# Inflation Alternative via the Gravitational Field of a Singularity

Michael Zlotnikov

###### Abstract

We explore the scenario that the observable universe emerged from the
ergosphere of a negative mass ring singularity, and all content of the
universe travels at the same group velocity close to the speed of light on a
geodesic trajectory along the axis of rotation of the singularity. In
appropriate coordinate parametrization and evaluated on the trajectory, we
find that the metric tensor in the vicinity of the trajectory exhibits a
conformal scale factor $\bm{a}(\eta)$ with contraction and subsequent
expansion properties that solve the horizon problem. We then introduce a
static flow of gravitating radiation along the trajectory (perturbatively with
respect to the mass scale of the singularity) to model a homogeneous radiation
dominated universe. Solving the Einstein field equations with a physically
motivated ansatz of metric perturbation then reveals that the effective
conformal scale factor indeed grows asymptotically with the same power law as
expected in a conventional radiation dominated universe.

##  1 Introduction

The horizon problem [1] is concerned with the fact that different regions of
the Cosmic Microwave Background (CMB) [2, 3], emitted at the surface of last
scattering, possess surprisingly uniform temperature. Assuming a generic
cosmic evolution, these regions could not have been in causal contact with
each other, and yet their almost equal temperatures suggest that they must
have thermalized at some point in the past. A possible solution to this
problem is the inflationary universe paradigm [4, 5], in which the universe
undergoes a period of accelerated expansion at early times, taking space-time
regions that previously interacted with each other out of causal contact,
which then re-enter the cosmological horizon upon later cosmic evolution. The
process of inflation is typically assumed to be driven by a space-filling
inflaton field (see for example [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16,
17, 18]). At the end of inflation all other fields are thinned out and the
universe is re-populated with standard model constituents through the decay
process of the inflaton field called reheating [19, 20].

In this work we explore an alternative way to resolve the horizon problem by
assuming that the observable universe has been emitted from the ergosphere of
a negative mass Kerr-like solution of the Einstein field equations,111Negative
mass regions are commonplace in exact solutions of General relativity, and can
be argued to be physically meaningful. See e.g. [21] for a motivation. and all
content of the universe travels at approximately the same group velocity close
to the speed of light along a radial geodesic parallel to the Kerr spin axis.
The naked singularity in this setting is hidden from view because the universe
is opaque at early times. The relative motion with respect to the singularity
manifests itself locally on the geodesic in terms of an effective cosmological
scale factor with specific contraction and expansion properties solving the
horizon problem.

Subsequently, introducing a static flow of gravitating radiation, with energy
density parametrically smaller than the mass of the singularity, we show that
a leading order perturbative correction to the metric tensor leads to the same
asymptotic late conformal time scaling of the effective scale factor as
expected in a Friedmann–Lemaître–Robertson–Walker (FLRW) radiation dominated
universe. The transition to a matter dominated universe could then be made in
a conventional formalism, without invoking a singularity in the far past.

This paper is organized as follows:

In section 2 we recall the basic tools of General Relativity to be used in the
following.

Section 3 introduces an appropriate cylindrical change of coordinates,
parameterizing the vicinity of a radial geodesic trajectory parallel to the
axis of rotation in a negative mass Kerr space-time. Evaluating the metric on
the geodesic trajectory and expressing the solution in conformal time reveals
that at leading order in the vicinity of the trajectory the metric possesses
an effective conformal scale factor $\bm{a}$. The scale factor $\bm{a}$ starts
at unity, contracts to a parametrically small number, depending on the Kerr
spin parameter and mass $a/M\to 0$, and subsequently asymptotically expands
back to unity in the case of an empty universe. The initial contraction and
subsequent expansion of the scale factor explains how seemingly separated
regions entering the cosmological horizon during the expansion phase were in
causal contact previously.

In section 4 we perturbatively add a static flow of gravitating radiation
traveling along the radial geodesic. With an appropriate perturbation ansatz
for the metric tensor, we solve the Einstein equations to leading order in the
vicinity of the radial trajectory and small Kerr spin parameter $a/M\to 0$.
The resulting correction to the effective conformal scale factor turns out to
be given by an integral over ratios of hypergeometric functions. At late
conformal time $\eta\gg M$, the efective conformal scale factor then turns out
to scale as $\sim\eta$, which is the expected scaling for a radiation
dominated universe.

We conclude in section 5, and offer a few suggestions for further
investigation that goes beyond the scope of this paper.

##  2 Preliminaries

In this section we recall the basic tools of General Relativity (see e.g.
[22]), to be applied in subsequent sections. We use the Einstein index
summation convention $a^{\mu}b_{\mu}\equiv\sum^{3}_{\mu=0}a^{\mu}b_{\mu}$
throughout.

Given a set of coordinates $x^{\mu}$ describing a space-time, along with the
corresponding metric tensor $g_{\mu\nu}$ (lowering indices
$x_{\mu}=g_{\mu\nu}x^{\nu}$) and inverse metric tensor $g^{\mu\nu}$ (raising
indices $x^{\mu}=g^{\mu\nu}x_{\nu}$), the Christoffel symbols are determined
as follows

$\displaystyle\Gamma^{\sigma}{}_{\mu\nu}=\frac{1}{2}g^{\sigma\beta}\left(\frac{\partial
g_{\beta\mu}}{\partial x^{\nu}}+\frac{\partial g_{\beta\nu}}{\partial
x^{\mu}}-\frac{\partial g_{\mu\nu}}{\partial x^{\beta}}\right).$ (2.1)

The Ricci curvature tensor $R_{\mu\nu}$ and Ricci scalar $R$ are then
constructed as

$\displaystyle R_{\mu\nu}=\frac{\partial\Gamma^{\beta}{}_{\mu\nu}}{\partial
x^{\beta}}-\frac{\partial\Gamma^{\beta}{}_{\beta\nu}}{\partial
x^{\mu}}+\Gamma^{\beta}{}_{\beta\beta}\Gamma^{\beta}{}_{\mu\nu}-\Gamma^{\beta}{}_{\mu\beta}\Gamma^{\beta}{}_{\beta\nu}~{}~{}~{}~{}~{}\text{and}~{}~{}~{}~{}~{}R=g^{\mu\nu}R_{\mu\nu}\,.$
(2.2)

These quantities enter the Einstein field equations, along with energy
momentum tensor $T^{\mu\nu}$

$\displaystyle G^{\mu\nu}\equiv R^{\mu\nu}-\frac{R}{2}g^{\mu\nu}+\Lambda
g^{\mu\nu}=\kappa T^{\mu\nu},$ (2.3)

where $G^{\mu\nu}$ is the Einstein tensor, and $\Lambda$ is a potentially non-
zero cosmological constant. In the following we will be working in geometrized
units with the Einstein gravitational constant set to unity $\kappa=1$.

For physical matter/radiation space-time content, the covariant divergence of
the energy momentum tensor must vanish

$\displaystyle 0=\nabla_{\nu}T^{\mu\nu}=\frac{\partial T^{\mu\nu}}{\partial
x^{\nu}}+\Gamma^{\mu}{}_{\sigma\nu}T^{\sigma\nu}+\Gamma^{\nu}{}_{\sigma\nu}T^{\mu\sigma}.$
(2.4)

Free falling particles in a given space-time follow geodesic trajectories that
are solutions to the geodesic equation

$\displaystyle\frac{d^{2}x^{\mu}}{d\tau^{2}}+\Gamma^{\mu}{}_{\beta\beta}\frac{dx^{\beta}}{d\tau}\frac{dx^{\beta}}{d\tau}=0,$
(2.5)

where $\tau$ is the proper time (or affine parameter) parameterizing the
trajectory. Sometimes, it is also convenient to employ the proper time
equation

$\displaystyle-d\tau^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}$ (2.6)

to solve for a geodesic trajectory.

##  3 Empty Universe

We start with the negative mass region of the Kerr metric [23] in Boyer-
Lindquist coordinates, employing the parameter $M>0$:222Compared to the
regular Kerr black hole metric, this differs by the sign in front of the mass
parameter $M$, and is equally an exact solution of the vacuum Einstein field
equations.

$\displaystyle-d\tau^{2}=$
$\displaystyle-\frac{\Delta_{r}}{\Sigma_{r}}\left(dt-
ad\phi\sin^{2}(\theta)\right)^{2}+\Sigma_{r}\left(\frac{dr^{2}}{\Delta_{r}}+d\theta^{2}\right)+\frac{\sin^{2}(\theta)}{\Sigma_{r}}\left(adt-d\phi\left(a^{2}+r^{2}\right)\right)^{2},$
(3.1)

where

$\displaystyle\Delta_{r}=a^{2}+2Mr+r^{2}~{}~{}~{}\text{and}~{}~{}~{}\Sigma_{r}=a^{2}\cos^{2}(\theta)+r^{2},$
(3.2)

and $a$ is the spin parameter. Since there is no relative minus sign in the
$\Delta_{r}$ factors, this space-time does not exhibit any black hole
horizons. Therefore, particles can escape from the ergosphere to infinity,
which will be an important feature in the following.

We work towards understanding how the metric behaves locally, when evaluated
on an outgoing radial geodesic trajectory and expressed in conformal
coordinates at leading order in the vicinity of the trajectory. This shall
facilitate a comparison with a conventional cosmological metric of an
unperturbed universe.

As a first step, we introduce a cylindrical type of coordinates, along with an
expansion around small cylindrical radius, to describe the vicinity of the
straight line trajectory. To that end, we perform the following change of
coordinates $(r,\theta)\to(\bm{r},z)$ for $z\geq 0$, and an expansion around
small $\bm{r}\ll M$:333Here, small cylindrical radius $\bm{r}$ corresponds to
small $\theta$, while the original spherical radius $r$ approaches $z$ in that
limit.

$\displaystyle r=$
$\displaystyle\,z+\bm{r}^{2}\frac{(M+z)}{2\left(a^{2}+z(2M+z)\right)}+\bm{r}^{4}\frac{\left((2M+3z)a^{4}+2z^{2}(4M+z)a^{2}-z^{4}(2M+z)\right)}{8\left(a^{2}+z^{2}\right)^{2}\left(a^{2}+z(2M+z)\right)^{2}}+\mathcal{O}\left(\bm{r}^{5}\right),$
$\displaystyle\theta=$
$\displaystyle\,\bm{r}\frac{1}{\sqrt{a^{2}+z(2M+z)}}+\bm{r}^{3}\frac{a^{4}+(3M-z)za^{2}-z^{3}(3M+2z)}{6\left(a^{2}+z^{2}\right)^{2}\left(a^{2}+z(2M+z)\right)^{3/2}}+\mathcal{O}\left(\bm{r}^{5}\right),$
(3.3)

resulting in the following differential relations

$\displaystyle dr$
$\displaystyle=\,dz+\bm{r}\frac{M+z}{a^{2}+z(2M+z)}d\bm{r}-\bm{r}^{2}\frac{-a^{2}+2M^{2}+z^{2}+2Mz}{2\left(a^{2}+z(2M+z)\right)^{2}}dz$
$\displaystyle\,+\bm{r}^{3}\frac{(2M+3z)a^{4}+2z^{2}(4M+z)a^{2}-z^{4}(2M+z)}{2\left(a^{2}+z^{2}\right)^{2}\left(a^{2}+z(2M+z)\right)^{2}}d\bm{r}+\mathcal{O}\left(\bm{r}^{4}\right),$
(3.4) $\displaystyle d\theta$
$\displaystyle=\frac{d\bm{r}}{\sqrt{a^{2}+z^{2}+2Mz}}-\frac{dz(M+z)\bm{r}}{\left(a^{2}+z(2M+z)\right)^{3/2}}+\frac{d\bm{r}\left(a^{4}+(3M-z)za^{2}-z^{3}(3M+2z)\right)\bm{r}^{2}}{2\left(a^{2}+z^{2}\right)^{2}\left(a^{2}+z(2M+z)\right)^{3/2}}$
$\displaystyle-\bm{r}^{3}\frac{3a^{6}+\left(M^{2}+12zM+4z^{2}\right)a^{4}+z^{2}\left(12M^{2}+6zM-z^{2}\right)a^{2}-z^{4}\left(5M^{2}+6zM+2z^{2}\right)}{2\left(a^{2}+z^{2}\right)^{3}\left(a^{2}+z(2M+z)\right)^{5/2}}z\,dz$
$\displaystyle+\mathcal{O}\left(\bm{r}^{4}\right).$ (3.5)

Introducing these coordinates into the metric (3.1), we obtain444The change of
variables (3) is set up specifically such that there is no $dzd\bm{r}$ term at
order $\mathcal{O}(\bm{r}^{1})$, the scale factor in front of all spatial
coordinates is homogeneous at $\mathcal{O}(\bm{r}^{0})$, and there is no non-
trivial $d\phi^{2}$ term at order $\mathcal{O}(\bm{r}^{4})$.

$\displaystyle-d\tau^{2}$
$\displaystyle=-\frac{a^{2}+z(2M+z)}{a^{2}+z^{2}}dt^{2}+\frac{a^{2}+z^{2}}{a^{2}+z(2M+z)}\left(d\bm{r}^{2}+\bm{r}^{2}d\phi^{2}+dz^{2}\right)$
(3.6)
$\displaystyle+\bm{r}^{2}M\left(\frac{\left(z^{2}(M+z)-a^{2}(M+3z)\right)dt^{2}+4az\left(a^{2}+z^{2}\right)dtd\phi}{\left(a^{2}+z^{2}\right)^{2}\left(a^{2}+z(2M+z)\right)}+\frac{z^{3}-a^{2}(2M+3z)}{\left(a^{2}+z(2M+z)\right)^{3}}dz^{2}\right.$
$\displaystyle\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{\left(a^{4}(M+6z)+2a^{2}z^{2}(5M+2z)-z^{4}(3M+2z)\right)}{\left(a^{2}+z^{2}\right)\left(a^{2}+z(2M+z)\right)^{3}}d\bm{r}^{2}\right)+dzd\bm{r}\,\mathcal{O}(\bm{r}^{3})$
$\displaystyle+\bm{r}^{4}\frac{2aM(a-z)(a+z)\left(a^{2}(M+3z)+3z^{2}(M+z)\right)}{\left(a^{2}+z^{2}\right)^{3}\left(a^{2}+z(2M+z)\right)^{2}}dtd\phi+\mathcal{O}(\bm{r}^{4}).$

The first line in (3.6) contains the leading order metric and features
homogeneous scaling of generic cylindrical spatial coordinates by a
$z$-dependent factor. The subsequent lines in (3.6) contain sub-leading metric
terms sufficient to ensure that the overall metric remains a vacuum solution
of the Einstein field equations (2.3) (with $\Lambda=0$) in the vicinity
($\bm{r}/M\to 0$) of the trajectory of interest to order
$\mathcal{O}(\bm{r}^{2})$:

$\displaystyle R_{\mu\nu}-\frac{R}{2}g_{\mu\nu}=\mathcal{O}(\bm{r}^{2}).$
(3.7)

Note in particular, that a $dzd\bm{r}\,\mathcal{O}(\bm{r}^{3})$ term as well
as several $\mathcal{O}(\bm{r}^{4})$ terms are neglected in (3.6), since they
contribute to (3.7) at order $\mathcal{O}(\bm{r}^{2})$ or higher and are
therefore irrelevant for our purposes.

What is the utility of formulating the metric as in (3.6)?

By the first postulate of General Relativity, the relativity principle, a
point-like observer at any point in any space-time perceives the metric in
their immediate vicinity as the flat Minkowski metric, such that local physics
is governed by Special Relativity. In other words, at any point in any
physical space-time one can find normal coordinates, such that the leading
term is given by the flat Minkowski metric, and first derivatives of the
metric with respect to all coordinates vanish. Therefore in our case of
interest, a point-like observer traveling from the ergosphere to infinity
along a radial trajectory finds themselves in such normal coordinates at any
given instant. If we desire to keep track of how the normal coordinates at a
given point relate to the respective normal coordinates at a different point
visited previously, a fruitful course of action is to relax the condition of
normality for the single coordinate aligned with the path of trajectory, and
by doing so treat coordinates at all visited points on an equal footing. The
form of the metric (3.6) achieves precisely that, while maintaining
homogeneous scaling of spatial coordinates at leading order in small
$\bm{r}\ll M$. Normality is relaxed for the $z$-direction, which parametrizes
different points on the trajectory, while normality is maintained in
$t,\bm{r},\phi$ coordinates (perturbatively in $\bm{r}\ll M$, and up to
constant rescaling at fixed $z$), with their first derivatives vanishing
locally. This is analogous to the situation in the FLRW metric of cosmology,
where the scale factor $\bm{a}(t)$ breaks the normality of the metric in the
$t$-direction, while the rest of the coordinates are normal (up to constant
rescaling at fixed $t$), and spatial coordinates experience homogeneous
scaling.

So far, we have chosen a particular parametrization (3.6) for the space-time
metric in the vicinity of a radial trajectory. To compare this metric with a
typical cosmological metric of an unperturbed universe, we evaluate the metric
along the radial trajectory parametrized in conformal time $\eta$.

Due to the asymptotic flatness of the Kerr space-time, all sub-leading terms
in (3.6) are guaranteed to vanish for increasing $z$ values. We therefore
concentrate on the leading order metric in (3.6) to model the observable
universe, and transform the time variable to conformal time $t=f(\eta)$,
$dt=f^{\prime}(\eta)d\eta$:

$\displaystyle-d\tau^{2}=$
$\displaystyle-\frac{a^{2}+z(2M+z)}{a^{2}+z^{2}}f^{\prime}(\eta)^{2}d\eta^{2}+\frac{a^{2}+z^{2}}{a^{2}+z(2M+z)}\left(d\bm{r}^{2}+\bm{r}^{2}d\phi^{2}+dz^{2}\right)+\mathcal{O}(\bm{r}^{2}),$
(3.8)

with the appropriate function $f(\eta)$ to be determined. In these variables,
the zero’th component of the geodesic equation (2.5) reads

$\displaystyle\frac{d^{2}x^{0}}{d\tau^{2}}+\Gamma^{0}{}_{\beta\beta}\frac{dx^{\beta}}{d\tau}\frac{dx^{\beta}}{d\tau}=\frac{2M(a-z)(a+z)\eta^{\prime}(\tau)z^{\prime}(\tau)}{\left(a^{2}+z^{2}\right)\left(a^{2}+z(2M+z)\right)}+\frac{f^{\prime\prime}(\eta)\eta^{\prime}(\tau)^{2}}{f^{\prime}(\eta)}+\eta^{\prime\prime}(\tau)=0.$
(3.9)

After dividing by $\eta^{\prime}(\tau)$, this directly integrates to

$\displaystyle\eta^{\prime}(\tau)=\frac{c\left(a^{2}+z(\tau)^{2}\right)}{\left(a^{2}+z(\tau)(2M+z(\tau))\right)f^{\prime}(\eta(\tau))},$
(3.10)

with an integration constant $c$. To find $z^{\prime}(\tau)$ in the case of a
massive particle, we use (3.8) with
$\bm{r}^{\prime}(\tau)=0,\,\phi^{\prime}(\tau)=0$, plug in (3.10), and solve
selecting the positive square root solution

$\displaystyle-1=$
$\displaystyle-\frac{a^{2}+z(2M+z)}{a^{2}+z^{2}}f^{\prime}(\eta)^{2}\eta^{\prime
2}+\frac{a^{2}+z^{2}}{a^{2}+z(2M+z)}z^{\prime
2}~{}~{}~{}\Rightarrow~{}~{}~{}z^{\prime}(\tau)=\sqrt{c^{2}-1-\frac{2Mz(\tau)}{a^{2}+z(\tau)^{2}}}.$

Note that in order for $z^{\prime}(\tau)$ to be real valued on the whole range
of $0\leq z(\tau)\leq\infty$, the integration constant $c$ must satisfy
$c\geq\sqrt{1+M/a}$. Therefore, we can set $c$ to e.g. the minimum admissible
value

$\displaystyle
c=\sqrt{1+\frac{M}{a}}~{}~{}~{}~{}~{}~{}\Rightarrow~{}~{}~{}~{}~{}~{}z^{\prime}(\tau)=\sqrt{\frac{M}{a}-\frac{2Mz(\tau)}{a^{2}+z(\tau)^{2}}}.$
(3.11)

Finally, to obtain $z^{\prime}(\eta)$ we use the chain rule
$z^{\prime}(\tau)=z^{\prime}(\eta)\eta^{\prime}(\tau)$, plug in (3.10, 3.11)
and solve

$\displaystyle
z_{m}^{\prime}(\eta)=\sqrt{\frac{M(a-z_{m})^{2}}{(a+M)\left(a^{2}+z_{m}^{2}\right)}}\frac{a^{2}+z_{m}(2M+z_{m})}{a^{2}+z_{m}^{2}}f^{\prime}(\eta),$
(3.12)

where we used the subscript $m$ to emphasize that this is the solution for a
massive particle.

To find $z^{\prime}(\eta)$ in case of a massless particle, we use (3.8) with
$d\bm{r}=0,\,d\phi=0,\,d\tau=0$, choose the positive square root solution and
obtain

$\displaystyle
z_{\gamma}^{\prime}(\eta)=\frac{a^{2}+z_{\gamma}(2M+z_{\gamma})}{a^{2}+z_{\gamma}^{2}}f^{\prime}(\eta),$
(3.13)

where the subscript $\gamma$ emphasizes that this is the solution for a
massless particle.

Recall that we introduced the change of time coordinate $t=f(\eta)$ in (3.8)
in order for $\eta$ to be conformal time. We can enforce this when we evaluate
the metric on the trajectory parametrized by $\eta$, such that $z=z(\eta)$. In
conformal time, we expect e.g. the equality $-g_{00}=g_{33}$ between metric
components. Therefore, choosing the positive square root solution, we define
$f(\eta)$ such that

$\displaystyle-
g_{00}=\frac{a^{2}+z(2M+z)}{a^{2}+z^{2}}f^{\prime}(\eta)^{2}=\frac{a^{2}+z^{2}}{a^{2}+z(2M+z)}=g_{33}~{}~{}~{}\Rightarrow~{}~{}~{}f^{\prime}(\eta)=\frac{a^{2}+z(\eta)^{2}}{a^{2}+z(\eta)(2M+z(\eta))}.$
(3.14)

We will see shortly that to model an effective inflation alternative we should
take the limit $a\ll M$. In that limit the massive and massless relations
(3.12) and (3.13) become equivalent, meaning that massive particles must
travel close to the speed of light to successfully emerge from the ergosphere.
Therefore, in the following we focus on the massless case without loss of
generality.

Plugging (3.14) into (3.13), we observe

$\displaystyle z^{\prime}(\eta)=1\,,~{}~{}~{}\text{the simplest solution
being}~{}~{}~{}z(\eta)=\eta.$ (3.15)

Thus, transformed to conformal time and evaluated on the trajectory of
interest, the leading order metric reads

$\displaystyle-d\tau_{\text{local}}^{2}=$
$\displaystyle\frac{a^{2}+\eta^{2}}{a^{2}+\eta(2M+\eta)}\left(-d\eta^{2}+d\bm{r}^{2}+\bm{r}^{2}d\phi^{2}+dz^{2}\right)+\mathcal{O}(\bm{r}^{2}),$
(3.16)

where all finite factors have been evaluated to their appropriate
$\eta$-dependence, while differentials $d\eta,\,d\bm{r},\,d\phi,\,dz$ refer to
local space-time properties along the trajectory at a given moment in
conformal time $\eta$.

Metric (3.16) is analogous to a conventional FLRW metric in conformal time,
with an effective conformal scale factor

$\displaystyle\bm{a}(\eta)=\sqrt{\frac{a^{2}+\eta^{2}}{a^{2}+\eta(2M+\eta)}}.$
(3.17)

The scale factor equals $\bm{a}(0)=1$ at the initial ergosphere surface, and
falls to a minimum of $\bm{a}(a)=\sqrt{{a}/({a+M})}$ when conformal time
reaches the value of the Kerr spin parameter $\eta=a$. Subsequently, the scale
factor starts growing and asymptotes back to $\bm{a}(\infty)=1$. There is a
brief period of accelerated expansion at the beginning of the growth phase,
see Figure 1 for a qualitative plot. However, the fact that the scale factor
starts at unity and decreases in magnitude initially is sufficient to ensure
prior causal contact between regions reentering the horizon during the later
expansion phase. The smaller the spin parameter $a$, the smaller the scale
factor becomes at its lowest point. Therefore, we concentrate on the small
spin case $0<a\ll M$ in the following, which ensures that any amount of
relative expansion starting from the lowest point
$\bm{a}(a)=\sqrt{{a}/({a+M})}$ can be accommodated. This demonstrates how non-
trivial time evolution of the cosmological scale factor solving the horizon
problem can be achieved dynamically e.g. by traversing the gravitational field
created by a super-galactic ring singularity in the far past, instead of the
conventional approach of producing inflation by introducing space-filling
energy-momentum at the particular time when the scale factor is evolving.

So far, no gravitating energy momentum content is traveling along the geodesic
trajectory, so that the above metric evolves into the flat Minkowski metric
for $\eta\to\infty$. In the next section we add gravitating radiation emerging
from the ergosphere ($z=0$ surface), to model a radiation dominated
unperturbed universe in this setting.

Figure 1:  Acceleration $\frac{d^{2}}{dt^{2}}\bm{a}$ of the scale factor
(3.17) as a function of the scale factor $\bm{a}$ itself, evaluated at
$M=1,~{}a=10^{-1}$. The curve starts at $\bm{a}(a)=\sqrt{a/(a+M)}$, where it
assumes its maximum value $\frac{1}{2}Ma^{-3}\sqrt{a/(a+M)}$. After a brief
period of positive acceleration, the expansion starts decelerating, until it
asymptotically comes to a halt when the scale factor reaches unity
$\bm{a}(\infty)=1$.

##  4 Radiation Dominated Universe

In this section we introduce a gravitating radiation energy density emerging
from the ergosphere and flowing along a radial outgoing geodesic parallel to
the axis of rotation. We assume the energy momentum flow to be constant at the
time scale of interest, and are therefore looking for a static perturbative
correction to the metric tensor.

The ergosphere collapses to a point if the Kerr spin parameter $a$ is set to
zero exactly, in which case the flow of radiation should vanish. Therefore we
assume that the emerging radiation density is proportional to $a$:

$\displaystyle\rho_{\gamma,0}\propto\frac{a}{M}.$ (4.1)

Furthermore, as observed in the previous section, the smaller the Kerr spin
parameter $a$ is, the more the effective scale factor (3.17) contracts before
the expansion phase. We can accommodate any relative amount of expansion from
the lowest point if we consider the small spin parameter regime $0<a\ll M$.

Introducing a non-zero energy momentum flow of radiation causes the geometry
of space-time to shift. In order to account for these changes, we take the
following ansatz for the perturbation of metric (3.6):

$\displaystyle-d\tau^{2}$
$\displaystyle=-\frac{a^{2}+z(2M+z)}{a^{2}+z^{2}}dt^{2}\left(1-\frac{a}{M}(1+\beta)\,h_{1}(z)+\frac{a\bm{r}^{2}}{M^{3}}h_{3}(z)\right)+2dtdz\frac{a\bm{r}^{2}}{M^{3}}h_{2}(z)$
(4.2)
$\displaystyle+\frac{a^{2}+z^{2}}{a^{2}+z(2M+z)}\left(d\bm{r}^{2}+\bm{r}^{2}d\phi^{2}+dz^{2}\right)\left(1+\frac{a}{M}h_{1}(z)+\frac{a\bm{r}^{2}}{M^{3}}h_{4}(z)\right)$
$\displaystyle+\bm{r}^{2}M\left(\frac{\left(z^{2}(M+z)-a^{2}(M+3z)\right)dt^{2}+4az\left(a^{2}+z^{2}\right)dtd\phi}{\left(a^{2}+z^{2}\right)^{2}\left(a^{2}+z(2M+z)\right)}+\frac{z^{3}-a^{2}(2M+3z)}{\left(a^{2}+z(2M+z)\right)^{3}}dz^{2}\right.$
$\displaystyle\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{\left(a^{4}(M+6z)+2a^{2}z^{2}(5M+2z)-z^{4}(3M+2z)\right)}{\left(a^{2}+z^{2}\right)\left(a^{2}+z(2M+z)\right)^{3}}d\bm{r}^{2}\right)+dzd\bm{r}\,\mathcal{O}(\bm{r}^{3})$
$\displaystyle+\bm{r}^{4}\frac{2aM(a-z)(a+z)\left(a^{2}(M+3z)+3z^{2}(M+z)\right)}{\left(a^{2}+z^{2}\right)^{3}\left(a^{2}+z(2M+z)\right)^{2}}dtd\phi+\mathcal{O}(\bm{r}^{4}),$

where $\beta$ is a constant and $h_{1}(z),h_{2}(z),h_{3}(z),h_{4}(z)$ are
perturbation functions to be determined. We focus on the case where purely
spatial deformations are homogeneous. As discussed above, each perturbation is
accompanied by a factor of $\frac{a}{M}$, making sure that for $a=0$ the
perturbations vanish along with the emerging radiation. We are seeking
$h_{i}(z)$ with $i=1,2,3,4$ solutions of the Einstein field equations at
leading order in $a/M\to 0$ as well as $\bm{r}/M\to 0$. We assume that at
leading order $\mathcal{O}(a^{1},\bm{r^{0}})$ the metric deformation can be
parametrized by a single function $h_{1}(z)$ similar to a scale factor;
however, for more generality we allow a relative constant factor $-(1+\beta)$
in the $dt^{2}$ metric component. Note that we have introduced a $dt\,dz$
perturbation into the metric at the sub-leading order
$\mathcal{O}(\bm{r}^{2})$ to account for non-zero energy momentum flow in the
$z$-direction without breaking the diagonal form of the leading order metric.

Radiation fluid typically possesses non-zero pressure. However, in our case we
are instead interested in radiation propagating at the speed of light in the
$z$-direction. This is equivalent to energy momentum of a superposition of
plane waves and takes the familiar form

$\displaystyle T^{\mu\nu}=\rho(z)U^{\mu}U^{\nu}\,,\text{ with four-velocity
}U^{\mu}=\left(\frac{1}{\sqrt{-g_{00}}},0,0,\frac{1}{\sqrt{g_{33}}}\right),$
(4.3)

where we allow the energy density $\rho(z)$ to vary statically with $z$ along
the geodesic trajectory. Here, the four-velocity $U^{\mu}$ is lightlike
$g_{\mu\nu}U^{\mu}U^{\nu}=0$ for $\bm{r}/M\to 0$ by construction.

At leading order in $\bm{r}/M\to 0$, the covariant derivative vanishing
condition (2.4) leads to a single non-trivial equation:

$\displaystyle\nabla_{\nu}T^{0\nu}\propto\nabla_{\nu}T^{3\nu}\propto\partial_{z}\left[\rho(z)\left(ah_{1}(z)+M\right)\left(M-a(1+\beta)h_{1}(z)\right)\right]=0.$
(4.4)

This directly integrates to555Note that $\rho$ has units of $[\rho]=L^{-2}$ as
appropriate in the geometrized units convention.

$\displaystyle\rho=\frac{\rho_{\gamma,0}}{\left(ah_{1}+M\right)\left(M-a(1+\beta)h_{1}\right)}=\frac{a}{M\left(ah_{1}+M\right)\left(M-a(1+\beta)h_{1}\right)},$
(4.5)

where the initial energy density (4.1) appears as an integration constant.
Making use of (4.5), the other two components of (2.4) vanish or are sub-
leading in $a\ll M$

$\displaystyle\nabla_{\nu}T^{1\nu}=0\,,~{}~{}~{}\nabla_{\nu}T^{2\nu}=\mathcal{O}(a^{2},\bm{r}^{0}).$
(4.6)

We utilize (4.2), (4.3) and (4.5) in the Einstein field equations (2.3). The
single non-trivial off-diagonal equation expanded to leading order
$\mathcal{O}(a^{1},\bm{r}^{0})$ fixes $h_{2}$ in terms of other quantities

$\displaystyle
R^{03}-\frac{R}{2}g^{03}=T^{03}~{}~{}~{}~{}~{}\Rightarrow~{}~{}~{}~{}~{}h_{2}=\frac{z}{4M+2z}.$
(4.7)

All other off-diagonal elements either vanish exactly, or are of sub-leading
order $\mathcal{O}(a^{2},\bm{r}^{0})$ and are therefore irrelevant for our
purposes.

The diagonal elements $R^{\mu\mu}-\frac{R}{2}g^{\mu\mu}=T^{\mu\mu}$ (no
summation over $\mu$) of the Einstein equations (2.3) at order
$\mathcal{O}(a^{1},\bm{r}^{0})$ (and an order $\mathcal{O}(a^{1},\bm{r}^{-2})$
term for $\mu=3$) then lead to the following three non-trivial linearly
independent constraints

$\displaystyle 0=$ $\displaystyle
M^{3}(3M+4z)h_{1}-z(2M+z)\left(M^{3}h_{1}^{\prime}+z^{2}+(2M+z)\left(M^{2}zh_{1}^{\prime\prime}+4zh_{4}\right)\right),$
(4.8) $\displaystyle 0=$ $\displaystyle
z(2M+z)\left((2M+z)\left[2z\left(h_{3}+h_{4}\right)-\beta
M^{2}zh_{1}^{\prime\prime}\right]+2(1+\beta)M^{3}h_{1}^{\prime}\right)$
$\displaystyle-2M^{3}h_{1}((1+\beta)(M+z)+z),$ (4.9) $\displaystyle 0=$
$\displaystyle
z(2M+z)\left(z\left(2\left(h_{3}+h_{4}\right)(2M+z)-z\right)-(1+\beta)M^{3}h_{1}^{\prime}\right)$
$\displaystyle-M^{3}h_{1}((2\beta+5)M+2(2+\beta)z).$ (4.10)

Note that no derivatives of $h_{3}$ and $h_{4}$ functions make an appearance,
which allows us to simply solve two out of the three equations algebraically:

$\displaystyle h_{3}=$
$\displaystyle\frac{z(2M+z)\left(z\left((2\beta+1)M^{2}(2M+z)h_{1}^{\prime\prime}+z\right)-(4\beta+3)M^{3}h_{1}^{\prime}\right)+M^{3}h_{1}((4\beta+1)M+4(1+\beta)z)}{4z^{2}(2M+z)^{2}}\,,$
$\displaystyle h_{4}=$
$\displaystyle\frac{M^{3}h_{1}(3M+4z)-z(2M+z)\left(M^{3}h_{1}^{\prime}+z\left(M^{2}(2M+z)h_{1}^{\prime\prime}+z\right)\right)}{4z^{2}(2M+z)^{2}}.$
(4.11)

The third equation then reduces to a differential equation in $h_{1}$ only:

$\displaystyle 0=z(2M+z)\left(z\left(z-\beta
M^{2}(2M+z)h_{1}^{\prime\prime}\right)+3(1+\beta)M^{3}h_{1}^{\prime}\right)+3M^{4}h_{1}.$
(4.12)

Despite its simple appearance, this equation is rather complicated and,
according to Mathematica, its general solution can be given in terms of
integrals over hypergeometric functions

$\displaystyle h_{1}=$ $\displaystyle z^{\frac{5\beta-
R_{1}+3}{4\beta}}(2M+z)^{\frac{R_{2}-\beta-3}{4\beta}}\left[z^{\frac{R_{1}}{2\beta}}\,_{2}F_{1}\left({\scriptstyle{\frac{R_{1}+R_{2}}{4\beta},\frac{R_{1}+R_{2}}{4\beta}+1}\atop{\frac{R_{1}}{2\beta}+1}};-\frac{z}{2M}\right)\times\right.$
(4.13) $\displaystyle\times$
$\displaystyle\left(c_{1}+\int_{a}^{z}dx\frac{8(7\beta(\beta+2)+3)x^{\frac{3\beta-
R_{1}-3}{4\beta}}(2M+x)^{\frac{3-3\beta-
R_{2}}{4\beta}}}{MH(x)}\,_{2}F_{1}\left({\scriptstyle{\frac{R_{2}-R_{1}}{4\beta},\frac{R_{2}-R_{1}}{4\beta}+1}\atop{1-\frac{R_{1}}{2\beta}}};-\frac{x}{2M}\right)\right)$
$\displaystyle+$
$\,{}_{2}F_{1}\left({\scriptstyle{\frac{R_{2}-R_{1}}{4\beta},\frac{R_{2}-R_{1}}{4\beta}+1}\atop{1-\frac{R_{1}}{2\beta}}};-\frac{z}{2M}\right)\times$
$\displaystyle\times$
$\displaystyle\left.\left(c_{2}-\int_{a}^{z}dx\frac{8(7\beta(\beta+2)+3)x^{\frac{3\beta+R_{1}-3}{4\beta}}(2M+x)^{\frac{3-3\beta-
R_{2}}{4\beta}}}{MH(x)}\,_{2}F_{1}\left({\scriptstyle{\frac{R_{1}+R_{2}}{4\beta},\frac{R_{1}+R_{2}}{4\beta}+1}\atop{\frac{R_{1}}{2\beta}+1}};-\frac{x}{2M}\right)\right)\right],$

where integration constants $c_{1},\,c_{2}$ can be used to alter the lower
boundaries of integration on each of the two integrals respectively.666By
default we have chosen lower boundaries of integration to be equal to the Kerr
spin parameter $a$ for $c_{1}=c_{2}=0$, such that the metric deformation is
smallest at the minimum of the effective scale factor (3.17). Abbreviations
$R_{1},\,R_{2}$ denote

$\displaystyle
R_{1}=\sqrt{\beta(25\beta+42)+9}~{}~{}~{},~{}~{}~{}R_{2}=\sqrt{\beta(\beta+18)+9}\,,$
(4.14)

and the function $H(x)$ is given by

$\displaystyle H$
$\displaystyle(x)=x\Big{(}\beta\left(\beta\left(8\beta-3R_{1}+7R_{2}+8\right)+2\left(7R_{2}-5R_{1}\right)\right)+3\left(R_{2}-R_{1}\right)\Big{)}\times$
(4.15) $\displaystyle\times$
$\,{}_{2}F_{1}\left({\scriptstyle{\frac{R_{2}-R_{1}}{4\beta}+1,\frac{R_{2}-R_{1}}{4\beta}+2}\atop{2-\frac{R_{1}}{2\beta}}};-\frac{x}{2M}\right)\,_{2}F_{1}\left({\scriptstyle{\frac{R_{1}+R_{2}}{4\beta},\frac{R_{1}+R_{2}}{4\beta}+1}\atop{\frac{R_{1}}{2\beta}+1}};-\frac{x}{2M}\right)$
$\displaystyle+$
$\,{}_{2}F_{1}\left({\scriptstyle{\frac{R_{2}-R_{1}}{4\beta},\frac{4\beta-
R_{1}+R_{2}}{4\beta}}\atop{1-\frac{R_{1}}{2\beta}}};-\frac{x}{2M}\right)\left[4(7\beta(\beta+2)+3)R_{1}M\,_{2}F_{1}\left({\scriptstyle{\frac{R_{1}+R_{2}}{4\beta},\frac{R_{1}+R_{2}}{4\beta}+1}\atop{\frac{R_{1}}{2\beta}+1}};-\frac{x}{2M}\right)\right.$
$\displaystyle-$
$\displaystyle\left.x\Big{(}\beta\left(\beta\left(8\beta+3R_{1}+7R_{2}+8\right)+2\left(5R_{1}+7R_{2}\right)\right)+3\left(R_{1}+R_{2}\right)\Big{)}\,_{2}F_{1}\left({\scriptstyle{\frac{R_{1}+R_{2}}{4\beta}+1,\frac{R_{1}+R_{2}}{4\beta}+2}\atop{\frac{R_{1}}{2\beta}+2}};-\frac{x}{2M}\right)\right].$

In order for the perturbations to make physical sense, the sub-leading terms
$h_{2},\,h_{3},\,h_{4}$ may not grow large for $z\to\infty$, while the leading
term $h_{1}$ is expected to grow with $z$. Making use of standard
hypergeometric expansions as well as the initial differential equation (4.12),
it is straightforward to determine that for $\beta\neq 0$:777The degenerate
case $\beta=0$ causes the $h_{1}$ solution to scale with a power of $z$ not
consistent with the expectation of a radiation dominated universe. Therefore,
we discard that case as unphysical.

$\displaystyle
h_{1}|_{z\to\infty}=\frac{1}{2M^{2}\beta}z^{2}+{{\scriptstyle\mathcal{O}}}(z^{2})\,,$
(4.16)

while equations (4.7) and (4) then reveal

$\displaystyle h_{2}|_{z\to\infty}$
$\displaystyle=\frac{1}{2}+{{\scriptstyle\mathcal{O}}}(z^{0})\,,$
$\displaystyle h_{3}|_{z\to\infty}$
$\displaystyle=\frac{1}{4}\left(3+\frac{1}{\beta}\right)+{{\scriptstyle\mathcal{O}}}(z^{0})\,,$
(4.17) $\displaystyle h_{4}|_{z\to\infty}$
$\displaystyle=-\frac{1+\beta}{4\beta}+{{\scriptstyle\mathcal{O}}}(z^{0})\,,$

where the small ${{\scriptstyle\mathcal{O}}}(x^{n})$ notation denotes terms
that grow strictly slower than $x^{n}$. For the leading order perturbation to
be positive at large $z$ (4.16), we must enforce $\beta>0$. The sub-leading
perturbations (4.17) indeed do not grow with $z$, as required.

Next, we evaluate the metric on the radial geodesic trajectory, transformed to
locally conformal time $t=f(\eta),\,dt=f^{\prime}(\eta)d\eta$. As in the
previous section, we focus on the leading order metric only to model the
observable universe

$\displaystyle-d\tau^{2}=$
$\displaystyle-\frac{a^{2}+z(2M+z)}{a^{2}+z^{2}}f^{\prime}(\eta)^{2}\left(1-\frac{a}{M}(1+\beta)\,h_{1}(z)\right)d\eta^{2}$
(4.18)
$\displaystyle+\frac{a^{2}+z^{2}}{a^{2}+z(2M+z)}\left(1+\frac{a}{M}h_{1}(z)\right)\left(d\bm{r}^{2}+\bm{r}^{2}d\phi^{2}+dz^{2}\right)+\mathcal{O}(\bm{r}^{2}).$

To find $z^{\prime}(\eta)$ in case of a massless particle, we use (4.18) with
$d\bm{r}=0,\,d\phi=0,\,d\tau=0$, choose the positive square root solution and
obtain888The relative minus sign in the square root looks like it may pose a
problem. However, note that we introduced $h_{1}$ as a perturbation that is
small thanks to the $a/M\to 0$ factor. In particular, we assume that $M$ is
large enough and $a$ small enough such that $h_{1}(z)\ll M/a$ for all $z$ of
interest.

$\displaystyle
z^{\prime}(\eta)=\frac{a^{2}+z(2M+z)}{a^{2}+z^{2}}f^{\prime}(\eta)\sqrt{\frac{1-\frac{a}{M}(1+\beta)\,h_{1}(z)}{1+\frac{a}{M}h_{1}(z)}}.$
(4.19)

Solving $-g_{00}=g_{33}$ for $f^{\prime}(\eta)$ when we evaluate the metric on
the trajectory $z=z(\eta)$ as in (3.14), we find

$\displaystyle
f^{\prime}(\eta)=\frac{a^{2}+z^{2}}{a^{2}+z(2M+z)}\sqrt{\frac{1+\frac{a}{M}h_{1}(z)}{1-\frac{a}{M}(1+\beta)\,h_{1}(z)}}.$
(4.20)

Plugging (4.20) into (4.19), the relation simplifies analogously to (3.15):

$\displaystyle z^{\prime}(\eta)=1\,,~{}~{}~{}\text{the simplest solution
being}~{}~{}~{}z(\eta)=\eta.$ (4.21)

Therefore, transformed to conformal time and evaluated on the trajectory of
interest, the perturbed leading order metric reads

$\displaystyle-d\tau^{2}=$
$\displaystyle\frac{a^{2}+\eta^{2}}{a^{2}+\eta(2M+\eta)}\left(1+\frac{a}{M}h_{1}(\eta)\right)\left(-d\eta^{2}+d\bm{r}^{2}+\bm{r}^{2}d\phi^{2}+dz^{2}\right)+\mathcal{O}(\bm{r}^{2}).$
(4.22)

This means, that due to the introduction of the gravitating flow of radiation
along the radial geodesic, the effective scale factor (3.17) is deformed to

$\displaystyle\bm{a}(\eta)=$
$\displaystyle\sqrt{\frac{a^{2}+\eta^{2}}{a^{2}+\eta(2M+\eta)}\left(1+\frac{a}{M}h_{1}(\eta)\right)}.$
(4.23)

From (4.16) we see that for late $\eta$ this implies

$\displaystyle\bm{a}(\eta)|_{\eta\gg M}\approx\left(\frac{a}{2\beta
M^{3}}\right)^{\frac{1}{2}}\eta.$ (4.24)

For a radiation dominated universe we expect $\bm{a}(t)\propto t^{1/2}$ and
the transformation to conformal time therefore requires
$t^{\prime}(\eta)^{2}\propto t(\eta)$, which is solved by
$t(\eta)\propto\eta^{2}$. The result (4.24) is thus precisely consistent with
our expectation for a radiation dominated universe. The free parameter $\beta$
can be used to fit the proportionality factor to observation. See Figure 2 for
a visualization of the effective scale factor.

Figure 2:  Comparison of effective scale factors (3.17) and (4.23), for
demonstration evaluated at $c_{1}=c_{2}=0,\,M=1,\,a=10^{-1},\,\beta=1$. Note
that, due to only a single order of magnitude difference in $M$ and $a$ for
visualization purposes, the radiation contribution appears dramatically
exaggerated.

##  5 Conclusion and Outlook

In this work we have explored the scenario that our observable universe has
been emitted from the ergosphere in a negative mass Kerr space-time, and is
traveling along a radial geodesic parallel to the axis of rotation of the ring
singularity. In appropriate cylindrical coordinates $\bm{r},z$, assuming large
enough mass parameter $M$ such that the observable universe is well described
by the leading terms in $\bm{r}/M$ at large $z$, we have shown that the metric
on the trajectory exhibits a particular conformal scale factor. The inherent
contraction and expansion properties of the scale factor solve the horizon
problem. Subsequently, we introduced a gravitating stream of radiation flowing
statically along the trajectory, to model a radiation dominated universe. With
an appropriate ansatz for metric perturbation, we then demonstrated that the
resulting asymptotic growth of the scale factor indeed matches the expected
result in a radiation dominated universe. As the features stemming from the
black hole in the past die off asymptotically at large conformal time, the
transition to a matter dominated universe could then be made utilizing the
conventional FLRW metric.

One objection to working with the negative mass Kerr space-time is the
existence of unbounded modes in linear time-dependent metric perturbations
[24, 25], which make the space-time unstable. However, just the presence of an
instability alone does not preclude a meta-stable state that may have a life-
time of the age of the universe. Additionally, the fact that we are
specifically working with only a patch of the space-time in the vicinity of
the trajectory along the axis of rotation means that, especially in the case
when radiation is present, a different global completion of the metric may
exist that does not exhibit the same instability.

The sub-leading terms in metric (3.6) vanish (or asymptote to a constant in
metric (4.2), negligible with respect to the growing leading terms) for large
$z$, which is achieved for late conformal time $\eta$ on the trajectory.
However, at early conformal time $\eta\approx 0$ the sub-leading terms can be
relevant away from the absolute center of the trajectory. Therefore, it would
be interesting to investigate if the sub-leading terms can imprint some
signature onto the streaming radiation at early times, which could be measured
at late times. Technically, the initial change of variables (3) is not
completely unique. Choosing a slightly different coordinate transformation can
change the early conformal time behavior of sub-leading terms and therefore
potentially imprint different signatures onto the radiation. Similarly, a re-
parametrization of the $z$-coordinate to any monotonously growing function
starting at zero and linearly asymptoting to infinity may possibly change the
quantitative details but not the qualitative picture.

Furthermore, the ansatz of metric perturbation (4.2) is physically motivated
with the target form of the resulting leading order metric in mind, but not
unique. It would be interesting to explore modifications of the ansatz, to
potentially determine an exhaustive equivalence class of qualitatively similar
perturbative solutions.

In the conventional inflationary paradigm, quantum fluctuations of the
inflaton field and the metric tensor blow up to macroscopic size and seed the
inhomogeneities that lead to structure formation such as galaxies. It would be
interesting to investigate if introducing quantum fluctuations in the flowing
radiation as well as the metric tensor in the scenario discussed in this paper
can similarly lead to an appropriate amount of inhomogeneity to seed structure
formation.

## Acknowledgments

This work was supported by DOE grant DE-SC0011941.

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Valentina Pansanella, Virginia Morini, Tiziano Squartini and Giulio Rossetti
11institutetext: Scuola Normale Superiore, Pisa, Italy
11email<EMAIL_ADDRESS>22institutetext: Computer Science
department, University of Pisa, Italy
22email<EMAIL_ADDRESS>33institutetext: IMT School for Advanced
Studies, Lucca, Italy
33email<EMAIL_ADDRESS>44institutetext: KDD Laboratory, ISTI,
National Research Council, Pisa, Italy
44email<EMAIL_ADDRESS>
# Change my Mind: Data Driven Estimate of Open-Mindedness from Political
Discussions

Valentina Pansanella 11 Virginia Morini 22 Tiziano Squartini 33 Giulio
Rossetti 44

###### Abstract

One of the main dimensions characterizing the unfolding of opinion formation
processes in social debates is the degree of open-mindedness of the involved
population. Opinion dynamic modeling studies have tried to capture such a
peculiar expression of individuals’ personalities and relate it to emerging
phenomena like polarization, radicalization, and ideology fragmentation.
However, one of their major limitations lies in the strong assumptions they
make on the initial distribution of such characteristics, often fixed so as to
satisfy a normality hypothesis.

Here we propose a data-driven methodology to estimate users’ open-mindedness
from online discussion data. Our analysis - focused on the political
discussion taking place on Reddit during the first two years of the Trump
presidency - unveils the existence of statistically diverse distributions of
open-mindedness in annotated sub-populations (i.e., Republicans, Democrats,
and Moderates/Neutrals). Moreover, such distributions appear to be stable
across time and generated by individual users’ behaviors that remain
consistent and underdispersed.

###### keywords:

Online Debates, Open-Mindedness, Political Polarization, Opinion Dynamics

##  1 Introduction

One of the most debated and analyzed phenomena on online social networks is
the tendency to observe political polarization [5, 19, 1, 20], i.e., the
divergence of political attitudes to ideological extremes not aiming at
reaching any form of synthesis. Indeed, social media blurred the boundaries of
communication and democratized content diffusion. However, the fact that we
can potentially engage with a lot of different information does not mean that
we can engage with and be influenced by opposing or even mildly different
stances [25].

Furthermore, humans are far from being perfectly rational individuals and are
affected by a series of cognitive biases111Cognitive biases cheat sheet:
shorturl.at/afjpM, last visited June 2022. in the process of forming their
belief system. Among these, it is worth mentioning confirmation bias [24, 14],
the human tendency to ignore content that counters their prior beliefs by a)
choosing to interact with like-minded individuals [17] b) ignoring
interactions with the “opposite faction” [8]. Unfortunately, by connecting
with like-minded individuals and ignoring contrasting views, users can easily
compromise their open-mindedness [35], exacerbating and polarizing their views
even more, and, in the end, giving rise to the formation and persistence of
echo chambers [32].

One of the most exploited approaches for understanding the effects of
different kinds of biases on public opinions - especially political opinions -
is through mathematical models of opinion formation [29], where parameters
incorporate psychological factors (e.g., cognitive biases) affecting
individual opinion evolution. Despite having some advantages, opinion dynamics
models lack empirical validation [26]. However, thanks to the advent of the
Internet - and with the rise of social media - an increasing part of human
interactions leave a massive digital footprint that can be exploited to study
the dynamics of opinion formation and diffusion. Following such reasoning,
since different models or different parameter values can predict different,
even opposite, effects of biases on opinions [16], there is a crucial need for
empirical works to study and quantify socio-psychological and external drivers
of the dynamics.

In this work, we move in this direction by studying how being politically
polarized affects users’ open-mindedness over time. In detail, we focus on a
twenty-months discussion on r/politics (i.e., the largest political subreddit
on Reddit), computing the probability of changing one’s political leaning over
time and estimating the distribution of open-mindedness within the user
population and its temporal evolution. Firstly, this analysis aims to
understand how open-mindedness is distributed within the population and over
time. A second aim is to understand if there is a substantial difference in
the level of open-mindedness in relation to the political leaning, i.e., if
more extreme individuals are more close-minded than moderate ones or vice
versa. Our results suggest that the different subpopulations’ open-mindedness
distributions are stable in time and statistically different:
Moderates/Neutrals and Republicans showing more nuanced close-mindedness
patterns w.r.t. Democrats. Finally, our analysis also underlines how Reddit
users tend to be consistent in their open-mindedness attitude across time,
showing, on average, low degrees of variance and dispersion.

The rest of the paper is organized as follows. In Section 2, we discuss the
literature involving open-mindedness modeling both from an opinion dynamics
and data-driven point of view. Section 3 discusses how we leverage Reddit data
in order to model opinion evolution over time. Then, in Section 4, we propose
an algorithm to estimate users’ open-mindedness and we longitudinally study
its evolution in three well-characterized sub-populations. Finally, Section 5
concludes the paper with a brief summary of the main results and discusses
limitations and directions for future work.

##  2 Related work

In the process of opinion formation cognitive biases play a role because
people usually feel discomfort when encountering divergent opinions, thus
tending to select information that confirm their pre-existing beliefs [24].
One of the first attempts at understanding the effects of such kind of
cognitive bias is the model by Deffuant, and Weisbuch [8]. In this case, the
bias is introduced with a single parameter, namely “confidence bound”, which
indicates that agents only influence each other if their opinion distance is
below a certain threshold. The main result of this model is that if the
population is “close-minded”, polarization arises. So it appears that
cognitive biases may be enough to justify opinion polarization, even without
considering additional factors, such as the network structure or the
information environment. However, the effects of open-mindedness alone and in
interplay with other factors have been extensively studied through opinion
dynamics models, extending the baseline [8] with additional agent-level
characteristics [15], considering multi-dimensional opinions [7] or adding
repulsive behaviors [4]. Despite conclusions from these models appear to be
realistic and seem to explain why polarization arises and resists in many
social networks, there is a need for empirical works unveiling the level of
open-mindedness within polarizing topics. Indeed, to the best of our
knowledge, there are still no works leveraging the vast amount of big data
traces left from online social networks to tackle such a task. With respect to
modelling works, simulation results have been compared to election data [34],
survey data [30], social experiments data [31], or social media data [36] in
order to manually tune parameters or learn them with automated approaches [18,
28, 13, 33]. Different data-driven works [11, 12, 10, 23, 27, 1] have shown
how the polarization of opinions around political issues led the people to
enter in closed polarized systems insulated from rebuttal, i.e., echo
chambers. Interestingly, authors in [9] found that echo chambers do not arise
around topics not politically polarized (e.g., TV shows, food), thus
suggesting that the echo chamber phenomenon and users’ open-mindedness might
be strictly related to the discussed topic.

##  3 Analyzing the Political Debate Online

Data Collection. For the purpose of this work, we built a dataset of political
discussions on Reddit. Reddit is a popular social platform that allows users
to post content to individual forums called subreddits, each dedicated to a
specific topic. Such a categorized structure makes it easy to find users
involved in specific debates. To assess the open-mindedness of users over
time, we decide to select a quite controversial domain, i.e., Politics. Among
the thousands of subreddits talking about politics we choose
r/politics222https://www.reddit.com/r/politics/ “r/politics is for news and
discussion about US politics.”, since it is the largest political subreddit
and further, it is not aligned with a specific ideology but rather is visited
regularly by users having different political beliefs. Notice that, as
highlighted in the subreddit description11footnotemark: 1, r/politics mainly
refers to political discussion in the US. Thanks to the Pushshift API [3], we
collected all posts and comments shared on the subreddit from May 2018 to
December 2019, i.e., about one year and a half of Donald Trump’s presidency -
covering all discussions of 1,089795 users. As shown in figure 1a, the numbers
of users who participated in the debate tend to increase over time, and
further, more than 60% of users are stable across contiguous months, meaning
that they continue posting or commenting at least for two consecutive months.
The code and the data used for this work are available in a dedicated GitHub
repository333https://github.com/ValentinaPansanella/OpenMindednessReddit.git.

Ideology Estimate. To assess if the open-mindedness of users evolves over
time, we have to establish the ideology of users in different time periods.
Since we are dealing with users debating political issues in the US, we try to
categorize them with respect to the US two-party system: Republican and
Democratic.
For such a purpose, we model the task of predicting users’ political alignment
as a text classification problem. In other words, we leverage users’ posts to
measure their degree of alignment with Republican and Democrat ideologies. To
accomplish this task, we leverage an LSTM model that we trained on Reddit US
political texts in our previous works [22, 21]. In detail, to train such a
model we defined a ground truth composed of Reddit posts belonging to
subreddits known to be either Pro-Trump or Anti-Trump (i.e., r/The_Donald for
the first group and r/Fuckthealtright and r/EnoughTrumpSpam for the second).
Accordingly, we modeled the text classification task as a binary problem.
During model selection, we perform a 3-fold Cross-Validation trying different
hyper-parameters configurations and obtaining the best performances on the
validation set (i.e., average accuracy of $82.9\%$) using GloVe word
embeddings and $128$ LSTM units444For further details on the model selection
and evaluation steps, the reader should refer to our previous works [22, 21].

(a)

(b)

Figure 1: a) Top: For each month, the number of users who participated in the
debate. Bottom: For each month, the percentage of users that are stable across
contiguous months. b) Authors’ leaning distribution in the whole time period.

Consequently, we apply the model to the r/politics dataset in order to infer
posts leaning for all the population. Notice that we apply it separately for
each month in order to assess users’ ideology stability over time. For each
textual content, we obtain model predictions ranging from $0$ to $1$ (i.e.,
the model confidence), where $1$ means that the post aligns itself with
Republican (specifically Pro-Trump) ideologies while $0$ with Democrats
(specifically Anti-Trump) ones. Then, for each user $u$ who participated in
the debate in each month $m$ we compute his leaning score, $L_{u,m}$, as the
average value of his monthly posts leaning as follows:

$L_{u,m}=\frac{\sum_{p\in P_{u,m}}\text{Prediction\\_Score}(p)}{|P_{u,m}|}$
(1)

where $P_{u,m}$ is the set of posts shared by a user $u$ in each month $m$.
Since we are interested in assessing if users with polarized opinions tend to
move to more moderate positions, we discretized such leanings into three
intervals: Democrat if $L_{u,m}\leq$ 0.4; Republican if $L_{u,m}\geq$ 0.6;
while Neutral if $0.4<L_{u,m}<0.6$. By adding a third label, we make sure to
capture users with highly polarized ideologies. Figure 1b shows the authors’
leaning score distribution for the whole time period obtained by averaging
users’ monthly scores. Such a distribution confirms what observed in a recent
work [6] that focus the analysis on the r/politics subreddit too: Republican
users (530,909) outnumber Democrat ones (185,256) and Neutrals users (373,630)
show a tendency towards republicans beliefs.

Network Definition. Given the users for which we inferred their ideology, we
define their interaction networks for each month to take into account the
evolution of leanings in time. The resulting networks have Reddit users as
node sets, $V$, and as edges the set of pair $(u,v)\in V$ for which a reply of
$u$ to a $v$’s post or a comment exists. We set each edge weight to represent
the total number of comments exchanged between two users. Also, we label users
(i.e., nodes) with their discretized leaning score $L_{u,m}$ (i.e.,
Republican, Neutral and Democrat). In Table 1, we provide the main averaged
network statistics across the 20 considered months.

$N$ | $N_{R}$ | $N_{D}$ | $N_{N}$ | $E$ | $\left\langle k\right\rangle$ | $r$
---|---|---|---|---|---|---
183296 | 88285 | 36385 | 58626 | 1271080 | 6.97 | 0.020
Table 1: Network statistics averaged across the 20 considered months: number
of users $N$, divided in Republican $N_{R}$, Democrat $N_{D}$ and Neutral
$N_{N}$, number of edges $E$, network average degree $\left\langle
k\right\rangle$, and network assortativity $r$ with respect to the political
leaning. Figure 2: For each ideology, users’ transition probabilities over
contiguous months.

##  4 Political Leaning: Stability and Open-Mindedness

Ideology Stability over Time. As a preliminary analysis, we try to understand
how users’ ideologies evolve. In detail, we are interested in exploring if
users are stable and consistent with the same ideology or instead tend to
change opinions according to specific events. We model such an issue in terms
of transition probabilities for such a purpose. In other words, for each user,
we compute his probability $p_{ij}$ to move from _state_ $i$ to _state_ $j$
over contiguous months. Notice that, in this scenario, _state_ stands for the
user’s ideology (i.e., Republican, Neutral or Democrat).
In Figure 2 we show, for each ideology, the probability of users to change or
remain in their states (i.e., opinions) over contiguous months. At a first
glance, we can observe a similar behavior for all the political ideologies:
users tend to be rooted in their position over months, with a probability to
remain in their state ranging around 0.60 for Democrat and Neutral leanings
and around 0.70 for Republican. Moreover, Republican users - when changing
their state - foster a more neutral position instead of moving to Democrat
beliefs (differently from Democrats). These attitudes reflect the extreme
polarization that characterizes the US political debate in the Trump Era [2].
However, for all the considered ideologies, we also notice an evident
fluctuation of opinions in favor of Democrats and Neutrals around November
2018, precisely when (i.e., 6 November) the Democratic Party won control of
the US House from the Republican Party. Indeed, this was a decisive moment in
which Democrats won the seats needed to take the House after capturing
districts where President Trump was unpopular.

if $u\in V_{t+1}$ then

Procedure to estimate $\widehat{{x}_{u}}(t+1)$ and $\widehat{CB}_{u}$

15: $N_{u,t}=\\{v|(u,v)\in E_{t}\\}$; $|N_{u,t}|=n$

$X_{N_{u,t}}[1...n]=\text{Array of opinions of nodes }v\in N_{u,t}$

if $N_{u,t}\neq\emptyset$ then

18: $\text{\text{Sort }}X_{N_{u,t}}[1...n]\text{ {by} }d_{u,v}\text{ \text{in
ascending order.}}$

$\widehat{{X}_{u}}(t+1)[0...n]\text{ array of estimated opinions}$

$\widehat{{X}_{u}}(t+1)[0]=x_{u}(t)$

21: $E=[0...n]\text{ array of estimation errors}$

$E[0]=1.0$

for i=1; i=n; i++ do

24: $x_{v}=X_{N_{u,t}}[i]$

$\widehat{{X}_{u}}(t+1)[i]=\frac{\widehat{X}_{u}(t+1)[i-1]+x_{v}}{2}$

$E[i]=|\widehat{{X}_{u}}(t+1)[i]-x_{u}(t+1)|$

27: $min_{e}=E[n]$

for i=n; i=0; i– do

$e=E[i]$

30: if $e\leq min_{e}$ then

$min_{e}=e$

$j=i$

33: end if

end for

end for

36: $\widehat{x_{u}}(t+1)=\hat{X}_{u}(t+1)[j]$

$\widehat{CB}=|x_{u}(t)-X_{N_{v,t}}[j]|$

end if

39:end if

Algorithm 1 Confidence bound estimation algorithm. $G_{t}=$ Weighted
undirected interaction network at time t; $V_{t}=$ set of nodes at time t;
9:$E_{t}=$ set of weighted edges at time t; $x_{u}(t)$ = opinion of agent $u$
at time $t$; $d_{u,v}$ = $|x_{u}(t)-x_{v}(t)|$ = opinion distance between
$u,v\in V$ at time t; 12:$\widehat{CB}$ = estimated confidence bounds.

Estimating Open-mindedness. In order to understand the levels of open-
mindedness involved in the process of changing one’s political leaning, we
started by assuming a simple process of opinion evolution at the individual
level, based on a very well known model of opinion formation [8]. In opinion
dynamics models, agents update their opinions after interacting with their
neighbors according to simple mathematical rules. For example, in [8] agents
average their opinion with the opinion of their interacting peer, which is
randomly chosen from the pool of their neighbors, if and only if their opinion
distance is below a certain threshold representing the open-mindedness of the
population. The hypothesis that open-mindedness is a characteristic trait of
an entire population and not a characteristic that varies from individual to
individual is strong and probably unrealistic. For this reason, in the present
work, we assume a Deffuant-like process of opinion update (i.e., users
averaging their opinions with the opinions of their interacting partners in a
pairwise fashion) and provide a data-driven time-aware estimate of individual-
level open-mindedness. To estimate users’ tendency to be influenced by their
neighbors, we developed a simple approach (see Alg. 12 for implementation
details) that takes as input the weighted interaction network at time $t$ and
the opinions of the agents at time $t$ and time $t+1$. In the estimation
procedure, we select each node $u$ for which we have both opinions $x_{u}(t)$
and $x_{u}(t+1)$ (Alg. 12 line 1) so that we can estimate how much the
interactions that happened in that time step influenced the opinion change and
therefore obtaining an estimate for the level of bounded confidence. After
selecting $u$, we order all the interacting partners (the neighbours of the
node in the snapshot network) from the closer to the further by the opinion
distance absolute value $d_{u,v}(t)=|x_{u}(t)-x_{i}(t)|$ (Alg. 12 line 6).
Then we compute an estimate $\widehat{x_{u}}(t+1)$ by iteratively averaging
the new estimate with the interacting neighbours (Alg. 12 line 13). The final
estimated value $\widehat{x_{u}}(t+1)$ is the one that minimizes the error
with respect to the real value $x_{u}(t+1)$ (Alg. 12 lines 15-22). Finally, we
compute the confidence bound as the distance in absolute value with the
neighbor that represents the point of minimum in the estimation errors
sequence (Alg. 12 line 25). With the proposed approach, we can compute an
estimate only for the subset of users present in two consecutive observations
(i.e., months) and have at least one link (i.e., interaction) in the snapshot
graph.

Figure 3: Estimated open-mindedness $\widehat{CB}$ distribution from May-2018
to December-2019. Distributions of the estimated confidence bound
$\widehat{CB}$ over the whole time period. All distributions are positively
skewed and constant over time.

Figure 3 underlines that the distribution of the estimated open-mindedness
$\widehat{CB}$ is stable during the considered time period in the analyzed
discussion. Moreover, it also highlights that the majority of the analyzed
users is close-minded, i.e., their confidence bound is $\widehat{CB}\leq 0.2$,
which is considered to be a sufficient condition for the population to become
polarized in the long term, according to [8]. This means that most of the
users participating in this discussion can hardly be influenced by neighbors
holding distant opinions, even if they interact with these users during the
considered time period, like in the case of this discussion, where the network
has a low assortativity with respect to the political leaning (see network
assortativity in Table 1). However, we can also see that the distributions at
each time step have a very high variance, allowing the presence of individuals
having a level of open-mindedness close to $1.0$ \- indicating that some of
the users can also be influenced by people holding very different opinions and
changing their expressed political leaning accordingly. The distribution of
the estimated confidence bound for the opinions is less skewed between October
and November 2018, i.e., around the US House of Representatives elections.
Such behavior confirms the data-driven analysis based on transition
probabilities performed in Section 3: in this time window, Republicans, which
normally have a highly skewed distribution, seemed to be somehow more open-
minded, and their average confidence bound is higher.

Figure 4: Estimated open-mindedness distribution in the period September -
December 2018. Distributions of the estimated confidence bound $\widehat{CB}$
for the different political leanings: Democrat (blue), Neutral (green), and
Republican (red) from September to December 2018.

In figure 4 we reported the estimated distribution for Democrat, Neutral, and
Republican users (we took the orientation at time t and estimated the
confidence bound $\widehat{CB}$ between time t and t+1). Only three months are
present in this figure, i.e., the three months around the US House election,
but conclusions still hold for the other months considered in this work. We
can see from figure 4 that there are differences in the distribution of open-
mindedness when we consider political orientation. Both neutral and republican
distributions are positively skewed, and the distribution has a long right
tail. Also, the Democrats distribution has a right tail, but much less than
the others. From our analysis, it appears that Neutral individuals are also
the most close-minded, while Democrats have a wider range of confidence bound
levels. Their distribution is not as skewed as the others, and many users have
a very high level of bounded confidence, i.e., they change their opinion
significantly over contiguous time periods under the influence of their
neighbors. Republicans, like Neutrals, have a positively skewed confidence
bound distribution, even with a higher average confidence bound. We performed
a 2-sample KS-test comparing the distributions of the estimated confidence
bound for each political leaning (e.g., Democrat vs. Republican
$\widehat{CB}$) within each time step obtaining a p-value $\approx 0.0$,
supporting the conclusions that distributions are different for the three
political leanings. Finally, while we can say that population-level open-
mindedness is reasonably constant over time, i.e., we have the same mix of
open-minded and close-minded individuals participating in the discussion, we
do not have information about how variable open-mindedness may be at the
individual level. In this analysis, each user may have a different value of
open-mindedness at each time step, making not only the overall distribution
heterogeneous but also the distribution at the individual-level. To understand
how much individual bounded confidence may vary, we computed the standard
deviation of our estimate for each observed user - figure 5a. Reddit users’
open-mindedness tendencies appear stable in time, showing a characteristic low
standard deviation, $\sigma$. Such a result is also confirmed by the
distribution of the Fano dispersion index (figure 5b) - i.e., the ratio
between variance $\sigma^{2}$ and mean value, $\mu$, of the estimated
individual open-mindedness scores. The observed Fano values, prevalently
distributed in the range $0<\frac{\sigma^{2}}{\mu}<1$, identify under-
dispersed behaviors, thus expressing consistent patterns of stability.

(a)  (b)

Figure 5: Users’ open-mindedness stability analysis. (a) Distribution of
individuals’ open-mindedness standard deviations; (b) Distribution of
individuals’ open-mindedness dispersion indexes (variance over mean value).

##  5 Conclusions

In this work, we introduced a methodology to estimate individuals’ open-
mindedness from online discussion data. When applied to study the political
debate that took place on Reddit during the first two years of the Trump
presidency, our proxy unveiled the existence of characteristic distributions
for well-defined sub-populations: Moderates/Neutrals and Republicans being
more close-minded on average than Democrats. The proposed longitudinal
analysis also unveiled that the observed Reddit users tend to maintain a
stable behavior for what concerns their open-mindedness, exhibiting low
variance and underdispersion.

Indeed the current study, like all empirical ones, suffers from limitations.
In particular, it leverages a data-driven estimate of the political leaning
that can be subject to errors and cannot be fully validated on ground truth
external data. Moreover, although Reddit users tend to be particularly
involved in political discussions, the population variability in time and the
sparsity of observation data do not allow to estimate the open-mindedness of
less active individuals. For each month, we were not able to estimate the
open-mindedness of $40-50\%$ of nodes since we have no information on the
opinion at time $t+1$ or the user does not have neighbors on the network at
time $t$.

As future works, we plan to enhance the proposed estimation procedure,
allowing for asymmetric open-mindedness. Additionally, we will investigate the
interplay of open-mindedness and known polarization phenomena (e.g., the
presence of echo chambers) in order to better characterize the role of
different individuals in their emergence.

### Acknowledgments

This work was supported by the scheme ’INFRAIA-01-2018-2019: Research and
Innovation action’, Grant Agreement n. 871042 ’SoBigData++: European
Integrated Infrastructure for Social Mining and Big Data Analytics, by the
CHIST-ERA grant CHIST-ERA-19-XAI-010, by MUR (grant No. not yet available),
FWF (grant No. I 5205), EPSRC (grant No. EP/V055712/1), NCN (grant No.
2020/02/Y/ST6/00064), ETAg (grant No. SLTAT21096), BNSF (grant No. o
КП-06-ДОО2/5 )

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