Consider a classical theory of a scalar field $\varphi$ coupled to the scalar curvature $R$, through an arbitrary function $U\left(\varphi \right)$, in the action:

$S={\displaystyle \int {\text{d}}^{4}x\sqrt{-g}}\left[U\left(\varphi \right)R+\frac{1}{2}{\partial }^{\mu }\varphi {\partial }_{\mu }\varphi -V\left(\varphi \right)+L_m\right]$(2)

We shall treat $S$ as a classical, effective, phenomenological action that could be good enough to do tree level (“on-shell”) physics. If need arises, we shall consider non polynomial forms for the effective potential and the NMC: $V\left(\varphi \right)$ and $U\left(\varphi \right)$. Such profiles have been used as “tracker-field” potentials [5] [6] [9] [10] and can be justified in particle physics models with dynamic symmetry breaking or non-perturbative effects. In particular they can arise from non-perturbative effects that lift flat directions in supersymmetric gauge theories as well as in modulii fields in string theory [13] [20] (see also [21] ). Strictly speaking, we would consider it premature to justify the forms for these functions at this formative stage on the basis of fundamental physics. For our purpose, we simply regard Equation (2) as the action of a parametrized Brans-Dicke fluid. Any arbitrary (cosmological) constant in the theory is included in $V\left(\varphi \right)$. $L_m\equiv L_w+L_{\psi }$ is the contribution from the rest of the matter fields. It includes a contribution from a Dirac fermion field:

$L_{\psi }\equiv \bar{U}{\left(\varphi \right)}^{-1}\left[\frac{1}{2}\left[\bar{\psi }{\stackrel{\leftarrow }{D}}_{\mu }{\gamma }^{\mu }\psi -\bar{\psi }{\gamma }^{\mu }{\stackrel{\to }{D}}_{\mu }\psi \right]-m\left(\varphi \right)\bar{\psi }\psi \right]$(3)

Here $D_{\mu }$ is the spin covariant derivative [22] . This Fermion action has an overall scaling function $\bar{U}{\left(\varphi \right)}^{-1}$. Such an arbitrary scaling can be absorbed in a redefinition of the Dirac field: $\psi \to {\psi }^{\prime }\equiv \psi /{\bar{U}}^{1/2}$ provided $\bar{U}$ were bounded. Indeed, by such a scaling (for a bounded $\bar{U}$), the $\varphi$ dependence in $L_{\psi }$ can be constrained to the mass munction $m\left(\varphi \right)$. The anti-symmetrized derivative appearing in the fermion lagrangian ensures the cancellation of the derivative of a real, non-vanishing, bounded scaling function. ¹ If $\bar{U}{\left(\varphi \right)}^{-1}$ vanishes at any point, the Dirac action identically vanishes. There is no dynamics for the Dirac field at such a point. The invariance of the Dirac Lagrangian under arbitrary phase change: $\delta \psi =iϵ\psi$ implies the vanishing divergence of the current:

$J^{\mu }=\bar{U}{\left(\varphi \right)}^{-1}\bar{\psi }{\gamma }^{\mu }\psi$(4)

and the conservation of the charge:

$Q\equiv {\displaystyle {\int }_{\Sigma }\bar{U}{\left(\varphi \right)}^{-1}\bar{\psi }{\gamma }^o\psi }$(5)

The Dirac particles would be confined to regions where $\bar{U}$ is bounded i.e. $\bar{U}{\left(\varphi \right)}^{-1}$ is non-vanishing. The only effect of a scaling function $\bar{U}{\left(\varphi \right)}^{-1}$ that vanishes outside a compact domain is to confine the fermions within the domain. Inside the domain the scaling can be identically absorbed in a rescaling of the fermion field. In what follows we shall essentially do this. This leads to conditions similar to those that arise in non-topological soliton solutions in the Lee-Wick model [17] - [19] . One could just as well have followed the Lee-Wick model in which confinement occurs for fermions that are not on-shell outside a domain on account of a large coupling with the Higgs field.

Our foremost task would be to demonstrate the vanishing of the covariant divergence of the stress tensor of the rest of the matter fields described by $L_w$ in Equation (2). Requiring the action to be stationary under variations of the metric tensor and the fields $\varphi$ & $\psi$ gives the equations of motion:

$U\left(\varphi \right)\left[R^{\mu \nu }-\frac{1}{2}{g}^{\mu \nu }R\right]=-\frac{1}{2}\left[T_w^{\mu \nu }+T_{\varphi }^{\mu \nu }+{\Theta }^{\mu \nu }+2U{\left(\varphi \right)}^{;\mu ;\nu }-2g^{\mu \nu }U{\left(\varphi \right)}_{;\lambda }^{;\lambda }\right]$(6)

$g^{\mu \nu }{\varphi }_{;\mu ;\nu }+\frac{\partial V}{\partial \varphi }-R\frac{\partial U}{\partial \varphi }+\frac{\partial m}{\partial \varphi }\overline{{\psi }^{\prime }}{\psi }^{\prime }=0$(7)

${\gamma }^{\mu }{D}_{\mu }{\psi }^{\prime }+m\left(\varphi \right){\psi }^{\prime }=0$(8)

$D_{\mu }\overline{{\psi }^{\prime }}{\gamma }^{\mu }-m\left(\varphi \right)\overline{{\psi }^{\prime }}=0$(9)

Here $T_w^{\mu \nu }$, ${\Theta }_{\nu }^{\mu }$ are the energy momentum tensors constructed from $L_w$ and $L_{\psi }$ respectively, and

$T_{\varphi }^{\mu \nu }\equiv {\partial }^{\mu }\varphi {\partial }^{\nu }\varphi -g^{\mu \nu }\left[\frac{1}{2}{\partial }^{\lambda }\varphi {\partial }_{\lambda }\varphi -V\left(\varphi \right)\right]$(10)

$L_w$ is taken to be independent of $\varphi$ and $\psi$. The two Dirac Equations (8) and (9) ensure $L_{\psi }$ to be null. The scalar field equations are therefore independent of $\bar{U}\left(\varphi \right)$. The Fermion stress tensor is simply:

${\Theta }_{\nu }^{\mu }\equiv -\frac{1}{2}\left[\overline{{\psi }^{\prime }}{\stackrel{\leftarrow }{D}}_{\nu }{\gamma }^{\mu }{\psi }^{\prime }-\overline{{\psi }^{\prime }}{\gamma }^{\mu }{\stackrel{\to }{D}}_{\nu }{\psi }^{\prime }\right]$(11)

The covariant divergence of (6) is easily seen to reduce to:

${\Theta }_{\nu ;\mu }^{\mu }=\frac{\partial m}{\partial \varphi }{\partial }_{\nu }\varphi \overline{{\psi }^{\prime }}{\psi }^{\prime }$(12)

Thus there is a violation of equivalence principle as far as the Dirac field is concerned. However, in a region where the scalar field gradient vanishes or for a $\varphi$ independent fermion mass, the covariant divergence of the fermion field stress tensor vanishes. To see how the equivalence principle strictly holds for the rest of the matter fields, i.e.: $T_{w;\nu }^{\mu \nu }=0$, consider the covariant divergence of Equation (6). From the contracted Bianchi identity satisfied by the Einstein tensor, we obtain

$U{\left(\varphi \right)}_{,\nu }\left[R^{\mu \nu }-\frac{1}{2}{g}^{\mu \nu }R\right]=-\frac{1}{2}\left[T_{w;\nu }^{\mu \nu }+t_{;\nu }^{\mu \nu }+{\Theta }_{;\nu }^{\mu \nu }\right]$(13)

with

$t^{\mu \nu }\equiv T_{\varphi }^{\mu \nu }+2U{\left(\varphi \right)}^{;\mu ;\nu }-2g^{\mu \nu }U{\left(\varphi \right)}_{;\lambda }^{;\lambda }$(14)

Using the identity:

$U{\left(\varphi \right)}^{;\rho }{R}_{\rho \alpha }=U{\left(\varphi \right)}_{;\lambda ;\alpha }^{;\lambda }-U{\left(\varphi \right)}_{;\alpha ;\lambda }^{;\lambda }$

and Equation (12), Equation (13) reduces to

$-\frac{1}{2}U{\left(\varphi \right)}^{,\mu }R=-\frac{1}{2}\left[T_{w;\nu }^{\mu \nu }+T_{\varphi ;\nu }^{\mu \nu }+{\partial }^{\mu }m\overline{{\psi }^{\prime }}{\psi }^{\prime }\right]$(15)

Finally, using the equation of motion for the scalar field (7), all the $\varphi$ dependent terms cancel the left hand side—giving the vanishing of the covariant divergence of the (w-) matter stress energy tensor. This in turn ensures that equivalence principle holds—in turn assuring the geodesic law.

Next, we write down the expression for a conserved pseudo energy momentum tensor [23] . Defining

$\text{A}\equiv \sqrt{-g}{g}^{\sigma \rho }\left[{\Gamma }_{\sigma \rho }^{\alpha }{\Gamma }_{\alpha \beta }^{\beta }-{\Gamma }_{\beta \rho }^{\alpha }{\Gamma }_{\alpha \sigma }^{\beta }\right]$(16)

$\text{B}\equiv \left[U\text{A}-\sqrt{-g}{g}^{\sigma \rho }{\Gamma }_{\sigma \alpha }^{\alpha }{U}_{,\rho }+\sqrt{-g}{g}^{\sigma \rho }{\Gamma }_{\sigma \rho }^{\alpha }{U}_{,\alpha }\right]$(17)

and $\hat{\text{B}}\equiv \text{B}+\sqrt{-g}{L}_{\varphi +\psi }$, the conserved pseudo energy momentum vector defined over a hypersurface Σ is:

$P_{\mu }\equiv {\displaystyle {\int }_{\Sigma }\text{d}\Sigma }\left[\sqrt{-g}{T}_{w\mu }^o-\hat{\text{B}}{\delta }_{\mu }^o-\frac{\partial \hat{\text{B}}}{\partial g_{,o}^{\tau \beta }}{g}_{,\mu }^{\tau \beta }-\frac{\partial \hat{\text{B}}}{\partial {\varphi }_{,o}}{\varphi }_{,\mu }-{\bar{\psi }}_{,\mu }\frac{\partial \hat{\text{B}}}{\partial {\bar{\psi }}_{,o}}-\frac{\partial \hat{\text{B}}}{\partial {\psi }_{,o}}{\psi }_{,\mu }\right]$(18)

This is also expressible as a surface integral over a 2 dimensional boundary of Σ:

$P_{\mu }=-{\displaystyle \int \left(\frac{\partial \hat{\text{B}}}{\partial g_{,j}^{\mu \nu }}{g}^{0\nu }\right)\text{d}{\Sigma }_j}$(19)

Thus in a generalised Brans-Dicke theory, the generalised energy momentum is determined by the metric tensor and its derivatives on a 2-dimensional boundary.

We consider classical solutions to Equations (6)-(9) for a fixed number of fermions. As demonstrated in [17] - [19] and [24] [25] the thermodynamic potentials for the fermion field are assumed to be adequately described in terms of the chemical potential and the temperature. Denoting the fermion density by $S_f$, Equation (7) reads:

$\square \varphi +\frac{\partial V}{\partial \varphi }-R\frac{\partial U}{\partial \varphi }+\frac{\partial m}{\partial \varphi }{S}_f=0$(20)

The trace of Equation (6) gives:

$U\left(\varphi \right)R=-\left(3U^{″}\left(\varphi \right)+\frac{1}{2}\right){\varphi }^{,\alpha }{\varphi }_{,\alpha }+2V\left(\varphi \right)+\frac{1}{2}m{S}_f-3U^{\prime }\left(\varphi \right)\square \varphi$(21)

Substituting in Equation (20) gives:

$\square \varphi +\frac{U^{\prime }\left(\varphi \right)\left(3U^{″}\left(\varphi \right)+\frac{1}{2}\right)}{1+3\frac{{U^{\prime }}^2}{U}}{\varphi }^{,\alpha }{\varphi }_{,\alpha }+\frac{V^{\prime }+m^{\prime }S-\frac{U^{\prime }}{U}\left(2V+\frac{1}{2}mS\right)}{1+3\frac{{U^{\prime }}^2}{U}}$(22)

Consider a familiar NMC theory with $U\left(\varphi \right)=\beta /8\pi -\xi {\varphi }^2/2$ and $V\left(\varphi \right)\equiv {\Lambda }_0/8\pi$ a constant. We define natural units in which $\beta$ takes on a unit value. The field equations simply read:

$\square \varphi +\xi R\varphi =0$(23)

and

$G_{\mu \nu }+{\Lambda }_O{g}_{\mu \nu }=-8\pi T_{\mu \nu }$(24)

with

$T_{\mu \nu }={\varphi }_{,\mu }{\varphi }_{,\nu }-1/2g_{\mu \nu }{\varphi }_{,\rho }{\varphi }^{,\rho }-\xi {\varphi }^2\left(R_{\mu \nu }-1/2g_{\mu \nu }R\right)+\xi g_{\mu \nu }\square {\varphi }^2-\xi {\varphi }^2{}_{;\mu ;\nu }$(25)

In terms of $\Phi \equiv {\varphi }^2$, the scalar field Equation (23) reads

$\square \Phi =-2\xi R\Phi +\frac{1}{2\Phi }{\partial }_{\mu }\Phi {\partial }^{\mu }\Phi$(26)

while the trace of Equation (24) is simply:

$-R+4{\Lambda }_0=-4\pi \left[-\frac{{\Phi }_{,\rho }{\Phi }^{,\rho }}{2\Phi }+2\Phi R\xi +6\xi \square \Phi \right]$(27)

The wave equation for the scalar field thus reduces to:

$\square =\frac{4{\Lambda }_0-R}{4\pi \left(1-6\xi \right)}$(28)

We shall follow the dynamics of $\varphi$ in a FRW metric:

$\text{d}{s}^2=\text{d}{t}^2-a^2\left(t\right)\left[\frac{\text{d}{r}^2}{1-k{r}^2}+r^2\left(\text{d}{\theta }^2+{\sin }^2\theta \text{d}{\phi }^2\right)\right]$(29)

For a homogeneous, time dependent scalar field, the above equations reduce to:

$\frac{3}{8\pi }\left[{\left(\frac{\stackrel{\dot{}}{a}}{a}\right)}^2+\frac{k}{a^2}\right]-\frac{{\Lambda }_0}{8\pi }=\frac{{\stackrel{\dot{}}{\Phi }}^2}{8\Phi }+3\xi \left[{\left(\frac{\stackrel{\dot{}}{a}}{a}\right)}^2+\frac{k}{a^2}\right]\Phi +3\xi \left(\frac{\stackrel{\dot{}}{a}}{a}\right)\stackrel{\dot{}}{\Phi }$(30)

and

$\ddot{\Phi }+3\left(\stackrel{\dot{}}{a}/a\right)\stackrel{\dot{}}{\Phi }=\frac{4{\Lambda }_0-R}{4\pi \left(1-6\xi \right)}$(31)

We look for solutions with the scale factor having a power law evolution: $a\left(t\right)=h{t}^{\alpha }$, with $h,\alpha$ arbitrary constants. The equation for Φ integrates exactly for all $\alpha \ne 1/3$ in terms of integration constants $K_o,C$:

$\Phi =A{t}^2/2+K_0{t}^{1-3\alpha }+C$(32)

where

$A=\frac{{\Lambda }_0}{\pi \left(1-6\xi \right)\left(1+3\alpha \right)}$(33)

(For $\alpha =1/3$ one has $K_0\ln \left(t\right)$ instead of the second term. From what just follows, this is not feasible as a late time solution). On substituting in Equation (20) it is found that it is possible to have $k\ne 0$ solutions for large times only if $\alpha \ge 1$. For $\alpha >1$, the solutions are independent of the curvature constant (i.e. hold for all $k$). The scale $\alpha$ directly relates to the parameter $\xi$ as: $\xi =-{\left(4\alpha -2\right)}^{-1}$. Similarly, for $\alpha =1$, $\xi =-{\left(2+2k/h\right)}^{-1}$. For all these cases, scalar condensates develop to have their energy cancell the effect of ${\Lambda }_0$:

$T_{\mu \nu }\to -\frac{1}{8\pi }{\Lambda }_0{g}_{\mu \nu }+O\left(t^{-2}\right)$(34)

The cancellation occurs over time scales large as compared to the time scales determined by dimensional parameters in the action. If one merely uses typical particle physics time scales, this implies that the stress energy of the scalar field would compensate the cosmological constant for the entire history of the universe except the very early history. It is the coupling of the scalar field to the spacetime curvature that is primarily responsible for the above instability of the scalar field which in turn leads to an efficient damping of the cosmological constant. Unfortunately, the effective value of the gravitational constant goes as $G_{eff}={\left(\beta +8\pi \left|\xi \right|{\varphi }^2\right)}^{-1}$. This becomes unacceptably small at large times. This is anexample of a “no-go” situation [3] . This behaviour was noted in [15] [16] for a flat $k=0$ model. We conclude that the presence of spatial curvature does not alter the results. It is also straightforward to demonstrate that the results are independent of the presence of the Planck sacle determined by ${\beta }^{-1}\equiv 8\pi G_0\equiv M_{plank}^{-2}$. Indeed in the absence of $\beta$ in the NMC function $U\left(\varphi \right)$, the FRW scale evolves as above: $a\left(t\right)\propto t^{\alpha }$ with $\alpha$ constrained as before in the presence of $\beta$, the stress tensor rapidly cancells the cosmological constant while the effective gravitational constant again approaches zero over time scales large in comparison to the particle physics time scales.

The above behaviour of a power law behaviour of the FRW scale factor and the cancellation of the vacuum energy has been generalized to scalar tensor theories within a large class of scalar field effective potentials [21] . There have also been attempts to use the above results to constrain dimensional parameters in the theory (see eg [26] ). One equates the observed gravitational constant to that calcuted above using conservative age estimates of the universe. Early universe nucleosynthesis constraints and recent Viking radar experiments severly constrain the NMC in a scalar tensor theory. The particle physics parameters that follow turn out to be rather unnatural. Instead we propose and explore the possibility that the above analysis successfully accounts for a NMC diverging with time and that an effective gravitational constant over most of the universe is indded vanishing. However, this is not the constant which ought to be equated with the Newtonian gravitational constant. Suspecting the problem to lie in the assumption of homogeneity, it has been suggested [15] [16] to look for a stabilization of $G_{eff}$ in an inhomogeneous model.

Taking a cue from the above, we consider a theory described by Equations (2) & (3). We shall be interested in dynamic NMC’s that diverge, as described above, outside bounded domains. It is convenient to recast the equations in terms of the effective planck length parameter which goes to zero as the NMC diverges. One such reparametrization of the scalar field is $\varphi \to M^2/\varphi$. The NMC becomes $U\left(\varphi \right)\approx \beta -\xi M^4/{\varphi }^{-2}$ [21] . With this reparametrization, the results of the last section translate to the vanishing of the effective gravitational and cosmological constants together with the vanishing of $\varphi$ over time scales large in comparison to any particle physics time scales in the theory. In other words we consider an equivalent problem that has the NMC divergent at any point that we take as $\varphi =0$ without any loss of generality. Thus $U\left(\varphi =0\right)\to \infty$. The model has a flat, Minkowski spacetime as a fixed point global solution with $\varphi =0$ for an arbitrary equation of state of matter. However, there are non-trivial solutions to the model as well [27] . We are particularly interested in solutions that have $\varphi \to 0$ outside compact domains but not in the interior. This would mean that the gravitational constant would vanish in the exterior. However, this does not imply doing away with gravitation between such domains. To see this, consider the Einstein tensor that follows from Equations (2) and (6):

$G_{\mu \nu }={\left[U\left(\varphi \right)\right]}^{-1}\times sourceterms$(35)

From the expansion of the metric about a flat space metric: $g_{\mu \nu }={\eta }_{\mu \nu }+h_{\mu \nu }$, in appropriate (harmonic) coordinates, one gets:

$\square h_{\mu \nu }={\left[U\left(\varphi \right)\right]}^{-1}\times sourceterms$(36)

The retarded solution reads:

$h_{\mu \nu }\left(x,t\right)={\displaystyle \int \frac{{\text{d}}^3{x}^{\prime }}{\left|x-x^{\prime }\right|}}{\left[U\left(\varphi \right)\right]}^{-1}\times sourceterms\left(x^{\prime },t-\left|x-x^{\prime }\right|\right)$(37)

Thus as long as the effective gravitational constant at the retarded source point $\left(x^{\prime },t-\left|x-x^{\prime }\right|\right)$ is bounded, the vanishing of the effective gravitational constant at the field point is of no consequence for the above solution. All one has to discover is the means of having the NMC locked to a universal value in non-trivial domains. The mechanism we invoke is to have the non-minimally coupled scalar field locked at the minimum of its effective potential [14] .

We recall studies on non-trivial configuration that have $\varphi$ vanishing outside, and $N_f\gg 0$ $\psi$ fermions trapped inside, compact domains in the Lee-Wick model [17] - [19] . With the fermion number conserving, the minimum energy configuration of such solutions are non-topological soliton solutions [NTS’s] in the theory. We recite the essential properties of such an NTS. Lee and Wick considered a theory with a Lagrangian:

$L=\frac{1}{2}{\partial }_{\mu }\varphi {\partial }^{\mu }\varphi -V\left(\varphi \right)+\bar{\psi }{\gamma }^{\mu }{\partial }_{\mu }\psi -m\bar{\psi }\left[1-\frac{\varphi }{{\varphi }_0}\right]\psi$(38)

where

$V\left(\varphi \right)=\frac{1}{2}{m}_{\varphi }^2{\varphi }^2{\left[1-\frac{\varphi }{{\varphi }_0}\right]}^2$(39)

With Fermion number conserved in this theory, solutions representing a degenerate distribution of fermions with a total fermion number $N_f$, at $\varphi ={\varphi }_0$, would remain confined there if the total energy of a fermion is less than its on-shell energy at $\varphi =0$. The NTS is a ball of radius $r_b$ inside which the scalar field is locked at $\varphi \approx {\varphi }_0$. Across the surface of the ball of thickness $~m_{\varphi }^{-1}$, the scalar field transits to $\varphi =0$. As a matter of fact, the spherical shape of the distribution follows from degeneracy of the fermion ensemble. For a degenerate fermion distribution, the fermion density is determined in terms of the chemical potential that occurs as a lagrange multiplier for the conserved fermi number of the ensemble. This determines the volume for a fixed conserved number of degenerate fermions. A positive definite surface energy would break the degeneracy of volume preserving (i.e. fermion number conserving) deformations of the distribution—choosing a configuration of minimum surface area. This gives the minimum energy configuration to be a sphere. In terms of the chemical potential $\mu$, the fermion density goes as ${\rho }_f~{\mu }^4$. The fermion number density goes as $n_f~{\mu }^3$. With the total energy and fermion number: $E_f={\rho }_{f}v~{\mu }^4{r}_b^3$, $N_f=n_{f}v~{\mu }^3{r}_b^3$, the chemical potential eliminates to give $E_f~N_f^{4/3}/r_b$. For a large enough $r_b$, the surface tension is given by the expression: $s~m_{\varphi }{\varphi }_0^2/6$, the total surface energy being $E_s=4\pi s{r}_b^2$. For a given $N_f$, one may vary the radius to seek the minimum of the volume and surface energies to give $E_f=2E_s$. The mass of the soliton is expressed as $M=E_f+E_s=3E_s=3E_f/2=12\pi s{r}_b^2$. This allows one to express:

$N_f~s^{3/4}{r}_b^{9/4};M~s^{1/3}{N}_f^{8/9};r_b~s^{-1/3}{N}_f^{4/9}$(40)

Thus with a given surface tension, one could get arbitrarily massive NTS’s by increasing the fermion number. However, the above flat spacetime analysis can not be extended past the Schwarzschild limit. In the Lee-Wick model, the Schwarzschild limit can be determined as the criteria for stability against gravitational collapse. As a matter of fact, gravitational instability occurs even before this limit is reached [28] . We shall use the Schwarzschild limit as a benchmark to keep sufficiently clear-off and as a convenience that ensures that the flat spacetime analysis holds. This limit is simply obtained by equating the radius of the NTS to the Schwarzschild radius for a critical mass ball $M_c$: equating $r_b=2G{M}_c$, and with the expression for the mass of a ball, one gets $M_c~{\left(48\pi G^{2}s\right)}^{-1}$. For $s~{\left(30Gev\right)}^3$, $M_c$ is roughly ${10}^{15}{M}_{\odot }$ with the critical radius some 10² Pc.

One can have NTS’s as large as some 10’s of kilo parsecs. This would require a class of low mass fermions that could keep the NTS from collapsing. In a hot big bang cosmology, any species of particles that decoupled from equilibrium when it was relativistic, ought to have the same relic density as say the relic photons or neutrinos: some ~200 per cc. To get ~10 to 100 KPc NTS’s, it would suffice to have $s~{\left(Mev\right)}^3$. This gives $N_f~{10}^{72}$ to 10⁷⁵ with the mass of the ball some 12 orders of magnitude smaller than the the Schwarzschild (critical) mass for $s~{\left(Mev\right)}^3$. We conclude that for such a surface tension and for NTS’s even as large as hundreds of kilo parsecs, one can consistently ignore the spacetime curvature in such solutions. In the following, we do this whenever convenience demands.

In the model described by Equation (3), fermions $\psi$ are confined to $\varphi \ne 0$ domains. It is easy to follow [27] [29] to demonstrate the existence of non-trivial solutions. In the “weak (gravitational) field approximation”, that would justify retaining only a first order deviation from a flat metric, the metric can be expressed in terms of the spherical [Schwarzschild] coordinates:

$\text{d}{s}^2={\text{e}}^{2u}\text{d}{t}^2-{\text{e}}^{2\bar{v}}\text{d}{r}^2-r^2\left[\text{d}{\theta }^2+{\sin }^2\theta \text{d}{\phi }^2\right]$(41)

We look for a static solution describing the scalar field trapped to a value $\varphi ={\varphi }_{in}$ in the interior of a sphere of radius $r_b$ and making a transition across a thin surface to $\varphi =0$ outside. The fermi gas trapped inside the soliton is described, in the Thomas Fermi approximation [17] - [19] , by the following distribution in momentum space: $n_k=\theta \left(k-k_f\right)$, $k$ being the momentum measured in an appropriate local frame that depends on $r$, and $k_f$ the fermi momentum. The fermion energy density is given by:

$W=\frac{2}{8{\pi }^3}{\displaystyle \int {\text{d}}^{3}k{n}_k{ϵ}_k}$(42)

with ${ϵ}_k=\sqrt{k^2+m{\left({\varphi }_{in}\right)}^2}$. The fermion number density ${\nu }_f$ and the non-vanishing components of fermion stress energy tensor are:

${\nu }_f=\frac{2}{8{\pi }^3}{\displaystyle \int {\text{d}}^{3}k{n}_k}$(43)

$T_t^t=W$

$T_r^r=T_{\theta }^{\theta }=T_{\phi }^{\phi }=T_{\rho }^{\rho }\equiv -T=-\frac{2}{8{\pi }^3}{\displaystyle \int {\text{d}}^{3}k{n}_k\frac{k^2}{3{ϵ}_k}}$(44)

The trace of the stress tensor is just:

$T_{\mu }^{\mu }=W-3T=m\left({\varphi }_{in}\right)S$(45)

with S the scalar density:

$S=\frac{2}{8{\pi }^3}{\displaystyle \int {\text{d}}^{3}k\frac{n_k}{{ϵ}_k}m\left({\varphi }_{in}\right)}$(46)

Defining $8\pi G_{in}\equiv U{\left({\varphi }_{in}\right)}^{-1}$ as the effective interior “gravitational constant”, the metric field equation in the interior can be expressed in the above spherical coordinates as:

$r^2{G}_t^t={\text{e}}^{-2\bar{v}}-1-2{\text{e}}^{-2\bar{v}}r\frac{\text{d}\bar{v}}{\text{d}r}=-8\pi G_{in}{r}^2\left[W+V\left(in\right)\right]$(47)

$r^2{G}_r^r={\text{e}}^{-2\bar{v}}-1+2{\text{e}}^{-2\bar{v}}r\frac{\text{d}u}{\text{d}r}=8\pi G_{in}{r}^2\left[T-V\left(in\right)\right]$(48)

$r^2{G}_{\theta }^{\theta }={\text{e}}^{-2\bar{v}}\left[r^2\frac{{\text{d}}^{2}u}{\text{d}{r}^2}+\left[1+r\frac{\text{d}u}{\text{d}r}\right]r\frac{\text{d}}{\text{d}r}\left(u-\bar{v}\right)\right]=8\pi G_{in}{r}^2\left[T-V\left(in\right)\right]$(49)

The scalar field satisfies:

${\varphi }_{;\mu }^{;\mu }+V^{\prime }+m^{\prime }\left(\varphi \right)S-U^{\prime }R=0$(50)

Taking the trace of the Einstein tensor $G_{\mu }^{\mu }$, the Ricci scalar $R$ substitutes in Equation (24) to give the following radial equation for the scalar field in the weak field limit:

$\frac{{\text{d}}^2\varphi }{\text{d}{r}^2}+\frac{2}{r}\frac{\text{d}\varphi }{\text{d}r}+F\left(U\right){\left[\frac{\text{d}\varphi }{\text{d}r}\right]}^2=\frac{\text{d}W}{\text{d}\varphi }$(51)

where

$\frac{\text{d}W}{\text{d}\varphi }\equiv \left[V^{\prime }+m^{\prime }\left(\varphi \right)S-\frac{U^{\prime }}{U}\left(\frac{mS}{2}+2V\right)\right]/\left[1+\frac{3{U^{\prime }}^2}{U}\right]$(52)

and

$F\left(U\right)\equiv \frac{U^{\prime }}{U}\left(\frac{1}{2}+3U^{\prime }\right)/\left[1+\frac{3{U^{\prime }}^2}{U}\right]$(53)

A sufficient condition for existence for a large NTS with ${\varphi }_{in}~\text{constant}$ is the vanishing of $W^{\prime }$ at ${\varphi }_{in}$. A convenient way to arrange this is to have $V\left(\varphi \right)$ to have a minima at ${\varphi }_{in}$, where $U^{\prime }/U$ is small, and to have fermions massless at ${\varphi }_{in}$.

The zero of $W^{\prime }$ then coincides with that for $V^{\prime }$. For a profile of $V$ described in Figure 1 , and $U\left(\varphi \right)=M_p^2+M^4/{\varphi }^2$ (with ${\varphi }^2\gg M^4/M_p^2$), the profile of $W$ closely follows that of $V$ except near $\varphi \approx 0$. The divergence of the NMC at $\varphi =0$ ensures the vanishing of $W^{\prime }\left(\varphi =0\right)$. The divergence of the NMC and in turn that of $F\left(U\right)$ also ensures that $\varphi =0$ is a solution with the Ricci scalar vanishing for arbitrary $V\left(\varphi =0\right)$. A typical non-trivial NTS solution has the scalar field locked to $\varphi =0$ outside a compact spherical domain of radius $r_b$ and transits to $\varphi ={\varphi }_{in}$ across the boundary:

$1-\frac{\varphi }{{\varphi }_{in}}\to \exp \left[-V^{″}\left({\varphi }_{in}\right)\left(r_b-r\right)\right]$

The transition zone is thus related only to $V^{″}\left({\varphi }_{in}\right)$ for large $r_b$. The radius $r_b$ is determined by the energetics of the ball and is governed by the number of trapped fermions inside the ball. The pressure of the degenerate fermions keeps the soliton from collapsing. As for the Lee-Wick NTS (see also [28] ) the total energy is expressible as the sum of (i) the volume energy, (ii) the surface energy and (iii) the energy of the fermions:

$E=\frac{4\pi }{3}V\left({\varphi }_{in}\right)r_b^3+4\pi s{r}_b^2+\alpha \frac{N_f^{4/3}}{r_b};\alpha \equiv \frac{1}{2}{\left(\frac{3}{2}\right)}^{5/3}{\pi }^{1/3}~O\left(1\right)$(54)

For a vanishing $V\left({\varphi }_{in}\right)$, the minimum of (ii) and (iii) give the total energy $E~N_f^{8/9}{\left(4\pi s\right)}^{1/3}$, with an energy per fermion $~{\left(4\pi s\right)}^{1/3}/N_f^{1/9}$. For $V\left({\varphi }_{in}\right)>0$, large $r_b,N_f$ solutions have energy $E~4{\left(4\pi V\right)}^{1/4}{N}_f/3$, with the energy per fermion roughly independent of the number of fermions. Finally, for negative $V\left(\varphi \right)$, NTS’s exist only for a small fermion number $\left({\bar{N}}_f\right)$ and for small enough $\left|V\right|$. In this last case, the existence of NTS’s requires the surface term to dominate over the volume term. This is possible only for small $r_b,{\bar{N}}_f$. The energy per fermion is again $~{\left(4\pi s^{\prime }\right)}^{1/3}/{\bar{N}}_f^{1/9}$. Energy per fermion for a large fermion number configuration is smaller, the closer the minimum of $\left|V\right|$ is to zero.

This can form the basis for a prescription to sort out the cosmological constant problem. We first recall that in standard GTR, the cosmological consatnt problem is sought to be solved by first making the cosmological constant dynamical: i.e. a constant of motion. This is done by using (for example) Witten’s [11] prescription by Coleman, Hawking and Weinberg [3] [30] [31] . One adds to the action a term:

$I_F=-\frac{1}{48}{\displaystyle \int {\text{d}}^{4}x\sqrt{-g}{F}_{\mu \nu \rho \sigma }{F}^{\mu \nu \rho \sigma }}$

with $F_{\mu \nu \rho \sigma }$ the exterior derivative of a three form gauge field $A_{\nu \rho \sigma }$:

$F_{\mu \nu \rho \sigma }={\partial }_{\left[\mu A_{\nu \rho \sigma }\right]}$

using the fact that $F_{\mu \nu \rho \sigma }$ is totally antisymmetric, it is proportional to the permutation tensor. The field equation for $A_{\nu \rho \sigma }$ implies that the proportionality factor is a constant. This reduces the action to:

$I_F=\frac{1}{2}{c}^2{\displaystyle \int {\text{d}}^{4}x\sqrt{-g}}$

With $c$ a constant of integration. This makes the effective cosmological constant a constant of integration. This being the case, then in a quantum theory we expect the wave function of the universe to be a superposition of states with different values of the cosmological constant. Hawking [30] has proposed that the application of conditional probability that homosapiens would come into being and study physics in a universe then solves the cosmological constant. However, such an anthropic principle is not a dynamical principle in the first place. Even if the evolution of an intelligent species were related to a dynamic prescription, it would be difficult to implement it in standard GTR where every value of an effective cosmological constant corresponds to a distinct FRW solution that can not be dynamically compared with a solution with a different cosmological constant.

In the structure that we have described, the situation is simpler. One can start with the action Equation (2) and consider NTS’s with a large average fermion density in the universe. Every NTS is a ball that has an effective gravitational constant vanishing in its exterior and having a unique value in its interior determined by the NMC at ${\varphi }_{in}$ where the effective potential is minimized. The effective cosmological constant vanishes in the exterior and the value of $V\left({\varphi }_{in}\right)$ contributes to the interior volume energy of the NTS. One can now play the same game as played in standard GTR and introduce $I_F$ as before to make $V\left({\varphi }_{in}\right)$ dynamical. Consider the wave function of the universe containing a large (conserved) number of fermions. We can expect the state vector of the universe to be a superposition of states with different $V\left({\varphi }_{in}\right)$. From the previous considerations, the minimum energy per fermion would be for the state that has $V\left({\varphi }_{in}\right)$ vanishing. The minimization of energy is a dynamic principle that may or may not have anything to do with the existence of apes!! With all NTS’s having common asymptotics for an arbitrary $V\left({\varphi }_{in}\right)$, a dynamical solution that uses an energy minimization prescription makes sense and is easy to implement.

It is interesting to note that were we to ensure bounded NMC inside domains as large as hadrons, one would again get an effective gravity model with no cosmological constant problem. Consider for example a theory described by Equation (42) in which the confined fermionic degrees of freedom were associated with baryons. With the NMC diverging at say $\varphi =0$ and the fermions confined to hadronic size microscopic $\varphi \ne 0$ domains, there would be effective gravitational effects for baryons. A “cosmological constant” as large as ${\Lambda }_{QCD}\approx {\left(0.2-0.3Gev\right)}^4$ would merely make a small contribution to the mass of a baryon and could be absorbed in its net mass. The vanishing of the covariant divergence of the matter stress energy tensor implies the geodesic law of motion for material particles. The geodesic law in turn implies the equality of the inertial mass and the passive gravitational mass. The gravitating (active) gravitational mass, however, would depend upon the number and binding energy of non-baryonic matter of the source. Considering the relatively large error bars in the determination of the gravitational constant in a typical Cavendish experiment, the change in the active gravitational mass would not be discernible. However, it is claimed [1] that the equality of the active and passive gravitational masses is established in lunar ranging experiments to a very high accuracy. For this reason (not withstanding our doubts on the claimed accuracy and interpretation of the above experiments) we feel that the large NTS’s described earlier offer a better promise for a problem free gravity theory.