#### Knotty inflation and the dimensionality of spacetime

Eur. Phys. J. C
Knotty inflation and the dimensionality of spacetime
Arjun Berera 2
Roman V. Buniy 1
Thomas W. Kephart 0
Heinrich Päs 4
João G. Rosa 3
0 Department of Physics and Astronomy, Vanderbilt University , Nashville, TN 37235 , USA
1 Schmid College of Science, Chapman University , Orange, CA 92866 , USA
2 Tait Institute, School of Physics and Astronomy, University of Edinburgh , Edinburgh EH9 3FD , UK
3 Departamento de Física da Universidade de Aveiro and CIDMA , Campus de Santiago, 3810-183 Aveiro , Portugal
4 Fakultät für Physik, Technische Universität Dortmund , 44221 Dortmund , Germany
We suggest a structure for the vacuum comprised of a network of tightly knotted/linked flux tubes formed in a QCD-like cosmological phase transition and show that such a network can drive cosmological inflation. As the network can be topologically stable only in three space dimensions, this scenario provides a dynamical explanation for the existence of exactly three large spatial dimensions in our Universe.
1 Introduction
Although the question of why our Universe has exactly three
(large) spatial dimensions is one of the most profound
puzzles in cosmology – especially in view of quantum gravity
scenarios such as string theory which assume nine or ten
space dimensions at the fundamental level – it is actually
only occasionally addressed in the literature [
1–22
]. In this
paper we propose a topological explanation for the
dimensionality of space-time based on the idea that inflation is
driven by a tightly knotted network of flux tubes generated
in a cosmological phase transition and the fact that knots are
topologically stable only in exactly three space dimensions
(“knotty inflation”).
The main idea of the model is connected to the fact that, in
non-Abelian gauge theories analogous to QCD,
chromoelectric flux tends to become confined and the resulting tube-like
structures can be treated as effective one-dimensional objects
or strings. The formation of flux tubes is behind the basic
models of hadronization in QCD, with flux strings
connecting quark–antiquark pairs giving rise to a linear potential
that confines them into mesons and prevents the existence of
free colored charges. A similar description with more
complicated flux tube shapes also describes the formation of baryons
and glueballs, the latter denoting bound states of pure gauge
fields.
During the phase transition from a generic quark–gluon
plasma to a hadron gas, a large number of flux tubes may
fill up the whole Universe and form an intricate network of
both open and closed flux tubes, with some similarities to
what is known as the ‘spaghetti vacuum’ [23–27], potentially
with a large linking or crossing number density provided
the flux tube density is sufficiently large. A related idea has
been suggested recently in [28]. It has been shown that such
a network is stabilized by a topological conservation law
[29,30].
This network will tend to relax into a tight configuration
that fills the whole space such that each tube minimizes its
energy, similarly to the description of glueballs in terms of
knotted/linked flux tubes proposed in [31–33]. The network
will then decouple from the Hubble flow.
As a result, the energy stored in this tight network will
provide an effective cosmological constant which is
homogeneous on scales large compared to the typical flux tube
width and on average isotropic if the flux tubes are randomly
oriented. This effective cosmological constant will thus
trigger a period of inflation in the early Universe that lasts until
the network decays through reconnection, tube breaking or
other quantum effects. Network breaking will proceed until
a gas of hadrons and radiation is formed and the standard
cosmological evolution begins.
Of all possible dimensionalities of space, our mechanism
picks out three as the only number of dimensions that can
inflate and thus become large. String theory and various types
of Kaluza–Klein theories are explicit examples where
different spacetime dimensionalities are possible. For example,
in string theory, which is an intrinsically higher-dimensional
theory, gauge theories are confined to hypersurfaces of p
spatial dimensions known as D p-branes, to which open strings
are attached. This concept has been employed in a wide
range of extra-dimensional scenarios, where fields are
confined to lower-dimensional slices of a higher-dimensional
spacetime.
The main difference between the dynamics of confinement
in 3+1 and higher-dimensional gauge theories lies in the fact
that flux tube knots and links will only be topologically stable
(or metastable) in three spatial dimensions, being otherwise
able to unknot/unlink in the extra dimensions. Given that this
is an essential feature for the formation of a tight network of
flux that decouples from the Hubble flow and provides an
effective cosmological constant, confinement can only lead
to inflation for gauge theories living in three-dimensional
hyperspaces.
The argument can be used in two different scenarios: First,
in string theory, particles represented as open strings are
attached to n-dimensional hypersurfaces (“branes”). Only
particles confined to 3-branes will produce a topologically
stable network, and only these branes will inflate. (Note that
in such a case an additional mechanism to stabilize the extra
dimensions is nevertheless required, although this is beyond
the scope of the present work.) Alternatively, the entire
Universe could have n dimensions. In this case of all possible
Universes only Universes with three dimensions will be able
to inflate. Thus this scenario can explain why our Universe
has only three large spatial dimensions in the absence of other
sources of vacuum energy.
It is the purpose of this Letter to outline the main features
of this mechanism and discuss the necessary properties that
a flux tube network must exhibit in order to yield a
successful inflationary model in three dimensions. While leaving a
detailed modeling of the network’s evolution and
observational predictions of this scenario for a future work, we will
describe the most important physical processes that drive the
network’s formation, inflationary dynamics and subsequent
decay, giving quantitative estimates for the relevant time and
energy scales.
2 Flux tubes in gauge theories
Understanding confinement in QCD and gauge theories in
general is one of the most important problems in particle
physics. Due to the strong coupling and non-perturbative
nature of confinement, this is an intrinsically hard problem
that can currently only be studied accurately in the
context of numerical simulations on the lattice. It is
nevertheless widely accepted that the confinement of chromoelectric
charges, such as quarks and anti-quarks in QCD, is
associated with a squeezing of chromoelectric flux lines into
tubelike structures connecting the charges. These flux tubes then
(1)
(2)
(3)
give rise to a potential energy that grows linearly with the
distance between pairs of quarks and anti-quarks, therefore
leading to their confinement in mesons, baryons and
potentially other hadronic structures. Flux tubes can be described
e.g. in terms of Abrikosov–Nielsen–Olesen vortices [34,35]
in dual superconductor models of confinement (see [36] for
a review).
It was pointed out in [31–33] that closed flux tubes may
have non-trivial topologies, including knotted tubes and links
between distinct closed flux tubes. The degree of
“knottedness” of each configuration is then associated with a
conserved topological charge, analogous e.g. to the (Abelian)
magnetic helicity that characterizes conducting fluids in
magnetohydrodynamics [37,38]. Any such topologically
nontrivial flux tube configuration will necessarily relax into
an equilibrium state by minimizing its length and
therefore its energy. These equilibrium states correspond to the
tightest knotted or linked configurations with a given
topological charge and number of flux quanta. This thus
motivates us to consider the description employed in [31–33],
where the chromoelectric field F0i (where i = 1, 2, 3 in
three spatial dimensions) is confined in knotted/linked
tubelike structures that carry one quantum of chromoelectric
flux.
Just like in standard Big Bang cosmology, where it is
difficult to generate seed magnetic fields, here we also assume that
our QCD analog gauge theory does not generate magnetic
fields. We assume there are particles with analog
chromoelectric charges, but no analog chromomagnetic monopoles.
One could include this generalization, but it would unduly
complicate the analysis.
The relevant (static) Lagrangian density is given by
1
L = 2 tr F0i F 0i
+ trλ
E
(π a2) − ni F0i ,
up to the addition of a constant energy density as in the MIT
bag model [39,40], where the last term enforces flux
conservation across the tube sections through a Lagrange multiplier
λ, with ni denoting the unit vector normal to the tube’s cross
section. We expect the flux tube radius a to be given by the
confinement scale , such that a ∼ −1 in natural units,
which we will employ in our discussion henceforth. This
yields the equations of motion for the gauge field:
D0(F0i − λni ) = 0,
Di (F0i − λni ) = 0,
the solution of which is given by a constant chromoelectric
field
F0i = π aE2 ni ,
which vanishes outside the flux tube. The energy density
inside each flux tube is then given by
1 tr 2E 4
ρE = 2 (π a2)2 ∼ 2π 2 ,
up to O(1) factors that will not affect our discussion.
3 Network formation
Let us then hypothesize that there exists a confining gauge
theory, analogous to QCD but with a generic high-energy
confinement scale , and which lives on a three-dimensional
brane within a higher-dimensional compact space. For
simplicity, we will denote the particles charged under this gauge
group as quarks and gluons, although one must bear in mind
that these are not the known QCD fields but rather novel
degrees of freedom that must decouple from the low-energy
effective theory.
In the early Universe, for temperatures above a certain
critical value Tc ∼ , these quarks and gluons are
essentially free, forming a plasma that we will assume is close
to thermal equilibrium and dominates the energy balance in
the Universe. This plasma has an energy density ρR(T ) =
(π 2/30)g∗T 4, where g∗ denotes the number of relativistic
degrees of freedom, which includes at least the massless
gluons and the light Standard Model fields of the theory. Due to
expansion, the temperature of the plasma will decrease until it
becomes lower than the critical value, at which point all the
free gluons become confined in both open and closed flux
tubes, the former connecting quark–antiquark pairs. Given
that
ρR(Tc) ∼ g∗ 4 ∼ ρE,
we expect that, at the confining phase transition, a large
number of flux tubes is formed within each Hubble volume. Such
a large density will naturally lead to a large number of
crossings between the flux tubes and, hence, to a large number of
knots and links of different configurations. This is analogous
to the behavior observed when a string is tumbled inside a box
of fixed volume, where the probability of knotting increases
very quickly above a certain critical string length and which
effectively corresponds to a critical density [41]. The flux
tube network will then be endowed with a non-trivial
topology, which can be metastable in three spatial dimensions and
that, as we describe below, can be sufficiently long-lived to
drive a period of inflation in the early Universe.
(4)
(5)
4 Inflation
After the phase transition, Hubble expansion will tend to
stretch the flux tubes, while maintaining their radius (set by
the strong non-perturbative dynamics), and increase the size
of the gaps in between them. However, as argued above, the
presence of knots and links makes the network behave
differently from a system of isolated flux tubes, endowing it
with a rigidity and making it try to relax into a tight
equilibrium configuration. Flux tubes will then shrink and approach
each other in order to maximize the fraction of the
spatial volume that they can occupy given the topological
constraints.
The number of knots and links in the network will decrease
due to the decay processes that we describe below. These will
make the network progressively less tight, with the tightest
configuration occupying an increasingly smaller fraction of
the spatial volume.
We then expect a network to remain tight if
τtight
H −1
τdecay,
(6)
i.e. if it relaxes into a tight configuration more quickly than
expansion and knots/links are stable on the Hubble time
scale. If the network is sufficiently dense at the phase
transition, i.e. if the gaps between flux tubes are not much wider
than the tube radius, we expect τtight −1, and the
network can in principle relax into a tight configuration within
a Hubble time for H . In this case, the tube width
is also much smaller than the Hubble radius during
inflation, so that the network is essentially homogeneous on
nearhorizon and super-horizon scales. It should also be, on
average, isotropic if quarks and gluons are randomly distributed
in the thermal plasma at the transition. Since the background
inflationary dynamics is dictated by the near- and
superhorizon properties of the dominant fluid, we may consider
a homogeneous and isotropic expansion as a first
approximation. The underlying anisotropic structure of the knotted
flux tube network may nevertheless give rise to small
deviations from isotropy that could e.g. be analyzed within the
framework developed in [42]. One should also note that, as
long as the gaps in the network never attain super-Hubble
sizes, the process of tightening can occur at sub-luminal (and
even non-relativistic) flux tube speeds, being consistent with
causality.
Let us see that a tight network will behave as an
effective cosmological constant as long as the above time scale
hierarchy is satisfied. The “tightness” of the network can be
evaluated in terms of the fraction of the spatial volume
occupied by the flux tubes, f , which will in general depend on
the space and time coordinates. On average, we expect a
network that is homogeneous and isotropic on scales larger than
the tube radius, such that f = f (t ) at leading order. The
network thus has an average energy density ρ = f ρE, with
the first law of thermodynamics yielding
d f
d(ρ V ) = dV (ρE V )dV + f ρEdV = − pdV ,
(7)
where we neglected decay processes and consequent heat
transfer. From this we conclude that the network has an
effective pressure:
p = −
1 d log f
1 + 3 d Ne
ρ ,
(8)
where Ne denotes the number of e-folds of expansion, and
hence we have an equation of state parameter w = p/ρ
−1 for f˙/ f 3 H . Note that, as discussed above, we expect
the network to be dense at the phase transition and hence close
to a tight configuration that fills a significant fraction of the
spatial volume, with f 1.
The network will initially relax into a tight configuration
with d log f /d Ne > 0, yielding a phantom-like equation of
state w < −1 [43]. Although this phantom behavior often
leads to instabilities [
44, 45
], it is nevertheless considered
in dynamical dark-energy models, as well as in dark-matter
scenarios with bulk viscosity [
46
], and it would be interesting
to consider this behavior in the context e.g. of axionic strings
[
47
].
The network will then subsequently remain close to a tight
configuration, with a fixed or at most slowly varying filling
fraction f . The rigidity of the network, which results from
the presence of a large density of knots and links, will thus
decouple it from the Hubble flow and generate an effective
cosmological constant.
As the number of knots and links decreases due to
network decay, f will then decrease such that w −1, thus
yielding a period of inflation with an equation of state
analogous to that of a canonical slowly rolling scalar field. Note
that, in the absence of the topological charge that
maintains the tightness of the network, the filling fraction would
decrease quickly and no accelerated expansion could be
obtained.
To better understand this, note that, in the absence of a
non-trivial topology, a network of long non-relativistic tubes
has an equation of state p = − 13 ρ, as a result of the fact
that only the tube length increases with expansion, while its
cross-section remains fixed due to flux conservation. On the
other hand, a two-dimensional network of small string loops
(or also e.g. a two-dimensional braid of long tubes) at rest will
behave, up to small distortions, as a rubber sheet or domain
wall, with an equation of state p = − 23 ρ. If we then add a
stack of two-dimensional chain mails and link them together
to form three-dimensional space filling chain mail, we get
p = −ρ, as long as the network is stable and the network
density remains constant.
For completeness it should be mentioned that the
expansion law for the flux tubes could also be modified to
p = −(1/3 + β)ρ for some nonzero β by conformal
symmetry breaking, similar to treatments in cosmic B
fields.
Notice that the number of knots in a comoving volume is
not constant in this scenario. As the Universe expands, the
gaps in the network would tend to expand in the absence
of knots or links, which would dilute the network. Knotting
prevents this dilution by keeping the gaps between
neighboring tubes at an approximately fixed size, thus
maintaining the average energy density. In order to sustain
inflation in such a scenario an inflating region necessarily has
to get filled with more and more flux tubes as it expands.
Again, there are two possible realizations how this could
happen: flux tubes could either be pulled into the inflating
patch from neighboring regions or produced inside the patch
itself.
In the first case, if the three dimensions where the knotted
network lives are infinite in extent, this would allow
inflation to occur everywhere, since there would be an infinite
supply of flux tubes that may relax into a tight
configuration. On the other hand, for compact three dimensions the
network is necessarily finite and inflation will only occur
in the regions that tighten faster and, in the process, pull
in flux tubes from neighboring (non-inflating) regions. This
may pose constraints on the initial size of our three
dimensions such that at least the small patch that later became the
presently observable Universe has inflated. Note that it is not
mandatory that the flux tubes from outside the patch
completely get pulled inside the initial inflationary patch. If they
only get partially pulled in, it will just mean that the regions
around the initial patch then join the initial patch in inflating.
The key point is once there is an overdensity of flux tubes
that initiate inflation, they act as an attractor for more flux
tubes.
In the above picture, flux tubes are seen as (semi)classical
objects that arise from the confinement of gluons in the
thermal bath after the phase transition and there is no
mechanism that, in this case, can create or destroy flux tubes
afterwards. Flux tube production is, however, not prohibited by
any conservation law, and the topologically metastable
network could be seen as a “false vacuum” configuration of the
chromoelectric field after the phase transition. In this case,
expansion does not preserve the number of flux tubes in a
comoving volume but rather the false vacuum field
structure, which contains the knotted network of tube-like field
configurations.
Just as in scalar field models of inflation, in this case
gravitational energy is converted into vacuum energy with the
same field structure as more space is created by expansion,
as for the vacuum-energy description of the present-day
cosmological constant. Although more speculative in nature, this
alternative description would allow a single patch to inflate
independently of the initial extension of the flux tube
network, being in this sense more attractive.
In both of the above scenarios, inflation can only occur
in three dimensions and lead to a uniform Universe with
three large dimensions at late times. The Friedmann equation
yields
(9)
during inflation, which implies that H < for a
subPlanckian confinement scale, with e.g a phase transition
at the grand unification scale, ∼ 1016 GeV, yielding
H/ ∼ 10−3 for f 1. This shows that the flux tube
network will in general be on average homogeneous and
isotropic on super-horizon scales, since the tube radius a ∼
−1 H −1 ∼ 6π 2/ f (MP/ ) −1, and that relaxation
may in principle occur in less than a Hubble time, as required
above.
5 Network decay and reheating
A knot or link between two flux tubes is only classically stable
if these are unable to intersect and either reconnect or pass
through each other. Such intercommutations lead to the
wellknown scaling behavior in cosmic string networks, which has
been observed in several examples of non-interacting strings
(see e.g. [
48,49
]). Adjacent tubes may, however, repel each
other as a result of the surface currents that support their flux.
This has been observed in type II superconductors, where
repulsion leads to the formation of hexagonal lattices as first
predicted by Abrikosov (see [50] for a review). Evidence for
a repulsive interaction between non-Abelian flux tubes has
also been found in the SU(2) Yang–Mills theory [
51
].
Flux tubes with small velocities will be classically
forbidden from overcoming this repulsive barrier and thus from
intersecting and reconnecting. This is e.g. supported by
twodimensional simulations of vortex collisions, where it has
been shown that colliding vortices only overlap above a
critical velocity [
52
]. We expect typical velocities to be indeed
small in our scenario due to the large number density of
mutually repelling flux tubes formed during the phase transition,
as discussed above. We note that relativistic tubes can
overcome the repulsive barrier and reconnect [
53
], but this is not
the relevant regime for the present scenario.
Although classically forbidden, reconnection may
nevertheless occur through quantum processes similar to those
responsible for the metastability of knotted/linked glueballs
in [31–33].
Two flux tubes may unknot/unlink by quantum tunneling.
We can estimate this by treating the intersection between two
flux tubes as a non-relativistic particle of mass m ∼ ρEa3 ∼
/2π 2 that tunnels through a potential barrier of length ∼ a
and height V . This yields a typical lifetime in units
of the Hubble time:
C
τt H = v (a H ) exp
2D
π
V
,
(10)
where v 1 is the typical tube velocity and C, D are O(1)
factors parametrizing the estimate’s uncertainty.
Flux tubes can also break through the well-known
Schwinger effect [
54
], where Q Q¯ pairs are produced in the
approximately constant chromoelectric field enclosed within
each flux tube. For Nf quark flavors of mass m Q , using the
fact that the string tension κ = π a2ρE ∼ 2/(2π ), we
obtain for the lifetime of a flux tube of length l
τb H ∼
This implies that the lifetime for string breaking can be
parametrically large if there are no light quarks in the spectrum
of the gauge theory, m Q , even for a H −1 and
flux tubes as long as the Hubble radius. Thus, the
confining gauge theory responsible for a sufficiently long period
of inflation in the proposed scenario has to be distinct from
QCD, since in the latter case knots and links between flux
tubes will quickly decay through the production of up and
down quark–antiquark pairs.
Given the exponential suppression of the string breaking
and tunneling rates, it is thus not difficult to envisage
scenarios where the network remains tight for the 50–60 e-folds of
accelerated expansion required to explain the present flatness
and homogeneity of the Universe.
String breaking will eventually lead to a system of
unknotted/unlinked flux tubes, and thus to a gas of hadrons. These
may be unstable and decay into radiation, including in
principle the Standard Model particles. The details of this process
depend, of course, on how the confining gauge theory at a
high-energy scale is related to the low-energy physics, but
it is clear that the proposed model naturally includes a
mechanism for reheating the Universe after inflation. It is also
possible for a significant amount of radiation to be produced
during the inflationary period, if e.g. small loops unlink from
the network and decay while the network is still tight, thus
potentially leading to a warm inflation model [
55,56
].
6 Cosmological perturbations
The flux tube network is, on average, homogeneous on scales
larger than the tube radius, which as shown above can be
parametrically below the Hubble radius. Nevertheless,
fluctuations in the energy density of the flux tube network may arise
on super-horizon scales and become imprinted on the
curvature of space-time, thus seeding the temperature anisotropies
in the Cosmic Microwave Background and the Large Scale
Structure of the present Universe.
Curvature perturbations in a perfect fluid driving a period
of accelerated expansion propagate with an imaginary sound
speed cs2 = d p/dρ = w < −1/3, which would make the
system unstable to small fluctuations as opposed to the
canonical scalar field models of inflation. However, the flux tube
network has a non-vanishing rigidity as a result of the
presence of numerous knots and links. The network thus behaves
more like an elastic solid than a perfect fluid, similarly to
the networks of topological defects considered in [
57–62
].
The anisotropic stress inherent to elastic solids allows for
the propagation of both longitudinal and transverse waves
and the associated sound speed cs2 = w + (4/3)μ/(ρ +
p) can be real for a sufficiently large shear modulus μ
[63].
Inflationary models with elastic solids may lead to a nearly
scale-invariant spectrum of both curvature and tensor
perturbations with interesting differences from scalar field models
[
63
]. Firstly, despite the absence of non-adiabatic modes,
perturbations evolve on super-horizon scales due to the
presence of anisotropic stress, although maintaining the relevant
scaling between different super-horizon modes. The
overall amplitude of the spectrum thus depends on the details
of reheating, which may potentially yield distinctive
observational signatures of our model, where reheating proceeds
through the decay of a knotted/linked network. Anisotropic
stress will also act as a source for tensor perturbations,
potentially modifying the primordial gravitational waves spectrum
with respect to canonical models.
Although concrete observational predictions require a
detailed modeling of the properties and dynamics of the flux
tube network, which is beyond the scope of this Letter, it
is worth mentioning that a red-tilted curvature spectrum can
only be obtained in elastic solid models of inflation for a
slowly varying equation of state. We expect this to be the
case in our model since a knotted network will be close to, but
not exactly in, a tight configuration, as a result of the
opposing effects of Hubble expansion, relaxation and unknotting
events.
7 Conclusion
A knotted/linked flux tube network formed in a QCD-like
phase transition can provide a natural source of inflation
and is one of the few scenarios not requiring a
fundamental scalar field. Furthermore, this model may explain why we
live in three large spatial dimensions, since knotted/linked
tubes are topologically unstable in higher-dimensional
spacetimes. This picture may also be applied to a model of dark
energy, which would eliminate the need for an ultra-light
scalar field.
Although exact solutions in such models are unavailable,
appropriate approximations should be enough to establish
the main qualitative features. A key advantage of this model
is that its underlying building block, the Abelian or
nonAbelian flux tube, is a quantity that has been extensively
studied and there are many tools and methodologies
available to further explore it. We will present a more detailed
analysis based on the Abelian Abrikosov–Nielsen–Olesen
model elsewhere [
64
].
Acknowledgements AB is supported by STFC. The work of TWK was
supported by US DOE Grant DE-SC0010504. HP acknowledges kind
hospitality and support at Vanderbilt University and by the
Alexander von Humboldt-Foundation. JGR is supported by the FCT
Investigator Grant no. IF/01597/2015 and partially by the
H2020-MSCARISE-2015 Grant no. StronGrHEP-690904 and by the CIDMA Project
no. UID/MAT/04106/2013. Some of this work was developed at the
Isaac Newton Institute and we thank them for their kind hospitality.
Open Access This article is distributed under the terms of the Creative
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ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
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