# Comparing Classical-Quantum Portfolio Optimization with Enhanced Constraints

Salvatore Certo<EMAIL_ADDRESS>Deloitte Consulting, LLP  Anh Dung Pham
Deloitte Consulting, LLP  Daniel Beaulieu  Deloitte Consulting, LLP

###### Abstract

One of the problems frequently mentioned as a candidate for quantum advantage
is that of selecting a portfolio of financial assets to maximize returns while
minimizing risk. In this paper we formulate several real-world constraints for
use in a Quantum Annealer (QA), extending the scenarios in which the algorithm
can be implemented. Specifically, we show how to add fundamental analysis to
the portfolio optimization problem, adding in asset-specific and global
constraints based on chosen balance sheet metrics. We also expand on previous
work in improving the constraint to enforce investment bands in sectors and
limiting the number of assets to invest in, creating a robust and flexible
solution amenable to QA.

Importantly, we analyze the current state-of-the-art algorithms for solving
such a problem using D-Wave’s Quantum Processor and compare the quality of the
solutions obtained to commercially-available optimization software. We explore
a variety of traditional and new constraints that make the problem
computationally harder to solve and show that even with these additional
constraints, classical algorithms outperform current hybrid solutions in the
static portfolio optimization model.

##  I Introduction

While quantum computers are theorized to solve problems currently intractable
on classical computers, the current devices are noisy, small, and are limited
with the types of algorithms feasible in today’s devices. While gate-based
machines in the Noisy Intermediate Scale Quantum (NISQ) era struggle to find
appropriate feasible applications, quantum annealers have less constraints and
appear to be the most promising in near-term industrial implementations.

Quantum annealing follows the adiabatic theorem, which states that if the time
evolution is long enough, a quantum system will stay in the ground state of
its Hamiltonian. For a quantum system starting in a ground state of a
hamiltonian $H_{b}$ and evolving to the ground state of the problem
hamiltonian $H_{p}$, the system can be described at time $t$ out of the total
run time $T$ as:

$\mathcal{H}(t)=(1-u(t))\mathcal{H}_{b}+u(t)\mathcal{H}_{p}$ (1)

where $u(t)=t/T$. Quantum annealing allows for interesting problems to be
solved by formulating the problem into the Hamiltonian $H_{p}$. At the end of
the annealing process, the system should be in the ground state and the
solution to our problem found.

Combinatorial optimization problems are perfect candidates for this annealing
process, as the problem Hamiltonian is easy to construct and is diagonal in
the computational basis, thereby having the ground state being the exact
readout of the qubits. A well-known combinatorial optimization problem is that
of portfolio optimization, where the task is to choose the assets in the right
proportion as to maximize return while minimizing risk.

This is an especially relevant problem not only for financial institutions,
but for a variety of industries as non-financial assets can also be used to
construct the problem bouland2020prospects herman2022survey mugel2020use . It
is also easy to formulate, the necessary data is readily available, and the
solutions are particularly significant for industrial applications. The
problem follows that of Markowitz’s Modern Portfolio Theory markowitz  that
the optimal portfolio is constructed with assets that maximize return for a
given level of risk.

This problem can be mapped to a Quadratic Unconstrained Binary Optimization
(QUBO) problem defined by the problem hamiltonian:

$\mathcal{H}_{p}=-\mu^{T}w+q*w^{T}\Sigma w+k(\sum_{n}{w_{n}}-1)^{2}$ (2)

where $\mu$ is the vector of logarithmic returns, $w$ is a vector of $n$
assets, where $w_{n}\in[0,1]$ the fraction of the total budget invested in
each asset $n$, $\Sigma$ is the matrix specifying the covariance of assets,
and $q$ is the given level of risk appetite for the investor. Typically, one
can adjust $q$ to map the target level of risk to the problem.

As QA require the cost function to be unconstrained, we can add soft
constraints with appropriate Lagrange Multipliers to enforce conditions. For
instance, we will want the total available budget to be invested, and so we
can add to $\mathcal{H}_{p}$ the budget constraint $k(\sum_{n}{w_{n}}-1)^{2}$
where we set $k$ to be sufficiently large enough to ensure the constraint is
satisfied.

Portfolio optimization models have been implemented in various forms on QA
cohen2020portfolio cohen2020portfolio_60 . Some simple implementations include
binary decision variables for each asset specifying whether or not to include
them in the portfolio, while others go a step beyond and create variables for
the exact number of shares or proportion of the asset to purchase given the
available budget. In this paper we extend the latter model in several ways.
First we introduce slack variables as an alternative to enforce minimum and
maximum investments in particular sectors, showing that this formulation has
several benefits that make it advantageous compared to other methods. We then
show how we can construct global and local constraints based on quantitative
features like balance sheet metrics to construct portfolios that not only
maximize return but also have the desired characteristics found in fundamental
analysis of stocks.

As current QA do not have the number of qubits nor the required connectivity
between them to implement large-scale models directly on annealers, we explore
the use of D-Wave’s hybrid models, specifically for implementing the QUBO
representation of the model in their hybrid Binary Quadratic Model (BQM) as
well as using integer variables and explicit constraints in their hybrid
Constrained Quadratic Model (CQM). We note that the QUBO formulations we
present will be especially advantageous once the annealing hardware matures
and the number of qubits increase, but also show the benefit of hybrid solvers
in the current era. We therefore provide direct comparisons between both
implementations and finally compare the results against a leading optimization
software package CPLEX.

##  II QUBO Model and Constraints

First we should note that in the model from Equation 2, instead of $w_{n}$
being the weight invested in asset $n$, it can instead be formulated to
include the current stock price with the goal being to specify how many shares
of each asset we wish to hold, considering our total budget $B$. This would
transform our cost function to:

$\displaystyle\mathcal{H}_{p}=-\mu^{T}w+q*w^{T}\Sigma
w+k(\sum_{n}{s_{n}p_{n}}-B)^{2}$ (3) $\displaystyle w=\frac{s_{n}p_{n}}{B}$

where $s_{n}$ is the number of shares invested in asset $n$, $p_{n}$ the
current asset price, and B the dollar value of our available funds.

This formulation ensures that we are only investing in bundles (or even lots)
and not fractional shares. While this is important, in reality only small
investors would need to ensure that the proportion of funds invested in each
asset is achievable in whole shares, as large portfolios can always be
constructed to an exact number of whole shares based on the weighted
proportion of that asset to the portfolio. In any case, the rest of this paper
can be reformulated for whole shares or round lots with a few substitutions of
variables.

In the subsequent sections we provide formulations for common real-world
constraints for the problem. These constraints, once appropriated mapped for
use in a QUBO problem, will be added to Equation 2 with penalty coefficients
to enforce the condition as necessary.

###  II.1 Binary to Integer Mapping

As our goal is to construct the cost function with binary only integers as to
be amenable to QA, we need a way to convert our binary variables to integer
ones. The easiest and most straightforward way to use a binary encoding
hybrid_holding  and a specified bit depth $N_{d}$ and degree of precision $P$
in our weights:

$w_{n}=\frac{1}{P}\sum_{d=0}^{N_{d}-1}2^{d}x_{n,d}$ (4)

where $x_{n,d}$ is a binary variable for each asset and bit. Here $P$ will be
the level of precision we want in our weights of assets to invest in. A $P$ of
100 will have each asset being invested in proportions of 1%, while a $P$ of
10,000 will allow us to have the smallest proportional weights of 1 basis
point. Based on our level of chosen precision, we can then choose $N_{d}$ such
that $2^{N_{d}}>P$.

A common condition in real life scenarios is to ensure there is a sufficient
level of diversification in the constructed portfolio. While we could add an
additional penalty term with a slack variable (similar to what will be
described in subsequent sections), a preferable method found in
palmer2021quantum  is to create the mapping:

$w_{n}=\frac{1}{P}\left(\sum_{d=0}^{N_{d}-2}2^{d}x_{n,d}+Mx_{n,N_{d}-1}\right)$
(5)

where $M=Pw^{max}_{n}-2^{N_{d}-1}-1$ and $N_{d}$ is chosen such that
$2^{N_{d}-1}-1\leq Pw^{max}_{n}$. For example, if $P=1000$ and
$w^{max}_{n}=2.5\%$, then $N_{d}=5$ and M = 10. This mapping directly imposes
an upper limit on the weight of each asset, and has the advantage that there
are no unnecessary bit variables included with the caveat that we will have to
specify this mapping for different asset-specific bounds.

###  II.2 Minimum and Maximum Sector Bands

The mapping specified above has also been used to create sector bands, i.e.
the maximum and minimum proportion of each industry sector we would like our
constructed portfolio to have. One approach to doing this palmer2021quantum
was to distribute equally the maximum sector bands across the individual
assets. This has a couple disadvantages, one being that by doing so we are
forcing the portfolio to include every asset in that sector with a high enough
minimum band. Another disadvantage is that we should allow for the flexibility
of allowing a subset of stocks to fill the entire quota for that sector, while
also satisfying the separate asset-specific maximum investment amount. For
instance, if we would only like 8% of our portfolio to be from the healthcare
sector while preventing one asset from being more than 2% of our portfolio, we
should still allow for the possibility of only 4 healthcare stocks being
selected for that sector. In our results this formulation allows for more
flexibility and more realistic input bands for sectors and assets.

We can therefore introduce more concrete investment bands on sectors by
introducing slack variables and creating soft constraints for maximum sector
allocation:

$\sum_{s}\left(\sum_{n}w_{n,s}+s_{s,u}-U_{s}\right)^{2}$ (6)

and minimum sector allocation:

$\sum_{s}\left(\sum_{n}w_{n,s}-s_{s,l}-L_{s}\right)^{2}$ (7)

where $s_{s}\geq 0$, $U_{s}$ and $L_{s}$ is our maximum and minimum percent
allocation for that sector, and $w_{n,s}$ is asset $n$ in sector $s$. Here
$s_{s,u}$ and $s_{s,l}$ are binary encoded integer-slack variables.

In Figure 1 we show the results of running the portfolio optimization model on
the entire S&P 500 with certain minimum and maximum investments per sector on
a QA. As compared to results elsewhere, our implementation allowed for much
tighter investment bands, more flexibility, and the hybrid solver was able to
satisfy all constraints formulated as penalty terms in the objective function.

Figure 1: Results for constraining sectors using the Hybrid BQM solver. The
red lines are the minimum and maximum allowable investments per sector and the
blue bars are the optimal portfolio composition found.

###  II.3 Balance Sheet Constraints

In many scenarios, we would like to consider more aspects of an asset rather
than their historical returns and volatilities we use to construct our
portfolio. This can arise from the desire to have the portfolio exhibit
certain characteristics the investor desires as well as to hedge against
biased historical data that we use to formulate our model. This would be
highly beneficial to asset managers who believe in fundamental analysis of
stocks and would like their portfolios to represent their belief in various
economic indicators.

An example we used for our portfolios was that of the current ratio, which is
a measure of a firm’s ability to cover short-term obligations, typically
within the next year. Companies with current ratios greater than 1 can be
thought of as more financially solvent than those with ratios less than 1.
Here we show two different realizations of this belief: one where we impose
local constraints that each asset should have a minimum current ratio, and
another global constraint where our entire portfolio should have a minimum
average. The former would result from the belief that we do not want any stock
in our portfolio that does not have a strong asset-liability ratio, while the
later would allow for some assets with lower current ratios as long as the
entire portfolio meets a minimum level.

The local constraint could be solved by filtering the data before modelling,
however its advantage is realized as a soft constraint where the investor has
a bias for assets above a certain threshold, but does not want to preclude
consideration of investment entirely. Therefore this constraint is most
powerful not as a hard constraint but rather as an extension of an investor’s
preferences.

The local constraint is:

$\sum_{n}\left(C_{n}w_{n}-s_{n,c}-C_{min}w_{n}\right)^{2}$ (8)

where $C_{n}$ is asset $n$’s current ratio, $s_{n,c}\geq 0$ is the binary
encoded integer slack variable, and $C_{min}$ is our minimum required ratio.

From this we can also derive the global constraint:

$\left(C^{T}w-s_{c}-C_{min}\right)^{2}$ (9)

where again $s_{c}\geq 0$.

Our results show how effective both quadratic constraints are when run with
appropriate Lagrange Multipliers. Both constraints offer flexibility to
investors depending on their preference for assets being included entirely or
for a specific characteristic of the portfolio. A combination of both these
constraints is also feasible, with investors preferring a minimum balance
sheet metric for an asset while still wanting the constructed portfolio to
have a targeted measure.

Figure 2: Distribution of current ratio for the local and global constraint on
the S&P 500. The left graph shows the soft constraint penalized assets whose
current ratio was less than 2, the effectiveness of which we can see by the
red dots showing those assets were not selected as investments. The right
graph allows for assets with current ratio less than 2, as long as the
weighted average of the ratio is greater than 2.

###  II.4 Cardinality Constraint

Another constraint we considered is the cardinality constraint found in the
Limited Asset Markowitz (LAM) model. This constraint is significant for
several reasons. The first is that the requirement of limiting the number of
assets to select is relevant to portfolio managers and individual investors
alike. Transaction costs, monitoring overhead, and many other considerations
typically require an investor to limit the number of assets to choose. To
limit the number of assets requires binary variables, transforming the
traditional model into a Mixed Integer Quadratic Programming (MIQP) problem
2014cesarone  which is known to be NP-Hard.

To limit the number of assets to $l$, we can first introduce a binary variable
$b_{n}$ for each stock which is set to 1 if it is invested in and 0 otherwise:

$\displaystyle\left(w_{n}-b_{n}\right)<=0,\forall{n}\in N$ (10)
$\displaystyle\sum_{n}b_{n}<=l$

To make these constraints amenable to a QUBO problem, we will have to
introduce additional binary-encoded integer slack variables $v$ with
appropriate Lagrange Multipliers:

$\displaystyle(\sum_{n}b_{n}+v-l)^{2}$ (11)
$\displaystyle\sum_{n}(w_{n}-b_{n})$

We can see that the first constraint will force the number of binary variables
to be less than or equal to the upper limit on the number of assets, and the
second forces the binary variable $b_{n}$ to be 1 if asset $n$ is invested in
as this will decrease the cost function. In this way there may be times that
when an asset is not invested in the binary variable will still be 1, however
as we are setting an upper constraint on the number of assets, this will
suffice to ensure that if an asset is indeed selected, the binary variable
will always be 1.

##  III Model Comparisons

As current QA do not have the qubit capacity to handle our QUBO model
directly, we can instead employ the use of D-Wave’s hybrid solvers. These
solvers use a mix of classical optimization techniques and the annealers in
combination to find feasible solutions. While the exact methodology of this
hybrid approach is proprietary, these solvers open the possibility of solving
very large problems while still leveraging the power of current annealers.

The Binary Quadratic Model (BQM) accepts as input binary variables and an
objective function. It is equivalent to the standard QUBO model and the
formulations we have presented thus far were inputted directly into the
solver. The Constrained Quadratic Model (CQM) accepts both integer and binary
variables as well as explicit equality and inequality constraints. This not
only makes the constraints we desire easier to handle, but the performance of
solver is significant on results run for the full dataset of the S&P 500. For
the classical optimizer, we employed the use of CPLEX, which is one of the
leading Integer and Mixed Integer optimization suites available.

In figure 3 we show the annualized return and the volatility for the
portfolios constructed from all models (BQM, CQM, and CPLEX) for different
values of $q$. The only constraints we explicitly stated was that the full
budget was used and an asset could not be more than 2.5% of the total
portfolio, however we also ran the CQM model with the volatility as a
constraint, which is shown in purple. We can see that for low values of $q$
(higher volatility), the CQM model significantly outperformed BQM. This is
most likely due to the native support for integers and bounds on those
variables. For higher values of $q$ however, the BQM solver outperformed CQM.
The cutoff appeared to be at $q=25$, where above that coefficient the
increased emphasis on the quadratic terms favored the BQM solver which
appeared to handle quadratic penalty coefficients better. We hypothesize this
is most likely the result of the use of the annealer in the algorithm, which
can natively support binary variables and quadratic terms. The CQM with target
volatility generally underperformed the other models and the results were not
as consistent.

Figure 3: Comparison of results from the BQM and CQM models for different
values of q. We used a range of q values from .1 to 500. The crossover point
found was q = 25, after which the BQM dealt better with the higher values in
the quadratic terms and found higher returns for the same volatility as CQM.
Explicitly stating the target volatility as a constraint, showed in purple.
did not outperform the standard CQM model.

In Figure 4 we compare the implementations of all the constraints we have
discussed thus far in both the BQM and the CQM with a $q$ value of 1 used for
all experiments. It is evident that the CQM hybrid capabilities at enforcing
explicit constraints while minimizing the objective function are superior to
BQM for this integer-valued portfolio optimization problem. Again, the caveat
is that with a larger $q$ value or smaller annualized volatility constraint,
the BQM solver can potentially outperform CQM. While we leave it to further
research to experiment with the endless combinations of constraints and
objectives possible, we posit that the CQM would most likely outperform BQM
for most instances.

In the comparison of classical and quantum solutions, we have ran the model
with varying levels of constraints and thus difficulty for analyzing the
performance of all approaches. In Figure 3 we can see that for the standard
mean-variance portfolio optimization problem, the classical solution found the
efficient frontier with minimal effort. Even with the wide variety of real-
world constraints we added to the model in Figure 4, the CPLEX solver
outperformed all others in terms of Sharpe Ratios.

Many have proposed portfolio optimization as a prime candidate for quantum
advantage; however the real-world constraints we have discussed thus far show
that at least in the static integer-valued case, it is unlikely to outperform
classical solutions. Classical MIQP solvers have been matured for decades and
can include millions of variables; it would be surprising that in the vast
majority of cases it would underperform others in optimizing portfolios.
Furthermore, many have stated that adding the target volatility as a
constraint makes the problem non-convex and thus would allow QA an advantage
compared to classical algorithms in the near future. While this may be true,
it is again hard to foresee a real-world scenario in which this is a critical
constraint, as it is just as easy to enforce the risk appetite with the $q$
penalty coefficient from Equation 2 in classical solvers as it is required in
QA.

Figure 4: Comparison of Sharpe Ratios for the variety of constraints we have
considered across the different models.

##  IV Conclusion

Here we have shown multiple ways to add flexible and realistic constraints to
the portfolio optimization problem. Importantly, we have formulated them for
use in a QUBO problem amenable to quantum annealers. We demonstrated their
effectiveness on the full S&P 500 dataset using D-Wave’s hybrid solvers. We
have also provided direct comparisons for both D-Wave’s hybrid solvers and
classical solvers, showing the performance of the solvers with current
hardware and real-world constaints.

##  V Disclaimer

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## References

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# Re-evaluation of $\Omega_{k}$ of the normalised Friedmann-Lemaître-
Robertson-Walker model: Implications for Hubble constant determinations

Ahmet M. Öztaş  Physics Engineering, Hacettepe University, TR-06800 Ankara,
Turkey  Michael L. Smith  Umeå University, SE-907 38 Umeå Sweden
<EMAIL_ADDRESS>
###### Abstract

The description of spacetime is an fundamental problem of cosmology. We
explain why the current assignments of spacetime geometries for $\Omega_{k}$
of the Friedmann-Lemaître-Robertson-Walker (FLRW) model are probably incorrect
and suggest more useful descriptions. We show that $\Omega_{k}$ represents not
only curvature but the influence of matter density on the extent of spacetime
between massive objects. Recent analyses of supernovae type Ia (SNe Ia) and
HII/GEHR data with the FLRW model present the best fits with a small value for
$\Omega_{m}$ and a large $\Omega_{k}$. These results are consistent with our
Universe exhibiting sparse matter density and quasi-Euclidean geometry and the
small $\Omega_{m}$ value agrees with Big Bang nucleosynthesis calculations. We
suggest the geometry of our current Universe is better described by a value
for $\Omega_{k}\approx$1 rather than 0. As an example we extend the FLRW model
towards the Big Bang and discover a simple explanation of how matter creation
developed into the currently geometrically flat Universe with sparse,
homogeneous, isotropic matter and energy distributions. Assigning
$\Omega_{k}\approx 1$ to describe quasi-Euclidean spacetime geometry is also
useful for estimating $H_{0}$ and should help resolve the tension surrounding
current estimates by different investigators.

###### keywords:

(cosmology:) cosmological parameters, (cosmology:) early universe , cosmology:
theory
PACS: 98.80.-k , 98.80.Bp , 98.80.Es , 98.80.Jk

††journal: New Astronomy

##  1 Introduction

A central problem of cosmology is explaining the interplay of the three
fundamentals; matter, energy and spacetime. Though we think we understand
spacetime there is no consensus about what constitutes spacetime, how it got
here or how it functions. Some people believe spacetime evolved entirely from
the Big Bang (BB), some not. Some think the spacetime between galaxies is
itself increasing with time and the rate now accelerating, some not. The broad
applicability of the Friedmann-Lemaître-Robertson-Walker (FLRW) model is also
often used to bolster the argument that spacetime itself is expanding but the
model is really noncommittal.

A major problem is explaining why our Universe seems geometrically flat,
homogeneous and isotropic in matter and light when assayed to great distance.
The notion that our Universe is really absolutely flat is now questioned [1,
2]. Einstein argued that a universe containing matter, spacetime geometry must
always be roughly elliptical, with or without the cosmological constant
$\Lambda$, and is currently quasi-Euclidean [3]. The results from observations
of supernovae type Ia (SNe Ia) emissions and some interpretations of the
cosmic microwave background (CMB) indicate to some observers quasi-Euclidean
and endless spacetime [4, 5, 6]. Re-examination of the CMB signals suggests to
others that spacetime geometry is elliptically curved and the Universe closed;
the disagreement is considered a crisis [2].

The properties of the term for spacetime, $K$ of the second Friedmann
relationship, are followed by many the past three decades [7]. Since $K$ is
but the constant of integration some admit we really don’t know what it
represents. Krizek and Somer [8] have recently shown that some of the current
assumptions about the geometries assigned to values of $K$ are simply wrong.
An example is the impossible situation of hyperbolic spacetime curvature which
is only possible in the presence of negative matter density. The problems with
such assignments coupled with the straight-forward argument of Einstein has
lead us to rethink the current method used for evaluation of critical SNe Ia
data and the interpretation of normalised term for spacetime, $\Omega_{k}$.

There is also ongoing disagreement, termed tension, over the value of the
local Hubble constant $H_{0}$ [9, 10, 11]. Part of the blame for this and the
crisis over spacetime geometry is the common notion that our universe
exhibiting flat, or nearly flat geometry, demands an $\Omega_{k}\approx 0$.
($\Omega_{k}$ and $\Omega_{m}$ as described in subsection 2.3.) Using proper
regression analyses with several public collections of SNe Ia data we have
shown that the FLRW cosmology without the normalised cosmological constant,
$\Omega_{\Lambda}$, is the preferred model with values of $\Omega_{k}$
approaching 1 and with $H_{0}$ less than 70 km s-1 Mpc-1 where Mpc is
megaparsec [12, 13, 14]. We have more recently found an $\Omega_{k}$
approaching 1 from evaluating HII and giant extragalactic HII regions (GEHR)
data [15]. Results from the Freedman group using emissions from the tip of the
the red giant branch [16] and detection of a bineutron star collision by the
LIGO consortium [17] both point to a $H_{0}$ of about 70 km s-1 Mpc-1. Our
reinterpretation of the $\Omega_{k}$ assignments supports a nearly flat and
slightly elliptical Universe geometry for the current epoch, consistent with a
new evaluation [2]. We shall show that one is not forced to find
$\Omega_{k}\approx 0$ to explain a quasi-Euclidean Universe geometry and a
value for $H_{0}\leq 70$ is very possible.

To explain why the geometry of our Universe seems flat with homogeneous,
isotropic matter and energy distributions at large scales, early investigators
[18, 19] argue the early Universe suffered exponential expansion from the BB
in an inflationary process. One version is now the chaotic inflation model
[20, 21] and another version has become the multiverse hypothesis [22].
Despite, or perhaps because of these many variations, the inflationary
universe explanation is currently popular as evidenced by the many reviews
[23]. The key mechanism of inflation involves some form of energy storage in
the early Universe followed by rapid release - which may or may not be related
to the dark energy supposedly responsible for our accelerating Universe
expansion. Some people however, strongly object to this theory [24, 25]. An
outstanding problem is how did it all end, that is, why and by what mechanism
stopped inflation, or is inflation eternal [26]. Inflationary theory has
developed many variations, any one which fits hand-selected observations and
there are so many versions that it is impossible to fairly test all [27].

We remind the reader that the properties assigned to $K$ are only suggestions
and subject to revision when confronted with data. Here we extend the meaning
of $K$, in the absence of a significant $\Lambda$, to represent everything
else in the universe, other than matter, which means primarily spacetime. As
an application of our finding we then employ the FLRW model to trace the
normalised matter parameters, $\Omega_{rm}+\Omega_{m}$, where $\Omega_{rm}$
represents the input of relativistic matter, and $\Omega_{k}$ values during
particle creation in an early universe. Because of the interdependence of the
matter and spacetime parameters we find the value for $\Omega_{k}$ fluctuated
wildly in the early Universe and the behaviour more interesting than
previously allowed. We trace the values of $\Omega_{rm}+\Omega_{m}$ and
$\Omega_{k}$ after particle creation and show why these are important to
understand the current dilute matter and energy distributions and our nearly,
but not quite flat, Universe geometry. We think our introductory view allows
more thorough study of inflationary theory and helps resolve the current
tension about the $H_{0}$ value.

##  2 Theory

We conservatively consider only energy, confirmed matter types and
4-dimensional spacetime after the Planck-inflationary epoch(s); the standard
model of particle physics without the necessities of $\Lambda$ or
supersymmetry.

###  2.1 The cosmological constant problem

The cosmological constant has been a fashionable topic the past two decades.
Since the initial announcements supporting a positive $\Lambda$ there are many
supportive opinions and reviews [28, 29]. Here are the many reasons we do not
incorporate $\Lambda$ in our analysis.

The energy associated with $\Lambda$ is volume dependent and tiny so the total
$\Lambda$ energy would have been minuscule in the early Universe and the
effect of $\Lambda$ is only expected to become noticeable at $\approx 3.5$
billion years [30]. The value of $\Omega_{\Lambda}$ was investigated,
beginning with a value currently favoured by astronomers
($\Omega_{\Lambda}\approx 0.7$), then tracing this towards the BB. It was
discovered that $\Omega_{\Lambda}$ must become incredibly large, $\gg 1$, with
increasing lookback demanding either $\Omega_{k}\ll-1$ or $\Omega_{m}<0$ and
both requirements lack reality [31]. Another way of putting this is when
$\Omega_{\Lambda}\gg 1$ as required by normalisation for accelerating Universe
expansion, $\Omega_{m}+\Omega_{k}$ must become $\ll 0$. Solutions with
$\Omega_{k}<0$ denoting hyperbolic curvature or $\Omega_{m}<0$ demanding
negative matter density are unrealistic and of mathematical interest only [8].

The SNe Ia data are subject to random and systematic errors; the signals are
source dependent and evolve over time [32, 33] and the random errors of
distance measurements, as $D_{L}$, are huge [14]. According to some
investigators when the local anisotropy is taken into account, evidence for
accelerating Universe expansion disappears along with the need for dark energy
[34, 35]. Being the major force in the Universe dark energy should be testable
but a sensitive laboratory experiment failed to detect any such force [36].
The value for $\Lambda$ suffers a difference from the required vacuum energy
by a factor of $10^{30}$ or more, the vacuum catastrophe as noted by some
early proponents [37, 38], but this has never been properly explained away.
Some variations of the dark energy hypothesis and some criticisms have been
addressed [39, 40]. Problems with $\Lambda$ have even been noted in the
popular press eliciting rebuttal from major proponents [41].

###  2.2 SNe Ia data analysis

####  2.2.1 The SNe Ia data analysis problem

The case for a real $\Lambda$ and a flat universe geometry with $\Omega_{k}=0$
began in the last millennia with three publications. The first was a review
article which made the idea of a real $\Lambda$ seem very interesting [7].
Later that decade two groups published results from investigations of
emissions from distant SNe Ia which were interpreted as support for a real
$\Omega_{\Lambda}$ and hence a real $\Lambda$ with $\Omega_{k}=0$ describing
flat universe geometry [42, 43]. Both groups analysed the SNe Ia data using
not the luminosity distances, $D_{L}$, to SNe Ia but values termed distance
magnitude, often simply mag, based on the relationship

$mag=5\hskip 2.84526ptlog(D_{L})+25.$ (1)

Neither group correlated $D_{L}$ with expansion factors or recession
velocities, for the independent variables, but the observed redshifts, $z$.
The incorrect use of mag and redshift rather than $D_{L}$ vs. the expansion
factor (or recession velocity) has been noted [15]. That article explains in
detail the many faults of this common method and the rationale for reanalysis
of SNe and HII/GEHR data. Using log-transformed data, such as equation (1),
often leads to erroneous findings [44, 45].

Our results of correlating $D_{L}$ vs. the expansion factor using the SNe Ia
data first published by the High-Z Supernova Search Team [42], are shown in
Table 1. The value of $H_{0}$ for both determinations is under 63 km s-1
Mpc-1. Similar results from analyses of more recent, larger sets of SNe Ia
data have been reported [12, 13, 14].

Table 1: Analysis of $D_{L}$ vs. expansion factor with 37 observations and the
local group (1,0) using the robust regression technique [15] For the reduced
$\chi^{2}$ where N is the number of data pairs (38) and FP the number of free
parameters, the better fitting model exhibits lower reduced $\chi^{2}$. The
best fit curve is presented in Figure 5

.  Model $\Omega_{m}$ $\Omega_{k}$ or $\chi^{2}$/(N-FP) $H_{0}$
$\Omega_{\Lambda}$ km s-1 Mpc-1 $\Omega_{m}+\Omega_{\Lambda}=1$ 0.37$\pm$0.13
0.63$\pm$0.13 1.50 62.9 $\Omega_{m}+\Omega_{k}=1$ 0.01$\pm$0.23 0.99$\pm$0.23
1.44 62.6

####  2.2.2 The Hubble correlation

Real distance and recession velocity data should be analysed in a manner
consistent with that originally used by Hubble to calculate $H_{0}$ [46]. Use
of that analytical technique, with tip of the red giant branch data, yielded a
value of $\approx 70$ km s-1 Mpc-1 [16] similar to other independently
determined values [17]. The FLRW model may be used, however, and the analysis
conforms with the requirements of physics when $D_{L}$ and expansion factor
data are correlated.

###  2.3 Current spacetime curvature assignments

The development of our Universe on the largest scale can be approximated by
the two Friedmann expressions relating the expansion factor, a , with matter
density $\rho$, pressure $p$, $c$ lightspeed and $\ddot{a}$ acceleration with
the first equation as

$\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}\left(\rho+\frac{3p}{c^{2}}\right)\,,$
(2)

and the second equation being

$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi
G}{3}\rho-\frac{Kc^{2}}{a^{2}}=H^{2}\,,$ (3)

with values of $H,\rho,p,K$ and $a$, all greater than zero for our Universe
with $\Lambda=0$ during the first epochs. This $K$ is usually assigned a value
describing constant spacetime curvature in the presence of conserved matter
density as

$\displaystyle\text{when}\hskip 2.84526ptK\begin{cases}&0<\hskip
17.07164pt\text{elliptical,}\\\ &=0\hskip 19.91692pt\text{Euclidean,}\\\
&<0\hskip 19.91692pt\text{hyperbolic.}\end{cases}$ (4)

The assignments are sometimes described with different terms; $K>0$, closed;
$K=0$, flat; $K<0$, open and Einstein considered nearly flat to be quasi-
Euclidean. We remind the reader that $K$ is unit-less, so gives no hint of
meaning and the properties above have been assigned and never really verified.
It may be that $K$ simply indicates that portion of the universe which is not
matter rather than only curvature.

More detailed descriptions of the various meanings and conditions commonly
assigned to $K$ are presented in detail [47] and with more geometry [48]. The
case where $K=0$ is a special situation and has problems due to over-
interpretation [8]. Also note our Universe has never been observed to display
hyperbolic geometry; geometric spacetime geometry is unrealistic in the
presence of ordinary matter.

One often considers the Universe of general relativity suffering spherical
expansion as described by the Robertson-Walker metric

$ds^{2}=-dt^{2}+a^{2}(t)\left[\frac{dr^{2}}{1-kr^{2}}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}\right]\,,$ (5)

where $r,\hskip 2.84526pt\theta$ and $\phi$ are the usual designations
associated with spherical geometry. For practical reasons the various terms of
this model are normalised according to the following conditions

$\displaystyle\rho_{s}=$ $\displaystyle\frac{\rho_{i}}{\rho_{c}},$
$\displaystyle\Omega_{rm},\Omega_{m}=$ $\displaystyle\frac{8\pi
G}{3H^{2}}\rho_{s}>0,$ $\displaystyle\Omega_{r}=$ $\displaystyle\frac{8\pi
G}{3H^{2}}\rho_{r},$ $\displaystyle\Omega_{k}=$
$\displaystyle-\frac{Kc^{2}}{a^{2}H^{2}}\,,$ (6)

where $\rho_{i}$ represents the densities of the various massive particle
forms; small relativistic particles such as neutrinos, with tiny mass, are
considered differently from massive particles. Above $\rho_{c}$ is the
critical particle density and the ratio of $\rho_{i}$ to $\rho_{c}$ is
important because $\frac{\rho_{i}}{\rho_{c}}>1$ encourages eventual universe
collapse. Here $\rho_{r}$ is radiation density, $\Omega_{r}$ the normalised
radiation density with $\Omega_{rm}$ and $\Omega_{m}$ the normalised matter
densities suffering relativistic and non-relativistic velocities,
respectively. During the early Universe, with a significant influence of
relativistic massive particles, the value for $\Omega_{rm}$ cannot be treated
in the same manner as non-relativistic, baryonic matter, $\Omega_{m}$. All
types of pure energy density, from radio to hard $\gamma$-rays including
exotic entities are grouped together as $\rho_{r}$ and $\Omega_{r}$ and we
remind the reader that pure energy has never been shown or observed to exert
gravity.

When describing our current epoch we employ the normalised relationship
without $\Omega_{\Lambda}$, for the many reasons given above, as

$\Omega_{m}+\Omega_{k}+\Omega_{r}=1\,,$ (7)

where $\Omega_{rm}$, consisting mainly of neutrinos, exerting a negligible
effect during the current epoch, is also ignored. (While the neutrino
population is large the gravitational effect is negligible.)

We think some presumptions around the nature of $\Omega_{k}$ are wrong. We
remind the reader that there is no firm rule demanding the properties often
assigned to $\Omega_{k}$ and $K$. We cannot also presume a direct
correspondence between $K$ and $\Omega_{k}$, because the very act of
normalisation automatically considers matter density. Nowadays many
astrophysicists claim the current $\Omega_{m}\approx$ 0.25 to 0.30 with
$\Omega_{\Lambda}\approx 0.75$ and an $\Omega_{k}\approx 0$ means a
geometrically flat spacetime. Flat spacetime geometry cannot possibly be true
with such a large value for $\Omega_{m}$, with or without $\Omega_{\Lambda}$
[3]. The assignment of Euclidean geometry to $K=0$ (and $\Omega_{k}=0$) comes
from later investigators and was not considered by Friedmann. For these
reasons it is best to be wary of evaluating $K$ or $\Omega_{k}$ at 0 [8]. It
is not always obvious if reports refer to spacetime curvature in terms of $K$
or $\Omega_{k}$, from here on we are interested in $\Omega_{k}$ and the reader
may trace our argument to $K$ for themselves.

###  2.4 Radiation and gravitational attraction

Since there is no experimental evidence nor robust observational report that
the photon has measurable mass or exerts gravitational attraction, the
gravitational effect of $\Omega_{r}$ in reality is negligible. One should
realise that a photon following a geodesic is evidence that matter can bend
spacetime but is not evidence that light can bend spacetime. (There is a study
attempting to measure the gravitational pull of the photon which reports
difficulties with large thermal effects and systematic error [49]). One may
use terms for radiation in the Einstein stress-energy tensor but this is not
evidence that massless particles elicit gravitational attraction as we have
pointed out [50].

###  2.5 Suggested spacetime geometry curvature properties

Spacetime geometry as $\Omega_{k}$ in the presence of matter cannot be either
hyperbolic nor Euclidean but only elliptical. Here we propose these spacetime
geometries associated with $\Omega_{k}$ for our Universe as

$\displaystyle\text{when}\hskip
2.84526pt\text{$\Omega_{k}$}\begin{cases}&=1,\hskip 28.45274pt\text{flat,
Euclidean}\\\ &<1,\hskip 28.45274pt\text{elliptical}\end{cases}$ (8)

and do not consider the geometries of other possible universe types.

Here are some reasons for our proposal. Consider the value for the limiting
case before particle creation when $\Omega_{k}=1$ and $\Omega_{m}=0$. In the
early universe $\Omega_{m}\approx 0$ leaving $\Omega_{k}\approx 1$ as per
equation (7). With but little gravity our Universe geometry would have been
quasi-Euclidean, $H$ very large, allowing energy to travel anywhere probably
at near lightspeed. On the other hand in a system cramped with matter, such as
a black hole with $\Omega_{m}\geq 1$, $\Omega_{k}$ will be $\leq 0$, with
little spacetime between particles and this small world must exhibit highly
elliptical geometry.

There are three important properties of our Universe and $\Omega_{k}$,
$\Omega_{m}$ consistent with our proposal. First, it is a long distance
between stars and between galaxies; spacetime is nearly devoid of matter.
Second, spacetime appears quasi-Euclidean for an observer distant from a
massive object. Third, an $\Omega_{k}$ near 1 and a small $\Omega_{m}$, as
presented in Table 1, indicates great spacetime between objects and slightly
elliptical universe geometry. There is also an important non-observation -
there are no reports of hyperbolic spacetime curvature.

Extending this reasoning further, we think the value of $\Omega_{k}$ indicates
the relative importance of spacetime itself as well as geometry. Normalisation
really means just that - the values for $\Omega_{rm}$, $\Omega_{m}$ together
with $\Omega_{k}$ reflect the average matter densities and the average
spacetime between massive objects as well as spacetime geometry.

##  3 Results

As an example of usefulness of this concept we apply the new assignments of
$\Omega_{k}$ geometry to the first few seconds after the BB. We first evaluate
the FLRW model with increasing values of $\Omega_{k}$ as a function of a with
$\Omega_{m}$ approaching 0 with constant $H$ and evaluate $\Omega_{k}$ again
as $\Omega_{r}$ disappears. We next apply our new understanding of
$\Omega_{k}$ to the problem of the geometry of the early Universe and to the
observations of homogeneity and isotropy of matter and energy distributions in
our Universe.

###  3.1 The FLRW model applied to the current and early universe

Equation (9) presents a relationship from the FLRW conditions which can be
used for testing cosmological models of the current epoch with SNe,
$\gamma$-ray burst and HII/GEHR emissions. The values for $D_{L}$ represent
standard candle distances while the associated redshifts are used to calculate
the expansion factor, a = 1/(1+z). For tests using the FLRW model without
$\Omega_{\Lambda}$ the relationship correlating distance with recession
velocity is

$D_{L}=\frac{c}{aH_{0}\sqrt{\lvert\Omega_{k}\rvert}}\text{sinn}\Bigg{[}\sqrt{\lvert\Omega_{k}\rvert}\int_{a_{1}}^{1}\frac{da}{a\sqrt{\frac{\Omega_{r}}{a^{2}}+\frac{\Omega_{m}}{a}+\Omega_{k}}}\Bigg{]}.$
(9)

Here sinn is sin when $\Omega_{k}<0$ or sinh when $\Omega_{k}>0$. As
instructed by Carroll et al. and by Riess et al. [7, 42] equations (9) and
(11) should be evaluated with $\sqrt{|\Omega_{k}|}=1$ when presuming
$\Omega_{k}=0$ for a flat universe geometry, which is an obvious
inconsistency. We rather suggest that assigning the value of $\Omega_{k}=1$
for Euclidean geometry means just that and should be used in equations (9) and
(11) when presuming flat or quasi-Euclidean spacetime geometry. In the case
where $\Omega_{k}=0$ it really means that $\frac{\rho}{\rho_{c}}=1$ and
equations (9) and (11) cannot be evaluated. (The situation where $K=0$ was
never addressed by Friedmann [8].) When $\Omega_{r}$ is included in equation
(9) during problem presentation it is always dropped during analysis of
astronomical data, the reason given as the effect of radiation during our
current epoch simply being too small to affect measurements and astronomical
observations.

Allowing the gravitational effect of radiation as unimportant leaves us the
simpler situation of

$\Omega_{m}+\Omega_{k}=1\,,$ (10)

with only two normalised parameters describing the Universe. For the condition
where $\Omega_{m}=0$ applies for a universe without matter, this relationship
must allow $\Omega_{k}=1$ as describing an empty universe with Euclidean
geometry while presuming a real $\Omega_{r}$ but no matter and is only useful
for speculation (Appendix B).

If we allow $\Omega_{m}=0$ and $\Omega_{k}=1$ to describe our early Universe
just after the BB, expression (10) collapses to 1 so is not terribly
informative. We think $\Omega_{m}\approx 0$ a reasonable assumption during
that epoch; the Universe before massive particle creation consisted primarily
of high energy but not matter. To get around this problem we evaluate a
universe with $\Omega_{m}$ decreasing from 0.1 to 0.001 using equation (9)
without the $\Omega_{r}$ term and results presented as Figure 1. The distance
allowed for light travel increases drastically with decreasing $\Omega_{m}$.
That is, $D_{L}$ approaches extreme dependence on the expansion factor with
decreasing matter density but at a of 1 most any distance is allowed. This
property of spacetime, as expressed in allowed light flight, is expected as
elliptical geometry relaxes towards quasi-Euclidean. An $\Omega_{k}$ of 1 is
consistent with a flat universe devoid of gravity with radiation everywhere in
a universe without bounds.

Figure 1: $D_{L}$ vs. a with $H_{0}=c=1$ for a universe with $\Omega_{m}$ from
0.1 to 0.001, left to right, with corresponding values of $\Omega_{k}$,
calculated with equation (9) and $\Omega_{r}=0$.

To consider $D_{L}$ for a universe before particle creation we must drop
$\Omega_{m}$ and consider $\Omega_{r}$. (For massive particles $G$ obviously
applies, but for a system consisting of only high energy species, such as
photons, Newton’s $G$ is irrelevant.) $D_{L}$ becomes highly dependent on the
expansion factor, with the parameters $\Omega_{r}$ and $\Omega_{k}$ as

$D_{L}=\frac{c}{aH_{0}\sqrt{|\Omega_{k}|}}\text{sinn}\Bigg{[}\sqrt{|\Omega_{k}|}\int_{a_{1}}^{1}\frac{da}{a\sqrt{\frac{\Omega_{r}}{a^{2}}+\Omega_{k}}}\Bigg{]}.$
(11)

A similar situation to Figure 1 is observed in Figure 4 (Appendix B) when the
expression (11) is used to estimate $D_{L}$ as $\Omega_{r}$ approaches 0,
supposing that radiation does have some gravitational influence (which has
never been observed). As the influence of radiation wanes the values for
allowed light travel is again very dependent on the expansion factor. In other
words, in the early universe devoid of gravity and $\Omega_{k}=1$, travel by
light is unbounded in all directions.

###  3.2 Spacetime response to particle genesis

As an example of how our new realisation can be applied we use $\Omega_{m}$
and $\Omega_{k}$ as tools exploring the characteristics of the Universe just
after the BB. We present the results of our calculations in broad terms
concentrating on the nature of spacetime, as $\Omega_{k}$, correlating with
the current understanding of particle creations. We only calculate initial
particle creations as rapid auto-catalytic events, but other mechanisms were
probably involved.

It is now thought the first epoch was dominated by only high energy species
including some exotic particles followed by quark-hadron matter creations a
few moments later [51]. Three phase transitions are firmly established; first
the electroweak (EW) transition at 100 to 200 GeV, next the quantum
chromodynamic (QCD) transition at 150 to 180 MeV, third the epoch of stable
electron-positron creation (EP) at about 170 keV [52, 53]. During the energy
dominated epoch, matter was completely absent with $\Omega_{rm}+\Omega_{m}=0$
therefore $\Omega_{k}=1$. The energy content, if considered as matter
equivalent, would probably have exhibited a density, $\rho_{r}>\rho_{c}$ in a
small volume, but being only high-energy radiation $\rho_{r}$ exhibits no
gravity and $\Omega_{r}$ was immaterial with respect to spacetime curvature.
Therefore $\Omega_{k}=1$ during this epoch means expansion at near lightspeed
in a geometrically flat Universe.

###  3.3 Spacetime geometry, matter and energy dispersal

The EW transition was quickly followed by the QCD transition as particle
creation began. Considering the conditions for equation (9), $\Omega_{rm}$ and
$\Omega_{m}$ were still tiny, reflecting few particles, and because these
parameters are normalised $\Omega_{k}$ remains $\approx 1$. At that instant
the positive value for $\Omega_{k}$ represented not only slight elliptical
curvature but also a large spacetime contribution to the Universe in which
massive particles were still rare.

The creation of the first particles perturbed the dense energy ensemble
spurring further particle creations in a catalytic manner. As the Universe
continued to cool below the EW and QCD transitions, more massive particles
were rapidly created close to the first initiating particles. The local values
for matter density, $\rho_{s}$, $\Omega_{rm}$ and $\Omega_{m}$ rapidly and
drastically increased along with the compensatory decrease of $\Omega_{k}$.
The creation processes were probably drastically prolonged due to both
particle-antiparticle annihilation and reduction of spin degrees of freedom
number from around 60 to about 15 shortly followed by the quark-hadron
transitions requiring significant entropy input [51]. Spin redistribution
would have prolonged the entropy input and allowed relatively constant
temperature.

We model the creation of massive particles where $n\gamma$ represents the
large number of high-energy species just after the BB with the forward rate
described by

$n\gamma\hskip 2.84526pt+\hskip 2.84526ptinitiation\hskip
2.84526ptparticles\hskip 2.84526pt\overset{k_{1}}{\Rightarrow}\hskip
2.84526ptmassive\hskip 2.84526ptparticles$ (12)

and the reverse reaction described by

$massive\hskip 2.84526ptparticles\hskip
2.84526pt\overset{k_{-1}}{\Rightarrow}\hskip 2.84526ptn\gamma$ (13)

where the forward rate constant, $k_{1}$, is a slightly larger than the
reverse, $k_{-1}$ and the term massive particles includes both relativistic
and non-relativistic matter. The increase in matter concentration, primarily
quarks and hadrons $[m]$, with respect to time, $t$, can be written as

$\frac{[dm]}{dt}=k_{1}[\gamma][m]-[m]^{2}$ (14)

where $[\gamma]$ is the local, high-energy species concentration. For
simplicity we model only two species, high-energy $\gamma$ species and massive
particles $[m]$.

Since massive particles were initially rare, the second-order annihilation
would have been negligible; the above is simplified by ignoring the second
term. The rate of $[\gamma]$ disappearance can be approximated by

$\frac{-d[\gamma]}{dt}=k_{1}[\gamma][m].$ (15)

If we now also use estimates for the initial concentrations of $m$ and
$\gamma$ as $[m_{i}]$ and $[\gamma_{i}]$ we can write the reaction(s) as an
auto-catalytic event by generalizing with an overall rate constant,
$\mathbf{k}$ not necessarily the same value as $k_{1}$, with

$\frac{-d[\gamma]}{dt}=\mathbf{k}\Bigg{(}[\gamma][\gamma_{i}]+[\gamma_{i}][m_{i}]-[\gamma]^{2}\Bigg{)}.$
(16)

Rearranging equation 16 and integrating both sides we have

$-\mathbf{k}\int
dt=\int\frac{d[\gamma]}{[\gamma][\gamma_{i}]+[\gamma_{i}][m_{i}]-[\gamma]^{2}}.$
(17)

The forward reaction creating matter is described by the relationships

$[m]=\frac{[m_{i}]+[\gamma_{i}]}{1+\frac{[\gamma_{i}]}{[m_{i}]}\text{e}^{-([m_{i}]+[\gamma_{i}])\mathbf{k}t}}\approx\frac{[\gamma_{i}]}{1+\frac{[\gamma_{i}]}{[m_{i}]}\text{e}^{-[\gamma_{i}]\mathbf{k}t}}\,,$
(18)

where the approximation is made on the right-hand side, for the condition
where $[\gamma_{i}]\gg[m_{i}]$. This is simply a general equation for the
auto-catalytic creation of most any quark, hadron or lepton which may have
been initiated by most any particle type. Here we only consider the overall
reaction(s) by high-energy species creating massive particles as an auto-
catalytic process occurring with a nearly constant concentration of high-
energy species where $[\gamma_{i}]$ is at least $200>[m_{i}]$. Some exotic
matter would have been created more rapidly after the BB increasing the
initiating particle concentration but we ignore this effect on overall $[m]$
to keep our model simple.

Because quark types probably had different formation rates a proper
description of quark creations would be a linear combination of several
equations with different rate constants. Since the critical temperatures for
both quark and hadron creations are similar but not identical, the rate
equations describing quark-hadron creations should also be included in a more
detailed description. All of this means the time available for quark-hadron
creations from high energy species would have been prolonged.

Stable leptons would have also been created to preserve electroneutrality, but
later at lower energies only slightly affecting gravitational attraction.
While the Universe was expanding there was simultaneous dilution of both $[m]$
and $[\gamma_{i}]$. A simple version of all these events is modelled by

$[m]=\Bigg{(}\frac{[\gamma_{i}]}{1+\frac{[\gamma_{i}]}{[m_{i}]}\text{e}^{-[\gamma_{i}]\mathbf{k}t}}\Bigg{)}\div\bigg{(}2\pi^{2}r^{3}\bigg{)}\,,$
(19)

where the denominator represents the time-dependent dilution of this process
by the growing hypersphere of the early Universe.

Using equation (19) we illustrate the rapid increase of
$\Omega_{rm}$+$\Omega_{m}$ as these species are being simultaneously diluted
in an expanding universe in Figure 2. We allow $\Omega_{rm}$+$\Omega_{m}=[m]$
with $[\gamma_{i}]$ being about $10^{7}$ times that of $[m_{i}]$ and presume a
k of 0.01 as the forward rate constant describing the inefficient creation of
matter in an energetic, matter-antimatter-photon-gluon ensemble. The initial
lag, when $\Omega_{m}=0$ and $\Omega_{k}=1$, represents that time when the
Universe consisted only of high-energy species too energetic to create stable
matter, exhibiting little gravitational attraction, hence the Universe just
after the BB was geometrically quasi-Euclidean.

After this short period, the catalytic particles initiated rapid increases of
matter density and $\Omega_{rm}$+$\Omega_{m}$ followed by both concentrations
declining with time. The somewhat gentle nature of the curve increases in
Figure 2 may have been even more gradual than illustrated, since $k_{-1}$ was
on the order of $k_{1}$ at high energies. The increase of
$\Omega_{rm}$+$\Omega_{m}$ may have also exhibited a flattened maximum if the
range of various overall rate constants, k, was large. Another reason for a
complicated increase of $\Omega_{rm}$+$\Omega_{m}$ is the quark-hadron
transitions which occurred quickly after quark creations, also affecting
matter characteristics and $\Omega_{rm}$+$\Omega_{m}$ concentrations. Very
general descriptions of particle creation with concurrent dilution into the
expanding Universe, as reflected by the probable developments of
$\Omega_{rm}$+$\Omega_{m}$ and $\Omega_{k}$, are illustrated by Figures 2 and
3.

###  3.4 FLRW model reflecting matter creation and dispersal

In Figure 2 we present values for $\Omega_{rm}$+$\Omega_{m}$ climbing toward a
value normalised to $\approx 1$, as the diagonally directed curve, as if the
universe expanded at lightspeed. The situation where the universe expanded at
lightspeed is probably unrealistic, so we present other solutions where the
universe expansion occurred at fractions of lightspeed and note that
$\Omega_{rm}$+$\Omega_{m}$ could have become much greater than 1 before
relaxing towards much smaller values.

Figure 2 also illustrates that matter creation would have occurred rather
quickly via auto-catalysis. If $\Omega_{rm}$+$\Omega_{m}$ was $>1$ even for a
short time, $\Omega_{k}$ would necessarily have been $<0$ and spacetime
geometry became drastically elliptical. The effect on $\Omega_{k}$ is shown in
Figure 3 where minimum values are often less than 0. This is shown in this
figure where points are plotted rather than curves drawn and where the
expansion rates are significantly slower than lightspeed. Reversal towards the
big crunch was avoided because the massive particles were then travelling at
relativistic velocities or nearly so, hence $H$ was large. This is consistent
with proposals for the mechanism whereby the Universe avoided gravitational
collapse [54]. Nevertheless, a short period with $\Omega_{rm}$+$\Omega_{m}>1$,
effecting a considerable drag on particle velocity, is consistent with a $H$
at recombination not too different than $H_{0}$.

The value for $\Omega_{rm}$+$\Omega_{m}$ in Figure 2 decreases rapidly towards
zero as matter was diluted in the expanding Universe. The small current value
of $\Omega_{m}$ predicted here is consistent the results of BB calculations by
[55] and values when SNe Ia data ensembles are used for analysis [12, 13, 14].
This also means that spacetime geometry fluctuated rapidly during particle
creation. This wild change of universe geometry randomised particle and energy
trajectories producing the homogeneity and isotropy we now observe. Also note
that the smooth curve increase followed by the slow decline of
$\Omega_{rm}$+$\Omega_{m}$ after the long period of particle creation means
detection of the primordial gravitational wave(s) will be more difficult than
if matter was created in an instantaneous manner.

Figure 2: History of $\Omega_{rm}+\Omega_{m}$ the first moments after the BB
allowing universe expansion at velocities from c to 0.3c using equation (19)
with k of 0.01. Time metric can be from $\mu s$ to a few seconds after the BB.
The values of $\Omega_{rm}+\Omega_{m}$ at lightspeed are presented as the
points lying on the diagonal of the distance/time plane with the maximum
normalised to $\Omega_{rm}+\Omega_{m}=1$ when $t=r$. Figure 3: History of
$\Omega_{k}$ the first moments after the BB allowing universe expansion at
velocities from lightspeed to 0.3c using equation (19) with k of 0.01. Time
from $\mu s$ to seconds after the BB. The values of $\Omega_{k}$ for expansion
at c are presented as the points lying on the diagonal of the distance/time
plane with the minimum at $\Omega_{k}=0$ when $t=r$.

One interesting result from our calculations is that with the dilution of
matter during universe expansion at a rate close to that examined here,
$\Omega_{m}$ eventually drops to less than 0.03 in agreement with studies of
particle creation [55, 56]. A second interesting finding is that our Universe
exhibits quasi-Euclidean, gently elliptical and closed spacetime geometry as
the inescapable result of particle dilution.

##  4 Conclusion

The notion that $\Omega_{k}$ alone determines spacetime curvature and that
$\Omega_{k}=0$ describes flat Universe geometry has long lead people astray.
The historic assignments of 0 for flat geometry, $-K$ for elliptical and $+K$
for hyperbolic, appeals to the human sense of symmetry but is not useful in
practice. Nowadays many astrophysicists report the current $\Omega_{m}\approx
0.25$ to $0.30$ with $\Omega_{\Lambda}\approx 0.70$ to $0.75$ and
$\Omega_{k}\approx 0$ describes a geometrically flat Universe. But how can
flat geometry be consistent with a large value for $\Omega_{m}$ given that a
large $\Omega_{\Lambda}$ is probably not true? Also, since our Universe can
never be described with hyperbolic curvature it is unnecessary to place an
$\Omega_{k}=0$ between elliptical and a hyperbolic geometry which can only
exist in one’s imagination. Such nagging problems have lead us to re-evaluate
the nature of $\Omega_{k}$.

We have demonstrated for our early Universe that the key value for
$\Omega_{k}$ is 1 which indicates Euclidean geometry and values approaching 1
describe quasi-Euclidean. We also suggest that $\Omega_{k}$ estimates not only
the average large-scale geometric curvature but also the average distance
between massive objects. That is, a value of $0<\Omega_{k}<1$ means more than
elliptical curvature, it also means the average distance between matter
ensembles, galaxies and galactic groups, is significant - which describes our
current reality.

Realisation that one is not forced to find $\Omega_{k}\approx 0$ to explain a
quasi-Euclidean universe geometry simplifies analysis and helps cosmologists
understand heretofore puzzling observations and results. In a universe nearly
devoid of matter, such as our current epoch, flat geometry is the expected
limit. There is no obvious limit for $\Omega_{m}$ in smaller systems, which
may locally be $\gg 1$ meaning that $\Omega_{k}\ll 0$ with little space
between particles. When $\Omega_{k}$ is 0, this indicates
$\Omega_{rm}+\Omega_{m}$ is equal to the critical matter density and the
Friedmann approximation cannot be evaluated nor $D_{L}$ determined for this
situation. When $\Omega_{k}<0$ the system suffers matter density greater than
critical but must still exhibit elliptical curvature and little spacetime
between objects. Much smaller worlds, for instance the spacetime of a black
hole, may suffer $\Omega_{k}<0$ and certainly elliptical spacetime geometry.

This new explanation of $\Omega_{k}$ greatly simplifies many heretofore
confusing results, for instance FLRW models evaluated using SNe Ia,
$\gamma$-ray burst and HII/GEHR data [14, 15]. Our analyses are also
consistent with recent concerns that an accelerating Universe expansion and
the $\Lambda$CDM model are not supported by hard data [36] or the searching
analyses by others [57, 58]. One obvious test of a current $\Omega_{k}\approx
1$ would be independent calculation of $\Omega_{k}$ for our Universe similar
to that recently published by [2] but calculated without resorting to
$\Omega_{\Lambda}$. Calculations using SNe and tip of the red giant branch
data should be made allowing $\Omega_{k}$ and $H_{0}$ as free parameters with
and without $\Omega_{\Lambda}$ to check our ideas. We think the results will
help resolve the current tension about the Hubble constant value.

We describe the first moments of our Universe leading to a flat, homogeneous
Universe without the need for a special event other than the hot BB. We have
shown that strong fluctuations of spacetime geometry occurred a few seconds
after the BB and may be the events responsible for our current condition. Our
explanation and calculations of a simple autocatalytic model are consistent
with the results of BB nucleosynthesis calculations. After particle creation,
matter and energy densities were diluted as the Universe expanded, allowed
$\Omega_{k}$ to approach 1 and spacetime to assume quasi-Euclidean geometry.
On the other hand, a phase of inflationary expansion around the Planck epoch
prior to the epoch of massive particle creation is not ruled out by anything
presented here. Realising this allows discussion to return towards simpler
versions of inflationary theory.

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##  Appendix A Deceleration parameter

The parameter $q$ is often used in cosmology describing the declining
expansion rate of a universe and termed the deceleration parameter. For
reasons given above we are interested only in $\Omega_{m}$ and $\Omega_{k}$
and ignore $\Omega_{\Lambda}$ and $\Omega_{r}$ for this derivation. The
derivation below is presented for students.

We first expand the Taylor series at $a$, the expansion factor, around any
time $t^{\prime}$ which may be the current or most any time, past or future,
as

$a(t)=a(t^{\prime})+\frac{da(t)}{dt}\Bigg{|}_{t^{\prime}}(t-t^{\prime})+\frac{1}{2}\frac{d^{2}a(t)}{dt^{2}}\Bigg{|}_{t^{\prime}}(t-t^{\prime})^{2}+\dots$
(20)

To derive $q$ we divide the series by $a(t^{\prime})$ and get following
expression

$\frac{a(t)}{a(t^{\prime})}=1+\frac{\dot{a}(t^{\prime})}{a(t^{\prime})}+\frac{1}{2}\frac{\ddot{a}(t^{\prime})}{a(t^{\prime})}(t-t^{\prime})^{2}.$
(21)

This series is well known in cosmology as

$\frac{a(t)}{a(t^{\prime})}=1+H(t^{\prime})(t-t^{\prime})-\frac{1}{2}q(t^{\prime})H(t^{\prime})^{2}(t-t^{\prime})^{2}+\dots$
(22)

Here $H(t^{\prime})={\frac{\dot{a}(t^{\prime})}{a(t^{\prime})}}$ is the
Hubble-Lemaître parameter and $q(t^{\prime})$ the deceleration parameter at
$t=t^{\prime}$ is

$q(t^{\prime})=-\frac{a(t^{\prime})\ddot{a}(t^{\prime})}{\dot{a}(t^{\prime})^{2}}=-\frac{1}{H(t^{\prime})^{2}}\frac{\ddot{a}(t^{\prime})}{a(t^{\prime})}\,,$
(23)

when $t^{\prime}$ is the current time we have

$q(t_{0})=-\frac{1}{H_{0}^{2}}\frac{\ddot{a}(t_{0})}{a(t_{0})}.$ (24)

The negative sign in this equation implies that if a(t) is positive (i.e. the
universe expansion is accelerating) the deceleration parameter must be
negative. We will now relate $q(t_{0})$ as $q_{0}$ to the Friedman equations
without $\Lambda$, but with Newton’s G and $\rho$ matter density and $p$
pressure; relations 2 and 3, as the right-hand side of equation (25) below

$q_{0}=-\left(\frac{a}{\dot{a}}\right)^{2}\frac{\ddot{a}}{a}=-\Bigg{[}\frac{-\frac{4\pi
G}{3}\left(\rho+\frac{3p}{c^{2}}\right)}{\frac{8\pi
G}{3}\rho-\frac{Kc^{2}}{a^{2}}}\Bigg{]}.$ (25)

This relationship allows us to eventually express $q_{0}$ in terms of $H$,
$\Omega_{m}$ and $\Omega_{k}$. For the current epoch we can ignore the
radiation pressure as being minuscule, we have the slightly simplified

$q_{0}=-\left(\frac{a}{\dot{a}}\right)^{2}\frac{\ddot{a}}{a}=-\Bigg{[}\frac{-\frac{4\pi
G\rho}{3}}{\frac{8\pi G}{3}\rho-\frac{Kc^{2}}{a^{2}}}\Bigg{]}.$ (26)

We can further simplify this to

$\frac{1}{q_{0}}=-\Bigg{[}\frac{\frac{8\pi
G}{3}\rho-\frac{Kc^{2}}{a^{2}}}{-\frac{4\pi
G\rho}{3}}\Bigg{]}=2-\frac{{\frac{Kc^{2}}{a^{2}}}}{{\frac{4\pi G\rho}{3}}}.$
(27)

We now rewrite this relationship in terms of $\Omega_{k}$, $\Omega_{m}$ and
$H^{2}$ as

$\frac{1}{q_{0}}=2+\frac{\Omega_{k}H^{2}}{\frac{\Omega_{m}H^{2}}{2}}=2+\frac{\Omega_{k}}{\frac{\Omega_{m}}{2}}\,,$
(28)

or as a single term for $q_{0}$ we have

$\frac{1}{q_{0}}=\frac{2(\Omega_{m}+\Omega_{k})}{\Omega_{m}}.$ (29)

We remember that $\Omega_{m}+\Omega_{k}=1$ without much influence from
$\Omega_{r}$, we have

$q_{0}=\frac{\Omega_{m}}{2}.$ (30)

So deceleration is independent of the value for $\Omega_{k}$, whether one
believes a flat universe is described by $\Omega_{k}=0$ or $1$.

In the absence of other parameters, with $\Omega_{m}+\Omega_{k}=1$ we arrive
at

$q(t_{0})=\frac{\Omega_{m}}{2}.$ (31)

The deceleration parameter value is only dependent on $\Omega_{m}$ and
consistent with a universe containing matter with the current geometry
described as elliptical.

##  Appendix B Light and the early Universe

While equations (9) and (11) are useful for modelling the current epoch we can
test these as $\Omega_{m}$ and $\Omega_{r}$ approach zero with $\Omega_{k}$
approaching 1, to get an idea of the properties of light travelling in the
early Universe near the instant of massive particle creation.

Figure 4: $D_{L}$ vs. a with $H_{0}=c=1$ for a universe with $\Omega_{r}$ from
0.1 to 0.001, left to right, with corresponding values of $\Omega_{k}$,
calculated with equation (11).

Again as $D_{L}$ approaches extreme dependence on the expansion factor with
decreasing high energy density and as a approaches 1 most any distance is
allowed as displayed in Figure 4. An $\Omega_{k}$ of 1 is consistent with a
universe devoid of gravity with radiation everywhere in a spacetime without
bounds.

##  Appendix C Graph SNe Ia data

Data including standard errors calculated from the first set of 37 SNe Ia
emissions presenting claims for an accelerating Universe expansion [42].
Location of our earth at (1,0) is bottom right is used as data pair number 38
which is the one data pair without error. Note the very large errors of
distant emissions which were damped in the mag vs. z analysis by the High-Z
Supernova Search Team thus negating the validity of their analysis.

Figure 5: $D_{L}$ (as Mpc) vs. the expansion factor, parameters from this
curve are listed in Table 1 as the $\Omega_{m}+\Omega_{k}=1$ model.