Higgs Inflation in f(Φ,R) Theory
aa r X i v : . [ a s t r o - ph . C O ] J a n Higgs Inflation in f (Φ , R ) Theory
Girish Kumar Chakravarty ∗ , Subhendra Mohanty † and Naveen K. Singh ‡ Theory Division, Physical Research Laboratory,Navrangpura, Ahmedabad 380 009, India (Dated: September 14, 2018)
Abstract
We generalize the scalar-curvature coupling model ξ Φ R of Higgs inflation to ξ Φ a R b to studyinflation. We compute the amplitude and spectral index of curvature perturbations generatedduring inflation and fix the parameters of the model by comparing these with the Planck+WPdata. We find that if the scalar self coupling λ is in the range (10 − − . a in therange (2 . − .
6) and b in the range (0 . − .
22) at the Planck scale, one can have a viableinflation model even for ξ ≃
1. The tensor to scalar ratio r in this model is small and our modelwith scalar-curvature couplings is not ruled out by observational limits on r unlike the pure λ Φ theory. By requiring the curvature coupling parameter to be of order unity, we have evaded theproblem of unitarity violation in scalar-graviton scatterings which plague the ξ Φ R Higgs inflationmodels. We conclude that the Higgs field may still be a good candidate for being the inflaton inthe early universe if one considers higher dimensional curvature coupling.Keywords: Higgs Inflation; CMB spectrum; Non-minimal coupling; Jordan frame; Einstein frame;Perturbations.
PACS numbers: 98.80.Cq, 14.80.Bn ∗ email: [email protected] † email: [email protected] ‡ email: [email protected] . INTRODUCTION The idea that the universe through a period of exponential expansion, called inflation[1–9] has proved useful for solving the horizon and flatness problems of standard cosmologyand in addition providing an explanation for the scale invariant super-horizon perturbationswhich are responsible of generating the CMB anisotropies and formation of structures inthe universe. A successful theory of inflation requires a flat potential where a scalar fieldacquires a slow-roll over a sufficiently long period to enable the universe to expand by atleast 60 e-foldings during the period of inflation. There is a wide variety of particle physicsmodels which can provide the slow roll scalar field ’inflaton’ for inflation [10]. From theobservations of CMB anisotropy spectrum by COBE and WMAP [11] it is not yet possibleto pin down a specific particle physics model as the one responsible for inflation. In the lightof recent discoveries by CMS [12] and ATLAS [13] it is of interest to consider the StandardModel Higgs boson as the candidate for inflaton. On the face of it the idea does not workas the inflaton quartic coupling should be of the order λ ∼ − to explain the amplitudeof CMB perturbations measured by WMAP [11] while the 125 GeV Higgs has a quarticcoupling λ ∼ .
13 at the electroweak scale which can however go down to smaller valuesat the Planck scale due to renormalization [14–19]. However just from the standard modelrenormalization one cannot have the Higgs coupling λ ∼ − over the entire range of therolling field (10-1 ) M P during inflation and the standard slow roll inflation with a Higgsfield does not give the observed amplitude and spectrum of density perturbations [20]. Ifthe Higgs and top mass are fine tuned then there can be a small kink in the Higgs potentialand the universe trapped in this false vacuum can undergo a period of inflation [21–23].Later a way out of fine tuning the scalar self coupling to unnaturally small values wasfound out [24–27] and it was shown that if one couples the scalar field to the Ricci scalar ξ Φ R then the effective potential in the Einstein frame becomes a slow roll one with theeffective scalar coupling being λ/ξ and the amplitude of the density perturbations constrainthis ratio rather than λ , hence ξ can be increased as large as required to get the desiredself-coupling λ . Density perturbations from inflation in the curvature coupled theories werecalculated in [28, 29]. The equivalence of the density perturbation in Jordan and Einsteinframe was shown by Komatsu and Futamase [30] who also calculated the tensor perturbationsand showed that the tensor to scalar ratio is generically small in ξ Φ R model.Bezrukov and Shaposhnikov [31] revived the large curvature coupling model to motivatethe idea that the standard model Higgs field could serve as the inflaton in the early universe.The amplitude and spectral index of density perturbations observed by WMAP can begenerated by the Higgs field with self coupling λ ∼ . ξ ∼ [31–36]. This large value of ξ needed however is seen as a problem as at the time of inflation theHiggs field is at the Planck scale and hence graviton-scalar scatterings due to the curvaturecoupling of the scalar would become non-unitary [37]. Ways of solving the unitarity violationproblem in the Higgs inflation models have been explored in [39–43].In this paper we assume that the dominant interaction between Higgs field and gravityis through operators of the form L = ξ ( H † H ) a/ R b M a +2 b − p . (1)2his form (1) of Higgs Curvature interaction has been mentioned in the Ref. [44]. Thecomplete dynamics of the Higgs field involves the role of the Goldstone modes as has beenstudied in detail in [45–47]. The multifield dynamics of the Goldstone modes gives rise tosizable non-gaussianity. We will study the dynamics of the Higgs mode and impose a chargeconservation and CP symmetry such that the Goldstone modes of the Higgs field do notacquire vevs. We will take the background Higgs field to be H = (cid:18) (cid:19) (2)where Φ is the Higgs mode with mass 126 GeV. Our inflation model falls in the class ofinflation in f (Φ , R ) theories studied in Ref. [48]. Our motivation is that we use the Higgsquartic coupling λ ( H † H ) where the standard model value of λ ( µ ∼ M P ) can lie in therange λ = (10 − − .
1) depending on the value of top quark mass [18, 19] or on new physics[49]. We take curvature coupling ξ to be unity and check the possibility of generating theobserved density perturbations from Higgs inflation by varying parameters a , b and λ . Thenon minimal coupling ξ has been taken unity in order to improve the unitarity behaviourwhich increases the natural cutoff scale Λ from Λ ≃ M p ξ ≃ to Λ ≃ M p ≃ .We derive the curvature perturbation during inflation in two different ways. We de-rive the perturbations of modified Einsteins field equation in the Jordan frame in presenceof the Higgs-curvature interaction terms and derive the amplitude and spectral index ofcurvature perturbation. We find that to generate the Planck+WP preferred amplitude∆ R = 2 . +0 . − . × − and spectral index n s = . ± . λ = 10 − we shouldhave a ∼ . , b ∼ .
49 ( and for λ = 0 . a ∼ . , b ∼ . ξ = 1.In the ξ Φ R theory we can always make a conformal transformation to the Einstein frameso one can compute the density perturbations either in Einstein frame or Jordan frame andthe gauge invariant curvature perturbations should be same in both the frames [28]. In ourcase with the ξ Φ a R b coupling we find that no conformal transformation exists which can ingeneral remove this term (i.e go to an Einstein frame). We find that in the general ξ Φ a R b theory such a conformal transformation is only possible if the metric is quasi-de Sitter. Theaccurate comparision with the experimental data should be made however with the Jordanframe results.Calculation of the curvature perturbation in both Einstein and Jordan frame for the ξ Φ R theory has been done previously in Ref. [28, 51–53]. In section (2) we derive the curvatureperturbations and tensor perturbation in our theory in the Jordan frame and in section (3)we make a conformal transformation to go to the Einstein frame and compute the curvatureperturbations. Finally in the last section (5) we compare the results of the two frames anddiscuss the viability of our considered Higgs inflation model.
2. CALCULATION IN THE JORDAN FRAME
In this section we introduce a scalar-gravity interaction term f (Φ , R ) in the action andcalculate physical quantities related to the inflationary density perturbations such as thespectral index, curvature perturbation and tensor-to-scalar ratio. We start with the action3or a scalar field interacting with gravity of the form S = Z d x √− g " − f (Φ , R )2 κ + 12 g µν ∂ µ Φ ∂ ν Φ + V (Φ) , (3)where we take, 1 κ f (Φ , R ) = 1 κ R + ξ Φ a R b M a +2 b − p ; V (Φ) = λ Φ , (4)where κ = 1 /M p and ξ is a dimensionless coupling constant. Varying the action (3) w.r.t g µν and Φ we obtain the field equations, G µν = F R µν − f g µν − ▽ µ ▽ ν F + g µν (cid:3) F = κ (cid:18) ▽ µ Φ ▽ ν Φ − g µν ▽ ρ Φ ▽ ρ Φ − V g µν (cid:19) , (5) (cid:3) Φ = V , Φ − f , Φ κ , (6)where F = ∂f /∂R = 1 + ξb Φ a R b − M a +2 b − p . For the unperturbed background FRW metric diag ( − , a ( t ) , a ( t ) , a ( t )) and scalar fieldΦ = φ ( t ), the above Eqs. (5) and (6) reduce to the form3 F H + 12 ( f − RF ) + 3 H ˙ F = κ (cid:18)
12 ˙ φ + V ( φ ) (cid:19) (7) − F ˙ H − ¨ F + H ˙ F = κ ˙ φ (8)¨ φ + 3 H ˙ φ + V ,φ − f ,φ κ = 0 . (9)Now we assume the second term of F i.e. ξbφ a R b − M a +2 b − p is dominant for some values of a and b .We find this assumption to be valid while solving numerically for the values of a and b inour model which give rise to the experimentally observed density perturbations as discussedin the section (4). From Eq. (7), under this assumption and considering the slow rollparameters which are defined in Eq. (28) as small, the Hubble parameter in the Jordanframe turns out to be of the form H = λ b √ (cid:2) ξ (2 − b ) (cid:3) b (cid:18) φM p (cid:19) − a b M p . (10)From Eq. (9) under the slow roll assumption we get˙ φ = − λφ H h − a − b ) i . (11)4 .2. Scalar field and metric perturbations Now we perturb Eqs. (5) and (6) by perturbing the scalar field Φ = φ ( t ) + δφ ( x, t ) andthe metric as ds = − (1 + 2 α ) dt − a ( t )( ∂ i β ) dtdx i + a ( t ) ( δ ij (1 + 2 ψ ) + 2 ∂ i ∂ j γ ) dx i dx j , (12)where, α , ψ , β and γ are scalar perturbations. We derive the Einstein equations for the f ( R, φ ) theory [54, 55] keeping the first order terms in the metric and scalar field perturba-tions. The component δG is given by △ a ( t ) ψ + HA = − F " (cid:18) H + 3 ˙ H + △ a ( t ) (cid:19) δF − H ˙ δF + 12 (cid:0) κ V ,φ − f ,φ (cid:1) δφ + κ ˙ φ ˙ δφ + (3 H ˙ F − κ ˙ φ ) α + ˙ F A , (13)and taking the difference δG ii − δG we get˙ A + 2 HA + (cid:18) H + △ a ( t ) (cid:19) α = 12 F " δF + 3 H ˙ δF − (cid:18) H + △ a ( t ) (cid:19) δF + 4 κ ˙ φ ˙ δφ + (cid:0) − κ V ,φ + f ,φ (cid:1) δφ − F ˙ α − ˙ F A − (cid:16) κ ˙ φ + 3 H ˙ F + 6 ¨ F (cid:17) α (14)where A = 3( Hα − ˙ ψ ) − △ χ/a ( t ) and χ = a ( t )( β + a ˙ γ ). Here, in arriving the Eqs. (13)and (14), the leading order Eqs. (7) and (8) are also used. The other components δG i and δG ij ( i = j ) of the first order perturbed Einstein equation can be written as Hα − ˙ ψ = 12 F " κ ˙ φδφ + ˙ δF − HδF − ˙ F α , (15)and ˙ χ + Hχ − α − ψ = 1 F (cid:16) δF − ˙ F χ (cid:17) (16)respectively. The equation of motion of scalar perturbation is¨ δφ +3 H ˙ δφ + " − △ a ( t ) + (cid:18) V ,φ − f ,φ /κ (cid:19) δφ = ˙ φ ˙ α + (cid:16) φ + 3 H ˙ φ (cid:17) α + ˙ φA + 12 F ,φ (cid:18) δRκ (cid:19) . (17)where δR = − " ˙ A + 4 AH + (cid:18) △ a ( t ) + 3 ˙ H (cid:19) α + 2 △ a ( t ) ψ (18)5ow we analyze the curvature perturbation R = ψ − Hδφ/ ˙ φ by choosing a gauge where δφ = 0 and δR = 0. This sets R = ψ and moreover we have δF = 0 via δF = ( ∂F/∂φ ) δφ +( ∂F/∂R ) δR . Under this gauge the Eq. (15) gives, α = ˙ R H + ˙ F / (2 F ) (19)and hence from Eq. (13) we get A = − H + ˙ F / (2 F ) △ a ( t ) R + (cid:16) H ˙ F − κ ˙ φ (cid:17) ˙ R F (cid:16) H + ˙ F / (2 F ) (cid:17) . (20)Using Eq. (8) and Eq. (14), we obtain˙ A + H + ˙ F F ! A + 3 ˙ F F ˙ α + F + 6 H ˙ F + κ ˙ φ F + △ a ( t ) ! α = 0 . (21)Now we may write the differential equation for curvature perturbation by using the aboveEqs. (19), (20) and (21) as ¨ R + ( a ( t ) Q s )˙ a ( t ) Q s ˙ R + k a ( t ) R = 0 , (22)where, Q s = ˙ φ + 3 ˙ F / (2 κ F ) (cid:16) H + ˙ F / (2 F ) (cid:17) . (23)In arriving Eq. (22), Eq. (8) is again used. Now one may re-write the Eq. (22) in terms ofvariables ω = a √ Q s and σ k = ω R as σ ′′ k + (cid:18) k − ω ′′ ω (cid:19) σ k = 0 , (24)where prime denotes the derivative with respect to the conformal time defined as dη = dt/a ( t ) and ω ′′ ω = a ′′ ( t ) a ( t ) + a ′ ( t ) a ( t ) Q ′ s Q s + 12 Q ′′ s Q s − (cid:18) Q ′ s Q s (cid:19) (25)under quasi de-Sitter expansion a ( η ) = − Hη (1 − ǫ ) and hence a ′′ ( t ) a ( t ) = η (cid:2) ǫ (cid:3) and a ′ ( t ) /a ( t ) = a ( t ) H . Therefore we have ω ′′ ω = 1 η h ν R − i (26)6here ν R = 94 h ǫ + ǫ − ǫ + ǫ ) i . (27)In arriving at the above expression we have defined ǫ = − ˙ HH , ǫ = ¨ φH ˙ φ , ǫ = ˙ F HF , ǫ = ˙ E HE ; (28) E = F + 3 ˙ F κ ˙ φ = Q s (1 + ǫ ) ˙ φ / ( F H ) . (29)Here ǫ i are slow roll parameters and ˙ ǫ i terms have been neglected. Equation (24) then hassolutions in the Hankel functions of order ν R σ = p π | η | e i (1+2 ν R ) π/ (cid:2) c H (1) ν R ( k | η | ) + c H (2) ν R ( k | η | ) (cid:3) (30)Applying the Bunch-Davies boundary condition σ ( kη → −∞ ) = e ikη / √ k we fix the inte-gration constants c = 1 and c = 0. Using the relation H ν ( k | η | ) = − iπ Γ( ν ) (cid:16) k | η | (cid:17) − ν for thesuper-horizon modes kη →
0, we obtain the expression for the power spectrum for curvatureperturbations is defined as P R = 4 πk (2 π ) |R| ≡ ∆ R (cid:18) ka ( t ) H (cid:19) n R − (31)The amplitude of the curvature power spectrum turns out to be∆ R = 1 √ Q s (cid:18) H π (cid:19) (32)and the spectral index is n R − − ν R ≃ − ǫ − ǫ + 2 ǫ − ǫ ≃ − ǫ . (33)Using slow roll parameters, Eq. (23) can be simplified to the form Q s ≃ F ǫ M p with κ ˙ φ F H << ǫ which will be justified later in section (4). In our model of f (Φ , R ) couplingwe find ǫ ≈ − ǫ , ǫ ≈ − ǫ and these relations are used in the calculation of perturbationamplitude and spectral index. Plugging the values H and ˙ φ from Eqs. (10) and (11) intoEq. (28), the slow roll parameters can be written as ǫ = b − ( a − − b ) (1 − b ) /b ( a + 2 b − λ ( b − /b ξ /b (cid:18) φM p (cid:19) a +2 b − b (34) ǫ = b − ( a + 6 b −
4) (2 − b ) (1 − b ) /b ( a + 2 b − λ ( b − /b ξ /b (cid:18) φM p (cid:19) a +2 b − b . (35)7or our model, we can write the expressions for the amplitude of power spectrum and thenumber of e-folding as∆ R = b [(2 − b ) /λ ] − b M a − b p ξ − b φ − a +2 b − b a − ( a + 2 b − π (36)and N J = Z φ f φ J H ˙ φ dφ = b [(2 − b ) /λ ] b − b ξ − b a + 2 b − (cid:18) φM p (cid:19) − a − bb (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) φ J φ f (37)respectively. Here φ J and φ f are the values of scalar field φ at the beginning and the end ofinflation respectively. We define the perturbation of metric as follows g µν = ¯ g µν + h µν and g µν = ¯ g µν + h µν , (38)where ¯ g µν is background metric and h ij = − a ( t ) h ij , h i = 1 a ( t ) h i , h = − h . (39)Now to get the equation of tensor pertubation, we set h i = h = 0 in the calculation.From the decomposition theorem, the non zero spatial components h ij are traceless anddivergenceless, i.e., h ii = 0 , ∂ i h ij = 0 . (40)Using Eqs. (39) and (40), we obtain δR = 0 , δR i = 0 , (41) δR ij = − a ( t ) ▽ h ij + 12 ¨ h ij − ˙ a a ˙ h ij + 2 (cid:18) ˙ aa (cid:19) h ij , δR = 0 . (42)So, perturbing Eq. (5), we obtain12 F a ¨ D ij + (cid:18)
12 ˙
F a + 32 a ˙ aF (cid:19) ˙ D ij − F ▽ D ij = h aa ˙ F − F (cid:18) ˙ aa (cid:19) − ¨ aa F + f
2+ ¨ F + κ (cid:16) ˙ φ − V (cid:17) i a D ij , (43)8here D ij = h ij /a . The right hand side of Eq. (43) vanishes by Eqs. (7) and (8). Thus wehave ¨ D ij + ( a F ) ˙ a F ˙ D ij + κ a D ij = 0 . (44)In the terms of polarization tensors e ij and e ij , the tensor D ij is written as D ij = D e ij + D e ij . (45)For gravity wave propagating in ˆ z direction, the components of polarization tensor are givenby e xx = − e yy = 1 , e xy = e yx = 1 , e , iz = e , zi = 0 . (46)So the Eq. (44) can be written as¨ D λ + ( a F ) ˙ a F ˙ D λ + κ a D λ = 0 , (47)where λ ≡ ,
2. Now substituting z = a √ F and v k = zD λ M P / √
2, we get v ′′ λ + (cid:18) k − z ′′ z (cid:19) v λ = 0 , (48)where, prime ′ is derivative with respect to conformal time. Summing over all polarizationstates, the Eq. (48) provides us the amplitude of power spectrum of D λ as P T = 4 × (cid:18) M p (cid:19) κ π a F v λ ≃ π (cid:18) HM P (cid:19) F . (49)So, the ratio of the amplitude of tensor perturbations to scalar perturbations r for f (Φ , R )theories is given by r ≃ κ Q s F ≃ ǫ . (50)
3. CALCULATION IN THE EINSTEIN FRAME
Starting with the considered action S = Z d x √− g h − M p R (cid:18) ξ Φ a R b − M a +2 b − p (cid:19) + 12 ∂ µ Φ ∂ µ Φ + λ Φ i (51)we perform a conformal transformation of the metric g µν to the Einstein frame metric˜ g µν which is defined as ˜ g µν ( x ) = Ω ( x ) g µν ( x ) , (52)9here Ω = 1 + ξ Φ a R b − M pa +2 b − . (53)The Ricci scalar transform as R = Ω h ˜ R + 6 ˜ (cid:3) ΩΩ −
12 ˜ ∂ µ Ω ˜ ∂ µ ΩΩ i . (54)For quasi de-Sitter space we can ignore the second and third terms in the bracket in Eq.(54) which is justified in the Eq. (67). For this slow roll case, we can write Eq. (53) inEinstein frame as Ω = 1 + ξ β Φ α ˜ R β M α +2 βp , (55)where, α = a/ (2 − b ) and β = ( b − / (2 − b ). Now we write the action (51) in term of newfield φ E , which is related to the field Φ by the relation dφ E d Φ = 1Ω Ω + 3 α ξ ′ (cid:18) Φ M p (cid:19) α − ! / , (56)where ξ ′ = ξ β ( ˜ R/M p ) β . This leads the action in term of φ E as follows S E = Z d x (cid:18) − M p R + 12 ˜ D µ φ E ˜ D µ φ E + U ( φ E ) (cid:19) , (57)where U ( φ E ) = 1Ω λ φ E ) . (58)For Φ >> M P /ξ ′ /α , Eq. (56) can be integrated to giveΦ = M p ξ ′ /α exp r φ E M p α − ! . (59)Considering ˜ g µν = diag ( − M ( t ) , ˜ a ( t ) , ˜ a ( t ) , ˜ a ( t )) and varying the action (57) with respectto M ( t ) or a ( t ) and setting M = 1 in the final equation which corresponds FRW metric, weget the Friedmann equation 12 ˜ H − ζ − M p λ (cid:18) βα (cid:19) = 0 , (60)where ζ = 12 β/α ˜ H M p ! β/α ξ β ) α exp r
23 ( α − φ E αM p ! . (61)10ere we have neglected all the derivative terms of Hubble parameter ˜ H . This corresponds toslow roll condition, i.e., ˙ φ E is much smaller than potential term. We may write the Hubbleparameter from Eq. (60) as˜ H = M p [(1 + 2 β/α ) λ ] α α +4 β ) √ ξ β ) α +4 β exp "r (cid:18) − αα + 4 β (cid:19) φ E M p . (62)Now using Eq. (62) and (58) we obtain U ( φ E ) = 14 M p λ αα +4 β ξ − β ) α +4 β (cid:18) βα (cid:19) − βα +4 β exp h r (cid:18) − αα + 4 β (cid:19) φ E M p i . (63)Here we have taken the approximation exp( q φ E M p ) >> φ E >> M p . We now computethe spectral index and curvature perturbation using above potential (63). The slow rollparameters for large φ E >> M p comes out to be ǫ = M p (cid:18) U ′ U (cid:19) = 43 (cid:18) a + 2 b − a + 4 b − (cid:19) ; η = M p (cid:18) U ′′ U (cid:19) = 83 (cid:18) a + 2 b − a + 4 b − (cid:19) (64)and the curvature perturbation∆ R = 3 H πU ′ ( φ E )= 18 √ π (cid:18) y + 22 y − x + 4 (cid:19) x +2 y +42 x λ y − x +42 x ξ − x (cid:18) xy (cid:19) exp − r yφ E xM p ! , (65)where x = a + 4 b − y = a + 2 b − n s = 1 − ǫ + 2 η .The number of e-folding is calculated as N E = Z φ E φ Ee U ( φ E ) U ′ ( φ E ) dφ E = − r (cid:18) xy (cid:19) (cid:18) φ E − φ Ee M p (cid:19) (66)For φ E ∼ M p and φ Ee ∼ M p , the number of e-folding is found to be around 60. Theslow roll parameters ǫ , η and the Hubble parameter H are nearly independent of λ and are ∼ . ∼ .
04 and ∼ . × − M p respectively.Now from Eqs. (53) and (59), we can calculate the order of terms like ¨Ω / Ω and ( ˙Ω / Ω) for φ >> M p ξ /α . For λ = 10 − and ξ = 1,¨ΩΩ ∼ U M p ( ǫ + √ ǫ ( η − ǫ )) = 4 . × − M p and ˙ΩΩ ! ∼ U M p ǫ = 3 . × − M p (67)11 . − − − − a 3 .
385 3 .
026 2 .
735 2 .
494 2 . .
277 0 .
439 0 .
571 0 .
679 0 . .
939 3 .
904 3 .
877 3 .
852 3 . φ E = 13 M p with ξ = 1 fordifferent values of λ . respectively, whereas the value of curvature scalar ˜ R = 12 ˜ H at the same values of parameteris 4 . × − M p . Thus our approximation (i.e. for quasi de-Sitter space we can ignore thesecond and third terms in the bracket in Eq. (54)) made is consistent and may be checkedfor other values of a and b .We now use the measured values of these CMB anisotropy parameters to get the numericalvalues for the parameters ( a, b, ξ, λ ).
4. COMPARISON WITH DATA
From the Planck+WP measurements [50] we know that the curvature perturbation ∆ R =2 . +0 . − . × − , spectral index n R = 0 . ± . r < . CL ). For inflation to solve the horizon and flatness problems of standardhot big bang cosmology the number of e-foldings in the Eintein frame N E is required tobe about 60. From eqn. (66) we see that to get 60 e-foldings, the scalar field φ E shouldroll from 13 M p to 1 M p . We compute the curvature perturbation (65) and spectral index inthe Einstein frame and equate the expressions with the Planck+WMAP values to computethe parameters a and b for different values of λ and assume that the curvature couplingparameter ξ = 1. Our results for the correlated set of parameters λ, a, b at φ E = 13 M P which give the measured values of ∆ R and n s are shown in the Table (I). We see thatcompared to the ξφ R models with large ξ the small deviations of a and b from 2 and 1respectively can result in a large change in ξ which is 1 in our model compared to the earliercurvature coupling models where ξ ∼ .Next we equate the curvature perturbations and spectral index in the Jordan frame fromEq. (36) and Eq. (33) with the Planck+WMAP data to evaluate the values of the parameters λ, a and b (keeping ξ = 1) . The scalar field values Φ in the Jordan frame correspondingto φ E = 1 M p and 13 M P for different values of λ are displayed in Table (II). Using thesevalues of the range of the roll in Φ we see that the number of e-foldings N J in the Jordanframe, corresponding to N E = 60 is N J ∼
830 . The values of the parameters λ, a and b which give the required curvature perturbation and spectral index are shown in the Table(II). The slow roll parameters are found to be ǫ ≃ − ǫ ≃ .
007 and ǫ ≃ − ǫ ≃ − .
013 forchosen range of λ . The calculated value for the tensor to scalar ratio and Hubble parameterare r ≃ .
002 and H ∼ − M p respectively.The values of F = 1 + ξb Φ a R b − M a +2 b − p are found to be ∼ i.e much larger than unity andhence our assumption of dropping the unity in the expression for F is justified. Also wefind that the order of the term κ ˙ φ F H ∼ − is much smaller than 6 ǫ ∼ − as assumed insection (2.2). 12 . − − − − φ f | ( φE =1 Mp ) . M p . M p . M p . M p . M p φ J | ( φE =13 Mp ) . M p . M p . M p . M p . M p a 3 . . . . . . . . . . . . . . . ξ = 1 and φ J | φE =13 Mp for different values of λ . We find that in the limit a ≃ b ≃ R and n R are obtainedfor λ ∼ . ξ ∼ . Our Jordan frame calculation in this limit isconsistent with the results of [31, 33–35] who do the calculation in the Einstein frame.
5. DISCUSSION AND CONCLUSION
We have generalised the curvature coupling models of Higgs inflation to study inflationwith a scalar field for a λ Φ potential and a curvature coupling of the form ξ Φ a R b M a +2 b − p . It maybe possible to generate a tree level term of this form by choosing a suitable Kahler potentialin a f ( R ) supergravity theory [57–59].We find that for ξ = 1 and λ in the range (10 − − . a and b fall in the ranges (2 . − .
6) and (0 . − .
22) respectively. We discoveran interesting symmetry related to the numerical value of a and b which give the correctamplitude and spectral index. We find that for any value of λ the values of a and b whichgive the required density perturbations satisfy the relation ( a +2 b ) ≃ ξ Φ a R b M a +2 b − p has no dimensional couplings and isscale invariant.It has been shown that the Higgs self coupling can go from λ = 0 .
13 at the electroweakscale for the 125 GeV Higgs to λ = 10 − at the Planck scale by tuning the top mass or byintroducing extra interactions [18, 19, 49]. This leads us to conclude that the Higgs fieldmay still be a good candidate for being the inflaton in the early universe if one considers ageneralised curvature-Higgs coupling of the form ξ Φ a R b .The tensor to scalar ratio r in this model is small and the λ Φ with scalar curvaturecouplings is not ruled out by observational limits on r unlike the pure λ Φ theory [11, 56].We find that the values of ( a, b ) computed with Jordan and Einstein frame calculations ofthe curvature perturbation and spectral index are comparable but are not identical becauseunlike the ξ Φ R theory, in the ξ Φ a R b theory it is not possible in general to go to an Einsteinframe with a conformal transformation. If the space is quasi de-Sitter however such antransformation given by Eq. (55) is possible but the results will differ in the two framesdue to the slow roll approximation. Finally, by requiring the curvature coupling parameterto be of order unity, we have evaded the problem of unitarity violation in scalar-gravitonscatterings [37] which plagued the ξ Φ R Higgs inflation models [31, 33–35].13 cknowledgment
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