# On dark stars, Planck cores
and the nature of dark matter

Igor Nikitin
Fraunhofer Institute for Algorithms and Scientific Computing
Schloss Birlinghoven, 53757 Sankt Augustin, Germany

<EMAIL_ADDRESS>
###### Abstract

Dark stars are compact massive objects, described by Einstein gravitational
field equations with matter. The type we consider possesses no event horizon,
instead, there is a deep gravitational well with a very strong redshift
factor. Observationally, dark stars can be identified with black holes. Inside
dark stars, Planck density of matter is reached, Planck cores are formed,
where the equations are modified by quantum gravity. In the paper, several
models of dark stars with Planck cores are considered, resulting in the
following hypothesis on the composition of dark matter. The galaxies are
flooded with low-energetic radiation from the dark stars. The particle type
can be photons and gravitons from the Standard Model, can also be a new type
of massless particles. The model estimations show that the extremely large
redshift factor $z\sim 10^{49}$ and the emission wavelength $\lambda_{0}\sim
10^{14}$m can be reached. The particles are not registered directly in the
existing dark matter experiments. They come in a density sufficient to explain
the observable rotation curves. The emission has a geometric dependence of
density on radius $\rho\sim r^{-2}$, producing flat rotation curves. The
distribution of sources also describes the deviations from the flat shape. The
model provides a good fit of experimental rotation curves. Outbreaks caused by
a fall of an external object on a dark star lead to emission wavelength
shifted towards smaller values. The model estimations give the outbreak
wavelength $\lambda\sim 1$m compatible with fast radio bursts. The paper
raises several principal questions. White holes with Planck core appear to be
stable. Galactic rotation curves in the considered setup do not depend on the
matter type. Inside the galaxy, dark matter can be of hot radial type. At
cosmological distances, it can behave like the cold uniform type.

Keywords: Planck stars, RDM-stars, TOV-stars, dark matter

Figure 1: On the left and in the center: an RDM-star – a black hole, coupled
to radial flows of dark matter. On the right: experimental rotation curves for
three galaxies. Image from [3], data from [4].

##  1 Introduction

Dark stars, also known as quasi black holes, boson stars, gravastars,
fuzzballs, are solutions of general theory of relativity, which first follow
the Schwarzschild profile and then are modified. Outside they are similar to
black holes, inside they are constructed differently, depending on the model
of matter used. An overview of these models can be found in the paper by
Visser et al. “Small, dark and heavy: but is this a black hole?” [1]. A recent
advance has been reported by Holdom and Ren in their paper “Not quite a black
hole” [2]. Our contribution to this family are RDM-stars [3], quasi black
holes coupled to Radial Dark Matter. A typical configuration of an RDM-star is
shown on Fig.1 on the left and in the center. It is a stationary solution,
including T-symmetric superposition of ingoing and outgoing radially directed
flows of dark matter.

An RDM-star can be used as the simplest model of a spiral galaxy. In the limit
of weak gravitational fields, the dark matter flows radially converging
towards the center of the galaxy produce a typical geometric dependence of
mass density on the radius $\rho\sim r^{-2}$, which corresponds to constant
orbital velocity $v=Const$, flat rotation curve. It is a qualitatively correct
behavior for many experimental rotation curves at large distances, see Fig.1
on the right. We will show that a distribution of RDM-stars in the galaxy also
allows to describe correctly the deviations of rotation curves from the flat
shape. The model of RDM-stars fits very well the experimental rotation curves
by Sofue et al. [4, 5, 6, 7, 8] and Salucci et al. [9, 10, 11, 12, 13].

In strong gravitational fields, RDM-stars behave interestingly. First of all,
the event horizon, typical for real black holes, is erased. Instead, a deep
gravitational well is formed, where the values of the redshift become
enormously large. As a result, for an external observer the star looks black,
like a real black hole. Simultaneously, the mass density increases rapidly,
reaching and exceeding the Planck value.

This is where Planck stars come into play. This model is based on the
calculations in quantum loop gravity, performed for a scalar field cosmology
by Ashtekar et al. [14, 15, 16]. According to these calculations, the mass
density has a quantum correction: $\rho_{X}=\rho(1-\rho/\rho_{c})$, where the
critical density $\rho_{c}\sim\rho_{P}$ is of the order of Planck value,
$\rho$ is the nominal density before the correction and $\rho_{X}$ is the
effective density participating in Einstein field equations. As a result of
this correction, $\rho=\rho_{c}$ corresponds to $\rho_{X}=0$, at the critical
density the gravity is effectively switched off, while $\rho>\rho_{c}$
corresponds to $\rho_{X}<0$, in excess of critical density the effective
negative mass appears (exotic matter), with gravitational repulsion (quantum
bounce phenomenon). In the Planck star model by Rovelli, Vidotto [17], Barceló
et al. [18], a collapse of a star leads to the quantum bounce, is replaced by
extension, as a result, the black hole turns white.

In this paper we consider a stationary version of a Planck star, stabilized
under the pressure of the external matter (Planck core). We will consider two
stationary spherically symmetric models with a Planck core in the center. The
subject is related to the stability of white holes, earlier investigated in
papers by Ori and Poisson [19], Eardley [20], Zel’dovich, Novikov and
Starobinskij [21]. It is also related to the origin of fast radio bursts and
gives an unusual viewpoint on the nature of dark matter.

The paper is organized as follows. In Section 2 the model of RDM-stars is
considered. The main computations have been performed in the author’s paper
[22], here a short overview of the results is given. In Section 3 the model of
TOV-stars with Planck core is presented. In Section 4 the nature of dark
matter according to the considered models is discussed. A theoretically
interesting question on stability of white holes is considered in the
Appendix.

##  2 RDM-stars with Planck core

### RDM-stars and rotation curves of galaxies.

RDM-star geometry can be used as a simplest model of dark matter distribution
in spiral galaxies. Let us consider dark matter flows radially converging
towards the center of a galaxy, displayed on Fig.1 center, in the limit of
weak gravitational fields. The one-line calculation

$\displaystyle\rho_{dm}\sim r^{-2},\ M_{dm}\sim r,\ v^{2}=GM_{dm}/r=Const$ (1)

evaluates mass density, enclosed mass function and orbital velocity of stars.
Adding a concentrated mass in the center, $v^{2}=GM_{0}/r+Const$, the rotation
curve described by the sum of Keplerian and constant terms can be obtained.
The real rotation curves, displayed on Fig.1 right, possess a similar
structure, with Keplerian behavior at small distances and flat shape at large
distances. The red line shows a segment 2-20kpc where the rotation curve for
the Milky Way can be considered as approximately flat, with the Sun position
at 8kpc. These plots show that the real rotation curves deviate from a simple
sum of Keplerian and constant terms, revealing additional structures,
oscillations. On the other hand, the model with a single RDM-star in the
center of the galaxy is also a simplification. Further we consider a model of
distributed RDM-stars, able to capture the additional structures. Then we
perform a calculation in the limit of strong gravitational fields to analyze
the interior structure of an RDM-star.

The detailed description of rotation curves in the RDM-model is based on two
assumptions: (1) all black holes are RDM-stars; (2) their density is
proportional to the concentration of the luminous matter in the galaxy. As a
result, the dark matter mass density can be represented by the integral

$\displaystyle\rho_{dm}(x)=\int
d^{3}x^{\prime}\,b(|x-x^{\prime}|)\,\rho_{lm}(x^{\prime}),\ b(r)=1/(4\pi
L_{KT})/r^{2}.$ (2)

Here $\rho_{lm}$ is the density of luminous matter, the kernel $b(r)$
represents a contribution of a single RDM-star and $L_{KT}$ is a parameter of
length dimension, regulating a coupling between the dark and the luminous
matter. This form of coupling has been proposed earlier in a context of a
different model in works by Kirillov and Turaev [23, 24]. The physical meaning
of the $L_{KT}$ parameter is the radius at which the enclosed mass of dark
matter equals to the mass of the luminous matter, to which it is coupled:
$M_{dm}(L_{KT})=M_{lm}$.

### The detailed rotation curve of Milky Way,

known also as Grand Rotation Curve (GRC), has been constructed on the basis of
various experimental data by Sofue et al. [4, 5, 6, 7, 8]. This curve is
presented on Fig.2 by data points with errors. Here one can see several
structures, including Keplerian contribution of the central black hole (BH),
inner and outer bulges (LM1,2), galactic disk (LM3), followed by dark matter
contribution (DM) and background outer part (bgr). The red line with the
marked Sun position represents the same 2-20kpc approximately flat interval as
on the previous figure. This part appears to be relatively small due to a much
larger range of distances involved in the analysis.

In paper [8], the distribution of luminous matter in the bulges is described
by exponential spheroid model, representing the mass density by an exponent
$\rho_{lm}\sim\exp(-r/a)$. For the galactic disk Freeman’s model [25] is used,
with the surface mass density described by similar exponent
$\rho_{lm}\sim\delta(z)\exp(-r/R_{D})$. Taking these distributions, the
integral (2) and the resulting rotation curve $v(r)$ can be evaluated
analytically. The lengthy explicit expressions per every structure are given
in [22], also used there as basis functions for the fitting procedure. For
stability of the fit, the relative coupling of dark matter to different
structures of luminous matter has been fixed as shown in Table 1. The
$\lambda$-constants are used as multiplicative factors to integrals (2). Since
the different galactic structures may possess a different density and
different population of black holes, we can select different coupling
constants for them. This procedure is equivalent to a readjustment of the
corresponding $L_{KT}$-parameters, while we prefer to use a single
$L_{KT}$-parameter and adjust the individual couplings by relative
$\lambda$-factors. Three scenarios have been considered in Table 1, the first
one assigns all dark matter coupling to the galactic disk, the second one
introduces equal coupling among all structures, the third one describes a
prevailing coupling for the central structures.

The result of the fit is shown by curves on Fig.2. The green line represents
the total rotation curve, a quadratic sum over all structures. It has almost
the same shape for all three scenarios. Also, the separated contributions of
different structures are shown. They depend on the scenario, e.g., the third
scenario with prevailing dark matter coupling to central structures also shows
a considerable contribution of dark matter in the center. Table 2 presents the
obtained fitting parameters – the total masses and geometric sizes of the
structures. In the considered modeling, the dark matter halo is sharply cut at
the radius $R_{cut}$, further providing Keplerian fall of the outer part of
the rotation curve, followed by its linear increase due to the uniform
background density. Interestingly, the parameters, characterizing the outer
part of the rotation curve, the total mass of dark matter halo
$M_{dm}(R_{cut})$ and the background density $\rho_{bgr}$, appear to be
approximately the same for the considered three scenarios.

Figure 2: Detailed rotation curve for Milky Way, fitted by RDM-model. Blue points with error bars – data from [8]. Green curve – fit by RDM-model from [22] for three coupling scenarios (s1-3). Contributions of different galactic structures are also shown. Table 1: GRC fit: coupling coefficients for 3 scenarios $\lambda_{KT}$ | s1 | s2 | s3
---|---|---|---
$\lambda_{smbh}$ | 0 | 1 | $10^{3}$
$\lambda_{1}$ | 0 | 1 | $10^{2}$
$\lambda_{2}$ | 0 | 1 | 2
$\lambda_{disk}$ | 1 | 1 | 1
Table 2: GRC fit: the results*

$par$ | s1 | s2 | s3
---|---|---|---
$M_{smbh}$ | $3.6\times 10^{6}$ | $3.6\times 10^{6}$ | $3.2\times 10^{6}$
$M_{1}$ | $5.5\times 10^{7}$ | $5.2\times 10^{7}$ | $3.6\times 10^{7}$
$a_{1}$ | $0.0041$ | $0.0039$ | $0.0036$
$M_{2}$ | $9.7\times 10^{9}$ | $8.6\times 10^{9}$ | $8.2\times 10^{9}$
$a_{2}$ | $0.13$ | $0.13$ | $0.13$
$M_{disk}$ | $3.2\times 10^{10}$ | $2.7\times 10^{10}$ | $3.5\times 10^{10}$
$R_{D}$ | $2.4$ | $2.5$ | $2.8$
$L_{KT}$ | $7.0$ | $6.3$ | $12.0$
$R_{cut}$ | $58$ | $45$ | $53$
$M_{dm}(R_{cut})$ | $2.7\times 10^{11}$ | $2.5\times 10^{11}$ | $2.6\times 10^{11}$
$\rho_{bgr}$ | $646$ | $653$ | $649$

* masses in $M_{\odot}$, lengths in $kpc$, density in $M_{\odot}/kpc^{3}$

Figure 3: Other galaxies: universal rotation curve, fitted by RDM model. The
points with error bars – data from [9]. Green curves – fit by RDM-model from
[22]. The data and the fits for different luminosity bins $mag$ are separated.

### Other galaxies

can be modeled with a concept of a Universal Rotation Curve (URC) introduced
by Salucci et al. [9, 10, 11, 12, 13]. It represents averaged experimental
rotation curves of more than 1000 galaxies. Before averaging, the galaxies are
subdivided to bins over the magnitude $mag$ and the curves $v(r,mag)$ are
normalized to the values at optical radius: $v/v_{opt}$, $r/R_{opt}$. Here,
$v_{opt}=v(R_{opt})$ and the optical radius of the galaxy $R_{opt}=3.2R_{D}$
is defined as a distance, under which 83% of the luminous mass is located. The
averaging smooths the individual features of the curves, their local minima
and maxima. The resulting experimental curves appear to be more smooth and are
shown by points with errors on Fig.3.

On these plots, the radius and velocity are presented in a linear scale,
rather than the logarithmic one used in previous plots. As a result, the
earlier described central structures are shrinked to a single unresolved
central contribution. The modeling is accordingly simplified, preserving only
the central and the disk contributions. The basis functions are explicitly
written in [22], the result of the fit is presented by green curves on Fig.3.

The presented plots show that the model of distributed RDM-stars, based on the
Newtonian weak field limit and the proportionality assumptions above, provides
a good fit of the experimental rotation curves, for both GRC and URC types.

Figure 4: RDM-star model in strong fields. A typical solution in different
coordinates (see text).

### RDM-star model in strong fields.

The system to solve is combined from Einstein gravitational field equations
and geodesic equations:

$\displaystyle G^{\mu\nu}=8\pi G/c^{4}\cdot T^{\mu\nu},\
u^{\nu}\nabla_{\nu}u^{\mu}=0,\ \nabla_{\mu}\rho u^{\mu}=0.$ (3)

We consider the model with T-symmetric non-interacting superposition of
ingoing and outgoing flows of dark matter. Therefore, geodesic equations can
be applied separately for every flow, described by velocity field $u^{\mu}$
and intrinsic mass density $\rho$. A static spherically symmetric metric is
chosen:

$\displaystyle
ds^{2}=-Adt^{2}+Bdr^{2}+Dr^{2}(d\theta^{2}+\sin^{2}\theta\;d\phi^{2}),$ (4)

where the profile $A(r)>0$ describes the redshift and time delay effects,
$B(r)>0$ measures geometric deformation in radial direction. $D=1$ can be put
by convention, so that $r$ is aerial radial coordinate, the area of $r$-sphere
is $4\pi r^{2}$. Time $t$ is measured by the clock of a distant observer,
where $A\to 1$ can be set. Energy-momentum tensor is taken in a form

$\displaystyle
T^{\mu\nu}=\rho(u_{+}^{\mu}u_{+}^{\nu}+u_{-}^{\mu}u_{-}^{\nu}),\ u_{\pm}=(\pm
u^{t},u^{r},0,0),$ (5)

a sum of T-symmetric radial flows of non-interacting dust matter.

The equations (3) have been solved in [3, 22]. Geodesic equations possess
analytical solution

$\displaystyle 4\pi\rho=c_{1}/\left(r^{2}u^{r}\sqrt{AB}\right),$ (6)
$\displaystyle u^{t}=c_{2}/A,\ u^{r}=\sqrt{c_{2}^{2}+c_{3}A}/\sqrt{AB},$

in $G=c=1$ normalization. The integration constants $c_{1-3}$ will be
considered in details later. The Einstein equations have a form

$\displaystyle rA^{\prime}=-A+AB+4c_{1}B\sqrt{c_{2}^{2}+c_{3}A},$ (7)
$\displaystyle
rB^{\prime}=B/A\left(A-AB+4c_{1}c_{2}^{2}B/\sqrt{c_{2}^{2}+c_{3}A}\right),$

they can be solved numerically. The typical solution is shown on Fig.4 left,
in $(A,B)$-coordinates. Initially, near the point $x_{1}$, the curve has a
hyperbolic form, typical for Schwarzschild solution. The difference starts
near the point $x_{2}$, where the Schwarzschild solution goes to infinity, the
event horizon is formed. In the considered solution, the dark matter acts like
a barrier, preventing the formation of the horizon. The solution then goes
rapidly towards very small values of $A$ and $B$, where it exhibits a strong
redshift and possesses a small proper length. Then the solution goes to large
values of $A$, a strong blueshift. The same solution is shown in the central
part of this figure, in logarithmic coordinates, and on the right part,
presenting a Misner-Sharp enclosed mass function:

$\displaystyle x=\log r,\ a=\log A,\ b=\log B,\ M=(1-B^{-1})\,r/2.$ (8)

In these coordinates the equations obtain the form more convenient for a
numerical solution

$\displaystyle a^{\prime}_{x}=-1+e^{b}+c_{4}e^{b-a}\sqrt{1+c_{5}e^{a}},$ (9)
$\displaystyle b^{\prime}_{x}=1-e^{b}+c_{4}e^{b-a}/\sqrt{1+c_{5}e^{a}},$ (10)
$\displaystyle c_{4}=4c_{1}c_{2},\ c_{5}=c_{3}/c_{2}^{2}.$ (11)

The convenience follows from the resolution of singularities, typical for
polynomial formulation, so that the resulting equations can be easily solved,
e.g., by Mathematica NDSolve algorithm. Also each term in these equations has
a clearly defined range of domination, so that normally only one term in the
equation is active. This simplifies the asymptotic analysis of the system.

The behavior of the mass function on Fig.4 right shows that the solution is
bounced off horizon line $M=r/2$, then falls very rapidly. This fall is
related to the phenomenon of mass inflation, described in the paper by
Hamilton and Pollack [26]. There is a positive feedback loop in black hole
solutions with counterstreaming matter flows: (1) increasing energy of the
crossing flows leads to (2) increasing pressure, that leads to (3) increasing
gravity, that leads again to (1). As a result, an accumulation of very large
mass in the counterstreaming region happens. For the considered solutions, the
function $M(r)$ decreases with decreasing $r$. To explain this property, one
can imagine spherical shells of positive mass consequentially removed from the
star. Finally, the mass arrives to a negative central value, a concentrated
negative mass. It corresponds to the well known Schwarzschild singularity of
naked type and explains the appearance of a blueshift region in the solution.
On the other hand, the singularity is coated in a massive shell appearing due
to the mass inflation phenomenon, so that the total mass of the system remains
positive. Also, below we will introduce a quantum gravity cutoff in the model,
which will remove the naked singularity with most of surrounding structures.

The integration constants $c_{1,2}>0$, while $c_{3}=u_{\mu}u^{\mu}=-1,0,+1$
can take three discrete values, corresponding to the type of dark matter
particles: massive, null or (theoretically) tachyonic. Interestingly, the
solution in strong fields ($A\ll 1$) depends on the matter type very weakly,
since the corresponding term $c_{3}A$ in the equations becomes small. Solution
in weak fields ($A\sim 1$) depends, at first, on the parameter $c_{5}$ that
defines asymptotic radial velocity of the dark matter: for $c_{5}<-1$, the
massive radial flow has a turning point, the matter cannot escape; $c_{5}>-1$,
possible for all matter types, the matter can escape to large distances, the
case further considered:

$\displaystyle c_{6}=c_{4}\sqrt{1+c_{5}},\ c_{7}=c_{4}/\sqrt{1+c_{5}},\
\epsilon=(c_{6}+c_{7})/2.$ (12)

The parameter $\epsilon$ defines an asymptotic gravitating density
$\rho_{grav}=\rho_{\mbox{\footnotesize\it eff}}+p_{\mbox{\footnotesize\it
eff}}$. The effective density and pressure, produced by counterstreaming dark
matter flows, are defined as components of energy-momentum tensor
$T_{\mu}^{\nu}=\mathop{\mbox{diag}}(-\rho_{\mbox{\footnotesize\it
eff}},p_{\mbox{\footnotesize\it eff}},0,0)$, where

$\displaystyle\rho_{\mbox{\footnotesize\it eff}}=c_{4}/(8\pi
r^{2}A)/\sqrt{1+c_{5}A},\ p_{\mbox{\footnotesize\it eff}}=c_{4}/(8\pi
r^{2}A)\cdot\sqrt{1+c_{5}A}.$ (13)

In the weak field limit $A\sim 1$ we obtain $\rho_{grav}=\epsilon/(4\pi
r^{2})$, $M_{grav}=\epsilon r$, as in (1). This makes $\epsilon$ a directly
measurable parameter, in physical units $\epsilon=(v/c)^{2}$, where $v$ is the
orbital velocity of stars at large distances from the galaxy center, for Milky
Way $v\sim 200$km/s, $\epsilon\sim 4\cdot 10^{-7}$.

Figure 5: On the left: quantum gravity cutoff in RDM-model for Milky Way
scenario. In the center: a mechanism for generating FRB in RDM-model. On the
right: simultaneous analysis of rotation curves and FRBs in RDM-model. Images
from [22].

### Quantum gravity cutoff.

Further, we omit index eff in the formulae, assuming that the effective
density and pressure are always considered. Also, for definiteness, we fix the
dark matter to null type (NRDM). The resulting model is equivalent to a
perfect fluid with the following equation of state (EOS):

$\displaystyle\rho=p_{r},\ p_{t}=0,$ (14)

there is a relativistic relation between mass density and radial pressure,
while the transverse pressure is switched off. The formulae (14) become

$\displaystyle\rho=p_{r}=\epsilon/(8\pi r^{2}A).$ (15)

Further, for illustration, we consider the solution for the Milky Way galaxy
with a concentrated RDM-star in the center. Fig.5 left shows the corresponding
metric profiles. The solution starts in point 1 far away from the center, then
in point 2 attempts to go to the Schwarzschild regime. We remind that the
scale is logarithmic and the metric profiles jump many orders of magnitude
near the point 2. Then they fall into abyss due to the red supershift
phenomenon. Much earlier than the $A$-profile reaches minimum in point 3, the
Planck density is achieved $\rho\sim\rho_{P}$. At this point we stop the
solution and place a Planck core below it. Since the $B$-profile is also very
small at this point and according to the formula (8), the Planck core
possesses negative total mass, whose repulsive force supports the whole system
in equilibrium. In further computations, only the order of the magnitude is
important, on necessity, corrections can be applied via phenomenological
factors [22]. Taking into account that $\rho_{P}=l_{P}^{-2}$ in the units
used, where $l_{P}$ is Planck length, also that redshift factor falls rapidly
at almost constant $r\sim r_{s}$, the value in the cutoff point becomes

$\displaystyle A_{QG}=\epsilon\,(l_{P}/r_{s})^{2}/(8\pi).$ (16)

For the Milky Way, substituting the estimation of the $\epsilon$-parameter
above and the known gravitational radius $r_{s}$ of the central black hole
from Ghez et al. [27], we obtain $A_{QG}^{1/2}=1.7\cdot 10^{-49}$. This value
will be important for our further calculations.

### RDM-stars as sources of Fast Radio Bursts.

The common property for all dark star solutions is the presence of high
energetic phenomena and strong redshift in their depths. Therefore, high
energy photons created in these phenomena on the way out can be shifted to a
long wave diapason. This makes dark stars natural candidates for sources of
FRBs, the powerful flashes of extragalactic origin, registered in radio band.
The lowest FRB frequency of 111 MHz has been reported by Fedorova and Rodin
[28], the highest of 8 GHz – by Gajjar et al. [29]. Detailed experimental
characteristics of FRBs can be found in frbcat catalogue by Petroff et al.
[30], there is also a catalogue of existing FRB theories frbtheorycat by
Platts et al. [31]. At the time of this writing, 118 distinct FRB sources have
been registered and 59 FRB theories have been created.

A particular scenario with an RDM-star generating an FRB has been considered
in [22]. An object of an asteroid mass falls onto the RDM-star. The
gravitational field inside the star acts as an accelerator with super-strong
ultrarelativistic factor $\gamma=A_{QG}^{-1/2}\sim 10^{49}$. The nucleons $N$
composing the asteroid enter in the inelastic collisions with particles $X$
forming the Planck core, producing the excited states of a typical energy
$E(X^{*})\sim\sqrt{2m_{X}E_{N}}$. The high-energy photons formed by the decay
of $X^{*}$ with energy $E(\gamma,in)\sim E(X^{*})/2$ are subjected to super-
strong redshift factor $\gamma^{-1}=A_{QG}^{1/2}\sim 10^{-49}$. The
$\gamma$-factors do not compensate each other due to the presence of the
square root in the formula. Thus, the outgoing energy
$E(\gamma,out)\sim\sqrt{m_{X}m_{N}/(2\gamma)}$, the wavelength
$\lambda_{out}=\sqrt{2\lambda_{X}\lambda_{N}\gamma}$. Taking $\lambda_{X}\sim
l_{P}$, we obtain a formula for FRB wavelength

$\displaystyle\lambda_{out}=2(2\pi)^{1/4}\sqrt{\lambda_{N}r_{s}}\,/\epsilon^{1/4},$
(17)

containing only Compton wavelength of nucleon $\lambda_{N}$ and
$(r_{s},\epsilon)$-parameters of the RDM-star. Interestingly, Planck values
are canceled out of the formula. Further, taking $\lambda_{N}=1.32\cdot
10^{-15}$m, $r_{s}=1.2\cdot 10^{10}$m, $\epsilon=4\cdot 10^{-7}$, the
wavelength and frequency of FRB are

$\displaystyle\lambda_{out}=0.5\textrm{m},\ \nu_{out}=0.6\textrm{GHz},$ (18)

that falls in the observed range 0.111-8GHz of FRB frequencies.

Further evaluations can be found in [22]. A snowball mechanism is introduced
for generating a sequence of excited states, which produces the energy
spectrum of photons cut from above by the computed $E_{out}$ value. The
spectrum is open towards low energy values, however, the increasing scatter
broadening dilutes the signal there. A common mechanism of stimulated emission
(aka LASER) can generate a short pulse of coherent radiation, by the scheme
displayed on Fig.5 center. Other parameters, such as pulse width and pulse
delay, spectral and beam efficiency, as well as polarization, repetition and
periodicity, observed for some FRBs, have been also discussed in [22]. Most of
these parameters are insensitive to the nature of the FRB source, being
imparted by local environment and/or interstellar/intergalactic medium on the
way of signal propagation. These parameters can be described by the known
source independent astrophysical mechanisms, such as scatter broadening and
signal dispersion, as well as scenarios with an FRB source passing through a
planetary system or an asteroid belt.

The estimation above has been made for a simplified scenario with a
concentrated RDM-star in the center of the galaxy with Milky Way alike
parameters. Fig.5 right shows more possibilities. The coordinates
$(r_{s},\epsilon)$ are the gravitational radius and the parameter defining the
contribution of a particular black hole (=RDM-star) to the galactic dark
matter halo, $\epsilon=GM\lambda/(c^{2}L_{KT}N)$, where $M,\lambda,L_{KT}$ are
parameters from the fit of the galactic structures explained at the beginning
of this section, $N$ is the number of black holes in the structure. The band
shows detected FRB frequencies between the lines (a) and (b), according to
(17). Two horizontal lines show two classes of solutions, supermassive and
stellar black holes, according to the scenarios considered above: (c) s2, (d)
s3, (e) corresponds to the minimal velocity value $v\sim 100$km/s on the plots
Fig.2, (f) $\epsilon=4\cdot 10^{-7}$ divided to $N=10^{9}$ stellar black
holes, (g) the same with $N=10^{6}$, the estimations of the number of stellar
black holes are from Wheeler and Johnson [32] and references therein.

The main conclusion from the analysis of the plot on Fig.5 right is that the
band and the horizontal lines have an intersection, therefore the model of
RDM-stars is able to describe simultaneously the rotation curves and FRBs.
Moreover, two solution classes exist, stellar and supermassive black holes.
The plot is constructed on the basis of Milky Way data, extracted from its
highly detailed rotation curve, and is valid for galaxies of similar
structure. It would be interesting to populate it with data from other
galaxies, that depends on the availability of rotation curves with a
comparable detalization.

##  3 TOV-stars with Planck core

In this section we consider Tolman-Oppenheimer-Volkoff (TOV) stars. It is well
known system, described by EOS

$\displaystyle w\rho=p_{r}=p_{t},$ (19)

differing from RDM-stars EOS (14) by the presence of two components of
transverse pressure
$T^{\mu}_{\nu}=\mathop{\mbox{diag}}(-\rho,p_{r},p_{t},p_{t})$, equally
distributed with the radial one (isotropic pressure). Parameter $w$ regulates
the composition and temperature of the star. Small values $w=kT/(mc^{2})$
correspond to an ideal gas of massive particles of a given temperature. In
this section we will mainly consider an ultrarelativistic plasma or photon
gas, corresponding to the value $w=1/3$. The Einstein equations have a form
(see, e.g., Blau [33]):

$\displaystyle w\rho^{\prime}_{r}=-(\rho M/r^{2})(1+w)(1+4\pi
r^{3}w\rho/M)(1-2M/r)^{-1},$ (20) $\displaystyle M^{\prime}_{r}=4\pi
r^{2}\rho,\ h^{\prime}_{r}=4\pi r(1-2M/r)^{-1}\rho(1+w),$ (21)

where the metric coefficients are chosen as

$\displaystyle A=e^{2h}f,\ B=f^{-1},\ f=1-2M/r.$ (22)

A consequence of this system is so the called hydrostatic equation

$\displaystyle r(p+\rho)A^{\prime}_{r}+2Arp^{\prime}_{r}=0,$ (23)

possessing an analytical solution

$\displaystyle 4\pi w\rho=k_{1}A^{k_{2}}$ (24)

with constants

$\displaystyle k_{1}=4\pi\rho_{1}w,\ k_{2}=-(1+1/w)/2,\ k_{3}=\log k_{1}.$
(25)

The system can be rewritten in logarithmic variables (8), to the form
convenient for a numerical solution:

$\displaystyle a^{\prime}_{x}=-1+e^{b}+2e^{2x+k_{2}a+b+k_{3}},$ (26)
$\displaystyle b^{\prime}_{x}=1-e^{b}+(2/w)e^{2x+k_{2}a+b+k_{3}}.$ (27)

They are complemented by initial data $a_{1}=0$,
$b_{1}=-\log(1-2M_{1}/r_{1})$, where $w=1/3$, $k_{2}=-2$,
$k_{3}=\log(4\pi\rho_{1}/3)$, $\rho_{1}$ and $M_{1}$ at a large $r_{1}$ are
given. The typical solution is shown on Fig.6 left. The coordinates are
$x=\log r$ and $\mathop{\mbox{arcsinh}}M$, the last one possesses asymptotics
$\pm\log|2M|$ at large $|M|$, convenient to display all features of a solution
in a single plot.

Figure 6: TOV-star solutions. On the left: solutions with negative central
mass, in the center: with positive central mass, with phenomenon of
Zel’dovich-Novikov-Starobinskij explained, on the right: a typical behavior of
metric coefficients.

Usually a regular solution is investigated, satisfying a condition $M=0$ in
the center. This solution is shown by a thick line on the figure. We
investigate what happens if this condition is relaxed. If a solution with the
same $\rho_{1}$ is started with smaller $M_{1}$, below the regular line, it
simply ends below this line in a negative central value. More interesting, if
the solution is started above the regular line, it will not end in a positive
central value and will not cross a horizon. Instead, it bounces off the
horizon, goes through the mass inflation and ends in an even more negative
central value. These solutions are clearly singular in the center, however,
they are of interest to us, since the quantum gravity cutoff considered below
can remove these singularities, replacing them with a regular Planck core.

For completeness we also consider a case of Fig.6 center, when the solution is
started above the horizon line, physically under the horizon. It similarly
bounces off the horizon from inside and goes to the positive central value. If
one reverts the integration, the solution started under the horizon from a
positive mass Schwarzschild singularity will stay inside the horizon. This
phenomenon was discovered by Zel’dovich, Novikov, Starobinskij [21]
investigating the formation of white holes under the influence of matter
ejected from the central singularity. The system is described by similar
equations and the result is that the ejected matter never leaves the horizon
and the white hole under the described circumstances cannot explode. This
effect (internal ZNS instability) is one of instability types inherent to
white holes, the other one (external Eardley instability) will be considered
below in the Appendix. Mainly, in this paper, we consider a dual solution,
possessing negative mass Schwarzschild singularity and evolving outside of the
instability region.

The plot on Fig.6 right shows the typical evolution of metric coefficients,
appearing to be very similar to such plots for RDM-stars. An important
difference is that the redshift fall and the mass inflation for a TOV-star
appear to be much more moderate in comparison with an RDM-star of similar
parameters. Table 3 shows the scenario with a stellar mass compact object in a
cosmic microwave background, described by TOV equations. Although the
variation of metric coefficients and enclosed mass in physical units is very
large, it is still much smaller than the analogous variations for RDM-stars.

Table 3: TOV-star, scenario with a stellar mass compact object in cosmic microwave background model parameters |  $M_{1}=10M_{\odot}$, $w=1/3$, $\rho_{1}=\rho_{cmb}=4\cdot 10^{-14}$ J/m3
---|---
starting point of |  $r_{1}=10^{6}$m, $a_{1}=0$, $b_{1}=0.0299773$,
the integration | $M_{1}/M_{\odot}=10$
|  $r_{2}=29532.4$m, $a_{2}=-54.2719$, $b_{2}=53.7265$,
supershift begins |  $M_{2}/M_{1}-1=-3.64729\cdot 10^{-23}$,
|  $r_{2}-2GM_{2}/c^{2}=1.37139\cdot 10^{-19}$m
|  $r_{3}=20638.1$m,
supershift ends |  $a_{3}=-107.522$, $b_{3}=-104.685$,
| $\log_{10}(-M_{3}/M_{\odot})=46.3087$
minimal radius |  $r_{4}=1.62\cdot 10^{-35}$m,
(Planck length), |  $a_{4}=-17.6594$, $b_{4}=-195.278$,
end of the integration | $M_{4}=1.728\ M_{3}$
Table 4: micro TOV-star, the critical case model parameters |  $w=1/3$, $\rho_{1}=\rho_{cmb}=4\cdot 10^{-14}$ J/m3, $a_{QG}=-146.264$
---|---
starting point of |  $r_{1}=1.13042\cdot 10^{-2}$m, $a_{1}=0$, $b_{1}=0.0100503$,
the integration |  $M_{1}=7.61132\cdot 10^{22}$kg
|  $r_{2}=1.13042\cdot 10^{-4}$m,
supershift begins |  $a_{2}=-73.6529$, $b_{2}=73.0876$,
|  $r_{2}-2GM_{2}/c^{2}=2.04973\cdot 10^{-36}$m
|  $r_{3}=7.89967\cdot 10^{-5}$m,
supershift ends |  $a_{3}=-146.264$, $b_{3}=-143.408$,
| $\log_{10}(-M_{3}/M_{\odot})=54.7085$
minimal radius |  $r_{4}=1.62\cdot 10^{-35}$m,
(Planck length), |  $a_{4}=-75.7824$, $b_{4}=-214.619$,
end of the integration | $M_{4}=1.72898\ M_{3}$

Considering this scenario in more details, we see that $r_{2}-r_{s}\sim
10^{-19}$m, the object comes very close to the gravitational collapse. This is
a distance where the matter terms, initially weak, representing cosmic
microwave background, are amplified and start to dominate in the equation.
Although this is the result of a purely classical model, quantum
considerations can change this number.

Further, $\log_{10}(-M_{3}/M_{\odot})\sim 46$, in comparison with the mass of
the observable universe: $\log_{10}(M_{uni}/M_{\odot})\sim 23$. Thus, the
considered compact object contains a core of negative mass, by absolute value
much greater than the mass of the universe, compensated by the coat of TOV
matter with almost the same positive mass. A similar computation for RDM-model
gives an even larger number: $\log_{10}(-M_{3}/M_{\odot})\sim 10^{5}$.

These enormous numbers could be the result of model idealization. Their origin
is the unrestrained phenomenon of mass inflation. It can be changed if a (non-
gravitational) interaction between the counterstreaming flows and
corresponding corrections to EOS will be taken into account. Also, the
considered solutions are stationary and can take enormous amount of time to
form. A qualitative interpretation of the obtained solutions is that a
permission of negative mass (Planck core) leads to a polarization of the
solution to the parts with highly positive and highly negative masses, almost
compensating each other in the result.

The other origin of large numbers is Planck density: $\rho_{P}=5\cdot
10^{96}$kg/m3. Straightforward estimation for the Planck density core of only
$R=1$mm radius gives the mass $M=(4/3)\pi R^{3}\rho_{P}=2\cdot 10^{88}$kg,
gravitational radius $R_{s}=2GM/c^{2}=3\cdot 10^{61}$m, much larger than the
mass and the radius of the observable universe $M_{uni}=10^{53}$kg,
$R_{uni}=4\cdot 10^{26}$m. Such a core will immediately cover the universe by
its gravitational radius, with a large margin. To place such objects in our
universe, a mechanism for mass compensation is necessary. For instance, the
one of this paper, effectively negative masses created by quantum gravity and
coated by positive mass shells until the equilibrium with a moderate mass
value is reached.

Enormous reserve of energy hiding inside TOV-stars can fuel extremely high-
energetic phenomena. Figuratively speaking, if such a bubble bursts somewhere,
the consequences can be felt throughout the universe. Thus, it is natural to
consider these objects as potential sources of FRBs and we will do this, at
first considering the quantum gravity cutoff and formation of Planck core in
the center of TOV-star.

### Quantum gravity cutoff.

Setting $w=1/3$ in a solution of TOV hydrostatic equation (24), obtain
$\rho\sim A^{-2}$. Therefore for the considered scenario with cosmic microwave
background: $\rho_{P}/\rho_{cmb}=A_{QG}^{-2}$. Taking $\rho_{P}=4.633\cdot
10^{113}$J/m3 and $\rho_{cmb}=4.19\cdot 10^{-14}$J/m3 from Longair [34], in
energetic units, have $A_{QG}=(\rho_{cmb}/\rho_{P})^{1/2}=3\cdot 10^{-64}$,
$a_{QG}=-146$. The question now is whether such value can be reached. For RDM-
stars with physically interesting parameters the redshift fall is enormous and
the Planck density can be always reached before achieving the minimum in
$a$-dependence. For a TOV-star, the redshift fall and associated density
increase in solutions are moderate. The solutions shown in Fig.6 right and
Table 3 pass the minimum $a_{3}$ before reaching $a_{QG}$ and the Planck
density for these solutions is not reached. The necessary condition for
formation of the Planck core is $a_{3}<a_{QG}$. To investigate a satisfaction
of this condition, we have performed the following numerical experiment.
Keeping the outer density fixed to $\rho_{cmb}$, we changed the solution mass,
or associated parameter $x_{10s}=\log_{10}r_{s}$, where $r_{s}$ is
Schwarzschild radius in meters, in the range $x_{10s}\in[-10,10]$. After the
integration of TOV-equations, we detected the minimum $a_{3}$ and found that
it is well approximated by linear dependence $a_{3}=-128.089+4.60519\cdot
x_{10s}$. As a result, $a_{3}<a_{QG}$ condition is satisfied at
$r_{s}<r_{s,crit}=0.11$mm (micro TOV-stars), $M<7.6\cdot 10^{22}$kg,
approximately Moon’s mass. The critical case is shown in Table 4. The equality
$a_{3}=a_{QG}$ and the resulting $r_{2}\sim r_{s}$ confirms this computation.

### TOV-stars as sources of Fast Radio Bursts.

Let us consider a photon of initially Planck energy, $E_{in}\sim E_{P}$,
$\lambda_{in}\sim l_{P}$, on the surface of the Planck core. After applying
the redshift, the outgoing wavelength
$\lambda_{out}=l_{P}A_{QG}^{-1/2}=0.9$mm. Experimentally it is
$\lambda_{exp}=37.5$mm, for the highest 8 GHz FRB detection of FRB121102
source [29]. The deviation factor $\lambda_{exp}/\lambda_{out}\sim 40$ can
still be considered as a good hit, taking into account 127 orders of
difference in the input density parameters $\rho_{cmb}/\rho_{P}$. Technically,
it can be compensated by an attenuation factor $E_{in}=E_{P}/N$, the initial
photon is $N\sim 40$ times weaker than Planck energy. A part of this factor
can be related with $(1+z)$ cosmological redshift of the source, $z\sim
0.2-0.3$, the remaining factor to explain is $N\sim 30$.

The analytical formula for the wavelength is also interesting:
$\lambda_{out}=l_{P}(\rho_{cmb}/\rho_{P})^{-1/4}$, or, in Planck units, simply
$\lambda_{out}=\rho_{cmb}^{-1/4}$, depending only on the cosmic microwave
background density.

Consideration of other FRB parameters proceeds similar to [22]. Most of the
parameters depend not on the source, but on its environment and propagation
medium of the signal. Here we consider one question: can the bursts repeat?
For the critical case $r_{s}=r_{s,crit}$ and isotropic estimation of the total
burst energy from Cao et al. [35], there is an inner reserve of energy for
$7.6\cdot 10^{22}\textrm{kg}\cdot c^{2}/(10^{32-34}\textrm{J})\sim 10^{6-8}$
bursts. The energy can be also refilled from the environment, e.g., a
companion, an asteroid belt, etc. In this refilling, when the threshold
$r_{s}>r_{s,crit}$ is passed, the conditions for Planck core existence
disappear. This can trigger the FRB, that will return the system to
$r_{s}<r_{s,crit}$ state.

In summary, TOV-stars can also be the sources of FRB, or may represent a
species of these signals. Differently to FRB from RDM-star, triggered by the
fall of an asteroid, TOV-star signals can be autogenerated, possessing also a
mechanism for autonomous oscillations around the critical state.

### Comparison of different models.

While both RDM- and TOV-stars can generate FRBs, the asymptotically flat
rotation curves are generated only by RDM-stars. Only they possess the
necessary $\rho\sim r^{-2}$ dependence, while the considered TOV solutions
have $\rho\to Const>0$ asymptotics.

On the other hand, in Barranco et al. [36] a different solution of the TOV
system has been investigated, possessing $\rho\sim r^{-2}$ dependence. It is a
well known analytical self-similar solution, whose existence follows from
scale-invariance of the system, see, e.g., the work by Visser and Yunes [37].
Due to the appropriate density profile, this solution can be used to describe
the rotation curves, a configuration known as isothermal dark matter halo. In
addition, the asymptotic velocity on this solution appears to be
$(v/c)^{2}=2w/(1+w)$. The experimentally observed velocities are non-
relativistic, achievable only for small $w$. From here a conclusion is drawn,
that the dark matter composing the galaxies “must be cold”.

If one uses RDM model instead of TOV, a physically different system with the
absence of transverse pressure is formed. Here all types of dark matter
produce the same density profile $\rho\sim r^{-2}$ and the same asymptotically
flat rotation curves. The value of the orbital velocity is defined by the
parameter $(v/c)^{2}=\epsilon$, while the type of the matter by the other
parameter $c_{5}$. As a result, the consideration of rotation curves in the
RDM model does not impose a restriction on the type of dark matter in the
galaxies.

Self-similar solutions of the TOV system form a very special class, different
from the ones considered in this section. The regular type solutions we
consider look like a ball of almost constant density, with a little bump of
density in the center, due to self-gravitation. The singular solutions we
consider have the same outer asymptotics, just possess a concentrated negative
mass or a regular Planck core in the center. Self-similar solutions possess
such a strong self-gravitation, that the whole solution shape is changed, also
at large distances. This is possible only at a very large mass of solution.
Especially, for the photon gas we consider, the mass should be enormous to
make the light condense under its own gravitation. The computation shows
$\rho_{1}^{*}=\epsilon^{*}/(4\pi r_{1}^{2})$, $M_{1}^{*}=\epsilon^{*}r_{1}$,
$\epsilon^{*}=2w/(1+6w+w^{2})$, in geometrical units, for self-similar
solution. With $w=1/3$, at $r_{1}=3.1\cdot 10^{21}$ m, the outer range of the
Milky Way galaxy, it is $\rho_{1}^{*}=0.21$ J/m3, $M_{1}^{*}=4.5\cdot
10^{17}M_{\odot}$, being compared with $\rho_{cmb}=4\cdot 10^{-14}$ J/m3,
$M_{cmb}=(4\pi/3)\rho_{cmb}r_{1}^{3}=2.8\cdot 10^{4}M_{\odot}$. Thus, the mass
characteristics of the system we consider are 13 orders of magnitude below the
formation of self-similar solutions.

The other question is an ability of Planck stars directly generate FRBs,
investigated by Barrau, Rovelli, Vidotto in [38]. The BRV model considers a
collapse of primordial matter to a black hole going through the quantum bounce
to the eruption of the white hole. The eruption appears at a delayed time due
to strong gravitation. The time of recollapse depends on the mass of the star
and is estimated to $t=0.2M^{2}$, in Planck units. Equating it with Hubble
time, the mass and the size of Planck stars are estimated, created at the Big
Bang and exploding “today”: $M=(5t_{H})^{1/2}=1.2\cdot 10^{23}$kg,
$r_{s}=2M=(20t_{H})^{1/2}=0.2$mm. This estimation comes close to the critical
size of TOV-stars $r_{s,crit}=0.11$mm obtained in our model.

The BRV model predicts an observable FRB signal at $\lambda\sim r_{s}\sim
0.2$mm. The cosmological redshift correction can be also applied. The result
is numerically similar to our model ($\lambda\sim 0.9$mm), although obtained
in a completely different setup: recollapse of Planck stars vs redshift of
photons of initially Planck energy that arise in stationary TOV solutions with
the Planck core in thermal equilibrium with CMB. Our prediction
$\lambda\sim\rho_{cmb}^{-1/4}$ and the BRV formula
$\lambda\sim(20t_{H})^{1/2}$ coincide up to a numerical factor $\sim 4.7$, if
one takes into account cosmological constraints
$\Omega_{cmb}=\rho_{cmb}/\rho_{crit}=4.2\cdot 10^{-5}$,
$\rho_{crit}=3H^{2}/(8\pi)$, $t_{H}=1/H$.

In the original BRV model of Planck stars only non-repeating FRBs are
possible. The work by Barceló et al. [18] proposes repeating recollapses and
final stabilization of an object due to dissipative effects. Such a stationary
object can be equivalent to the RDM- and TOV-stars discussed here. After its
formation, it can produce both repeating and non-repeating FRBs depending on
the environment.

The further paper by Barceló et al. [39] seems to “close” the topic of Planck
stars, referring to Eardley instability of the white hole part. Due to this
instability, the white holes under the influence of external radiation would
turn into black holes, not having time to emit the FRB. Below, in Appendix, we
will bring a contra-argument, showing that Eardley instability can be
eliminated if the core of the white hole possesses negative mass. Physically,
it can be the Planck core, formed as the result of quantum gravity corrections
when the Planck density is reached. Therefore the models from the Planck star
family, as well as the FRB estimates based on them, avoid the white hole
instabilities in a self-consistent way.

##  4 Discussion: What is dark matter made of?

In this section we consider three hypotheses on the composition of dark
matter, based on the considered dark star models.

### Hypothesis 1:

the galactic dark matter can be cold, can be hot, producing the same rotation
curves.

It follows from the solution properties of the RDM model, the orbital velocity
depends only on intensity factor $\epsilon$, not on matter constitution
(cold/hot, M/N/T cases, controlled by the other constant $c_{5}$). It can be a
new type of particles, which can be sterile for interactions with the known
matter sectors, i.e., enter only in gravitational interactions with them. It
can be almost sterile, i.e., other interactions allowed at high energies in
Planck cores, while extremely weak at low energies outside.

One more fascinating possibility is that the dark matter is composed of known
particles, placed in an unusual condition. Let consider a photon of Planck
energy, emitted from the surface of the Planck core of an RDM-star:
$E_{in}\sim E_{P}$, $\lambda_{in}\sim l_{P}$. Applying the redshift
$A_{QG}^{1/2}\sim 10^{-49}$, have $\lambda_{out}=l_{P}A_{QG}^{-1/2}\sim
10^{14}$m. It is an extremely large wavelength, about 4 light days, 16 times
the Sun-Pluto distance. Such longwave photons can not be registered by usual
means, e.g., via radio telescopes. Although the energy of every such photon is
extremely small, they come in numbers providing the necessary mass density to
explain the rotation curves of the galaxies. The detailed consideration shows
that at the Planck core one Planck energy particle per Planck area per Planck
time is emitted, that corresponds to Planck density and pressure on its
surface. After that the factor $A_{QG}^{1/2}$ is applied twice, for redshift
and gravitational time dilation, then the geometrical $(r_{s}/r)^{2}$ factor
corresponds to the measured halo density $\rho=p_{r}=1/l_{P}^{2}\cdot
A_{QG}\cdot(r_{s}/r)^{2}=\epsilon/(8\pi r^{2})$, where for the redshift factor
the formula (16) is used.

In the considered scenario the particles should be massless. For massive
particles the Compton length must be greater than $\lambda_{out}\sim
10^{14}$m, obviously, excluding lightest neutrino species and other massive
particles. Those particles do not overcome the gravitational barrier and
remain bounded inside the RDM-star. From the Standard Model, the only
appropriate particles for this scenario can be photons and gravitons.
Scenarios with massive particles should have a larger starting energy to
overcome the barrier.

Figure 7: Illustration to hypothesis 2. On the left: the mass shells for
massive, null and tachyonic particles. In the center: a difference between Big
Bang and Planck core light cones structure. On the right: the world lines of
dark matter particles captured by a wormhole.

### Hypothesis 2:

the emission of galactic dark matter from a Planck core is T-symmetric, in
future and in past directions.

We remind that an RDM-star contains two T-symmetric flows, ingoing and
outgoing ones. Fig.7 left shows the mass shells for momentum $P$ or velocity
$u$ vectors. There is a one-sheet tachyonic shell, containing both ingoing and
outgoing directions, and two-sheet massive/null shells, where these directions
are separated. In any case, we assume that all mass shells become completely
occupied at the Planck core. The reason can be an extremely high temperature,
in Planck range $T\sim T_{P}$, the one achievable at Big Bang. It is so hot
there, that the vicinity of the Planck core becomes insensitive to the
external thermodynamical time arrow and develops an own, T-symmetric
thermodynamics. An important difference in this context is that RDM
singularity and Planck core are timelike, while Big Bang singularity is
spacelike. Different orientation of light cones can lead to the absent time
arrow (recovered T-symmetry) near the Planck core and its presence near/after
the Big Bang. This difference is shown in Fig.7 center, the Big Bang light
cones have only the upper part, while the Planck core light cones have both,
T-symmetrically occupied parts.

One technical remark about $P_{0}<0$ parts of the mass shells. Although they
formally correspond to negative energy and seem to be related with negative
mass exotic matter, really they just correspond to T-conjugated flows of the
same particles as $P_{0}>0$ counterparts. To verify this, consider
T-reflection, that reverts $P^{\mu}$ and $u^{\mu}$ vectors, as well as
orientation of the world lines, while preserves the action $A=m\int
d\tau|x^{\prime}_{\mu}x^{\prime\mu}|^{1/2}$ and the energy-momentum tensor
$T^{\mu\nu}=\rho u^{\mu}u^{\nu}$, which are only physically important.

One more exotic possibility for T-symmetric emission is that the world lines
of dark matter are captured by a geometry of wormhole, as shown on Fig.7
right. The ingoing flows from one universe become outgoing in the other
universe and vice versa. In a stationary scenario, their T-symmetric
superposition can be chosen.

Independently on the detailed properties of the considered models, the
emission of T-symmetric type is necessary due to simple physical reasons. If
only outgoing flows would be present, the total mass of a dark matter halo
could not be greater than the mass of (quasi) black holes, from where it
originates. Experimentally, the halo mass is much greater than the mass of
black holes. In the considered setup, ingoing and outgoing flows compensate
each other and allow for arbitrary ratio between halo and black hole masses.

Figure 8: Illustration to hypothesis 3. On the left: evolution of photon gas
in standard cosmology. In the center: the same with RDM-stars. On the right:
Swiss cheese model with galaxies filled by hot dark matter, surrounded by cold
dark matter, in expanding universe.

### Hypothesis 3:

the cosmological dark matter mimics cold type.

The common opinion is that dark matter both in the galaxies and in between
them is cold, i.e., is composed of massive non-relativistic particles. The hot
cosmological dark matter would lead to a different expansion rate of the
universe. Let us consider an evolution of uniform photon gas in standard
cosmology, as shown schematically on Fig.8 left. There is an initial flash,
then the energy and the number density of photons fall in the expanding
universe. For cold dark matter only the density falls. This makes a difference
to the evolution of the energy-momentum tensor.

On the other hand, the distribution of dark matter photons in the RDM model is
different, see Fig.8 center. Their initial energy at Planck core is fixed:
$E\sim E_{P}$, the exit energy is also fixed by the local $A_{QG}^{1/2}$
factor. If the resulting distribution will possess a constant temperature,
then in the long-range evolution it will behave like cold dark matter.

The other possibility is that EOS of cosmological dark matter is not identical
to the galactic one. There is a class of Swiss cheese models, where the
galaxies and their halos do not change their size and structure under
cosmological expansion and move as a whole. The cosmological expansion acts
only on the level where the matter distribution can be considered as uniform.
The galaxies coated in massive halos can behave like macro-particles of cold
dark matter, as shown on Fig.8 right. Internally they can be filled with hot
radiation, externally produce the same gravitational fields as cold massive
particles.

As we have mentioned earlier, while fitting the Milky Way rotation curve, the
experimental data are compatible with the presence of a cut of dark matter
halo at $R_{cut}\sim 50$kpc. While this cut was taken in the model just
phenomenologically, the physical mechanisms for it can be constructed. Two-
phase distribution, with hot radial dark matter inside the galaxy joined to
cold uniform dark matter outside, can be used. It resembles a known phenomenon
of termination shock on the border of the Solar system, where the solar wind
meets the interstellar medium. This can be modeled similarly in galactic
scales. A suitable mechanism for the termination can be any kind of
interaction of dark matter particles in the ingoing/outgoing flows and the
outer medium. In particular, it can be an absorption or a scatter of longwave
dark photons by the intergalactic medium.

##  5 Conclusion

In this paper we have experimented with the insertion of Planck core in
several earlier known astrophysical models. What becomes possible as a result
of such modification:

(1) RDM solutions can be properly continued to the strong field mode. These
are stationary solutions describing black holes, coupled to the radial flows
of dark matter. In weak fields, such configuration of dark matter can be used
as a model of spiral galaxies possessing realistic rotation curves. In this
model, the geometric dependence of the density on the distance $\rho\sim
r^{-2}$, typical for the RDM configuration in a single center approximation,
gives flat rotation curves, while assuming the coupling of all black holes in
the galaxy to RDM, deviations of the rotation curves from the flat shape are
also described. In strong fields, a peculiar phenomenon of erasing the event
horizon occurs; instead, a spherical region of super-strong redshift is
formed. This phenomenon is accompanied by the effect of mass inflation by
Hamilton-Pollack, in a thin layer near the gravitational radius a very large
positive mass is accumulated, approximately compensated by the negative mass
of the Planck core. Outside, such an object, an RDM-star, is perceived as a
Schwarzschild black hole of limited mass.

(2) when an external body, for example, an asteroid, falls on an RDM-star, a
flash of high-energy photons occurs, then the super-strong redshift of the
RDM-star moves the flash frequency to the radio band. This process can be
considered as a mechanism for generating fast radio bursts. The calculations
lead to the formula for the wavelength
$\lambda_{out}=2(2\pi)^{1/4}(\lambda_{N}r_{s})^{1/2}/\epsilon^{1/4}$, where
$\lambda_{N}$ is the Compton wavelength of the nucleons that make up the
asteroid, $r_{s}$ is the gravitational radius of the RDM-star,
$\epsilon=(v/c)^{2}$ is the parameter determining the orbital velocity of
stars $v$ in the galaxy. Evaluation with the parameters of the Milky Way
galaxy gives the wavelength $\lambda_{out}=0.5$ m and the frequency
$\nu_{out}=0.6$ GHz, in the range 0.111 … 8 GHz for the observed FRB
frequencies.

(3) the Tolman-Oppenheimer-Volkoff system of equations describing the
equilibrium of isotropic matter, for the Planck core located in the center,
also has an interesting structure of solutions. For the matter, we consider
photon gas stitched with cosmic microwave background at infinity. As a result
of the calculation, a stationary solution with the parameters of a micro black
hole $r_{s}<0.11$ mm is obtained. With a smooth change in the external
boundary condition (slow accretion of external matter), autonomous
oscillations arise in the system, accompanied by self-generation of photonic
flashes. In this model, a formula for the outgoing wavelength is
$\lambda_{out}=l_{P}(\rho_{cmb}/\rho_{P})^{-1/4}$, numerically
$\lambda_{out}=0.9$ mm, $\nu_{out}=333$ GHz, that fall close to the observed
FRB range.

(4) white holes become stable. We have examined the dynamics of white holes in
the Ori-Poisson model and showed that the insertion of negative mass core
eliminates both Eardley and Zel’dovich-Novikov-Starobinskij types of
instability.

Based on the considered models, in frames of this work, we have proposed three
hypotheses about the composition of astrophysical dark matter.

(1) In galaxies, the dark matter can be cold or hot, massive, null or even
tachyonic, producing the same rotation curves. Particularly, it can be
composed of massless particles with initially Planck energy, finally
redshifted to the extremely large wavelength $\lambda_{out}\sim 10^{14}$m.
More particularly, it can be composed of low energy photons with such
wavelength.

(2) The emission of dark matter particles happens in a T-symmetric way, in
future and in past directions. This allows to explain the abundance of dark
matter in comparison with the masses of its sources.

(3) At cosmological distances, the dark matter behaves like it is cold.
Several mechanisms for this behavior have been proposed.

## Acknowledgements

The author thanks the organizers and participants of the XXIII Bled Workshop
“What comes beyond the Standard models?” for fruitful discussions. The author
also thanks Kira Konich for proofreading the paper.

## Appendix: Stability of white holes

The model by Ori-Poisson [19], describing a white hole or, more precisely, its
juncture with a black hole, can be considered as a simplification of the
stationary models presented in this paper. Instead of continuous superposition
of ingoing and outgoing shells of matter, there are precisely two shells, one
ingoing, one outgoing. The model is not stationary, it has a white hole as the
initial state and a black hole as the final state. The advantage of the model
is the existence of an analytical solution. The system is usually considered
unstable, but we will now show that it is not.

In addition to the previously described internal ZNS instability, there is
external instability found in the work by Eardley [20]. In Ori-Poisson
formulation, the instability can be illustrated by the Penrose diagram on
Fig.9 left. The original white hole (1) explodes, completely releasing its
mass into an outgoing null shell (2). There is an ingoing null shell (3) of
originally small energy. This shell cannot enter the white hole, in principle
(like a shell that cannot exit from the interior of a black hole). Instead, it
slows down at the horizon and after a long wait (one can substitute here
Hubble time, for instance) receives a super-strong blue shift. It forms a thin
super-energetic “blue sheet”. Upon its collision with the outgoing shell, an
analytically computable rearrangement of energy occurs. As a result, a
negligible part of the initial energy comes out (4). The ingoing blue sheet
disappears under the horizon of a newly formed black hole (5).

Figure 9: Penrose diagrams for white holes in Ori-Poisson model. On the left:
low efficient white hole eruption, in the center: no signal case, on the
right: T-symmetric case.

In Barceló et al. [39] an even more asymmetric scenario is considered,
depicted on Fig.9 center. Here not only the ingoing, but also the outgoing
shell disappears in the black hole. Absolutely nothing comes out of this
system towards an external observer.

One can immediately ask, how can it be that the black holes are stable, but
the white holes are not? What about T-symmetry? As a resolution of this
paradox, a T-symmetric scenario can be constructed, according to the principle
“for each incoming blue sheet, there is the same outgoing”. It is displayed on
Fig.9 right. Warning: the negative mass is required in the scenario. The
original white hole (1) explodes, emitting more energy than its own mass in
the form of an outgoing blue sheet (2). It leaves behind the core of negative
mass (2’). After the collision between equal blue sheets (2) and (3), a shell
of the same energy as the ingoing one comes out (4). The negative mass of the
core is compensated by the incoming blue sheet, a black hole of the same mass
as the original white hole is formed (5).

The details of the computation are the following. The solution consists of 4
Schwarzschild’s patches, marked ABCD on the figures. Their masses are

$m_{A}=M-E,\ m_{B}=M-dm,\ m_{C}=m_{0},\ m_{D}=M,$ (28)

where $M$ is the mass of the white hole with the ingoing shell, $dm$ is the
mass of the ingoing shell, $E$ is the mass/energy of the outgoing shell
($G=c=1$), $m_{0}$ is a remainder.

The computation is based on Dray-’t Hooft-Redmount (DTR) relation [40]:

$f_{A}f_{B}=f_{C}f_{D},\ f_{i}=1-2m_{i}/R,\ i=A,B,C,D,$ (29)

where $R$ is the radius at shells intersection. Relative parameters are
introduced:

$\xi=R/(2M)-1,\ \alpha=dm/M,\ \beta=m_{0}/M,\ \eta=E/M,$ (30)

where $\xi$ measures the relative distance to the horizon, $\eta$ is the
relative efficiency of white hole eruption. DTR relation results in the
expression for the efficiency:

$\eta=(1-\alpha-\beta)\,\xi/(\alpha+\xi).$ (31)

As a side remark, in [19] a different global structure linking Schwarzschild’s
patches with a cosmological model was used, however, this does not influence
the obtained efficiency formula. Now, let us consider the exponential
evolution of the shell

$\xi=\xi_{0}\exp(-\tau/(4M)).$ (32)

Here $\tau=2t$ is the total time for the ingoing shell to reach the point of
collision and for the outgoing shell to reach the distant observer, therefore
a double factor in the formula. To bring in some values, for $r_{s}=1.2\cdot
10^{10}$m it is an exponential process where the distance is halved every
minute and for $\tau=13.8\cdot 10^{9}$years the distance factor takes an
enormously small value $\xi\sim\exp(-10^{16})$, practically insensitive to the
starting $\xi_{0}$.

Considering (31) for $\beta=0$, as in original Ori-Poisson paper, in the limit
$0<\xi\ll\alpha\ll 1$, obtain $\eta\sim\xi/\alpha\sim\exp(-10^{16})$, a
vanishingly small efficiency of eruption.

Calculation with $\beta<0$ reveals a different class of solutions:
$\beta\sim-\alpha/\xi$ results in $\eta\sim 1$, 100% efficiency, while for
$\eta=\alpha$, $E=dm$, the T-symmetric case is reached selecting

$\beta=-(\alpha^{2}-\xi+2\alpha\xi)/\xi\sim-\alpha^{2}/\xi.$ (33)

The result of this computation demonstrates that in Ori-Poisson model Eardley
instability can be eliminated if the system has a core of negative mass. Being
combined with the earlier obtained result, both types of instability, Eardley
and ZNS, can be removed from the white hole models by the introduction of a
Planck core. A noticeable numerical difference between Ori-Poisson energetic
parameters in comparison with those in the other considered models appears
mainly due to the under-exponent Hubble time delays for the processes, that
run continuously in the RDM and TOV models.

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Throughput Maximisation in Ultra-wideband Hybrid-amplified Links

Henrique Buglia,1 Eric Sillekens,1  Lidia Galdino,2 Robert Killey1 and Polina Bayvel1

1 Optical Networks Group, Dept. of Electronic & Electrical Engineering, University College London, U.K.

2 Corning Optical Communications, Ewloe, U.K

A semi-analytical, real-time nonlinear-interference model including ASE noise in hybrid-amplified links is introduced. Combined with particle-swarm optimisation, the capacity of a hybrid-amplified 10.5-THz 117x57-km link was maximised, increasing throughput by 12% versus an EDFAs-only configuration.

§ INTRODUCTION

To enable the exponential growth of data transmission required by new internet services, ultra-wideband (UWB) transmission technologies have been widely explored in optical fibre communication systems. Among these technologies, the use of hybrid-amplification schemes was almost universally applied in all the recent field trials achieving record throughputs over single-mode fibre (SMF), as shown in [1]. The increased throughput with hybrid amplification is due to the lower noise figure achievable with distributed Raman amplification in comparison to that possible using Erbium-doped fibre amplifiers (EDFA) alone. Distributed amplification also allows a reduction in the launch power and correspondingly - in the nonlinear interference (NLI) noise. To optimise this type of system, pump optimisation is required to find the best wavelength and power allocation to maximise the system throughput. To achieve this, real-time estimation of the nonlinear interference (NLI) noise in Raman-amplified links is required, enabled by recently-developed semi-analytical expressions of the Gaussian noise (GN) model in the presence of Raman amplification (RA) [2, 3, 4, 5]. Together with this, the amplified spontaneous emission (ASE) noise generated by Raman amplifiers needs to be taken into account in this model to obtain a complete estimation of the total received signal-to-noise ratio (SNR).

In this paper, for the first time, we have included the ASE noise generated by Raman amplifiers in the model [2, 3, 4]. This new step allowed the design of a capacity-achieving backward-pumped hybrid amplifier.  Using particle swarm optimisation (PSO) algorithm to optimise the total launch power, pump powers and pump wavelengths resulted in a maximum system throughput within a 10.5 THz transmission bandwidth, corresponding to the utilisation of the C- and L-band, where backward Raman pumps are placed in the S-band. The results are finally compared with the same system optimised to operate with EDFAs alone. This work represents the first complete (ASE + NLI) semi-analytical model which allows the calculation of the received SNR & throughput in real time for any UWB Raman-amplified link setup.

§ THE GN MODEL IN THE PRESENCE OF RA AND THE RAMAN ASE NOISE GENERATION

For an ideal transceiver, the SNR for the $i$-th channel at the end of the span, after amplification, can be estimated as $\text{SNR}_{i}^{-1} \approx \text{SNR}_{\text{NLI},i}^{-1} + \text{SNR}_{\text{ASE},i}^{-1}$, where
$\text{SNR}_{\text{NLI},i}$, $\text{SNR}_{\text{ASE},i}$ originate from fibre nonlinearity and amplifier noise, respectively. In this work, ideal amplification was assumed, so that the transmitted power is fully recovered at the receiver. For Raman amplifiers, the evolution of the channel of interest (COI) power along the fibre distance ($P_i(z)$) is coupled with the ASE noise, generated by the remaining channels and the pumps, and is written as

\begin{equation}
\vspace*{-.20cm}
\small
\begin{aligned}
\pm  & \frac{\partial P_i}{\partial z} =  - \sum_{k=i+1}^{\text{N}_{\text{ch}}} \frac{f_k}{f_i} g(|\Delta f|) (P_k + P_{\ASE,k}) P_i - \sum_{p:f_i>f_p} \frac{f_p}{f_i} g(|\Delta f|) (P_p + P_{ASE,p}) P_i + \\ &\sum_{k=1}^{i-1} g(|\Delta f|) (P_k + P_{\ASE,k}) P_i + \sum_{p:f_i<f_p} g(|\Delta f|) (P_p + P_{\ASE,p}) P_i - \alpha_i P_i,
\end{aligned}
\label{eq:diff_Raman}
\end{equation}

where, $P_i$, $f_i$ are the power and frequency of the COI, $P_k$, $f_k$ are the power and frequency of the remaining WDM channels, $P_p$, $f_p$ are the power and the frequency of the pumps, $g_r(|\Delta f|)$ is the polarisation averaged, normalized (by the effective core area $A_\text{eff}$) Raman gain spectrum for a frequency separation $|\Delta f| = |f_i - f_j|$, $j=k,p$ and $\alpha_i$ is the frequency-dependent attenuation coefficient. The symbol $\pm$ represents the pump under consideration, i.e., $+$ for forward pump and $-$ for backward pump configurations. $P_{\ASE,i}$, $P_{\ASE,k}$ and $P_{\ASE,p}$ are the ASE noise respectively in the COI, channel $k$ and pump $p$. The $\text{SNR}_{\text{NLI},i}$ is calculated in closed form using the model in [2, 3], which is derived from the integral ISRS GN model [6], by using a suitable semi-analytical expression to express the normalised power profile evolution along the fibre distance, given by $\rho(z,f_i) = P_i(z)/P_i(0)$ (see Appendix A of  [2]), where $P_i(z)$ is obtained from Eq (<ref>). The ASE noise generated in channel $i$ ($P_{\ASE,i}$) is obtained as [7]

\begin{equation}
\vspace*{-.1cm}
\begin{aligned}
&\frac{\partial P_{ASE,i}}{\partial z} = - \sum_{k= i + 1}^{\text{N}_{\text{ch}}}\frac{f_k}{f_i} g(|\Delta f|) (P_k + P_{ASE,k}) (P_{ASE,i} + 2h\kappa B_if_i) -\sum_{p: f_i > f_p}\frac{f_p}{f_i} g(|\Delta f|) (P_p + P_{ASE,p}) (P_{ASE,i} + 2h\kappa B_if_i) + \\
& + \sum_{k=1}^{i-1} g(|\Delta f|) (P_k+ P_{ASE,k}) (P_{ASE,i} + 2h\kappa B_if_i) + \sum_{p: f_i < f_p} g(|\Delta f|) (P_p+ P_{ASE,p}) (P_{ASE,i} + 2h\kappa B_if_i) - \alpha_i P_{ASE,i},
\end{aligned}%
\label{eq:diff_Raman_ase}
\end{equation}

with $\kappa = 1 + \eta = 1/(1 - exp(-h\Delta  f/k_B/T))$, where $\eta$ is the phonon occupancy factor, $h$ is the Planck constant, $T$
is the temperature of the system and $k_B$ is Boltzmann’s constant. Note that Eq. (<ref>) should be written for each channel $k$ and pump $p$ by replacing $i=p,k$, whereas Eq. (<ref>) is written for each channel $k$ by replacing $i=k$. Together, they represent a set of $2N_{ch} + N_{p}$ coupled differential equations describing the signal evolution and the ASE noise generation in Raman-amplified links. This equation is solved for each span, where the accumulated ASE noise at the end of each span is used as the initial condition for the following span. For hybrid-amplified links, the ASE generated by distributed Raman amplification, obtained from Eq. (<ref>), is amplified by the ideal EDFA gain ($G_i$) placed at the end of the fibre.
The total ASE noise is then given by $P_{\ASE,i}^{\prime \prime} = G_i P_{\ASE,i} + P_{\ASE,i}^\prime$, where $P_{\ASE,i}^\prime = 2(G_i-1)n_{sp} hf_iB_i$, with $n_{sp} \approx \text{NF}/2$ the spontaneous emission factor and $\text{NF}$ the EDFA noise figure. The per-channel SNR contribution from the total ASE noise is then calculated as $\text{SNR}_{\text{ASE},i} = P_i/P_{\text{ASE,i}}^{\prime \prime}$.

§ TRANSMISSION SETUP AND SYSTEM OPTIMISATION

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(a) Fibre attenuation coefficient and Raman gain spectrum for the fibre used; and optimised pump wavelength allocation. (b) Optimised hybrid amplifier gain for backwards distributed Raman amplification stage (red) and ideal lumped (EDFA) stage (blue).

The transmission system is assumed to use a hybrid amplification technology, consisting of two stages: a backward-distributed Raman amplifier followed by an ideal EDFA. It was assumed to amplify a WDM signal with $N_{\text{ch}}$=105 Nyquist-spaced channels centred at 1571 nm, where each channel was modulated at the symbol rate of 100 GBd, with Gaussian symbols. This setup results in a total bandwidth of 10.5 THz, ranging from 1530 nm to 1615 nm, corresponding to the full utilisation of the C- and L- bands.
Signal transmission was evaluated over 117 x 57 km (a total distance of 6669 km). The transmission optical fibre under consideration is assumed to have wavelength-dependent attenuation and the Raman gain profile compliant with Fig. <ref>.a, and an effective area of $150~\mu\text{m}^2$, resulting in a nonlinear coefficient $\gamma = 0.55\text{ W}^{-1} \text{km}^{-1}$.
Dispersion parameters considered are $D = 21~\text{ps }\text{nm}^{-1}\text{km}^{-1}$, $S = 0.067~\text{ps }\text{nm}^{-2}\text{km}^{-1}$. The NFs of the EDFAs were assumed to be 5 dB and 6 dB, for the C- and L- bands, respectively. For the distributed Raman amplification, pumps were placed in the S-band, as shown in Fig. <ref>.a, and their wavelengths and powers, as well as the total transmitted launch power are optimised to maximise the system throughput.

The optimised hybrid amplifier was designed, based on a PSO algorithm, where a spectrally uniform launch power profile and 6 backward pumps, limited to $500~\text{mW}$ each and placed in the S-band (1470 nm - 1520 nm) are considered. The total launch power and the pump wavelengths and powers are optimised to maximise the cost function $C = \sum_{i = 1}^{N_{\text{ch}}} 2\cdot\log_2(1+\SNR_i)$, such that the total throughput is maximised. The PSO algorithm has 7 variables to be optimised (6 pumps + total launch power). The number of particles was chosen to be 50 with a maximum of 50 iterations selected as the stopping criterion. For the algorithm bounds, we let the total channel launch power vary between 15 dBm and 25 dBm, and the power of each pump at the end
of the fibre from 0 mW to 500 mW. The optimisation resulted in a total launch power of 20.4 dBm, corresponding to 0.49 dBm per channel, and 3 pumps with non-negligible power, with wavelengths 1470 nm, 1499 nm and 1502 nm, and powers of 433 mW, 107 mW and 113 mW, respectively as shown in Fig. <ref>.a. The optimised hybrid amplifier gain is shown in Fig. <ref>.b where the lumped (EDFA) gain is assumed to be ideal to completely recover the transmitted power.

§ THROUGHPUT MAXIMISATION RESULTS

unbounded coords=jump,
title = (a) Hybrid Backward Raman Amplification,
legend style=font=, at=(rel axis cs:1,0), anchor=south east,

legend columns=3,
width=8.5cm, height = 5.5cm,
xlabel=Wavelength [nm],
ylabel=$\text{SNR}$ [dB],

xticklabel style=/pgf/number format/1000 sep=,
ylabel near ticks,
ytick distance=2,
xtick distance=20,

[anchor=west] (source) at (axis cs:1545,21.2)C;
[anchor=west] (source) at (axis cs:1587,21.2)L;

[anchor=west] (source) at (axis cs:1540,14.3);
(destination) at (axis cs:1540,17);

[] at (axis cs: 1540,18.2) 99.22 Tbit/s;
[] at (axis cs: 1540,17.3) 20.4 dBm;

[on background layer]
[green,opacity=0.1] (rel axis cs:0.1,0.0) rectangle (rel axis cs:0.475,1);
[blue,opacity=0.1] (rel axis cs:0.475,0)
rectangle (rel axis cs:0.97,1);

[Set1-B, very thick] table[x=wav,y=NLI] Data/SNR_Hybrid.txt;
[Set1-A, very thick] table[x=wav,y=ASE] Data/SNR_Hybrid.txt;
[Set1-C, very thick] table[x=wav,y=total] Data/SNR_Hybrid.txt;

unbounded coords=jump,
title = (b) Lumped Amplification,
legend style=font=, at=(rel axis cs:0,0.61), anchor=north west,

legend columns=1,
width=8.5cm, height = 5.5cm,
xlabel=Wavelength [nm],
ylabel=$\text{SNR}$ [dB],

xticklabel style=/pgf/number format/1000 sep=,
ylabel near ticks,
ytick distance=2,
xtick distance=20,

[anchor=west] (source) at (axis cs:1545,21.2)C;
[anchor=west] (source) at (axis cs:1587,21.2)L;

[anchor=west] (source) at (axis cs:1582,13.2);
(destination) at (axis cs:1584,16);

[] at (axis cs: 1586,17) 88.55 Tbit/s;
[] at (axis cs: 1586,16.2) 23 dBm;

[on background layer]
[green,opacity=0.1] (rel axis cs:0.1,0.0) rectangle (rel axis cs:0.475,1);
[blue,opacity=0.1] (rel axis cs:0.475,0)
rectangle (rel axis cs:0.97,1);

[Set1-B, very thick] table[x=wav,y=NLI] Data/SNR_lumped.txt;
[Set1-A, very thick] table[x=wav,y=ASE] Data/SNR_lumped.txt;
[Set1-C, very thick] table[x=wav,y=total] Data/SNR_lumped.txt;

Performance in terms of SNR for the system in Sec. <ref> operating with an optimised (a) hybrid amplification and (b) full lumped amplification.

Fig <ref>.a shows the transmission system performance of the capacity-achieving hybrid amplifier in terms of SNR. Because the majority of Raman gain occurs at the shorter wavelengths of the L band (see Fig. <ref>.b), the NLI noise is worst in this region, due to the increased power levels propagating through the fibre, which reduces the $\text{SNR}_{\text{NLI}}$. On the other hand, the better ASE performance of Raman amplifiers when compared to EDFA reduces the ASE noise in this same region increasing the $\text{SNR}_{\text{ASE}}$; this is because for those channels almost all the fibre loss is compensated with RA, whereas in the remaining part of the spectrum, greater EDFA gain is required.

To compare the benefits of using hybrid amplification schemes, Fig <ref>.b show the performance results in terms of SNR for the same transmission system described in Sec. <ref> using a fully lumped amplification scheme, i.e, without any distributed Raman pumps injected in the transmission fibre. The results are calculated using the model in [8] and the total launch power was also optimised to maximise the throughput, resulting in a value of 23 dBm, corresponding to 3.09 dBm per channel. The increased ASE noise values for this case when compared to the hybrid amplification case are a result of the lower performance of EDFAs in comparison to Raman amplifiers in terms of NF. In the case of NLI noise, despite pump powers also being injected into the transmission fibre for the hybrid amplification case, the increased channel launch powers optimising the system with lumped amplification make this case also worse in terms of NLI noise. Indeed, with optimised hybrid amplification, a total throughput of 99.22 Tbit/s was obtained with the total channel launch power of 20.4 dBm, while for the system with optimised lumped amplification, this value was 88.55 Tbit/s with 23 dBm total launch power.

§ CONCLUSIONS

The first semi-analytical, real-time nonlinear-interference model capable of predicting the performance of hybrid-amplified (EDFA + Raman) systems was developed. The key new step was the inclusion of the Raman ASE noise in this model. Enabled by this, the throughput of a 10.5-THz 6669-km link was maximised by using a particle-swarm optimisation (PSO) algorithm. A total throughput of 99.22 Tbit/s was achieved, corresponding to a 12 % increase when compared to the same system operating with EDFAs only.

Financial support under the EPSRC Programme Grant TRANSNET (EP/R035342/1) and EWOC (EP/W015714/1),  EPSRC studentship (EP/T517793/1), and the Microsoft 'Optics for the Cloud' Alliance are gratefully acknowledged.

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