Aspects of non-topological soliton stars in a class of induced gravity theories

We explore some interesting properties of soliton stars arising in a class of effective gravity theories having a Higgs scalar field that serves a dual role of generating an effective mass of a fermion field as well as generating an effective gravitational constant. With a suitable choice of non-minimal coupling, such solitons have the canonical effective gravitational constant in their exterior while in the interior the effective gravitational constant could be arbitrarily smaller. Such a choice enables an exact determination of gravitational effects on the total conserved energy of the soliton. While the external metric of such a soliton is the standard Schwarzschild metric, the interior entropy is much larger than that of a black hole of the same equilibrium temperature.


Introduction
There are numerous studies on non-trivial, non-topological soliton solutions involving scalar fields having typical Higgs coupling with fermions.These are rendered stable on account of a conserved number of fermions that are trapped inside a compact region.The Higgs coupling of the fermions with the scalar field is chosen to ensure that the fermions are massless inside the compact soliton and are trapped there as they do not have enough energy move around (i.e. to be 'on-shell') outside the soliton.i.e. -their energy in the interior is less than their large effective rest mass in the exterior region.These solutions have been extensively studied as * Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.models for objects as small as baryons, to stars with masses comparable to a typical galactic mass [1][2][3].For model parameters in use in hadron spectroscopy, soliton stars with masses comparable to typical neutron stars can also be conceived [4,5].
Such solitons could naturally arise as the Universe transits from an unconfined to the confined phase of a quark gluon plasma resulting in the formation of baryons, or in an earlier leftright symmetry breaking phase that leads to much larger 'Lee-Wick' and 'Lee-Pang' solitons.In the former case the scalar field potential's parameters are chosen to endow the solitons with a bag-pressure B ≈ (100 MeV) 4 and a surface-tension s ≈ (30 GeV) 3 /6 that are characteristic energies used in hadron spectroscopy.In the latter case, left-right symmetry breakdown is described by models that use an appropriate and similar Higgs structure, along with two chirally degenerate ground states such that in one ground state the left handed neutrinos are massive but the right handed ones are massless, while in the other ground state, the left handed neutrinos are massless but the right handed ones are massive.In this case, choosing the surface tension of the wall separating the two phases, s ⩽ (1.93 TeV) 3 , can account for large trapped right handed massless neutrino balls [6,7] with masses as large as a typical galactic mass.
A common feature of these models is that at temperatures much less than the mass of the Higgs particle and the mass of the fermions (in a region where the mass is non-vanishing), the Higgs effective potential does not get significantly modified and there is no production of either the Higgs particles or massive fermions.However, inside the soliton where the fermions are massless, copious amounts of massless fermions are produced that remain trapped there.This substantially affects the free energy of the system.
Apart from the above ideas that use a scalar field induced Higgs mechanism to account for fermion masses, there have also been proposals of incorporating the concept of spontaneous symmetry breaking to generate an effective gravitational constant itself in scalar-tensor theories of gravity [8].In this article we explore features of non-topological solitons in a theory where the scalar field, that provides mass to a fermion field, is also coupled to the scalar curvature of the underlining spacetime to generate an effective gravitational constant.If the Higgs potential has two local minimum values-i.e. two local ground states, as used in the Lee-Wick models, the non-topological solitons would be endowed with an effective gravitational 'constant' in its interior that would be quite distinct from its value in its exterior.This article studies some interesting properties of such configurations.Of particular interest would be models that could account for solitons having the canonical effective gravitational constant in the exterior that differs from its much (vanishingly) smaller value in its interior.Such a structure enables an exact determination not only of the metric and the conserved energy of a stable soliton, but also the entropy of the soliton for comparison with entropy determined by the exterior parameters-particularly in the limit that the soliton size approaches the Schwarzschild limit.

A generalized scalar tensor theory
General features of scalar-tensor theories of gravity were studied right from the earlier versions of Jordan and Brans-Dicke [9,10] in the 1960's to their more recent generalizations [11][12][13].All such renditions of non-minimally coupled theories are special cases of Horndeski's consolidated form of a general second order Scalar-Tensor theory in four dimensions [14].Variants of such general theories have appeared in studies of dark energy and 'tracker field' cosmological models.These studies include general (even non polynomial) expressions for an effective potential for a scalar field as well as for the function of the scalar field that nonminimally couples to the Ricci scalar of the underlying spacetime: typically of the form 1  2 ϵϕ 2 R or a more general U(ϕ)R.The matter part of the action is usually chosen to be independent of the scalar field and with only a minimal coupling with the metric.This leads to conservation of the stress tensor of matter fields and the geodesic motion for a pressureless fluid element of matter (dust) [15].To have the construction indistinguishable from general relativity, the scalar field must dynamically approach a constant value to give an effective gravitational 'constant'.This can be achieved either (i) requiring ϕ to rapidly evolve to a constant value as the Universe evolves-which places cosmological constraints that have been abundantly studied [16], or, (ii) fixing the scalar field to the constant value of the minimum of a Higgs potential.We consider the latter in a generalization of the model proposed by Zee [8] in which the action for the scalar tensor theory reads: Here ϕ is a scalar field, V(ϕ) its effective potential and U(ϕ) a function of the scalar field prescribing the non-minimal coupling of the scalar field with the Ricci scalar (R).f(ϕ) is yet another function multiplying the kinetic term of the scalar field.L w is the Lagrangian for other matter fields.The original version of the Scalar Tensor (Brans Dicke) theory used U(ϕ) = ϕ = ω/f(ϕ), however, we can explore more general functions.In particular the analysis presented herein would use the usual choice f(ϕ) = 1.
If the scalar field gets dynamically locked at the minimum of the scalar potential V(ϕ) at ϕ = ϕ o , it is easy to see that the action at ϕ = ϕ o , is indistinguishable from the canonical Einstein-Hilbert action in general relativity (with a cosmological constant) with the identification The equations of motion that follow from the variation of the above action with respect to the metric and the scalar field are: Here T µν w is the energy momentum tensor of the rest of the matter fields, chosen to be independent of ϕ.As usual the covariant derivative and its contravariant projection of a tensor '[ ]' are denoted as [ ] ;µ and [ ] ;µ respectively, and It is straightforward to demonstrate that with L w chosen to be independent of ϕ, the rest of the matter field satisfies equivalence principle expressed by way of the vanishing of the covariant divergence of the its stress energy tensor: T µν w;ν = 0.This follows from the Bianchi identity as well as the equation of motion of ϕ: equation (4).On the other hand, if the matter field itself has any ϕ dependence, we get: However, if the scalar field is anchored to a constant value in a region, for example at a minimum of its potential V(ϕ), its derivative vanishes in that region and the vanishing of covariant divergence of the matter stress tensor T µν w in that region is again assured.
There have been extensions of the action described by equation (1) in which one subtracts away a surface term given identically by the non-minimal function U(ϕ) convoluted with the trace of the second fundamental form K µ at the boundary ∂M of the manifold M. The resulting action reads: In a region where ϕ is constant, standard general relativity again follows.The surface term merely cancels the second derivative terms of U(ϕ) from equation (3).For a general nonminimal coupling term, such a subtraction is essential to define a consistent 'quasi conformal mass' in the theory [15,17,18].The addition of such a surface term again leads to a violation of the equivalence principle, giving instead: This would be non-vanishing even if the matter part of the action is independent of ϕ.However as before, in a region where ϕ is dynamically anchored to a constant value or in a Ricci-flat region (R µν = 0), the vanishing of the covariant divergence is again assured.Zee [8] used U(ϕ) ∝ ϕ 2 , f(ϕ) = 1, and a scalar potential V(ϕ) having a vanishing minimum at ϕ = ϕ o : and demonstrated that with the scalar field ϕ anchored at the minimum ϕ o of the potential, the theory is indistinguishable from general relativity at low energies.
For the purpose of this article, we consider a general U(ϕ) in the action: and consider a potential V(ϕ) having a zero that is also minima at ϕ = ϕ o , and another minima at ϕ = 0; V(0) ⩾ 0. The function U(ϕ), that has dimensions of mass squared, would then have distinct values at these two minima.We choose any function U(ϕ) such that the effective gravitational constant at ϕ = ϕ o , that we identify as the canonical gravitational constant, is much greater than the effective gravitational constant at ϕ = 0.This can be ensured for example, but not limited to, the choice of the function: where ϵ << ϕ o , is a vanishingly small (or zero) mass.When ϕ is anchored at the minima of V(ϕ) at ϕ = ϕ o , we identify as the square of the plank mass: with G N the canonical Newtonian gravitational constant.On the other hand, when ϕ is anchored at the other minimum of the potential at ϕ = 0, the effective gravitational constant would be negligibly smaller than the Newtonian value.
It would suffice to consider any function U(ϕ) for which For the choice of potential V(ϕ), keeping in mind results of similar potentials used by Lee, Wick, Cottingham, Chandra et al [1,3,5], we consider: This has two local minima at ϕ = 0 and ϕ = ϕ o with values V = B and V = 0 respectively.Scalar field excitations at these two minima would have masses o , these excitations would have almost the same mass.Non-trivial time independent configurations would arise that have the field ϕ locked at the minima of V(ϕ) at ϕ = 0 inside a sphere, transiting to the other minima at ϕ o outside the sphere across a thin wall of thickness ≈ m −1 ϕ and surface energy Here s ≈ m ϕ ϕ 2 o /6 is the surface tension of the boundary of the sphere.These standard results that have been established in a flat spacetime by Lee, Wick, Cottingham (ibid), suffer only minute negligible corrections in the 'thin-wall approximation' in the present case.With ϕ varying only across a thin shell and being constant in the interior and the exterior, the contribution to the surface energy due to varying U(ϕ) across the boundary would simply lead to a small, negligible and inconsequential correction in s.At ϕ = 0 in the interior, the effective gravitational constant is tuned to an arbitrarily small (or vanishing) value in comparison to its Newtonian value in the exterior with the chosen U(ϕ).
V(ϕ) having its two minima at ϕ = 0 &; ϕ = ϕ o is sufficient to establish existence of a non-trivial solution but not sufficient to assure its stability.As in the Lee-Wick model, stability is assured by a conserved number of massless fermions that are trapped inside the interior of the sphere as they do not have enough energy to be 'on-shell' outside.This in turn is assured by an appropriate Higgs coupling of the fermions with the scalar field.Thus L w would include, besides the usual Standard Model Lagrangian L SM , a fermion field having a ϕ-dependent Higgs coupling: Here nf = ψγ o ψ is the net fermion number density and µ is the chemical potential introduced as a Lagrange multiplier to account for a conserved number of fermions in the system.m f is chosen as the fermion mass parameter that varies from 0 to m ext as ϕ goes from 0 to ϕ o : Here D µ is the spin covariant derivative: Γ µ are the spin connection [Fock-Ivanenko] coefficients [19] that are determined by requiring: A non-minimally coupled Scalar Tensor theory, in what is referred to as the 'Jordan frame' can, by a conformal transformation of the metric, be related to a minimally coupled scalar field theory in an 'Einstein frame'.However the features of the 'Higgs' coupling of the scalar field to a fermion field and giving the latter distinctive effective mass at multiple minima of the effective potential, gives rise to domains containing a conserved number of fermions in the interior.This makes the Jordan frame the preferred frame.In any case, in the region where ϕ has a constant value ϕ o , the characteristics of Einstein and the Jordan frame would be indistinguishable.Features of such generalized Scalar-Tensor theories, along with the expression for a conserved energy momentum (pseudo-) vector for such an action, have been studied in detail in Bose and Lohiya [15].A domain in which ϕ = 0, the action would simply describe matter fields and a massless fermion field in a flat spacetime while in a domain having ϕ = ϕ o , the action would be indistinguishable from that of the Einstein action for matter fields and would be a region where the fermion field acquires a rest mass m ext .

A spherically symmetric soliton
The Lee-Wick type model with the characteristic non minimal coupling is described above in equations ( 10)- (12).To explore the properties of a spherically symmetric, time independent soliton we describe: (i) The expression for the metric; (ii) The expressions for the thermodynamic potential, free energy and entropy.

The metric
For the scalar field locked at the minimum of the potential V(ϕ) at ϕ = ϕ o outside a sphere of radius R = R o , and at the other minima at ϕ = 0 in the sphere's interior, the exterior geometry is described by the Schwarzschild metric that we write in isotropic coordinates for future convenience as: with ρ the radius in isotropic coordinates.As the effective gravitational constant in the interior is chosen to be vanishingly small in comparison to its value in the exterior (i.e. in the limit ϵ → 0) the interior metric would just be the flat metric expressed as: Matching the metric at the boundary The Schwarzschild radius in terms of the radial coordinates R = 2m, corresponds to ρ = m/2 in isotropic coordinates.In terms of the radial coordinates inside the soliton, the volume and surface area of the sphere are simply: The parameter m is the total conserved gravitational geometric mass determined by the total conserved energy at infinity.The total conserved (pseudo) energy inside a sphere of radius ρ ⩾ ρ o , can be easily determined to be [20]: This is an exact expression for the energy inside a sphere of radius ρ ≥ ρ o with an asymptotic mass M = m/G.As ρ o → m/2, i.e. the Schwarzschild radius in isotropic coordinates, the energy inside ρ = ρ o goes to zero.As ρ → ∞ the conserved energy is the total mass M which is just the mass in the absence of gravitation.Thus as ρ o → m/2 the entire conserved mass is accounted for by the gravitational field outside the Schwarzschild radius.

The thermodynamic potential
To determine the conserved energy as given in equation ( 18), one needs to determine M, the energy in the absence of gravitation, and use the second term as the complete exact correction in isotropic coordinates to get the total energy in the presence of gravitation.In the absence of gravitation, we consider the scalar field anchored at ϕ = 0 in the interior and at ϕ = ϕ o outside a sphere of volume V.At temperatures KT ≡ β −1 << m ϕ , m ext (being respectively the mass of the scalar field, and the fermion field at ϕ = ϕ o ), temperature corrections to the effective scalar potential would be negligible.However copious amount of massless fermion-anti fermion pairs would be produced inside the sphere.These Fermions would substantially alter the free energy.The contribution from every energy state k to the partition function from an equal number of fermion-antifermion pairs (for a vanishing chemical potential µ) is related to the contribution to the free energy by: This gives the total contribution to the free energy from all energy states as: Here the factor 4 comes from both the spin states of fermions and antifermions.However, to have a net conserved fermion number N, a non-vanishing chemical potential µ relates the partition function to the grand potential Ω and modifies the free energy F by a factor µN: The thermodynamic variables for radiation alone are defined in terms of this grand potential Ω Here S, E, n and P are the entropy, energy, fermion number density and pressure of the fermions enclosed inside the soliton of volume V and temperature T = β −1 respectively.With a uniform interior vacuum energy density (or negative pressure) B and the surface tension energy s, as rendered by the effective potential V(ϕ ), the total free energy of a spherical soliton gets modified to: with corresponding changes in pressure and other quantities as described below.The conserved fermion number is simply These integrals are exactly determined for all β to give: Requiring the pressure due to the surface tension to cancel the internal pressure-i.e. the net external pressure to vanish at equilibrium, which is in effect equivalent to minimising the free energy, gives: The total energy or mass of the soliton can be easily deduced from equations ( 27) and ( 29): This again is an exact expression valid for all β.The µ independent terms proportional to the volume in the expression for free energy F are described by a parameter α that determines a 'critical temperature' T c at α = 0: The scenario of phase transition begins at T > T c when the volume term in the expression for free energy is positive and the Universe is filled with a dispersed massless fermion phase and no effective gravitation as the scalar field is locked at its local minimum value ϕ = 0.At the critical temperature T = T c , the phase transition would commence with the formation, expansion and percolation of bubbles of ϕ = ϕ o , G eff = G N that would constrain the massless fermions to the old ϕ = 0 phase domains.During this phase, the temperature remains constant and the latent heat comprising in B is liberated as these bubbles expand while the temperature remains at T c .These bubbles would eventually coalesce-restraining the fermions inside ϕ = 0 domains that keep contracting-till they are stabilized by the surface term-eventually forming the solitons [21][22][23] From equation (31) it follows that the contribution to the internal energy from the chemical potential of the conserved number of trapped fermions and the thermodynamic potential can be deduced by subtracting the surface energy and the volume energy to give: The above equations determine properties of non-topological solitons when gravitational effects are not considered.In the current set up, gravitational effects are completely determined by the external metric while the grand potential of the fermions trapped in the interior is easily determined in the flat metric there.Continuing to ignore gravitation for the time being, at the critical temperature, the expressions for the radius and the conserved number of the soliton following from equations ( 27) and (29) reduce to here X ≡ βµ/π.For B ≈ (100 MeV) 4 ; and s ≈ (30 GeV) 3 /6 (the characteristic energies used in hadron spectroscopy), one gets β −1 c = T c ≈ 127MeV.At this critical temperature N ≃ 6.69 × 10 20 where we have used the fact that for large solitons with N > 10 25 the approximation X = µβ/π << 1 holds quite well.Such solitons are formed with radius and mass: Requiring the radius of the soliton to be greater than the Schwarzschild radius, R > 2G N M, gives the bound on the number of Fermions: N < 10 52 .As described later in this article, this bound is significantly reduced on accurate incorporation of gravitational effects.Staying well below this naive bound for N = 10 49 (say), gives a mass M ≈ 10 −3 M ⊙ and radius R ≈ 3 × 10 6 cm.As the temperature decreases, the size quickly approaches the size of a soliton containing a completely degenerate gas of fermions.This rapid approach to the degenerate state as the temperature falls below T c has been confirmed and reported by precise numerical calculations by Chandra and Goyal [5].On the other hand, for completely degenerate case (B = 0), we obtain results for cold soliton stars considered by Lee [3].The free energy and the fermion number are simply: This implies: Minimising this free energy under variation of R gives F min = 12π R 2 s.This is simply the result that also follows from the exact expression equation (31), that is valid for all temperatures, for the degenerate vacuum case B = 0.All that is required is that the fermion number be conserved.The Schwarzschild limit would thus be determined by the surface tension alone.With T D Lee's choice of s = 1/6 × (30 GeV) 3 , one naively gets the critical N c ≈ 10 76 with R c ≃ 1Ly and M c ≃ 10 13 M ⊙ .In particular, solitons with mass some 10 7 M ⊙ with N ≈ 10 69 would be well below the critical Schwarzschild limit.These expressions also reproduce solitons studied by Bob Holdom with the surface tension in the TeV range to give M c ≈ 1.3 × 10 8 /s(TeV 3 )M ⊙ .
Accurately incorporating gravitational effects significantly alters these bounds.Fortunately, exact inclusion of gravitational effects is quite straightforward.The conserved mass at infinity being just the expressions of these quantities as determined in the absence of gravitation.Both the free energy as well as the conserved mass energy expression are incremented by the gravitational term: The energy and the free energy of the soliton are equal at zero temperature.The effect of gravitation can be described in terms of the ratio of the radius of the soliton to the radius of minimum energy soliton in the absence of gravitation.For the degenerate case (B = 0), varying F (equation ( 35)) with respect the radius R gives the radius for the minimum energy soliton for a fixed conserved N: For any radius, we define R ≡ xR min to express the gravity free expression for energy as: Including gravitational effects gives the exact expression for the total energy as a function of x: It is convenient to express this in Solar mass units (M ⊙ ≈ 1.989 × 10 33 g as: with Normalising this with respect to the value used in hadron spectroscopy: s o ≡ (30 GeV) 3 /6, gives the values: The profile of E T can be deduced from figure 1.For each value of the conserved number of fermions N, the dotted line determines the total conserved energy of the soliton contained in an infinite sphere around it, as a function of the radius of the soliton that is determined by the radial parameter x.All curves give the energy of the soliton inside the soliton radius.Both these have a local minimum at x ≃ 1 that gives the radius of the stable soliton: R min .For each N, the respective curves describe the energy contained inside the radius R = R min x.Solitons with a sufficiently small radius collapse to the Schwarzschild singularity determined by the value of x for which E T vanishes.The metastable minimum at x ≃ 1, turns over to a point of inflection for N ≳ 10 76 .Thus the size of all solitons with N < 10 76 is much larger than their corresponding Schwarzschild radii.As an example, for N = 10 72 , The Schwarzschild horizon is at x ≈ .0596.The stable point at x = 1 is thus roughly 16.78 times the size of the horizon of a soliton of the same fermion number.For a stable soliton, the conserved mass as measured at infinity in units of M ⊙ , and radius in light years are: ( s o s ) 1  3 Ly.
For the non-degenerate case (B ̸ = 0) at zero temperature, eliminating the chemical potential µ again gives the expression for energy for a general radius for N trapped fermions: In the absence of the surface term, variation with respect to the radius R gives the radius and minimum energy of the soliton: For any radius, we define R ≡ x ′ Rmin to express the gravity free expression for energy including the surface term (equation (44)) as: where, again normalising with respect to the hadron spectroscopy parameter values s o ≡ (30 GeV) 3 /6, and B o = (100 MeV) 4 : For large enough N >> 10 20 , the surface term contribution becomes negligible and stability is determined largely by the volume term.Including gravitational effects as given in equation (40), gives: with εo ≡ (4π B) The profile of ĒT can be deduced from figure 2. As in the degenerate case, for each N, the Schwarzschild radius is determined by the value of x ′ for which ĒT vanishes.The metastable minimum for all N is again at x ′ ≈ 1, with a value ≈ 4/3, for all solitons with N < 10 58 .The same value turns over to a point of inflection for N ≳ 10 58 .Thus the size of all solitons with N < 10 58 is again much larger than their corresponding Schwarzschild radii.For such stable solitons, the expression for the radius (in km)and the conserved mass as measured at infinity in units of M ⊙ , are for different N values of trapped fermions at finite temperature as a function of the radial parameter x for the degenerate case, along with the dotted-gravity free case curve.As in figure 1, E T has again been scaled by N 8/9 .The figure is almost identical to figure 1: except for a small variation in the Schwarzschild limit where the curves touch the x axis.The point of inflexion at x ≈ 1 occurs at roughly N = 1.3 × 10 75 instead of 10 76 for the zero temperature case described in figure 1.The expression for energy in the absence of gravitation is: The expression for free energy including gravitational effects is: Under a change of scale: The expression for free energy becomes: Here the function E(y) and the constants in the above expression for the usual choice of parameters in hadron spectroscopy are: The free energy in equation ( 50) is determined in terms of the following factor (in grams): (4π s) The other dimensionless factors being: ] ≃ N 4.86 × 10 8 The values for s and B in hadron spectroscopy are used in figure 5 to find the limits of stable solitons N ⩽ 4.45 × 10 41.5 ; r c ≈ 1.93 × 10 −12 cm; M c ≈ 6.5 × 10 −21 M ⊙ .

Effect of varying temperature
The formation, evolution, and survival of non topological solitons with the typical temperature variation in standard cosmology has been addressed to by several authors [4,[21][22][23][24].Therein solitons in thermal equilibrium with free fermions were considered.Chemical equilibrium would set in between the fermions inside the soliton and the free fermions in the exterior when accretion and evaporation reactions are faster than the expansion rate of the Universe.At freezeout temperature T F , accretion and evaporation rates become smaller than the expansion rate of the Universe.Below the 'turnaround temperature' T U , evaporation may be faster than accretion-depending on model parameters.Evaporation followed by freezeout would determine the fate of the solitons.For the range of parameters in the Standard Model, it turns out that the smallest number of fermions that are required for the soliton to survive evaporation till freezeout is N S ≈ 10 46 .This is way larger than the fermion number of the largest stable soliton in the presence of gravity which is N ≈ 10 28 .N S .It even exceeds the total number of fermions available inside the particle horizon at the critical temperature which is N H ≈ 10 37 .For these reasons it was expected that any non-topological soliton that may be discovered would not be a relic of the phase transition in the early Universe but an outcome of a collapsing star.
However, if the epoch of the left-right symmetry breaking is induced by a similar first order phase transition, and massless right handed neutrinos are trapped inside a domain with ϕ = 0 that is separated from the domain with ϕ = ϕ o where the right handed neutrinos are extremely massive (KT << m νR ), evaporation of fermions would be dynamically suppressed.The bounds on the fermion number for a stable degenerate soliton can be read off from figures 1 and 3: N < 10 76,75 .With a vanishingly small effective gravitational constant at ϕ = 0 (the symmetric, false ground state), the horizon volume would be large and would not pose any impediment to the existence and survival of such large solitons.Dolgov and Holdom [6,7] have described the cosmic evolution of solitons containing trapped neutrinos.A gas of neutrino-anti neutrino pairs trapped inside a soliton, with N f = 0, would be expected to have a finite lifetime and explode due to annihilation on account of ν ν annihilating into electron positron pairs.However, this would not preclude the existence of large balls with a large net N f , for example in a model where the right handed neutrino is extremely massive outside the soliton and is massless inside.

Entropy
It is a matter of interest to compare the entropy of the solitons described here with the entropy of Black holes of the same mass as also, in case of solitons formed at critical temperature T c , entropy of black holes that could be in equilibrium at same temperature.The entropy being the logarithm of the measure of the number of ways a state can be realised, the ergodicity hypothesis suggests that the probability of finding a system in a given state would be proportional to the exponential of the entropy of that state.Such a comparison would keep sight of the classic work of Stephen Hawking on black holes and their equilibrium with a bath of same temperature [25].For a bath in thermal equilibrium at temperature T, if the volume of the box is greater than a critical volume: the state of maximum probability would be that of black body radiation with no black hole.
Here E is 1.25 times the energy of a black hole.However, for a radiation bath of volume , where V c is the Schwarzschild volume of a black hole at the equilibrium temperature, the most probable state would be that of a black hole in equilibrium with the radiation bath.For a volume V ⩽ V c the whole volume would undergo a gravitational collapse.The scenario of phase transition that gives rise to soliton formation involves fermions in a thermal bath being restricted to contracting volumes of the old phase as bubbles of new phase expand.This could lead to finite volumes in local thermal equilibrium for which one needs to compare the entropy of the soliton and that of a black hole in equilibrium at the same temperature.
The entropy of the soliton-fermion system is The gravitational increment ∆E that contributes equally to both the energy as well as the free energy cancels out in this expression.For large N, i.e. small X, the expression for entropy reduces identically to β c times the mass of the soliton.The entropy of these black holes is much smaller than the entropy of the solitons.If the volume of each domain is greater than V h , the production of black holes would be suppressed.This volume follows from equations ( 65) & (65): V h = 75 32 This is quite small.Thus for domains in equilibrium at temperatures β −1 c and volumes larger than V h , black hole formation in equilibrium with the bath would be suppressed in comparison to soliton formation in equilibrium.

Discussion and conclusion
The expressions for energy and free energy for all the solitons that have been discussed determine the minimum energy soliton for a given fermion number N. The stability against the soliton disintegrating into free fermions is determined by the mass of the free fermions-were they to escape in the exterior region.For the degenerate B = 0 solitons, the exponent of N in the expression for energy being less than one, establishes its stability against disintegrating into free particles for large enough N depending on the mass of the free fermion.For all solitons, the stable point transits to an inflexion point for large enough N even before the radius reaches the Schwarzschild limit and result in instability and gravitational collapse.
Such solitons would arise in the epoch of canonical phase transitions that lead to formation of hadrons.However, severe constraints are placed on the survival of such stable solitons on account of evaporation before the freezeout.The number of trapped fermions required for the survival of such solitons exceed the number contained in the particle horizon at critical temperature.Therefore, any solitons that may be observed could only be a result of collapsing stars-rather than being relics of an earlier phase transition.
On the other hand, if right handed neutrinos are massless inside a domain with ϕ = 0 and extremely massive at ϕ = ϕ o , solitons would in principle again arise if the epoch of the leftright symmetry breaking is governed by the symmetry being broken by a similar first order phase transition.For the purposes of this article, it is sufficient to have massless fermion species that may not interact with rest of the matter fields other than through gravity, or be extremely massive in the exterior of the soliton.Effective gravity may be a result of such a phase transition in the Universe that starts with a vanishing effective gravitational constant and makes a phase transition to a Universe with effective canonical gravity-trapping massless fermions of the earlier phase inside solitons.With the effective gravitational constant vanishingly small at the epoch of phase transition, the cosmological scale factor would evolve linearly and the constraints of fermions in a particle horizon would not be an impediment for the existence of such solitons.Such large solitons could be relics of the phase transition in the model discussed in this article.

Figure 1 .
Figure 1.Variation of energy/free energy [F = E] for the degenerate case for different values of trapped fermions at zero temperature as a function of the radial parameter x.The dotted line describes the energy as a function of the radial parameter in the absence of gravitation.E T vanishes at x = xc.(E T has been scaled by factor N 8/9 for comparison of the profile of the rest of the function for differing N).

Figure 2 .
Figure 2. Variation of energy / free energy [F = E] for the non-degenerate case for different values of trapped fermions at zero temperature as a function of the radial parameter x ′ .The dotted line describes the energy as a function of the radial parameter in the absence of gravity.E T vanishes at x ′ = xc.(ĒT has been scaled by factor N for comparison of the profile of the rest of the function for differing N).

Figure 3 .
Figure 3. Variation of energy [F = E]for different N values of trapped fermions at finite temperature as a function of the radial parameter x for the degenerate case, along with the dotted-gravity free case curve.As in figure1, ET has again been scaled by N 8/9 .The figure is almost identical to figure1: except for a small variation in the Schwarzschild limit where the curves touch the x axis.The point of inflexion at x ≈ 1 occurs at roughly N = 1.3 × 10 75 instead of 10 76 for the zero temperature case described in figure1.

Figure 4 .
Figure 4. Variation of energy [F = E] for different N values of trapped fermions at finite temperature as a function of the radial parameter x' for the non-degenerate case, along with the dotted-gravity free curve.As in figure 2, ĒT has again been scaled by N. The figure is almost identical to figure 2: except for a small variation in the Schwarzschild limit where the curves touch the x axis.

Figure 5 .
Figure 5. Variation of free energy for solitons at critical temperature for different values of trapped fermions as a function of the radial parameter x.The dotted curve describes the free energy in the absence of gravity.
entropy of a black hole of temperature β −1 c are given by the Hawking formulae as simply