Phenomenology of Hybrid Scenarios of Neutrino Dark Energy
aa r X i v : . [ a s t r o - ph ] O c t MPP-2008-90arXiv:0807.4930
Phenomenology of Hybrid Scenarios of NeutrinoDark Energy
Stefan Antusch ⋆ , Subinoy Das † and Koushik Dutta ¶ Max-Planck-Institut f¨ur Physik (Werner-Heisenberg-Institut), F¨ohringer Ring 6,D-80805 M¨unchen, Germany Center for Cosmology and Particle Physics, Department of Physics, New YorkUniversity, New York, NY, 10003, USAE-mail: ⋆ [email protected], † [email protected], ¶ [email protected] Abstract.
We study the phenomenology of hybrid scenarios of neutrino dark energy,where in addition to a so-called Mass Varying Neutrino (MaVaN) sector a cosmologicalconstant (from a false vacuum) is driving the accelerated expansion of the universetoday. For general power law potentials we calculate the effective equation of stateparameter w eff ( z ) in terms of the neutrino mass scale. Due to the interaction of thedark energy field (“acceleron”) with the neutrino sector, w eff ( z ) is predicted to becomesmaller than − z >
0, which could be tested in future cosmological observations.For the considered scenarios, the neutrino mass scale additionally determines whichfraction of the dark energy is dynamical, and which originates from the “cosmologicalconstant like” vacuum energy of the false vacuum. On the other hand, the field value ofthe “acceleron” field today as well as the masses of the right-handed neutrinos, whichappear in the seesaw-type mechanism for small neutrino masses, are not fixed. This, inprinciple, allows to realise hybrid scenarios of neutrino dark energy with a “high-scale”seesaw where the right-handed neutrino masses are close to the GUT scale. We alsocomment on how MaVaN Hybrid Scenarios with “high-scale” seesaw might help toresolve stability problems of dark energy models with non-relativistic neutrinos.
1. Introduction
The evidence for the existence of a dark sector in the universe has made the presentera of cosmology fascinating and challenging. At present, the microscopic natures ofdark energy and dark matter are still an open question, with both components onlyprobed gravitationally [1, 2, 3, 4]. Regarding dark matter, there are various particlephysics candidates which may belong to the class of Weakly Interacting Massive Particles(WIMPS) or which may interact only gravitationally. On the other hand, for particlephysics explanations of the observed dark energy, the main fundamental question iswhether dark energy is a cosmological constant or a dynamical field. While in theformer case the equation of state parameter w of dark energy is constant and equalto −
1, in the latter case it is a function of redshift z and in general differs from − henomenology of Hybrid Scenarios of Neutrino Dark Energy w from − ρ / DE ∼ . much smaller than the “theoretical expectation”. In this paper we will not addressthis “cosmological constant” problem, i.e. the suppression/cancellation of the variousgeneric contributions to dark energy which are too large, but assume that it is resolvedby some other mechanism.Our work is motivated by the intriguing observation that the energy density ofdark energy is close to another very small scale in particle physics, namely the one ofneutrino masses. The discovery of flavour conversion of neutrinos from various sources,interpreted within the framework of neutrino oscillations, points to two mass eigenvaluesof the light neutrinos above about 0 .
01 eV and 0 .
05 eV [6], while searches for neutrinomasses from Tritium β -decay and neutrinoless double β -decay yield an upper bound foreach mass eigenvalue of roughly 0 . ρ / DE remains valid in earlier cosmic epochs, also dark energy behavesdifferently from a cosmological constant. Compared to other dynamical models of darkenergy (e.g. quintessence [8, 9, 10]), in the MaVaN scenario one does not have to choosea mass for the dynamical field of the order of the present Hubble parameter H . In fact itcan be much larger, even of the order of the neutrino mass scale, due to the stabilisingeffect of the contribution of the relic neutrino density to the potential. As a result,the field adiabatically tracks the minimum of the effective potential whose evolution iscontrolled by the time evolution of neutrino number density.The MaVaN models have been investigated in many studies, e.g. regarding possibleexperimental signatures and constraints [11, 12, 13, 14, 15, 16, 17, 18, 19, 20] as well asmodel-building issues [21, 22, 23, 25, 26]. It has turned out that while there are manyinteresting possible signatures and attractive features, the scenario is tightly constrainedby the requirement of consistency with late time structure formation. In particular, ithas been pointed out [27] that if neutrinos are non-relativistic today, due to fifth forceeffects the neutrinos would cluster at late time and finally form “neutrino nuggets”which would spoil the dark energy behavior of the neutrino fluid.The goal of this paper is to investigate the phenomenology of a generalised scenariowhich we will refer to as MaVaN Hybrid Scenario, where in addition to a MaVaN sectora cosmological constant (from a false vacuum) is driving the accelerated expansion ofthe universe today. As we will show, for a generalised power law potential in the MaVaNsector the effective equation of state parameter w eff ( z ) as well as the fraction to whichthe dark energy is of dynamical nature is determined by the neutrino mass scale. We will henomenology of Hybrid Scenarios of Neutrino Dark Energy
2. The Original MaVaN Scenario
In the basic MaVaN scenario a singlet fermion N (right-handed neutrino) and adynamical real scalar field A (called “acceleron”) are introduced. Their dynamics givesrise to the acceleration of the universe and to neutrino masses which vary with time.The basic form of the Lagrangian is given by L ⊃ m D νN + 12 κAN N + H.c. + V ( A ) , (1)where m D is neutrino Dirac mass, κ is a dimensionless coupling in dark sector and V ( A )is the potential energy of the acceleron field. The Standard Model neutrino ν as well asthe singlet neutrino N are both written as two-component left-chiral Weyl spinor fields.If κA ≫ m D , i.e. if the Majorana mass of the right-handed neutrino is much larger thanthe neutrino Dirac mass, one can effectively integrate out N from the low energy theorywhich leads to the effective Lagrangian L ⊃ − m D κA νν + H.c. + V ( A ) . (2)The neutrino mass m ν = | m D κA | depends on the dynamics of A . Taking into account themasses of the non-relativistic relic neutrinos we now include a term proportional to therelic neutrino number density n ν to the effective potential, V eff = n ν (cid:12)(cid:12)(cid:12)(cid:12) m D κA (cid:12)(cid:12)(cid:12)(cid:12) + V ( A ) . (3)The effective potential V eff for the acceleron A has a minimum because the firstterm pushes A to larger values whereas as the second term V ( A ) is assumed to favoursmaller values of A (e.g. if V ( A ) ∝ log( A/ Λ) , or ∝ A α ). Typically, in this type of modelsa sub-eV scale is introduced in order to realize the desired values of V ( A ) (and finallyof V eff ( A )). ‡ As a consequence of the two contributions to the potential and since n ν decreases with time, the field value of the acceleron field A becomes smaller with time ‡ A scenario of quintessence cosmology with growing matter component (which may be applied, forinstance, to neutrinos) has recently been proposed without explicitly invoking any sub-eV scale [28, 29]. henomenology of Hybrid Scenarios of Neutrino Dark Energy A is above theHubble parameter). The combined A and neutrino fluid drives the accelerated expansionof the universe.We remark that unlike in conventional quintessence models the scalar field doesnot roll down an extremely flat potential but it rather gets stabilised by the finiteneutrino density. For this reason, the slope of the potential can be much higher thanin quintessence scenarios and the mass of the scalar field A can even be as high as theneutrino mass scale, whereas in quintessence models it has to be as tiny as the Hubblescale.Although the model can successfully explain the present acceleration of the universewith masses of the dynamical scalar field much larger than the Hubble scale (e.g. in thesub-eV range) and provide a connection between dark energy and the physics of neutrinomasses, it has also some shortcomings: For example, the potential is not stable underradiative corrections, unless new states appear with sub-eV masses, and it still hasto be rather flat in order to be consistent with present constraints on the equation ofstate parameter. Furthermore, the MaVaN scenario with non-relativistic neutrinos hasa problem with respect to the consistency with late time structure formation, as hasbeen pointed out in Ref. [27]. If neutrinos are non-relativistic today, due to fifth forceeffects the neutrinos could cluster at late time and finally form “neutrino nuggets” whichwould spoil the dark energy behavior of the neutrino fluid.However, it has been demonstrated that under certain conditions these problemsmight be overcome [21]. For example, in supersymmetric theories of neutrino darkenergy the potential can be stable under radiative corrections (for example in gaugemediated supersymmetry breaking where supersymmetry breaking is only transmittedto the dark sector via gravity such that the dark sector superpartners are light) andstability problems with respect to neutrino density perturbations can be avoided ifneutrinos are highly relativistic, although then the connection to neutrino masses isweakened since the two observed mass scales correspond to non-relativistic neutrinos.
3. Generalised Framework: The MaVaN Hybrid Scenario
After having reviewed the basics of the original MaVaN scenario in the last section,we will now set up the framework for our analysis. It consists of a generalised MaVaNpotential of power law type plus a constant vacuum energy contribution. In the followingwe will specify it explicitly and comment on how it may be realised in classes of modelsas well as on the connection to the (in principle) experimentally measurable neutrinomass scale.
In the following we will consider a rather general setup where neutrino dark energyis realised in a type of “hybrid dark energy” setup with the theory being in a false henomenology of Hybrid Scenarios of Neutrino Dark Energy § Motivated by the notion of providing a rather general framework for ourstudy, we will therefore consider an effective potential for neutrino dark energy of theform V ( A ) eff = V ( A ) + ρ (0) ν a (cid:18) A A (cid:19) β + V , (4)where V ( A ) = M − α A α . (5) A is the acceleron field k , ρ (0) ν is the energy density of the SM neutrinos today and a isthe scale factor. The main ingredients are a constant term V (from a false vacuum), apotential for A which is monotonically increasing with | A | (in our case a general powerlaw potential ∝ A α ) and an effective term n ν m ν , where we assume m ν ∝ /A β in orderto generalise the conventional seesaw relation. We note that the potential of Eq. (5) isnot the most general form of a potential that may arise in a realistic model. Instead ofthe monomial potential that we are considering, it can for instance also be a polynomialor logarithmic function of the field A , or a combination of both. To motivate the appearance of the constant vacuum energy in the “hybrid scenario”by an example, we may consider the following superpotential (after EW symmetrybreaking), W = λ ˆ A ( ˆ N − v N ) + y ˆ H u ˆ ν ˆ N + 12 m A ˆ A , (6)where hats indicate superfields. ˆ N denotes the SM singlet (right-handed neutrino)superfield(s), and the most relevant contributions to the scalar potential stem from theF-terms | F A | + | F N | = | λv N | + | λAN + m A A + ... | . The first term contributesthe constant vacuum energy while the second term gives a mass term ( ∼ | A | ) to theacceleron field. Together with the additional effective potential term n ν m ν ∼ n ν v u / | A | ,where v u is the vev of the Higgs field which couples to the neutrinos, the potential for theacceleron has a minimum for a non-zero field value of A . | F N | also gives a time-varyingmass to the right-handed neutrino, m N ∼ | A | .While the above superpotential may serve as a simple example which could realisethe generalised potential of Eq. (4) with α = 2 and β = 1, it should be kept in mindthat that there is in principle a large variety of possible models which can give rise toother values of α and β (as well as to other non-power-law forms of the potential). Forexample, different powers of A may appear in the superpotential or the neutrino massesmay originate from a different type of seesaw mechanism (e.g. from a so-called double § One such example model is the SUSY version of the MaVaN scenario proposed in [21]. k For simplicity we will take A as a real scalar field as in Section 2, unless stated otherwise. henomenology of Hybrid Scenarios of Neutrino Dark Energy n ν m ν ∼ n ν v u / | A | β , and so on. Furthermore, one mayanticipate that the parameters α and β in realistic values should be positive integers.However, it is also possible that fractional values of the exponents appear, for exampleif the kinetic terms in a model are non-canonical and therefore a field transformationhas to be performed in order to bring the kinetic terms back to canonical form. We willdiscuss the possible origin of the generalised potential in more detail in the Appendix. From neutrino oscillation experiments, two so-called mass squared differences are known:∆ m = m − m is known mainly from atmospheric neutrinos and ∆ m = m − m from solar neutrino data and from the KamLAND experiment. The experimental valuesare about ∆ m = 7 . × − eV , | ∆ m | = 2 . × − eV [6]. With this nomenclaturewe also know that m > m , but m can be the heaviest or the lightest of the masseigenvalues. For the sum ¯ m ν defined as¯ m ν = X i =1 m i (7)we know that there is a lower bound from oscillations (saturated if m ≈ m ≈ . m ≈ .
05 eV) and also an upper bound from neutrinoless double β experiments ofabout 1 . m ≈ m ≈ m ≈ . m ν ∈ [0 .
06 eV , . . (8)On the other hand, the lightest of the neutrino mass eigenvalues ( m or m ) can bearbitrarily small.One important question is whether this sum ¯ m ν is relevant for the MaVaN setup,or whether the mass of only one of the light neutrinos matters. To answer this question,we have to go to the full three family scenario where we distinguish two possible cases.In one possible situation (which we may call case I) ¯ m ν is indeed relevant. Thiscase applies when the vev of the field A gives masses to all the right-handed neutrinos N i such that the part of the Lagrangian relevant for neutrino masses reads for instance( X ij is a coupling matrix) L ⊃ X ij AN i N j + ( Y ν ) αi ν α H u N i . (9)The seesaw suppressed mass matrix of the light neutrinos is then given by( m ν ) αβ = v ( Y ν ) αi X − ij ( Y ν ) Tjβ
A . (10)If we diagonalise m ν , the effective potential reads V eff = X α n ν α ( ¯ m diag ν ) αα = n ν X α ( ¯ m diag ν ) αα = n ν ¯ m ν , (11)where we have assumed that n ν α = n ν for all generations α of neutrinos. henomenology of Hybrid Scenarios of Neutrino Dark Energy N i gets its mass from a different scalar field, e.g. from the vevs of fields A, B and C and that we have a Lagrangian of the form L ⊃ x AN N + 12 x BN N + 12 x CN N + X i ( y ν ) i ν i H u N i , (12)where x i are coupling constants. Now the light neutrino mass matrix would be given by( m ν ) αβ = v ( y ν ) x A + v ( y ν ) x B + v ( y ν ) x C . (13)In this example, only m would be relevant for the effective potential for A . ¶ Ifconsistency of the MaVaN scenario (for a particular form of the potential) would requirea very light neutrino mass scale ≪ .
01 eV, this could be realised in case II. The relevantmass would then be m for a so-called normal hierarchical spectrum or m for an invertedhierarchical spectrum.In the following, we will mainly be interested in case I, since it has a particularlyclose link to the observed neutrino masses which manifests itself in the upper and lowerexperimental bounds on today’s value of ¯ m ν given in Eq. (8). We will therefore presentour results using the notation for case I where the relevant neutrino mass scale is ¯ m ν .Most of the formulae can readily be adopted to case II, as long as neutrinos remainnon-relativistic.
4. Phenomenology of the MaVaN Hybrid Scenario
In this section we will investigate the dynamics and phenomenology of the MaVaNHybrid Scenario with a general power law potential V ( A ) = M − α A α . (14)The observed small neutrino masses are generated by a version of the seesaw mechanism,and we will assume a general dependence of the neutrino masses on A of the form¯ m ν ( A ) ∼ v A β , (15)where v is the vev of the SM Higgs field (which is proportional to the Dirac masses ofthe neutrinos). The notation ¯ m ν is introduced in section 3.3. As a first step of our analysis we consider the background dynamics of the scalar field A in the MaVaN Hybrid Scenario defined in section 3, which leads to an effective potentialfor A of the form V ( A ) eff = M − α A α + ρ (0) ν a ¯ m ν ( A )¯ m (0) ν + V , (16) ¶ Of course, analogously, one can realise the case where any of the mass eigenvalues of the lightneutrinos is relevant for the effective potential for A . henomenology of Hybrid Scenarios of Neutrino Dark Energy n (0) ν ¯ m (0) ν = ρ (0) ν and where the index “(0)” indicates todays values of theparameters. The second term in the potential originates from the coupling betweenneutrino and the scalar field in the effective Lagrangian. In the following discussions,if we do not mention the values of α and β , we will assume α = 2 and β = 1which corresponds to the “standard” MaVaN scenario, i.e. where the potential for A is quadratic and where the seesaw mechanism is of type I. We will also focus on the casewhere the neutrinos are non-relativistic today.The Friedmann equation in our scenario takes the form3 H M P l = ρ (0) CDM a + ρ (0) Baryons a + V ( A ) + ρ (0) ν a (cid:18) A A (cid:19) β + 12 ˙ A + V (17)and the scalar field A obeys the following modified Klein Gordon equation¨ A + 3 H ˙ A = − dV ( A ) dA − ρ (0) ν a d ¯ m ν dA ¯ m (0) ν . (18)We note that while the dark matter density ρ CDM redshifts as a − , the time dependenceof the neutrino energy density is non-trivial due to the time-dependence of A .Now we use adiabaticity , i.e. the feature of MaVaN scenario that the acceleron fieldadiabatically follows the minimum of the effective potential since long time back in thecosmic history. Consequently, the condition ∂V eff ∂A = 0 determines the evolution of thescalar field to be A = βρ (0) ν A β a α M − (4 − α ) ! α + β . (19)The present value of A can be found from the above equation and is given by A = βρ (0) ν α M − (4 − α ) ! /α , (20)which allows to derive the relation AA = (1 + z ) α + β , (21)where a = 1 / (1 + z ) has been used with z being the redshift. From this expression wecan calculate the kinetic energy of the field compared to the total energy density carriedby the field, ˙ A / ρ A = 1 . (cid:18) A M P l (cid:19) (cid:18) α + β (cid:19) (1 + z ) α + β . (22)We can see that if the field value of A today is smaller than the Planck scale M P l ,the kinetic energy of the field can consistently be neglected for the relevant time ofthe evolution of the dark energy + . In addition, a constant vacuum energy contributionfurther reduces the significance of the kinetic energy compared to the Hubble constant + On the other hand we will see that a high value of A . M P l is typically required with respect tothe stability issues for non-relativistic neutrinos (c.f. discussion in section 6). henomenology of Hybrid Scenarios of Neutrino Dark Energy V ( A ) to the acceleron-dependentneutrino fluid density in Eq. (16), we find that the ratio of the cosmological densityparameters of these two components is given byΩ ν Ω A = αβ , (23)where we have substituted 3 H Ω ν = ρ (0) ν a ( A A ) β and 3 H Ω A = V ( A ). We would liketo highlight the result that the ratio of the energy densities from the interaction termand from the potential term V ( A ) is constant in time. This feature, which is basicallya by-product of the adiabaticity condition, will be useful to derive constraints on themodel parameters from cosmological observations and it will lead to a particularly closeconnection to the neutrino mass scale in the MaVaN Hybrid scenario with power lawpotentials. Using the above results for the background evolution, we will now investigate how theparameters α , β and V of the model are constrained by cosmological observations, inparticularly regarding the dark energy equation of state, as well as by the present boundson (and eventually by a future measurement of) the neutrino mass scale ¯ m ν . ∗ We will start by deriving the effective dark energy equation of state parameter w eff ( z ). The main point here is that when the dark energy equation of state is extractedfrom cosmological data sensitive to the Hubble scale H ( z ), it is always assumed thatthe dark matter component redshifts as 1 /a . Massive neutrinos also contribute a smallfraction to the dark matter today. The total dark matter density today is then given by ρ (0) DM = ρ (0) CDM + ρ ν (0) DM , (24)where ρ (0) CDM is the amount of cold dark matter and where ρ ν (0) DM is calculated from theactual value of the neutrino mass ¯ m (0) ν (which may be determined more precisely byfuture experiments), i.e. ρ ν (0) DM = ρ (0) ν . However, as we have already mentioned, theneutrino masses vary with time and therefore this contribution does not redshift as1 /a . Consequently, an observer will not extract the intrinsic dark energy equation ofstate, but rather an effective one, w eff .More explicitly, the effective dark energy density (which defines the effectiveequation of state w eff ) is extracted from the Hubble equation (assuming for simplicity ∗ We would like to note that only the terrestrial bound on neutrino masses can be applied directlyto MaVaN models. Cosmological bounds on neutrino masses are generically relaxed for mass varyingneutrinos (see e.g. [32]) since the neutrino masses decrease with increasing z , as we will discuss below.In fact, one “smoking gun” signal of MaVaN scenarios would be a “cosmological upper bound” on thesum of neutrino masses (if they are assumed as constant in time) which is incompatible with todayslower bound from terrestrial experiments. henomenology of Hybrid Scenarios of Neutrino Dark Energy
10a flat universe)3 H M P l = ρ (0) CDM a + ρ (0) Baryons a + ρ ν (0) DM a + ρ effDE , (25)where it is (wrongly, but for a model-independent analysis unavoidably) assumed thatthe neutrino density redshifts like ordinary matter.We would like to add a few remarks to clarify the notation: ρ ν (0) DM (i.e. the value ofthe quantity ρ νDM today) is same as ρ (0) ν . However, in an earlier epoch, the true neutrinoenergy density ρ ν is not equal to the quantity ρ νDM = ρ ν (0) DM /a .To calculate ρ effDE , following [31], we rewrite Eq. (17) as3 H M P l = ρ (0) CDM a + ρ (0) Baryons a + ρ (0) νDM a − ρ (0) νDM a + ρ (0) ν a ¯ m ν ( A )¯ m (0) ν + V ( A ) + 12 ˙ A + V . (26)Comparing this relation with Eq. (25) yields ρ effDE = ρ (0) ν a ¯ m ν ( A )¯ m (0) ν − ρ (0) νDM a ! + V ( A ) + 12 ˙ A + V . (27)From ρ effDE the observed equation of state w eff can be calculated via the continuityequation dρ effDE dt = − H (1 + w eff ) ρ effDE . (28)Taking the time derivative of Eq. (27) and using Eq. (18) and the expression for ˙ ρ A , wefind dρ effDE dt = − H ρ (0) ν a (cid:18) ¯ m ν ( A )¯ m (0) ν − (cid:19) − H ˙ A . (29)We compare the above equation with Eq. (28) to obtain w eff = − V ( A ) + V − ˙ A ρ (0) ν a (cid:16) ¯ m ν ( A )¯ m (0) ν − (cid:17) + V ( A ) + V + ˙ A . (30)This is the general result for the effective equation of state parameter w eff ( z ) for general A -dependent neutrino masses ¯ m ν ( A ) and for a general acceleron potential V ( A ).We now discuss some of the characteristic feature of w eff : To start with, it canbe seen from the above expression that at present time ( z = 0), the ρ (0) ν -dependentterm disappears from the denominator. Furthermore, in the absence of the constantvacuum energy term V , the equation of state parameter at z = 0 is simply given by theconventional equation of state parameter for the scalar field A , w A = ˙ A / − V ( A )˙ A / V ( A ) . (31)Since the slow-roll approximation can be applied for field values less than M P l and sincethe kinetic energy ∼ ˙ A is much smaller than V ( A ) in this case, w A is very close to -1. henomenology of Hybrid Scenarios of Neutrino Dark Energy w eff for z = 0, adding the constantvacuum energy term V even makes its value closer to − z >
0, however, the neutrino density term in the denominator of Eq. (30) is non-zero. Furthermore, noting that in the MaVaN-type models of interest ¯ m ν ( A ) < ¯ m (0) ν ,the ρ ν -dependent term is negative which implies that w eff becomes less than − z > w eff ( z ) is defined by Eq. (28),with ρ effDE related to the Hubble parameter H ( z ) by Eq. (25). By definition, w eff ( z )contains information about whether the effective dark energy density is dynamical orbehaves like a cosmological constant at a given redshift z . In order to extract ρ effDE from H ( z ), additional information regarding the matter components has to be used. Forexample, SN data provides some information on H ( z ). However, ρ (0) CDM + ρ (0) Baryons isrequired as an input, in addition to the neutrino energy density ρ (0) νDM which depends onthe (in principle) measurable neutrino mass scale today. One possibility to obtain ρ (0) CDM and ρ (0) Baryons is to extract it from the CMB, where the neutrinos were relativistic and donot contribute to the matter density. On the other hand, if information on the matterdensity from other observations, e.g. from weak lensing, is used it may already contain ρ ν if the neutrinos are clustered on the relevant scales. We note that in order to confrontthe MaVaN Hybrid Scenarios with the whole variety of (future) data from cosmologicalobservations, such issues have to be taken into account carefully. Finally, in order totest a particular dark energy model of this type using all available cosmological data,one would have to go beyond the analysis of w eff ( z ) and fit all model parameters to thedata. We will now quantify the above general statements about w eff ( z ) for the more explicitscenario where the MaVaN potential has power law form, and investigate the furtherphenomenological consequences. We start with rewriting the general form of the effectiveequation of state in terms of the parameters of the MaVaN Hybrid Scenario, which (atthis stage) are α, β, and V (assuming further a fixed/measured value of ¯ m ν ). To do so,we use the results from our adiabatic background solution ρ ν = αβ ρ A (see Eq. 23) andthe relation ρ ν ( z ) = ρ (0) ν (1 + z ) αα + β or ¯ m ν ( z ) = ¯ m (0) ν (1 + z ) − βα + β , (32)which is plotted in Fig. (1) for illustration.As the neutrino mass scale increases with time, neutrinos were relativistic at someearlier epoch. For example, for αβ = 2, this happens around redshift z = 100. We notethat at this (or larger) redshift, the above-derived formulae do no longer apply sincewe have assumed non-relativistic neutrinos throughout. The field A remains stabilised henomenology of Hybrid Scenarios of Neutrino Dark Energy ♯ , however at values which in general differ from thosegiven in Eq. (19). During CMB formation time where z ∼ ρ (0) DE = V + ρ (0) A , (33)we find that ρ (0) ν ρ (0) DE = αβ (1 − γ ) , (34)where we have defined the parameter γ = V ρ (0) DE (35)for the fractional contribution of V to the total observed dark energy density today.After some straightforward calculations, we finally get the expression for the equationof state in terms of the parameters α, β, γ and as a function of redshift z , w eff ( z ) = − γ + (1 − γ )(1 + z ) αα + β γ + (1 + αβ )(1 − γ )(1 + z ) αα + β − αβ (1 − γ )(1 + z ) . (36)We can see from the above expression that for γ = 1, w eff ( z ) = − γ = 1 we find that w eff ( z ) deviates from − z >
0. The amount of deviation depends on the choice of parameters α, β, γ .We can also see that the value of the ¯ m (0) ν does not appear in Eq. (36). As we willshow in the next subsection, ¯ m (0) ν is determined in this setup by the parameters α, β and γ . In fact, we use this to trade the parameter γ for the measurable quantity ¯ m (0) ν and re-express w eff ( z ) as a function of ¯ m (0) ν . As mentioned above, the neutrino mass scale ¯ m (0) ν can be calculated as a function of themodel parameters α, β and γ . For non-relativistic neutrinos (corresponding roughly to m i & − eV for i = 1 , ,
3) the neutrino energy density is ρ (0) ν = ¯ m (0) ν n (0) ν . As we havementioned in section 3.3, we are considering case I where the acceleron field providesthe masses of all three right-handed neutrinos. Furthermore, the observed dark energydensity today is given by ρ (0) DE = V + V ( A ). Using these two equations one can showthat ¯ m (0) ν can be expressed in terms of γ as¯ m (0) ν = (1 − γ ) αβ ρ (0) DE n (0) ν . (37) ♯ See e.g. [24] for a discussion in the original MaVaN scenario which can readily be generalised to thehybrid case. henomenology of Hybrid Scenarios of Neutrino Dark Energy Α (cid:144) Β m (cid:143)(cid:143)(cid:143) Ν H z L(cid:144) m (cid:143)(cid:143)(cid:143) Ν H L = m (cid:143)(cid:143)(cid:143) Ν H L @ eV D Α (cid:144) Β Γ =
Figure 1.
The left panel shows the neutrino mass scale ¯ m ν = P i =1 m i , divided bytheir present value, as a function of redshift and of α/β . The right panel shows thevalue of γ = V /ρ (0) DE as a function of the neutrino mass scale today and of α/β . Plugging in the observed value ρ (0) DE ∼ × − eV and the (standard) theoreticalprediction for the neutrino number density today, n (0) ν ∼ . × − eV [22, 33], wefind ¯ m (0) ν ≈ − γ ) αβ eV . (38)This simple expression relates the present value of the neutrino mass to the quantity γ which specifies which fraction of dark energy is constant vacuum energy. For fixed γ ,the neutrino mass scale ¯ m (0) ν is predicted in this scenario.One immediate consequence of the above relation is that for γ = 0, i.e. withouta constant energy density contribution V , the predicted neutrino mass violates thethe present experimental bounds, unless either α is very small or β is very large. Inparticular, for the standard case α = 2 and β = 1, and for ¯ m (0) ν of the order 1 eV, V must contribute ∼
99% of todays dark energy of the total observed dark energy density(c.f. Fig. 1). On the other hand, for smaller α/β the value of γ decreases and a largerfraction of dark energy is dynamical. For example, for β = 1 , α = 0 . m (0) ν ∼ m (0) ν by eliminating the parameter γ , which yields w eff = − (cid:16) − βα (cid:16) ¯ m (0) ν eV (cid:17)(cid:17) + βα ( ¯ m (0) ν eV )(1 + z ) αα + β (cid:16) − βα ( ¯ m (0) ν eV ) (cid:17) + (cid:0) βα (cid:1) ( ¯ m (0) ν eV )(1 + z ) αα + β − ( ¯ m (0) ν eV )(1 + z ) . (39)In Fig. 2 the z -dependent effective equation of state parameter w eff is plotted for thecase α = 2, β = 1 and various values of ¯ m (0) ν . henomenology of Hybrid Scenarios of Neutrino Dark Energy - - - - - w e ff m (cid:143)(cid:143)(cid:143) Ν = m (cid:143)(cid:143)(cid:143) Ν = m (cid:143)(cid:143)(cid:143) Ν = m (cid:143)(cid:143)(cid:143) Ν = - - - - - - w e ff m (cid:143)(cid:143)(cid:143) Ν = m (cid:143)(cid:143)(cid:143) Ν = m (cid:143)(cid:143)(cid:143) Ν = m (cid:143)(cid:143)(cid:143) Ν = Figure 2.
The plots show the effective dark energy equation of state parameter w eff as a function of redshift for different values of the neutrino mass scale ¯ m ν = P i =1 m i .The plots show the case α = 2 , β = 1 and the left panel includes examples with¯ m (0) ν ≤ . m (0) ν ≥ . For a dark energy equation of state which varies with time, a model independentextraction of w eff ( z ) is difficult from the currently available data. Therefore, in mostanalyses of the present data, a constant w is assumed. One quantity which is used inthe literature to compare with the best fit result under the assumption of a constant w is the weighted (or averaged) equation of state parameter w avg defined in the followingway [31] w avg = R z w eff ( z )Ω effDE ( z ) dz R z Ω effDE ( z ) dz . (40)In Fig. 3, w avg is plotted as a function of ¯ m (0) ν for the case α = 2 , β = 1. TheWMAP 5 year 2 σ limit on the constant dark energy equation of state parameter is − . < w < .
142 for zero curvature and − . < w < .
14 when a nonzerovalue of the curvature is allowed [34]. From the plot we can see that even ¯ m (0) ν as largeas 1 .
5. The MaVaN Hybrid Scenarios with High Scale Seesaw Mechanism
One interesting result of the previous section is that with power law MaVaN potentialsas in Eq. (4), the predictions for the observables do not depend on the value of A today. With the masses M R of the right-handed neutrinos originating from terms in thepotential of the form L M R = 12 λAN N (41) henomenology of Hybrid Scenarios of Neutrino Dark Energy m (cid:143)(cid:143)(cid:143) Ν @ eV D - - - - - - - w a vg Figure 3.
Averaged equation of state parameter w avg as a function of neutrino massscale ¯ m ν = P i =1 m i . The plot show the case where α = 2 and β = 1. this means that in principle the masses of the right-handed neutrinos today ( ∼ λA )could be close to the GUT scale and we might realise the MaVaN Hybrid Scenario witha “high-scale” seesaw mechanism. Compared to the usual MaVaN scenario where theright-handed neutrino masses are very small and where tiny neutrino Yukawa couplingshave to be postulated, in the “high-scale” seesaw the neutrino Yukawa couplings canbe of O (1) and the smallness of the neutrino masses is explained by the large masses ofthe right-handed neutrinos.Recalling Eq. (20), the scales A and M in the considered scenarios are connectedto the neutrino mass scale ¯ m (0) ν by A M − αα = (cid:18) βα ¯ m (0) ν n (0) ν (cid:19) /α . (42)To check the adiabaticity condition for the case of large A , we consider the square ofthe effective mass of the acceleron field today which is given by m eff ≡ d V eff dA = α ( α + β ) ( β/α ) − α ρ (0) ν − α M α (4 − α ) . (43)Using Eq. (42) in Eq. (43) we can eliminate the dependence on M and obtain m eff = 1 A ( β ( α + β ) ¯ m (0) ν n (0) ν ) / . (44)Now, for adiabaticity to hold, m eff must be larger than the present value of the Hubblescale ∼ . × − eV . Assuming for example α = 2 and β = 1 (i.e. a simple mass termpotential for A and the standard seesaw relation), with A around the GUT scale ∼ GeV and ¯ m (0) ν = 1 . m eff is about 2 × − eV, which is considerably heavier thanthe Hubble scale. For the same choice of parameters and present vev of acceleron field,but with ¯ m (0) ν = 0 .
06 eV; m eff ∼ × − eV, still larger than the Hubble scale. Thisis in contrast to quintessence-type models, where the mass of the relevant scalar fieldhas to be below the present Hubble scale. henomenology of Hybrid Scenarios of Neutrino Dark Energy A of at least about 10 − eV. Leaving these model-building questionsaside, we will now comment on how the “high-scale” seesaw might help with respect tostability issues for non-relativistic neutrinos.
6. High Scale Seesaw and Stability Issues for Non-relativistic Neutrinos
In this section we will comment on the stability issue regarding the neutrino densityperturbations, which are typically considered as a serious problem for MaVaN scenarioswith non-relativistic neutrinos. Let us first briefly review the problem: Soon after theintroduction of the original MaVaN scenario [7], it has been pointed out [27] that theoriginal scenario faces a catastrophic instability for non-relativistic neutrinos due to theextra force carried between neutrinos and the acceleron field A . With the mass m A ofthe field A much larger than the Hubble scale, it has been argued that the adiabaticperturbations at scales between m − A < λ < H − are unstable.Let us now discuss how this problem might be overcome in the MaVaN HybridScenarios with a “high-scale” seesaw mechanism. Rather than attempting a fullnumerical analysis of the density perturbations of the neutrinos in this paper (whichwe will leave for a future study), we will provide qualitative arguments and analyticalestimates which will show how the MaVaN Hybrid Scenarios might help to suppressdangerous instabilities. We will start with a general consideration regarding the couplingbetween the neutrinos and the acceleron field, and then turn to analytical criteria forthe occurrence of the instabilities.One main difference between the MaVaN Hybrid Scenarios with power law typepotentials (where a “high-scale” seesaw mechanism is in principle possible) and thestandard MaVaN scenario is that in the former the present value A of the acceleronfield can be much larger than the eV scale, even close to the GUT or Planck scale. Onecan readily see that this might help with respect to the stability issue, when the heavyright-handed neutrino fields N are integrated out the theory. Then, the interaction termbetween the light neutrino(s) and the acceleron field becomes (for, e.g., β = 1) L int = ¯ ν L ν L y v A , (45)where A = A + δA with δA denoting the quantum fluctuations around the classicalvalue of the field A . With δA ≪ A , the term in Eq. (45) takes the form L int ≈ ¯ ν L ν L ¯ m (0) ν A δA , (46) henomenology of Hybrid Scenarios of Neutrino Dark Energy m (0) ν = y v A . In the usual MaVaN scenario the field value A is not much largerthan ¯ m (0) ν . In contrast to this, as we have discussed in the previous section, in the MaVaNHybrid Scenarios with power law potentials the present value of A can in principle beas large as the GUT scale (or Planck scale) and the coupling between neutrinos and A can be strongly suppressed (by a factor ¯ m (0) ν /A ).Let us now turn to the analytical criteria: First of all we note that when revisitingthe stability issues of the MaVaN scenario, in Ref. [35] Bjaelde et al. have arguedthat a negative value of the sound speed does not always indicate the occurance of aninstability. Due to the dragging force of cold dark matter, neutrino perturbations canremain stable even if the sound speed of the dark energy fluid becomes negative. Morespecifically, the dragging force due to the dark matter and baryons can stabilise theneutrino perturbations if the condition [35], (cid:18) Ω CDM + Ω b Ω ν (cid:19) (cid:18) GG eff (cid:19) (cid:18) δ CDM δ ν (cid:19) ≫ G eff = G f ( A ) M P l a k ( V ′′ ( A ) + ρ ν f ′ ( A )) ! , (48)and where the coupling function f ( A ) is defined as f ( A ) = 1¯ m ν d ¯ m ν dA , (49)with f ( A ) = − A for the case β = 1. Using the above definitions the stability conditionof Eq. (47) can be rewritten as f ( A ) < (cid:18) Ω CDM + Ω b − Ω ν ν M P l (cid:19) (cid:18) δ CDM δ ν (cid:19) . (50)From this equation we can see that a small value of Ω ν (i.e. a small neutrino mass scaletoday) as well as a larger value of δ CDM compared to δ ν can help to stabilise the neutrinoperturbations.Setting δ CDM /δ ν ∼ m (0) ν ∼ . f ( A ) has to be smaller than 2 M − P l or for ¯ m (0) ν ∼ .
06 eV, f ( A ) has to beless than 10 M − P l . In order to avoid the instability the present value of A should thereforebe larger than about M P l /
10 for ¯ m (0) ν ∼ .
06 eV and M P l / m (0) ν ∼ . m (0) ν ∼ .
06 eV this condition is well compatible with the adiabaticity condition inEq. (22) and the simple analytical consideration suggests that the instability problemmight be cured. On the other hand, for ¯ m (0) ν ∼ . eV the adiabaticity and the stabilityconditions cannot safely be satisfied simultaneously and we conclude that whether theinstability appears has to be checked numerically. †† †† We note that the calculation of sound speed (see e.g. [36]) does not depend on the present value of A and it is therefore still negative in our model, however, as we have mentioned before we are followingthe argument of [35] that despite a negative sound speed the instability can be avoided by the draggingforce of dark matter. henomenology of Hybrid Scenarios of Neutrino Dark Energy α = 2 , β = 1) the dynamical contribution to thetotal dark energy is rather small (typically a few percent), and the main contributionarises from a constant vacuum energy density. This means that even if the neutrinoswould cluster, the observable effects on the smoothness of dark energy are much smallerthan in the standard MaVaN scenario. Another related question is whether the scales onwhich neutrinos might cluster finally leads to any observable effect [37]. We conclude byremarking that although we have given some arguments which suggest that stabilityproblems for non-relativistic neutrinos could be resolved in certain MaVaN HybridScenarios with “high-scale” seesaw, this issue requires further investigations (which areleft for future studies).
7. Summary and Conclusions
Motivated by the intriguing proximity of the energy density of dark energy and theneutrino mass scale we have studied the phenomenology of hybrid scenarios of neutrinodark energy, where in addition to a so-called Mass Varying Neutrino (MaVaN) sector, acosmological constant (from a false vacuum) is driving the accelerated expansion of theuniverse today. Within the generalised framework we have focused on phenomenologicalissues such as on the connection to the neutrino mass scale and on its consequences forthe dynamical nature of dark energy.We have therefore calculated the effective equation of state parameter w eff ( z ) inthe MaVaN Hybrid Scenario where the effective potential for the dynamical real scalarfield (the “acceleron” field A ) has the following form V ( A ) eff = M − α A α + ρ (0) ν a (cid:18) A A (cid:19) β + V . (51)We found that, for the case of a power law potential in the MaVaN sector, w eff ( z ) isdetermined by the neutrino mass scale and by the parameters α and β (c.f. Eq. (39)).Due to the interactions of the dark energy field with the neutrino sector, w eff ( z ) ispredicted to become smaller than − z > α = 2 and β = 1) we found that compatibility with the terrestrial neutrino mass boundsrequires a large contribution of constant vacuum energy (c.f. Eq. (38)).Another interesting question, which we have investigated in the MaVaN HybridScenario with power law potentials is whether it is possible to realise neutrino darkenergy with a “high-scale” seesaw mechanism, where the right-handed neutrino massesare close to the GUT scale. We found that the field value of the “acceleron” field as wellas the masses of the right-handed neutrinos can indeed be large and in principle a hybridscenario of neutrino dark energy might be realised with a “high-scale” seesaw. We have henomenology of Hybrid Scenarios of Neutrino Dark Energy w eff ( z ) whichdepends only on the parameters α and β of the potential and on the neutrino massscale. The prediction that w eff ( z ) < − z >
8. Acknowledgements
We would like to thank Neal Weiner for useful discussions. This work was partiallysupported by The Cluster of Excellence for Fundamental Physics “Origin and Structureof the Universe” (Garching and Munich). SD is supported by US NSF CAREERgrant PHY-0449818 and DOE OJI program under grant DE-FG02-06ER41417. SDacknowledges the hospitality of Max-Planck-Institut f¨ur Physik (Werner-Heisenberg-Institut) in Munich where this work was initiated.
AppendixA. Some Remarks on the possible Origin of the MaVaN Hybrid Potential
A.1. Fractional Power Potential from Non-Canonical Kinetic Energy Terms
One possible origin of fractional powers in the potential in Eq. (5) is a non-canonicallynormalised real scalar field A ′ with the following Lagrangian L = A ′ n ′ M n ′ ( ∂ µ A ′ ) − V ( A ′ ) , (A.1)where V ( A ′ ) = M − α ′ A ′ α ′ .In the MaVaN Hybrid Scenario, the effective energy density of the field A ′ has theform V eff ( A ′ ) = A ′ n ′ M n ′ ( ∂ µ A ′ ) + M − α ′ A ′ α ′ + ρ (0) ν a (cid:18) A ′ A ′ (cid:19) β ′ , (A.2)where the last term arises form the smooth background energy density. We have assumed¯ m ν ( A ′ ) ∼ A ′ β ′ . henomenology of Hybrid Scenarios of Neutrino Dark Energy A , whichsatisfies 1 √ ∂ µ A = (cid:18) A ′ M (cid:19) n ′ / ∂ µ A ′ . (A.3)Integrating this equation we obtain the canonically normalised field A in terms A ′ , andthe effective potential in terms of the normalised field A becomes, V eff ( A ) = g α ′ M (4 − α ′ )2 M n ′ α ′ n ′ +2 A α ′ n ′ +2 + ρ (0) ν a (cid:18) A A (cid:19) β ′ n ′ +2 , (A.4)where g = ( n ′ + 2) n ′ +2 / n ′ +2 which can be easily absorbed in the redefinition of M .As a consequence of the non-canonically normalised scalar field we have obtainedfractional powers in the MaVaN Hybrid potential with α = α ′ n ′ +2 and β = β ′ n ′ +2 . A.2. Generalisation to Three Families: Possible Origin of V One motivation for introducing a constant term V to the potential arises insupersymmetric “hybrid-type” models of dark energy. Generalising the superpotential ofEq. (6) to three families, we may realise the situation that some of the “waterfall fields” N i are still in the false vacuum while others are in the true vacuum where | N i | ∼ v N i .A simple superpotential with this characteristic feature is the following W = λ ˆ A ( x i ˆ N i − v N i ) + y αi ˆ H u ˆ ν α ˆ N i . (A.5)In this case, the vacuum energy of the acceleron field does not account for the total darkenergy, but only for part of it. The other part stems from the N i which are in the falsevacuum leading each to a contribution of energy density λ v N i . It is also interestingto note that the N i which have non-zero vevs generate a mass term for A from thecontribution | F N | to the scalar potential. A.3. Higher Powers of A from the Superpotential
It is furthermore possible to generalise the setup of Eq. (A.5) such that higher powersof A arise in the potential. Such higher powers can emerge from a superpotential of theform W = λ A n M n − ( x i ˆ N i − v N i ) + y αi ˆ H u ˆ ν α ˆ N i . (A.6)The most relevant parts of the scalar potential (for a N i in the true vacuum) would thenbe given by | F A | ∼ v N i | A | n − and | F N i | ∼ | λA n M n − N i + y αi ˆ H u ˆ ν α | . This generalisation ofthe superpotential has two effects: Firstly, | F A | leads to a higher power of A, explicitly | A | n − , i.e. α = 2 n − | F N i | results inmasses of the right-handed neutrinos of the form m N i ∼ x i λ | A | n M n − which leads to massesof the light neutrinos ¯ m ν ∼ M n − | A | n v . This implies β = n in Eq. (4). 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