# Dark matter and space-time symmetry restoration

João Magueijo<EMAIL_ADDRESS>Theoretical Physics Group, The
Blackett Laboratory, Imperial College, Prince Consort Rd., London, SW7 2BZ,
United Kingdom

###### Abstract

We examine local physics in the presence of global variables: variables
associated with the whole of the spacelike surfaces of a foliation. These
could be the (pseudo-)constants of nature and their conjugate times, but our
statements are more general. Interactions between the local and the global
(for example, dependence of the local action on global times dual to
constants) degrades full space-time diffeomorphism invariance down to spatial
diffeomorphism invariance, and so an extra degree of freedom appears. When
these presumably primordial global interactions switch off, the local action
recovers full invariance and so the usual two gravitons, but a legacy matter
component is left over, bearing the extra degree of freedom. Under the
assumption that the preferred foliation is geodesic, this component behaves
like dark matter, except that 3 of its 4 local degrees of freedom are frozen,
forcing its rest frame to coincide with the preferred foliation. The non-
frozen degree of freedom (the number density of the effective fluid) is the
survivor of the extra “graviton” present in the initial theory, and keeps
memory of all the past global interactions that took place in a given location
in the preferred foliation. Such “painted-on” dark matter is best
distinguished from the conventional one in situations where the preferred
frame would be preposterous if all 4 degrees of freedom of dark matter were
available. We provide one example: an outflowing halo of legacy matter with
exact escape speed at each point and a very specific profile, surrounding a
condensed structure made of normal matter.

##  I Introduction

This work is both part of a project on variability of the laws of physics evol
; BHevol ; MachianCDM , and a repository of results likely applicable in the
broader context of theories with a similar structure (e.g. HL ; shinji ; Isham
; brownkuchar ; viqar ; StueckelDM ; Barvinsky , reviewed further in our
concluding Section). Suppose we foliate space-time into spatial leaves.
Suppose some dynamical variables are functions of these leaves as a whole: we
call them global variables. Finally, suppose that the local action depends
parametrically on the global variables. In such theories, while global
variables are interacting with the local physics (in a sense to be made
precise in this paper), the local Hamiltonian constraint is lost, signaling
the downgrading of full 4D diffeomorphism invariance to 3D spatial
diffeomorphism invariance on each leaf. An extra local degree of freedom
therefore appears (often blamed on an extra graviton mode).

In this paper we show in great generality that this degree of freedom persists
even after global interactions switch off, indeed it carries a memory of their
integrated past effects at a given point. However, this degree of freedom can
then be neatly separated so that the local action regains full diffeomorphism
invariance (and so, the standard two gravitons). The extra degree of freedom
is absorbed by an effective matter component. If the preferred frame is
geodesic (a major assumption to be made in this paper and dropped in its
sequel LIVCDM ), this component is equivalent to a pressureless fluid with 3
out of its 4 degrees of freedom frozen, pre-fixing its rest frame (identified
with the fixed original foliation). A connection is thus made with dark
matter, but this is “matter without matter”. In addition, the presence of a
preferred frame makes this effective dark matter distinguishable in principle
from conventional dark matter.

We show in detail how this works without reference to any concrete theory, but
a specific realization arises from taking for global variables the “constants”
of Nature and their dual clocks JoaoLetter ; JoaoPaper . A prototype for this
construction is the Henneaux-Teitelboim formulation unimod  of unimodular
gravity unimod1 ; UnimodLee1 ; alan ; daughton ; sorkin1 ; sorkin2 , where the
global variables are $\Lambda$ and 4-volume or unimodular time Bombelli ;
UnimodLee2 ; JoaoLetter ; JoaoPaper , but other pairs can be chosen pad ; pad1
; lomb ; vikman ; vikman1 ; vikaxion ; JoaoLetter ; JoaoPaper ; evol . If the
clocks appear in the local action (i.e. if there is evolution) then there are
global interactions and we fall within the remit of this paper.

##  II Preliminary definitions

We start with a few general definitions. We assume that the theory contains a
preferred foliation $\Sigma_{t}$ (coordinatized by $(t,x)$). This is pre-fixed
in a way that is to be further qualified later (when we introduce a metric in
the theory). Given $\Sigma_{t}$ we distinguish local and global variables
(collectively represented by $\\{q(x),p(x)\\}$ and $\\{\alpha,T_{\alpha}\\}$)
from the structure of the action:

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
dt\,V_{\infty}\dot{\alpha}T_{\alpha}+S_{0}$ (1) $\displaystyle=$
$\displaystyle\int
dt\,\left[V_{\infty}\dot{\alpha}T_{\alpha}+\int_{\Sigma_{t}}d^{3}x\,(\dot{q}(x)p(x)-{\cal
H}_{E})\right]$

where $V_{\infty}=\int_{\Sigma_{t}}d^{3}x$ is the volume of $\Sigma_{t}$,
either finite or with a limiting procedure implied. The global variables are
independent of $x$ on $\Sigma_{t}$, so they may be considered a property of
the whole leaf. In contrast, each point on the leaf $x$ is associated with
canonical pairs of local variables, $q(x)$ and $p(x)$. The global variables,
just as the local ones, are assumed to be intensive variables, which justifies
the factor of $V_{\infty}$ in the first term of (1).

The first term in (1) contains the global variables already in their canonical
Hamiltonian form. The global variables are assumed not to have a Hamiltonian
of their own. The remaining terms, collected in $S_{0}$, make up the “local”
action. Upon a Legendre transformation, $S_{0}$ splits into the local
variables’ canonical terms and the extended Hamiltonian density, ${\cal
H}_{E}$. The non-vanishing Poisson brackets therefore are:

$\displaystyle\\{q(x),p(y)\\}$ $\displaystyle=$ $\displaystyle\delta(x,y)$ (2)
$\displaystyle\\{\alpha,T_{\alpha}\\}$ $\displaystyle=$
$\displaystyle\frac{1}{V_{\infty}}.$ (3)

The local action $S_{0}$ is assumed to depend parametrically on the global
variables (i.e. it does not depend on any derivatives of $\alpha$ or
$T_{\alpha}$), so the same happens to ${\cal H}_{E}$. In what follows, the
dependence of ${\cal H}_{E}$ on the local variables can be generic, ranging
from ultra-local (where ${\cal H}_{E}$ depends only on the local variables at
that point) to local or perturbatively non-local (where it can depend on any
order of gradients at that point). Thus, $S_{0}$ would perhaps be better
dubbed “non-global”, rather than “local” action, to distinguish it from the
terms we wish to study. In contrast, the first term in (1) refers to variables
that exist globally, as a function $\Sigma_{t}$, not terms that depend non-
locally on local variables.

To isolate effects due to global interactions it is useful to define the non-
local Poisson bracket:

$\\{f,g\\}_{NL}=\sum_{(\alpha,T_{\alpha})}\frac{\partial
f}{\partial\alpha}\frac{\partial g}{\partial T_{\alpha}}-\frac{\partial
f}{\partial T_{\alpha}}\frac{\partial g}{\partial\alpha}$ (4)

where $f$ and $g$ are general functions of phase space and the sum is over all
global variable pairs. Then, many expressions which in local theories are zero
become proportional to the non-local Poisson bracket111The violations of
energy conservation studied in evol  (cf. Eq.82) are an example.. We can also
consider a corresponding “local” Poisson bracket, so that:

$\\{f,g\\}=\\{f,g\\}_{L}+\\{f,g\\}_{NL}.$ (5)

The equations of motion involve the total Hamiltonian $\mathbf{H}$, which is
the space integral of ${\cal H}_{E}$:

$\displaystyle{\bf H}$ $\displaystyle=\int_{\Sigma_{t}}d^{3}x\,{\cal
H}_{E}(x),$ (6)

From this extensive quantity we can define an intensive total Hamiltonian
density:

$\displaystyle{\cal H}_{T}$
$\displaystyle=\frac{{\mathbf{H}}}{V_{\infty}}=\frac{1}{V_{\infty}}\int_{\Sigma_{t}}d^{3}x\,{\cal
H}_{E}(x):$ (7)

The Hamilton equations for the local variables are the usual:

$\displaystyle\dot{q}(x)$ $\displaystyle=$ $\displaystyle\\{q(x),{\bf
H}\\}=\frac{\delta{\cal H}_{E}(x)}{\delta p(x)}$ (8) $\displaystyle\dot{p}(x)$
$\displaystyle=$ $\displaystyle\\{p(x),{\bf H}\\}=-\frac{\delta{\cal
H}_{E}(x)}{\delta q(x)}$ (9)

computed at fixed $\alpha$ and $T_{\alpha}$ (just as one fixes all variables
other than the pair involved). Hence, the evolution of local variables at a
given point only depends on the functional derivatives of the Hamiltonian
density ${\cal H}$ at that point. This is to be contrasted with the equations
of motion for the global variables. With our definition of ${\cal H}_{T}$ the
factors of $V_{\infty}$ cancel out giving:

$\displaystyle\dot{\alpha}$ $\displaystyle=$
$\displaystyle\\{\alpha,\mathbf{H}\\}=\frac{\partial{\cal H}_{T}}{\partial
T_{\alpha}}$ (10) $\displaystyle\dot{T}_{\alpha}$ $\displaystyle=$
$\displaystyle\\{T_{\alpha},\mathbf{H}\\}=-\frac{\partial{\cal
H}_{T}}{\partial\alpha}$ (11)

where only intensive variables appear. The evolution of global variables
therefore depends on the Hamiltonian density over the whole leaf. In this
dependence, however, regions are weighted by their fractional volume:

$\displaystyle\dot{\alpha}$ $\displaystyle=$
$\displaystyle\int_{\Sigma_{t}}\frac{d^{3}x}{V_{\infty}}\frac{\partial{\cal
H}_{E}}{\partial T_{\alpha}}$ (12) $\displaystyle\dot{T}_{\alpha}$
$\displaystyle=$
$\displaystyle-\int_{\Sigma_{t}}\frac{d^{3}x}{V_{\infty}}\frac{\partial{\cal
H}_{E}}{\partial\alpha},$ (13)

so that the larger the region, the more it weighs on the global variables.
This is important in any setting, dubbed Machian in MachianCDM , where a
global presiding Universe acts as a “reservoir” fully determining the global
variables, which can thus be seen as external to the equations of motion of
any local system not part of the reservoir.

##  III Symmetries and constraints

Our first task is to examine the impact of global variables on the local
space-time symmetries, the associated constraints and their algebra. We assume
that the theory has a metric222So far we have only assumed a spatial
foliation, so a conformal structure has been assumed., which upon an ADM split
adapted to $\Sigma_{t}$ leads to lapse $N$, shift $N^{i}$, and induced or
spatial metric $h_{ij}$. The extended Hamiltonian density takes the form:

${\cal H}_{E}=N{\cal H}+N^{i}{\cal H}_{i}+...$ (14)

where ${\cal H}$ is the Hamiltonian density, ${\cal H}_{i}$ is the momentum
density and the ellipsis denotes the possibility of other terms, to be
explored elsewhere. Under the local Poisson bracket, $\cal H$ and ${\cal
H}_{i}$ form the Dirac or hypersurface deformation algebra Dirac ;
DiracCanadian ; Thiemann ; Bojo :

$\displaystyle\\{{\cal H}_{i}(x),{\cal H}_{j}(y)\\}_{L}$ $\displaystyle={\cal
H}_{i}(y)\delta_{,j}(x,y)+{\cal H}_{j}(x)\delta_{,i}(x,y)$ (15)
$\displaystyle\\{{\cal H}_{i}(x),{\cal H}(y)\\}_{L}$ $\displaystyle={\cal
H}(x)\delta_{,i}(x,y)$ (16) $\displaystyle\\{{\cal H}(x),{\cal H}(y)\\}_{L}$
$\displaystyle=({\cal H}^{i}(x)+{\cal H}^{i}(y))\delta_{,i}(x,y)$ (17)

where $\delta_{,i}(x,y)=\frac{\partial}{\partial x^{i}}\delta(x,y)$ and ${\cal
H}^{i}=h^{ij}{\cal H}_{j}$. The relation to diffeomorphisms is well
established. It will later be useful to perform calculations using their
smeared version:

$\displaystyle\\{H_{i}(N^{i}),H_{j}(M^{j})\\}_{L}$
$\displaystyle=H_{i}([N,M]^{i})$ (18)
$\displaystyle\\{H_{i}(N^{i}),H(N)\\}_{L}$
$\displaystyle=H(N^{i}\partial_{i}N)$ (19) $\displaystyle\\{H(N),H(M)\\}_{L}$
$\displaystyle=H_{i}(h^{ij}(N\partial_{j}M-M\partial_{j}N)$ (20)

where $H(N)=\int N{\cal H}$ and $H_{i}(N^{i})=\int N^{i}{\cal H}_{i}$.

The fact that global variables select a foliation should put us on notice: we
should not start by assuming more than spatial diffeomorphism invariance on
leaves $\Sigma_{t}$, leaving the status of full space-time diffeomorphisms on
hold. (Even the assumption of spatial diffeomorphism invariance may have to be
questioned later, but let us see how far it can take us.) Note that regarding
spatial diffeomorphisms, $q$, $\alpha$, $N$ and $N^{i}$ are tensors, and that
$p$, $T_{\alpha}$, ${\cal H}$ and ${\cal H}_{i}$ are tensor densities of
weight 1, balancing out the weight of $d^{3}x$ or $V_{\infty}$, so that the
theory is spatially diffeomorphism invariant.

Since we assume spatial diffeomorphism invariance, $N^{i}$ should be allowed
to be a general function $N^{i}(t,x)$, so that its variation implies the local
momentum constraint:

${\cal H}_{i}(x)\approx 0.$ (21)

Its preservation requires the vanishing of 3 terms:

$\displaystyle\dot{\cal H}_{i}=\\{{\cal H}_{i},\mathbf{H}\\}=$
$\displaystyle\\{{\cal H}_{i},\mathbf{H}_{0}\\}_{L}+\\{{\cal
H}_{i},\mathbf{P}\\}_{L}+\\{{\cal H}_{i},\mathbf{H}\\}_{NL}$

where we split $\mathbf{H}$ as:

$\displaystyle{\bf H}_{0}$ $\displaystyle\equiv
H(N)=\int_{\Sigma_{t}}d^{3}x\,N{\cal H}(x)$ (22) $\displaystyle\mathbf{P}$
$\displaystyle\equiv H_{i}(N^{i})=\int_{\Sigma_{t}}d^{3}x\,N^{i}{\cal
H}_{i}(x).$ (23)

The first term becomes:

$\\{{\cal H}_{i}(x),\mathbf{H}_{0}\\}_{L}={\cal H}(x)\frac{\partial
N}{\partial x^{i}},$ (24)

(using (16) or (19)). The second term is zero on-shell (using (15) or (18) and
(21)). The third, non-local, term implies:

$\\{{\cal H}_{i},\mathbf{H}\\}_{NL}=\frac{\partial{\cal
H}_{i}}{\partial\alpha}\dot{\alpha}-\frac{\partial{\cal H}_{i}}{\partial
T_{\alpha}}\dot{T}_{\alpha}.$ (25)

due to (12) and (13), and must be zero separately. Except in trivial
situations this places constraints on the dependence of ${\cal H}_{i}$ on the
global variables. If the dependence on global variables can be factorized as:

${\cal H}_{i}=F(\alpha,T_{\alpha})\bar{\cal H}_{i}$ (26)

with $\bar{\cal H}_{i}$ independent of global variables, we have for any
function $f$ of phase space:

$\\{f,{\cal H}_{i}\\}_{NL}\approx 0.$ (27)

We will assume this from now onwards. (More trivially, the momentum constraint
may not depend on any global variables, just as it does not depend on the
metric in most theories.)

We are thus left with condition (24), which presents us with two possibilities
for consistency of spatial diffeomorphism invariance on $\Sigma_{t}$, and the
fate of the Hamiltonian constraint. Either we also have a local Hamiltonian
constraint, ${\cal H}(x)=0$, in which case $N$ can be general (and if so,
indeed this implies ${\cal H}=0$); or we do not, in which case $N$ cannot
depend on space. To investigate where we stand regarding ${\cal H}=0$ we
evaluate its would-be secondary constraint:

$\displaystyle\dot{\cal H}=\\{{\cal H},\mathbf{H}\\}=$ $\displaystyle\\{{\cal
H},\mathbf{H}_{0}\\}_{L}+\\{{\cal H},\mathbf{P}\\}_{L}+\\{{\cal
H},\mathbf{H}\\}_{NL}.$

The first terms vanishes on-shell (using (17) or (20) and (21)). The second
term gives:

$\\{{\cal H}(x),\mathbf{P}\\}_{L}\approx D_{i}(N^{i}{\cal H})$ (28)

using (16). Hence we get a master equation for the variation of the
Hamiltonian, assuming the diffeomorphism constraint:

$\boxed{\dot{\cal H}=D_{i}(N^{i}{\cal H})+\\{{\cal H},\mathbf{H}\\}_{NL}.}$
(29)

Regarding the last term, notice that (27) implies $\\{{\cal
H},\mathbf{P}\\}_{NL}=0$, but we are still left with:

$\\{{\cal H},\mathbf{H}\\}_{NL}=\\{{\cal
H},\mathbf{H}_{0}\\}_{NL}=\frac{\partial{\cal
H}}{\partial\alpha}\dot{\alpha}-\frac{\partial{\cal H}}{\partial
T_{\alpha}}\dot{T}_{\alpha}.$ (30)

The vanishing of this expression is therefore a sine qua non for a possible
${\cal H}=0$. We thus use its non-vanishing as a definition for global
interactions being switched on. If these are switched on:

$\\{{\cal H},\mathbf{H}\\}_{NL}\neq 0\implies\dot{\cal H}\neq 0\implies{\cal
H}\neq 0$ (31)

so that the consistency of spatial diffeomorphisms (cf. (24)) forces

$\partial_{i}N=0\implies N=N(t)$ (32)

and variation with respect to $N$ gives only a global, rather than a local,
Hamiltonian constraint:

$\mathbf{H}_{0}=0.$ (33)

This global condition does not generate any new secondary constraints, since:

$\displaystyle\dot{\mathbf{H}}_{0}$
$\displaystyle=\\{\mathbf{H}_{0},\mathbf{H}\\}=\\{\mathbf{H}_{0},\mathbf{P}\\}=0$
(34)

in view of (27). We are left with first class constraints corresponding to
spatial diffeomorphisms and a global time reparameterization invariance of
leaves $\Sigma_{t}$. Their associated constraints are the usual 4 global
constraints, but only 3 local ones. Hence we gain a local degree of freedom
while global interactions are switched on.

We will not dwell upon its significance while global interactions are switched
on, but rather investigate whether full diffeomorphism invariance could ever
be restored in such theories once global interactions switch off (as defined
by the negative of (31)) and if so, where the extra degree of freedom goes
under such circumstances.

##  IV Symmetry restoration post-global interactions

Equation (29) is very interesting. Recall that $\cal H$ is a (3D) spatial
density of weight 1, so the first term on its right hand side is the spatial
Lie derivative on $\Sigma_{t}$:

$\pounds_{{N^{i}}}{\cal H}=\partial_{i}(N^{i}{\cal H})=D_{i}(N^{i}{\cal H})$
(35)

where $D_{i}$ is the spatial covariant derivative. If ${\cal H}\neq 0$ and
global interactions are switched off, then the only time variations in $\cal
H$ are those due to the spatial diffeomorphisms associated with $N^{i}$ (this
generalizes the result found in MachianCDM : if $N^{i}=0$ the legacy
Hamiltonian is a constant in time).

We can recast this fact as the vanishing of a total time derivative defined as
the 4D Lie derivative with respect to the normal $n^{\mu}$ to $\Sigma_{t}$ and
the congruence $\cal C$ of its integral lines. With the ADM convention:

$\displaystyle n_{\mu}$ $\displaystyle=(-N,0)$ (36) $\displaystyle n^{\mu}$
$\displaystyle=\left(\frac{1}{N},\frac{-N^{i}}{N}\right)$ (37)

(where both $N$ and $N^{i}$ can a priori depend on $t$ and $x$) and noticing
that $N{\cal H}$ is a 4D density, we can work out the identity (true for any
4D density of weight 1 built from $N$ and a 3D density):

$\displaystyle\pounds_{n^{\mu}}(N{\cal H})=$ $\displaystyle
n^{\mu}\partial_{\mu}(N{\cal H})+(\partial_{\mu}n^{\mu})N{\cal H}$
$\displaystyle=$ $\displaystyle\dot{\cal H}-\partial_{i}(N^{i}{\cal
H})=\dot{\cal H}-D_{i}(N^{i}{\cal H})$ (38)

(in the algebraic manipulations we did not use $\partial_{i}N=0$). Then (29)
becomes:

$\boxed{\frac{D{\cal H}}{dt}\equiv\pounds_{n^{\mu}}(N{\cal H})=\\{{\cal
H},\mathbf{H}\\}_{NL}=\frac{\partial{\cal
H}}{\partial\alpha}\dot{\alpha}-\frac{\partial{\cal H}}{\partial
T_{\alpha}}\dot{T}_{\alpha}.}$ (39)

When global interactions are switched on the Hamiltonian is driven by a
source, leading to an expression which can be formally written as

${\cal H}(t,x)=\int_{\cal C}\left(\frac{\partial{\cal
H}}{\partial\alpha}\dot{\alpha}-\frac{\partial{\cal H}}{\partial
T_{\alpha}}\dot{T}_{\alpha}\right)$ (40)

that is the line integral of the global interactions along the congruence
$\cal C$ tangent to $n^{\mu}$ (this is just the integral in time if
$N^{i}=0$). When the interactions are switched off one is left with a
generically non-zero, space-dependent integration constant which is a memory
of all past interactions. The free evolution of this non-zero Hamiltonian is
equivalent to Lie dragging $N{\cal H}$ along the vector $n^{\mu}$ normal to
$\Sigma_{t}$.

It would therefore seem that we are left with a permanent blemish: full
diffeomorphism invariance once broken is gone forever, even after global
interactions switch off. This does not need to be the case. Let us assume that
the base theory $S_{0}$ in the absence of global interactions (past, present
and future) has full diffeomorphism invariance333This is to be contrasted with
the equivalent problem in Horava-Lifshitz theory HL ; shinji , subject to
global interactions HLPaolo  or otherwise. In that case there is nothing to
restore, $N(t)$ can never be replaced by $N(t,x)$, and the Hamiltonian
constraint remains essentially global.. After global interactions switch off,
defined as:

$\\{{\cal H},\mathbf{H}\\}_{NL}=0$ (41)

(for example due to $\partial{\cal H}/\partial T_{\alpha}=0$), we can rewrite
the Hamiltonian as:

$\bar{\cal H}={\cal H}(q(x),p(x);\alpha)+m\approx 0$ (42)

with $m(x,t)$ a 3D density evolving according to:

$\frac{Dm}{dt}=\dot{m}-\partial_{i}(N^{i}m)=0$ (43)

and initial conditions set at the $\Sigma_{t_{0}}$ leaf when global
interactions were switched off. These initial conditions contain the
integrated effect of all previous global interactions and can be formally
written as:

$m(t_{0},x)=-\int_{\Sigma_{-\infty}}^{\Sigma_{t_{0}}}\left(\frac{\partial{\cal
H}}{\partial\alpha}\dot{\alpha}-\frac{\partial{\cal H}}{\partial
T_{\alpha}}\dot{T}_{\alpha}\right)$ (44)

over the line $\cal C$ linking the two $\Sigma_{t}$ surfaces. The newly found
local Hamiltonian constraint (42) can be enforced by allowing $N$ to depend on
space:

$N(t)\rightarrow N(t,x)$ (45)

in the corresponding action $\bar{S}_{0}$ built from $\bar{\cal H}$. In turn,
as we will see, the evolution (43) can be enforced by giving $m$ a canonical
partner $\phi$ and adding to the momentum constraint a new term:

$\bar{\cal H}_{i}={\cal H}_{i}+m\,\partial_{i}\phi$ (46)

with $\partial_{i}\phi=0$ on $\Sigma_{t}$ enforced by a new constraint
$\lambda^{i}\partial_{i}\phi$. Then $\bar{S}_{0}$, as written in coordinates
adapted to foliation $\Sigma_{t}$, reads:

$\displaystyle\bar{S}_{0}=$ $\displaystyle\int
dt\int_{\Sigma_{t}}d^{3}x\,(\dot{q}(x)p(x)+\dot{\phi}(x)m(x)$
$\displaystyle-N(t,x)\bar{\cal H}-N^{i}(t,x)\bar{\cal
H}_{i}+\lambda^{i}\partial_{i}\phi),$ (47)

which splits neatly into:

$\displaystyle\bar{S}_{0}=$ $\displaystyle S_{0}+S_{m}.$ (48)

Here

$\displaystyle S_{0}=$ $\displaystyle\int
dt\int_{\Sigma_{t}}d^{3}x\,(\dot{q}(x)p(x)-N(t,x){\cal H}-N^{i}(t,x){\cal
H}_{i})$ (49)

is nothing but the local action we started from, but with $N=N(t,x)$, that is
with full diffeomorphism invariance restored. The price to pay for this is the
new term:

$\displaystyle S_{m}=$ $\displaystyle\int
dt\int_{\Sigma_{t}}d^{3}x\,(\dot{\phi}(x)m(x)-N(t,x)m$
$\displaystyle-N^{i}(t,x)m\partial_{i}\phi-\lambda^{i}\partial_{i}\phi)$ (50)

to be examined for the rest of the paper.

The blemish in diffeomorphism invariance is therefore contained. $S_{0}$ now
has all the properties of the base theory as if no global interactions ever
existed. If $S_{m}$ can be ignored, then 4D diffeomorphism invariance is
restored. This is true even ignoring $S_{m}$ just conceptually: when we
evaluate the number of degrees of freedom of gravity, we mean “in vacuum” in
the sense of ignoring the matter degrees of freedom. By separating the legacy
effects of past global interactions, we can call them a matter component. Then
the gravity action $S_{G}$ contained in $S_{0}$ has 4 first class constraints,
and so 2 gravitons as usual. $S_{m}$ adds degrees of freedom, just like any
other matter component adds degrees of freedom. We will now examine the
properties of the extra degree of freedom transferred to $S_{m}$.

##  V An equivalence principle for foliations

Focusing on $S_{m}$ and the consistency of the constraints it generates we
find that a central issue is how the supposedly “pre-fixed” $\Sigma_{t}$
interacts with the metric, and specifically the lapse function, once $N$ is
allowed to have spatial dependence on $\Sigma_{t}$ in the variational problem.
The question is whether $N$ can be made to remain solely a function of $t$ on-
shell, even when we allow it to vary in space in the variational problem, cf.
(45). This sets up a major fork in the formalism, and is related to the
geodesic nature of the preferred $\Sigma_{t}$.

In this paper we impose a “weak equivalence principle” for preferred
foliations, stating that the foliations remain geodesic even after the lapse
function’s local degrees of freedom are released. This can be phrased as:

$a_{\mu}=n^{\nu}\nabla_{\nu}n_{\mu}=0$ (51)

and is equivalent to the statement that the (normalized) normal $n^{\mu}$ is
proportional to a gradient:

$n_{\mu}=\partial_{\mu}T$ (52)

for some scalar $T$ (which could be any global variable $T_{\alpha}$ which has
dropped out of the dynamics). In turn this is equivalent to $N$ being only a
function of $t$ on the preferred $\Sigma_{t}$. More directly we could use (see
ADMReview  for an excellent primer):

$a_{\mu}=h_{\mu}^{\;\nu}\nabla_{\nu}\ln N$ (53)

with $h_{\mu\nu}=g_{\mu\nu}+n_{\mu}n_{\nu}$ the projector orthogonal to
$n_{\mu}$ (and intrinsic metric $h_{ij}$).

Thus, whilst $\Sigma_{t}$ had to be geodesic under the action of spatially
diffeomorphism invariant global interactions (cf. (32)), we are now
postulating that this remains true after they switch off. If $\Sigma_{t}$ is
geodesic, then our coordinates may be chosen so that $N=N(t)$, that is,
although we have allowed $N$ to depend on space in the variations leading to
the Hamiltonian constraint (cf. (45)), we can choose coordinates such that on-
shell $N=N(t)$. (This is not unusual. In a related example which we will find
in Section IX, the LTB metric has $N^{i}=0$ and yet it has a non trivial
momentum constraint, resulting from varying $N^{i}(t,x)$.)

It is not difficult to see that if we drop this assumption the rest of the
story is very different. If $\Sigma_{t}$ is non-geodesic, after global
interactions switch off we have on $\Sigma_{t}$:

$\displaystyle\dot{\cal H}$ $\displaystyle=\\{{\cal
H},\mathbf{H}\\}_{L}=\partial_{i}(N^{i}{\cal H})+\partial_{i}({\cal
H}^{i}N)+{\cal H}^{i}\partial_{i}N$ (54) $\displaystyle\dot{\cal H}_{i}$
$\displaystyle=\\{{\cal H}_{i},\mathbf{H}\\}_{L}={\cal
H}\,\partial_{i}N+\partial_{j}(N^{j}{\cal H}_{i})+{\cal
H}_{j}\partial_{i}N^{j}$ (55)

(where ${\cal H}^{i}=h^{ij}{\cal H}_{j})$ which can be most easily derived
using (18)-(20). These evolution equations are inherited by ${\cal H}_{m}$ and
${\cal H}^{m}_{i}$ upon rewriting:

$\displaystyle\bar{\cal H}$ $\displaystyle={\cal H}(q(x),p(x);\alpha)+{\cal
H}^{m}\approx 0$ (56) $\displaystyle\bar{\cal H}_{i}$ $\displaystyle={\cal
H}_{i}(q(x),p(x);\alpha)+{\cal H}_{i}^{m}\approx 0.$ (57)

Requiring that ${\cal H}^{m}_{i}\approx 0$ as we did in the last section
therefore requires $N=N(t)$ on-shell. Hence the geodesic assumption for the
preferred frame, the statement that there are coordinates for which $N=N(t)$,
and the constraint that an initially vanishing ${\cal H}_{i}$ remains zero in
the preferred frame, are all equivalent.

Dropping these equivalent assumptions is not only allowed: it is a very
interesting possibility. But it does lead to a totally different formalism and
we leave its development to a separate paper LIVCDM .

##  VI Effective dark matter

Returning to $S_{m}$ defined in (50) we start by making statements in
coordinates adapted to $\Sigma_{t}$. Action (50) implies the equations of
motion:

$\displaystyle\dot{m}=$ $\displaystyle\partial_{i}(N^{i}m)$ (58)
$\displaystyle\dot{\phi}=$ $\displaystyle N+N^{i}\partial_{i}\phi,$ (59)

which are an abridged version of (54) and (55), since

$\displaystyle{\cal H}_{m}$ $\displaystyle=m$ $\displaystyle{\cal H}_{i}^{m}$
$\displaystyle=m\,\partial_{i}\phi$

with the additional constraint ${\cal H}^{m}_{i}=0$. We see that (58) states
that $S_{m}$ produces the Lie dragging equation (43), as claimed in Section
IV. In addition, under the assumption that $\Sigma_{t}$ is geodesic, $N=N(t)$;
therefore (59) implies that the constraint $\partial_{i}\phi=0$ is preserved
by the time evolution, with on-shell identities $n_{0}=-N=-\dot{\phi}$ in
these coordinates, implying the covariant expression:

$n_{\mu}=-\partial_{\mu}\phi.$ (60)

We see that $\phi$, the conjugate variable to $m$, is (minus) the fluid proper
time variable for dust matter.

The only other equations of motion affected by $S_{m}$ are the gravitational
equations, in which $S_{m}$ enters via its implied stress energy tensor. In
coordinates adapted to $\Sigma_{t}$ this has only one non-vanishing component:

$T_{00}^{m}=\frac{-2}{N\sqrt{h}}\frac{\delta S_{m}}{\delta
g^{00}}=N^{2}\frac{m}{\sqrt{h}},$ (61)

and can therefore be written in covariant form as:

$T^{m}_{\mu\nu}=\rho_{m}(x)n_{\mu}n_{\nu},$ (62)

where

$\rho_{m}=\frac{m}{\sqrt{h}}$ (63)

is a 4D scalar (i.e. a density of weight zero). This is the stress-energy
tensor of a pressureless perfect fluid. Variation of $\bar{S}_{0}$ with
respect to the metric shows that $T_{\mu\nu}^{m}$ appears in the right hand
side of the gravitational equations as an additional term:

$\bar{T}_{\mu\nu}=T_{\mu\nu}+T^{m}_{\mu\nu}$ (64)

(where $T_{\mu\nu}$ refers to all the forms of matter contained in $S_{0}$),
so it has the usual active gravitational properties.

It is also independently conserved:

$\displaystyle\nabla_{\mu}T^{\mu\nu}_{m}=0$ (65)

as we now prove. First, $n^{\mu}$ is by construction geodesic, so:

$h^{\alpha}_{\;\;\nu}\nabla_{\mu}T^{\mu\nu}=\rho
n^{\mu}\nabla_{\mu}n^{\alpha}=0.$ (66)

Second, the Lie dragging equation (43) implies the other component of the
conservation equation, (constancy of the dust grains’ rest mass),
$\nabla_{\mu}(\rho n^{\mu})=0$. Recalling (37), we have:

$-n_{\nu}\nabla_{\mu}T^{\mu\nu}=\nabla_{\mu}(\rho
n^{\mu})=\frac{1}{\sqrt{-g}}(\dot{m}-(mN^{i})_{,i})=0$ (67)

that is, the relativistic conservation equation:

$\dot{\rho}+\rho\nabla_{\mu}n^{\mu}=0.$ (68)

##  VII Expression in a general frame

All of the above was done in coordinates adapted to the preferred
$\Sigma_{t}$, but we may not know this frame, so it is useful to write the
frame-selecting action $S_{m}$ in generic coordinates. Let us change notation,
replacing $n_{\mu}\rightarrow u_{\mu}$ in the previous section, so that
$u_{\mu}$ denotes the normal of the special foliation $\Sigma_{t}$, releasing
$n_{\mu}$ to stand for the normal to a generic ADM foliation. From the
expression for the “gamma factor” of a relativistic fluid brown :

$-n^{\mu}u_{\mu}=\sqrt{1+h^{ij}u_{i}u_{j}}\equiv\gamma(u_{i},h^{ij})$ (69)

we can write in a generic frame:

$\displaystyle u_{\mu}$ $\displaystyle=(N^{i}u_{i}-N\gamma,u_{i})$ (70)
$\displaystyle u^{\mu}$
$\displaystyle=\left(\frac{\gamma}{N},u^{i}-\frac{N^{i}\gamma}{N}\right)$ (71)

where $N$, $N^{i}$ and $h_{ij}$ are defined in the new foliation and we
define:

$u^{i}\equiv h^{ij}u_{j}$ (72)

not to be confused with the 4-D counterpart
$g^{i\mu}u_{\mu}=h^{ij}u_{j}-N^{i}\gamma/N$. Inserting this form of $u_{\mu}$
in:

$T^{m}_{\mu\nu}=\rho_{m}(x)u_{\mu}u_{\nu},$ (73)

we get the stress-energy tensor expression in any frame. This provides the
effect on the gravitational equations of motion in any coordinates or
foliation.

Likewise, the general form of the Hamiltonian and momentum can be read off
from (73), using relations:

$\displaystyle{\cal H}_{m}$
$\displaystyle=\sqrt{h}n^{\mu}n^{\nu}T^{m}_{\mu\nu}$ (74) $\displaystyle{\cal
H}^{m}_{i}$ $\displaystyle=\sqrt{h}n^{\mu}P^{\nu}_{\;i}T^{m}_{\mu\nu}.$ (75)

Recall that, although $\rho_{m}$ is a 4D scalar, $m(x)$ is a spatial density.
If $m$ and $h_{S}$ are the Hamiltonian and determinant of the spatial metric
in the original frame, and $m(u_{i})$ and $h$ those in the new one, we have:

$m(u_{i})=m\sqrt{\frac{h}{h_{S}}}.$ (76)

Using coordinates adapted to the new frame we have
$h_{ij}=g_{ij}+n_{i}n_{j}=g_{ij}$, whereas the induced metric in the old
(preferred) foliation is $h^{(S)}_{ij}=g_{ij}+u_{i}u_{j}$. Their determinants
are thus related by:

$h_{S}=h(1+h^{ij}u_{i}u_{j})$ (77)

so that:

$m(u_{i})=\frac{m}{(1+h^{ij}u_{i}u_{j})^{1/2}}.$ (78)

Using (74)-(75) and (69) we finally have:

$\displaystyle{\cal H}_{m}$
$\displaystyle=m(u_{i})(n^{\mu}u_{\mu})^{2}=m(1+h^{ij}u_{i}u_{j})^{1/2}$ (79)
$\displaystyle{\cal H}^{m}_{i}$
$\displaystyle=m(u_{i})(n^{\mu}u_{\mu})u_{i}=-mu_{i}.$ (80)

Given (60), in a general frame we should replace $\lambda^{i}\partial_{i}\phi$
in (50) by $\lambda^{i}(n_{i}+\partial_{i}\phi)$. We can also simply ignore
this constraint and write the action as

$\displaystyle S_{m}$ $\displaystyle=\int
dt\,d^{3}x\,(\dot{\phi}(x)m(x)-N{\cal H}_{m}-N^{i}{\cal H}^{m}_{i})$ (81)

where it is understood that $u_{i}$ is shorthand for:

$u_{i}\equiv-\partial_{i}\phi.$ (82)

This is just the Brown action brown  for a relativistic fluid, freezing out
the entropy and hydrodynamical velocity degrees of freedom444See 3.10 and 3.11
with 3.7 and 3.9 in Ref. brown . See also brownkuchar .. Hence the full legacy
of past evolution is just a form of cold dark matter (at least assuming that
$\Sigma_{t}$ is geodesic). However, as we now show, this is a very special
type of dark matter.

##  VIII Preferred geodesic frames and special dust

Freezing the entropy and hydrodynamical degrees of freedom makes sense and
completes our discussion of symmetries and degrees of freedom. We have found
that theories with global interactions have 3 local gravitational degrees of
freedom, having lost one local constraint (the Hamiltonian constraint). After
global interactions switch off the extra degree of freedom can be exported
into a matter-like component, since we can separate the action into an action
$S_{0}$ with full diffeomorphism invariance restored (and thus the usual two
gravitons), and an effective fluid action absorbing the extra degree of
freedom (which therefore is not a propagating “graviton”).

This is necessarily a very special fluid. A generic perfect fluid has 5
degrees of freedom: the number density, the entropy density and the freedom
the fluid has to move in 3 spatial directions. These correspond to 5
unconstrained pairs of variables (see e.g. brown ). Since we are absorbing
only one degree of freedom into this effective fluid, the fluid has to have 4
degrees of freedom “frozen”. The entropy degree of freedom is erased by the
dust equation of state $p=0$. The hydrodynamical degrees of freedom are frozen
by the assumption that $u_{\mu}$ is not dynamical, but is a pre-given
preferred frame (albeit geodesic, according to the assumptions made here). The
counting therefore matches.

Such “freezing” is better formalized by showing its equivalence to imposing
first class constraints on a generic fluid. The dust condition, like any other
equation of state, can be seen as a first class constraint on a more generic
fluid since dust remains (or can remain) dust under time evolution. In
addition we have to impose 3 conditions on the most general dust action,
$S_{d}$. Such action can be written (see brown ; brownkuchar ):

$\displaystyle S_{d}$ $\displaystyle=\int
dt\,d^{3}x\,(\dot{\phi}m+\dot{\alpha}^{A}(m\beta_{A})-N{\cal H}_{m}-N^{i}{\cal
H}^{m}_{i})$ (83)

with ${\cal H}_{m}$ and ${\cal H}^{m}_{i}$ still given by (79) and (80), but
with (82) replaced by:

$u_{i}=-(\partial_{i}\phi+\beta_{A}\partial_{i}\alpha^{A}).$ (84)

Here $\alpha^{A}$ and $\beta_{A}$ ($A=1,2,3$) are 6 space-time scalars
containing the hydrodynamical degrees of freedom. The $\alpha^{A}$ are the
Lagrangian or “comoving” coordinates of the fluid, and $m\beta_{A}$ are their
conjugate momenta. They evolve under brown ; brownkuchar :

$\displaystyle\dot{\alpha}^{A}$
$\displaystyle=\\{\alpha^{A},\mathbf{H}\\}=-\frac{N}{\gamma}u^{i}\partial_{i}\alpha^{A}+N^{i}\partial_{i}\alpha^{A}$
(85) $\displaystyle(m\beta_{A})^{.}$
$\displaystyle=\\{m\beta_{A},\mathbf{H}\\}=-\partial_{i}(Nm\beta_{A}u^{i})+\partial_{i}(N^{i}m\beta_{A}).$
(86)

Hence

$\beta_{A}=0$ (87)

is the required set of 3 constraints555This choice breaks the fluid symmetries
described in brown , but the matter will not be relevant here.. If
$\beta_{A}=0$ (and so also $\partial_{i}\beta_{A}=0$) the evolution equation
(86) implies $\dot{\beta}_{A}=0$, so the constraints are first class and do
remove or “freeze” the 3 hydrodynamical degrees of freedom of the fluid. The
other evolution equation (85) is just the transformation law between Eulerian
and Lagrangian coordinates. The $\alpha^{A}$ and $\beta_{A}$ do not contribute
to $u^{\mu}$, or to the extended Hamiltonian. They drop out of the remaining
dust equations of motion:

$\displaystyle\dot{\phi}$
$\displaystyle=\\{\phi,\mathbf{H}\\}=N\gamma+N^{i}\partial_{i}\phi$ (88)
$\displaystyle\dot{m}$
$\displaystyle=\\{m,\mathbf{H}\\}=-\partial_{i}(Nmu^{i})+\partial_{i}(N^{i}m)$
(89)

which are just (58) and (59) in a generic frame (i.e. not in the rest frame
$\Sigma_{t}$ or dust).

##  IX “Painted on” dark matter and testability

Hence, even with the geodesic assumption, a special frame remains in place.
The effective fluid absorbing the extra degree of freedom is not (and could
not be, as we know just from “counting”) the most general dust fluid, for
which the introduction of a foliation is just a convenient mathematical device
with no physical significance. The rest frame of the equivalent dust fluid is
pre-fixed, among all possible geodesic frames, so we still have what in
MachianCDM  was labelled “painted on” dark matter.

But how could we distinguish it from conventional dark matter? Unlike the case
in LIVCDM , where a non-geodesic $\Sigma_{t}$ leaves dramatic new imprints,
the distinction here is subtle. Apart from the fact that vortical motion could
never be assigned to our effective fluid666The flow is a gradient flow, a
requirement of Frobenius theorem applied to $\Sigma_{t}$., and barring pre-
knowledge of the preferred frame, we can only distinguish our effective matter
from conventional dark matter by the preferred frame being something that for
normal dark matter would be “outlandish”, in the sense that it would require
the initial conditions having an uncanny pre-cognition of what would come in
the future. We give a simple example, which may have phenomenological
significance in an improved form.

Consider the Lemaitre-Tolman-Bondi (LTB) formalism exactslns ; enqvist ;
Goncalves , starting from metric:

$ds^{2}=-dt^{2}+X^{2}(r,t)dr^{2}+A^{2}(r,t)d\Omega^{2},$ (90)

which only assumes spherical symmetry. This is a geodesic frame, since $N=1$.
Although $N^{i}=0$ there is a momentum constraint (obtained by varying $N^{i}$
around zero). If this frame is the preferred frame $\Sigma_{t}$, then ${\cal
H}_{i}^{m}=0$, and assuming all other forms of dust matter are comoving with
it (or vanish) in the region and epoch of interest, the momentum constraint
reads:

$\displaystyle\dot{A}^{\prime}-A^{\prime}\frac{\dot{X}}{X}$ $\displaystyle=0$
(91)

(where ${}^{.}\equiv\partial/\partial t$ and
${}^{\prime}\equiv\partial/\partial r$). This equation has a first integral
which places the metric in the form:

$ds^{2}=-dt^{2}+\frac{A^{\prime 2}}{1-K(r)}dr^{2}+A^{2}d\Omega^{2}.$ (92)

(We stress that this step would not be valid if ${\cal H}_{i}^{m}\neq 0$.) The
remaining equations then amount to the local Hamiltonian constraint:

$\displaystyle\frac{\dot{A}^{2}}{A^{2}}$ $\displaystyle=$
$\displaystyle\frac{F(r)}{A^{3}}-\frac{K(r)}{A^{2}}$ (93)

with:

$F^{\prime}=8\pi G\rho A^{\prime}A^{2}.$ (94)

The FRW metric and equations can be recognized within the LTB series by
setting $A=a(t)r$, $K=kr^{2}$ (with $a$ the expansion factor and $k=0,\pm 1$
the normalized intrinsic curvature) as well as:

$F(r)=\frac{8\pi G}{3}r^{3}(\rho_{tot}a^{3})=\frac{8\pi G}{3}r^{3}m_{tot}$
(95)

(where $m_{tot}=\rho_{tot}a^{3}$ refers to the constant of motion for the
total pressureless component). This suggests that the $t=$const. leaves in
these coordinates could be the preferred frame $\Sigma_{t}$ inherited from a
cosmological origin777The label $r$ is also fixed by the form of (95). We
could call this the Hubble gauge, noting that in general LTB has a residual
freedom of redefinition of $r$.. This is obviously true in the FRW
approximation of MachianCDM  (and of shinji  in the context of Horava-Lifshitz
theory HL ), but the LTB framework allows us to consider inhomogeneous
solutions. These could include primordial inhomogeneities in $m(x,t_{0})$, but
more simply result from situations where a primordial homogeneous
$m(x,t_{0})=m(t_{0})$ reacts to the condensation of structures made from other
forms of matter.

Let us therefore assume that the preferred frame is the LTB frame with $K=0$
matching a primordial and a present asymptotic FRW Universe. This frame
remains present even when primordial perturbations are added on, or large
scale structures condense at late times. As a test tube for the implications
of this statement, let other pressureless matter form a central black hole
with Schwarzchild radius $R_{s}=2GM$ creating a spherical hole in an otherwise
$k=0$ FRW Universe. If the LTB frame is $\Sigma_{t}$, obviously our effective
DM cannot condense, since $N^{i}=0$ makes Lie dragging (43) trivial. Hence the
effective matter will continue to expand even though other forms of matter
condensed, creating the first anomaly.

Eq. (43) implies $\dot{m}=0$ where $m$ is a spatial density, which is uniform
in Cartesians ($m=m_{0}$). By evaluating the relevant factors of $h$ (first
from Cartesians to polars, then to full LTB) we find that888We could also
derive this expressions from the LTB formula enqvist ; Goncalves :
$\rho_{m}=\frac{F^{\prime}_{m}(r)}{8\pi GA^{\prime}A^{2}}$ (96) using (95) and
noting that $F(r)$ is additive.:

$\rho_{m}(r,t)=\frac{m_{0}r^{2}}{A^{2}|A^{\prime}|}.$ (97)

In the inner region where the black hole dominates (or assuming $\Omega_{m}\ll
1$, so that the escavated mass $M$ always dominates Stephani ) we have the
vacuum solution exactslns :

$A(r,t)=\left(\frac{9R_{s}}{4}\right)^{1/3}(t+r-r_{0})^{2/3}$ (98)

in the geodesic outgoing coordinates (with $A^{\prime}>0$) associated with
$\Sigma_{t}$. We have adjusted constants so that the FRW solution is matched
at $r=r_{0}$, setting $t=0$ for the Big Bang time. The other matching
condition equates $M$ with the escavated mass taken out of the FRW model to
create the vacuole (e.g. Stephani ), or:

$\frac{8\pi G}{3}\rho_{c}a^{3}r_{0}^{3}=R_{s}\implies H^{2}A_{0}^{3}=R_{s}$
(99)

(where $\rho_{c}$ is the critical cosmic density, H is the Hubble parameter,
and $A_{0}=ar_{0}$).

Eqs. (97) and (98) imply:

$\rho_{m}=\frac{m_{0}r^{2}}{A^{3/2}\sqrt{R_{s}}}=\frac{2m_{0}r^{2}}{3R_{s}(t+r-r_{0})}$
(100)

and since this is a scalar we know the density in any coordinates (modulo the
fact that different coordinate charts may cover different regions). In
particular, we can transform to Schwarzchild coordinates
$ds^{2}=-dT^{2}(1-R_{s}/A)+dA^{2}/(1-R_{s}/A)+A^{2}d\Omega^{2}$, noting that
$\\{t,r\\}$ for this solution are the Lemaitre coordinates exactslns  (with
$A^{\prime}=\sqrt{R_{s}/A}$)999We recall the transformation $dt=dT-
dA/(\sqrt{x}(1-1/x))$ and $dr=-dT+\sqrt{x}dA/(1-1/x)$, where $x=A/R_{s}$. . In
fact, all we need is:

$\displaystyle r=-T+R_{s}\int^{A/R_{s}}\frac{dx\,\sqrt{x}}{1-\frac{1}{x}}$
(101)

(we spare the reader the explicit expression). The integral is logarithmically
divergent at the black hole horizon, $A=R_{s}$, which is the boundary of the
patch covered by the LTB outgoing coordinates; this leads to a divergence in
$\rho_{m}$. The effects on black hole formation could be interesting.

In the Newtonian region:

$r\approx-T+r_{0}+\frac{2}{3\sqrt{R_{s}}}A^{3/2}$ (102)

where we adjusted the integration constants so that this matches (98) with
$T\approx t$. Fixing $r$, we can recognize the Newtonian trajectory of
particles with exact escape speed, starting from delayed “Big Bangs”101010In
the conventional sense of times for which $A=0$ and $\dot{A}=\infty$
extrapolating back in time, erroneously using the Newtonian picture all the
way back.. Unlike with a Newtonian Hubble flow, where the explosion started at
the same time for all the shells and each shell is slowed down by the gravity
of all the inner shells, in our case the explosion started later for the inner
shells, and all shells feel the gravity due to the same dominant central mass
$M$. Specifically, we see a flow satisfying Newton’s laws for a central mass
$M$:

$T-T_{BB}(r)=\frac{2}{3\sqrt{R_{s}}}A^{3/2}$ (103)

with $T_{BB}=r_{0}-r$. The density profile in the Newtonian region is:

$\displaystyle\rho_{m}(A,T)$
$\displaystyle=\frac{m_{0}}{A^{3/2}\sqrt{R_{s}}}\left(-T+r_{0}+\frac{2}{3\sqrt{R_{s}}}A^{3/2}\right)^{2}$
(104)

with

$\displaystyle\frac{2}{3}\frac{A^{3/2}}{R_{S}^{1/2}}<T<r_{0}+\frac{2}{3}A^{3/2}/R_{S}^{1/2}$

so no longer do we have a homogeneous, time dependent profile as in the case
of a Hubble flow. Instead we can think of this profile as a halo with roughly:

$\rho_{m}(A,T)\approx\frac{m_{0}r_{0}^{2}}{A^{3/2}\sqrt{R_{s}}}$ (105)

(for $A\gg R_{s}$), that is with a roughly static profile $1/A^{3/2}$ for a
large period of its life time $\Delta T\approx r_{0}$. Eq. (104) complements
the details of this simplified picture.

This would be a very strange kind of “halo” if made of normal matter. It would
consist of dust particles moving outwards along geodesics which are on the
border between bounded and unbounded, each with a delayed starting time
according to a well defined rule. This would happen even when other types of
pressureless matter collapsed to produce a structure (presumably due to being
in a bounded Hubble flow from the start). Such “halo” would have a density
profile neither matching the Hubble flow it came from, nor the current
condensed system into each it is embedded. Barring conspiracy initial
conditions, where would such a structure come from if not from the presence of
a preferred cosmological geodesic frame?

This is just an example, and more elaborate models can be built, as we now
sketch. Depending on the size of the vacuole with regards to the Hubble
length, this structure might more closely resemble an ejected corona than a
halo. We could also consider non-homogenous initial $m(x)$, or situations
where $\rho_{m}$ is not subdominant inside condensed structures. Could this
help explain galaxy rotation curves, bypassing the shortcomings of more
conventional dark matter, such as the cusp-core problem?

##  X Conclusions

In this paper we investigated in a general setting the effects of breaking
full diffeomorphism invariance down to spatial diffeomorphism invariance on a
foliation, and then restoring it. A new local degree of freedom is created by
such partial symmetry breaking, which persists even after full symmetry is
restored. If the foliation breaking the symmetry is geodesic by construction,
this degree of freedom is equivalent to a dark matter component, which does
not need to be homogeneous. However, only one out of its four local degrees of
freedom is available. Phenomenological implications were discussed.

Our general setting involved the action upon local physics of global
variables, that is variables defined on the whole leaves of the foliation.
This is to be distinguished from perturbatively non-local interactions, such
as those with higher order derivatives. We stress that we are envisaging the
effects of variables that exist globally on $\Sigma_{t}$, rather than the more
traditional setting of terms in the action that depend non-locally on local
variables111111Indeed such perturbatively non-local terms could be bundled
into what we called local action, which should perhaps more rigorously be
dubbed “non-global”.. Global interactions break locality maximally and
introduce a preferred frame or foliation. They have a “Machian” flavour,
within a meaning backtracking to before the advent of General Relativity, as
pointed out in MachianCDM . Our considerations have nothing to do with
inertia, but they do draw on Mach’s approach MachBook ; MachReview  in that we
are entertaining direct instantaneous action of the whole Universe on its
parts.

Several types of theory fall within this structure. Our original motivation
was work on variability of the laws of physics (evol ; BHevol  and more
specifically MachianCDM ), itself grounded on unimodular-like theories unimod
; unimod1 ; UnimodLee1 ; alan ; daughton ; sorkin1 ; sorkin2 ; Bombelli ;
UnimodLee2 ; JoaoLetter ; JoaoPaper ; pad ; pad1 ; lomb ; vikman ; vikman1 ;
vikaxion ; JoaoLetter ; JoaoPaper . In this context the global variables are
the (pseudo-)constants of Nature and their canonically dual times, the latter
defined as in unimod  (and not on the past light cone of an observer, as often
done in the context of causal set theory). If time is evaluated on space-like
hypersurfaces, any time dependence of the local Hamiltonian gives physical
relevance to the foliation (just as, conversely, time independence and on-
shell space-time constancy of constants erases its physical relevance unimod
). Despite this declared motivation, the results derived here are likely
applicable with minimal modification to other theories with the same
structure. We mention Horava-Lifshitz theory (e.g. HL ; shinji ), some
approaches to time in Quantum Gravity (e.g. Sec. 4.3.2 of Isham  and
references therein; also brownkuchar ; viqar ) and more generally the issue of
gauge fixing in Quantum Gravity (e.g. StueckelDM ; Barvinsky ) and how it may
introduce a conceptual preferred frame.

Our results motivate much further work. The existence of a preferred frame
introduces an element of cosmic precognition common to all theories with a
global structure (e.g. pad ; pad1 ; padilla 121212The term “pre-cognition” in
relation to the cosmological constant was first used by Coleman Coleman .). As
we saw in Section IX, this can be used to distinguish the associated effective
matter from conventional dark matter, in the sense that the latter would need
to have its initial conditions incredibly fine tuned to reproduce what a
preferred geodesic frame naturally gives to our effective dust, with three of
its four degrees of freedom frozen. We gave an example in Section IX, but
could a more elaborate model serve as partial explanation, for example, of the
galaxy rotation curves? Astrophysical issues such as the flattening of the
galaxy rotation curves and the Tully-Fisher relation might then be truly
cosmological initial condition problems.

More dramatically, the preferred $\Sigma_{t}$ might not be geodesic as
explored in the sequel to this paper LIVCDM .

##  XI Acknowledgments

We thank Paolo Bassani, Claudia de Rham, Shinji Mukohyama and Tom Zlosnik for
discussions. This work was partly supported by the STFC Consolidated Grants
ST/T000791/1 and ST/X00575/1.

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