Higher order terms in the inflaton potential and the lower bound on the tensor to scalar ratio r
aa r X i v : . [ a s t r o - ph . C O ] D ec Higher order terms in the inflaton potential and the lower bound on the tensor toscalar ratio r
C. Destri ∗ Dipartimento di Fisica G. Occhialini, Universit`a Milano-Bicocca and INFN,sezione di Milano-Bicocca, Piazza della Scienza 3, 20126 Milano, Italia.
H. J. de Vega † LPTHE, Universit´e Pierre et Marie Curie (Paris VI) et Denis Diderot (Paris VII),Laboratoire Associ´e au CNRS UMR 7589, Tour 24, 5`eme. ´etage,Boite 126, 4, Place Jussieu, 75252 Paris, Cedex 05, France andObservatoire de Paris, LERMA. Laboratoire Associ´e au CNRS UMR 8112.61, Avenue de l’Observatoire, 75014 Paris, France.
N. G. Sanchez ‡ Observatoire de Paris, LERMA. Laboratoire Associ´e au CNRS UMR 8112.61, Avenue de l’Observatoire, 75014 Paris, France. (Dated: May 30, 2018)The MCMC analysis of the CMB+LSS data in the context of the Ginsburg-Landau approach toinflation indicated that the fourth degree double–well inflaton potential in new inflation gives anexcellent fit of the present CMB and LSS data. This provided a lower bound for the ratio r of thetensor to scalar fluctuations and as most probable value r ≃ .
05, within reach of the forthcomingCMB observations. In this paper we systematically analyze the effects of arbitrarily higher order terms in the inflaton potential on the CMB observables: spectral index n s and ratio r . Furthermore,we compute in close form the inflaton potential dynamically generated when the inflaton field is afermion condensate in the inflationary universe. This inflaton potential turns out to belong to theGinsburg-Landau class too. The theoretical values in the ( n s , r ) plane for all double well inflatonpotentials in the Ginsburg-Landau approach (including the potential generated by fermions) fallinside a universal banana-shaped region B . The upper border of the banana-shaped region B isgiven by the fourth order double–well potential and provides an upper bound for the ratio r . Thelower border of B is defined by the quadratic plus an infinite barrier inflaton potential and providesa lower bound for the ratio r . For example, the current best value of the spectral index n s = 0 . r is in the interval: 0 . < r < . Contents
I. Introduction II. Physical parametrization for inflaton potentials
III. Higher–order even polynomial double-well inflaton potentials IV. The quadratic plus the n th order double-well inflaton potential n → ∞ limit at fixed u . 14B. The double limit n → ∞ and u →
1. 15
V. The quadratic plus the exponential potential. b → ∞ at fixed u . 17B. The double limit b → ∞ and u →
1. 18
VI. Dynamically generated inflaton potential from a fermion condensate in the inflationary stage. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected]
VII. The Universal banana region B References I. INTRODUCTION
The current WMAP data are validating the single field slow-roll scenario [1]. Single field slow-roll models providean appealing, simple and fairly generic description of inflation [2, 3]. This inflationary scenario can be implementedusing a scalar field, the inflaton with a Lagrangian density L = a ( t ) (cid:20) ˙ ϕ − ( ∇ ϕ ) a ( t ) − V ( ϕ ) (cid:21) , (1.1)where V ( ϕ ) is the inflaton potential. Since the universe expands exponentially fast during inflation, gradient termsare exponentially suppressed and can be neglected. At the same time, the exponential stretching of spatial lengthsclassicalize the physics and permits a classical treatment. One can therefore consider an homogeneous and classicalinflaton field ϕ ( t ) which obeys the evolution equation¨ ϕ + 3 H ( t ) ˙ ϕ + V ′ ( ϕ ) = 0 , (1.2)in the isotropic and homogeneous Friedmann-Robertson-Walker (FRW) metric ds = dt − a ( t ) d~x , (1.3)which is sourced by the inflaton. Here H ( t ) ≡ ˙ a ( t ) /a ( t ) stands for the Hubble parameter. The energy density andthe pressure for a spatially homogeneous inflaton are given by ρ = ˙ ϕ V ( ϕ ) , p = ˙ ϕ − V ( ϕ ) . (1.4)The scale factor a ( t ) obeys the Friedmann equation, H ( t ) = 13 M P l (cid:20)
12 ˙ ϕ + V ( ϕ ) (cid:21) . (1.5)In order to have a finite number of inflation efolds, the inflaton potential V ( ϕ ) must vanish at its absolute minimum V ′ ( ϕ min ) = V ( ϕ min ) = 0 . (1.6)These two conditions guarantee that inflation is not eternal. Since the inflaton field is space-independent inflation isfollowed by a matter dominated era (see for example ref. [7]).Inflation as known today should be considered as an effective theory , that is, it is not a fundamental theorybut a theory of a condensate (the inflaton field) which follows from a more fundamental one. In order to describethe cosmological evolution it is enough to consider the effective dynamics of such condensates. The inflaton field φ may not correspond to any real particle (even unstable) but is just an effective description while the microscopicdescription should come from a Grand Unification theory (GUT) model.At present, there is no derivation of the inflaton model from a microscopic GUT theory. However, the relationbetween the effective field theory of inflation and the microscopic fundamental theory is akin to the relation betweenthe effective Ginsburg-Landau theory of superconductivity [4] and the microscopic BCS theory, or like the relationof the O (4) sigma model, an effective low energy theory of pions, photons and nucleons (as skyrmions), with thecorresponding microscopic theory: quantum chromodynamics (QCD).In the absence of a microscopic theory of inflation, we find that the Ginsburg-Landau approach is a powerfuleffective theory description. Such effective approach has been fully successful in several branches of physics when themicroscopic theory is not available or when it is very complicated to solve in the regime considered. This is the casein statistical physics, particle physics and condensed matter physics. Such GL effective theory approach permits toanalyse the physics in a quantitative way without committing to a specific model [4].The Ginsburg-Landau framework is not just a class of physically well motivated inflaton potentials, among themthe double and single well potentials. The Ginsburg-Landau approach provides the effective theory for inflation, withpowerful gain in the physical insight and analysis of the data. As explained in this paper and shown in the refs.[6]-[7], the analysis of the present set of CMB+LSS data with the effective theory of inflation, favor the double wellpotential. Of course, just analyzing the present data without this powerful physical theory insight, does not allow todiscriminate between classes of models, and so, very superficially and incompletely, it would seem that almost all thepotentials are still at the same footing, waiting for the new data to discriminate them.In the Ginsburg-Landau spirit the potential is a polynomial in the field starting by a constant term [4]. Linearterms can always be eliminated by a constant shift of the inflaton field. The quadratic term can have a positive or anegative sign associated to unbroken symmetry (chaotic inflation) or to broken symmetry (new inflation), respectively.As shown in refs. [6, 7] a negative quadratic term and a negligible cubic term in new inflation provides a very goodfit to the CMB+LSS data, (the inflaton starts at or very close to the false vacuum ϕ = 0). The analysis in refs.[6, 7]showed that chaotic inflation is clearly disfavoured compared with new inflation. Namely, inflaton potentials with V ′′ (0) < ϕ = 0.We can therefore ignore the linear and cubic terms in V ( ϕ ). I f we restrict ourselves for the moment to fourth orderpolynomial potentials, eq.(1.6) and V ′′ (0) < V ( ϕ ) = − m ϕ + 14 λ ϕ + m λ = 14 λ (cid:18) ϕ − m λ (cid:19) . (1.7)The mass term m and the coupling λ are naturally expressed in terms of the two energy scales which are relevantin this context: the energy scale of inflation M and the Planck mass M P l = 2 . GeV, m = M M P l , λ = y N (cid:18) MM P l (cid:19) . (1.8)Here y = O (1) is the quartic coupling.The MCMC analysis of the CMB+LSS data combined with the theoretical input above yields the value y ≃ . y turns out to be order one consistent with the Ginsburg-Landau formulation of the theoryof inflation [7].According to the current CMB+LSS data, this fourth order double–well potential of new inflation yields as mostprobable values: n s ≃ . , r ≃ .
051 [6, 7]. This value for r is within reach of forthcoming CMB observations [21].For the best fit value y ≃ .
26, the inflaton field exits the horizon in the negative concavity region V ′′ ( ϕ ) < M = 0 . × GeV for the scale of inflation and m = 1 . × GeV for the inflaton mass . (1.9)It must be stressed that in our approach the amplitude of scalar fluctuations | ∆ R k ad | = (4 . ± . × − allows usto completely determine the energy scale of inflation which turns out to coincide with the Grand Unification energyscale (well below the Planck energy scale). Namely, we succeed to derive the energy scale of inflation without theknowledge of the value of r from observations. M ≪ M P l guarantees the validity of the effective theory approach toinflation. The fact that the inflaton mass is m ≪ M implies the appearance of infrared phenomenon as the quasi-scaleinvariance of the primordial power.Since the inflaton potential must be bounded from below V ( ϕ ) ≥
0, the highest degree term must be even and witha positive coefficient. Hence, we consider polynomial potentials of degree 2 n where 1 < n ≤ ∞ .The request of renormalizability restricts the degree of the inflaton potential to four. However, since the theory ofinflation is an effective theory, potentials of degrees higher than four are in principle acceptable.A given Ginsburg-Landau potential will be reliable provided it is stable under the addition to the potential ofterms of higher order. Namely, adding to the 2 n th order potential further terms of order 2 n + 1 and 2 n + 2 shouldonly produce small changes in the observables. Otherwise, the description obtained could not be trusted. Since, thehighest degree term must be even and positive, this implies that all even terms of order higher or equal than fourshould be positive.Moreover, when expressed in terms of the appropriate dimensionless variables, a relevant dimensionless couplingconstant g can be defined by rescaling the inflaton field. This coupling g turns out to be of order 1 /N where N ∼ /N [5]. It is then natural to introduce as coupling constant y ≡ N g = O ( N ). This is consistent with the stability of the results in the above sense. Generally speaking, theGinsburg-Landau approach makes sense for small or moderate coupling.Odd terms in the inflaton field ϕ are allowed in V ( ϕ ) in the effective theory of inflation. Choosing V ( ϕ ) an evenfunction of ϕ implies that ϕ → − ϕ is a symmetry of the inflaton potential. At the moment, as stated in [7, 12], we donot see reasons based on fundamental physics to choose a zero or a nonzero cubic term, which is the first non-trivialodd term. Only the phenomenology, that is the fit to the CMB+LSS data, decides on the value of the cubic and thehigher order odd terms. The MCMC analysis of the WMAP plus LSS data shows that the cubic term is negligibleand therefore can be ignored for new inflation [6, 7]. CMB data have also been analyzed at the light of slow-rollinflation in refs. [8].In the present paper we systematically study the effects produced by higher order terms ( n >
4) in the inflationarypotential on the observables n s and r .We show in this paper that all r = r ( n s ) curves for a large class of double–well potentials of arbitrary high orderin new inflation fall inside the universal banana region B depicted in fig. 10. Moreover, we find that the r = r ( n s )curves for even double–well potentials with arbitrarily positive higher order terms lie inside the universal bananaregion B [fig. 10]. This is true for arbitrarily large values of the coefficients in the potential.Furthermore, the inflaton field may be a condensate of fermion-antifermion pairs in a grand unified theory (GUT) inthe inflationary background. In this paper we explicitly write down in closed form the inflaton potential dynamicallygenerated as the effective potential of fermions in the inflationary universe. This inflaton potential turns out to belongto the Ginsburg-Landau class of potentials considered in this paper. We find that the corresponding r = r ( n s ) curveslie inside the universal banana region B provided the one-loop part of the inflaton potential is at most of the sameorder as the tree level piece. Therefore, a lower bound for the ratio tensor/scalar fluctuations r is present for all potentials above mentioned. For the current best value of the spectral index n s = 0 .
964 [1, 7] the lower bound turnsout to be r > . combined with the value n s = 0 .
964 for the spectral index yieldsthe lower bound r > . n s (in disagreement with observations) r can bearbitrarily small within the GL class of inflaton potentials.The upper border of the universal region B tells us that r < .
053 for n s = 0 . B within the large class of potentials considered here0 . < r < .
053 for n s = 0 . . Interestingly enough r ≃ .
04 is within reach, although borderline for the Planck satellite [21].Among the simplest potentials in the Ginsburg-Landau class, the one that best reproduces the present CMB+LSSdata, is the fourth order double–well potential eq.(1.7), yielding as most probable values: n s ≃ . , r ≃ . n s and r including new inflation and in particular hilltop inflation [13–16].The question on whether a lower bound for r is found or not depends on whether the Ginsburg-Landau (G-L)effective field approach to inflation is used or not. Namely, within the G-L approach, the new inflation double wellpotential determines a banana shaped relationship r = r ( n s ) which for the observed n s value determines a lowerbound on r . The analysis of the CMB+LSS data within the G-L approach which we performed in refs. [6, 7] showsthat new inflation is preferred by the data with respect to chaotic inflation for fourth degree potentials, and that thelower bound on r is then present. Without using the powerful physical G-L framework such discrimination betweenthe two classes of inflation models is not possible and the lower bound for r does not emerge. Other references inthe field (i. e. [13, 19, 20]) do not work within the Ginsburg-Landau framework, do not find lower bounds for r andcannot exclude arbitrarily small values for r , much smaller than our lower bound r ≃ . r = r ( n s ) curves.Sec. III contains the 2 n th order double–well polynomial inflaton potentials with arbitrary random coefficients andtheir r = r ( n s ) curves. Sec. IV presents the n → ∞ limits of these polynomial potentials and we present in sec. V theexponential potential and its infinite coupling limit. In sec. V we compute the inflaton potential from dynamicallygenerated fermion condensates in a de Sitter space-time displaying their r = r ( n s ) curves. Finally, we present anddiscuss the universal banana region in sec. VII together with our conclusions. II. PHYSICAL PARAMETRIZATION FOR INFLATON POTENTIALS
We start by writing the inflaton potential in dimensionless variables as [12] V ( ϕ ) = M v (cid:18) ϕM P l (cid:19) , (2.1)where M is the energy scale of inflation and v ( φ ) is a dimensionless function of the dimensionless field argument φ = ϕ/M P l . Without loss of generality we can set v ′ (0) = 0. Moreover, provided V ′′ (0) = 0 we can choose withoutloss of generality | v ′′ (0) | = 1 / N = − Z φ end φ exit dφ v ( φ ) v ′ ( φ ) , (2.2)where φ exit is the inflaton field at horizon exit. To leading order in 1 /N we can take φ end to be the value φ min atwhich v ( φ ) attains its absolute minimum v ( φ min ), which must be zero since inflation must stop after a finite numberof efolds [7].Then, in chaotic inflation we have φ min = 0, with v ′ ( φ ) > φ >
0, while in new inflation we have φ min > v ′ ( φ ) < < φ < φ min . We consider potentials v ( φ ) that can be expanded in Taylor series around φ = φ min ,with a non-vanishing quadratic (mass) term.It is convenient to rescale the inflaton field in order to conveniently parametrize the higher order potential. Wedefine a coupling parameter g > v ( φ ) = 1 g v ( φ √ g ) (2.3)For a potential v ( u ) expanded in power series around u = 0 we write: v ( u ) = c ∓ u + X k ≥ c k k u k (2.4)Then, replacing u = φ √ g , (2.5)we find v ( φ ) = c g ∓ φ + X k ≥ g k/ − k c k φ k . (2.6)The positive sign in the quadratic term corresponds to chaotic inflation (in which case c = 0), while the negativesign corresponds to new inflation (in which case c is chosen such that v ( u ) vanishes at its absolute minimum).Clearly g plus the set of coefficients c k provide an overcomplete parametrization of the inflaton potential which wewill now reduce. In the case of chaotic inflation a convenient choice is c = 1, so that v ( φ ) = 12 φ + √ g c φ + g φ + X k ≥ g k/ − k c k φ k [chaotic inflation] (2.7)which represents a generic higher order perturbation of the trinomial chaotic inflation studied in refs. [6].In the case of new inflation, where φ min >
0, it is more convenient to set without loss of generality that u min =1 , φ min = 1 / √ g . In order to have appropriate inflation, u min = 1 must be the absolute minimum of v ( u ) and theclosest one to the origin on the positive semi–axis. That is, v ′ (1) = − X k ≥ c k = 0 (2.8)and then v (1) = 0 fixes from eq.(2.3) the constant term c in the potential c = 12 − X k ≥ c k k (2.9)We thus get for the inflaton potential v ( u ) = 12 (1 − u ) + X k ≥ c k k ( u k −
1) [new inflation] , (2.10)corresponding to v ( φ ) = 12 (cid:18) g − φ (cid:19) + X k ≥ c k k (cid:18) g k/ − φ k − g (cid:19) [new inflation] (2.11)For the coupling g and the field φ using eq.(2.5), g = 1 φ min = M P l ϕ min , u = φφ min = ϕϕ min . (2.12)From eq.(2.2) it now follows that the parameter g can be expressed as the integral y ( u ) = 8 Z uu min dx v ( x ) v ′ ( x ) , u ≡ √ g φ exit , (2.13)where, g = y ( u )8 N , (2.14)with u min = 0 for chaotic inflation and u min = 1 for new inflation. Eq.(2.13) can be regarded as a parametrizationof g and y ( u ) in terms of the rescaled exit field u . Clearly, as a function of u , g is uniformly of order 1 /N . g isnumerically of order 1 /N as long as y ( u ) is of order one. As we shall see, typically both u at horizon exit and y ( u )are of order one. We have 0 < u < < u < + ∞ for chaotic inflation.In what follows we therefore use y ( u ) instead of g as a coupling constant and make contact with eq.(2.1) bysetting ϕ = M P l s Ny u , V ( ϕ ) = 8 N M y v (cid:18)r y N ϕM
P l (cid:19) . (2.15)We can easily read from this equation the order of magnitude of ϕ and V ( ϕ ) since N ∼ , M is given by eq.(1.9)and u and y are of order one. Hence, ϕ ∼ M P l and V ( ϕ ) ∼ N M .As we will see below, the coupling y (or g ) is the most relevant coupling since it is related to the inflaton rescaling:the tensor–scalar ratio r and the spectral index n s vary in a more relevant manner with y than with the rest of theparameters c k , k ≥ y ( u ) has the following properties • y ( u ) > • y ′ ( u ) > u > • y ′ ( u ) < < u < u min = 1 in new inflation; • y ( u ) = 2 ( u − u min ) + O ( u − u min ) → u → u min ; • y ( u ) → ∞ as u → ∞ in chaotic inflation; • y ( u ) ≃ − v (0) log u → + ∞ as u → + in new inflation.In terms of this parametrization and to leading order in 1 /N , the tensor to scalar ratio r and the spectral index n s read: r = y ( u ) N (cid:20) v ′ ( u ) v ( u ) (cid:21) , n s − − r + y ( u )4 N v ′′ ( u ) v ( u ) (2.16)Notice that both n s − r are of order 1 /N for generic inflation potentials in this Ginsburg-Landau framework aswe see from eq.(2.16). Moreover, the running of the scalar spectral index from eq.(2.15) and its slow-roll expressionturn out to be of order 1 /N ∼ / ∼ × − ≪ dn s d ln k = − y ( u )32 N (cid:26) v ′ ( u ) v ′′′ ( u ) v ( u ) + 3 [ v ′ ( u )] v ( u ) − v ′ ( u )] v ′′ ( u ) v ( u ) (cid:27) . and therefore can be neglected [7]. Such small estimate for dn s /d log k is in agreement with the present data [1] andmakes the running unobservable for a foreseeable future.Since y = y ( u ) can be inverted for any 0 < u < u min , these two relations can also be regarded as parametrizations r = r ( y ) and n s = n s ( y ) in terms of the coupling constant y .We are interested in the region of the ( n s , r ) plane obtained from eq.(2.16) by varying y (or u ) and the otherparameters in the inflaton potential. We call B this region.From now on, we will restrict to new inflation.For a generic v ( u ) [with the required global properties described above] we can determine the asymptotic of B ,since they follow from the weak coupling limit y → y → ∞ . When y →
0, then u → u min = 1 and from the property above, r = 8 N + O ( u −
1) = 0 . . . . + O ( u −
1) (2.17)and n s = 1 − N + O ( u −
1) = 0 . . . . + O ( u − . (2.18)When y → ∞ we have in new inflation u → n s ≃ N log u −→ −∞ , r ≃ − N u log uv (0) −→ + . (2.19)We see that in the strong coupling regime r becomes very small and n s becomes well below unity. However, theslow-roll approximation is valid for | n s − | < n s < . r can be rewritten using eq.(2.13) in the suggestive form, r = 64 N y ( u ) (cid:20) d ln y ( u ) du (cid:21) − (2.20)Since 64 /N ∼ , r may be small only in case y ( u ) is large (the logarithmic derivative of y ( u ) has a milder effect forlarge y ( u ).) Therefore, we only find r ≪ strong coupling regime. Notice that ϕ is much smaller than M P l inthe strong coupling regime [eq.(2.12)].Let us now study large classes of physically meaningful inflaton potentials in order to provide generic bounds onthe region B of the ( n s , r ) plane within an interval of n s surely compatible with the WMAP+LSS data for n s , namely0 . < n s < .
99. To gain insight into the problem, we consider first the cases amenable to an analytic treatment,leaving the generic cases to a numerical investigation. As we will see below, the boundaries of the region B turn outto be described parametrically by the analytic formulas (2.23) and (4.6). A. The fourth degree double–well inflaton potential
The case when the V ( ϕ ) is the standard double–well quartic polynomial V ( ϕ ) = 14 λ (cid:18) ϕ − m λ (cid:19) has been studied in refs. [6, 7]. In the general framework outlined above we have for this case, v ( u ) = 14 ( u − = 14 − u + 14 u , λ = y N (cid:18) MM P l (cid:19) , m = M M P l . (2.21)By explicitly evaluating the integral in eq. (2.13) one obtains y ( u ) = u − − log u , (2.22)and then, from eq. (2.16) n s = 1 − N u + 1(1 − u ) ( u − − log u ) , r = 1 N u (1 − u ) ( u − − log u ) (2.23)where 0 ≤ u ≤ u min = 1. As required by the general arguments above, u is a monotonically decreasing function of y , ranging from u = 1 till u = 0 when y increases from y = 0 till y = + ∞ . In particular, when u → − , y vanishesquadratically as, y ( u ) u → − = 12 (1 − u ) . The concavity of the potential eq.(2.21) for the inflaton field at horizon crossing takes the value v ′′ ( u ) = 3 u − . We see that v ′′ ( u ) vanishes at u = 1 / √
3, that is at y = ln 3 − / . . . . . (This is usually called the spinodalpoint [17]). Therefore, v ′′ ( u ) > y < . . . . and v ′′ ( u ) < y > . . . . . (2.24)Our MCMC analysis of the CMB+LSS data combined with the theoretical model eq.(2.21) yields y ≃ .
26 [6, 7] deepin the negative concavity region v ′′ ( u ) < v ′′ ( u ) < y > . . . . is specific to new inflation eq.(2.21). v ′′ ( u ) can beexpressed as a linear combination of the observables n s and r as n s − r = y ( u )4 N v ′′ ( u ) v ( u )As expected in the general framework presented above, the limit u → − implies weak coupling y → + , that is, thepotential is quadratic around the absolute minimum u min = 1 and we find, n s y → = 1 − N , r y → = 8 N , u y → = 1 , (2.25)which coincide with n s and r for the monomial quadratic potential in chaotic inflation.In the limit u → + which implies y → + ∞ (strong coupling), we have u y → + ∞ = e − ( y +1) / → + and n s y ≫ = 1 − yN , r y ≫ = 16 yN e − y − . (2.26)Notice that the slow-roll approximation is no longer valid when the coefficient of 1 /N becomes much larger than unity.Hence, the results in eq.(2.26) are valid for y . N . We see that in this strong coupling regime (see fig. 1), r becomesvery small and n s becomes well below unity. However, the WMAP+LSS results exclude n s . . y ≫ r = r ( n s ) defined by eq.(2.23) is a single curvedepicted with dotted lines in fig. 1. It represents the upper border of the banana shaped region B in fig. 1.Notice that there is here a maximum value for n s , namely n maxs = 0 . . . . with r ( n maxs ) = 0 . . . . [7]. Thecurve r = r ( n s ) has here two branches : the lower branch r < r ( n maxs ) in which r increases with increasing n s andthe upper branch r > r ( n maxs ) in which r decreases with increasing n s . B. The sixth–order double–well inflaton potential
We consider here new inflation described by a six degree even polynomial potential with broken symmetry. Ac-cording to eq. (2.1) and eq. (2.3) we then have V ( ϕ ) = M g v (cid:18) √ g ϕM P l (cid:19) , v ( u ) = c − u + c u + c u . (2.27)where for stability we assume c ≥
0. Moreover, if we regard this case as a higher order correction to the quarticdouble–well potential, then c is positive.The inflaton potential eq.(2.27) is a particular case of eq.(2.4). The conditions eqs. (2.8) and (2.9) that the absoluteminimum of v ( u ) be at u min = 1 yields c + c = 1 , c = 12 − c − c (2.28)It is convenient to use b ≡ c as free parameter so that b ≥ c = 1 − b . Thus, v ( u ) = 12 (1 − u ) − − b − u ) − b − u ) = 112 (1 − u ) (3 + b + 2 b u ) (2.29)where b ≤ c ≥ y ( u ) = 8 Z u dx v ( x ) v ′ ( x ) = 23 ( u − −
13 (3 + b ) log u + (1 + b ) b log 1 + b u b (2.30)According to the general arguments presented above [see the lines below eq. (2.13)] one can verify that y ( u ) is amonotonically decreasing function of u for 0 < u <
1, where + ∞ > y > n s and the tensor–scalar ratio r are evaluated from eq. (2.16) as r = yN (cid:20) u (1 + b u )(1 − u ) (3 + b + 2 b u ) (cid:21) , n s = 1 − r + 3 y ( u ) N b u + 3 (1 − b ) u − − u ) (3 + b + 2 b u ) (2.31)Various curves r = r ( n s ) are plotted in fig. 1 for several values of b in the interval [0 ,
1] sweeping the region B . We seethat for increasing b [namely, for increasing sextic coupling and decreasing quartic coupling, see eq.(2.29)] the curvesmove down and right, sweeping the banana-shape region B depicted on fig. 1.Clearly, y is a variable more relevant than b . Changing y moves n s and r in the whole available range of values,while changing b only amounts to displacements transverse to the banana region B in the n s , r plane. In particular,for a given n s , r becomes smaller for increasing b . n s r c = 1−b = 1, c = b = 0c = 0 , c = 1b = 0.2, 0.4, 0.6, 0.8 by FIG. 1: We plot here r vs. n s for the broken–symmetry sixth–order inflaton potential eq.(2.29) setting N = 60. The curvesare obtained from eq. (2.31) with the sextic coefficient b ≡ c fixed to the values indicated in the figure. We see that y is the relevant coupling while b only varies r and n s transversely to the narrow banana-shape region. The two importantlimiting curves are shown: b = 0 corresponding to the fourth degree potential eq.(2.21) and b = 1 corresponding to the sixthdegree potential eq.(2.32). The uppermost point where all curves coalesce corresponds to the monomial quadratic potential n s = 0 . . . . , r = 0 . . . . for N = 60 [see eqs.(2.17)-(2.18)]. We see in fig. 1 two important limiting curves: the b → b → b = 0 the function v ( u )reduces to the fourth order double-well potential eq.(2.21) and we recover its characteristic curve r = r ( n s ). When b = 1 the potential has no quartic term and reduces to the quadratic plus sixth order potential: v ( u ) b → = 16 (1 − u ) (2 + u ) = 13 − u + 16 u . (2.32)In summary, the quadratic plus quartic broken–symmetry potential describes the upper/left border of the banana–shaped region B of fig. 1, while the quadratic plus sextic broken–symmetry potential describes its lower/right border.0 III. HIGHER–ORDER EVEN POLYNOMIAL DOUBLE-WELL INFLATON POTENTIALS
The generalization of the sixth order inflaton potential with broken symmetry to arbitrarily higher orders is nowstraightforward: V ( ϕ ) = M g v (cid:18) √ g ϕM P l (cid:19) , v ( u ) = 12 (1 − u ) + n X k =2 c k k ( u k − , (3.1)with the constraint eq.(2.8) n X k =2 c k = 1 (3.2)which guarantees that u = 1 is an extreme of v ( u ).We consider here the case when all higher coefficients c k are positive or zero : c k ≥ , k = 2 , . . . , n such that u min = 1 is the unique positive minimum. n s r
2n = 10c = c = c = 0 , c = 1c = 1 , c = c = c = 0 ξ , ξ , ξ , ξ = 0.001, 0.5, 0.999 FIG. 2: r vs. n s for the 10th. order even polynomial potential eq. (3.1) with n = 5 and the coefficients c k taking independentlythe values indicated. The relation to the numbers ξ k is given in eq. (3.3). The upper/left border curve c = 1 , c = c = c = 0corresponds to the fourth order potential eq.(2.21). The lower/right border curve c = 1 , c = c = c = 0 corresponds to thequadratic plus 10th order term potential eq.(4.1) for n = 5. These are the limiting curves of the banana B region. We determine the shape of the B region for arbitrary positive or zero values of the coefficients c k [subject to theconstraint (3.2)], performing a large number of simulations with different setups. After producing coefficients c k wenumerically computed the function y ( u ) following eq.(2.13) y ( u ) = 4 Z u dxx − x + P nk =2 c k k ( x k − − P nk =2 c k x k − and obtain the r = r ( n s ) curves from eq. (2.16) by plotting directly r vs. n s .Uniform distributions of coefficients are obtained by setting c k = n X j =1 log ξ j − log ξ k , k = 1 , , . . . , n (3.3)1 r n s r FIG. 3: A detail The banana region B in the ( n s , r ) plane for the quadratic plus 10th order polynomial as in fig. 2, but with thecurves split in two parts by the value r ( n maxs ). The upper panel shows the upper branches r > r ( n maxs ) in which r decreases with n s while the lower panel shows the lower branches r < r ( n maxs ) in which r increases with n s . The quadratic plus 10thorder polynomial thus provides the lower border of the banana region B setting the lower bound on r . This bound is here r > for the observed allowed range < n s < . where the numbers ξ k are independently and uniformly distributed in the unit interval. We used the parametrizationeq.(3.3) also when the ξ k are chosen according to other rules.For example, in figs. 2-3 we plot the results when n = 5, that is for the ten degree polynomial. In this case welet ξ , ξ , ξ and ξ take independently the values 0 . , . . c = 1 , c = c = c = 0 and c = 1 , c = c = c = 0.For higher values of n we extracted the numbers ξ k at random within the unit interval. In particular, for thehighest case considered, n = 50, we used three distributions: in the first, the ξ k were all extracted independently anduniformly over the unit interval; in the second we set log ξ k = 2 − k log ˜ ξ k and extracted the ˜ ξ k independently anduniformly; in the third we picked at random four ξ k freely varying and fixed to 1 the remaining 45 ones (that is wepicked at random four possibly non–zero c k , setting the rest to zero); the values of the four free ξ k were chosen atrandom in the same set of values (0 . , . , . n = 5 case. The results of these simulations are shown infig. 5.As evident from fig. 3, where the r = r ( n s ) curves are split in upper/lower branches with growing/decreasing r = r ( n s ) and especially from fig. 5, the case of the quadratic plus 2 n th order polynomial provides a bound to thebanana region B from below. That is, for any fixed value of n s , the quadratic plus 2 n th order polynomial providesthe lowest value for r .One sees from fig. 5 that some blue curves r = r ( n s ) go beyond the slashed red curve r = r ( n s ) for the quadraticplus u potential on the right upper border of the banana region B . Namely, the right upper border of the B regionis not given by the quadratic plus u potential while this potential provides the lower border of the B region.We performed many other tests with intermediate values of n and several other distributions, including other k − dependent distributions, with characteristic values for c k growing linearly with k or decreasing in a power–like orexponential way. In all cases, the results were consistent with those given above.It is also important to observe that the class of potentials considered, that is arbitrary even polynomials withpositive or zero couplings, is a class of weakly coupled models. This is evident from fig. 4, were n s is plotted vs.the coupling y , which remains of order one when n s decreases well below the current experimental limits. This weakcoupling is the reason why the addition of higher even monomials to these potentials causes only minor quantitativechanges to the shape of the r = r ( n s ) curves.2 y n s FIG. 4: n s vs. the coupling y within the same setup as in fig. 2. The inflaton potential V ( ϕ ) eq.(3.1) in the original inflaton field ϕ takes therefore the form V ( ϕ ) = 4 N M y ( u ) ( − y ( u )8 N ϕ M P l + n X k =2 c k k (cid:18) y ( u )8 N (cid:19) k ϕ k M kP l ) , Therefore, since the coupling y ( u ) is O (1) we have the 2 k -th term in the potential suppressed by the 2 k -th power of M P l as well as by the factor N k ∼ k .In particular, the quartic term y ( u ) c N (cid:18) MM P l (cid:19) ϕ possesses a very small quartic coupling since M ≪ M P l . Notice that these suppression factors are natural in the GLapproach and come from the ratio of the two relevant energy scales here: the Planck mass and the inflation scale M .When the GL approach is not used these suppression factors do not follow in general.The validity of the GL approach relies on the wide separation between the scale of inflation and the higher energyscale M P l (corresponding to the underlying unknown microscopic theory as discussed in ref. [5].) It is not necessaryto require ϕ ≪ M P l in the GL approach but to impose [5] V ( ϕ ) ≪ M P l and hence v ( φ √ g ) ≪ g . This last condition gives an upper bound for the inflaton field ϕ depending on the large argument behavior of v ( u ).We get for example: ϕ ≪ M P l for v ( u ) u →∞ ∼ u , ϕ ≪ M P l for v ( u ) u →∞ ∼ u . The validity of the effective GL theory relies on that separation of scales and the GL approach allows to determinethe scale of inflation as 0 . × GeV (at the GUT scale) and well below the Planck scale M P l using the amplitudeof the scalar fluctuations from the CMB data [6, 7].Inflaton potentials containing terms of arbitrary high order in the inflaton are considered in ref. [13], sec. 25.3.2without using the GL approach and within the small field hypothesis ϕ ≪ M P l . Smallness conditions on the expansioncoefficients are required in ref. [13]. This is actually not needed in the GL approach, whose validity relies only on thewide separation of scales between M and M P l , at least in the case of even polynomials with positive coefficients.
IV. THE QUADRATIC PLUS THE n TH ORDER DOUBLE-WELL INFLATON POTENTIAL
In order to find the observationally interesting right and down border of the banana we consider the quadratic plusthe 2 n th order potential for new inflation [10], v ( u ) = 12 (cid:0) − u (cid:1) + 12 n (cid:0) u n − (cid:1) . (4.1)3 n s r
2n = 100c = 1, c = 0 for 3<=k<=50c = 0 for 2<=k<50, c = 1 ξ k uniformely random in [0,1] ξ k2 k uniformely random in [0,1] n s r
2n = 100c > 0 for 4 random values of kc = 0 for 2<=k<50, c = 1c = 1, c = 0 for 3<=k<=50 FIG. 5: r vs. n s for the 100th. order polynomial potential eq.(3.1) for n = 50. The coefficients c k were chosen or extractedat random as indicated in the two panels. The two border curves of the banana region B are clearly indicated. The upperborder is the fourth order potential eq.(2.21) and the lower border is the quadratic plus the 2 n th order potential eq.(4.1). Thequadratic plus the 2 n th order potential always provides the lowest value for r at any fixed n s in its lower branch. In the upperpanel, all the coefficients c k were extracted independently from a flat distribution ranging from 0 to 100; in this case the curves accumulate near the quadratic plus quartic potential eq.(2.21). The upper panel is the generic case. In the lower panel, wepicked at random four possibly non-zero c k and fixed to zero the remaining 44 ones; in this case the curves accumulate nearthe quadratic plus 2 n th order potential eq.(4.1) with 2 n = 100. P S f r ag r e p l a c e m e n t s n = 3 n = 2 n = 5 n = 10 n = 20 n = 100 n = ∞ n = ∞ , u = 1FIG. 6: r vs. n s for the quadratic plus u n potential eq.(1.7) setting N = 60. The curves for the exponents n = 2 , , , , n = ∞ limits eqs.(4.6) and (4.11). Eq.(4.6) describes thelower bordering curve while eq.(4.11) describes the upper-right bordering curve. We see that for growing n the curves r vs. n s tend towards the limiting curves. The uppermost point where all curves coalesce corresponds to the monomial quadraticpotential n s = 0 . . . . , r = 0 . . . . [see eq.(2.25)]. As in the general case eq.(2.10), we choose the absolute minimum at u = 1. The customary relation eq.(2.13) takeshere the form [10], y ( u ) = 4 n Z u dxx n (1 − x ) + x n − − x n − where 0 < u < . (4.2)This integral can be expressed as a sum of n terms including logarithms and arctangents [9].In the weak coupling limit y → , n s and r take the values of the quadratic monomial potential eqs.(2.17)-(2.18)[7, 10]: n s − y → = − N = − . . . . , r y → = 8 N = 0 . . . . , (4.3)while in the strong coupling limit y → ∞ at fixed n, n s and r take the values n s ≃ N log u −→ −∞ , r ≃ − N nn − u log u −→ + , in accordance with the general formula eq.(2.19). In fig. 6 we plot r vs. n s for the potential eq.(4.1) and the exponents n = 5 , , , ,
500 and 5000. We see that for n → ∞ , r vs. n s tends towards a limiting curve. For y → y the left and lowerend of the curve is reached. However, the current CMB–LSS data rule out this strong coupling part of the curve for n s < . A. The n → ∞ limit at fixed u . Let us first compute y ( u ) eq.(4.2) for n → ∞ at fixed u . Since 0 < x < n →∞ x n = 0 . y ( u ) n →∞ = 4 n Z u dxx (cid:2) n (1 − x ) − (cid:3) = 2 (cid:20) u − − ln u + O (cid:18) n (cid:19)(cid:21) . Hence, eq.(4.2) becomes y ( u ) n →∞ = 2 (cid:0) − ln u − u (cid:1) where 0 < u < < y < + ∞ . (4.4)which is just twice the result found in the quartic double–well potential, eq. (2.22). Notice that v ( u ) eq.(4.1) in the n → ∞ limit becomes lim n →∞ v ( u ) = (cid:26) (1 − u ) for u < ∞ for u > . . (4.5)From eqs.(2.16), (4.1) and (4.4) we find for r and n s in the n → ∞ limit n s − n →∞ = − N u + 1(1 − u ) (cid:0) − ln u − u (cid:1) ,r n →∞ = 8 N u (1 − u ) (cid:0) − ln u − u (cid:1) . (4.6)Now, in the limiting cases u → u → at n = ∞ ), that is, the strong coupling limit y → ∞ and the the weakcoupling limit y →
0, respectively, we obtain from eqs.(4.6)lim u → n s ( n = ∞ ) − − N = −
140 = − . , lim u → r ( n = ∞ ) = 4 N = 115 = 0 . . . . , (4.7)lim u → n s ( n = ∞ ) = −∞ , lim u → r ( n = ∞ ) = 0 . However, as explained in sec. II, the slow-roll expansion is no more valid when | n s − | &
1. Moreover, the WMAP+LSSresults exclude n s . . u → lower part, namely 0 < r < /N = 0 . . . . . Theupper part is obtained in the double limit n → ∞ and u → n → ∞ and y → B. The double limit n → ∞ and u → . As we can see from fig. 6, when y varies from zero to infinity at fixed n , the potential eq.(4.1) covers the region0 < r < N , the point r = 8 /N corresponding to the small coupling limit y = 0.Notice however that the n → ∞ limit eqs.(4.4) and (4.6) only describe the region 0 < r < /N . In order to alsodescribe the small coupling region 8 /N > r > /N for n → ∞ , we have to take in eq.(4.2) the double limit u → and n → ∞ . This can be achieved by changing the integration variable in eq.(4.2) as x = t n , y ( u ) = 2 n Z τ dtt n (1 − t n ) + t − − t − n where τ ≡ u n , < τ < . Letting n → ∞ at fixed τ yields, n y ( u ) n →∞ , u → = Z τ dtt t − − ln t − t = ln τ + 12 ln τ + Li (1 − τ ) , (4.8)where Li ( s ) = − Z s dtt ln(1 − t ) , is the dilogarithmic function [9].6Then, in this double limit n → ∞ , u → γ ( τ ) ≡ n y ( u ) n →∞ , u → = ln τ + 12 ln τ + Li (1 − τ ) . (4.9)That is, τ and γ are fixed in this n → ∞ , u → < τ < , < γ < ∞ while y → y ( u ) n →∞ , u → = 2 γ ( τ ) n → u = τ n n →∞ = 1 + O (cid:18) n (cid:19) . (4.10)From eq.(2.16) the spectral index n s , and the ratio of tensor to scalar fluctuations r for fixed γ and τ take here( n = ∞ , y = 0 and u = 1) the following form, n s − n →∞ , u → = − γ ( τ ) N (1 − τ ) ( τ − − ln τ ) + 2 γ N ττ − − ln τ ,r n →∞ , u → = 8 γ ( τ ) N (1 − τ ) (1 − τ + ln τ ) . (4.11)We obtain from eqs.(4.11) in the limiting cases τ → τ → τ → n s − − N = −
140 = − . , lim τ → r = 4 N = 115 = 0 . . . . , lim τ → n s − − N = −
130 = − . . . . , lim τ → r = 8 N = 215 = 0 . . . . (4.12)Notice that n s and r for n → ∞ and then u → r and n s in the double limit n → ∞ , u → τ = u n → upper part, namely 8 /N = 0 . . . . > r > /N = 0 . . . . . The lower part, 0 < r < /N = 0 . . . . , is described by eqs.(4.6). r and n s given by eqs.(4.6)and (4.11) continuously match at n s = 0 . , r = 0 . . . . . However, the derivative dr/dn s is discontinuous at thispoint.There is here a quadratic relation between n s and r for r → − /N valid in the n = ∞ limit: (cid:18) r − N (cid:19) = − N (cid:18) n s − N (cid:19) " O r n s − N ! . (4.13)From eqs.(4.13) and (4.11) we get respectivelylim r → − /N drdn s = + ∞ , lim r → + /N drdn s = − , as we can see in fig. 6. V. THE QUADRATIC PLUS THE EXPONENTIAL POTENTIAL.
Since the exponential function contains all powers of the variable, it is worthwhile to consider it. As before, werestrict ourselves to potentials even in u : v ( φ ) = c ˆ g − φ + 12 ˆ g c (cid:16) e ˆ g φ − − ˆ g φ (cid:17) , (5.1)where ˆ g > c > c ensures that v ( φ ) vanishes at its absolute minimum φ = φ min = 1 / √ g . We find φ min = 1 √ g = r g log(1 + c ) , b ≡
12 log(1 + c ) > , g = ˆ g b and c = 12 (cid:20)(cid:18) c (cid:19) log(1 + c ) − (cid:21) u = φ/φ min the potential v ( u ) defined in general by eq.(2.3) takes here the form, v ( u ) = e − b (1 − u ) − b (1 − u )4 b (1 − e − b ) . (5.2)Expanding the potential eq.(5.2) in powers of u yields v ( u ) u → = 1 + e b (2 b − b ( e b − − u + b e b − u + O ( u ) . It is interesting to expand the potential in powers of b in order to make contact with the polynomial potentials of sec.II A-II B. We get from eq.(5.2) v ( u ) b → = 112 (1 − u ) (3 + b + 2 b u ) + O ( b )which is exactly the fourth order double–well potential eq.(2.21) to zeroth order in b and the sixth–order double–wellpotential eq.(2.29) to first order in b .The field u at horizon exit follows from the customary eq.(2.13) which takes here the form: y ( u ) = 2 b Z u dxx e − b (1 − x ) + 2 b (1 − x ) − e − b (1 − x ) − , (5.3)Changing the integration variable to w ≡ − e − b (1 − x ) , eq.(5.3) becomes y ( u ) = 2 (cid:0) − ln u − u (cid:1) + 2 b ln u − b Z − e − b (1 − u dww log(1 − w )2 b + log(1 − w ) . (5.4)The spectral index n s , and the ratio r are expressed from eq.(2.16) as, n s − − r + b y ( u ) N (4 b u + 1) e − b (1 − u ) − e − b (1 − u ) + 2 b (1 − u ) − ,r = 16 b N u y ( u ) " e − b (1 − u ) − e − b (1 − u ) + 2 b (1 − u ) − . (5.5)We study below eqs.(5.4)-(5.5) in the b → ∞ limit in the two regimes: b → ∞ with u fixed and b → ∞ with u → u min = 1. These are the limits investigated in secs. IV A and IV B for the quadratic plus u n potential,respectively.In fig. 7 we plot r vs. n s for the quadratic plus exponential potential eq.(5.1) and the values of the coefficient c = 0 . , . , , , and 10. We see that for growing c, r vs. n s tends towards a limiting curve. This curve is the lower border of the banana shaped region B . The upper border is determined by the fourth order potential eq.(2.21). A. The limit b → ∞ at fixed u . For large b we have in eqs.(5.4)-(5.5), e − b (1 − u ) ≪ b ≫ u < , and we find v ( u ) b →∞ = 12 (1 − u ) + O (cid:18) b (cid:19) for u < , lim n →∞ v ( u ) = + ∞ for u > ,y ( u ) b →∞ = 2 (cid:0) − ln u − u (cid:1) + O (cid:18) b (cid:19) ,n s − b →∞ = − N u + 1(1 − u ) (cid:0) − ln u − u (cid:1) + O (cid:18) b (cid:19) ,r b →∞ = 8 N u (1 − u ) (cid:0) − ln u − u (cid:1) + O (cid:18) b (cid:19) . (5.6)These equations for r vs. n s exactly coincide with eqs.(4.5)-(4.6) for the quadratic plus 2 n th order potential. Wehave therefore proved that the quadratic plus the u n potential and the quadratic plus exponential potential have identical limits letting n → ∞ in the former and b → ∞ in the latter, keeping always u fixed.8 P S f r ag r e p l a c e m e n t s b = 0 b = 0 . b = 4 b = 8 b = 40 b = 160 b = 1600 b = 1 . × b = 8 × b = ∞ , u = 1 b = ∞ FIG. 7: r vs. n s for the quadratic plus exponential potential eq.(5.1) with the coefficient 0 ≤ b ≤ ∞ and setting N = 60. Wesee that for growing b ≫ r vs. n s tends towards a limiting curve to the right and down of the banana shaped region B .This curve is the lower border of the region B . The upper border is determined by the fourth order potential eq.(2.21). Theuppermost point where all curves coalesce corresponds to the monomial quadratic potential n s = 0 . . . . , r = 0 . . . . [see eq.(2.25)]. B. The double limit b → ∞ and u → . It is useful to introduce here the variable τ ≡ e − b (1 − u ) hence u = 1 + log τ b → − for b → ∞ at fixed τ , < τ < . We then find from eq.(5.4) for b → ∞ and fixed τ ,2 b y ( u ) = 2 (cid:0) − ln u − u (cid:1) + 2 b ln u − b Z − τ dww log(1 − w ) + O (cid:18) b (cid:19) == ln τ + 12 ln τ + Li (1 − τ ) + O (cid:18) b (cid:19) , (5.7)We find in this limit from eq.(5.5) for r vs. n s , r b →∞ , u → = 8 γ ( τ ) N (1 − τ ) (1 − τ + ln τ ) ,n s − b →∞ , u → = − γ ( τ ) N (1 − τ ) ( τ − − ln τ ) + 2 γ ( τ ) N ττ − − ln τ ,γ ( τ ) ≡ c y ( u ) b →∞ , u → = ln τ + 12 ln τ + Li (1 − τ ) , (5.8)where we keep fixed γ . Eqs.(5.8) coincide with eqs.(4.9)-(4.11) for the quadratic plus u n potential.These results plus those in sec. V A prove that the quadratic plus u n potential and the quadratic plus exponentialpotential eq.(5.1) have identical limits letting n → ∞ in the former and b → ∞ in the latter.9 VI. DYNAMICALLY GENERATED INFLATON POTENTIAL FROM A FERMION CONDENSATE INTHE INFLATIONARY STAGE.
The inflaton may be a coarse-grained average of fundamental scalar fields, or a composite (bound state) or con-densate of fields with spin, just as in superconductivity. Bosonic fields do not need to be fundamental fields, forexample they may emerge as condensates of fermion-antifermion pairs < ¯ΨΨ > in a grand unified theory (GUT) inthe cosmological background [7].We investigate in this section an inflaton potential dynamically generated as the effective potential of fermions in theinflationary universe. We consider the inflaton field coupled to Dirac fermions Ψ through the interaction Lagrangian L = Ψ [ i γ µ D µ − m f − g Y ϕ ] Ψ . (6.1)Here g Y stands for a generic Yukawa coupling between the fermions and the inflaton ϕ . The fermion mass m f canbe absorbed in a constant shift of the inflaton field. The Dirac matrices γ µ are the curved space-time γ -matrices and D µ stands for the fermionic covariant derivative. P S f r ag r e p l a c e m e n t s s = 0 . s = 350 s = 440 s = 500FIG. 8: The effective potential v ( u ) generated from fermions eq.(6.9) vs. u for several values of the rescaled Yukawa coupling s . The steeper potential corresponds to the largest s . The shallower potential corresponds to s → For our purposes in this section, the inflationary stage can be approximated by a de Sitter space-time (that is, weneglect the slow decrease in time during inflation of the Hubble parameter H ). In this way, the effective potential offermions can be computed in close form with the result [7, 11], V f ( ϕ ) = V + 12 µ ϕ + 14 λϕ + H Q (cid:16) g Y ϕH (cid:17) , (6.2)where, Q ( x ) = x π (cid:8) (1 + x ) [ γ + Re ψ (1 + i x )] − ζ (3) x (cid:9) , x ≡ g Y ϕH , = x π " (1 + x ) ∞ X n =1 n ( n + x ) − ζ (3) = x π ∞ X n =1 ( − n +1 [ ζ (2 n + 1) − ζ (2 n + 3)] x n . (6.3)0 P S f r ag r e p l a c e m e n t s s → s = 500 s = 600 s = 700 s = 800FIG. 9: We plot here r vs. n s for the effective potential obtained from fermions in de Sitter stage eq.(6.7) for the physical valueof the parameter q eq.(6.11). For weak Yukawa coupling s ≪ r = r ( n s ) curve for the quadratic plus quarticpotential eq.(2.21). The r = r ( n s ) curves are inside the universal banana region [fig. 10] provided s ≤ We included in V f ( ϕ ) the renormalized mass µ and renormalized coupling constant λ which are free and finiteparameters. ψ ( x ) stands for the digamma function, γ for the Euler-Mascheroni constant and ζ ( x ) for the Riemannzeta function [9].Eq.(6.2) is the energy density for an homogeneous inflaton field ϕ coupled to massless fermions through the La-grangian eq.(6.1) in a de Sitter space-time.The power series of the function Q ( x ) has coefficients with alternating signs, but it can be readily verified that Q ( x ) > Q ′ ( x ) > x >
0. Moreover, to leading order we have Q ( x ) x →∞ = x π (cid:20) log x + γ − ζ (3) + O (cid:18) x (cid:19)(cid:21) . (6.4)The constant V in eq. (6.2) must be such that the potential V f ( ϕ ) fulfills eq. (1.6) producing a finite number ofinflaton efolds. We consider new inflation and choose µ = − m <
0. Hence V f ( ϕ ) has a double–well shape with theabsolute minimum at ϕ = ϕ min , with ϕ min a function of the free parameters of the potential.Expanding V f ( ϕ ) in powers of ϕ gives V f ( ϕ ) = V − m ϕ + 14 λ ϕ + 18 π [ ζ (3) − ζ (5)] ( g Y ϕ ) H + O (cid:0) g Y ϕ (cid:1) . where ζ (3) − ζ (5) = 0 . . . . > H → H = √ N H m , m = M M P l (6.5)where the dimensionless Hubble parameter H turns out to be of order one [7].As in the general description of section II, eq.(2.12) we introduce the dimensionless coupling constant g as g = M P l ϕ min g we can now form two other independent and positive dimensionless shape parameters, that is s ≡ g Y ϕ min H = g Y √ g M P l
H , q ≡ H m ϕ min = √ g H m M P l = √ g H M , x = s u . (6.6)We derive the dimensionless potential v ( u ) from eq.(6.2) using the general transformation equations (2.15). Weobtain, v ( u ) = gM V f ( ϕ min u ) = c − u + 14 c u + gM H Q ( s u ) (6.7)The parameters c and c are determined by requiring that v (1) = v ′ (1) = 0 as in sec. II. We thus obtain fromeq.(6.7) c = 1 − q s Q ′ ( s ) , c = 14 + 14 q s Q ′ ( s ) − q Q ( s ) (6.8)Inserting c and c into eq.(6.7) yields for the inflaton potential v ( u ) = gM V f ( ϕ min u ) = 12 (1 − u ) + 14 (cid:2) − q s Q ′ ( s ) (cid:3) ( u −
1) + q [ Q ( s u ) − Q ( s )]= 12 (1 − u ) + 14 (1 − b ) ( u −
1) + b F ( u, s ) , (6.9)where b ≡ q s Q ′ ( s ) ≥ , F ( u, s ) ≡ Q ( s u ) − Q ( s ) s Q ′ ( s ) (6.10)Notice that v ( u ) reduces to the quartic double–well potential (1 − u ) when s → q (that is, when g Y → b → s . This last limit means g Y → g Y M P l /H fixed.Only the interval 0 < u < u in this interval, F ( u, s ) is negativedefinite and is monotonically decreasing as a function of s . In particular, F ( u, s ) s → = 16 ( u − , F ( u, s ) s →∞ = 14 ( u −
1) + O (cid:18) s (cid:19) , < u < s → b we obtain again the sixth–order double–well potential of eq. (2.29) v ( u ) −→
112 (1 − u ) (3 + b + 2 b u ) , s → b , while b cancels out for large s and we get back the quartic double–well potential v ( u ) −→
14 (1 − u ) , s → ∞ at fixed b . The terms containing Q in the effective potential eqs.(6.2) and (6.9) represent the one–loop quantum contributions.They vanish when b = 0 while for b > < b ≤ Q ) pieces are of the order q Q ( s u ) compared with the tree-level pieces.We can compute q using eqs.(2.14), (6.5) and (6.6) with the result q = r N y (cid:18) H MM P l (cid:19) ≃ .
854 10 − ≪ , (6.11)where y ≃ . H ≃ . s ≫
1. We can therefore use the asymptotic behavior eq.(6.4) toestimate Q ( s u ) for large s . In order the one-loop part to be smaller or of the order of the tree level piece we mustimpose in the strong coupling regime s ≫ π s q ln s . ⇒ s . − / ≃ . (6.12)The one-loop potential eq.(6.9) is therefore reliable for s . s the one-loop piece is largerthan the tree level part and hence all higher order loops should be included too.2For s ∼ s q are negligible in eq.(6.9)and n s and r are thus given by eqs.(2.23). We find from eqs.(6.9) and (6.11) in the case s ∼ v ( u ) = 14 − u + 14 u + O (cid:18) H M P l (cid:19)
That is, the terms beyond u in the effective potential from the fermions are of the same order of magnitude as theloop corrections to inflation [7, 18] and can be neglected since ( H/M
P l ) ∼ − .We display in fig. 9 r vs. n s for various values of the Yukawa coupling s . Therefore, the banana region B in the( n s , r ) plane for the effective potential eq.(6.9) is the region limited by the curves for the potential for s ≤
500 andfor s → B for the effective potential eq.(6.9) is well above the lower border of the universal B region displayed in fig. 10.In summary, the r = r ( n s ) curves for the dynamically generated inflaton potential eq.(6.9) are inside the universalbanana region B for all values of the Yukawa coupling g Y that keep the result for this one-loop potential reliable.Namely, the one-loop piece is smaller or of the order of the tree level part. VII. THE UNIVERSAL BANANA REGION B In summary, we find that all r = r ( n s ) curves for double–well inflaton potentials in the Ginsburg-Landau spirit fall inside the universal banana region B depicted in fig. 10 for new inflation. Namely, • The fourth degree double–well potentials containing a cubic term studied in ref. [6]: v ( u ) = 14 + β − u − β u + 14 (1 + 2 β ) u , (7.1)where β ≥ β = 0. • The quadratic plus sixth-order potential eq.(2.29). • The even polynomial potentials with arbitrarily higher–order degrees and positive coefficients (sec. III). • The quadratic plus exponential potential (sec. V). • The inflaton potential dynamically generated from fermions (sec. VI).Potentials in the Ginsburg-Landau spirit have usually coefficients of order one when written in dimensionless variables.This is the case of the inflaton potentials v ( u ). In that case, we found that all r = r ( n s ) curves for double–wellpotentials fall inside the universal banana region B depicted in fig. 10. Moreover, for even double–well potentialswith arbitrarily large positive coefficients, their r = r ( n s ) curves lie inside the universal banana region B [fig. 10].The study of the dynamically-generated inflaton potential in sec. VI leads to analogous conclusions. This one-loopinflaton potential is reliable as long as the one-loop piece is smaller or of the same order than the tree level part. Insuch regime all the curves r = r ( n s ) produced by this fermion-generated potential lie inside the universal bananaregion B More generally, we see from eq.(2.20) that r ≪ y ≫
1. However, this strongcoupling regime corresponds to n s values well below the current best observed value n s = 0 . B corresponds to the limit binomial potential eq.(4.5) v ( u ) = 12 (1 − u ) for u < , v ( u ) = + ∞ for u > . and is described parametrically by eq.(4.6). We obtain such potential and such parametrization of r = r ( n s ) both asthe n → ∞ limit of the quadratic plus u n potential in sec. IV as well as the b → ∞ limit of the e b g φ potential insec V.The upper-right border of the universal banana-shaped region B is not given by eqs.(4.9) and (4.11) correspondingto the double limit n → ∞ and u → b → ∞ and u → r = r ( n s ) curves above the limiting curves for the quadratic plus u potential asdepicted in fig. 5.The upper-left border of the universal region B depicted in fig. 10 is given by the fourth order double–well potentialeq.(2.21) and it is described parametrically by eq.(2.23).The lower border of the universal region B is particularly relevant since it gives a lower bound for r for eachobservationally allowed value of n s . For example, the best n s value n s = 0 .
964 implies from fig. 10 that r > . B tells us the upper bound r < .
053 for n s = 0 . B . < r < .
053 for n s = 0 . . (7.2)Notice that these bounds on r are compatible with the experiments [1].The Ginsburg-Landau criterion applied to the inflaton potential eq.(2.11) leads to values of ( n s , r ) inside theuniversal banana-shaped region B . In principle, within the Ginsburg-Landau approach, one should choose potentials v ( u ) with small coefficients | c k | ≫ large and positive coefficients c k ≫ n s , r ) within the universal banana region B . Moreover, the double–well quartic potential eq.(7.1) also provides( n s , r ) inside the banana region B for arbitrarily large coefficient β [6].On the contrary, double–well potentials of degree larger than four with large negative coefficients are outside theGinsburg-Landau class and produce ( n s , r ) outside the region B . In particular, it is possible to produce in this way n s values compatible with the data together with arbitrarily small values for r by choosing large enough negativecoefficients c k .We find that the Ginsburg-Landau class of potentials is physically well motivated and therefore that the banana-shaped region B is a natural region to expect to observe ( n s , r ). Namely, taking into account the present data for n s we expect that r will be observed in the interval eq.(7.2). Anyhow, the fourth order double–well potential eq.(1.7)provides and excellent fit to the present CMB/LSS data and yields as most probable values: n s ≃ . , r ≃ . r are the best way to support the searching for CMB polarisation and the future missions on it. [1] E. Komatsu et al. (WMAP collaboration), Astrophys. J. Suppl. 180:330 (2009).G. Hinshaw et al. (WMAP collaboration), Astrophys. J. Suppl. 180:225 (2009).M. R. Nolta et al. (WMAP collaboration), Astrophys. J. Suppl. 180:296 (2009).E. Komatsu et al. (WMAP collaboration), arXiv:1001.4538.[2] Kolb EW and Turner MS, The Early Universe , Addison Wesley. Redwood City, C.A. 1990. Dodelson S,
Modern Cosmology ,Academic Press, 2003.[3] See for example: Hu W., Dodelson S.,
Ann. Rev. Astron. Ap.
40: 171 (2002); Lidsey J, Liddle A, Kolb E, Copeland E,Barreiro T, Abney M,
Rev. of Mod. Phys.
69: 373, (1997).[4] Statistical Physics, vol 9, E M Lifshitz, L P Pitaevsky, Pergamon Press, Oxford 1980, see secs. 142 part I and 45 part II.L. D. Landau, Zh. Eksp. Teor. Fiz., , 19 (1937) and , 545 (1937) and in Collected Papers of L. D. Landau , PergamonPress, Oxford, 1965. V. L. Ginsburg, Zh. Eksp. Teor. Fiz. , 739 and , 107 (1945). V. L. Ginsburg, L. D. Landau, Zh.Eksp. Teor. Fiz. , 1064 (1950). V. L. Ginsburg, About Science, Myself and Others , Part I, Chapters 5-7, IoP, Bristol,2005.[5] D. Boyanovsky, H. J. de Vega, N. G. S´anchez, Phys. Rev.
D73 , 023008 (2006).[6] C. Destri, H. J. de Vega, N. G. S´anchez, Phys. Rev.
D77 , 043509 (2008).[7] D. Boyanovsky, C. Destri, H. J. de Vega, N. G. S´anchez, arXiv:0901.0549, Int. J. Mod. Phys.
A 24 , 3669-3864 (2009).[8] F. Finelli, M. Rianna, N. Mandolesi, JCAP 0612 (2006) 006. M. Bridges, A.N. Lasenby, M.P. Hobson, MNRAS, 369, 1123(2006). Huffenberger, K. M. et al. Ap. J. 688, 1 (2008), Ap. J. 651, L81 (2006). H. K. Eriksen et al., ApJ, 656, 641 (2007).[9] A. P. Prudnikov, Yu. A. Brichkov, O. I. Marichev, Integrals and Series, Nauka, Moscow, 1981.[10] D. Boyanovsky, H. J. de Vega, C. M. Ho, N. G. S´anchez, Phys. Rev.
D75 , 123504 (2007).[11] D. Boyanovsky, H. J. de Vega, N. G. S´anchez, Phys. Rev.
D72 , 103006 (2005).[12] D. Cirigliano, H. J. de Vega, N. G. S´anchez, Phys. Rev.
D 71 , 103518 (2005).[13] D. Lyth, A. Liddle, ‘The Primordial Density Perturbation’, Cambridge University Press, 2009.[14] M. B. Hoffman, M. S. Turner, Phys. Rev. D64, 023506 (2001), W. H. Kinney, Phys. Rev. D66, 083508 (2002), R. Easther,W. H. Kinney, Phys. Rev. D67, 043511 (2003). A. Kosowsky, M. S. Turner, Phys. Rev.
D52 , R1739 (1995), L. A. Boyle,P. J. Steinhardt, N. Turok, Phys. Rev. Lett. 96 (2006) 111301. B. A. Powel, W. H. Kinney, JCAP 0708:006, (2007). W. H.Kinney, E. W. Kolb, A. Melchiorri, A. Riotto, Phys. Rev. D74 (2006) 023502. C. Y. Chen et al., Class. Quant. Grav. 21,3223 (2004). E. Ramirez, A. R. Liddle, Phys. Rev. D71, 123510 (2005). R. Easther, J. T. Giblin, Phys. Rev. D72, 103505(2005). M. Spalinski, JCAP0708:016,2007.[15] W. H. Kinney, Phys. Rev. D58, 123506 (1998). C. Savage, K. Freese, W. H. Kinney, Phys. Rev. D74, 123511 (2006). L.Alabidi, D. Lyth JCAP 0605 (2006) 016 and 0608 (2006) 013. L. Boyle, P J Steinhardt, arXiv:0810.2787.[16] K. Kadota, E. D. Stewart, JHEP (2003) 013. L. Boubekeur, D. Lyth JCAP 0507 (2005) 010. K. Kohri, C-M Lin, D. Lyth,JCAP 0712 (2007) 004. C-M Lin, K. Cheung, JCAP 0903 (2009) 012.[17] D. Boyanovsky, H. J. de Vega, D. J. Schwarz, hep-ph/0602002, Ann. Rev. Nucl. Part. Sci. , 441-500, (2006).[18] D. Boyanovsky, H. J. de Vega, N. G. S´anchez, Nucl. Phys. B747 , 25 (2006) and Phys. Rev.
D72 , 103006 (2005).[19] Kinney, W. H. et al. 2008, Phys. Rev. D78, 087302. n s r quartic double wellquadratic + u ∞ double well PSfrag replacementsQuadratic plus quartic potentialQuadratic plus u ∞ potentialFIG. 10: We plot here the borders of the universal banana region B in the ( n s , r )-plane setting N = 60. The curves arecomputed with the quadratic plus quartic potential eq.(2.21) and with the n = ∞ limit of the quadratic plus u n potentialeq.(4.1) (or the b = ∞ limit of the quadratic plus exponential potential eq.(5.1), which gives identical results) as given byeqs.(4.4)-(4.6) and eqs.(4.9) and (4.11). Notice that the lower part of the right border of B , 0 < r < /N = 0 . . . . corresponds to the limit n = ∞ at fixed u eq.(4.6). The upper part 4 /N < r < /N of the right border of B is not displayedhere. We display in the vertical full line the LCDM+r value n s = 0 . ± .
015 using WMAP5+BAO+SN data. The brokenvertical lines delimit the ± σσ