Neutrino Lump Fluid in Growing Neutrino Quintessence
aa r X i v : . [ a s t r o - ph . C O ] N ov Neutrino Lump Fluid in Growing Neutrino Quintessence
Youness Ayaita, ∗ Maik Weber, and Christof Wetterich
Institut f¨ur Theoretische Physik, Universit¨at HeidelbergPhilosophenweg 16, D–69120 Heidelberg, Germany (Dated: August 2, 2018)Growing neutrino quintessence addresses the why now problem of dark energy by assuming thatthe neutrinos are coupled to the dark energy scalar field. The coupling mediates an attractive forcebetween the neutrinos leading to the formation of large neutrino lumps. This work proposes aneffective, simplified description of the subsequent cosmological dynamics. We treat neutrino lumpsas effective particles and investigate their properties and mutual interactions. The neutrino lumpfluid behaves as cold dark matter coupled to dark energy. The methods developed here may findwider applications for fluids of composite objects.
I. INTRODUCTION
The observed accelerated expansion of the Universecan be described by a dark energy component [1, 2]. Itsenergy density dominates that of matter at present, whileit constituted a very small fraction of the energy budgetin earlier stages of the cosmic evolution [3, 4]. This “whynow” problem has motivated the idea of dark energy be-ing dynamically coupled to other cosmological species.It has been proposed that a dependence of the neutrinomass on the dark energy scalar field, the cosmon, maynaturally trigger the onset of accelerated expansion in re-cent times [5, 6]. The background evolution of the result-ing cosmological model, growing neutrino quintessence,is similar to the concordance model with a cosmologicalconstant Λ.Since the energy density in neutrinos is small, thecosmon-mediated attraction between neutrinos has to besubstantially stronger than the gravitational one in orderto be effective. This results in a fast formation of neutrinolumps of the size of clusters or larger at redshift aroundone. The dynamics of the perturbations in the cou-pled cosmon-neutrino fluid is complicated. In contrastto models of uncoupled or weakly coupled dark energy, amere analysis of the background equations together withlinear perturbation theory is insufficient. Linear pertur-bation theory breaks down even at large scales [7], andthe nonlinear evolution exerts significant backreaction ef-fects on the background evolution. This has lead to thedevelopment of a specifically designed N-body based sim-ulation method, which accounts for local cosmon pertur-bations, relativistic neutrino motion, and backreactioneffects [8]. These simulations are, so far, successful until z ≈
1, where a collection of spherical neutrino structureshas formed, cf. Fig. 1.Although it is numerically challenging to resolve theinternal dynamics of the neutrino lumps, these detailsmay not be crucial for the broad cosmological picture.In gravity, e. g., the detailed evolution inside galaxies orclusters is not relevant for the cosmological evolution. ∗ [email protected] FIG. 1. Neutrino structures in a simulation box of comovingsize L = 600 h − Mpc at redshift z = 1. Shown are regionswith a neutrino number density contrast above 5 [8]. Once the neutrino lumps have formed, one would like touse a picture of a pressureless fluid of neutrino lumps.In contrast to the universal properties of gravity, whereonly the total mass of a bound object matters, the under-standing of neutrino lumps needs more information. Themass of a lump with a given number of neutrinos is stillexpected to depend on the local value of the cosmon fieldˆ ϕ averaged in a region around the lump. This effectivecoupling of ˆ ϕ to the lumps induces an effective attrac-tive interaction between the lumps. Since the lumps arehighly nonlinear objects, the ˆ ϕ -dependence of the massis sensitive to the total number of neutrinos in the lumpand possibly even to additional properties of the lump.This work presents analytical and numerical studies ofthe properties of the neutrino lumps. We indeed find aneffective description. This opens the possibility for anapproximate and much simpler approach to the under-standing of the cosmological evolution of growing neu-trino quintessence for the period after the formation ofthe lumps.The paper is organized as follows. We collect somebasics of growing neutrino quintessence in Sec. II andmotivate the approach taken in this work. Section IIIdescribes the effective cosmological dynamics in the pres-ence of stable neutrino lumps. Starting from the basicidea of approximating lumps as particles, we eventuallydevelop a simplified simulation scheme of growing neu-trino quintessence. Some more technical aspects requiredfor this scheme are postponed to Sec. IV. The questionof stability of neutrino lumps is discussed in Sec. V. Weconclude in Sec. VI. II. FUNDAMENTALS AND MOTIVATION
After briefly summarizing the basics of growing neu-trino quintessence in Sec. II A, we explain the main ideaof this work. We give physical arguments why the neu-trino lumps may be approximated as nonrelativistic par-ticles. This forms the basis of the effective description ofthe cosmological dynamics presented in Sec. III.
A. Basics of growing neutrino quintessence
The cosmon-neutrino coupling is described by theenergy-momentum exchange ∇ λ T µλ ( ϕ ) = + βT ( ν ) ∂ µ ϕ, (1) ∇ λ T µλ ( ν ) = − βT ( ν ) ∂ µ ϕ, (2)where β is a dimensionless coupling parameter and T ( ν ) ≡ T µλ ( ν ) g µλ is the trace of the neutrino energy-momentumtensor. We work in units where 8 πG = 1 and use the met-ric convention ds = − (1 + 2Ψ) dt + a (1 − d x . Thistype of coupling corresponds to early proposals of cou-pled quintessence [9, 10]. On the particle physics level,the coupling is realized as a dependence of the (average)neutrino mass m ν on the cosmon field [6]: β = − d ln m ν dϕ . (3)For simplicity, we consider the case of a constant couplingparameter β as used in, e. g., [7, 8, 11]. Typical valuesare of order β ∼ − .When the cosmon rolls down its potential towardslarger values, a negative β implies a growing neutrinomass. As long as the neutrinos are highly relativistic( w ν ≈ / T ( ν ) = − ρ ν (1 − w ν ) is close tozero and hence the coupling is small. This changes oncethe neutrinos become nonrelativistic. The coupling thenstops the further evolution of the cosmon resulting in aneffective cosmological constant. In this way, the modeladdresses the “why now” problem of dark energy. As instandard quintessence models [12, 13], the energy densityof the dark energy scalar field ϕ decays similarly to theother species during most of the cosmological evolution,thereby alleviating the fine-tuning of the present amountof dark energy. The energy-momentum exchange, Eqs. (1) and (2), im-plies [14], in the Newtonian limit, an attractive force be-tween the neutrinos of order | F | ≈ | β ∇ ϕ | ≈ β | F gravity | . (4)We shall see that the interaction between neutrino lumpsis similar but with an effective coupling weaker than β . B. Lumps as nonrelativistic particles
The simulations of growing neutrino quintessence haveshown that, after a phase of rapid neutrino clustering, al-most all cosmic neutrinos are bound in roughly sphericallumps, cf. Fig. 1.Inside these lumps, the neutrinos have relativistic ve-locities [8, 11]. For the neutrino fluid alone, one thusobserves a large pressure such that a nonrelativistic treat-ment is not applicable. This is reflected in the equationof state w ν = p ν /ρ ν , which reaches w ν ≈ . z = 1 [8].Nevertheless, we argue that the lumps as static boundobjects behave as particles with vanishing internal pres-sure. The pressure induced by the neutrino motions iscancelled by a corresponding negative pressure of the lo-cal cosmon perturbations. Furthermore, the peculiar ve-locities of the lumps are nonrelativistic. This is similar toa gas of atoms at low velocities. Although the electronsmove at high velocities, their contribution to the pressureis cancelled by a contribution from the electromagneticfield.Whereas the total pressure of a lump vanishes, thecontributions of neutrinos and the cosmon perturbationdo not cancel locally. The neutrinos are rather concen-trated and hence their pressure contribution is restrictedto a small radius. The cosmon perturbation, in contrast,extends to larger distances, analogously to the gravita-tional potential around a massive object. The cancella-tion thus only refers to the integrated contributions at asufficiently large distance from the lump.In the following, we discuss this in more detail. Sincegravity is subdominant compared to the fifth force, cf.Eq. (4), it may be neglected for a simple discussion. Ongeneral grounds, one can show that a bound object hasvanishing pressure if three conditions are met:1. The object is described by a conserved energy-momentum tensor.2. The energy-momentum tensor vanishes outside avolume surrounding the object.3. The energy-momentum tensor is static.The argument is given in Sec. IV A. For the purpose ofillustration, we have numerically simulated an exemplaryspherical neutrino lump satisfying these idealized condi-tions. The staticity of the lump was realized by a hy-drodynamic balance equation, cf. Sec. V B. The neutrinopressure integrated to a comoving radius r from the cen-ter is given by a sum over particles pP ν ( r ) = Z r πr dr p g (3) T i ( ν ) i = X p γ p m p v p , (5)with the Lorentz factor γ p and the determinant of thespatial metric p g (3) ≈ a . The contribution of the cos-mon perturbation is P δϕ ( r ) = − Z r πr dr p g (3) (cid:20) | ∇ δϕ | a + V ′ ( ¯ ϕ ) δϕ (cid:21) , (6)where we have subtracted the pressure induced by thebackground field ¯ ϕ . Figure 2 shows the cancellation ofthe pressure contributions for large radii. As already ex- -0.003-0.002-0.001 0 0.001 0.002 0.003 0.004 0 10 20 30 40 50neutrinos + cosmonneutrinoscosmon PSfrag replacements physical radius ar [ h − Mpc] P ( r ) / M l FIG. 2. Integrated pressure contributions P ν (black dashed), P δϕ (black dotted), and their sum (red solid), normalized bythe lump mass M l . plained, the cosmon contribution is more extended thanthe neutrino contribution.In the cosmological context, the aforementioned con-ditions are met, at best, approximately and realistic neu-trino lumps will not be exactly pressureless. We shall nowdiscuss the three conditions. First, only the total energy-momentum tensor of neutrinos, local cosmon perturba-tion, and background cosmon is conserved. The back-ground field, however, cannot be attributed to the lump(otherwise, the second condition would not be satisfied).The energy-momentum tensor of the lump, defined to in-clude the neutrinos and the local cosmon perturbation,is thus not exactly conserved due to exchange betweenthe lump and the outside cosmon field. Finally, even fora virialized lump with a fixed number of neutrinos, theenergy-momentum tensor is not static. Due to the timeevolution of the outside cosmon field, the mass of theneutrinos and therefore the mass of the lump changes.One may argue that these effects are suppressed by thedifference in the relevant time scales for the dynamics ofthe lump and the cosmological evolution. Indeed, the vi-olations of staticity and energy-momentum conservation are proportional to the time derivative of the cosmon fieldaveraged on length scales much larger than the size of thelump. This is suppressed by the fact that the associatedtime scale is large as compared to the dynamical timescale of the lump. The effective description of growingneutrino quintessence presented in the next section as-sumes that the pressure of neutrino lumps approximatelyvanishes. III. EFFECTIVE DYNAMICS
The approach of this section is to treat the neutrinolumps as effective particles. We then merely have to char-acterize their mutual interactions and their influence onthe background as well as on the gravitational potential.A numerical treatment of the internal structure of thelumps will no longer be required.In Sec. III A, we shall describe how lumps can betreated as particles with an effective coupling. Sec-tion III B derives the equation of motion for these ef-fective particles and explains how to calculate the rele-vant potentials: the large-scale cosmon ˆ ϕ and the grav-itational potential ˆΨ. Finally, we explain in Sec. III Chow the results can be used to construct the simplifiedsimulation scheme for growing neutrino quintessence. A. Description of lumps
Let us introduce a comoving length scale λ , which islarger than the typical lump sizes but smaller than theirtypical distances (the mean distance between neighboringlumps is of order 100 h − Mpc). On scales larger than λ , a lump l at comoving coordinates x l looks effectivelypoint-shaped, T µνl ≈ A µν p g (3) δ (3) ( x − x l ) , (7)the amplitude A µν being given by the integrated localenergy-momentum tensor of the lump, A µν = Z d y p g (3) T µν local ( y ) . (8)We will see in Sec. IV A that this indeed reduces to thestandard one-particle case A µν ≈ M l γ u µ u ν , (9)where M l is the lump’s rest mass (consisting of a neutrinoand a cosmon contribution) and u µ is its four-velocity.The Lorentz factor is defined as γ = √− g u . In thebackground metric, we have γ = u . The result for A µν is a consequence of the approximate pressure cancellationdiscussed in Sec. II B.The interactions between the lumps are mediated bythe cosmon field ϕ . Given that the distances betweenthe lumps are greater than λ , it suffices to consider thesmoothed field (indicated by a hat)ˆ ϕ ( x ) = Z d y p g (3) W λ ( x − y ) ϕ ( y ) (10)with a suitable window W λ of size λ .Analogously to the fundamental coupling parameter β ,Eq. (3), we may define the effective coupling by β l = − d ln M l d ˆ ϕ . (11)The effective coupling may depend on the scale λ overwhich the field is averaged. Whereas the fundamentalcoupling β describes the dependence of the microscopicneutrino mass m ν on the local cosmon field ϕ , the ef-fective coupling β l measures the mass dependence of thetotal lump mass M l on the large-scale cosmon value ˆ ϕ .As the fundamental parameter β quantifies the force be-tween neutrinos, cf. Eq. (4), the effective parameter β l will determine the interactions between lumps.We next show quantitative results for the distributionof lumps and the effective couplings at z = 1. For thispurpose, we have performed 10 simulation runs with themethod and the parameters of Ref. [8]: fundamental cou-pling β = −
52, box size L = 600 h − Mpc, but withreduced resolution N cells = 128 . The positions of thelumps have been identified as local maxima of the neu-trino density field (cf. DENMAX halo finding [15]). Aglance at Fig. 1 shows that there is not much ambiguityin identifying lumps.Once a stable lump has formed, the number of boundneutrinos is approximately fixed (neglecting merging pro-cesses). It is thus natural to characterize different lumpsby their amount of neutrinos.We measure the effective couplings β l and the lumpmasses M l . The latter include a (dominant) neutrinocontribution M ( ν ) l and a somewhat smaller cosmon part M ( ϕ ) l . Integration over the comoving lump volume V l yields γ l M ( ν ) l = Z V l d x p g (3) ρ ν ≈ X particles p γ p m ν,p , (12) γ l M ( ϕ ) l = Z V l d x p g (3) ( ρ ϕ − ρ ˆ ϕ ) , (13)with ρ ϕ = ˙ ϕ + | ∇ ϕ | a + V ( ϕ ). The Lorentz factor γ depends on the velocities of the lumps or particles, re-spectively. The smoothed field ˆ ϕ is considered externalto the lump and thus subtracted.Figure 3 shows the abundance of lumps and the distri-butions of β l and M l . The couplings β l are measured bynumerical differentiation according to Eq. (11). The twolower figures show approximate functional dependenceson the neutrino amount with only relatively small statis-tical fluctuations. The effective coupling is systematicallyweaker than the fundamental coupling. This becomesmore pronounced with increasing neutrino number. -7 -6 -5 -4 -3 -2 PSfrag replacements threshold f l u m p a bund a n c e N ( f l > f ) i n V H -7 -6 -5 -4 -3 -2 PSfrag replacementsthreshold f lump abundance N ( f l > f ) in V H neutrino number fraction f l e ff e c t i v e c o up li n g β l / β -7 -6 -5 -4 -3 -2 PSfrag replacementsthreshold f lump abundance N ( f l > f ) in V H neutrino number fraction f l effective coupling β l /β neutrino number fraction f l l u m p m a ss M l / M ⊙ FIG. 3. Lump abundances, effective couplings, and lumpmasses as functions of the neutrino number fraction f l (num-ber of neutrinos in the lump normalized to the number ofneutrinos in the Hubble volume V H = H − ) at redshift z = 1.The error bars indicate the variance of lumps in the same bin. For the averaging scale λ , we have taken λ =30 h − Mpc. This is clearly smaller than the typical lumpdistances ∼ h − Mpc but larger than the neutrinoconcentration of the lumps. Concerning the cosmon field,there remains some ambiguity since we attribute onlythe cosmon perturbations at scales smaller than λ to thelumps. If λ is chosen larger, the pressure cancellation andthus the particle approximation are better, cf. Fig. 2, butthere may arise overlaps between spatially close lumps. B. Evolution equations
Within our effective description, the equation of mo-tion of a neutrino lump is derived from the standard one-particle action S = Z d x √− g T µνl g µν = − Z dτ M l ( ˆ ϕ ) (14)with the proper time τ and the smoothed cosmon field ˆ ϕ evaluated at the lump trajectory. Along the same linesas for the single neutrino case [8], we arrive at du µ dτ + Γ µρσ u ρ u σ = β l ∂ µ ˆ ϕ + β l u λ ∂ λ ˆ ϕ u µ . (15)The left-hand side describes gravity (expansion and grav-itational potential), the right-hand side is due to thecosmon-neutrino interaction. The (spatial) term β l ∇ ˆ ϕ is the cosmon-mediated fifth force analogous to Newto-nian gravity, cf. Eq. (4). The second contribution on theright-hand side reflects momentum conservation: A lumpis accelerated when it moves towards a direction whereit loses mass.In order to use this effective equation of motion, weneed to know the smoothed cosmon field ˆ ϕ , the gravita-tional potential Ψ, and the background evolution. In thefollowing, we shall describe how this is achieved.For the calculation of ˆ ϕ , we recall the coupled Klein-Gordon equation, separated in background and pertur-bation parts [8],¨¯ ϕ + 3 H ˙¯ ϕ + V ′ ( ¯ ϕ ) = − β ¯ T ( ν ) , (16)∆ δϕ − a V ′′ ( ¯ ϕ ) δϕ = β a δT ( ν ) . (17)In Eq. (17) we have neglected the gravitational potentialagainst the cosmon perturbation. The second equationis similar to the gravitational Poisson equation. A natu-ral choice for the cosmon potential V is the exponentialpotential V ( ϕ ) ∝ exp( − αϕ ) [16].Next, we will smooth the perturbation equation (17).For the left-hand side, it is straightforward to show bypartial integration that d ∆ δϕ ( x ) = Z d y p g (3) W λ ( x − y )∆ y δϕ ( y )= ∆ x Z d y p g (3) W λ ( x − y ) δϕ ( y )= ∆ δ ˆ ϕ ( x ) , (18)up to surface terms and neglecting the metric pertur-bations, p g (3) ≈ a . On the right-hand side, we write [ δT ( ν ) = ˆ T ( ν ) − ¯ T ( ν ) with the smoothed energy-momentumtensor of neutrinosˆ T ( ν ) ( x ) = Z d y p g (3) W λ ( x − y ) T ( ν ) ( y ) . (19) We next employ the relation (shown in Sec. IV B) β ˆ T ( ν ) ≈ X lumps l β l ˆ T l . (20)Here, the smoothed trace of the energy-momentum ten-sor of a lump ˆ T l can be calculated from the effective lumpenergy-momentum tensor, Eqs. (7) and (9), T l = T µνl g µν :ˆ T l = Z d y p g (3) W λ ( x − y ) T l ( y )= − M l γ l W λ ( x − x l ) . (21)With these results, the smoothed perturbation equationeventually reads∆ δ ˆ ϕ ( x ) − a V ′′ ( ¯ ϕ ) δ ˆ ϕ ( x ) = − a X lumps l β l M l γ l W λ ( x − x l ) − βa ¯ T ( ν ) . (22)Assuming that all neutrinos are bound in lumps, one has β ¯ T ( ν ) = − V phys X lumps l β l M l γ l (23)in some cosmological volume V phys .On scales larger than λ , the window W λ ( x − x l ) inEq. (22) may be replaced by a point ∝ δ (3) ( x − x l ). Foran approximate solution of Eq. (22) at distances largerthan λ from the sources, we thus use a sum of Yukawapotentials, δ ˆ ϕ ≈ X lumps l (cid:18) β l πa M l /γ l | x − x l | e − am ϕ | x − x l | + δϕ res ,l (cid:19) (24)with the scalar mass m ϕ ≡ V ′′ ( ¯ ϕ ). The residual term δϕ res ,l ( x − x l ) is needed to cancel the background part ∝ ¯ T ( ν ) on the right-hand side and to ensure δϕ = 0 in asimulation volume, similar to Ψ res ,l below.If the lumps are moving rather slowly compared to thespeed of light, γ l ≈
1, the two smoothed metric potentialsare equivalent, ˆΦ ≈ ˆΨ. Numerically, this relation is veri-fied on large scales [8]. Then, we write for the smoothedgravitational potential induced by lumps (with the sameapproximations as for ˆ ϕ )∆ ˆΨ( x ) ≈ a X lumps l M l δ (3) ( x − x l ) p g (3) − M l V phys ! . (25)The solution is (up to a constant)ˆΨ( x ) = − X lumps l (cid:18) πa M l | x − x l | + Ψ res ,l (cid:19) , (26)where the residual contribution can be given explicitly asΨ res ,l = M l a | x − x l | /V phys . The total gravitational po-tential also includes the matter-induced potential whichis calculated as usual. Taking into account relativisticcorrections would require the calculation of both poten-tials, ˆΨ and ˆΦ, cf. Ref. [8].In order to have a full description of the cosmologicaldynamics, we still need to describe the evolution of thecosmological background, i. e. the Hubble expansion H and the background cosmon ¯ ϕ . The background evolu-tion cannot be calculated without taking into account the backreaction due to the perturbation evolution [8, 17]. In-stead, the background and the perturbations have to beevolved simultaneously. In particular, one averages first β ¯ T ( ν ) as in Eq. (23) and inserts this into the backgroundpart of the Klein-Gordon equation (16). In every step,the perturbations enter the background equations via β l and M l . C. Simulation scheme
The methods developed in the previous sections allowfor a considerable simplification of the numerical treat-ment. Rather than evolving a large number of N-bodyparticles and the fields ϕ and Ψ on a grid, one now merelyhas to evolve a drastically reduced set of differential equa-tions. This becomes possible as soon as a collection ofstable neutrino lumps has formed (at about z ≈ z ≈ x l , with neutrino number factions f l , restmasses M l , and effective couplings β l . This is the start-ing point for the simplified scheme.Section III B collects a set of coupled differential equa-tions describing the cosmological evolution. These arethe equation of motion (15), the background Klein-Gordon equation (16) with its right-hand side (23) andthe usual Friedmann equations. They involve the aver-aged potentials at the lump positions, i. e. { δ ˆ ϕ ( x l ) } and { ˆΨ( x l ) } (and their gradients) as given by Eqs. (24) and(26). Finally, the mass change is computed according to dM l dt = − β l M l d ˆ ϕdt . All these equations have mutual de-pendences and can only be solved simultaneously. Colddark matter, if included, has to be treated with standardN-body techniques. The influence of neutrino lumps onthe matter component was studied in [8, 11, 18].The aforementioned equations are only complete to-gether with functional relations β l ( f l , ˆ ϕ ) and M l ( f l , ˆ ϕ ),cf. Fig. 3, known at all times. As a first approach, onemay assume a time-independent relation. This is rea-sonable if the lumps are virialized and hence their in-ner structure is approximately frozen. We will explorethe stability of individual lumps in Sec. V. Furthermore,the dependence on ˆ ϕ may be neglected if the derivative ∂β/∂ ˆ ϕ or the variation of ˆ ϕ are sufficiently small.It is not clear whether z ≈ E ( ν ) = R d x p g (3) ρ ν ( x ) ∝ ¯ ρ ν a ,shown in Fig. 4. For a & .
45, one observes a transition
PSfrag replacements scale factor a ¯ ρ ν a [ − M p c − ] FIG. 4. Stabilization of the energy in neutrinos. The dashedline shows the evolution calculated by the background equa-tions, the solid line is taken from a full simulation run. to a regime with a small constant slope. This would becompatible with a small monotonic change of the large-scale cosmon field and an effective lump mass dependingon this field, corresponding to the expectation of approx-imate mass freezing within neutrino lumps [19]. Thismay be taken as a hint that the neutrino lump fluid maybecome a reasonable picture for a & . IV. ENERGY-MOMENTUM TENSOR OFLUMPS
In Sec. III, we had to assume properties of the energy-momentum tensor associated with neutrino lumps. Thederivations will be provided in this section. An importantresult is the integrated amplitude A µν of a single lump’senergy-momentum tensor, see Eq. (8). The derivation inSec. IV A includes the vanishing of the total internal pres-sure in stable lumps. Next, in Sec. IV B, we will considerthe term β ˆ T ν , cf. Eq. (20), which sources the energy-momentum exchange between cosmon and neutrinos. A. Single lump
We now study a single neutrino lump described by itsenergy-momentum tensor T µν including contributions ofthe bound neutrinos and the local cosmon field. Thelump occupies a volume V ; its energy-momentum tensorvanishes outside. On scales much larger than the lumpsize, it is useful to consider the amplitude A µν = Z V d x p g (3) T µν . (27)We first switch to the rest frame of the lump where wewill show A µν = − M l δ µ δ ν . Let us therefor consider thedifferent components separately. Clearly, A = − M l by definition of the rest mass. A i = P i is the totalmomentum and thus vanishes in the rest frame, whereby A i = 0. It remains to show A ij = 0.We assume that the lump is approximately static, i. e.its energy-momentum content in a physical volume isconserved, ∂ (cid:0) a T µν (cid:1) ≈ , (28)neglecting the metric perturbations. Together with theenergy-momentum conservation equation,0 = ∇ λ T λj = ∂ T j + ∂ i T ij + 3 ˙ aa T j , (29)the staticity condition implies ∂ i T ij = 0.It is convenient to define the three-vector v =( T j , T j , T j ) for a given column j . We have just showndiv v = 0. From now on, we choose i = j = 1 for sim-plicity. The amplitude A can then be written as A = a Z dx Z dy dz v = a Z dx Z S x d S · v , (30)where S x is the slice of V normal to the x direction. Out-side the lump, we extend the area S x to a closed surface.We can equally integrate over this closed surface sincethere is no contribution outside the lump. We concludethat the integral vanishes since div v = 0 inside the en-closed volume. This implies A = 0. The derivation canequally be done for arbitrary i and j , whereby A ij = 0.In the presence of an external cosmon perturbation δ ˆ ϕ sourced by other lumps, the energy-momentum conser-vation used in Eq. (29) only applies to the full energy-momentum tensor T µν tot including the contribution due to δ ˆ ϕ . Spatial variations of ˆ ϕ on the scale of the lump are,however, small, such that ∂ i T i tot j ≈ ∂ i T ij = div v . Ifthe external field δ ˆ ϕ varies only slowly, the staticity con-dition, Eq. (28), applies to the total energy-momentumtensor as well.The straightforward generalization of the rest-frameresult gives the amplitude A µν = M l u µ u ν /γ as antic-ipated in Sec. III A. The lump, on scales larger thanits size, is described by a standard one-particle energy-momentum tensor T µν = 1 √− g Z dτ M l u µ u ν δ (4) ( x − x l ) (31)= 1 p g (3) M l γ u µ u ν δ (3) ( x − x l ) , (32)where we have used u = dx /dτ and γ = √− g u . B. Smoothed conservation equation
In the effective description, the two dynamic compo-nents are the collection of lumps (with the neutrino and a local cosmon contribution) and the cosmon field ˆ ϕ out-side the lumps, which mediates the interaction. Thisdiffers from the usual split in the neutrinos T µν ( ν ) and thecosmon T µν ( ϕ ) introduced in Sec. II A. The total energy-momentum content T µν tot can thus be expressed in twoways, T µν tot = T µν ( ν ) + T µν ( ϕ ) = T µν lumps + T µν ( ˆ ϕ ) . (33)The neutrino contribution is completely contained in T µν lumps . The cosmon field splits into ϕ = ˆ ϕ + δϕ loc ,and the contribution of the local perturbation δϕ loc isattributed to the energy-momentum tensor of the lumps.The part of the cosmon energy-momentum tensor notdepending on the local fluctuation δϕ loc is T µν ( ˆ ϕ ) = ∂ µ ˆ ϕ ∂ ν ˆ ϕ − g µν (cid:18) ∂ λ ˆ ϕ ∂ λ ˆ ϕ + V ( ˆ ϕ ) (cid:19) , (34)which corresponds to the standard form of a scalar-fieldenergy-momentum tensor.Only the total energy-momentum tensor is conservedand we want to investigate the energy-momentum flowbetween the components T µν ( ˆ ϕ ) and T µν lumps . This will yieldan effective coupling β l between the lumps and ˆ ϕ . Thefour-divergence of T µν ( ˆ ϕ ) is ∇ λ T µλ ( ˆ ϕ ) = (cid:0) ∇ λ ∇ λ ˆ ϕ − V ′ ( ˆ ϕ ) (cid:1) ∂ µ ˆ ϕ. (35)In order to evaluate the right-hand side, we employ theequation of motion of the full cosmon field ϕ inferredfrom Eq. (1): ∇ λ ∇ λ ϕ − V ′ ( ϕ ) = βT ( ν ) . (36)Smoothing this relation at the scale λ (cf. Sec. III A) atlinear order in δϕ loc and inserting into Eq. (35) yields ∇ λ T µλ ( ˆ ϕ ) = β ˆ T ( ν ) ∂ µ ˆ ϕ. (37)The right-hand side can be expressed in terms oflump properties by making use of the conservation equa-tion for the total energy-momentum tensor, ∇ λ T µλ ( ˆ ϕ ) = −∇ λ T µλ lumps . The part ∇ λ T µλ lumps can be analyzed in theeffective description where lumps are treated as pointparticles. The equation of motion (15) implies ∇ λ T µλ lumps ≈ − X lumps l β l T l ∂ µ ˆ ϕ. (38)Comparison with Eq. (37) yields β ˆ T ( ν ) = X lumps l β l ˆ T l , (39)which is the relation used in Sec. III B. V. ASPECTS OF STABILITY
The effective description of the cosmological dynam-ics outlined in Sec. III relies on the assumption of stablelumps. At the current stage of the comprehensive sim-ulation method [8], however, it is not possible to trackthe evolution of lumps after z ≈
1. In this section, wesketch some analytic arguments why stable lumps areexpected to form. We start with considerations concern-ing the angular momentum, Sec. V A, and construct anexplicit example of a static configuration using hydrody-namic equations in Sec. V B. In the following, we neglectthe metric perturbations.
A. Angular momentum
The cosmon-mediated fifth force felt by the neutrinos isstronger than but in some respects similar to gravity. Thefield equation for δϕ , Eq. (17), can be compared to theusual gravitational Poisson equation. The cosmon per-turbation δϕ thus plays the role of a potential – similarto the gravitational potential – in which a neutrino parti-cle moves. In contrast to the gravitational case, however,the particle changes its mass m ν = m ν ( ϕ ) while movingwith velocity u µ according to˙ m ν = − βm ν u λ ∂ λ ϕu . (40)The loss of mass when moving towards a minimum of thepotential implies, by momentum conservation, an addi-tional acceleration [8]. Hence, it has to be investigatedwhether neutrino lumps are unstable, i. e. continuouslyshrink to smaller sizes until they are stabilized, e. g., bythe degeneracy pressure [20].The cosmon-mediated fifth force, despite the mass vari-ation along a particle trajectory, shares an importantproperty with gravity: the conservation of angular mo-mentum. For example, a single particle moving in aspherically symmetric and static cosmon potential ϕ ( r )(in physical coordinates) has the conserved angular mo-mentum L = γ m ν r ˙ θ (41)in polar coordinates ( r, θ ) and with the Lorentz factor γ .The equation of motion, written for the radial momentum p r = γm ν ˙ r , then contains an angular momentum barrier,which prevents the particle from falling into the center.It reads ˙ p r = L γm ν r + βm ν γ dϕdr . (42)This is analogous to Newtonian gravity with an angularmomentum barrier ∝ L /r and an inward potential gra-dient. The only difference is the variation of m ν (and γ )along the particle’s trajectory. Since the mass decreases when approaching the center, this even amplifies the an-gular momentum barrier.Of course, these results for a test particle in a centralpotential need not generalize to a distribution of particlesforming a lump. There, we define a neutrino angularmomentum density l µνα ( ν ) as in special relativity, l µνα ( ν ) = x µ T να ( ν ) − x ν T µα ( ν ) , (43)which, without the cosmon-neutrino coupling, would sat-isfy a conservation equation ∇ α ( l ijα ( ν ) /a ) = 0 due to theconservation equation for T µν ( ν ) . Here, derivatives aretaken with respect to comoving coordinates. Definingthe total spatial neutrino angular momentum L ij ( ν ) ≡ Z d x p g (3) l ij ν ) , (44)the conservation equation for l ijα ( ν ) in the uncoupled casetranslates to the conservation law ∂ t (cid:16) a L ij ( ν ) (cid:17) = 0 . (45)With the coupling, Eq. (2), we instead obtain ∇ α (cid:16) a − l ijα ( ν ) (cid:17) = − a − βT ( ν ) (cid:0) x i ∂ j ϕ − x j ∂ i ϕ (cid:1) , (46)and thus ∂ t (cid:16) a L ij ( ν ) (cid:17) = ∂ t Z d x p g (3) a l ij ν ) (47)= Z d x p g (3) a βT ( ν ) (cid:0) x i ∂ j ϕ − x j ∂ i ϕ (cid:1) . (48)For a spherically symmetric lump and thus cosmon po-tential ϕ = ϕ ( t, r ), it is straightforward to show x i ∂ j ϕ − x j ∂ i ϕ = 0 . (49)In this case, the quantity a L ij ( ν ) is indeed conserved.This is related to the fact that a spherically symmetricscalar field does not carry spatial angular momentum, l ij ϕ ) = − ˙ ϕ (cid:0) x i ∂ j ϕ − x j ∂ i ϕ (cid:1) = 0 . (50)The conservation of the total angular momentum a L ij tot then reduces to the conservation of a L ij ( ν ) .Our considerations hold for an arbitrary isotropic andhomogeneous background metric. Thus, a does not needto be the cosmic scale factor but can also describe somelocal properties of the metric. Fluctuations of the metricaround the background metric as well as fluctuations ofthe cosmon around an averaged field as ˆ ϕ in Eq. (10) canbe added to the neutrino energy-momentum tensor inEq. (43). The right-hand side of Eq. (48) involves then ˆ ϕ instead of ϕ and β l instead of β , resulting in a reductionof the change of angular momentum. We conclude thatangular momentum conservation is similar to standardgravity. This constitutes a strong hint for a dynamicstabilization of the lump. B. Hydrodynamic balance
We will now study a neutrino lump within a hydrody-namic framework and derive a balance equation for a sim-ple class of lumps. For this purpose, we will employ mo-ments of the neutrino phase-space distribution function f ( t, x i , p j ) describing the distribution of particles withcomoving position x i and momentum p j = m ν u j . A dis-cussion of stability based on the Tolman-Oppenheimer-Volkoff equation can be found in Ref. [21]. For simplicity,we will restrict ourselves to first-order relativistic correc-tions in this section. The equations of motion for a neu-trino particle under the influence of the fifth force canthen be written as˙ x i = p i m ν , ˙ p j = (cid:18) − p k p k m ν (cid:19) βm ν ∂ j ϕ. (51)The fully relativistic equation in terms of the four-velocity u µ is presented in [8].We will consider the following moments of the phase-space distribution function f : n = Z d p f ( t, x , p ) , (52) nU i = Z d p p i am ν f ( t, x , p ) , (53) σ ij + nU i U j = Z d p p i am ν p j am ν f ( t, x , p ) . (54)The quantities n ( t, x ), U ( t, x ), and σ ij ( t, x ) are inter-preted as the number density, the locally averaged pecu-liar velocity, and the velocity dispersion tensor, respec-tively. Their evolution equations can be derived from theprinciple of particle conservation in phase-space, which isexpressed by the continuity equation˙ f + ∂ ( f ˙ x i ) ∂x i + ∂ ( f ˙ p j ) ∂p j = 0 . (55)The whole procedure is similar to the standard case ofgravity (cf. Ref. [22]) with the peculiarity of a varyingmass m ν = m ν ( ϕ ).Integrating over the momentum in Eq. (55) yields thezeroth moment ˙ n + ∂ r i ( nU i ) = 0 , (56)with ∂ r i = a − ∂/∂x i . A static number density profile,˙ n = 0, is realized if the microscopic motion adds locallyup to zero, U = 0. This is the case for a locally isotropicvelocity distribution. For this class of lumps, the equa-tion for ˙ U , which follows by taking the first moment ofEq. (55) and using the equations of motion (51), takes aparticularly simple form:˙ U i = − n ∂ r j σ ij + β∂ r i ϕ (cid:18) − σ n (cid:19) + 1 n σ ij β∂ r j ϕ, (57)with σ ≡ σ ii /
3. For a static lump, we demand ˙ U = 0 in addition to˙ n = 0. A glance at Eq. (57) shows that this requires acertain balance between the effective pressure ∝ ∂ r j σ ij ,generated by the microscopic neutrino motion, and thefifth force ∝ β∂ r i ϕ . Assuming spherical symmetry, thebalance equation reads1 n σ ′ = βϕ ′ (cid:16) − σ n (cid:17) , (58)with a prime denoting derivatives with respect to theradial coordinate. Here, we have used σ ij = σ δ ij . Solv-ing this equation together with the (radial) Klein-Gordonequation for the cosmon field yields static lump configu-rations. These lump configurations differ from the solu-tions discussed in Ref. [20] since the stabilizing pressureis now provided by the neutrino motion rather than bythe degeneracy pressure.At first sight, it is not clear whether this staticity con-dition constitutes a stable equilibrium. We perform anexemplary numerical check by simulating a single, iso-lated lump with the N-body technique [8]. Rather thanstarting with a static lump configuration by Eq. (58),we use a somewhat smaller velocity dispersion σ . Inthe subsequent evolution, the lump shrinks and the neu-trino pressure increases. Figure 5 shows how the neutrinoprofile becomes more concentrated and indeed stabilizes.The simulation evolves the lump in physical time t . Forconvenience, we have translated time intervals ∆ t to scalefactor intervals ∆ a by the Hubble parameter at z = 1. ∆ a = 0.00 ∆ a = 0.01 ∆ a = 0.10 PSfrag replacements physical radius ar [ h − Mpc] r a d i a l p r o fi l e FIG. 5. The radial profile ∝ πr n ( r ) of neutrinos inside theperturbed lump normalized to unity. The pressure cancellation between the contributionsof neutrinos and the cosmon perturbations, cf. Sec. II B,is established during the stabilization process. This isshown in Fig. 6. Similar to Fig. 2, we observe that thepressure cancellation is only established at rather largedistances from the lump. At smaller distances, a residualpositive pressure remains.0 -0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0 0.01 0.02 0.03 0.04 0.05r = 100 Mpc/hr = 20 Mpc/hneutrino contribution
PSfrag replacements ∆ a i n t e g r a t e dp r e ss u r e P ( r ) / M l FIG. 6. The total pressure integrated to r = 100 h − Mpc(red, solid) and 20 h − Mpc (dark red, dashed). For com-parison, we also plot the neutrino contribution covering allneutrinos (black, dotted).
VI. CONCLUSION
We have shown that a simplified, effective descriptionof the cosmological dynamics in the growing neutrinoquintessence model is possible. It bases upon describingstable cosmon-neutrino lumps as nonrelativistic particleswith an effective interaction. After the main idea wasgiven (Sec. II B), several aspects needed to be investi-gated.The first issue concerns the stability of the lumps.We have shown in a hydrodynamic analysis of spheri-cally symmetric lumps that the neutrino velocity disper-sion indeed stabilizes the lumps against the attractivecosmon-mediated fifth force (Sec. V B). On more generalgrounds, stability of the lumps is already expected by an-gular momentum conservation which holds similarly tothe gravitational case (Sec. V A). Stable lumps may thenbe characterized by the amount of bound neutrinos. Innumerical simulations of growing neutrino quintessence,we have found lumps containing a fraction up to & − of all neutrinos in the Hubble volume, reaching a massof ∼ solar masses (Sec. III A). The total number ofidentified lumps in the Hubble volume is of order 10 .Second, it is not clear a priori that the lumps can bedescribed as particles. The most important aspect here isthe vanishing of the total internal pressure. The neutri-nos, however, have reached high velocities and an equa-tion of state w ν ≈ .
1. We have shown that – underidealized conditions – the neutrino pressure is exactlycancelled by a negative pressure contribution from the local cosmon perturbations (Sec. IV). A numerical checkis given in Fig. 2. Under realistic conditions, the pressurecancellation may not hold exactly but to a good approx-imation. Approximate cancellation of neutrino and cos-mon pressure occurs at a characteristic radius r l that issubstantially larger than the radius of the neutrino coreof the lump. For an effective particle description, r l isthe size of the lump. A fluid description requires thatthe typical distance between lumps exceeds r l .Third and finally, a description of the cosmological dy-namics requires the equation of motion for the lumps andthe field equation for the smoothed field ˆ ϕ mediating theinteraction between the lumps. These equations havebeen derived in Sec. III. The decisive quantity character-izing the lump interaction is the effective cosmon-lumpcoupling β l . For small lumps, it approaches the funda-mental coupling β quantifying the cosmon-mediated fifthforce between neutrinos. For big lumps, the effective cou-pling β l is suppressed by a factor of two to three as com-pared to β . Since the attractive force is proportional tothe squared coupling, this corresponds to a suppressionof the attraction by one order of magnitude.The effective description of growing neutrinoquintessence complements sophisticated numericaltechniques as it provides physical insight into thedynamics. Furthermore, the effective description couldprove useful in understanding the evolution for redshift z <
1, where numerical simulations have not yet beensuccessful [8, 11]. Quantitative results for low redshiftsare needed to eventually confront growing neutrinoquintessence with observational constraints.The process of lump formation (in the redshift range z ≈ ACKNOWLEDGMENTS
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