# Cosmology without an Initial Singularity

R.N. Henriksen  Queen’s University at Kingston, Ontario, Canada
<EMAIL_ADDRESS>
###### Abstract

We present a model of the expanding Universe that begins in a zero mass/energy
vacuum state. The Universe results from the spontaneous breaking of the
electroweak symmetry, wherein the vacuum with positive expectation energy
(i.e. ‘dark energy’ ) produces the dark and visible matter and launches the
expansion. A Hubble-Lemâitre constant of $\sim 73.36km/sec/Mpc$ and the
equipartition red shift are part of the model. The current value of Einsein’s
$\Lambda\sim 1.29\times 10^{-56}cm^{-2}$. We require the age of the Universe
to be $\sim 28$ billion years. Therefore one may find galaxies, quasars and
black holes at higher redshifts than the maximum value now known, but
otherwise the description of the development of structure is not greatly
changed. A Dimensional argument based on the Buckingham theorem applied to the
Universe, is given in an appendix to justify the dark energy density $\propto
1/t^{2}$. Such behaviour produces rapid but not exponential Universal
expansion. We speculate regarding the nature of each Universe energy
component, due to a coincidence between the sine of the electroweak Weinberg
angle and the ratio of matter to dark energy.

Cosmology: Dark Energy, Vacuum State

##  1 Introduction

The current ‘concordance model’ of cosmology has a string of predictive
sucesses (e.g. Ostriker and Steinhardt (1995), Peter (2013), Planck
Collaboration (2020)), with the anticipation of Ostriker and Steinhardt (1995)
regarding the importance of the vacuum energy being remarkable. The origin of
the local gravitational structure in quantum fluctuations plus inflation, as
observed in the CMB and in the present epoch; is well described pedagogically
in Peter (2013). Predictions regarding primordial gravitational waves remain
as outstanding tests. Any competing cosmology must do at least as well.

However the existence of the initial singularity remains a part of the model
that defies our present physics. Moreover Roger Penrose (e.g. Penrose (2004),
Penrose (2010)) has persistently called attention to the cosmological problem
presented by the second law of thermodynamics. Assuming the Bekenstein-Hawking
entropy of a black hole, one requires the gravitational collapse degrees of
freedom to be minimized in order that the Universe begin in a low entropy
state. Together with the second law this implies the normal arrow of time.
This argues against the existence of primordial super massive black holes for
which there is no current evidence, although intermediate scale objects may be
possible (e.g.Bicknell& Henriksen (1979)).

Currently (e.g. Planck Collaboration (2020)) the equation of state of the dark
energy is consistent with the cosmological constant, although a time dependent
scalar field (‘Quintessence’) with a dominant self-interacting potential has
the same equation of state. The vacuum term should always be Lorentz
invariant, which requires the pressure to be the negative of the energy
density.

I wish to call attention in this article to an older and simpler idea
regarding the dark energy (e.g. Henriksen (1982), Olson and Jordan (1987),
Ratra and Peebles (1988) ) that may describe the initial state of the
Universe. This idea accepts that Einstein’s cosmological term $\Lambda$, once
translated to the energy momentum tensor part of the field equations, can be
regarded as the vacuum term and need not be constant in time. The mechanism
for the time dependence is left undetermined except for Dimensional (we use
capital ‘D’ to mean physical parameters and small ‘d’ for geometrical
dimensions). The Dimensional argument is not entirely without physical content
however, because of relations between Universe properties dictated by the
Buckingham theorem. Only the total mass/energy is conserved in such models, so
that the balance between matter and vacuum energy may evolve in time.

A ‘natural’ temporal variation is given Dimensionally as an inverse length
squared . This is explicitly (Henriksen (1982)) for the $\Lambda$ function

$\Lambda(t)=\frac{\lambda}{c^{2}t^{2}},$ (1)

where $t$ is cosmic time, $\lambda$ is a number and $c$ is the absolute speed
of light. One should note that this implies that an associated scale
$\ell=1\sqrt{\Lambda}$ is essentially equal to the apparent horizon at each
epoch. The scale $\ell$ is also roughly the scale at which the gravitational
binding energy of the matter is about equal to the vacuum energy.

A temporal variation of the vacuum energy has been discussed elsewhere, also
before the arrival of extensive cata on the CMB. Ratra and Peebles (Ratra and
Peebles (1988)) studied it as an early form of the inflaton field and Olson
and Jordan (Olson and Jordan (1987)) discussed the implied age of the Universe
in terms of various power laws. The latter paper gives many earlier references
111The reference to Henriksen 1982 was omitted due probably to appearing in an
European journal in which physical justifications are attempted. Ultimately
the inflaton field and the power laws remain essentially ad hoc, except for
Dimensional analysis and the Dirac argument that we use.

The observational inputs to our model amount to the ratio of all matter to
dark energy, $\Omega_{mv}$, the red shift of equipartition (Planck
Collaboration (2020)), and the temperature at the electroweak symmetry
breaking $T\sim 160GeV$. We iterate on the initial ratio of matter to photon
energy until the current scale factor agrees with an initial assumption, and
we iterate on the $\lambda$ parameter until the current ratio of matter to
dark energy is obtained. With these inputs and procedures; that age of the
Universe, the current Hubble-Lemaitre ‘constant’, and the current cosmological
‘constant’ are predicted more or less uniquely.

The thermodynamical evolution of the model is similar to the concordance
model, but there is no inflation (spatial flatness is here ensured by the
initial vacuum state) and the surface of last scattering is at a much later
epoch than currently believed (assuming the same red shift for the CMB) .

##  2 Physical Description of the Model

Our model (to be demonstrated below) is a Universe that begins in a unstable,
static state of zero total energy. At some epoch $t_{1}$ (which is
subsequently adopted as the unit of time) the vacuum component spontaneously
breaks the temporal symmetry, creating a rapid transition to a mixture of
matter (including dark matter) and ‘dark energy’. This dynamic state is one of
‘a priori’ spatially flat expansion with $\Omega_{m}+\Omega_{v}=1$ . The ratio
of matter density to dark energy can be fixed asymptotically to a value
$\Omega_{mv}\sim 0.46$ (from $\Omega_{m}=31.5\pm 1.3\%$ and
$\Omega_{m}+\Omega_{v}=1$; Abdullah, Klypin& Wilson (2020),Planck
Collaboration (2020)), by adjusting one model constant that defines the post
instability evolution dynamically. This scenario follows from the equations of
General Relativity, homogeneity, and assumption (1); taken to hold backwards
in time to the tr epoch of vacuum instability. The Universe may be in the zero
energy state for an indefinite earlier time because of the temporal symmetry
(i.e. static).

One can refer to this static, zero energy state as a ‘classical vacuum’, but
before $t_{1}$ we are beyond classical validity and we must necessarily
hypothesize a quantum vacuum. This would be comprised of various particle
fields as described by quantum field theory, with their characteristic
fluctuations and expectation values. A detailed discussion is beyond the scope
of this model, but this assumption does allow some useful speculations.

The question arises as to how the temporal symmetry (static vacuum) is broken.
This question is similar to determining the origin of the ‘big bang’. In our
case this event is an example of spontaneous symmetry breaking of the
metastable vacuum in the net zero energy Universe. The vacuum symmetry
breaking produces matter with positive energy (exciting it from the negative
energy state) and launches the expansion. This suggests that the observable
Universe expansion is a ‘pseudo’ Goldstone boson (see Lykken & Spiropulu
(2013) for an interesting qualitative presentation) at large scale.

The positive matter density must involve the gauge fields associated with the
initial symmetry. We need these gauge fields to be spontaneously mixed in the
presence of the Higgs field to yield massive particle excitations according to
the Brout, Englert, Higgs (BEH) mechanism. The descendants of these particles
provide the total matter content of the Universe.

The matter gauge fields that seem to fit with the future history of the
Universe (see below) are those associated with the electroweak gauge theory.
At a temperature $T\sim 160GeV$ the massless fields spontaneously mix to give
the massive $Z^{o}$ vector boson, and the massive $W^{\pm}$ conjugate vector
bosons in the presence of the Higgs scalar boson. These massive vector
particles decay quickly to hadrons and leptons and so begin the normal matter
evolution to low temperature. The mass distribution of these descendants are
presumably due to interaction with the Higgs field, whose excitations also
decay. There is a remarkable coincidence between $\Omega_{mv}$ and the
Weinberg angle defining the boson mixtures, that encourage us to speculate
about the nature of the matter and the dark energy. It may of course be simply
a coincidence, but it could be relevant to the necessary incomplete
matter/antimatter annihilation.

It is possible that the vacuum state is not static and some quintessence type
self-interacting scalar field is decaying and yielding the $1/t^{2}$ time
dependence from the Planck epoch to the instability. Perhaps it would begin
only at a GUT (Grand Unified Theory) spontaneous symmetry breaking. This would
have the advantage of connecting the earlier (Planck or GUT) value of
$\Lambda$ to the current value. However this simply displaces the problem in
an ‘ad hoc’ manner, and very little is certain about a hypothetical GUT.

Our model requires an older Universe (by a factor $2$), but the surface of
last scattering is at the usual redshift $z_{s}\approx 1087$. There may
consequently be time between equipartition and $z_{s}$ for stronger
perturbations to grow near the last scattering surface. These would presumably
be adiabatic and at physical scales above the Jeans length. The linear
analysis above the Jeans scale gives growth $\propto t^{0.57}$ with the
parameters of our model.

In Henriksen (1982) the usual FLRW formulation of cosmology was used. We
revisit and expand this treatment in the next section using the more intuitive
Misner ( Misner (1969) chapter 3) formulation of the field equations, assuming
flat FLRW symmetry. The zero energy initial state was already found in
Henriksen (1982), but not analyzed in terms of modern cosmological parameters.

In an appendix we give an argument based on the Buckingham theorem of
Dimensional analysis (Henriksen and Irwin (2019), see also Henriksen (2015))
applied to the Universe. This suggests numbers of order one at our epoch may
have always been of order one. If so, the inverse square temporal dependence
of the vacuum energy ($\Lambda$) follows. This is much like the argument that
Dirac used to justify a temporal variation in the gravitational constant.

##  3 Basic Formulation

We select the independent equations from (Misner (1969); see also Henriksen,
Emslie and Wesson (1983) and references therein) in the homogeneous form using
co-moving coordinates $\\{r,t\\}$;

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
dt^{2}-e^{\omega(t)}dr^{2}-R(r,t)^{2}(d\theta^{2}+(\sin{\theta})^{2}d\phi^{2}),$
(2) $\displaystyle\partial_{r}m(r,t)$ $\displaystyle=$ $\displaystyle 4\pi
R^{2}\rho\partial_{r}R,$ (3) $\displaystyle\partial_{t}\omega$
$\displaystyle=$
$\displaystyle-4\frac{\partial_{t}R}{R}-2\frac{\partial_{t}\rho}{p+\rho},$ (4)
$\displaystyle e^{-\omega}(\partial_{r}R)^{2}$ $\displaystyle-$
$\displaystyle(\partial_{t}R)^{2}=1-\frac{2m(r,t)}{R},$ (5)
$\displaystyle\partial_{t}m(r,t)$ $\displaystyle=$ $\displaystyle-4\pi
R^{2}p\partial_{t}R.$ (6)

We have set $G=c=1$; $m(r,t)$ is the total mass-energy inside a co-moving
spherical shell labelled $r$; $R$ is the circumferential radius; $\rho(t)$ is
the density of mass-energy; $p(t)$ is the total pressure. The spherical angles
$\theta$ and $\phi$ are constant (co-moving) in radial motion and $ds$ is the
metric distance. The last of these equations is normally satisfied identically
when the other equations are satisfied, but it is necessary when the scale
dependence of the various individual components of $\rho$ are to be found.

Our working equations follow in our assumed symmetry by setting

$\displaystyle m(r,t)$ $\displaystyle=$ $\displaystyle M(t)r^{3},$ (7)
$\displaystyle R(r,t)$ $\displaystyle=$
$\displaystyle\frac{S(t)}{S_{1}}r\equiv\tilde{S}(t)r,$ (8)

so that the Dimensions of $M$ are those of $\rho$. The scale factor $S$ is
Dimensionless, but it is convenient to involve an arbitrary value at the time
of the symmetry breaking instability $t_{1}$ . These expressions for the mass
and circumferential radius guarantee the geometric flatness of the model.

Moreover the energy density is taken to be

$\rho=\rho_{V}(t)+\rho_{\nu}(t)+\rho_{d}(t),$ (9)

where $\rho_{V}$ is the vacuum energy density, $\rho_{\nu}$ is the zero rest
mass (effectively or really; including neutrinos) energy density, and
$\rho_{d}$ is the zero pressure matter density, which we assume to include the
dark and baryonic matter. We normally write

$\rho_{m}\equiv\rho_{\nu}+\rho_{d}.$ (10)

The cosmological pressure at any epoch is therefore

$p=\frac{\rho_{\nu}}{3}-\rho_{V}.$ (11)

The vacuum energy is taken to be

$\rho_{V}=\frac{c^{4}\Lambda(t)}{8\pi
G},~{}~{}~{}~{}~{}~{}\Lambda(t)=\frac{\lambda}{c^{2}t^{2}},$ (12)

where we have restored the units temporarily to indicate that $\lambda$ is a
number..

We may write the equation of state of the matter as

$w(\tilde{S})=\frac{p_{m}}{\rho_{m}}\equiv\frac{\rho_{\nu}/3}{\rho_{m}}\equiv\frac{1}{3(1+\frac{\rho_{d}(1)}{\rho_{\nu}(1)}\tilde{S})},$
(13)

where $(1)$ indicates the value at $\tilde{S}=1$, if the epoch of symmetry
breaking is $t_{1}$. This time is a parameter that we use as a unit of time
subsequenly. We have used the usual scale dependences of the ‘photon’ density
and the dust density that follow from equation (6).

The initial ratio of the ‘dust’ density to the ‘zero rest mass’ density
appears as another parameter in the problem. However the initial value is
related to the value at any epoch by

$\frac{\rho_{d}(1)}{\rho_{\nu}(1)}=\frac{1}{\tilde{S}}\frac{\rho_{d}}{\rho_{\nu}}.$
(14)

Applying this at the current epoch and at equipartition we infer that

$\frac{\rho_{d}(1)}{\rho_{\nu}(1)}=\frac{z_{eq}}{\tilde{S}_{o}},$ (15)

which is very small. The value of $z_{eq}$ according to (Planck Collaboration
(2020)) is $3387$.

Given $z_{eq}$,there is now an iterative procedure necessary because before
integration of the governing equations (see equations 20 and 21) we must
assign an assumed $\tilde{S}_{o}$. The iteration is continued until the
integration arrives at our epoch (defined by an acceptable Hubble-Lemaitre
value coincident with an acceptable ratio of matter to vacuum density) with
the starting value of $\tilde{S}_{o}$. This procedure defines the global
quantities of the model as summarized below.

We continue formally by substituting the form for $m$ and $R$ from equations
(7) and (8) into equation (3) to find

$M(t)=\frac{4\pi}{3}\tilde{S}^{3}\rho_{V}(1+\Omega_{mv}),$ (16)

where we have again set $\tilde{S}\equiv S/S_{1}$ and

$\Omega_{mv}\equiv\frac{\rho_{m}}{\rho_{V}}.$ (17)

These are key variables in our analysis.

We can now substitute for the mass from equations (16) and (7) into equation
(5) together with the scaled form of $R$. This requires us to split the
equation into two equations

$\displaystyle e^{\omega}$ $\displaystyle=$ $\displaystyle\tilde{S}^{2},$ (18)
$\displaystyle\frac{\dot{S}^{2}}{S^{2}}$ $\displaystyle=$
$\displaystyle\frac{8\pi}{3}\rho_{V}(1+\Omega_{mv}).$ (19)

Equation (18) confirms the flatness of the metric. Recalling equation (12) we
write equation (19) in the useful form

$t~{}h=\sqrt{\frac{\lambda}{3}}\sqrt{1+\Omega_{mv}}\equiv
t\frac{\dot{S}}{S}\equiv t\frac{\dot{\tilde{S}}}{\tilde{S}},$ (20)

which defines the Hubble variable $h(t)$.

Equation (4) can now be written explicitly as an equation for $\Omega_{mv}(t)$
using equation (20) as

$t~{}\dot{\Omega}_{mv}=2(1+\Omega_{mv})-\sqrt{3\lambda}(1+w(\tilde{S}))\Omega_{mv}\sqrt{1+\Omega_{mv}},$
(21)

which must be solved together with equation (20).

Once solved, these two equations define a cosmology with some interesting
aspects. The iterative procedure provides the uniqueness of the model. to
emphasize this we give the iterative results here which are to be used in
additional discussion below.

The iteration yields $\rho_{d}(1)/\rho_{\nu}(1)=2.512\times 10^{-33}$ with an
initial $\tilde{S}_{o}=1.348\times 10^{36}$ yielding after iteration the
corresponding value $\tilde{S}_{o}=1.347\times 10^{36}$. This occurs at
$\ln{t_{o}/t_{1}}=41.324$, which gives the age of our epoch as $28.0Gyr$ if
$t_{1}$ is chosen to be $1s$. This agrees with a direct calculation from
equation (20) below. the value of $\lambda$ is given as
$\sqrt{3\lambda}=5.2134$ and $\Omega_{mv}(0)=0.4641$. These quantities require
a Hubble/Lemaitre value at the current epoch of $73.36km/s/Mpc$ and a current
$\Lambda=\lambda/(c^{2}t_{o}^{2})$ equal to $1.29\times 10^{-56}cm^{-2}$ all
assuming $t_{1}=1s$.

These values are remarkably close to concordance values with the exception of
the factor two in the age of the Universe. we discuss this further in the next
section. It should be emphasized that because inflation is absent in this
model, the initial vacuum fluctuations may not be as in the concordance model.
However all of the relevant physics and analysis leading to the explanation of
structure arising since the equipartition epoch is applicable to this model.
Only the extra time available is a major difference. There is a very small
creation of matter from the vacuum at any epoch where $\Omega_{mv}$ has not
attained its ultimate value (i.e. $\tilde{S}\neq\infty$).

##  4 Cosmological model details

In figure (1) we show $\Omega_{mv}(x)$ where $x=\ln(t)$. At $x=0$ (or
$t/t_{1}=1$) we have automatically $\tilde{S}(0)=1$. The unit of time $t_{1}$
is also a free parameter, but we have agreement with current global values
when $t_{1}=1s$. We have iterated $\lambda$ and $\tilde{S}_{o}$ until the
asymptotic value $0.4641$ at large $x$ is nearly that given by Planck
satellite measurements (Planck Collaboration (2020); see also Abdullah,
Klypin& Wilson (2020), where the number is approximately $0.461=0.316/0.685$).
A good value of the asymptotic $\Omega_{mv}$ at large enough $\tilde{S}$ is
found from equation (21 by setting $w=0$ as

$\Omega_{mv}(\infty)=\frac{2}{k^{2}}(1+\sqrt{1+k^{2}}),$ (22)

and hence

$k^{2}=4\frac{1+\Omega_{mv}(\infty)}{\Omega_{mv}(\infty)^{2}}.$ (23)

We have set $k=\sqrt{3\lambda}$.

The important behavioural feature shown in the left panel of figure (1) is the
abrupt rise of $\Omega_{mv}$ from the early zero energy value $-1$ (the
classical vacuum) at $x<0$, to the first asymptotic value equal to $\sim
0.332$ at $x\geq 1$. This is the symmetry breaking instability (any variation
of $\Omega_{mv}>-1$ initiates the instability according to equation (21)) that
we attribute to the electroweak symmetry breaking.

The right hand panel of the figure shows the subsequent rise to the current
value $\sim 0.4642$. This rise is not quite so abrupt as the initial
instability. It corresponds to the equation of state (EOS) declining towards
zero from the massless EOS value of $1/3$. Equation (21) shows that this leads
to a rapid increase in $\Omega_{mv}$.

Throughout this note we refer to the usual cosmic co-moving flat FLRW
coordinate time scale, although we calculate the logarithm. As many have
recognized the logarithmic time $x$ has a great advantage in recording the
history of the Universe. In these terms the zero energy vacuum state has
persisted for an infinity, before the world creating instability. This removes
awkward questions about ‘before the beginning’, if indeed the Universe unfolds
in logarithmic time. This is the case for equations (20) and (21) that define
our model.

The expansion of the Universe is indicated in the lower panels of figure (2).
It begins in the lower left panel with the sudden predominance of the vacuum
energy when $\Omega_{mv}$ fluctuates upwards from $-1$. Using our parameters
in equation (20) we find that $\tilde{S}\propto t^{2.0}$ in the early
asymptotic stage on the lower left panel, and $\tilde{S}\propto t^{2.1}$ in
the late asymptotic stage on the lower right panel. These correspond to
classical acceleration parameter $q=-\ddot{S}/(h^{2}S)$ of $q=-0.5$ and
$q=-0.524$ respectively. The latter value is the value at the current epoch,
which would be $-1$ for exponential expansion.

The upper row in figure (2) shows the Hubble/Lemaitre variable at the initial
instability on the left and near the current epoch on the right. The log time
at the current epoch as a result of the iterations is $x_{o}=41.324$, from
which all of our values follow.

We have assumed that the vacuum variation with $1/t^{2}$ begins with the
initial instability and continues driving the subsequent accelerating
expansion. We require in effect the entire Universe to behave as a coherent
excitation, namely as a long wavelength, scalar, pseudo Goldstone (Higgs)
Boson particle. This is hypothetically coincident with the electroweak
symmetry breaking. The wavelength of this ‘particle’ is $1/\sqrt{\Lambda}$,
which is equal approximately to the apparent horizon, given the inverse square
temporal variation of $\Lambda$.

After our iteration to obtain a self consistent $\tilde{S}_{o}$ and a current
value of $\Omega_{mv}$ we find $t_{o}h_{o}$ from equation (20) and from the
direct integration. The iterated values of $\lambda$ and $\Omega_{mv}(o)$ are
used on the RHS of equation (20). The resulting self consistent $h_{o}$ and
$t_{o}$ and $\Lambda_{o}=\lambda/t_{o}^{2}$ are, as stated previously,

$t_{o}=28.0Gyr,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}h_{o}=73.36km/s/Mpc,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Lambda_{o}=1.29\times
10^{-56}cm^{-2}.$ (24)

Given $z_{eq}=3387$ the coordinate time
$t_{eq}\approx(1/z_{eq})^{1/2.1}t_{o}\approx 0.58Gyr$. Thus there is a rather
longer time in which perturbations above the Jeans scale can grow into
observed structure. The linear growth rate using the standard formula
(e.g.Peacock (1999)) is however reduced to $t^{\sim 0.57}$ compared to
$t^{2/3}$ for a matter dominated Universe. We have used $S\propto t^{2.1}$.

It may be nevertheless that primordial black holes and/or galaxies will have
appeared quite soon after the recombination ($\sim 1.0Gyr$), having grown
linearly since $z_{eq}$, which is already an interval of $0.42Gyr$. That is,
the extra time may allow perturbations to go non linear and detach from the
expansion ‘earlier’ relative to us. Abundant such objects at high red shifts
would be one way of distinguishing this model from the concordance model. The
counter tendency is the rapid historical expansion rate. This may lower the
physical density of the detached objects.

The rapid expansion rate removes the horizon problem for homogeneity just as
does inflation. If $\tilde{S}\propto t^{2}$ as in the earlier asymptote, then
the particle horizon size at time $t>t_{ph}\geq t_{1}$ can be written as
(taking $\tilde{S}_{ph}=1$)

$R_{ph}(t)=ct^{2}(\frac{1}{t_{ph}}-\frac{1}{t}).$ (25)

Here $t_{ph}\neq 0$ is the earliest time from which light signals arrive at
$t$. This implies that a causally connected patch expands faster than the
apparent horizon of the Universe. However, in our model the vacuum metastable
state is already homogeneous (but for quantum fluctuations) whicheffectively
removes the problem.

During the zero energy vacuum era the entropy is essentially zero by the third
law of thermodynamics (plus no local gravitational degrees of freedom (cf
Penrose (2004),Penrose (2010)) and the model is homogeneous, although
presumably subject to quantum noise. This initial condition avoids two of the
criticisms of the big bang model, without inflation. Bouncing cosmological
models encounter the black hole entropy problem (Penrose (2004)), which has
driven a conformally based cycling model Penrose (2010).

At equipartion $\tilde{S}_{eq}\sim 2.3\times 10^{34}$. This implies formally
an enormous temperature, exceeding the Planck temperature
$kT_{P}/=m_{P}c^{2}$, (or $T_{P}\sim 1.4\times 10^{32}K$) at
$\tilde{S}_{1}=1$. This does not correspond to the expected temperature of
symmetry breaking, $T_{1}=159.5\pm 1.5$ GeV, that is $T_{1}\sim 1.85\times
10^{15}K$. This would place us in the region of Grand Unified Theory (GUT)
symmetry breaking (assuming the temperature can not in fact exceed the Planck
value) about which very little is certain.

However it is possible to delay the expansion in order to fit an electro weak
transition in the following fashion. We set $\Omega_{mv}(x=25)=0$ so that the
vacuum state continues until $t=e^{25}s\sim 2.28\times 10^{3}yrs$. At this
value of $x$ we give $\tilde{S}$ the same value that it would have in figure
(2) namely $\sim 5.745\times 10^{21}$. This delayed model is shown in figure
(3).

There is no essential change in our values for $t_{o}$, $h_{o}$ and
$\Lambda_{o}$. We find, due mainly to the different integration interval, that
the self consistent $\tilde{S}_{o}=1.295\times 10^{36}$, rather than the
previous $1.348\times 10^{36}$. We observe in the upper row of figure (3) the
early and late transitions in $\Omega_{mv}$, while the lower row shows the
current behaviour of $h(x)$ and $\tilde{S}(x0$.

The value $\tilde{S}(25)=5.745\times 10^{21}$. This implies a temperature
ratio relative to the recombination epoch ($z_{rec}=1087$) of
$\tilde{S}(rec)/\tilde{S}(25)=1.191\times 10^{33}/5.745\times 10^{21}$. This
yields $T(25)/T_{rec}=2.073\times 10^{11}$. The temperature $T_{rec}$ is
consistently $\sim 3000K$ if $T\propto 1/\tilde{S}$, whence $T(25)=6.227\times
10^{14}K$. This is within a factor $3$ of the electroweak transition
temperature. If we use the scale factor inside the vacuum of
$\tilde{S}_{vac}=3.811\times 10^{21}$, then $T(vac)=9.34\times 10^{14}$, a
factor of $2$ low. It seems that a slight adjustment of the instability to
slightly earlier times could remove even this discrepancy.

This resolves the temperature problem at the presumed elctroweak symmetry
breaking. Other quantities remain essentially the same and the age age of the
Universe remains the outstanding disagreement with the concordance model.

---
Figure 1:  The figure on the left shows the beginning of this Universe model as an instability in a zero energy vacuum that causes $\Omega_{mv}$ to transition rapidly from $-1$ to an early asymptotic value $\sim 0.332$. This rises rapidly again at a subsequent epoch of rapid matter creation beginning near $x=35$, as shown on the right panel. The value of $\sqrt{3\lambda}\approx 5.2134$ in order to give $\Omega_{mv}(o)\equiv\rho_{m}(o)/\rho_{V}(o)\approx 0.4642$. When $\Omega_{mv}=-1$ the total energy is zero. The figure on the right shows the delayed transition. On the left the vacuum $\tilde{S}\approx 0.663$ and on the right $\tilde{S}\approx 2.951\times 10^{30}$. |
---|---
|
Figure 2: In the upper row we show the Hubble/Lemaître variable as a function of $x$ near the initial instability on the left and approaching the current epoch on the right. The only era in which $h$ actually increases is during the instabiity. The lower row shows the corresponding scale factor near the transition epoch on the right and much later near the current epoch on the right. The initial variation is $S\propto t^{2.0}$ for $x<\sim 35$ on the left while in the asymptotic region on the right panel $S\propto t^{2.1}$.  |
---|---
|
Figure 3: In the upper row we show on the left $\Omega_{mv}$ as a function of
$x$ near the delayed instability, and the ‘equation of state’ driven
transition to the current epoch value on the right panel. The parameter
$\lambda$ is the same as in figure (1). The lower row shows the
Hubble/Lemaitre variable near the current epoch on the left panel and the
scale variable near the current epoch on the right.

##  5 Discussion

Our fundamental assumption is that the dark energy evolves rather slowly
($\rho_{V}\propto 1/t^{2}$) compared to the inflationary era. We have shown
that the best justification of this assumption is through Dimensional
analysis. The assumption also ensures that the vacuum scale is always similar
to the apparent horizon of the Universe, which suggests that he Universe is a
long wave length pseudo-Goldstone excitation.

Our model requires an older Universe than that of the concordance model. The
material Universe is due to an instability of a zero energy vacuum state which
we take to coincide with electroweak spontaneous symmetry breaking. The nature
of matter and the positive energy vacuum (‘dark energy’) in the Universe and
their relative importance therefore follows from this event. That is, we
assume that the electroweak symmetry breaking produces immediately the matter
to vacuum ratio $\rho_{m}/\rho_{v}\equiv\Omega_{mv}$.

We expect therefore; that the decay products of the $W^{\pm}$ and $Z^{0}$
vector bosons and their subsequent interactions, plus high energy photons and
their interactions, to comprise both the visible and dark material Universe
plus the ‘Dark Energy’ ($\rho_{V}$). The Higgs Boson is essential to the
symmetry breaking of the gauge invariance, and its decay products will also be
present. The details of this process are certainly beyond the scope of this
article, but there is a peculiar coincidence that should be noted.

The Weinberg or Weak Mixing Angle $\theta_{W}$ is given in the most recent
CoData compilation as $\sin^{2}{\theta_{W}}=0.22290$, which is
$\sin{\theta_{W}}=0.472$. This is at an energy of $91.2GeV$. The angle is not
strictly constant appearing to ‘run’ with energy. For example at $7~{}TeV$ the
ATLAS collaboration at the LHC gives $sin^{2}(\theta_{W})=0.23080\pm 0.00120$
or $\sin{\theta_{W}}=0.479$. Proceeding with the lower value (closer to the
electroweak energy of symmetry breaking), we have according to the theory,

$\sin{\theta_{W}}=\frac{\sqrt{M_{Z}^{2}-M_{W}^{2}}}{M_{Z}}\approx 0.472,$ (26)

where $M_{Z}\approx 91.2GeV$ and $M_{W}=80.38GeV$ are the corresponding boson
masses.

To within errors, this is almost the value of the current epoch matter to
‘dark energy’ ratio $\rho_{m}/\rho_{V}\approx 0.315/0.685=0.460$
($0.32/0.68=0.4705$). The current number is a little smaller than what we have
used for our calculations namely $0.4642$, although that value is within
observational divergences and can be refined. This suggests that difference in
the decay chains of the bosons normalized by the decay of the $Z^{0}$ boson,
yields the matter to dark energy ratio in the Universe.

Note that the matter includes the dark matter, so the dark matter must be
present in the difference between the $Z^{0}$ and $W^{\pm}$ evolutionary
sequences. Perhaps one way that the matter antimatter symmetry inherent in the
bosons can be broken is if the $Z^{0}$ decay chain interacts asymmetrically
with the $W$ decay chain. The different masses of the bosons may lead to
differences in equilibrium abundance of the decay products. The dark energy by
contrast appears to be due solely to the decay chain beginning with the
production of the $Z^{0}$ bosons.

Astrophysically, the dominant change in this model is the age of the Universe.
This may allow structure to grow at higher redshifts then currently
anticipated. Searches for fully formed galaxies or Quasars at ever higher red
shifts might provide a decisive test of this model. Philosophically, the
initial zero energy vacuum appears as the matter/antimatter balance that
yields nothing. Fortunately it is unstable.

##  6 Acknowledgements

I thank Judith Irwin for her constant encouragement, criticism and help.

##  7 Appendix A:Cosmological Numbers

In Henriksen and Irwin (2019)) it is argued that the Buckingham theorem
confirms that statements about the Universe should amount to expressions
between Dimensionless ‘numbers’. With $n=9$ relevant quantities
($h,\bar{\rho},\Lambda,T_{b},\delta T_{b},R_{AH},t,c,G$) and $m=4$ independent
Dimensions (length, time, mass, temperature) for the homogeneous classical
Universe we expect a basic function of five ($n-m$; e.g. Henriksen (2015))
variables. We omit the Mond possibility here, but see Henriksen and Irwin
(2019). We have written $R_{AH}$ for the apparent horizon and $T_{b}$, $\delta
T_{b}$ for the background temperature and temperature fluctuation.

The most general expression involving all independent ‘cosmic numbers’ (not
all need ultimately appear) would be some single continuous ‘cosmic’ function

$f(K,V,F,T,R)=0,$ (27)

where a complete (but not unique) set of ‘numbers’ at some epoch follow from
our cosmic catalogue as ($t$ is the epoch age)

$\displaystyle K$ $\displaystyle\equiv$ $\displaystyle\frac{8\pi
G\bar{\rho}}{3h^{2}},$ (28) $\displaystyle V$ $\displaystyle\equiv$
$\displaystyle\frac{\Lambda c^{2}}{3h^{2}},$ (29) $\displaystyle F$
$\displaystyle\equiv$ $\displaystyle\frac{\delta T_{b}}{T_{b}},$ (30)
$\displaystyle T$ $\displaystyle\equiv$ $\displaystyle ht,$ (31)
$\displaystyle R$ $\displaystyle\equiv$ $\displaystyle\frac{R_{AH}}{ct}.$ (32)

All of these numbers are of $O(1)$ 222$K$ is $\Omega_{m}$ but the numerical
factor of 3 and the factor $8\pi$, which factors can not be realized entirely
by Dimensional argument according to currently accepted values, except the
number $F$. It is of $O(10^{-5})$ and can be eliminated from the relation (27)
by a zeroth order MacLaurin expansion in $F$. It is the only number that
corresponds to a definite epoch and certainly contains information about the
history of the Universe. However the MacLaurin expansion suggests that it does
not affect directly the global properties of later epochs (ignoring sub
structure of the homogeneous Universe). Moreover the apparent horizon of a
flat Universe is essentially $c/h$ (e.g. Rindler (2006)) so that $R$ is not
independent of $T$. This leaves the numbers $K$, $V$ $T$ as arguments of the
cosmic function, which each contain the Hubble variable $h(t)$ so that the
cosmic function becomes

$\tilde{f}(K,V,T)=0.$ (33)

We note that the fractional expansion rate $h$ varies as $1/t$ in most
cosmological models (including this one and excepting de Sitter). This makes
$T$ and normally $K$ ‘universal’ constants. In order to avoid making our epoch
special we need only assume that $V$ is also constant. In fact on general
grounds, we expect the numbers $K$, $V$ and $T$ to be close to unity at all
epochs. Otherwise very big numbers (or rather their reciprocals) or very small
numbers, would be eliminated as was the number $F$.

This argument requires $h\propto 1/t$, and $\bar{\rho}\propto 1/t^{2}$ as is
known, but also $\Lambda\propto 1/t^{2}\propto\bar{\rho}$. The counter
argument to this conclusion is based on the anthropic principle, but this is
not readily testable. In principle the values of the numbers at different
epochs are measurable.

We have used only the Buckingham theorem, a reasonable choice of the set of
Universe characteristics (a ‘catalogue’ of properties), and a conviction that
important numbers should be $O(1)$, in order to reach the form of the cosmic
function. We have also assumed that our epoch is not unique. There is no use
of dynamics to this point.

However as an example, the Friedman equation in the concordance model gives
with zero curvature

$h\equiv\sqrt{\frac{8\pi G}{3}\bar{\rho}+\frac{\Lambda c^{2}}{3}},$ (34)

which can be written as ($h\neq 0$) the familiar

$1-(V+K)=0.$ (35)

This indicates a degeneracy between the numbers $V$ and $K$ when $T$ is non
zero, and the Universe is spatially flat. Agreement with the general form (33)
follows trivially by multiplying equation(35) by T. One does not normally
resolve the function (33) without additional information, in this case General
Relativity. However once one assumes that $T$ is constant, the degeneracy
between $K$ and $V$ is inevitable although its form is not. In the concordance
model $K=0.32$ and $V=0.68$ although more recently (Abdullah, Klypin& Wilson
(2020), Planck Collaboration (2020)) these are $K=0.315$ and $V=0.685$.

## References

  * Abdullah, Klypin& Wilson (2020) Abdullah, M.H., Klypin, A. & Wilson, Gillian, Astrophysical Journal, 901, # 2, 2020
  * Berestetskii, Lifshitz and Pitaevskii (1971) Berestetskii, V.B., Lifshitz, E.M.,and Pitaevskii, L.P., ”Relativistic Quantum Field Theory”, Pergamon (Addison-Wesley), Oxford, p36, 1971
  * Bicknell& Henriksen (1979) Bicknell, G.V. & Henriksen, R.N., Astrophysical Journal, 232, 670, 1979
  * Henriksen (1982) Henriksen, R.N., Physics Letters B, 119, 85, 1982
  * Henriksen, Emslie and Wesson (1983) Henriksen, R.N., Emslie, G.A. and Wesson, P.S., Phys. Rev.D, 27, 1219, 1983
  * Henriksen (2015) Henriksen, R.N., ”Scale Invariance”,Wiley-VCH, Weinheim, Germany, 2015
  * Henriksen and Irwin (2019) Henriksen, R.N. and Irwin, J.A., ”A ‘Numbers ’Approach to Astronomical Correlation”,ArXiv190208704, 2019
  * Lykken & Spiropulu (2013) Lykken, J. & Spiropulu, Maria, Physics Today, 66, # 12, 28, 2013
  * Misner (1969) Misner, C.W., in ”Astrophysics and General Relativity”, v1,Chrétien,M.,Deser, S. and Goldstein, J. (eds), Gordon and Breach, London, chpt 3, 1969
  * Oesch,Brammer, van Dokkum et al. (2016) Oesch, P.A., Brammer,G., van Dokkum, P.G. and 15 others, Ap.J.. 819, 129,2016
  * Olson and Jordan (1987) Olson, T. and Jordan, T., Phys. Rev. D, 35, 3258, 1987
  * Ostriker and Steinhardt (1995) Ostriker, J.P. and Steinhardt, P., Nature, 377, 600, 1995
  * Peacock (1999) Peacock, J. A., ”Cosmological Physics”, Cambridge University Press, Cambridge, UK, 1999
  * Penrose (2004) Penrose, R., ”The Road to Reality”, BCA(Jonathan Cape), G.B., chpt. 27, 2004
  * Penrose (2010) Penrose, R., ”Cycles of Time”, The Bodley Head , London, 2010
  * Peter (2013) Peter, P., arXiv1303.2509, 2013
  * Planck Collaboration (2020) Planck Collaboration, Astron. and Astrophys., 641, A6, 2020
  * Ratra and Peebles (1988) Ratra, B. and Peebles, P.J.E., Phys. Rev. D, 37,3406,1988
  * Rindler (2006) Rindler, W.,”Relativity”. second Ed., Oxford University Press, New York, chpt. 17, 2006


# A Software Tool for ”Gluing” Distributions

Peter Werner<EMAIL_ADDRESS>Institut für Physik, Universität
Oldenburg, 26111 Oldenburg, Germany

###### Abstract

When performing Monte-Carlo simulations, distributions are sometimes
determined only for sub-intervals of the desired total range. In such cases, a
frequent problem is to connect, or glue, individual distributions to obtain
the final result. Most prominent examples, where this is usually necessary,
are certain large-deviation simulation techniques. However, there are multiple
approaches to do this, depending on the data and individual requirements.
Here, a software tool is presented, containing multiple algorithms, to aid
with this task. An introduction to the available methods is presented together
with a short tutorial using exemplary data.

###### Contents

  1. I Introduction
    1. I.1 How to obtain the tool?
    2. I.2 How to run the tool?
  2. II Gluing Algorithms
    1. II.1 Least-Squares-Method
    2. II.2 Ferrenberg-Swendsen-Method
  3. III Tutorial
    1. III.1 Raw unbiased data
    2. III.2 Histogrammed unbiased data
    3. III.3 Raw biased data
    4. III.4 Histogrammed biased data
  4. IV Final Remarks
  5. A Analytical solution for LSQRS method weight factors

##  I Introduction

In large-deviation Monte-Carlo simulations Hartmann (2002, 2014); Börjes  _et
al._ (2019); Werner and Hartmann (2021) usually multiple runs are performed
that are targeting different ranges for the quantity of interest to obtain its
distribution in the desired regime of rare events. One important part of the
data post-processing is to connect the individual histograms to one final
distribution. The software tool that is showcased here was originally
developed with this specific use case in mind, but also extends beyond. It can
be useful for any variant of Monte-Carlo simulation, even without a relation
to large-deviations, and experimental data alike.

In the following, it is first explained how to obtain and run the tool.
Subsequently, there is a description of the implemented algorithms. Finally, a
tutorial is given that demonstrates the core functionality using exemplary
data.

###  I.1 How to obtain the tool?

The tool is available through the repository https://gitlab.uni-
oldenburg.de/ag-computerphysik/distribution-gluer , from where it can either
be cloned via git or directly downloaded by a web-browser. It comes in the
form of a python3-script.

###  I.2 How to run the tool?

To run the script, a recent version of Anaconda
https://www.anaconda.com/products/individual is recommended. Otherwise, the
following libraries must be installed in a local python3 environment: numpy,
gmpy2, matplotlib and scipy. As a starting point, typing

    ΨΨ  python3 distribution_gluer.py --help
    ΨΨ

into a command line will display an overview of all options. The displayed
help-output also serves as the main reference for the program.

##  II Gluing Algorithms

The tool can handle raw data, which is naively just a file containing one
column of real valued numbers, or data that is already processed into a
histogram. For raw data, the tool can determine the corresponding histograms
by itself. In any case, as a result of numerical simulations Hartmann (2015);
Newman and Barkema (1999) or obtained otherwise, let there be $N$ histograms
$H_{\Theta_{i}}(x)$ with $i=1,\dots,N$ for an arbitrary bin center value $x$.
The corresponding errors are denoted by $\sigma_{\Theta_{i}}(x)$. The value
$\Theta_{i}$ is a simulation parameter on which the histograms depend, usually
the temperature or a temperature like parameter. For example in large
deviation simulations Hartmann (2002, 2014); Börjes  _et al._ (2019); Werner
and Hartmann (2021), these are typically used to control the extend by which
the simulation is steered towards rare events. Here, $\Theta_{i}$ is referred
to as a pseudo-temperature, even though it can be an actual temperatures as
well. This dependence can, but does not have to, result in a bias that needs
to be accounted for prior to connecting the distributions by multiplying with
a factor $f(\Theta,x)$:

$\displaystyle H_{i}(x)$ $\displaystyle=H_{\Theta_{i}}(x)f(\Theta_{i},x)$ (1)
$\displaystyle\sigma_{i}(x)$
$\displaystyle=\sigma_{\Theta_{i}}(x)f(\Theta_{i},x).$ (2)

This factor could be just $f(\Theta,x)=1$ in case all pseudo-temperatures are
the same $\Theta_{i}=\Theta$, but the histograms were determined over
different intervals (e.g. as it is the case for Wang-Landau sampling Wang and
Landau (2001); Vogel  _et al._ (2014)). Or when the overall quantity of
interest is the energy-state-density of a physical system following the
Boltzmann-distribution, the factor would be $f(\Theta,x)=\exp{(x/\Theta)}$.

At this point in time, the tool offers two ways, namely the Least-Squares
(LSQRS) and the Ferrenberg-Swendsen (FS) Ferrenberg and Swendsen (1989)
method, to connect histograms/distributions, which are described briefly in
the following.

###  II.1 Least-Squares-Method

The LSQRS method tries to minimize the squared difference

$\Delta(g_{i},g_{j})=\sum_{x\in\mathcal{M}_{i,j}}\frac{(g_{i}H_{i}(x)-g_{j}H_{j}(x))^{2}}{\sigma_{i}^{2}(x)+\sigma_{j}^{2}(x)}$
(3)

between two histograms $i,j=1,\dots,N$ and $i\neq j$, where
$\mathcal{M}_{i,j}$ is the set of bins $x$ for which both histograms have a
non-zero count, i.e. $H_{i}(x)\neq 0\wedge H_{j}(x)\neq 0$. Additionally, the
weights $g_{i},g_{j}\neq 0$ have to fulfill the side condition of a normalized
total distribution. There is an analytical solution to eq. (3) (see appendix
A) resulting in a weighting factor $g_{i}$ for each histogram. Subsequently,
these are used to determine the final probability density and uncertainty

$\displaystyle p(x)$
$\displaystyle=\frac{\sum_{i=1}^{N}\frac{g_{i}H_{i}(x)}{(g_{i}\sigma_{i}(x))^{2}}}{\sum_{j=1}^{N}\frac{1}{(g_{j}\sigma_{j}(x))^{2}}}$
(4) $\displaystyle\sigma_{p}(x)$
$\displaystyle=\frac{1}{\sqrt{\sum_{i=1}^{N}\frac{1}{(g_{i}\sigma_{i}(x))^{2}}}}.$
(5)

###  II.2 Ferrenberg-Swendsen-Method

The FS method assumes a particular bias of
$f(\Theta_{i},x)=\exp{(x/\Theta_{i})}$ on the histograms $H_{\Theta_{i}}(x)$,
which is in contrast to the LSQRS method that works with any form of bias.
According to Ferrenberg and Swendsen (1989), the final distribution is
obtained by solving the equations

$\displaystyle p(x)$
$\displaystyle=\frac{\sum_{i=1}^{N}H_{\Theta_{i}}(x)}{\sum_{j=1}^{N}e^{-\frac{x}{\Theta_{j}}+h_{j}}\sum_{\tilde{x}}H_{\Theta_{j}}(\tilde{x})}$
(6) $\displaystyle e^{-h_{i}}$
$\displaystyle=\sum_{x}p(x)e^{-\frac{x}{\Theta_{i}}}$ (7)

numerically for $h_{i}$ ($i=1,\dots,N$) with an iterative procedure. The
uncertainty on $p(x)$ is given by Ferrenberg and Swendsen (1989)

$\sigma_{p}(x)=\frac{p(x)}{\sqrt{\left(\sum_{i=1}^{N}H_{\Theta_{i}}(x)\right)}}.$
(8)

##  III Tutorial

This tutorial assumes basic knowledge in using a command line
interpreter/shell (e.g. bash). The tool comes with example data contained in
the archive file example_data.zip, which should be unpacked before trying the
following commands. The data can either be in raw form or already be
histogramed. For reference on the specific file formatting, see the -f option
of the help-output. In order to process raw data the -c option has to be
selected, which will let the tool create the histograms by itself (see
https://numpy.org/doc/stable/reference/generated/numpy.histogram˙bin˙edges.html
for technical reference; the bins=’auto’ argument is used). For convenience
also the -P option is set in the following commands, yielding a plot of the
result. The displayed figure shows in the bottom plot the original histograms
and in the top one, depending on the gluing method, for LSQRS the individual
reweighted histograms and for FS the final distribution. The legend indicates
the corresponding pseudo-temperatures when provided. The final distribution is
written to the output file output.txt as specified by the -o option. If no
gluing method is set, the tool defaults to LSQRS.

###  III.1 Raw unbiased data

In order to combine all files in the directory raw_unbiased_data into one
distribution, the command

    ΨΨ  python3 distribution_gluer.py -f \
    ΨΨ  example_data/raw_unbiased_data/* -c -P \
    ΨΨ  -o output.txt
    ΨΨ

can be used.

Figure 1: Program output plot example for raw unbiased data. Top:
Glued/Rescaled distributions using the LSQRS method. Bottom: Original
histograms determined by the gluing tool.

An example of how the displayed plot might look like is given in Fig. 1.
To improve the result, separate binning intervals can be specified using the
-b BINNING_SEPERATOR1 BINNING_SEPERATOR2 … option. Each of the separator
variables marks the interval edges for which the internal binning algorithm is
run independently. This can be helpful, when there are regions with fewer data
points:

    ΨΨ  python3 distribution_gluer.py -f \
    ΨΨ  example_data/raw_unbiased_data/* -c -P \
    ΨΨ  -o output.txt -b -3 3
    ΨΨ

It is also possible to completely ignore any standard deviation on the
histogram bins with the -i option:

    ΨΨΨΨ  python3 distribution_gluer.py -f \
    ΨΨΨΨ  example_data/raw_unbiased_data/* -c -P \
    ΨΨΨΨ  -o output.txt -b -3 3 -i
    ΨΨ

Sometimes it is useful to cut off outlying values from the individual
histograms, before applying the LSQRS-method. The -s STD_MULTIPLE option will
discard any bins that are more than the specified multiple of the standard
deviation away from the mean:

    ΨΨ  python3 distribution_gluer.py -f \
    ΨΨ  example_data/raw_unbiased_data/* -c -P \
    ΨΨ  -o output.txt -b -3 3 -s 3
    ΨΨ

###  III.2 Histogrammed unbiased data

Like with the case of raw unbiased data, it is possible to combine data that
is already processed into histograms:

    ΨΨ  python3 distribution_gluer.py -f \
    ΨΨ  example_data/histogrammed_unbiased_data/* \
    ΨΨ  -P -o output.txt
    ΨΨ

###  III.3 Raw biased data

For biased data, pseudo temperatures have to be provided (option -p PSEUDO1
PSEUDO2 …), where a bias of the form $f(\Theta_{i},x)=\exp{(x/\Theta_{i})}$ is
assumed. The FS method is selected via the -g FS option. In the following
command, also a file containing specific bin edges (option -B BINNING_FILE) is
provided:

    ΨΨ  PSEUDO={-10.0,10.0,-2.0,2.0,-4.0,4.0,\
    ΨΨ  Ψ-6.0,6.0,-8.0,8.0,inf};\
    ΨΨ  python3 distribution_gluer.py -f \
    ΨΨ  $(eval echo example_data/raw_biased_data\
    ΨΨ  /pseudo_$PSEUDO.dat) \
    ΨΨ  -p $(eval echo $PSEUDO) \
    ΨΨ  -B example_data/raw_biased_data\
    ΨΨ  /bin_edges.txt -c -P -o output.txt -g FS
    ΨΨ

###  III.4 Histogrammed biased data

Again, it is also possible to work with histograms of biased data directly:

    ΨΨ  PSEUDO={-10.0,10.0,-2.0,2.0,-4.0,4.0\
    ΨΨ  ,-6.0,6.0,-8.0,8.0,inf};\
    ΨΨ  python3 distribution_gluer.py -f \
    ΨΨ  $(eval echo example_data\
    ΨΨ  /histogrammed_biased_data\
    ΨΨ  /pseudo_$PSEUDO.dat) \
    ΨΨ  -p $(eval echo $PSEUDO) -P \
    ΨΨ  -o output.txt -g FS
    ΨΨ

Figure 2: Program output plot example for biased histogrammed data. Top: Glued
final distribution using the FS method. Bottom: Original biased histograms at
different pseudo-temperatures $\Theta_{i}$ provided to the gluing tool.

An example of the program output plot for this command is displayed in Fig. 2.
Biased histograms can also be treated with the default LSQRS method using:

    ΨΨ  PSEUDO={-10.0,10.0,-2.0,2.0,-4.0,4.0\
    ΨΨ  Ψ,-6.0,6.0,-8.0,8.0,inf}; \
    ΨΨ  python3 distribution_gluer.py -f \
    ΨΨ  $(eval echo example_data\
    ΨΨ  /histogrammed_biased_data\
    ΨΨ  /pseudo_$PSEUDO.dat) \
    ΨΨ  -p $(eval echo $PSEUDO) -P -o output.txt
    ΨΨ

##  IV Final Remarks

The tool is in continuous development and any comments and suggestions for
improvement are very welcome. The author wishes to thank Alexander K. Hartmann
for carefully reading the manuscript and testing the program.

##  Appendix A Analytical solution for LSQRS method weight factors

Rewriting eq. (3) with relative weights $g_{ij}:=g_{j}/g_{i}$ (relative
weights have two indices and absolute weights one) yields

$\tilde{\Delta}(g_{ij}):=\sum_{x\in\mathcal{M}_{i,j}}\frac{(H_{i}(x)-g_{ij}H_{j}(x))^{2}}{\sigma_{i}^{2}(x)+\sigma_{j}^{2}(x)}.$
(9)

The minimal squared difference must satisfy the condition
$\frac{\partial\tilde{\Delta}}{\partial g_{ij}}\overset{!}{=}0$, resulting in

$g_{ij}=\frac{\sum_{x\in\mathcal{M}_{i,j}}\frac{H_{i}(x)H_{j}(x)}{\sigma_{i}^{2}(x)+\sigma_{j}^{2}(x)}}{\sum_{x^{\prime}\in\mathcal{M}_{i,j}}\frac{H^{2}_{j}(x^{\prime})}{\sigma_{i}^{2}(x^{\prime})+\sigma_{j}^{2}(x^{\prime})}}.$
(10)

For relative weights between non-overlapping histograms (i.e.
$\mathcal{M}=\emptyset$) eq. (10) is not applicable. However, these can be
calculated in a chain like manner. When $g_{ij}$ and $g_{jk}$ with $i\neq
j\neq k$ are known and the relative weight of interest $g_{ik}$ can not be
calculated directly via eq. (10), it is still possible to use

$g_{ik}=\frac{g_{k}}{g_{i}}=\frac{g_{j}}{g_{i}}\frac{g_{k}}{g_{j}}=g_{ij}g_{jk}.$
(11)

By fixing one of the absolute weight factors (e.g. $g_{i}$) such that the
total distribution is normalized, all other weights $g_{j}$ are determined by

$g_{j}=g_{ij}g_{i}.$ (12)

## References

  * Hartmann (2002) A. K. Hartmann, Phys. Rev. E 65, 056102 (2002).
  * Hartmann (2014) A. K. Hartmann, Phys. Rev. E 89, 052103 (2014).
  * Börjes  _et al._ (2019) J. Börjes, H. Schawe, and A. K. Hartmann, Phys. Rev. E 99, 042104 (2019).
  * Werner and Hartmann (2021) P. Werner and A. K. Hartmann, Phys. Rev. E 104, 034407 (2021).
  * (5) https://gitlab.uni-oldenburg.de/ag-computerphysik/distribution-gluer, .
  * (6) https://www.anaconda.com/products/individual, .
  * Hartmann (2015) A. K. Hartmann,  _Big Practical Guide to Computer Simulations_, 2nd ed. (WORLD SCIENTIFIC, 2015) https://www.worldscientific.com/doi/pdf/10.1142/9019 .
  * Newman and Barkema (1999) M. E. J. Newman and G. T. Barkema,  _Monte Carlo methods in statistical physics_ (1999).
  * Wang and Landau (2001) F. Wang and D. P. Landau, Phys. Rev. Lett. 86, 2050 (2001).
  * Vogel  _et al._ (2014) T. Vogel, Y. W. Li, T. Wüst, and D. P. Landau, Phys. Rev. E 90, 023302 (2014).
  * Ferrenberg and Swendsen (1989) A. M. Ferrenberg and R. H. Swendsen, Phys. Rev. Lett. 63, 1195 (1989).
  * (12) https://numpy.org/doc/stable/reference/generated/numpy.histogram_bin_edges.html, .