$\alpha$-attractor-type Double Inflation

We study the background dynamics and primordial perturbation in $\alpha$-attractor-type double inflation. The model is composed of two minimally coupled scalar fields, where each of the fields has a potential of the $\alpha$-attractor type. We find that the background trajectory in this double inflation model is given by two almost straight lines connected with a turn in the field space. With this simple background, we can classify the property of the perturbations generated in this double inflation model just depending on whether the turn occurs before the horizon exit for the mode corresponding to the scale of interest or not. In the former case, the resultant primordial curvature perturbation coincides with the one obtained by the single-field $\alpha$-attractor. On the other hand, in the latter case, it is affected by the multifield effects, like the mixing with the isocurvature perturbation and the change of the Hubble expansion rate at the horizon exit. We obtain the approximated analytic solutions for the background, with which we can calculate the primordial curvature perturbation by the $\delta N$ formalism analytically, even when the multifield effects are significant. We also impose observational constraints on the model parameters and initial values of the fields in this double inflation model based on the Planck result.


I. INTRODUCTION
Cosmic inflation is the simplest scenario for the origin of primordial perturbations in our Universe, whose predictions are verified by the observations of the cosmic microwave background (CMB) and large scale structure (LSS) of the Universe [1][2][3][4] (see e.g. [5,6] for reviews). Actually, the recent Planck observations confirm that the primordial curvature perturbations are almost scale-invariant and Gaussian [7,8]. These observations are consistent with the predictions of the simplest class of inflation models given by a single scalar field with a canonical kinetic term and a sufficiently flat potential, and couples minimally to gravity. Regardless of the phenomenological success, it is still nontrivial to embed such a single-field slow-roll inflation model into a more fundamental theory (see [9], for a review). In this context, the so-called α-attractor models [10][11][12][13][14] have been actively studied so far. Phenomenologically, this class of models predict common values of the spectral index n s and the tensorto-scalar ratio r, well matching observational data. Theoretically, the attractive property of these models is attributed to the conformal symmetry and they are successfully embedded into supergravity through the hyperbolic geometry [15,16] (see also [17][18][19][20][21][22] for other interesting cosmological scenarios based on the hyperbolic geometry). Furthermore, recently, the models with certain values of α were shown to be derived from string/M theory setups based on the seven-disk manifold [23,24].
However, since scalar fields such as moduli fields are ubiquitous in the fundamental theories like supergravity and string/M theory, it is natural to consider the multi-field ex-tension of the α-attractor models. From the viewpoint of the primordial perturbations, the curvature perturbation can evolve even on sufficiently large scales caused by the mixing with the isocurvature perturbation in multiple inflation, which gives rich phenomenology [25,26]. Among the multiple inflation models, the so-called double inflation composed of two minimally coupled massive scalar fields is a simple and concrete model including the multi-field effects on the primordial perturbation [27][28][29] and have been actively studied (see, e.g., [30][31][32]). The T-model, which is one of the simplest realization of the α-attractor and has a plateau-like potential, is shown to appear out of a massive scalar field with a noncanonical kinetic term with a pole [14] with a field redefinition so that the new field has a canonical kinetic term. Therefore, as a simple multi-field extension of the α-attractor, in this paper, we consider the α-attractor-type double inflation, which is composed of two minimally coupled scalar fields and each of the field has a potential of the α-attractor-type 1 . For other interesting works considering the multi-field extension of the α-attractor models in different contexts, see [34][35][36][37][38][39][40][41][42] 2 .
The rest of this paper is organized as follows. In Sec. II, we present our α-attractor-type multiple inflation model in general form. We then analyze the background dynamics for the case of double inflation, which we focus in what follows, in Sec. III, showing that the analytic solutions based on the slowroll approximation can approximate the numerical results very 1 Although we concentrate on the phenomenology of the α-attractor-type double inflation and do not consider the link of the setup with fundamental theories in this paper, it might be possible to derive this model based on the seven-disk manifold [33]. 2 It is known that the Starobinsky inflation model [43] is included in the α-attractor, as so-called the E-model [11]. The multi-field extension of the Starobinsky inflation have been also actively studied recently, see, e.g., [44][45][46][47][48][49][50][51][52]. arXiv:1810.06914v1 [hep-th] 16 Oct 2018 well. In Sec. IV, we calculate the primordial curvature perturbation in this double inflation model that can be verified by the recent CMB observations. We show that the δ N-formalism, where we can calculate the primordial curvature perturbation analytically in this model, can reproduce the numerical results very well. After imposing the constraints on the model parameters and initial values of the fields based on the Planck result in Sec. V, we summarize in Sec. VI with conclusions and discussions. Some technical issues in this model related with the background dynamics based on a potential without the symmetric property, the excitation of the heavy field at the turn, and the primordial non-Gaussianity are discussed in Appendix A, Appendix B, and Appendix C, respectively. In this paper, when we show plots, all quantities with dimensions are normalized by M Pl , unless otherwise mentioned.

A. α-attractor inflation
It is well known that inflation models of canonical scalar field are described by the following action: where g is the determinant of the spacetime metric g µν , R is the scalar curvature and M Pl is the Planck mass. The αattractor models motivated by the conformal symmetry include various potentials whose asymptotic shape looks like for |φ | √ αM Pl , where ∓ in the parentheses above denotes − for φ > 0 and + for φ < 0. Here, Λ is a parameter with mass dimension, while γ and α are dimensionless parameters. If most of the last 60 e-folds of inflation is driven in the region, where the potential is given by (2.1), this class of models are shown to have the attractor property on inflational observables independent of γ. Actually, the α-attractor models, with α not too large, generically predict that the spectral index n s and tenor-to-scalar ratio r defined later are given by the number of e-folding of inflation N at leading order in 1/N as with N ≡ ln(a end /a), where a end is the scale factor at the end of inflation. In order to satisfy this condition, we assume that α is not significantly different from unity, which includes the so-called conformal attractor of α = 1. Although Λ in Eq. (2.1) does not appear in the expression of n s and r, it controls the amplitude of the curvature perturbation P R , which we will also define later.

B. Multiple Inflation Model
In this paper, since we are interested in α-attractor-type multiple inflation, we consider the following action with n scalar fields ϕ I (I = 1, 2, · · · , n): We use Einstein's implicit summation rule for the scalar field indices I, J, · · · and all the field indices can be lowered by the field-metric δ IJ . For the potential, we are interested in the case that each V I takes the form given by Eq. (2.1) for |ϕ I | M Pl .
To be concrete, we may adopt the simplest T-model potential as V I [10], which has two parameters λ I and M I . Before starting the analysis, let us consider the singlefield T-model potential more and discuss the possible parameter regime for the α-attractor-type double inflation based on Eq. (2.4) with I = 1, ϕ 1 = φ , λ 1 = λ , M 1 = M. For |φ | M, it is approximated by which shows the form of the α-attractor given by Eq. (2.1) with Λ = (λ 1/4 M)/ √ 6, γ = 4, α = M 2 /(6M 2 Pl ). Then, the single-field T-model predicts n s and r given by Eq. (2.2) as long as φ stays at this region during most of the last 60 e-folds of inflation. On the other hand, for |φ | M, the potential behaves just as the mass term driving the chaotic inflation [53], With M M Pl , since most of the last 60 e-folds of inflation is given by the potential (2.6), the prediction on n s and r in this model becomes indistinguishable from the ones in the chaotic inflation. Since the conventional double inflation based on two massive scalar fields have been actively studied, we do not consider the case with M M Pl in this paper. Therefore, we assume that one of ϕ I is larger than M in most of the last 60 e-folds of inflation throughout this paper.
Notice that although we concentrate on the model based on T-model, as long as the parameters are chosen so that each V I takes the form of the α-attractor-type potential given by (2.1) when the observable mode is generated, most results shown in this paper are also obtained by the α-attractor-type multiple inflation based on different models, like E-model [11].
In what follows, we focus on the case of two scalar fields (double inflation), which is simple but includes the interesting multi-field effects on the primordial perturbations. In concluding remarks, we mention briefly about some properties expected in multiple inflation.

III. BACKGROUND DYNAMICS OF α-ATTRACTOR TYPE DOUBLE INFLATION
In what follows, we will analyze α-attractor type double inflation with T-model potential (2.4). We describe two scalar fields as ϕ 1 = φ and ϕ 2 = χ. In the main text of this paper, we concentrate on the potential symmetric between φ and χ so that M 1 = M 2 = M and λ 1 = λ 2 = λ . For more general cases without imposing M 1 = M 2 nor λ 1 = λ 2 , we discuss the background dynamics in Appendix A, and we find that the phenomenologically essential aspects of the α-attractor-type double inflation can be extracted sufficiently by this simple symmetric potential.
In Fig. 1, we plot the shape of the potential of the αattractor-type double inflation based on the symmetric twofield T-model. From the symmetric property of the potential, we can restrict ourselves to the region χ ≥ φ ≥ 0 without loss of generality. In Fig. 2, we depict the gradient of the potential. In this section, we analyze the background dynamics. We show that the analytic solutions based on the slow-roll approximation can approximate the numerical results very well.

A. Basic equations for background dynamics
Suppose that the background Universe is homogeneous and isotropic with flat Friedmann-Lemaître-Robertson-Walker (FLRW) metric where a(t) is the scale factor whose evolution is governed by the Friedmann equations where H ≡ȧ/a is the Hubble expansion rate and a dot denotes a derivative with respect to the cosmic time t. With ,I denoting a derivative with respect to ϕ I , the equations of motion for the homogeneous scalar fields ϕ I arë For later use, following Ref. [25], we introduce the adiabatic and entropic directions based on the speed of the fieldṡ σ and the angle Θ characterizing the direction of the background trajectory, wherė The adiabatic direction is along the background fields' evolution, while the entropic direction is orthogonal to the adiabatic one. The unit vectors corresponding to these directions n I and s I , respectively are defined by which are related with each other throughṅ I =Θs I ,ṡ I = −Θn I . By projecting Eqs. (3.3) onto the adiabatic direction and entropic direction, we can obtain the evolution equations forσ and Θ, respectively as with V ,σ ≡ V ,I n I , V ,s ≡ V ,I s I .

B. Numerical results on background dynamics
Before showing numerical results, let us first give a brief overview of the background dynamics in the α-attractor-type double inflation model, which can be easily expected from the potential form (Fig. 1). For the initial values of the fields ϕ I ini satisfying χ ini ≥ φ ini M, the slow-roll approximation becomes valid soon, where the background trajectory starts following the gradient shown in Fig. 2. In the first phase, unless we fine-tune φ ini χ ini , the motion towards χ = 0 soon becomes negligible compared with that towards φ = 0 and the background trajectory becomes almost a straight line towards φ = 0. We expect a slow-roll inflation in this phase. Then, when φ becomes smaller and approaches to 0, the background fields are captured by the potential valley around φ = 0 and changes the direction of the motion to χ = 0. Θ defined by Eq. (3.4) changes from π to (3/2)π. After the direction has changed completely to χ = 0, the background trajectory becomes a straight line again towards χ = 0. This phase is characterized again as a single-field slow-roll inflation driven by χ and it continues until χ becomes close to 0. In this paper, for simplicity, we call these three evolutionary periods as the first inflationary phase, the turning phase, and the second inflationary phase, respectively, although inflation may not be necessarily interrupted in the turning phase.
We perform numerical calculations to show the typical behavior of the background dynamics in the α-attractor-type double inflation. We adopt N defined by Eq. (2.2) as the temporal variable. As an example here, we set the model parameters M and λ as as while we choose the initial data as Here we have assumed the slow-roll solution from the beginning. Since it is an attractor, the numerical result does not depend on the choice ofφ I ini significantly. In Fig. 3, we first show the background trajectory of two fields, which confirms the above overview. This confirms our expectation mentioned at the end of Sec. III A that the motion of the background fields is towards φ = 0 with almost constant χ in the early stage, and after φ reaches 0, it changes the direction towards χ = 0 with keeping φ = 0. Actually, until N ∼ 65 the motion towards φ = 0 is more dominant compared with that towards χ = 0, which we have called the first inflationary phase. On the other hand, after then the motion is completely towards χ = 0, which we have called the second inflationary phase.
In Fig. 4, we plot the time evolution of φ (left) and χ (right). In the calculations, we regard that the inflation ends when the slow-roll parameter ε = 1 and assign N = 0 to that time. For the plots related with the time evolution, we use −N as the temporal variable so that the time evolves from left to right. Later we will obtain analytic solutions which approximate these phases well.
In the left panel in Fig. 5, we plot the time evolution of H, where there is a gap between the first and second inflationary phases around N ∼ 65, This is a unique characteristics of the α-attractor-type double inflation, which we will discuss analytically later. In the right panel in Fig. 5, we plot the time evolution ofΘ/H = dΘ/d(−N) around the turn between the first and second inflationary phases. In this paper, formally, we define the turning phase as the period satisfyingΘ/H > 0.05.

C. Analytic solutions based on slow-roll approximation
Here, we present analytic solutions for the first and second inflationary phases in the α-attractor-type double inflation, where the kinetic terms in Eq. (3.1) and the second derivative in Eqs. (3.3) are neglected. Then, we check the validity of the approximated solutions by comparing them with the numerical results.

Second inflationary phase
First we show the approximated solution for the second inflationary phase. Although it has been already given as a single-field α-attractor in Ref. [10], it gives hints to analyze the first inflationary phase. Soon after the second inflationary phase starts, where φ is trapped at 0, the potential can be approximated as for χ M . Therefore, as long as χ M, the first term dominates in Eq. (3.9) and together with the slow-roll approximation, the Hubble expansion rate in the second inflationary phase is given by H/M Pl ( For the example we performed numerical calculations in Sec. III B, with M and λ given by Eq. (3.7), we find H/M Pl 4.3 × 10 −6 , which is consistent with the second plateau shown in the left panel in Fig. 5. We can also obtain the analytic solution describing the evolution of χ based on the slow-roll approximation. In the second inflationary phase, the equation of motion for χ is approximately given by About N 2 , it was shown in Ref. [10] that the integration constant corresponding to this is well approximated by the number of e-folding at the end of inflation, that is, 0, in the singlefield α-attractor. Therefore, since the end of second inflationary phase coincides with the end of inflation in the α-attractortype double inflation, we can also set N 2 = 0. This is roughly explained by Eq. (3.10), with which we can see that χ becomes 0 slightly before N = 0 if N 2 = 0. In the right panel in Fig. 4, we confirm that the evolution of χ is well approximated by Eq. (3.10) with N 2 = 0. This is mainly because the duration when χ is in the region χ > M is much longer than that in the region χ < M in the second inflationary phase, as we can see from the same plot. For later use, we evaluate the time evolution of the slow-roll parameters in this phase here. Under the slow-roll approximation, which gives the following expression for the slow-roll parameters defined by the derivatives of the potential The slow-roll parameters obtained in Eqs. (3.11) and (3.12) are related with the ones defined in terms of the Hubble expansion rate as when the time evolution of φ compared with that of χ is negligible.

First inflationary phase
Next, in a similar way as the previous subsubsection, we obtain the approximated solutions for the first inflationary phase. With ϕ I ini mentioned previously, in the early stage, we can approximate the potential as for φ M , and χ M . Therefore, by considering only the first term in Eq. (3.14) and making use of the slow-roll approximation, the Hubble expansion rate in the first inflationary phase is given by H/M Pl ( For the example we performed numerical calculations in Sec. III B, with M and λ given by Eqs. (3.7) and (3.8), H/M Pl 6.1 × 10 −6 , which is consistent with the numerical results shown in the left panel in Fig. 5. The gap of the Hubble expansion rate between the first and second inflationary phases is a typical feature in the α-attractor-type double inflation. We can also discuss the analytic solution describing the evolution of ϕ I in the first inflationary phase. Together with the slow-roll approximation, the equations of motion for the scalar fields are given by  17) where N ini is N at initial time. The time evolution of φ in the first inflationary phase can be also written as Applying the same logic that fixed the integration constant N 2 appeared in Eq. (3.10) to N 1 , we can regard N 1 as the number of e-folding at the end of the first inflationary phase. In order to evaluate N 1 for given ϕ I ini , we assume that the turning phase is instantaneous and N = N 1 corresponds to not only the end of We choose the initial values as φ ini = 4.65M Pl and χ ini = 5.50M Pl . In both panels, the blue solid lines denote the numerical results, while the red dotted lines denote the analytic results based on Eqs. (3.16) for the first inflationary phase and Eq. (3.10) for the second inflationary phase. In the left panel, we plot the analytic result only for the first inflationary phase. For the analytic results, we choose the integration constants so that N ini is given by Eq. (3.19) and N 2 = 0. the first inflationary phase, but also the onset of the second inflationary phase. Under this "instantaneous turn" assumption, N 1 becomes the number of e-folding of the second inflationary phase and from Eq. (3.10), this depends on the value of χ when the second inflationary phase starts, which we denote χ 1 . Since we can also regard χ 1 as the value of χ at the end of the first inflationary phase, from Eqs. (3.16) and (3.17), χ 1 can be estimated as Here, we have assumed that the first inflationary phase ends at φ = 0 and the expression (3.16) and (3.17) are valid until that time. Making use of Eq. (3.10) with N = N 1 and N 2 = 0, and Eq. (3.18) we can express N 1 and N ini in terms of ϕ I ini as (3.20) Regardless of the fact that Eqs. (3.19) and (3.20) are based on a couple of assumptions, in Fig. 4, we confirm that the evolution of ϕ I is well approximated by Eqs. (3.16) and (3.17) with N ini given by Eqs. (3.20). This is again explained by that the duration when φ is in the region φ > M is much longer than that in the region φ < M in the first inflationary phase, as we can see in the left panel in Fig. 4. Now that we have obtained the time evolution of ϕ I in the first inflationary phase, we can express the relation between φ and χ along the background trajectory originating from ϕ I ini and Θ, introduced in Eq. (3.4) as The information given by Eqs. (3.21) and (3.22), confirmed numerically, explain our expectation mentioned at the end of Sec. III A that the motion of the background fields is towards φ = 0 with almost constant χ in the early stage in a more quantitative way.
In the first inflationary phase, after φ becomes smaller than M, the potential becomes of the form where m is defined by Eq. (2.6). With this potential, the behavior of the background trajectory around φ = 0, which we have called the turning phase, when Θ roughly changes from π to (3/2)π, is classified into two cases. In one case, φ asymptotes smoothly to 0 during the turn, while the motion towards χ = 0 becomes gradually important, which was shown in the right panel in Fig. 5. In the other case, the turn is accompanied by the oscillations in the φ direction caused by the excitation of the heavy field. As we have mentioned, since the duration when φ is in the region φ > M is much longer than the one in the region φ < M, the existence of the turning phase or whether the excitation of the heavy field occurs or not does not affect the expression (3.15) describing the background dynamics in the first inflationary phase. Regardless of this, as we will discuss in detail in Appendix B, the perturbations generated in inflation before the turn are affected if there is heavy field excitation. After this turning phase, the second inflationary phase, about which we have already discussed in the previous subsubsection, starts.

Correspondence between the initial field values and number of e-folding
Before moving to the part of perturbations, since we have obtained intuitive understanding of the integration constants N 1 , N ini given by Eqs. (3.19) and (3.20), it is helpful to discuss more on the correspondence between the initial values ϕ I ini and two e-folds N 1 , N ini . Later in this paper, when we discuss the perturbations, we assume that the pivot scale of the recent CMB observations exits the horizon scale at N = N * = 60, where * denotes that the quantity is evaluated when the scale exits the horizon scale, that is, 60 e-folds before the inflation ends. This means that for given M and λ , the region in ϕ I ini space with N ini < 60 is ruled out (the gray region in Fig. 6). For the remaining region in ϕ I ini space that satisfies N ini > 60, since the property of the perturbations exiting the horizon scale in the first inflationary phase and that in the second inflationary phase are completely different, it is useful to classify the region based on in which phase the scale of interest exits the horizon scale. In the left panel in Fig. 6, the classification is based on N 1 defined by Eq. (3.19). From the discussion in the previous subsection, we expect that if N 1 < 60, N = N * = 60 is in the first inflationary phase (the light red region), while if N 1 > 60, N = N * = 60 is in the second inflationary phase (the light blue region). Since N 1 is obtained analytically with the "instantaneous turn" assumption, in order to be more precise, in the right panel of Fig. 6 we classify the region by specifying in which phase the mode exits the horizon scale with given ϕ I ini based on the numerical calculations. This shows that even though some small purple region corresponding to near N 1 = 60 in the left panel is not clearly seen, in most part the classification based on N 1 is correct. As we have already explained, we think that the minor disagreement comes from the existence of the finite turning phase and the period when φ is in the region φ < M in the first inflationary phase. We have adopted M = √ 3M Pl here and it is worth mentioning that these plots are independent of λ as N ini and N 1 do not depend on λ .

IV. DYNAMICS OF THE PRIMORDIAL CURVATURE PERTURBATION
In this section, we calculate the primordial curvature perturbation in the α-attractor-type double inflation model that is verified by the recent CMB observations. We also show that the δ N-formalism, where we can calculate the primordial curvature perturbation analytically, can reproduce the numerical results very well.

A. Perturbations of fields
We decompose the scalar fields ϕ I into the background valuesφ I plus the perturbations δ ϕ I as In the following, we will simply write the background homogeneous values as ϕ I as long as this does not give confusions. Here, we restrict ourselves to the linear perturbation and work with the spatially-flat gauge given by where the scalar-type degrees of freedom are completely encoded in δ ϕ I owing to the Hamiltonian and momentum constraints The quadratic action for δ ϕ I can be obtained by expanding the action (2.3) to quadratic order in perturbations. In terms of [54,55] and the conformal time τ ≡ dt/a(t), it becomes where a prime denotes a derivative with respect to τ, the matrix µ IJ corresponds to the (squared) mass matrix given by Varying the action with respect to v I and moving to the Fourier space, we obtain the linear equations of motion, where k is the comoving wave number. If we choose the initial time τ 0 when the corresponding mode is on sufficiently small scales satisfying |a /a|, |µ IJ | k 2 and thus Eqs. (4.2) can be approximated as it is legitimate to choose the Bunch-Davies vacuum [56] as an initial condition of our calculation: Here e I (k) are independent unit Gaussian random variables and with to be ensemble average they satisfy

B. Curvature perturbation and isocurvature perturbation
In order to compare the prediction of linear perturbation in the α-attractor-type double inflation with the CMB observations, it is necessary to relate δ ϕ I to other perturbed quantities describing the metric perturbations, which are also the physical degrees of freedom. One of the useful quantities is R that is gauge-invariant and coincides with curvature perturbation in the comoving slice [57]. It can be shown that R is related to the perturbation of the fields in the spatially-flat gauge δ ϕ I as 3 where δ σ is the adiabatic perturbation defined based on the unit adiabatic vector n I given the first equation of (3.5) as The time derivative of Eq. (4.4) is written aṡ where Ψ is the Bardeen potential [58] that is related to B in the spatially-flat gauge given by Eq. (4.1) as and δ s is the entropy perturbation defined in terms of the unit entropic vector s I given by Eq. (3.6) as On sufficiently large scales, in the absence of the anisotropic stress, which holds in the current model, the first term in Eq. (4.6) is negligible. However, if the entropy perturbation is not suppressed and if the background trajectory changes the direction, i.e.,Θ = 0, the curvature perturbation evolves even on sufficiently large scales. In this paper, assuming that the curvature perturbation at the end of inflation is connected to the adiabatic perturbation at late time, we will follow the evolution of curvature perturbation in the inflation and estimate the amplitude at the end of inflation through the power spectrum defined by From the Planck result [7], P R at the pivot scale k * = 0.05Mpc −1 is constrained as (2.198 +0.076 −0.085 ) × 10 −9 with 1σ errors. In order to constrain model parameters more, we also calculate the spectral index of P R at the same pivot scale defined by which is constrained as (0.9655 ± 0.0062) with 1σ errors [7].
In single-field slow-roll inflation, P R and n s are expressed by where the quantities with * are evaluated when the pivot scale exits the horizon scale. As we will see later, for some cases, the estimation based on Eqs. (4.11) is valid, while for the other cases, the multi-field effects during inflation give modifications from the estimation based on Eqs. (4.11). We also calculate the tensor-to-scalar ratio r defined by where is the power spectrum of the primordial gravitational waves at the same scale [59]. The upper bound of r from the Planck result is r < 0.10 with 2σ and in single-field slow-roll inflation, r is expressed by r = 16ε * .
In this paper, in order to see how the entropy perturbation affects the dynamics of curvature perturbation during inflation, we will also follow the evolution of entropy perturbation during the inflation based on the power spectrum defined by where S is the isocurvature perturbation defined so that it becomes dimensionless and has a similar normalization factor to the curvature perturbation. One technical thing for calculating P R , P S in double inflation is that we must treat v φ and v χ as two independent stochastic variables for the modes well inside the horizon as Eqs. (4.3). To reflect this independent property, as pointed out by Ref. [29], we must perform two numerical computations. One computation with an initial condition that v φ is in the Bunch Davies vacuum and v χ = 0 giving the solutions R = R 1 and S = S 1 corresponds to e 1 (k). Another computation with an initial condition that v φ = 0 and v χ is in the Bunch-Davies vacuum giving the solutions R = R 2 and S = S 2 corresponds to e 2 (k). Then, we can obtain the correct power spectra in terms of R 1 , R 2 , S 1 , and S 2 , as

C. Typical Examples
In double inflation models, the background trajectories depends on ϕ I ini , from which the resultant inflationary observables like P R , n s , r depend on ϕ I ini . Here, with fixed M and λ given by Eqs. (3.7) and (3.8), we follow the dynamics of the comoving curvature perturbation numerically and obtain P R , n s and r at the end of inflation for the pivot scale of the recent CMB observations, with some representative ϕ I ini . In this paper, as we mentioned, we assume that the scale exits the horizon scale at N = N * = 60. To show the typical examples in double inflation, we consider the following four initial values:  Fig. 7. We also depict their background evolutions in the right panel. In the case A, the horizon exit occurs in the first inflation phase, while it happens in the second inflation phase for the cases C and D. The case B is the marginal. The left panel in Fig. 8 shows the time evolution of P R and P S in the case A. In this example, from the left panel in Fig. 7, the scale exits the horizon scale in the second inflationary phase and since the turn occurs when the scale stays at deep inside the horizon scale, the effect of the turn is suppressed compared with k 2 from Eqs. (4.2). Therefore, the resultant P R , n s and r are expected to coincide with the ones in the α-attractor based on the single-field T-model given by χ with φ = 0. Indeed, with this ϕ I ini , we obtain P R 2.213 × 10 −9 , n s 0.9667, r 0.0017. From Eqs. (4.11) and making use of the relations H 2 (λ M 4 )/(108M 2 Pl ), Eqs. (3.11) and (3.12) with N * = 60 and Eqs. (3.13), we can see that these really coincide with the ones in single-field model. These are consistent with the Planck result as we have chosen M and λ so that the resultant P R , n s and r are so in the singlefield model. The right panel in Fig. 8 shows the time evolution of P R and P S in the case B. In this example, from the left panel in Fig. 7, the scale exits the horizon scale in the turning phase and around that time, the behavior of R R is slightly modified by the mixing with S . However, since the mixing occurs during just one or two e-foldings around the term, the modification on P R , n s and r from those in single-field model is small. Actually, with this ϕ I ini , we obtain P R 2.213 × 10 −9 , n s 0.9665, r 0.0017, which are still consistent with the Planck result. Comparing these two, since the interval between the horizon exit and the turn is longer, the enhancement of the curvature perturbation is smaller in the case D. Fig. 9 shows the time evolution of P R and P S in the case C (left) and the case D (right). In these examples, from the left panel in Fig. 7, the scale exits the horizon scale in the first inflationary phase and during the turn when the scale is on superhorizon scales, the curvature perturbation is sourced by the isocurvature perturbation. At the end of inflation, in the case C, we obtain P R 3.286 × 10 −9 , n s 0.917, r 0.0017, while in the case D, we obtain P R 3.352 × 10 −9 , n s 0.974, r 0.0021. Comparing these two cases, since the isocurvature perturbation decays on superhorizon scales in the first inflationary phase, the effect of the isocurvature perturbation is more significant for the case C, where the interval between the horizon exit and the turn is shorter. This explains the fact that the deviation of n s from 1 becomes larger for the case C, where the single-field model predicts n s 0.967. On the other hand, P R is larger for the case D, which can be explained as follows. Since the first inflationary phase can be regarded as the single-field inflation with φ , Eq. (4.11) is valid when we estimate P R * . Comparing the two ϕ I ini , from Fig. 7, since the values of φ * and H * are larger for the case D, P R * is also larger for the case D. Although P R experiences the enhancement by the end of inflation caused by the effect of isocurvature perturbation, which is larger for the case C, the difference at N = N * = 60 can not be compensated. Regardless of this discussion on the amplitude, as a result, for the cases C and D, P R is too large to be consistent with the Planck result.
Next, in Fig. 10, we show the numerical results of P R and n s by changing φ ini but fixing the parameters M, λ and the initial value χ ini as before. Here, for the constraint on P R we adopt the 1σ shown below Eq. (4.9), while for the constraints on n s we adopt the 2σ constraints shown in Ref. [7]. Since we find that r is always sufficiently small to be consistent with the Planck result, we do not plot the resultant r. For 0 ≤ φ ini /M Pl ≤ 4.58, the scale exits the horizon scale in the second inflationary phase, where P R and n s are same as the one in the single-field model. Since both P R and n s are consistent with the Planck result with ϕ I ini in this region, we omit to show the result for φ ini /M Pl < 4.00 in Fig. 10. For 4.58 < φ ini /M Pl < 4.71, the scale exits the horizon scale near the turn, where the influence of the isocurvature perturbation on the curvature perturbation is expected to be small. Actually, the observational constraints of P R and n s give φ ini /M Pl < 4.71 and φ ini /M Pl < 4.70, respectively, which shows that for φ ini in the most part of this region, the resultant P R and n s are consistent with Planck result. For 4.71 ≤ φ ini /M Pl ≤ 5.50, the scale exits the horizon scale in the first inflationary phase, where because of the influence of the isocurvature perturbation and the change of the Hubble expansion rate at N = N * = 60, the modification of P R and n s from those in the single-field model is significant. For φ ini in this region, the observational constraints on P R and n s give 4.84 ≤ φ ini /M Pl and φ ini /M Pl ≥ 5.42, respectively. Although it seems that multi-field effects are so complicated that we must rely on numerical calculations for the quantitative understanding, as we will show in the next subsection, we can reproduce the results analytically with a very high accuracy.

D. δ N-formalism
In the previous subsection, we have found two regions in the initial values φ ini with fixed M , λ , and χ ini /M Pl in order to obtain a consistent result with the CMB data. In one part, we can evaluate P R and n s based on the single-field model consisting of only the second inflationary phase. In the other part, P R is enhanced compared with that in the single-field model caused by the mixing with isocurvature perturbation on super horizon scales as well as the change of H at the horizon exit. We also numerically calculated P R and n s to estimate these multi-field effects there. Here, we provide a simple way to understand these multi-field effects based on the δ Nformalism [60][61][62], which turns out to be very powerful in the α-attractor-type double inflation model. According to the δ N-formalism, on super horizon scales, R evaluated at some time t = t f coincides with the perturbation of the number of e-foldings from an initial spatially-flat hypersurface at t = t i to a final comoving hypersurface at t = t f , i.e., where H (t, x) is the inhomogeneous Hubble expansion rate. Since the local expansion on sufficiently large scales behaves like a locally homogeneous and isotropic Universe (separate universe approach) [63,64], we can calculate δ N on large scales using the homogeneous equation of motion. Suppose that we consider inflation with multiple scalar fields ϕ I and choose the initial time t i to be during inflation, when some scale exits the Hubble horizon, k i = aH| t i , and t f as some time (t f > t i ) during or after inflation when R has become constant. Then, we can regard N as a function of the configuration of the fields on the spatially-flat hypersurface at t = t i , ϕ I (t i , x) and t f , Expanding the above expression in terms of the fields' fluctuations on the spatially-flat hypersurface δ ϕ I at t = t i , we obtain Here, N ,I is the derivative of the unperturbed number of efoldings N(t f ,t i ), with respect to the unperturbed values of the fields at t i . Moving to the Fourier space and if we take ϕ I as uncorrelated stochastic variables with scale invariant spectrum of massless scalar fields in de-Sitter spacetime at t = t i much before the turn so that P ϕ I (t i ) ≡ P(t i ) = H 2 /(4π 2 )| t=t i , we can obtain where for the last equality, we have used the slow-roll approximation. Applying Eq. (4.10) to Eq. (4.14), we can also obtain the spectral index n s by the δ N-formalism and when the slowroll approximation is valid, it is given by [61] n s (t Next, we apply the δ N-formalism to the α-attractor-type double inflation model, especially for the cases with significant multi-field effects, where the scale exits the horizon scale during the first inflationary phase. From the discussion in Sec. III C, if we choose t i as t ini and t f as the end of inflation, N ini given Eqs. (3.19) and (3.20) serves as N and ϕ I ini serves as ϕ I in the δ N-formalism. Regardless of this, since the scale exits the horizon scale not at t ini but at t * , the correct choice to calculate P R and n s by the δ N-formalism is t i = t * . Therefore, for this purpose, it is necessary to express N * in terms of ϕ I * and in general multi-field inflation models we need to integrate the background equations numerically. However, in this model, for a given set of ϕ I ini , we have already specified the background trajectory in the first inflationary phase as Eqs. (3.16) and (3.17), with which we can obtain where we have used Eqs. (3.19) and (3.20) to express N ini in terms of ϕ I ini and we can substitute N * = 60. Just repeating the discussion in Sec. III C with the replacements N ini → N * and ϕ I ini → ϕ I * , we can express N * in terms of ϕ I * as with ϕ I * given by Eqs. (4.16) and (4.17). Then, by just regarding N * as N and ϕ I * as ϕ I in Eqs. (4.14) and (4.15), we can calculate P R and n s by the δ N-formalism analytically. Actually, P R at the end of inflation is given by where N * ,φ * and N * ,χ * are given by Similarly, n s at the end of inflation is given by where under the slow-roll approximation, the quantities ε * , V * ,φ * φ * and V * ,χ * χ * can be expressed as From these, as long as the mode exits the horizon scale in the first inflationary phase, based on the δ N-formalism, the resultant P R and n s can be expressed analytically in terms of M, λ and ϕ I ini . The plots in Fig. 10 show that this method can reproduce the numerical results very well for ϕ I ini in the region giving n s consistent with the Planck result.

V. OBSERVATIONAL CONSTRAINTS
With the tendency explained in Sec. IV C in mind, we impose observational constraints on the α-attractor-type double inflation model based on the Planck result. For this, we take into account the constraints on P R and n s only, as that on r does not give an additional constraint. As in the previous sections, if we assume that the initial velocity of the fields obey the slow-roll approximation, the predictions on P R and n s in this model depend on M, λ , and ϕ I ini . Here, in order to constrain them, first we fix M and specify possible λ for given Here, we impose the observational constraints for the case with M = √ 3M Pl and first we perform numerical calculations with various ϕ I ini and fixed λ given by Eqs. (3.7) and (3.8). The left panel in Fig. 11 shows the region of ϕ I ini giving n s consistent with the Planck result 4 . The region is composed of the two parts, that is, the light blue and light red ones. In the light blue region, P R can be calculated based on the singlefield model, while in the light red part, the resultant P R is modified from that in the single-field model by ϕ I ini dependent multi-field effects. For fixed χ ini , the dependence of the resultant n s on φ ini become similar to the right panel in Fig. 10. belongs to the light red part, while (C) belongs to the white region between the light blue and light red parts. This is because, as we explained in Sec. IV C, the interval between the horizon exit and the turn is longer for ϕ I ini in the light red part, the multi-field effects on n s is smaller. In the light blue region the resultant P R can be calculated based on the single-field model and is consistent with the Planck result, while in the light red part the resultant P R is modified from that in the single-field model by ϕ I ini dependent multi-field effects. (Right) Contourplot of P R in the light red region in the left panel. We normalize the resultant P R by that in the single-field model and only the dark blue region gives P R consistent with the Planck result.
Since we have only considered the constraint on n s so far, in order to specify the allowed region for this set of M and λ , we further consider the constraint on P R . As we mentioned before, when P R can be calculated based on the single-field model, λ = 2.20 × 10 −10 gives P R = 2.213 × 10 −9 , which is consistent with the Planck result. Therefore, the light blue region in the left panel in Fig. 11 is allowed. On the other hand, when the multi-field effects are important, the resultant P R depends on ϕ I ini . The right panel in Fig. 11 shows the resultant P R normalized by that in the single-field model for given ϕ I ini in the light red region in the left panel in Fig. 11 as a contourplot, which briefly shows ϕ I ini dependence of P R . We find that the maximum value is about 1.6. For fixed χ ini , the behavior of P R becomes similar to the left panel in Fig. 10. Since the Planck result admits P R up to about 3% greater than P R = 2.213 × 10 −9 , only the dark blue region in the contourplot is allowed.
So far, we have imposed observational constraints on ϕ I ini with fixed M and λ , while our original purpose here is to impose the observational constraints on ϕ I ini and λ with fixed M. One straightforward way is simply repeating numerical calculations with varying λ , which takes a considerable amount of time. Contrary to this, we make use of the fact that some ingredients of the plots in Fig. 11 are λ independent. We begin with considering the boundary corresponding to the lower bound for the sufficient total number of e-folding during inflation, which is written as the gray line in the left panel in Fig. 12. Although we have obtained this line by solving the background equation of motion numerically in order to be precise, as in the discussion in Sec. III D, the total number of e-folding is well approximated by N ini defined by Eq. (3.19) and (3.20). Since the correspondence between ϕ I ini and N ini for given M is independent of λ , we expect that the location of this boundary does not depend on λ .
Next, we consider the boundary between the part where the resultant P R coincides with that in the single-field model and the part where that is modified by the multi-field effects, which is written as the blue line in the left panel in Fig. 12. In the left panel in Fig. 11, we have obtained this by specifying the line giving n s = 0.952 based on the numerical calculation on the perturbations with fixed λ . However, in Sec. III D, we have already shown that this boundary is qualitatively understood as N 1 = 60, where N 1 is defined by Eq. (3.19). Since N 1 is also independent of λ for given M and ϕ I ini , we expect that the location of this boundary does not depend on λ , either. From the discussion in Sec. IV C, if the pivot scale of the recent CMB observations exits the horizon scale in the turning phase like the example (B) there, the multi-field effects are still sufficiently small, where the resultant n s is consistent with the Planck result. Therefore, in order to be more precise, instead of N 1 = 60, we regard the boundary as that between the regions where the scale exits the horizon scale in the turning phase and in the first inflationary phase specified by the numerical calculation on the background, which is shown as the blue line in the left panel in Fig. 12. Compared with the corresponding boundary shown in the left panel in Fig. 11, we see that this method can reproduce the result from the numerical calculations on the perturbations with a very high accuracy. Combining the above two discussions, the locations of the boundaries of the light blue region in the left panel in Fig. 11 are λ independent. Therefore, the remaining thing for ϕ I ini in the light blue region, where the resultant n s is consistent with the Planck result, is to assign λ so that the resultant P R that cincides with the one in the single-field model is consistent with the Planck result. Within the regime consistent with the Planck result, λ must be between 4.52% smaller than the "fiducial value", 2.20 × 10 −10 , and 2.76% greater than that. Let us move on to the region where the resultant P R is modified from that in the single-field model and first think the left boundary of the light red region in the left panel in Fig. 11. This line corresponds to n s = 0.952, the lower bound of n s according to the Planck result. In the left panel in Fig. 11 we have obtained it by specifying the line giving n s = 0.952 from the numerical calculation on the perturbations with fixed λ . For this, we make use of the fact that the analytic δ Nformalism works well when the mode exits the horizon scale in the first inflationary phase sufficiently before the turn, as shown in Sec. IV D. Actually, this boundary in the left panel in Fig. 12 written as a red line is obtained by specifying the line giving n s = 0.952 based on the analytic δ N-formalism. Compared with the plots in Fig. 11, we see that the result from the numerical calculations on the perturbations can be reproduced with a very high accuracy. Although we do not show explicitly the whole plot corresponding to the right panel in Fig. 11, we can also reproduce the numerical result on P R by the analytic δ N-formalism very well. In the part giving n s consistent with Plnck result, we show that the two power spectra P R obtained by two different methods differ at most 3%.
Once we confirm the validity of the analytic δ N-formalism in the light red part in the left panel in Fig. 11, we can discuss λ dependence on P R and n s with ϕ I ini in this part quantitatively. Based on the analytic δ N-formalism, n s is given by Eq. (4.22) and λ appears only through V * and V * ,IJ , which are eventually cancelled. Therefore, although we have specified the line giving n s = 0.952, which is the left boundary of the light red region in the left panel in Fig. 11, or equivalently the red line in the left panel in Fig. 12 with fixed λ , the location of this line does not depend on λ . To complete the discussion, we assign λ for given ϕ I ini in the light red part in the left panel in Fig. 11, which was shown to give n s consistent with the Planck result independent of λ so that the resultant P R is also consistent with the Planck result. For this, we can use the fact that ϕ I ini dependent enhancement factor in this part obtained in the right panel in Fig. 11 is independent of λ . This can be seen by that based on the δ N-formalism, P R is given by Eq. (4.19) and λ appears only through V * ∝ λ , while P R in the single-field model also depends on λ only through V * ∝ λ . Then, the desirable λ for given ϕ I ini is just the "fiducial value" of λ giving P R consistent with the Planck result in the single-field model, divided by ϕ I ini dependent enhancement factor obtained in the right panel in Fig. 11.
Combining all the discussions in this subsection, in the right panel in Fig. 12, we summarize the observational constraints with M = √ 3M Pl . We have specified the region giving n s consistent with the Planck result in ϕ I ini space, and assigned there λ so that the resultant P R is also consistent with the Planck result. In the plot, we set the "fiducial value" of λ giving consistent P R in the single-field model to be 2.20 × 10 −10 . Since the "fiducial value" can be 4.52% smaller and 2.76% greater in the single-field model, λ shown in the contourplot can also vary within this range. Although we have imposed the constraint on λ for each ϕ I ini with M = √ 3M Pl , practically, there is no way for us to know the initial values of the fields. Then, this result shows that the observational constraint on λ , where it is 2.10 × 10 −10 < λ < 2.26 × 10 −10 in the single field inflation, becomes weaker so that 1.31×10 −10 < λ < 2. Here we impose the observational constraints on λ and ϕ I ini with different M and to see how the constraints change. As an example with M smaller than M = √ 3M Pl considered before, we consider the case with M = M Pl , while as an example with M larger than M = √ 3M Pl , we consider the case with M = √ 6M Pl . When we change M, we choose the "fiducial value" of λ that gives P R consistent with the Planck result in the single-field model so that it does not change P R given by Eqs. Before starting the analysis, it is worth summarizing the boundaries of the parts in ϕ I ini space shown in the left panel in Fig. 12. The black line denotes just χ ini = φ ini above which we only consider from the symmetric property of the potential. The gray line denotes the lower bound for the region with sufficient total number of e-folding during inflation, which is approximately given by N ini = 65. The red and blue lines denote n s = 0.951, and we first specified them based on numerical calculations on perturbations. However, as we mentioned above, the blue line is approximately given by N 1 = 60 and the red line is obtained based on the analytic δ N-formalism. Although the above is not sufficient for the accurate specification of the gray and blue lines, it turns out that we can reproduce the results with a very high accuracy by solving the background equations of motion numerically. Here, we obtain the plots corresponding to the one in the right panel in Fig. 12 with different M in a similar form as the previous subsection, but based on the quicker way without relying on the numerical calculations on perturbations. Although we have not checked the full region in ϕ I ini giving n s consistent with Planck result presented in the following plots, along a line with some fixed χ ini , we see that the two power spectra P R obtained by the numerical calculations on the perturbations and the analytic δ N-formalism differ at most 1% for the case with M = M Pl and 6% for the case with M = √ 6M Pl . First, in order to see M dependence of the background dynamics, we show the plots corresponding to the one in the left panel in Fig. 6 in Fig. 13 with M = M Pl (left) and M = √ 6M Pl (right). From Eqs. (3.19), for given φ ini , χ ini giving N 1 = 60 is always larger than the one giving N ini = 60. On the other hand, in this paper, we avoid the situation with M M Pl so that the approximated background solution is valid, where we will discuss later in this section about the estimation of this condition. We find that as long as this condition is satisfied, both χ ini giving N 1 = 60 and the one giving N ini = 60 with fixed φ ini increase as M increases. Then, even if some ϕ I ini gives N 1 > 60 and is in the light blue region for small M, if we increase M, above some M, this ϕ I ini no longer gives N 1 > 60. But since χ ini giving N 1 = 60 is always larger than the one giving N ini = 60 for given φ ini , there is some regime M, where ϕ I ini does not give N 1 > 60, but still gives N ini > 60 and is in the light red part. If we further increase M, this ϕ I ini eventually can not give N ini > 60, either and is in the gray part.
The left panel in Fig. 14 shows the observational constraints with M = M Pl in a similar form as the one in the right panel in Fig. 12 with M = √ 3M Pl and we set the "fiducial value" of λ giving consistent P R in the single-field model to be 6.60 × 10 −10 . Since λ can be 3.99% smaller and 3.32% greater in the single-field model, λ shown in the contourplot can also vary within this range. Compared with the case with M = √ 3M Pl , the most evident difference is that in the part where the multi-field effects modify P R from that in the single-field model, there is region not consistent with the Planck result around λ ∼ 4.0 × 10 −10 . This is because in this region, the resultant n s is larger than the upper bound of the 2σ constraints in the Planck result, 0.9771. We also see that the maximum value of ϕ I ini dependent enhancement factor of P R from that in the single-field model (about 1.8) is larger than the case with M = √ 3M Pl (about 1.6 with M = √ 3) as well as the width of the region in ϕ I ini space giving n s smaller than the lower bound of the 2σ constraints in the Planck result, 0.9521 is narrower than the case with M = √ 3M Pl . Regardless of detailed discussion of ϕ I ini dependence, as in the discussion in the last part of Subsec. V A, practically, there is no way for us to know the initial values of the fields. Then, this result shows that the observational constraint on λ , where it is 6.34 × 10 −10 < λ < 6.82 × 10 −10 in the single field inflation, becomes weaker so that 3.52 × 10 −10 < λ < 6.82 × 10 −10 in this double inflation with M = M Pl .
The difference of the features of the left panel in Fig. 14 with those of the right panel in Fig. 12 can be explained by the excitation of the heavy field during the turn in multi-field inflation (see e.g. [66][67][68][69][70][71]). As we discuss in Appendix B, by analyzing the background dynamics of the turning phase, when H < (2/3)m, or equivalently M < (2 √ 6/3)M Pl , we find that the background trajectory experiences oscillations during the turn. Actually, in the left panel in Fig. 16, we show the time evolution of φ around the turn with M = M Pl . In this model, the efficiency of the heavy field excitation depends on the dynamics of φ at 0 < φ < M in the first inflationary phase, where the potential can not be approximated by neither Eq. (3.14) nor (3.23). As in the right panel in Fig. 16, we find that the efficiency of the heavy field excitation monotonously increase if M decreases. Then, if we decrease M further, we expect that the wider region is excluded by the constraint on n s and the maximum value of ϕ I ini dependent enhancement factor of P R from that in the single-field model is larger. Notice that in the left panel in Fig. 14, in the part, where the multi-field effects modify the resultant P R and n s from that in the singlefield model, we change λ from the fiducial value, which gives different m and H at the turn. However, since both H and m scale as ∝ √ λ with fixed M, the ratio m/H, parametrizing the efficiency of the heavy field excitation does not change. Therefore, we expect that the region excluded by too large n s does not change with λ giving P R consistent with the Planck result. On the other hand, the larger ϕ I ini dependent enhancement factor of P R from that in the single-field model and narrower region in ϕ I ini space giving too small n s is the typical behavior of the sharp turn accompanied with the heavy field excitation (see, [70]). The right panel in Fig. 14 shows the observational constraints with M = √ 6M Pl in a similar form as the one in the right panel in Fig. 12 with M = √ 3M Pl and we set the "fiducial value" of λ giving consistent P R in the single- field model to be 1.10 × 10 −10 . Since the "fiducial value" can be 4.86% smaller and 2.39% greater in the single-field model, λ shown in the contourplot can also vary within this range. In this case, compared with the case with M = √ 3M Pl , the maximum value of ϕ I ini dependent enhancement factor of P R from the one in the single-field model is smaller (about 1.4) as well as the width of the region in ϕ I ini space giving n s smaller than the lower bound of the 2σ constraints in the Planck result, 0.9521 is wider than the case with M = √ 3M Pl , which can be explained by the lower efficiency of the heavy field excitation. As in the case with M = √ 3M Pl there is no region giving n s larger than the upper bound of the 2σ constraints in the Planck result. Similar to the cases with M = √ 3M Pl and M = M Pl , we can consider the constraints on λ taking into account the fact that there is no way for us to know the initial values of the fields, practically. Then, this result shows that the observational constraint on λ , where it is 1.05 × 10 −10 < λ < 1.13 × 10 −10 in the single field inflation, becomes weaker so that 7.48 × 10 −11 < λ < 1.13 × 10 −10 in this double inflation with M = M Pl .
If we increase M further, although there is no heavy field excitation, we must care other effects. As we mentioned, in this paper, in order for the α-attractor-type double inflation occurs, we must avoid the situation with M M Pl . Here, we discuss the upper bound of M based on ϕ I ini = (0, χ ini ) giving N = N * = 60. When the approximated background solution of the α-attractor-type double inflaton is valid, this is given by Eqs. (3.19) and ( As a very rough order estimation, if M becomes as large as M = M χ ini, * , the condition χ M is no longer valid during the last 60 e-foldings in the inflation, which means that the inflationary background is not described by this double inflation. Furthermore, since χ anal ini, * is obtained by assuming that χ M holds during the last 60 e-foldings in the inflation, the actual upper bound of M should be smaller than M χ ini, * . On the other hand, we can obtain actual ϕ I ini = (0, χ ini ) giving N = N * = 60 by solving the background equations numerically, which we denote χ num ini, * . We find that ∆χ ini, * ≡ χ num ini, * − χ anal ini, * increases as we increase M.
We finalize this section by briefly explaining what happens if we further increase M based on ∆χ ini, * /χ num ini, * . At M = 3M Pl , we find that ∆χ ini, * /χ num ini, * 0.0042. Although this difference seems not so significant, the two power spectra P R obtained by the numerical calculations on the perturbations and the analytic δ N-formalism differ at most as much as 10% in the part corresponding to the light red part in the left panel in Fig. 11. This is because in the analytic δ N-formalism, the accurate specification of N * given by Eq. (4.18) is indispensable. Regardless of this, with this M, the main property of the α-attractor-type double inflation that the background trajectory is composed of two almost straight lines and can be specified once we fix ϕ I ini still holds. Then, with any ϕ I ini giving the horizon exit of the mode in the second inflationary phase, it occurs at χ num ini, * and because of this attractor property the resultant P R and n s are constant. How about ϕ I ini giving the horizon exit of the mode in the first inflationary phase, where the multi-field effects modify the resultant P R and n s from that in the single-field model? With such ϕ I ini , the fact that the modification depends on the interval between the horizon exit and the turn, and if the interval is shorter, the modification is larger will not change. Then if we calculate P R and n s with such ϕ I ini with fixed M and λ , we expect that similar results as shown in the plots in Fig. 11 are obtained, although the analytic δ N-formalism no longer works efficiently. If we further increase M, at M = 6M Pl , we find that ∆χ ini, * /χ num ini, * 0.041. With this M, the direction of the background trajectory starts changing much before φ = 0 and it is no longer regarded as two almost straight lines. With M above this, the two-field model no longer possesses the good property of the α-attractor-type double inflation discussed in this paper and the calculation and understanding of P R and n s become much more complicated, as in the conventional double inflation [27][28][29][30][31][32].

VI. CONCLUSIONS AND DISCUSSIONS
Recently, the α-attractor models have been very actively studied. Not only this class of models phenomenologically explain the observational results 1 − n s ∼ 1/N and r ∼ 1/N 2 , they can be derived from fundamental theories like supergravity or string theory. Since scalar fields are ubiquitous in these theories, it is natural to consider the multi-field extension of the α-attractor models. Among the multi-field inflation, the so-called double inflation composed of two minimally coupled massive scalar fields is the simplest toy model including sufficient ingredients giving rich phenomenology with multifield effects in the primordial perturbation. On the other hand, the common property of the potential in the α-attractor is that it has a plateau for the large field values and especially in the T-model, the potential is obtained by stretching the massive potential. Therefore, as a simple multi-field extension of the α-attractor, in this paper, we have considered the α-attractortype double inflation, which is composed of two minimally coupled scalar fields ϕ I = (φ , χ) and each of the field has a potential of the α-attractor-type. Although our analysis is based on the symmetric two-field T-model, where the potential is given by Eq. (2.4), we expect that most results shown in this paper are also obtained by other models as long as the asymptotic forms of the potentials are α-attractor-type.
About the background dynamics, first we showed the numerical results and explained the brief overview. If the initial values of the fields ϕ I ini satisfies φ ini < χ ini , in the early stage, the motion of the fields is towards φ = 0 with almost a straight trajectory. Then, when φ approaches to 0, the background fields are captured by the potential valley around φ = 0 and changes the direction of the motion to χ = 0. After the direction has changed completely to χ = 0, the background trajectory again becomes a straight line towards χ = 0. In this paper, briefly, we called these three phases as the first inflationary phase, the turning phase and the second inflationary phase, respectively. Then, we obtained the analytic solutions describing the first and second inflationary phases based on the slow-roll approximation. Since the second inflationary phase can be regarded as single-field α-attractor-type inflation driven solely by χ, the approximated solutions for this phase are just the conventional α-attractor-type ones. On the other hand, the approximated solutions for the first inflationary phase, given by Eqs. (3.16) and (3.17) are new. Especially, the integration constant N ini appeared there, or equivalently N 1 defined by Eq. (3.18) are shown to be regarded as the total number of e-folding during inflation and the number of efolding at the end of the first inflationary phase. As we showed in Fig 6, in terms of N ini and N 1 , we can roughly classify the region in ϕ I ini space based on whether the total number of e-folding during inflation is sufficient and if so, in which phase the pivot scale of the recent CMB observations exits the horizon scale. This classification gives intuitive understanding of the property of the the perturbations generated by this double inflation model. Then, we have considered the primordial perturbation in the α-attractor-type double inflation. Since we have assumed that the curvature perturbation at the end of inflation is connected to the adiabatic perturbation at late time, the power spectrum P R , the spectral index n s and the tensor-to-scalar ratio r constrained by the Planck result are given by Eqs. (4.9), (4.10), and (4.12), respectively. In multiple inflation, the curvature perturbation can evolve even on sufficiently large scales caused by the mixing with the isocurvature perturbation if the background trajectory changes the direction. In the αattractor-type double inflation, whether the mixing with the isocurvature perturbation affects the curvature perturbation or not depends on whether the mode corresponding to the scale of interest exits the horizon scale before the turn or not. In this model, since we obtained the analytic expression for the background dynamics, this can be seen by the similar figure as Fig 6 with appropriate M. With fixed M, λ and some sets of ϕ I ini , we performed the numerical calculations on perturbations to obtain P R , n s and r. We find that r is always sufficiently small to be consistent with the Planck result, which is the basic property of α-attractor, also holds in this double inflation model. On the other hand, since we obtained the approximated analytic solutions for the background, we can calculate P R and n s based on the δ N-formalism analytically, given by Eqs. (4.19) and (4.22), even when the multi-field effects are significant. We find that in ϕ I ini space, as long as the mode exits the horizon scale sufficiently before the turn and the resultant n s is consistent with the Planck result, the analytic method can reproduce the numerical results very well.
After that, we have imposed observational constraints on the α-attractor-type double inflation model based on the Planck result. More precisely, we imposed the observational constraints by specifying possible λ for given ϕ I ini if there exist for given M. The results with M = √ 3M Pl , M = M Pl and M = √ 6M Pl were summarized in the right panel in Fig. 11 and in the two panels in Fig. 14, respectively, which were the main results of this paper. Although we have obtained the ϕ I ini dependent constraint on λ in these plots, practically, there is no way for us to know the initial values of the fields. Based on this viewpoint, these results shows that the observational constraint on λ for given M in this double inflation becomes weaker compared with the single field inflation and this double inflation allow smaller λ . This result suggests that α attractor-type multiple inflation with more than two scalar fields could relax the constraints on the coupling constant λ and the mass scale M furthermore.
Although we can obtain these plots by numerical calculations on the perturbations, as an alternative method with which we can intuitively understand the results, the analytic δ N-formalism played a crucial role. Especially, the facts that N * in Eq. (4.18) does not depend on λ and P R depends on λ linearly as shown in Eq. (4.19) make the analysis simpler. As we can see from them, although the qualitatively they look similar, quantitatively there are some differences depending on M. With smaller M, the width of the part in ϕ I ini space giving n s smaller than the lower bound of the Planck result is narrower, while the maximum value of ϕ I ini dependent enhancement factor of P R from that in the single-field model is larger. Another evident difference is that there is a part in ϕ I ini space giving n s is larger than the upper bound of the Planck result only with M = M Pl . These were shown to be explained by the excitation of the heavy field depending on the ratio between m defined by Eq. (2.6) and H at the turn, where it is more efficient with smaller M. We also discussed the regime of M, where the main property of the α-attractor-type double inflation discussed in this paper and we found that when M becomes as large as M = 6M Pl , the background trajectory can no longer be regarded as the two almost straight lines connected with a turn and the two-field model no longer possesses the good property of the α-attractor-type double inflation.
As we have mentioned above, since most of the results in this paper approximated by analytic solutions are based on the α-attractor-type asymptotic form of the potential like Eq. (2.5), we expect that the results like sufficiently small r are not restricted to the symmetric two-field T-model, but also applicable to other class of α-attractor-type double inflation. As we have shown in Appendix C, we also think that the result that the primordial non-Gaussianity is sufficiently small to be consistent with the Planck result is the generic prediction of the α-attractor-type double inflation. On the other hand, as we have discussed in Appendix B, since we used the information of the full potential in the symmetric two-field T-model to estimate the excitation of the heavy field, in principle, by investigating the detailed spectrum of the curvature perturbation around the scale corresponding to the turn, we can distinguish models that are all classified to α-attractor-type double inflation. In this paper, as a first study of the α-attractor-type double inflation, we have concentrated on the possibility that the primordial perturbations observed by CMB are generated by this double inflation, implicitly assuming that the scale corresponding to the mode exits the horizon scale around the turn is within or at least near the regime observable by CMB. Regardless of this, we can also consider the case where the scale the exits the horizon scale around the turn is far below the CMB scales. In this case, we do not have any stringent observational constraints on the excitation of the heavy fields and the change in the Hubble expansion rate accompanied by the turn. Then, the model without imposing the symmetry between the two fields considered in Appendix A can give quantitatively interesting phenomenology. It might be interesting to consider the possibility that the primordial black holes, which accounts for the cold dark matter of the current Universe can be produced in the α-attractor-type double inflation. Finally, in this paper, we have just concentrated on the phenomenological aspect of the α-attractor-type double inflation. It is worth trying to see if this class of models are obtained out of fundamental theories. We would like to leave these topics for future works.
Appendix A: Background dynamics in α-attractor-type double inflation with an asymmetric potential In the main text, we have concentrated on the α-attractortype double inflation based on the two-field T-model with a symmetric potential satisfying M 1 = M 2 = M and λ 1 = λ 2 = λ . Here, we briefly summarize the background dynamics of a more general α-attractor-type double inflation model that is still based on the two-field T-model, but without imposing M 1 = M 2 nor λ 1 = λ 2 . For simplicity, we call the model considered in the main text as the symmetric model, while the one we discuss from now on as the asymmetric model. Since even in the asymmetric model, the potential is symmetric with φ → −φ and χ → −χ, we can restrict ourselves to the re-gion φ ≥ 0, χ ≥ 0, without loss of generality. Furthermore, for simplicity, we do not consider the cases M 1 M Pl nor M 2 M Pl , which makes us easy to analyze the background dynamics of double inflation, as we will see. Then, from the similar discussion in Sec. III, in the early stage, the potential is approximated by Actually, by considering only the first two terms in Eq. (A1) and making use of the slow-roll approximation, the Hubble expansion rate in the first inflationary phase in this asymmet- On the other hand, in this asymmetric model, we can show that these trajectories are separated by the line which means that in ϕ I ini space, the slope of the line (A6) depends on M 2 /M 1 , while its χ ini -intercept depends on ( √ λ 2 M 2 )/( √ λ 1 M 1 ). In Fig. 15, we show the plots similar to Fig. 2  In this case, the motion towards χ = 0 becomes important as φ approaches to 0 and Θ changes from π to (3/2)π smoothly.
Although it is necessary to confirm that the slow-roll approximation holds all the way to φ ∼ 0 numerically, the evolution of φ in the smooth turn with m < H is described by Eq. (B1). The case corresponding to Fig. 4 is an example giving a smooth turn.
On the other hand, if H m does not hold, the second derivative of φ in Eq. (3.3) is no longer negligible. Especially if m 2 /H 2 > 9/4, with another integration constant θ 0 , the evolution of φ is given by which describes dumped oscillations. If the integration constant φ 0 is large, or equivalently the kinetic energy of φ is sufficiently large around φ = 0, the background trajectory experiences oscillations during the turn, which can be interpreted as the excitation of the heavy field.
In the examples considered in Sec. V B, with M = M Pl , there are oscillations of φ during the turn as shown in the left panel in Fig. 16. This excitation of heavy field affects the resultant P R and n s , as discussed in Sec. VB. In the current model, the parameter dependence of the efficiency of heavy field excitation depends on the dynamics of φ at 0 < φ < M in the first inflationary phase, where the potential can not be approximated by neither Eq. (3.14) nor Eq. (3.23). Here, we numerically estimate the parameter dependence of the efficiency of the heavy field excitation. As we mentioned in Sec. V B, when we change M, we also change the "fiducial value" of λ that gives P R consistent with the Planck result in the single-field model so that λ M 2 is fixed. From Eq. (2.6), we also fix m = ( √ 2λ M)/6 6.1 × 10 −4 M Pl so that it covers the fiducial values of λ considered in Sec. V. Here, we investigate the efficiency of the heavy field excitation in terms of the maximum value of |φ | after it passes 0 by changing M with χ ini giving the turn about N N * = −60. However, it is worth mentioning that as long as χ * is much greater than M, we expect that χ ini dependence is weak, as H * √ V * /( √ 3M Pl ) is almost constant with χ * M. The right panel in Fig. 16 shows M dependence of the efficiency of the heavy field excitation. We see that by decreasing M, the heavy field excitation occurs around M ∼ 10 0.2 M Pl , and the efficiency monotonously increases when M decreases. Notice that M giving m 2 /H 2 = 9/4 based on H given by the slow-roll approximation (B1) gives (2 √ 6)/3M Pl , which is very close to 10 0.2 M Pl . Appendix C: Primordial non-Gaussianity in α-attractor-type double inflation In the main text, we imposed the observational constraints on λ and ϕ I ini for a given M in the α-attractor-type double inflation model based on the Planck result of P R and n s . On the other hand, for the region in ϕ I ini space where the multi-field effects modify the resultant P R from that in the single-field model, while the resultant n s is still consistent with the Planck result, we can obtain P R and n s analytically based on the δ N-formalism with a high accuracy. Since we can easily calculate the primordial bispectrum, which includes the leading signal of non-Gaussianity, based on the δ N-formalism, here we calculate it to see if additional constraints are obtained by the primordial non-Gaussianity on this model or not.
By extending the relation between R and δ ϕ I in the δ N-formalism given by Eq. (4.13), we obtain On the other hand, the bispectrum of the curvature perturbation B R in the Fourier space is defined by where the left hand side of Eq. (C2) can be evaluated using Eq. (C1) as Here, denotes the convolution and perms. denotes the remaining two other permutations. In Eq. (C3), roughly speaking, the first term in the right hand side denotes the contribution from the intrinsic non-Gaussianity of the field perturbations, giving nonlocal-type bispectrum, while the second term denotes the contribution from the nonlinear dynamics of the background field on super horizon scales giving the local-type bispectrum [65].
Observational constraints on the primordial bispectrum are given by the nonlinear parameter f NL defined by [72] Since the kinetic terms in this model are canonical and the former contribution is expected to be small [72], by concentrating on the local-type primordial bispectrum, we obtain the following expression of f loc NL in the δ N-formalism f loc NL = (N * ,φ * ) 2 N * ,φ * φ * + (N * ,χ * ) 2 N * ,χ * χ * ((N * ,φ * ) 2 + (N * ,χ * ) 2 ) 2 , where N * ,φ * and N * ,χ * are given by Eq. (4.20) and N * ,φ * φ * and N * ,χ * χ * are given by We find that for the three cases with M = M Pl , M = √ 3M Pl and M = √ 6M Pl we considered in Sec. V, in the part in ϕ I ini space, f loc NL does not depends on the parameter M or the initial values ϕ I ini so much and takes the value 0.013 < f loc NL < 0.016, which is consistent with the Planck result [8]. This result shows that the primordial non-Gaussianity cannot give additional constraints on the α-attractor-type double inflation other than the ones considered in the main text.