The Hidden Flat Like Universe: Starobinsky-like inflation induced by f(T) gravity

We study a single fluid component in a flat like universe (FLU) governed by $f(T)$ gravity theories, where $T$ is the teleparallel torsion scalar. The FLU model, regardless the value of the spatial curvature $k$, identifies a special class of $f(T)$ gravity theories. Remarkably, the FLU $f(T)$ gravity does not reduce to teleparallel gravity theory. In large Hubble spacetime the theory is consistent with the inflationary universe scenario and respects the conservation principle. The equation of state (EoS) evolves similarly in all models $k=0, \pm 1$. We study the case when the torsion tensor is made of a scalar field, which enables to derive a quintessence potential from the obtained $f(T)$ gravity theory. The potential produces Starobinsky-like model naturally without using a conformal transformation, with higher orders continuously interpolate between Starobinsky and quadratic inflation models. The slow-roll analysis shows double solutions so that for a single value of the scalar tilt (spectral index) $n_{s}$ the theory can predict double tensor-to-scalar ratios $r$ of $E$-mode and $B$-mode polarizations.


I. INTRODUCTION
The General Relativity (GR) theory has succeeded to confront astrophysical observations for long time. It predicts perfectly the perihelion shift of Mercury, time delay in the solar system, even in strong field regimes such as binary pulsars it amazingly predicts the orbit shrink due to gravitational radiation by these systems. Also, it matches the cosmological observations providing an expanding universe but decelerating. However, later astronomical observations of hight-redshift Type Ia supernovae have provided an evidence for a late acceleration phase of an expanding universe [21]. This implies the existence of a repulsive gravity which the GR fails to provide. The repulsive gravity has been imposed in GR by recalling the cosmological constant back to life. Unfortunately, there is no evidence supporting this assumption. Some authors showed that formulation of the GR using the Levi-Civita connection (i.e. Riemannian geometry) gives only one side of gravity, the attractive one. Nevertheless, other geometries with higher qualities may give the other one, the repulsive side [24]. It is known that Levi-Civita connection plays the role of the "displacement field " in the GR, so we expect different qualities when using another connection; Weitzenböck; the teleparallel connection [25]. Actually, this geometry provides an alternative tool for deeper studies of the gravity. In what follows we give the structure of Weitzenböck spacetime. This space is described by as a pair (M, h i ), where M is an n-dimensional smooth manifold and h i (i = 1, · · · , n) are n independent vector fields defined globally on M . The vector fields h i are called the parallelization vector fields. In the four dimensional manifold the paralleliza- * waleed.elhanafy@bue.edu.eg † nashed@bue.edu.eg tion vector fields are called the tetrad field. They are characterized by where ∂ ν = ∂ ∂x ν and Γ µ λν define the nonsymmetric affine connection [26].
Equation (1) implies the metricity condition. Also, the curvature tensor of the connection (2) vanishes identically. The metric tensor g µν is defined by where η ij = (+, −, −, −) is the metric of Minkowski spacetime. We note that, the tetrad field h i µ determines a unique metric g µν , while the inverse is incorrect. The torsion T and the contortion K tensor fields are We next define the torsion scalar of the teleparallel equivalent to general relativity (TEGR) as where the tensor S α µν is defined as which is skew symmetric in the last two indices. Similar to the f (R) theory one can defines the action of f (T ) theory as where M Pl is reduced Planck mass, which is related to the gravitational constant G by M Pl = √ 8πG/c 2 . Assuming the units in which G = c = = 1, in the above equation h = √ −g = det (h a µ ), Φ A are the matter fields. The variation of (7) with respect to the field h i µ requires the following field equations [3] where and T µ ν is the energy-momentum tensor.
Recent applications of the f (T ) in cosmology show an interesting results [16]. For example, avoiding the big bang singularity by presenting a bouncing solution [6,7]. Also, f (T ) cosmology provides an alternative tool to study inflationary models [1, 2,10,12,18]. Moreover, the problems in f (T ) theories are discussed [11,19].
The work is organized as follows: In Section II, We use the modified version of the Friedmann equations according to f (T ) treatment to describe a flat like universe. This treatment does not restrict the space to be globally flat, i.e. k = 0. We derive a pair of a scale factor a(t) and a function f (T ) as an inverse power series of the square root of teleparallel torsion scalar. In Section III, we discuss the physical consequences of the obtained results. We show that the theory indicates an inflationary universe at early time. The study of the EoS of a single-fluid component universe shows a similar behaviour in the k = 0, ±1 models. Also, we examine the validity of the conservation principle of this theory. In Section IV, we assume the case when the torsion is made of a scalar field. We show that potential is a generalized form of Starobinsky model. However, it contentiously interpolates between Starobinsky and polynomial chaotic inflation models. We classify each term in the potential corresponds to the order of expansion of f (T ). The classification shows that the lowest possible potential contains an inverse square-law potential suggested to represent a quintessence field. We study possible solutions of the slow-roll parameters of this theory. In Section VI, we discuss testable predictions of the theory according to the recent released Planck and BICEP2 data [4,20].

II. COSMOLOGICAL MODIFICATIONS OF f (T )
We first assume that our universe is an isotropic and homogeneous, which directly gives rise to the tetrad given by Robertson [22]. This can be written in spherical polar coordinate (t, r, θ, φ) as follows: where a(t) is the scale factor, L 1 = 4 + kr 2 and L 2 = 4 − kr 2 . The tetrad (9) has the same metric as FRW metric.

A. Modified Friedmann equations
Applying the f (T ) field equations (8) to the FRW universe (9), assuming an isotropic perfect fluid so the energy-momentum tensor is T µ ν = diag(ρ, −p, −p, −p). The f (T ) field equations (8) read where i = 1, 2, 3 and H(:=ȧ/a) is the Hubble parameter, the dot expresses the derivative with respect to time t. Equations (10) and (11) are the modified Friedmann equations in the f (T )-gravity in its generalized form. Substituting from the vierbein (9) into (5), we get the torsion scalar B. Searching for a flat-like universe We find that the first bracket in (11) is identical to (10) but opposite which provides a good chance to hunt a cosmological constant matter like perfectly, while extra terms of (11) enable evolution away from the cosmological constant. By careful look to (11) we find that these extra quantities can be split into a free k term and k de-pendent term 1 . The inflationary models imply a nearly vanishing curvature density parameter Ω k := − k a 2 H 2 . The very small value of Ω k could be due to the large value of Hubble parameter H at early time. But this does not imply a global flatness by taking k = 0 universe model. In our exploration here searching for a flat like universe, without imposing a vanishing k, we would examine the vanishing of the third bracket of (11) Note that this treatment cannot be applied in the TEGR theory, i.e. f (T ) = T , as the coefficient of the 4k/a 2 in (11) will be a unit. So this would not provide us with the a condition similar to (13) to examine a flat like universe.
In this case one is obliged to assume a global spatial flatness by putting k = 0 to study flat universe model. Thus we expect that this treatment enables to study higher order gravity beyond the TEGR or GR scale in a flat like universe. As f (T ) in FRW spacetime is a function of time f (T → t), one easily can show that Substituting from (14) into (13), then solve to f (T → t) we get: where Λ and λ are integration constants. We later show that the constant Λ can be perfectly interpreted as cosmological constant. In order to reduce the dependence on k, we follow the same manner when dealing with the quantities combined with k. So, in the above equation, we take by solving to a(t) we get the scale factor where a 0 is an arbitrary constant of integration, with an initial condition H 0 = H(t 0 ). One should mention that the scale factor is independent of the choice of k. On the other hand, we do not expect or accept that the three world models k = 0, ±1 be completely coincide, then the function f (T → t) should depend on the choice of k in its final form. Substituting from (17) into (5) we get the torsion scalar 1 one should note that f (T ) may contain k dependent quantities.
Substituting from (17) into (15) we have f (T ) as a function of t as where c n are constant coefficients. The zeroth term of (19) derives the Friedmann equations (10) and (11) to produce a cosmological constant like DE, so fix the value c 0 to the cosmological constant. The time dependent series produces the evolution away from the cosmological constant. Using (18), we can reexpress (19) in terms the torsion scalar as inverse power-law of T as where α n are known constant coefficients.

III. COSMIC EVOLUTION
The FRW spacetime that is governed by f (T ) gravity is determined by the scale factor a(t) and the f (T → t).
As we mentioned before the scale factor (17) is independent of k, while (19) depend on k as it should be. This allows to study different evolution scenarios according to the choice of k, but similar. We next examine the cosmic evolution assuming a single fluid component that are given by (10) and (11).

A. The large Hubble-spacetime
We study the large H-regime, i.e. at early time universe. The Hubble parameter corresponds to the scale factor (17) is given by where the modified Friedmann equations are symmetric under H → −H. We easily find that the universe is always accelerating as the deceleration parameter q := aä a 2 = −5/3. Nevertheless, the matter density (10) of this theory in the large H-spacetime is given as ρ → 3 , which agrees with the predictions of the vacuum density with a correction term. This is consistent with the inflationary universe scenario at this stage.

B. Single-fluid equation of state
In this f (T ) theory we assume only a single fluid component to describe the matter of the universe. As we mentioned in the subsection II B, there is a good chance to hunt negative pressure matter in the f (T ) framework with expected deviation from the cosmological constant behaviour. We now examine the nature and possible evolutions of this single-fluid component. Using (10) and (11), we get a time dependent EoS parameter (ω := p/ρ). The dynamical behaviour shows a similar asymptotic behaviour regardless that the universe is flat or not. It is only affected by the order of expansion of (19). We find that the EoS asymptotic behaviour increases by a quantized value 2 9 as {−1, − 7 9 , − 5 9 , ...} as the expansion order of (19) increases. We study the EoS evolution when the f (T → t) is up to O(t 3 ) during different phases in the three models k = 0, ±1. Although, we used a single-fluid component, its dynamical evolution allows the EoS parameter initially to start from ω < −1 (phantom). The evolution of the EoS shows almost the same behaviour in the three models, see plots of Figure 1. Then it evolves to pressureless cold dark matter (CDM) (ω = 0), radiation (ω = 1/3) and possibly stiff-matter (ω = 1), while it shows a quintessence fate as ω → −1/3 as t → ∞ in all models.

C. Conservative universe
In the case of the cosmological constant (i.e. EoS parameter is ω = −1) that is required to describe the accelerating universe, one cannot get an evolutionary scenario for this dark component. In addition, its continuity equationρ + 3H(ρ + p) = 0, shows thatρ = 0, i.e. density is constant. Nevertheless, the density should decreases as universe expands. This conflict implies a continuous creation of matter in order to keep the density of the expanding universe constant. This case is similar to the steady state cosmology which breaks the conservation principle. In this f (T ) theory we used only one component to describe the evolution of the universe, with a negative pressure matter dominating most of different epochs during evolution. Moreover, we get a conservation universe. This can be obtained by substituting from (17) and (19) into (10) and (11), which shows that the continuity equation is always verified.
As we showed in Subsection II A that this theory is consistent with inflationary scenario universe. At this early time the scalar fields play the key role to interpret the inflation stage. On the other hand, the torsion plays the main role in the teleparallel geometry. We next consider the role of the torsion as added value quality of the spacetime when formed by a scalar field and its role at the early universe time.

IV. TORSION POTENTIAL
In order to get the idea of the torsion potential, we first discuss the physical meaning of the connection coefficients as the displacement field. So we use (4) to reexpress the Weitzenböck connection (2) as By careful look to the above expression of the new displacement field, it consists of two terms. The first is the Levi-Civita connection which consists of the gravitational potential (metric coefficients, g µν ) and its first derivatives w.r.t. the coordinates. Where the second term is the contortion (21) which consists of the tetrad vector fields and its first derivatives w.r.t. the coordinates. In this sense we find the first term contributes to the displacement field as the usual attractive force of gravity, while the second term contributes as a repulsive force. Now we can see how teleparallel geometry adds a new quality (torsion or contortion) to the spacetime allowing repulsive side of gravity to showup.

A. Torsion potential of a scalar field
We consider here the physical approach to form the torsion from a scalar field ϕ(x). In the view of the above discussion, we can treat the contortion in the Weitzenböck connection (21) as a force. Accordingly, it is required to construct the contortion (or torsion) from a tensor and it first-order derivatives. Then, this tensor now plays the role of the potential of the torsion. We follow the approach that has been purposed by [27], by introducing sixteen fields t µ i that are called "torsion potential ". These fields form a quadruplet basis vectors, so we write the following linear transformation: the torsion potential t µ i and its inverse are satisfying the conditions: Finally, this enables to express the torsion as [27] T α The torsion potential t µ i can be reformed by a physical scalar, vector or tensor fields. This may have a great interest in physical applications. Now we take the case when the torsion potential is formed by a scalar field ϕ(x) with a Minkowskian background, then we take where ϕ is a non-vanishing scalar field. Then the torsion is expressed as Using the above equations and (6), the teleparallel torsion scalar (5) can be written in terms of the scalar field ϕ as where ϕ ,µ = η µα ϕ ,α . The above treatment shows that the torsion acquires dynamical properties and it propagates through space. One should mention here that same results can be achieved by applying the so called Einstein λ-transformations of the connection coefficients [8] Γ α µν = Γ α µν − δ α ν ϕ ,µ .
Noting that the above transformation can be obtained directly by the conformal transformation of the tetrad fields h µ i = e ϕ h µ i ,h i µ = e −ϕ h i µ [9]. Actually this approach, indeed, has a geometrical framework as well. Where the connection is a semi-symmetric one, this case has been studied by many authors [17], so it can be written as Accordingly, the semi-symmetric torsion and the contortion that are associated to the connection (27) takes the form (23) and (24), respectively.

B. Generalized Starobinsky potential by f (T )
In the following treatment we take the flat model case k = 0. Also, we define the constant τ 0 := t 0 + 3 2H0 . Using (18), (25) and solve to ϕ we write where ϕ 0 is the constant of integration. The above equations allows to express the time t mathematically in terms of the scalar field ϕ. So all the dynamical expressions can be expressed in terms of the scalar field.
The relation between the scalar fields T and ϕ has to be investigated. Using (18) and (28) we can express the torsion scalar field T in terms of the scalar field ϕ as This relation is the source code to reproduce Starobinsky universe naturally without imposing a conformal transformation. We might write T (ϕ) = −ξΩ 2 (ϕ), the new scalar field Ω(ϕ) := e ± √ 2/3ϕ , where the constant ϕ 0 can be always absorbed in the coefficient ξ. Now we may recognize that the scalar fields T and Ω are related by Ω(T ) ∝ √ −T and both act as a canonical scalar fields at strong coupling conditions. Nevertheless, the scalar field ϕ is related to Ω as ϕ = ± 3 2 log Ω(T ), so it acts as a canonical normalized scalar field. Now we study the dynamics of the scalar filed ϕ by considering the lagrangian density of a homogeneous scalar field, i.e. that depends only on time, which can be expressed as where the coupling constant of the torsion scalar field is too small, the first term in the above equation represents the kinetic term of the torsion scalar field, as usual, while V (ϕ) represents its potential energy. Using Noether's theorem gives the energy momentum tensor as which reads the scalar field pressure as By differentiating (28) w.r.t. time, we can write the kinetic term in the above equation aṡ Also, by using (19) and (17), the pressure (11) can be reexpressed in terms of the scalar field ϕ as where β n are known constant coefficients. Substituting from (31) and (32) into (30), we evaluate the potential of the scalar field ϕ as where V 0 = Λ 16π + 3 4 e 2 √ 2/3(ϕ−ϕ0) . According the value of λ and the order of expansion, the above equation may give a Starobinsky-like inflation model [23], where the potential blows up at ϕ < 0 and inflation can occur only at ϕ > 0. In contrast, it gives a quadratic or Higgs-like inflation model where the potential allows inflation to occur at both ϕ < 0 and ϕ > 0, see plots of Figure 2.

C. Classified torsion scalar potentials
In the previous subsection we applied a new technique to induce the scalar potential (33) by the f (T ) gravity (19). In this theory we obtained a power series potential of e − √ 2/3ϕ where ϕ represents the inflaton field. So the theory may cover different classes of inflationary models. We next examine the potentials correspond to the order of the expansion of (19). In this way, and with the help of the evaluated EoS induced by the fluid (10) and (11), we might be able to compare these potentials to the already known inflaton potentials. The triple (19), EoS and (33) would enable us to give a physical classification of these potentials.

The V0-model
The modified Friedmann equations (10) and (11) of the f (T ) gravity theories pay attention to the choice of f (T ) = const. as it acts perfectly as the cosmological constant. This is exactly the case here when assume the zeroth order of (19) solution. We first evaluate the above mentioned triple as where −∞ < ϕ < ∞. It is clear the f 0 (T ) acts perfectly as cosmological constant dark energy (ΛDE) with EoS ω = −1. Also, it worths to mention that the V 0 is the lowest possible potential in this theory that produces a quasi de Sitter model. The first term in (36) is the vacuum energy density (cosmological constant) of the false vacuum spacetime, while the second term associated to the vacuum potential represents a phase transition potential dragging the universe away from false vacuum (ϕ = 0 state) to a true vacuum (ϕ = 0 state) at its minimum effective potential, see Figure 2 (n = 0). However, the EoS still ω 0 = −1 during the whole stage. Now, we investigate the phase transition potential in (36). Hence we can write V 0 ∝ 1/Ω 2 which gives an inverse square law potential. This type potentials represent the original class of quintessence fields. Indeed, the later expression shows a minimum effective potential only at ϕ > 0 or ϕ < 0 with a potential barrier at Ω = 0, while (36) allows the full range −∞ < ϕ < ∞. But as the inflation occurs only in a single plateau, then the two expression are identical at ϕ > 0.

The V1-model
We take the expansion up to the first order giving the triple The f 1 (T ) ∝ 1 √ −T which is usually used to identify the CDM. In this order the EoS increases by 2/9 than ΛDE. The potential V 1 reduces to V 0 where λ = 0. However, the nonvanishing values of λ show an interesting pattern, there plots appear in Figure 2 (n = 1), they predict the similar behaviour at ϕ > 0 where the potentials are nearly flat at the false vacuum (ϕ = 0) slowly rolls to its effective minimum at the true vacuum (ϕ = 0). At ϕ < 0, the the potentials predict different behaviours according to the value of λ. The very small negative or positive values of λ the potentials are nearly flat. The large positive λ produces Starobinsky-like model where the potential blows up exponentially. Nevertheless, the large negative λ turns the potential to a quadratic-like model.

The V2-model
With higher order expansion we have these triple This case still in the quintessence range where ω 2 → −5/9. The potential V 2 reduces to V 0 when λ = 0. The non-vanishing values of λ produce almost the same behaviour at ϕ > 0 while it produces different behaviours at ϕ < 0. In contrast to the V 1 , the large negative λ allows Starobinsky-like model to dominate the ϕ < 0 epoch while the large positive λ turns the potential to a quadratic-like model, see Figure 2 (n = 2), at ϕ < 0. Its plot shows two different minima allowing inflation in both |ϕ| > 0. The behaviour is similar to Higgs potential but with no reflection symmetry about the line ϕ = ϕ 0 .

The V3-model
We study one more case where the triple are given by The evolution of the EoS for this case has been studied in this work in Subsection III B. The interest in this case is motivated by the study of the so-called "tracker field" when assuming the inflation ends when ω 3 → −1/3 which is the case here [28], recall Figure 1. The potential V 3 reduces to V 0 when λ vanishes, its nonvanshing values show general behaviours similar to V 1 . Whereas, Figure 2 (n = 3) shows that, in particular for large negative λ, the false vacuum is separated by a broad barrier. However, the top of the barrier is quite flat. The decay of the false vacuum is followed by slow-roll inflation allowing a tunneling event from the high energy false vacuum. We will see later how this type of potentials affect the slow roll solution. We find this model fulfills the requirements of [5] to perform well fitting both Planck and BICEP2 data.

The V n≥4 -models
For the V 4 -model we find it similar to V 2 -model but with strong flat plateau at ϕ > 0 of the false vacuum. At ϕ < 0 the symmetry of negative and positive large values of λ no longer valid. The V n>4 -models, the symmetry of large ±λ holds again but they alter their roles, see plots of Figure 2 (n = 5, 6, 7). Moreover, the asymptotic EoS goes to increase by 2/9 with a radiation limit for ω 6 while ω 9 gives stiff matter all higher order expansions give unknown matter with ω > 1. The overall, picture shows similar behaviours for all the models at ϕ > 0 while they interpolate between Starobinsky and polynomial potentials at ϕ < 0 with different details as mentioned above. It is known that the PICEP2 data excludes the small tensor-to-scalar ratio models such as Starobinsky model [4]. In contrast, the Planck data restricts the tensor-toscalar ratio to be small so it excludes large inflationary models such as even polynomials (i.e. Higgs model) [20]. In this theory we find that the inflationary potential interpolate between these two different classes of inflation. These results lead to investigate the inflationary parameters within this theory.

V. SINGLE-SCALAR FIELD WITH DOUBLE SLOWLY-ROLLING SOLUTIONS
We use the standard slow-roll approximation technique to analyze the solution. Assuming that the inflation epoch is dominated by the scalar field potential only. This defines two slow-roll parameters as The prime denotes the derivative with respect to (w.r.t) the scalar field ϕ. Consequently, the slow-roll inflation is valid where ǫ ≪ 1 and |η| ≪ 1. In the V 0 case, we evaluate the slow roll parameters We can find that the slow roll parameters are related as where this relation is not only independent of the values of Λ and ϕ 0 but also it allows a vanishing cosmological constant to exist without affecting the generality of the relation (49). The slow-roll parameters can be tested when define the two cosmological parameters of the inflation: The spectral index n s := 1 − 6ǫ + 2η and the tensor-to-scalar ratio r := 16ǫ of the power spectrum. The observations of Planck and BICEP2 agree on the value of the spectral index n s ∼ 0.963. So, in this model, the parameter reads n s = 1 + 2η 0 − 9πη 2 0 . This quadratic equation in η 0 has two solutions. The first has a negative value of η 0 ∼ −0.014 which leads to a small ǫ 0 ∼ 2.85 × 10 −4 . The second solution of η 0 has a positive value of ∼ 0.086 which gives ǫ 0 ∼ 1.39 × 10 −3 . In order to interpret this result, we again refers to [5], it is suggested that when a negative value of η 0 is observed near the peak of ϕ, it would need to be offset by a positive value of η 0 at some later time over a comparable field range in order to get ǫ 0 to be small again during the period of observable inflation. Now we discuss how the tensor-to-scalar ratio and the spectral index of the power spectrum are affected by these double slow roll solutions. Surely both negative and positive values of η 0 give the same spectral index n s ∼ 0.963. Nevertheless, we can get two simultaneous tensor-to-scalar ratios: the first is too small r ∼ 0.005 and another larger value of ∼ 0.022. Similar analysis in the higher order potential model leads to same proportionality ǫ n ∝ η 2 n where the proportionality constant depends on the value of the cosmological constant and the value of ϕ 0 . When the ratio ǫ n /η 2 n ∼ 1 at the end of the inflation phase, the parameters (n s , r) approach two solutions (0.963, 0.004) and (0.963, 0.369). Finally, we find that the slow roll treatment of this theory can perform both Planck and BICEP2 data. Recent treatments of finding double attractors in the inflationary models [13][14][15] could be useful to analyze the obtained results.

VI. DISCUSSIONS AND FINAL REMARKS
The study of the modified Friedmann equations of f (T ) gravity has shown a built-in negative pressure matter, this can be explained due to repulsive force of the contortion in the displacement field (Weitzenböck connection). The negative pressure has initiated the idea of an inflationary epoch at early time universe. The inflationary models imply a nearly vanishing curvature density parameter Ω k = − k a 2 H 2 . The very small value of Ω k could be due to the large value of Hubble parameter H at early time. But this does not imply a global flatness by taking k = 0 universe model. We have studied an interesting case we call it flat like universe. We have shown that this case cannot be covered by the TEGR or GR theories. The flat like universe assumption enables us to derive compatible pair of a scale factor and f (T ) gravity. The obtained solution of f (T ) has been expressed as a power series of time or equivalently as inverse power series of √ −T . We have shown that the theory is consistent with inflationary scenario of the universe. We have taken the physical approach when the torsion potential is made of a scalar field ϕ. This enabled us to study the early universe in terms of a single self-interacting scalar field. The theory produces Starobinsky model naturally with no need to apply a conformal transformation. We have used the triple of f n (T ), EoS ω n and potential V n . Accordingly, We gave a complete picture of the universe due to different n-th orders expansion. We found that the n = 0 model produces a quasi de Sitter universe. The higher order models predict similar behaviour at ϕ > 0, while in the ϕ < 0 limit the models continuously interpolate between Starobinsky and quadratic potential models. We have studied the slow-roll solution of the theory. The n = 0 model shows double slow roll parameters, which produces two values of the tensor-to-scalar ratio r of a single value of the spectral index n s of the power spectrum. At the end of the inflation epoch the theory predicts two values of r ∼ 0.004 and ∼ 0.369 for the same value of n s ∼ 0.963. The theory uses a single-scalar field model but it is capable to predict small r of Planck as well as a large value of BICEP2.