Reconstruction of potentials of the hybrid inflation in the light of primordial black hole formation
RReconstruction of potentials of thehybrid inflation in the light ofprimordial black hole formation
Ki-Young Choi, a Su-beom Kang, b and Rathul Nath Raveendran a a Department of Physics, Sungkyunkwan University, 16419 Korea b Sungkyunkwan University, 16419 KoreaE-mail: [email protected], [email protected], [email protected]
Abstract.
The large enhancement of the primordial power spectrum of the curvatureperturbation can seed the formation of primordial black hole, that can play as a dark mattercomponent in the Universe. In multi-filed inflation models, the curved trajectory of thescalar fields in the field space can generate a peak in the power spectrum on small scalesdue to the existence of the isocurvature perturbation. Here we show that a potential can bereconstructed from a given power spectrum, which is made of a scale-invariant one on largescales and the other function with a peak on small scales. In multi-field inflation models thereconstructed potential may not be unique and we can find different potentials from a givenpower spectrum. a r X i v : . [ a s t r o - ph . C O ] F e b ontents δN formalism 44 Reconstruction of a potential in two-field model 5 The observation of the large scale structures and the anisotropy of the cosmic microwavebackground (CMB) allowed the precise determination of the primordial power spectrum atlarge scales [1]. Cosmic inflation can naturally produce the required spectrum in additionto resolving the problems in the standard big bang model. However on small scales, thepower spectrum is constrained weakly only by the formation of primordial black hole (PBH),ultracompact minihalo, or dark matter (DM) annihilation for some specific models [2, 3].The PBHs can form through the collapse of large fluctuations of the density in the earlyUniverse when a given scale enters the horizon [4–6]. The PBHs can evaporate via Hawkingradiation, however those with the mass larger than 5 × g can survive until today. Thepresent remnant of PBHs can contribute to the non-baryonic component of dark matter [7].See the recent review [8] for PBH as a candidates for dark matter.For the formation of PBH, typically the amplitude of the power spectrum need to belarger than around 10 − , that is 10 bigger than that at the CMB scales, P CMB ∼ − .Several models were suggested to generate the large enhancement at small scales with asingle scalar field, such as features in the potential, running mass of the inflaton, hilltopmodels, and inflection point [9–17]. However, for a single field inflation, the slow-roll mustbe violated at scales between the CMB scale and PBH mass scales. In this case, a numericalcalculation is needed to evaluate the power spectrum properly.In Ref. [18], Hertzberg et al. provided a method to reconstruct the inflaton potential froma given power spectrum within canonical singled field model. They applied this method tothe formation of PBHs and confirmed again that the slow-roll conditions need to be violatedin order to generate a significant spike in the spectrum.The multi-field models which can generate PBHs include hybrid inflation, double in-flation, and a curvaton field [19–25]. For non-canonical kinetic terms, the accurate analyticprediction was derived for the formation of PBHs in multifiled case [26].In this paper, we propose a method to reconstruct a potential of multi-field scalarmodel from the given power spectrum. We consider a primordial power spectrum made ofa scale-invariant one on large scales and the other with a peak on small scales in the lightof generating PBHs. We use a hybrid-type potential and show that a few different examplesof the reconstructed potential. We find that the reconstructed potential may not be unique.– 1 –nce we know the potential, it is possible to obtain the power spectrum numerically bysolving the exact equations of motions. Lastly, we present these numerical results to supportour analytical arguments.In Sec. 2, we summarize the background evolution and the perturbations during in-flation with two scalar fields, and in Sec. 3 we give the formulation of the δN formalismapproximation. In Sec. 4, we introduce a method to reconstruct the potential from a givenpower spectrum and show the analytical results with numerical calculation. We conclude inSec. 5. In this section, we briefly revisit the evolutions of relevant background quantities and scalarperturbations (for more details, see [27, 28]). We shall focus on models with two real scalarfields, φ and ψ , described by the action S [ φ I ] = − (cid:90) d x √− g (cid:34) (cid:88) I =1 ∂ µ φ I ∂ µ φ I + V ( φ I ) (cid:35) , (2.1)where φ I = { φ, ψ } . The equations of motion of the fields can be written as¨ φ I + 3 H ˙ φ I + V φ I = 0 , (2.2)where H = ˙ a/a is the Hubble parameter and V φ I = d V / d φ I . Two Friedmann equationsdescribing the evolution of the scale factor are given by H = 13 M (cid:20) (cid:16) ˙ φ + ˙ ψ (cid:17) + V ( φ, ψ ) (cid:21) , (2.3a)˙ H = − M (cid:16) ˙ φ + ˙ ψ (cid:17) . (2.3b)In terms of the e-folding number, N t , defined as N t = ln ( a/a i ), where a i is the scale factorat a suitably chosen time, the field equations are d φdN t + (3 − (cid:15) H ) dφdN t + V φ H = 0 ,d ψdN t + (3 − (cid:15) H ) dψdN t + V ψ H = 0 , (2.4)where (cid:15) H ≡ − H dHdN t is the Hubble slow roll parameter.Since there are two fields involved, evidently, apart from the curvature perturbation,isocurvature perturbation also arises. In the spatially flat gauge, for instance, the Mukhanov-Sasaki variables associated with the curvature and the isocurvature perturbations v σ and v s are given by v σ = a (cos θ δφ + sin θ δψ ) , (2.5) v s = a ( − sin θ δφ + cos θ δψ ) , (2.6)where cos θ = ˙ φ/ ˙ σ , sin θ = ˙ ψ/ ˙ σ and ˙ σ = ˙ φ + ˙ ψ . The curvature and the isocurvatureperturbations are defined as R = v σ /z and S = v s /z , respectively, with z = a ˙ σ/H [28].– 2 –t is convenient to introduce the adiabatic and entropy vectors E Iσ and E Is in the fieldspace, defined as E Iσ = (cos θ, sin θ ) , (2.7) E Is = ( − sin θ, cos θ ) , (2.8)where I = { φ, ψ } . The equations governing the gauge invariant Mukhanov-Sasaki variables v σ and v s can be expressed as [28] v (cid:48)(cid:48) σ + (cid:18) k − z (cid:48)(cid:48) z (cid:19) v σ = 1 z ( z ξ v s ) (cid:48) , (2.9a) v (cid:48)(cid:48) s + (cid:18) k − a (cid:48)(cid:48) a + a µ s (cid:19) v s = − z ξ (cid:16) v σ z (cid:17) (cid:48) , (2.9b)where ξ = − a V s / ˙ σ and the quantity µ s is given by µ s = V ss − (cid:18) V s ˙ σ (cid:19) , (2.10)with the subscript φ or ψ indicating differentiation with respect to the fields. Also, thequantities V σ , V s and V ss are given by V σ = E Iσ V I , V s = E Is V I and V ss = E Is E Js V IJ , withimplicit summations assumed over the repeated indices I and J .As we know, the perturbations considered are quantum in nature. We can quantise theperturbations by promoting the variables to quantum operators as [28]ˆ v σ = f σ ˆ a + f (cid:63)σ ˆ a † + g σ ˆ b + g (cid:63)σ ˆ b † , (2.11)ˆ v s = f s ˆ a + f (cid:63)s ˆ a † + g s ˆ b + g (cid:63)s ˆ b † , (2.12)where f σ,s and g σ,s are the solutions of Eq. (2.9) , (ˆ a, ˆ b ) and (ˆ a † , ˆ b † ) are the annihilation andcreation operators. Vacuum states are defined asˆ a | (cid:105) = ˆ b | (cid:105) = 0 . (2.13)When the modes are very deep inside the Hubble radius, the equations of motion governingthe set of variable ( f σ , f s ) and ( g σ , g s ) are decoupled and we shall set the initial conditions,as usual, by the Minkowski-like vacuum as f σ ( η ) = g s ( η ) = e − ikη √ k , (2.14) f s ( η ) = g σ ( η ) = 0 . (2.15)The two scalar power spectra can be expressed as [28, 29] P R = k π | f σ | + | g σ | z ,P S = k π | f s | + | g s | z . (2.16)– 3 – Power spectrum with δN formalism Using the δN -formalism [30–33], we can evaluate the curvature perturbation on the hyper-surfaces of constant energy density, ζ , on super-horizon scales with the perturbation of thee-folding number N defined as N ( t e , t ∗ , x ) ≡ (cid:90) t e t ∗ Hdt. (3.1)The integral is evaluated from an initial flat hyper-surface at t = t ∗ to a final uniform densityhyper-surface at t = t e . The e-folding number N ( t e , t ∗ , x ) can be a function of the field athorizon exit at t = t ∗ and its perturbation can be expanded in terms of the filed perturbations δφ ( t ∗ , x ) and δψ ( t ∗ , x ), ζ (cid:39) δN = ∂N∂φ ∗ δφ ∗ + ∂N∂ψ ∗ δψ ∗ . (3.2)Here we assumed the slow-roll and ignored the dependence on the time derivative of the filed˙ φ and ˙ ψ . The field perturbation satisfies the two-point correlation function (cid:104) δφ ∗ ( k ) δφ ∗ ( k ) (cid:105) = (2 π ) δ (3) ( k + k ) 2 π k P ∗ ( k ) , P ∗ ( k ) ≡ H ∗ π , (3.3)where H ∗ is evaluated at Hubble exit k = a ∗ H ∗ . The similar relation is applied to δψ ∗ . Thenthe power spectrum of the curvature perturbation, P ζ , is defined as (cid:104) ζ ( k ) ζ ( k ) (cid:105) = (2 π ) δ (3) ( k + k ) 2 π k P ζ ( k ) . (3.4)From Eq. (3.3) and Eq. (3.4), we obtain the power spectrum of the curvature perturbationas P ζ = H ∗ π (cid:34)(cid:18) ∂N∂φ ∗ (cid:19) + (cid:18) ∂N∂ψ ∗ (cid:19) (cid:35) . (3.5)In the slow-roll limit of φ field, the number of e-foldings can be written by [29] N ( φ ∗ , ψ ∗ ) = − M (cid:90) φ e φ ∗ VV φ dφ, (3.6)where φ e and ψ e are functions of φ ∗ and ψ ∗ . The partial derivatives of the e-folding numberare ∂N∂φ ∗ = 1 M (cid:20)(cid:18) VV φ (cid:19) ∗ − (cid:18) VV φ (cid:19) e ∂φ e ∂φ ∗ (cid:21) , ∂N∂ψ ∗ = − (cid:18) VV φ (cid:19) e ∂φ e ∂ψ ∗ . (3.7)We note that the comoving curvature perturbation R coincides with the perturbationon hypersurfaces of constant energy density ζ on scales far out side the horizon k (cid:28) aH [34].In other words, P ζ ≈ P R (3.8)on super horizon scales. – 4 – Reconstruction of a potential in two-field model
We are interested to reconstruct a potential that produces a power spectrum with a peakon small scales. For this we consider a power spectrum composed of two parts, almost-scaleinvariant one and the other with a peak given by P ζ ( k ) = P s ( k ) + P p ( k ) , (4.1)with P s ( k ) ≡ A s (cid:18) kk p (cid:19) n s − . (4.2)Here k p = 0 .
05 Mpc − is the pivot scale used by Planck, and A s (cid:39) . × − and n s (cid:39) .
96 [1].On large scales around CMB observation, P s ( k ) is dominant and gives almost scale-invariantpower spectrum for k (cid:39) − − − , however on small scales the peak spectrum isdominant P s (cid:28) P p . In the light of the PBH formation, we consider that P p (cid:38) P s aroundscales of the peak.It is known that, the power spectrum that peaks at a scale k c = 10 Mpc − can generatestochastic background of gravitational waves which peaks in the frequency band targetedby the future interferometer LISA [35]. By following this, in our numerical calculations,we choose to work with the parameters such that the power spectrum peaks at the scale k c = 10 Mpc − .In the two-field inflation models, we expect that the scale invariant power spectrum P s comes from the φ field, and the one with a peak from the ψ field, i.e. using δN -formalism, P s = H ∗ π (cid:18) ∂N∂φ ∗ (cid:19) , P p = H ∗ π (cid:18) ∂N∂ψ ∗ (cid:19) . (4.3)From these two equations, we will reconstruct the potential of two scalar fields. However, inmulti-filed case, the reconstructed potential from the power spectrum may not be unique [36].In the followings, we choose a potential of the type of hybrid inflation given by V = V [1 + f ( φ ) + g ( φ ) h ( ψ )] , (4.4)where 1 (cid:29) f ( φ ) + g ( φ ) h ( ψ ) during inflation to ensure the vacuum-domination.The trajectories on the field space can be labelled by the integral of motion along thetrajectory [29] C = (cid:90) g ( φ ) f φ ( φ ) dφ − (cid:90) h ψ ( ψ ) dψ = F ( φ ) − H ( ψ ) , (4.5)where we defined F ( φ ) ≡ (cid:90) g ( φ ) f φ ( φ ) dφ, H ( ψ ) ≡ (cid:90) h ψ ( ψ ) dψ. (4.6)Since the integral of motion connects the field values at the horizon exit and the end ofinflation by F ( φ ∗ ) − H ( ψ ∗ ) = F ( φ e ) − H ( ψ e ) , (4.7)– 5 –e can find the relation of the partial derivatives g ∗ f φ ∗ = g e f φ e (cid:18) ∂φ e ∂φ ∗ (cid:19) − h ψ e (cid:18) ∂ψ e ∂φ ∗ (cid:19) , − h ψ ∗ = g e f ψ e (cid:18) ∂φ e ∂ψ ∗ (cid:19) − h ψ e (cid:18) ∂ψ e ∂ψ ∗ (cid:19) , (4.8)where we used a notation g ∗ = g ( φ ∗ ), g e = g ( φ e ), f φ ∗ = df ( φ ) dφ (cid:12)(cid:12)(cid:12) φ = φ ∗ , and etc. In addition tothis, if we know the condition of ending inflation E ( φ e , ψ e ) = 0 , (4.9)then, in principle, we can obtain φ e and ψ e in terms of φ ∗ and ψ ∗ by solving Eq. (4.7) andEq. (4.9) together.From the potential in Eq. (4.4), the e-folding number in the slow-roll regime can beevaluated as N ( φ ∗ , ψ ∗ ) (cid:39) − M (cid:90) φ e φ ∗ dφf φ ( φ ) , (4.10)where we assumed f φ ( φ ) (cid:29) g φ ( φ ) h ( ψ ). Then we obtain M ∂N∂φ ∗ = − (cid:18) ∂φ e ∂φ ∗ (cid:19) f φ e + 1 f φ ∗ ,M ∂N∂ψ ∗ = − (cid:18) ∂φ e ∂ψ ∗ (cid:19) f φ e . (4.11)For simplicity we assume that the inflation ends by the condition given by only ψ e E ( φ e , ψ e ) = E ( ψ e ) = 0 , (4.12)and ψ e is independent of any φ ∗ and ψ ∗ . In this case, using Eq. (4.8), Eq. (4.11) becomes M ∂N∂φ ∗ = 1 f φ ∗ (cid:18) − g ∗ g e (cid:19) ,M ∂N∂ψ ∗ = 1 g e h ψ ∗ . (4.13)Now, by matching this with the given power spectrum in Eq. (4.3), we obtain equations P s = H ∗ π M (cid:20) f φ ∗ (cid:18) − g ∗ g e (cid:19)(cid:21) , P p = H ∗ π M (cid:20) g e h ψ ∗ (cid:21) , (4.14)where N ∗ ≡ N ( φ ∗ , ψ ∗ ). We solve these equations with the equations of motion of the fieldsEq. (2.4) to reconstruct functions f ( φ ) , g ( φ ), and h ( ψ ) in the potential. In the following sub-sections, we consider two cases for them and present the power spectra which are calculatedusing the equations Eq. (2.9) and Eq. (2.16) numerically.– 6 – .1 Case 1: Gaussian peak Here we we consider a input power spectrum with a peak defined as: P p = H ∗ π (cid:104) δ + βe − α ( N t − N tc ) − e λ ( N t − N t ) (cid:105) . (4.15)Note that N is defined from the end of inflation. However, we also use the notation of N t ,which is defined from some initial time of inflation with a relation N t = N tot − N with N tot the e-folding number between some initial time and the end. It is evident from the aboveexpression that, the power spectrum is Gaussian near N t = N tc and it decreases exponentiallynear to the end of inflation when N t > N t .From this input power spectrum in Eq. (4.15), we try to reconstruct a potential bysolving the relations in Eq. (4.14) from δN formalism. By comparing both equations, we canobtain h ψ ( N t ) from P p as h ψ ( N t ) = 1 M g e (cid:104) δ + βe − α ( N t − N tc ) − e λ ( N t − N t ) (cid:105) . (4.16)In order to reconstruct a potential h ( ψ ) which can produce the above power spectrum, weconsider that the function h ψ in Eq. (4.4) is proportional to ψ n − with integer n larger orequal to 2. In this case, h ( ψ ) is simply h ( ψ ) = 1 n (cid:18) ψκ (cid:19) n , (4.17)where κ is a constant with the same dimension as ψ . Then, Eq. (4.17) directly gives therelation ψ ( N t ) as ψ ( N t ) = ψ c (cid:18) h ψ ( N t ) h ψ ( N tc ) (cid:19) n − , (4.18)where ψ c ≡ ψ ( N tc ). In the above equation, for convenience, we have rewritten κ as κ = ψ c ( h ψ ( N tc ) ψ c ) /n . From above expression we expect that, the ψ is nearly constant at initialposition and slowly evolves towards a minimum and then increases. For this evolution, wefind that, the second derivative of ψ cannot be neglected for a short duration near the pointwhere ψ begins to roll down towards the minimum and also near the point where the firstderivative of ψ is zero which happens at ψ = ψ c . We check this deviation of slow roll conditionof ψ field in our numerical calculations as well. Then, from the equation of motion, Eq. (2.4),the function g ( φ ) is obtained in terms of N t , as g ( φ ) = − H V h ψ ( N t ) (cid:20) d ψdN t + (3 − (cid:15) H ) dψdN t (cid:21) , (cid:39) − h ψ ( N t ) (cid:20) d ψdN t + 3 dψdN t (cid:21) , (4.19)where in the second line we ignored the subdominant (cid:15) H , however we included the secondderivative d ψ/dN t . Using the explicit form of ψ ( N t ), we find g ( φ ) = − ng e h ( φ )3( n − (cid:26) n [ λEx ( φ ) + 2 N d ( φ ) G ( φ )] + ( n −
1) [ λ (3 + λ ) Ex ( φ ) + 2 α (1 + N d ( φ )(3 − αN d ( φ ))) G ( φ )] N ψ ( φ ) (cid:27) (4.20)– 7 – . . . . . φ/M Pl − . − . − . − . − . − . . . g ( φ ) / g e Figure 1 : Evolution of g ( φ ) (left) and the trajectory of the fields on the potential V ( φ, ψ )(right) for case 1 with n = 2.where N d [ φ ] ≡ N t ( φ ) − N tc G ( φ ) ≡ βe − α ( N t ( φ ) − N tc ) Ex ( φ ) ≡ e λ ( N t ( φ ) − N t ) (4.21)The e-folding number N t can be replaced as a function of φ from the equation of motionin Eq. (2.4) once we know the evolution of φ in terms of N t . As a specific choice, let us usea small field inflation with a function f ( φ ), f ( φ ) = − (cid:18) φµ (cid:19) . (4.22)By solving the slow-roll equation of motion, we find the evolution of the field φ , φ = φ i e M µ ( N t − N ti ) = φ i e M µ ( N i − N ) , (4.23)and the function g ( φ ) is obtained by replacing N t with φ ( N t ).For the numerical calculation, we choose the power spectrum has a peak at k c =10 Mpc − corresponding to N tc = 40. We also assume that the pivot scale exit the Hubbleradius at N tp = 10. Using these information and also assuming slow roll condition, we fix, µ = 10 ,V = 24 π φ ( N p ) A s /µ . (4.24)In addition, we choose the smallest scale relevant for the perturbations in CMB is k p =1 Mpc − and this scale exit the Hubble radius at N tp = N tp + log k p k p . As we have mentionedearlier, on small scales the peak spectrum is dominant, i.e. P s (cid:28) P p and also P p (cid:38) P s around scales of the peak. These information leads to the constraints δ < µ φ ( N tp ) ,β > µ √ φ ( N tp ) . (4.25)– 8 – − k (Mpc − ) − − − − − − − − − P R ( k ) Inputn=2n=3
Figure 2 : Left: The power spectrums calculated numerically from the potential in Case1 with n = 2 (blue), n = 3 (green), and the input power spectrum (orange). We used α = 1 / , β = 4 × , δ = 12, N tc = 40 and N t = 65, with λ = 5.For the numerical calculations, we choose to work with α = 1 / , δ = 12 , β = 4 × . (4.26)The remaining parameters are the initial field values used at N t = 0 which are fixed as φ i = φ ( N tp ) e − N tp /µ , ψ i = 10 − . In Fig. 1 (Left), we show the function g ( φ ) for the function f ( φ ) used in Eq. (4.22). Theexplicit form of the potential is obtained substituting Eq. (4.17) and Eq. (4.20) in Eq. (4.4),which is complicated and not shown here. Instead, in Fig. 1 (Right), we show the potentialand the trajectory of the fields in the plane of ( φ/φ i , ψ/ψ i ). It is interesting to note that,during the initial stages of inflation, the function g ( φ ) is very small. This is expected sincethe potential is dominated by the field φ alone. In the case of n = 2 the square of the massof ψ is proportional to the function g ( φ ). Around N t = N tc , g ( φ ) is linear in N t and changesits sign from positive to negative. This means that the field ψ becomes tachyonic. Finally,the square of mass decreases exponentially and this leads inflation to end when the (cid:15) H = 1.In Fig. 2, we show the power spectrums for this case: the input power spectrum (orangedashed) and the power spectrums from the reconstructed potential with n = 2 (blue solid), n = 3 (green solid). We can see that the input power spectrum matches the power spectrumobtained numerically quite well within an error of 10%. In the previous case, we have been able to reconstruct the potential by considering theGaussian power spectrum and the function h ψ ∝ ψ n . Let us now try to reconstruct thepotential by assuming the same h ψ but a different type of peak power spectrum which is– 9 – . . . . . . . . φ/M Pl − . − . − . − . . . g ( φ ) / g e Figure 3 : Evolution of g ( φ ) (left) and the trajectory of the fields on the potential V ( φ, ψ )(right) for case 2 with n = 2 . We used the same parameters as in Fig. 2 except γ =1 / , N t = 57, and λ = 3. − k (Mpc − ) − − − − − − − − − P R ( k ) Inputn=2n=3
Figure 4 : The power spectrums calculated numerically from the reconstructed potential inCase 2 with n = 2 (red), n = 3 (purple) and the input power spectrum (orange). We usedthe same parameters as in Fig. 2 except γ = 1 / , N t = 57, and λ = 3.given by P p = H ∗ π (cid:104) δ + β sech [ γ ( N t − N tc )] − e λ ( N t − N t ) (cid:105) . (4.27)The task is to reconstruct g ( φ ) by using the Eq. (4.19). Following the same method used incase 1, it is straight forward to obtain g ( φ ) as g ( φ ) = − ng e h ( φ )3( n − (cid:26) nλ Ex ( φ ) + λEx ( φ ) [( n − λ ) N ψ ( φ ) + 2 nβγS [ φ ] T [ φ ]]+ γβS ( φ ) (cid:2) nβγS ( φ ) T ( φ ) + ( n − N ψ ( φ )(2 γS ( φ ) − γ + 3 T ( φ )) (cid:3) (cid:27) (4.28)– 10 –here S ( φ ) = 2( φ/φ c ) γµ / + ( φ/φ c ) − γµ / ,T ( φ ) = ( φ/φ c ) γµ − ( φ/φ c ) − γµ ( φ/φ c ) γµ + ( φ/φ c ) − γµ . (4.29)In Fig. 3, we show the evolution of g ( φ ) and the trajectory of the fields on the potential V ( φ, ψ ) for case 2 with n = 2. As one can see from this figure, the overall behavior of thefunction g ( φ ) and the evolution of the inflationary trajectory are similar as in the case 1. InFig.4, we show the power spectrum for the reconstructed potential in Case 2 with n = 2 (red), n = 3 (purple)with the input power spectrum (orange). Here we used γ = 1 / , N t = 57,and λ = 3. It is clear from the figure that the given spectrum match the numerical resultsvery well. The primordial black hole can be produced from the enhanced primordial power spectrum ofthe curvature perturbation on small scales, and may play as dark matter. In the literature,there have been several trials to calculate an enhanced power spectrum from a given potentialin single field or multi-field inflation models. In this paper, however, we suggest new methodto reconstruct a potential from a given power spectrum. With this way, we could findpotentials that have a peak on small scales, which is large enough to generate primordialblack holes.In this work, we used the input power spectrum composed of a nearly scale-invariant oneon large scales and the other with a peak on small scales. To reconstruct a potential with twocanonical scalar fields with φ and ψ , we have used a hybrid type potential. Guided by the δN formalism, we matched each power spectrum to the contribution from each field and solvedthem. We have been able to solve them numerically and could reconstruct a potential. Weevaluated the scalar power spectra in reconstructed models and confirmed that the resultingspectra are quite compatible with the input spectrum.In the reconstructed models, the scale-invariant power spectrum is generated when thefield φ is dominant while the field ψ is nearly constant. After this period, the field ψ startsmoving towards its minimum and then bounces at a critical value to increase. During thisbounce, the power spectrum is dominated by the ψ field and the enhance peak is generated.We should also mention that, though the correct form of the power spectrum for smallscales is yet to be understood, for illustration, we have chosen two types of power spectrum,Gaussian and hyperbolic. It is also important to note that, in this work, we have focussedon constructing small-field hybrid type of potentials. We believe that, using our methodsdiscussed in this work, one can explore more complex models beyond the models we haveconstructed here. Acknowledgments
K.-Y.C. and R.N.Raveendran were supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government (MEST) (NRF-2019R1A2B5B01070181). S.Kang was supported by Korea Initiative for fostering University of Research and InnovationProgram of the National Research Foundation (NRF) funded by the Korean government(MSIT) (No.2020M3H1A1077095). – 11 – eferences [1] Y. Akrami et al. [Planck], Astron. Astrophys. (2020), A10doi:10.1051/0004-6361/201833887 [arXiv:1807.06211 [astro-ph.CO]].[2] T. Bringmann, P. Scott and Y. Akrami, Phys. Rev. D (2012), 125027doi:10.1103/PhysRevD.85.125027 [arXiv:1110.2484 [astro-ph.CO]].[3] K. Y. Choi, J. O. Gong and C. S. Shin, Phys. Rev. Lett. (2015) no.21, 211302doi:10.1103/PhysRevLett.115.211302 [arXiv:1507.03871 [astro-ph.CO]].[4] Y. B. ;. N. Zel’dovich, I. D., Soviet Astron. AJ (Engl. Transl. ), (1967), 602[5] S. Hawking, Mon. Not. Roy. Astron. Soc. (1971), 75[6] B. J. Carr, Astrophys. J. (1975), 1-19 doi:10.1086/153853[7] D. N. Page, Phys. Rev. D (1976), 198-206 doi:10.1103/PhysRevD.13.198[8] A. M. Green and B. J. Kavanagh, [arXiv:2007.10722 [astro-ph.CO]].[9] P. Ivanov, P. Naselsky and I. Novikov, Phys. Rev. D (1994), 7173-7178doi:10.1103/PhysRevD.50.7173[10] E. D. Stewart, Phys. Lett. B (1997), 34-38 doi:10.1016/S0370-2693(96)01458-X[arXiv:hep-ph/9606241 [hep-ph]].[11] K. Kohri, C. M. Lin and D. H. Lyth, JCAP (2007), 004 doi:10.1088/1475-7516/2007/12/004[arXiv:0707.3826 [hep-ph]].[12] K. Kohri, D. H. Lyth and A. Melchiorri, JCAP (2008), 038doi:10.1088/1475-7516/2008/04/038 [arXiv:0711.5006 [hep-ph]].[13] L. Alabidi and K. Kohri, Phys. Rev. D (2009), 063511 doi:10.1103/PhysRevD.80.063511[arXiv:0906.1398 [astro-ph.CO]].[14] J. Garcia-Bellido and E. Ruiz Morales, Phys. Dark Univ. (2017), 47-54doi:10.1016/j.dark.2017.09.007 [arXiv:1702.03901 [astro-ph.CO]].[15] C. Germani and T. Prokopec, Phys. Dark Univ. (2017), 6-10 doi:10.1016/j.dark.2017.09.001[arXiv:1706.04226 [astro-ph.CO]].[16] H. Motohashi and W. Hu, Phys. Rev. D (2017) no.6, 063503doi:10.1103/PhysRevD.96.063503 [arXiv:1706.06784 [astro-ph.CO]].[17] G. Ballesteros and M. Taoso, Phys. Rev. D (2018) no.2, 023501doi:10.1103/PhysRevD.97.023501 [arXiv:1709.05565 [hep-ph]].[18] M. P. Hertzberg and M. Yamada, Phys. Rev. D (2018) no.8, 083509doi:10.1103/PhysRevD.97.083509 [arXiv:1712.09750 [astro-ph.CO]].[19] J. Fumagalli, S. Renaux-Petel, J. W. Ronayne and L. T. Witkowski, [arXiv:2004.08369[hep-th]].[20] J. Yokoyama, Astron. Astrophys. (1997), 673 [arXiv:astro-ph/9509027 [astro-ph]].[21] M. Kawasaki, N. Sugiyama and T. Yanagida, Phys. Rev. D (1998), 6050-6056doi:10.1103/PhysRevD.57.6050 [arXiv:hep-ph/9710259 [hep-ph]].[22] J. Yokoyama, Phys. Rev. D (1998), 083510 doi:10.1103/PhysRevD.58.083510[arXiv:astro-ph/9802357 [astro-ph]].[23] M. Kawasaki, N. Kitajima and T. T. Yanagida, Phys. Rev. D (2013) no.6, 063519doi:10.1103/PhysRevD.87.063519 [arXiv:1207.2550 [hep-ph]].[24] S. Clesse and J. Garc´ıa-Bellido, Phys. Rev. D (2015) no.2, 023524doi:10.1103/PhysRevD.92.023524 [arXiv:1501.07565 [astro-ph.CO]]. – 12 –
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