# Gravity of gluonic fluctuations and the value of the cosmological constant

Kris Mackewicz  University of Chicago  Craig Hogan  University of Chicago

###### Abstract

We analyze the classical linear gravitational effect of idealized pion-like
dynamical systems, consisting of light quarks connected by attractive gluonic
material with a stress-energy $p=-\rho c^{2}$ in one or more dimensions. In
one orbit of a system of total mass $M$, quarks of mass $m<<M$ expand apart
initially with $v/c\sim 1$, slow due to the gluonic attraction, reach a
maximum size $R_{0}\sim\hbar/Mc$, then recollapse. We solve the linearized
Einstein equations and derive the effect on freely falling bodies for two
systems: a gluonic bubble model where uniform gluonic stress-energy fills a
spherical volume bounded by a 2D surface comprising the quarks’ rest mass, and
a gluonic string model where a thin string connects two pointlike quarks. The
bubble model is shown to produce a secular mean outward residual velocity of
test particles that lie within its orbit. It is shown that the mean
gravitational repulsion of bubble-like virtual-pion vacuum fluctuations agrees
with the measured value of the cosmological constant, for a bubble with a
radius equal to about twice the pion de Broglie length. These results support
the view that the gravity of standard QCD vacuum fluctuations is the main
source of cosmic acceleration.

##  I Introduction

A widely repeated calculation for the value of the cosmological constant,
based on summing the zero point fluctuations of quantum fields, gives a
famously wrong answer Weinberg (1989); Padmanabhan (2003); Weinberg (2008):
the sum of zero-point mode-fluctuation energies up to a UV cutoff at mass
scale $M$ leads to a cosmological constant $\Lambda$ equivalent to a mass
density of order $\rho_{\Lambda}\sim M^{4}c^{3}/\hbar^{3}$, which leads to a
gravitational cosmic acceleration rate of order
$H_{\Lambda}\sim(M/m_{P})^{2}/t_{P}$, or a cosmological constant

$\Lambda\sim(M/m_{P})^{4}/t_{P}^{2},$ (1)

where $t_{P}=\sqrt{\hbar G/c^{5}}$ denotes the Planck time. For $M$ equal to
the Planck mass $m_{P}=\sqrt{\hbar c/G}$, the predicted value of $\Lambda$ is
larger than the observed value by about 122 orders of magnitude.
ExperimentsAdelberger  _et al._ (2009); Kapner  _et al._ (2007) rule out
proposed modifications of gravity or quantum field fluctuation amplitudes with
a cutoff at the milli-eV mass scale that would give the correct value of
$\Lambda$.

For this reason, it is widely agreed that there must be a basic conceptual
error in the way this calculation is formulated. There needs to be a basic
symmetry of quantum gravity that makes the gravitation of vacuum field
fluctuations nearly vanish, and also a mechanism to account for the nonzero
measured value of the actual cosmological constant.

One possibility is that symmetries of quantum geometry make $\Lambda$ exactly
vanish for pointlike particles, but allow a small nonzero $\Lambda$ from the
gravity of nonlocal vacuum fluctuation states of interacting fields. In this
case, the value of $\rho_{\Lambda}$ would be much less than the Planck value
quoted above, suppressed by a power of the field energy scale. A long-studied
example is the hypothesis Zel’Dovich (1967); Zeldovich (1968); Schutzhold
(2002); Bjorken (2003); Brodsky and Shrock (2011); Bjorken (2010); Klinkhamer
and Volovik (2009); Poplawski (1990); Hogan (2020) is that the cosmological
constant arises from quantum fluctuations in the strong interaction vacuum.

Studies of this hypothesis have generally sought to compute the expected low-
energy energy momentum tensor from the system of QCD quantum fields. In this
paper, we instead analyze the system geometrically, using classical
gravitational models. We estimate the gravitational effect of QCD field
fluctuations by analyzing simple idealized classical systems whose energy-
momentum structure resembles that of pions, the lowest-energy QCD excitations.
The energy-momentum of these systems is dominated by the kinetic energy of
pointlike quarks and massless gluons, and the nonlocal self-attractive
interaction of the gluons. We then use these systems to estimate the
gravitational effects of pion-like vacuum fluctuations, and show that they
approximately agree with the measured cosmic acceleration.

In our simple models, fluctuations of gluonic tension produce secular
repulsive gravitation. The energy-momentum tensor of a homogeneous condensate
of massless gluons in localized virtual fluctuations takes a form proportional
to the metric, with pressure and density related by $p=-\rho c^{2}$ in one or
more dimensions. For more than one dimension, this equation of state violates
the strong energy condition, so its gravitational effect is repulsive. In
field language, this behavior for gluonic fluctuations in strongly-interacting
QCD vacua arises from the gravitational effect of a trace anomalySchutzhold
(2002). The one-dimensional case is also familiar from early models of pions
which modeled strong interactions as strings.

Like early phenomenological models of hadrons, our analysis does not provide a
rigorous connection to QCD field degrees of freedom. However, it provides a
simple classical model for gravitational effects of nonlinear QCD
fluctuations, and shows how they depend critically on nonlocal causal
coherence of field states in more than one dimension. It provides physical
insights into how the gravity of vacuum fluctuations works at a microscopic
level, in particular the reason for the small value of the cosmological
constant. Simply put, the QCD-bubble model predicts that cosmic acceleration
has about the same magnitude (with opposite sign) as Newtonian gravitational
acceleration at the edge of a proton.

##  II Causally coherent gravity of gluonic fluctuations

We study the dynamics and gravity of two idealized models with different
geometries. The first model, shown in Fig. (1), is a bubble: a spherical
volume of gluonic matter is approximated by a uniform isotropic tension and
density with the Lorentz-invariant relationship

$p=-\rho c^{2}$ (2)

in three dimensions, bounded by a uniform shell of dustlike quark material of
constant total mass. The other model, discussed in the Appendix, is a more
traditional idealized model of pions, where a straight gluonic string with
$p=-\rho c^{2}$ in one dimension joins two light pointlike quarks. The two
systems have similar dynamics: they start at small radius with a large
$\gamma$ factor, expand to a maximum size determined by the masses of the
quarks and the tension of the gluons, then recollapse.

As discussed below, tension in more than one dimension is required for gravity
to produce cosmic acceleration, so we focus on the bubble model. Since it is
spherically symmetric, gravity outside the bubble is simply a Schwarzschild
metric. Inside the bubble, the effect of the quarks on a test particle
resembles displacements by a null shock on a causal diamond, whose outwards
and inwards gravitational displacements cancel over a whole orbit. The main
gravitational effect in the interior is from the gluonic matter.

We find that gravity inside the bubble produces a mean repulsive residual
velocity, in the sense that on average it causes test particles within the
orbit to accelerate systematically apart from each other in the radial
direction. Ultimately this unique behavior can be traced to the exotic nature
of the source, whose mass-energy, dominated for much of its orbit by the
gluonic matter with $p=-\rho c^{2}$, violates the strong energy condition.
(The string model, which does not violate this condition, also creates
repulsive gravitational impulses along some directions, but not in a global
average.)

We then adapt the classical model to estimate the mean gravitational effect of
QCD vacuum fluctuations. Gravitationally repulsive virtual gluonic material is
borrowed from vacuum, so its gravitational effect only extends over a compact
causal diamond with a radius $\sim\hbar/m_{\pi}c$ determined by the pion mass
$m_{\pi}$. Since all of space in a sense lies “inside a virtual bubble”, this
model leads to a simple picture of how cosmic repulsion works. Within the
causal diamond of a fluctuating bubble, test bodies on one side of the bubble
accelerate away from the center, and if quantum gravity is causally coherent,
also away from the entire future light cone beyond the center. The
acceleration is approximately the Newtonian gravitational acceleration for a
mass with the bubble density and bubble radius, rather than a cosmic radius.
As discussed further below, virtual bubbles thus create gravitational
fluctuations whose secular gravitational repulsion is much smaller than
Eq.(1):

$\Lambda\sim(m_{\pi}/m_{P})^{6}/t_{P}^{2}.$ (3)

As shown below, the bubble estimate approximately agrees with the measured
cosmological constant for parameters similar to physical pions: for mass
$M=m_{\pi}$, it requires a bubble radius $R_{0}\sim 2.0\ \hbar/m_{\pi}c$,
about two femtometers. Such close agreement is remarkable, since the model is
idealized in several important ways. For example, a smaller radius would be
expected from the fact that real QCD fluctuations do not have a maximally-
repulsive isotropic equation of state; their gravity would be expected to
behave like something in between the bubble and the string.

The model provides a well controlled connection, albeit still idealized,
between the measured properties of pions and the measured cosmological
constant. The rather close agreement, based on a simple correspondence
argument and a highly idealized model system, suggests that if gravitational
states of the field vacuum are coherent in causal diamonds, an absolute value
for the cosmological constant can in principle be derived from properties of
Standard Model fields. Realistic numerical studies of gravitational effects
from the QCD vacuum would not require a theory of quantum gravity, but would
require a coherent nonlocal calculation of expected mass-energy flows in the
vacuum state.

Figure 1: Spacetime diagram of the gluonic bubble model. The diamond
represents a bubble of total mass $M$ filled with gluonic material $g$ with
$p=-\rho c^{2}$ and gravitational timescale $T_{0}$, with a spherical quark
shell $q$ of mass $m<<M$ on a nearly-null trajectory. The shell first
propagates outwards to maximum radius $R_{0}$, then collapses inwards,
separated by a small nonrelativistic reversal region. The timelike world line
represents a freely falling body. The gravity of the gluonic matter produces
an outwards residual velocity $\delta v_{g}\sim+R_{0}^{2}/cT_{0}^{2}$ during
the time a body spends within the bubble (Eq. 9).

##  III Gluonic bubble model

###  III.1 Bubble model with light quarks

The idealized “gluonic bubble” model (Fig. 1) is a spherical ball of total
mass $M$, filled with uniform gluon gas of the unique Lorentz-invariant form
$p_{0}=-\rho_{0}c^{2}$, surrounded by a thin sheet of pressureless quark dust
of total mass $m$ on the surface. It captures the nonlocal, nonabelian self-
tension of the gluon fields in an idealized way that complements the 1D string
model more commonly used for pions. It allows for a solution of the Einstein
equations and derivation of gravitational effects for an isotropic pressure in
3D. We call it a bubble model to differentiate from the bag model, an
idealized picture of a stable nucleon in a confining vacuum. The bubble model,
like the string model, is an idealized picture of the dynamical mass-energy of
QCD fields in a pion-like state, designed to approximate the virtual
fluctuating energy flows of the QCD vacuum.

The quarklike surface of the bubble is dust, that is, it has no tension or
pressure, and is infinitesimally thin. Its mass is constant as it expands, so
the mass density thins out, and the inwards acceleration from the constant
gluonic tension increases. The equation of motion is thus not the same as the
string model, but the solutions are similar. For light quarks $m<<M$, the bulk
of the orbit is relativistic inwards or outwards motion. There is a brief
turnaround near maximum expansion where the velocities are much less than $c$.

###  III.2 Gravitational velocity kick from a bubble orbit with light quarks

In the bubble model, there is no gravitational radiation, so the outgoing and
incoming parts of the orbit are identical under time reversal. The inwards and
outwards shocks from the passage of the quark surface identically cancel, so
there is no residual gravitational effect of the quark surface on the motion
of test particles, apart from those of a “background” Schwarzschild solution
of mass $M$, which is the space-time outside the bubble.

However, worldlines that pass through the interior of the bubble’s causal
diamond accumulate outwards acceleration while they are inside. The mean
gravitational effect on test bodies during the time that they pass within the
volume of the bubble leaves behind an outwards “residual velocity” whose mean
cumulative effect resembles cosmic acceleration.

It is a well known result in general relativity that the gravitational
acceleration at radius $r$ relative to the center of a homogeneous sphere is

$a(r)=\dot{v}=-(4\pi/3)Gr(\rho+3p/c^{2}).$ (4)

This Newtonian weak field limit is valid for a system much smaller than the
Schwarzchild radius of the contained mass. For empty space outside the sphere,
the solution is Schwarzchild so it approaches flat space at large radii. In
the opposite limit where matter uniformly fills a large volume, the exact
solutions are FRW cosmologies.

The effect of general relativity is captured by the last term, the Newtonian
gravitational effect of pressure. The large negative pressure within the
volume of a gluon bubble leads to a net positive acceleration or gravitational
repulsion at radius $r$,

$a_{g}(r)=\dot{v}=+r/T_{0}^{2},$ (5)

where we have defined a gravitational timescale

$T_{0}\equiv(8\pi G\rho_{0}/3)^{-1/2},$ (6)

for a gluonic bubble of maximum radius $R_{0}$ and density

$\rho_{0}=(M-m)(4\pi R_{0}^{3}/3)^{-1}.$ (7)

In the light quark limit $m<<M$, we can ignore the short turnaround part of
the orbit. A worldline at radius $r$ spends a time $\tau_{g}(r)=2(R_{0}-r)/c$
inside the bubble. The outward velocity accumulated at radius $r$ during this
time is

$\delta v(r)_{g}=\tau_{g}(r)a(r)_{g}=+2r(R_{0}-r)/cT_{0}^{2}$ (8)

This quantity vanishes both at $r=R_{0}$ and $r=0$, so there is no residual
velocity kick for a world line on the maximal boundary of the bubble or at the
origin. In between, the residual velocity is positive, with a maximum value at
$r=R_{0}/2$,

$\delta v_{g}(r=R_{0}/2)=+R_{0}^{2}/cT_{0}^{2}.$ (9)

After a bubble orbit, two particles on opposite sides of the center are moving
apart by the sum of their two kicks. Several sample trajectories are shown in
Fig. (2).

Figure 2: Trajectories of test particles (blue/purple) accelerating within the
causal diamond (red) of a bubble with $m<<M$ and $R_{0}=1$, according to Eq.
(5). The acceleration increases with increasing $r$, but the total time inside
the bubble decreases with increasing $r$, so the maximum residual velocity
(Eq. 8) occurs at $r=R_{0}/2$. For illustration, the motion of the test
particles is exaggerated in this plot by a factor of $R_{0}^{2}/T_{0}^{2}$
compared to that of the bubble wall, or about 40 orders of magnitude for QCD
fluctuations.

In the classical solution, this outwards velocity kick is combined with an
inwards acceleration accumulated while a world line lies outside the bubble,
where the metric is a Schwarzschild solution of mass $M$. The source in this
regime includes the gravity of the (mostly relativistic) quark material as
well as the total mass of the gluonic material. As noted above, the quark
contribution vanishes in the limit of light quarks, since the inwards and
outwards shocks cancel.

###  III.3 Gluonic bubble solution for $m\neq 0$

To rigorously solve for the dynamics of the bubble, we will apply the Israel
junction conditions following Barrabès and Israel (1991). We will begin by
considering gluing two general spacetimes, then restrict ourselves to the
spherically symmetric case, and finally further restrict to the weak field
regime (still allowing for the possibility of relativistic velocities). In
this section we work in units where $c=1$ to avoid keeping track of factors of
$c$ when raising and lowering tensor indices with the metric. Consider two
distinct spacetime manifolds $\mathcal{M}_{+},\mathcal{M}_{-}$ with associated
metrics $g^{+}_{\alpha\beta}(x^{\mu}_{+}),g^{-}_{\alpha\beta}(x^{\mu}_{-})$.
The two spacetimes are bounded by hypersurfaces $\Sigma_{+},\Sigma_{-}$ with
induced metrics $g^{+}_{ab},g^{-}_{ab}$ ($a,b=1,2,3$). We can glue the
spacetimes together by making the identification
$\Sigma_{+}=\Sigma_{-}=\Sigma$ with intrinsic coordinates $\xi^{a}$.

We can construct a tetrad of vectors $n^{\mu},e^{\mu}_{(a)}$ ($\mu=0,1,2,3$)
(which can be defined in both spacetime regions) satisfying

$\left.n_{\mu}n^{\mu}|_{+}=n_{\mu}n^{\mu}\right|_{-}=1\quad,\quad\left.n_{\mu}e^{\mu}_{(a)}\right|_{\pm}=0$
(10)

where the vectors $e^{\mu}_{(a)}$ are adapted to the hypersurface $\Sigma$
such that

$g_{ab}^{\pm}=g_{\alpha\beta}\left.e^{\alpha}_{(a)}e^{\beta}_{(b)}\right|_{\pm}$
(11)

We can parametrically define $\Sigma$ such that $\Phi\equiv R(t)-r=0$ on the
hypersurface. This gives a natural identification of hypersurfaces of $\Phi>0$
with $\mathcal{M}_{+}$ and hypersurfaces of $\Phi<0$ with $\mathcal{M}_{-}$.
We can then define $n^{\mu}$ to be normal to surfaces of constant $\Phi$ such
that $n_{\mu}=\alpha^{-1}\partial_{\mu}\Phi$, where $\alpha$ is chosen to
ensure normalization.

One can show that by appropriate choice of intrinsic coordinates $\xi^{a}$ we
can make $g^{+}_{ab}(\xi)=g^{-}_{ab}(\xi)=g_{ab}(\xi)$. However, there will be
a discontinuous jump in the normal extrinsic curvature defined by

$K_{ab}=-n_{\mu}e^{\nu}_{(b)}\nabla_{\nu}e^{\mu}_{(a)}$ (12)

In Newtonian gravity, this gives rise to the familiar jump in the normal
derivative of the Newtonian potential. The induced surface stress-energy can
be related to the jump in the extrinsic curvature by an analog of the Einstein
equations

$-8\pi\left(S_{ab}-\frac{1}{2}g_{ab}S\right)=\left[K_{ab}\right]$ (13)

where $[F]$ denotes $(F_{+}-F_{-})\left.\right|_{\Sigma}$, i.e. the difference
in $F$ across the hypersurface $\Sigma$. The full stress-energy tensor
restricted to the hypersurface $\Sigma$ is then given by

$T^{\mu\nu}_{\Sigma}=-S^{ab}e^{\mu}_{(a)}e^{\nu}_{(b)}|\alpha|\delta(\Phi)$
(14)

Now, we will restrict our attention to spherically symmetric spacetimes. In
Eddington-Finkelstein coordinates, the metric can be written as

$ds^{2}=e^{\psi}du(fe^{\psi}du+2\zeta dr)+r^{2}d\Omega$ (15)

where $u=t-\zeta r^{*},dr^{*}/dr=1/f$ and $f(u,r)=1-2m(u,r)/r$. $\zeta=\pm 1$
denotes whether the hypersurface $\Sigma$ is moving outward (increasing $r$)
or inward. The Einstein equations then give us differential equations for the
functions $m,\psi$.

$\displaystyle\partial_{u}m$ $\displaystyle=4\pi r^{2}T^{r}_{\,u}$ (16)
$\displaystyle\partial_{r}m$ $\displaystyle=-4\pi r^{2}T^{u}_{\,u}$ (17)
$\displaystyle\partial_{r}\psi$ $\displaystyle=4\pi rT_{rr}$ (18)

For the bubble model being considered, the stress-energy inside of the shell
is that of a de-Sitter spacetime with positive cosmological constant and
stress-energy proportional to the spacetime metric.

$T^{dS}_{\mu\nu}=-\rho g_{\mu\nu}$ (19)

In the Eddington-Finkelstein coordinates we find

$T^{u}_{\,u}=T^{r}_{\,r}=T^{\theta}_{\,\theta}=T^{\phi}_{\,\phi}=-\rho$ (20)

Therefore we get that $T_{rr}=0,T^{r}_{\,u}=0$, which by eq. (18),(16) imply
$\psi=0,f=f(r)$. Solving eq. (17) assuming the exterior region to be
Schwarzschild, we find

$\displaystyle f_{+}=1-\frac{8\pi}{3}\rho r^{2}$ (21) $\displaystyle
f_{-}=1-\frac{2M}{r}$ (22)

Now we may explicitly define the tetrad in terms of the chosen coordinates. We
will switch back to using the more familiar $t,r$ coordinates.

$\displaystyle
n_{\mu}=-\frac{1}{(f-f^{-1}\dot{R}^{2})^{1/2}}\left(\delta^{r}_{\mu}+\dot{R}\delta^{t}_{\mu}\right)$
(23)

where the over-dot signifies the derivative with respect to coordinate time
$t$. The intrinsic metric for the timelike spherical shell is given by

$ds^{2}_{\Sigma}=-d\lambda^{2}+r^{2}d\Omega^{2}$ (24)

where $\lambda$ is the proper time of a co-moving observer on the shell. Then
we can choose the rest of our tetrad vectors to be

$\displaystyle e^{\mu}_{(1)}$
$\displaystyle=\frac{1}{(f-f^{-1}\dot{R}^{2})^{1/2}}\left(\delta^{\mu}_{t}+\dot{R}\delta^{\mu_{r}}\right)$
(25) $\displaystyle e^{\mu}_{(2)}$ $\displaystyle=\delta^{\mu}_{\theta}$ (26)
$\displaystyle e^{\mu}_{(3)}$ $\displaystyle=\delta^{\mu}_{\phi}$ (27)

One can compute the extrinsic curvature using eq. (12), which produces the
following stress-energy on the spherical shell:

$-S^{ab}=\sigma\delta^{a}_{1}\delta^{b}_{1}$ (28)
$T^{\mu\nu}_{\Sigma}=|\alpha|\sigma e^{\mu}_{(1)}e^{\nu}_{(1)}\delta(r-R(t))$
(29)

where $\sigma$ is the surface energy density of the shell given by

$\sigma=-\zeta\frac{[m]}{4\pi r^{2}}$ (30)

The form of eq. (29) is that of a pressureless dust in the rest frame of the
shell. In the weak-field limit this reduces to the usual form

$T^{\mu\nu}_{\Sigma}\approx\frac{\gamma m}{4\pi
r^{2}}v^{\mu}v^{\nu}\delta(r-R(t))$ (31)

where $v^{\mu}=dx^{\mu}/dt,\gamma=(1-\dot{R}^{2})^{-1/2}$. The evolution of
the shell radius $R(t)$ is determined by conservation of stress-energy and is
given by

$\left[{\rm
sgn}(n^{\nu}\partial_{\mu}r)(f+(dR/d\lambda)^{2})^{1/2}\right]=-\frac{M}{r}$
(32)

Using the definition for $n^{\mu}$ and $f_{\pm}$ we find

$\Gamma m=M-\frac{4}{3}\pi r^{3}\rho$ (33)

where $\Gamma$ is defined by

$2\Gamma=(f_{+}+(dR/d\lambda)^{2})^{1/2}+(f_{-}+(dR/d\lambda)^{2})^{1/2}$ (34)

In the weak field limit with $dR/d\lambda>>1$ this reduces to

$\Gamma\approx\gamma=(1-\dot{R}^{2})^{-1/2}$ (35)

which is consistent with the expected mass/energy conservation law.

Next, let us define the glued metric over the entire spacetime by

$\tilde{g}_{\mu\nu}=g^{+}_{\mu\nu}\Theta(\Phi)+g^{-}_{\mu\nu}\Theta(-\Phi)$
(36)

where $\Theta(x)$ is the heaviside step function, and $\Phi$ again
parameterizes the hypersurface $\Sigma$. Other quantities with an over tilde
are defined to have a similar meaning. Since the metric can be made continuous
along $\Sigma$, we get

$\displaystyle\partial_{\alpha}\tilde{g}_{\mu\nu}$
$\displaystyle=\partial_{\alpha}g^{+}_{\mu\nu}\theta(\Phi)+\partial_{\alpha}g^{-}_{\mu\nu}\theta(-\Phi)+[g_{\mu\nu}]\delta(\Phi)\partial_{\alpha}\Phi$
$\displaystyle=\widetilde{\partial_{\alpha}g_{\mu\nu}}$ (37)

A direct consequence of eq. (37) is that the Christoffel symbols suffer a step
discontinuity, but there is no $\delta(\Phi)$ contribution. Therefore, the
radial acceleration experience by an observer crossing the shell does not
produce an instantaneous displacement kick.

However, the Riemann tensor does contain such a delta function contribution,
indicating that two nearby test bodies would experience an instantaneous
relative velocity kick.

$R^{\alpha}_{\,\,\beta\mu\nu}=\tilde{R}^{\alpha}_{\,\,\beta\mu\nu}-2[\Gamma^{\alpha}_{\beta[\mu}]n_{\nu]}\alpha\delta(\Phi)$
(38)

From this point forward we shall operate in the weak-field limit exclusively
and assume $\gamma>>1,m<<M$. The full stress-energy tensor for a gluonic
bubble connected to a shell of mass $m>0$ is given by eq. (19) and (31)

$T_{\mu\nu}=-\rho g_{\mu\nu}\Theta(R(t)-r)+\frac{\gamma m}{4\pi
R(t)^{2}}\delta(r-R(t))v_{\mu}v_{\nu}$ (39)

where $\rho$ is the energy density of the gluonic region which acts as a
perfect fluid with $p=-\rho$ and $v_{\mu}=(-1,\dot{R},0,0)$ is the four
velocity of the shell. Conservation of stress-energy gives the equation of
motion for the surface of the bubble:

$\ddot{R}=-\frac{4\pi\rho R(t)^{2}}{m\gamma^{3}}$ (40)

This still assumes the bubble is small so the gravitational effect on the wall
is negligible compared to the gluon tension.

Finding a closed-form analytic solution to this differential equation is quite
difficult. It is easier to find an approximate solution for $\gamma(t)$ during
the initial era and the turn-around. The total mass of the system is conserved
and is given by

$M=\frac{4}{3}\pi\rho R(t)^{3}+\gamma m$ (41)

For highly relativistic initial velocity, we have $\dot{R}\approx
1,R(t)\approx t$. During this portion of the evolution, we have

$\gamma(t)\approx\gamma_{0}-\frac{4}{3m}\pi\rho t^{3}$ (42)

Near the turn-around point, the shell will become sub-relativistic. The
equation of motion then approximates to

$\ddot{R}\approx-\frac{4\pi\rho R_{0}^{2}}{m}$ (43)

where $R_{0}$ is the maximum value of $R$ at the turn-around. The solution is

$R(t)\approx R_{0}-\frac{2\pi\rho R_{0}^{2}}{m}(t-t_{0})^{2}$ (44)
$\gamma(t)\approx 1+\frac{8\pi^{2}\rho^{2}R_{0}^{4}}{m^{2}}(t-t_{0})^{2}$ (45)

where $R(t_{0})=R_{0}$. Before the turnaround, both $R(t),\gamma(t)$ are
monotonic functions of time, so we can smoothly connect the cubic and
quadratic regions for $\gamma(t)$. Numerical solutions to eq. (40) are plotted
in Fig. 3. The corresponding boost factor $\gamma$ is plotted in Fig. 4.

Figure 3: Trajectory of the shell $R(t)$ (solution to eq. (40)) for fixed
bubble tension and variable initial boost factor $\gamma_{0}$. For
$\gamma_{0}\sim 1$, the trajectory is no longer predominantly nearly null, and
the turnaround is less abrupt. The units of the spatial axis have been re-
scaled by $\hbar/Mc$ while the units of the time axis have been re-scaled by
$\hbar/Mc^{2}$. Figure 4: $\gamma$ boost factor
$\gamma=(1-\dot{R}^{2})^{-1/2}$ of the shell for fixed bubble tension and
variable initial boost factor $\gamma_{0}$. The early and late time behavior
are approximated in eq. (42) and (45).

In the limit of zero quark mass, the period of a single orbit of the bubble is
given by $T=2R_{0}$. For a test body which begins at rest at position $r$, the
time spent inside of the bubble interior in the zero quark mass limit is given
by $\tau=2(R_{0}-r)$. For the case $0\neq m<<M$, the period is given by

$T=2\int_{0}^{R_{0}}\frac{dR}{|\dot{R}|}$ (46)

where $\frac{4}{3}\pi\rho R_{0}^{3}=M-m$. From eq. (41) we find that

$\dot{R}=\pm\left(1-\frac{m^{2}}{(M-(4/3)\pi\rho R^{3})^{2}}\right)^{1/2}$
(47)

Plugging this into eq. (46), making a variable substitution, and substituting
$\gamma_{0}=M/m$ we get

$T=2R_{0}\int_{0}^{1}\left(1-\frac{1}{\gamma_{0}^{2}(1-(1-\gamma_{0}^{-1})x^{3})^{2}}\right)^{-1/2}dx$
(48)

Taylor expanding for $\gamma_{0}>>1$ the orbit period is approximately

$T\approx
2R_{0}\left(1+\frac{1}{3\gamma_{0}}+\frac{1}{27\gamma_{0}^{2}}(\sqrt{3}\pi+9{\rm
ln}3+6{\rm ln}\gamma_{0})\right)$ (49)

The time spent inside of the bubble by a test body to leading order in $m/M$
is given by

$\tau\approx 2(R_{0}-r)\left(1+\frac{m}{3M}\right)$ (50)

Meanwhile, the radial acceleration experienced by a test body scales as

$\dot{v}=\frac{2M}{R_{0}^{3}}\left(1-\frac{m}{M}\right)$ (51)

Therefore the accumulated residual velocity after a single orbit is given by

$\delta v\approx\frac{4M}{R_{0}^{3}}r(R_{0}-r)\left(1-\frac{2m}{3M}\right)$
(52)

Since the true quark masses would yield $\gamma_{0}\sim 10$, there will be a
small but finite correction to the estimate for the radius of the bubble
needed to produce the observed cosmic acceleration computed in section IV B.

##  IV Cosmic acceleration

###  IV.1 Gravity of virtual fluctuations

In our model of the gravitational effect of the vacuum, we will assume that
virtual point particles have no gravitational mass. The whole gravitational
effect of virtual QCD fluctuations lies within bubbles, and is dominated by
gluons. Since a virtual fluctuation has zero mean energy and “borrows” energy
only causally, the only gravitational effect is internal to the causal diamond
occupied by the gluonic field fluctuationBrodsky and Shrock (2011). Thus,
nonlocal gluonic fluctuations produce a residual velocity between particles
approximated by the classical bubble solution, given by Eq. (9) for each
fluctuation on scale $R_{0}$.

Consider gravitational repulsion from a space-filling vacuum of virtual pion-
like bubbles as a model for how the physical cosmological constant is produced
by QCD vacuum fluctuations. A small secular acceleration comes from the
accumulation of small mutually repulsive velocity kicks within each orbit. The
characteristic acceleration time is

$T_{\Lambda}=R_{0}/\delta v_{g}\sim cT_{0}^{2}/R_{0},$ (53)

that is, it is larger than the gravitational timescale $T_{0}$ by a factor
$cT_{0}/R_{0}$. In Planck units, $T_{0}^{2}\sim M^{-4}$ and $R_{0}\sim
M^{-1}$, which leads to a “cosmic” acceleration rate

$T_{\Lambda}^{-1}\sim M^{3}.$ (54)

This approximately agrees with the observed value of $\Lambda$. As explained
in more detail below, it is much smaller than the value that would correspond
to a universe filled with gluonic plasma of density $\rho_{0}$, or with
thermal or quantum field excitations on the same scale, which is the standard
estimate as in Eq. (1),

$T_{0}^{-1}\sim M^{2}.$ (55)

The difference between the residual effect of a fluctuation, and the residual
effect of a volume uniformly filled with the same material, arises because the
gravitational acceleration from fluctuations comes only from the mass of
material on the bubble scale $R_{0}$, instead of a volume with a gravitational
radius $\sim cT_{0}$.

In this microscopic physical picture of how cosmic acceleration works, test
particle trajectories, which are shown in Fig. (2) in flat-space coordinates,
correspond to geodesics of the emergent, slightly curved cosmological metric.

###  IV.2 Cosmic acceleration from virtual bubbles

The bubble model allows a more precise comparison of virtual bubble parameters
with measured cosmic acceleration. Fits to cosmological dataTanabashi  _et
al._ (2018); Prat  _et al._ (2022) yield an estimated value $\Lambda_{0}$ that
corresponds to acceleration on a cosmic scale with a rate

$T_{\Lambda}^{-1}\equiv\sqrt{\frac{\Lambda}{3}}=1.0\times 10^{-61}\
t_{P}^{-1}\ \sqrt{\frac{\Lambda}{\Lambda_{0}}}.$ (56)

(A pure-vacuum cosmology would have a Hubble radius and event horizon radius
$c/H_{\Lambda}=cT_{\Lambda}.$) We now evaluate the bubble parameters for which
this mean cosmic acceleration matches the mean repulsive acceleration of test
particles in the bubble model.

For virtual fluctuation states coherent on causal diamonds, the physical
picture is that a test particle inside any virtual bubble accelerates away
from the entire universe on the opposite side of the bubble’s center at the
mean rate given by the bubble model. The mean acceleration is given by the
mean repulsive velocity impulse over a bubble orbit (Eq. 8), divided by the
duration of the orbit $2R_{0}/c$, which we equate with cosmic acceleration:

$T_{\Lambda}^{-1}=(1/2)\langle\delta v_{g}/R_{0}\rangle_{B},$ (57)

where $\langle\rangle_{B}$ denotes a volume average over the world lines that
pass through the bubble. Since the impulse accounts for a whole orbit, an
average that gives equal weight to each element of the bubble 3-volume also
accounts for the time average of the fluctuating acceleration:

$\langle\delta v_{g}(r)\rangle_{B}=\frac{\int_{0}^{R_{0}}drr^{2}\delta
v_{g}(r)}{\int_{0}^{R_{0}}drr^{2}}$ (58)

This weighting yields

$T_{\Lambda}^{-1}=3R_{0}/20cT_{0}^{2},$ (59)

so writing the result in Planck units,

$T_{0}^{-2}\equiv 8\pi G\rho_{0}/3=2M^{4}(R_{0}M)^{-3},$ (60)

we obtain

$T_{\Lambda}^{-1}=(3/10)M^{3}(R_{0}M)^{-2}.$ (61)

Apart from the numerical coefficient, with $R_{0}M\sim 1$ this is the same
result as the simple estimate in Eq. (54).

Combining these results, the predicted cosmological constant from virtual
gluonic bubbles in Planck units is:

$\Lambda_{bubble}=3\mathcal{H}^{2}M^{6},$ (62)

where

${\mathcal{H}}\equiv(3/10)(R_{0}M)^{-2}.$ (63)

For bubbles with the physical pion mass ($M=m_{\pi_{0}}=135{\rm MeV}$), we
find

$\frac{\Lambda_{bubble}(M=m_{\pi})}{\Lambda_{0}}=\left(\frac{R_{0}m_{\pi}c}{2.0\
\hbar}\right)^{-4}.$ (64)

That is, for pion-mass fluctuations to give the right cosmological constant,
the one parameter in this simple model— the size of a bubble in units of the
de Broglie wavelength for its mass— needs to be

$R_{0}=2.0\ \hbar/m_{\pi}c,$ (65)

for their mean gravity to produce the observed cosmic acceleration. This
idealized model shows quantitatively how a cosmological constant close to the
observed value results from the gravitational effect of vacuum fluctuations in
QCD fields, as long as the quantum states of the fields and their gravity are
coherent on causal diamonds.

###  IV.3 Improvements on the idealized bubble model

The comparison of cosmological and microscopic measurements in Eqs. (64),(65)
is precise, but it is not accurate: it is based on a highly idealized model
system and is not expected to produce exact agreement with the physical
cosmological constant. In the real QCD vacuum, coherent gluonic wave states
have a more complex 4D structure than the bubble model. The quantum wave
function of virtual gluonic matter is not a homogeneous sphere, that of quarks
is not a thin shell, actual pion states are not radially homogeneous, and
virtual stress is not isotropic as in the bubble. A more stringlike gluon
state, which has less repulsive gravity, would require a smaller value of
$R_{0}$ to match the observed cosmic acceleration. The estimate just given
also does not allow for finite quark mass, $m\neq 0$, but since physical quark
masses have $m<<m_{\pi}$, this difference produces only a small fractional
change, as shown in the solution above (Eq. 52).

There are also ambiguities in our idealized application of the correspondence
principle to virtual orbits, which depend on how quantum gravity actually
works in detail. For example, the volume average taken above (Eq. 58) uses the
mean acceleration of bodies relative to the center of the bubble over an
orbit, but it might be more accurate to include a directional projection of
the component of radial acceleration onto the opposite hemisphere of the
causal diamond. Such a projection factor would change the answer by a small
numerical factor. In principle, nonlocally-coherent gravitational effects of
vacuum QCD fluctuations could be better approximated with an explicit
calculation of nonlinear quantum field dynamics.

###  IV.4 Remarks

####  IV.4.1 Why QCD?

It is natural to ask, why QCD? What’s special about its vacuum, compared to
the other fields, that make it source the cosmological constant?

In a coherent relational model of locality, there are straightforward physical
reasons why the gravitational effect of vacuum fluctuations for most standard
model fields should vanish. The classical gravity of a null particle with
momentum $p$ is simple: it creates a null shock with a displacement
$\delta\tau=Gp/c^{4}$, with an observable portion that depends on the location
of the particle relative to observerMackewicz and Hogan (2022). A zero-point
field vacuum excitation, generated by a creation operator on an infinite plane
wave mode, creates a completely delocalized state, so according to the
correspondence principle, there is no observable gravitational effect:
essentially, everything “moves together”.

This argument applies to the Standard Model fields whose interactions and
correlations fall off in the infrared. As noted above, the vacuum fluctuations
of gluons are uniquely different from those of other forces. The “IR slavery”
of the QCD vacuum confines baryons into bags, and leads to a finite range at
the Fermi scale for strong Yukawa interactions mediated by pions. The same
effect makes QCD vacuum gravity different from that of the other forces: the
vacuum fluctuations of QCD correspond to coherent localized bubbles of energy
flow on the Fermi scale, so the argument just given for delocalized vacuum
states does not apply. Renormalization required for quantum field theory fails
to account correctly for gravitational entanglement of causal structure with
long wavelength modesHollands and Wald (2004); Stamp (2015), so it is
plausible that the IR slavery of vacuum QCD fluctuations leads to different
gravitational effects from other fields.

Outside of hadrons, the QCD vacuum at low temperature is a coherent
condensate, whose mean gravitating density is negligibleBrodsky and Shrock
(2011). Its fluctuations resemble the lightest resonant excitations, pions,
which are spatially extended but localized. Nearly all of their virtual energy
comes from the massless gluon field. In the gluonic bubble model, fluctuations
in the tensile gluon interaction energy produce a small but cosmologically
detectable repulsive gravity. Estimates of the effective equation of state
from field theorySchutzhold (2002); Bjorken (2003, 2010); Klinkhamer and
Volovik (2009); Poplawski (1990) reproduce the estimate from our bubble model.

The other non-Abelian forces of the Standard Model, the weak interactions, are
mediated by massive particles with a short range, and the nonlocal space-time
correlations of their vacuum fluctuations are qualitatively different from
QCD. For these, the “zero momentum mode” of fluctuations takes the form of a
globally spatially uniform scalar condensate with homogeneous fluctuations
around the minimum of an effective potential. This Higgs condensate, whose
order parameter describes the low-temperature vacuum expectation value of the
effective potential, apparently has zero gravitationWeinberg (1989). As
explained in the Appendix, the difference in gravitational effect from the
space-filling QCD vacuum can be understood from an exponential suppression of
the trace anomaly at weaker coupling strengthSchutzhold (2002).

####  IV.4.2 Causal coherence of virtual fluctuations

The bubble model illustrates classically how a fluctuation could have a
durable macroscopic physical effect if positional relationships among world
lines are determined by coherent causal diamonds. In such an emergent
relational holographic picture, classical locality emerges as a consistent
approximation on large scales, based on relationships of a causal diamond with
those it is nested in. Exact relational positions within causal diamonds are
indeterminate.

In our model, systematic secular effects of fluctuations are assumed to lead
to a durable effect on the classical metric. This hypothesis leads to the
assumption used in our estimate of mean acceleration of test particles,
relative to the center of the bubble. On average, the acceleration applies to
test particles in relation to the future light cones on the opposite side of a
pion-like causal diamond vacuum fluctuation. The coherence propagates local
coherent acceleration to the future light cone of a microscopic causal
diamond, which leads to coherent acceleration of the same magnitude on a
cosmic scale.

The bubble model illustrates concretely how causal coherence of virtual
fluctuation states is connected with the small nonzero value of the
cosmological constant. According to this scenario, the wildly wrong estimate
of vacuum fluctuation density in Eq. (1) results from an incorrect physical
interpretion of vacuum energy that does not take directional causal coherence
of virtual states into account; it arises from the incorrect model of locality
built into a particular interpretation of field theory.

Careful studies of entanglement and decoherence in virtual field fluctuations
confirm the need to account for causal consistency to avoid apparent paradoxes
with nonrelativistic quantum thought-experimentsDanielson  _et al._ (2022);
Belenchia  _et al._ (2019). The decomposition into modes that are used to
construct a vacuum state— the differentiation between radiation and vacuum—
depends on the choice of Cauchy surfaces used to describe the system. Apparent
paradoxes are resolved when field states are measured on Cauchy surfaces that
correspond to correlated measurements.

Similar causal coherence of primordial virtual fluctuations has recently been
used in a model that explains some observed anomalies of cosmological
anisotropy at large angular separation Hogan and Meyer (2022); Hogan  _et al._
(2023). In that context, directional hemispherical coherence leads to a causal
“shadow” in primordial virtual correlation, which is observed as a symmetry of
temperature correlations at large angles.

####  IV.4.3 Why now?

Typically, field-based models of cosmic acceleration require introduction of
new fields with new, arbitrary and very small dimensionless parameters, in
some cases accompanied by an anthropic explanationWeinberg (1989). In the
gluonic-bubble scenario, $\Lambda$ is not an independent parameter, but should
have a precisely calculable value from Standard Model field fluctuations and
standard semiclassical gravity.

In principle, this scenario roughly accounts for the well-known puzzle
sometimes nicknamed the “why now” coincidence— the fact that the timescale
associated with fundamental cosmic acceleration coincides with the current age
of the universe, which in turn presumably is determined by astrophysical
timescales, such as those determined by stellar evolution. The very long
evolution timescale of stars and other astrophysical systems in Planck units
originates mainly from the cube of the nucleon massHogan (2000):

$T_{astro}/t_{P}\sim(m_{P}/m_{proton})^{3}.$ (66)

(Additional dimensionless factors that are numerically less significant, such
as the electromagnetic coupling and electron/nucleon mass ratio, depend on the
specific astrophysical system.) The exponentially large dimensionless number
$m_{P}/m_{proton}$, which expresses the weakness of gravity on a nuclear
scale, appeared mysterious to Planck, Eddington and Dirac, but now has a
natural interpretation in the context of modern unified field theory, because
of the logarithmic running of the QCD coupling constant with energy
scaleWilczek (1999). In any case, because nucleon masses are determined by the
same scale that fixes masses of pions and QCD vacuum fluctuation bubbles, the
astrophysical timescale $T_{astro}$ naturally (roughly) coincides with the
bubble model for $T_{\Lambda}$, since they originate from the same large
dimensionless number.

##  V Conclusion

The gluonic bubble model demonstrates the classical gravitational coupling of
a single pion-like oscillation with geometry. Our proposal for the
cosmological constant is that fluctuations in the QCD vacuum have a similar
relationship with gravity. Delocalized zero-point fluctuations of field vacua
contribute nothing to the mean density that couples to gravity, but locally
coherent fluctuations have a net repulsive effect that mimics a uniform
cosmological constant. This effect occurs for the strong interactions in
particular because of gluonic tension, represented in our toy model by highly
tensile gluonic gas. Our model shows how this works geometrically in classical
systems, and why a nonlocally coherent 4D structure is needed to obtain a net
repulsive gravitational effect.

Although the bubble model adopted here is a simplified idealization of real
QCD vacuum states, the essential elements that create the cosmological
constant of the magnitude estimated here— nonlocal directional causal
coherence of vacuum states, and a tension from the strong nonabelian self-
interactions of gluon fields— must also appear in the states of the physical
QCD vacuum. In this scenario, the absolute value of the physical cosmological
constant can in principle be calculated exactly from a nonlinear computation
of spacelike correlations of 4D mass-energy flows in the virtual QCD vacuum.
Such a calculation would allow more precise tests than the approximate
agreement obtained here with a highly idealized picture.

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##  VI Appendix

###  VI.1 Gravity of a gluonic string model

####  VI.1.1 Nonrotating string model

We now consider the gravitational effect if the gluonic material does not
uniformly fill a causal diamond as it does in the bubble model. Consider an
extended particle of total mass $M$ represented by two “quarks” of mass $m<<M$
connected by a one dimensional string with mass per length
$\mu=\mu_{0}M^{2}(c/\hbar)$, where $\mu_{0}$ is a dimensionless parameter that
characterizes the gluonic tension. The system has zero angular momentum, so
the quarks travel on radial trajectories connected by the straight string. It
starts with zero length, stretches to length $L_{M}=\hbar/\mu_{0}Mc$, then re-
contracts under the string tension.

For a classical model of strongly interacting gluonic quantum field
excitations like pions, we can take the parameter $\mu_{0}$ to be less than
but of the order unity. Values $\mu_{0}<<1$ correspond to longer strings,
which break apart into many shorter strings as new quark pairs are created.
Values $\mu_{0}>1$ describe systems smaller than the de Broglie wavelength, so
they do not correspond to physical quantum states.

As in the bubble model, very light quarks of mass $m<<M$, with a negligible
fraction of the total mass, start with very high gamma-factor $\gamma=M/2m$
relative to the center, so the trajectories are nearly null. Their kinetic
energy is converted to string energy as they travel. Eventually $\gamma\approx
1$, and for a time short compared to $L_{M}/c$, they enter a subrelativistic
regime where the string tension turns them around. For pion-like systems with
$m<<M$, the short subrelativistic regime is a small fraction of the whole
trajectory, so the main effects do not depend strongly on the value of $m$.

Figure 5: Causal diagram of gravitational radiation from a single gluonic
string orbit. For $M>>m$, displacement is concentrated near anisotropic,
spherical null shocks.

####  VI.1.2 Gravitational waves

As shown in more detail in the linear solution below, a gluonic string
produces gravitational waves (Fig. 5). A simple estimate of the radiation rate
from dimensional arguments shows how the rate of energy loss scales with $M$.
In one orbit, a fully relativistic string with deficit angle $\delta\theta$
radiates a fraction

$\delta\theta\sim\delta M/M\sim\mu G/c^{2}\sim\mu\hbar/cm_{P}^{2}$ (67)

of its energy as gravitational radiation. For a gluonic string of length
$L_{M}=\hbar/\mu_{0}Mc$ with $m<<M$, the decay rate by gravitational radiation
in Planck units is about

$t_{P}/\tau_{d}\sim\mu_{0}^{2}(M/m_{p})^{3}.$ (68)

The string model displays roughly the same energy flow between QCD
fluctuations and the gravitational vacuum as the bubble model, but it does not
produce the same mean classical repulsion as gravity from bubble-like
fluctuations.

####  VI.1.3 Linear string solution for $m\geq 0$

#### i. Equations of motion

For all computations in this section we will work in units where $c=1,G=1$ and
restore units when needed. The energy-momentum tensor

$T^{ab}=T^{ab}_{s}+T^{ab}_{p}$ (69)

consists of two parts: that of particles (representing quarks),

$T^{ab}_{p}=m\delta(x)\delta(y)(\delta(z-z_{1}(t))v^{a}_{1}u^{b}_{1}+1\leftrightarrow
2),$ (70)

whose four-velocity is

$u_{a}=\gamma v_{a}=\gamma_{1,2}(1,0,0,\dot{z}_{1,2}),$ (71)

and that of a string (representing gluons),

$T^{ab}_{s}=\mu\delta(x)\delta(y)\mathcal{T}^{ab}\Theta(z-z_{2}(t))\Theta(z_{1}(t)-z),$
(72)

where $\mu$ is the constant mass per unit length of the string,
$\mathcal{T}^{ab}\equiv t^{a}t^{b}-z^{a}z^{b}$, and $\Theta(x)$ is the
heaviside step function which is $1$ for $x>0$ and vanishes elsewhere.

The equations of motion are similar for the two models, the string and the
bubble. They are not identical: in the bubble, the quark surface mass density
decreases with radius, so the inward acceleration at the edge for constant $p$
increases with radius. The difference between solutions is small for $m<<M$,
since the turnaround happens within a small fractional change in radius.

In the string model, conservation of total stress-energy $\nabla_{a}T^{ab}=0$
gives an equation of motion for the positions of the quarks $z_{1,2}(t)$,

$\frac{d}{dt}(\gamma_{1,2}\dot{z}_{1,2})=\mp\frac{\mu}{m}\equiv\mp\alpha.$
(73)

The solution for a uniformly accelerated relativistic particle with constant
(proper) acceleration $\alpha$, initial 3-velocity $v_{0}$, and initial
position $z_{0}=0$ is given by

$z_{1,2}(t)=\pm\frac{1}{\alpha}[\gamma_{0}-(1+(\gamma_{0}v_{0}-\alpha
t)^{2})^{1/2}].$ (74)

The acceleration parameter is given by $\alpha=\mu/m$. Note that the
turnaround time (or equivalently, the duration of time for which the particle
is non-relativistic) is approximately $\delta t\sim 1/\alpha\sim
L/\gamma_{0}$.

In the string model, we will see later that the peaks in the “radiative” part
of the curvature occur at the retarded time associated with the turnaround of
each particle. The period of the trajectory is $T=2\gamma_{0}v_{0}/\alpha$.
Therefore the fraction of time during which the trajectory is non-ultra-
relativistic is $\delta t/T\sim 1/\gamma_{0}v_{0}\sim 1/\gamma_{0}<<1$.
However, we will see later that the relative acceleration experienced by
nearby test bodies scales like $\dot{v}\sim\alpha^{2}$, so that $\delta
v\sim\dot{v}\delta t\sim\alpha$.

#### ii. Gravitational effect

To determine the gravitational effect of such a system, we begin with the
linearized Einstein equations in the Lorenz gauge $\nabla^{a}\bar{h}_{ab}=0$,
where $\bar{h}_{ab}$ is the trace reversed metric perturbation.

$\nabla_{c}\nabla^{c}\bar{h}_{ab}=-16\pi T_{ab}$ (75)

The solution to this equation is found by integrating over the intersection of
the source world sheet with the past light cone.

$\bar{h}_{ab}=4\int_{\Lambda}\frac{T_{ab}(t^{\prime},\vec{x}^{\prime})}{|\vec{x}-\vec{x}^{\prime}|}d^{3}x^{\prime}$
(76)

where $\Lambda$ denotes the past light cone of the event $(t,\vec{x})$. Since
the delta function depends both explicitly on $\vec{x}^{\prime}$ and
implicitly through $t_{\rm ret}$, we need to use the Jacobian of the
coordinate transformation to evaluate the integrals of the delta functions.

The metric perturbation associated with the low mass particles is

$\bar{h}^{p}_{ab}=\frac{4m\gamma_{1}v^{1}_{a}v^{1}_{b}}{\alpha_{1}|\vec{x}-\vec{X}_{1}(t_{\rm
ret})|}+\frac{4m\gamma_{2}v^{2}_{a}v^{2}_{b}}{\alpha_{2}|\vec{x}-\vec{X}_{2}(t_{\rm
ret})|}$ (77)

where

$\alpha_{1,2}=1-\hat{n}_{1,2}\cdotp\frac{d\vec{X}_{1,2}}{dt}(t_{\rm ret})$
(78) $\hat{n}_{1,2}=\frac{\vec{x}-\vec{X}_{1,2}(t_{\rm
ret})}{|\vec{x}-\vec{X}_{1,2}(t_{\rm ret})|}$ (79)

From this point forward, evaluation at retarded time will be understood unless
explicitly stated otherwise. The metric perturbation associated with the
string is

$\bar{h}^{s}_{ab}=4\mu{\rm
ln}\left(\frac{z-z_{2}+\sqrt{s^{2}+(z-z_{2})^{2}}}{z-z_{1}+\sqrt{s^{2}+(z-z_{1})^{2}}}\right)\mathcal{T}_{ab}$
(80)

where $s^{2}=x^{2}+y^{2}$ denotes the transverse distance from the string. We
note here that while in the case of the infinite string the spacetime is flat
with an angular deficit, the dynamic string does in fact produce curvature. We
are most interested in the leading order in $1/r$ behavior of the metric
perturbation. In this limit, $\hat{n}\rightarrow\hat{x}$ and $t_{\rm
ret}\approx t-r+{\rm cos}\theta z_{1,2}(t_{\rm ret})$.

$\bar{h}^{s}_{ab}=\frac{4\mu}{r}(z_{1}-z_{2})\mathcal{T}_{ab}+\mathcal{O}\left(\frac{1}{r^{2}}\right)$
(81)
$\bar{h}^{p}_{ab}=\frac{4m}{r}\left(\frac{\gamma_{1}v^{1}_{a}v^{1}_{b}}{1-{\rm
cos}\theta\dot{z}_{1}(t_{\rm ret})}+1\leftrightarrow
2\right)+\mathcal{O}\left(\frac{1}{r^{2}}\right)$ (82)

In the far field limit, the retarded time can be solved explicitly as a
function of $u=t-r$ and $\theta$.

$t^{1,2}_{\rm ret}=\frac{1}{\alpha{\rm
sin}^{2}\theta}\left(\vphantom{\sqrt{u^{2}}}u\alpha\pm\gamma_{0}{\rm
cos}\theta\right.\\\ \left.\mp\left|{\rm cos}\theta\right|\sqrt{({\rm
sin}^{2}\theta+(u\alpha\pm\gamma_{0}{\rm cos}\theta)^{2})}\right)$ (83)

The linearized Riemann tensor is given in terms of the metric perturbation by

$R_{abcd}=2\nabla_{[a}\nabla_{|[d}h_{c]|b]}$ (84)

The geodesic deviation equation tells us how nearby test bodies move relative
to one another for small separations.

$t^{c}\nabla_{c}(t^{d}\nabla_{d}D^{a})=-R^{a}_{\;cbd}t^{c}t^{d}D^{b}$ (85)

Since we are primarily interested in the far field behavior of the curvature,
we need only consider derivatives which act on the time dependent terms in eq.
(81) and (82). One can show that $\partial t_{\rm ret}/\partial
t=1/\alpha_{1,2}$ and $\nabla t_{\rm ret}=-\hat{n}/\alpha_{1,2}$. In other
words, $\nabla_{a}f(t_{\rm ret})=-K_{a}\dot{f}(t_{\rm ret})/\alpha_{1,2}$,
where $K_{a}=-\nabla_{a}u=t_{a}+r_{a}$ is a radially out-going null vector. In
this limit, the linearized Riemann tensor simplifies to

$R_{abcd}\approx 2K_{[a}K_{|[d}\ddot{h}_{c]|b]}$ (86)

The relevant components of the Riemann tensor for eq. (85) are plotted in Fig.
6 as a function of the retarded time coordinate $u=t-r$. Note the sharp spikes
in curvature, which occur at the turnaround points of the quarks as viewed by
a stationary observer.

Figure 6: Components of the Riemann curvature tensor from eq. (85) plotted as
a function of retarded time $u=t-r$ for
$\gamma_{0}=10,\alpha=10,\theta=\pi/4$. The curvature spikes at the retarded
times associated with the turnaround of each quark. Due to the highly
relativistic motion, the turnaround is no longer simultaneous as seen by an
observer away from the equator ($\theta=\pi/2$).

The effective stress-energy tensor for the gravitational waves is given by

$T^{\rm
GW}_{ab}=\frac{1}{32\pi}\left\langle\bar{h}_{cd},_{a}\bar{h}^{cd},_{b}-\frac{1}{2}\bar{h},_{a}\bar{h},_{b}\right.\\\
-\left.\vphantom{\frac{1}{2}}\bar{h}^{cd},_{d}\bar{h}_{cb},_{a}-\bar{h}^{cd},_{d}\bar{h}_{ca},_{b}\right\rangle$
(87)

where $\langle...\rangle$ denotes an average over several wavelengths of the
radiation. Although this object is not gauge invariant (and therefore its
physical interpretation is unclear), the total integrated flux of energy to
null infinity as defined by

$P=-\lim_{r\rightarrow\infty}\int T_{0a}dS^{a}$ (88)

is gauge invariant and is therefore a physically meaningful quantity. It turns
out that the dynamics of the string do not contribute directly to the stress
energy of the gravitational waves in this setup, although the string does
indirectly contribute by determining the dynamics of the quarks. Restoring
units, the total integrated power in the string model scales like

$P=f(\gamma_{0})\alpha^{2}m^{2}Gc^{-1}$ (89)

where $f(\gamma_{0})$ is plotted in Fig. 7 on a log-linear scale. We see that
the asymptotic scaling of $P$ is logarithmic in $\gamma_{0}$, hence confirming
the previous assumption that the power does not strongly depend on the mass of
the quarks in the $m<<M$ limit. For a pion mass of $135$ MeV and quark masses
of about $3-5$ MeV, we get $\gamma_{0}=M/2m\sim 10^{1}$ and $f(\gamma_{0})\sim
10^{1}$. In agreement with the estimate given by eq. (68), the timescale of
decay is

$\tau_{d}^{-1}=\frac{P}{M}\sim 10^{1}\mu_{0}^{2}(M/m_{p})^{3}\tau_{p}^{-1}$
(90)

Figure 7: Plot of $f(\gamma_{0})$ in eq. (89) on a log-linear scale,
demonstrating an asymptotically logarithmic scaling of the power.

###  VI.2 Strong energy condition, quantum trace anomaly, and dimensional
dependence

We now explore how the particular nature of the QCD interactions can allow for
a repulsive gravitational effect. Consider a congruence of timelike geodesics
described by a vector field $u_{a}$. The expansion, shear, and twist of the
congruence are defined by

$\displaystyle\theta$ $\displaystyle=q_{ab}\nabla^{a}u^{b}$ (91)
$\displaystyle\sigma_{ab}$
$\displaystyle=q_{a}^{\,c}q_{b}^{d\,}\nabla_{(c}u_{d)}-\frac{1}{3}\theta
q_{ab}$ (92) $\displaystyle\omega_{ab}$
$\displaystyle=q_{a}^{\,c}q_{b}^{\,d}\nabla_{[c}u_{d]}$ (93)

where $q_{ab}$ denotes the spatial part of the metric that is orthogonal to
$u_{a}$, and $(a,b),[a,b]$ denote symmeterization and antisymmeterization of
indices, respectively. Raychaudhuri’s equation tells us how the expansion
evolves with time.

$\frac{d\theta}{d\tau}=-\frac{1}{3}\theta^{2}-\sigma_{ab}\sigma^{ab}+\omega_{ab}\omega^{ab}-R_{ab}u^{a}u_{b}$
(94)

Here $R_{ab}$ denotes the Ricci tensor. As is typically done, we will ignore
the twist. For a congruence which initially has zero expansion and shear, we
have (after using the Einstein field equations to replace $R_{ab}$ with
$T_{ab}$)

$\left.\frac{d\theta}{d\tau}\right|_{t=0}=-8\pi\left(T_{ab}-\frac{1}{2}g_{ab}T\right)u^{a}u^{b}$
(95)

where $T=g^{ab}T_{ab}$, which evaluates to $T=-T_{00}+T_{11}+T_{22}+T_{33}$ in
global inertial coordinates. For a perfect fluid in 3 spatial dimensions (in
units where $c=1$),

$T_{ab}={\rm diag}(\rho,p,p,p),$ (96)

so eq. (95) becomes

$\left.\frac{d\theta}{d\tau}\right|_{t=0}=-4\pi(\rho+3p)$ (97)

Therefore if the perfect fluid obeys an equation of state $p<-\rho/3$ (such
that the strong energy condition $R_{ab}u^{a}u^{b}>0$ is violated), the
expansion of the congruence will initially be positive, leading to a repulsive
gravitational effect. As the expansion and shear grow, the RHS of eq. (94)
will eventually reach zero, but the sign of the rate of change of expansion
can never become negative.

Now consider another fundamental constraint: a perfect fluid composed purely
of a classical non-abelian gauge field (e.g., massless noninteracting point
particles, such as photons), the equation of state is $p=\rho/3$, which does
not violate the strong energy condition. The form of the field strength tensor
in classical Yang Mills theory is such that the trace of the stress-energy
tensor always vanishes, similar to classical Maxwell theory. However, quantum
interactions between the gluons and quarks and the self-interaction of gluons
lead to what is known as a trace anomaly, i.e. a non-vanishing of the trace of
the quantum stress energy tensor.

In order for the strong energy condition to be violated by quantum
interactions, Eq. (91) tells us that the equation of state must satisfy
$\rho+T/2<0$, or $T<-2\rho$. For a perfect fluid, we have $T=-\rho+3p$. This
puts a constraint on the magnitude of the trace anomaly if QCD vacuum
fluctuations are to serve as the source of accelerating cosmic expansion. It
has been hypothesized Schutzhold (2002) that this trace anomaly can produce
the violation of the strong energy condition needed to produce a repulsive
gravitational effect.

The vanishing of the trace of the stress-energy tensor is synonymous with
conformal invariance of the action of a field theory, i.e. $S=\int
d^{4}x\sqrt{-g}\mathcal{L}$ is invariant under
$g_{\mu\nu}\rightarrow\Omega(x^{\alpha})g_{\mu\nu}$. In ref. Schutzhold (2002)
it is shown that this conformal invariance is preserved under the standard QFT
renormalization procedures for free fields in flat space but not for self-
interacting fields in flat space or free fields in curved spacetimes. It is
estimated there that the self-interacting gluonic field gives rise to a
negative energy density and pressure proportional to the renormalization group
$\beta$ function, with a corresponding cosmological constant that is many
orders of magnitude too large.

According to our interpretation, where the gravitational effect of virtual
fluctuations is evaluated using a causally coherent bubble model, the
magnitude is about right. This result is consistent with the idea that active
gravity entangles long wavelength field modes with causal structure, which is
not accounted for in standard renormalization proceduresHollands and Wald
(2004); Stamp (2015).

Although the sharp boundary and spherical symmetry of the bubble model are
artificial, we conjecture that the physical gravitational effect is indeed
microscopic, localized at the Fermi scale. Unlike the model sketched in ref.
Schutzhold (2002), the cosmic acceleration is a true cosmological constant,
with a value determined entirely by the properties of the stable QCD ground
state vacuum. These arguments suggest that at very high temperatures in the
early universe, when QCD had significantly weaker interactions, the value was
much smaller, but there are no observable consequences of this variation.

Ref. Schutzhold (2002) also shows that free fields in curved space gives a
positive energy density with a cosmological constant that is many orders of
magnitude too small, which supports our conjecture that fluctuations of the
other Standard Model fields do not contribute significantly to the
gravitational energy of the vacuum. The effects of the anomaly are
nonperturbative, and are exponentially suppressed for coupling constants that
are not of order unity.

Finally, we address why the bubble model of the pion behaves differently from
the string model for producing a repulsive gravity. Although the string model
of the pion is commonly used as a toy model in other contexts, it does not
work as a gravitational model because of the dimensionality of the matter
distribution. In order to violate the strong energy condition in $D$ spatial
dimensions, the fluid must satisfy

$T_{00}+\frac{1}{2}T=\rho+\frac{1}{2}(-\rho+Dp)<0\rightarrow\rho+Dp<0$ (98)

Therefore, a pressure $p=-\rho$ in $D=1$ spatial dimensions (i.e. a straight
string) cannot violate the strong energy condition: more than one dimension of
tension is needed to account for cosmic acceleration.