Neutrino Lumps in Quintessence Cosmology

Neutrinos interacting with the quintessence field can trigger the accelerated expansion of the Universe. In such models with a growing neutrino mass the homogeneous cosmological solution is often unstable to perturbations. We present static, spherically symmetric solutions of the Einstein equations in the same models. They describe astophysical objects composed of neutrinos, held together by gravity and the attractive force mediated by the quintessence field. We discuss their characteristics as a function of the present neutrino mass. We suggest that these objects are the likely outcome of the growth of cosmological perturbations.

The mechanism responsible for the onset of the accelerating phase in quintessence cosmology remains undetermined. Explaining the emergence of an accelerating phase in recent cosmological times constitutes one of the most difficult challenges of quintessence models -the coincidence problem. A possible trigger for the acceleration has been proposed recently [1,2], arising through the interaction of the quintessence field with a matter component whose mass grows with time. This matter component may be identified with neutrinos [1,2,3]. In the proposed scenario the neutrinos remain essentially massless until recent times. When their mass eventually grows close to its present value, their interaction with the quintessence field (the cosmon) almost stops its evolution. The potential energy of the cosmon becomes the dominant contribution to the energy density of the Universe. Cosmological acceleration ensues.
For the coupled neutrino-cosmon fluid the squared sound speed c 2 s may become negative -a signal of instability [3]. Indeed, the sign of c 2 s oscillates in the accelerating phase for one of the proposed models [2]. A natural interpretation of this instability is that the Universe becomes inhomogeneous with the neutrinos forming denser structures. Within the linear approximation the neutrino fluctuations can be followed in these models until a redshift around one, when the neutrino overdensities become nonlinear [4]. One suspects that some form of subsequent collapse of these fluctuations will result into bound neutrino lumps. In this letter we present static, spherically symmetric solutions of the Einstein equations that describe such structures and study their characteristics. Astrophysical objects composed of neutrinos have also been studied in [5,6].
We assume that the energy density of the Universe involves a gas of weakly interacting particles (neutrinos). The mass m of the particles depends on the value of a slowly varying cosmon field φ [7]. For the field equation we approximate the neutrino energy-momentum tensor as T µ ν = diag(−ρ, p, p, p). The cosmology of [1,2] also assumes the presence of another gas of particles (dark matter) whose mass is independent of φ.
We consider stationary, spherically symmetric configurations, with metric For the neutrinos we assume a Fermi-Dirac distribution, with locally varying density -the Thomas-Fermi approximation. The local chemical potential satisfies µ(r) = µ 0 / B(r) [6,8]. Stable configurations are prevented from collapsing by the pressure generated through the exclusion principle. We concentrate on vanishing temperature of the neutrino gas. We do not expect qualitative changes of our solution for a non-zero temperature. For simplicity we consider one neutrino species, with the generalization to degenerate neutrino masses being straightforward. We parametrize the particle mass by a dimensionless functionm, defined according to m(φ) = σm (φ −φ)/M , with σ an arbitrary energy scale and M = (16πG) −1/2 ≃ 1.72 × 10 18 GeV. Hereφ is a fixed reference value, close to the present value of the quintessence field. Hence, σ is of the order of the present neutrino mass, in the eV range or somewhat below. For concreteness, we consider a cosmon potential of the form (1). However, the effect of the potential on our solutions is negligible. For this reason, the predicted astrophysical objects are largely independent of the form of the potential, and depend mainly on the interaction between dark energy and neutrinos. The present cosmological value of φ is given by the requirement that U (φ) constitute about 3/4 of the critical energy density U (φ) ≃ 10 −11 (eV) 4 . The cosmological value of φ is taken as the asymptotic value φ as of our local solutions for large r, obeying (φ as −φ)/M =φ as ≃ (1/a) [25.3 + ln C + 4 ln (σ/eV)].
The equations of motion become more transparent if we define the dimensionless variablesφ = (φ −φ)/M andr = σ 2 r/M . All other dimensionful quantities are multiplied with appropriate powers of σ, in order to form dimensionless quantities denoted as tilded.
We useB = B/μ 2 0 = Bσ 2 /µ 2 0 andμ(r) = 1/ B (r). We define the radiusR of the compact object by the value ofr at which the fermionic density becomes negligible. The physical radius is The mass of the object is given by its Schwarzschild radiusR s . Forr → ∞ we have B = 1/A = 1 −R s /r. In units of the solar mass, the mass of the neutrino lump is Another important characteristic is the total neutrino number, which we assume to be conserved. It is The field equations read [8] where a prime denotes a derivative with respect tor. We also havẽ forμ ≥m, andñ =ρ =p = 0 forμ <m. Finally,Ũ (φ) = C exp −aφ . We need four initial conditions for the system of equations (3). Two of them are imposed by the regularity of the solution atr = 0:φ ′ (0) = 0, A(0) = 1. The value ofB(0) is the only free integration constant. Since AB(r → ∞) = 1 one has AB(r → ∞) = (µ 0 /σ) −2 . As a result, the choice ofB(0) determines the chemical potential and, therefore, the total number of neutrinos in the lump. Finally,φ(0) must be chosen so that φ(r → ∞) reproduces correctly the present valueφ as of the cosmological solution. (We assume that the time scale of the cosmological solution is very large and neglect the time dependence ofφ(r → ∞).) We consider two types of models, distinguished by the dependence of the particle mass on the field: Model I assumesm(φ) = −1/φ [2], with the fieldφ taking negative values. [1], where a different convention for M is used.) In both cases we are interested in values of the field nearφ = 0. For model II we can chooseφ such thatφ as = 0, implying that σ = m ν (t 0 ) equals the present neutrino mass. One infers for the quintessence potential ln C = −25.3 − 4 ln (σ/eV). The parameter a is fixed by requiring that during the early stages of the cosmological evolution the dark energy be subleading and track the radiation or the dark matter. During the radiation and matter dominated epochs, the dark energy follows a "tracker" solution with a con-stant density parameter Ω h,early = n/(2a 2 ), where n = 3 (4) for matter (radiation) [1]. Observations require a to be large, typically a > ∼ 7 [9]. We use a = 7 in the following. The future of our Universe is described by a different attractor, for which the dark energy dominates. Our present era coincides with the transition between the two cosmic attractors. The influence of the neutrinos on the evolution of the cosmon field is determined by the second term in the r.h.s. of eq. (1). Demanding that today this term be equal to the first term, that arises from the potential, fixes the present neutrino fraction to the value Ω ν (t 0 ) = −(b/a)Ω h (t 0 ) [1]. For a realistic cosmology with present dark energy fraction Ω h (t 0 ) ≃ 3/4 one has to adjust b to the neutrino mass. For one dominant neutrino species we have b = −a(36 eV/m ν (t 0 )). For model I we need to know how closẽ φ as is to zero, with σ = −φ as m ν (t 0 ). As compared to model II, we have now an effectivẽ φ-dependent b(φ) = −1/φ, which results in the conditionφ as = −(1/a)m ν (t 0 )/(36 eV) or σ = (1/a)m 2 ν (t 0 )/(36 eV) . In fig. 1 we present a typical solution describing a static astrophysical object in model I. The chemical potential has the valueμ 0 ≃ 2.9. The scalar field becomes more negative near the center of the solution, so that the neutrinos become lighter there. The asymptotic value isφ as = −0.02, which corresponds to m ν (t 0 ) ≃ 5 eV. The pressure and density of the fermionic gas vanish forr ≥R ≃ 0.91. The mass of the object can be deduced from the asymptotic form of A or B forr → ∞. We findR s ≃ 0.12. The total fermionic number isÑ ≃ 0.88. The form of the solutions in model II is similar to the one depicted in fig. 1.
The variation of the chemical potential results in a whole class of solutions, depicted by the solid line in fig. 2. We display the dimensionless Schwarzschild radiusR s as a function of the dimensionless radius of the objectR. There is a maximal value for the mass, denoted by the end of the thick line. The continuation of the curve has the form of a spiral and is depicted by a thinner line. This branch is unstable to perturbations that can lead to gravitational collapse [10]. In order to demonstrate this fact, we plot in the same figureR s as a function ofÑ /6 (dotted line). This curve has two branches. The one depicted by a thinner line corresponds to the thinner line of the curveR s (R). There are two possible values ofR s that correspond to the same value of the total neutrino number N . The value on the thinner line has a larger value ofR s and results in a larger mass. The corresponding configuration is unstable towards one with the sameÑ located on the thicker line. A striking feature is the existence of neutrino lumps with arbitrarily small mass. They correspond to the lower left corner of the figure, where bothR andR s vanish. For such objects the contribution from gravity is negligible and their existence is a consequence of the attractive force mediated by the scalar field. Such configurations are not generic, but depend crucially on the assumed form ofm(φ). A completely different form of solutions appears in model II. In fig. 2 we also depict the gravitational potential Φ(r) = −R s /(2r) at a distancer =R equal to the radius of the astrophysical object.
The functionR s (R) in model II displays a different behaviour. In fig. 3  The crucial qualitative difference with model I concerns the form of the solutions with low values ofR s . In model I forR s → 0 we haveR → 0, while in model II we haveR → ∞. The attractive interaction mediated by the scalar field in model II is not sufficiently strong to lead to bound objects with a small fermion number. Gravity must play a role for compact objects to exist. As |b| increases the dependence ofm onφ becomes more pronounced. The effective neutrino mass in the interior of a compact object can become smaller without a large variation ofφ (and a significant energy cost through the field derivative term). This has two significant effects: a) Objects with smallerÑ andR can exist. As a result the bending of the curveR s (R) for lowR s takes place for smaller R. b) The configurations that are gravitationally unstable (indicated by the spiral in the upper part of the curve) are shifted toward larger values ofR s . The reason is that the neutrinos are essentially massless in the interior of of such configurations, carrying only  The curvesR s (R) in model II with b = −500 and −50 are also depicted in fig. 3. We have not managed to determine numerically a maximal value ofR s , as objects with huge values ofR s ,R (larger by more than twenty orders of magnitude than the ones depicted) are possible. For comparison we note than in model I we have a maximal value (R s ) max = 0.80 with a corresponding radius (R) max = 2.0. In fig. 3 we observe minimal values of the radius, (R) min = 0.54 for b = −50 and (R) min = 0.054 for b = −500.
The corresponding values of the Schwarzschild radius are (R s ) min = 1.1 × 10 −5 and (R s ) min = 1.1 × 10 −8 , respectively. It is apparent that for smallR s we have the scaling behaviourR ∼ b −1 ,R s ∼ b −3 . This can be understood by noticing that in the limit A ′ , B ′ → 0, A → 1, and for negligible dŨ /dφ, the factors of b in eq. (3) can be eliminated through the redefinitions bφ →φ, br →r. In fig. 3 we also depict the surface gravitational potential Φ(R) = −R s /(2R) as a function ofR for the cases b = −500 and −50.
In fig. 4 we display the size R of the astrophysical objects as a function of the present neutrino mass m ν ≡ m ν (t 0 ). Restoring physical units requires the scale σ. We use for model I a = 7 or (σ/eV) 1/2 ≃ 0.063(m ν /eV). The functionR s (R) has a very mild dependence onφ as for 0 ≤φ as ≤ 0.02 (see fig. 2). For givenR the variation of σ (or equivalently m ν ) produces a class of astrophysical objects of variable physical size. They all generate the same surface gravitational potential Φ = −R s /(2R). In fig. 4 we depict three such classes. The first two contain objects with strong gravitational potentials, while the last one contains objects that generate weaker fields. Solutions withR → 0 produce curves parallel to those in fig. 4, but located closer to the lower left corner. In the same figure we also depict two solutions of model II. In this model the neutrino mass is uniquely determined by the value of b. The two points in fig. 4 correspond to the minimal values ofR for b = −50 and −500. These areR = 0.54 andR = 0.054, respectively. Recently, a first investigation of the coupled fluctuations of dark matter, neutrinos, baryons and the cosmon field has been performed for the models within the linear approximation [4]. For a specific model with a present average neutrino mass of 2.1 eV, the neutrino fluctuations grow nonlinear at a redshift around one. The typical size of these fluctuations is large, in the range of superclusters and beyond. A further investigation of the fate of these neutrino lumps will have to follow their collapse due to the scalarmediated attractive interaction and gravity. This should generate the distribution of the integration constants of the present solution, like the characteristic mass and size of the lumps.
Our study demonstrates that the presence of instabilities in quintessence cosmologies with a variable neutrino mass may have interesting astrophysical consequences. After a sufficiently long time, these instabilities may lead to the formation of stable bound neutrino lumps. Their radius and mass within the family of allowed solutions (for given m ν ) depend on the details of the dynamical formation mechanism. Since in the models of [1,2] the neutrinos remain free streaming until a rather recent cosmological epoch (say, z = 5), one may expect a large typical size of the neutrino lumps (more than 100 Mpc). At the present stage of the investigations it is not clear if such lumps have already decoupled from the cosmological expansion -for this, the perturbations have to grow nonlinearor if this will happen only in the future. In the extreme case of an early formation of a population of lumps with subgalactic size, they could even play the role of dark matter. The detection of lumps could proceed directly through their gravitational potential, or indirectly through their attraction for baryons. Quintessence cosmologies may provide surprises for structures on very large scales.