistic corrections to the potentials relativisticNbody1 (); relativisticNbody2 ().
From a non–perturbative perspective, the spherically symmetric exact solutions of Einstein’s equations generically known as Lemaître–Tolman–Bondi (LTB) dust models provide an idealized, but useful, toy model description of inhomogeneous configurations of astrophysical and cosmological interest (see comprehensive reviews of these models in kras1 (); kras2 (); BKHC2009 ()). While a nonzero term can be easily incorporated into the dynamics of these exact solutions, these solutions have been widely used to model large scale CDM density voids to fit observational data without assuming the existence of dark energy or a cosmological constant (see reviews in bisnotwal (); marranot ()). Moreover, if we assume that , LTB models become an inhomogeneous generalization of the –CDM model describing exact non–perturbative CDM inhomogeneities in a –CDM background favored by observations (see Romano ()). In fact, observational data also fit LTB models with and an FLRW background that is not necessarily the usual –CDM background LLTB ().
Introducing a representation based on covariant scalars (q–scalars part1 ()) and their associated “exact fluctuations” allows for a clear study of important properties of LTB models: their phase space evolution as a dynamical system sussDS2 (); sussmodes (), their radial asymptotics RadAs (), the nature and evolution of density profiles RadProfs (), as well as their use to probe theoretical formalisms of spacetime averaging sussBR () (see review in part1 ()) and gravitational entropy susslar (). It is important to remark that in these references the exact fluctuations were called “exact perturbations”, which may not be a convenient name because the term “perturbation” is commonly understood to refer to approximated (not exact) quantities.
It is a well known fact (see extensive work in part2 ()) that the q–scalars and their fluctuations, in their local and non–local versions, fully determine the dynamics of LTB models recast in terms of evolution equations, analogous to those of linear perturbations on an FLRW background. This resemblance can be reframed in precise unambiguous terms by a rigorous correspondence maps that give rise to a rigorous covariant and gauge invariant perturbation formalism. In particular, it can be shown that the density fluctuation can be expressed in terms of exact covariant expressions that generalize the density growing and decaying modes of linear dust perturbations sussmodes (). Also, the non–local density fluctuation provides a precise covariant characterization of the intuitive notion of the “density contrast”, a concept loosely, and often incorrectly, employed in many astrophysical and cosmological applications of LTB models.
In the present paper we extend the abovementioned studies by establishing equivalences between the perturbative CPT and GIC quantities and exact inhomogeneities defined through the exact fluctuations. Throughout this paper, we are careful to stress the fact that the evolution equations for the fluctuations do not describe “small” deviations from a FLRW background, but the evolution of exact quantities of an exact solution of GR (LTB models). Yet we show how, in a suitable linear regime, these fluctuations reduce to the spherically symmetric linear perturbations of the GIC and CPT formalisms. This result is summarised in Table I of Sec. IX. In verifying this correspondence we consider the comoving gauge of CPT dust perturbations (as LTB models are defined in a comoving frame). We argue that the exact fluctuations represent a generalisation of the usual perturbation scalars to the nonlinear regime, as first suggested in part2 () and here extended to the case . Analysing such generalisations is important to determine the fate of small fluctuations throughout the nonlinear stages of structure formation, a regime poorly explored in relativistic cosmology. We also show that the time conservation of the spatial curvature perturbation of CPT theory can be expressed (up to linear terms) in terms of time preserved quantities of LTB models.
The paper is organized as follows. In section II we introduce the LTB models in terms of the q–scalars formed from the standard fluid flow covariant LTB scalars: the energy density , the Hubble expansion , and the spatial curvature (where is the threedimensional Ricci scalar). In section III we define the fluctuations as exact deviations between the q–scalars and the standard covariant scalars . In section IV we introduce the non–local fluctuations as exact deviations with respect to the “q–averages”, which are the non–local functionals associated with the q–scalars . Asymptotic non–local fluctuations are discussed in section V. The conditions that define a linear regime in LTB exact fluctuations are given in section VI. The comparison between all fluctuations in the linear regime with the CPT formalism is described in section VII, while the correspondence with the GIC perturbations is discussed in section VIII. We summarise and discuss our results in the final Section IX, where we present a useful perturbationtofluctuation dictionary in Table I. The relation between the LTB metric variables that we used and the standard ones is given in Appendix A. We examine in Appendix B the Darmois matching conditions that are used for the rigorous definition of an FLRW background for the exact fluctuations, while the form of LTB metric functions in the linear regime are discussed in Appendix C.
Ii LTB dust models in the q–scalar representation.
A convenient parametrization of LTB dust models is given by the following useful FLRW–like metric (the relation with the standard metric variables is given in Appendix A):
(1) 
where the scale factors and satisfy:
(2)  
(3) 
while the functions and are defined further ahead (see Eq. (6)). The subindex will denote henceforth evaluation at an arbitrary fiducial hypersurface , which can be taken as the present cosmic time. Notice that we have chosen the radial coordinate so that .
The standard approach to LTB models is based on using the solutions (whether analytic or numerical) of (2) to determine the metric functions and in order to compute all relevant quantities. We follow here a different approach, based on a set of useful alternative variables called “q–scalars”, constructed with the standard covariant scalars sussmodes (); part1 (); part2 () ^{1}^{1}1The connection between these integral definitions and a weighted proper volume average is discussed in section IV. See a comprehensive discussion in part1 (); sussBR ().
(4) 
where and is the energy density. The homogeneous expansion is , with the gradient projected in the hypersurfaces orthogonal to . Also is the spatial curvature of these hypersurfaces, with the threeRicci scalar. The scalars and in Eq. (4) are related through the following “exact fluctuations” ^{2}^{2}2We discuss in detail the notion of an “exact fluctuation” in the following section. The q–scalars and the exact fluctuations are directly related to curvature and kinematic scalars part1 (). The domain of integration in the integrals in (4) and (5a)–(5c) is a spherical comoving domain parametrized by , where is a symmetry center. See part1 (); part2 (); sussmodes () for a comprehensive discussion on the definition and properties of these variables.
(5a)  
(5b)  
(5c) 
The q–scalars and the exact fluctuations and satisfy the following scaling laws derived from the energy conservation equation and the components of the Einstein equations part1 (); part2 (); sussmodes ():
(6)  
(7) 
These are complemented by the algebraic constraints (analogous to the “Hamiltonian” and spatial curvature constraints)
(8)  
(9) 
where the subindex denotes evaluation at an arbitrary fixed and we have introduced, together with , in (7), the relative exact fluctuation ^{3}^{3}3The term “perturbation” was used in part1 (); part2 (); sussmodes () only to denote the dimensionless quotient fluctuations , while and in (5b) and (5c) were called “fluctuations”. In this article the term “exact fluctuations” will denote both the and the . We consider as basic set of exact fluctuations the quantities because they provide a straightforward link to perturbation formalisms in the literature in which only the density perturbation is constructed in the dimensionless quotient form (5a) (inspired on the intuitive notion of the density contrast). Besides this point, the exact relative fluctuations and constructed as in (10)can become ill–defined (they diverge) if or (which appear in the denominator) vanish, which can occur in physically interesting scenarios in LTB models, for example: occurs at the “bounce” from expansion to collapse in collapsing models, or necessarily holds along a comoving “boundary” layer separating comoving regions in which switches sign.:
(10) 
Any LTB model becomes fully determined, either analytically (if or in certain cases with sussDS2 ()) or numerically (the general case ), and can be uniquely specified by selecting a value of and, as free parameters or initial conditions, any two of the initial value functions .
The analytic forms (6)–(10) are exact solutions of the evolution equations constructed from the variables and sussmodes (); part2 (),
(11a)  
(11b)  
(11c)  
(11d) 
subject to the algebraic constraints (8)–(9), which will hold for all once we solve them by specifying initial conditions at arbitrary . Combining the evolution equations (11a)–(11b) leads to the second order equation
(12) 
which is an exact generalization of the well known evolution equation of linear dust perturbations in the comoving gauge lyth:liddle (). The constraints (8)–(9) allow for the construction of systems equivalent to (11a)–(11d), but based on alternative set of variables and/or relative fluctuations (see examples for the case in equations (21a)–(21d) of part2 ()).
Iii Local exact fluctuations.
It is intuitively clear that we can identify in the system (11a)–(11d) the subset of evolution equations (11a)–(11b) for FLRW–like “background variables” , as these are identical to FLRW evolution equations for their equivalent FLRW scalars (although scalars also carry a spatial dependence). On the other hand the subset (11c)–(11d) corresponds to the evolution equations of the exact fluctuations defined in (5a)–(5c).
iii.1 The notion of an “exact fluctuation”.
The connection between “exact fluctuations” and “perturbations” requires further clarification given the common use of these terms in the literature. Consider for example the relation that follows from (5a): this is an exact relation, and thus it does not require a small parameter expansion to describe departures of from , since both and are exact LTB scalars (in other words: we have not assumed and need not assume a small ). The same argument goes for the relations between vs and vs that follow from (5b) and (5c). Hence, we call “exact fluctuations” in order to distinguish them from the common usage of the term “perturbations” in standard formalisms of Cosmological Perturbation Theory, namely: quantities defining a “perturbed” spacetime that is “almost FLRW”, meaning that it represents a “small departure” from a suitable known background FLRW spacetime through a linearization procedure applied to characteristic quantities (metric, scalars, vectors, tensors) of the latter. In our case the “exact fluctuations” relate an exact LTB model (the spacetime ) to a precise exact FLRW background defined by suitable scalars (the ) of the same LTB model once we choose a given comoving domain . In other words, the evolve in an identical way to their FLRW equivalents when evaluated at a specific coordinate .
iii.2 Fluctuationtoperturbation correspondence maps and gauge invariance.
The set of exact fluctuations in (4) and (5a)–(5c) are covariant local objects, as they provide the exact deviation between the covariant scalars and their corresponding q–scalars (which are also covariant sussmodes (); part2 ()) along every concentric 2–sphere labeled by constant that marks the boundary of an integration domain (a spherical comoving region). This is illustrated in Figures 1 and 2. We remark that the definition of exact fluctuations can easily be extended to the non–spherical Szekeres models sussbol (). Their role as exact fluctuations can be defined rigorously through a covariant and gaugeinvariant formalism (see part2 ()).
Since any LTB model () and any dust FLRW spacetime () share the same comoving geodesic 4–velocity, spherical comoving coordinates and dust source, the appropriate correspondence mapping that defines the exact fluctuations is furnished rigorously by associating to each comoving domain of the LTB model () the unique FLRW dust spacetime defined by the continuity of the 3–metric and extrinsic curvature of the common “interface” hypersurface (world–tube generated by comoving observers at fixed arbitrary )^{4}^{4}4In previous papers, e.g. part2 (), this mapping was denoted by a ”perturbation” mapping, but we prefer to call it fluctuation–to–perturbation mapping to avoid the semantic problem emanating from the fact that the term ”perturbation” is used to describe approximate quantities.. As shown in Appendix B, this is equivalent to the conditions for a smooth match of and at an arbitrary , which implies the continuity of the q–scalars and the FLRW scalars at for all
(13) 
where denotes evaluation at fixed . are given by Eq. (6) and an over bar will hereafter denote FLRW scalars. It is important to remark that this identification of and is strictly a rigorous and precise procedure to define an FLRW background and q–perturbations for every of a generic LTB model: it does not require that we undertake an actual matching of the domain and in the form of a “Swiss Cheese” configuration (see Figures 1 and 2 and reference part2 () for a comprehensive discussion).
The gauge invariance (GI) of the exact fluctuations follows from the Stewart–Walker lemma Stewart:1974uz (): a GI “perturbation” is any nonzero quantity in that vanishes in the background for a given perturbation formalism in which and have been defined and related through suitable mappings (as for example the map introduced in part2 () summarized above). It is important to bear in mind that the conditions in Eq. (13) do not involve the continuity of the usual covariant scalars , as and thus and hold in general for an arbitrary part2 () (see Figure 2 and Appendix B). As a consequence, the GI criterion based on Stewart’s lemma does not require to vanish at any fixed , but to vanish in the FLRW spacetime characterized by the scalars in that has been mapped through the Darmois conditions (13). In particular, if we impose as supplementary conditions besides (13), that hold for any given fixed finite ; then we are forcing . This leads to “Swiss Cheese” models of exact fluctuations (see Figure 2 and Appendix B).
Iv Non–local exact fluctuations.
We can also consider (4) as the correspondence rule of a linear functional for a single (but arbitrary) domain . This functional is the “q–average”, which assigns to each scalar and each arbitrary fixed comoving domain the real number part1 (); part2 () ^{5}^{5}5The q–average is the proper volume average of with weight factor over the comoving domain with . A detailed comparison with the standard proper volume average emerging from Buchert’s formalism is given in part1 (); part2 () (see also sussBR ()).
(14) 
where . It is important to remark that the q–averages and the q–scalars are different objects: the q–averages are non–local because is a single real number assigned to the whole domain , and thus must be treated as an effective constant for inner concentric domains with (see Figures 1c, 1d, 3a and 3d), whereas is a function of the domain boundary and thus it smoothly varies for these inner domains (see Figure 1). Hence, they only coincide at the domain boundary: of every (see also comprehensive discussion on this in part1 (); part2 ()).
Since the average is a non–local quantity, we can construct non–local exact fluctuations in an analogous way as the exact fluctuations and in (5a)–(5c) part2 () as follows:
(15) 
such that
(16) 
where and depend on . The relations involving the gradients of given by (5a)–(5c) for are only valid for . The non–local nature of the exact fluctuations (15) follows from the fact that they compare (for all ) the local values with a non–local quantity assigned by Eq. (14) to the whole domain (see Figure 2).
iv.1 Evolution equations and background variables as averages.
Combining (15) and (16) we obtain the relation between exact fluctuations and their non–local analogues:
(17a)  
(17b)  
(17c) 
which upon substitution in (11a)–(11d) yields an analogous set of evolution equations:
(18a)  
(18b)  
(18c)  
(18d)  
where we omitted the domain indicator in the q–averages to simplify notation. The system (18a)–(18d) must be supplemented by the evolution equations (11a)–(11b), since appears explicitly in (18d), and by the algebraic constraints
(19a)  
(19b) 
which are analogous to (8) and (9). It is straightfoward to derive, from (18a)–(18d), a second order equation for :
(20) 
which is analogous to Eq. (12). Notice that (18a)–(18d), (19a)–(19b) and (20) reduce to (11a)–(11d), (8)–(9) and (12) at the domain boundary for which and exactly coincide and thus and hold for all . The fact that (18a)–(18b) (which involve averages) are formally the same evolution equations as (11a)–(11b) follows from the fact that back–reaction vanishes for the q–average (see part1 ()).
As with the evolution equations (11a)–(11d) for local exact fluctuations, we can also identify in (18a)–(18d) the subset of FLRW–like evolution equations (18a)–(18b) for the background variables and the subset (18c)–(18d) of evolution equations for the exact (now non–local) fluctuations . Hence, these variables also give rise to a covariant and gauge invariant perturbation formalism (see part2 ()) that is analogous to that of the local fluctuations. Notice that holds for every (see Figures 1 and 2), and hence the Darmois matching conditions in Eq. (13) now identify an FLRW background spacetime through the q–average of covariant scalars over domains .
iv.2 The density contrast.
It is worth recalling that the non–local exact density fluctuation defined in (15)–(16) provides a rigorous and covariant (and GI) definition for the “density contrast” in a domain , as it compares the local density at each point with the FLRW background density identified by the q–average of the density in this domain (see Figures 1 and 2). Therefore, equations (18a)–(18d), as well as (20), provide the evolution of the exact, non–perturbative, density contrast. Notice, however, that the sign of is opposite to that of for a given density profile:
This sign difference follows from the fact that compares with , which remains fixed inside , whereas is proportional to the gradient (from (5a)) and compares and , which are both varying at inner points, these differences are clearly displayed in Figures 1, 2 and 3.
V Exact fluctuations on an asymptotic FLRW background.
We have considered so far exact fluctuations (local and non–local) that are “confined” in bounded concentric comoving domains (Figures 1 and 2). For LTB models converging in the asymptotic radial direction to FLRW models, the metric functions and covariant scalars take the following forms as ^{6}^{6}6The conditions for LTB models to be asymptotic to a FLRW background spacetime in the spacelike radial direction at every time slice were discussed in RadAs (). The radial rays in time slices orthogonal to are spacelike geodesics of the LTB metric, hence for a well defined radial coordinate the proper length along these curves is a monotonic function of , and thus the proper radial asymptotic limit of any scalar is given by . Only the case was examined in RadAs (), but the results easily extend to the case . If LTB models can be radially asymptotic to FLRW models that are spatially flat (Einstein de Sitter ) or with negative spatial curvature (open FLRW ), as “closed” FLRW lack an asymptotic radial range. If then convergence to a FLRW model with is possible (see sussDS2 ()).
(21a)  
(21b)  
(21c)  
(21d) 
where and are the scale factor and covariant scalars of the asymptotic FLRW model . Evidently, the asymptotic conditions (21b) and (21c) clearly identify the variables as GI exact fluctuations (they vanish as ) and as GI background variables in an asymptotic FLRW background. The spatial curvature requires special considerations, since is not GI if the asymptotic FLRW background is spatially flat ( holds while is in general nonzero for finite ).
v.1 Asymptotic non–local exact fluctuations.
Non–local exact fluctuations can also be defined for asymptotic domains ( for but ) in LTB models admitting radial convergence to FLRW. These fluctuations are depicted in Figure 3. We will denote these non–local fluctuations by the subindex label (which stands for “asymptotic”), as in this case becomes the global asymptotic average of in the the whole time slice^{7}^{7}7The q–average of covariant scalars coincides with their standard average from Buchert’s formalism in the radial asymptotic limit of LTB models that converge in this limit to an FLRW spacetime, as the back–reaction term vanishes (see proof in sussBR ()).:
(22)  
so that the and the have the same asymptotic limits given by the asymptotic FLRW scalars , leading to
(23a)  
(23b)  
(23c) 
which depend on and and are (in general) nonzero for finite , though (from the limit (21d))the fluctuations above do vanish in the limit (see Figure 3).
It follows readily from (22) and (23a)–(23c) that the asymptotic exact fluctuations are GI perturbations (they vanish in ), while the asymptotic q–averages are the GI background variables ( is only a GI variable when the asymptotic FLRW model is not spatially flat).
The evolution of non–local q–perturbations for an asymptotic FLRW background can be fully determined by applying (22) and (23a)–(23c) to (18c)–(18d) and to the spatial curvature perturbation constraint in (19a)–(19b), leading to:
(24a)  
(24b)  
(24c) 
where and (which are determined by the background subsystem (18a)–(18b)) take the following analytic forms
(25) 
with . As (18c)–(18d), this system must be supplemented by (11a)–(11b) to determine . As for the nonlocal fluctuations, we can derive the following second order equation for :
(26) 
which is equivalent to (20). In particular, if the asymptotic FLRW background is spatially flat () we can use explicit analytic expression for the background variables (25) in (23c)–(24c) and (26).
We remark that the asymptotic non–local fluctuation provides, for LTB models which radially converge to an asymptotic FLRW background, a covariant and GI description of the density contrast with respect to the asymptotic FLRW background.
Vi Linear regime in LTB models.
The exact fluctuations (local and non–local) that we have introduced provide an exact non–linear measure of the deviation of LTB dynamics with respect to a domain dependent FLRW background. In order to compare these objects with linear perturbations used in the literature we need to define a linear regime involving specific evolution times in which this deviation is also linear.
Let (for ) be the covariant scalars characterizing an FLRW background on a given domain (bounded through Eq. (13) or asymptotic). The necessary and sufficient conditions for a linear regime follow by assuming that an arbitrarily small positive number exists, such that for all along a domain (all for asymptotic domains) in a fiducial time slice (say ), the following relations hold
(27) 
where and denotes order (or linear first order deviations) for suitable expansions (see Appendix C). Since (choice of radial coordinate), then (5a)–(5c) and (17a)–(17c) together with Eq. (27) imply that the following quantities are all for all
(28a)  
(28b)  
(28c) 
As a consequence of Eq (27), it is also straightforward to show (see proof in Appendix C) that a time range containing exists such that the metric variables and in (1) satisfy at all
(29) 
and thus, from the scaling laws (6)–(9), the relations (28a)–(28c) hold for this time range
(30a)  
(30b)  
(30c) 
which implies that and are quantities because of the evolution equations (11a)–(11d) and 18a–(18d). In fact, we can identify the linear regime in terms of a linear deviation between LTB and FLRW metric functions through Eq. (29) and a linear deviation between and from the background scalars through Eqs. (28c) and (30c). On the other hand, products of all quantities in these evolution equations are of (at least) quadratic order , and thus are negligible in the linear regime.
Considering the characteristic features of the linear regime, it is important to emphasise the following points:

The general evolution of LTB models is non–linear. Hence, the linear regime is only valid for a restricted evolution time range of an LTB model in which the fluctuations and relations we presented above remain of (this time range is defined rigorously in Appendix C). The linear regime is usually defined with respect to a spatially flat dust FLRW background (Einstein de Sitter of –CDM) at initial times after the last scattering surface. However, it can also be defined with respect to a spatially curved background.

Under a linear regime, the non–linear second order equations for the density fluctuations (12), (20) and (26) are reduced to,
(31a) (31b) (31c) all of which match (at ) the well known linear evolution equation for dust perturbations in the synchronouscomoving gauge as discussed in the next section. As a consequence, these density fluctuations in their linear regime can be expressed as the linear superposition in terms of the growing and decaying density modes . The explicit analytic form of the functions are given in sussmodes (). In particular, for a spatially flat FLRW background at early times (so that the effect is negligible), we have from equations (27), (36) and (38) of sussmodes ()
(32) with . This is a solution of (31a)–(31c) (notice that the other fluctuations and take the same form as at linear order). The generalisation of the linear density modes to the exact non–linear regime (for the case ) is discussed extensively in sussmodes ().

It is worthwhile comparing the linear limit of the fluctuations we have introduced with those obtained by Zibin zibin () for LTB models that are “close” to an Einstein de Sitter FLRW background (). The linear limit of in (32) is formally identical to Zibin’s equation (A1) in the Appendix of zibin (), and the linear expansion of the exact growing mode obtained in sussmodes () (first term in the right hand side of (32)) exactly coincides with Zibin’s equation (A3) in zibin (). The direct relation between this growing mode linear expansion and the small deviations from spatial flatness expressed in terms of in (32) is what Zibin calls “curvature fluctuation” and motivates his comment that “the curvature perturbation consists of just the growing mode”. However, this quantity is not the complete curvature perturbation (see also next section).
In the following sections we will use the properties of the