Testing a Conjecture on Cosmology and Dark Energy

Abstract

The consequences of the strand conjecture are explored on cosmological scales. The starting point is the realization that strands fluctuating at the Planck scale appear to explain both the Lagrangian of the standard model of particle physics, extended with massive mixing neutrinos, and, at sub-galactic distances, the Hilbert Lagrangian of general relativity. Both Lagrangians arise without any modification, with particle masses, mixing angles and coupling constants that are unique and calculable.

On cosmological scales, fluctuating strands lead to many effects that can be tested against observations. Above all, the formation of an expanding cosmological horizon is predicted. The resulting horizon temperature, horizon luminosity and horizon entropy are compatible with observations. Inside the horizon, the strand conjecture predicts the appearance of empty space, particles and black holes. In particular, the conjecture yields a natural model for matter, radiation and dark energy. The three predicted density values are compatible with observations. The density of dark energy is predicted to be small and constant over time. The value $w$ in the equation of state of dark energy is predicted to be negative. Inflation is found not to have occurred. No contradictions between the strand conjecture and observations are found.

Full text

Testing a Conjecture
on Cosmology and Dark Energy
Christoph Schiller *
28 December 2021
Abstract
The consequences of the strand conjecture are explored on cosmological scales. The
starting point is the realization that strands fluctuating at the Planck scale appear to
explain both the Lagrangian of the standard model of particle physics, extended with
massive mixing neutrinos, and, at sub-galactic distances, the Hilbert Lagrangian of
general relativity. Both Lagrangians arise without any modification, with particle
masses, mixing angles and coupling constants that are unique and calculable.
On cosmological scales, fluctuating strands lead to many effects that can be tested
against observations. Above all, the formation of an expanding cosmological hori-
zon is predicted. The resulting horizon temperature, horizon luminosity and horizon
entropy are compatible with observations. Inside the horizon, the strand conjecture
predicts the appearance of empty space, particles and black holes. In particular, the
conjecture yields a natural model for matter, radiation and dark energy. The three pre-
dicted density values are compatible with observations. The density of dark energy is
predicted to be small and constant over time. The value win the equation of state of
dark energy is predicted to be negative. Inflation is found not to have occurred. No
contradictions between the strand conjecture and observations are found.
Keywords: general relativity; cosmological constant; dark energy; strand conjecture.
*Motion Mountain Research, 81827 Munich, Germany, cs@motionmountain.net, ORCID 0000-0002-8188-6282.
1

1 The quest for the origin of dark energy
The physical origin of dark energy is a subject of intense research. Unravelling its details is
important for astronomy and for particle physics.
The so-called strand conjecture proposes a fundamental principle that describes nature at the
Planck scale. The conjecture describes all physical systems as made of the same extended compo-
nents. In particular, space, particles and horizons are described as made of fluctuating strands of
Planck radius. The strand conjecture appears to reproduce both general relativity and the standard
model of particle physics, including mixing massive neutrinos, without any deviations. In both do-
mains, the conjecture makes numerous predictions that allow testing it. So far, no contradictions
between experiments and the strand conjecture arose. In the domain of fundamental physics, the
conjecture appears to be complete: no observation is unexplained. In particular, strands appear to
explain and determine the constants of the standard model – masses, mixing angles and coupling
constants. Only this last result justifies the exploration of strands in the domain of cosmology.
As argued in the following, applying the strand conjecture to cosmology yields specific exper-
imental predictions, in particular about dark energy. The predictions, highlighted by Pr. n, are
deduced step by step, proceeding with care. They all follow from the fundamental principle. The
topic of dark matter will be covered in a subsequent paper.
2 The origin of the strand conjecture
The strand conjecture goes back to the so-called string trick or belt trick that Dirac used in his
lectures. From around 1929 onwards, Dirac explained spin 1/2 as a result of tethered rotation,
even though he never published anything about it [1]. Tethers were the first hint that nature might
be built from unobservable extended constituents. Several decades later, in 1980, Battey-Pratt
and Racey understood that also the complete Dirac equation could be deduced from unobservable
extended constituents whose crossings can be observed [2]. Then, in 1987, Kauffman conjectured
a direct relation between the canonical commutation relation – and thus ~– and a crossing switch
of such unobservable tethers [3]. These connections were rediscovered in the early twenty-first
century. Building on these results, crossing switches of unobservable extended constituents were
found to describe the complete standard model of particle physics, including the gauge groups,
the particle spectrum, masses, mixing angles and coupling constants [4, 5]. It thus appeared that
every quantum effect can be described as being due to unobservable extended constituents whose
crossings can be observed.
In addition, the surface dependence of black hole entropy [6,7] and the discovery of maximum
power and force [8–15] led to deduce all black hole properties and Einstein’s field equations from
crossing switches of unobservable extended constituents [4]. It thus appeared that every gravita-
tional effect can be seen as being due to unobservable extended constituents. Because the term
‘string’ had acquired a different meaning in the meantime, the alternative term strand appeared
more appropriate.
2

Strand conjecture:
The fundamental Planck-scale principle of the strand conjecture
Observation:
A fundamental event, localized in space within two Planck lengths
t t + ∆t
W=~
∆l≥2p~G/c3
∆t≥2p~G/c5
S=kln 2
Figure 1: The fundamental principle of the strand conjecture defines the simplest obser-
vation in nature, the almost point-like fundamental event. Every event results from a skew
strand crossing switch, at a given position in three-dimensional space. The strands them-
selves are not observable; they are impenetrable and best imagined with Planck radius. The
crossing switch defines all fundamental constants. The double Planck length limit and the
double Planck time limit arise, respectively, from the smallest and from the fastest crossing
switch possible. (See also Appendix A.) In the following, cosmology is deduced from the
fundamental principle.
3 The strand conjecture
In the strand conjecture all physical systems in nature – matter, radiation, space and horizons – are
built from fluctuating strands of Planck radius. Strands are defined as smooth one-dimensional
curves embedded in three-dimensional space with a bending radius everywhere larger than the
Planck length. This definition leads to the strand conjecture, to be detailed in the following:
⊲Space is a strand network. Horizons are strand weaves. Particles are strand tan-
gles. Strands are unobservable; however, crossing switches of strands are. Crossing
switches determine the Planck units, as illustrated in Figure 1.
Strands have no observable properties: they have no colour, no tension, no mass, no energy.
Strands cannot be cut and have no ends. It is easiest (but not fully correct) to imagine strands as
having a Planck-size radius. Strands cannot interpenetrate; they never form a real crossing. When
3

Observation:
Nothing
(for long
observation
times)
Virtual pairs
(for short
observation
times)
The strand conjecture:
The vacuum
time average
of crossing
switches
Figure 2: An illustration of the strand conjecture for a flat vacuum: for sufficiently long
time scales, the lack of crossing switches leads to a vanishing energy density; for short time
scales, particle-antiparticle pairs, i.e., rational tangle-antitangle pairs, arise.
the term ‘crossing’ is used in the present context, only the two-dimensional projection shows a
crossing. In three dimensions, strands are always at a distance. Crossing switches only arise via
strand deformation, as illustrated in Figure 1. This allows stating:
⊲In the strand conjecture, all physical observables – including action, energy, velocity,
momentum, mass, length, surface, volume, force, entropy, all field intensities and
quantum numbers – arise from strand crossing switches.
In short: all physical observables emerge from strands. The following sections give a summary of
how crossing switches of fluctuating strands produce general relativity and quantum effects. Then
the implications and consequences of crossing switches in the domains of cosmology are explored.
First, however, some conceptual issues are clarified.
4 Measurements and minimum time
The fundamental principle can be used to deduce the three known gauge interactions from the
three Reidemeister moves [4,5]. The three Reidemeister moves are called twists, pokes and slides.
The electromagnetic interaction arises from twist exchange. This result explains the fundamental
principle: in twist exchange, the sign of a crossing changes orientation. And in nature, all mea-
surements and all observations are due to the electromagnetic interaction. This is exactly what the
fundamental principle tells: all measurements and all observations are due to crossing switches.
4

smallest
area
2
lPl
Black hole horizon
The strand conjecture, side view top view
2
lPl
Observed
horizon:
a thin spherical cloud

rst ring (black)

rst ring
(black) n=1
additional
crossing
Figure 3: The strand conjecture for a Schwarzschild black hole is shown: the black hole
horizon is a cloudy or fuzzy surface produced by the crossing switches of the strands woven
into it. Due to the additional crossings on the side of the observer, the number of microstates
per smallest area is larger than 2.
The fundamental principle appears to allow arbitrary fast crossing switches. This would imply
that no minimum time arises and is defined. However, an arbitrary fast crossing switch does not
yield an electromagnetic signal: no electromagnetic wave can have a wavelength shorter than
a Planck length. Therefore, an arbitrary fast crossing switch is not observable. Only crossing
switches that take more than one (corrected) Planck time are observable.
5 From strands to black holes and their horizons
This section summarizes how strands lead to black holes and all their properties [4, 16,17].
The strand conjecture posits that horizons are one-sided weaves of fluctuating strands. One-
sided means that all strands leave the horizon on the observer side. A simplified illustration of a
black hole is given in Figure 3, both as a cross-section and as a top view. All strands arrive from
far away, are woven into the horizon, and leave again to large distances. If strands are imagined as
having Planck radius, the strand weave is maximally tight.
Maximally tight weaves allow determining the energy of a spherical horizon. Energy Ehas
the dimension action per time. Because every crossing switch is associated with an action ~,
the black hole horizon energy is found by determining the number Ncs of crossing switches per
unit time. Crossing switches propagate across the horizon weave. For a maximally tight weave,
5

the propagation speed is one smallest possible crossing per one shortest possible switch time:
switch propagation thus occurs at the speed of light c. In the time Tthat light would take to
circumnavigate a black hole horizon of radius R, all crossings of the horizon switch, yielding the
black hole energy
E=Ncs
T=c4
4G
4πR2
2πR =c4
2GR . (1)
Strands thus reproduce the known relation between energy and radius of a Schwarzschild black
hole.
A maximally tight weave also determines the number of microstates per black hole horizon
area. Figure 3 shows that for each circular or ring area that contains just one crossing, the effective
number Nof microstates above that area is larger than 2. This excess occurs because of the neigh-
bouring strands that sometimes cross above that smallest area. The additional crossing probability
depends on where the neighbouring strand leaves the black hole horizon; this yields
N= 2 + 1
2! +1
3! +1
4! +... +1
n!+... = e = 2.718281... (2)
In this expression deduced from strands, the term 2is due to the two options at the very bottom of
the smallest area. The next term 1/2! arises from the strand leaving the neighbouring ring shown
in Figure 3. The subsequent terms are due to the subsequent rings.
Expression (2) states that N= e >2is the average number of strand microstates for each
smallest area, i.e., for each corrected Planck area AcPl = 4 G~/c3. Each corrected Planck area on
a black hole horizon thus contains more than 1 bit of information.
The total entropy of the black hole is
S=kln Ntotal ,(3)
where kis the Boltzmann constant and Ntotal the total number of microstates of the horizon. The
full horizon area Ais composed of corrected Planck areas. The product of the number of states
for every corrected Planck area yields the total number of microstates:
Ntotal =NA/AcPl .(4)
Inserting the result (2), due to strands, yields
Ntotal = eA/(4 G~/c3).(5)
This total number of horizon microstates can then be inserted into expression (3) for the entropy.
The resulting value for the black hole entropy is
S
k=A
4G~/c3.(6)
This is the expression discovered by Bekenstein [6].
6

T  oc
t
 

e

o
fo
te
s
t
Curved space
O
to
Figure 4: An illustration of the strand conjecture for a curved vacuum. The strand configu-
ration is intermediate between that of a horizon and that of a flat vacuum. The black strands
differ in their configuration from those in a flat vacuum: they are tangled. The configura-
tion implies that such a region of space has non-vanishing curvature, energy, entropy and
temperature.
The graviton
   
w




Figure 5: The strand conjecture for the graviton: a twisted pair of strands automatically has
spin 2, boson behaviour, and vanishing mass. A gravitational wave is a coherent superposi-
tion of a large number of gravitons.
The ratio of black hole energy and twice the entropy determines the temperature:
TBH =~c
4πk
1
R.(7)
In short, strands thus reproduce all thermodynamic properties of black holes. For example, black
hole evaporation arises from strands or tangles that detach from the horizon. So far however, these
properties are not comparable with observations.
7

6 From strands to general relativity
This section summarizes how strands lead to general relativity [4, 16].
In 1995, Jacobson showed [18] that the thermodynamic properties of black holes imply Ein-
stein’s field equations of general relativity. He started with the entropy–area relation S=A kc3/4G~,
the temperature–acceleration relation T=a~/2πkc, and the relation between heat and entropy
δQ =T δS . He introduced them into the relation
δE =δQ , (8)
which is valid only in case of a horizon, and derived the first principle of horizon mechanics
δE =c2
8πG a δA . (9)
The left-hand side can be rewritten, using the energy-momentum tensor Tab, as
ZTabkadΣb=c2
8πG a δA , (10)
where dΣbis the general surface element and kis the Killing vector that generates the horizon.
The right-hand side can be rewritten, using the Raychaudhuri equation, as
ZTabkadΣb=c4
8πG ZRabkadΣb,(11)
where Rab is the Ricci tensor that describes space-time curvature. This equality between the
integrals implies
Tab =c4
8πG (Rab −(R/2 + Λ) gab ),(12)
where Ris the Ricci scalar and the cosmological constant Λappears as an undetermined constant
of integration. These are Einstein’s field equations of general relativity.
In short, the field equations result from thermodynamics of space. Like every horizon and
every black hole, also curved space is a thermodynamic system. Indeed, curved space and hori-
zons can be transformed into each other by a change of coordinate system. These results can be
summarized by stating that space is made of microscopic degrees of freedom and curvature and
gravity are due to microscopic degrees of freedom. Figure 4 gives an impression of curved space,
and Figure 5 of the graviton. Given that strands lead to the thermodynamic properties of black
holes, strands realize Jacobson’s argument [18], as well as his later thoughts [19]: strands produce
the field equations of general relativity [4, 16].
7 From strands to quantum theory and particle physics
This section summarizes how strands lead to quantum field theory and particle physics.
8

The strand conjecture:
!"#$
c
%&$
phase
t
$'$&
()
$ 
*
$&"$
o
+ ,&%
--(!"
-.(

,'$-
p&%//(#
(
0
d$!
-(
0
Observation:
crossing
p'-
$
-p(!-p(!
A fermion
Figure 6: In the strand conjecture, the wave function and the probability density are due,
respectively, to crossings and crossing switches at the Planck scale. The wave function
arises as the time average of crossings in fluctuating tangled strands; a Hilbert space also
arises. The probability density arises as the time average of the crossing switches in a
tangle. The tethers – connections that continue up to large spatial distances – generate spin
1/2 behaviour under rotations and fermion behaviour under particle exchange. The tangle
model ensures that fermions are massive and move slower than light.
As explained in detail in previous publications [4, 5, 20], strands allow deducing quantum
theory and the standard model of particle physics. In flat space, the fundamental principle, to-
gether with Dirac’s belt trick, implies that tangled fluctuating strands describe particles and wave
functions. The wave function is the time average of the strand crossing density. For fluctuating
rational, i.e., unknotted tangles, the crossing switch density yields the probability density. Tangles
also yield the Hilbert space, the quantum phase, interference, and the freedom in the definition
of phase. Tangles imply spin 1/2 and, above all, Dirac’s equation. The general connection is il-
lustrated in Figure 6. Tangles also reproduce fermion behaviour and entanglement. Tangles are
fully equivalent to quantum theory. No extension or deviation arises. The tendency to keep the
number of crossing switches as low as possible, the principle of fewest crossing switches, leads to
the principle of least action. Strands thus visualize quantum theory: every quantum effect is due
to crossing switches. And every crossing switch is a quantum effect.
Rational, i.e., unknotted tangles reproduce the known spectrum of elementary particles [4, 5,
20]. Each massive elementary particle is represented by an infinite family of rational tangles made
of two or three strands. The family members differ only by the number of attached braids. Each
braid corresponds to a Higgs boson. Three generations of quarks and of leptons arise. W, Z and
Higgs bosons are massive. Tangles for the massless gauge bosons, the photon and the gluons,
9

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=?
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t2
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=?
1
on
>=
?
1
on
76<vv
4
76<vv
4
76<vv
4
>?
:318?A 5@5
ctron
Figure 7: An illustration of two Feynman diagrams of QED in the tangle model.
also arise. Rational tangles also describe particle mass: more complex tangles imply higher mass.
Particle mixing is described. Strands reproduce the quark model, including the mass sequence of
mesons and hadrons, the CP violation and the mixing of mesons.
Interactions are due to deformations of tangle cores. Such deformations can be classified.
This classification is based on the Reidemeister moves and yields the gauge groups U(1), broken
SU(2), and SU(3). No other gauge group nor any combined gauge group is possible. Emission of
particles, as well as particle-antiparticle creation and annihilation also arise. These results produce
all Feynman diagrams of the standard model – including the examples illustrated in Figure 7.
10

Due to topological reasons, additional elementary particles, additional gauge groups, and further
Feynman diagrams are explicitly excluded. Perturbative quantum field theory, including quantum
electrodynamics and quantum chromodynamics, is reproduced completely.
In short, the strand conjecture reproduces all observations in high-energy physics and provides
numerous predictions for experiments [4, 5,20]. Strands allow calculating the fundamental con-
stants of the standard model: masses, mixing angles and coupling constants. Discovering a single
effect beyond the standard model would falsify the strand conjecture.
8 Strand predictions about local physics: particles, gravity, and Planck limits
A number of predictions deduced from the strand conjecture in previous papers [4, 5, 16, 20] are
important in the following. They concern observations at sub-galactic scales.
Pr. 1 In the domain of high energy physics, strands predict the lack of deviations from the stan-
dard model of particle physics with massive Dirac neutrinos with PMNS mixing. In the
strand conjecture, all particles are represented by rational tangles of strands. This tangle
model of quantum particles reproduces the known elementary particles and interactions
without any additions or modifications.
The tangle model leaves no room for the axion, for WIMPs, the inflaton, or for any other
non-standard elementary particle or field conjectured in the past. Finding a new elemen-
tary particle or any effect beyond the standard model of particle physics with at least two
massive Dirac neutrinos would falsify the tangle model and the strand conjecture. All this
is not in contrast with observations so far [21].
Strands also predict the lack of new gauge symmetries, of any other symmetries, or of new
energy scales. A future discovery of any such property would falsify the strand conjecture.
Strands do not allow such additional structures in particle physics. So far, none has been
observed.
Pr. 2 In the domain of gravitation, the strand conjecture predicts that local power and luminosity
values are limited by c5/4G, local force values are limited by c4/4G, local mass flow rates
are limited by c3/4G, and mass-to-length ratios are limited by c2/4G. In short, strands
predict the lack of measurable deviations from general relativity at sub-galactic distances.
These predictions, including the factor 4, agree with all observations and calculations so
far. A future discovery of any deviation from the field equations, i.e., from the Hilbert
action of general relativity, at sub-galactic scales would falsify the strand conjecture.
Pr. 3 In the domain of quantum gravity, strands predict that no length and no effect of scales
smaller than the corrected Planck length p4G~/c3can be observed. Similarly, strands
predict that no time intervals and no effect of time intervals smaller than the corrected
Planck time p4G~/c5can be observed. Elementary particle energies are predicted not
to exceed the corrected Planck energy p~c5/4G. These predictions explicitly include
11

the factor 4. A future discovery of any trans-Planckian effect would falsify the strand
conjecture.
Pr. 4 Strands imply that all fundamental constants of particle physics – coupling constants, par-
ticle masses and mixing angles – are constant over time and space. This agrees with obser-
vation, despite occasional claims of the contrary. Finding a variation of the fundamental
constants would falsify the strand conjecture.
Pr. 5 Strands imply that all fundamental constants of particle physics – coupling constants, par-
ticle masses and mixing angles – can be calculated. Finding a difference between the
observed and the calculated values of the fundamental constants would falsify the strand
conjecture.
In particular, strand imply that elementary particle masses are much smaller than the
Planck mass p~c/4G. Strands thus explain the ratio 1042 between the electric and grav-
itational force between an electron and a positron. Strands imply the SU(3) symmetry of
the strong interactions and the quark model. The masses of the baryons and the properties
of the strong nuclear interaction explain – in principle – the fusion of hydrogen to helium.
Pr. 6 In the tangle model of particles, electric charge is represented by topologically chiral tan-
gles [4, 20]. All particle tangles are free of magnetic charge; such tangles are not possible.
Strands thus predict the lack of magnetic monopoles in nature. Strands thus solve the mag-
netic monopole problem, because the tangle model has no room for them. This agrees
with observations. Finding a magnetic monopole would falsify the strand conjecture.
Pr. 7 Strands predict that electric and magnetic fields are limited via maximum force c4/4G
and smallest electric charge e/3by the expression E≤3c4/4Ge = 5.7·1062 V/mand
B≤3c3/4Ge = 1.9·1054 T. The factor 4 is part of the prediction.
Pr. 8 Strands also predict the lack of cosmic strings, wormholes, and regions of negative energy
Pr. 9 In 1967, Sakharov showed that the observed matter-antimatter asymmetry in the universe
requires three properties of particle interactions [22]. In modern terms, the three properties
are (1) violation of baryon number conservation, (2) C and CP violation, and (3) a non-
equilibrium situation during the expansion of the universe. In the strand conjecture, like in
all cosmological models, the third property holds automatically. The second property also
holds, as discussed in a previous paper [4]: strands predict a unitary mixing matrix with a
CP phase for quarks and one for leptons. Also the first property holds; adding or taking a
strand from a tangle is a non-perturbative process that does not conserve baryon number.
In addition, strands imply that there might be a further effect: the tangling of the universal
strand at the very beginning breaks chirality. All this together yields a baryon—antibaryon
asymmetry.
In particular, the first property, baryon non-conservation, is realized by the tangle model
of quarks and leptons: rearranging the strands of a quark and combining it with a vacuum
strand, allows forming a lepton [4]. This non-perturbative process – different from the
12

perturbative processes described with Feynman diagrams – appears to model baryon non-
conservation with strands. This description should help resolve the ongoing discussion of
whether the standard model is sufficient or not to explain the observed baryon–antibaryon
asymmetry, and the relative importance of baryogenesis and leptogenesis. The answer
should arise when performing simulations of fluctuating strands. If such simulations dis-
agree with observations, the strand conjecture is falsified.
Pr. 10 In the strand conjecture, both gravity and particle physics are possible only in 3 dimen-
sions. Finding evidence for more or fewer dimensions would falsify the strand conjecture.
In short, all these predictions can be condensed into the general prediction that there is no unknown
fundamental physics in nature. This prediction agrees with all known experiments on microscopic,
macroscopic and astrophysical scales. Building on this correspondence between the strand con-
jecture and observations, it makes sense to explore the consequences of strands in the domain of
cosmology.
9 The main observations about the cosmos
At night, the sky is dark. The observation has several well-known reasons. First, there is a max-
imum age in nature, about t0= 13.8(1) ·109a≈0.43 Es [23]. Secondly, precise experiments
show that also observable distances are limited: the universe is enclosed by a cosmological (par-
ticle) horizon at a finite distance. Thirdly, stars and galaxies are surrounded by a vacuum that
allows seeing the dark sky. Fourthly, observations also show that, on average, all matter recedes
from all other matter, and that the recession speed increases with distance; the universe is expand-
ing. Therefore, light from distant, ancient sources is red-shifted.
In an expanding universe, the cosmological (particle) horizon is defined as that surface inside
which something has already been observed [24]; the cosmological horizon is thus defined by the
observable photons that are arriving from the big bang. The cosmological horizon and all matter
inside it are increasing in distance with time. The expansion of the universe is confirmed by all
observations so far. Also all consequences deduced from the expansion – such as the cosmic
background radiation and the details of nucleosynthesis – agree with observations.
In ΛCDM cosmology, the distance to the present cosmological particle horizon is measured to
be about 3.3(1) ·ct0[23], or 14.4 Gpc. The cosmological horizon is observed to recede at speeds
larger than the speed of light. In other words, the cosmological horizon is larger than the Hubble
radius. Egan and Lineweaver estimated the entropy of the cosmological horizon to be around
10122 k[25]. They estimated the entropy inside the observable universe to be around 10103 k, not
taking into account dark energy.
In ΛCDM cosmology, inside the universe, matter, radiation and space are expanding. The
expansion rate, the so-called Hubble constant H0, is measured using star distances and redshifts.
The modern value is 71(3) km/(s Mpc) or, in SI units, 2.3(1)/Es [23]. The uncertainty is large,
because, in 2020, the value deduced from the cosmic background radiation 67.4(5) km/(s Mpc)
13

Cosmological
horizon (dark)
Physical space or
cosmic vacuum (whit
HIJ
mKLH MN LHPRHSU VKWXHL
uPYKPZSHL RY[KPLRJ
M\RH[]K\SH^ SMWKSSU
Wu[]HL^
_KY MP K]H[KZH
Background
space (
Z[HUI`
PMY VnURaWKS^
PMY M\RH[]K\SH
b
osmological
t
HYnH[R
The present universe
Particle
tangle;
mKLH M
f
YKPZSHL
strands
Figure 8: In the strand conjecture, the universe is limited by a cosmological (particle)
horizon, as schematically illustrated here. Physical space (white) matches the background
space (grey) only inside the horizon. Physical space only exists inside the cosmic horizon.
and the value deduced from more local measurements 73.8(1.1) km/(s Mpc) differ by 10% [26].
This difference is the subject of intense research. The small but measurable acceleration of the
expansion is explored below, from Section 17 onwards.
At large scales, the universe is observed to be homogeneous and isotropic. The observed mass-
energy density of conventional matter inside the universe is Ωmatter = 0.045 [21]. The observed
energy density of photons inside the universe is Ωphotons = 0,000048. The observed energy
density of neutrinos inside the universe is 0.0009 <Ωneutrinos <0.048.
The observed baryon–antibaryon asymmetry in the universe is 6.3·10−10 [27]. Other numbers
that characterize the universe are the density ratios of various nuclei. All these ratios are due to
particle physics and are not discussed in the following.
The universe is observed to have 3 dimensions also at galactic and at cosmological scales. At
cosmological scales – within measurement errors – the universe is observed to be flat. Also, no
singularities of any kind, no cosmic strings, and no different vacuum states have been observed.
All the observations just mentioned – about the horizon and about the interior of the universe
– must be reproduced by the strand conjecture; otherwise, it is falsified. Obviously, some values
depend on observational constraints, for example, the present age of the universe. But taking these
constraints as input, all other values must follow.
14

10 Strand cosmology: the horizon and its interior
The observed expansion of the universe can be derived from the field equations of general rel-
ativity. Given that the strand conjecture reproduces general relativity [4, 16], it also reproduces
the expansion of the universe. However, the strand conjecture goes further. In fact, it suggests a
simple cosmological model:
⊲The universe is a single strand.
For the present universe, this conjecture is illustrated in Figure 8. In the conjecture, this universal
strand is woven into the cosmological particle horizon, leaves the horizon somewhere, continues
into the interior, forms tangles and thus particles, and then continues again back to the horizon in
another direction. There, the strand becomes again part of the weave that forms the cosmological
particle horizon, until the strand again leaves into the interior at another location of the sky. In
short,
⊲The cosmological horizon is a one-sided weave of strands all leaving towards the
inside.
Such a one-sided weave is a true horizon. It implies:
Pr. 11 Nothing can be observed behind the cosmological horizon.
This prediction is made for any observer inside the universe – and only such observers are possible
in the strand model. In short, the strand conjecture implies that all observers see a cosmological
horizon. And they all see a different one. It is predicted that no observer can observe an effect
whatsoever that requires an origin beyond the horizon. Any observation to the contrary would
falsify the strand conjecture.
Strands divide the universe into two: the cosmological horizon and its interior. The cosmolog-
ical horizon is a weave and provides a limit to the interior. The universal strand forms an interior
network that contains untangled vacuum strands and tangles of strands, i.e., matter and radiation,
as illustrated in Figure 8. In short,
Pr. 12 Strands imply the existence of space and particles inside the cosmological horizon.
This agrees with observation. In the strand conjecture, woven,untangled and tangled strand seg-
ments are the three structures that make up the whole universe; they form, respectively, horizons,
vacuum and matter.
Despite the limitations due to the circularity of definitions discussed in Appendix A, the strand
conjecture for the universe implies numerous additional checks and testable predictions about
cosmology. They are explored in the following.
15

Strand tangle model:
ghij
cosmological
klqh
r
on
The early expanding universe
Observations:
ghij xz
j
qx{j
of crossing
swit
|k
j}
~l
~j
Figure 9: In the strand conjecture for the early universe, the universe increases in complex-
ity over time and thereby forms a boundary: the cosmological horizon. When the universal
strand increases in complexity, both the crossings on the horizon and the number of strands
in the interior of the universe – the so-called cosmological strands – increase in number.
Note the reckless use of background space in this illustration.
11 Strand predictions about uniqueness, time, expansion, and the horizon
Pr. 13 The strand conjecture implies that there is just one universe. First of all, strands do not
allow separating or distinguishing "different universes". In addition, given that strands
allow calculating the fundamental constants of the standard model uniquely [4], strands
again imply that there is just one universe. No "other universes" with different values of
the coupling constants are predicted to exist: such alternatives are not consistent with the
strand conjecture and thus impossible. This conclusion agrees with observation. Even if
the universe were made of several strands, these arguments would remain valid.
Pr. 14 Strands predict the lack of different vacua, with different values of the coupling constants
or different values of any other parameter of the standard model [4,16]. The strand con-
jecture forbids this option. This also implies the lack of domain walls in the universe. This
agrees with observations.
Pr. 15 The strand conjecture for the history of the early universe is shown schematically in Fig-
ure 9. The strand conjecture implies that the universe expands, including its horizon and
its contents. The expansion is observed. Strands therefore explain why the sky is dark at
night. When asking what ‘drives’ the expansion of the universe in the strand conjecture,
an answer that is not too wrong would be the following: At cosmological scales, tangle
16

fluctuations tend to form more complex tangles with a larger probability than simpler ones.
Once the cosmological horizon had formed, its strand fluctuations – which include evap-
oration from the horizon – continued to drive the cosmological horizon outwards. This
answer has to be taken with a grain of salt though, because time itself and space itself also
result from the tangling of the universal strand.
Strands thus imply that the expansion of the universe is tied to the existence of a cosmo-
logical horizon. In the strand conjecture, there is no expansion without a cosmological
horizon, and vice versa. This is consistent with observation.
Pr. 16 The tangling of the fluctuating universal strand defines cosmological time. Cosmological
time is intrinsically related to cosmological expansion.
A fluctuating universal strand implies that its tangling on a cosmological scale increases
in complexity with time. There is a degree of circularity in this statement, as it assumes
the existence of time, and time itself arises from strands. However, as argued in Appendix
A, observers cannot describe observations without time; a certain degree of circularity is
unavoidable. The circularity implies that an axiomatic presentation is not possible, but a
consistent and correct description is.
In particular, cosmological tangledness has all the properties expected from a global time
coordinate: it is one-dimensional, has a smallest interval – the corrected Planck time – and
increases continuously. Strands imply that the universe has a finite age. Strands imply
that all components of the universe have a finite age and that the age is the same for all
components. This is indeed observed.
The present age of the universe is due to our human fate; the age is not a fundamental
parameter of the strand conjecture and is not predicted by it, as expected.
Pr. 17 Strands imply, as Figure 9 illustrates, that there is no sharp beginning of time. A time
coordinate can only be defined when the universe has a size of a few Planck lengths or
more. Equivalently, time can only be defined when a cosmological horizon has formed.
These statements do not contradict observations.
Pr. 18 Given that the universe is made of one strand, and given that the tangling defines cosmo-
logical time, it is predicted that no effect in the universe is due to causes that precede the
big bang. (This is true even if it were made of several strands.)
If the universe is made of a single strand, it is an unknot. If it is made of several strands, it
would be a link.
Thus, the universe could in principle be cyclic, but there is no way to prove or test this. In
particular, strands imply that in a cyclic universe, there is no effect from one cycle to the
next.
Pr. 19 As illustrated schematically in Figure 8 and in Figure 9, the strand conjecture implies
the existence of a cosmological horizon at a finite distance. The horizon limits observable
17

distances. A limit to distances and a cosmological horizon are indeed observed. Again, the
present distance to the horizon is due to our human fate; the distance is not a fundamental
parameter of the strand conjecture and is not predicted by it.
Pr. 20 In the strand conjecture, all horizons are one-sided weaves. If they were not one-sided,
they would not be horizons.
Strands thus predict the lack of matter, energy and space behind a horizon for the ob-
server belonging to the horizon. This is predicted both for black hole horizons and for the
cosmological particle horizon.
The prediction of an actual, true void behind all horizons arises from the specific com-
bination of gravity and quantum effects that is provided by strands. The prediction is in
full contrast with classical cosmology, as derived from pure general relativity. Despite
this contradiction, the prediction is not in contrast with observation, as nothing behind the
cosmological horizon can be detected. In fact, the prediction that nothing exists behind the
cosmological horizon will be of importance in the discussion on inflation and on FLRW
models.
In short, the strand conjecture confirms the basic cosmological observations about our universe.
12 Strand predictions about vacuum, topology, dimensionality, singularities and
shape
Pr. 21 In the strand conjecture, an infinite flat vacuum is impossible, because the situation is
undefined. An infinite flat vacuum is an idealization. This agrees with observation.
Pr. 22 Because strands form tangles, as illustrated in Figure 8 and in Figure 9, and because tangles
represent quantum particles, the strand conjecture implies that the universe is not empty,
but filled with particles and physical space. For example, strands imply that we live neither
in a pure de Sitter space nor in a pure anti-de Sitter space. This agrees with observation.
Pr. 23 Crossings and tangling are not possible in other dimensions. The strand conjecture implies
that the interior of the universe has trivial topology with three spatial dimensions also on
galactic and cosmological scales.
The quantum of action does not allow determining dimensionality at scales below the
Planck scale. In the strand conjecture, dimensionality is an intrinsic property. No other
number of dimensions than three is possible. This agrees with observation.
Pr. 24 Strands imply that there has never been a situation in which the universe had an infinite
density or an infinite temperature T. In the strand conjecture, these observables are
limited by the respective Planck values:
≤c5
4~Gand kT ≤rc5~
4G,(13)
18

where kis the Boltzmann constant. Strands predict the lack of an initial singularity, and
also the lack of any other kind of singularity. This also implies the lack of a "big rip". So
far, these statements do not contradict observations.
Pr. 25 A cosmological horizon that rotates against the average matter inside it is a theoretical
possibility. Also more complex relative motions can be imagined. Strands predict that for
statistical reasons, these options do not arise. However, no data is yet available on this
issue.
Pr. 26 Strands also predict that a non-spherical horizon is impossible: Again, for statistical rea-
sons, a spherical horizon is vastly more probable than a horizon of any other shape. This
agrees with the data.
Pr. 27 Strands imply that the interior of the universe – all matter and radiation – is, on average,
flat, homogeneous and isotropic. This is observed. This will be explored in detail in the
next section.
In short, the strand conjecture confirms the general observations made about space in the universe.
13 Strand predictions about matter’s origin, expansion speed and density limits
In contrast to first impression, quantum effects can play a role at galactic and cosmological scales,
as several authors have pointed out [28–30]. The following sections explore this field of enquiry.
Pr. 28 In strand cosmology, matter, radiation and space arise at the cosmological horizon. The
strands that make up matter, radiation and space ‘start’ at the cosmological horizon: they
leave the horizon and enter the interior of the universe. The final effect is that the cosmo-
logical horizon is predicted to shine.
⊲All particles in the universe were radiated from the horizon.
This statement cannot be checked directly, but conclusions from it can.
Pr. 29 Whenever a matter tangle arises at or near the horizon, the horizon departs further. For
every strand switch that occurs inside the horizon, something similar will occur on the
horizon. This process ‘pushes’ the horizon further away.
This motion of the distant horizon is not limited by c, because it is not measured locally.
Pr. 30 Due to the superluminal expansion of the horizon and of space, strands also imply that
distant matter appears to depart faster than c. This superluminal motion of the horizon
and of matter near it is observed [24]: redshifts above 1 for distant matter are routinely
observed in astronomy.
19

Pr. 31 A cosmological horizon allows deducing a limit for matter density if the surface through
which matter appears is chosen carefully. General relativity limits the luminosity, or en-
ergy flow, by P≤c5/4G= 0.91(1) ·1052 W– provided that the surface is physical and
closed, i.e., that every point of the surface can be assigned to a physical observer.
In this way, strands appear to imply that also the luminosity of the Hubble radius is limited
by c5/4G. This luminosity limit has been frequently explored in the literature [9–13].
The luminosity limit leads to an energy density limit when the age t0of the universe is
included. The limit energy density is given, for flat space, by twice their product:
ρ0≤3
8πG (t0)2≈8.6·10−27 kg/m3.(14)
This is the so-called critical density [23]. The value corresponds to a few atoms per cubic
metre. It arises directly from the power limit.
Pr. 32 The power or luminosity limit of the universe, or equivalently, its energy density limit, can
be checked in various ways: for photons, for neutrinos, for baryons, and for their sum. For
photons, the luminosity of the horizon can be estimated using the Stefan-Boltzmann law
P=σAT 4– where σ= 5.67 ·10−8W/m2K4. In the universe, almost all photons are in
the cosmic background radiation [21]. Even using the observed temperature T0= 2.7 K
of the cosmic background radiation as horizon temperature – an overestimate by 30 orders
of magnitude, as we will see shortly – the resulting luminosity is about a thousand times
smaller than nature’s power limit. In other words, despite the considerable size of the
cosmological horizon, at most one crossing switch every thousand Planck times on it is
due to photons. Photons thus obey the power limit.
For neutrinos, the resulting value for the energy density is again small compared to the
corrected Planck limit. With an estimated temperature T0= 1.9 K of the neutrino back-
ground [21], the resulting neutrino luminosity is orders of magnitude smaller than the
power limit.
For baryons, the number and the density of baryons in nature, for flat space, is also pre-
dicted to be limited by the critical density. The baryon number limit is
N0,baryons ≤c3t0
2G mbaryon
.(15)
Using t0= 13 800 million years, the expression yields a predicted numerical value of
N0,baryons ≤5.4·1079. The limit is clearly larger than the usual estimates from observa-
tions [21, 23].
More precisely, in ΛCDM cosmology, the measured baryon density value is Ωbh2=
0.022 = 0.049 Ωcrit ≈4.2·10−28 kg/m3. The observed baryon density is indeed be-
low the density limit naively predicted by the strand conjecture. Observations thus respect
the baryon density limit.
20

As a further check, also the sum of all particle luminosities is observed to be below the
critical density. They do not exceed maximum power. On the other hand, known particles
alone do not reach the critical density.
As a note, one could argue that the radius of the surface for the power limit should be
the distance to the cosmological horizon, not the Hubble radius. In ΛCDM cosmology,
the cosmological horizon is 3.3 times further away than the Hubble radius (4.4·1026 m
instead of 1.3·1026 m). If the power limit is applied to the cosmological horizon radius,
there is a contradiction between the limit and the observed density. However, the surface
defined by the cosmological horizon is not physical, in contrast to the Hubble radius.
Therefore, the power limit does not apply to the cosmological horizon and no useful limit
density can be deduced.
Pr. 33 With the changes in temperature over time that are deduced below, strands imply that the
expansion rate of the universe first decreased because of the increase of matter content.
Afterwards, when the temperature was so low that matter was not generated any more, the
expansion rate increased because of dark energy. This agrees with conventional cosmol-
ogy.
Pr. 34 Strands appear to predict that in the present, dark-energy-dominated era, the horizon has
a low temperature. Therefore, the matter density of the universe decreases roughly as
ρ∼1/R3. This prediction agrees with the ΛCDM model.
In short, the predictions of the strand conjecture about the basic properties of the universe either
agree or at least do not contradict observations. However, no prediction for the value of the ex-
pansion speed has been deduced yet. No compelling way to approach the issue has been derived
yet. In short, the understanding of the present speed of cosmic expansion, 3.3c, requires an addi-
tional idea. This is still a subject of research. It remains an open challenge to simulate the history
of the universe, in particular of the early universe, using numerical or analytical methods. Such
simulations will also allow checking whether strand fluctuations lead to a sufficient amplitude for
the density fluctuations required to start galaxy formation.
14 Strand predictions for the cosmological horizon structure
Pr. 35 In the strand conjecture, the cosmological horizon is a maximally tight weave. A horizon
made from a loose weave makes no operational sense: it is indistinguishable by measure-
ments from a horizon that is closer and tighter.
Pr. 36 Observations show that in Planck units, the present cosmological horizon has a radius of
about 4.3·1026 m, or about 1.3·1061 smallest lengths.
In the strand conjecture, the cosmological horizon therefore contains about 1.7·10122
crossings. As a result, the cosmological horizon has entropy and temperature. This deduc-
tion cannot be tested directly, but indirect tests are possible.
21

Two tacit assumptions need to be verified. First: is the cosmological horizon tight, like a
black hole horizon? As mentioned, there is no physical difference between a tight horizon
and a loose horizon with lower strand density. Second: is there indeed one tether per
smallest area? Again there is no physical difference between a large horizon with few
tethers and a smaller horizon with the maximum tether density.
Pr. 37 In other terms, strands imply that the number of tethers per horizon area that leave towards
the interior is constant over time. The horizon tether density does not decay over time. For
the same reason, the horizon tethers density does not increase over time.
Pr. 38 Inside the universe, the most numerous strand crossings are those due to empty space.
These crossings are due to the tethers leaving the horizon. As mentioned, the cosmological
horizon has one tether per minimum area.
The number of tethers of the cosmological horizon yields an upper limit for the crossing
number inside the universe. The upper limit is based on the assumption that two random
strands cross only once on the horizon and only cross at most once more in its interior.
Figure 9 also makes the point.
Pr. 39 If all matter and radiation arises at the horizon, the number of strand segments making up
the horizon must be compatible with the number of strand segments forming the particles
in its interior. In the strand conjecture, it is assumed that, on average, a strand is only rarely
part of two different particle tangles.
Therefore, the maximum number of particles made of one strand (such as photons) that
can be observed inside the horizon must be smaller than the number of tethers from the
horizon. In the present universe, the observed number of photons is estimated to be around
1090 [21]. This number is indeed much smaller than the maximum number of tethers inside
the universe, which is about 10122.
Also (twice) the maximum number of particles made of two strands (such as quarks or
gravitons) that can be observed inside the horizon, must be smaller than the number of
tethers from the horizon. For quarks, this is the case; for gravitons, there is no good
observational estimate. In his work, Page [31] estimated that more than 10113 gravitons
exist. Another estimate starts from the fact that gravitons carry, together, much less energy
than photons, and have, on average, a 1025 times larger wavelength [32]. This kind of
estimate yields more than 10110 gravitons in the universe. The number of gravitons does
not exceed the maximum possible value.
Finally, (thrice) the maximum number of particles made of three strands (such as neutrinos
or charged leptons) that can be observed inside the horizon must also be less than the
number of tethers from the horizon. The higher number of these, the number of neutrinos
in the universe, is expected to be very similar to the number of photons [31]. The total
number of observed particles is thus indeed smaller than the (maximum) number of tethers
from the cosmological horizon.
22

Pr. 40 In any case, the strand conjecture implies that most particles inside the universe are gravi-
tons because they arise naturally during expansion. This agrees with expectations [32]. On
the other hand, the topic is not simple, as the discussion on the nature of dark energy will
show below.
Pr. 41 The above predictions of the strand conjecture imply
⊲Cosmic vacuum arises from the horizon and is criss-crossed by strands.
This is the central property of the cosmic vacuum in the strand conjecture.
The structure of the cosmic vacuum merits a detailed investigation. The first task is a compar-
ison with Minkowski vacuum.
In the strand conjecture for infinite flat space with vanishing vacuum energy, the vacuum strand
density vanishes. Also entropy and temperature vanish. For example, in an infinite and flat uni-
verse, there are no effects analogous to the Tolman temperature. The lack of vacuum strands
confirms that flat infinite space, the Lorentz vacuum, is not an actual model of nature, but an ideal-
ized limit case. In such an infinite and flat space – with or without vacuum energy – strands imply
that gravitational and inertial mass are equal because both mass effects arise from the same strand
process, namely the belt trick. In this limit, the equivalence principle is thus valid exactly.
The next task is to explore how the statements on Minkowski space are modified for the case
of a universe with an expanding cosmological particle horizon. This will be done in the following.
Pr. 42 In the strand conjecture, energy is a quantum effect. In addition, in the strand conjecture,
every energy density is a crossing switch density per time. The strand model of the uni-
verse implies that the vacuum, even if flat, is criss-crossed by tethers coming from the
cosmological horizon. These strands cross inside the universe. As a result,
⊲Dark energy is a natural consequence of the strand model: it is due to the cross-
ing switches in the cosmic vacuum. Because the density of crossing switches
is low, dark energy has a small positive value.
This central result of the strand conjecture is already implied in Figure 9. The next step is
to deduce quantitative predictions from this qualitative statement.
In short, in the strand conjecture, a vacuum arises at the horizon. The expansion increases the
number of tethers in the interior and leads to a small dark energy density. Also matter arises in the
interior of the universe, once the tethers tangle up. This occurs rarely; therefore, the matter density
is small compared to the maximum imaginable value. The strand conjecture suggests that matter
is only created when the universe is small and hot. All this is not in contrast with usual cosmology.
23

15 Strand predictions about the temperature of the horizon and the vacuum
Pr. 43 Figure 9 also implies that during the expansion of the cosmological horizon, new strands
and new crossings continuously appear in the cosmic vacuum. In addition, due to the
expansion, the strands continuously rearrange. In other words, together with vacuum
energy,
⊲Strands imply a non-vanishing entropy and temperature of the cosmic vacuum.
The cosmic vacuum is thus a bath. So far however, these properties have not been ob-
served.
Pr. 44 Strands imply that the naive (absolute value of the) present temperature of the cosmologi-
cal horizon, for de Sitter space, is given by
|Tc0|=~c
2πk
1
R(16)
where Ris the radius of the horizon. It is natural to assume that de Sitter space is an
approximation for the present universe. Inserting the radius of the present cosmological
horizon, Rc0= 4.4·1026 m, or 3.3ctimes 13 800 million years, yields
|Tc0|= 8.5·10−31 K,(17)
The resulting temperature of roughly 10−30 Kcorresponds to a kinetic energy of around
10−34 eV. This vanishingly small horizon temperature confirms older research [33–35];
it is not in contradiction with observations. Strands thus imply that the temperature of
the horizon is so low that it will probably never be checked by direct, local experiments.
Also the temperature value is so low that it will not be detected by cosmological measure-
ments. The temperature of the present cosmic background radiation, 2.7 K, is much larger
and will mask the horizon temperature. This masking also occurs for neutrinos. Even
if the situation were different for gravitons, the fact would be purely academic, as single
gravitons cannot be detected [36, 37].
Pr. 45 The horizon temperature, properly speaking, is negative, as argued by Klemm [38] and by
Cvetiˇ
c et al. [39]. The entropy flow that results points towards the horizon. The sign and
small value of the horizon temperature do not appear to influence the matter far inside the
horizon.
Pr. 46 Strands imply and predict that the (absolute value of the) vacuum temperature of the inte-
rior of the universe is and has always been equal or larger than the (absolute value of the)
temperature of the horizon. The (absolute value of the) horizon temperature is predicted
to be an effective lower bound for vacuum temperature. The two temperatures are equal if
they are in equilibrium. In general, they are not. This agrees with expectations.
24

Pr. 47 It might be worth noting that the ‘black hole’ luminosity of the horizon, despite its size,
is much lower than the limit c5/4G. The reason is that the temperature of the horizon
decreases with the radius as 1/R; as a result, the total emitted power, which depends on
T4, is extremely low and continues to decrease. The present luminosity of the horizon is
negligible. In the past, however, the luminosity was much higher.
Pr. 48 Both the negative horizon temperature and the strand model appear to imply that there is
no measurable horizon relic radiation from the times when the horizon was much smaller
and hotter. However, this prediction cannot be checked by experiments, as the involved
numbers are too small.
In short, strands imply that the vacuum temperature and the horizon temperature can differ, due
to non-equilibrium effects. Strands also imply that the horizon temperature decreases with the
expansion of the universe, and so does the vacuum temperature. The precise effects of the contin-
uous addition of a colder vacuum to the already existing warmer vacuum must still be worked out.
The corresponding effect for photons could be a guide.
16 Qualitative strand predictions about horizon and interior entropy
Pr. 49 Strands imply that the cosmological horizon is a tight weave. As a consequence of weave
tightness, the expression for entropy of a black hole, which is also a tight weave, can be
used for the entropy SCH of the cosmological horizon. Using A= 4πR2and R=... m,
one gets
SCH =A kc3
4G~≈10122k . (18)
This entropy value agrees with other published estimates [25]. The concavity of the cos-
mological horizon, which contrasts with the convex shape of a black hole horizon, should
not change the result.
Pr. 50 Strands imply that the entropy of the cosmological horizon has the same value as or more
than the entropy of the interior. This yields
Sinterior ≤O(1)A kc3
4G~≈10122k . (19)
Since the universe is mainly empty, the interior entropy is predicted to be mainly the
entropy of the cosmic vacuum. In other words, strands predict the existence of entropy of
dark energy.
Pr. 51 Once the interior entropy is known, one can deduce an entropy density of the cosmic
vacuum, or of dark energy, given by
sinterior =Sinterior
V≤1044k/m3.(20)
This is an astonishingly high limit value. For comparison, a gas at standard conditions has
an entropy of about 1026k/m3.
25

In short, strands imply that the entropy density of the cosmic vacuum is positive and considerable.
The next step is to deduce quantitative statements.
17 Accelerated expansion: observations and usual description
Observations of the distance-velocity relation of distant galaxies, using type Ia supernovae, lead
to the conclusion that the expansion rate of the universe is accelerating. The acceleration of the
expansion is a small effect that requires elaborate precise astronomical measurements at cosmo-
logical scales. At sub-galactic scales, including the solar system and all laboratories, no effects of
the acceleration of the expansion are observed.
The most common description for the acceleration is that of ΛCDM cosmology: acceleration
is due to dark energy, which is described by a non-vanishing, positive cosmological constant Λ
that is a fundamental constant of nature. The cosmological constant Λis part of the field equations
(12) and of the Lagrangian of general relativity. Measurements yield the value [23]
Λ = 1.1·10−52 m−2.(21)
This value is small: it is close to the inverse square of the Hubble radius.
Cosmological measurements about dark energy also find an energy per pressure value of
E/p =w=−1(22)
within measurement errors [23], consistent with a constant value of dark energy over time. The
constancy of Λhas been measured over an age span that covers most of the age of the universe.
Apart from the distance-velocity relation, most other effects of the cosmological constant Λ
are seen in simulations of the history of the universe. It might be that Λis related to the rotation
curves of galaxies; however, this issue is not settled. In any case, no effect of Λis predicted to be
observable at sub-galactic distances or in everyday life.
In short, in the ΛCDM model, the value of Λis an unexplained, fundamental constant of
nature.
18 Proposed explanations of expansion acceleration
Over the years, researchers regularly have tried to understand the value of Λ. In the concordance
model of cosmology, the value is constant over time. The explanation attempt by Verlinde is based
on the thermodynamics of space-time [28]. Other examples are the proposals by Hossenfelder [41]
and by Padmanabhan [29]. None of these explanation proposals has gained general acceptance.
Several alternative and more radical explanations for the acceleration of the expansion of the
universe also exist, triggered by the closeness between Λand the inverse squared Hubble radius. In
the first alternative, the measured acceleration is seen as a consequence of a peculiar (accelerating)
observer status of the Earth [42]. This alternative explanation is not yet ruled out by observations
[43, 44].
26

In a second, similar alternative, the acceleration is seen as an artefact of the inhomogeneities
of the universe. If the Earth is located in a so-called (partial) void, an apparent acceleration can
arise, as explained by Wiltshire [45, 46].
In a third alternative, certain researchers question the measurement data that led to the conclu-
sion of acceleration [47, 48]. They argue that the conclusion is due to a bias in the selected set of
type Ia supernovae that is used to determine distance values.
In a general fourth alternative, various authors have proposed changes to general relativity
itself. These approaches have been compared and discussed in detail elsewhere [49]. Most of these
proposals are not discussed in the present work, because the agreement with data is questionable.
Many proposals disagree with general relativity – because they contradict maximum force, as
discussed in reference [15] – or disagree with the standard model of particle physics. Therefore
these proposals also disagree with the strand conjecture.
A fifth alternative is formed by the models built on quintessence [50]. These approaches
are less radical and just assume an evolution of Λwith time. A slow evolution of Λmight be
compatible with data.
In short, so far, all observations failed to determine the origin of Λ. Equally, many models
failed to calculate the value of Λ. If the strand conjecture is correct, it must be able to distinguish
between ΛCDM cosmology, the proposed explanations of Λ, and the mentioned five alternatives.
The strand conjecture must make specific, testable predictions, and must explain the origin, value
and time-dependency of Λ. It is useful to start with some general considerations.
19 Strand predictions about the vacuum: energy, entropy, temperature and their
time dependency
Pr. 52 Figure 9, showing the history of the universe, illustrates that the density of strands in the
cosmic vacuum is determined by the horizon. The cosmic vacuum can be seen as a result
of the tethers starting at the horizon. In particular, this implies that the strand density in
the cosmic vacuum is non-vanishing,flat,homogeneous and isotropic across the interior
of the universe. All this is observed. Therefore, both vacuum entropy density svac and
vacuum energy density uvac are predicted to be homogeneous and isotropic – as long as
matter and radiation do not disturb it. There is no contradiction between this prediction
and observations.
Pr. 53 The vacuum entropy density svac and the vacuum energy density uvac are related.
uvac =O(1) ·svac Tvac .(23)
This relation predicts that also the vacuum temperature is homogeneous and isotropic. So
far, measurements have not allowed any check of this statement – because the vacuum
temperature is around 10−30 K– but the result is expected.
The next task is to deduce the time dependency and the values of all the vacuum quantities from
strands. The following arguments are based, as before, on the number of tethers leaving from the
27

cosmological horizon: it is assumed that one tether starts at each smallest horizon area AcPl =
4G~/c3.
Pr. 54 First of all, the change in strand density in the cosmic vacuum can be estimated from the
average number and the average length of strands, both of which depend on the cosmo-
logical horizon surface and volume. Given a horizon radius R, the average strand length
grows as 3R/4, and the strand number grows as R2. The total strand length thus grows as
R3. As a result, the strand density – i.e., strand length per volume – is thus independent of
the size Rof the horizon. In other words, the strand density is constant over time.
Pr. 55 The value of the strand density can be estimated from the geometry of the situation. The
result is surprising: there is, within a factor O(1), a strand in every Planck volume. In other
terms, strands make the astonishing prediction that vacuum is essentially full of strands.
in contrast to the low density illustrated in Figure 8 and Figure 9.
Pr. 56 Secondly, the crossing density NC/V has to be explored, and in particular, its change with
radius R. The close-by crossing density, of strands passing each other close by, does not
depend on R: it is constant. The total crossing density, of strands passing each other at any
distance, will grow as R4/R3=R. In short, even though strand density is constant, the
crossing density either is constant or increases; its change depends on the distance range
of the crossings under consideration.
Pr. 57 Again, the value of the close-by strand crossing density can be estimated from the geom-
etry of the situation. Again, there is, within a factor O(1), a crossing in every smallest
possible volume. Strands thus predict that vacuum is essentially full of strand crossings.
Pr. 58 Strands allow estimating the change of the crossing switch density. To do this, the average
time taken by a crossing switch needs to be determined. In particular, it must be understood
how this average time depends on the horizon size R. Only crossings of close-by strands
can switch with a finite probability. The probability depends on the speed and amplitude
of the average strand motion. In a highly dense strand vacuum, these two quantities will
not depend on R. Thus the crossing switch density of close-by strands is expected to be
constant in time.
In other terms, as long as close-by crossings dominate, the energy density Λof the vacuum
is constant in time. The same is predicted for the entropy density and the temperature of
the vacuum. It needs to be noted that this result also assumes that vacuum is the dominant
structure in the universe, i.e., that matter and radiation play a negligible role. This is indeed
the case at the present age of the universe.
In short, strands imply that
⊲The cosmological constant Λis a large-scale quantum effect.
Pr. 59 Strands should also allow estimating the present value of the vacuum energy density Λ.
Close-by crossings will switch only rarely, due to the fully packed vacuum. The process is
28

rare; as a result, Λis very small. This is as expected, and is compatible with observations
[23].
However, at present, there does not appear to be a good way to estimate the probability of
a crossing switch in a fully packed vacuum. Nevertheless, a few results arise.
Pr. 60 Strands imply that dark energy is due to the horizon. But strands imply more:
⊲Strands imply that dark energy is not different from cosmic vacuum.
Strands imply that there is no cosmic vacuum without dark energy.
Pr. 61 Strands also imply something about the nature of dark energy:
⊲Dark energy resembles extremely cold particles – gravitons, photons, neutrinos,
etc. – with cosmological-sized wavelength.
This result is a genuine strand prediction. It is not in contradiction with observation. So
far, there is no research literature on this option.
Pr. 62 In other terms, strands imply that dark energy is not fundamentally different from known
matter and radiation. However, strands predict that at cosmological scales, different types
of particles cannot be distinguished from each other. Strands thus predict
⊲Dark energy has no specific particle aspect that allows determining the involved
particle types.
All this does not disagree with observations, but it challenges habits of thought.
Pr. 63 In conventional physical language, the cosmological horizon has a temperature and radi-
ates; the radiated particles are real, as in every thermal radiation. However, the strand
conjecture modifies this conclusion:
⊲There is no difference between real and virtual particle at cosmological scales.
The cosmic expansion can thus be said to generate a mixture of real and virtual particles.
At cosmological scales, the two cases are indistinguishable in the strand model of dark
energy.
Pr. 64 Summing up the previous three predictions, strands imply that at cosmological wave-
lengths, there is no difference between vacuum and particles. This unexpected result in
the infrared limit complements the older, similar strand result in the ultraviolet: at Planck
scales, strands do not allow distinguishing vacuum and matter. The equivalence in the ul-
traviolet has been known since a long time [40], but the equivalence in the infrared appears
to be new. It might be surprising that neither limit contradicts observations.
29

Pr. 65 Dark energy, like any other type of cosmic energy density, is characterized by the dimen-
sionless factor win the equation of state p=w that relates the pressure pand the energy
density . The factor wdescribes the type of behaviour of the energy density.
For photons and for relativistic matter, whas the value 1/3; for slow (non-relativistic)
matter, such as baryons, whas the value 0. For a fundamental constant, whas the value
−1.
In the strand conjecture, the number win the equation of state p=w is negative because
the vacuum effectively pushes the horizon away.
In the ΛCDM model, the equation of state for dark energy is w=−1, i.e., the cosmo-
logical constant is indeed constant. Observations confirm this wvalue for dark energy to
within 10% [23].
Pr. 66 The value of the vacuum energy density cannot be estimated – yet. It is not clear whether
strands imply a measurable or a negligible value of Λ. Also the time dependence is not
completely clear. A comparison with the literature [51–53] is not yet possible.
In short, strands naturally allow the modelling of dark energy in all its properties.
20 Strand predictions about inflation
Pr. 67 Because in the strand model, flatness, homogeneity and isotropy arise naturally, strands
have no necessity for inflation to explain these observations. The same holds for density
fluctuations.
Given that in the strand conjecture, matter arises at the horizon, strands imply that there
is no re-entry into the cosmological horizon. Strands predict that no stars, no galaxies, no
matter – nothing – ever re-enters the horizon. Again, this is an argument against inflation.
Because strands do not allow fields outside those of the standard model, strands also ex-
clude every field mechanism proposed for inflation in the past [54].
In short, for three reasons, strands imply that inflation did not occur. This conclusion does not
contradict observations but is hard to test in detail.
21 Discussion
The exploration has led to several conclusions.
⊲Strands are compatible with a very small or with vanishing vacuum energy.
⊲Strands imply that vacuum energy density remains constant with increasing cosmo-
logical radius.
⊲Strands predict that w=p/ is negative and compatible with the value −1.
⊲Strands imply that the value of the vacuum energy density Λis hard to estimate.
30

These results are disappointing. Still, it could well be that there are errors in the discussion so far.
A check makes sense.
First of all, the universe could be made of several strands. Exploring the option, one finds that
it does not affect the numerical predictions.
Secondly, the number of horizon strands leaving towards the interior of the universe need not
match those of a Schwarzschild black hole. The number of tethers could be less than one per
minimal area. In this case, the entropy would still be proportional to area, but the proportionality
constant could be different from that of black holes. Again, this would have no effect.
Thirdly, the ratio between tether number and horizon area could be smaller, or even change
over time – increasing or decreasing. In this case, the entropy of the horizon could depend on the
area in a different way. This would also change temperature and energy. But the option seems
impossible: every strand in the horizon has a tether.
A fourth option needs to be explored. Could the time dependence of Λdepend on the setting?
Could it be that Λis constant when a measurement is local, such as in quantum theory, but that it
decays when a measurement is global? In other terms, can Λbe, at the same time, both constant
and decaying? Or is Λunmeasurably small throughout? So far, the strand conjecture has difficul-
ties in making hard statements. Strands appear to allow a constant, an increasing and a decaying
cosmological constant Λ. In the case that Λis constant, strands do not yet yield a simple way of
calculating it. In the case that Λif found to decay, strands suggest Λ = 1/R2.
22 Outlook
From one fundamental principle the strand conjecture deduces the standard model of particle
physics with massive mixing neutrinos, without any modification [4]. From the same fundamen-
tal principle, at sub-galactic scales, the strand conjecture deduces general relativity, the lack of
singularities, as well as the existence of a maximum force value c4/4Gand a maximum power
value c5/4G[16]. The agreement of these consequences with all observations so far encourages
the exploration of larger scales.
In the domain of cosmology, fluctuating strands predict an expanding cosmological (particle)
horizon. Strands starting at the cosmological horizon and transversing the interior form matter,
radiation and dark energy. Matter and radiation are due to tangled strands, dark energy is conjec-
tured to consist of crossing switches of untangled vacuum strands.
The strand conjecture implies that the cosmological horizon has entropy and temperature. The
expansion of the universe results from the ever-increasing tangling of the interior strand segments.
The expansion rate first decreased because of the increase in matter content and then increased
because of dark energy.
Applying the power limit c5/4Gto a surface inside the Hubble radius leads to an upper limit of
energy density for the universe. All observed energy densities, taken together, are lower than the
limit. Strands imply that, at present, the matter density decreases with time, as observed, because
no new matter is being created any more.
31

In the strand conjecture, dark energy can be seen as being made of real and virtual partiles with
cosmological wavelength. At those wavelengths, strands also predict that particle types cannot be
distinguished from each other. Strands predict that dark energy has a low-density value, deter-
mined by the size, entropy and temperature of the cosmological horizon. Strands further suggest
that, at present, the density of dark energy is constant. Strands imply that dark energy manifests
itself as fluctuations of space. Future investigations should check these conclusions in all possible
ways.
The strand conjecture can be tested in the domain of cosmology with simulations of the col-
lective dynamics of large numbers of strand segments. This exploration will allow calculating the
time-dependence and spatial variation of matter and radiation densities. Also the topic of dark
matter and galaxy rotation curves must be explored. This will be done in a subsequent article.
Further comparisons with cosmological measurements will then be possible.
23 Acknowledgments and declarations
The author thanks Thomas Racey, Pavel Kroupa, Gary Gibbons, Isabella Borgogelli Avveduti,
Philip Mannheim, David Thornton and Silke Klemm for discussions. The work was partially
supported by the Klaus Tschira Foundation. The author declares the lack of competing interests.
Appendix A On the circularity of the fundamental principle
Any strand crossing switch is assumed to take place in space. On the other hand, space, distances
and all physical observables are assumed to arise from strands. The circularity can greatly be
reduced – but not fully eliminated – with a more precise formulation.
In detail, crossing switches take place in a local background space, a space defined by the
observer to allow descriptions of observations. In contrast, physical space, including physical
distances and physical observables, arises from strand fluctuations and their crossing switches.
When space is flat, background space and physical space are identical. When space is curved,
background space is (usually) the local tangent space of physical space.
The strand conjecture stresses that a description of nature without a local background space
and time is impossible. Every observation and every measurement of an object or a system requires
the use of a local background space introduced by the observer. (There are many possible choices
for the background – as many as there are possible observers.)
The need for a local background space is due to a fundamental contrast between nature and its
precise description. The properties of nature and the properties required for a precise description
differ; in fact, they contradict each other. A precise description of nature requires axioms, sets,
elements, functions and points in (local background) space and time. In contrast, due to the un-
certainty relation, at the Planck scale, nature itself does not provide the possibility to define points
in (physical) space or time; they are emergent. Due to the uncertainty relation, neither sets nor
elements exist in nature at the most fundamental level. As a result, an axiomatic description of
32

nature is impossible. Axiomatic descriptions only exist for parts of physics, when the description
and nature itself are approximated to have the same properties. Axiomatic descriptions are possi-
ble in quantum theory, in electrodynamics, in special relativity, or in general relativity. Axiomatic
descriptions are impossible when the uncertainty relation is applied to space itself, thus not when
describing all of nature at the same time. In particular, axiomatic descriptions are impossible for
quantum gravity.
Because of the impossibility of an axiomatic description of nature and of quantum gravity,
any complete description of nature must include a limited degree of circularity, in particular in its
definition of time and space. By using background space, the strand conjecture indeed introduces
such a limited amount of circularity: physical space is defined with the help of particles – e.g.,
with rulers made of matter or light – and particles are defined with the help of physical space –
for example, with energy and spin that are localized in three spatial dimensions. Both physical
space and particles arise in background space. At first sight, this eliminates the circularity. But
background space itself arises as an approximation of local physical space. Therefore, a small
amount of circularity remains. Despite this residual circularity, strands do allow a description of
nature.
The main example of the difference between an axiomatic description and a complete, consis-
tent and partially circular description is the dimensionality of space. The number of spatial dimen-
sions is not a consequence of the fundamental principle or of some axiom; the number of spatial
dimensions is included in the fundamental principle right from the start. The number of dimen-
sions is included at the start because it is the only consistent choice: Only three dimensions allow
crossing switches, allow particle tangles, allow spin 1/2, allow Dirac’s equation, allow deducing
U(1), broken SU(2) and SU(3), and allow deducing Einstein’s field equations. Only a background
space with three dimensions allows the existence of tangles and allows a precise description of
nature.
In short, due to the use of a local background space, the strand conjecture cannot be tested
by asking whether it is an axiomatic description of nature. No unified description of nature can
be axiomatic. Every unified description and every model of quantum gravity must be somewhat
circular. Nevertheless, like any unified description, the strand conjecture can be tested by asking
whether it is a correct, complete and consistent description of nature. So far, it has passed all
experimental and theoretical tests [4, 5, 16, 20].
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