Do massive neutron stars end as invisible dark energy objects?
Abstract
Astronomical observations reveal a gap in the mass spectrum of relativistic objects:
neither black holes nor neutron stars having masses in the range of 2  5 have ever been observed.
Based on the solution of the TOV equation modified to include a universal scalar field we argue that
all moderate and massive neutron stars should end invisible dark energy objects (DEOs).
Triggered by the baryonic matter interaction, a phase transition from normal compressible nuclear matter
into an incompressible quarksuperfluid is shown to occur at roughly times the nuclear density.
At the transition front, the scalar field is set to inject energy at the maximum possible rate via a nonlocal interaction potential
This energy creates a global confining bag, inside which a sea of freely moving quarks is formed in line with the asymptotic freedom
of quantum chromodynamics.
The transition front, creeps from insidetooutside to reach the surface
of the object on the scale of Gyrs or even shorter, depending on its initial compactness. Having reached
then the total injected dark energy via turns NSs into invisible DEOs.
While this may provide an explanation for the absence of stellar BHs with and NSs with , it also
suggests that DEOs might have hidden connection to dark matter and dark energy in cosmology.
Keywords: Relativity: general, black hole physics — neutron stars — superfluidity — QCD — dark energy — dark matter
1 Turbulent superfluidity in Neutron stars
The interiors of pulsars and NSs most likely are made of superfluids governed by triangular lattice of quantized vortices as prescribed by the OnsagerFeynman
equation: here denote the velocity field, the vector of lineelement, the reduced Planck constant and the mass of the superfluid particle pair, respectively.
Accordingly, the core of the Crab pulsar, should have approximately
neutron and protonvortices (Fig. 1).
Let the evolution of the number density of vortex lines, , obey the following advectiondiffusion equation:
(1) 
where denote the transport velocity at the cylindrical radius and dissipative coefficient in the local frame of reference, respectively. When , then the radial component of in cylindrical coordinates reads: In the case of the Crab; this implies that approximately neutron vortices must be expulsed/annihilated each second, and therefore the object should switch off after up to yr, depending on the underlying mechanism of heat transport (see Baym, 1995; Link, 2012, and the references therein).
On the other hand, recent numerical calculations of superfluids reveal generation of large amplitude Kelvin waves
that turn superfluids turbulent (see Baranghi, 2008; Baggaley & Laurie, 2014; Dix, 2014, and the references therein).
It is therefore unlikely that trillions of Kilometerlong neutron and protonsvortices inside pulsars and NSs would behave differently.
In this case, should be replaced by a mean turbulent velocity with being an upper
limit
In analogy with normal massive luminous stars, massive and highly compact NSs appear to also switchoff earlier than their less massive
counterparts (Hujeirat 2016). For alternative models explaining the abovementioned mass gap see
Belczynski (2012), Chapline (2014) and the references therein.
2 The onset of incompressibilitiy
Modeling the internal structure of cold NSs while constraining their masses and radii to observations, would require their central densities to inevitably be much higher than the nuclear density : a density regime in which all EOSs become rather uncertain and mostly acausal (see Hempel et al., 2011, and the references therein). This however can be viewed as a consequence of the considerable reduction of the compressibility of the nuclear matter at . To clarify this argument: Assume that the energy density and the pressure at have reached the critical state, at which the particle involved communicate with each other at the the maximum possible speed, e.g. the speed of light. This corresponds to the EOS: In this case, the the chemical potential equation reads:
Let the fast communicating particles occupy the finite central volume where is an arbitrary small radius.
The particles involved practically form a fluid portion that cannot accept compression anymore, as otherwise the causality condition would be violated.
The number density here would saturate around and yields a maximum local pressure
With this and , the fluid portion inside is practically incompressible.
On the other hand, the energy inside is uniform and the involved particles share the same energy, i.e.
where But as then local pressure must vanish.
This means that the validity of calculating the pressure from the chemical potential alone breaks down.
As a consequence, using this formula in this regime would give rise to unrealistically high central energy densities and most likely would violate causality.
Moreover, the regularity condition imposed on the pressure at r=0 enforces the supranuclear dense fluid inside to also be nearly incompressible.
To explain this point: since the gradient of the pressure vanishes at the RHS of the TOVequation (see Eq. 9) must vanish as well.
This is feasible, if the enclosed mass becomes vanishingly small i.e. for
On such small length scales and, in the absence of local or exotic feeding mechanisms, gravity alone cannot enforce
to increase faster than as Therefore the spatial variation of inside remains limited, which means that and therefore the formation of the density plateau around becomes inevitable.
Under these conditions, computing the local pressure from the chemical potential alone would yield an unrealistic
The usual adopted strategy to escape this pressuredeficiency is to enforce an unfounded inwardincrease of as resulting
therefore in unreasonably large central densities.
3 The onset of quarksuperfluidity
Another possible solution, which we propose here, runs as follows:

The nuclear matter at indeed reaches the compressibility limit and can be welldescribed by the stiffest EOS

A pure incompressible nuclear matter has a constant chemical potential and therefore the validity of computing the local pressure from the chemical potential alone breaks down. In such flows a nonlocal pressure for controlling the dynamics of the nuclear fluid is required.

The transition from compressible into pureincompressible fluid phase might be provoked by the onset of a scalar field  matter interaction, which become active, once a critical density, is surpassed. The interaction potential, generates a nonlocal negative pressure which is capable of supporting the fluidconfiguration against its own selfgravity.

The onset of interaction has a runaway character: injects dark energy, which in turn enforces the transition front to creep from insidetooutside to abruptly terminate at the surface of the object.
Indeed, beyond , shortrange repulsive interactions between particles mediated by the exchange of
vector mesons most likely will dominate the dynamics of nuclear matter and would enhance the asymptotic
convergence of the EOSs towards (see Haensel et al., 2007; Camenzind, 2007, and the references therein).
The chemical potential here increases linearly with the number density of the baryonic matter
This regime is classified here as HState and depicted in redcolor in Fig. (2).
Recalling that central densities, in NSs increase with their masses, but upperbounded by to fit the observed mass function (see Lattimer, 2011, and the references therein), we conclude that the linear correlation must terminate at a certain critical density where attains a global maximum Baym & Chin (1976).
On the other hand, in an ever expanding universe, the eternalstate of matter should be the one at which the internal energy reaches a
global minimum in spacetime (zerotemperature, zeroentropy and where Gibbs energy per baryon is lowest; henceforth the LState).
Taking into account that inside the object together with the a posteriori results (see Fig. 5),
we conclude that and therefore the local pressure
must vanish as well. In this case a nonlocal pressure must be generated in order to oppose
selfcollapse
If the transition layer between the H and Lstates is of finite width in the space, then here may be positive, negative and/or discontinuous.
However, the case should be excluded, as it implies that the eternal state of matter would be more energetic than the HState, which is
a contradiction by constrcution.
Similarly, the case is forbidden as it would violate energy conservation
(; adding more particles yields a smaller pressure).
Moreover, let us rewrite the TOV equation in terms of :
(2) 
Obviously, as a negative would destabilize the hydrostatic equilibrium, unless external sources are included, e.g. bag energy and/or external fields.
Therefore, although a first order phase transition may not be completely excluded, a crossover phase transition
into an incompressible superfluid phase with would be more likely.
Here, and on both sides of the transition front are equal and, with the help of an
external field, both and across the front can be made even continuous (Fig. 2)).
In the present study, the simultaneous occurrence of the onset of baryonic matter interaction with the crossover phase transition
is necessary in order to generate a nonlocal pressure with capable of opposing compression exerted by the surrounding
curved spacetime.
In the regime such a pressure may nicely resemble a nonlocal bag energy of quarks in the continuum.
In the presence of , the chemical potential per particle at would be upperlimited by the energy required for quarkdeconfinement.
In this case, the corresponding Gibbs function reads:
(3) 
Based on our test calculations, an interaction potential obeying a power law distribution of the type:
turns out to be optimal for maximizing the compactness of the compact object, i.e,
where corresponds to the dynamical Schwarzschild radius (Fig. 5).
Subtituting and in Eq. (3) at , then reduces to:
(4) 
The Gibbs function here may accept several minima at though doesn’t necessary vanish.
However and should be excluded, as they are energetically unfavorable for smooth crossover phase transitions to occur.
On the other hand, by varying a set of true minima could be found.
One way to constrain is to relate it to the canonical energy scale characterizing the effective coupling of quarks, i.e.
(see Bethke, 2007, and the references therein). Indeed, as shown in Fig (3), attains a zerominimum at
for
The question to be addressed here is whether the abovementioned localized analysis would apply for the whole object as well?
Indeed, the injected dark energy, via enforces the spacetime embedding the whole object to be
increasingly curved, thereby maximizing the compression of the fluid in front of up to the critical limit and sets into an outward motion.
The enclosed dark energy via grows with radius as i.e. faster than the growth of the baryonic mass, thereby enabling the object to reach a maximum compactness precisely at
Note that the cases with and should be excluded. In the former case, the resulting objects must have collapsed
into BHs with which have not been observed. The latter case is not supported by observation either as the surfaces of these
massive NSs would continue to be dominated by a normal luminous matter.
Behind a sea of freely moving quarks is formed, though globally confined by the strongly curved spacetime surrounding the object,
which acts as a global confining bag for the quarks.
Note that, unlike the constant bag energy model of quarks, where the enclosed deconfinement energy scales linearly with the number of
3quarks flavors , the injected dark energy in the present model scales as This extraenergy may be viewed as a
mechanism for further enhancing the gluon likefield embedding the quarkcontinuum.
We may examine the conditions of coupling of particles in this pure quarksea by setting and taking to be the number of quark flavors in the effective quarkgluon coupling constant:
(5) 
Relating to Fermi momentum and use to infer the Fermi wave number we obtain
However, noting that the sea of quarks is incompressible in which communication between particles is mediated with the speed of light, we conclude that the value of should attain its true minimum, which is expected to be much smaller than Nevertheless, the present value of still ensures that quarks are in the safe energy regime, where they move almost freely in line with the asymptotic freedom of quantum chromodynamics QCD (see Bethke, 2007, and the references therein).
4 Governing equations and solution method
Our investigation here is based on numerical solving the TOV equation modified to include scalar fields The modified stress energy tensor reads:
(6) 
The superscripts ”0” and ”” correspond to baryonic and scalar field tensors:
(7) 
here is the 4velocity, the subindices is a background metric of the form:
(8) 
where are functions of the radius.
Assuming the configuration to be in hydrostatic equilibrium, then the GR field equations, reduce into the generalized TOV equations:
(9) 
where is the total enclosed mass:
and where
here denotes the interaction potential of the scalar field with the baryonic matter, i.e., the rate at which dark energy
is injected into the system and is the timederivative of
Our reference object is a NS with with a radius where is the Schwarzschild radius.
is assumed to be spatially and temporarily constant, whereas is set to obey the powerlaw distribution:
and are constant parameters that are chosen so to fulfill the
a posteriori requirement: for In most of the cases considered here, is set to
be identical to the canonical energy scale at which mometum transfer between quarks saturates, i.e., GeV.
The fluid in the post transition phase is governed by the EOS:
For a given central density, the solution procedure adopted here is based on integrating the equations for the pressure, enclosed mass and
pseudogravitational potential from insidetooutside, using either the first order Euler or fourth order RungeKutte integration methods.
5 Results & discussions
The herepresented model of DEOs is motivated by the following three unresolved theoretical and observational problems in the astrophysics of NSs:

Why neither NSs nor BHs have ever been observed in the massrange 2  5

Most sophisticated EOS used to model the internal structure of NSs are based on central densities that are far beyond the nuclear density: a density regime of great uncertainty.

How NSs end their life in an ever expanding universe and whether there is a hidden connection between the missing massive NSs and dark matter on the one hand and with dark energy in the universe on the other hand.
In this paper we argue that the formation of DEOs may provide answers to these unresolved problems. This scenario could be summarized as follows:

The very central regions of NSs are made of superfluid nuclear matter and that these would be the first to be evacuated from vortex lines and all other removable energies that do not contribute significantly to the pressure. The nuclear fluid here is governed by the stiff EOS:

In order to escape collapse into a BH with the chemical potential in the very central regions cannot grow indefinitely, and it must terminate at a certain critical value, where the fluid is set to undergo a phase transition.
Based on minimum energy consideration, a crossover phase transition into an incompressible quarksuperfluid has been shown to be energetically a favorable transition. 
We have shown that in the presence of a universal scalar field the injected dark energy is capable of provoking a phase transition into INQSF that roughly occurs at The action of the injected energy is equivalent to generating a gluonlike field, or enhance the available gluonfield through forming a global energy bag in the continuum, inside which quarks move almost free in line with the asymptotic freedom of quantum chromodynamics.

We have shown that the transition front creeps from insidetooutside on the scale of Gyrs, forming a sea of quarks behind the front. Indeed, the very slow outwards propagation of grants the NS ample of time to stably react to all possible conditions associated with the phase transition, including a global redistribution of mass inside (Fig. 6).

We have shown that an interaction potential of the type is capable of maximizing the compactness of the object (Fig. 5).

Having reached the surface of the NSs, these object become DEOs. Their interiors are made solely of INQSFs with constant chemical potential. The spacetime inside DEOs has been identified to be fairly flat, whereas it promptly becomes extraordinary curved across their surfaces (Fig. 5). Inside DEOs, the nuclear fluid has a vanishing binding energy and therefore mimicking the configuration of an ultragiant hadron trapped in a strongly curved spacetime.

According to the herepresented scenario, all visible pulsars and NSs must contain incompressible quarksuperfluid cores supported and confined by a dark energy component which is induced by a scalar field of universal origin. The gravitational significance of the injected dark energy in these cores depends strongly on their evolutionary phase and in particular on their ages and initial compactness. Accordingly, young NSs should be less massive than old ones, and the very old NSs should turn invisible by now.
To quantify the massenhancement by let be the mass of the NS at its birth and being the mass enhancement due to . Requiring then the following inequality holds:(10) or equivalently,
(11) where denote the total energy, the density in units of g/cc and the baryoinc mass of the NS in units of respectively.
Thus, NSs are born with and by interacting with they become more massive and more compact to finally reach at the end of their luminous phase, which would last for approximately yr or less, depending on their initial compactness. Thus, a NS with initial compactness will have to double its mass to become a DEO (Fig. 4 and Fig. 7).
According to the present scenario, the HulseTaylor pulsar should have an INQSF core, though the dark energy component is gravitationally insignificant due to its young age, and therefore the size of its INQSFcore must be still small. Assuming the baryon mass of the pulsar to remain constant as it evolve on the cosmological time scale, then the pulsar will turn into invisibleDEO in roughly one Gyr. This would imply that the onset of baryon interaction should occur at roughly four times the nuclear density, which is in the range of the herepredicated critical density (Fig. 3). On the other hand, the extramass resembles the lower energy limit required for deconfining the sea of quarks, i.e., the energy needed for generating a see of quark antiquark pairs. Similar to quarks in hadrons, the sea of quarks inside DEOs can never be observed as free objects in the sky.
Recalling that the effective potential of the gluonfield inside individual hadrons is on the average predicted to increases with radius as and that
the spatial variation of the coefficient of the Schwarzschild metric on comparable length scales is negligibly small
(), we conclude that gluonfields do not accept stratification by gravitational fields.
Therefore as in the present DEOmodels is dominant and increases with radius, the
sea of quarks inside DEOs is in a purely incompressible state and cannot accept stratification (see in Fig. 5).
In such gravitationally bounded incompressible fluidconfigurations, not only that but the classical repulsive pressure
must vanish also and should be replaced by a nonlocal pressure,
in order to avoid the formation of BHs with
Unlike EOSs in compressible normal plasmas, classical EOSs in incompressible superfluids are nonlocal. In the latter case,
constructing a communicator that merely depends on local exchange of information generally would not be sufficient for efficiently coupling
different/remote parts of the fluid in a physically consistent manner. A relevant example is the solution of the TOVequation for classical
incompressible fluids .
In this case, the pressure depends, not only on the global compactness of the object, but it becomes even acausal whenever the global compactness is enhanced.
This is similar to the case when solving the incompressible NavierStokes equations, where an additional Laplacian operator for describing the spatial variation of a nonlocal scalar field is constructed to generate a pseudopressure (; actually a Lagrangian multiplier), which, again, does not respect causality Hujeirat & Thielemann (2009).
Indeed, DEOs made of incompressible quarksuperfluids would be stable also against massenhancement from outside. Let a certain amount of
baryonic matter, be added to the object from outside. Then the relative increase of compared to
scales as: where
is the average density of the newly settled matter. Unless
which is forbidden under normal astrophysical conditions, the star would react stably. However, in the case of superEddington accretion or merger,
the newly settled matter must first decelerate, compressed and subsequently becomes virially hot, giving rise therefore to
On the other hand, such events would lower the confinement stress at the surface
and would turn the quantum jump of the energy density at which falls abruptly from approximately erg/cc at down to zero outside it, into an extraordinary steep pressure gradient in the continuum.
While such actions would smooth the strong curvature of spacetime across they
would enable DEOs to eject quark matter into space with ultrarelativistic speeds, which is forbidden.
Nonetheless, even if this would occur instantly, then the corresponding time scale would be
of order where is the jump width in centimeters and c is the speed of light. Relating to the average spacing between
two arbitrary particles this yields s, which is many orders of magnitude shorter
than any known thermal relaxation time scale between arbitrary luminous particles.
Although electromagnetic activities and jets have not been observed in dark matter halos, they are typical events for systems containing black holes. Recalling that supermasive GBECs are dynamically unstable Hujeirat (2012), our results here address the following two possibilities:

If the onset of baryon interaction indeed occurs at then the majority of the first generation of stars and the massive stars formed in the subsequent early epochs must have ended as pulsars and NSs, rather than collapsing into stellar BHs with . In this case, dark matter halos most likely should be DEOrich clusters. These clusters must have been extraordinary luminous in the early universe, but became inactive and dark after the nuclear matter in the interiors of NSs converted into the INQSFphase, subsequently sweeping away all sorts of luminous matter in their surroundings due to their inability to accrete normal matter. The enormous surface stress confining the sea of quarks in the interiors of DEOs render their surfaces impenetrable for normal matter, hence these objects behave as noninteracting objects.

The average repulsive forces governing clusters of DEOs most likely would enforce approaching luminous matter to deviate from facetoface collisions and therefore stay inactive, though nbody and SPHnumerical calculations are needed here to verify this argument.
Finally we note that, similar to the gluon field confining and governing the dynamics of almost massless quarks in hadrons, the induced energy enhancement of the gluonlike field in DEOs cannot surpass the limit, beyond which they collapse to form BHs with Moreover, the enclosed dark energy injected via scales as : this outlines an upper limit for the increase of confining energy with radius in DEOs, beyond which they undergo a selfcollapse. However, whether this limit applies for the potential of the gluon field inside hadrons is not clear at the moment and demands further investigations.
In a subsequent article, we discuss the compatibility and physically consistency of the herepresented
internal structures of DEOs with the bimetric formulation of spacetime in general relativity proposed by Rosen (1977).
Acknowledgment The author thanks Johanna Stachel, Friedel Thielemann, James Lattimer, George Chapline, Tsvi Piran, Juergen Berges, Jan Martin Pawlowski, Max Camenzind, Ravi Samtaney and Matthias Hempel for the very helpful comments and useful discussions on various aspects of this article.
Footnotes
 pagerange: Do massive neutron stars end as invisible dark energy objects?–LABEL:lastpage
 pubyear: 2002
 The rotational energy associated with the outwardtransported vortex lines from the central regions are turbulently redistributed in the outer shells and should not necessary suffer a complete annihilation.
 An incompressible fluid with has a negative local pressure. Therefore an acausal nonlocal pressure is necessary for stabilizing the configuration.
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