# Splitting the spacetime: A systematic analysis of foliation dependence in
cosmic averaging

Pierre Mourier  Asta Heinesen

###### Abstract

It is a fundamental unsolved question in general relativity how to
unambiguously characterize the effective collective dynamics of an ensemble of
fluid elements sourcing the local geometry, in the absence of exact
symmetries. In a cosmological context this is sometimes referred to as the
averaging problem. At the heart of this problem in relativity is the non-
uniqueness of the choice of foliation within which the statistical properties
of the local spacetime are quantified, which can lead to ambiguity in the
formulated average theory. This has led to debate in the literature on how to
best construct and view such a coarse-grained hydrodynamic theory. Here, we
address this ambiguity by performing the first quantitative investigation of
foliation dependence in cosmological spatial averaging. Starting from the aim
of constructing slicing-independent integral functionals (volume, mass,
entropy, _etc_.) as well as average functionals (mean density, average
curvature, _etc_.) defined on spatial volume sections, we investigate
infinitesimal foliation variations and derive results on the foliation
dependence of functionals and on extremal leaves. Our results show that one
may only identify fully foliation-independent integral functionals in special
scenarios, requiring the existence of associated conserved currents. We then
derive bounds on the foliation dependence of integral functionals for general
scalar quantities under finite variations within physically motivated classes
of foliations. Our findings provide tools that are useful for quantifying,
eliminating or constraining the foliation dependence in cosmological
averaging.

######

  1. 1 Introduction
  2. 2 Covariant averaging over spatial foliations
  3. 3 Infinitesimal variation of the foliation
    1. 3.1 Freedom of parameterisation of the foliation
    2. 3.2 Variation of integral quantities with respect to the foliation
      1. 3.2.1 Example: Rest mass
      2. 3.2.2 Example: Volume
    3. 3.3 Variation of averaged quantities with respect to the foliation
      1. 3.3.1 Example: Entropy
      2. 3.3.2 Example: Minimally differing frames
  4. 4 Finite foliation changes and quantitative bounds on foliation dependence
    1. 4.1 Correspondence between hypersurfaces of different foliations
    2. 4.2 Simplifying assumptions and notations
    3. 4.3 The difference of integral functionals between leaves of two foliations
      1. 4.3.1 $\Delta I(S)$ in terms of covariant 4-integration
      2. 4.3.2 $\Delta I(S)$ in terms of 3-integration in adapted coordinates
    4. 4.4 Bounds for foliations with small relative tilts
      1. 4.4.1 The small tilts assumption
      2. 4.4.2 Bounding the distance between two tilted slices
      3. 4.4.3 Resulting bounds on the norm of $\Delta I(S)$
      4. 4.4.4 Consequences for the foliation dependence of averages
    5. 4.5 Bounds for constant–proper time foliations
      1. 4.5.1 The family of constant–proper time foliations
      2. 4.5.2 Resulting bounds on the norm of $\Delta I(S)$
      3. 4.5.3 Example: proper time foliations synchronised near the last scattering epoch
  5. 5 Summary and discussion
  6. A Complements to the derivation and alternative forms of the bounds on finite variations of spatial integrals and averages
    1. A.1 General time coordinate and relation between determinants
    2. A.2 Alternative form of the local integrand and of the global bounds on the variation of spatial integrals and averages
    3. A.3 The proper-time distance between the slices as an integral along a spatial curve

##  1 Introduction

The formulation of the theory of relativity came with the remarkable insight
that proper time varies between observers, and, consequently, the Newtonian
notion of a unique time parameterization of physical phenomena was abandoned.
In concrete applications of relativity to the modelling of physical systems,
it is nevertheless practical to introduce a time parametrization. Foliations
of spacetime into a set of spatial leaves that are labelled by a time
coordinate, also known as ‘$3+1$’ decompositions, appear very commonly in
applications of general relativity (and alternative theories of gravity), and
especially in the cosmological context. Such foliations allow for an initial
value formulation of general relativity [1, 2].

Due to the non-uniqueness of time parameterization in general relativity,
there is a broad freedom in choosing the foliation of a given spacetime. The
non-uniqueness of the foliation is not a problem _per se_ : it may be seen as
an advantage that there is the freedom to consider a foliation where the
physical phenomena of interest are more easily described. Once one applies
_operations_ that are tied to a particular slicing, such as the integration or
averaging of a field over the leaves of the foliation, the choice of foliation
is important. In this paper, it will be our goal to systematically examine and
quantify the impact of the choice of foliation in spatial averaging with a
focus on applications for cosmology, and we thus give a brief review of the
cosmological _averaging problem_ here.

Cosmology typically aims at describing overall, statistical, or average
properties of the Universe as a whole as a function of _cosmic time_. Thus,
the concept of $3+1$ foliations is inherent in the questions posed in
cosmology. The formulation of a large-scale effective cosmological theory can
be approached either (i) by making an _ansatz_ for the large-scale metric
(that must be assumed to be a meaningful object) and for the large-scale
matter content and seeking solutions within this ansatz — possibly also
allowing for smaller-scale perturbations thereof; or (ii) by explicitly
averaging over small-scale dynamics to _derive_ the large-scale evolution laws
of the locally defined spacetime.

The route in (i) is most often applied. It works well when the small and
intermediate scales of gravitational phenomena can effectively be ignored in
the large-scale evolution of the Universe, and it becomes simple when there is
a notion of large-scale symmetries. When assuming homogeneity and isotropy
over spatial sections of the Universe at its largest scales, one arrives at a
Friedman-Lemaître-Robertson-Walker (FLRW) metric description by this route.
The Lambda Cold Dark Matter ($\Lambda$CDM) paradigm of cosmology is founded on
this approach, where structures are described as perturbations around a large-
scale (and background) FLRW model.

The alternative route in (ii) is more involved, since it starts with the local
spacetime description — which must necessarily be complicated by accounting
for the hierarchy of scales and nonlinearity of gravitational phenomena in the
Universe — as the basis for deriving the dynamics of the largest scales. This
step also involves the construction of an appropriately defined _coarse-
graining_ or averaging operation. If the assumptions of decoupling of the
physics at scales comparable to the size of the visible Universe from physics
at smaller scales hold, and if large-scale homogeneity and isotropy apply,
then one should, by such a procedure, arrive at the same FLRW metric
description as for the approach in (i). However, if these conditions are not
satisfied, then the results of the route in (ii) is expected to differ from
the usual applications of the procedure in (i) — such a difference is called a
_backreaction_ effect of the dynamics of inhomogeneous structures. Despite of
its empirical success, the $\Lambda$CDM paradigm is subject to model anomalies
on a wide range of scales [3, 4, 5, 6, 7], and continues to face the
fundamental challenges of the nature of the dark energy and dark matter
sources. There is a debate regarding whether exploring the route in (ii) could
help resolve the anomalies and interpretational challenges of the $\Lambda$CDM
paradigm; see for instance [8, 9].

The problem of how to formally approach the route in (ii) may be denoted as
the _averaging problem_ or the _fitting problem_ , and was first discussed in
detail in early works by [10, 11, 12]. The most widely studied procedure for
cosmological averaging is the Buchert averaging scheme for scalar quantities
[13, 14, 15, 16], but there have been numerous contributions on
complementary/alternative procedures, _e.g._ , [17, 18, 19, 20, 21, 22]. A
variety of applications, mainly following the Buchert averaging scheme, and
generalisations thereof, have directly addressed the impact of the choice of
foliation in these schemes, either quantitatively or in qualitative
discussions [23, 24, 25, 26, 27, 28, 29]. Very briefly summarized, these
papers point towards a variation of the cosmological averages performed in a
given space-time with the choice of foliation, which is indeed natural: the
choice of foliation is defining the averaging domains and therefore also the
final values of the computed averages. Such a variation may be worsened by
additional dependences of the averaging scheme in the choice of slices, _e.g._
if the quantities to be averaged or the spatial boundaries of the domain
depend on the foliation too. Without such extra dependences, it may be
possible to select from physical constraints a broad class of foliations
within which averaged observables remain approximately invariant, as argued
qualitatively in [28]. The foliation dependence in cosmological averaging is
not in itself a problem, but if left uncontrolled, it can introduce ambiguity
in the average cosmological theory.

Foliation dependence in cosmological averaging has not yet been quantified
systematically in the literature. In this paper, we present covariant and
broadly applicable methods for quantifying, eliminating and/or constraining
the foliation dependence in cosmological averaging. Although the methods are
investigated with the cosmological averaging problem in mind, they have broad
applications to foliation studies in general relativity and differential
geometry.

In section 2, we present the general scalar averaging formalism that we use to
quantify foliation dependence throughout this paper. We then consider
foliation (in)dependence in an exact way, in section 3, through calculus of
variation, for integral and average functionals. We also investigate how
foliations might be singled out uniquely from their extremal properties. In
section 4, we consider bounds on foliation dependence for finite variations,
which are relevant when considering physically motivated restricted classes of
cosmic foliations. Finally, in section 5, we summarize and discuss our results
and their potential application to the foliation dependence of large-scale
backreaction terms and effects in cosmological averaging.

##  2 Covariant averaging over spatial foliations

We now define a scalar averaging scheme, relevant for averaging scalar
functions over 3-dimensional (spatial or null) slices as embedded within the
four-dimensional spacetime manifold $\mathcal{M}$. We use the very general
$3+1$ averaging formalism presented in [22], building upon [20, 21, 30] — see
also the related development in [31] — and we consider spacetime domains that
are selected by a window function of the form:

$W_{A,A_{0},B,B_{0},\bm{V}}=-V^{\mu}\nabla_{\mu}(\mathcal{H}(A_{0}-A))\mathcal{H}(B_{0}-B)\;,$
(2.1)

where the _foliation scalar_ $A$ defines the foliation and is assumed to have
time-like or null gradient, the _boundary scalar_ $B$ determines the spatial
boundaries. The constants $A_{0}$ and $B_{0}$ single out a leaf of the
foliation $A=A_{0}$ and the spatial boundary $B=B_{0}$. We shall restrict
ourselves to the case where $B$ is independent of $A$ and its derivatives,
_i.e._ , we consider a fixed spatially bounded $3+1$ dimensional tube in
spacetime.

The _volume measure vector_ $\bm{V}$ determines the integration measure on the
selected 3-dimensional surfaces. We shall allow $\bm{V}$ to be any time-like
vector field, and the integration measure will thus generally not coincide
with Riemannian volume measure on the leaves. Allowing for a non-Riemannian
volume measure is convenient for certain types of fluid intrinsic averaging
[14], and shall also be convenient in the present paper for making explicit
which properties of the averaged expressions are related to the domain
selection and which are related to the volume measure.

For the examples of applications considered in this paper we mainly have in
mind spatial foliations ($\bm{\nabla}A\cdot\bm{\nabla}A<0$), which are
relevant for formulating averaged evolution equations in time, i.e., viewing
the averaging problem as an initial value problem. We shall however also be
interested in considering null-foliations ($\bm{\nabla}A\cdot\bm{\nabla}A=0$)
in some cases. When $A$ defines space-like surfaces, we will usually be
interested in restricting $B$ to have spatial gradient (we may in this case
think of $B$ as a radial coordinate in a suitable coordinate system). When $A$
defines null-surfaces, we will sometimes be interested in $B$ having time-like
gradient, i.e., a light cone truncated at a time-like hypersurface.

For the unit Heaviside step-function, $\mathcal{H}$, we use the right
continuous convention $\mathcal{H}(0)=1$ throughout. In the following we shall
omit the subscripts on $W_{A,A_{0},B,B_{0},\bm{V}}$ except for those relating
to the foliation and refer to the window function (2.1) as $W^{A}_{A_{0}}$,
leaving the choice of boundaries and volume measure implicit.

We define the integral over a scalar $S$ over the spacetime domain singled out
by $W^{A}_{A_{0}}$ in the following way

$I(S)_{A_{0}}^{A}\equiv\int_{\mathcal{M}}d^{4}x\sqrt{g}SW^{A}_{A_{0}}\;,$
(2.2)

where $g$ is the _modulus_ of the determinant of the spacetime metric
$\mathbf{g}$ (the components of which are noted $g_{\mu\nu}$),
$g\equiv|\det(g_{\mu\nu})|=-\det(g_{\mu\nu})$. We then define the volume
$\mathcal{V}_{A_{0}}^{A}$ of the domain and the volume average
$\braket{S}_{A_{0}}^{A}$ of an arbitrary scalar $S$, respectively, as

$\mathcal{V}_{A_{0}}^{A}=I(1)_{A_{0}}^{A}\qquad;\qquad\braket{S}_{A_{0}}^{A}\equiv\frac{I(S)_{A_{0}}^{A}}{\mathcal{V}_{A_{0}}^{A}}\;.$
(2.3)

The derivative of the integral (2.2) with respect to $A_{0}$ is given by [22]:

$\frac{\partial I(S)_{A_{0}}^{A}}{\partial
A_{0}}=-\int_{\mathcal{M}}d^{4}x\sqrt{g}V^{\nu}\nabla_{\nu}(\mathcal{H}(A_{0}-A))\frac{\nabla_{\mu}(SV^{\mu}\mathcal{H}(B_{0}-B))}{V^{\sigma}\nabla_{\sigma}A}\;,$
(2.4)

and the analogous derivative of the average (2.3) is given by:

$\displaystyle\frac{\partial\braket{S}_{A_{0}}^{A}}{\partial A_{0}}$
$\displaystyle=\frac{1}{I(1)_{A_{0}}^{A}}\frac{\partial
I(S)_{A_{0}}^{A}}{\partial
A_{0}}-\frac{\braket{S}_{A_{0}}^{A}}{I(1)_{A_{0}}^{A}}\frac{\partial
I(1)_{A_{0}}^{A}}{\partial A_{0}}$
$\displaystyle{}=-\frac{1}{I(1)_{A_{0}}^{A}}\int_{\mathcal{M}}{d^{4}x\sqrt{g}\,V^{\nu}\nabla_{\nu}(\mathcal{H}(A_{0}-A))\frac{\nabla_{\mu}\left(SV^{\mu}\mathcal{H}(B_{0}-B)\right)-\braket{S}_{A_{0}}^{A}\nabla_{\mu}(V^{\mu}\mathcal{H}(B_{0}-B))}{V^{\sigma}\nabla_{\sigma}A}}\;.$
(2.5)

When $A$ is a time-function, meaning that $\bm{\nabla}A\cdot\bm{\nabla}A<0$,
these derivatives may be thought of as describing the time-evolution of the
integral/average.

##  3 Infinitesimal variation of the foliation

In this section we consider local extrema of the integrals and averages
defined above, when they are viewed as functionals of the foliation. We thus
compute stationarity conditions of the integrals/averages under variation of
the foliation scalar $A$. Such stationarity conditions have at least two
useful applications. Firstly, we are interested in identifying foliation
independent quantities. Foliation-independence is equivalent to stationarity
of the given functional for _all_ foliations. Secondly, we may be interested
in finding a foliation (or leaf of a foliation) that extremise a specific
integral or average functional. Extremal leaves of such functionals can be
thought of as generalisations of paths of shortest distance, and they may be
thought of defining natural hypersurfaces in specific contexts. Thus, such
extremals may provide an interesting way of defining leaves/foliations
uniquely. Moreover, the corresponding extremum value of the functional would
be a natural foliation independent measure of this functional, generalising
the shortest distance (minimised over all possible paths) as a preferred
measure of distance between two points.

We leave the foliation scalar unconstrained in the variation. Thus the leaves
associated with the solutions to the resulting stationarity conditions may in
principle be space-like, time-like, null, or of a varying nature (depending on
the point) amid these three possibilities; and the nature of the solution will
have to be checked in each case. Analogous constrained stationarity conditions
can be derived by imposing local or global constraints of physical interest,
such as constraining the foliation scalar to have an everywhere null gradient
(e.g. when discussing light cones) or to be a proper-time function for a given
4-velocity field. We avoid such considerations in this paper for the sake of
simplicity.

We shall consider stationarity conditions either for a single leaf or for an
entire foliation, the latter being more restrictive. We consider cases where
the integrated scalar $S$ of interest is possibly dependent on $\bm{\nabla}A$,
but not $A$ or higher-order derivatives, and where the volume measure vector
$\bm{V}$ is possibly dependent on $A$ and $\bm{\nabla}A$111We consider
possible dependence of $\bm{V}$ on $\bm{\nabla}A$ since we want to allow for
cases where $\bm{V}$ is normal to the hypersurfaces defined by $A$. We also
consider possible dependence of $S$ on $\bm{\nabla}A$, the reason being that
factors of $\bm{V}\cdot\bm{\nabla}A$ arise naturally for time-derivatives as
seen in the commutation rules (2.4) and (2). For analysing higher-order
derivatives or adapted scalar curvature, dependence on second-order
derivatives $\bm{\nabla}\bm{\nabla}A$ in $S$ would need to be considered in
the variation..

###  3.1 Freedom of parameterisation of the foliation

Before considering the variation of the average/integral functionals in
generality, we shall consider the trivial subset of variations that define a
map of the foliation onto itself. We shall require that the functional is
invariant under such mappings. Consider a given foliation
$\mathcal{F}=\\{\Sigma_{A=A_{0}}\\}$, where $A$ is a spacetime scalar
representing the foliation, $A_{0}$ is a parameter defined over some range
$A_{0,1}\leq A_{0}\leq A_{0,2}$ selecting a leaf of the foliation, and where
$\Sigma_{A=A_{0}}$ represents a leaf of the foliation. The transformation

$A\mapsto f(A)\quad,\quad A_{0}\mapsto f(A_{0})\;,$ (3.1)

where $f$ is a strictly monotonic function, defines a map from the foliation
onto itself $\mathcal{F}\mapsto\mathcal{F}$. We thus denote the choice of $f$
the _freedom of parameterisation_ of the foliation.

We can define an averaging operation independent of this parameterisation by
requiring that a change of parametrisation leaves the integral (2.2) invariant
when $S$ is itself independent of the parametrisation of the foliation. Thus,
we require that transformations $A\mapsto f(A)$, $A_{0}\mapsto f(A_{0})$ lead
to mappings of the integral onto itself $I(S)_{A_{0}}^{A}\mapsto
I(S)_{f(A_{0})}^{f(A)}=I(S)_{A_{0}}^{A}$, where $S$ is an arbitrary scalar
independent of the parametrisation of the foliation $\mathcal{F}$. Demanding
that $I(S)_{A_{0}}^{A}$ remains invariant under the gauge transformations
$A\mapsto f(A)$, $A_{0}\mapsto f(A_{0})$ is equivalent to requiring
stationarity of $I(S)_{A_{0}}^{A}$ for all representations $A$ of the
foliation $\mathcal{F}$ under variations $A\mapsto\tilde{A}=A+\delta
f(A)\,,\,A_{0}\mapsto\tilde{A}_{0}=A_{0}+\delta f(A_{0})$, where $\delta f(A)$
(and its derivatives) is an infinitesimal scalar function of $A$.

For a generic integral operation (2.2) the integral evaluated at the
representation $\tilde{A}$ of the foliation can be expressed in terms of the
integral evaluated in the representation $A$ as follows

$\displaystyle I(S)_{\tilde{A}_{0}}^{\tilde{A}}$
$\displaystyle=\int_{\mathcal{M}}d^{4}x\sqrt{g}\,SV^{\mu}(\tilde{A},\bm{\nabla}\tilde{A})\,\nabla_{\mu}(\tilde{A})\,\delta_{K}(\tilde{A}_{0}-\tilde{A})\mathcal{H}(B_{0}-B)$
$\displaystyle{}=\int_{\mathcal{M}}d^{4}x\sqrt{g}\,SV^{\mu}(\tilde{A},\bm{\nabla}\tilde{A})\nabla_{\mu}A\delta_{K}(A_{0}-A)\mathcal{H}(B_{0}-B)\;,$
(3.2)

where the second line follows from
$\partial_{A}\tilde{A}\;\delta_{K}(\tilde{A}_{0}-\tilde{A})=\delta_{K}(A_{0}-A)$,
where $\delta_{K}$ is the Dirac delta function. We have made the functional
dependence of $\bm{V}$ explicit by writing
$V^{\mu}(\tilde{A},\bm{\nabla}\tilde{A})$. Performing the first order
functional expansion of $V^{\mu}(\tilde{A})$ around the representation $A$ and
plugging it into (3.1) we have

$\displaystyle\left.\delta I(S)_{A_{0}}^{A}\right|_{\mathcal{F}}$
$\displaystyle\equiv I(S)_{\tilde{A}_{0}}^{\tilde{A}}-I(S)_{A_{0}}^{A}$
$\displaystyle=\int_{\mathcal{M}}d^{4}x\sqrt{g}\,S\left(\frac{\partial
V^{\mu}}{\partial A}\,\delta f(A)+\frac{\partial
V^{\mu}}{\partial\nabla_{\nu}A}\,\nabla_{\nu}\delta
f(A)\right)\nabla_{\mu}A\,\delta_{K}(A_{0}-A)\mathcal{H}(B_{0}-B)$
$\displaystyle=\delta f(A_{0})\,I\left(\frac{S\,\frac{\partial
V^{\mu}(A)}{\partial
A}\nabla_{\mu}A}{V^{\kappa}(A)\nabla_{\kappa}A}\right)_{A_{0}}^{A}+\frac{\partial\delta
f(A_{0})}{\partial A_{0}}\,I\left(\frac{S\frac{\partial
V^{\mu}(A)}{\partial\nabla_{\nu}A}\nabla_{\nu}A\nabla_{\mu}A}{V^{\kappa}(A)\nabla_{\kappa}A}\right)_{A_{0}}^{A}\;,$
(3.3)

where $\left.\delta I(S)_{A_{0}}^{A}\right|_{\mathcal{F}}$ denotes the first
order variation of $\delta I(S)_{A_{0}}^{A}$ under variations of $A,A_{0}$
that maps the foliation $\mathcal{F}$ onto itself. We require (3.1) to vanish
for any choice of $S$. Since $\delta f(A_{0})$ and its derivative can be
chosen arbitrarily and independently on a given leaf $A=A_{0}$, the following
two conditions,

$\frac{\partial V^{\mu}(A)}{\partial
A}\nabla_{\mu}A=0\qquad\text{and}\qquad\frac{\partial
V^{\mu}(A)}{\partial\nabla_{\nu}A}\nabla_{\nu}A\nabla_{\mu}A=0\;,$ (3.4)

must be independently satisfied. Loosely speaking, (3.4) states that the
volume measure (as determined by $\bm{V}$) can depend only on the _direction_
defined by $\bm{\nabla}A$, but not on its norm or the values of $A$.

###  3.2 Variation of integral quantities with respect to the foliation

We shall now derive the stationarity conditions for the integral functional
(2.2) under variation of the hypersurface scalar222We are here varying the
_domain_ of integration, by varying the spatial slice. This is in contrast to
most variational problems, where a field living on a fixed domain is varied.
Note that the functional $I(S)_{A_{0}}^{A}$ is differentiable even though it
contains a delta-function in $A$, since the variation of the delta-function is
defined through partial integration. $A$. Physical integral functionals of
interest may for instance be volume or mass functionals.

We write the first order variation of the integral $I(S)_{A_{0}}^{A}$ (2.2) as
a function of the variation $A\rightarrow A+\delta A$ as

$I(S)_{A_{0}}^{A}\rightarrow I(S)_{A_{0}}^{A+\delta A}=I(S)_{A_{0}}^{A}+\delta
I(S)_{A_{0}}^{A}\;,$ (3.5)

with

$\displaystyle\delta I(S)_{A_{0}}^{A}$
$\displaystyle=\int_{\mathcal{M}}d^{4}x\sqrt{g}\,\delta\\!\left(SW^{A}_{A_{0}}\right)$
$\displaystyle=-\int_{\mathcal{M}}d^{4}x\delta
A\sqrt{g}\,\delta_{K}(A_{0}-A)\nabla_{\mu}(SV^{\mu}\mathcal{H}(B_{0}-B))$
$\displaystyle{\phantom{={}}}+\int_{\mathcal{M}}d^{4}x\delta(\nabla_{\mu}A)\sqrt{g}\,\delta_{K}(A_{0}-A)\mathcal{H}(B_{0}-B)\left[\frac{\partial}{\partial(\nabla_{\mu}A)}(SV^{\nu}\nabla_{\nu}A)-SV^{\mu}\right]$
$\displaystyle=-\int_{\mathcal{M}}d^{4}x\delta
A\sqrt{g}\,\delta_{K}(A_{0}-A)\nabla_{\mu}(SV^{\mu}\mathcal{H}(B_{0}-B))$
$\displaystyle{\phantom{={}}}+\int_{\mathcal{M}}d^{4}x\delta(\nabla_{\mu}A)\sqrt{g}\,\delta_{K}(A_{0}-A)\mathcal{H}(B_{0}-B)\nabla_{\nu}A\left[\frac{\partial}{\partial(\nabla_{\mu}A)}(SV^{\nu})\right]\;,$
(3.6)

which we can formally rewrite as follows using partial integration:

$\displaystyle\delta I(S)_{A_{0}}^{A}=$
$\displaystyle-\int_{\mathcal{M}}d^{4}x\delta
A\sqrt{g}\frac{\partial}{\partial
A}(\delta_{K}(A_{0}-A))\mathcal{H}(B_{0}-B)\nabla_{\mu}A\nabla_{\nu}A\,\frac{\partial}{\partial(\nabla_{\mu}A)}(SV^{\nu})$
$\displaystyle-\int_{\mathcal{M}}d^{4}x\delta
A\sqrt{g}\,\delta_{K}(A_{0}-A)\,\nabla_{\mu}\\!\left(\frac{\partial}{\partial(\nabla_{\mu}A)}(SV^{\nu}\nabla_{\nu}A)\mathcal{H}(B_{0}-B)\right)\;.$
(3.7)

Finally, we rewrite the first term in (3.2) to obtain

$\displaystyle\delta I(S)_{A_{0}}^{A}$
$\displaystyle=\int_{\mathcal{M}}d^{4}x\delta\left(\nabla_{\sigma}A\right)\sqrt{g}\delta_{K}(A_{0}-A)\frac{Z^{\sigma}}{Z^{\kappa}\nabla_{\kappa}A}\mathcal{H}(B_{0}-B)V^{\nu}\nabla_{\nu}A\nabla_{\mu}A\,\frac{\partial
S}{\partial(\nabla_{\mu}A)}$
$\displaystyle{\quad}+\int_{\mathcal{M}}d^{4}x\delta
A\sqrt{g}\delta_{K}(A_{0}-A)\,\nabla_{\sigma}\\!\left(\frac{Z^{\sigma}}{Z^{\kappa}\nabla_{\kappa}A}\mathcal{H}(B_{0}-B)V^{\nu}\nabla_{\nu}A\nabla_{\mu}A\,\frac{\partial
S}{\partial(\nabla_{\mu}A)}\right)$
$\displaystyle{\quad}-\int_{\mathcal{M}}d^{4}x\delta
A\sqrt{g}\,\delta_{K}(A_{0}-A)\,\nabla_{\mu}\\!\left(\frac{\partial}{\partial(\nabla_{\mu}A)}(SV^{\nu}\nabla_{\nu}A)\mathcal{H}(B_{0}-B)\right)\;,$
(3.8)

where $\bm{Z}$ is an arbitrary vector field chosen such that
$\bm{Z}\cdot\bm{\nabla}A\neq 0$ (this is for instance guaranteed for a time-
like $\bm{Z}$, if $A$ has an everywhere time-like or null gradient), and where
the requirement of gauge-invariance of the averaging operation (3.4) has been
used. The variation $\delta A$ as evaluated on a given surface can be
considered independent from its derivatives away from the surface along some
direction $\bm{Z}$. Thus we arrive at the two independent constraints for
stationarity of $I(S)_{A_{0}}^{A}$ around the surface $A=A_{0}$:

$\displaystyle\qquad\qquad\qquad\qquad\delta
I(S)_{A_{0}}^{A}=0\quad\forall\,\delta A\qquad\Leftrightarrow$
$\displaystyle\left\\{\begin{array}[]{@{}ll@{}}&\left.\mathcal{H}(B_{0}-B)\nabla_{\mu}A\frac{\partial
S}{\partial(\nabla_{\mu}A)}\right|_{A_{0}}=0\\\ &\text{and}\\\
&\left.\nabla_{\sigma}\\!\left(\frac{Z^{\sigma}}{Z^{\kappa}\nabla_{\kappa}A}\mathcal{H}(B_{0}-B)V^{\nu}\nabla_{\nu}A\nabla_{\mu}A\frac{\partial
S}{\partial(\nabla_{\mu}A)}-\frac{\partial(SV^{\nu}\nabla_{\nu}A)}{\partial(\nabla_{\sigma}A)}\mathcal{H}(B_{0}-B)\right)\right|_{A_{0}}=0\,,\end{array}\right.$
(3.12)

where we have made use of the assumption $V^{\nu}\nabla_{\nu}A\neq 0$. The
first condition above is nothing but the requirement for $S$ to be invariant
under a change of the foliation $\mathcal{F}=\\{\Sigma_{A=A_{0}}\\}$. The
conditions for stationarity are independent on the arbitrary choice of
$\bm{Z}$ despite of the apparent dependence of $\bm{Z}$ in the second
condition of (3.12). 333This can be seen by expanding $\bm{Z}$ in a component
proportional to $\bm{\nabla}A$ and a component orthogonal to $\bm{\nabla}A$,
and noting that all contributions orthogonal to $\bm{\nabla}A$ vanish due to
the first condition of (3.12) and to its spatial derivatives along the
$\\{A=A_{0}\\}$ hypersurface. Requiring that $I(S)_{A_{0}}^{A}$ is stationary
for all surfaces $A_{0}$ results in the following conditions:444We could have
considered a domain with a small width in time, by replacing the delta
function in (2.1) by a narrow step-function. In that case we arrive at (3.2)
with the delta-function replaced by a step function. In this case the
expression equivalent to (3.2) is not singular, but the “boundary condition”
$\nabla_{\mu}A\;\partial S/\partial(\nabla_{\mu}A)=0$ would still need to be
fulfilled separately in order for there to be extrema.

$\displaystyle\delta I(S)_{A_{0}}^{A}=0\quad\forall\,\delta
A,A_{0}\qquad\Leftrightarrow\qquad\left\\{\begin{array}[]{@{}ll@{}}&\mathcal{H}(B_{0}-B)\nabla_{\mu}A\frac{\partial
S}{\partial(\nabla_{\mu}A)}=0\\\ &\text{and}\\\
&\nabla_{\sigma}\\!\left(\frac{\partial(SV^{\nu}\nabla_{\nu}A)}{\partial(\nabla_{\sigma}A)}\mathcal{H}(B_{0}-B)\right)=0\,.\end{array}\right.$
(3.16)

Note that the stationarity requirements (3.12) and (3.16) for the integral
$I(S)_{A_{0}}^{A}$ are _local_ conservation equations. Note also, that when
$S$ is dependent on the representation $A$ of the foliation such that
$\nabla_{\mu}A\;\partial S/\partial(\nabla_{\mu}A)\neq 0$, no stationary
foliations exist for $I(S)_{A_{0}}^{A}$, when allowing for all possible
variations of the foliation555Sometimes we might be interested in considering
restricted subclasses of variations, e.g. variations within a class of proper
time-foliations $\tau\rightarrow\tau+\delta\tau$ of a 4-velocity $\bm{u}$,
that by construction satisfy $\bm{u}\cdot\bm{\nabla}\delta\tau=0$. In such
cases stationary foliations for certain functionals might exist for
$\nabla_{\mu}A\;\partial S/\partial(\nabla_{\mu}A)\neq 0$..

It can be shown from (2.4) and the conditions (3.12) that stationarity of
$I(S)_{A_{0}}^{A}$ implies the vanishing of its derivative with respect to
$A_{0}$

$\delta I(S)_{A_{0}}^{A}=0\quad\forall\,\delta
A\qquad\Rightarrow\qquad\left.\frac{\partial I(S)_{A_{0}}^{A}}{\partial
A_{0}}\right|_{A_{0}}=0\;.$ (3.17)

Similarly (3.16) implies

$\delta I(S)_{A_{0}}^{A}=0\quad\forall\,\delta
A,A_{0}\qquad\Rightarrow\qquad\frac{\partial I(S)_{A_{0}}^{A}}{\partial
A_{0}}=0\;.$ (3.18)

These results are expected since (infinitesimal) constant translations of $A$
at fixed $A_{0}$, which are part of the class of variations $\delta A$, can
also be seen as a translation in time within the original foliation. In fact,
the results (3.17) and (3.18) are general and apply to integral functionals
with arbitrary functional dependence on
$\bm{\nabla}A,\bm{\nabla}\bm{\nabla}A,..,\bm{\nabla}^{(n)}A$.

##### We now consider the special case where $S$ and $\bm{V}$ are independent
of the foliation.

In particular, we let $S$ and $\bm{V}$ be independent on the direction vector
$\bm{\nabla}A$ of the foliation, and it follows that

$\displaystyle\left.\nabla_{\nu}A\frac{\partial
V^{\nu}}{\partial(\nabla_{\mu}A)}\right|_{A_{0}}=0\quad\text{and}\quad\left.\frac{\partial
S}{\partial(\nabla_{\mu}A)}\right|_{A_{0}}=0\quad\text{and}\quad$ (3.19)
$\displaystyle\left.\nabla_{\sigma}\left(\nabla_{\nu}A\frac{\partial
V^{\nu}}{\partial(\nabla_{\mu}A)}\right)\right|_{A_{0}}=0\quad\text{and}\quad\left.\nabla_{\sigma}\left(\frac{\partial
S}{\partial(\nabla_{\mu}A)}\right)\right|_{A_{0}}=0\;,$

in the case of extremisation for a single leaf; or,

$\nabla_{\nu}A\frac{\partial
V^{\nu}}{\partial(\nabla_{\mu}A)}=0\quad\text{and}\quad\frac{\partial
S}{\partial(\nabla_{\mu}A)}=0\;,$ (3.20)

for extremisation for the entire foliation. In these cases, the first
condition of (3.12) and of (3.16) respectively are automatically satisfied.
Moreover, the condition for stationarity for a single hypersurface simplifies
to

$\delta I(S)_{A_{0}}^{A}=0\quad\forall\,\delta
A\qquad\Leftrightarrow\qquad\left.\nabla_{\sigma}\left(SV^{\sigma}\mathcal{H}(B_{0}-B)\right)\right|_{A_{0}}=0\;,$
(3.21)

and the stationarity condition for all surfaces reduces to

$\delta I(S)_{A_{0}}^{A}=0\quad\forall\,\delta
A,A_{0}\qquad\Leftrightarrow\qquad\nabla_{\sigma}\left(SV^{\sigma}\mathcal{H}(B_{0}-B)\right)=0\;.$
(3.22)

The requirements (3.19) or (3.20) are indeed natural in many cases666Note that
the case of $\bm{V}$ being the unit normal to the hypersurfaces defined by $A$
automatically satisfies (3.19) and (3.20) when $S$ is independent of the
foliation.. In the case where $\bm{V}$ is independent of the foliation, the
stationarity requirement (3.21) for a single surface is only dependent on the
foliation through the surface of evaluation, and the requirement (3.22) for
stationarity for an entire foliation does not depend on the foliation at all.
In this case, if (3.22) is satisfied for a particular foliation, then it is
satisfied for any possible foliation.

##### We remark that the stationarity conditions for integral functionals are
restrictive.

In particular, the stationarity requirement (3.22) for an entire foliation is
extremely restrictive. The foliation is stationary only when $S\bm{V}$ is a
conserved current. The stationarity requirement (3.21) for a single leaf is
more flexible: obtaining stationarity in a given foliation amounts to being
able to collect points for which $\bm{\nabla}(S\bm{V})=0$ is satisfied to
construct a space-like (or null-like) leaf. We emphasise that the stationarity
conditions (3.21) and (3.22) are derived under assumptions. For instance, they
are derived under the assumption of no functional dependence of
$I(S)_{A_{0}}^{A}$ on second or higher order derivatives of $A$ (thus, for
integrals over foliation-adapted curvature degrees of freedom the results
derived in the present section do in general not apply). We also emphasise,
that extrema which are not stationary points can exist in the form of infimums
or in the form of local extrema introduced by a “boundary” in the solution
space.

####  3.2.1 Example: Rest mass

Consider a conserved local rest mass current

$M^{\mu}=\varrho u^{\mu},\qquad\nabla_{\mu}M^{\mu}=0\;,$ (3.23)

where $\varrho$ is a rest mass density of a fluid with 4-velocity $\bm{u}$. We
might seek to define a total rest mass of spatial domains as volume integrals
over $\varrho$.

Let us consider the case where we take $\bm{V}$ to coincide with the fluid
4-velocity: $\bm{V}=\bm{u}$. This is a natural choice when averaging local
quantities intrinsic to the fluid [14, 28], and it allows for the integrated
$\varrho$, $I(\varrho)_{A_{0}}^{A}$, to properly define a total rest mass
[22]. We will assume that we can define boundaries $B$ intrinsic to the fluid
flow through parallel transport $\bm{u}\cdot\bm{\nabla}B=0$, such that the
spatial boundary is comoving with the fluid flow. This ensures the
preservation of the fluid content of the averaging domain over time, and will
accordingly allow for the preservation of its total rest mass.

Accordingly, we define a total rest-mass associated with the (fluid-comoving)
averaging domain in a given foliation $A$ as

$M_{A_{0}}^{A}=I(\varrho)_{A_{0}}^{A}=\int_{\mathcal{M}}d^{4}x\sqrt{g}\,\varrho\,W^{A}_{A_{0}}\;,$
(3.24)

with

$W^{A}_{A_{0}}=-u^{\mu}\nabla_{\mu}(\mathcal{H}(A_{0}-A))\mathcal{H}(B_{0}-B)\;.$
(3.25)

The mass (3.24) can be shown to be conserved over time ($\partial
M_{A_{0}}^{A}/\partial A_{0}=0$) for any choice of foliation $A$ [22].
Furthermore, we have from (3.2) that

$\delta M_{A_{0}}^{A}\equiv\delta
I(\varrho)_{A_{0}}^{A}=-\int_{\mathcal{M}}d^{4}x\delta
A\sqrt{g}\,\delta_{K}(A_{0}-A)\nabla_{\mu}(\varrho
u^{\mu}\mathcal{H}(B_{0}-B))\;,$ (3.26)

and we recover the condition (3.22), $\nabla_{\mu}(\varrho
u^{\mu}\mathcal{H}(B_{0}-B))=0$, for the stationarity of $M_{A_{0}}^{A}$ with
respect to the foliation. Using the conservation (3.23) of the rest mass
current and the comoving boundary assumption, $\nabla_{\mu}(\varrho
u^{\mu}\mathcal{H}(B_{0}-B))=\nabla_{\mu}(\varrho
u^{\mu})\mathcal{H}(B_{0}-B)-\varrho
u^{\mu}\nabla_{\mu}B\,\delta_{K}(B_{0}-B)$ always vanishes, implying that the
mass is stationary for all foliations and thus foliation independent.

The rest mass definition (3.24) and its stationarity still hold if
$\bm{\nabla}A$ is light-like777The induced volume measure
$\bm{V}\cdot\bm{\nabla}A\sqrt{g}$ is in this case the generalisation of the
induced volume measure on the null-cone considered in [30]. The present
integration measure reduces to that of [30] when $\bm{V}$ is taken to be a
time-like unit vector field which is hypersurface-forming.. Thus, this mass is
not only invariant with respect to a choice of spatial hypersurface; it is
also invariant under a change from spatial to light-like hypersurfaces,
providing a potentially interesting correspondence between light-cone and
spatial hypersurface averaging.

####  3.2.2 Example: Volume

Suppose we want to consider extremal foliations for the volume
$\mathcal{V}_{A_{0}}^{A}\equiv I(1)_{A_{0}}^{A}$ of a domain lying within the
hypersurfaces. We consider variations $A\rightarrow A+\delta A$ that vanish
($\delta A=0$) on the boundary $B=B_{0}$. This implies slightly weaker
stationarity requirements, removing the conditions imposed at the domain
boundary since only the interior region has to be constrained.

The problem of finding the extremal volume enclosed by a fixed spatial
boundary, can be thought of as a higher dimensional generalisation of finding
the shortest path between two fixed spacetime events. This is a well-studied
problem in the literature, at least for the Riemannian volume measure (see
e.g. [32] and references therein for the case of Riemannian geometry and [33]
and references therein for Lorentzian manifolds). It is nevertheless worth
recalling here as an illustrative example for the present discussion, as the
volume functional is of great importance in cosmology.

##### We first consider the case where $\bm{V}$ is the unit normal to the
hypersurfaces.

In this case, we recover the Riemannian volume measure on the surfaces. We
write

$V_{\mu}=n_{\mu}=\frac{-\nabla_{\mu}A}{\mathcal{N}},\qquad\mathcal{N}\equiv(-g^{\alpha\beta}\nabla_{\alpha}A\nabla_{\beta}A)^{1/2}\;,$
(3.27)

where $\bm{n}$ is the future-pointing unit normal to the hypersurfaces. In
this case the condition (3.20) is satisfied, and the stationarity requirement
reduces to (3.22). This simplifies further due to the fixed value of $A$ at
the boundary, removing the condition at $B=B_{0}$,
$\bm{V}\cdot\bm{\nabla}B\,\delta_{K}(B_{0}-B)=0$. It follows that a foliation
extremises $I(1)^{A}_{A_{0}}$ if and only if the extrinsic curvature scalar of
each hypersurface vanishes inside the domain,
$\bm{\nabla}\cdot\bm{V}=\bm{\nabla}\cdot\bm{n}=0$. We thus recover the well-
known condition for the stationarity of this volume functional.

Suppose that we can find a foliation for which this condition is satisfied. We
want to determine whether such an extremal foliation is a maximum, minimum, or
a saddle-point. For this purpose we use the identities

$\frac{\partial\mathcal{N}}{\partial\nabla_{\mu}A}=V^{\mu},\qquad\frac{\partial
V^{\nu}}{\partial\nabla_{\mu}A}=-\frac{1}{\mathcal{N}}h^{\mu\nu},\qquad
h^{\mu\nu}\equiv V^{\mu}V^{\nu}+g^{\mu\nu}\;,$ (3.28)

to compute the second variation of the volume

$\displaystyle\delta^{2}\mathcal{V}_{A_{0}}^{A}\equiv\delta^{2}I(1)_{A_{0}}^{A}$
$\displaystyle=\int_{\mathcal{M}}d^{4}x\sqrt{g}\mathcal{H}(B_{0}-B)\delta^{2}\left(\delta_{K}(A_{0}-A)\mathcal{N}\right)$
$\displaystyle{}=\int_{\mathcal{M}}d^{4}x\sqrt{g}\mathcal{H}(B_{0}-B)\left(\frac{\partial^{2}\delta_{K}(A_{0}-A)}{\partial
A^{2}}\mathcal{N}\delta A^{2}\right.$
$\displaystyle\left.\quad\;\;+\,2\,\frac{\partial\delta_{K}(A_{0}-A)}{\partial
A}\frac{\partial\mathcal{N}}{\partial\nabla_{\mu}A}\delta A\nabla_{\mu}(\delta
A)+\delta_{K}(A_{0}-A)\frac{\partial^{2}\mathcal{N}}{\partial\nabla_{\nu}A\partial\nabla_{\mu}A}\nabla_{\nu}(\delta
A)\nabla_{\mu}(\delta A)\right)$
$\displaystyle{}=\int_{\mathcal{M}}d^{4}x\sqrt{g}\mathcal{H}(B_{0}-B)\delta_{K}(A_{0}-A)\frac{\partial^{2}\mathcal{N}}{\partial\nabla_{\nu}A\partial\nabla_{\mu}A}\nabla_{\nu}(\delta
A)\nabla_{\mu}(\delta A)$
$\displaystyle{}=-\int_{\mathcal{M}}d^{4}x\sqrt{g}\mathcal{H}(B_{0}-B)\delta_{K}(A_{0}-A)\frac{1}{\mathcal{N}}h^{\mu\nu}\nabla_{\nu}(\delta
A)\nabla_{\mu}(\delta A)\;,$ (3.29)

where the second last equality follows from partial integration of the term
involving $\mathcal{N}\delta A^{2}\,(\partial^{2}\delta_{K}(A_{0}-A)/\partial
A^{2})$, the condition for stationarity $\bm{\nabla}\cdot\bm{V}=0$, and
(3.28). The last equality follows from (3.28). Since $h^{\mu\nu}$ is positive
semi-definite, we have $\delta^{2}\mathcal{V}_{A_{0}}^{A}\leq 0$, with
equality when the perturbation $\delta A$ depends solely on $A$ (with
$\bm{\nabla}\delta A\propto\bm{\nabla}A$). Such perturbations map the
foliation onto itself, and have to vanish in the present case due to the
boundary condition $\left.\delta A\right|_{B=B_{0}}=0$. Thus
$\delta^{2}\mathcal{V}_{A_{0}}^{A}<0$ for any non-zero perturbation, and hence
the extremal foliation _maximises_ the volume. This is an expected consequence
of the Lorentzian signature of the metric, while a Riemannian signature would
induce a minimisation of the volume.

##### We next consider the case where $\bm{V}$ is independent of the
foliation.

For variations of $A$ that are zero on the boundary $B=B_{0}$, all foliations
are extrema of the volume if and only if $\bm{\nabla}\cdot\bm{V}=0$, and it
follows that the volume is foliation independent if this condition is
satisfied. Furthermore, we trivially have
$\delta^{2}\mathcal{V}_{A_{0}}^{A}=0$ for all foliations in this class. The
volume has no local extremum, on the other hand, if
$\bm{\nabla}\cdot\bm{V}\neq 0$. This is the case for a fluid proper volume
measure, i.e. $\bm{V}=\bm{u}$ where $\bm{u}$ is the 4-velocity of a fluid
source [22, 28], if any expansion or contraction of the fluid occurs within
the integration domain.

Choosing $\bm{V}$ as a conserved current defines a foliation-independent
“volume”. We can consider again the example of a conserved rest mass current
$\bm{M}=\varrho\bm{u}$ (3.23) from the above subsection, and set
$\bm{V}=\varrho\bm{u}$. In this example, the conserved “volume” is simply the
total rest mass as defined in (3.24), and averages with such a window function
are mass-weighted [34, 22]. The foliation-independence is still, in principle,
restricted to the case $\left.\delta A\right|_{B=B_{0}}=0$, but is recovered
for any deformation with the additional requirement of a domain boundary
comoving with the rest mass current, $\bm{u}\cdot\bm{\nabla}B=0$.

##### A few remarks are in order on extremal leaves.

In this section we have commented on volume extremising foliations. However,
we remark that _a single leaf_ that extremises volume – or another integral
functional of physical interest – could be used as a preferred surface for the
initial value problem in cosmology. A preferred initial surface may in turn be
extrapolated to form a foliation, by for instance propagating the initial
surface along a physically motivated 4-velocity field. Since stationarity
requirements for a single leaf are easier to satisfy than for a full
foliation, single leaf extrema may be explored in cases where it is not
possible to identify a full extremal foliation.

###  3.3 Variation of averaged quantities with respect to the foliation

We will now derive stationaity conditions for the average functional (2.3)
under variations of the hypersurface scalar $A$, analogous to the above
results for the integral functional. Examples of physical average functionals
of interest in cosmology are average density, expansion rate, and spatial
curvature degrees.

We write the first order variation of the average (2.3) under the variation
$A\rightarrow A+\delta A$ as

$\braket{S}_{A_{0}}^{A}\rightarrow\braket{S}_{A_{0}}^{A+\delta
A}=\braket{S}_{A_{0}}^{A}+\delta\braket{S}_{A_{0}}^{A}.$ (3.30)

The variation of the average can be expressed through the variation of
integral quantities in the following way

$\delta\braket{S}_{A_{0}}^{A}=\frac{\delta
I(S)_{A_{0}}^{A}}{I(1)_{A_{0}}^{A}}-\braket{S}_{A_{0}}^{A}\frac{\delta
I(1)_{A_{0}}^{A}}{I(1)_{A_{0}}^{A}},$ (3.31)

and we can plug in (3.2) to obtain stationarity conditions for averaged
quantities. The conditions for demanding stationarity for the entire foliation
are

$\displaystyle\qquad\qquad\qquad\qquad\delta\Braket{S}_{A_{0}}^{A}=0\quad\forall\,\delta
A,A_{0}\qquad\Leftrightarrow$
$\displaystyle\left\\{\begin{array}[]{@{}ll@{}}&\mathcal{H}(B_{0}-B)V^{\nu}\nabla_{\nu}A\nabla_{\mu}A\left(\frac{\partial}{\partial(\nabla_{\mu}A)}(S)\right)=0\\\
&\text{and}\\\
&\nabla_{\sigma}\left(\frac{\partial(SV^{\nu}\nabla_{\nu}A)}{\partial(\nabla_{\sigma}A)}\,\mathcal{H}(B_{0}-B)\right)-\Braket{S}_{A_{0}}^{A}\nabla_{\sigma}\left(\frac{\partial(V^{\nu}\nabla_{\nu}A)}{\partial(\nabla_{\sigma}A)}\,\mathcal{H}(B_{0}-B)\right)=0\;.\end{array}\right.$
(3.35)

The condition for stationarity of an average functional for a single leaf can
be similarly derived from (3.2), providing an analogous criterion to (3.12)
for integral functionals. We do not include this condition for simplicity
given that such a situation will not be of interest for the investigations
discussed below.

The global constraint equation

$\delta\Braket{S}_{A_{0}}^{A}=0\quad\forall\,\delta
A,A_{0}\qquad\Rightarrow\qquad\frac{\partial\Braket{S}_{A_{0}}^{A}}{\partial
A_{0}}=0$ (3.36)

must be satisfied in order to obtain stationarity for a single slice of a
foliation selected by $A=A_{0}$ and for stationarity for the entire foliation
respectively. The necessary condition (3.36) is analogous to the integral
condition (3.18). As for (3.18), the condition (3.36) can be shown to hold for
averages arising from a general window function with arbitrary functional
dependence on $\bm{\nabla}A,\bm{\nabla}\bm{\nabla}A,\dots,\bm{\nabla}^{(n)}A$.

##### We now consider the special case where $S$ and $\bm{V}$ are independent
of the foliation.

In this case, we have that the constraints (3.20) are satisfied. In the case
where the constraints (3.20) are fulfilled, stationarity conditions for the
entire foliation (3.35) reduce to

$\delta\Braket{S}_{A_{0}}^{A}=0\quad\forall\,\delta
A,A_{0}\qquad\Leftrightarrow\qquad\nabla_{\sigma}\left(SV^{\sigma}\mathcal{H}(B_{0}-B)\right)-\Braket{S}_{A_{0}}^{A}\nabla_{\sigma}\left(V^{\sigma}\mathcal{H}(B_{0}-B)\right)=0\;.$
(3.37)

Using (3.36), we see that the above condition is equivalent to
$\Braket{S}_{A_{0}}^{A}$ being constant together with
$\left(S-\Braket{S}_{A_{0}}^{A}\right)V^{\sigma}$ being a conserved current
comoving with the boundaries of the domain, such that
$\nabla_{\sigma}\left(\left(S-\Braket{S}_{A_{0}}^{A}\right)V^{\sigma}\mathcal{H}(B_{0}-B)\right)=0$
is satisfied. The latter boundary condition can be neglected if we consider
variations that are fixed on the boundary.

##### We remark that the conditions for stationarity of average functionals
are very restrictive.

The existence of extramal foliations are conditioned on the existence of a
locally-conserved current. Considering the case $\bm{V}=\bm{u}$, where
$\bm{u}$ is a fluid four velocity field with an associated rest mass density
$\varrho$, we note that a natural conserved current is $\varrho\bm{u}$.
However, since $\Braket{\varrho}_{A_{0}}^{A}\geq 0$ with equality only when
$\varrho=0$ everywhere, it follows that $S=\varrho$ cannot generate
stationarity solutions to (3.37), except for the trivial case $\varrho=0$.

We note that solutions found in this section are valid only for the specified
functional dependence of the average functional on the foliation. As discussed
for integral functionals in section 3.2, there might exist extremals that are
not stationary points, occuring as infimums or as local extrema on the
boundary of a set of allowed foliations.

####  3.3.1 Example: Entropy

The study of entropy is a rich topic in gravitational physics, and in
cosmology it has found various applications, for instance in the
characterisation of initial conditions and inflationary scenarios [35], cyclic
universe models [36], and structure formation [37]. Here, we focus on the
following entropy measure, inspired by the Kullback-Leibler relative
information entropy [38, 39]:

$\mathcal{S}_{A_{0}}^{A}\equiv
I\left(S\ln\left(\frac{S}{\braket{S}_{A_{0}}^{A}}\right)\right)_{A_{0}}^{A}.$
(3.38)

where $S$ must be a field in which gravity induces an increased clustering or
inhomogeneity. Physically relevant substitutions for $S$ include rest mass
densities $\varrho$ as in [39], expansion scalars $\theta$ (in case of
positive expansion everywhere), and proper time measures $\tau$. The variation
of $\mathcal{S}$ with the foliation reads

$\displaystyle\delta\mathcal{S}_{A_{0}}^{A}$ $\displaystyle=\delta
I\left(S\ln\left(S\right)\right)_{A_{0}}^{A}-\delta\left(I\left(S\right)_{A_{0}}^{A}\ln\left(\braket{S}_{A_{0}}^{A}\right)\right)$
$\displaystyle{}=\delta
I\left(S\ln\left(S\right)\right)_{A_{0}}^{A}-\left(\ln\left(\braket{S}_{A_{0}}^{A}\right)+1\right)\delta
I\left(S\right)_{A_{0}}^{A}+\braket{S}_{A_{0}}^{A}\delta
I\left(1\right)_{A_{0}}^{A}\;.$ (3.39)

##### We restrict our investigation to the case where $\bm{V}$ is independent
of the foliation.

In this case, the condition (3.20) applies, and using (3.2), the stationarity
condition becomes

$\delta\mathcal{S}_{A_{0}}^{A}=0\quad\forall\quad\delta
A,A_{0}\qquad\Leftrightarrow\qquad\left\\{\begin{array}[]{@{}ll@{}}&\left.\left(S\ln\left(\frac{S}{\braket{S}_{A_{0}}^{A}}\right)+\braket{S}_{A_{0}}^{A}-S\right)V^{\mu}\nabla_{\mu}B\right|_{B=B_{0}}=0\\\
&\text{and}\\\
&\ln\left(\frac{S}{\braket{S}_{A_{0}}^{A}}\right)\nabla_{\mu}\left(SV^{\mu}\right)-\left(S-\braket{S}_{A_{0}}^{A}\right)\nabla_{\mu}V^{\mu}=0,\end{array}\right.$
(3.40)

Assuming that $S$ is hypersurface-forming, the foliation into constant-$S$
hypersurfaces, $A=S$, satisfies (3.40) since $S$ is then by construction
homogeneous over each slice, and $S=\braket{S}_{A_{0}}^{A}$. It follows that
$\mathcal{S}$ is stationary with value $\mathcal{S}=0$ for this foliation.
This point of stationarity is guaranteed to be a unique global minimum when
$S$ is a strictly positive function (as is the case for a rest mass density)
since

$\frac{\mathcal{S}_{A_{0}}^{A}}{I\left(1\right)_{A_{0}}^{A}}=\Braket{S\ln\left(\frac{S}{\braket{S}_{A_{0}}^{A}}\right)}_{A_{0}}^{A}=\Braket{\frac{S}{\braket{S}_{A_{0}}^{A}}\ln\left(\frac{S}{\braket{S}_{A_{0}}^{A}}\right)}_{A_{0}}^{A}\braket{S}_{A_{0}}^{A}\geq
0\;,$ (3.41)

where the inequality follows from Jensen’s inequality for the function
$x\mapsto x\ln(x)$, with equality only if $S$ is constant over the $A=A_{0}$
hypersurface. For a hypersurface-forming $S$, such an extremal-foliation
exists irrespective of how $\bm{V}$ and $B$ are defined. However, to satisfy
non-singularity of the integration measure ($\bm{V}\cdot\bm{\nabla}A\neq 0$)
for this foliation, we would have to demand $\bm{V}\cdot\bm{\nabla}S\neq 0$.

Note that in case $\bm{\nabla}S\cdot\bm{\nabla}S<0$ is not fulfilled
everywhere, the constant-$S$ solution does not define a spatial foliation.
This situation may occur for the choices of $S$ suggested above,
$S\in\\{\varrho$, $\theta$, $\tau\\}$, hypersurface-forming property of $S$
must always be checked. Moreover, in general, the constant-$S$ foliation needs
not be the unique solution to (3.40), i.e. there may be other local extrema.

##### We now consider the special case where $S\bm{V}$ is a conserved current.

In this case, we have $\nabla_{\mu}(SV^{\mu})=0$. This includes the physical
example where $\bm{V}$ is a 4-velocity field and $S$ is the associated rest
mass density. In this case, the stationarity conditions (3.40) reduce to

$\delta\mathcal{S}_{A_{0}}^{A}=0\quad\forall\quad\delta
A,A_{0}\qquad\Leftrightarrow\qquad\left\\{\begin{array}[]{@{}ll@{}}&\left.\left(S\ln\left(\frac{S}{\braket{S}_{A_{0}}^{A}}\right)+\braket{S}_{A_{0}}^{A}-S\right)V^{\mu}\nabla_{\mu}B\right|_{B=B_{0}}=0\\\
&\text{and}\\\
&(\braket{S}_{A_{0}}^{A}-S)\nabla_{\mu}V^{\mu}=0\;.\end{array}\right.$ (3.42)

For $\bm{\nabla}\cdot\bm{V}=0$ the second condition of (3.42) is automatically
satisfied for any foliation. Hence, in this case, the entropy is foliation
independent and has a constant value (which is zero if and only if $S$ is a
constant of spacetime), up to the boundary condition, i.e. the first condition
of (3.42). The latter can be accounted for by either imposing
$\bm{V}\cdot\bm{\nabla}B=0$ or by simply keeping the foliation fixed at the
boundaries. Note that this case of $\bm{\nabla}\cdot\bm{V}=0$ also implies
$\bm{V}\cdot\bm{\nabla}S=0$, and thus a foliation defined by $A=S$ (where
$\mathcal{S}$ would realise its global minimum at $\mathcal{S}=0$) would
result in a singular volume element.

For $\bm{\nabla}\cdot\bm{V}\neq 0$ everywhere, (3.42) is equivalent to
$S=\braket{S}_{A_{0}}^{A}$, i.e., the global minimum of $\mathcal{S}$
corresponding to the constant-$S$ foliation is the only local extremum. It
follows for instance that, for an everywhere expanding fluid with 4-velocity
$\bm{V}=\bm{u}$ and rest mass density $S=\varrho$, the constant-$\varrho$
foliation is the unique minimiser for the entropy.

####  3.3.2 Example: Minimally differing frames

Suppose that we have a physical time-like vector field $\bm{u}$ in our
cosmological theory in the frame of which averaged quantities would be
desirable. This would for instance mean averaging in the rest frame of a fluid
source if $\bm{u}$ represents its 4-velocity. This 4-velocity field can have
vorticity, which will prevent defining hypersurfaces that are orthogonal to
its flow lines. In this case we may ask whether there is a unique foliation
(defined by a scalar $A$), or a family of foliations, such that their normal
vector field $\bm{n}$, given by

$n^{\mu}=\frac{-\nabla^{\mu}A}{(-g^{\nu\kappa}\nabla_{\nu}A\nabla_{\kappa}A)^{1/2}}\;,$
(3.43)

are maximally close to $\bm{u}$ by some measure. In cases where such a
foliation could be defined, this would provide a natural frame for definining
averages as close to the frame of $\bm{u}$ as possible. The tilt between the
two normalised time-like vector fields

$\gamma=-n^{\mu}u_{\mu},\qquad\gamma\geq 1\;,$ (3.44)

is a natural local scalar measure of their closeness, where $\gamma=1$ if and
only if $\bm{n}=\bm{u}$. We define the measure of “statistical closeness” of
the vector fields $\bm{n}$ and $\bm{u}$ over the domain defined by
$\\{A=A_{0}\,,\,B\leq B_{0}\\}$, as

$\Braket{\gamma}_{A_{0}}^{A}=\frac{\int_{\mathcal{M}}d^{4}x\sqrt{g}W_{A_{0},\bm{n}}^{A}\gamma}{\int_{\mathcal{M}}d^{4}x\sqrt{g}W_{A_{0},\bm{n}}^{A}}=\frac{\int_{\mathcal{M}}d^{4}x\sqrt{g}W_{A_{0},\bm{u}}^{A}}{\int_{\mathcal{M}}d^{4}x\sqrt{g}W_{A_{0},\bm{n}}^{A}}\;\geq
1\;,$ (3.45)

with

$W_{A_{0},\bm{n}}^{A}=-n^{\mu}\nabla_{\mu}(\mathcal{H}(A_{0}-A))\mathcal{H}(B_{0}-B)\;;\qquad
W_{A_{0},\bm{u}}^{A}=-u^{\mu}\nabla_{\mu}(\mathcal{H}(A_{0}-A))\mathcal{H}(B_{0}-B)\;,$
(3.46)

being the window functions with volume measure defined with respect to the
normal vector $\bm{n}$ (3.43) and the vector field $\bm{u}$, respectively.
Plugging in $\bm{V}=\bm{n}$ and $S=\gamma$ in (3.35), we have that the first
condition of (3.35) is automatically satisfied while the second condition of
(3.35) becomes

$\delta\Braket{\gamma}_{A_{0}}^{A}=0\quad\forall\,\delta
A,A_{0}\qquad\Leftrightarrow\qquad\left\\{\begin{array}[]{@{}ll@{}}&\Braket{\gamma}_{A_{0}}^{A}\nabla_{\sigma}n^{\sigma}-\nabla_{\sigma}u^{\sigma}=0\\\
&\text{and}\\\
&\left.\left(\Braket{\gamma}_{A_{0}}^{A}n^{\sigma}-u^{\sigma}\right)\nabla_{\sigma}B\right|_{B=B_{0}}=0\;,\end{array}\right.$
(3.47)

where the second condition in (3.47) is a boundary condition that needs to be
satisfied only if we include variations of the foliation at the boundary of
the domain as well as on its interior. A necessary condition for (3.47) is
$\partial_{A_{0}}\Braket{\gamma}_{A_{0}}^{A}=0$. In solving (3.47), we can
thus consider ${\left\langle{\gamma}\right\rangle^{A}_{A_{0}}}$ as a constant
parameter.

Investigations of the general solution to Eq. (3.47) is beyond the scope of
this paper. However, we consider here the special case where $\bm{u}$ is
divergence-free, $\bm{\nabla}\cdot\bm{u}=0$. The first condition of (3.47)
then reduces to $\bm{\nabla}\cdot\bm{n}=0$. Finding a solution to the first
condition of Eq. ((3.47) in this case amounts to examining the existence of
zero extrinsic scalar curvature foliations. The problem of extremising the
averaged tilt (3.45) thus becomes equivalent to extremising the volume as in
section 3.2.2. This is because
$\int_{\mathcal{M}}\mathrm{d}^{4}x\sqrt{g}\,W^{A}_{A_{0},\bm{u}}$ becomes
foliation independent, so that we are extremising the inverse of the
Riemannian volume (as defined by
$\int_{\mathcal{M}}\mathrm{d}^{4}x\sqrt{g}\,W^{A}_{A_{0},\bm{n}}$). From the
results in section 3.2.2 we know that stationary points for the Riemannian
volume are maximal, meaning that the stationary points for the averaged tilt
(3.45) are minimal in this case.

##  4 Finite foliation changes and quantitative bounds on foliation dependence

The results of section 3 show that, while foliation independent statements can
be made for special cases, most integral or average functionals are foliation
dependent. Nevertheless, it may be possible in many cases of interest to
quantify the level of foliation dependence. The aim of this section is to
determine quantitative bounds on the foliation dependence in scenarios
relevant for cosmological models.

###  4.1 Correspondence between hypersurfaces of different foliations

In the following, we will consider two different foliations $\mathcal{F}$ and
$\mathcal{F}^{\prime}$ corresponding to the respective level sets of two
scalars with past-pointing time-like gradients888Due to the signature of the
Lorentzian metric tensor, a time function that increases towards the future
has a past-pointing gradient., $A$ and $A^{\prime}$, where the transformation
$A\mapsto A^{\prime}$ needs not be infinitesimal.

In order to make comparisons of leaves $\Sigma_{0}\equiv\Sigma_{A=A_{0}}$ and
$\Sigma_{0}^{\prime}\equiv\Sigma_{A^{\prime}=A^{\prime}_{0}}$ of the two
foliations in a meaningful way, we must ensure that the two slices correspond
to the “same time” in some sense. We shall specifically require that
$\Sigma_{0}$ and $\Sigma_{0}^{\prime}$ intersect at at least one event within
the bounded spatial region determined by the tube
$\mathcal{T}_{B_{0}}\equiv\\{x\in\mathcal{M}\,/\,B(x)\leq B_{0}\\}$. This
ensures a notion of synchronisation at at least one point within the domain of
interest. It prevents in particular artificial differences due to the
comparison of two different parametrisations of the same foliation.

We can always choose a parametrization such that $A^{\prime}_{0}\equiv A_{0}$,
such that
$\Sigma_{0}^{\prime}=\Sigma_{A^{\prime}=A^{\prime}_{0}}=\Sigma_{A^{\prime}=A_{0}}$
for all $A_{0}$ This can be achieved by using the freedom of reparametrization
of $\mathcal{F}^{\prime}$ as per transformations of $A^{\prime}$ of the class
(3.1). The requirement of intersection of the pairs of corresponding slices
from the foliations will suffice in what follows, even though it does not
always uniquely specify the parametrization999To obtain, if necessary, a
unique determination of the parametrization $A^{\prime}$ given $\mathcal{F}$,
$\mathcal{F}^{\prime}$, and $A$, one could, for instance, specify the
intersection point of each pair of slices $\Sigma_{A=A_{0}}$,
$\Sigma_{A^{\prime}=A_{0}}$) by requiring that $A^{\prime}=A$ everywhere along
a given time-like curve within the domain $\mathcal{T}_{B_{0}}$. Such a curve
could correspond to the worldline of a given (_e.g._ geocentric) observer.
$A^{\prime}$ for $\mathcal{F}^{\prime}$ from the parametrization choice $A$
for $\mathcal{F}$. It already ensures, in particular, that $A^{\prime}$ must
be chosen as equal to $A$ (_i.e._ , the transformation $A\mapsto
A^{\prime}=f(A)$ reduces to the identity) in the case
$\mathcal{F}=\mathcal{F}^{\prime}$ mentioned above.

###  4.2 Simplifying assumptions and notations

In this section, we shall only consider cases where $S$ and $\bm{V}$ are
invariant under deformations $A\mapsto A^{\prime}$ of the foliation. As has
been argued in the above sections, defining $\bm{V}$ as a physical vector
field independent of the foliation is natural for many purposes — a natural
choice of $\bm{V}$ could for instance be the 4-velocity field of a physical
matter source. In most cases we will also be interested in averaging physical
scalars that are independent of the foliation101010In some cases, factors of
$\bm{V}\cdot\bm{\nabla}A$ appear naturally, when computing derivatives of
global quantities, as seen in (2.4) and (2). We can still consider scalars
that have dependence on $\bm{V}\cdot\bm{\nabla}A$ in this framework, when we
restrict the deformation of the foliation $A$ to have gradient orthogonal to
$\bm{V}$. Such a class of deformation is in fact of physical interest, _e.g._
in the context of proper time foliations (see Sec. 4.5 below)., as for
instance the rest mass density or expansion rate of a physical matter source.

The boundaries of the domain and their propagation between slices, as
determined by the scalar $B$, are already considered to be set independently
of the foliation as part of our averaging scheme. Here, we shall make the
simplifying assumption that the domain propagation follows the flow of the
volume-measure vector: $\bm{V}\cdot\bm{\nabla}B=0$. We set $\bm{V}$ to be
unitary, $\bm{V}\cdot\bm{V}=-1$; a non-normalized vector field (corresponding
to weighted volume averages) could formally be absorbed into the scalar $S$ to
be averaged,
$S\bm{V}=\left[S(\bm{V}\cdot\bm{V})^{1/2}\right]\times\left[(\bm{V}\cdot\bm{V})^{-1/2}\,\bm{V}\right]$.
The above assumption and normalization convention are again compatible with
the choice of $\bm{V}$ as a source fluid’s $4$-velocity; in this case with a
fluid-comoving domain propagation. This is indeed one of the main applications
we have in mind for this section (see, _e.g._ , [28, 14]).

We denote as $\bm{n}$ the future-pointing unit normal vector field to the
foliation $\mathcal{F}$, _i.e._ satisfying

$\bm{n}=-N\,\bm{\nabla}A\quad;\quad
N=(-\nabla_{\mu}A\,\nabla^{\mu}A)^{-1/2}\;,$ (4.1)

where $N$ is the lapse function associated with $A$. We similarly define
$\bm{n^{\prime}}$ as the unit normal associated with $\mathcal{F}^{\prime}$
and $N^{\prime}$ as the lapse associated with $A^{\prime}$, with
$\bm{n^{\prime}}=-N^{\prime}\,\bm{\nabla}A^{\prime}$.

###  4.3 The difference of integral functionals between leaves of two
foliations

We now consider the difference between integrals of a given scalar $S$ over
leaves of two foliations $\mathcal{F}$ and $\mathcal{F}^{\prime}$: $\Delta
I(S)\equiv{I\\!\left({S}\right)^{A^{\prime}}_{A_{0}}}-{I\\!\left({S}\right)^{A}_{A_{0}}}$,
where $S$ is any foliation-independent scalar. Note that we are using the
short hand notation for the difference, $\Delta I(S)$, where the dependence on
the foliations and the leaves selected by $A_{0}$ is implicit. We shall
rewrite this difference in a way that will be convenient for defining upper
bounds for its norm.

####  4.3.1 $\Delta I(S)$ in terms of covariant 4-integration

We consider a given $A_{0}$ and the corresponding pair of intersecting slices
$\Sigma_{0}=\Sigma_{A=A_{0}}$, $\Sigma_{0}^{\prime}=\Sigma_{A^{\prime}=A_{0}}$
from the two foliations $\mathcal{F}$, $\mathcal{F}^{\prime}$, obeying the
above assumptions. While the roles played by $\mathcal{F}$ and
$\mathcal{F}^{\prime}$ are formally symmetric, we will consider $\mathcal{F}$
as the reference foliation in which the spatial integrals
${I\\!\left({\xi}\right)^{A}_{A_{0}}}$ or averages
${\left\langle{\xi}\right\rangle^{A}_{A_{0}}}$ of various scalars $\xi$ are
known. $\mathcal{F}^{\prime}$, on the other hand, will be considered as an
arbitrary other spatial foliation that may be subjected to certain conditions,
such as having a small tilt everywhere with respect to $\mathcal{F}$. We will
keep $\mathcal{F}^{\prime}$ fully general in the present subsection. We assume
that $\mathcal{M}$ is a path-connected manifold, hence so are $\Sigma_{0}$ and
$\Sigma_{0}^{\prime}$ according to the global hyperbolicity assumption.

In the following, we will make use of the flowlines of $\bm{V}$ as a
diffeomorphism between the domains of $\Sigma_{0}$ and $\Sigma_{0}^{\prime}$
that are within the tube $\mathcal{T}_{B_{0}}$. While this mapping is
covariantly defined, it will be convenient to introduce an associated set of
spatial coordinates. We do so by arbitrarily choosing a coordinate basis
$(X^{i},\,i=1,2,3)$ on one of the slices $\Sigma_{A=A_{1}}$ of $\mathcal{F}$
(or some open subset of $\Sigma_{A=A_{1}}$ containing
$\Sigma_{A=A_{1}}\cap\mathcal{T}_{B_{0}}$). The three coordinates $X^{i}$,
assumed to span $\mathbb{R}^{3}$ without loss of generality, can then be
extended into an incomplete (spatial) set of coordinates in spacetime by
requiring them to be constant along the flow lines of $\bm{V}$:
$\bm{V}\cdot\bm{\nabla}(X^{i})=0,\,\forall i\in\\{1,2,3\\}$. The spacetime
tube $\mathcal{T}_{B_{0}}=\\{x\in\mathcal{M}\,/\,B(x)\leq B_{0}\\}$ with
$\bm{V}\cdot\bm{\nabla}B=0$, then corresponds to a given compact domain in the
space of the spatial coordinates $(X^{i})$; in other words
$\mathcal{H}(B_{0}-B)$ is a function of $(X^{i})$. One may then introduce any
time coordinate to complete $(X^{i})$ into a spacetime coordinate set — again
for convenience in the below calculations. We shall complete it into a
synchronous–comoving coordinate set adapted to the $4-$vector field $\bm{V}$
by introducing a proper time $\tau$ of $\bm{V}$, i.e., a function satisfying
$\bm{V}\cdot\bm{\nabla}\tau=1$ (see section 2.4.2 of [14]). We can use the
residual freedom in the definition of $\tau$ to demand that $\tau=0$ on
$\Sigma_{0}$. This then uniquely specifies111111Although we still simply
denote it as $\tau$ for convenience, note that the scalar function uniquely
defined in this way is specific to $\Sigma_{0}$; it will define a different
function if another $A_{0}$ is considered. $\tau$ among the proper-time
parametrizations of the family of worldlines of $\bm{V}$.

With the simplifying assumptions from Sec. 4.2, the difference $\Delta
I(S)={I\\!\left({S}\right)^{A^{\prime}}_{A_{0}}}-{I\\!\left({S}\right)^{A}_{A_{0}}}$
can be expressed covariantly as:

$\\!\\!\\!\\!\Delta
I(S)=\int_{\mathcal{M}}d^{4}x\,\sqrt{g}\,S\,\mathcal{H}(B_{0}-B)\left[V^{\mu}\nabla_{\mu}(\mathcal{H}(A^{\prime}-A_{0}))-V^{\mu}\nabla_{\mu}(\mathcal{H}(A-A_{0}))\right].$
(4.2)

The spacetime functions $A$, $A^{\prime}$ and $\tau$ are all nondecreasing
along each of the worldlines of $\bm{V}$. Hence, these three scalars are
nondecreasing functions of each other for any constant value of the
$\bm{V}$-comoving spatial coordinates $X^{i}$. This implies that slices of
both foliations $\mathcal{F}$ and $\mathcal{F}^{\prime}$ can be parametrized
by $\tau$. In particular, $\Sigma_{0}^{\prime}$ can be characterized as:
$P\in\Sigma_{0}^{\prime}\Leftrightarrow\tau(P)=\tau_{0}(X^{i}(P))$ (for any
spacetime point $P$), where the values taken by the scalar field $\tau$ on
$\Sigma_{0}^{\prime}$ define a smooth function of the spatial
coordinates121212The space-like hypersurface $\Sigma_{0}^{\prime}$ from
$\mathcal{F}^{\prime}$, like $\Sigma_{A=A_{1}}$ from $\mathcal{F}$, intersects
each of the (time-like) flow lines of $\bm{V}$ exactly once. Hence, the flow
of $\bm{V}$ defines a diffeomorphic parametrization of the points of
$\Sigma_{0}^{\prime}$ by their spatial coordinates $(X^{i})$ within the domain
of interest. that we denote as $\tau_{0}(X^{i})$. In turn, $\Sigma_{0}$ is
simply parametrized as $P\in\Sigma_{0}\Leftrightarrow\tau(P)=0$. Using these
parametrizations, the strictly monotonous relations between $A$, $A^{\prime}$
and $\tau$ also imply that $\mathcal{H}(A^{\prime}-A_{0})$ and
$\mathcal{H}(A-A_{0})$ in Eq. (4.2) can be rewritten as, respectively:

$\mathcal{H}(A^{\prime}-A_{0})=\mathcal{H}(\tau-\tau_{0}(X^{i}))\quad\mathrm{and}\quad\mathcal{H}(A-A_{0})=\mathcal{H}(\tau)\;.$
(4.3)

Eq. (4.2) then becomes

$\Delta
I(S)=\int_{\mathcal{M}}d^{4}x\,\sqrt{g}\,S\,\mathcal{H}(B_{0}-B)(V^{\mu}\nabla_{\mu}\tau)\left[\delta_{K}(\tau-\tau_{0}(X^{i}))-\delta_{K}(\tau)\right].$
(4.4)

####  4.3.2 $\Delta I(S)$ in terms of 3-integration in adapted coordinates

Using the definition $V^{\mu}\nabla_{\mu}\tau=1$ and choosing the spacetime
coordinate system $x^{\mu}=(\tau,X^{i})$ as the coordinates of integration in
(4.4), we can integrate this expression over $\tau$:

$\Delta
I(S)=\int_{\mathbb{R}^{3}}d^{3}X\,\mathcal{H}(B_{0}-B)\left[\big{(}\sqrt{g}\,S\big{)}_{(\tau=\tau_{0}(X^{i}),X^{i})}-\big{(}\sqrt{g}\,S\big{)}_{(\tau=0,X^{i})}\right]\,,$
(4.5)

where the subscript coordinates refer to an evaluation point, that is,
$f_{(\tau,X^{i})}$ would be a function $f$ evaluated at the event of
coordinates $(\tau,X^{i})$.

The metric determinant modulus $g$ appearing in Eq. (4.5) above is the one
obtained in the coordinate system $(\tau,X^{i})$. To recover a more
coordinate-independent expression, one may introduce the (positive)
determinant $b=\det(b_{ij})$ of the local spatial projector orthogonal to
$\bm{V}$, $\mathbf{b}={b_{\mu\nu}\,dx^{\mu}\otimes
dx^{\nu}}=b_{ij}\,dX^{i}\otimes dX^{j}$, $i,j=1,2,3$, with $b_{\mu\nu}\equiv
g_{\mu\nu}+V_{\mu}V_{\nu}$. This determinant remains invariant under a change
of the time coordinate, and it coincides with the value taken by $g$ in the
coordinate system $(\tau,X^{i})$. (We generalize the choice of time coordinate
and show the relation between the two determinants in Appendix A.1.) The
occurrences of $\sqrt{g}$ in Eq. (4.5) may thus equivalently be replaced by
$\sqrt{b}$. The associated volume $3-$form $\sqrt{b}\,d^{3}X$ is then
invariant both under a change of time coordinate and under a time-independent
change of the spatial coordinates $X^{i}$ (_i.e._ , under a relabelling of the
flow lines of $\bm{V}$). This volume $3$-form corresponds to the (manifestly
covariant) Hodge dual $\star\underline{\bm{V}}$ to the $1$-form
$\underline{\bm{V}}\equiv V_{\mu}\,dx^{\mu}$ associated to $\bm{V}$, and may
be interpreted as the infinitesimal spatial volume element in the local
$\bm{V}$-orthogonal frames (see [14, sec. 4 and Appendix D] and [22]). In
cases where $\bm{V}$ is hypersurface-forming, $\mathbf{b}$ would correspond to
the Riemannian spatial metric tensor induced by $\mathbf{g}$ on the
$\bm{V}$-orthogonal hypersurfaces, and $\sqrt{b}\,d^{3}X$ would then simply be
the associated spatial volume form.

To compute the difference in $\sqrt{b}\,S$ between two points along a given
flow line of $\bm{V}$ that appears in Eq. (4.5), let us first write the
evolution equation along $\bm{V}$ of the $\sqrt{b}$ factor, in the adapted
coordinates $(\tau,X^{i})$:

$V^{\mu}\partial_{\mu}\\!\left(\sqrt{b}\right)=\partial_{\mu}\\!\left(\sqrt{b}\,V^{\mu}\right)=\partial_{\mu}\\!\left(\sqrt{g}\,V^{\mu}\right)=\sqrt{g}\,\nabla_{\mu}V^{\mu}=\sqrt{b}\,\nabla_{\mu}V^{\mu}\,,$
(4.6)

where we again made use of two relations holding in these coordinates:
$\sqrt{b}=\sqrt{g}$ and $V^{\mu}=(1,0,0,0)$. The first and last sides of Eq.
(4.6), and hence their equality, are nevertheless independent of the choice of
the time coordinate. The evolution rate of the volume measure $\sqrt{b}$,
which is additionally invariant under a change of $\bm{V}$-comoving spatial
coordinates $(X^{i})$, is thus given by

$\frac{1}{\sqrt{b}}\,V^{\mu}\partial_{\mu}\\!\left(\sqrt{b}\right)=\frac{1}{\sqrt{b}}\,\left.\frac{\mathrm{d}}{\mathrm{d}\tau}\right|_{X^{i}}\\!\left(\sqrt{b}\right)=\nabla_{\mu}V^{\mu}\,,$
(4.7)

where the operator $\left.(\mathrm{d}/\mathrm{d}\tau)\right|_{X^{i}}$
corresponds to a derivative along $\bm{V}$ with respect to $\tau$. Using the
coordinates $(\tau,X^{i})$, we can now explicitly integrate Eq. (4.7) into

$\big{(}\sqrt{b}\big{)}_{(\tau=\tau_{1},X^{i})}=\big{(}\sqrt{b}\big{)}_{(\tau=0,X^{i})}\,\exp\left(\int_{\tau=0}^{\tau_{1}}(\nabla_{\mu}V^{\mu})_{(\tau,X^{i})}\;d\tau\right)\,,$
(4.8)

for any value $\tau_{1}$ of $\tau$. The main integrand in expression (4.5) for
$\Delta I(S)$ is then rewritten as:

$\big{(}\sqrt{b}\,S\big{)}_{(\tau=\tau_{0}(X^{i}),X^{i})}-\big{(}\sqrt{b}\,S\big{)}_{(\tau=0,X^{i})}=\big{(}\sqrt{b}\big{)}_{(\tau=0,X^{i})}\;\psi(X^{i})\;,$
(4.9)

with

$\displaystyle\psi(X^{i})\equiv\left[\exp\left(\int_{\tau=0}^{\tau_{0}(X^{i})}(\nabla_{\mu}V^{\mu})_{(\tau,X^{i})}\;d\tau\right)-1\right]S_{(\tau=0,X^{i})}$
$\displaystyle\quad+\exp\left(\int_{\tau=0}^{\tau_{0}(X^{i})}(\nabla_{\mu}V^{\mu})_{(\tau,X^{i})}\;d\tau\right)\;\int_{\tau=0}^{\tau_{0}(X^{i})}(V^{\mu}\partial_{\mu}S)_{(\tau,X^{i})}\;d\tau\;.$
(4.10)

With this rewriting, the expression for $\Delta I(S)$ finally reduces to

$\Delta
I(S)=\int_{\mathbb{R}^{3}}d^{3}X\,\mathcal{H}(B_{0}-B)\,\big{(}\sqrt{b}\big{)}_{(\tau=0,X^{i})}\,\psi(X^{i})={I\\!\left({\bar{\psi}}\right)^{A}_{A_{0}}}\,,$
(4.11)

where $\psi(X^{i})$ is extended into a spacetime scalar $\bar{\psi}$ by
defining the latter as equalling $\psi(X^{i})$ on some Cauchy hypersurface
(say $\Sigma_{0}$) and satisfying $\bm{V}\cdot\bm{\nabla}\bar{\psi}=0$; _i.e._
, $\bar{\psi}_{(\tau,X^{i})}=\psi(X^{i})\;\forall\tau$. With a slight abuse of
notation, one may simply write $\Delta
I(S)={I\\!\left({\psi(X^{i})}\right)^{A}_{A_{0}}}$.

We give in Appendix A.2 an alternative form of $\psi(X^{i})$ re-expressed in
terms of the local current $\nabla_{\mu}\left(SV^{\mu}\right)$. This form
allows for a more direct connection with our results on exact
foliation-(in)dependence of Sec. 3.2 and for alternative bounds on finite
variations of spatial integrals and averages to the ones presented below.

###  4.4 Bounds for foliations with small relative tilts

Cosmology is typically studied under the assumption of small (nonrelativistic)
relative velocities between relevant observers in spacetime. In this section,
we shall therefore consider bounds on integrals relevant for a class of
foliations which are close to the reference field $\bm{V}$ (and to each other)
in terms of their relative Lorentz factors.

####  4.4.1 The small tilts assumption

We define the local Lorentz factors between $\bm{V}$, $\bm{n}$ and
$\bm{n^{\prime}}$:

$\gamma_{\bm{V},\bm{n}}\equiv-\bm{V}\cdot\bm{n}\geq
1\quad;\qquad\gamma_{\bm{V},\bm{n^{\prime}}}\equiv-\bm{V}\cdot\bm{n^{\prime}}\geq
1\quad;\qquad\gamma_{\bm{n},\bm{n^{\prime}}}\equiv-\bm{n}\cdot\bm{n^{\prime}}\geq
1\;,$ (4.12)

and introduce the local decomposition of $\bm{n^{\prime}}$ with respect to
$\bm{V}$:

$\bm{n^{\prime}}=\gamma_{\bm{V},\bm{n^{\prime}}}\left(\bm{V}+\bm{v^{\prime}}\right)\;,\qquad
v^{\prime\mu}V_{\mu}=0\;,$ (4.13)

where $\bm{v^{\prime}}$ automatically satisfies
$v^{\prime\mu}v^{\prime}_{\mu}=1-\gamma_{\bm{V},\bm{n^{\prime}}}^{-2}$.

The key assumption that we shall use in this subsection is that both $\bm{n}$
and $\bm{n^{\prime}}$ are close to $\bm{V}$, that is, that the tilt velocities
$\sqrt{1-\gamma_{\bm{V},\bm{n}}^{-2}}$ and
$\sqrt{1-\gamma_{\bm{V},\bm{n^{\prime}}}^{-2}}$ (
$=\sqrt{v^{\prime\mu}v^{\prime}_{\mu}}$ ) are small, _i.e._ , globally bounded
by a small parameter $v_{1}\ll 1$. This implies in particular that the
relative tilt between slices of the two foliations also remain small and
globally bounded: $\sqrt{1-\gamma_{\bm{n},\bm{n^{\prime}}}^{-2}}\leq 2v_{1}$.
We then also have $\gamma_{\bm{V},\bm{n}}\leq(1-v_{1}^{2})^{-1/2}\simeq 1$,
and we can introduce the global small parameter $v_{0}\equiv
v_{1}(1-v_{1}^{2})^{-1/2}\simeq v_{1}$, satisfying everywhere
$v_{1}\leq\gamma_{\bm{V},\bm{n}}\,v_{1}\leq v_{0}\ll 1$ as the main
characteristic spatial velocity to be used in the below bounds.

####  4.4.2 Bounding the distance between two tilted slices

From the expression of $\psi$ above, Eq. (4.10), or its rewriting in Eq.(A.4),
it is clear that in addition to assuming upper limits on the local $S$– and
$\bm{V}$–based variables $|S|$ (on $\Sigma_{0}$), $|V^{\mu}\partial_{\mu}S|$,
$|\nabla_{\mu}(SV^{\mu})|$ and/or $|\nabla_{\mu}V^{\mu}|$, one also needs to
be able to bound the proper-time parametrization function $|\tau_{0}(X^{i})|$
of $\Sigma_{0}^{\prime}$ for all points. This quantity provides a measure of
the distance between the two slices $\Sigma_{0}$ and $\Sigma_{0}^{\prime}$,
and in this section we shall provide a bound of $|\tau_{0}(X^{i})|$ in terms
of the small velocity parameter $v_{0}$.

Figure 1: The spatial slices $\Sigma_{0}$ and $\Sigma_{0}^{\prime}$, the
spatial curve $\mathscr{C}\subset\Sigma_{0}$, and the main points and vector
fields used for expressing and bounding the time-like distance $\tau_{0}$
between the two slices. $P$ is an arbitrary point of $\Sigma_{0}$, while
$P_{0}$, at coordinates $(\tau=0,X^{i}_{0})$, belongs to the intersection
$\Sigma_{0}\cap\Sigma^{\prime}_{0}$ of both slices under consideration:
$\tau_{0}(X^{i}_{0})=0$. Note that for this schematic representation which is
not specifically concerned with causality, we use a Riemannian picture of
orthogonality for easier visualisation.

Consider any given point $P$ — of coordinates $(\tau=0,X^{i})$ for a certain
$(X^{i})$ — within the integration domain on $\Sigma_{0}$. One can draw a
geodesic spatial curve $\mathscr{C}$ within $\Sigma_{0}$ joining $P$ to a
reference point $P_{0}$ — say of coordinates $(\tau=0,X^{i}_{0})$ — within the
integration domain on $\Sigma_{0}$ where $\tau_{0}(X^{i}_{0})=0$. That is,
$P_{0}$ is taken as an intersection point of the two slices:
$P_{0}\in\Sigma_{0}\cap\Sigma_{0}^{\prime}\cap\mathcal{T}_{B_{0}}$ which we
assumed to be non-empty. (See Fig. 1 for an illustration of this geometric
setup.) $\mathscr{C}$ can be parametrized by its unit space-like
$\bm{n}$–orthogonal tangent vector $\bm{K}$, and the associated affine
parameter $\lambda$. Setting $\lambda=0$ at $P_{0}$, $\lambda$ then runs from
$0$ at $P_{0}$ to $\mathcal{L}$ at $P$, where $\mathcal{L}$ is the total
proper length of the curve within $\Sigma_{0}$. The coordinates of the point
at parameter $\lambda$ along $\mathscr{C}$ can then be parametrized as
$x^{\mu}(\lambda)=(\tau=0,X^{i}(\lambda))$. These definitions allow us to
perform spatial integrations along $\mathscr{C}$, using $P_{0}$ as a reference
point where $\tau_{0}=0$, writing for instance

$\tau_{0}(X^{i})=\int_{\lambda=0}^{\mathcal{L}}\,\frac{\mathrm{d}}{\mathrm{d}\lambda}\\!\left(\tau_{0}(X^{i}(\lambda))\right)\,d\lambda\;.$
(4.14)

In the following we will use the short-hand notation
$\tau_{0}(\lambda)\equiv\tau_{0}(X^{i}(\lambda))$.

We may extend the tangent vector $\bm{K}$ of $\mathscr{C}$ into a vector field
$\bm{\tilde{L}}$ on the congruence generated by $\bm{V}$ from $\mathscr{C}$ —
_i.e._ , along all worldlines of $\bm{V}$ that intersect $\Sigma_{0}$ at a
point on $\mathscr{C}$ — by Lie dragging $\bm{K}$ along $\bm{V}$:
$\mathscr{L}_{\bm{V}}\bm{\tilde{L}}\equiv\nabla_{\bm{V}}\bm{\tilde{L}}-\nabla_{\bm{V}}\bm{\tilde{L}}=0$,
_i.e._ ,
$(\mathrm{d}/\mathrm{d}\tau)\tilde{L}^{\nu}|_{X^{i}}=\tilde{L}^{\mu}\partial_{\mu}V^{\nu}$,
with $\bm{\tilde{L}}=\bm{K}$ at $\tau=0$, along any given
$\mathscr{C}$–intersecting worldline of $\bm{V}$. Along these same worldlines,
we can then introduce a unit space-like vector $\bm{L}$ built from
$\bm{\tilde{L}}$:

$L^{\mu}\equiv\frac{b^{\mu}_{\
\nu}\tilde{L}^{\nu}}{b_{\mu\nu}\tilde{L}^{\mu}\tilde{L}^{\nu}}\quad;\quad\bm{L}\cdot\bm{L}=b_{\mu\nu}L^{\mu}L^{\nu}=1\;.$
(4.15)

The $\bm{V}$–orthogonal projection $b^{\mu}_{\ \nu}\tilde{L}^{\nu}$ is indeed
nonvanishing, given that $\bm{\tilde{L}}$ is not parallel to $\bm{V}$ at
$\tau=0$ (as $\bm{K}\cdot\bm{K}=1$ while $\bm{V}$ is time-like) and that this
property is preserved by the Lie dragging. Note that, on points of
$\mathscr{C}$, $\bm{K}$ coincides with $\bm{\tilde{L}}$ (by construction), but
_a priori_ not with the projected vector $\bm{L}$, since $\bm{K}$ needs not be
orthogonal to the local $\bm{V}$.

The derivative of $\tau_{0}$ along $\mathscr{C}$ involved in Eq. (4.14) is
then obtained as

$\frac{d\tau_{0}(\lambda)}{d\lambda}=\left(\frac{\bm{\tilde{L}}\cdot\bm{n^{\prime}}}{\gamma_{\bm{V},\bm{n^{\prime}}}}\right)_{(\tau=\tau_{0}(\lambda),X^{i}(\lambda))}\quad.$
(4.16)

It can also be re-expressed as follows:

$\frac{d\tau_{0}(\lambda)}{d\lambda}=(\bm{V}\\!\cdot\bm{K})_{\lambda}+\sqrt{1+(\bm{V}\\!\cdot\bm{K})_{\lambda}^{2}}\left[(\bm{L}\cdot\bm{v^{\prime}})_{(\tau=\tau_{0}(\lambda),X^{i}(\lambda))}\;F(\tau_{0}(\lambda),\lambda)+G(\tau_{0}(\lambda),\lambda)\right]\;,$
(4.17)

where
$(\bm{V}\\!\cdot\bm{K})_{\lambda}\equiv(\bm{V}\\!\cdot\bm{K})_{(\tau=0,X^{i}(\lambda))}$
is $\bm{V}\\!\cdot\bm{K}$ evaluated at the current point on $\mathscr{C}$ and
where

$\displaystyle F(\tau_{1},\lambda)$
$\displaystyle\equiv\exp\left[\int_{\tau=0}^{\tau_{1}}{\,(\Theta_{\mu\nu}\,L^{\mu}L^{\nu})_{(\tau,X^{i}(\lambda))}\;d\tau}\right]\;;$
(4.18) $\displaystyle G(\tau_{1},\lambda)$
$\displaystyle\equiv\int_{\tau=0}^{\tau_{1}}{\,(\bm{L}\cdot\bm{a})_{(\tau,X^{i}(\lambda))}\;F(\tau,\lambda)\,d\tau}\;,$
(4.19)

for any $\tau_{1}$. The derivation of these results, Eqs. (4.16)–(4.19), is
given in Appendix A.3. In the above expressions for $F$ and $G$, we have used
the decomposition of the covariant derivative of $\underline{\bm{V}}$ with
respect to $\bm{V}$ [40]:

$\nabla_{\mu}V_{\nu}=-V_{\mu}\,V^{\rho}\nabla_{\rho}V_{\nu}+b^{\rho}_{\,\mu}b^{\sigma}_{\,\nu}\nabla_{\rho}V_{\sigma}=-V_{\mu}\,a_{\nu}+\Theta_{\mu\nu}+b^{\rho}_{\,[\mu}b^{\sigma}_{\,\nu]}\nabla_{\rho}V_{\sigma}\,.$
(4.20)

This defines the expansion tensor of $\bm{V}$, with components
$\Theta_{\mu\nu}\equiv
b^{\rho}_{\,(\mu}b^{\sigma}_{\,\nu)}\nabla_{\rho}V_{\sigma}$; its vorticity
tensor, of components
$b^{\rho}_{\,[\mu}b^{\sigma}_{\,\nu]}\nabla_{\rho}V_{\sigma}$; and its
acceleration vector $\bm{a}$, with components $a^{\mu}\equiv
V^{\nu}\nabla_{\nu}V^{\mu}$.

We can right away use the above global smallness assumption on the relative
tilts between $\bm{n}$, $\bm{n}^{\prime}$ and $\bm{V}$ to derive bounds on the
terms $|\bm{V}\\!\cdot\bm{K}|$ and
$\sqrt{1+(\bm{V}\cdot\bm{K})^{2}_{\lambda}}\,\left|\bm{L}\cdot\bm{v^{\prime}}\right|$
appearing in Eq. (4.17). We first note that
$|\bm{V}\\!\cdot\bm{K}|=|h_{\mu\nu}V^{\mu}K^{\nu}|\leq\sqrt{h_{\mu\nu}V^{\mu}V^{\nu}}=\sqrt{\gamma_{\bm{V},\bm{n}}^{2}-1}=\gamma_{\bm{V},\bm{n}}\sqrt{1-\gamma_{\bm{V},\bm{n}}^{-2}}$,
where $h_{\mu\nu}\equiv g_{\mu\nu}+n_{\mu}n_{\nu}$ are the components of the
spatial projector (which is also the induced metric) on the leaves of
$\mathcal{F}$. Using this, the second above term obeys the following
inequality:

$\displaystyle\sqrt{1+(\bm{V}\cdot\bm{K})^{2}_{\lambda}}\,\left|\bm{L}\cdot\bm{v^{\prime}}\right|_{(\tau,X^{i}(\lambda))}$
$\displaystyle{}={}\sqrt{1+(\bm{V}\cdot\bm{K})^{2}_{\lambda}}\,\left|b_{\mu\nu}L^{\mu}v^{\prime\nu}\right|_{(\tau,X^{i}(\lambda))}$
$\displaystyle\leq(\gamma_{\bm{V},\bm{n}})_{(\tau=0,X^{i}(\lambda))}\sqrt{(v^{\prime\mu}v^{\prime}_{\mu})_{(\tau,X^{i}(\lambda))}}\;.$
(4.21)

With the small tilt velocities $v_{1}$, $v_{0}$ satisfying everywhere
$\sqrt{1-\gamma_{\bm{V},\bm{n}}^{-2}}\leq v_{1}$,
$\sqrt{v^{\prime\mu}v^{\prime}_{\mu}}\leq v_{1}$ and
$\gamma_{\bm{V},\bm{n}}v_{1}\leq v_{0}$, both of the above terms from Eq.
(4.17) are everywhere smaller than $v_{0}$.

##### We shall now assume the existence of a global bound on the norm of the
expansion tensor.

That is, we assume that there exists a bound on
$(\Theta^{\mu}_{\;\nu}\,\Theta^{\nu}_{\;\mu})^{1/2}$ that applies throughout
the part of $\mathcal{M}$ under consideration (_i.e._ , a certain portion of
$\mathcal{T}_{B_{0}}$). This is equivalent to assuming a global bound
$\bar{H}$ on all eigenvalues of the (diagonalizable) matrix
$\Theta^{\mu}_{\;\nu}$ over the local tangent spaces orthogonal to $\bm{V}$,
so that for any point $P$ in the spacetime region of interest, for each
eigenvalue $\theta_{k}(P)$ ($k=1,2,3$) of $\Theta^{\mu}_{\;\nu}$ at $P$ over
the tangent space orthogonal to $\bm{V}$ at $P$,
$\left|\theta_{k}(P)\right|\leq\bar{H}$. Note that these eigenvalues are real,
and covariantly defined. This implies, in particular, that the norm of the
volume expansion rate
$\nabla_{\mu}V^{\mu}=b^{\mu\nu}\nabla_{\mu}V_{\nu}=\Theta^{\mu}_{\;\mu}$ is
also globally bounded: $\left|\nabla_{\mu}V^{\mu}\right|\leq 3\bar{H}$. We
also set the acceleration of $\bm{V}$ to zero (see the discussion below for
the case of a nonzero acceleration), so that $G(\tau_{0}(\lambda),\lambda)=0$.

We shall now use the above quantities to set bounds on $\tau_{0}(X^{i})$. Let
us first consider the case $\tau_{0}(X^{i})>0$. This correspond to
$\Sigma_{0}^{\prime}$ lying to the future of the $\bm{V}$-comoving observer at
$P$. If $\tau_{0}(\lambda)$ changes sign along $\mathscr{C}$, it will cross
$0$ again at some $\lambda=\lambda_{0}>0$, corresponding to another point in
$\Sigma_{0}^{\prime}\cap\Sigma_{0}\cap\mathcal{T}_{B_{0}}$.131313This requires
the integration domain $\Sigma_{0}\cap\mathcal{T}_{B_{0}}$ to be path-
connected, and geodesically convex or at least star-shaped with respect to
$P_{0}$. If this is not satisfied, spatial paths can be drawn within an
extended flow-tube $\overline{\mathcal{T}}_{\\!\\!B_{0}}$ of $\bm{V}$,
encompassing $\mathcal{T}_{B_{0}}$, such that
$\Sigma_{0}\cap\overline{\mathcal{T}}_{\\!\\!B_{0}}$ satisfies these
properties while keeping its diameter as small as possible (_e.g._ ,
$\Sigma_{0}\cap\overline{\mathcal{T}}_{\\!\\!B_{0}}$ could be taken as the
smallest sphere containing $\Sigma_{0}\cap\mathcal{T}_{B_{0}}$). The assumed
global bounds on tilts, $\Theta_{\mu\nu}$, or the integration domain’s
diameter, then simply are understood to hold on a portion of
$\overline{\mathcal{T}}_{\\!\\!B_{0}}$ rather than only the corresponding one
of $\mathcal{T}_{B_{0}}$, while the domain of spatial integration is unchanged
($\Sigma_{0}\cap\mathcal{T}_{B_{0}}$ or
$\Sigma_{0}^{\prime}\cap\mathcal{T}_{B_{0}}$). If $\Sigma_{0}$ is itself not
geodesically convex _e.g._ due to punctures, when referring to spatial curves
within $\Sigma_{0}$ the meaning of ‘geodesic’ has to be extended to a non-
necessarily unique path taken as short as possible within $\Sigma_{0}$ (for
instance, we do not need $\bm{K}$ to satisfy the spatial geodesic equation
within $\Sigma_{0}$). We then only need to consider the part of $\mathscr{C}$
beyond this point, and we may simply replace the integration bounds
$[\lambda=0,\,\lambda=\mathcal{L}]$ in integrals like Eq. (4.14) by
$[\lambda=\lambda_{0},\,\lambda=\mathcal{L}]$. Otherwise, we can simply set
$\lambda_{0}=0$. We thus ensure in either case that
$\tau_{0}(\lambda)>0\;\forall\lambda\in\,]\lambda_{0},\mathcal{L}]$, with
$\tau_{0}(\lambda=\lambda_{0})=0$.

From the bound $v_{0}$ on $\left|(\bm{V}\cdot\bm{K})_{\lambda}\right|$ and on
$\sqrt{1+(\bm{V}\cdot\bm{K})^{2}_{\lambda}}\,\left|\bm{L}\cdot\bm{v^{\prime}}\right|_{(\tau,X^{i}(\lambda))}$
(for any $\tau$), and with $G(\tau_{0}(\lambda),\lambda)=0$, Eq. (4.17) gives

$\frac{d}{d\lambda}\tau_{0}(\lambda)\leq
v_{0}\,\left[1+F(\tau_{0}(\lambda),\lambda)\right]\;,$ (4.22)

remembering that $F>0$ by definition. One moreover has

$\Theta_{\mu\nu}\,L^{\mu}L^{\nu}=(b_{\mu\rho}L^{\rho})\,\Theta^{\mu}_{\;\nu}\,(b^{\nu}_{\,\sigma}L^{\sigma})\leq\bar{H}\left(b_{\mu\nu}L^{\mu}L^{\nu}\right)=\bar{H}\;,$
(4.23)

where we have used again the orthogonality of $\bm{V}$ and its expansion
tensor, implying
$\Theta_{\mu\nu}=b_{\mu}^{\;\rho}\,\Theta_{\rho\nu}=\Theta_{\mu\sigma}b^{\sigma}_{\,\nu}$.
Injecting the above inequality into Eq. (4.18) implies
$F(\tau_{0}(\lambda),\lambda)\leq\exp\\!\left(\bar{H}\,\tau_{0}(\lambda)\right)$
for any $\lambda\in[0,\mathcal{L}]$, given that $\tau_{0}(\lambda)\geq 0$. Eq.
(4.22) then becomes,

$\\!\\!\\!\\!\frac{d}{d\lambda}\tau_{0}(\lambda)\leq
v_{0}\,\left[1+\exp\\!\left(\bar{H}\,\tau_{0}(\lambda)\right)\right],\;i.e.,\;\frac{d}{d\lambda}\\!\left(\ln\\!\left[1+\exp\\!\left(-\bar{H}\,\tau_{0}(\lambda)\right)\right]\right)\geq-
v_{0}\bar{H}.$ (4.24)

This may then be integrated between $\lambda_{0}$, where
$\tau_{0}(\lambda_{0})=0$, and any $\lambda\geq\lambda_{0}$, to give

$\ln\\!\left[1+\exp\\!\left(-\bar{H}\,\tau_{0}(\lambda)\right)\right]\geq\ln(2)-v_{0}\bar{H}(\lambda-\lambda_{0})\;,$
(4.25)

that is,

$\exp\\!\left(-\bar{H}\,\tau_{0}(\lambda)\right)\geq
2\,\exp\\!\left(-v_{0}\bar{H}(\lambda-\lambda_{0})\right)-1\,.$ (4.26)

If the above right-hand side is nonnegative, which is ensured for all
$\lambda\in[\lambda_{0},\mathcal{L}]$ provided
$v_{0}\bar{H}(\mathcal{L}-\lambda_{0})<\ln(2)$, we then obtain the following
bound on $\tau_{0}(X^{i})=\tau_{0}(\lambda=\mathcal{L})$ as a function of the
tilt velocity $v_{0}$:

$0\leq\tau_{0}(X^{i})\leq-\frac{1}{\bar{H}}\ln\\!\left[2\,\exp\\!\left(-v_{0}\bar{H}(\mathcal{L}-\lambda_{0})\right)-1\right]\;.$
(4.27)

With the slightly stronger assumption
$v_{0}\bar{H}(\mathcal{L}-\lambda_{0})\leq\eta$ for a certain value
$\eta<\ln(2)$, we can use the convexity of the function
$x\mapsto-\ln\\!\left(2e^{-x}-1\right)$ for $0\leq x<\ln(2)$ to write
$-\ln\\!\left[2\,\exp\\!\left(-v_{0}\bar{H}(\mathcal{L}-\lambda_{0})\right)-1\right]\leq\alpha\,v_{0}\bar{H}(\mathcal{L}-\lambda_{0})$,
with $\alpha\equiv-\eta^{-1}\ln\\!\left(2e^{-\eta}-1\right)$. This then gives
a bound on $\tau_{0}(X^{i})$ that is linear in $v_{0}$ and independent of
$\bar{H}$ (apart from setting the above condition on
$v_{0}\bar{H}(\mathcal{L}-\lambda_{0})$):

$0\leq\tau_{0}(X^{i})\leq\alpha\,v_{0}\,(\mathcal{L}-\lambda_{0})\;.$ (4.28)

The (positive) numerical factor
$\alpha=-\eta^{-1}\ln\\!\left(2e^{-\eta}-1\right)$ yields for instance
$\alpha\simeq 2.11$ for $\eta=1/10$, and converges to $\alpha=2$ for
$\eta\rightarrow 0$.

The factor $\mathcal{L}-\lambda_{0}$ ($\leq\mathcal{L}$) in the above bound
and condition, is simply the length of a reduced curve $\tilde{\mathscr{C}}$,
the (still spatially geodesic) part of $\mathscr{C}$ parametrized by
$\lambda\in[\lambda_{0},\mathcal{L}]$, _i.e._ , the part of $\mathscr{C}$
joining the point of coordinate $x^{\mu}(\lambda_{0})$ to $P$. It is in fact
clear that this bound may be written in terms of the length of the shortest
among all curves on $\Sigma_{0}$ that join $P$ to a point of
$\Sigma_{0}\cap\Sigma_{0}^{\prime}$. Since this simply amounts to a
redefinition (if necessary) of $\mathscr{C}$ and of its length $\mathcal{L}$,
we may simply rewrite the above bound as

$0\leq\tau_{0}(X^{i})\leq\alpha\,v_{0}\,\mathcal{L}\;,$ (4.29)

under the assumption

$v_{0}\bar{H}\mathcal{L}\leq\eta<\ln(2)\;,$ (4.30)

where $\mathcal{L}$ is now interpreted as the length of the shortest curve on
$\Sigma_{0}$ joining $P$ to $\Sigma_{0}\cap\Sigma_{0}^{\prime}$ as above. This
length may be significantly smaller than that of a geodesic on $\Sigma_{0}$
joining $P$ to a given arbitrary point
$P_{0}\in\Sigma_{0}\cap\Sigma_{0}^{\prime}$ in cases where the $\Sigma_{0}$
and $\Sigma_{0}^{\prime}$ slices intersect multiple times (_e.g._ ,
periodically).

The symmetric case $\tau_{0}(X^{i})<0$, corresponding to $\Sigma_{0}^{\prime}$
lying to the past of the $\bm{V}$-comoving observer at $P$, can be handled in
a similar way. One may first assume that
$\tau_{0}(\lambda)<0\,\forall\lambda\in]0,\mathcal{L}]$ with
$\tau_{0}(\lambda=0)=0$ up to a suitable redefinition of $\mathscr{C}$,
$\mathcal{L}$ (and the parameter $\lambda$) as above. The bound $v_{0}$ on
$\left|(\bm{V}\cdot\bm{K})_{\lambda}\right|$ and on
$\sqrt{1+(\bm{V}\cdot\bm{K})^{2}_{\lambda}}\,\left|\bm{L}\cdot\bm{v^{\prime}}\right|_{(\tau,X^{i}(\lambda))}$,
along with Eq. (4.23) (writing this time
$\Theta_{\mu\nu}\,L^{\mu}L^{\nu}\geq{-\bar{H}}$), can then again be used to
bound the derivative of $\tau_{0}(\lambda)$ as given by Eq. (4.17), this time
from below (_cf._ Eqs. (4.22) and (4.24)):

$\frac{d}{d\lambda}\tau_{0}(\lambda)\geq-
v_{0}\,\left[1+F(\tau_{0}(\lambda),\lambda)\right]\geq-
v_{0}\,\left[1+\exp\\!\left(-\bar{H}\,\tau_{0}(\lambda)\right)\right]\;.$
(4.31)

Proceeding similarly to the $\tau_{0}(X^{i})>0$ case above, this results in

$0\geq\tau_{0}(X^{i})\geq-\alpha\,v_{0}\,\mathcal{L}\;,$ (4.32)

still assuming $v_{0}\bar{H}\mathcal{L}\leq\eta<\ln(2)$, with the same
numerical factor $\alpha=-\eta^{-1}\ln\\!\left(2e^{-\eta}-1\right)$ as above.
In this bound, $\mathcal{L}$ is to be interpreted in the same way as discussed
for Eqs. (4.29)-(4.30) above.

Combining Eqs. (4.29), for the $\tau_{0}(X^{i})>0$ case, and (4.32), for the
$\tau_{0}(X^{i})<0$ case, implies that in all cases141414This bound remains of
course valid in the $\tau_{0}(X^{i})=0$ case, even considering that
$\mathcal{L}$ may then be set to $0$.,

$\left|\tau_{0}(X^{i})\right|\leq\alpha\,v_{0}\,\mathcal{L}\;,$ (4.33)

under the condition (4.30), with $\alpha=\alpha(\eta)$ as above and still the
same interpretation for $\mathcal{L}$ as discussed above.
$\left|\tau_{0}(X^{i})\right|$ corresponds to the distance between the slices
$\Sigma_{0}$ and $\Sigma_{0}^{\prime}$ as measured along the flow line of
$\bm{V}$ passing through $P$. The spatial curve length $\mathcal{L}$ depends
on the point $P\in\Sigma_{0}\cap\mathcal{T}_{B_{0}}$ considered, but it may
itself be globally bounded for all points $P$ in this domain by a finite
length $\bar{\mathcal{L}}$. The latter may for instance be defined as the
diameter of the domain within $\Sigma_{0}$,

$\bar{\mathcal{L}}=\bar{\mathcal{L}}_{1}\equiv\max_{P_{1},P_{2}\,\in\,\Sigma_{0}\cap\mathcal{T}_{B_{0}}}\mathrm{d}(P_{1},P_{2})\;,$
(4.34)

or as the maximal distance along $\Sigma_{0}$ of a point in the domain to the
intersection of the two slices within the domain,

$\bar{\mathcal{L}}=\bar{\mathcal{L}}_{2}\equiv\max_{P\,\in\,\Sigma_{0}\cap\mathcal{T}_{B_{0}},\,P_{0}\,\in\,\Sigma_{0}\cap\mathcal{T}_{B_{0}}\cap\Sigma_{0}^{\prime}}\;\mathrm{d}(P,P_{0})\;.$
(4.35)

Above, $\mathrm{d}(P,P^{\prime})$ is the spatial distance along $\Sigma_{0}$
between the points $P$ and $P^{\prime}$, that is, the proper length of the
shortest curve on $\Sigma_{0}$ joining $P$ and $P^{\prime}$. The existence of
the above maxima is guaranteed by the compactness of the spatial domain
$\Sigma_{0}\cap\mathcal{T}_{B_{0}}$ (and consequently of
$\Sigma_{0}\cap\mathcal{T}_{B_{0}}\cap\Sigma_{0}^{\prime}$, which is non-
empty). $\bar{\mathcal{L}}_{2}$ is always a smaller (or equal) length than
$\bar{\mathcal{L}}_{1}$, and may be much smaller, but it could be harder to
determine in practice, and it generally depends on $\Sigma_{0}^{\prime}$.

We may then require

$v_{0}\bar{H}\bar{\mathcal{L}}\leq\eta<\ln(2)\;,$ (4.36)

for a certain $\eta$, ensuring that the condition (4.30) is satisfied for all
points $P\in\Sigma_{0}\cap\mathcal{T}_{B_{0}}$. Eq. (4.33) then implies that
for all such points, of coordinates $(\tau=0,X^{i})$,

$\left|\tau_{0}(X^{i})\right|\leq\alpha\,v_{0}\,\bar{\mathcal{L}}\;.$ (4.37)

This global bound means in particular that, within the spacetime tube
$\mathcal{T}_{B_{0}}$ delineating the domain of interest, the distance between
the slices $\Sigma_{0}$ and $\Sigma_{0}^{\prime}$ (along $\bm{V}$) is
everywhere much smaller than the spatial size of the domain as measured in the
reference slice $\Sigma_{0}$, if $v_{0}\ll 1$.

##### Case of a nonvanishing $4$-acceleration / velocity dispersion.

In the more general case, the $4$-acceleration $\bm{a}$ of $\bm{V}$ cannot be
everywhere neglected. When for instance $\bm{V}$ represents the $4$-velocity
of a source fluid, $\bm{a}$ can correspond to non-gravitational accelerations,
which we would consider negligible apart from extremely small fractions of the
volume of the domain considered, but it can also arise from the effective
modelling of velocity dispersion within the source fluid by effective pressure
terms. We can then assume some global bound $\bar{a}$ (with dimension time-1
or length-1) on $\sqrt{\bm{a}\cdot\bm{a}}=\sqrt{b_{\mu\nu}a^{\mu}a^{\nu}}$ and
follow a similar derivation as above from Eq. (4.17), noting that
$|G(\tau_{0}(\lambda),\lambda)|\leq\bar{a}\,\int_{0}^{\tau_{0}(\lambda)}\exp{(\bar{H}\tau)}\,d\tau=\bar{a}/\bar{H}\left[\exp{(\bar{H}\tau_{0}(\lambda))}-1\right]$.
For a small enough $\bar{a}$ so that $\bar{a}\bar{\mathcal{L}}$ is at most of
order unity and under a possibly more stringent, $\bar{a}$–dependent
constraint on $v_{0}\bar{H}\mathcal{L}$ than previously, bounds on
$|\tau_{0}|$ now involving $\bar{a}\bar{\mathcal{L}}$ can be similarly
obtained.

For instance, let us assume that
$\zeta(\bar{a}\bar{\mathcal{L}}-v_{0}\bar{H}\bar{\mathcal{L}})\leq K$ for some
constant $K>0$, where $\zeta$ is the nonnegative, nondecreasing function
defined by $\zeta(x)\equiv(e^{x}-1)/x$. The inverse function $\zeta^{-1}$ can
be defined and is nondecreasing as well, and the above condition is
equivalently rewritten as $\bar{a}\bar{\mathcal{L}}\leq
v_{0}\bar{H}\bar{\mathcal{L}}+\zeta^{-1}(K)$. Assuming additionally that
$v_{0}\bar{H}\bar{\mathcal{L}}$ is bounded by some $\eta$ small enough such
that $v_{0}\bar{H}\bar{\mathcal{L}}\leq\eta<1/(2K)$, one can show that

$\left|\tau_{0}(X^{i})\right|\leq\left(\frac{-\ln{(1-2K\eta)}}{K\eta}\right)\,\times
K\times v_{0}\,\bar{\mathcal{L}}\;.$ (4.38)

Note that the above requirements imply again that $\eta<\ln{2}$, and thus
$K>1/(2\ln{2})$. The prefactor $[-\ln{(1-2K\eta)}]/[K\eta]$ in the above
expression is always larger than $2$ (corresponding to its $K\eta\ll 1$
limit), but remains of order unity if $2K\eta$ is not too close to $1$.
Consequently, for the above bound on $|\tau_{0}|$ to be significant, $K$
should not be very large, and $\bar{a}\bar{\mathcal{L}}$ should accordingly be
at most of order unity (with $\zeta^{-1}(K)\sim\ln{K}$ for large $K$). For
instance, $\bar{a}\bar{\mathcal{L}}\simeq 1$ would still allow for taking
$K\lesssim 2$. Note that one can take _e.g._ $K=1$ if the $4$-acceleration is
small enough that $\bar{a}\bar{\mathcal{L}}\leq
v_{0}\bar{H}\bar{\mathcal{L}}\leq\eta<1/2$ (as $\zeta^{-1}(1)=0$).

In practice, the above requirements may be too restrictive, or the bound too
large, to be used directly.151515Considering for instance, in the late
Universe, the natural case of $\bm{V}$ representing the $4$-velocity field of
a non-relativistic (nearly dust) matter fluid, the main contribution to its
$4$-acceleration over most of the volume is expected to arise from velocity
dispersion within the fluid acting as effective pressure forces which
contribute to oppose gravitational collapse in bound structures. The magnitude
of such a $4$-acceleration from effective pressure can then be estimated as
being at most of the same order as the (Newtonian) gravitational acceleration
within those virialized domains. A typical present-day value for these
accelerations in the outskirts of galaxies or within galaxy cluster haloes,
for instance, would then be of the order of
$10^{-10}\,\mathrm{m}.\mathrm{s}^{-2}$ (_e.g._ , [41, 42]), or a little
smaller. This value of $\bar{a}$ accumulated over a large cosmological domain
with a diameter $\bar{\mathcal{L}}$ of a Hubble length, would correspond to
$\bar{a}\bar{\mathcal{L}}$ of about $10^{-1}$. A more pessimistic estimate of
the maximum $\bar{a}$ of the $4$-acceleration local amplitude, _e.g._
accounting for the central regions of dense clusters or for the haloes of very
massive elliptic galaxies, and still with $\bar{\mathcal{L}}\simeq 1/H_{0}$,
may then lead to an $\bar{a}\bar{\mathcal{L}}$ of order unity or beyond. The
regions where this may occur, however, would occupy a very small volume
fraction at such scales and may accordingly be avoided by the spatial path
and/or neglected in the volume-weighted spatial integrals considered. This can
be considered as being due to accumulating proper-time differences along
$\Sigma_{0}$ when $4$-accelerations have a specific consistent spatial
orientation (corresponding to making the Cauchy-Schwarz bound
$|\bm{L}\cdot\bm{a}|\leq{\sqrt{\bm{a}\cdot\bm{a}}\,\sqrt{b_{\mu\nu}L^{\mu}L^{\nu}}\;\left(=\sqrt{\bm{a}\cdot\bm{a}}\right)}$
in $G$ into an equality, with no changes of sign). However, especially when
$\bm{V}$ models a physical fluid $4$-velocity, and regardless of $\bm{a}$
corresponding to non-gravitational forces or to effective velocity dispersion,
a residual $4$-acceleration is only expected around localized overdensities,
hence small parts of the domain. The acceleration vector is moreover expected
to have radial orientations around the centers of those overdensities, leading
to compensating signs in $\bm{L}\cdot\bm{a}$ over paths crossing such regions.
In such a case, we may thus assume that the fluid’s $4$-acceleration only
contributes small corrections to the above ($\bm{a}=\bm{0}$) bounds. It can
moreover be argued that the small spatial regions where $\bm{a}$ may be non-
negligible could be avoided by the spatial path $\mathscr{C}$ up to a small
increase in its total length, while the cases where the endpoint $P$ of the
path, spanning the whole integration domain, falls within such a dense region
may be neglected for their small contribution to the volume-weighted spatial
integrals $I(S)$ in either slice. When $\bm{V}$ arises from a different, more
geometric construction, we shall simply assume it to be a geodesic vector
field for simplicity. Accordingly, in the following, we will adopt either of
these simplifying assumptions and use the bounds obtained in the vanishing-
acceleration case above — with small corrections being _e.g._ encompassed into
taking a slightly pessimistic value for $\eta$ or $v_{0}$, if necessary.

####  4.4.3 Resulting bounds on the norm of $\Delta I(S)$

For any given $X^{i}$ in the domain considered, the above bound on
$|\tau_{0}(X^{i})|$, Eq. (4.33), results in a constraint on the volume ratios
$(\sqrt{b})_{(\tau=\tau_{1},X^{i})}/(\sqrt{b})_{(\tau=0,X^{i})}=\exp\left(\int_{\tau=0}^{\tau_{1}}(\nabla_{\mu}V^{\mu})_{(\tau,X^{i})}\,d\tau\right)$
that appear in $\psi(X^{i})$, Eq. (4.10), for $\tau_{1}\equiv\tau_{0}(X^{i})$.
Together with the global bound $\left|\nabla_{\mu}V^{\mu}\right|\leq
3\bar{H}$, Eq. (4.33) implies

$\left|\int_{\tau=0}^{\tau_{1}}(\nabla_{\mu}V^{\mu})_{(\tau,X^{i})}\,d\tau\right|\leq
3\bar{H}|\tau_{1}|\leq 3\bar{H}|\tau_{0}(X^{i})|\leq
3\,\alpha\,v_{0}\bar{H}\mathcal{L}\;,$ (4.39)

still assuming the condition (4.30) to hold. Noting that
$|\exp(x)-1|\leq\exp(|x|)-1$ for any real number $x$, and then successively
using the monotonicity and the convexity of the exponential function, the
above relation gives

$\left|\,\exp\left(\int_{\tau=0}^{\tau_{1}}(\nabla_{\mu}V^{\mu})_{(\tau,X^{i})}\,d\tau\right)-1\,\right|\leq
e^{3\bar{H}|\tau_{0}(X^{i})|}-1\leq
e^{3\,\alpha\,v_{0}\bar{H}\mathcal{L}}-1\leq\tilde{\alpha}\,v_{0}\bar{H}\mathcal{L}\,.$
(4.40)

The numerical factor $\tilde{\alpha}$ defined as
$\tilde{\alpha}\equiv\eta^{-1}(e^{3\;\\!\alpha\;\\!\eta}-1)=2\eta^{-1}(e^{\eta}-1)(e^{2\eta}-2e^{\eta}+4)/(2-e^{\eta})^{3}$,
yields for instance $\tilde{\alpha}\simeq 8.8$ for $\eta=1/10$, and converges
to $\tilde{\alpha}=6$ for $\eta\rightarrow 0$.

Taking $\tau_{1}=\tau_{0}(X^{i})$ and applying this inequality to $\psi$ in
Eq. (4.10) then gives:

$\displaystyle|\psi(X^{i})|$
$\displaystyle\leq\tilde{\alpha}\,v_{0}\bar{H}\mathcal{L}\left|S_{(\tau=0,X^{i})}\right|+(1+\tilde{\alpha}\,v_{0}\bar{H}\mathcal{L})\left|\tau_{0}(X^{i})\right|\\!\times\\!\\!\max_{\tau\,\in\,[0,\,\tau_{0}(X^{i})]}\left|\left(V^{\mu}\partial_{\mu}S\right)_{(\tau,X^{i})}\right|$
$\displaystyle\leq\tilde{\alpha}\,v_{0}\bar{H}\mathcal{L}\left|S_{(\tau=0,X^{i})}\right|+\alpha(1+\eta\tilde{\alpha})\,v_{0}\,\mathcal{L}\times\max_{\tau\,\in\,[0,\,\tau_{0}(X^{i})]}\left|\left(V^{\mu}\partial_{\mu}S\right)_{(\tau,X^{i})}\right|\;.$
(4.41)

The numerical factor $\alpha(1+\eta\tilde{\alpha})$ in the second term yields
for instance $\alpha(1+\eta\tilde{\alpha})\simeq 4.0$ for $\eta=1/10$ and
converges to $\alpha(1+\eta\tilde{\alpha})=2$ for $\eta\rightarrow 0$.

The range $\tau\,\in[0,\,\tau_{0}(X^{i})]$ for the maximum simply corresponds
geometrically to taking a maximum over the segment of a flow line of $\bm{V}$
that joins the two slices $\Sigma_{0}$ and $\Sigma_{0}^{\prime}$. The interval
$[0,\tau_{0}(X^{i})]$ (which should be read as $[\tau_{0}(X^{i}),0]$ in case
$\tau_{0}(X^{i})<0$) is part of the larger interval
$[-\alpha\,v_{0}\,\mathcal{L},\,\alpha\,v_{0}\,\mathcal{L}]$ from Eq. (4.33),
and we may then use

$\max_{\tau\,\in\,[0,\,\tau_{0}(X^{i})]}\left|\left(V^{\mu}\partial_{\mu}S\right)_{(\tau,X^{i})}\right|\leq\max_{\tau\,\in\,[-\alpha\,v_{0}\,\mathcal{L},\,\alpha\,v_{0}\,\mathcal{L}]}\left|\left(V^{\mu}\partial_{\mu}S\right)_{(\tau,X^{i})}\right|\;,$
(4.42)

to get rid of the remaining dependence of Eq. (4.41) on the specific foliation
$\mathcal{F}^{\prime}$. Assuming from now on the global condition (4.36),
$v_{0}\bar{H}\bar{\mathcal{L}}\leq\eta$, we can moreover replace the two
$X^{i}$-dependent factors $\mathcal{L}$ in Eq. (4.41) by the global length
bound $\bar{\mathcal{L}}$. With these two remarks, inserting the above Eq.
(4.41) into Eq. (4.11) provides the following bound on the variation of the
spatial integral of the scalar $S$ between the two slices under the condition
(4.36):

$\displaystyle\left|\Delta
I(S)\right|\leq{I\\!\left({\left|\psi(X^{i})\right|}\right)^{A}_{A_{0}}}$
$\displaystyle\;\;\leq\tilde{\alpha}\,v_{0}\bar{H}\bar{\mathcal{L}}\;{I\\!\left({\left|S\right|}\right)^{A}_{A_{0}}}+\alpha(1+\eta\tilde{\alpha})\,v_{0}\,\bar{\mathcal{L}}\;{I\\!\left({\max_{\tau\,\in\,[-\alpha\,v_{0}\,\mathcal{L},\,\alpha\,v_{0}\,\mathcal{L}]}\left|\left(V^{\mu}\partial_{\mu}S\right)_{(\tau,X^{i})}\right|}\right)^{A}_{A_{0}}}\,.$
(4.43)

Above, we have kept the $X^{i}$-dependent distance $\mathcal{L}$ in the second
spatial integral, but it may as well be replaced by the global size
$\bar{\mathcal{L}}$ (since
$[-\alpha\,v_{0}\,\mathcal{L},\,\alpha\,v_{0}\,\mathcal{L}]\subset[-\alpha\,v_{0}\,\bar{\mathcal{L}},\,\alpha\,v_{0}\,\bar{\mathcal{L}}]$)
to give a simpler, though _a priori_ weaker bound. We also note that, from Eq.
(4.10), we could instead have written

$\left|\int_{\tau=0}^{\tau_{0}(X^{i})}\left(V^{\mu}\partial_{\mu}S\right)_{(\tau,X^{i})}\,d\tau\right|\leq\int_{\tau=-\alpha\,v_{0}\,\mathcal{L}}^{\alpha\,v_{0}\,\mathcal{L}}\left|V^{\mu}\partial_{\mu}S\right|_{(\tau,X^{i})}\,d\tau\;,$
(4.44)

and thus the second spatial integral in Eq. (4.43) could alternatively be
replaced by:

${I\\!\left({\int_{\tau=-\alpha\,v_{0}\,\mathcal{L}}^{\alpha\,v_{0}\,\mathcal{L}}\left|V^{\mu}\partial_{\mu}S\right|_{(\tau,X^{i})}\,d\tau}\right)^{A}_{A_{0}}}\;,\quad\text{
or
by:}\quad\;\;{I\\!\left({\int_{\tau=-\alpha\,v_{0}\,\bar{\mathcal{L}}}^{\alpha\,v_{0}\,\bar{\mathcal{L}}}\left|V^{\mu}\partial_{\mu}S\right|_{(\tau,X^{i})}\,d\tau}\right)^{A}_{A_{0}}}\,,$
(4.45)

where we have omitted the prefactor
$\alpha(1+\eta\tilde{\alpha})\,v_{0}\,\bar{\mathcal{L}}$ in front of this
integral.

Another variant of the above bound, Eq. (4.43), is provided in Appendix A.2 as
Eq. (A.7). It is based on an alternative writing of $\psi(X^{i})$ and requires
some knowledge about the local variable $\nabla_{\mu}\left(SV^{\mu}\right)$,
rather than $S$ and $V^{\mu}\partial_{\mu}S$ as in the above.

The bound obtained here, Eq. (4.43), ensures that $|\Delta
I(S)|/I(|S|)^{A}_{A_{0}}\ll 1$ provided the ‘tilt-weighted’ domain size
$v_{0}\bar{\mathcal{L}}$ ($\ll\bar{\mathcal{L}}$) can be considered much
smaller than the characteristic lengths (or times) associated with
$\bar{H}^{-1}$ and with
$\big{|}S_{(0,X^{i})}\big{|}\,/\,|(V^{\mu}\nabla_{\mu}S)_{(\tau,X^{i})}|$. At
least in the cases where $S$ does not change sign on $\Sigma_{0}$ (_e.g._ ,
for an energy or mass density, or for $S=1$, or as a known property of
$\Sigma_{0}$), this expresses a direct constraint on the relative variation of
its integral, $|\Delta I(S)/I(S)_{A_{0}}^{A}|$.

In general, this bound, as well as its alternative form in Eq. (A.7), require
that the global upper limit $\bar{H}$ on the local directional expansion rates
exists over the appropriate spacetime region, and that it and/or the domain
size and tilt velocity be sufficiently small for the condition (4.36) to be
satisfied. As mentioned, the bounds become more stringent if one can even
ensure that $v_{0}\bar{H}\bar{\mathcal{L}}\ll 1$. In a cosmological setup, and
in the case where $\bm{V}$ can be associated with the $4$-velocity of a matter
fluid source component, one can expect the typical expansion rate within the
domain to be of the order of the Hubble parameter at a characteristic time of
$\Sigma_{0}$. If local fluctuations of the expansion rate do not substantially
exceed this value, then $v_{0}\bar{H}\bar{\mathcal{L}}\ll 1$ is ensured up to
a domain size of the order of the associated Hubble length (and possibly
larger for $v_{0}$ small enough). This assumption might be violated in rapidly
collapsing, overdense regions. However, as discussed earlier about the
$4$-acceleration, which is also expected to only have potentially non-
negligible values in strong overdensities, such regions typically occupy very
small spatial volumes and may accordingly be avoided and neglected in the
spatial integrals, allowing for a tighter $\bar{H}$ bound in the remaining
domain. This may as well be applied to the tilt velocities (interpreted as
peculiar velocities of the source fluid in each slice), considering that a
tighter threshold may be imposed on the tilts everywhere outside small
overdense regions. This would allow for an even smaller $v_{0}$ to be picked
while neglecting the contributions of the remaining overdensities as small
corrections to the result.

####  4.4.4 Consequences for the foliation dependence of averages

We can now derive a bound on the difference of scalar averages between slices
of two foliations, while making use of the above results on bounds on the
difference of scalar integrals.

We first obtain a bound on the modulus of the volume difference
$\Delta\mathcal{V}\equiv\mathcal{V}_{A_{0}}^{A^{\prime}}-\mathcal{V}_{A_{0}}^{A}$,
by taking $S=1$ in Eq. (4.43), from which we have:

$|\Delta\mathcal{V}|=|\Delta
I(1)|\leq\tilde{\alpha}\,v_{0}\bar{H}\bar{\mathcal{L}}\,\mathcal{V}_{A_{0}}^{A}\leq\tilde{\alpha}\eta\,\mathcal{V}_{A_{0}}^{A}\;.$
(4.46)

Here, we do assume non-weighted averages, _i.e._ , that $\bm{V}$ is indeed
normalized without a need to include its normalization into a redefinition of
$S$.

Considering again any given foliation-independent scalar $S$, the difference
of its average between slices of the two foliations, $\Delta\langle
S\rangle\equiv{\left\langle{S}\right\rangle^{A^{\prime}}_{A_{0}}}-{\left\langle{S}\right\rangle^{A}_{A_{0}}}$,
is given by

$\displaystyle\Delta\langle S\rangle$
$\displaystyle=\frac{{I\\!\left({S}\right)^{A^{\prime}}_{A_{0}}}}{\mathcal{V}_{A_{0}}^{A^{\prime}}}-\frac{{I\\!\left({S}\right)^{A}_{A_{0}}}}{\mathcal{V}_{A_{0}}^{A}}=\frac{\Delta
I(S)}{\mathcal{V}_{A_{0}}^{A^{\prime}}}-{I\\!\left({S}\right)^{A}_{A_{0}}}\;\frac{\Delta\mathcal{V}}{\mathcal{V}_{A_{0}}^{A}\;\mathcal{V}_{A_{0}}^{A^{\prime}}}$
$\displaystyle{}=\frac{\Delta
I(S)-S_{0}\,\Delta\mathcal{V}}{\mathcal{V}_{A_{0}}^{A^{\prime}}}=\frac{\Delta
I(S-S_{0})}{\mathcal{V}_{A_{0}}^{A}+\Delta\mathcal{V}}\;,$ (4.47)

where we introduced the short-hand notation
$S_{0}\equiv{\left\langle{S}\right\rangle^{A}_{A_{0}}}$ for the average of $S$
within the reference slice $\Sigma_{0}$, seen as a constant number. We can
then apply the integral bound in Eq. (4.43) to the shifted scalar $S-S_{0}$ to
obtain the following bound:

$|\Delta
I(S-S_{0})|\leq\tilde{\alpha}\,v_{0}\bar{H}\bar{\mathcal{L}}\;{I\\!\left({\left|S-S_{0}\right|}\right)^{A}_{A_{0}}}+\alpha(1+\eta\tilde{\alpha})\,v_{0}\,\bar{\mathcal{L}}\;{I\\!\left({\max_{\tau\,\in\,[-\alpha\,v_{0}\,\mathcal{L},\,\alpha\,v_{0}\,\mathcal{L}]}\left|\left(V^{\mu}\partial_{\mu}S\right)_{(\tau,X^{i})}\right|}\right)^{A}_{A_{0}}}\,.$
(4.48)

Injecting this into Eq. (4.47) and using Eq. (4.46), we thus obtain the
following bound on the variation of the average of $S$ between the slices:

$\displaystyle|\Delta\langle S\rangle|$
$\displaystyle\leq\frac{1}{1-\tilde{\alpha}\eta}\left[\tilde{\alpha}\,v_{0}\bar{H}\bar{\mathcal{L}}\;{\left\langle{\left|S-{\left\langle{S}\right\rangle^{A}_{A_{0}}}\right|}\right\rangle^{A}_{A_{0}}}\right.$
$\displaystyle\qquad\qquad\qquad\left.{}+\alpha(1+\eta\tilde{\alpha})\,v_{0}\,\bar{\mathcal{L}}\;{\left\langle{\max_{\tau\,\in\,[-\alpha\,v_{0}\,\mathcal{L},\,\alpha\,v_{0}\,\mathcal{L}]}\left|\left(V^{\mu}\partial_{\mu}S\right)_{(\tau,X^{i})}\right|}\right\rangle^{A}_{A_{0}}}\right]\;,$
(4.49)

provided $\tilde{\alpha}\eta<1$, which is ensured for $\eta$ small enough
since $\tilde{\alpha}\eta\sim 6\eta$ when $\eta\rightarrow 0$.

In the same way as for the bounds on $\Delta I(S)$ in Eqs. (4.43)–(4.45), we
could have replaced in the above the maximum of
$\left|V^{\mu}\partial_{\mu}S\right|$ by its time integral (along the
worldline of $\bm{V}$) over the same interval, _i.e._ ,

${\left\langle{\max_{\tau\,\in\,[-\alpha\,v_{0}\,\mathcal{L},\,\alpha\,v_{0}\,\mathcal{L}]}\left|\left(V^{\mu}\partial_{\mu}S\right)_{(\tau,X^{i})}\right|}\right\rangle^{A}_{A_{0}}}\quad\mapsto\quad{\left\langle{\int_{\tau=-\alpha\,v_{0}\,\mathcal{L}}^{\alpha\,v_{0}\,\mathcal{L}}\left|V^{\mu}\partial_{\mu}S\right|_{(\tau,X^{i})}\,d\tau}\right\rangle^{A}_{A_{0}}}\;,$
(4.50)

and $\mathcal{L}$ (appearing in the interval bounds
$\pm\,\alpha\,v_{0}\,\mathcal{L}$) could be replaced by the global
$\bar{\mathcal{L}}$ in either expression.

In Eq. (A.8) of Appendix A.2, an alternative form for the bound on
$\Delta\langle S\rangle$ is presented. This form is rather based on the local
quantities $\nabla_{\mu}\left(SV^{\mu}\right)$ and $\nabla_{\mu}V^{\mu}$
thanks to a rewriting of $\psi(X^{i})$, following the same logic as for the
alternative form of the bound on $\Delta I(S)$ mentioned hereabove and also
presented in Appendix A.2.

Eq. (4.49) and its variants provide upper limits for the possible changes in
the spatial average of the scalar $S$ over the domain of interest when going
from $\Sigma_{0}$ to another, intersecting slice that may belong to any other
foliation $\mathcal{F}^{\prime}$, of normal vector $\bm{n^{\prime}}$, provided
it (like $\mathcal{F}$) obeys the global small tilt condition
$(\gamma_{\bm{V},\bm{n^{\prime}}}^{2}-1)^{1/2}\leq v_{0}$. This independence
in $\Sigma_{0}^{\prime}$ is fully achieved if the result is expressed in terms
of $\bar{\mathcal{L}}$ instead of $\mathcal{L}$ and $\bar{\mathcal{L}}$ is for
instance defined as the diameter of the averaging domain in $\Sigma_{0}$ (see
Eq. (4.34)). It is also achieved if some intersection point $P_{0}$ between
$\Sigma_{0}$ and any possible choice of $\Sigma_{0}^{\prime}$ is kept fixed
_via_ the requirements on the parametrization of $A^{\prime}$, and
$\mathcal{L}$ is replaced by the (_a priori_ larger) value of the spatial
distance along $\Sigma_{0}$, $\mathrm{d}(P,P_{0})$, between the current point
$P$ (of coordinates $(\tau=0,X^{i})$) and $P_{0}$.

As for the variation of integrals $\Delta I(S)$ in Sec. 4.4.3 above, these
bounds are expressed as a function of $v_{0}\bar{H}\bar{\mathcal{L}}$ and
require this factor to obey the constraint (4.36); hence, similar remarks on
setting $\bar{H}$ and/or $v_{0}$ apply. The bounds on the variation
$\Delta\langle S\rangle$ still depend on the local evolution rate
$V^{\mu}\nabla_{\mu}S$ or non-conserved current $\nabla_{\mu}(SV^{\mu})$; but
we note in this case that their additional dependence is on the local
fluctuations of $S$ on the reference slice, rather than just
${\left\langle{S}\right\rangle^{A}_{A_{0}}}$.

###  4.5 Bounds for constant–proper time foliations

In cosmology, the age of the Universe is typically measured with respect to
the proper time of fundamental observers. However, the choice of proper time
function is associated with a calibration freedom, and a family of proper time
functions in general exist for a given 4-velocity field. In the present
section, we shall investigate bounds of integrals formulated within classes of
proper time foliations. In particular, we give as an example proper time
foliations that are calibrated within the epoch of last scattering.

####  4.5.1 The family of constant–proper time foliations

We consider proper time foliations of a single 4-velocity field, as an example
of a set of foliations that are natural to compare. Let us consider the class
of proper time functions of a given 4-velocity field. We shall use this (unit)
$4$-velocity field as our volume measure vector $\bm{V}$, and as above use
this $\bm{V}$ to define the propagation of the boundaries of the tube
$\mathcal{T}_{B_{0}}$. We say that $\tau$ is a proper time function of
$\bm{V}$ if it satisfies the equation

$V^{\mu}\nabla_{\mu}\tau=1.$ (4.51)

Let $\Sigma_{\text{init}}$ be an initial reference hypersurface chosen at
convenience, and let $\tau=\tau_{\text{ref}}$ be the proper time function
satisfying $\tau_{\text{ref}}(\Sigma_{\text{init}})=$ constant. All solutions
to Eq. (4.51) can be expressed as

$\tau=\tau_{\text{ref}}+\xi\,,$ (4.52)

where $\xi$ is a given function satisfying the transport rule
$V^{\mu}\nabla_{\mu}\xi=0$. If the function $\xi$, describing the distance of
a given solution $\tau$ to the reference time function $\tau_{\text{ref}}$,
can be bounded on a single hypersurface, then it can be bounded throughout the
tube $\mathcal{T}_{B_{0}}$. In the following, we shall consider bounds that
are relevant for when we can assume that we can bound $\xi$ on the initial
surface $\Sigma_{\text{init}}$ such that $|\xi|\leq\delta T$ everywhere on
$\Sigma_{\text{init}}$. It then follows immediately from the above transport
rule that $|\xi|\leq\delta T$ globally within $\mathcal{T}_{B_{0}}$. This
scenario is thus substantially simpler than the setup considered in subsection
4.4 above where the main difficulty was bounding the proper time distance
between the two slices considered. With such a bound already ensured, we do
not need to assume small tilts between the slices of the constant–proper time
foliations.

####  4.5.2 Resulting bounds on the norm of $\Delta I(S)$

We now consider an arbitrary scalar function $S$, and two intersecting leaves
$\Sigma_{0}$ and $\Sigma_{0}^{\prime}$ of the foliations $\mathcal{F}$ and
$\mathcal{F}^{\prime}$ corresponding to the level sets of $\tau_{\text{ref}}$
and $\tau^{\prime}$, where $\tau^{\prime}$ is an arbitrary member of the class
of solutions (4.52). We may choose
$\tau_{\text{ref}}(\Sigma_{0})=0=\tau^{\prime}(\Sigma_{0}^{\prime})$ without
loss of generality, using the gauge freedom of shifting $\tau^{\prime}$ by an
additive constant if necessary. Let $(X^{i})$ again be a set of three
$\bm{V}$-comoving spatial coordinates. The values $\tau_{0}(X^{i})$ taken by
$\tau_{\text{ref}}$ on the $\Sigma_{0}^{\prime}$ hypersurface are equivalently
given by the values of $-\xi$ on that same surface since
$\tau_{\mathrm{ref}}=\tau^{\prime}-\xi$; and it follows that
$|\tau_{0}(X^{i})|\leq\delta T$ within $\mathcal{T}_{B_{0}}$. From this global
upper bound on $|\tau_{0}(X^{i})|$ we can formulate an upper bound on
$|\psi(X^{i})|$ from Eq. (4.10), which reads, in the
$(\tau_{\mathrm{ref}},X^{i})$ coordinate system:

$\displaystyle\left|\psi(X^{i})\right|$
$\displaystyle\leq\left(\exp\left(\max_{\pm\delta
T}\left|\int_{\tau=0}^{\pm\delta
T}\left|\nabla_{\mu}V^{\mu}\right|_{(\tau,X^{i})}\;d\tau\right|\right)-1\right)\left|S_{(\tau_{\mathrm{ref}}=0,X^{i})}\right|$
$\displaystyle\qquad\quad+\exp\left(\max_{\pm\delta
T}\left|\int_{\tau=0}^{\pm\delta
T}\left|\nabla_{\mu}V^{\mu}\right|_{(\tau,X^{i})}\;d\tau\right|\right)\times\max_{\pm\delta
T}\left|\int_{\tau=0}^{\pm\delta
T}\left|V^{\mu}\partial_{\mu}S\right|_{(\tau,X^{i})}\;d\tau\right|\;.$ (4.53)

Assume that we can define a global upper bound $3\bar{H}$ on the volume
expansion rate, time-averaged along $\bm{V}$ for $\tau_{\mathrm{ref}}$
spanning $[-\delta T,0]$ or $[0,\delta T]$:

$\frac{1}{\delta T}\left|\int_{\tau=0}^{\pm\delta
T}\,\left|\nabla_{\mu}V^{\mu}\right|_{(\tau,X^{i})}\;d\tau\right|\leq
3\bar{H}\;.$ (4.54)

This may arise still as a consequence of $3\bar{H}$ holding as a bound on the
local expansion rate everywhere over the spacetime range considered, as in
subsection 4.4 above; but in the present case only the less restrictive time-
averaged bound is required. With this assumption, Eq. (4.53) becomes

$\left|\psi(X^{i})\right|\leq\left(e^{3\bar{H}\,\delta
T}-1\right)\left|S_{(\tau_{\mathrm{ref}}=0,X^{i})}\right|+e^{3\bar{H}\,\delta
T}\times\max_{\pm\delta T}\left|\int_{\tau=0}^{\pm\delta
T}\left|V^{\mu}\partial_{\mu}S\right|_{(\tau,X^{i})}\;d\tau\right|\;.$ (4.55)

Similarly to subsection 4.4, the exponential terms above may as well be
replaced by affine expressions in $\bar{H}\,\delta T$ as $e^{3\bar{H}\,\delta
T}\leq 1+\eta^{-1}(e^{3\eta}-1)\,\bar{H}\,\delta T$ by requesting
$\bar{H}\,\delta T$ to remain always smaller than a certain threshold value
$\eta$. The above local constraints on $|\psi(X^{i})|$ then bound the
variation of the integral of $S$, $\Delta I(S)=I(\psi(X^{i}))$, as $|\Delta
I(S)|\leq I(|\psi(X^{i})|)$. This translates as well into a bound on the
variation of averages, $\Delta\langle S\rangle$, along the same lines as in
Sec. 4.4.4 above. These bounds are useful when it is possible to constrain
$(V^{\mu}\partial_{\mu}S)_{(\tau,X^{i})}$ over the bounding time span
$\pm\delta T$ between the leaves. When $\delta T$ is much smaller than typical
values of the time scales set by
$\left|(\nabla_{\mu}V^{\mu})_{(\tau,X^{i})}\right|^{-1}$ and
$\left|S_{(\tau=0,X^{i})}\right|/\left|(V^{\mu}\partial_{\mu}S)_{(\tau,X^{i})}\right|$
within the subdomain of $\mathcal{T}_{B_{0}}$ defined by $-\delta
T\leq\tau_{\text{ref}}\leq\delta T$, then $|\Delta I(S)|/I(|S|)_{A_{0}}^{A}$
is much smaller than 1. This directly constrains the relative variation
$|\Delta I(S)/I(S)_{A_{0}}^{A}|$ at least when $S$ has a constant sign along
the reference slice.

####  4.5.3 Example: proper time foliations synchronised near the last
scattering epoch

We consider a class of proper time foliation scalars (4.52) of a physical
matter congruence $\bm{V}=\bm{u}$ which are synchronised at the epoch of _last
scattering_. The epoch of last scattering defines a natural initialisation
epoch in cosmology, and this epoch may thus naturally be used for
synchronizing the proper time of physical observers. This epoch does however
cover a spacetime region with a finite width in cosmic time. In the physics of
recombination, the visibility function quantifies the probability of the
streaming of photons as protons and electrons combine into hydrogen. The
visibility function might then be used to quantify an epoch of last
scattering, after which most photons are freely streaming. The full width at
half maximum of the visibility function in $\Lambda$CDM cosmology is $\sim
10^{5}$ years, and can be used to define the duration of the last scattering
epoch. Consider the sub-class of proper time functions (4.52) of $\bm{u}$
which can all be initialized within the epoch of last scattering. That is, we
set $\tau_{\text{ref}}$ as defined from an initialization hypersurface
$\Sigma_{\mathrm{init}}$ that is fully contained within the spacetime region
of last scattering, and all remaining proper time functions considered are
also required to have an (“initial”) level set contained within the region of
last scattering. These proper time functions accordingly satisfy the global
constraint $\left|\xi\right|\leq\delta T$ with $\delta T\sim 10^{5}$ years. We
might think of this class of synchronisation hypersurfaces as defining a set
of equally preferred calibrations of a cosmic age function.

Let us investigate how integral quantities computed at the present epoch,
defined as the $\tau=t_{0}$ slice in each foliation for the fixed present-day
age $t_{0}$, differ between foliations of this set. Let typical values of the
matter fluid’s expansion rate $|\nabla_{\mu}u^{\mu}|$ around the present-epoch
Universe be of the order of $3H_{0}$ with the Hubble constant $H_{0}\sim
10^{-10}$ years-1, and let its local fluctuations reach at most a factor of a
few times this value (say, at most $\sim 10H_{0}$), such that
$\left|\int_{\tau=0}^{\pm\delta
T}\left|\nabla_{\mu}u^{\mu}\right|_{(\tau,X^{i})}d\tau\right|\leq
3\bar{H}\,\delta T\lesssim{10^{-4}}$, with $3\bar{H}\lesssim 10H_{0}$. Let us
see for instance how the present-epoch volume of the Universe is bounded
within members of this set of proper time foliations. We accordingly take
$S(\tau,X^{i})=1$, and the bound (4.53) becomes

$\left|\psi(X^{i})\right|\leq\exp\left(\max_{\pm\delta
T}\left|\int_{\tau=0}^{\pm\delta
T}\left|\nabla_{\mu}V^{\mu}\right|_{(\tau,X^{i})}\;d\tau\right|\right)-1\lesssim\exp\left(3\bar{H}\,\delta
T\right)-1\approx 3\bar{H}\,\delta T\lesssim 10^{-4}\;.$ (4.56)

Thus, the difference in the present-epoch volume $\mathcal{V}_{0,\tau}\equiv
I(1)^{\tau}_{t_{0}}$, as measured in two members of the class of proper time
foliations synchronised at the last scattering epoch, is bounded as

$\left|\Delta\mathcal{V}_{0}\right|=\left|\Delta I(1)\right|\leq
I(|\psi(X^{i})|)^{\tau_{\text{ref}}}_{t_{0}}\lesssim 3\bar{H}\,\delta
T\,I(1)^{\tau_{\text{ref}}}_{t_{0}}=3\bar{H}\,\delta
T\,\mathcal{V}_{0,\tau_{\mathrm{ref}}}\lesssim
10^{-4}\times\mathcal{V}_{0,\tau_{\mathrm{ref}}}\;.$ (4.57)

Hence, the volume of the present-epoch Universe is well-defined at the level
of ${}\sim 10^{-4}$ in the class of proper time foliations calibrated within
the region of last scattering. This is due to the time scale set by the
duration of last scattering through the width of the visibility function being
much smaller than the typical time scale associated with volume expansion in
the present-epoch Universe.

##  5 Summary and discussion

The $3+1$ foliation of spacetime in general relativity is a powerful tool for
casting Einstein’s equations into an initial value problem and for
constructing coarse-grained variables by integration operations defined on the
leaves. This is in particular used in approaches for cosmological averaging,
as in [17, 18, 13, 14, 20, 21, 19, 15, 16]. It has been noted that average
quantities that appear in such formalisms have dependence on the foliation
within which they are formulated [23, 24, 25, 27, 26, 28, 29].

In the present paper, we go beyond these previous investigations and consider
the $3+1$ foliation problem in a broad context. We are hence not addressing a
particular metric model nor a perturbative setting, but are treating the
problem of foliation dependence in relativistic integrals and averages of
scalar variables over arbitrary bounded regions of 3-dimensional
hypersurfaces, in presence of a generic nonsingular spacetime metric. We view
the integral and averaged quantities as functionals of the general scalar
function defining the foliation. Our systematic analysis of these functionals
considering infinitesimal variations of foliations revealed that globally
foliation-independent functionals do exist but must be associated with a
locally-conserved current. Thus, the only physically relevant exactly
preserved quantities are total rest masses within a bounded domain or other
quantities that are per construction preserved within the individual volume
elements. This is not in contradiction with the gauge-independent scheme
recently proposed in [29] in the context of perturbation theory, since there,
the coordinate transformations considered are not affecting the spacetime
foliation, and thus actual foliation changes are by construction absent in the
authors’ scheme.

We additionally examined choosing foliations specifically to leave a certain
functional invariant under infinitesimal deformations, _i.e._ , foliations or
individual slices which extremize a given functional, as in the well-known
case of extremal-volume slices. Such extremals provide examples of ways to
uniquely specify a foliation and thus to eliminate the ambiguity of foliation
choice. We briefly discussed, as specific applications, the extremization of
entropy functionals and the selection of slices with a minimal average tilt
with respect to a given vortical (hence not hypersurface-orthogonal) fluid
flow.

Since strictly foliation-independent integral/average quantities are rare, as
might have been expected, in the second half of our paper we investigated the
bounding of foliation dependence of integrals and averages under finite
changes of foliations. There we showed that we can in some cases set upper