# Minimum length universe in a Robertson Walker geometry

Dragana Pilipović  Department of Physics, University of Illinois Chicago

###### Abstract

A minimum length universe is enforced via a diffusive Markovian field.
Requiring the invariance of the proper time functional under spacetime
transformations generated by such fields provides us with many new terms added
to the Christoffel Connection, Equations of Motion, and Friedmann Equation.
Under the Robertson-Walker (RW) metric, we obtain species-specific energy-
density evolution, now a function of both the parameters defining the
diffusive fields as well as the RW scale factor.

The math leads us to a relationship between temporal diffusion, universe
expansion acceleration, and the speed of universe expansion, along with a
derivation of the cosmological constant as a function of both minimum length
and the scalar field expectation value of the Higgs mechanism.

The theory gives us a means of relating the Hubble parameter to its diffusive
ML ’equivalent’, and a means to express the effective equation of state in
terms of the deceleration parameter, q. We find that q can take on a range of
higher values without invoking the need for a dark matter species, and that
the universe inhabits a vacuum or matter dominated regime as a function of q.

I. INTRODUCTION

Robertson Walker Minimum Length (RWML) theory corresponding to a negative
deceleration parameter, $q$, and a positive Hubble parameter, $H$, requires
Markovian ML fields to transform at a different scale to that of spacetime.
While both spacetime and ML fields are, independently, conformally Minkowski
in a local frame ($r=0$), relative to each other the two conformal spaces
scale at a non-zero power of the RW scale factor. In simultaneously
incorporating the effects of RW geometry and ML, the scale factor can no
longer be conformally ’transformed away’.

RWML theory opens up some new possibilities and introduces some new concepts.
It allows for a greater possible range of the deceleration parameter and the
effective universe equation of state, with corresponding matter vs. vacuum
dominated regimes. RWML also introduces a new diffusion parameter $D$, the
diffusion ’equivalent’ of $H$, and a potentially new species, the Higgs
vacuum.

Specifically, we set out to answer the following questions:

a) why is the vacuum energy density so small relative to what one might expect
given large scalar field expectation values in Quantum Field Theory (QFT)?

b) what is the origin of the small positive energy density of the vacuum?

c) how can we explain the relative ratios of energy density parameters across
species, specifically why is $\rho_{M}/\rho_{V}\sim 0.5$ while
$\rho_{R}/\rho_{M}\sim 10^{-3}$ [1]?

d) is it possible to have a deceleration parameter, $q$, which is greater in
magnitude than allowed by Robertson-Walker (RW) theory ($q=-\frac{1}{2}$)
without requiring a new (dark matter) species with $w<-1$?

RWML suggests the following answers:

a) ML temporal diffusion is the driver of both the universe expansion and its
acceleration. But also, at the minimum possible vacuum energy, the expansion
velocity is directly proportional to the QFT scalar field expectation value,
$\bar{v}$. ML theory in a RW geometry thus gives us the relationship between
temporal diffusion, $\bar{v}$, and the acceleration of universe expansion. The
resulting RW scale factor is at present extremely large, giving us a very
small vacuum energy.

b) ML spatial diffusion gives the vacuum energy density a minimum positive
value proportional to $\bar{v}^{2}$. The RWML theory further shows us that the
vacuum energy density is an irreducible quadratic function of $H$ and $D$ over
the reals, with $H>0$, but only if the ML fields transform at a different
power of the scale factor relative to that of spacetime. At this minimum
possible positive vacuum energy density, the velocity of expansion is positive
and defined by spatial diffusion, and hence $\bar{v}$.

c) Given the present energy density ratios across the species, we can estimate
that the magnitude of the rate of spatial diffusion for radiation is roughly
three times that of the rate of diffusion for matter.

d) Under RWML theory we find that - without invoking any new species - the
universe has matter-dominated and vacuum-dominated regimes as a function of
the acceleration parameter $f,f\equiv-q$. We find an open111While we work in a
local, and thus geometrically flat spacetime, under RWML the universe can
still be open or closed as a relative function of $D$ and $H$. and mostly
vacuum-dominated universe for $f$ in the neighborhood of one, with an
effective equation of state described by $w\approx-0.7$. Lower values for $f$
put the universe in the approximately constant $D$ regime with an exactly
matter-dominated effective equation of state, $w=0$, for the choice of
$f=\frac{1}{2}$.

RWML further suggests that were the acceleration parameter to grow well beyond
one, universe has a choice of one of two paths. One path allows for a
’peaceful’ transition between matter and vacuum dominated states as long as
$H$ remains positive. The second path is a universe ’dooms-day’ scenario
providing a degenerate set of solutions at higher values of the acceleration
parameter but for $H<0$.

II. MOTIVATION FOR RWML UNIVERSE

While RW theory without ML diffusion gives us a curious static solution to the
Friedmann equations by requiring that $w=-\frac{1}{3}$, this is not a problem
in an expanding universe. However, the ad hoc addition of the cosmological
constant in the case of a mostly flat ($\kappa\approx 0$) universe makes the
RW theory incomplete. On the other hand, ML diffusion under a Minkowski metric
($\eta$) gives us no means by which we can relate the metric expansion and its
acceleration to the ML diffusive process. Each theory - on its own - lacks the
exact characteristics of the other.

ML diffusion under an RW metric ($g$) describes an universe naturally
characterized by minimum positive energy under a flat geometry (we work in
$txyz$ coordinates, a local frame $r=0$, and a mostly positive RW metric
$g_{\mu\nu}=(-1,a^{2},a^{2},a^{2})$):

$\displaystyle(R^{ML}_{\mu\nu})_{\eta}=(0,9c^{2}\sigma^{4},9c^{2}\sigma^{4},9c^{2}\sigma^{4})\rightarrow(R^{RWML}_{\mu\nu})_{g}$
(1) $\displaystyle c^{2}\sigma^{4}=\frac{\pi g_{\sigma}^{4}\bar{v}^{2}}{72}\
\approx\ 55\ GeV^{2}\ $ (2)

The spatial diffusion parameters, $c\sigma^{2}$, take on the values defined by
the Higgs mechanism in QFT, where $g_{\sigma}\approx 0.38$ is the rate of
diffusion corresponding to the Higgs field,222Also the third generation of
fermions and the QCD regime. See D. Pilipović, et al. forthcoming papers on
QFT/Higgs aspects of ML theory. and $\bar{v}\approx 246\ GeV$ is the scalar
field expectation value.

By working in the RW geometry, we are able to relate the curvature created by
ML diffusion, and captured by the spatial diffusion parameter
$D\equiv\frac{c\sigma^{2}}{a^{2}}$, to the Hubble parameter
$H\equiv\frac{\dot{a}}{a}$, as well as define a relationship between the
effective equation of state parameter $w$ and the acceleration parameter.

III. POSITIVE EXPANSION ACCELERATION

The resulting RWML universe exhibits an expansion acceleration driven by
temporal diffusion parameter $c_{\epsilon}$, with the following relationship
to the RW scale factor $a_{t}$ in the case of a vacuum dominated universe
($w=-1$):

$\displaystyle\frac{3}{2}c_{\epsilon}=\frac{\ddot{a}}{a}+(\frac{\dot{a}}{a})^{2}=(f+1)H^{2},$
$\displaystyle H\equiv\frac{\dot{a}}{a},f\equiv(\frac{\ddot{a}}{a})^{-1}H^{2}$
(3)

Note that if we can approximately express the acceleration term in (3) using
the acceleration factor $f$, the above gives us an exponential scale factor.
For an acceleration parameter close to $\frac{1}{2}$, hence
$\frac{\ddot{a}}{a}\approx\frac{1}{2}(\frac{\dot{a}}{a})^{2}$, we have
$a_{t}\sim e^{\sqrt{c_{\epsilon}}t}$.

For a positive temporal diffusion parameter, equation (3) puts a lower bound
on the acceleration parameter, $f>-1$. However, entirely based on (3), there
appears to be no upper bound on $f$: even for a constant $c_{\epsilon}$, the
increase in $f$ can be directly offset with a decrease in
$H^{2}\sim\frac{1}{f+1}$.

As is shown in later section 4. (see equation (11)), at the point of minimum
energy density for an approximately constant $D$, we obtain a relationship
between the Hubble parameter $H$ and the spatial diffusion parameters
$c\sigma^{2}$, giving us further a relationship between temporal diffusion,
spatial diffusion, and the acceleration of the universe expansion:

$\displaystyle H|_{\rho_{min}}=\frac{3}{4}\frac{c\sigma^{2}}{a^{2}}\ \ \
\Rightarrow\ \ \
\frac{3}{2}c_{\epsilon}=\frac{\ddot{a}}{a}+\frac{9}{16}\frac{c^{2}\sigma^{4}}{a^{4}}$
(4) $\displaystyle(\frac{\ddot{a}}{a})_{0}\approx\frac{1}{2}H_{0}^{2}\ \ \
\Rightarrow\ \ \ ({c_{\epsilon}})_{0}\approx H_{0}^{2}$ (5)

The approximate present value for the temporal diffusion parameter in (5)
above is based on the often favored approximation for the acceleration
parameter, $f\approx\frac{1}{2}$. We can thus establish the scale of the
temporal diffusion parameter:

$\displaystyle c_{\epsilon}\sim 10^{-84}\ GeV^{2}$ (6)

The introduction of an RW metric under ML is critical in establishing the
relative scales between the temporal and spatial diffusion parameters: while
$c\sigma^{2}\sim 10\ GeV$, $\sqrt{c_{\epsilon}}\sim 10^{-42}\ GeV$. These
scales do not changes as we allow for higher values of the acceleration
parameter $f$, or varying universe regimes.

Furthermore, ML under a Minkowski metric gives us a torsionless universe with
zero temporal diffusion, but ML under RW metric gives us an approximately
torsionless universe given the extremely small temporal diffusion ($R_{0i}\sim
c_{\epsilon}\sigma_{\epsilon}^{2}H$).

IV. SPECIES’ ENERGY DENSITY RATIOS

In addition to giving us the scale factor and temporal diffusion relationship
in (3) for a vacuum dominated universe, equations of motion,
$\nabla_{\mu}T^{\mu\nu}=0$, also formulate the evolution of species’ energy
density. For a universe with a roughly flat geometry, $\kappa<<a^{2}$, we
have:

$\displaystyle\frac{\dot{\rho}_{w}}{{\rho}_{w}}=-3\\{(1+w)\frac{\dot{a}}{a}+\frac{c_{w}\sigma_{w}^{2}}{a^{2}}(2+3w)\\}$
(7)

A static solution to (7) corresponding to the Hubble parameter value in (4)
defines the new Higgs-vacuum, with an equation of state given by
$w=-\frac{11}{15}\approx-0.7333$. The additional possible static solution
remains the physical vacuum devoid of diffusion, with $c_{V}\sigma_{V}^{2}=0$
and $w=-1$.

If we further allow matter to take on the same value for diffusion rate as the
Higgs, $(g_{\sigma})_{M}\approx g_{\sigma}$, roughly imposing species energy
density ratios $\rho_{M}/\rho_{V}\sim 0.5$ and $\rho_{R}/\rho_{M}\sim
10^{-3}$, we find that the magnitude of diffusion rate for radiation is
roughly three times that of matter:
$\frac{|(g_{\sigma})_{R}|}{(g_{\sigma})_{M}}\approx 3.1$, given that initial
density is uniform across species..

In contrast to a universe with two static solutions, consider the case where
$\frac{\dot{a}}{a}<<\frac{c_{w}\sigma_{w}^{2}}{a^{2}}$. Now a specie’s energy
density evolution is dominated by the species-specific spatial rates of
diffusion, $\frac{c_{w}\sigma_{w}^{2}}{a^{2}}$, which are further functions of
the specie’s rates of diffusion
($c_{w}\sigma_{w}^{2}\sim(g_{\sigma})_{w}^{2}$). In this case,333As is
explained in Section 6.4, the spatial diffusion parameter $\sigma$ is a
function of the scale factor $a_{t}$, and hence should be brought inside the
time integral. For rough back-of-the-envelope calculations here, we keep
things simple instead.

$\displaystyle\rho_{V}=(\rho_{V})_{0}\
e^{3c_{V}\sigma_{V}^{2}\int\frac{dt}{a^{2}}}$ (8)
$\displaystyle\frac{\rho_{M}}{\rho_{V}}\approx(\frac{\rho_{M}}{\rho_{V}})_{0}\
e^{-3(2c_{M}\sigma_{M}^{2}+c_{V}\sigma_{V}^{2})\int\frac{dt}{a^{2}}}\approx\frac{1}{2}$
(9)
$\displaystyle\frac{\rho_{R}}{\rho_{M}}\approx(\frac{\rho_{R}}{\rho_{M}})_{0}\
e^{-3(3c_{R}\sigma_{R}^{2}-2c_{M}\sigma_{M}^{2})\int\frac{dt}{a^{2}}}\approx
10^{-3}$ (10)

By setting the rate of diffusion for matter to roughly equal that of a vacuum,
we can estimate $\int\frac{dt}{a^{2}}\approx\frac{ln2}{9\sqrt{55}}\approx
0.01$, given that initial density is uniform across species. In this case
also, the magnitude of the radiation rate of diffusion appears to be roughly
three times the rate of diffusion for matter:
$\frac{|(g_{\sigma})_{R}|}{(g_{\sigma})_{M}}\approx 3.3$.

V. UNIVERSE UNDER A SLOW MOVING $D$

For an approximately constant diffusion parameter $D$, we minimize the energy
density relative to $H$. This scenario corresponds to $f\lessapprox 0.7998$,
hence it includes the typically considered case of $f\approx\frac{1}{2}$. In
the next section we will consider the higher-valued acceleration parameters
which allow us to properly minimize energy density to obtain a deeper
understanding of how universe regimes relate to the acceleration parameter.

For an RW metric with a zero curvature parameter $\kappa$, hence a flat RW
geometry444A zero-curvature RW geometry corresponds to a conformally Minkowski
space and a geometrically flat universe. in accordance with both a vacuum-
dominated universe and the current experimental observations, the cosmological
constant acquires a minimum positive value defined by $D^{2}$ and directly
related to the QFT (Higgs mechanism) scalar field expectation value:

$\displaystyle\frac{8\pi G}{3}\rho=(H-\frac{3}{4}D)^{2}+\frac{63}{16}D^{2}$
(11)

where $\rho$ is the total energy density. For vacuum, matter, and radiation
being the only physical species, at the present values based on observations,
we have $\rho=\rho_{V}+\rho_{M}+\rho_{R}\approx\frac{3}{2}\rho_{V}$, giving us
the present value for the cosmological constant:

$\displaystyle\Lambda=\frac{63}{8}\frac{c^{2}\sigma^{4}}{a^{4}}=\frac{7\pi}{64}\frac{g_{\sigma}^{4}\bar{v}^{2}}{a^{4}}$
(12)

Thus, for slow moving $D$, the universe has a small vacuum energy density with
a positive Hubble parameter,

$\displaystyle H=\frac{\dot{a}}{a}=\frac{3}{4}\frac{c\sigma^{2}}{a^{2}}>0$
(13)

By applying the Friedmann Equations, we can solve for the equation of state
parameter $w$ as a function of $f$, and thus obtain the effective species for
the chosen acceleration parameter under the particular theory in question. In
case of the RW theory, with a choice of $f=\frac{1}{2}$, we find that
$w=-\frac{2}{3}$ is the effective species, corresponding to expected mostly
vacuum and matter mix. It is interesting that the equation of state for the ML
Higgs vacuum matches the RW effective universe equation of state.

In contrast, RWML theory with an approximately constant $D$ and with
$f=\frac{1}{2}$ gives us $w=0$ as the solution. In this case, RWML Friedmann
Equations describe a matter-dominated universe.

Clearly the above approach is flawed if we cannot approximate the diffusion
parameter as constant. In fact, for $f\gtrapprox 0.7998$ not only can we not
assume that $D$ is approximately constant, but the minimum energy density
state becomes well defined, albeit degenerate.

VI. ACCELERATION AND UNIVERSE REGIMES

On two separate occasions above we have taken a simplified approach to the
diffusion parameters by setting them constant. However, the two differential
equations (3) and (13), as well as the requirement on how spatial diffusion
parameter $\sigma$ must transform from Minkowski to RW geometry (see section
VII.D.), tell us that this cannot be so, and that the diffusion parameters
must be functions of time. Equipped with the knowledge of what’s to come (see
section VI.), the ratio of the spatial diffusion parameters relative to the
temporal diffusion parameter grows with time:

$\displaystyle c_{\epsilon}\sim\frac{(f+1)}{a^{12}},\ \ \
{c\sigma^{2}}\sim\frac{1}{a^{4}},\ \ \
\frac{c\sigma^{2}}{\sqrt{c_{\epsilon}}}\sim\frac{a^{2}}{\sqrt{f+1}}$ (14)

Both $H$ and $D$ are functions of time. In fact, since $\dot{H}=-(1-f)H^{2}$
and $D\sim\frac{1}{a^{6}},\dot{D}=-6HD$, we see that in the neighborhood of
$f=1$ it is $H$ which can be assumed relatively constant. By minimizing the
energy density, $\dot{\rho}=0$, with the additional help of RWML Friedmann
equations, we obtain both the relationship between the diffusive and Hubble
parameters, $D$ and $H$, as well as the relationship between the acceleration
parameter and the equation of state parameter, $f$ and $w$. We find that for
the acceleration parameter in the neighborhood of one, we obtain a universe
with an effective $w\approx-0.7$ and thus well approximated by the RW theory.

Allowing the acceleration parameter to grow under RWML provides us with two
different paths for $D$ (and thus also $H$): the first path corresponds to an
effective equation of state with $w\approx-0.7$ in the neighborhood of $f=1$.
This path also allows for a smooth transition from a mostly vacuum dominated
universe to matter dominated, with a bounded energy density described by
$H=\frac{3}{4}D$, as long as the Hubble parameter remains positive.

The second path gives us an unacceptable answer for $f=1$ by requiring that
$D=0$: $D$ is an inverse function of $a$, and as such can get extremely small,
but never zero. As acceleration parameter grows, this second path describes a
dooms-day scenario with a negative Hubble parameter and an infinitely growing
energy density.

Finally, we must consider at what point the universe might transition from
open to closed. While we are assuming a geometrically flat universe by working
in local coordinates ($r=0$), the presence of ML fields introduces a curvature
which allows the universe to be open or closed despite the geometric flatness.
We find that the turning point occurs at $D=\frac{H}{3}$ with a corresponding
acceleration parameter $f=3$: for lesser values of the diffusion parameter or
acceleration we have an open universe and otherwise closed.

VII. SUPPORTING DERIVATIONS
A. Markovian fields in Minkowski geometry

We enforce ML scale by introducing stochastic behavior in spacetime. Each
point in spacetime, $x$, under minimum length scale obtains a stochastic term:
$x^{\mu}\rightarrow x^{\mu}+\sigma\delta\xi_{x}^{\mu}$. The Markovian field
$\delta\xi_{x}^{\mu}$ represents the uncertainty at $x$ due to minimum length
scale. It scales with the square root of the minimum length scale:
$\delta\xi^{\mu}\sim\sqrt{\delta t}$, consistent with the behavior of
stochastic variables.

The second moment of such a field thus scales with $\delta t$, and is unique
to each point in spacetime:

$\displaystyle\langle\delta\xi_{x}^{\alpha}\
\delta\xi_{y}^{\beta}\rangle=\delta^{4}(x-y)\ g^{\alpha\beta}\ \delta t$ (15)

The derivative of a stochastic field is not a ’typical’ derivative at all, but
rather a difference of two ’new’ Markovian fields which we will refer to as
the ’kinetic Markovian fields’. For example, for a partial derivative in the
direction of $\delta x^{\mu}$ we have

$\displaystyle\partial_{\mu}\delta\xi_{x}^{\alpha}=\frac{\delta\xi_{x+\delta
x^{\mu}/2}^{\alpha}-\delta\xi_{x-\delta x^{\mu}/2}^{\alpha}}{\delta x^{\mu}}=$
$\displaystyle\ \ \ \ (\delta\zeta_{x+\delta
x/2})^{\alpha}_{\mu}-(\delta\zeta_{x-\delta x/2})^{\alpha}_{\mu}$ (16)

The ’new’ field $\delta\zeta_{x}$ is also a Markovian field, but one which
scales with the square root of the inverse of minimum length scale,
$(\delta\zeta_{x})^{\alpha}_{\mu}\sim\frac{1}{\sqrt{\delta x^{\mu}}}$. Its
second moment now follows:

$\displaystyle\langle(\delta\zeta_{x})_{\mu}^{\alpha}\
(\delta\zeta_{y})_{\nu}^{\beta}\rangle|_{\mu=\nu}=\delta^{4}(x-y)\
g^{\alpha\beta}\ \frac{1}{\delta x^{\mu}},$ $\displaystyle\ \ \ \ \ \ \
\langle(\delta\zeta_{x})_{\mu}^{\alpha}\
(\delta\zeta_{y})_{\nu}^{\beta}\rangle|_{\mu\neq\nu}=0$ (17)

_B. ML connection_

Regardless of the metric of choice or the theory of choice, the single
universal law, $\mathcal{U}^{2}=-1$, defines the action $I$555See [3], pg.
106-107 for a detailed derivation of the proper time functional used here. for
a free particle moving along a world line. We introduce Markovian fluctuations
into the classical theory of General Relativity by allowing for a Markovian
(stochastic) translation, with the requirement that $I$ must be conserved:

$\displaystyle I=-\frac{1}{2}\int
d\tau\frac{dx^{\mu}}{d\tau}g_{\mu\nu}\frac{dx^{\nu}}{d\tau},$ $\displaystyle\
\ \ \ \ \ \ \ x^{\mu}\rightarrow
x^{\mu}+\delta\overline{x}^{\mu}+\sigma\delta{\xi}^{\mu},\ \ \
\delta\xi^{\mu}\sim\mathcal{N}(0,\sqrt{\delta t})$ (18) $\displaystyle
I\rightarrow-\frac{1}{2}\int
d\tau\\{\frac{d(x^{\mu}+\delta\overline{x}^{\mu}+\sigma\delta\tilde{\xi}^{\mu})}{d\tau}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
(g_{\mu\nu}+(\partial_{\rho}g_{\mu\nu})(\delta\overline{x}^{\rho}+\sigma\delta\tilde{\xi}^{\rho})$
$\displaystyle\ \ \
+\frac{1}{2}(\partial_{\alpha}\partial_{\beta}g_{\mu\nu})\sigma^{2}\mathnormal{h}^{\alpha\beta}_{\rho}\delta\overline{x}^{\rho})\frac{d(x^{\nu}+\delta\overline{x}^{\nu}+\sigma\delta\tilde{\xi}^{\nu})}{d\tau}\\}$
(19) $\displaystyle\langle\delta I\rangle=\int d\tau
g_{\mu\rho}\delta\overline{x}^{\rho}\\{\frac{d^{2}x^{\mu}}{d\tau^{2}}+\tilde{\Gamma}^{\mu}_{\alpha\beta}\frac{dx^{\alpha}}{d\tau}\frac{dx^{\beta}}{d\tau}\\}=0$
(20)
$\displaystyle\tilde{\Gamma}^{\mu}_{\alpha\beta}={\Gamma}^{\mu}_{\alpha\beta}-\mathcal{H}^{\mu}_{\alpha\beta}$
(21)

The new Christoffel connection now includes the ’standard’ Christoffel
connection defined entirely by the metric, but also new terms due to ML
fluctuations:

$\displaystyle\Gamma^{\mu}_{\alpha\beta}=\frac{1}{2}g^{\mu\sigma}(\partial_{\alpha}g_{\beta\sigma}+\partial_{\beta}g_{\sigma\alpha}-\partial_{\sigma}g_{\alpha\beta})$
(22)
$\displaystyle\mathcal{H}^{\mu}_{\alpha\beta}=\frac{1}{2}g^{\mu\delta}\\{\frac{1}{2}\sigma^{2}\mathnormal{h}^{\eta\sigma}_{\delta}(\partial_{\eta}\partial_{\sigma}g_{\alpha\beta})$
$\displaystyle\ \ \ \ \
+(\partial_{\alpha}\sigma^{2})(\partial_{\sigma}g_{\eta\beta})\mathnormal{h}^{\eta\sigma}_{\delta}+(\partial_{\alpha}\sigma)(\partial_{\beta}\sigma)\mathnormal{h}^{\eta\sigma}_{\delta}g_{\eta\sigma}$
$\displaystyle\ \ \ \ \
+2\sigma^{2}a^{\sigma\eta}_{\delta\alpha}(\partial_{\sigma}g_{\eta\beta})+(\partial_{\alpha}\sigma^{2})a^{\eta\sigma}_{\delta\beta}g_{\eta\sigma}$
$\displaystyle\ \ \ \ \
+\sigma^{2}\mathnormal{k}^{\eta\sigma}_{\delta\alpha\beta}g_{\eta\sigma}\\}$
(23)
$\displaystyle\kappa^{\mu}_{\alpha\beta}\equiv\frac{1}{2}g^{\mu\delta}\sigma^{2}\mathnormal{k}^{\eta\sigma}_{\delta\alpha\beta}g_{\eta\sigma}$
(24)
$\displaystyle\mathcal{A}^{\mu}_{\alpha\beta}\equiv\frac{1}{2}g^{\mu\delta}\sigma^{2}a^{\sigma\eta}_{\delta\alpha}(\partial_{\sigma}g_{\eta\beta})$
(25)
$\displaystyle\Sigma^{\mu}_{\alpha\beta}\equiv\frac{1}{4}g^{\mu\delta}\sigma^{2}\mathnormal{h}^{\eta\sigma}_{\delta}(\partial_{\eta}\partial_{\sigma}g_{\alpha\beta})$
(26)

where $h,k,a$ are tensors corresponding to various two-point functions
described in detail in 6.3. and 6.4. below. For an isotropic vacuum diffusion
parameters must be constant in space: $\sigma(x)=\sigma$, but are not required
to be constant in time. In fact, as we shall see, the spatial diffusion
parameter $\sigma$ is in fact a function of time, with
$\sigma\sim\frac{1}{a^{2}}$. However, we also find that all terms with
$\partial_{t}\sigma$ are either $O(\sigma_{\epsilon}^{4})$ or $O(DH^{2})$ and
can be approximated away.666See the Appendix for detail on these terms.

_C. ML Under Minkowski Metric_

For ML in a Minkowski geometry, the problem simplifies further:

$\displaystyle
g_{\mu\nu}=\eta_{\mu\nu}\Rightarrow{\Gamma}^{\mu}_{\alpha\beta}=0,\mathcal{H}^{\mu}_{\alpha\beta}=\kappa^{\mu}_{\alpha\beta},\tilde{\Gamma}^{\mu}_{\alpha\beta}=-\kappa^{\mu}_{\alpha\beta}$
(27)

Equations (22)-(26) above assume a tensorial formulation of the various
necessary two-point functions for the ML fields, as follows:

$\displaystyle\langle\delta{\xi}^{\mu}\delta{\xi}^{\nu}\rangle\equiv\mathnormal{h}^{\mu\nu}_{\rho}\delta\overline{x}^{\rho},\
\ \ \
\langle\partial_{\alpha}\delta{\xi}^{\mu}\partial_{\beta}\delta{\xi}^{\nu}\rangle\equiv\mathnormal{k}^{\mu\nu}_{\rho\alpha\beta}\delta\overline{x}^{\rho},$
$\displaystyle\ \ \ \ \ \ \ \
\langle\delta{\xi}^{\mu}\partial_{\alpha}\delta\tilde{\xi}^{\nu}\rangle\equiv\mathnormal{a}^{\mu\nu}_{\rho\alpha}\delta\overline{x}^{\rho}$
(28)
$\displaystyle\langle\frac{d\delta{\xi}^{\mu}}{d\tau}\frac{d\delta{\xi}^{\nu}}{d\tau}\rangle=\langle\partial_{\alpha}\delta{\xi}^{\mu}\partial_{\beta}\delta{\xi}^{\nu}\rangle\frac{dx^{\alpha}}{d\tau}\frac{dx^{\beta}}{d\tau}=$
$\displaystyle\ \ \ \ \ \ \ \
\mathnormal{k}^{\mu\nu}_{\rho\alpha\beta}\frac{dx^{\alpha}}{d\tau}\frac{dx^{\beta}}{d\tau}\delta\overline{x}^{\rho},$
(29)
$\displaystyle\langle\delta{\xi}^{\mu}\frac{d\delta\tilde{\xi}^{\nu}}{d\tau}\rangle=\langle\delta\tilde{\xi}^{\mu}\partial_{\alpha}\delta{\xi}^{\nu}\rangle\frac{dx^{\alpha}}{d\tau}=\mathnormal{a}^{\mu\nu}_{\rho\alpha}\frac{dx^{\alpha}}{d\tau}\delta\overline{x}^{\rho}$
(30)
$\displaystyle\mathnormal{h}^{\mu\nu}_{\rho}=\mathnormal{h}^{\nu\mu}_{\rho},\mathnormal{k}^{\mu\nu}_{\rho\alpha\beta}=\mathnormal{k}^{\nu\mu}_{\rho\beta\alpha}$
(31)
$\displaystyle[{x}^{\mu}]=[\delta\overline{x}^{\mu}]=L,[\sigma]=L^{\frac{1}{2}},[\delta\tilde{\xi}^{\mu}]=L^{\frac{1}{2}},$
$\displaystyle\ \ \ \ \ \ \
[\mathnormal{h}^{\mu\nu}_{\rho}]=L^{0},[\mathnormal{k}^{\mu\nu}_{\rho\alpha\beta}]=L^{-2},[\mathnormal{a}^{\mu\nu}_{\rho\alpha}]=L^{-1}$
(32)

We parametrize the two point functions, with the help of (15)-(17) above. We
continue treating space as isotropic, but we separate out the diffusive
effects in space vs. time, and we do this by folding the square of the
magnitude of diffusion into the definitions of the tensors $h,k,a$.777Thus,
the dimension of each of these now becomes
$[\mathnormal{h}^{\mu\nu}_{\rho}]=L^{1},[\mathnormal{k}^{\mu\nu}_{\rho\alpha\beta}]=L^{-1},[\mathnormal{a}^{\mu\nu}_{\rho\alpha}]=L^{0}$.
We thus rewrite (15)-(17), to define the two point functions at some point in
spacetime $x$ as follows:

$\displaystyle\sigma^{2}\langle\delta\xi^{\alpha}_{x}\delta\xi^{\beta}_{x}\rangle|_{\alpha=\beta=i}\rightarrow\sigma^{2}\eta^{ij}|_{i=j}\delta
t\equiv h^{ij}_{0}|_{i=j}\delta t$ (33)
$\displaystyle\sigma^{2}\langle\delta\xi^{\alpha}_{x}\delta\xi^{\beta}_{x}\rangle|_{\alpha=\beta=0}\rightarrow\sigma_{\epsilon}^{2}\eta^{00}\sum_{i}\delta
x^{i}\equiv h^{00}_{i}\delta x^{i}$ (34)
$\displaystyle\sigma^{2}\langle\partial_{\mu}\delta\xi^{\alpha}_{x}\partial_{\nu}\delta\xi^{\beta}_{x}\rangle|_{\alpha=\beta=i,\mu=\nu=0}\rightarrow\sigma^{2}\eta^{ij}|_{i=j}\frac{2}{\delta
t^{2}}\delta t$ $\displaystyle\ \ \ \ \equiv k^{ij}_{000}|_{i=j}\delta t$ (35)
$\displaystyle\sigma^{2}\langle\partial_{\mu}\delta\xi^{\alpha}_{x}\partial_{\nu}\delta\xi^{\beta}_{x}\rangle|_{\alpha=\beta=i,\mu=\nu=k}\rightarrow\sigma^{2}\eta^{ij}|_{i=j}\frac{2}{\delta
x^{k}\delta t}\delta t$ $\displaystyle\ \ \ \ \equiv
k^{ij}_{0kl}|_{i=j,k=l}\delta t$ (36)
$\displaystyle\sigma^{2}\langle\partial_{\mu}\delta\xi^{\alpha}_{x}\partial_{\nu}\delta\xi^{\beta}_{x}\rangle|_{\alpha=\beta=0,\mu=\nu=0}\rightarrow\sigma_{\epsilon}^{2}\eta^{00}\frac{2}{\delta
t\delta x^{i}}\delta x^{i}$ $\displaystyle\ \ \ \ \equiv k^{00}_{i00}\delta
x^{i}$ (37)
$\displaystyle\sigma^{2}\langle\partial_{\mu}\delta\xi^{\alpha}_{x}\partial_{\nu}\delta\xi^{\beta}_{x}\rangle|_{\alpha=\beta=0,\mu=\nu=k}\rightarrow\sigma_{\epsilon}^{2}\eta^{00}\frac{2}{\delta
x^{k}\delta x^{i}}\delta x^{i}$ $\displaystyle\ \ \ \ \equiv
k^{00}_{ikl}|_{k=l}\delta x^{i}$ (38)

From (33) and (34), we can directly identify tensor $h$, which is entirely
defined by Markovian fields. Tensor $k$ has both spacetime and Markovian field
dependencies888Specifically, for tensor $k^{\alpha\beta}_{\rho\mu\nu}$,
indecies $\alpha,\beta$, and $\rho$ reference Markovian fields or minimum
length scale characteristic of the Markovian fields, while $\mu,\nu$ are
spacetime indecies corresponding to the partial derivatives. and can be
generalized with an appropriate choice of diffusion parameters, shown below as
$c$ and $c_{\epsilon}$. Finally, to remove any confusion about index summation
and to simplify notation, we will label the indecies explicitly in terms of
$t,x,y,z$ from here on, to finally obtain:999Here we excluded the tensors
which give zero contributions to the connection in both Minkowski and RW
geometry.

$\displaystyle h^{tt}_{x}=h^{tt}_{y}=h^{tt}_{z}=-\sigma_{\epsilon}^{2}$ (39)
$\displaystyle h^{xx}_{t}=h^{yy}_{t}=h^{zz}_{t}=\sigma^{2}$ (40)
$\displaystyle\mathnormal{k}^{tt}_{xtt}=\mathnormal{k}^{tt}_{xxx}=\mathnormal{k}^{tt}_{xyy}=\mathnormal{k}^{tt}_{xzz}=-2\mathnormal{c}_{\epsilon}\sigma_{\epsilon}^{2}$
(41)
$\displaystyle\mathnormal{k}^{xx}_{ttt}=\mathnormal{k}^{xx}_{txx}=\mathnormal{k}^{xx}_{tyy}=\mathnormal{k}^{xx}_{tzz}=2\mathnormal{c}\sigma^{2}$
(42)

Note that the diffusion parameters $c\sigma^{2}$ and
$c_{\epsilon}\sigma_{\epsilon}^{2}$ effectively represent the diffusion
acceleration rates in space and time, respectively. Finally, it is not obvious
in the Minkowski geometry, but ML under the RW metric shows us that all the
tensors as well as the diffusion acceleration rates are defined not only by
the nature of diffusion, but also by the geometry of choice.

Continuing to develop the math of ML under Minkowski metric, after some
algebra we obtain

$\displaystyle\kappa^{t}_{tt}=\kappa^{t}_{xx}=\kappa^{t}_{yy}=\kappa^{t}_{zz}=-3\mathnormal{c}\sigma^{2}$
(43)
$\displaystyle\kappa^{x}_{tt}=\kappa^{x}_{xx}=\kappa^{x}_{yy}=\kappa^{x}_{zz}=\mathnormal{c}_{\epsilon}\sigma_{\epsilon}^{2}$
(44)

Finally, with the help of Einstein’s Equations, our results correspond to a
maximally symmetric universe:

$\displaystyle
R^{\alpha}_{\sigma\mu\nu}=\kappa^{\alpha}_{\mu\beta}\kappa^{\beta}_{\nu\sigma}-\kappa^{\alpha}_{\nu\beta}\kappa^{\beta}_{\mu\sigma},$
$\displaystyle\ \ \ \ \ \ \ \
R_{\sigma\nu}=R^{\rho}_{\sigma\rho\nu}=\kappa^{\alpha}_{\alpha\beta}\kappa^{\beta}_{\nu\sigma}-\kappa^{\alpha}_{\nu\beta}\kappa^{\beta}_{\alpha\sigma}$
(45) $\displaystyle
R_{tx}=R_{ty}=R_{tz}=3cc_{\epsilon}\sigma\sigma_{\epsilon},\ \ \ \
R_{tt}=3c_{\epsilon}^{2}\sigma^{4}_{\epsilon}$ (46) $\displaystyle
R_{xx}=R_{yy}=R_{zz}=9c^{2}\sigma^{4}+2c_{\epsilon}^{2}\sigma^{4}_{\epsilon}\approx
9c^{2}\sigma^{4}$ (47) $\displaystyle
R_{xy}=R_{yz}=R_{xz}=-\kappa^{x}_{yy}\kappa^{y}_{xx}=-c_{\epsilon}^{2}\sigma^{4}_{\epsilon}$
(48)

_D. ML under RW geometry_

Under RW geometry we must adjust the explicit parametrization of the tensorial
two-point functions to take into consideration the scale factor. To do this we
need to formulate how the tensors and diffusion parameters transform from
Minkowski geometry to the RW geometry. In the absence of Markovian fields this
transformation is already well defined. The metric transformation - from
Minkowski geometry (denoted by $\alpha,\beta,\mu,\nu$, etc.) to the RW
geometry (denoted by $\hat{\alpha},\hat{\beta},\hat{\mu},\hat{\nu}$) - must
satisfy

$\displaystyle g_{\hat{\alpha}\hat{\beta}}=\frac{\partial x^{\alpha}}{\partial
x^{\hat{\alpha}}}\frac{\partial x^{\beta}}{\partial
x^{\hat{\beta}}}\eta_{\alpha\beta}=(-1,a^{2},a^{2},a^{2})$ $\displaystyle\ \ \
\ \ \ \ \Rightarrow\ \ \ x^{\hat{i}}=\frac{1}{a}x^{i}$ (49)

The scale factor is only a function of time, giving us the simple relationship
between the new (RW) and the old (Minkowski) spatial coordinates.

The tensor transformation in the absence of Markovian fields is also readily
available,

$\displaystyle T^{\alpha\beta...}_{\mu\nu...}\longrightarrow
T^{\hat{\alpha}\hat{\beta}...}_{\hat{\mu}\hat{\nu}...}=\frac{\partial
x^{\hat{\alpha}}}{\partial x^{\alpha}}\frac{\partial x^{\hat{\beta}}}{\partial
x^{\beta}}...\frac{\partial x^{\mu}}{\partial x^{\hat{\mu}}}\frac{\partial
x^{\nu}}{\partial x^{\hat{\nu}}}...T^{\alpha\beta...}_{\mu\nu...}$ (50)

But for a tensor which is exclusively a function of Markovian fields, we have:

$\displaystyle Q^{\alpha\beta...}_{\mu\nu...}\longrightarrow
Q^{\hat{\alpha}\hat{\beta}...}_{\hat{\mu}\hat{\nu}...}=$ $\displaystyle\ \ \ \
\frac{\partial\delta\xi^{\hat{\alpha}}}{\partial\delta\xi^{\alpha}}\frac{\partial\delta\xi^{\hat{\beta}}}{\partial\delta\xi^{\beta}}...\frac{\partial\delta\xi^{\mu}}{\partial\delta\xi^{\hat{\mu}}}\frac{\partial\delta\xi^{\nu}}{\partial\delta\xi^{\hat{\nu}}}...Q^{\alpha\beta...}_{\mu\nu...}$
(51)

The last remaining task - for now - is to determine how the Markovian fields
in the new metric relate to the Markovian fields in the old metric.

Let’s consider the two-point function for Markovian fields in space,
$\langle\delta\xi^{x}\delta\xi^{x}\rangle\sim h^{xx}_{t}$. We already know
that in the Minkowski geometry this two-point function scales with
$\eta^{xx}$, which is just one. But in the RW geometry each of the fields
first obtains a factor of $\frac{1}{a}$ for ’residing’ in space, and further
the RW two point function obtains an additional factor of
$g^{\hat{x}\hat{x}}=\frac{1}{a^{2}}$.101010Note that this tells us that
tensors defined entirely by Markovian fields experience an effective local
geometry given by $g^{eff}_{\alpha\beta}=(-1,a^{4},a^{4},a^{4},a^{4})$. Hence,
we venture a guess:

$\displaystyle h^{\hat{x}\hat{x}}_{\hat{t}}=\frac{1}{a^{4}}h^{xx}_{t}\ \ \
\Rightarrow\ \ \
\frac{\partial\delta\xi^{\hat{x}}}{\partial\delta\xi^{x}}=\frac{1}{a^{2}}$
(52)

More generally, $h^{\alpha\beta}_{\rho}$ has indexes $\alpha,\beta$ which
observe Markovian field transformations, and $\rho$ which is specific to how
$\delta x^{\rho}$ transforms. $k^{\alpha\beta}_{\rho\mu\nu}$ has the same
indexes as $h$, with the addition of $\mu,\nu$ which follow the ordinary rules
of transformation. By capturing the transformation of infinitesimal change in
spacetime111111Hence, minimum length. from one geometry to another with some
power of the scale factor, $a^{z}$ \- where $z$ is to be
defined,121212Originally I have treated this transformation as equivalent to
that of a Markovian field. However, this was unnecessary in the absence of a
gravitational source. we are now ready to fully formulate the transformation
of tensors $h,k$:

$\displaystyle h^{\alpha\beta}_{\rho}\longrightarrow
h^{\hat{\alpha}\hat{\beta}}_{\hat{\rho}}=\frac{\partial\delta\xi^{\hat{\alpha}}}{\partial\delta\xi^{\alpha}}\frac{\partial\delta\xi^{\hat{\beta}}}{\partial\delta\xi^{\beta}}\frac{\partial\delta
x^{\rho}}{\partial\delta x^{\hat{\rho}}}h^{\alpha\beta}_{\rho}$ (53)
$\displaystyle k^{\alpha\beta}_{\rho\mu\nu}\longrightarrow
k^{\hat{\alpha}\hat{\beta}}_{\hat{\rho}\hat{\mu}\hat{\nu}}=\frac{\partial\delta\xi^{\hat{\alpha}}}{\partial\delta\xi^{\alpha}}\frac{\partial\delta\xi^{\hat{\beta}}}{\partial\delta\xi^{\beta}}\frac{\partial\delta
x^{\rho}}{\partial\delta x^{\hat{\rho}}}\frac{\partial x^{\mu}}{\partial
x^{\hat{\mu}}}\frac{\partial x^{\nu}}{\partial
x^{\hat{\nu}}}k^{\alpha\beta}_{\rho\mu\nu}$ (54) $\displaystyle\frac{\partial
x^{i}}{\partial x^{\hat{i}}}=a,\ \ \ \frac{\partial\delta
x^{i}}{\partial\delta x^{\hat{i}}}=a^{z},\ \ \
\frac{\partial\delta\xi^{\hat{i}}}{\partial\delta\xi^{i}}=\frac{1}{a^{2}}$
(55)

From this point we work entirely in the RW geometry and can drop the use of
’hats’ in the indecies. The full set of tensors necessary for the formulation
of Equations of Motion, Einstein, and Friedmann Equations is given by:

$\displaystyle h^{tt}_{x}=h^{tt}_{y}=h^{tt}_{z}=-\sigma^{2}_{\epsilon}a^{z}$
(56) $\displaystyle h^{xx}_{t}=h^{yy}_{t}=h^{zz}_{t}=\frac{\sigma^{2}}{a^{4}}$
(57)
$\displaystyle\mathnormal{k}^{tt}_{xtt}=\mathnormal{k}^{tt}_{ytt}=\mathnormal{k}^{tt}_{ztt}=-2\mathnormal{c}_{\epsilon}\sigma_{\epsilon}^{2}a^{z}$
(58)
$\displaystyle\mathnormal{k}^{tt}_{xxx}=\mathnormal{k}^{tt}_{xyy}=\mathnormal{k}^{tt}_{xzz}=-2\mathnormal{c}_{\epsilon}\sigma_{\epsilon}^{2}a^{2+z}$
(59)
$\displaystyle\mathnormal{k}^{xx}_{ttt}=\mathnormal{k}^{yy}_{ttt}=\mathnormal{k}^{zz}_{ttt}=\frac{2c\sigma^{2}}{a^{4}}$
(60)
$\displaystyle\mathnormal{k}^{xx}_{txx}=\mathnormal{k}^{xx}_{tyy}=\mathnormal{k}^{xx}_{tzz}=\frac{2\mathnormal{c}\sigma^{2}}{a^{2}}$
(61)

During the formulation of the Equations of Motion in the absence of a
gravitational source, all $a^{z}$ terms cancel. Hence, RWML theory in the
absence of a gravitational source is indifferent to the manner in which
$\delta x^{\rho}$ transforms from Minkowski space to RW space. To make all
transformations consistent, we set $z=2$, to obtain:

$\displaystyle\kappa^{t}_{tt}=-3\frac{c\sigma^{2}}{a^{2}}$ (62)
$\displaystyle\kappa^{t}_{xx}=\kappa^{t}_{yy}=\kappa^{t}_{zz}=-3\mathnormal{c}\sigma^{2}$
(63)
$\displaystyle\kappa^{x}_{tt}=\kappa^{y}_{tt}=\kappa^{z}_{tt}=c_{\epsilon}\sigma^{2}_{\epsilon}$
(64)
$\displaystyle\kappa^{x}_{xx}=\kappa^{x}_{yy}=\kappa^{x}_{zz}=\mathnormal{c}_{\epsilon}\sigma^{2}_{\epsilon}a^{2}$
(65)
$\displaystyle\Sigma^{t}_{xx}=\Sigma^{t}_{yy}=\Sigma^{t}_{zz}=-\frac{3}{2}\frac{\kappa\sigma^{2}}{a^{2}}$
(66)
$\displaystyle\Sigma^{x}_{xx}=\Sigma^{x}_{yy}=\Sigma^{x}_{zz}=-\frac{\sigma^{2}_{\epsilon}}{2}(a\ddot{a}+\dot{a}^{2})$
(67) $\displaystyle\Gamma^{t}_{xx}=\Gamma^{t}_{yy}=\Gamma^{t}_{zz}=a\dot{a}$
(68)
$\displaystyle\Gamma^{x}_{tx}=\Gamma^{y}_{ty}=\Gamma^{z}_{tz}=\frac{\dot{a}}{a}$
(69)

The elements of the new connection are now given by:

$\displaystyle\tilde{\Gamma}^{t}_{tt}=3\frac{c\sigma^{2}}{a^{2}}$ (70)
$\displaystyle\tilde{\Gamma}^{t}_{xx}=\tilde{\Gamma}^{t}_{yy}=\tilde{\Gamma}^{t}_{zz}=a\dot{a}+3c\sigma^{2}+\frac{3}{2}\frac{\kappa\sigma^{2}}{a^{2}}$
(71)
$\displaystyle\tilde{\Gamma}^{x}_{tt}=\tilde{\Gamma}^{y}_{tt}=\tilde{\Gamma}^{z}_{tt}=-c_{\epsilon}\sigma^{2}_{\epsilon}$
(72)
$\displaystyle\tilde{\Gamma}^{x}_{tx}=\tilde{\Gamma}^{y}_{ty}=\tilde{\Gamma}^{z}_{tz}=\frac{\dot{a}}{a}$
(73)
$\displaystyle\tilde{\Gamma}^{x}_{xx}=\tilde{\Gamma}^{x}_{yy}=\tilde{\Gamma}^{x}_{zz}=-c_{\epsilon}\sigma^{2}_{\epsilon}a^{2}+\frac{\sigma^{2}_{\epsilon}}{2}(a\ddot{a}+\dot{a}^{2})$
(74)

As already discussed in VII.B., we must consider how the spatial diffusive
parameter $\sigma$ transforms from Minkowski to RW geometry. It is useful to
note that $\sigma$ has the same dimension as $\xi^{\alpha}$:
$[\sigma]=L^{\frac{1}{2}}$, and thus we would expect that it transforms in the
same manner. We can define its parameter transformation as
$\sigma=\sigma_{Minkowski}\ \ \ \longrightarrow\ \ \
\sigma=\frac{\sigma_{Minkowski}}{a^{y}}$, where we still need to obtain the
power of the scale factor, $y$. If we are correct about our intuitive guess
with regards to ML field transformations, we ought to find that $y=2$.

The Friedmann Equations show us that a physical universe with a positive
Hubble parameter for a slow moving $D$ relative to $H$
($|\dot{D}|<<|\dot{H}|$), is not possible for $y<2$. We thus set $y=2$, the
lowest possible value in obtaining a physical universe which matches
experimental observations, giving us $\sigma\sim\frac{1}{a^{2}}$.

Equipped with the above, we can formulate the Reimann tensor to ultimately
obtain the components of the Ricci tensor and the Ricci scalar:

$\displaystyle
R_{tt}=-3\frac{\ddot{a}}{a}+9\frac{\dot{a}}{a}\frac{c\sigma^{2}}{a}+O(c_{\epsilon}\sigma_{\epsilon}^{4})$
(75) $\displaystyle R_{xx}=R_{yy}=R_{zz}=$ $\displaystyle\ \ \ \
a\ddot{a}+2\dot{a}^{2}+2\kappa-6\frac{\dot{a}}{a}c\sigma^{2}+9\frac{c^{2}\sigma^{4}}{a^{2}}$
$\displaystyle\ \ \ \
+O(\frac{\kappa}{a^{2}})+O(c_{\epsilon}\sigma_{\epsilon}^{4})$ (76)
$\displaystyle
R=6(\frac{\ddot{a}}{a}+(\frac{\dot{a}}{a})^{2})-27\frac{\dot{a}}{a}\frac{c\sigma^{2}}{a^{2}}+27\frac{c^{2}\sigma^{4}}{a^{4}}$
$\displaystyle\ \ \ \
+O(\frac{\kappa}{a^{2}})+O(c_{\epsilon}\sigma_{\epsilon}^{4})$ (77)

_E. Expansion acceleration_

The new RWML connection allows us to obtain equations of motion for each
species. In fact, there are two such equations, one giving us the evolution of
species-specific energy density through time, and the other giving us the
gradient of pressure. In an isotropic vacuum we expect the gradient of
pressure to be zero. This second equation for both RW theory without diffusion
and ML theory in Minkowski geometry is trivial. But here we learn that, in
fact, temporal diffusion cannot be assumed away. While it appears to scale as
extremely small, it is the driving force behind the positive acceleration of
the universe expansion.

Assuming an isotropic universe, the pressure-specific equation of motion for
RWML gives us:

$\displaystyle c_{\epsilon}a^{2}=\frac{2w}{1+4w}(a\ddot{a}+\dot{a}^{2})$ (78)

With this we obtain Equation (4) for the case of a vacuum-dominated universe
($w=-1$). Note that matter ($w=0$) does not participate in temporal
acceleration.

_F. Species-specific energy density_

The more complete equation131313See the Appendix for even more detail.
describing the species-specific energy density evolution is given by
$\nabla_{\mu}T^{\mu t}$:

$\displaystyle\frac{\dot{\rho}_{w}}{{\rho}_{w}}=-3\\{(1+w)\frac{\dot{a}}{a}+\frac{c_{w}\sigma_{w}^{2}}{a^{2}}(2+3w(1+\frac{\kappa}{2ca^{2}})\\}$
$\displaystyle\ \ \ \
+O((\frac{\dot{a}}{a})^{2}\frac{c_{w}\sigma_{w}^{2}}{a^{2}})$ (79)

_G. RWML Friedmann Equations_

The first and second Friedmann Equations in RWML theory are given by:

$\displaystyle\frac{8\pi
G}{3}\rho=(\frac{\dot{a}}{a})^{2}-\frac{3}{2}\frac{\dot{a}}{a}\frac{c\sigma^{2}}{a^{2}}+\frac{9}{2}\frac{c^{2}\sigma^{4}}{a^{4}}+O(\frac{\kappa}{a^{2}})$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \approx
H^{2}-\frac{3}{2}HD+\frac{9}{2}D^{2}$ (80) $\displaystyle-{8\pi
G}p=2\frac{\ddot{a}}{a}+(\frac{\dot{a}}{a})^{2}-\frac{15}{2}\frac{\dot{a}}{a}\frac{c\sigma^{2}}{a^{2}}+\frac{9}{2}\frac{c^{2}\sigma^{4}}{a^{4}}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+O(\frac{\kappa}{a^{2}})$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \
\approx(2f+1)H^{2}-\frac{15}{2}HD+\frac{9}{2}D^{2}$ (81)

With a particular choice for $H$ and $f$, we can solve for $w$ ($p=w\rho$).
For example, setting
$H=\frac{\dot{a}}{a}=\frac{3}{4}\frac{c\sigma^{2}}{a^{2}}$, the value which
minimizes the total energy density for a slow moving $D$, and choosing
$f=\frac{1}{2}$, we obtain $w=0$. Under RWML theory, Friedmann Equations for a
fast moving Hubble parameter relative to the diffusive parameter describe a
matter-dominated universe.

Alternatively we can consider the case where the Hubble parameter does not
change in value. This occurs when $\dot{H}=(f-1)H^{2}=0$, thus when the
acceleration parameter equals one. In this case the energy density is
minimized when $D=\frac{H}{6}$, giving us $w=-\frac{5}{7}\approx-0.7$, the
universe with the effective equation of state as we understand it to be
presently, mostly a $2:1$ mix of vacuum and matter.

In the last example here, we can consider a static universe described by the
Friedmann Equations of RWML theory, with both zero velocity and acceleration.
Just as in the case of RW theory, the only species which can accommodate this
scenario is curvature, with $w=-\frac{1}{3}$.

_H. Effective universe equation of state and deceleration parameter_

If we assume that $D$ is approximately constant, we can minimize the energy
density relative to $H$, to obtain a minimum energy density defined by
$D^{2}$:

$\displaystyle\frac{\partial\rho}{\partial H}=0\ \ \ \Rightarrow\ \ \
H=\frac{3}{4}D,\ \ \ \frac{8\pi G}{3}\rho=\frac{63}{16}D^{2},$ $\displaystyle\
\ \ \ \ \ \ f=\frac{1}{2}(1-21w)$ (82) $\displaystyle f=0.5\ \ \ \Rightarrow\
\ \ w=0$ (83)

On the other hand, for $H$ approximately constant, we can minimize the energy
density relative to $D$:

$\displaystyle\frac{\partial\rho}{\partial D}=0\ \ \ \Rightarrow\ \ \
D=\frac{1}{6}H,\ \ \ \frac{8\pi G}{3}\rho=\frac{7}{8}H^{2},$ $\displaystyle\ \
\ \ \ \ \ f=\frac{1}{16}(1-21w)$ (84) $\displaystyle f=1\ \ \ \Rightarrow\ \ \
w\approx-\frac{5}{7}\approx-0.7$ (85)

However, we can be more precise. Both $H$ and $D$ are functions of time as
follows:

$\displaystyle\frac{\ddot{a}}{a}\equiv f(\frac{\dot{a}}{a})^{2}\ \ \
\Rightarrow\ \ \ \dot{H}=-(1-f)H^{2}$ (86)
$\displaystyle\sigma\sim\frac{1}{a^{2}}\ \ \ \Rightarrow\ \ \
D\sim\frac{1}{a^{6}}\ \ \ \Rightarrow\ \ \ \dot{D}=-6HD$ (87)
$\displaystyle\frac{8\pi G}{3}\dot{\rho}=-36H(D^{2}-\frac{1}{36}(7-f)HD$
$\displaystyle\ \ \ \ \ \ \ +\frac{1}{27}(1-f)H^{2})$ (88)
$\displaystyle\dot{\rho}=0\Rightarrow
D^{\pm}\equiv\frac{H}{72}\\{7-f\pm\sqrt{f^{2}+178f-143}\\}$ (89)
$\displaystyle f\geqslant 0.7998$ (90) $\displaystyle f\approx 1\ \ \
\Rightarrow\ \ \ D^{+}\approx\frac{H}{6},\ \ \ w\approx-0.7$ (91)
$\displaystyle f\rightarrow\infty\Rightarrow D^{+}\rightarrow\frac{4}{3}H,\
D^{-}\rightarrow(-\infty)H$ (92)

We see that for the acceleration parameter in the neighborhood of $1$ we
obtain a universe solidly in the mostly vacuum-dominated region and well
approximated by the RW theory. We also see that for the energy density to have
a well defined minimum with a real density parameter, the acceleration
parameter must be greater than or equal to $0.7998$.

Finally, consider how the choice of the local minimum determines whether the
universe is flat, open, or closed, as defined by the relative value of energy
density to the critical energy density:

$\displaystyle\rho_{c}\equiv\frac{3H^{2}}{8\pi G},\ \ \
\rho_{d}\equiv\frac{3D^{2}}{8\pi G},\ \ \
\Omega_{d}\equiv\frac{\rho_{d}}{\rho_{c}}=\frac{D^{2}}{H^{2}}$ (93)
$\displaystyle\rho=\rho_{c}-\frac{3}{2}\sqrt{\rho_{d}\rho_{c}}+\frac{9}{2}\rho_{d}+\frac{3\kappa}{8\pi
Ga^{2}}$ (94)
$\displaystyle\Omega-1=\frac{9}{2}\Omega_{d}-\frac{3}{2}\sqrt{\Omega_{d}}+\frac{\kappa}{H^{2}a^{2}}$
(95)

Note that we can now have a geometrically flat universe with $\kappa\approx 0$
along with a closed or open universe as a function of the diffusive energy
density relative to the critical density. For a geometrically flat universe we
have:

$\displaystyle\Omega>1\ \ \ \Rightarrow\ \ \ D>\frac{1}{3}H,\ \ \ f>3$ (96)
$\displaystyle\Omega<1\ \ \ \Rightarrow\ \ \ D<\frac{1}{3}H,\ \ \ f<3$ (97)

VIII. ACKNOWLEDGEMENTS

I would like to thank Maja Burić, Adam Peterson, and Pavel Wiegmann for
valuable discussions and extremely thoughtful questions and comments. I would
also like to thank Rick Cavanaugh and the CMS Collaboration for their
encouragement and support. While this paper has greatly benefited from such
assistance, any remaining errors are my own.

APPENDIX A: ML CONNECTION

$\displaystyle I=-\frac{1}{2}\int
d\tau\frac{dx^{\mu}}{d\tau}g_{\mu\nu}\frac{dx^{\nu}}{d\tau},$ $\displaystyle\
\ \ \ \ x^{\mu}\rightarrow
x^{\mu}+\delta\overline{x}^{\mu}+\sigma\delta{\xi}^{\mu},\delta\xi^{\mu}\sim\mathcal{N}(0,\sqrt{\delta
t})$ (98) $\displaystyle I\rightarrow-\frac{1}{2}\int
d\tau\\{\frac{d(x^{\mu}+\delta\overline{x}^{\mu}+\sigma\delta\tilde{\xi}^{\mu})}{d\tau}(g_{\mu\nu}+(\partial_{\rho}g_{\mu\nu})$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
(\delta\overline{x}^{\rho}+\sigma\delta\tilde{\xi}^{\rho})$ $\displaystyle\ \
\ \ \
+\frac{1}{2}(\partial_{\alpha}\partial_{\beta}g_{\mu\nu})\sigma^{2}\mathnormal{h}^{\alpha\beta}_{\rho}\delta\overline{x}^{\rho})\frac{d(x^{\nu}+\delta\overline{x}^{\nu}+\sigma\delta\tilde{\xi}^{\nu})}{d\tau}\\}$
(99) $\displaystyle\langle\delta I\rangle=-\frac{1}{2}\int
d\tau\\{(\partial_{\alpha}g_{\mu\nu})\delta
x^{\alpha}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}$ $\displaystyle\ \ \ \
\ \
+\frac{1}{2}\sigma^{2}(\partial_{\alpha}\partial_{\beta}g_{\mu\nu})\langle\delta\xi^{\alpha}\delta\xi^{\beta}\rangle\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}$
$\displaystyle\ \ \ -\partial
x^{\nu}(\partial_{\alpha}g_{\mu\nu})\frac{dx^{\alpha}}{d\tau}\frac{dx^{\mu}}{d\tau}-\partial
x^{\nu}g_{\mu\nu}\frac{d^{2}x^{\mu}}{d\tau^{2}}$ $\displaystyle\ \ \ \ \ \
-\partial
x^{\mu}(\partial_{\alpha}g_{\mu\nu})\frac{dx^{\alpha}}{d\tau}\frac{dx^{\nu}}{d\tau}-\partial
x^{\mu}g_{\mu\nu}\frac{d^{2}x^{\nu}}{d\tau^{2}}$ $\displaystyle\ \ \
+\sigma\langle\delta\xi^{\beta}(\partial_{\alpha}\sigma\xi^{\nu})\rangle(\partial_{\beta}g_{\mu\nu})\frac{dx^{\alpha}}{d\tau}\frac{dx^{\mu}}{d\tau}$
$\displaystyle\ \ \ \ \ \
+\sigma\langle\delta\xi^{\beta}(\partial_{\alpha}\sigma\xi^{\mu})\rangle(\partial_{\beta}g_{\mu\nu})\frac{dx^{\alpha}}{d\tau}\frac{dx^{\nu}}{d\tau}$
$\displaystyle\ \ \
+\langle(\partial_{\alpha}\sigma\xi^{\mu})(\partial_{\beta}\sigma\xi^{\nu})\rangle
g_{\mu\nu}\frac{dx^{\alpha}}{d\tau}\frac{dx^{\beta}}{d\tau}\\}$ (100)
$\displaystyle\langle\delta I\rangle=\int d\tau g_{\mu\rho}\delta
x^{\rho}\\{\frac{d^{2}x^{\mu}}{d\tau^{2}}$ $\displaystyle\ \ \ \ \ \
+\frac{dx^{\alpha}}{d\tau}\frac{dx^{\beta}}{d\tau}[\frac{1}{2}g^{\mu\sigma}(\partial_{\alpha}g_{\beta\sigma}+\partial_{\beta}g_{\sigma\alpha}-\partial_{\sigma}g_{\alpha\beta})$
$\displaystyle\ \ \ \ \ \
-\frac{1}{2}g^{\mu\delta}(\frac{1}{2}\sigma^{2}\mathnormal{h}^{\eta\sigma}_{\delta}(\partial_{\eta}\partial_{\sigma}g_{\alpha\beta})+(\partial_{\alpha}\sigma^{2})(\partial_{\sigma}g_{\eta\beta})\mathnormal{h}^{\eta\sigma}_{\delta}$
$\displaystyle\ \ \ \ \ \
+(\partial_{\alpha}\sigma)(\partial_{\beta}\sigma)\mathnormal{h}^{\eta\sigma}_{\delta}g_{\eta\sigma}+2\sigma^{2}a^{\sigma\eta}_{\delta\alpha}(\partial_{\sigma}g_{\eta\beta})$
$\displaystyle\ \ \ \ \ \
+(\partial_{\alpha}\sigma^{2})a^{\eta\sigma}_{\delta\beta}g_{\eta\sigma}+\sigma^{2}\mathnormal{k}^{\eta\sigma}_{\delta\alpha\beta}g_{\eta\sigma})]\\}$
(101)

APPENDIX B: TWO-POINT FUNCTIONS IN RW GEOMETRY

$\displaystyle\frac{\partial x^{i}}{\partial x^{\hat{i}}}=a,\ \ \
\frac{\partial\delta x^{i}}{\partial\delta x^{\hat{i}}}=a^{z},\ \ \
\frac{\partial\delta\xi^{\hat{i}}}{\partial\delta\xi^{i}}=\frac{1}{a^{2}}$
(102) $\displaystyle
h^{\hat{\alpha}\hat{\beta}}_{\hat{\rho}}=\frac{\partial\delta\xi^{\hat{\alpha}}}{\partial\delta\xi^{\alpha}}\frac{\partial\delta\xi^{\hat{\beta}}}{\partial\delta\xi^{\beta}}\frac{\partial\delta
x^{\rho}}{\partial\delta x^{\hat{\rho}}}h^{\alpha\beta}_{\rho}$ (103)
$\displaystyle\Rightarrow\ \ \
h^{\hat{x}\hat{x}}_{\hat{t}}=\frac{1}{a^{2}}\frac{1}{a^{2}}\sigma^{2}=\frac{\sigma^{2}}{a^{4}},$
$\displaystyle\ \ \ \ \ \ \ \
h^{\hat{t}\hat{t}}_{\hat{x}}=a^{z}(-\sigma_{\epsilon}^{2})=-\sigma_{\epsilon}^{2}a^{z}$
(104) $\displaystyle
k^{\hat{\alpha}\hat{\beta}}_{\hat{\rho}\hat{\mu}\hat{\nu}}=\frac{\partial\delta\xi^{\hat{\alpha}}}{\partial\delta\xi^{\alpha}}\frac{\partial\delta\xi^{\hat{\beta}}}{\partial\delta\xi^{\beta}}\frac{\partial\delta
x^{\rho}}{\partial\delta x^{\hat{\rho}}}\frac{\partial x^{\mu}}{\partial
x^{\hat{\mu}}}\frac{\partial x^{\nu}}{\partial
x^{\hat{\nu}}}k^{\alpha\beta}_{\rho\mu\nu}$ (105) $\displaystyle\Rightarrow\ \
\
k^{\hat{x}\hat{x}}_{\hat{t}\hat{t}\hat{t}}=\frac{1}{a^{2}}\frac{1}{a^{2}}2c\sigma^{2}=\frac{2c\sigma^{2}}{a^{4}},$
$\displaystyle\ \ \ \ \ \ \ \
k^{\hat{x}\hat{x}}_{\hat{t}\hat{y}\hat{y}}=\frac{1}{a^{2}}\frac{1}{a^{2}}aa2c\sigma^{2}=\frac{2c\sigma^{2}}{a^{2}}$
(106) $\displaystyle\Rightarrow\ \ \
k^{\hat{t}\hat{t}}_{\hat{x}\hat{t}\hat{t}}=a^{z}(-2c_{\epsilon}\sigma_{\epsilon}^{2})=-2c_{\epsilon}\sigma_{\epsilon}^{2}a^{z},$
$\displaystyle\ \ \ \ \ \ \ \
k^{\hat{t}\hat{t}}_{\hat{x}\hat{y}\hat{y}}=a^{z}aa(-2c_{\epsilon}\sigma_{\epsilon}^{2})=-2c_{\epsilon}\sigma_{\epsilon}^{2}a^{2+z}$
(107)

APPENDIX C: CONNECTION TENSORS IN RW GEOMETRY

To make the derivations clearer, we work with the two-point functions $h$ and
$k$ prior to folding $\sigma$ or $\sigma_{\epsilon}$ into their definitions:

$\displaystyle\partial_{x}^{2}g|_{r=0}=2\kappa(0,a^{2}),\ \ \
\partial_{t}^{2}g|_{r=0}=2(0,\dot{a}^{2}+a\ddot{a}),$ $\displaystyle\ \ \ \ \
\ \ \ \ \ \ \partial_{x}g|_{r=0}=0$ (108)
$\displaystyle\sigma\sim\frac{1}{a^{y}},\ \ \
\partial_{t}\sigma=-y\sigma\frac{\dot{a}}{a},\ \ \
\partial_{t}\sigma^{2}=-2y\sigma^{2}\frac{\dot{a}}{a},$ $\displaystyle\ \ \ \
\ \ \ \ \ \ \ \partial_{t}\sigma_{\epsilon}=0$ (109)
$\displaystyle\Sigma^{\mu}_{\alpha\beta}\equiv\frac{1}{4}g^{\mu\delta}\sigma^{2}\mathnormal{h}^{\eta\sigma}_{\delta}(\partial_{\eta}\partial_{\sigma}g_{\alpha\beta})$
$\displaystyle\ \ \ \ \ \ \ \
+\frac{1}{2}g^{\mu\delta}(\partial_{\alpha}\sigma^{2})(\partial_{\sigma}g_{\eta\beta})\mathnormal{h}^{\eta\sigma}_{\delta}$
$\displaystyle\ \ \ \ \ \
+\frac{1}{2}g^{\mu\delta}(\partial_{\alpha}\sigma)(\partial_{\beta}\sigma)\mathnormal{h}^{\eta\sigma}_{\delta}g_{\eta\sigma}$
(110)
$\displaystyle\Rightarrow\Sigma^{t}_{\alpha\beta}=\frac{3}{4}g^{tt}\sigma^{2}\mathnormal{h}^{xx}_{t}(\partial_{x}^{2}g_{\alpha\beta})$
$\displaystyle\ \ \ \ \ \ \ \
+\frac{3}{2}g^{tt}(\partial_{\alpha}\sigma)(\partial_{\beta}\sigma)\mathnormal{h}^{xx}_{t}g_{xx}$
(111)
$\displaystyle\Rightarrow\Sigma^{t}_{tt}=-\frac{3}{2}y^{2}\frac{\sigma^{2}}{a^{2}}(\frac{\dot{a}}{a})^{2}=O(\frac{\sigma^{2}}{a^{2}}H^{2}),$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \
\Sigma^{t}_{xx}=-\frac{3}{2}\frac{\kappa\sigma^{2}}{a^{2}}=O(\sigma^{2}\frac{\kappa}{a^{2}})$
(112)
$\displaystyle\Rightarrow\Sigma^{x}_{\alpha\beta}=\frac{1}{4}g^{xx}\sigma_{\epsilon}^{2}\mathnormal{h}^{tt}_{x}(\partial_{t}^{2}g_{\alpha\beta})+0+0$
(113) $\displaystyle\ \ \ \Rightarrow\Sigma^{x}_{tt}=0,\ \ \
\Sigma^{x}_{yy}=-\frac{1}{2}\sigma_{\epsilon}^{2}a^{z}(\frac{\ddot{a}}{a}+(\frac{\dot{a}}{a})^{2})$
(114)
$\displaystyle\kappa^{\mu}_{\alpha\beta}\equiv\frac{1}{2}g^{\mu\delta}\sigma^{2}\mathnormal{k}^{\eta\sigma}_{\delta\alpha\beta}g_{\eta\sigma}$
(115)
$\displaystyle\Rightarrow\kappa^{t}_{tt}=\frac{3}{2}g^{tt}\sigma^{2}\mathnormal{k}^{xx}_{ttt}g_{xx}=-3\frac{c\sigma^{2}}{a^{2}},$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \
\kappa^{t}_{xx}=\frac{3}{2}g^{tt}\sigma^{2}\mathnormal{k}^{yy}_{txx}g_{yy}=-3{c\sigma^{2}}\
\ \ $ (116)
$\displaystyle\Rightarrow\kappa^{x}_{tt}=\frac{1}{2}g^{xx}\sigma_{\epsilon}^{2}\mathnormal{k}^{tt}_{xtt}g_{tt}=\frac{c_{\epsilon}\sigma_{\epsilon}^{2}a^{z}}{a^{2}},$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \
\kappa^{x}_{yy}=\frac{1}{2}g^{xx}\sigma_{\epsilon}^{2}\mathnormal{k}^{tt}_{xyy}g_{tt}=c_{\epsilon}\sigma_{\epsilon}^{2}a^{z}\
\ \ $ (117)
$\displaystyle\mathcal{A}^{\mu}_{\alpha\beta}\equiv\frac{1}{2}g^{\mu\delta}\sigma^{2}a^{\sigma\eta}_{\delta\alpha}(\partial_{\sigma}g_{\eta\beta})\
\ \ \Rightarrow\ \ \ \mathcal{A}|_{r=0}=0$ (118)
$\displaystyle\Gamma^{\mu}_{\alpha\beta}=\frac{1}{2}g^{\mu\sigma}(\partial_{\alpha}g_{\beta\sigma}+\partial_{\beta}g_{\sigma\alpha}-\partial_{\sigma}g_{\alpha\beta})$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \Rightarrow\ \ \
\Gamma^{t}_{xx}=a\dot{a},\ \ \ \Gamma^{x}_{xt}=\frac{\dot{a}}{a}$ (119)
$\displaystyle\tilde{\Gamma}^{\mu}_{\alpha\beta}={\Gamma}^{\mu}_{\alpha\beta}-{\kappa}^{\mu}_{\alpha\beta}-\mathcal{A}^{\mu}_{\alpha\beta}-{\Sigma}^{\mu}_{\alpha\beta}$
(120)
$\displaystyle\Rightarrow\tilde{\Gamma}^{t}_{tt}=3\frac{c\sigma^{2}}{a^{2}}(1+\frac{y^{2}}{2c}(\frac{\dot{a}}{a})^{2})=3\frac{c\sigma^{2}}{a^{2}}+O(\frac{\sigma^{2}}{a^{2}}H^{2})$
$\displaystyle\ \ \ \ \ \ \ \ \approx 3\frac{c\sigma^{2}}{a^{2}},$ (121)
$\displaystyle\Rightarrow\tilde{\Gamma}^{t}_{xx}=a\dot{a}+3\frac{c\sigma^{2}}{a^{2}}+\frac{3}{2}\frac{\kappa\sigma^{2}}{a^{2}}=a\dot{a}+3\frac{c\sigma^{2}}{a^{2}}+O(\sigma^{2}\frac{\kappa}{a^{2}})$
$\displaystyle\ \ \ \ \ \ \ \ \approx a\dot{a}+3\frac{c\sigma^{2}}{a^{2}}$
(122)
$\displaystyle\Rightarrow\tilde{\Gamma}^{x}_{tt}=-\frac{c_{\epsilon}\sigma_{\epsilon}^{2}a^{z}}{a^{2}},\
\ \ \tilde{\Gamma}^{x}_{xt}=\frac{\dot{a}}{a},$ $\displaystyle\ \ \ \ \ \ \ \
\ \ \
\tilde{\Gamma}^{x}_{yy}=\frac{1}{2}\sigma_{\epsilon}^{2}a^{z}(\frac{\ddot{a}}{a}+(\frac{\dot{a}}{a})^{2})-c_{\epsilon}\sigma_{\epsilon}^{2}a^{z}$
(123)

APPENDIX D: RWML EQUATIONS OF MOTION

Taking $z=2$ from here on, we have

$\displaystyle\nabla_{\mu}{T^{\mu\nu}}=\partial_{\mu}T^{\mu\nu}+\tilde{\Gamma}^{\alpha}_{\alpha\beta}T^{\beta\nu}+\tilde{\Gamma}^{\nu}_{\alpha\beta}T^{\alpha\beta}=0,$
$\displaystyle\ \ \ \ \ \ \ \ T_{\mu\nu}=\rho(1,wa^{2},wa^{2},wa^{2})$ (124)
$\displaystyle\partial_{t}T^{tt}+(2\tilde{\Gamma}^{t}_{tt}+3\tilde{\Gamma}^{x}_{xt})T^{tt}+3\tilde{\Gamma}^{t}_{xx}T^{xx}=0$
(125)
$\displaystyle\Rightarrow\partial_{t}\rho=-3\rho\\{(1+w)\frac{\dot{a}}{a}+$
$\displaystyle\ \ \ \ \
\frac{c\sigma^{2}}{a^{2}}(2+\frac{2}{c}(\frac{\dot{a}}{a})^{2}+3w(1+\frac{\kappa}{2ca^{2}})\\}$
(126) $\displaystyle
4\tilde{\Gamma}^{x}_{xx}T^{xx}+\tilde{\Gamma}^{x}_{tt}T^{tt}=0$ (127)
$\displaystyle\Rightarrow\ \ \
c_{\epsilon}(1+4w)=2w(\frac{\ddot{a}}{a}+(\frac{\dot{a}}{a})^{2})$ (128)

APPENDIX E: RWML Ricci tensor, scalar, and Friedmann Equations

$\displaystyle
R_{\mu\nu}=\partial_{\alpha}\tilde{\Gamma}^{\alpha}_{\mu\nu}-\partial_{\nu}\tilde{\Gamma}^{\alpha}_{\alpha\mu}+\tilde{\Gamma}^{\alpha}_{\alpha\beta}\tilde{\Gamma}^{\beta}_{\mu\nu}-\tilde{\Gamma}^{\alpha}_{\nu\beta}\tilde{\Gamma}^{\beta}_{\alpha\mu}$
(129) $\displaystyle\Rightarrow\
R_{tt}=\partial_{\alpha}\tilde{\Gamma}^{\alpha}_{tt}-\partial_{t}\tilde{\Gamma}^{\alpha}_{\alpha
t}+\tilde{\Gamma}^{\alpha}_{\alpha\beta}\tilde{\Gamma}^{\beta}_{tt}-\tilde{\Gamma}^{\alpha}_{t\beta}\tilde{\Gamma}^{\beta}_{\alpha
t}$ $\displaystyle\ \ \ \ \ \ \ \ \ \
=R_{tt}^{RW}+3\tilde{\Gamma}^{x}_{xt}\tilde{\Gamma}^{t}_{tt}+3\tilde{\Gamma}^{x}_{xx}\tilde{\Gamma}^{x}_{tt}$
$\displaystyle\ \ \ \ \ \ \ \ \ \
=-3\frac{\ddot{a}}{a}+9\frac{\dot{a}}{a}\frac{c\sigma^{2}}{a^{2}}+O(\sigma_{\epsilon}^{4})$
(130) $\displaystyle\Rightarrow\
R_{xx}=\partial_{\alpha}\tilde{\Gamma}^{\alpha}_{xx}-\partial_{x}\tilde{\Gamma}^{\alpha}_{\alpha
x}+\tilde{\Gamma}^{\alpha}_{\alpha\beta}\tilde{\Gamma}^{\beta}_{xx}-\tilde{\Gamma}^{\alpha}_{x\beta}\tilde{\Gamma}^{\beta}_{\alpha
x}$ $\displaystyle\ \ \ \ \ \ \ \ \ \
=R_{xx}^{RW}-(\partial_{t}+\tilde{\Gamma}^{y}_{yt})(\kappa^{t}_{xx}+\Sigma^{t}_{xx})$
$\displaystyle\ \ \ \ \
+\tilde{\Gamma}^{t}_{tt}\tilde{\Gamma}^{t}_{xx}+2\tilde{\Gamma}^{x}_{xx}\tilde{\Gamma}^{x}_{xx}$
$\displaystyle\ \ \ \ \ \ \ \ \ \
=a\ddot{a}+2(\dot{a})^{2}+2\kappa-6(y-1)\frac{\dot{a}}{a}c\sigma^{2}$
$\displaystyle\ \ \ \ \
-\frac{3}{2}(2y+1)\frac{\dot{a}}{a}\frac{\kappa\sigma^{2}}{a^{2}}+\frac{9}{2}\frac{c^{2}\sigma^{4}}{a^{2}}(2+\frac{\kappa}{ca^{2}})$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+O(c_{\epsilon}\sigma_{\epsilon}^{2})$ (131)

Setting $y=2$, the minimum value to give us a universe with an energy density
which has a local minimum at a positive Hubble parameter, we obtain the Ricci
scalar, and Friedmann Equations:

$\displaystyle R=g^{\mu\nu}R_{\mu\nu}=-R_{tt}+\frac{3}{a^{2}}R_{xx}=$
$\displaystyle\ \ \ \ \
6(\frac{\ddot{a}}{a}+(\frac{\dot{a}}{a})^{2}+\frac{\kappa}{a^{2}})-27\frac{\dot{a}}{a}\frac{c\sigma^{2}}{a^{2}}+27\frac{c^{2}\sigma^{4}}{a^{4}}$
(132) $\displaystyle R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=8\pi GT_{\mu\nu}$ (133)
$\displaystyle\Rightarrow\ R_{tt}+\frac{1}{2}R=8\pi G\rho,\ \ \
R_{xx}-\frac{1}{2}Ra^{2}=8\pi G\rho wa^{2}$ (134)
$\displaystyle\frac{\ddot{a}}{a}\approx f(\frac{\dot{a}}{a})^{2},\ \ \
H\equiv\frac{\dot{a}}{a},\ \ \ D\equiv\frac{c\sigma^{2}}{a^{2}}$ (135)
$\displaystyle\Rightarrow\frac{8\pi
G}{3}\rho=H^{2}-\frac{3}{2}HD+\frac{9}{2}D^{2}+O(\frac{\kappa}{a^{2}})+O(\sigma_{\epsilon}^{4})$
(136) $\displaystyle\Rightarrow-\frac{8\pi
G}{3}w\rho=\frac{1}{3}(2f+1)H^{2}-\frac{5}{2}HD+\frac{3}{2}D^{2}$
$\displaystyle\ \ \ \ \ +O(\frac{\kappa}{a^{2}})+O(\sigma_{\epsilon}^{4})$
(137)

where $f=-q$ is the acceleration parameter, the negative of the typically used
deceleration parameter $q$.

BIBLIOGRAPHY

* (1) H. Georgi, Lie algebras in particle physics. From isospin to unified theories, (1982).
* (2) P. Wiegmann, Lecture notes in Quantum Field Theory, (University of Chicago, Chicago, 2016).
* (3) S. Carroll, Spacetime and geometry: an introduction to general relativity, (Pearson Education, Essex, 2013).
* (4) M. Peskin, D. Schroeder An Introduction to quantum field theory, (Addison-Wesley, Reading, USA, 1995).


# Temporal graph-based approach for behavioural entity classification

Francesco Zola  Vicomtech Foundation,
Basque Research and
Technology Alliance (BRTA)
20009 San Sebastian, Spain
<EMAIL_ADDRESS>Lander Segurola  Vicomtech Foundation,
Basque Research and
Technology Alliance (BRTA)
20009 San Sebastian, Spain
<EMAIL_ADDRESS>Jan Lukas Bruse  Vicomtech Foundation,
Basque Research and
Technology Alliance (BRTA)
20009 San Sebastian, Spain
<EMAIL_ADDRESS>Mikel Galar Idoate  Institute of Smart Cities,
Public University of
Navarre,
31006 Pamplona, Spain
<EMAIL_ADDRESS>
###### Abstract

Graph-based analyses have gained a lot of relevance in the past years due to
their high potential in describing complex systems by detailing the actors
involved, their relations and their behaviours. Nevertheless, in scenarios
where these aspects are evolving over time, it is not easy to extract valuable
information or to characterize correctly all the actors.

In this study, a two phased approach for exploiting the potential of graph
structures in the cybersecurity domain is presented. The main idea is to
convert a network classification problem into a graph-based behavioural one.
We extract these graph structures that can represent the evolution of both
normal and attack entities and apply a temporal dissection approach in order
to highlight their micro-dynamics. Further, three clustering techniques are
applied to the normal entities in order to aggregate similar behaviours,
mitigate the imbalance problem and reduce noisy data. Our approach suggests
the implementation of two promising deep learning paradigms for entity
classification based on Graph Convolutional Networks.

###### Index Terms:

Cybersecurity analysis, Behavioural classification, Temporal graph analysis,
Clustering, Graph-based structure

Tipo de contribución: Investigación en desarollo

##  I Introduction

Cybersecurity has become a critical aspect for many companies as a
vulnerability or a breach in its infrastructure can generate a considerable
loss of value, in economical, reputational, digital, psychological and
societal terms[1]. The increasing amount of cyber threats and their
significant impacts, has led to increased efforts for preventing and reducing
their risks [2]. In this sense, machine learning and derived techniques such
as deep learning are promising means to obtain new insights from available
data in order to quantify cyber risks and to optimize cybersecurity operations
[3].

These techniques are principally used for performing anomaly detection [4], an
operation that can be used for risk detection as well as a preventive
approach. In particular, anomaly detection has shown very promising results
analyzing data as graphs [5]. Graph structures help integrate both structured
and unstructured data in a representation of entities and the relationships
among them.

Nevertheless, in real use cases, entity behaviour can evolve such as entities
disappearing, emerging or simply changing their dynamics. In these cases,
considering a static graph is not sufficient to describe a complex system, not
least as with this monolithic approach small interactions between entities can
be obscured by more frequent ones - generating a loss of classification
quality. In such a skewed scenario, machine learning models tend to focus on
macro-dynamics and may falsely assign one static behaviour for each entity as
generated by the overall status.

In this work, we propose a two phased approach for addressing these issues. In
the first phase, called Manipulation phase, graph-based structures are
extracted from the initial network traffic dataset defining key concepts such
as entity, edges and entity behaviour using a temporal dissection approach.
Then, still in the first phase, a clustering operation is applied for reducing
noisy data and addressing the class imbalance problem. In the second phase,
called Learning phase, deep learning models are trained with these graph data
in order to detect attack/malicious behavioural entities. In this work,
operations related to the Manipulation phase are introduced and three
clustering techniques are compared using the UNSW15 dataset [6]. Further,
detailed guidelines for the Learning phase are provided.

##  II Methodology

###  II-A Problem description

In several cases, when the aim is to perform a network traffic classification,
the most common and straightforward idea is to consider the information as a
time series and then apply Artificial Intelligence paradigms for classifying
these flows in order to discover malicious/anomalous patterns. In this study,
a two phased approach based on converting this time series analysis into a
graph-based behavioural classification is proposed.

During the first phase, Manipulation phase, the initial dataset is manipulated
in order to extract graph information. The main idea is to split this initial
dataset in fixed time interval called ”snapshot”, and, in each of them,
extract graph-based structures defining entities and edges. In this sense, it
is important to perform this temporal dissection operation in order to uncover
micro-dynamics and highlight how the entities relations change within the
time. Once the graph structures are ready, a clustering algorithm is applied
in order to aggregate similar behaviours, i.e. similar entities, for reducing
the overall amount of information. This operation is very important,
especially when the initial dataset is characterized by noisy data or it is
affected by a strong imbalanced problem. In these cases, applying the
clustering only over the majority class helps to reduce those problems. In
this research, $3$ clustering techniques are compared: Density-Based Spatial
Clustering of Applications with Noise (DBSCAN [7]), Ordering Points To
Identify the Cluster Structure (OPTICS [8]) and Hierarchical DBSCAN
(HDBSCAN[9]) All these operations allow to create the appropriate input for
the next phase (Learning phase), which is focused on training $2$ graph deep
learning models, both belonged to the Graph Convolutional Network (GCN)
family, for classifying the node behaviours and detect the malicious/anomalous
ones.

It is important to remark that this work is not focused on classifying or
predicting directly the node’s communication as presented in [10, 11], neither
on developing another framework based on graph deep learning models [12], but
the key idea is to convert network flow classification into a behavioural
entity task, and to investigate how to manipulate the initial data for
generating new insights and improve the final classification. To the best of
our knowledge, this is the first work that combine temporal graph dissection,
clustering operation and planning the usage of graph deep learning for
detecting malicious entities in cybersecurity domain.

###  II-B Behavioral Node Identification

In order to extract and define behavioural entities within the traffic dataset
a pre-processing operation called Behavioral Node Identification is
introduced. In particular, knowing that each record of the dataset represents
a communication between two nodes, we map that communication as an edge in the
graph. These edges connect an entity source and an entity destination defined
respectively by the combination of the source IP (srcip) and source port
(srcport), and the destination IP (dstip) and the destination port (dstport)
given as features for the considered record.

Figure 1: Behavioral Node Identification example

Once obtained the graph, it is important to assign a label to each node in
order to define its behavior in the network. In this sense, the information
provided by the labelled records of the network traffic dataset are used to
label the nodes’ behaviour in the network. The possible nodes’ behaviours are
two: a normal behaviour identified with the labelled $0$ and an anomalous, or
attack, behaviour identified with the label $1$. Each node behaviour is
defined by analyzing the nodes’ connections, both the incoming and outgoing
(Figure 1). If the malicious connections represent the majority of the node’s
connections, the node takes the label of anomalous behaviour ($1$), as for
example node 3 in the Figure 1. On the other hand, the node takes the label of
normal behaviour ($0$), as for example node 1 in Figure 1. In case of draw,
the node takes the label of normal behaviour ($0$), as for example node $4$.

###  II-C Temporal graph

The Node Behavioural Identification can be applied considering the whole
network traffic dataset for creating an unique static and monolithic graph,
however, with this approach all the dynamics and small interaction between the
entities will be overwhelmed by the more frequently and more heavy - in terms
of exchanged data - ones. Furthermore, this approach increases the amount of
data in the graph requiring more computational resources, making hard the real
application of the solution. For example, if a model is trained with a
monolithic graph created with $\sim 10$ hours data, for the testing phase, it
needs of a comparable structure, i.e. a graph obtained in other $\sim 10$
hours data. This can compromise the usability of the trained model, since in
anomaly detection the timing is fundamental in order to mitigate promptly an
attack applying countermeasures and avoid the worsening of the situation.

To address these issues, and learn the micro behaviours in the network and
detecting the small dynamics within, in this research, the initial dataset is
divided in fixed temporal intervals (or temporal snapshot), and in each one of
it, it is performed an operation of Node Behaviour Identification, as
described in the Section II-B. Basically, all the communications in a temporal
snapshot are analyzed in order to create a photograph of the network status
through a relational graph where the behaviours’ entities and their relations
are drawn.

###  II-D Clustering Algorithms

In cybersecurity domain, the network traffic datasets are usually
characterized by class imbalanced problem. In fact, it is easier to find and
generate traffic related with normal activities rather than attack
connections. This imbalance problem in the population, between normal and
attack entities, can generate loss of classification quality when supervised
classification algorithms are used, since the model will be trained with a
skewed dataset favoring the classification of a the major-represented class
over the under-represented one [13]. In this scenario, in order to reduce the
amount of information in each temporal snapshot, a clustering algorithm is
used. The idea is to apply this operation in order to aggregate entities with
similar behaviours and homogenize the final population. In particular, we
propose to apply this operation just over the normal behavioural entities in
order to reduce their population.

A variety of clustering algorithms can be separated based on the criteria used
to create the groups [14], the more common are the hierarchical clustering,
the distribution-based clustering, the density-based clustering and the
centroid-based clustering. In this work, the analysis is focused over the
density-based clustering which represents a group of algorithms where the main
parameter to define the clusters is the density difference between zones of
the space [15]. This cluster’s category uses unsupervised learning methods
that identify distinctive groups in the data, based on the idea that a cluster
in a data space is a contiguous region of high point density, separated from
other such clusters by contiguous regions of low point density[15]. The data
points in the separating regions of low point density are typically considered
noise/outliers [16].

In particular, in this analysis, DBSCAN, OPTICS and HDBSCAN techniques are
used. The first $2$ algorithms define dense regions using a minimum number of
points that must belong to the cluster (minPts) and the maximum distance from
one point to another for both to be considered neighbours ($\epsilon$). On the
other hand, DBSCAN paradigm intends to automatically find a clustering that
gives the best stability over $\epsilon$.

After the application of the cluster algorithm, the shape of each temporal
graph changes, and so, the corresponding adjacency matrix changes. This
aggregation promotes the creation of new clustered entities in which their
behaviours are computed by combining (averaging) the single behaviour of the
aggregated entities.

##  III Validation

###  III-A Experiment

In this work, UNSW-NB15 dataset111https://www.unsw.adfa.edu.au/unsw-canberra-
cyber/cybersecurity/ADFA-NB15-Datasets/ is used as input for the Manipulation
phase. As described in [6], this dataset was created with the aim to improve
the existing benchmark datasets, which are not able to describe the
comprehensive representation of network traffic and attack scenarios. The
UNSW-NB15 dataset contains real normal and synthetic abnormal network traffic
generated in the synthetic environment of the University of New South Wales
(UNSW) cyber security lab. The UNSW-NB15 dataset is characterized by $9$
attack families and normal traffic generated in two distinct capture-day. In
particular, in this study, the connections are labelled as $0$ if they are
normal traffic or $1$ if they are attack traffic.

The Manipulation phase starts fixing the value for the temporal dissection
operation, in particular, for this work, $600$ seconds ($10$ minutes) is
chosen. Then, from each of the extracted temporal snapshots a behavioural
graph is created in order to detect the entities and their relations within
the network. For each node, a feature vector that describes its behaviour, is
extracted and it is used as input for the clustering operation. This
clustering process allows us to reduce the amount of data and to mitigate the
imbalance problem as presented in Section II-C. In particular, $3$ clustering
algorithms - OPTICS, DBSCAN and HBDSCAN - are applied separately, studying how
their parameters affect their effectiveness. In particular, the OPTICS and the
DBSCAN algorithm are implemented both with minPts value of $2$ and with $3$
different $\epsilon$ values: $0.2$, $0.5$ and $0.8$. The HDBSCAN instead is
applied again with minPts equals $2$ and with a minimum size of clusters of
$5$, and a euclidean metric to compute the distance between the instances is
used.

###  III-B Results

Choosing fixed temporal intervals of $600$ seconds, $147$ temporal snapshots
are generated, as shown in Figure 2. Each temporal snapshot is characterized
by a very large number of nodes, where the amount of normal entities overwhelm
the amount of anomalous one. In fact, in all the extracted $147$ temporal
snapshots, the average number of normal nodes in a snapshot is more than
$17,000$, for an overall amount of $2,583,605$, while the average attack
population is about $545$, for a global value of $80,174$. Furthermore, in
several temporal snapshots there are not attack nodes, as for example between
the $17$th and the $77$th snapshots.

Figure 2: Initial population in each temporal size

Table I shows the overall clustered population generated by each algorithm.
This overall value is computed by summing the number of normal nodes in all
temporal snapshots after the execution of the cluster algorithms, where the
low point density regions are discarded, as they are considered as
noise/outliers.

It should be noted that from an initial overall amount of $2,583,605$ normal
nodes, the cluster algorithms reduce them in a range between $479,938$ and
$127,346$, i.e. reducing the class of about $81$% - $95$% (Figure 3).
Nevertheless, for the OPTICS datasets, the new normal population still
represent more than $80$% of the whole population. These cluster methods are
not applied over the attack class, as explained in Section II-C, so the final
attack population is invariant ($80,174$ nodes)

Table I shows also that a fixed value of $\epsilon$ parameter generates
different results according to the considered clustering algorithm. In
particular, increasing the $\epsilon$ value increases the ability of the
OPTICS algorithm to aggregate the samples in dense regions, reducing the
number of noise/outliers. The same highest value of $\epsilon$ produces a
reverse effect in the DBSCAN, in fact, it decreases its ability to create
dense regions, increasing the number of noise/outliers and reducing the final
population.

Method | $\epsilon$ |  | normal entities
---
after clustering
| clustered normal entities
---
over the whole dataset
OPTICS | 0.2 | 369,931 | 82.19 %
0.5 | 447,026 | 84.79 %
0.8 | 479,938 | 85.69 %
DBSCAN | 0.2 | 141,935 | 63.90 %
0.5 | 137,903 | 63.24 %
0.8 | 127,346 | 61.37 %
HDBSCAN | - | 176,645 | 68.78 %
TABLE I: Clustering effects Figure 3: Clustered Population

###  III-C Learning guidelines

Once the Manipulation phase is ended, the information is ready for training
learning models and achieve the attack behavioural classification task related
with the Learning phase. In particular, in order to fully exploit the
extracted graph relations highlighted in the first phase, it is planned to use
the Graph Convolutional Network. These models, introduced in [17], are able to
learn the local and global structural patterns of a graph through a
convolution operation as well as happens in the Convolution Neural Network
(CNN).

The convolution operation can be defined through the Equation 1, where
$x\in\mathbb{R}^{n}$ represents a scalar vector (the scalar vector for every
node in the graph), $U$ is the matrix of eigenvectors of the Laplacian of the
graph and $g_{\theta}(\Lambda)$ represents the filter in the Fourier domain.
Nevertheless, solving this equation can be computationally complex and
unreachable, even more prohibitively expensive for large graphs.

$y=g_{\theta}*x=Ug_{\theta}(\Lambda)U^{T}x$ (1)

In this sense, in [18] a promising solution to the Equation 1 is obtained
parametrizing the term $g_{\theta}(\Lambda)$ as a polynomial function that can
be computed recursively. In particular, Chebyshev polynomials with $k$ degree
are used. Kipf et al. [17] demonstrated that a good approximation can be
reached by truncating the Chebichev polynomial to get a linear polynomial ($k$
= 1) and by performing a renormalization trick in order to avoid numerical
instabilities and vanishing gradients.

For the validation of the Learning phase, the idea is to exploit the
highlighted graph-based relations applying $2$ GCNs models, the simple one
introduced in [17], and its higher version based on a different approximation
of the Chebychev polynomials ($k>1$), in order to highlight how the relational
information affect the final classification.

##  IV Conclusions

In this research, a two phased approach for converting a network traffic
analysis into a behavioural entity classification task is presented. In
particular, this work introduces and validates the first phase of this
approach called Manipulation phase, and draws several guidelines for the next
phase, called Learning phase

The main idea of the first phase is to understand and manipulate correctly the
input data regardless the chosen classification model. In particular, a method
for converting the network traffic time series into graph-based structures is
presented, as well as a temporal dissection operation used to uncover micro-
dynamics that in a complex system can be overwhelmed by the most recurrent
ones. Furthermore, $3$ clustering techniques - OPTICS, DBSCAN and HDBSCAN -
for reducing noisy data and aggregate similar behaviour, are tested and
compared using a relevant cybersecurity dataset (UNSW15). A preliminary study
shows how their effectiveness changes according to several parameters, as well
as the DBSCAN is more proned to create regions with lower density that are
considered as noise/outliers. Finally, $3$ GCNs’ architectures are presented
as relevant candidate for implementing the Learning phase

## acknowledgement

This work has been partially supported by the Basque Country Government under
the ELKARTEK program, project TRUSTIND (KK-2020/00054).

## References

  * [1] I. Agrafiotis, J. R. Nurse, M. Goldsmith, S. Creese, and D. Upton, “A taxonomy of cyber-harms: Defining the impacts of cyber-attacks and understanding how they propagate,” _Journal of Cybersecurity_ , vol. 4, no. 1, p. tyy006, 2018\.
  * [2] A. Khraisat, I. Gondal, P. Vamplew, and J. Kamruzzaman, “Survey of intrusion detection systems: techniques, datasets and challenges,” _Cybersecurity_ , vol. 2, no. 1, p. 20, 2019.
  * [3] I. H. Sarker, A. Kayes, S. Badsha, H. Alqahtani, P. Watters, and A. Ng, “Cybersecurity data science: an overview from machine learning perspective,” _Journal of Big Data_ , vol. 7, no. 1, pp. 1–29, 2020.
  * [4] S. Omar, A. Ngadi, and H. H. Jebur, “Machine learning techniques for anomaly detection: an overview,” _International Journal of Computer Applications_ , vol. 79, no. 2, 2013.
  * [5] L. Akoglu, H. Tong, and D. Koutra, “Graph based anomaly detection and description: a survey,” _Data mining and knowledge discovery_ , vol. 29, no. 3, pp. 626–688, 2015.
  * [6] N. Moustafa and J. Slay, “Unsw-nb15: a comprehensive data set for network intrusion detection systems (unsw-nb15 network data set),” in _2015 military communications and information systems conference (MilCIS)_. IEEE, 2015, pp. 1–6.
  * [7] M. Ester, H.-P. Kriegel, J. Sander, X. Xu _et al._ , “A density-based algorithm for discovering clusters in large spatial databases with noise.” in _Kdd_ , vol. 96, no. 34, 1996, pp. 226–231.
  * [8] M. Ankerst, M. M. Breunig, H.-P. Kriegel, and J. Sander, “Optics: ordering points to identify the clustering structure,” _ACM Sigmod record_ , vol. 28, no. 2, pp. 49–60, 1999.
  * [9] R. J. Campello, D. Moulavi, and J. Sander, “Density-based clustering based on hierarchical density estimates,” in _Pacific-Asia conference on knowledge discovery and data mining_. Springer, 2013, pp. 160–172.
  * [10] T. Oba and T. Taniguchi, “Graph convolutional network-based suspicious communication pair estimation for industrial control systems,” _arXiv preprint arXiv:2007.10204_ , 2020.
  * [11] L. Zheng, Z. Li, J. Li, Z. Li, and J. Gao, “Addgraph: Anomaly detection in dynamic graph using attention-based temporal gcn.” in _IJCAI_ , 2019, pp. 4419–4425.
  * [12] J. Jiang, J. Chen, T. Gu, K.-K. R. Choo, C. Liu, M. Yu, W. Huang, and P. Mohapatra, “Anomaly detection with graph convolutional networks for insider threat and fraud detection,” in _MILCOM 2019-2019 IEEE Military Communications Conference (MILCOM)_. IEEE, 2019, pp. 109–114.
  * [13] X. Guo, Y. Yin, C. Dong, G. Yang, and G. Zhou, “On the class imbalance problem,” in _2008 Fourth international conference on natural computation_ , vol. 4. IEEE, 2008, pp. 192–201.
  * [14] D. Xu and Y. Tian, “A comprehensive survey of clustering algorithms,” _Annals of Data Science_ , vol. 2, no. 2, pp. 165–193, 2015.
  * [15] H.-P. Kriegel, P. Kröger, J. Sander, and A. Zimek, “Density-based clustering,” _Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery_ , vol. 1, no. 3, pp. 231–240, 2011.
  * [16] J. Sander, _Density-Based Clustering_. Boston, MA: Springer US, 2010, pp. 270–273. [Online]. Available: https://doi.org/10.1007/978-0-387-30164-8\\_211
  * [17] T. N. Kipf and M. Welling, “Semi-supervised classification with graph convolutional networks,” in _International Conference on Learning Representations (ICLR)_ , 2017.
  * [18] M. Defferrard, X. Bresson, and P. Vandergheynst, “Convolutional neural networks on graphs with fast localized spectral filtering,” in _Advances in neural information processing systems_ , 2016, pp. 3844–3852.