Cosmon as the Modulon: Non-Gaussianity from Dark Energy
aa r X i v : . [ h e p - ph ] O c t Cosmon as the Modulon: Non-Gaussianity from Dark Energy
Chian-Shu Chen , ∗ and Chia-Min Lin , † Physics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan 300 Department of Physics, National Tsing Hua University, Hsinchu, Taiwan 300 Institute of Physics, Academia Sinica, Taipei, Taiwan 115 (Dated: Draft September 26, 2018)In this paper, we show that the idea of growing neutrino can result in a modulated reheatingeffect and produce detectable non-Gaussianity in the model where the Higgs triplet from type IIseesaw mechanism plays the role of the inflaton in chaotic inflation.
PACS numbers:
Introduction : Inflation [1] not only solves many problems (for example, horizon problem and flatness problem) ofconventional hot big bang model but could also produce the primordial density perturbation which is the seed forstructure formation and cosmic microwave background (CMB) temperature fluctuations. Intuitively, the mechanismfor producing density fluctuation from inflation is that during inflation the quantum fluctuations of the inflaton whenstretched outside the horizon becomes classical perturbations which is different in different patches of the universeseparated by the horizon. Each patch could be regarded as a “separate universe” and evolves in the same way. However,inflation ends in different “time” for each universe and this result in a primordial density (curvature) perturbation.The primordial curvature perturbation ζ (on uniform-density slices) is given by (with a suitable coordinate choice) dl = a ( t ) e ζ dx i dx j ∼ a ( t )(1 + 2 ζ ) dx i dx j . (1)Here ζ describes the difference between the perturbed universe and unperturbed universe. During inflation, since a ∼ e N , therfore intuitively one may think δN = ζ . This has been rigorously proved and is called the δN formalism[2–5]. This relation is true up to nonlinear orders. For example, if the primordial density perturbation is from thefluctuation of the inflaton φ , we can write δN = N φ δφ + 12 N φφ ( δφ ) + · · · (2)where subscript denotes derivative with respective to the corresponding argument. Since δφ ∼ H/ π is Gaussian, thesecond term represents non-Gaussianity. Future experiments (like PLANCK) can probe the second (or even higher)order effects for the primordial curvature perturbation which is parameterized as δN = ζ = ζ g + 35 f NL ζ g + · · · . (3)Here ζ g is the linear (Gaussian) term and f NL is the (local) nonlinear parameter. From Eqs. (2) and (3), we couldobtain f NL = 56 N φφ N φ . (4)Generally, for single-field slow-roll inflation, f NL is too small to be detected in the near future [6]. Therefore adetection will force us to go for more complicated (inflation) models.In Ref. [7], we proposed the idea that the Higgs triplet ∆ in type II seesaw mechanism could play the role of theinflaton for chaotic inflation. The potential form is V = 12 M ∆ , (5)where the quartic terms are ignored . The number of e-folds is hence given by N = 1 M P Z VV ′ dφ = 1 M P φ ∗ [email protected] † [email protected] There is an argument given in [8] about why we may neglect the quartic terms. which is independent of the mass. Here M P is the reduced Planck mass. From Eq. (2) and (6), we can see ζ ∼ H ∗ ∼O (10) M ∆ which is fixed to be O (10 − ) by CMB observation. Here H ∗ means the Hubble parameter at horizon exit. Ifthe primordial density perturbation is from the quantum fluctuation of the inflaton, this would imply that the inflatonmass m ∆ is fixed to be around 10 GeV and the primordial density perturbation would be gaussian. However, it ispossible that the primordial density perturbation is from some other mechanism and we could have H ∗ < ∼ − . Forexample, in the case of modulated reheating scenario [9], the inflaton decay width is determined by some light field σ (called the modulon). During inflation the quantum fluctuation of the modulon will “modulate” the decay widthof the inflaton hence when inflation decays after inflation this will contribute to the primordial density perturbation.The number of e-folds is related to the decay width Γ via N = −
16 ln Γ . (7)In this case if the contribution of δN from inflaton is negaligible, similar to Eq. (2) we could write δN = N σ δσ + 12 N σσ ( δσ ) + · · · . (8)Therefore to first order we obtain δN = − δ ΓΓ (9)and similar to Eq. (4), we can easily obtain 65 f NL = 6 (cid:18) − ΓΓ σσ Γ σ (cid:19) . (10)One possibility of the dependence of the decay width is that the inflaton mass is determined by the field valueof the modulon. For a Higgs triplet, this kind of dependence was proposed in Ref. [10, 11] for another purpose andthe light field is called the cosmon. In this model the cosmon field would make the neutrino mass growing (growingneutrino) through the varying mass of the Higgs triplet via type II seesaw mechanism. The result is in the currentuniverse the cosmon field will be freezed due to growing neutrino and the scalar potential of the cosmon would becomethe dark energy we observe today. In this paper, we will show that cosmon field in the early universe could play therole of the modulon in the Higgs triplet inflation and produce detectable primordial non-Gaussianity. The constraintto the Higgs triplet mass can also be liberated. The seesaw conception used in particle physics to understand thesmallness of neutrino mass through the high energy physics can apply to connect the early universe (inflation ∼ GrandUnification scale) and current universe (dark energy ∼ eV scale) in our model. Construction : The type-II seesaw mechanism contains one complex SU (2) L triplet scalar ∆ with hypercharge Y = 2in addition to the standard model Higgs doublet H [13]. The Higgs triplet ∆ can interact with left-handed leptonsthrough the coupling, Y ij L TiL
Ciτ ∆ L jL + H.c., here i is the flavor index, C is the charge conjugation, and τ is thePauli matrix. There is also a trilinear term, µH T iτ ∆ † H + H.c. in the potential ( µ is the dimensionful parameter).The coexistence of both two terms breaks the lepton number by two units. After taking the minimal condition of thepotential, the 3 × m ν ij = Y ij µv M = Y ij v ∆ . (11)Here v and v ∆ are the vacuum expectation values of standard model Higgs and the triplet ∆ respectively. We continuethe idea that the type-II seesaw scalar triplet ∆ as the inflaton for chaotic inflation [7] and combine the proposal inRef. [11] that the mass of ∆ depends on the field value of cosmon field σ is assumed in the following way, M = c ∆ M GUT (cid:20) − τ exp( − ǫ σM P ) (cid:21) . (12)Here M GUT is the grand unification scale, and c ∆ and τ are the order one parameters. The potential of the cosmonfield is given by V ( σ ) = M P e − ασ/M P (13) We study the possible role of the cosmon in the early universe in a different set up in [12]. with α > ∼
10 (from early dark energy constraint [14]). This term will result in a tracker behavior of the cosmon fieldafter inflation. The evolution of σ is given by [15]¨ σ + 3 H ˙ σ = − ∂V∂σ + β ( σ ) M ( ρ ν − p ν ) (14)where β ( σ ) ≡ ǫτ exp( − ǫ σM P )1 − τ exp( − ǫ σM P ) (15)We can write β in the form β ( σ ) = M P σ − σ t for σ t ≡ − M P ln τǫ when σ is close to σ t . When σ approaches σ t Eq. (13)would behave like a cosmological constant and becomes the dark energy. For ασ t /M P ∼
276 the cosmological constanthas a value compatible with observation. This implies ǫ = − α ln τ /
276 therefore if we choose α = 10 and ln τ = 1, wewould have ǫ = − .
05 which implies a mild dependence of M ∆ on σ through Eq. (12). Furthermore it is pointed outin Ref. [11] that the detail form of σ -dependence of M ∆ is not important as long as a Taylor expansion is applicablearound σ ≈ σ t . We will use those values in the following context.Since ∆ is the inflaton and can decay into several channels such as ∆ → νν, HH, ZZ, and σσ to reheat the universe.One should note that the decay mode of ∆ → σσ is possible via the nonrenormalizable couplings which are from theexpansion of Eq. (12), read L eff = − c ∆ ǫ τ ( M GUT M P ) σ ∆ . (16)The main decay widths of ∆ are given by Γ ∆ ( ν i ν j ) = Y ij π (1 + δ ij ) M ∆ , (17)Γ ∆ ( HH ) = M v πv , (18)Γ ∆ ( ZZ ) = g m Z v πM ∆ cos θ W v , (19)and Γ ∆ ( σσ ) = c ǫ πτ ( M GUT M P ) ρ ∆ M . (20)The decay product produced through the coupling in Eq. (16) might explain the hint of the need for an extra, dark,relativistic energy component in recent analyses [16–21]. The energy density of the new degree of freedom is usuallynormalized to neutrino energy density ρ ν in a convenient way with the “effective number of equivalent neutrinos” N νeff defined by ρ ν = ρ γ
78 ( 411 ) / N νeff . (21) N νeff = 4 . ± . H (the Hubble constant) [16], and the current observed primordial Helium mass fraction prefers a larger value Y p = 0 . ± . . ) ± . . ) than standard BBN prediction Y p = 0 . ± . ρ σ s | BBN < ∼ ρ ν s | BBN . (22) A possible derivation of the exponential mass dependence on the cosmon field associated with supersymmetry breaking is obtained in[12].
Here s is the entropy density and ρ σ | decay = Br (∆ → σ ) M ∆ n ∆ with Br and n ∆ denote branching ratio of andnumber density of inflaton ∆. And since n ∆ s = T R M ∆ we have ρ σ s | BBN = ρ σ s | T R ( T BBN T R ) = 34 Br (∆ → σ ) T BBN (23)and ρ ν s | BBN ∼ × T BBN
19 (24)for the two sides of Eq. (22). Therefore we roughly have the constraint for the Br (∆ → σσ ) < ∼ O ( ). The boundcan be easily satisfied in Eqs. (17) - (20). It also can be understood that the decay rate is proportional to the Hubbleparameter squared and decreases faster than the universe expansion rate. Inflaton will not decay completely intoradiation and reheat the universe if the four-point interaction (Eq. (16)) is the dominant decay. Primordial density perturbations : If the cosmon σ is light during inflation, it is subject to fluctuations similar tothe inflaton, namely, δσ ∼ H ∗ / π . This would lead to fluctuations of the decay width by the variation of the inflatonmass M ∆ , and may contribute to the primordial density perturbation. The potential of the cosmon field is given byEq. (5) and (12): V ( σ ) = 12 c ∆ M GUT (cid:20) − τ exp( − ǫ σM P ) (cid:21) ∆ . (25)There is another term in the potential given by Eq. (13) but it is subdominant and negligible (for a wide range of σ )during inflation.The condition of the cosmon being “light” (during inflation) is given by | V ′′ /H | ∼ | ǫβ | ≪ | ǫ | ∼ .
01, Eq. (26) can be satisfied if | β | < ∼ ζ ∼ δ ΓΓ ∼ δM ∆ M ∆ = | β end | δσM P ∼ − . (27)Here | β end | means | β | at the end of inflation. We could see that for | β end | > ∼ δσM P ∼ H ∗ /M P < ∼ − . Notethat the condition that ζ inf subdominant would require | β end | > ∼
1. Therefore we require 1 < ∼ | β end | < ∼ . In thiscase, the primordial density perturbation is dominated by the fluctuation of the cosmon field which would play therole of dark energy in the current universe.Actually we can also consider | β end | < ∼
1. In this case, σ would be slow-rolling until inflation ends. Accordingto Eq. (27), the contribution of primordial curvature perturbation is subdominant. However, it is still possible togenerate sizeable non-Gaussianity [22, 23]. We will discuss this in the following section. Non-Gaussianity : From Eq. (10) we can obtain65 f NL = 6(1 + O (1) ǫβ ) , (28)where the order one factor depends on different decay widths in Eqs. (17) - (20). From here we can see that larger β implies smaller f NL . This may be intuitively understood by the following argument. If we require ζ ∝ βδσ ∼ − ,large β implies small δσ which means the nonlinear (non-Gaussian) effect ∝ ( δσ ) is small. In the case where thecontribution of the primordial density perturbation is dominated by the fluctuation of the cosmon field, we have | ǫ | ∼ .
01 and | β | > ∼ f NL = 5 which may be detected in the near future by PLANCK satellite.If the contribution of ζ from modulated reheating is subdominant, from Eq. (3), it can be shown that f NL wouldbe reduced by a factor of β / (1 + β ) . For example, if β ∼ .
5, we would have f NL ∼ O (1) which is close to themarginal value of experimental sensitivity in the near future. The non-Gaussianity produced is still larger than thecase that we only have chaotic inflation without modulated reheating. Isocurvature and leptogenesis : As we have shown in [7] the baryon asymmetry of the universe can be obtained vialeptogenesis if two triplet scalars exist. In our model, if the primordial density perturbation is dominated by theflucturation of the cosmon field through modulated reheating, it is possible to generate a large baryonic isocurvature
P=-7 P=-3P=-10 2 4 6 8 10 - - S B (cid:144) Ζ FIG. 1: The ratio of isocurvature to primordial density perturbations versus the parameter K and P = − , − , − νν, HH, ZZ decay modes respectively. perturbation. Let’s consider the possibility of isocurvature perturbation induced from the lepton asymmetry in thisconstruction.The CP violation is generated through the interference between the tree level and self-energy correction of thetriplet scalar decay, given by ǫ ≈ Im[ µ µ ∗ P k , l (Y Y ∗ )]8 π ( M − M ) (cid:16) M ∆ Γ ∆ (cid:17) . (29) µ , are the cubic couplings involving the triplet and two powers of the Higgs-doublet and indices 1 , , . The parameter K is defined by K = Γ ∆ /H ( T = M ∆ )with H ( T ) | T = M ∆1 = q π g ∗ M M P (here we assume M ∆ < M ∆ ) and g ∗ ∼
100 is the effective number of masslessparticles. After solving the Boltzmann equations that involve decay, inverse decay, and annihilation processes, thebaryon asymmetry can be approximated by n B s ∼ . × − ǫ × ( K + 9) − / (30)for 0 < K <
10 [24]. Let M ∆ = 3 × GeV, µ , ∼ GeV, Y (1 , ij ∼ . m ν ∼ − ∼− eV, and K = 5,the n B /s ≈ − as desired. We take ǫ ∝ M − and K ∝ M P ∆ where P is integer and depends on the decay widthsgiven in Eqs. (17) - (20). The baryon-isocurvature fluctuation can be expressed as S B ≡ δ ( n b /s ) n b /s = ζ (cid:20) − − P K ( K + 9) − (cid:21) . (31)The observational constraint on the uncorrelated baryon isocurvature is | S B /ζ | < ∼ O (1) [21, 25]. We show thecontributions to S B from the decay modes of ∆ as the function of K in Fig. 1. It indicates the model is well withinconstraint. For the case of primordial density perturbations from modulated reheating is subdominant we expect the S B is smaller. Conclusion : In this paper, we investigated the possible cosmological consequences of inflation driven by a Higgstriplet of type II seesaw mechanism if the dark energy is from the growing neutrino mechanism. Interestingly, inthis setup, we found that the primordial curvature perturbation and/or non-Guassianity may be from the quantumfluctuations of the cosmon field which would cause the dark energy we observe today and cosmon would play the roleof the modulon. If the contribution of the modulated reheating to the curvature perturbation dominates, there is anissue of baryon isocurvature perturbation. In this case, we investigated the allowed region of the parameter spaceand found the constraint is not very severe. Futhermore, if isocurvature perturbation is found in future experiments,it would provide an interesting constraint to our model. If the curvature perturbation from the modulon is sub-dominate, there is no issue about isocurvature perturbation. However, sizable non-Gaussianity may still be generated. We ignore the σσ mode as it is supposed to be subdominant. Acknowledgments
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