The Branch-Cut Cosmology: evidences and open questions

1,2]César A. Zen Vasconcellos*

3,4]Peter O. Hess

5]José de Freitas Pacheco

1]Dimiter Hadjimichef

6]Benno Bodmann

C.A. Zen Vasconcellos et al

[1]Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, Brazil

[2]International Center for Relativistic Astrophysics Network (ICRANet), Pescara, Italy

[3]Universidad Nacional Autónoma de Mexico (UNAM), México City, México

[4]Frankfurt Institute for Advanced Studies (FIAS), Hessen, Germany

[5]Observatoire de la Côte d'Azur, Nice, France

[6]Universidade Federal de Santa Maria (UFSM), Santa Maria, Brazil

*César A. Zen Vasconcellos<EMAIL_ADDRESS>
In this work we present a brief review of the branch-cut cosmology as well as we approach topics not yet investigated. Starting with an already explored topic, the theme of the primordial singularity suppression, we approach essential themes for a better understanding of the scope of branch-cut cosmology. More precisely, based on recent contributions, we advance a better understanding of the branch-cut scenarios as well as essential topics for modern cosmology such as the flatness, homogeneity and horizon problems from the classical point of view of the branch-cut cosmology.

§ BRANCH CUT COSMOLOGY: A BRIEF REVIEW

The main proposal of the branch-cutting model is, — in addition to suppressing the primordial singularity —, to consistently propose a contraction phase of the universe, before the big bang explosion and to restore the physical conditions of the expansion period of the universe, without introducing any special feature such as inflation or bouncing, based in turn just on an evolutionary
global topological phase transition description of the universe[We recently found an article that, although following a very different line of investigation, also seeks to describe the origin of the universe through a topological phase transition. An interesting point of contact of this proposition with the branch-cut cosmology, is that the author describes the birth of the universe using a global topological phase transition with a complex manifold where the time, $\tau$, is considered as a complex variable. However, before the big bang, differently from the branch-cut cosmology it is a purely imaginary variable so that the space can be considered as  Euclidean <cit.>.].

In this sense, the primary idea of branch-cut cosmology is to offer a description of the early evolutionary universe
without resorting to an ad hoc mechanism that assumes an entire universe created out of essentially nothing, based on the
“materialization” of negative energy states, assuming the energy of the gravitational field could become negative without limits [10].
Or, as proposed also in an ad hoc way, in the original version of the bouncing model, a non-singular bounce mechanism
that occurs at densities well below the Planck scale where quantum gravity effects are small <cit.>.

The branch-cut classical model of the universe replaces the cosmological singularity with a finely-tuned region, — purely geometric in nature, which occurs, in the classical version, for densities well below the Planck density scale, where quantum gravity effects may be disregarded  —,  that allows the smooth transition between the contraction and expansion phases of the universe, thus enabling the resolution of fundamental problems of cosmology <cit.>.  However, as shown recently, the impossibility of packaging energy and entropy according to the Bekenstein criterion <cit.> in a finite size makes the transition phase very peculiar, imposing a topological leap between the two phases or a transition region similar to a wormhole, with space-time shaping itself topologically in the format of a helix-shape like as proposed by the branch-cut cosmology around a branch-point, preserving the fundamental conservation laws during this transition <cit.>.

By means of the composition of the multiverse proposal developed by  4 of a hypothetical set of multiple universes, existing in parallel and the technique of analytical continuation in complex analysis applied to the  FLRW metric [5, 6, 7, 8],
a new set of Friedmann-type equations for a complexified version of the  $\Lambda$CDM  ($\Lambda\neq 0$) model were obtained [1, 2, 3]:

\begin{equation}
\Biggl(\frac{\frac{d}{dt} \ln^{-1}[\beta(t)]}{\ln^{-1}[\beta(t)]  } \Biggr)^2   =    \frac{8 \pi G}{3} \rho(t)
-  \frac{kc^2}{\ln^{-1}[\beta(t)]} + \frac{1}{3} \Lambda \, ;  \label{NFE1}
\end{equation}

\begin{equation}
\Biggl( \frac{\frac{d^2}{dt^2} \ln^{-1}[\beta(t)]}{\ln^{-1}[\beta(t)] } \Biggr)   =  - \frac{4 \pi G}{3} \Big(\rho(t) + \frac{3}{c^2} p(t) \Bigr)
+  \frac{1}{3} \Lambda
,  \label{NFE2}
\end{equation}

where $\Lambda$ represents the cosmological constant. These equations are
referred  as the first (<ref>) and second (<ref>) new Friedmann-type  equations analytically continued to the complex plane. The
analytically continued energy-stress conservation law in the expanding universe is given by:

\begin{equation}
\frac{d }{dt}\rho(t) + 3 \Big(\rho(t) + \frac{p(t)}{c^2}  \Bigr) \Biggl( \frac{\frac{d}{d t} \ln^{-1}[\beta(t)]}{\ln^{-1}[\beta(t)]}\Biggr)  =  0 \, .  \label{ECln}
\end{equation}

This closed set of equations relate a new complex cosmic scale factor, $\ln^{-1}[\beta(t)]$, the energy density, $\rho(t)$, and the pressure, $p(t)$, for a flat, open and closed universe ($k=0, 1, -1$)
(for comparison with the conventional treatment see <cit.>). The unique feature of this representation of the universe lies in
the restricted superposition of multiverses leading to a single multi-leaf universe in the imaginary domain with a branch-cut type connection around a branch-point while in the real domain the multiverses are disconnected. In this sense, the only domain that allows a plausible physical connected interpretation and allows causality to accomplish is the imaginary domain.

The complex scale factors $\ln^{-1}[\beta(t)]$, where $\beta(t)$ represents a regularization function[This notation indicates the reciprocal of $\ln[\beta(t)]$, not its inverse.] allows to define the  analytically continued Hubble rate $H[\beta(t)]$ as

\begin{equation}
H[\beta(t)]  \equiv \Biggl[  \frac{\frac{d}{dt} \ln^{-1}[\beta(t)]}{\ln^{-1}[\beta(t)]} \Biggr]  \, ,
\end{equation}

\begin{eqnarray}
\dot{H}[\beta(t)] & = &  - H^2[\beta(t)] \Biggl( 1  -  \frac{1}{H^2[\beta(t)]} \Biggl[ \frac{\frac{d^2}{dt^2} \ln^{-1}[\beta(t)]}{\ln^{-1}[\beta(t)]} \Biggr]  \Biggr)  \nonumber \\
& \equiv & H^2[\beta(t)] (1 + q_{\rm{ac}}(t)) \,  .
\end{eqnarray}

From this expression,

\begin{equation}
q_{\rm{ac}}(t) \equiv  \frac{1}{H^2[\beta(t)]} \Biggl[ \frac{\frac{d^2}{dt^2} \ln^{-1}[\beta(t)]}{\ln^{-1}[\beta(t)]} \Biggr] \, ,
\end{equation}

$q_{\rm{ac}}$ defines the analytically continued deceleration parameter which provides a relationship between the density of the
branch-cut universe and the critical density ($\rho_{\rm{cr}}$), i.e., the density corresponding to  $k = 0$
for the radiation- and matter-dominated eras [1, 2, 3]).

Following a similar technical procedure (see 1), we
arrive at the following complex conjugated Friedmann's-type  equations:

\begin{equation}
\Biggl(\frac{\frac{d}{dt} \ln^{-1}(\beta^*(t^*))}{\ln^{-1}(\beta^*(t^*))} \Biggr)^2   =     \frac{8 \pi G}{3} \rho^*(t^*)
-  \frac{kc^2}{\ln^{-2}(\beta^*(t^*))} + \frac{1}{3} \Lambda^* ,   \label{NCFE1}
\end{equation}

\begin{equation}
\Biggl( \frac{\frac{d^2}{dt^{*2}} \ln^{-1}(\beta^*(t^*))}{\ln^{-1}(\beta^*(t^*)) } \Biggr)    =   - \frac{4 \pi G}{3} \Biggl(  \rho^*(t^*)  +  \frac{3}{c^2} p^*(t^*)  \Biggr)
+  \frac{1}{3} \Lambda^*  .   \label{NCFE2}
\end{equation}

The corresponding complex conjugated expression for the energy-stress conservation law in the expanding universe is given by

\begin{equation}
\frac{d \rho^*(t^*)}{d t^*} + 3 \Big(\rho^*(t^*) + \frac{p^*(t^*)}{c^2}  \Bigr) \Biggl( \frac{\frac{d}{d t^*} \ln^{-1}[\beta^*(t^*)]}{\ln^{-1}[\beta^*(t^*)]}\Biggr)  =  0 \, .  \label{CECln}
\end{equation}

Similar procedures allow to obtain complex conjugate analytically continued expressions for the Hubble rate, deceleration parameter as well as
complex and complex conjugated expressions for the analytically continued
Ricci scalar and the Ricci curvature.

The set of complex equations makes possible the description of the past-present-future evolution of the universe, in which the different Riemann sheets represent space-temporal separations between the different instants of time. In this sense, although outlined in the form of an helix, as the universe evolves, the different instants of time correspond to Riemann sheets that are progressively under construction, thus differentiating the branch-cut model from the Block Universe <cit.>, and therefore presenting some similarity with the Evolutionary Block Universe <cit.>.

At this point it is important to reaffirm that these equations are not a simply direct generalization of the conventional Friedmann's equations based on the real FLRW single-pole metric, nor a simple parametrization of $a(t)$. Due to the non-linearity of the Einstein's equations, such a direct generalization would not be formally consistent. The present formulation as stated earlier, is the outcome of complexifying the FLRW metric and is a result of the solution of a Riemann sum of equations associated to infinitely many poles (in tune with Hawking's assumption of infinite number of primordial universes that occurred simultaneously) arranged along a line in the complex plane with infinitesimal residues (for the details see [1, 2, 3]).

In the following, we discuss the singularity, flatness, homogeneity and horizon problems from the classical point of view of the branch-cut cosmology and we make a few comments with other predictions.

§ FUNDAMENTAL QUESTIONS OF MODERN COSMOLOGY

In the following, we discuss some fundamental problems of modern cosmology and propositions of branched cosmology in shedding some light on these questions.

§.§ The singularity problem and the branch-cut cosmology

The analytically continued
Ricci scalar, $R_{[\rm{ac}]}  = g_{[\rm{ac}]}^{\mu \nu} R_{[\rm{ac}] \mu \nu}$,
where $R_{[\rm{ac}]\mu \nu}$ defines the analytically continued ([ac]) Ricci curvature tensor, becomes

\begin{equation}
R_{[\rm{ac}]}   =  6 \Biggl[  \Biggl(\! \frac{\frac{d^2}{dt^2}{\ln^{-1}[\beta(t)]}}{\ln^{-1}[\beta(t)]}  \Biggr)  +  \Biggl( \frac{\frac{d}{dt}{\ln^{-1}[\beta(t)]}}{\ln^{-1}[\beta(t)]}   \Biggr)^2
+  \frac{k}{\ln^{-2}(\beta(t))} \Biggr].
\end{equation}

From this expression, we conclude that the new scale factor, $ \ln^{-1}[\beta(t)]$,   the solely dynamical degree of freedom in the analytically continued FLRW metric,
shapes the curvature of a hypothetical universe with characteristic parameters analytically continued to the complex plane.
In addition to branch cuts, in the branch-cut cosmology there are `singularities', — the branch points —, but at the same time there are multiple points that configure continuous paths in the Riemann sheets. This enables continuous solutions off the primordial singularity, which, in general relativity, are inescapable, allowing this way the analytically continued Ricci scalar curvature do not bend to infinity at the Planck scale, thus eliminating, on the complex plane, the presence of essential singularities.
For this to happen, the presumption at the level of a local continuity prevails, i.e.,  that there is some neighborhood of the branch point, let's call it $z_0$, close enough although not equal to $z_0$, where one can find a small region around (local patches) where $\ln^{-1}[\beta(t)]$ is single valued and continuous. The cuts in the branch cut, — shaped by the $\beta(t)$ function —,  defines the range of  $\ln^{-1}[\beta(t)]$,
crucial to model the generation of the structures observed today via primordial fluctuations.
This statement can be visualized by means of the
representation of the parameter $\ln^{-1}[\beta(t)]$ in polar form with $\beta(t) =  \kappa(t) e^{i n \theta}$, and with the parameter $\kappa(t)$ directly related to the complex analytically  continued Ricci scalar curvature and  to the complex analytically continued radius of the universe [1, 2, 3]. Recent calculations based on the Bekenstein criterion have shown that the domain of the beta function is much more extensive than the Planck dimensions, allowing the passage through a kind of `classical tunneling' from the primordial universe, nested to ours, of primordial cosmological matter <cit.>.

On the left, the characteristic plots of the Riemann surface $R$ associated with the imaginary and the real parts of the function $\ln[\beta(t)]$ that describes the scaling in time of the branch-cut universe (the reciprocal of $\ln^{-1}[\beta(t)]$). On the extreme left, the plot of the imaginary part shows connected glued domains: the various branches of the function are glued along the copies of each upper half plane with their copies on the corresponding lower half plane.  Each two copies can be visualized as two levels of a continuously spiralling parking garage, from “level" $\ln z = \ln \kappa(r) + i \theta $ for instance to the “level" $\ln z = \ln \kappa(r) + i (\theta + 2 \pi) $ or to the “level $ \ln z = \ln \kappa(r) + i (\theta - 2 \pi) $, and so on. As a final result we have a connected Riemann surface with infinitely many “levels",  $ \ln z = \ln \kappa + i (\theta \pm 2 n \pi) $, extending clockwise or counterclockwise both upward and downward. For simplicity, the design is limited to a few Riemann sheets. The transition region between the negative and positive sectors corresponds to the predominantly descriptive domains of quantum mechanics and general relativity.
The figure next to the previous one on the left represents
the real part of $\ln[\beta(t)]$, with $\tau = |\beta (t)| = \sqrt{\kappa^2_{x} + \tau^2_{y}}$ decomposed in two components in the form $\kappa = (\kappa_{x},\kappa_{y})$ (in a temporal scale of billions of years) and shows a set of multiverses. This procedure allows obtaining complex solutions of Friedmann's-type integral equations  of an evolutionary universe in which the spacetime fabric develops continuously along Riemann sheets that circumvent the branch-cut, thus avoiding discontinuations of the general relativity equations. In short, the corresponding solutions describe a branch cut universe, with a cut from $-\chi(t)$ to $\chi(t)$,  which can be thought as stressed before as a sum of infinity single-poles arranged along a line in the complex plane with infinitesimal residues.  The figures on the right show the characteristic plots of the Riemann surface $R$ associated to the imaginary and real parts of the scale factor of the analytically continued FLRW metric $\ln^{-1}[\beta(t)]$.

This representation allows to map the behavior of the
$\ln^{-1}[\beta(t)]$ parameter in terms of level curves that describe the slope and variations of a hypothetical topological contour.  In a visualization of $\ln[\beta(t)]$ shown in [1, 2, 3],
— the reciprocal of the new complex scale factor which represents  a linear scaling factor in
time, bringing to time a complex nature, with an imaginary component[As historically stated by [12], in general relativity and by [13], in statistical mechanics.] —,
the Riemann surface describes a spiral curve around a vertical line corresponding to the origin of the complex plane (Figure  (<ref>)).
We extended the graphical representation of the previous article showing the real part of this function, where some of the multiverses are represented (Figure  (<ref>)). Evidently, the presence of multiverses in the real part is associated with the peculiar characteristics of the proposal in which the superposed multiple
universes as stressed before correspond to a convenient mathematical device. The representation of the real part of the new scale factor does not contain
Riemann sheets,
the analytical varieties of complex dimension and therefore the `multi-universes' representation is still present, although with their singularities infinitesimally separated. In complex analysis, contour integrals are made in the complex sector, simply by summing the values of the complex residues inside the contour.
In Figure (<ref>) we plot the imaginary and real parts of $\ln^{-1}[\beta(t)]$ for which similar conclusions to the previous case can be drawn.
The actual surfaces of
the images shown in Figure (<ref>)
extends arbitrarily far both horizontally and vertically the representation sector.

In short, the present treatment presents a solution for the presence of singularities in general relativity where the Riemann curvature tensor for instance is replaced by an equivalent tensor continued to the imaginary sector that behaves continuously in the real domain and is complemented on the imaginary axis by discontinuity jumps of $2\pi$.

§.§ Inflation

Alan Guth has suggested that our universe is product of inflation [10, 14], an approach based on the existence of states with negative pressure, which effects can be seen in the  Friedmann equations [5]
combined with the physics of a scalar field, the  inflaton field. The corresponding
energy-momentum tensor of the inflaton field $\phi$ is defined as [10, 14]

\begin{equation}
T^{\mu \nu}  = \partial^{\mu} \phi \partial^{\nu} \phi - g^{\mu \nu} \Bigl[ \frac{1}{2} \partial_ {\lambda} \phi \partial^{\lambda} \phi + V(\phi)\Bigr].
\end{equation}

The energy density and pressure in the inflation model are then given by [10]

\begin{eqnarray}
\rho(t) & = & T^{00} = \frac{1}{2} \dot{\phi}^2 + \frac{1}{2} \Bigl(\nabla_i \phi  \Bigr)^2 + V(\phi)\,, \nonumber \\
p(t) & = & \frac{1}{3} \sum_{i=1}^3 T_{ii} = \frac{1}{2} \dot{\phi}^2
- \frac{1}{6} \Bigl(\nabla_i \phi  \Bigr)^2 - V(\phi).
\end{eqnarray}

Here $\rho(t)$ is the energy density, $p(t)$ is the pressure, and $V(\phi)$ is the potential energy density. Accordingly, any state dominated by the potential energy of a scalar field will have negative pressure. As a fundamental result, Friedmann's equations then indicate that a negative pressure may contribute to the acceleration of the expansion of the universe, emulating this way a “repulsive” form of gravity.
The physical state in the original version of the model involved the so-called inflaton scalar field around a local minimum, a false vacuum, as it behaves, albeit temporarily, as the lowest possible energy density state. For most inflationary models the energy density of the universe $\rho(t)$ is approximately constant leading to exponential expansions of the scale factor $a(t) \sim e^{\chi t}$ (late-time asymptotic behavior), with $\chi = $ const. Thus, the scale factor of evolution of the universe, grows exponentially during the inflation period, causing the quantum fluctuations of a hypothetical scalar inflaton field to be stretched to macroscopic scales; when leaving the horizon, these fluctuations are frozen. In the later stages of the domain of radiation and matter, these fluctuations re-enter the horizon and thus enables the initial conditions for the formation of structures.  These conditions would lead in the classical case to a completely stable state due to the separation potential barrier from lower energy states. And in the quantum case, — contrary to expectations of a tunneling process which could cause possibly the ending of inflation —, it was found that “the randomness of bubble formation when the false vacuum decayed would produce disastrously (sic) large inhomogeneities" [10]. In  Guth's own words [10], a “graceful exit” problem was solved by the invention of the new inflationary universe model by [15] and [16], in which inflation is driven by a scalar field perched on a plateau of the potential energy diagram.

Inflation allows solving at once the monopoles, horizon and flatness  problems, among others in the Standard Big Bang (SBB) cosmology.
Guth's original model, despite controversies, still remains as an important component of modern cosmology and has inspired countless authors, as Andrei Linde, Paul Steinhardt, Andy Albrecht and others (see for example [17]), as well as a plethora of models in subsequent years to overcome inconsistencies or unexplained features of SSB, despite this theoretical framework being still one of the most accepted model describing the central features of the observed universe.
Most of these inflation models, — introduced in an ad hoc way similarly to the original version of the inflationary mechanism, as well as based on general relativity —, differ fundamentally from each other in terms of the shape of the potential $V(\phi)$ (see, for example, [17, 18]).

In this respect, the Cosmic Microwave Background (CMB) measurements by the Planck satellite <cit.>, has offered an unprecedented opportunity to constrain the inflationary theory, particularly with regard to the scalar spectrum and spectral index of the early universe. These measurements resulted in a certain degree of ordering of the numerous proposals presented so far and at the same time provided researchers with additional constrains for the inflation realization potential $V(\phi)$, regarding in special the spectra
of scalar and tensor fluctuations of the primordial universe, responsible for generating the structures observed today (the scalar spectrum) and the cosmological background of gravitational waves (the tensor spectrum).

§.§ The Flatness Problem, inflation and the FLRW cosmology

Among unfathomable features of the universe, some stand out. For example, according to general relativity, mass bends space and time. In this context, a universe like ours would be expected to completely curve on itself, with “positive” or “negative” curvature (using a conventional three-dimensional image, like a “ball” in the first case or a “saddle” in the second).

The spatial geometry of the universe has been recently measured by the Planck Collaboration <cit.> to be nearly flat with high accuracy[If $\Omega_0$ is less than 1, the universe is open and will continue to expand forever; if $\Omega_0$ is greater than 1, the universe is closed and may eventually stop its expansion and re-collapse; if $\Omega_0$ is exactly equal to 1, then the universe is flat and contains enough matter to stop its expansion, but probably not enough to re-collapse.], — the main reason why the flatness problem is understood as a fine tuning problem —,
a puzzle that has no explanation in classical FLRW cosmology, giving rise to the so-called flatness problem.
The flatness problem concerns the value of the ratio $\Omega_{tot} = \rho_{tot}/\rho_c$, where $\rho_{tot}$ is the average total mass density of the universe (including the vacuum energy associated with the cosmological constant $\Lambda$) and $\rho_c$ is the critical density, the density corresponding to $k = 0$.

In the original version of the FLRW model, during the
radiation-dominated and matter-dominated eras, from the first Friedmann's equation, the scale factor $a(t)$ has the following time dependence

\begin{equation}
\left\{
\begin{array}{l}
\quad a(t) = \Bigl(\frac{H_0}{2}\Bigr)^{1/2} t^{1/2}  \quad \mbox{(radiation-dominated era)}\, ; \\ \quad \mbox{and} \\ \quad a(t) = \Bigl(\frac{3H_0}{2}\Bigr)^{2/3} t^{2/3}   \quad \mbox{(matter-dominated era)}\, ,
\end{array}
\right.
\end{equation}

with the density of the universe $\rho(t)$ and critical density $\rho_c(t)$ (for $k=0$) given by:

\begin{equation}
\rho(t)  \! = \! \frac{3}{8 \pi G} \Biggl\{ \! \Biggl(\frac{\dot{a}(t)}{a(t)} \Biggr)^2 \! \! + \frac{k}{a^2(t)} \! \Biggr\} \, ,
\! \quad \! \mbox{and} \! \quad \! \rho_c(t)  \! = \! \frac{3}{8 \pi G} \Biggl(\frac{\dot{a}(t)}{a(t)} \Biggr)^2 \!.
\end{equation}

From these equations it results:

\begin{eqnarray}
\frac{\rho(t) - \rho_c}{\rho_c} & = & \frac{\rho(t)}{\rho_c} - 1 = \Omega_{tot} - 1 = \frac{k}{a^2(t)}
\nonumber \\
& = &  \left\{ \begin{array}{l}
\frac{2k}{H_0}t^{-1} \quad \mbox{(radiation-dominated era)} \\ \\
\frac{2k}{3H_0}t^{-4/3}  \quad \mbox{(matter-dominated era)}  \\
\end{array} . \right. \label{Flat}
\end{eqnarray}

Thus, in the limit where $t \to 0$, the FLRW cosmology results are inconsistent with the observational predictions.
However, while the extraordinary flatness of the early universe as measured by the Planck Collaboration <cit.> has no explanation in classical FLRW cosmology, it is a `natural' prediction for inflationary cosmology. During the inflationary period, instead of $\Omega_{tot}$
being driven away from one as described by FLRW cosmology, it  is driven toward one, with exponential swiftness:

\begin{equation}
\Omega_{tot} - 1 \propto e^{-2 H_{inf} t} \quad  \mbox{leading to} \quad \Omega_0 = 1.02 \pm 0.02.,
\end{equation}

where $H_{inf}$ is the Hubble parameter during inflation.

However, as previously mentioned, relevant questionings to this formulation persist, such as resorting to an ad hoc mechanism that assumes an entire universe created out of essentially nothing, based on the “materialization” of negative energy states, assuming the energy of the gravitational field could become negative without limits [10]. The same can be said about the standard cosmology that presents some formal and conceptual inconsistencies, as seen above and as we will see below. In this sense, as we intend to demonstrate in the continuation, the branch-cut cosmology presents itself as a descriptive alternative of the evolutionary universe whose potentialities require a deeper approach, which we intend to do in this contribution.

§.§ Patch and horizon sizes scaling and causality in the standard cosmology

In the conventional FLRW cosmology, the cosmic scale factor $a(t)$ and the Hubble parameter  both characterize as a function of time a smooth and isotropic universe. The cosmic scale factor $a(t)$ is a dimensionless quantity that describes the size variation of a patch of space due to expansion or contraction of the universe and $H(t) = \frac{\dot{a}(t)}{a(t)}$ in turn measures the expansion rate of the universe. Assuming the observable universe corresponds to a patch of space with radius $R(t^{\prime})$ at time $t^{\prime}$, at time $t$
the corresponding patch size in the standard cosmology is sized up by $a(t)$

\begin{equation}
R(t)  \Rightarrow  \frac{a(t)}{a(t^{\prime})} R(t^{\prime})\, ,
\end{equation}

and the reciprocal of $H(t)$ measures the Hubble radius or horizon size:

\begin{equation}
H^{-1}(t)  =  a(t) \int_0^t \frac{dt^{\prime}}{a(t^{\prime})}\, ,  \label{a&H}
\end{equation}

which represents the maximal region that is “causally connected” through interactions with light or any other particles.  The horizon size thus defines the maximum distance to be traveled by light in the age of the universe and therefore the maximum observation distance of any event since the big bang.

In the radiation dominated era, with $a(t) \sim t^{1/2}$, from the FLRW metric [5, 6, 7, 8], since  light propagates along world
lines for which the spacetime interval is zero,

\begin{equation}
ds^2 = c^2 dt^2 - a^2(t) \Bigl(\frac{dr^2}{1 - k r^2}  + r^2 \bigl(d\theta^2 + \sin^2\theta d \phi^2 \bigr)\Bigr) = 0 \, , \label{rd}
\end{equation}

we get

\begin{equation}
H^{-1}(t)  \! = \!  a(t) \!\! \int_0^t \frac{c dt^{\prime}}{a(t^{\prime})} \! = \! a(t) \!\! \int_0^{r_H} \frac{dr}{\sqrt{1 - kr^2}}
\! \sim \!  t^{1/2} \!\! \int_0^t \frac{c dt^{\prime}}{t^{\prime 1/2}} \! = \! 2 ct \, . \label{md}
\end{equation}

For the matter-dominated era, $H^{-1}(t) = 3 ct$. Thus, in standard cosmology, this simple calculation shows that causality could be established only if a signal could propagate at about 2 or 3 times the speed of light, a proposition that clearly contradicts the known laws of physics.

A key issue in cosmology concerns the geometric scaling properties of the horizon size. To address this topic, we consider the energy-stress conservation law, in the standard approach:

\begin{eqnarray}
&& \frac{1}{\rho(t)}\frac{d}{dt}\rho(t)    +    3 \Biggl( 1 + \frac{p(t)}{c^2\rho(t)}   \Biggr) \frac{1}{a(t)} \frac{da(t)}{dt} = 0 \, ,  \nonumber \\
&& \Rightarrow \quad \frac{d}{dt}\ln  \bigl(\rho(t)\bigr)  +
3 \Biggl(1 + \frac{p(t)}{c^2\rho(t)}  \Biggr)  \frac{d}{dt} \ln a(t) =  0.  \nonumber \end{eqnarray}

From this equation it results

\begin{equation}
\int_{\rho_0}^{\rho(t)} d \ln \bigl( \rho(t) \bigr)  =  - 2 \int \Biggl\{ \frac{3}{2}\Biggl(1 + \frac{p(t)}{c^2 \rho(t)}  \Biggr) \Biggr\} d\ln a(t) \, ,
\end{equation}

\begin{equation}
\epsilon(t) \equiv \frac{3}{2} \Biggl(1 + \frac{p(t)}{c^2 \rho(t)}  \Biggr). \label{r0}
\end{equation}

In this equation $\epsilon(t)$ represents the dimensionless thermodynamics connection between the energy density $\rho(t)$ and the pressure $p(t)$ of a perfect fluid thus enabling the fully description of the equation of state (EoS) of the system. Positive pressure corresponds to $\epsilon > 3/2$,  negative pressure to  $\epsilon < 3/2$ and for a universe
dominated by a cosmological constant, $\epsilon \to 0$.
In the limit in which the
dimensionless thermodynamics connection obeys
$\epsilon(t) \to  \epsilon = $ constant, the integral (<ref>) reduces to[The value of $\epsilon$ is nearly constant over long epochs, varying rapidly from one constant value to another
when the dominant form of energy changes. For example, in the standard big bang model,
the universe is radiation-dominated (corresponding to $\epsilon = 2$) for the first 50,000 yrs after the big bang and dust-dominated (corresponding to $\epsilon = 3/2$) for the remaining 13.8 Gyrs.]

\begin{eqnarray}
\ln (\rho(t)/\rho_0) & = & - 2  \lim_{\epsilon(t) \to \epsilon} \int \epsilon(t) d \ln a(t) \nonumber \\
& \simeq & \! - 2 \epsilon \ln a(t)  \Rightarrow \ln  \Bigl(\! a(t) \! \Bigr)^{-2\epsilon}
\!\!\!\!\!\!\!  \Rightarrow \! \rho(t) \simeq \frac{\rho_0}{a^{2 \epsilon}(t)}. \label{rhoscaling}
\end{eqnarray}

From this equation, when going back in time considering the primordial universe, if $\epsilon$ is positive, the universe's density increases, while for negative $\epsilon$, its density decreases.
From the first Friedmann's equation in the FLRW cosmology, for $k=0$, combined with the scaling behaviour of $\rho(t)$ (see Equation (<ref>)),  we

\begin{eqnarray}
H^2 & = & \Biggl(\frac{\dot{a}(t)}{a(t)} \Biggr) = \frac{8\pi G}{3} \rho(t) \sim \frac{8 \pi G}{3} \frac{\rho_0}{a^{2 \epsilon}(t)}  \nonumber \\
& \Rightarrow & H^{-1} = \sqrt{\frac{3}{8 \pi G \rho_0}} a^{\epsilon}(t). \label{scalingH}
\end{eqnarray}

From Equation (<ref>), we have seen that the curvature $R(t)$ of the universe scales with the parameter $a(t)$ and from Equation (<ref>), that the horizon size,
$H^{-1}(t)$ scales as  $a^{\epsilon}(t)$ so their ratio continuously decreases when extrapolating $a(t)$ backwards in time.
This scaling difference between the curvature of the universe and horizon size
represents a fundamental limitation of standard cosmology, generating a non-causal behavior of the patch size and the horizon size.

§.§ Flatness, Horizon, Inhomogeneity, and Smoothness Problems
in the Branch-cut Cosmology

In the branch-cut cosmology,
from the first Friedmann's equation, extended to the complex plane, the following expression, for $\Lambda = 0$ and $k \neq 0$ then holds

\begin{equation}
\Biggl( \frac{\frac{d}{dt} \ln^{-1}[\beta(t)]}{\ln^{-1}[\beta(t)]  } \Biggr)^2       =    \frac{8 \pi G \rho(t)}{3}
-  \frac{kc^2}{\ln^{-2}[\beta(t)]}  \, ,
\end{equation}

\begin{equation}
\Biggl( \frac{d}{dt} \ln^{-1}[\beta(t)] \Biggr)^2       =    \frac{8 \pi G}{3}\rho(t)  \ln^{-2}[\beta(t)]
-  kc^2  \, . \label{FE}
\end{equation}

Solving equation (<ref>) we get the solutions:

\begin{equation}
\ln^{-1}[\beta(t)]  =     \left\{ \begin{array}{l}
\sqrt{\ln^{-2}[\beta(t_P)]
+ \frac{1}{ \ln^{2}(\beta_0)} \sqrt{\frac{2 \pi G \rho_0}{3}} \Bigl( t - t_P  \Bigr)
} \\  \mbox{(radiation-dominated era)} \\ \\
\sqrt[2/3]{ \ln^{-3/2}[\beta(t_P)]
+  \frac{1}{ \ln^{\, 3/2}(\beta_0)}  \sqrt{6 \pi G \rho_0} \Bigl( t - t_P  \Bigr)
}  \\  \mbox{(matter-dominated era)}  \\
\end{array}\, . \label{SFE} \right.
\end{equation}

From expression (<ref>), we obtain the following expressions for the density of the universe, $\rho(t)$, and the critical density, $\rho_c(t)$:

\begin{equation}
\rho(t) = \frac{3}{8 \pi G} \Biggl\{ \Biggl( \frac{\frac{d}{dt} \ln^{-1}[\beta(t)]}{\ln^{-1}[\beta(t)]  } \Biggr)^2
+  \frac{kc^2}{\ln^{-2}[\beta(t)]} \Biggr\} \, ,\label{Rho}
\end{equation}

\begin{equation}
\rho_c(t) = \frac{3}{8 \pi G} \Biggl( \frac{\frac{d}{dt} \ln^{-1}[\beta(t)]}{\ln^{-1}[\beta(t)]  } \Biggr)^2    \, . \label{Rhoc}
\end{equation}

Combining (<ref>), (<ref>) and  (<ref>) we finally obtain:

\begin{eqnarray}
\frac{\rho(t) - \rho_c}{\rho_c}  =  \Omega_{tot} - 1
& = & \frac{k}{\Bigl( \ln^{-1}[\beta(t)]\Bigr)^2}  \nonumber \\ & = &
\left\{\!  \begin{array}{l}
\frac{k}{\ln^{-2}[\beta(t_P)]
+ \frac{\sqrt{2 \pi G \rho_0}}{\sqrt{3}\ln^{2}(\beta_0)}(t - t_P)}
\\ \mbox{(radiation-dominated era)} \\
\frac{k}{\bigl(\ln^{-3/2}[\beta(t_P)]
+  \frac{ \sqrt{6 \pi G \rho_0}}{ \ln^{\, 3/2}(\beta_0)} (t - t_P)\bigr)^{4/3}}
\\ \mbox{(matter-dominated era)}  \\
\end{array} \right.
\end{eqnarray}

This result looks at a first glance quite similar to the corresponding result of the original FLRW model (see Equation (<ref>)). Once again care must be taken in this interpretation.

[fill=yellow!30] (0,0) – (6,0) arc[start angle=0, end angle=60,radius=6cm] – (0,0);

[black] (-0.5,0.0)
[black] (3.1,-0.3)
[black] (0.5,2.80)
[black] (5.5,2.80)

[clip] (0,0) circle(1.2);

[red] (0,-1.5)–(0,1.5);
[] at (0.4,0) $2R(t_0)$;

In the upper figure: an arc of a circle containing a portion of space representing schematically the relationship between the patch size and the horizon size,  with the sides of the arc labeled by the scale factor $a(t)$. The curved portion of the arc of circle represents the linear size of a patch of space at the present time $t_0$, represented by $2R(t_0)$. On the below figure: the diameter $2R(t_0)$ of a circle of radius $R(t_0)$, associated to the presently observed universe.

In the classical scenario of the branch-cut cosmology, the universe evolves continuously from the negative complex cosmological time sector, prior to a primordial singularity, to the positive one, circumventing continuously a branch-cut, and no primordial singularity occurs in the imaginary sector, only branch-points. The branch-cut universe involves a continuous sum of an infinite number of infinitesimally (originally) separated poles, surrounding a primordial branch-point, arranged along a line in the complex plane with infinitesimal residues.  And similarly to the primordial branch-point singularity, the argument of the resulting analytic function, can be mapped from a single point in the domain to multiple points in the range.

As stressed before, the flatness problem concerns the value of
the ratio $\Omega_{tot} = \rho_{tot}(t)/\rho_c(t)$.
The problem of the flatness of the universe is related, additionally, to the  cosmic curvature factor <cit.>,
a time-dependent and dimensionless quantity that characterize the apparent
spatial curvature
of the universe,
$\Omega_c$, defined in the FLRW cosmology

\begin{equation}
\Omega_c \equiv - \frac{k H^{-2}(t)}{a^2(t)}  \quad \mbox{which scales as} \quad \sim \frac{a^{2\epsilon}(t)}{a^2(t)}.
\end{equation}

Moreover, in the standard cosmology, the horizon problem
arises exactly because the patch corresponding to the observable universe was never causally connected in the past <cit.>.  As we mention previously, in the present time ($t = t_0$), the patch size, $R(t_0)$, and the horizon size, $H^{-1}(t_0)$, are equal, i.e., $R( t_0) = H^{-1}(t_0)$. In earlier times, the ratio between the horizon size to the patch size decreases monotonically extrapolating back in time as $a(t) \rightarrow 0$ in the form

\begin{equation}
\frac{H^{-1}(t)}{a(t)} \sim
\frac{a^{\epsilon}(t)}{a(t)} = a^{\epsilon - 1}(t) \quad
\mbox{and} \quad  \lim_{a(t) \rightarrow 0}{a(t)^{\epsilon - 1}} \rightarrow 0, \label{aepsilona}
\end{equation}

with the horizon size approaching zero faster than the patch size.
According to the CBM measurements, the density and temperature were almost uniform throughout the primordial patch (last CBM surface scattering). Explaining the uniformity of the CMB at length scales greater than the size of the horizon at the last scattering surface grounded on the CBM measurements
and at all previous times fundamentally constitutes the horizon problem <cit.>. Moreover, the CMB measurements also reveal a spectrum of small amplitude density fluctuations, nearly scale-invariant whose explanation constitutes the inhomogeneity problem <cit.>.

As we stressed before, this non-causal behavior of the patch size and the horizon size, — since their ratio continuously decreases when extrapolating $a(t)$ backwards in time —, represents one of the fundamental limitations of standard cosmology. Combined with the primordial singularity, where any trace of causality completely disappears, these two factors represent the main roots for solving the problems of the cosmic singularity, horizon, inhomogeneity and flatness of the universe. The
smoothness problem of the universe is related in turn to the
cosmic anisotropy factor, $\Omega_a$, a time-dependent, dimensionless quantity that characterizes the apparent anisotropy, due to small temperature fluctuations in the primordial blackbody radiation:

\begin{equation}
\Omega_a \equiv \frac{\sigma^2 H^{-2}}{a^6 (t)} \quad \mbox{which scales as}
\quad \sim \frac{a^{2\epsilon}(t)}{a^6(t)};
\end{equation}

in this case, the patch size approaches zero faster than the horizon size.

The solution of the branch-cut-type cosmology corresponds as we have seen to the reciprocal of a complex  multi-valued function, the natural complex logarithm function $\ln[\beta(t)]$.
Taking the ratio

\begin{equation}
\frac{\Bigl(\ln^{-1}[\beta(t)]\Bigr)^{2\epsilon}}{\Bigl(\ln^{-1}[\beta(t)]\Bigr)^2} = \ln^{2(1 - \epsilon)}[\beta(t)] \, ,
\end{equation}

\begin{equation}
\lim_{\ln[\beta(t)] \rightarrow 0}{\ln[\beta(t)]^{2(1 - \epsilon)}(t)} \rightarrow   \Biggl\{ \begin{array}{l}
0 \quad for \quad 1 - \epsilon > 0 ; \\
\infty \quad for \quad 1 - \epsilon < 0 .
\end{array}
\end{equation}

Going from the present to the past of the branch-cut universe, as we return to the beginning of the expansion phase,
as the patch size decreases with a linear dependence on $\ln[\beta(t)]$, light travels through geodesics on each Riemann sheet,  circumventing continuously the branch-cut,  and although the horizon size scale with $\ln^{\epsilon}[\beta(t)]$, the length of the path to be traveled by light compensates for the scaling difference between the patch and horizon sizes. Making an analogy with a continuously spiralling parking garage, everything happens as if two cars left a certain level of the garage at the same time, so that one car, slower, followed a straight ramp while the other car, faster, followed the longer, circular route of the garage, and both arrived at their destination at the same time.

Under these conditions, causality between the horizon size and the patch size may be achieved through the accumulation of branches in the transition region between the present state of the universe and the past events.

This topic, the number of ramifications accumulated to achieve causality was recently addressed in <cit.>. Ijjas and Steinhardt <cit.> have shown that
quantum fluctuations responsible for temperature variations observed in the Cosmic Microwave Brackground (CBM)
are produced with roughly 60 e-folds of contraction in the FLRW cosmic scale factor $a(t)$ remaining
before the bounce when the horizon size is exponentially large compared to the
Planck scale.

In a recent calculation we showed that the number of `turns' or Riemann sheets to achieve causality is $\sim 10^{60}$, a number comparable to 60 e-folds of contraction in the bouncing model <cit.>. More precisely, the e-folds of bouncing contractions are comparable to the running-number or circulation sheets or number of branches in between the present state of the universe and the region where causality is achieved, the cosmic microwave background radiation's surface of last scatter.

However, the impossibility of packaging energy and entropy according to the Bekenstein Criterion in a finite size makes the transition phase very peculiar, imposing a topological leap between the two phases or a transition region similar to a wormhole, with space-time shaping itself topologically in the format of a helix-shape like as proposed by branch-cut cosmology around a branch-point. Topological wormholes are said to essentially constitute `tunnels' in spacetime, in which two topological sheets, at the top and at the bottom, are connected via a wormhole. The two topological sheets can represent two different universes or two sections of the same universe which, without the presence of the wormhole, would be separated by a very non-causal large distance.

Branch-cut cosmology has similarities with wormholes, in the sense the $\ln[\beta(t)]$ function corresponds to a helix-like superposition of cut-planes, the Riemann sheets, with an upper edge cut in the $n$-th plane joined with a lower edge of cut in the ($n + 1$)-th plane. In this sense, the topology of a wormhole is reproduced successively between pairs of upper-edge and lower-edge cut-planes. $\ln[\beta(t)]$ in turn maps an infinite number of  Riemann sheets onto horizontal strips, which represent in the branch-cut cosmology the time evolution of the time-dependent horizon sizes. The patch sizes in turn maps progressively the various branches of the $\ln[\beta(t)]$ function which are glued along the copies of each upper-half plane with their copies on the corresponding lower-half planes. In the branch-cut cosmology, the cosmic singularity is replaced by a family of Riemann sheets in which the scale factor shrinks to a finite critical size, — the range of $\ln^{-1}[\beta(t)]$, associated to the cuts in the branch cut, shaped by the $\beta(t)$ function —, well above the Planck density.

§.§ Scenarios for the Branch-cut cosmology

Before we conclude this contribution, we summarize below the present stage of the scenarios recently outlined <cit.> for the branch-cut cosmology which are
sketched in an artistic representation
(see Fig. (<ref>)), with a branch-point and a branch-cut on the left figure, and no primordial singularity on the right one[Figures based on an artistic impression originally developed by ESO / M. 9.].

The figures show artistic representations of the cosmic contraction and expansion phases of the branch-cut universe evolution with two scenarios. In the representation sketched on the left figure, the branch-cut universe evolves from negative to positive values of the imaginary cosmological time $t_i$, circumventing continuously the branch-cut and no primordial singularity occurs, only branch points.  The right figure sketches the second scenario, where the branch cut and branch point disappear after the realisation of the imaginary time $t_i$ by means of a Wick rotation, which is replaced here by the real and continuous thermal time (temperature), $T$. In this second scenario, a mirrored parallel evolutionary universe, adjacent to ours, is nested in the structure of space and time, with its evolutionary process going backwards in the cosmological thermal time negative sector. The figures were based on artistic impressions [9].

Two scenarios have been outlined in the branch-cut cosmology.
The first scenario is characterized by a branch-point and a branch-cut, with the suppression of a primordial singularity, in which the universe continuously evolves from the negative imaginary cosmological-time sector, —
where the contraction of the universe takes place —,
to the positive one, —  where the expansion of the universe occurs <cit.>.  This scenario, due to the absence of a singularity, may be characterized by a transition region with dimensions that surpass Planck's dimensions according to Bekenstein Criterion, separating the contraction and expansion phases <cit.>.

In the second scenario, the branch-cut and the branch-point disappear after realization of imaginary time through a Wick rotation, which is replaced here by the real and continuous thermal time (temperature) <cit.>. In this scenario,
the connection between the negative and positive imaginary cosmological-time sectors is broken as a result of the Wick rotation, and a mirrored parallel evolutionary universe adjacent to our own is nested in the fabric of space and time, with its evolutionary process oriented in the opposite direction to that corresponding to the positive imaginary-time sector <cit.>.

Singularity means there is no way for spacetime to start smoothly. Branch-cut cosmology, alternatively, proposes an absolutely non-temporal beginning in the imaginary sector, a configuration of pure space, through a Wick rotation that replaces the imaginary component of time with the temperature, the thermal time, that flows in the opposite direction of the arrow of time in the expansion phases of the first and second scenarios.
In the first scenario, in the contraction phase, before entering into the expansion phase, the temperature and entropy of the branch-cut universe must reach values consistent
with the corresponding ones in the expansion phase. For this to happen, the temperature of the universe in the contraction phase must increase, but the entropy must decrease, as determined by thermodynamics, reversing this way the arrow of time.
In the contraction sector of the first scenario, as the transition region approaches, there occurs a progressive decrease in the entropy and an increase in the temperature of the universe, so there is a critical region, whose dimensions are determined by the Bekenstein Criterion <cit.>, where the entropy reaches its minimum value and the temperature in contra-position its maximum value.
In the expansion sector, the opposite occurs where, having as a starting point the minimum values of entropy and maximum values of the temperature of the universe, progressively the entropy starts  to increase and the temperature to decrease. In this sense, in the contraction region, the arrow of time points in the opposite direction of the evolving orientation of the universe, while in the expansion region it points in the same direction. On the other hand, thermal time points in the evolutionary direction of the universe in the region of contraction and in the opposite direction in the region of expansion.

Going back to the original conception, the arrow of time is associated with the direction of entropy increase and therefore oriented towards the expansion of the universe. However, in the contraction phase of the universe, entropy decreases and the arrow of time therefore has, in this conception, its direction inverted. One way to overcome such a discrepancy in order to maintain the “past-to-future”  global orientation in the branch-cut cosmology is to define a time arrow oriented towards the decreasing of entropy in the contraction sector of the universe, and the conventional conception in the expansion phase. This proposition would be valid for both scenarios of the branch-cut cosmology, as in the first scenario there is a contraction sector followed by an expansion phase of the universe, while in the second scenario all sectors are associated with the expansion of both, ours and the mirror-universe. In this conception, the cosmological arrow of time is determined as the direction in which “time”, from the macroscopic point of view, flows globally. This conception differs from that of George Ellis <cit.>, who identifies the arrow of time with the locally determined direction in which time flows at any time in the evolution of the universe. In the branch-cut cosmology, in turn, this local component is identified with the corresponding real component of the complexified-time that results from the complexification of the FWLR metric [2].

§ FINAL REMARKS

The limitations presented by conventional cosmology arise not only from the fact that the ratio of the horizon size to the patch size shrinks when extrapolating back in time but essentially from the existence of singularities that break down this ratio and thus make it impossible to restore causality. In the branch-cut cosmology, this problem is overcome by the presence of cuts, defining this way the range of $\ln^{-1}[\beta(t)]$, making the scale factor shrinks as stressed before to a finite critical size.
Furthermore, the present formulation assumes the description of the background evolution of the branch-cut cosmological scenarios in leading order by classical equations of motion.

The anisotropy associated with the smoothness problem of small temperature fluctuations has a peculiar solution that still lacks a more consistent explanation. This is because, in the final phase of the contraction process, choosing the epsilon parameter with a negative value (dark matter?), the scaling behavior of the ratio  $\frac{a^{2\epsilon_{DM}}(t)}{a^6(t)}$ is reversed, simultaneously assuming the remaining ratios $\frac{a^{\epsilon}(t)}{a(t)}$ are locked in.

This finite size limitation of the range, well above the Planck limit,  implies in the branch-cut cosmology <cit.> a leap, of a classical nature, between the primordial phases of contraction and expansion, a topic that needs further investigation in the future. A quantum approach, more in line with the nature of this evolutionary domain, following a recent proposal [3], is under investigation. Other aspects to be investigated refer to a more consistent evolution mapping of the cut-planes, the consequences of adopting a non-symmetric approach regarding the dimensionless thermodynamics connection $\epsilon(t)$, a different ordering of values of $\epsilon(t)$, and the role of dark matter in the evolution of the branch-cut universe. Cosmic Microwave Background (CMB) measurements by the Planck satellite <cit.> offer on the other hand an unprecedented opportunity to constrain and test the branch-cut model, particularly regarding the fluctuations associated to the energy density primordial spectrum, the seeds of all structures in the early universe. There are also questions regarding the multiverse content. For example, the conjuncture of a thermal contact of primordial multi-universes, immersed in a thermal bath, subjected to a contraction crunch, and the thermal evolution consequences to our universe after decoupling. And the ensuing questions, as the homogeneity of the primordial radiation and average density of matter as key elements for the formation of the structures observed today in the univers, among others. Evidently, most of these questions should not be limited to a classical description of the evolutionary universe, as their nature is intrinsically quantum. These topics are presently under investigation.

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# Estimation of von Neumann entropy of different light sources using array
detectors

R. P. George  Departement of Physics, Indian Institue of Technology Jodhpur,
Rajasthan,342037, India  Tomis  Departement of Physics, Indian Institue of
Technology Jodhpur, Rajasthan,342037, India  V. Narayanan  Departement of
Physics, Indian Institue of Technology Jodhpur, Rajasthan,342037, India
Interdisciplinary Research Platform- Quantum Information and Computation(IDRP-
QIC), Indian Institue of Technology Jodhpur, Rajasthan,342037, India
Corresponding author<EMAIL_ADDRESS>Subhashish Banerjee  Departement of
Physics, Indian Institue of Technology Jodhpur, Rajasthan,342037, India
Interdisciplinary Research Platform- Quantum Information and Computation(IDRP-
QIC), Indian Institue of Technology Jodhpur, Rajasthan,342037, India

###### Abstract

We employ a method involving an array detector to measure the transverse
spatial variation of the von Neumann Entropy (VNE) associated with the
polarization state of light for different light sources including the coherent
light from a diode laser, the chaotic light from an LED, the fluorescent
emission from a dye widely used as a contrast agent and also the downconverted
output from the process of Spontaneous Parametric Down Conversion which has
widespread applications in Quantum Optics as a source of entangled photon
pairs. Additionally, we studied the variation of the Stokes parameters as well
as von Neumann entropy with pixel binning in the array detector output.

###### doi:

††journal: jocn

##  1 Introduction

The von Neumann Entropy[1] associated with the polarization of light has
several interpretations; in accordance with the thermodynamic definition for
entropy, it can be considered as a measure of randomness or from an
information standpoint, it can be seen as the difference in the amount of
information [2] between pure and mixed states of the same intensity. In the
context of polarization, it exhibits an inverse relationship with the degree
of polarization [1,3] , which is an expression of polarimetric purity. The von
Neumann or polarization entropy can be thought of as another measure of
polarimetric purity with its value tending to zero when the polarization state
is pure and approaching unity when it is completely mixed.

The von Neumann entropy is derived from the normalized coherency matrix [1]
which can uniquely characterize any polarization state. This connects it with
the concept of Stokes polarimetry [4] and opens up new possibilities in the
field of polarimetric imaging which has been seeing a lot of interest lately.
In this work we have developed a simple yet scalable setup to measure the
transverse spatial variation of the polarization von Neumann Entropy across
the dimensions of the illuminated area of the array detector.

Additionally, we extracted the transverse spatial variations of all normalized
Stokes parameters as well as the sum of their squares. We also studied the
effects of pixel binning on VNE and square sum values. The light from a diode
laser at 405 nm, an LED, a fluorescent dye and the polarization entangled SPDC
output from a BBO crystal phasematched for type-II downconversion using a 405
nm pump were studied using our setup.

This has applications in several fields, primarily as a means of studying the
variations in polarization brought about by a medium [5-14] during the
propagation of polarized EM waves through said medium and also in Lidar
imaging, remote sensing and astronomy [15]. Polarimetry based studies have
been extensively investigated for the measurement of magnetic field in Laser
produced plasmas [16].

##  2 Stokes Parameters and polarization von Neumann entropy

The concept of Stokes polarimetry has been vastly explored and reported in
exceptional details [1, 3-6]. The four Stokes parameters are sufficient to
fully describe any polarization state; pure or otherwise. They are derived
from a set of projective measurements made on the unknown state of light. A
set of six measurements are commonly adopted in polarimetric measurement [6]
to yield the normalized Stokes parameters $s_{0},s_{1},s_{2}$ and $s_{3}$

$\displaystyle s_{0}$ $\displaystyle=\dfrac{I_{H}+I_{V}}{I_{H}+I_{V}},$
$\displaystyle s_{1}=\dfrac{I_{H}-I_{V}}{I_{H}+I_{V}},$ (1) $\displaystyle
s_{2}$ $\displaystyle=\dfrac{I_{A}-I_{D}}{I_{A}+I_{D}},$ $\displaystyle
s_{3}=\dfrac{I_{R}-I_{L}}{I_{R}+I_{L}}.$

Here $I_{V},I_{H},I_{A},I_{D},I_{R}$ and $I_{L}$ correspond to the six
intensity measurements along vertical, horizontal, anti-diagonal, diagonal,
right circular and left circular bases respectively. Using a combination of
Stokes parameters and Pauli’s spin matrices [1], it is possible to represent
the polarization state of light using a coherency matrix [1]

$\Phi=\dfrac{1}{2}\sum_{i=0}^{3}s_{i}\sigma_{i}=\dfrac{1}{2}\begin{bmatrix}s_{0}+s_{1}&s_{2}-is_{3}\\\
s_{2}+is_{3}&s_{0}-s_{1}\end{bmatrix}.$ (2)

Here, $\sigma_{i}$ represent the four Pauli matrices and $\Phi$ represents the
coherency matrix. A coherency matrix is a 2x2 matrix that is positive semi-
definite, Hermetian and contains all measurable second order information about
the 2D polarization state.

Once the coherency matrix is realized, it is possible to arrive at the von
Neumann entropy associated with the polarization state using the following
relation [1]

$S_{VN}=-tr\\{\Phi log_{2}\Phi\\}.$ (3)

Alternatively, the von Neumann entropy can be arrived at from the normalized
eigenvalues of the coherency matrix $\Phi$, represented by $\lambda_{1}$ and
$\lambda_{2}$

$S_{VN}=-\sum_{i=1}^{2}\\{\lambda_{i}log_{2}\lambda_{i}\\}.$ (4)

When the value of $S_{VN}$ approaches 0, the state is very nearly pure, i.e.
is almost completely polarized and becomes mixed or unpolarised as $S_{VN}$
nears unity.

Alternatively, one can turn to the degree of polarization ($\mathcal{P}$) as
another metric to quantify state purity from the Stokes parameters using
[1,2,6]

$\mathcal{P}=\dfrac{\sqrt{s_{1}^{2}+s_{2}^{2}+s_{3}^{2}}}{s_{0}}.$ (5)

While $\mathcal{P}$ describes a different attribute as compared to the von
Neumann entropy, they are related by the expression [1]

$S_{VN}=-[\dfrac{1+\mathcal{P}}{2}log_{2}(\dfrac{1+\mathcal{P}}{2})+\dfrac{1-\mathcal{P}}{2}log_{2}(\dfrac{1-\mathcal{P}}{2})].$
(6)

This shows a one to one correspondence between the von Neumann entropy and the
degree of polarization, validating the use of $S_{VN}$ as a quantifier of
polarimetric purity.

##  3 Measuring the Stokes Parameters using an array detector

Much like single pixel Stokes polarimetry, a combination of waveplates and
linear optics need to be adopted for this purpose. The Electron Multiplied
Charge Coupled device (EMCCD, Mode; PRO-1024BX:Excelon) was employed as an
array detector in this investigation. A very weak incident photon signals are
amplified greatly to a level above the inherent noise with the EMCCD and this
feature opens up several applications for the device. Adopting the basic setup
for polarimetry, it is possible to examine the spatial variation of
polarization of a very weak optical signals using the EMCCD, as shown in Fig.
1. The intensity values from each detector pixel can be used to derive the
Stokes Parameter values for the same. Furthermore knowing the Stokes
parameters, we can derive the coherency matrix corresponding to every pixel
and ultimately the von Neumann entropy corresponding to each pixel.

An efficient method would be to utilize the available detector area on the
EMCCD by employing some sort of polarizing beam splitter like a Glan-Taylor
polarizer or a Wollaston prism and splitting the beam into orthogonally
polarized components that are allowed to fall on spatially separated regions
of the Sensor array. Of course finding the Stokes parameters and all the
associated quantities for each and every pixel will be computationally
intensive and as a consequence, time consuming. We therefore will focus on the
two regions of interest corresponding to the orthogonally polarized modes
emergent from the polarizing beam splitter. Furthermore, the device allows for
binning of intensity data from individual pixels and the effect of this
binning on Polarization parameter values needed examination. The intensity
values from each pixel is binned into groups with a user defined number of
pixels in each group. The advantage of using the EMCCD is that the necessary
information can be extracted post experiment and processed using a simple
program written in Matlab as the EMCCD supports the export of experimental
data with a communicating computer.

Our main setup relied on a Glan Taylor polarizing (GTP) beam splitter to
separate the orthogonal components of the attenuated laser beam and while the
EMCCD was aligned to image the direct propagating mode, the other mode was
redirected onto the EMCCD sensor using a mirror. We also employed a setup with
a Wollaston prism replacing the Glan Taylor polarizer, negating the need for
alignment mirrors just before the EMCCD.

Figure 1: The measurements are made in three steps with different waveplates.
In the experimental setup, L1, L2 and L3 are lenses, ND refers to Neutral
Density filter used to attenuate the light intensity, M1 and M2 are mirrors
and GTP is a Glan Taylor Polarizer.

A total of three images are acquired using the EMCCD with the waveplates set
to different settings corresponding to the required measurements, as shown in
Fig. 1. A Matlab program was written to read the output files from the EMCCD
and to process the output files, intensity profiles from the region of
interest were recorded appropriately. The resulting data is a set of six
intensity valued arrays, with each array corresponding to the intensity values
recorded by each pixel within the region of interest for the corresponding
measurement setting. According to the formulae to calculate the Stokes
parameters the program then performs the required mathematical operations with
the acquired arrays. Once the Stokes parameters corresponding to each pixel in
the region of interest are obtained and stored, the program will move on to
the process of using the freshly obtained Stokes parameters to compile the set
of coherency matrices also corresponding to every pixel from the region of
interest. Finally, with the coherency matrices generated, the program will
derive the von Neumann entropy values corresponding to every pixel in the
region of interest.

Also, we had decided to go with three different bin sizes (hardware bins) for
the experiment and study the variation across these. The first involved no
binning resulting in a 1024 by 1024 array of intensity values corresponding to
the flux incident on each physical pixel. The second case involved binning the
intensity of 4 pixels into one (2x2 binning), resulting in a 512 by 512 array
of intensity values. The final case would give a 256 by 256 intensity array
after binning 16 pixels into one (4x4 binning). Binning has the added effect
of improving the signal to noise ratio at the expense of spatial resolution.
This became apparent when the same acquisition settings that could be
considered the bare minimum for a proper signal-to-noise ratio (SNR) in the
case of no binning, caused the sensor to saturate in the 4x4 binned state.
This forced us to adopt acquisition settings, in terms of exposure time and
gain, such that the sensor will not saturate in the maximum binned state but
at the cost of poor SNR ratio in the 1024 by 1024 unbinned state. In the
coming sections we will discuss our findings using the different light
sources.

##  4 Diode Laser

The laser we used was a vertically polarized CW diode laser (iflex-2000,
Qioptiq) with output peaking at 405 nm and with about 2 nm of Full-width half
maxima (FWHM). This source was used to calibrate the setup because of the very
well defined polarization of its output. Our array detector, being an EMCCD,
requires the use of a combination of several neutral density filters to
attenuate the laser output down to an intensity that will not damage our
EMCCD. A combination of GTP with mirrors was utilized to image the orthogonal
components of the beam using EMCCD.

The effect of the reduced SNR shows in our calibration experiment where
parameters like the s1 value, which should be close to negative unity but
shows a spatial average of around half of that. However, for both binned
cases, the values are very much in agreement with the theory and we attribute
this to the improved SNR associated with the binning process. We had to
compromise SNR in the case of the unbinned measurement to maintain uniform
measurement parameters of exposure time and EMCCD gain across all binning
conditions. The device sensor quickly exceeds its safe limit when binning is
applied if SNR of the unbinned measurement is kept high using large gain
values or increased exposure times. The colormaps shown in Fig. 2. depict the
profile of von Neumann entropy for the diode laser and it is evident from the
scale how minute the variations are in this case. It is also clear from Fig.
2. how binning reduces the resolution. Fig. 3 shows how binning affects the
Stokes parameter values as well as von Neumann entropy and the square sum of
the Stokes parameters. The plot exhibits the inverse proportionality of the
squared sum of the Stokes parameters to the von Neumann entropy for different
binning conditions.

(a) 4$\times$4 binning

(b) 2$\times$2 binning

(c) 1$\times$1 binning

Figure 2: The spatial variation in the von Neumann entropy for different bin
sizes as shown in each map title. These correspond to the case of light from
the diode laser. The variations, although apparently drastic, are actually
spread over a very narrow range.

(a)

(b)

Figure 3: Plots (a) The variation of each Stokes parameter with different
binning settings for the diode laser; (b) The variation of VNE and squared sum
across different binning conditions.

##  5 Light Emitting Diode

The next source was a partially polarized LED with output peaking at 450 nm
and FWHM of around 30 nm (recorded via ocean optics spectrometer, figure not
shown). A linear polarizer was used to make the output partially polarized
along with a set of Neutral density filters so as to reduce the intesity to a
level that is safe for the EMCCD sensor. The light from an LED is known to be
unpolarized and using the linear polarizer allowed us to get a better
understanding of the effectiveness of the measurement. The basic setup
remained more or less the same apart from some slight realignment.

The result of measuring the polarization parameters in this case was that
while the linear polarizer introduced a certain degree of linear polarization
to the light, it did not completely polarize the LED output. This shows up in
the values of both square sum of the Stoke’s parameters $s_{1},s_{2}$ and
$s_{3}$, which comes out to be a little over 0.7 and the von Neumann entropy
hovers around 0.4 across the region of interest in the case of maximum binning
which gives best SNR as shown in Fig. 5. This is in agreement with our
calibration experiment. The effect of improved SNR is also apparent in Fig. 4,
where the range of variation across the profile improves with binning.

(a) 4$\times$4 binning

(b) 2$\times$2 binning

(c) 1$\times$1 binning

Figure 4: The spatial variation in the von Neumann entropy for different bin
sizes as shown in each map title. These correspond to the region of interest
in the case of LED light. The variations are once again spread over a very
narrow range.

(a)

(b)

Figure 5: Plots (a) The variation of each Stokes parameter with different
binning settings for the LED; (b) Exhibits the inverse proportionality of the
squared sum of the Stokes parameters to the von Neumann entropy for different
binning conditions.

##  6 Fluorescein dye

We employed a 405 nm diode laser to pump a solution of the organic
fluorophore, Fluorescein which emits across a broad range from about 490 nm to
640 nm. The emission bandwidth was restricted using a 577 nm optical filter
and collected and imaged using a system of lenses aligned to form an image at
the EMCCD sensor. The polarimetry setup was introduced prior to the EMCCD and
the Stokes measurements were recorded to study the transverse spatial
variation of the fluroscent beam. Fluorescein is an organic fluorescent dye
with widespread applications in bio-imaging as an indicator or tracer.
Examining the transverse spatial variation of polarization entropy of this
contrast agent might bring out otherwise hidden information about the sample
under examination. The technique we employed can easily be modified to work
with a bio-imaging setup. As is evident from Fig. 6, the von Neumann entropy
varies very little and the presence of polarization change inducing structures
or processes will exhibit a noticeable change. In Fig. 7, both VNE and the
square sum of the Stokes parameters appear not to change very much across
different binning conditions. This can be attributed to the highly unpolarized
nature of the dye emission.

(a) 4$\times$4 binning

(b) 2$\times$2 binning

(c) 1$\times$1 binning

Figure 6: The spatial variation in the von Neumann entropy for different bin
sizes as, shown in each map title. These correspond to the region of interest
in the case of fluorophore emission output.

(a)

(b)

Figure 7: Plots (a) The variation of each Stokes parameter with different
binning settings for the fluorescent dye; (b) Exhibits the inverse
proportionality of the squared sum of the Stokes parameters to the von Neumann
entropy for different binning conditions.

##  7 Spontaneous Parametric Down Conversion

Type-II Spontaneous parametric down conversion (SPDC) [17] output has a cross
section in the shape of twin rings intersecting at two points, as shown in
Fig. 10. While the rings are orthogonally polarized w.r.t each other, their
points of intersection at any given instant can have either vertical or
horizontal polarization. At these points of intersection, the polarizations
across these points will always be anti-correlated but it is impossible to
predict the polarization at any one point at any given instant. We wanted to
see if this randomness would reflect in the von Neumann entropy measurement of
these points of intersection.

To measure the polarization properties at the points of intersection, we first
had to image the twin rings of SPDC without any of the optics associated with
polarimetric in the way. Once SPDC was confirmed, a telescope type optical
system was introduced between the EMCCD and nonlinear crystal to allow for
space in between to introduce the polarimetry specific optics. Custom
apertures were used to isolate the points of intersection of the twin rings in
the image. Once again, the variations across our sensor area are minimal but
become slighly more pronounced at higher binning stages, as shown in Fig. 8.
In Fig. 9, the variation of the squared sum of the Stokes parameters is more
drastic than that of VNE, hinting at an inverse but nonlinear relationship
between the two parameters.

(a) 4$\times$4 binning

(b) 2$\times$2 binning

(c) 1$\times$1 binning

Figure 8: The spatial variation in the von Neumann entropy for different bin
sizes as shown in each map title. These correspond to the region of interest
in the case of SPDC output.

(a)

(b)

Figure 9: (a) The variations of the stokes parameters and (b) The variation of
VNE and square sum across different stages of binning in the case of SPDC.
Figure 10: The image of SPDC rings recorded using EMCCD.

##  8 Results and Discussion

We have studied the polarization properties across the beam profile of several
light sources. The highly polarized laser output was found to have the least
von Neumann entropy associated with polarization, while the emission from
fluorescein dye, along with SPDC output showed the highest values. However,
the von Neumann entropy values of a few small regions within the laser beam
spot appear to take on very small negative values while the majority of of the
area shows values just above zero. While we expect the small positive VNE
values for the highly polarized output of the diode laser, the slight negative
deviations can be attributed to experimentally induced effects.

The effect of binning on the VNE of the different sources, as shown in Fig.
11(b), suggests that more accurate values are obtained with higher binning;
the well defined linear polarization of the diode laser can be thought to have
less disorder or entropy and the lowest value of VNE is obtained at the
highest possible stage of binning. A similar argument can be made for the
partially polarized output from the LED. The emission of the Fluorescein dye,
being somewhat unpolarized, showed not much variation in its VNE values across
different stages of binning and also the least amount of variation within the
region of interest. The correlation between degree of polarization and VNE is
evident in Fig. 11(a) where the unpolarized emission of the dye shows the
highest value of VNE of all the sources. The VNE of SPDC is also high. This is
because where the dual rings of SPDC intersect, the polarization toggles
randomly between horizontal and vertical and our measurements show that this
gives it a fairly high VNE value which also varies more across the region, as
compared to that of the dye.

The range of variation of the Stokes parameter $s_{1}$, across the region of
interest for the different sources shows the expected results with the laser
values varying across a very narrow range centred close to -1, which is as
expected for the state of polarization. The standout feature in the plot is
the large range of variation of SPDC output derived $s_{1}$ centred at around
zero which we attributed to the fast toggling between the horizontal and
vertical states of polarization at the points of intersection.

(a)

(b)

(c)

(d)

Figure 11: (a) The range of variation of VNE within the region of interest for
the different sources, (b) A comparison between different sources in terms of
how the VNE varies across different stages of binning, (c) The range of
variation of $s_{1}$ within the region of interest for all the investigated
sources and (d) The variation of square sum of Stokes parameters entropy
across different binning conditions for all the investigated sources.

##  9 Conclusions

To conclude, we have measured the Stokes parameters as well as von Neumann
entropy and their transverse spatial variation across the illuminated area
using our simple setup for the light from a CW diode laser, an LED, a
fluorescent dye and the entangled output of SPDC. Furthermore, we studied the
effects of binning on all measured parameters which we found to be in step
with the improvement in SNR or visibility that the process of binning brings
about. There is a clear trend of increasing von Neumann entropy with
decreasing polarization purity which is as expected but there is also a
decrease of the von Neumann entropy with degree of binning. These results can
be incorporated positively in the field of Synthetic Aperture Radar
polarimetry which employs polarization entropy concepts to map and lock on to
targets spread out over a large area.

## Acknowledgments

VN and SB acknowledge the support from Interdisciplinary Cyber Physical
Systems (ICPS) programme of the Department of Science and Technology (DST),
India, Grant No.:DST/ICPS/QuST/Theme-1/2019/6. Tomis would like to acknowledge
CSIR for the research funding. RP acknowledges DST for the finanancial
assitance given for the research.

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