COSMOS-E'-Gtachyon from String Theory

Eur. Phys. J. C (2016) 76:278 DOI 10.1140/epjc/s10052-016-4072-2

Regular Article - Theoretical Physics

COSMOS-e’-GTachyon from string theory

Sayantan Choudhury1,a, Sudhakar Panda2,3 1 Department of Theoretical Physics, Tata Institute of Fundamental Research, Colaba, Mumbai 400005, India 2 Institute of Physics, Sachivalaya Marg, Bhubaneswar, Odisha 751005, India 3 Harish-Chandra Research Institute, Allahabad 211019, India

Received: 23 December 2015 / Accepted: 10 April 2016 / Published online: 19 May 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com

Abstract In this article, our prime objective is to study the expressions for inflationary observables in the context of the inflationary paradigm in the context of the generalized multi-field tachyons. tachyon (GTachyon) living on the world volume of a non- BPS string theory. The tachyon action is considered here Contents is modified compared to the original action. One can quantify the amount of the modification via a power q instead of 1 Introduction ...... 2 1/2 in the effective action. Using this set-up we study infla- 2 GTachyon in string theory ...... 4 tion by various types of tachyonic potentials, using which 3 Variants of tachyonic models in string theory .... 7 we constrain the index q within, 1/2 < q < 2, and a spe- 3.1 Model I: inverse cosh potential ...... 9 cific combination (∝ α M4/g ) of the Regge slope α,the s s 3.2 Model II: logarithmic potential ...... 10 string coupling constant g and the mass scale of tachyon s 3.3 Model III: exponential potential Type-I ..... 11 M , from the recent Planck 2015 and Planck+BICEP2/Keck s 3.4 Model IV: exponential potential Type-II (Gaus- Array joint data. We explicitly study the inflationary con- sian) ...... 13 sequences from single field, assisted field and multi-field 3.5 Model V: inverse power-law potential ..... 14 tachyon set-ups. Specifically for the single field and assisted 4 Cosmological dynamics from GTachyon ...... 15 field cases we derive the results in the quasi-de Sitter back- 4.1 Unperturbed evolution ...... 15 ground in which we will utilize the details of cosmological 4.2 Dynamical solution for various phases ..... 21 perturbations and quantum fluctuations. Also we derive the 5 Inflationary paradigm from GTachyon ...... 29 expressions for all inflationary observables using any arbi- 5.1 Computation for single field inflation ...... 30 trary vacuum and the Bunch–Davies vacuum. For the single 5.1.1 Condition for inflation ...... 30 field and the assisted field cases we derive the inflationary 5.1.2 Analysis using slow-roll formalism ... 32 flow equations, new sets of consistency relations. Also we 5.1.3 Basics of tachyonic perturbations .... 37 derive the field excursion formula for the tachyon, which shows that assisted inflation is on the safe side compared 5.1.4 Computation of scalar power spectrum .. 40 to the single field case to validate the effective field theory 5.1.5 Computation of tensor power spectrum .46 framework. Further we study the features of the CMB angu- 5.1.6 Modified consistency relations ...... 51 lar power spectrum from TT, TE and EE correlations from 5.1.7 Field excursion for tachyon ...... 55 scalar fluctuations within the allowed range of q for each 5.1.8 Semi-analytical study and cosmological of the potentials from the single field set-up. We also put parameter estimation ...... 61 constraints from the temperature anisotropy and polarization 5.1.9 Analyzing CMB power spectrum ..... 81 spectra, which shows that our analysis is consistent with the 5.2 Computation for assisted inflation ...... 84 Planck 2015 data. Finally, using the δN formalism we derive 5.2.1 Condition for inflation ...... 84 5.2.2 Analysis using slow-roll formalism ... 92 5.2.3 Basics of tachyonic perturbations .... 96 Sayantan Choudhury: Presently working as a Visiting (Post-Doctoral) 5.2.4 Computation of scalar power spectrum .. 99 fellow at DTP, TIFR, Mumbai On lien from Harish-Chandra Research Institute, Allahabad, India. 5.2.5 Computation of tensor power spectrum . 101 5.2.6 Modified consistency relations ...... 102 a e-mails: [email protected]; [email protected] 5.2.7 Field excursion for tachyon ...... 105 123 278 Page 2 of 130 Eur. Phys. J. C (2016) 76 :278

5.2.8 Semi-analytical study and cosmological quences have been studied in the following topics over many parameter estimation ...... 110 years: 5.3 Computation for the multi-field inflation ....111 5.3.1 Condition for inflation ...... 112 1. Inflationary paradigm [75–86,92], 5.3.2 Analysis using slow-roll formalism ...113 2. Primordial non-Gaussianity and CMB aspects [96–104], 5.3.3 The δN formalism for Multi tachyons . . 116 3. Reheating scenario and particle creation [105,106], 5.3.4 Computation of scalar power spectrum . . 118 4. Late time cosmic acceleration (dark energy) [107–110]. 5.3.5 Computation of tensor power spectrum . 122 5.3.6 Analytical study for the multi-field model 123 In this work, we introduce the most generalized version of 6 Conclusion ...... 125 the tachyon effective action in which we are interested to References ...... 127 study the cosmological consequences from the non-canonical higher-dimensional effective field theory operators, originating from string theory. Technically we rename such a non- 1 Introduction canonical field a generalized tachyon (GTachyon), which name we will frequently use throughout the rest of the paper. 2 The primordial inflationary paradigm is a very sublime The prime motivation of writing this paper is the following: thought aimed to explain various aspects of the early universe, in which perturbations occur and matter is created. • To update the present status of non-canonical interaction For recent developments see Refs. [1–10]. The success of pri- for stringy tachyon field appearing within the framework mordial inflation can be gauged by the current observations of string theory after releasing the Planck 2015 data. arising from the cosmic microwave background (CMB) radi- • To explicitly study the role of tachyon field and its generation [11–13]. The observations from Planck have put inter- alized version, GTachyon, to explain the observed tem- estingly tight bounds on a number of cosmological observ- perature anisotropy and polarization in the CMB angular ables related to the perturbations, which also determines power spectrum. any departure from the Gaussian perturbations and the con- • To study the specific role of the most generalized ver- straint on the tensor-to-scalar ratio, r, which can potentially sion of non-canonical higher-dimensional effective field reveal the scale of New Physics within any given effective theory Wilsonian operators. field theory set-up. But a big issue may crop up in model • To give a broad overview of the present constraints on the discrimination and also in the removal of the degeneracy inflationary paradigm from the most generalized version of cosmological parameters obtained from CMB observa- of the tachyon string theory. tions. Non-canonical interactions in the effective theory set- • To test the explicit dynamical features of various tachy- up [14–23] are among the possibilities through which one onic potentials obtained from string theory and also to can address this issue.1 The natural source for such non- learn about the specific structural form of effective field canonical interactions is string theory. The most promising theory operators for a specific type of potential. example is the Dirac–Bonn–Infeld (DBI) effective action for the tachyon field [48–51]. In this paper, we will only focus Throughout the paper we have taken the following assump- on tachyons in weakly coupled Type-IIA/IIB string theory tions: [52,53]. The phenomenon of tachyon condensation was introduced 1. The tachyon field T is minimally coupled to the Einstein in Refs. [48,49], where Type-IIA/IIB string theory and the gravity sector. tachyon instability on D-branes have been studied elabo- 2. The initial condition for inflation is fixed via Bunch– rately. The rolling tachyon [54–56] in weakly coupled Type- Davies (BD) vacuum [111]. For completeness we also IIA/IIB string theory may be described in terms of an effec- present the results obtained from an arbitrary vacuum tive field theory for the tachyon condensate in Refs. [57– (AV). For a classical initial condition the amplitude of 73]. The cosmological implications of the tachyon were studied in [74–86]. For more details see also Refs. [87– 2 Since the Gtachyon contains non-minimal interactions, it is naturally 95]. In the context of cosmology, the detailed conse- expected that it will sufficiently modify the consistency relations. Also due to such non-canonical interactions it is expected that the amount of primordial non-Gaussianity is getting enhanced by a sufficient amount, 1 The other possibilities are: non-minimal interactions between the by breaking the non-Gaussian consistency relations known in cosmo- matter and gravity sector [(example: SM Higgs with Einstein grav- logical literature. For completeness it is important to mention here ity [24,25]], addition of higher derivative terms in the gravity sector that we have not investigated the issues related to primordial non- (example: Starobinsky model [26]), modification of the gravity sector Gaussianity yet in this paper. We will report on these issues in our through extra dimensions (example: 5D membrane models [27–47]). future work. 123 Eur. Phys. J. C (2016) 76 :278 Page 3 of 130 278

the primordial gravitational waves would be very tiny and nS = 0.9569 ± 0.0077 (within 3σ C.L.), (1.3) practically undetectable, therefore this can be treated as α = / = . +0.014 ( . σ . .), S dnS dlnk 0 011−0.013 within 1 5 C L the first observable evidence of quantum theory gravity, (1.4) such as string theory. However, apart from the impor- κ = 2 / 2 = . +0.015 ( . σ . .). tance and applicability of the quantum version of a S d nS dlnk 0 029−0.016 within 1 5 C L Bunch–Davis vacuum on its theoretical and observa- (1.5) tional ground, it is still not at all well understood from the previous works in this area whether the quantum Planck (2015)+BICEP2/Keck Array joint data: [121] Bunch–Davies vacuum is the only possible source gen- erating a large value of the tensor-to-scalar ratio during the inflationary epoch or not. One of the prime possi- r(k) ≤ 0.12 (within 2σ C.L.), (1.6) bilities coming from the deviation from the quantum + . ln(1010 P ) = 3.089 0 024 (within 2σ C.L.), (1.7) Bunch–Davies vacuum and consideration of the quantum S −0.027 = . ± . ( σ . .), non-Bunch–Davies or an arbitrary vacuum in the present nS 0 9600 0 0071 within 3 C L (1.8) non-canonical picture may also be responsible for the αS = dnS/dlnk =−0.022 ± 0.010 generation of a large tensor-to-scalar ratio and large non- (within 1.5σ C.L.), (1.9) Gaussianity during inflation. Within the framework of + . κ = d2n /dlnk2 = 0.020 0 016 effective field theory, such an arbitrary vacuum is com- S S −0.015 ( . σ . .). monly identified with the α-vacuum [112–118], which within 1 5 C L (1.10) has a string theoretic origin. 3. We would like to point out that the tachyon mode appears The plan of the paper is as follows: in the quantization of the open string struck to the non- BPS brane. The effective action of this tachyon field is • In Sect. 2, we have discussed the role of tachyon in non- constructed on the assumption that the tachyon field cou- BPS barnes in weakly coupled Type-IIA/IIB string theory ples only to the graviton of the closed string sector with and also introduced the GTachyon in the effective action fixed vacuum expectation value for the dilation field. The of string theory. open string tachyon condensation phenomenon fits in • In Sect. 3, we have introduced and studied the features well with this assumption. from variants of the tachyonic potentials inspired by non- 4. The sound speed is cS < 1 in general for non-canonical BPS branes in string theory. interactions [14–23], which is the most promising ingre- • In Sect. 4, we have studied the cosmological dynamics dient for a tachyonic set-up, to generate simultaneously from GTachyons, in which we have explicitly discussed a detectable value of the tensor-to-scalar ratio and large the unperturbed evolution and dynamical solution in var- non-Gaussianity. This fact can be more clearly visualized ious phases of the universe. when we go to an all higher order expansion in slow-roll • In Sect. 5, we have explicitly studied inflationary or more precisely by taking the exact solution of the mode paradigm from single, assisted and multi-field Gtachyons. functions for scalar and tensor fluctuations as obtained Particularly for the single field and assisted field cases in from cosmological perturbation theory by appropriately the presence of GTachyon, we have derived the inflation- choosing the Bunch–Davies case or any arbitrary initial ary Hubble flow and potential dependent flow equations, conditions for inflation. new sets of consistency relations, which are valid in the 5. UV cut-off of the effective field theory is fixed at UV ∼ slow-roll regime and field excursion formula for tachyon 18 Mp, where Mp = 2.43×10 GeV is the reduced Planck in terms of the inflationary observables. mass. But in principle one can fix the scale between GUT • In Sect. 6, we have mentioned the future prospects of the scale and the reduced Planck scale, i.e., GUT <UV ≤ present work and summarized the context of the present Mp. But in such a situation UV acts as a regulating work. parameter in the effective field theory [23,119]. 6. Within the region of N(=Ncmb) ≈ O(50–70) e- In this paper, we have explored various cosmological conse- foldings, we will use the following constraints in the quences from the GTachyonic field. We start with the basic background of CDM model for: introduction of tachyons in the context of non-BPS string theory, where we also introduce the GTachyon field, in the Planck (2015)+WMAP-9+high L(TT) data: [120] presence of which the tachyon action is modified and one can quantify the amount of the modification via a super- ( ) ≤ . ( σ . .), r k 0 11 within 2 C L (1.1) script q instead of 1/2. This modification exactly mimics 10 ln(10 PS) = 3.089 ± 0.036 (within 2σ C.L.), (1.2) the role of effective field theory operators and studying the 123 278 Page 4 of 130 Eur. Phys. J. C (2016) 76 :278 various cosmological features from this theory, one of our the δN formalism, we have derived the expressions for infla- final objectives is to constrain the index q and a specific tionary observables in the context of multi-field tachyons and ∝ α 4/ α combination ( Ms gs) of the Regge slope parameter , demonstrated the results for inverse cosh potential for com- the string coupling constant gs and mass scale of tachyon pleteness. Ms, from the recent Planck 2015 and Planck+BICEP2/Keck Array joint data. To serve this purpose, we introduce various types of tachyonic potentials: the inverse cosh, loga- 2 GTachyon in string theory rithmic, exponential and inverse polynomials, using which we constrain the index q. To explore this issue in detail, we In this section we explicitly study the world volume actions start with the characteristic features of the each potentials. for non-BPS branes which finally govern their cosmological Next we discuss the dynamics of GTachyon as well as usual dynamics. For the sake of simplicity in this discussion we tachyon for single, assisted and multi-field scenario. Next neglect the contribution from the fermions and concentrate we have explicitly studied the inflationary paradigm from only on the massless bosonic fields for the non-BPS branes. a single field, assisted field and multi-field tachyon set-up. The world volume action for non-BPS branes is described Specifically for the single field and assisted field cases we by the sum of the Dirac–Born–Infeld (DBI) and the Wess– have derived the results in the quasi-de Sitter background Zumino (WZ) terms in Type-IIA/IIB string theory. The effec- in which we have utilized the details of: (1) cosmological tive action for DBI in a non-BPS p-brane is given by [122– perturbations and quantum fluctuations for scalar and ten- 124] sor modes, (2) the slow-roll prescription. In this context we (p) =−T p+1σ −φ − Z + F , have derived the expressions for all inflationary observables SDBI p d e det μν μν (2.1) using any arbitrary vacuum and also for Bunch–Davies vacuum. For single field and assisted field case in the presence of where the metric has signature (−, +, +, +) and Zμν is GTachyon we have derived the inflationary Hubble flow and defined as potential dependent flow equations, new sets of consistency Zμν = Gμν + α ∂μT ∂ν T. (2.2) relations valid in the slow-roll regime and also derived the expression for the field excursion formula for the tachyon Here T is the dimensionless tachyon field whose properties in terms of the inflationary observables from both of the have been discussed later in detail. Also in Eq. (2.1) Tp rep- solutions obtained from arbitrary and Bunch–Davies initial resents the brane tension defined as [122–124] conditions for inflation. Particularly the derived formula for √ T = ( π)−p −1 the field excursion for GTachyon can be treated as an one p 2 2 gs (2.3) of the important probes through which one can distinguish between various tachyon models and also check the valid- and α represents the Regge slope parameter in string theory. ity of effective field theory prescription and compare the Here Type-IIA/IIB string theory contains the non-BPS Dp- results obtained from assisted inflation as well. The results branes [125], which have precisely those dimensions which obtained in this context explicitly show that assisted inflation BPS D-branes do not have explicitly. This implies that Type- is better compared to single field inflation from the tachyon IIA string theory has non-BPS Dp-branes for only odd p portal, provided the number of identical tachyon fields is and Type-IIB string theory has non-BPS Dp-branes only for required to be large to validate effective field theory pre- even p in the present context. Additionally, it is important to scription. Next using the explicit form of the tachyonic poten- note that gs characterizes the string coupling constant. Also tials we have studied the inflationary constraints and quan- in Eq. (2.1), Gμν is defined via the following transformation tify the allowed range of the generalized index q for each of equation: the potentials. Hence using each specific form of the tachy- M N onic potentials in the context of the single field scenario, we Gμν = G MN∂μ X ∂ν X . (2.4) have studied the features of CMB angular power spectrum from TT, TE and EE correlations from scalar fluctuations In Eq. (2.4) M, N characterize the ten-dimensional (D = 10) within the allowed range of q for each potentials. We also indices, which run from 0, 1,...,9; σ μ(0 ≤ μ ≤ p) denotes put the constraints from the Planck temperature anisotropy the world volume coordinates of the Dp brane. Also it is and polarization data, which shows that our analysis is consis- important to note that, in this discussion, G MN represents tent with the data. We have additionally studied the features the ten-dimensional (D = 10) background metric for Type- of the tensor contribution in the CMB angular power spec- IIA/IIB string theory. trum from TT, BB, TE and EE correlations, which will give It is important to mention here that, in the context of more interesting information in near future while the signa- non-BPS D-brane, the stringy tachyon comes from only one ture of primordial B-modes can be detected. Further, using specific sector of string theory and consequently it is a real 123 Eur. Phys. J. C (2016) 76 :278 Page 5 of 130 278 scalar field using which we will explain the cosmological not at all arbitrary for the present set-up but the structural dynamics in this article for p = 3. Additionally it is also form of the metric is restricted in the specific sense that important to note that Type-IIA/IIB string theories contain it has to satisfy the sets of background field equations in unstable non-BPS D-branes in their spectra. The most eas- this context. Also in our discussion the transverse compo- iest way to define these types of D-branes in the context nent of the fluctuations of the D-membrane is described of IIA/IIB string theory is to start the computation with a by (9 − p) scalar fields Xi , where the index i runs from ¯ coincident BPS Dp-Dp-brane pair in Type-IIB/IIA string p + 1 ≤ i ≤ 9, and the gauge field Aμ describes the theory, and then take a specific orbifold of the string theory fluctuations along the longitudinal direction of the mem- F by (−1) L , where FL signifies the specific contribution to brane. the space-time fermion number from the left-moving sec- Before writing down the total effective action for non- tor of the world-sheet string theory. Now in this context the BPS D-brane in string theory it is important to mention that Ramond–Ramond (RR) fields are odd under (−1)FL trans- the non-BPS p-brane has an extra tachyon field appearing in formation and consequently all the Ramond–Ramond (RR) both of the Dirac–Born–Infeld (DBI) and the Wess–Zumino fields of Type-IIB/IIA string theory are projected out by the (WZ) stringy effective actions. The corresponding effective same amount via (−1)FL projection. As a result the twisted actions can be written in the non-BPS string theory set-up as sector stringy states then give us back the RR fields of Type- [122–124] IIA/IIB string theory in the present context. Most impor- (p) + −φ FL =− p 1σ − Z + F tantly, here the (−1) projection reverses the signature of SDBI d e det μν μν the RR charge and consequently it transforms a BPS Dp- × (T,∂μT, Dμ∂ν T ), (2.5) brane to a D¯ p-brane and vice versa. This further implies that due to its operation on the open string states on a Dp–D¯ p- (p) = p+1σ ∧ ∧ F , SWZ d C dT e (2.6) brane stringy system will do the job of conjugate operation on the Chan–Paton factor by the action of exchange operator where the field C contains Ramond–Ramond (RR) fields and σ1 in this context. Thus technically the modding out opera- the leading term has a (p + 1)-differential form. This also tionontheDp–D¯ p-brane by exactly the amount of (−1)FL mimics the role of a source term for the membrane and its eliminates all the open string states which carry a Chan– presence is explicitly required for consistency of the specific Paton factor σ2 and σ3, as both of them anti-commute with version of the field theory like anomaly cancellation within the exchange operator σ1. Additionally it is important to note the set-up of string theory. It is important to mention here that this operation finally keeps the open string states which that we have the world volume action for a Dp-brane in the are characterized via the Chan–Paton factors I and σ1. Finally (p + 1)-dimensional case, where p characterizes the spatial all such operations give us a non-BPS Dp-brane in the present dimension and 1 stands for time. The WZ term plays an context. Although in this discussion we are only interested important role since the BPS Dp-brane is charged under a in the non-BPS D-branes, the most important characteristic (p + 1) rank RR gauge field. Consequently the total action feature that distinguishes the physics of non-BPS D-branes is therefore the DBI action together with the WZ action. For from BPS D-branes is that the mass spectrum of open strings the non-BPS Dp-brane, the brane is charged under a rank p on a non-BPS D-brane contains a single mode of negative RR gauge field. As a result the WZ action consists of a wedge mass squared besides an infinite number of other modes of product of this p form and additionally a one form dT , where positive definite mass squared. This negative mass squired T is the tachyon field. Also it is important to note that for the mode is identified with the tachyonic mode which is exactly non-BPS case Fμν is explicitly defined as [122–124] equivalent to a particular linear combination of the two tachy- I I Fμν = Bμν + 2πα Fμν + ∂μY ∂νY onic modes living on the original brane–antibrane pair stringy F I J I system that survives the previously mentioned (−1) L pro- + (G IJ + BIJ) ∂μY ∂νY + GμI + BμI ∂νY jection and contains exactly the same mass squared contri- I + (G I ν + BI ν) ∂μY , (2.7) bution. In our analysis for the sake of simplicity we have neglected and T represents the dimensionless tachyon field and the contribution from the antisymmetric Kalb–Ramond two characterizes the generalized functional in non-BPS p- form field from the effective action but the gauge invari- membrane. In Eq. (2.7), the rank-2 field strength tensor Fμν ance of the action requires the presence of all such anti- is defined as symmetric tensor contributions in the original version of the string effective action. In the present context, it is Fμν = ∂[μ Aν] (2.8) important to note that Gμν can be physically interpreted as the induced metric on the membrane. Additionally it and Bμν represents a rank-2 Kalb–Ramond field and some- is important to note that the background metric G MN is times this can be interpreted as the pullback of BMN onto 123 278 Page 6 of 130 Eur. Phys. J. C (2016) 76 :278 the D-brane world volume. Using this specific action, men- (p) =− p+1σ ( ) − Z SD d V T det μν tioned in Eq. (2.7), we can compute the source contributions and terms for various closed string fields produced by the p+1 =− d σ V (T ) −det Gμν + α ∂μT ∂ν T membrane. Additionally, it is important to note that, on a √ non-BPS Dp-membrane world volume we have an infinite p+1 μν =− d σ −gV(T ) 1 + α g ∂μT ∂ν T . tower of massive fields, a U(1) gauge field Aμ with the restric- tion on gauge indices, 0 ≤ μ ≤ p, and a set of scalar fields (2.14) Y I , one for each direction y I transverse besides the tachy- In a more generalized prescription Eq. (2.14) can be modified onic field to the D-brane. Here it is important to note that ( + ) ≤ ≤ into the following effective action: p 1 I D, where D is 9 for superstring theory and √ (pq) =− p+1σ − ( ) + α μν∂ ∂ q , 25 for bosonic string theory. Within the present set-up the SD d gV T 1 g μT ν T tachyon field is defined in such a way that for (2.15)

T = 0, F = Tp, (2.9) which we identified with the most generalized Gtachyon action in string theory. Here for p = 3, i.e., for the D3 brane, the a constraint condition is explicitly satisfied. For a non- Eq. (2.14) refers to the following crucial issues: BPS p membrane the field content C as appearing in the Wess–Zumino (WZ) action contains the Ramond–Ramond • Here p = 3, q = 1/2 corresponds to the exact tachyonic (RR) fields but careful observation clearly indicates that the behavior of the effective action and it is commonly used leading order contribution in C is characterized by the p form to describe the cosmological dynamics, in the effective action. Also it is important to mention here • Here p = 3, q = 1 corresponds to the single field behav- that, for constant tachyon background T , the Wess–Zumino ior in cosmological dynamics where the kinetic term of (WZ) effective action automatically vanishes in the present the tachyon field is non-canonical. In this case the non- context as canonical contribution in the effective action is given by αV (T ), where V (T ) is the tachyon effective potential. dT = 0, (2.10) This situation is different from the usual single field models of inflation where the kinetic term is canonical within and for such a specific background the generalized functional the framework of string theory. • = , / < < = , < / can be recast in the following form: For p 3 1 2 q 1orp 3 q 1 2 or p = 3, q > 1 it contains various non-trivial fea- (T,∂μT, Dμ∂ν T ) = V (T ), (2.11) tures in cosmological dynamics. In this case the effective action is significantly different from the usual tachyon where V (T ) represents the effective tachyon potential within action as appearing in the context of string theory. In which the contribution from the membrane tension is already this case the effective action describes a huge class of taken. Consequently Eq. (2.5) can be recast in the following effective field theories of inflationary models which can simplified form: be embedded within the framework of tachyon in string ( ) theory. In the present context, the generalized factor p =− p+1σ −φ ( ) − Z + F . μν q SDBI d e V T det μν μν V (T ) 1 + α g ∂μT ∂ν T with exponent q can be (2.12) treated as Wilsonian operators as appearing in the context of effective field theory. For an example let us con- For non-BPS string theory in the constant dilaton background sider a situation where we treat the Regge slope param- the purely tachyonic part of the action, after inclusion of the eter α to be small. In this case the generalized factor μν q massless fields on the Dp-membrane world volume around V (T ) 1 + α g ∂μT ∂ν T takes the following struc- the tachyon vacuum, is given by ture: μν q V (T ) 1 + α g ∂μT ∂ν T (p) p+1 S =− d σ V (T ) −det Zμν + Fμν + ∂μY I ∂νY I . D q = ( ) q α μν∂ ∂ k (2.13) V T Ck g μT ν T k=0 After neglecting the contribution from the massless fields μν = V (T ) 1 + q α g ∂μT ∂ν T for non-BPS string theory the tachyonic part of the effective action describing the Dp-brane world volume is given by q(q − 1) μν 2 + α g ∂μT ∂ν T +··· , (2.16) [122–124] the following simplified form: 2 123 Eur. Phys. J. C (2016) 76 :278 Page 7 of 130 278

where ··· contains higher order terms which are sup- N N ( ) = ( ) = ( ) = ( ). pressed by the powers of the Regge slope parameter α. VE T V Ti V T NV T (2.23) (= , , ,..., ) i=1 i=1 Here for each value of k 0 1 2 q the expansion q k factor Ck α mimics the role of Wilson coefficients. In the next section we will discuss the various aspects of tachyonic potential V (T ) and also mention the various mod- Here we would like to point out that the tachyon mode appears els of tachyonic potential that can be derived from a string in the quantization of the open string struck to the non-BPS theory background. brane. The effective action of this tachyon field is constructed on the assumption that the tachyon field couples only to the graviton of the closed string sector with fixed vacuum expec- 3 Variants of tachyonic models in string theory tation value for dilation field. The open string tachyon condensation phenomenon fits in well with this assumption. On general string theoretic grounds, one can argued that at ( ) For a multi-tachyonic field scenario one can generalize the the specified minimum T0 of the effective potential V T tachyonic part of the non-BPS action as stated in Eqs. (2.14) vanishes [122–124], i.e., and (2.15): V (T0) = 0. (3.1) N √ (p) =− p+1σ − ( ) + α μν∂ ∂ , Consequently the world volume action vanishes identically SD d gV Ti 1 g μTi ν Ti i=1 and in this situation the gauge field mimics the role of (2.17) Lagrange multiplier field. Finally this imposes a constraint on the non-BPS set-up such that the gauge current also vanishes N √ (pq) =− p+1σ − ( ) + α μν ∂ ∂ q . identically. This implies that all the states which are charged SD d gV Ti 1 g μTi ν Ti i=1 under this gauge field are to disappear from the spectrum. (2.18) Also it is important to mention another important feature of the tachyonic potential in which it admits kink profile for In this case one can introduce a total effective potential of the stringy tachyonic field. Also on an unstable non-BPS p- ( ) tachyonic fields VE T , which can be expressed in terms of brane tachyon condensation occurs to form a kink profile and N component tachyonic fields as finally it forms a stable BPS (p−1) brane configuration. This N kink profile for the tachyon is expected to give a δ-function VE (T ) = V (Ti ), (2.19) from dT - contribution and thus to reproduce the standard WZ i=1 term in the resulting D(p − 1)-brane. Most importantly, the which is very useful to study the cosmological dynamics for kink solution effectively reduces the dimension of the world the p = 3 case, i.e., for the D3 brane. volume by one. Although finding an explicit form of the Also for the assisted case one can assume all N multi- tachyonic potential is a very difficult, string theory predicts tachyonic fields are identical to each other, i.e., the approximated form of the tachyonic potential. Addition- ally, we also assume that the tachyonic potential V (T ) sat- Ti = T ∀ i = 1, 2, 3,...,N. (2.20) isfies the following properties to describe the cosmological dynamics for the non-BPS D3 brane set-up: Consequently in such a prescription Eqs. (2.17) and (2.18) can be rewritten as 1. The tachyonic potential at T = 0 satisfies N √ (p) p+1 μν S =− d σ −gV(T ) 1 + α g ∂μT ∂ν T 4 D Ms = V (T = 0) = λ = , (3.2) i 1 ( π)3 √ 2 gs p+1 μν =− d σ −gNV(T ) 1 + α g ∂μT ∂ν T , where g is the string coupling constant and M signifies (2.21) s s the mass scale of the tachyonic string theory. For multi- N √ (pq) =− p+1σ − ( ) + α μν ∂ ∂ q tachyonic field and assisted case Eq. (3.2) is modified as SD d gV T 1 g μT ν Ti i=1 √ p+1 μν q N N =− d σ −gNV(T ) 1 + α g ∂μT ∂ν T . VE (T = 0) = V (Ti = 0) = λi (2.22) i=1 i=1 N M4 In this case the total effective potential of the tachyonic fields = s , (3.3) ( ) ( π)3 (i) VE T can be recast as i=1 2 gs 123 278 Page 8 of 130 Eur. Phys. J. C (2016) 76 :278

N N V (T > 0)>0 (3.11) VE (T = 0) = V (Ti = T = 0) = λi = Nλ i=1 i=1 where represents differentiation with respect to the M4 N = s . (3.4) tachyon field T . For the multi-tachyonic and assisted (2π)3g s cases Eq. (3.11) can be recast as (i) Here gs represents the string coupling constant for the N ( > ) ith field content, which are not same for all N tachy- ( > ) = dV Ti 0 > , VE T 0 0 (3.12) onic degrees of freedom. For the sake of simplicity, here dTi i=1 we also assume that the mass scale associated with N N ( > ) N ( > ) tachyons for multi-tachyonic case and assisted case is ( > ) = dV Ti 0 = dV T 0 VE T 0 identical for all degrees of freedom. On the other hand, dTi dT i=1 i=1 for the assisted case we assume all couplings are exactly = NV (T > 0)>0. (3.13) identical and consequently we get an overall factor of N multiplied with the result obtained for the single tachyonic field case. 4. At the asymptotic case of the single tachyonic field |T |→ 2. Inflation generally takes place at an energy scale: ∞ the potential should satisfy

1/4 = 1/4( ˜ ) ∝ λ1/4 Vinf V T0 (3.5) V (|T |→∞) → 0. (3.14) ˜ with the single tachyon field fixed at T ∼ T0, and within ˜ For the multi-tachyonic and assisted cases Eq. (3.13) can the set-up of string theory T0 is identified to be the mass scale of the tachyon by in the following fashion: be recast thus:

˜ N T0 ∼ Ms. (3.6) VE (|T |→∞) = V (|Ti |→∞) → 0, (3.15) For the multi-tachyonic field case Eqs. (3.5) and (3.6)are i=1 modified as N VE (|T |→∞) = V (|Ti |→∞) / / N 1 4 N 1 4 i=1 1/4 = 1/4 = ( ˜ ) ∝ λ = (| |→∞) → . Vinf VE V T0i i NV T 0 (3.16) i=1 i=1 1/4 N M4 5. Also one can consider that the tachyonic potential con- = s , (3.7) = ( π)3 (i) tains a global maximum at T 0, i.e., i=1 2 gs N ( = )< ˜ ˜ V T 0 0 (3.17) T0 ∼ T0i = Ms. (3.8) i=1 for which the value of the potential is given by Eq. (3.2). Similarly, for the assisted case Eqs. (3.5) and (3.6)are Similarly for the multi-tachyonic and assisted cases modified as Eq. (3.18) can be recast thus:

/ / N 1 4 N 1 4 / / N 2 1 4 = 1 4 = ( ˜ ) ∝ λ d V (Ti > 0) Vinf VE V T0i i ( = ) = < , VE T 0 2 0 (3.18) i=1 i=1 dT i=1 i 4 1/4 / M N N 2 N 2 = (Nλ)1 4 = s , (3.9) d V (Ti > 0) d V (T > 0) (2π)3g V (T = 0) = = s E dT 2 dT 2 i=1 i i=1 N N = ( = )< . ˜ ˜ ˜ ˜ NV T 0 0 (3.19) T0 ∼ T0i = T0 = N T0 = NMs. (3.10) i=1 i=1 In the next subsections we mention variants of the tachy- 3. For T > 0 the first derivative of the single tachyonic onic potentials which satisfy the various above mentioned potential is always positive, i.e., characteristics. 123 Eur. Phys. J. C (2016) 76 :278 Page 9 of 130 278 λ T 3.1 Model I: inverse cosh potential V (T ) =− sech3 T 2 T 0 0 For single field case the first model of the tachyonic potential − T 2 T . is given by [55,56,84,126] sech tanh (3.29) T0 T0 λ V (T ) = , (3.20) Now to find the extrema of the potential we substitute cosh T T0 V (T ) = 0 (3.30) and for the multi-tachyonic and assisted cases the total effective potential is given by which give rise to the following solutions for the tachyonic field: ⎧ ⎪ N λ ⎪ i ⎪ , for Multi tachyonic, |T |=2mT0π, (2m + 1) T0π (3.31) ⎪ Ti ⎪ i=1 cosh ⎨⎪ T0i N λ where m ∈ Z. Further substituting the solutions for VE (T ) = ⎪ tachyonic field in Eq. (3.29) we get ⎪ T ⎪ i=1 cosh ⎪ T0 ⎪ Nλ ⎩⎪ = , for Assisted tachyonic. λ cosh T V (|T |=2mT0π) =− , (3.32) T0 T 2 (3.21) 0 λ V (|T |=(2m + 1) T0π) =− , (3.33) T 2 Here the potential satisfies the following criteria: 0 and at these points the value of the potential is computed • At T = 0 for the single field tachyonic potential: as V (T = 0) = λ (3.22) λ V (|T |=2mTπ) = = λ sech (2m π) , 0 cosh (2m π) and for the multi-tachyonic and assisted cases we have (3.34) λ N V (|T |=(2m + 1) T π) = 0 cosh ((2m + 1) π) VE (T = 0) = λi , (3.23) = λ sech ((2m + 1) π) . (3.35) i=1 N V (T = 0) = λ = Nλ. (3.24) It is important to note for the single tachyonic case that for E i λ> (| |= π, ( + ) π) > i=1 0, V T 2mT0 2m 1 T0 0, i.e., we get maxima on the potential and for the assisted case the • At T = T0 for the single field tachyonic potential: results are the same, provided the following replacement occurs: λ V (T = T0) = = 0 (3.25) cosh(1) λ → Nλ, (3.36)

and for the multi-tachyonic and assisted cases we have and finally for the multi-tachyonic case we have N λ i λi Ti Ti VE (T = 0) = = 0, (3.26) V (T ) =− sech tanh , (3.37) cosh(1) i T T T i=1 0i 0i 0i λ T N λ λ ( ) =− i 3 i i N V Ti 2 sech VE (T = 0) = = = 0. (3.27) T T0i cosh(1) cosh(1) 0i i=1 T T − sech i tanh2 i . (3.38) T T • For the single field tachyonic potential: 0i 0i Now to find the extrema of the potential we substitute λ T T V (T ) =− sech tanh , (3.28) T0 T0 T0 V (Tj ) = 0 ∀ j = 1, 2,...,N (3.39) 123 278 Page 10 of 130 Eur. Phys. J. C (2016) 76 :278

V (T ) which gives rise to the following solutions for the jth E ⎧ ⎪ N 2 2 tachyonic field: ⎪ λ Ti Ti + , , ⎪ i ln 1 for Multi tachyonic ⎪ T0i T0i ⎪ i=1 ⎨ N 2 2 = T T |Tj |=2mT0 j π, (2m + 1) T0 j π ∀ j = 1, 2,...,N λ ln + 1 ⎪ T T ⎪ i=1 0 0 (3.40) ⎪ ⎪ T 2 T 2 ⎩⎪ = Nλ ln + 1 , for Assisted tachyonic. T0 T0 where m ∈ Z. Further substituting the solutions for (3.46) tachyonic field in Eq. (3.38) we get Here the potential satisfies the following criteria: λ j V (|T |=2mT π) =− , (3.41) j 0 j 2 • At T = 0 for the single field tachyonic potential: T0 j λ (| |=( + ) π) =− j , ( = ) = λ V Tj 2m 1 T0 j 2 (3.42) V T 0 (3.47) T0 j and for the multi-tachyonic and assisted cases we have and at these points the value of the total effective potential is computed as N VE (T = 0) = λi , (3.48) = N N λ i 1 (1) j N V = V (|Tj |=2mT j π) = E 0 cosh (2m π) ( = ) = λ = λ. j=1 j=1 VE T 0 i N (3.49) i=1 N = sech (2m π) λ j , (3.43) • At T = T0 for the single field tachyonic potential: j=1 N V (T = T ) = λ, (3.50) (2) = (| |=( + ) π) 0 VE V Tj 2m 1 T0 j = j 1 and for the multi-tachyonic and assisted cases we have N λ j = N cosh ((2m + 1) π) j=1 VE (T = T0) = λi , (3.51) = N i 1 = sech ((2m + 1) π) λ . (3.44) N j ( = ) = λ = λ. j=1 VE T T0 i N (3.52) i=1

It is important to note for the multi-tachyonic case that • For the single ﬁeld tachyonic potential: for λ j > 0, V (|Tj |=2mT0 j π, (2m + 1) T0 j π) > 0, ( ) i.e., we get maxima on the potential V Tj as well as in 2T λ T T ( ) V (T ) = ln 1 + ln , (3.53) VE T . T 2 T T 0 0 0 2λ T T V (T ) = + + 2 . 3.2 Model II: logarithmic potential 2 1 3ln ln (3.54) T0 T0 T0

For single field case the second model of the tachyonic poten- Now to find the extrema of the potential we substitute tial is given by [79] V (T ) = 0 (3.55) T 2 T 2 V (T ) = λ ln + 1 , (3.45) T0 T0 which give rise to the following solutions for the tachyonic field: and for the multi-tachyonic and assisted cases the total effec- T T = 0, T , 0 . (3.56) tive potential is given by 0 e 123 Eur. Phys. J. C (2016) 76 :278 Page 11 of 130 278

Further substituting the solutions for the tachyonic ﬁeld and at these points the value of the total effective potential in Eq. (3.54) we get is computed as

( = ) →∞, N N V T 0 (3.57) ( ) λ V 1 = V (T = 0) = λ , (3.71) ( = ) = 2 , E j j V T T0 2 (3.58) j=1 j=1 T0 N N T0 2λ (2) V T = =− , (3.59) V = V (Tj = T0 j ) = λ j , (3.72) e 2 E T0 j=1 j=1 N N ( ) T j 1 and at these points the value of the potential is computed V 3 = V T = 0 = 1 + λ . (3.73) E j e e2 j as j=1 j=1

V (T = 0) = λ, (3.60) It is important to note for the multi-tachyonic case that V (T = T ) = λ, (3.61) λ > ( = )> ( = T0 j )< 0 for j 0, V T T0 j 0 and V T e 0, T 1 i.e., we get both maxima and minima on the potential V T = 0 = λ 1 + . (3.62) e e2 V (Tj ) as well as in VE (T ).

It is important to note for the single tachyonic case that 3.3 Model III: exponential potential Type-I λ> ( = )> ( = T0 )< for 0, V T T0 0 and V T e 0 i.e., we get both maxima and minima on the potential. For For single ﬁeld case the third model of the tachyonic potential the assisted case the results are the same, provided the is given by [83] following replacement occurs: ( ) = λ − T , λ → λ, V T exp (3.74) N (3.63) T0

and ﬁnally for the multi-tachyonic case we have and for the multi-tachyonic and assisted cases the total effective potential is given by 2λ T T T ( ) = i i i + i , ⎧ V Ti 2 ln 1 ln (3.64) T T0i T0i ⎪ N 0i ⎪ λ − Ti , , λ ⎪ i exp for Multi tachyonic 2 i Ti 2 Ti ⎪ T i ( ) = + + . ⎨⎪ i=1 0 V Ti 2 1 3ln ln (3.65) T T0i T0i N 0i VE (T ) = T ⎪ λ exp − ⎪ T ⎪ i=1 0 Now to ﬁnd the extrema of the potential we substitute ⎩⎪ = Nλ exp − T , for Assisted tachyonic. T0 V (Tj ) = 0 ∀ j = 1, 2,...,N (3.66) (3.75)

which give rise to the following solutions for the jth Here the potential satisfies the following criteria: tachyonic field: • At T = 0 for the single field tachyonic potential: T j T = 0, T , 0 ∀ j = 1, 2,...,N. (3.67) j 0 j e V (T = 0) = λ (3.76) Further substituting the solutions for the tachyonic field in Eq. (3.65) we get and for the multi-tachyonic and assisted cases we have

N V (Tj = 0) →∞, (3.68) VE (T = 0) = λi , (3.77) 2λ j V (Tj = T0 j ) = , (3.69) i=1 T 2 0 j N λ T0 j 2 j VE (T = 0) = λi = Nλ. (3.78) V Tj = =− , (3.70) 2 i=1 e T0 j 123 278 Page 12 of 130 Eur. Phys. J. C (2016) 76 :278

• At T = T0 for the single ﬁeld tachyonic potential: behavior of the potential. For the assisted case the results are the same, provided the following replacement occurs: λ V (T = T0) = (3.79) e λ → Nλ, (3.92) and for the multi-tachyonic and assisted cases we have and ﬁnally for the multi-tachyonic case we have N 1 VE (T = T0) = λi , (3.80) λ T e V (T ) =− i exp − i , (3.93) i=1 i T0 T0i N λ λ T ( = ) = 1 λ = N . ( ) = i − i . VE T T0 i (3.81) V Ti 2 exp (3.94) e e T T0i i=1 0i

• For the single field tachyonic potential: Now to find the extrema of the potential we substitute λ T V (T ) = 0 ∀ j = 1, 2,...,N (3.95) V (T ) =− exp − , (3.82) j T0 T0 λ T which gives rise to the following solutions for the jth V (T ) = exp − . (3.83) 2 tachyonic field: T0 T0

Now to ﬁnd the extrema of the potential we substitute Tj →∞. (3.96)

( ) = , V T 0 (3.84) Further substituting the solutions for the tachyonic field in Eq. (3.94) we get which gives rise to the following solution for the tachyonic field: V (Tj →∞) → 0, (3.97) T →∞. (3.85) and also at the points Tj = 0, T0 j we have Further substituting the solutions for the tachyonic field λ in Eq. (3.83) we get ( = ) = j , V Tj 0 2 (3.98) T0 j ( →∞) → , λ V T 0 (3.86) ( = ) = j , V Tj T0 j 2 (3.99) eT0 j and also at the points T = 0, T0 we have and at these points the value of the total effective potential λ ( = ) = , is computed as V T 0 2 (3.87) T0 λ N V (T = T ) = , (3.88) ( ) 0 2 V 1 = V (T →∞) → 0, (3.100) eT0 E j j=1 and at these points the value of the potential is computed N N (2) = ( = ) = λ , as VE V Tj 0 j (3.101) j=1 j=1 ( →∞) → , N N V T 0 (3.89) ( ) 1 V 3 = V (T = T ) = λ . (3.102) V (T = 0) = λ, (3.90) E j 0 j e j λ j=1 j=1 V (T = T ) = . (3.91) 0 e It is important to note for the multi-tachyonic case that for λ > ( = , )> It is important to note for the single tachyonic case that for j 0, V T 0 T0 j 0, i.e., we get an asymptotic behavior of the potential V (T ) as well as in V (T ). λ>0, V (T = 0, T0)>0, i.e., we get an asymptotic j E 123 Eur. Phys. J. C (2016) 76 :278 Page 13 of 130 278 3.4 Model IV: exponential potential Type-II (Gaussian) λ 2 2 ( ) =−2 − T − 2T . V T 2 exp 1 2 (3.112) T0 T0 T0 For single field case the first model of the tachyonic potential is given by [127] Now to find the extrema of the potential we substitute T 2 ( ) = , V (T ) = λ exp − , (3.103) V T 0 (3.113) T0 which gives rise to the following solution for the tachy- and for the multi-tachyonic and assisted cases the total effec- onic field: tive potential is given by T = 0, ∞. (3.114) ⎧ N ⎪ T 2 ⎪ λ exp − , for Multi tachyonic Further substituting the solutions for the tachyonic field ⎪ T ⎪ i=1 0 ⎨⎪ in Eq. (3.112) we get N 2 ( ) = T VE T ⎪ λ exp − 2λ ⎪ T V (T = 0) =− , (3.115) ⎪ i=1 0 2 ⎪ T0 ⎪ 2 ⎩ = Nλ exp − T , for Assisted tachyonic. ( →∞) → T0 V T 0 (3.116) (3.104) and also additionally for T = T0 we have Here the potential satisfies the following criteria: 2λ V (T = T0) = , (3.117) eT 2 • At T = 0 for single field tachyonic potential: 0 and at these points the value of the potential is computed ( = ) = λ V T 0 (3.105) as

and for the multi-tachyonic and assisted cases we have V (T →∞) → 0, (3.118) V (T = 0) = λ, (3.119) N λ V (T = ) = λ , V (T = T0) = . (3.120) E 0 i (3.106) e i=1 N It is important to note for the single tachyonic case that ( = ) = λ = λ. VE T 0 i N (3.107) for λ>0, V (T = 0, T0)>0, i.e., we get maxima i=1 on the potential. For the assisted case the results are the same, provided the following replacement occurs: • At T = T0 for the single field tachyonic potential: λ → Nλ, (3.121) λ V (T = T0) = (3.108) e and finally for the multi-tachyonic case we have and for the multi-tachyonic and assisted cases we have λ 2 ( ) =−2 i Ti − Ti , V Ti 2 exp (3.122) T0i T0i N ( = ) = 1 λ , λ 2 2 VE T T0 i (3.109) 2 i Ti 2Ti e V (Ti ) =− exp − 1 − . i=1 2 2 T0i T0i T0i N 1 Nλ (3.123) V (T = T ) = λ = . (3.110) E 0 e i e i=1 Now to find the extrema of the potential we substitute • For the single field tachyonic potential: V (Tj ) = 0 ∀ j = 1, 2,...,N (3.124) λ 2 ( ) =−2 T − T , which give rise to the following solutions for the jth V T 2 exp (3.111) T0 T0 tachyonic field: 123 278 Page 14 of 130 Eur. Phys. J. C (2016) 76 :278

Tj = 0, ∞. (3.125) Here the potential satisﬁes the following criteria:

Further substituting the solutions for the tachyonic ﬁeld • At T = 0 for the single ﬁeld tachyonic potential: in Eq. (3.123) we get V (T = 0) = λ, (3.134) λ ( = ) =−2 , V Tj 0 2 (3.126) T0 and for the multi-tachyonic and assisted cases we have V (Tj →∞) → 0, (3.127) N VE (T = 0) = λi , (3.135) Additionally for the point Tj = T0 j we have i=1 N 2λ j ( = ) = , VE (T = 0) = λi = Nλ. (3.136) V Tj T0 j 2 (3.128) eT0 j i=1

and at these points the value of the total effective potential • At T = T0 for the single ﬁeld tachyonic potential: is computed as λ V (T = T0) = (3.137) N 2 (1) = ( →∞) → , VE V Tj 0 (3.129) j=1 and for the multi-tachyonic and assisted cases we have N N N (2) = ( = ) = λ , 1 VE V Tj 0 j (3.130) VE (T = T0) = λi , (3.138) j=1 j=1 2 i=1 N N N (3) 1 1 Nλ V = V (Tj = T0 j ) = λ j . (3.131) ( = ) = λ = . E e VE T T0 i (3.139) j=1 j=1 2 2 i=1

It is important to note for the multi-tachyonic case that • For single field tachyonic potential: for λ j > 0, V (T = 0, T0 j )>0, i.e., we get maxima ( ) ( ) 3 on the potential V Tj as well as in VE T . 4λT V (T ) =− , (3.140) 4 2 3.5 Model V: inverse power-law potential T 4 1 + T 0 T0 λ 6 λ 2 For the single field case the first model of the tachyonic poten- 32 T 12 T V (T ) = − . tial is given by [86,128] 4 3 4 2 T 8 1 + T T 4 1 + T 0 T0 0 T0 λ V (T ) = , (3.132) (3.141) 4 + T 1 T 0 Now to find the extrema of the potential we substitute

and for the multi-tachyonic and assisted cases the total effec- V (T ) = 0, (3.142) tive potential is given by ⎧ which give rise to the following solution for the tachyonic ⎪ N ⎪ λi ﬁeld: ⎪ , for Multi tachyonic ⎪ 4 ⎪ i=1 1 + Ti ⎪ T0i = , ∞. ⎨⎪ T 0 (3.143) N ( ) = λ VE T ⎪ ⎪ 4 Further substituting the solutions for the tachyonic ﬁeld ⎪ T ⎪ i=1 1 + ⎪ T0 in Eq. (3.141) we get ⎪ λ ⎪ = N , for Assisted tachyonic. ⎩⎪ 4 1+ T ( = ) = , T0 V T 0 0 (3.144) (3.133) V (T →∞) → 0 (3.145) 123 Eur. Phys. J. C (2016) 76 :278 Page 15 of 130 278

and also additionally for T = T0 we have Tj = 0, ∞. (3.154) λ V (T = T ) = , (3.146) 0 2 Further substituting the solutions for the tachyonic ﬁeld T0 in Eq. (3.152) we get and at these points the value of the potential is computed as V (Tj = 0) = 0, (3.155) V (Tj →∞) → 0, (3.156) V (T →∞) → 0, (3.147) = V (T = 0) = λ, (3.148) Additionally for the point Tj T0 j we have λ λ V (T = T0) = . (3.149) ( = ) = j , 2 V Tj T0 j 2 (3.157) T0 j It is important to note for the single tachyonic case that and at these points the value of the total effective potential for λ>0, V (T = 0, T0)>0, i.e., we get maxima on the potential. For the assisted case the results are the is computed as same, provided the following replacement occurs: N ( ) V 1 = V (T →∞) → 0, (3.158) λ → Nλ, (3.150) E j j=1 N N and ﬁnally for the multi-tachyonic case we have (2) = ( = ) = λ , VE V Tj 0 j (3.159) = = 4λ T 3 j 1 j 1 ( ) =− i i , V Ti (3.151) N N 4 2 (3) 1 4 Ti = ( = ) = λ . T 1 + VE V Tj T0 j j (3.160) 0i T0i 2 j=1 j=1 6 2 32λi T 12λi T ( ) = i − i . It is important to note for the multi-tachyonic case that V Ti 3 2 4 4 λ > ( = , )> T 8 1 + Ti T 4 1 + Ti for j 0, V T 0 T0 j 0, i.e., we get maxima 0i T0i 0i T0i on the potential V (Tj ) as well as in VE (T ). (3.152)

Now to ﬁnd the extrema of the potential we substitute: 4 Cosmological dynamics from GTachyon

V (Tj ) = 0 ∀ j = 1, 2,...,N (3.153) 4.1 Unperturbed evolution

which give rise to the following solutions for the jth For p = 3 non-BPS branes the total tachyonic model action tachyonic ﬁeld: can be written as

⎧ ⎪ √ M2 ⎪ 4σ − p − ( ) + α μν∂ ∂ , , ⎪ d g R V T 1 g μT ν T for Single ⎪ 2 ⎪ ⎨⎪ √ M2 N = 4σ − p − ( ) + α μν∂ ∂ , , ST ⎪ d g R V Ti 1 g μTi ν Ti for Multi (4.1) ⎪ 2 ⎪ i=1 ⎪ ⎪ √ M2 ⎪ 4 p μν ⎩⎪ d σ −g R − NV(T ) 1 + α g ∂μT ∂ν T , for Assisted, 2

123 278 Page 16 of 130 Eur. Phys. J. C (2016) 76 :278 and in a more generalized situation Eq. (4.1) is modiﬁed as

⎧ ⎪ √ M2 ⎪ 4 p μν q ⎪ d σ −g R − V (T ) 1 + α g ∂μT ∂ν T , for Single, ⎪ 2 ⎪ ⎨⎪ √ M2 N (q) 4 p μν q S = d σ −g R − V (Ti ) 1 + α g ∂μTi ∂ν Ti , for Multi, (4.2) T ⎪ 2 ⎪ i=1 ⎪ ⎪ √ M2 ⎪ 4 p μν q ⎩⎪ d σ −g R − NV(T ) 1 + α g ∂μT ∂ν T , for Assisted, 2 where Mp is the reduced Planck mass, Mp = 2.43 × It clearly appears that, for q = 1/2, the result is per- 18 10 GeV. By varying the action as stated in Eqs. (4.1) and fectly consistent with Eq. (4.6). Further using the perfect (4.2), with respect to the metric gμν we get the following fluid assumption the energy-momentum stress tensor can be equation of motion: written as Tμν ⎧ Gμν = (4.3) ⎪ (ρ + p)uμuν + gμν p, for Single, 2 ⎪ Mp ⎪ ⎨ N where Gμν is defined as = (ρ + ) (i) (i) + , , Tμν ⎪ i pi uμ uν gμν pi for Multi 1 ⎪ = Gνν = Rμν − gμν R (4.4) ⎪ i 1 2 ⎩ N (ρ + p)uμuν + gμν p , for Assisted, and the energy-momentum stress tensor Tμν is defined as √ (4.8) 2 δ( −gLTachyon) Tμν =−√ where for the assisted case we assume that the density and − δ μν (4.5) g g pressure of all identical tachyonic modes are the same. Here uμ signifies the four velocity of the fluid, which is defined as where LTachyon is the tachyonic part of the Lagrangian for non-BPS set-up. Explicitly using Eq. (4.1) the energy- momentum stress tensor can be computed as ⎧ ⎪ α∂ ∂ ⎪ μT ν T μν ⎪ V (T ) − gμν + α g ∂μT ∂ν T , , ⎪ μν 1 for Single ⎪ 1 + α g ∂μT ∂ν T ⎪ ⎨⎪ N α∂ ∂ μTi ν Ti μν Tμν = ( ) − μν + α ∂μ ∂ν , , (4.6) ⎪ V Ti μν g 1 g Ti Ti for Multi ⎪ 1 + α g ∂μT ∂ν T ⎪ i=1 i i ⎪ ⎪ ⎪ α ∂μT ∂ν T ⎪ ( ) − + α μν∂ ∂ , , ⎩ NV T μν gμν 1 g μT ν T for Assisted 1 + α g ∂μT ∂ν T and similarly in a more generalized situation using Eq. (4.2) the energy-momentum stress tensor can be computed as

⎧ ⎪ α∂ ∂ ⎪ 2q μT ν T μν q ⎪ V (T ) − gμν 1 + α g ∂μT ∂ν T , for Single, ⎪ + α μν∂ ∂ 1−q ⎪ 1 g μT ν T ⎪ ⎨ N α∂ ∂ (q) 2q μTi ν Ti μν q Tμν = V (Ti ) − gμν 1 + α g ∂μTi ∂ν Ti , for Multi, (4.7) ⎪ + α μν∂ ∂ 1−q ⎪ i=1 1 g μTi ν Ti ⎪ ⎪ α∂ ∂ ⎪ 2q μT ν T μν q ⎪ NV(T ) − gμν 1 + α g ∂μT ∂ν T , for Assisted. ⎩ μν 1−q 1 + α g ∂μT ∂ν T

123 Eur. Phys. J. C (2016) 76 :278 Page 17 of 130 278 ⎧ ∂ ⎪ μT Next using Eqs. (4.10) and (4.11) one can write down the ⎪ , for Single, ⎪ − αβ ∂ ∂ expression for the equation of state parameter: ⎪ g α T β T ⎪ ⎨⎪ N ∂ ⎧ μTi uμ = , for Multi, (4.9) ⎪ p = α ˙ 2 − , , ⎪ αβ ⎪ T 1 for Single ⎪ i=1 −g ∂α Ti ∂β Ti ⎪ ρ ⎪ ⎪ ⎪ N∂μT ⎪ ⎪ , . ⎨ N N ⎪ for Assisted p ⎩ αβ w = i = α ˙ 2 − , , −g ∂α T ∂β T ⎪ Ti 1 for Multi ⎪ ρi ⎪ i=1 i=1 ⎪ ⎪ Np Further comparing Eqs. (4.6) and (4.8) the density ρ and the ⎩⎪ = N αT˙ 2 − , , ρ 1 for Assisted pressure p for the tachyonic field can be computed as (4.14) ⎧ ⎪ V (T ) ⎪ , for Single, ⎪ ˙ ⎪ 1 − α T 2 and for the generalized case using Eqs. (4.12) and (4.13) ⎪ ⎨⎪ N ( ) one can write down the expression for the equation of state ρ = V T , , ⎪ for Multi (4.10) parameter: ⎪ 1 − αT˙ 2 ⎪ i=1 ⎪ ( ) ⎧ ⎪ NV T ˙ ⎩ , for Assisted, ⎪ p α T 2 − 1 ˙ ⎪ = , , 1 − α T 2 ⎪ for Single ⎪ ρ 1 − α (1 − 2q)T˙ 2 ⎪ ⎨⎪ and N N αT˙ 2 − 1 w = pi = i , , ⎪ ˙ 2 for Multi ⎧ ⎪ ρi 1 − α (1 − 2q)T ⎪ i=1 i=1 i ⎪ − ( ) − α ˙ 2, , ⎪ ⎪ V T 1 T for Single ⎪ α ˙ 2 − ⎪ ⎪ Np N T 1 ⎨⎪ ⎩⎪ = , for Assisted. N ρ 1 − α(1 − 2q)T˙ 2 = ˙ p ⎪ − V (T ) 1 − α T 2, for Multi, (4.11) ⎪ (4.15) ⎪ i=1 ⎩⎪ −NV(T ) 1 − αT˙ 2, for Assisted. Further using the spatially flat k = 0 FLRW metric defined through the following line element: Similarly for the generalized situation comparing Eqs. (4.7) and (4.8) the density ρ and pressure p for tachyonic field can be computed as ds2 =−dt2 + a2(t)dx2, (4.16) ⎧ ⎪ V (T ) ⎪ 1 − α (1 − 2q)T˙ 2 , for Single, ⎪ ˙ 1−q the Friedmann equations can be written as ⎪ 1 − α T 2 ⎪ N ⎨ V (T ) i − α( − ) ˙ 2 , , 2 ρ = − 1 1 2q Ti for Multi a˙ ρ ⎪ − α ˙ 2 1 q 2 = = , ⎪ i=1 1 Ti H (4.17) ⎪ a 3M2 ⎪ ( ) p ⎪ NV T − α( − ) ˙ 2 , , ⎩ − 1 1 2q T for Assisted − α ˙ 2 1 q a¨ (ρ + 3p) 1 T H˙ + H 2 = =− , 2 (4.18) (4.12) a 6Mp

ρ and where the density and pressure p is computed in Eqs. (4.10)– (4.11). Also H is the Hubble parameter, deﬁned as ⎧ q ⎪ ˙ 2 ⎪ −V (T ) 1 − α T , for Single, ⎪ ( ) ˙ ⎪ 1 da t a ⎨ N H = = . (4.19) q a(t) dt a = − ( ) − α ˙ 2 , , p ⎪ V Ti 1 Ti for Multi ⎪ = ⎪ i 1 ⎪ q On the other hand, varying the action as stated in Eq. (4.1) ⎩ − ( ) − α ˙ 2 , . NV T 1 T for Assisted with respect to the tachyon ﬁeld the equation of motion can (4.13) be written as 123 278 Page 18 of 130 Eur. Phys. J. C (2016) 76 :278

⎧ √ α ¨ ( ) ⎪ 1 αβ T ˙ dV T ⎪ √ ∂μ −gV(T ) 1 + g ∂α T ∂β T = + 3Hα T + , for Single, ⎪ −g 1 − αT˙ 2 V (T )dT ⎪ ⎨ √ α ¨ ( ) 1 αβ Ti ˙ dV Ti 0 = √ ∂μ −gV(Ti ) 1 + g ∂α Ti ∂β Ti = + 3Hα Ti + , for Multi, (4.20) ⎪ −g 1 − αT˙ 2 V (T )dT ⎪ i i i ⎪ √ α ¨ ( ) ⎪ 1 αβ T ˙ dV T ⎩ √ ∂μ −gV(T ) 1 + g ∂α T ∂β T = + 3Hα T + , for Assisted. −g 1 − αT˙ 2 V (T )dT

Similarly, in the most generalized case, varying the action as stated in Eq. (4.2) with respect to the tachyon ﬁeld the equation of motion⎧ can be written as ¨ ˙ ⎪ 1 √ αβ q 2qα T T dV (T ) ⎪ √ ∂μ −gV(T ) 1 + g ∂α T ∂β T = + 6qα H + , for Single, ⎪ − − α ˙ 2 − α( − ) ˙ 2 ( ) ⎪ g 1 T 1 1 2q T V T dT ⎨ ¨ ˙ 1 √ αβ q 2qα Ti Ti dV (Ti ) 0 = √ ∂μ −gV(Ti ) 1 + g ∂α Ti ∂β Ti = + 6qα H + , for Multi, ⎪ −g 1 − αT˙ 2 1 − α(1 − 2q)T˙ 2 V (T )dT ⎪ i i i i ⎪ ¨ ˙ ⎪ 1 √ αβ q 2qα T T dV (T ) ⎩⎪ √ ∂μ −gV(T ) 1 + g ∂α T ∂β T = + 6qα H + , for Assisted. −g 1 − αT˙ 2 1 − α(1 − 2q)T˙ 2 V (T )dT (4.21)

Further using Eqs. (4.10), (4.11) and (5.328) the expression for the adiabatic sound speed cA turns out to be ⎧ ⎪ ⎪ ⎪ ⎪ p˙ 2 dV (T ) ⎪ = −w 1 + , for Single, ⎪ ρ˙ α ˙ V (T )dT ⎪ 3H T ⎪ ⎨⎪ N N c = p˙i 2 dV (Ti ) (4.22) A ⎪ = −w 1 + , for Multi, ⎪ ρ˙ i α ˙ ( ) ⎪ = i = 3H Ti V Ti dTi ⎪ i 1 i 1 ⎪ ⎪ ⎪ ⎪ N p˙ 2 dV (T ) ⎩⎪ = −wN 1 + , for Assisted, ρ˙ 3HαT˙ V (T )dT and for the generalized case using Eqs. (4.10), (4.11) and (4.21) the expression for the adiabatic sound speed turns out to be: ⎧ ⎪ ⎪ ⎪ ⎪ ( −α( − q)T˙ 2) ( ) ⎪ −w 1 + 1 1 2 dV T ⎪ 3qHαT˙ V (T )dT ⎪ ! ! , for Single, ⎪ ( − ) ( −α( − q)T˙ 2) ( ) ( −α( − q)T˙ 2) ( ) ⎪ 1 − 1 1 − 1 2q w 1 + 1 1 2 dV T − 1 1 2 dV T ⎪ q (1−q) 6qHαT˙ V (T )dT 6qHαT˙ V (T )dT ⎪ ⎪ ⎪ ⎪ ⎧ ⎫ ⎨ 1−α(1−2q)T˙ 2 ( ) ⎪ −w + i dV Ti ⎪ N ⎨ i 1 ˙ ( ) ⎬ c = 3qHα Ti V Ti dTi A ⎪ , , ⎪ ! for Multi ⎪ ⎪ ( − ) 1−α(1−2q)T˙ 2 ( ) 1−α(1−2q)T˙ 2 ( ) ⎪ ⎪ i=1 ⎩ 1 − − 1 2q w + i dV Ti − i dV Ti ⎭ ⎪ 1 1 ( − ) i 1 α ˙ ( ) α ˙ ( ) ⎪ q 1 q 6qH Ti V Ti dTi 6qH Ti V Ti dTi ⎪ ⎪ ⎪ ⎪ ⎪ (1−α(1−2q)T˙ 2) ( ) ⎪ −wN 1 + dV T ⎪ 3qHαT˙ V (T )dT ⎪ ! ! , for Assisted. ⎪ ( − ) ( −α( − q)T˙ 2) ( ) ( −α( − q)T˙ 2) ( ) ⎩ 1 − 1 1 − 1 2q w 1 + 1 1 2 dV T − 1 1 2 dV T q (1−q) 6qHαT˙ V (T )dT 6qHαT˙ V (T )dT (4.23) It is important to mention here that, substituting q = 1/2inEq.(4.23) one can get back the result obtained in Eq. (4.22). Similarly in the present context the effective sound speed cS is deﬁned as

123 Eur. Phys. J. C (2016) 76 :278 Page 19 of 130 278

⎧ ⎪ ∂ ⎪ p √ ⎪ ∂T˙ 2 = −w, , ⎪ ∂ρ for Single ⎪ ⎪ ∂T˙ 2 ⎪ ⎪ ⎨⎪ ∂p N i N ∂T˙ 2 cS = i = − w , , (4.24) ⎪ ∂ρ i for Multi ⎪ i ⎪ i=1 ∂ ˙ 2 i=1 ⎪ Ti ⎪ ⎪ ∂ ⎪ p √ ⎪ ∂T˙ 2 = −w , , ⎩⎪ N ∂ρ N for Assisted ∂T˙ 2 and for the generalized case the effective sound speed cS is deﬁned as ⎧ ⎪ ∂p ⎪ ∂ ˙ 2 2(q − 1)(1 + w) ⎪ T = 1 + , for Single, ⎪ ∂ρ 1 + (1 − 2q)(2w + 1) ⎪ ∂T˙ 2 ⎪ ⎪ ⎪ ∂p ⎨ N i N ∂T˙ 2 2(q − 1)(1 + w ) c = i = + i , , (4.25) S ⎪ ∂ρ 1 for Multi ⎪ i 1 + (1 − 2q)(2wi + 1) ⎪ i=1 ∂T˙ 2 i=1 ⎪ i ⎪ ⎪ ∂ ⎪ p ⎪ ∂ ˙ 2 2(q − 1)(1 + w) ⎪ N T = N 1 + , for Assisted. ⎩ ∂ρ 1 + (1 − 2q)(2w + 1) ∂T˙ 2 Finally comparing Eqs. (4.22), (4.23), (4.24) and (4.25) we get the following relationship between adiabatic and effective sound speed in tachyonic ﬁeld theory: ⎧ ⎪ ⎪ p˙ 2 dV (T ) ⎪ = c 1 + , for Single, ⎪ ρ˙ S α ˙ ( ) ⎪ 3H T V T dT ⎪ ⎨⎪ N N p˙i 2 dV (Ti ) cA = = 2 + , , (4.26) ⎪ cS,i 1 ˙ for Multi ⎪ ρ˙i 3Hα T V (Ti )dTi ⎪ i=1 i=1 i ⎪ ⎪ ⎪ N p˙ 2 dV (T ) ⎩⎪ = cS 1 + , for Assisted, ρ˙ 3HαT˙ V (T )dT and for the generalized case we get ⎧ ⎪ 2 −α( − ) ˙ 2 ⎪ cS 1 1 2q T dV (T ) ⎪ 1 + ⎪ + (1−2q) 2 − 3qHα T˙ V (T )dT ⎪ 1 ( − ) c 1 ⎪ 1 q S , , ⎪ ( − ) for Single ⎪ 1 2q 2 ⎪ ( − ) c 1−α (1−2q)T˙ 2 ( ) 1−α (1−2q)T˙ 2 ( ) ⎪ 1 − + 1 q S + dV T − dV T ⎪ q 1 1 ( − q) 1 ˙ ( ) ˙ ( ) ⎪ 1+ 1 2 c2 −1 6qHα T V T dT 6qHα T V T dT ⎪ (1−q) S ⎪ ⎪ ⎪ ⎧ ⎫ ⎪ ⎪ 2 −α( − ) ˙ 2 ⎪ ⎪ ⎪ c , 1 1 2q T ( ) ⎪ ⎪ ⎪ S i + i dV Ti ⎪ ⎪ ⎪ (1−2q) 1 α ˙ ( ) ⎪ ⎪ N ⎨ 1+ c2 −1 3qH Ti V Ti dTi ⎬ ⎪ (1−q) S,i ⎨⎪ , for Multi, ⎪ (1−2q) 2 −α( − ) ˙ 2 −α( − ) ˙ 2 ⎪ = = ⎪ ( − ) c , 1 1 2q T ( ) 1 1 2q T ( ) ⎪ cA i 1 ⎪ 1 − + 1 q S i + i dV Ti − i dV Ti ⎪ ⎪ ⎩ q 1 1 (1−2q) 1 α ˙ ( ) α ˙ ( ) ⎭ ⎪ + c2 − 6qH Ti V Ti dTi 6qH Ti V Ti dTi ⎪ 1 (1−q) S,i 1 ⎪ ⎪ ⎪ ⎪ 2 −α( − ) ˙ 2 ⎪ cS% & 1 1 2q T dV (T ) ⎪ 1 + ˙ ( ) ⎪ c2 3qHα T V T dT ⎪ + (1−2q) S − ⎪ 1 (1−q) N 1 ⎪ ⎪ ⎡ ⎧ ⎫ ⎤ , for Assisted. ⎪ ⎪ ⎪ ⎪ ⎨⎪ (1−2q) ⎬⎪ ⎪ ⎢ c2 −α( − ) ˙ 2 −α( − ) ˙ 2 ⎥ ⎪ ⎢ 1 (1−q)% S & 1 1 2q T dV (T ) 1 1 2q T dV (T ) ⎥ ⎪ ⎣ − 1 1 + 1 + ˙ ( ) − ˙ ( ) ⎦ ⎪ q ⎪ c2 ⎪ 6qHα T V T dT 6qHα T V T dT ⎩⎪ ⎩⎪ + (1−2q) S − ⎭⎪ N 1 (1−q) N 1 (4.27) 123 278 Page 20 of 130 Eur. Phys. J. C (2016) 76 :278

1 Let us mention other crucial issues which we use through- |T˙ | >> √ (4.37) out the analysis performed in this paper: α

and for the generalized q = 1/2 case additionally we 1. At early times the tachyonic field satisfies the follow- have to satisfy another constraint: ing small field criteria to validate the effective field theory prescription within the framework of tachyonic string 1 theory: (1 − 2q)|T˙ | >> √ . (4.38) α |T | << 1, (4.28) 6. Further using Eqs. (4.36), (4.37) and (4.38)inEqs.(4.14) Mp and (4.15), the equation of state parameter w can be 1 |T˙ | << √ (4.29) approximated by α ⎧ ⎪ p = α ˙ 2, , ⎪ T for Single ⎪ ρ and for the generalized q = 1/2 case additionally we ⎪ ⎨⎪ have to satisfy another constraint: N p N w = i = α ˙ 2, , (4.39) ⎪ Ti for Multi ⎪ ρi 1 ⎪ i=1 i=1 (1 − 2q)|T˙ | << √ . (4.30) ⎪ ⎪ Np α ⎩⎪ = NαT˙ 2, , ρ for Assisted 2. At early times using Eqs. (4.28), (4.29) and (4.30)in = / Eqs. (4.14) and (4.15), the equation of state parameter and for the generalized q 1 2 case we have w ⎧ can be approximated by ⎪ p 1 ⎪ = , for Single, ⎪ ρ ( − ) ⎪ 2q 1 w ≈−1. (4.31) ⎪ ⎨ N N w = pi 1 ⎪ = , for Multi, 3. At late times the tachyonic field satisfies the following ⎪ ρi (2q − 1) ⎪ i=1 i=1 small field criteria within the framework of tachyonic ⎪ ⎪ Np N string theory: ⎩ = , for Assisted. ρ (2q − 1) |T | (4.40) ∼ 1, (4.32) Mp Here one can control the parameter q and N to get the |T˙ | ∼ √1 desired value of equation of state parameter w, which is 2 (4.33) Mp α necessarily required to explain the cosmological dynamics. and for the generalized q = 1/2 case additionally we 7. Additionally, it is important to note that, within the set- have to satisfy another constraint: up of string theory, the tachyonic modes can form cluster 1 on small cosmological scales. Consequently tachyonic (1 − 2q)|T˙ |∼√ . (4.34) α string theory can be treated as a unified prescription to explain the inflationary paradigm and dark matter. 4. At late times using Eqs. (4.32), (4.33) and (4.34)in 8. Reheating and creation of matter particles in a class of Eqs. (4.14) and (4.15), the equation of state parameter specific models where the minimum of the tachyon poten- ( ) →∞ w can be approximated by tial V T is at T , which is a pathological issue in the present context because the tachyon field in such w ≈ 0. (4.35) a type of string theories does not participate in oscillations.3 To solve this crucial pathological problem in the 5. There might be another interesting possibility appear in present context, one can think about a particular phys- the present context, where the tachyonic modes satisfy ical situation where the universe is initially dominated the large field criteria, represented by the following con- by a inflationary epoch and this can be explained via the straint: energy density of the tachyon condensate as mentioned

|T | 3 >> 1, (4.36) Here it is important to note that the oscillations are necessarily Mp required to explain the reheating phenomenon. 123 Eur. Phys. J. C (2016) 76 :278 Page 21 of 130 278

in the introduction of this article and according to this of spatially flat FLRW metric and in the presence of the proposal the set-up will always remain dominated by Einstein–Hilbert term in the gravity sector. Below we explic- the tachyons. To resolve this pathological issue it may itly show that the solution for the tachyonic field can happen that the tachyon condensation phenomenon is explain various phases of the universe starting from inflation potentially responsible for a short period of an inflation- to the dust formation. To study the cosmological dynam- ary epoch prior, which occurs at a Planckian mass scale, ics from the tachyonic string field theoretic set-up let us 18 Mp ∼ 2.43 × 10 GeV and also one needs to require start with the following solution ansatz of the tachyon a second stage of inflationary epoch just followed by field:

⎧ ⎛ ⎞ ⎪ 2t − ⎪ 1 exp bT 1 1 t ⎪ √ ⎝ 0 ⎠ = √ tanh , for Single, ⎪ ⎪ α exp 2t + 1 α bT0 ⎪ bT0 ⎛ ⎞ ⎪ ⎨ N N 2t − N 1 exp bT 1 1 t T˙ = T˙ = √ ⎝ 0i ⎠ = √ tanh , for Multi, (4.41) ⎪ i ⎪ α 2t + α bT0i ⎪ i=1 i=1 exp 1 i=1 ⎪ ⎛ ⎞ bT0i ⎪ 2t − ⎪ N exp bT 1 N t ⎪ √ ⎝ 0 ⎠ = √ , , ⎩⎪ tanh for Assisted α exp 2t + 1 α bT0 bT0

the previous one which occurs at the vicinity of the GUT which will satisfy the equation of motion of the tachyon field scale (1016 GeV). This directly implies that the tachyon as stated in Eqs. (5.328) and (4.21) respectively. Here b is a serves no crucial purpose in the post inflationary epoch new parameter of the theory which has inverse square mass till at the very later stages of its cosmological evolution dimension, i.e., [M]−1. Consequently the argument of the on time scales. All these crucial pathological problems hyperbolic functions, i.e., ( t ) is dimensionless. From var- bT0 do not appear in the context of the well-known hybrid ious cosmological observations it is possible to put strin- inflationary set-up where the complex tachyon field has gent constraint on the newly introduced parameter b. Further a specific minimum value at the sub-Planckian (

123 278 Page 22 of 130 Eur. Phys. J. C (2016) 76 :278

⎧ ⎡ ⎤ ⎪ ˙ sinh t ⎪ ⎢ 1 V t 3V bT0 ⎥ ⎪ ⎣ + coth + ⎦ , for Single, ⎪ bT V bT t M ⎪ 0 0 cosh p ⎪ bT0 ⎪ ⎡ ⎤ ⎪ ⎨ N ˙ t ⎢ 1 V t 3V sinh bT ⎥ 0 = ⎣ + coth + 0i ⎦ , for Multi, (4.43) ⎪ bT i V bT i t Mp ⎪ i=1 0 0 cosh ⎪ bT0i ⎪ ⎡ ⎤ ⎪ ⎪ ˙ t ⎪ ⎢ 1 V t 3V sinh bT ⎥ ⎪ N ⎣ + coth + 0 ⎦ , for Assisted, ⎩⎪ bT0 V bT0 cosh t Mp bT0 and for the generalized case we get ⎧ ⎡ 1 ⎤ ⎪ 3 V sinh t ⎪ ⎢ ˙ t bT0 ⎥ ⎪ 1 1 V t cosh bT ⎪ ⎢ + + 0 ⎥ , , ⎪ ⎣ coth ⎦ for Single ⎪ bT0 2q V bT0 − ( − ) 2 t ⎪ Mp 1 1 2q tanh ⎪ bT0 ⎪ ⎪ ⎡ 1 ⎤ ⎪ 3V t ⎨⎪ sinh N ⎢ ˙ cosh t bT0i ⎥ = ⎢ 1 + 1 V t + bT0i ⎥ , , 0 ⎪ ⎣ coth ⎦ for Multi (4.44) ⎪ bT i 2q V bT i 2 t ⎪ i=1 0 0 Mp 1 − (1 − 2q)tanh ⎪ bT0i ⎪ ⎪ ⎡ 1 ⎤ ⎪ ⎪ 3 V sinh t ⎪ ⎢ ˙ cosh t bT0 ⎥ ⎪ 1 1 V t bT ⎪ N ⎢ + coth + 0 ⎥ , for Assisted. ⎪ ⎣ ⎦ ⎩ bT0 2q V bT0 M 1 − (1 − 2q)tanh2 t p bT0

The solutions of Eqs. (4.43) and (4.44)aregivenby ⎧ ⎡ ⎤ ⎪ 2 ⎪ λ 1 ⎪ ⎣ √ !⎦ , for Single, ⎪ λ ⎪ cosh t 1 + 3 bT0 t − tanh t ⎪ bT0 2Mp bT0 bT0 ⎪ ⎡ ⎤ ⎪ 2 ⎨ N N λ i ⎣ 1 ⎦ V (t) = Vi (t) = √ ! , for Multi, (4.45) ⎪ t 3λbT0i t t ⎪ i=1 i=1 cosh 1 + − tanh ⎪ bT0i 2Mp bT0i bT0i ⎪ ⎪ ⎡ ⎤ ⎪ 2 ⎪ Nλ 1 ⎪ ⎣ √ !⎦ , for Assisted, ⎪ λ ⎩ cosh t 1 + 3 bT0 t − tanh t bT0 2Mp bT0 bT0 and for the generalized case we get ⎧ ⎡ ⎤ ⎪ 2 ⎪ λ 1 ⎪ ⎣ √ !⎦ , for Single, ⎪ 2q λ ⎪ t + 3 bT0 t − t ⎪ cosh 1 2M bT tanh bT ⎪ bT0 p ⎡0 0 ⎤ ⎪ 2 ⎪ N N ⎨ λ 1 V (t) = i ⎣ √ !⎦ , for Multi, V (t) = i 2q λ (4.46) ⎪ t + 3 bT0i t − t ⎪ i=1 i=1 cosh 1 tanh ⎪ bT0i 2Mp bT0i bT0i ⎪ ⎡ ⎤ ⎪ 2 ⎪ ⎪ Nλ 1 ⎪ ⎣ √ !⎦ , for Assisted, ⎪ 2q λ ⎩ t + 3 bT0 t − t cosh 1 M bT tanh bT bT0 2 p 0 0

123 Eur. Phys. J. C (2016) 76 :278 Page 23 of 130 278 where to get the analytical solution from the generalized case we assume that the time scale of our consideration satisfies the following constraint: − 1 t << bT tanh 1 (4.47) 0 2q − 1 which is valid for all values of q except q = 1/2. For the usual tachyonic case and for the generalized situation we use the following normalization condition: V (t = 0) = V (0) = λ (4.48) to fix the value of arbitrary integration constant. Further using the explicit solution for the tachyonic field as appearing in Eq. (4.42), we can write the time as a function of tachyonic field as ⎧ %√ & ⎪ αT ⎪ −1 , ⎪ bT0 cosh exp for Single ⎪ bT0 ⎪ ⎨⎪ %√ & N α = −1 T , , t ⎪ bT0i cosh exp for Multi (4.49) ⎪ bT i ⎪ i=1 0 ⎪ %√ & ⎪ α ⎪ −1 T ⎩⎪ NbT0 cosh exp , for Assisted, bT0 and further substituting Eq. (4.49)inEqs.(4.45) and (4.46) we get the following expression for the potential as a function of tachyonic field: ⎧ % & ⎡ ⎤ ⎪ √ 2 ⎪ αT 1 ⎪ λ exp − ⎣ √ √ ! √ !!⎦ , for Single, ⎪ λ α α ⎪ bT0 1 + 3 bT0 cosh−1 exp T − tanh cosh−1 exp T ⎪ 2Mp bT0 bT0 ⎪ ⎡ ⎤ ⎪ % √ & 2 ⎨ N α Ti 1 V (T ) = λ exp − ⎣ √ √ ! √ !!⎦ , for Multi, ⎪ i λ α α ⎪ bT i 3 bT0i −1 Ti −1 Ti ⎪ i=1 0 1 + cosh exp − tanh cosh exp ⎪ 2Mp bT0i bT0i ⎪ ⎡ ⎤ ⎪ % √ & 2 ⎪ ⎪ α T 1 ⎪ Nλ exp − ⎣ √ √ ! √ !!⎦ , for Assisted, ⎩⎪ λ α α bT0 1 + 3 bT0 cosh−1 exp T − tanh cosh−1 exp T 2Mp bT0 bT0 (4.50) and for the generalized case we get ⎧ ⎡ ⎤ % & 2 ⎪ √ ⎪ 2q αT ⎢ 1 ⎥ ⎪ λ exp − ⎣ √ √ ! √ !! ⎦ , for Single, ⎪ λ α α ⎪ bT0 + 3 bT0 −1 2q T − −1 2q T ⎪ 1 M cosh exp bT tanh cosh exp bT ⎪ 2 p 0 0 ⎪ ⎡ ⎤ ⎪ % √ & 2 ⎨ N 2q α Ti ⎢ 1 ⎥ V (T ) = λ exp − ⎣ √ √ ! √ !! ⎦ , for Multi, ⎪ i λ α α ⎪ bT i 3 bT0i −1 2q Ti −1 2q Ti ⎪ i=1 0 1 + cosh exp − tanh cosh exp ⎪ 2Mp bT0i bT0i ⎪ ⎡ ⎤ ⎪ % & 2 ⎪ √ ⎪ 2q αT ⎢ 1 ⎥ ⎪ Nλ exp − ⎣ √ √ ! √ !! ⎦ , for Assisted. ⎩⎪ λ α α bT0 1 + 3 bT0 cosh−1 exp 2q T − tanh cosh−1 exp 2q T 2Mp bT0 bT0 (4.51)

123 278 Page 24 of 130 Eur. Phys. J. C (2016) 76 :278

Next we use the following redefinition in the tachyonic field: √ αT T → , (4.52) √bT0 T0 αT T → . (4.53) bT0i T0i Hence using the redefinition, the potential as stated in Eqs. (4.50) and (4.51) can be re-expressed as ⎧ ⎡ ⎤ ⎪ 2 ⎪ T 1 ⎪ λ − ⎣ √ √ ! √ !!⎦ , , ⎪ exp for Single ⎪ T 3λbT0 −1 α T −1 α T ⎪ 0 1 + cosh exp − tanh cosh exp ⎪ 2Mp bT0 bT0 ⎪ ⎪ ⎡ ⎤ ⎪ 2 ⎨ N Ti 1 V (T ) = λ exp − ⎣ √ √ ! √ !!⎦ , for Multi, ⎪ i λ α α ⎪ T i 3 bT0i −1 Ti −1 Ti ⎪ i=1 0 1 + cosh exp − tanh cosh exp ⎪ 2Mp bT0i bT0i ⎪ ⎪ ⎪ ⎡ ⎤ ⎪ 2 ⎪ T 1 ⎪ Nλ exp − ⎣ √ √ ! √ !!⎦ , for Assisted, ⎩⎪ λ − α − α T0 1 + 3 bT0 cosh 1 exp T − tanh cosh 1 exp T 2Mp bT0 bT0 (4.54) and for the generalized case we get ⎧ ⎡ ⎤ ⎪ 2 ⎪ ⎪ 2qT ⎢ 1 ⎥ ⎪ λ exp − ⎣ √ √ ! √ !!⎦ , for Single, ⎪ λ α α ⎪ T0 + 3 bT0 −1 2q T − −1 2q T ⎪ 1 2M cosh exp bT tanh cosh exp bT ⎪ p 0 0 ⎪ ⎪ ⎡ ⎤ ⎪ 2 ⎨ N 2qTi ⎢ 1 ⎥ V (T ) = λ exp − ⎣ √ √ ! √ !!⎦ , for Multi, ⎪ i λ α α ⎪ T i 3 bT0i −1 2q Ti −1 2q Ti ⎪ i=1 0 1 + cosh exp − tanh cosh exp ⎪ 2Mp bT0i bT0i ⎪ ⎪ ⎪ ⎡ ⎤ ⎪ 2 ⎪ ⎪ 2qT ⎢ 1 ⎥ ⎪ Nλ exp − ⎣ √ √ ! √ !!⎦ , for Assisted. ⎪ λ α α ⎩ T0 1 + 3 bT0 cosh−1 exp 2q T − tanh cosh−1 exp 2q T 2Mp bT0 bT0 (4.55)

Now let us explicitly study the limiting behavior of the potential V (T ) in detail, which is appended below:

• At T << T0 and Ti << T0i limiting case one can use the following approximation for the usual tachyonic case: t ln cosh << 1, (4.56) bT0 t ln cosh << 1. (4.57) bT0i

Using this approximation one can use the following expansion: t 1 t 2 ln cosh ≈ , (4.58) bT 2 bT 0 0 t 1 t 2 ln cosh ≈ . (4.59) bT0i 2 bT0i

123 Eur. Phys. J. C (2016) 76 :278 Page 25 of 130 278

• Hence using the solution obtained for the tachyonic field and for the generalized case we have as stated in Eq. (4.42) we get ⎧ ⎪ 2qT ⎪ λ exp − , for Single, ⎧ ⎪ 2 ⎪ T0 ⎪ T0 t ⎪ ⎪ , , ⎨⎪ N N ⎪ for Single 2qT ⎪ 2 bT0 ( ) = ( ) = λ − i , , ⎪ V T ⎪ V Ti i exp for Multi ⎪ ⎪ T0i ⎨ N N 2 ⎪ i=1 i=1 T t ⎪ T (t) = T (t) = 0i , for Multi, ⎪ ⎪ i ⎩⎪ λ − 2qT , , ⎪ = = 2 bT0i N exp for Assisted ⎪ i 1 i 1 T0 ⎪ ⎪ NT t 2 (4.66) ⎩ 0 , for Assisted, 2 bT0 (4.60) where the behavior of these types of potentials has been elaborated in the earlier section. • >> >> and by inverting Eq. (4.78) the associated time scale can At T T0 and Ti T0i limiting case one can use the be computed as following approximation for the usual tachyonic case: ⎧ t >> , ⎪ 2T ln cosh 1 (4.67) ⎪ , , bT0 ⎪ bT0 for Single ⎪ T0 t ⎪ >> . ⎨⎪ ln cosh 1 (4.68) N bT0i = 2T , , t ⎪ bT0i for Multi (4.61) ⎪ T0i ⎪ i=1 Using this approximation one can use the following ⎪ ⎪ expansion: ⎪ 2T , . ⎩ NbT0 for Assisted T0 t 1 t t ln cosh ≈ ln exp ≈ , bT0 2 bT0 bT0 Consequently we have (4.69) % & t ≈ 1 t ≈ t . t 2T 2T ln cosh ln exp Single: tanh ≈ tanh ≈ , bT0i 2 bT0i bT0i bT0 T0 T0 (4.70) (4.62) % & Hence using the solution obtained for the tachyonic field t 2T 2T Multi: tanh ≈ tanh ≈ , as stated in Eq. (4.42) we get bT0i T0i T0i ⎧ (4.63) t % & ⎪ , for Single ⎪ b t 2T 2T ⎪ Assisted : tanh ≈ tanh ≈ . ⎪ ⎨ N N bT0 T0 T0 T (t) = t Nt ⎪ Ti (t) = = , for Multi, (4.64) ⎪ 2b b ⎪ i=1 i=1 ⎪ ⎩⎪ Nt and finally for the usual tachyonic case the potential V (T ) , for Assisted, b can be approximated by (4.71) ⎧ ⎪ T ⎪ λ exp − , for Single, and by inverting Eq. (4.71) the associated time scale can ⎪ ⎪ T0 ⎪ be computed as ⎨ N N ( ) = ( ) = λ − Ti , , ⎧ V T ⎪ V Ti i exp for Multi , , ⎪ = = T0i ⎪ bT for Single ⎪ i 1 i 1 ⎨⎪ ⎪ N ⎪ T ⎩ Nλ exp − , for Assisted, t = bTi , for Multi, (4.72) T ⎪ 0 ⎩⎪ i=1 (4.65) NbT, for Assisted.

123 278 Page 26 of 130 Eur. Phys. J. C (2016) 76 :278

Consequently we have – At T = 0 for the single field tachyonic potential: ( = ) →∞, t V T 0 (4.80) Single: tanh ≈ 1, (4.73) bT0 and for the multi-tachyonic and assisted cases we t have Multi: tanh ≈ 1, (4.74) bT 0i VE (T = 0) →∞. (4.81) : t ≈ . Assisted tanh 1 (4.75) – At T = T0 for the single field tachyonic potential: bT0 2 4Mp and finally for the usual tachyonic case the potential V (T ) V (T = T0) = (4.82) 3eb2T 2 can be approximated by 0 ⎧ and also for the multi-tachyonic and assisted cases ⎪ 4M2 T 2 T ⎪ p 0 − , , we have ⎪ 2 exp for Single ⎪ 3b2T T T0 ⎪ 0 N 2 ⎪ 4M ⎨ N N 2 2 p 4Mp T0i Ti V (T = T ) = , V (T )= V (T ) = exp − , for Multi, E 0 2 (4.83) ⎪ i 2 2 ⎪ 3b2T Ti T0i = 3eb T ⎪ i=1 i=1 0i i 1 0i ⎪ ⎪ 2 2 2 ⎪ 4Mp N T T 4M N ⎩ 0 exp − , for Assisted, ( = ) = p . 3b2T 2 T T VE T T0 (4.84) 0 0 3eb2T 2 (4.76) 0 For the generalized case one can repeat the same and for the generalized case we have computation with the following redefinition of the ⎧ b parameter of tachyonic field theory: M2 2 ⎪ 4 p T0 2qT − − ⎪ exp − , for Single, 2 (− ) → 2. ⎪ 3b2T 2 T T b exp 2q b (4.85) ⎪ 0 0 ⎪ ⎨ N N 2 2 4Mp T0i 2qTi – For single field tachyonic potential: V (T ) = V (T ) = exp − , for Multi, ⎪ i 2 2 ⎪ = = 3b T Ti T0i ⎪ i 1 i 1 0i 2 ⎪ 4Mp T T ⎪ 4M2 N 2 V (T ) =− exp − 2 + , (4.86) ⎩⎪ p T0 − 2qT , , 2 3 2 exp for Assisted 3b T T0 T0 3b2T T T0 0 2 (4.77) 4Mp T V (T ) = − 2 4 exp 3b T T0 where the scale of inflation for the usual tachyonic case T T 2 is fixed by: × 6 + 4 + . (4.87) T0 T0 ⎧ / ⎪ λ1 4, for Single Now to find the extrema of the potential we substitute ⎪ ⎪ / ⎨ N 1 4 ( ) = / V T 0 (4.88) 1 4 ∝ λ , , Vinf ⎪ i for Multi (4.78) ⎪ which give rise to the following solution for the tachy- ⎪ i=1 ⎩ / onic field: (Nλ)1 4, for Assisted, T =−2T0, ∞. (4.89) and for the generalized case it is fixed by Further substituting the solutions for the tachyonic ⎧ % & / field in Eq. (4.87) we get ⎪ 2 1 4 ⎪ 4Mp ⎪ , for Single, e2 M2 ⎪ 2 2 p ⎪ 3b T0 V (T =−2T0) = , (4.90) ⎪ 6b2T 4 ⎪ 0 ⎪ 1/4 ⎨ N 2 ( →∞) → / 4M V T 0 (4.91) 1 4 ∝ p Vinf , for Multi, (4.79) ⎪ 3b2T 2 and also additionally for T = T we have ⎪ i=1 0i 0 ⎪ ⎪ % & 2 ⎪ 1/4 44Mp ⎪ 4M2 N V (T = T ) = , (4.92) ⎪ p 0 2 4 ⎩ , for Assisted. 3eb T0 3b2T 2 0 and at these points the value of the potential is com- Here the potential should satisfy the following criteria: puted as 123 Eur. Phys. J. C (2016) 76 :278 Page 27 of 130 278

V (T →∞) → 0, (4.93) Further substituting the solutions for the tachyonic 2 2 ﬁeld in Eq. (4.98) we get e Mp V (T =−2T ) = , (4.94) 0 2 2 2 2 3b T e Mp 0 V T =−2T = , (4.101) 2 j 0 j 2 4 4Mp 6b T V (T = T ) = . (4.95) 0 j 0 2 2 3b T0 V (Tj →∞) → 0. (4.102)

It is important to note for the single tachyonic case Additionally for the point Tj = T0 j we have that: 2 • for b2 > 0, V (T =−2T , T )>0, i.e., we 44Mp 0 0 V (T = T ) = , (4.103) j 0 j 2 4 get maxima on the potential. 3eb T0 j • for b2 < 0, V (T =−2T , T )<0, i.e., we 0 0 and at these points the value of the total effective get minima on the potential. potential is computed as and for the assisted case the results are the same, N provided the following replacement occurs: ( ) V 1 = V (T →∞) → 0, (4.104) λ → λ, E j N (4.96) j=1 and finally for the multi-tachyonic case we have N N 2 2 ( ) e Mp V 2 = V (T =−2T )= , (4.105) 2 E j 0 j 2 2 4Mp T T 3b T V (T ) =− exp − i 2 + i , j=1 j=1 0 j i 2 3 3b Ti T0i T0i N 2 (3) 4NMp (4.97) V = V (Tj = T0 j ) = . (4.106) E 3b2T 2 4M2 j=1 0 ( ) = p − Ti V Ti 4 exp 3b2T T0i It is important to note for the multi-tachyonic case i 2 that: Ti Ti 2 × 6 + 4 + . (4.98) • for b > 0, V (T =−2T0 j , T0 j )>0, i.e., T T 0i 0i we get maxima on the potential Vj (T ) as well in ( ) Now to find the extrema of the potential we substitute: VE T . • 2 < ( =− , )< for b 0, V T 2T0 j T0 j 0, i.e., V (T ) = 0 ∀ j = 1, 2,...,N (4.99) j we get minima on the potential Vj (T ) as well in ( ) which give rise to the following solutions for the jth VE T . tachyonic field: Next using Eq. (4.17) the Hubble parameter can be T =−2T , ∞. (4.100) j 0 j expressed in terms of the usual tachyonic potential V (T ) as ⎧ ⎡ ⎤ 2 ⎪ ⎪ λ 1 ⎪ ⎣ √ !⎦ , for Single, ⎪ 2 λ ⎪ 3Mp 1 + 3 bT0 t − tanh t ⎪ 2Mp bT0 bT0 ⎪ ⎡ ⎤ ⎪ 2 ⎨⎪ N N λ 1 H 2(t) = H (t) = i ⎣ √ !⎦ , for Multi, (4.107) ⎪ i 2 λ ⎪ 3M + 3 bT0i t − t ⎪ i=1 i=1 p 1 tanh ⎪ 2Mp bT0i bT0i ⎪ ⎡ ⎤ ⎪ 2 ⎪ Nλ 1 ⎪ ⎣ √ !⎦ , for Assisted, ⎩⎪ 2 λ 3Mp 1 + 3 bT0 t − tanh t 2Mp bT0 bT0

123 278 Page 28 of 130 Eur. Phys. J. C (2016) 76 :278 and for the generalized case we have ⎧ ⎡ ⎤ 2 ⎪ ⎪ λ ⎣ 1 ⎦ ⎪ √ ! , for Single ⎪ 2(2q−1) λbT ⎪ 2 t + 3 0 t − t ⎪ 3M cosh 1 2M bT tanh bT ⎪ p bT0 p 0 0 ⎪ ⎡ ⎤ ⎪ 2 ⎨ N λ i ⎣ 1 ⎦ H 2(t) = √ ! , for Multi, (4.108) ⎪ 2(2q−1) 3λbT ⎪ 2 t + 0i t − t ⎪ i=1 3M cosh 1 2M bT tanh bT ⎪ p bT0i p 0i 0i ⎪ ⎡ ⎤ ⎪ 2 ⎪ ⎪ Nλ 1 ⎪ ⎣ √ !⎦ , for Assisted. ⎪ 2(2q−1) λbT ⎩ 2 t + 3 0 t − t 3M cosh 1 2M bT tanh bT p bT0 p 0 0 Let us study the limiting situation from the expression obtained for the Hubble parameter explicitly:

• In the T << T0 and Tj << T0 j limiting situation the Hubble parameter can be approximated for the usual tachyonic case as: ⎧ ⎪ λ ⎪ , for Single, ⎪ 3M2 ⎪ p ⎪ ⎨ N λ H 2(t) = i , for Multi, (4.109) ⎪ 3M2 ⎪ i=1 p ⎪ ⎪ Nλ ⎩⎪ , , 2 for Assisted 3Mp

and for the generalized case we have ⎧ ⎪ λ ⎪ , for Single, ⎪ 2(2q−1) ⎪ t ⎪ 3M2 cosh ⎪ p bT0 ⎪ ⎨⎪ N λi 2( ) = , for Multi, H t ⎪ 2(2q−1) (4.110) ⎪ i=1 3M2 cosh t ⎪ p bT0i ⎪ ⎪ Nλ ⎪ , for Assisted. ⎪ 2(2q−1) ⎩ 3M2 cosh t p bT0

In this limiting situation the solution for the scale factor a(t) from Eq. (4.109) can be expressed as ⎧ √ ⎪ ⎪ λ ⎪ ainf exp √ (t − tinf ) , for Single, ⎪ 3M ⎪ p ⎪ ⎨ N λi a(t) = ainf exp (t − tinf ) , for Multi, (4.111) ⎪ 3M2 ⎪ i=1 p ⎪ √ ⎪ ⎪ Nλ ⎩⎪ ainf exp √ (t − tinf ) , for Assisted, 3Mp

which is exactly the de Sitter solution required for inﬂation. Here at the inﬂationary time scale t = tinf the scale factor is given by

ainf = a(t = tinf ). (4.112) 123 Eur. Phys. J. C (2016) 76 :278 Page 29 of 130 278

On the other hand for the generalized case we have the following solution for the scale factor a(t):

⎧ √ ⎪ λ t ⎪ bT0 t 1 3 2 t ⎪ ainf exp √ sinh 2 F1 , q; ;−sinh , for Single, ⎪ bT 2 2 bT ⎪ 3Mp 0 0 tinf ⎪ ⎨⎪ √ N λ bT t t t ( ) = √ i 0i 1, ; 3;− 2 , , a t ⎪ ainf exp sinh 2 F1 q sinh for Multi (4.113) ⎪ 3M bT0i 2 2 bT0i ⎪ i=1 p tinf ⎪ √ ⎪ λ t ⎪ N bT0 t 1 3 2 t ⎩⎪ ainf exp √ sinh 2 F1 , q; ;−sinh , for Assisted, bT 2 2 bT 3Mp 0 0 tinf

which replicates the behavior of quasi-de Sitter solution which is exactly the dust like solution required for the during inﬂation. For the q = 1/2 case it exactly follows formation of dark matter. Here at the inﬂationary time = the de Sitter behavior. scale t tdust the scale factor is given by

adust = a(t = tdust). (4.117) • In the T >> T0 and Tj >> T0 j limiting situation the Hubble parameter can be approximated for the usual tachyonic case as: On the other hand for the generalized case we have the following solution for the scale factor a(t): ⎧ ⎪ 4 ⎧ ⎪ , ⎪ 4q t ⎪ 2 for Single ⎪ − ⎪ 9t ⎪ adust exp Ei 2q ⎪ ⎪ 3 bT0 ⎨⎪ N ⎪ ! ⎪ 2 4 ⎪ −Ei tdust − 2q , for Single, H (t) = , for Multi, (4.114) ⎪ bT0 ⎪ 2 ⎪ ⎪ = 9t ⎪ ⎪ i 1 ⎪ N q ⎪ ⎨⎪ 4 t − ⎪ 4N adust exp Ei 2q ⎩ , , ( ) = 3 bT0i for Assisted a t ⎪ i=1 9t2 ⎪ ! ⎪ − tdust − , , ⎪ Ei bT 2q for Multi ⎪ 0i and for the generalized case we have ⎪ q ⎪ N4 t ⎪ adust exp Ei − 2q ⎧ ⎪ 3 bT ⎪ 4 ⎪ ! 0 ⎪ , for Single ⎩⎪ ⎪ 2(2q−1) −Ei tdust − 2q , for Assisted, ⎪ 9t2 1 exp t bT0 ⎪ 2 bT0 ⎪ (4.118) ⎪ N ⎨ 4 2( ) = , for Multi, H t ⎪ 2(2q−1) ⎪ i=1 2 1 t which replicates the behavior of quasi-dust like solution. ⎪ 9t 2 exp bT ⎪ 0i For the q = 1/2 case it exactly follows the dust like ⎪ ⎪ 4N ⎪ , for Assisted. behavior. ⎩⎪ 2(2q−1) 9t2 1 exp t 2 bT0 (4.115) 5 Inflationary paradigm from GTachyon In this limiting situation the solution for the scale factor It is a very well known fact that during cosmological infla- a(t) from Eq. (4.114) can be expressed as tion, quantum fluctuations are stretched on the scales larger ⎧ / than the size of the horizon. Consequently, they are frozen ⎪ t 2 3 ⎪ , , until they re-enter the horizon at the end of inflationary phase. ⎪ adust for Single ⎪ tdust In the present context a single tachyonic field drives the infla- ⎪ ⎨ N 2/3 tionary paradigm, which finally gives rise to the large-scale ( ) = t a t adust , for Multi, (4.116) perturbations with a quasi-scale invariant primordial power ⎪ t ⎪ i=1 dust ⎪ spectrum corresponding to the scalar and tensor modes. Devi- ⎪ 2/3 ⎩⎪ t ations from the scale invariance in the primordial power spec- adust , for Assisted, tdust trum can be measured in terms of the slow-roll parameters 123 278 Page 30 of 130 Eur. Phys. J. C (2016) 76 :278 which we will explicitly discuss in the next subsections. By Similarly, in the most generalized case, detailed computation we explicitly show that at lowest order ( ) of the primordial spectrum the scalar perturbations is exactly V T − ( + )α ˙ 2 > . − 1 1 q T 0 (5.6) the same as that obtained for the usual single field inflation- 2 − α ˙ 2 1 q 3Mp 1 T ary set-up. For completeness the next to leading order cor- rections to the cosmological perturbations are also derived, Here Eq. (5.6) implies that to satisfy inflationary constraints which finally give rise to the sufficient change in the cos- in the slow-roll regime the following constraint always holds mological consistency relations with respect to the results good: obtained for the usual single field inflationary set-up. Hence we apply all the derived results for the tachyonic inflationary 1 models as explicitly mentioned in the previous section and T˙ < , (5.7) study the CMB constraints by applying the recent Planck α(1 + q) 2015 data. 9 T¨ < 3H T˙ < H. (5.8) α(1 + q) 5.1 Computation for single field inflation Consequently the field equations are approximated by 5.1.1 Condition for inflation dV (T ) 6qα H T˙ + ≈ 0. (5.9) For single field tachyonic inflation, the prime condition for V (T )dT inflation is given by Also for both cases in the slow-roll regime the Friedmann a¨ (ρ + 3p) equation is modified as H˙ + H 2 = =− > , 2 0 (5.1) a 6Mp V (T ) H 2 ≈ . 2 (5.10) which can be re-expressed in terms of the following con- 3Mp straint condition in the context of single field tachyonic inflation: Further substituting Eq. (5.10)inEqs.(5.5) and (5.9) we get √ ( ) ( ) ( ) V T 3 ˙ 2 3V T ˙ dV T 1 − α T > 0. (5.2) α T + ≈ 0, (5.11) 2 − α ˙ 2 2 M V (T )dT 3Mp 1 T √ p V (T ) dV (T ) q √ αT˙ + ≈ . 6 ( ) 0 (5.12) Here Eq. (5.6) implies that, to satisfy inflationary constraints 3Mp V T dT in the slow-roll regime, the following constraint always holds good: Finally the general solution for both cases can be expressed ( ) 1 in terms of the single field tachyonic potential V T as ˙ 2 √ T < , (5.3) / α 3α T V 3 2(T ) 3 1 t − t ≈− T , i d ( ) (5.13) 6 Mp Ti V T T¨ < 3H T˙ < H. (5.4) α 6qα T V 3/2(T ) t − t ≈−√ T . i d ( ) (5.14) 3Mp Ti V T Consequently the field equations are approximated by Let us now re-write Eqs. (5.332) and (5.333), in terms of the dV (T ) 3Hα T˙ + ≈ 0. (5.5) string theoretic tachyonic potentials as already mentioned in V (T )dT the last section. For the q = 1/2 situation we get

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⎧ ⎧ ⎡1 1 ⎤ ⎪ ⎪ ⎪ ⎪ Ti T ⎪ ⎨⎪ ⎢ sech + 1 1 − sech ⎥ ⎪ α 2 √ T0 T0 ⎪ T0 ⎢ ⎥ ⎪ 3λ ln ⎢1 1 ⎥ ⎪ 2Mp ⎪ ⎣ ⎦ ⎪ ⎪ T + − Ti ⎪ ⎩ sech 1 1 sech ⎪ T0 T0 ⎪ ⎫ ⎪ ⎡1 1 ⎤ ⎪ ⎪ ⎪ Ti − T ⎪ ⎪ ⎢ sech T sech T ⎥ ⎬ ⎪ −1 ⎢ 0 0 ⎥ ⎪ +2tan ⎣ 1 ⎦ , for Model 1, ⎪ ⎪ ⎪ 1 + sech Ti sech T ⎭⎪ ⎪ T0 T0 ⎪ ⎪ ⎛ ⎞ ⎨⎪ T α 2 √ ln T ln i + 1 − ≈ T0 ⎝ T0 T0 ⎠ t ti ⎪ − 3λ ln , for Model 2, (5.15) ⎪ 2Mp Ti T + ⎪ ln T ln T 1 ⎪ 0 0 ⎪ ⎪ 2αT 2 √ ⎪ − 0 λ − T − − Ti , , ⎪ 3 exp exp for Model 3 ⎪ Mp 2T0 2T0 ⎪ % & % & ⎪ ⎪ α T 2 √ T 2 T 2 ⎪ − 0 λ − i − − , , ⎪ 3 Ei 2 Ei 2 for Model 4 ⎪ 4Mp 2T 2T ⎪ 0 0 ⎪ ⎧ ⎛ ⎞ ⎫ ⎪ ⎪ 2 √ ⎨ 4 + 4 + 2 4 4⎬ ⎪ α T T0 T T T T ⎪ 0 λ ⎝ ⎠ + + 0 − + 0 , , ⎪ 3 ⎩ln 1 1 ⎭ for Model 5 ⎩ 8Mp 4 + 4 + 2 Ti T T0 Ti Ti and for any arbitrary q we get the following generalized result: ⎧ ⎧ ⎡1 1 ⎤ ⎪ ⎪ ⎪ 1 ⎪ Ti T ⎪ ⎨ ⎢ sech + 1 1 − sech ⎥ ⎪ 3qα T 2 λ ⎢ T0 T0 ⎥ ⎪ 0 ⎢1 1 ⎥ ⎪ ⎪ln ⎪ Mp 3 ⎪ ⎣ ⎦ ⎪ ⎩⎪ T + − Ti ⎪ sech T 1 1 sech T ⎪ 0 0 ⎪ ⎫ ⎪ ⎡1 1 ⎤ ⎪ ⎪ ⎪ Ti − T ⎪ ⎪ ⎢ sech T sech T ⎥ ⎬ ⎪ −1 ⎢ 0 0 ⎥ ⎪ +2tan ⎣ 1 ⎦ , for Model 1, ⎪ ⎪ ⎪ 1 + sech Ti sech T ⎭⎪ ⎪ T0 T0 ⎪ ⎪ ⎛ ⎞ ⎨⎪ 1 T 2 ln T ln i + 1 3qα T λ T0 T0 t − ti ≈ − 0 ln ⎝ ⎠ , for Model 2, (5.16) ⎪ ⎪ Mp 3 ln Ti ln T + 1 ⎪ T0 T0 ⎪ 1 ⎪ 2 ⎪ 12qα T λ T Ti ⎪ − 0 exp − − exp − , for Model 3, ⎪ M 3 2T 2T ⎪ p 0 0 ⎪ 1 % & % & ⎪ α 2 λ 2 2 ⎪ 3q T0 Ti T ⎪ − Ei − − Ei − , for Model 4, ⎪ 2M 3 2T 2 2T 2 ⎪ p 0 0 ⎪ ⎧ ⎛ ⎞ ⎫ ⎪ 1 ⎪ 2 ⎨ 4 + 4 + 2 4 4⎬ ⎪ 3qα T λ T0 T T T T ⎪ 0 ⎝ ⎠ + + 0 − + 0 . . ⎩⎪ ⎩ln 1 1 ⎭ for Model 5 4Mp 3 4 + 4 + 2 Ti T T0 Ti Ti

123 278 Page 32 of 130 Eur. Phys. J. C (2016) 76 :278

Further using Eqs. (5.15), (5.16) and (5.10) we get the fol- 5.1.2 Analysis using slow-roll formalism lowing solution for the scale factor in terms of the tachyonic field for the usual q = 1/2 and for a generalized value of q Here our prime objective is to define slow-roll parameters for as: tachyon inflation in terms of the Hubble parameter and the ⎧ single field tachyonic inflationary potential. Using the slow- ⎪ α T V 2(T ) ⎪ exp − dT , for q = 1/2, roll approximation one can expand various cosmological ⎪ 2 ( ) ⎨ Mp Ti V T observables in terms of small dynamical quantities derived a = ai × ⎪ from the appropriate derivatives of the Hubble parameter ⎪ 2qα T V 2(T ) ⎩⎪ exp − dT , for any arbitrary q. and of the inflationary potential. To start with here we use M2 V (T ) p Ti the horizon-flow parameters based on derivatives of Hub- (5.17) ble parameter with respect to the number of e-foldings N, defined as Finally re-writing Eq. (5.334), in terms of the string theoretic tend tachyonic potentials as already mentioned in the last section N(t) = H(t) dt, (5.20) for q = 1/2, we get t ⎧ ⎡ ⎛ ⎞⎤ ⎪ 2 T where tend signifies the end of inflation. Further using ⎪ α T λ tanh 2T ⎪ ⎣ 0 ⎝ 0 ⎠⎦ , , ⎪ exp 2 ln for Model 1 Eqs. (5.5), (5.9), (5.10) and (5.335) we get ⎪ M Ti ⎪ p tanh ⎪ 2T0 ⎪ ⎡ ⎛ ⎞⎤ ⎪ T ⎪ 2 T i + ˙ ⎪ α T λ ln T ln T 1 dT T ⎪ ⎣− 0 ⎝ 0 0 ⎠⎦ , , = ⎪ exp 2 ln for Model 2 ⎪ 2M Ti T ⎪ p ln ln + 1 dN H ⎪ T0 T0 ⎧ ⎨ ⎪ 2H α 2λ ⎪ − , = / , a ≈ a × T Ti T ⎨⎪ for q 1 2 i ⎪ exp 0 exp − − exp − , for Model 3, 3α H 3 ⎪ 2 ⎪ Mp T0 T0 = ⎪ ⎪ ˙ 2 ⎪ % & % & ⎪ 2H 1 − α (1 − 2q)T ⎪ 2 2 2 ⎩ − , q, ⎪ α T λ T T 3 for any arbitrary ⎪ exp − 0 Ei − i − Ei − , for Model 4, 3α H 2q ⎪ 2 2 2 ⎪ 4Mp T T ⎪ 0 0 (5.21) ⎪ % & ⎪ ⎪ α T 4λ 1 1 ⎩⎪ exp − 0 − , for Model 5, 8M2 T 2 T 2 ˙ p i where H > 0 which makes always T > 0 during the infla- (5.18) tionary phase. Further using Eq. (5.21) we get the following differential operator identity for tachyonic inflation: and for any arbitrary q we get ⎧ ⎡ ⎛ ⎞⎤ 1 d d d ⎪ T = = ⎪ α 2λ tanh ⎪ 2q T 2T0 H dt dN dlnk ⎪ exp ⎣ 0 ln ⎝ ⎠⎦ , for Model 1, ⎪ 2 ⎧ ⎪ 3Mp tanh Ti ⎪ 2T0 ⎪ 2H d ⎪ ⎪ − , for q = 1/2, ⎪ ⎡ ⎛ ⎞⎤ ⎨ α 3 ⎪ T 3 H dT ⎪ 2 ln T ln i + 1 = ⎪ qα T λ T0 T0 ⎪ exp ⎣− 0 ln ⎝ ⎠⎦ , for Model 2, ⎪ ˙ 2 ⎪ 2 ⎪ 2H 1 − α (1 − 2q)T d ⎪ Mp Ti T + ⎩ − , . ⎪ ln T ln T 1 for any arbitrary q ⎪ 0 0 3α H 3 2q dT ⎨⎪ ≈ × 2qα T 2λ T T a ai ⎪ exp 0 exp − i − exp − , for Model 3, ⎪ 2 ⎪ Mp T0 T0 (5.22) ⎪ ⎪ ⎪ % & % & ⎪ qαT 2λ T 2 2 ⎪ − 0 − i − − T , , ⎪ exp Ei Ei for Model 4 Next we define the following Hubble slow-roll parameters: ⎪ 2M2 T 2 T 2 ⎪ p 0 0 ⎪ ⎪ % & ⎪ ⎪ qα T 4λ 1 1 H ⎪ − 0 − . . 0 = , (5.23) ⎩ exp 2 2 2 for Model 5 4Mp T Ti H (5.19) dln|i | i+ = , i ≥ 1 (5.24) 1 dN Hence using Eqs. (5.15), (5.16), (5.18) and (5.19) one can study the parametric behavior of the scale factor a(t) with where H is the Hubble parameter at the pivot scale. Fur- respect to time t and expected to be as like exact de Sitter ther using the differential operator identity as mentioned in or quasi-de Sitter solution during the inflationary slow-roll Eq. (5.22) we get the following Hubble flow equation for phase, as explicitly derived in the previous section. tachyonic inflation for i ≥ 0: 123 Eur. Phys. J. C (2016) 76 :278 Page 33 of 130 278 ⎧ ⎪ 2H d ⎪− i , for q = 1/2, ⎨ α 3 1 di di 3 H dT = = + = (5.25) i 1 i ⎪ ˙ 2 H dt dN ⎪ 2H 1 − α (1 − 2q)T di ⎩− , for any arbitrary q. 3α H 3 2q dT For a realistic estimate from the single field tachyonic inflationary model substituting the free index i to i = 0, 1, 2in Eqs. (5.23) and (5.25) we get the contributions from the first three Hubble slow-roll parameter, which can be depicted as ⎧ 2 ⎪ 2 H 3 ˙ 2 ⎪ = α T , for q = 1/2 ⎪ 3α H 2 2 ⎪ ⎪ ˙ ⎨ 2 dln|0| H H − α ( − q)T˙ 2 = =− = 2 1 1 2 (5.26) 1 2 ⎪ dN H ⎪ 3α H 2 2q ⎪ ⎪ ⎪ 3 2q ⎩ = α T˙ 2 , for any arbitrary q, 2 1 − α(1 − 2q)T˙ 2 ⎧ ⎪ ¨ ⎪ 2 1 = T , = / ⎪ 2 ˙ for q 1 2 ⎪ 3α 1 H H T ⎪ ⎨⎪ | | ¨ dln 1 H 2 1 − α(1 − 2q)T˙ 2 2 = = + 21 = 1 (5.27) dN H H˙ ⎪ α ⎪ 3 1 2q H ⎪ ⎪ ¨ ⎪ 2T ⎩⎪ = , for any arbitrary q, H T˙ 1 − α(1 − 2q)T˙ 2

... | | ¨ ¨ 2 ˙ 2 = dln 2 = 1 H − H − H + H 3 ˙ 3 3 ˙ 4 4 dN 2 H 2 H H H 2 H 2 H

1 ⎧ ... ⎪ 2 2T ⎪ 1 2 = + − 2 , for q = 1/2, ⎪ α 2 ˙ 1 ⎨ 3 H H T 2 2 = ... (5.28) ⎪ 2 T 2 α( − ) ¨ 2 ⎪ ˙ 2 + − 4 1 2q T ⎪ 21 1 − α (1 − 2q)T H 2T˙ 1 2 ⎩⎪ 2 = 2 + H , for any arbitrary q. α ˙ 2 2 3 2q H 1 − α (1 − 2q)T 1 − α (1 − 2q)T˙ 2 It is important to note that:

˙ 2 • In the present context 1 is characterized by the part of the total tachyonic energy density T . Inflation occurs when 1 < 1 and ends when 1 = 1, which is exactly the same as the other single field slow-roll inflationary paradigm. • The slow-roll parameter 2 characterizes the ratio of the field acceleration relative to the frictional contribution acting on it due to the expansion. ˙ ¨ ... • The third slow-roll parameter 3 is made up of both 1 and 2. More precisely, 3 is made up of T , T and T . This clearly implies that the third slow-roll parameter 3 carries the contribution from the part of total tachyonic energy density, field acceleration relative to the frictional contribution and rate of change of field acceleration. • The slow-roll conditions stated in Eqs. (5.3) and (5.4) are satisfied when the slow-roll parameters satisfy 1 << 1, 2 << 1 and 3 << 1. This also implies that in the slow-roll regime of the tachyonic inflation product of the two slow-roll parameters are also less than unity. For an example from Eq. (5.38) it is clearly observed that to satisfy the slow-roll condition we need to have additionally 23 << 1.

Now for the sake of clarity, using Hamilton–Jacobi formalism, the Friedman equations and conservation equation can be rewritten as

123 278 Page 34 of 130 Eur. Phys. J. C (2016) 76 :278 ⎧ ⎪ 9α α ⎪ H (T ) 2 − H 4(T ) + V 2(T ), for q = 1/2, ⎪ 4 ⎪ 4 4Mp ⎪ − ⎪ 2 2 1 q ⎨ 4 H (T ) 1−α(1−2q)T˙ 2 H 2(T ) 1− 0 ≈ α 4( ) (5.29) ⎪ 9 H T 2q ⎪ ⎪ 2 2 ⎪ V (T ) 4(1 − 2q) H (T ) 1 − α(1 − 2q)T˙ 2 ⎪ − 1 − , for any arbitrary q, ⎩ 2 α 4( ) 3Mp 9 H T 2q and ⎧ ⎪ 3α ⎨⎪ − H 2(T )T˙ , for q = 1/2, 2 H (T ) ≈ (5.30) ⎪ 3α 2q ⎩ − H 2(T )T˙ , for any arbitrary q. 2 1 − α(1 − 2q)T˙ 2

Further using the deﬁnition of the ﬁrst Hubble slow-roll parameter 1 in Eq. (5.29) we get ⎧ 1/2 ⎪ 2 V (T ) ⎪ H 2(T ) − (T ) = , q = / , ⎨ 1 1 2 for 1 2 3 3Mp − (5.31) ⎪ 1 1 q V (T ) (1 − 2q) ⎩⎪ H 2(T ) − (T ) ≈ − (T ) , q, 1 1 2 1 1 for any arbitrary 3q 3Mp 3q where for the arbitrary q we have used the following constraint condition: − α( − ) ˙ 2 ≈ . 1 2 1 342q T 5 1 (5.32) <<1

Now as in the slow-roll regime of tachyonic inﬂation 1(T )<<1, consequently one can expand the exponents appearing in the left hand side of Eq. (5.31), which leads to the following simpliﬁed expression: ⎧ ⎪ 1 V (T ) ⎪ H 2(T ) − (T ) + O(2(T )) = , q = / , ⎨⎪ 1 1 1 2 for 1 2 3 3Mp (5.33) ⎪ 1 − q V (T ) (1 − 2q) ⎩⎪ H 2(T ) − (T ) + O(2(T )) ≈ − (T ) , q. 1 1 1 2 1 1 for any arbitrary 3q 3Mp 3q It is important to mention here that:

• = / O(2( )) The result for q 1 2 implies that except for the second order correction term in slow-roll i.e., 1 T the rest of the contribution exactly matches with the known result for the single field slow-roll inflationary models. But in the non-slow- roll limiting situation truncating at second order in slow-roll is not allowed and in that case for correct computation one needs to consider the full binomial series expansion. • = 1 For q 2 , i.e., for any other arbitrary q in the slow-roll regime of tachyonic inflation we have allowed the second order O(2( )) = 1 correction term in slow-roll i.e., 1 T as appearing for q 2 . But the final result implies significant deviation from the result that is well known for single field slow-roll inflationary models in the slow-roll regime. As mentioned earlier in the non-slow-roll limiting situation truncating at a certain order in slow-roll is not allowed and in that case for correct = 1 computation one needs to consider the full binomial series expansion.Also for q 2 case, the right hand side of Eq. (5.33) gets modified in the presence of slow-roll parameter 1.

Our next objective is to express the Hubble slow-roll parameters in terms of the tachyon potential dependent slow-roll parameters. To serve this purpose let us start with writing the expression for the derivatives of the potential in terms of the Hubble slow-roll parameters. Allowing up to the second order contribution in the Hubble slow-roll parameters we get [86]

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( ) − 2 + 2 2 V T 1 3 1 6 M =− 61 , (5.34) p V (T )H(T ) 1 − 2 3 1 2 2 2 2 V (T ) 1 − 1 + 1 − 1 − M4 = 6 3 6 3 3 p ( ) 2( ) 1 2 V T H T 1 − 2 3 1 2 51 − − 3 + 2 3 + 3 − 2 , (5.35) 2 − 2 1 2 1 3 1 Further using Eqs. (5.36), (5.37), (5.38), (5.34) and (5.35) one can re-express the Hubble slow-roll parameters in terms of the potential dependent slow-roll parameter as ⎧ ⎪ 2 2( ) ⎪ Mp V T V ⎪ = ≡¯V , for q = 1/2, ⎨ 2α V 3(T ) V (T )α 1 ≈ (5.36) ⎪ 2 2 ⎪ Mp V (T ) V ¯V ⎩ = ≡ , for any q, 4qα V 3(T ) 2qV(T )α 2q

⎧ % & 2 2 ⎪ Mp V (T ) V (T ) 2 (3V − ηV ) ⎪ 3 − 2 = = 2 (3¯ −¯η ) , for q = 1/2, ⎨⎪ α V 3(T ) V 2(T ) V (T )α V V 2 ≈ % & (5.37) ⎪ 2 ⎪ 2 2( ) ( ) (3V − ηV ) ⎪ Mp V T V T q 2 ⎩ √ 3 − 2 = = (3¯V −¯ηV ) . for any q 2qα V 3(T ) V 2(T ) V (T )α q

⎧ % & 4 2 4 ⎪ Mp V (T )V (T ) V (T )V (T ) V (T ) ⎪ 2 − 10 + 9 ⎪ V 2(T )α 2 V 2(T ) V 3(T ) V 4(T ) ⎪ ⎪ ⎪ ⎪ ξ 2 − η + 2 ⎪ 2 V 5 V V 36 V ¯ 2 2 ⎪ = = 2ξ − 5η¯V ¯V + 36¯ , for q = 1/2, ⎨⎪ V 2(T )α 2 V V ≈ 3 2 ⎪ % & (5.38) ⎪ 4 2 4 ⎪ Mp V (T )V (T ) V (T )V (T ) V (T ) ⎪ √ 2 − 10 + 9 ⎪ 2( )α 2 V 2(T ) V 3(T ) V 4(T ) ⎪ 2qV T ⎪ ⎪ ⎪ ξ 2 − η + 2 ξ¯ 2 − η¯ ¯ + ¯2 ⎩⎪ 2 V 5 V V 36 V 2 V 5 V V 36 V = √ = √ , for any q, 2qV2(T )α 2 2q ,η ,ξ2 ,σ3 where the potential dependent slow-roll parameters V V V V are defined as 2 2 Mp V (T ) V = , (5.39) 2 V (T ) V (T ) η = M2 , (5.40) V p V (T ) V (T )V (T ) ξ 2 = M4 , (5.41) V p V 2(T ) % & V 2(T )V (T ) σ 3 = M6 , (5.42) V p V 3(T ) which is exactly similar to the expression for the slow-roll parameter as appearing in the context of single field slow-roll inflationary models. However, for the sake of clarity here we introduce new sets of potential dependent slow-roll parameters for tachyonic inflation by rescaling with the appropriate powers of αV (T ):

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2 2 V Mp V (T ) ¯V = = , (5.43) αV (T ) 2αV (T ) V (T )

2 η Mp V (T ) η¯ = V = , (5.44) V αV (T ) αV (T ) V (T ) ξ 2 M4 ( ) ( ) ¯ 2 V p V T V T ξ = = , (5.45) V α 2V 2(T ) α 2V 2(T ) V 2(T ) % & σ 3 M6 2( ) ( ) 3 V p V T V T σ¯ = = . (5.46) V α 3V 3(T ) α 3V 3(T ) V 3(T )

Further using Eqs. (5.39)–(5.46) we get the following operator identity for tachyonic inflation: ⎧ ⎪ ¯ 1/4 ⎪ 2 V − 2¯ d , = / , ⎨⎪ Mp 1 V for q 1 2 1 d d d V (T )α 3 dT = = ≈ (5.47) H dt dN dlnk ⎪ 1/4 ⎪ 2¯V Mp 1 d ⎩⎪ 1 − ¯V , for any arbitrary q. V (T )α 2q 3q dT Finally using Eq. (5.47) we get the following sets of flow equations in the context of tachyonic inflation: ⎧ 1/4 ⎪ d¯V 2 ⎨⎪ = 2¯V (η¯V − 3¯V ) 1 − ¯V , for q = 1/2, d1 dN 3 = (5.48) dN ⎪ ¯ ¯ 1/4 ⎩⎪ 1 d V V 1 = (η¯V − 3¯V ) 1 − ¯V , for any q, 2q dN q 3q ⎧ ⎪ 1/4 ⎪ ¯ η¯ − ¯2 − ξ¯ 2 − 2¯ , = / , ⎨⎪ 2 10 V V 18 V V 1 V for q 1 2 d2 3 = (5.49) dN ⎪ 1/4 ⎪ 2 2 ¯ 2 1 ⎩ 10¯V η¯V − 18¯ − ξ 1 − ¯V , for any q, q V V 3q ⎧ / ⎪ 1 4 ⎪ σ¯ 3 − ¯3 + ξ¯ 2 η¯ − ξ¯ 2 ¯ + ¯2 η¯ − η¯2 ¯ − 2¯ , = / , ⎨ 2 V 216 V 2 V V 7 V V 194 V V 10 V V 1 V for q 1 2 d(23) 3 = (5.50) dN ⎪ σ¯ 3 − ¯3 + ξ¯ 2 η¯ − ξ¯ 2 ¯ + ¯2 η¯ − η¯2 ¯ 1/4 ⎩⎪ 2 V 216 V 2 V V 7 V V 194 V V 10 V V 1 √ 1 − ¯V , for any q, 2q 3q where we use the following consistency conditions for the rescaled potential dependent slow-roll parameters: ⎧ ⎪ 1/4 ⎪ 2 ⎨ 2¯V (η¯V − 3¯V ) 1 − ¯V , for q = 1/2, d¯ 3 V = (5.51) ⎪ 1/4 dN ⎪ 1 ⎩ 2¯V (η¯V − 3¯V ) 1 − ¯V . for any q 3q ⎧ ⎪ 1/4 ⎪ ¯ 2 2 ⎨ ξ − 4¯V η¯V 1 − ¯V , for q = 1/2, dη¯ V 3 V = (5.52) dN ⎪ 1/4 ⎪ ¯ 2 1 ⎩ ξ − 4¯V η¯V 1 − ¯V . for any q, V 3q ⎧ ⎪ 1/4 ⎪ σ¯ 3 + ξ¯ 2 η¯ − ξ¯ 2 ¯ − 2¯ , = / , ¯ 2 ⎨ V V 1 V for q 1 2 dξ V V V 3 V = (5.53) dN ⎪ 1/4 ⎪ 3 ¯ 2 ¯ 2 1 ⎩ σ¯ + ξ η¯V − ξ ¯V 1 − ¯V . for any q V V V 3q

123 Eur. Phys. J. C (2016) 76 :278 Page 37 of 130 278 % & ⎧ ( ) / k 1 V T ⎪ 2 1 4 N(k) ≈ 71.21 − ln + ln ⎪ σ¯ 3 (η¯ − ¯ ) − ¯ , = / , k 4 M4 σ¯ 3 ⎨ V V 12 V 1 V for q 1 2 0 p d V 3 = 1 V (T) 1 − 3w ρ dN ⎪ 1/4 + + int reh , ⎪ 3 1 ln ln ⎩ σ¯ (η¯V − 12¯V ) 1 − ¯V , for any q. 4 ρ 12(1 + w ) ρ V 3q end int end (5.59) (5.54) ρ ρ In terms of the slow-roll parameters, the number of e-foldings where end is the energy density at the end of inflation, reh = can be re-expressed as is an energy scale during reheating, cSk0 a0 H0 is the ( ) ⎧ 1 present Hubble scale, V T corresponds to the potential ⎪ 3α Tend H(T ) energy when the relevant modes left the Hubble patch dur- ⎪ √ T ⎪ d = ⎪ 2 T 1 ing inflation corresponding to the momentum scale cSk ⎪ 1 = w ⎪ T a H cSkcmb, and int characterizes the effective equation ⎪ 3α end H(T ) ⎪ = √ dT of state parameter between the end of inflation and the energy ⎪ 2 ¯ ⎪ T V scale during reheating. Further using Eq. (5.59)inEq.(5.57) ⎪ ⎪ α T V 2(T ) we get the following expression: ⎪ ≈ dT, for q = 1/2 ⎨ M2 V (T ) p Tend % & N(T ) = 1 ( ) ⎪ T k 1 V T ⎪ 3α end H(T ) N ≈ 71.21 − ln + ln ⎪ 2q √ dT k 4 M4 ⎪ 2 0 p ⎪ T 1 ⎪ 1 V (T) 1 − 3wint ρreh ⎪ Tend ( ) + ln + ln , (5.60) ⎪ = α H T ρ ( + w ) ρ ⎪ 3 q √ dT 4 end 12 1 int end ⎪ T ¯ ⎪ V ⎪ √ ⎪ 2qα T V 2(T ) which is very useful to fix the number of e-foldings within ⎩⎪ ≈ dT. for any q 2 ( ) < < Mp Tend V T 50 N 70 for tachyonic inflation. (5.55) where Tend characterizes the tachyonic field value at the end 5.1.3 Basics of tachyonic perturbations of inflation t = tend. It is important to mention here that in the single field tachyonic inflationary paradigm the field In this subsection we explicitly discuss the cosmological lin- value of the tachyon at the end of inflation is computed from ear perturbation theory within the framework of tachyonic the following condition: inflation. Let us clearly mention that here we have various ways of characterizing cosmological perturbations in the , |η |, |ξ 2 |, |σ 3 | ≡ . maxφ=φe V V V V 1 (5.56) context of inflation, which finally depend on the choice of gauge. Let us do the computation in the longitudinal guage, Let T denote the value of tachyonic field T at which a length where the scalar metric perturbations of the FLRW back- scale or more precisely the modes crosses the Hubble radius ground are given by the following infinitesimal line element: during inflation, which is given by the momentum at pivot = scale k a H.Herea and H signify the scale factor ds2 =−(1 + 2(t, x)) dt2 and Hubble parameter at the horizon crossing scale or at the + 2( ) ( − ( , )) δ i j , pivot scale. Then the definition of the number of e-foldings a t 1 2 t x ijdx dx (5.61) as stated in Eqs. (5.335) and (5.55)gives where a(t) is the scale factor, (t, x) and (t, x) charac- aend N = N(T) = ln , (5.57) terizes the gauge invariant metric perturbations. Specifically, a the perturbation of the FLRW metric leads to the perturbation in the energy-momentum stress tensor via the Einstein where aend is the scale factor at the end of inflation. Then using Eq. (5.57) the corresponding horizon crossing momen- field equation or equivalently through the Friedmann equa- tum scale or the pivot scale can be computed as tions. For the perturbed metric as mentioned in Eq. (5.61), the perturbed Einstein field equations can be expressed for cSk = a H = aend H exp (−N) . (5.58) the q = 1/2 case of the tachyonic inflationary set-up as

Now at any arbitrary momentum scale the number of e- k2 1 foldings, N(k), between the Hubble exit of the relevant 3H H(t, k) + (˙ t, k) + =− δρ, (5.62) 2( ) 2 modes and the end of inﬂation can be expressed as a t 2Mp 123 278 Page 38 of 130 Eur. Phys. J. C (2016) 76 :278

k2 1 (¨ t, k) + 3H H(t, k) + (˙ t, k) + H(˙ t, k) + 2H˙ (t, k) + ((t, k) − (t, k)) = δp, (5.63) 2( ) 2 3a t 2Mp αV (T ) T˙ (˙ t, k) + H(t, k) =− δT, (5.64) ˙ 2 1 − α T 2 Mp (t, k) − (t, k) = 0. (5.65)

Similarly, for any arbitrary q the perturbed Einstein ﬁeld equations can be expressed as k2 1 3H H(t, k) + (˙ t, k) + =− δρ, (5.66) 2( ) 2 a t 2Mp k2 1 (¨ t, k) + 3H H(t, k) + (˙ t, k) + H(˙ t, k) + 2H˙ (t, k) + ((t, k) − (t, k)) = δp, (5.67) 3a2(t) 2M2 p α V (T ) 1 − α (1 − 2q)T˙ 2 T˙ (˙ t, k) + H(t, k) =− δT, ˙ 1−q 2 1 − α T 2 Mp (5.68) (t, k) − (t, k) = 0. (5.69) Here (t, k) and (t, k) are the two gauge invariant metric perturbations in the Fourier space, deﬁned via the following transformation: (t, x) = d3k (t, k) exp(ik.x), (5.70) (t, x) = d3k (t, k) exp(ik.x). (5.71)

Additionally, it is important to note that in Eq. (5.65), the two gauge invariant metric perturbations (t, k) and (t, k) are equal in the context of minimally coupled tachyonic string ﬁeld theoretic model with an Einstein gravity sector. In Eqs. (5.62) and (5.63) the perturbed energy density δρ and pressure δp are given by ⎧ ⎪ ( )δ αV (T ) T˙ δT˙ + T˙ 2(t, k) ⎪ V T T + , = / , ⎪ / for q 1 2 ⎪ − α ˙ 2 − α ˙ 2 3 2 ⎪ 1 T 1 T ⎪ ⎪ 6 7 ⎨⎪ V (T ) 1 − α (1 − 2q)T˙ 2 δT − 4α (1 − 2q)V (T )T˙ δT˙ δρ = − (5.72) ⎪ − α ˙ 2 1 q ⎪ 1 T ⎪ ⎪ ⎪ ˙ ˙ ˙ ˙ ⎪ 2α (1 − q)V (T ) 1 − α (1 − 2q)T 2 T δT + T 2(t, k) ⎪ + , , ⎩ 2−q for any arbitrary q 1 − αT˙ 2 and ⎧ ⎪ αV (T ) T˙ δT˙ + T˙ 2(t, k) ⎪ ˙ 2 ⎪ −V (T ) 1 − α T δT + , for q = 1/2, ⎨⎪ 1 − αT˙ 2 δp = (5.73) ⎪ ⎪ q 2qα V (T ) T˙ δT˙ + T˙ 2(t, k) ⎪ − ( ) − α ˙ 2 δ + , . ⎩ V T 1 T T 1−q for any arbitrary q 1 − αT˙ 2 Similarly after the variation of the tachyoinic ﬁeld equation motion we get the following expressions for the perturbed equation of motion:

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⎧ ¨ ˙ ˙ ˙ 2 ˙ 2 2 2 ⎪ 2α T T δT + T (t, k) Mp 1 − α T k 2k ⎪ δT¨ + 3HδT˙ + + − 3H˙ (t, k) − (t, k) ⎪ − α ˙ 2 αV (T ) a2 a2 ⎪ 1 T ⎪ ⎪ ⎪ − (¨ , ) + (˙ , ) + (˙ , ) + ˙ ( , ) + 2( , ) ⎪ 3 t k 4H t k H t k H t k 4H t k ⎪ ⎪ ⎪ ! ⎪ −α ˙ 2 2 ⎪ ˙ 3 2V (T ) ˙ 2 ˙ ˙ ˙ Mp 1 T V (T ) V (T ) ⎪ − 6Hα T − 1 − α T (t, k) − (t, k) + 3(t, k) T − − , for q = 1/2, ⎨⎪ α V (T ) α V (T ) V 2(T ) 0 ≈ 1−q ⎪ 2αT¨ T˙ δT˙ + T˙ 2(t, k) M 1 − αT˙ 2 2 2 ⎪ δ ¨ + δ ˙ + + p k − ˙ ( , ) − 2k ( , ) ⎪ T 3H T ( − ) 3H t k t k ⎪ − α ˙ 2 2 1 q α V (T ) 1 − α (1 − 2q)T˙ 2 a2 a2 ⎪ 1 T ⎪ ⎪ ( ) 2(1−q) ⎪ − (¨ , ) + (˙ , ) + (˙ , ) + ˙ ( , ) + 2( , ) − α ˙ 3 − 2 V T − α ˙ 2 ⎪ 3 t k 4H t k H t k H t k 4H t k 6H T 1 T ⎪ q α V (T ) ⎪ ⎪ ( − ) % & ⎪ ˙ 2 2 1 q 2 ⎪ Mp 1 − α T V (T ) V (T ) ⎩⎪ ×(t, k) − (˙ t, k) + 3(˙ t, k) T˙ − √ − , for any q. 2qα V (T ) V 2(T ) (5.74)

Further we will perform the following steps throughout the where δ p¯ characterizes the non-adiabatic or entropic con- next part of the computation: tribution in the first order linearized cosmological pertur- • First of all we decompose the scalar perturbations into bation. In the present context δ p¯ can be expressed as two components-(1) entropic or isocurvature perturba- δ ¯ = ˙, tions which can be usually treated as the orthogonal pro- p p (5.79) jective part to the trajectory and (2) adiabatic or curva- where characterizes the relative displacement between ture perturbations which can be usually treated as the hypersurfaces of uniform pressure and density. Addi- parallel projective part to the trajectory. tionally, it is important to note that Eq. (5.375) signi- • If inflation is governed by a single scalar field then we fies the change in the curvature perturbation on the uni- deal with adiabatic or curvature perturbations.Onthe form density hypersurfaces on the large scales. Also from other hand for multiple scalar fields we deal with entropic Eq. (5.375) it is clearly observed that the contribution or isocurvature perturbations. from the time evolution of the adiabatic or curvature per- • In the present context the inflationary dynamics is gov- turbations are directly proportional to the non-adiabatic erned by a single tachyonic field, which implies the sur- contribution which comes from significantly from the viving part of the cosmological perturbations are gov- isocurvature part of pressure perturbation δp and are com- erned by the adiabatic contribution. pletely independent of the specific mathematical struc- • Within the framework of first order cosmological per- ture of the gravitational field equations in the context of turbation theory we define a gauge invariant primordial Einstein gravity framework. In a generalized prescrip- curvature perturbation on the scales outside the horizon: tion the pressure perturbation in arbitrary gauge can be H decomposed into the following two contributions: ζ = − δρ. (5.75) ρ˙ δ = 2 δρ + δ ¯, p cS p (5.80) • Next we consider the uniform density hypersurface in 2 where cS is the effective sound speed, which is mentioned which in the earlier section of the paper. • Now let us consider a situation where the pressure pertur- δρ = 0. (5.76) bation is completely made up of adiabatic contribution from the cosmological perturbation on large cosmologi- Consequently the curvature perturbation is governed by cal scales. Consequently we get ζ = . (5.77) δ p¯ = p˙ = 0 ⇒ ζ = Constant, (5.81)

• Further, the time evolution of the curvature perturbation which is consistent with the single field slow-roll condi- can be expressed as tions in the context of the tachyonic inflationary set-up. Finally, in the uniform density hypersurfaces, the curva- δ p¯ ζ˙ = H , (5.78) ture perturbation can be written in terms of the tachyonic ρ + p field fluctuations on spatially flat hypersurfaces as 123 278 Page 40 of 130 Eur. Phys. J. C (2016) 76 :278 δT ζ =−H . (5.82) T˙

5.1.4 Computation of scalar power spectrum

In this subsection our prime objective is to compute the primordial power spectra of scalar quantum ﬂuctuations from tachyonic inﬂation and study the cosmological consequences from the previously mentioned string theory originating tachyonic potentials in the light of Planck 2015 data. To serve this purpose let us start with the following canonical variable vk, which can be quantized with the standard techniques: vk ≡ zMp ζk, (5.83) where ζk is the curvature perturbation in the momentum space, which can be expressed in terms of the curvature perturbation in position space through the following Fourier transformation: 3 ζ(t, x) = d k ζk(t) exp(ik.x). (5.84)

Also z is defined as ⎧ √ ⎪ α ( ) ˙ ( ) ( ) ⎪ 3 a t T = a t = a t ¯ , = / , ⎨⎪ 2 1 2 V for q 1 2 ( ) √ 1 − αT˙ 2 cS cS = a t ρ + = z p ⎪ √ √ (5.85) cS HMp ⎪ 6qαa(t)T˙ 1 + (1 − 2q)αT˙ 2 a(t) 2 a(t) 2¯ ⎩⎪ = 1 = √ V , for any q. ˙ 2 1 − αT˙ 2 1 − (1 − 2q)α T cS 2qcS Next we use conformal time η instead of using the time t, which is defined via the following infinitesimal transformation: dt = a dη (5.86) using which one can redefine the Hubble parameter in conformal coordinate system as 1 da(η) H(η) = = aH(t). (5.87) a(η) dη Further we derive the equation of motion of the scalar fluctuation by extremizing the tachyonic model action as 2 2 d 2 2 1 d z + c k − vk = 0 (5.88) dη2 S z dη2 where − 4 Higher order slow52 roll correction 3 2 1 dz d 1 d2 d 2 1 + 1 1 1 d z 2 z dη dη 2 dη2 1 dη = 2a2 H 2 + + η2 2 2 z d 3 1 − 1 9 − 2 ⎧ 3 1 3 1 ⎪ ¯ 2¯ ¯ 2 ⎪ 1 dz d V + 1 d V d V ⎪ 2 z dη dη 2 dη2 1 dη ⎪ 2a2 H 2 + + ⎪ 3 1 − 2 ¯ 9 − 2 ¯ 2 ⎪ 3 V 1 V ⎪ 3 ⎪ = 2 2 + ¯ − η¯ + ( ¯ −¯η )2 +¯ ( ¯ −¯η ) + 1 ξ¯ 2 − η¯ ¯ + ¯2 +··· , = / , ⎪ a H 2 8 V 3 V 3 V V V 3 V V 2 V 5 V V 36 V for q 1 2 ⎪ 2 ⎨⎪ ¯ 2¯ ¯ 2 1 dz d V + 1 d V d V = 1 z dη dη 2 dη2 1 dη ⎪ a2 H 2 + + ⎪ 2 2 2 ⎪ 3q − 1 ¯ 36q 1 ⎪ 1 3q V 1 − ¯V ⎪ 3q ⎪ 9 1 3 1 1 ⎪ = a2 H 2 2 + √ − ¯ − √ η¯ + (3¯ −¯η )2 + ¯ (3¯ −¯η ) ⎪ V V V V ( )3/2 V V V ⎪ 2q 2q 2q 2q 2q ⎪ ⎩⎪ + √1 2ξ¯ 2 − 5η¯ ¯ + 36¯2 +··· , for any q, 2 2q V V V V (5.89)

123 Eur. Phys. J. C (2016) 76 :278 Page 41 of 130 278 and the factor aH can be expressed in terms of the conformal As a result the previously mentioned integration constants time η as = π , = are fixed at the values C1 2 C2 0. Consequently the ⎧ solution of the mode function for scalar fluctuations takes the ⎪ − 1 ( +¯ ) +··· , = / , following form: ⎨ 1 V for q 1 2 1 aH ηπ η ≈ (5.90) (1) ⎪ 1 ¯V vk(η) = − Hν (−kcSη) . (5.97) ⎩ − 1 + +··· , for any arbitrary q. 2 aH 2q On the other hand, the solution stated in Eq. (5.94) determines Further replacing the factor aH in Eq. (5.89), we finally get the future evolution of the mode including its super-horizon the following simplified expression:

⎧ 2 ⎪ 1 3 1 ⎪ + 4¯ −¯η − +··· , for q = 1/2, ⎨⎪ η2 V V 1 d2z 2 4 ≈ (5.91) z dη2 ⎪ 2 ⎪ 1 3 1 3 1 1 ⎩ + + √ ¯V − √ η¯V − +··· , for any q. η2 2 2q 2q 2q 4

dynamics at cSk << aH or |kcSη| << 1orkcSη → 0 and Now for further simplification in the computation of scalar this is due to power spectrum we introduce a new factor ν, which is defined − η −ν as (1) i kcS ⎧ lim Hν (−kcSη) = (ν) . (5.98) kc η→0 π 2 ⎪ 3 S ⎨⎪ + 4¯V −¯ηV +··· , for q = 1/2, 2 ν ≈ ⎪ Consequently the solution of the mode function for scalar ⎪ 3 1 3 1 ⎩ + + √ ¯V − √ η¯V +··· , for any q. fluctuations takes the following form: 2 2q 2q 2q 1 −ν (5.92) ηπ i −kcSη vk(η) = − (ν) . (5.99) 2 π 2 Hence using Eq. (5.100)inEq.(5.88), we get the following simplified form of the equation of motion: % & Finally combining the results obtained in Eqs. (5.94), (5.97) 2 ν2 − 1 and (5.128) we get d 2 2 4 + c k − vk(η) = 0, (5.93) dη2 S η2 ⎧ √ ⎪ −η C H (1) (−kc η) ⎪ 1 ν S ⎪ and the most general solution of Eq. (5.123)isgivenby ⎪ ⎪ + (2) (− η) , , √ ⎪ C2 Hν kcS for AV (1) (2) ⎨ vk(η) = −η C1 Hν (−kcSη) + C2 Hν (−kcSη) , 1 vk(η) = ηπ ⎪ − (1) (− η) , +| η| >> , ⎪ Hν kcS for BD kcS 1 (5.94) ⎪ 2 ⎪ 1 ⎪ −ν where C1 and C2 are two arbitrary integration constants, ⎪ ηπ i −kcSη ⎩ − (ν) , for BD +|kcSη| << 1, which can be fixed from the appropriate choice of the bound- 2 π 2 (1) (2) ary conditions. Additionally Hν and Hν represent the (5.100) Hankel function of the first and second kind with rank ν.Now where AV and BD signify the arbitrary vacuum and the to impose the well known Bunch–Davies boundary condition Bunch–Davies vacuum respectively. Finally the two point at early times we have used: function from a scalar fluctuation for both AV and BD can (1) lim Hν (−kcSη) be expressed as η→−∞ kcS 1 2 π H 1 2 1 1 ζ ζ = δT δT = v v = √ exp (ikcSη) exp i ν + , (5.95) k k ˙ k k 2 2 k k π −η 2 2 T z Mp (2) 2π 2 lim Hν (−kcSη) = ( π)3δ3( + ) ( ), kc η→−∞ 2 k k ζ k (5.101) S 1 k3 2 1 π 1 where the primordial power spectrum for the scalar modes at = √ exp (−ikcSη) exp −i ν + . π −η 2 2 any arbitrary momentum scale k can be written for both AV (5.96) and BD with q = 1/2as

123 278 Page 42 of 130 Eur. Phys. J. C (2016) 76 :278

⎧ ⎧ 8 8 2 ⎪ ⎪ 2ν−3 (− η )3−2ν 2 8 (ν) 8 ⎪ ⎪ 2 k cS H 8 8 ⎪ ⎪ 8 8 , for |kcSη| << 1, ⎪ ⎪ 8c ¯ (1 +¯ )2π 2 M2 8 3 8 ⎪ ⎪ S V V p 2 ⎪ ⎪ 8 8 ⎪ ⎨ − ν 8 82 ⎪ 2ν−3 2 2 ( +¯ )1−2ν 2 8 (ν) 8 ⎪ 2 cS 1 V H for BD, ⎪ ⎪ 8 8 , for |kcSη|=1, 3 3 2 ⎨ ⎪ 8¯ π 2 M2 8 3 8 k Pζ (k) k |vk| ⎪ V p 2 ζ (k) ≡ = = ⎪ (5.102) π 2 π 2 2 2 ⎪ ⎪ (1) 2 2 z Mp ⎪ ⎪ (−kηc )3 H 2|Hν (−kc η) |2 ⎪ ⎩⎪ S S , |kc η| >> , ⎪ 2 2 for S 1 ⎪ 8cS¯V (1 +¯V ) π M ⎪ p ⎪ 8 8 ⎪ 8 (1) (2) 82 ⎪ (−kηc )3 H 2 8C Hν (−kc η) + C Hν (−kc η)8 ⎪ S 1 S 2 S ⎩⎪ , for AV, ¯ ( +¯ )2π 2 2 4cS V 1 V Mp and similarly the primordial power spectrum for the scalar modes at any arbitrary momentum scale k can be written for both AV and BD with any arbitrary q as ⎧ ⎧ 8 8 2 ⎪ ⎪ 2ν−3 3−2ν 2 8 8 ⎪ ⎪ 2 q (−kηcS) H 8 (ν) 8 ⎪ ⎪ 8 8 , for |kc η| << 1, ⎪ ⎪ 2 8 3 8 S ⎪ ⎪ ¯ + 1 ¯ π 2 2 ⎪ ⎪ 4cS V 1 V Mp 2 ⎪ ⎪ 2q ⎪ ⎪ − ν ⎪ ⎨ − ν 1 2 8 8 ⎪ 22ν−3qc2 2 1 + 1 ¯ H 2 8 82 ⎪ S 2q V 8 (ν) 8 ⎪ ⎪ 8 8 , for |kc η|=1, for BD ⎪ ⎪ ¯ π 2 2 8 3 8 S 3 3 2 ⎨ ⎪ 4 V Mp k Pζ (k) k |vk| ⎪ 2 ζ (k) ≡ = = ⎪ ( ) (5.103) π 2 π 2 2 2 ⎪ ⎪ (− η )3 2| 1 (− η) |2 2 2 z Mp ⎪ ⎪ q k cS H Hν kcS ⎪ ⎪ , for |kcSη| >> 1. ⎪ ⎩⎪ 2 ⎪ 4c ¯ 1 + 1 ¯ π M2 ⎪ S V 2q V p ⎪ 8 8 ⎪ 8 82 ⎪ (− η )3 2 (1) (− η) + (2) (− η) ⎪ q k cS H 8C1 Hν kcS C2 Hν kcS 8 ⎪ , , ⎪ 2 for AV ⎩ ¯ + 1 ¯ π 2 2 2cS V 1 2q V Mp where the effective sound speed cS is given by ⎧ 1 1 ⎪ 2 2 ⎪ 1 − = 1 − ¯V , for q = 1/2, ⎨⎪ 3 1 3 c = S 2 1 (5.104) ⎪ 1 − 1 − ¯V ⎪ 3 1 = 3q , . ⎩⎪ ( − ) ( − ) for any q 1 + 2 1 2q 1 + 1 2q ¯ 3q 1 3q2 V Now starting from the expression for primordial power spectrum for the scalar modes one can compute the spectral tilt at any arbitrary momentum scale k for both AV and BD with q = 1/2as

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dlnζ (k) dlnζ (k) nζ (k) − 1 ≡ = dlnk dN ⎧ ⎧ ⎪ ⎪ 1/4 ⎪ ⎪ 2 2 ⎪ ⎪ (3 − 2ν) 1 − ¯V (η¯V − 3¯V ) 1 − ¯V +··· , for |kcSη| << 1 ⎪ ⎪ 3 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1/4 ⎪ ⎪ 2 2 ⎪ ⎪ (3 − 2ν) 1 − ¯V (η¯V − 3¯V ) 1 − ¯V +··· , for |kcSη|=1, ⎪ ⎨ 3 3 ⎪ ⎪ for BD, ⎪ ⎪ 1/4 ⎪ ⎪ 2 2 ⎪ ⎪ ¯V − ¯V (η¯V − 3¯V ) + 2(η¯V − 3¯V ) 1 − ¯V ⎪ ⎪ 3 3 ⎪ ⎪ ⎪ ⎪ ⎨⎪ ⎪ (− η) (1) (− η) − (1) (− η) ⎪ cS Hν−1 kcS Hν+1 kcS ≈ ⎩⎪ + +··· , for |kc η| >> 1, (1) S (5.105) ⎪ Hν (−kc η) ⎪ S ⎪ / ⎪ 2 2 1 4 ⎪ ¯ − ¯ (η¯ − 3¯ ) + 2(η¯ − 3¯ ) 1 − ¯ ⎪ V V V V V V V ⎪ 3 3 ⎪ ⎪ (− η) (1) (− η) − (1) (− η) ⎪ cS C1 Hν−1 kcS Hν+1 kcS ⎪ + ⎪ (1) (2) ⎪ C Hν (−kc η) + C Hν (−kc η) ⎪ 1 S 2 S ⎪ ⎪ (2) (2) ⎪ (−cSη)C2 Hν− (−kcSη)−Hν+ (−kcSη) ⎩⎪ + 1 1 +··· , . (1) (2) for AV C1 Hν (−kcSη)+C2 Hν (−kcSη) and for both AV and BD with any arbitrary q as dlnζ (k) dlnζ (k) nζ (k) − 1 ≡ = dlnk dN ⎧ ⎧ ⎪ ⎪ 1/4 ⎪ ⎪ ( − ν) − 1 ¯ (η¯ − ¯ ) − 1 ¯ +··· , | η| << , ⎪ ⎪ 3 2 1 V V 3 V 1 V for kcS 1 ⎪ ⎪ 3q 3q ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1/4 ⎪ ⎪ ( − ν) − 1 ¯ (η¯ − ¯ ) − 1 ¯ +··· , | η|= , ⎪ ⎨⎪ 3 2 1 V V 3 V 1 V for kcS 1 ⎪ 3q 3q ⎪ , ⎪ ⎪ / for BD ⎪ ⎪ ¯ 1 1 2 1 4 ⎪ ⎪ V − ¯ (η¯ − 3¯ ) + (η¯ − 3¯ ) 1 − ¯ ⎪ ⎪ V V V V V V ⎪ ⎪ 2q 3q q 3 ⎪ ⎪ ⎪ ⎪ (− η) (1) (− η) − (1) (− η) ⎨⎪ ⎪ cS Hν−1 kcS Hν+1 kcS ⎩⎪ + +··· , for |kc η| >> 1, = (1) S (5.106) ⎪ Hν (−kcSη) ⎪ ⎪ ¯ 1/4 ⎪ V 1 1 2 ⎪ − ¯V (η¯V − 3¯V ) + (η¯V − 3¯V ) 1 − ¯V ⎪ 2q 3q q 3 ⎪ ⎪ ( ) ( ) ⎪ (−c η) C H 1 (−kc η) − H 1 (−kc η) ⎪ S 1 ν−1 S ν+1 S ⎪ + ⎪ (1) (− η) + (2) (− η) ⎪ C1 Hν kcS C2 Hν kcS ⎪ ⎪ ⎪ (− η) (2) (− η) − (2) (− η) ⎪ cS C1 Hν−1 kcS Hν+1 kcS ⎪ + +··· , for AV. ⎩⎪ (1) (2) C1 Hν (−kcSη) + C2 Hν (−kcSη) One can also consider the following approximations to simplify the ﬁnal derived form of the primordial scalar power spectrum for the BD vacuum with the |kcSη|=1 case:

123 278 Page 44 of 130 Eur. Phys. J. C (2016) 76 :278

1. We start with the Laurent expansion of the Gamma function: 1 1 π 2 1 γπ2 (ν) = − γ + γ 2 + ν − γ 3 + + 2ζ(3) ν2 + O(ν3). (5.107) ν 2 6 6 2

where γ being the Euler–Mascheroni constant and ζ(3) characterizing the Riemann zeta function of order 3 originating in the expansion of the gamma function. 2. Hence using the result of Eq. (5.107)forq = 1/2 and for arbitrary q we can write: ⎧ ⎪ π 2 ⎪ 1 − γ + 1 γ 2 + 3 + ¯ −¯η ⎪ 4 V V ⎪ 3 + 4¯ −¯η 2 6 2 ⎪ 2 V V ⎪ ⎪ 1 γπ2 3 2 ⎪ − γ 3 + + 2ζ(3) + 4¯ −¯η +··· , for q = 1/2, ⎪ V V ⎪ 6 2 2 ⎨⎪ 1 ! − γ (ν) = (5.108) ⎪ 3 + 1 + √3 ¯ − √1 η¯ ⎪ 2 2q 2q V 2q V ⎪ ⎪ π 2 ⎪ + 1 γ 2 + 3 + 1 + √3 ¯ − √1 η¯ ⎪ V V ⎪ 2 6 2 2q 2q 2q ⎪ ⎪ γπ2 2 ⎪ 1 3 3 1 3 1 ⎩ − γ + + 2ζ(3) + + √ ¯V − √ η¯V +··· , for any q. 6 2 2 2q 2q 2q

3. In the slow-roll regime of inﬂation all the slow-roll parameters satisfy the following constraint:

¯V << 1, (5.109)

|¯ηV | << 1, (5.110) |ξ¯ 2 | << , V 1 (5.111) |¯σ 3 | << . V 1 (5.112)

Using these approximations the primordial scalar power spectrum can be expressed as 2 2 CE H ζ, ≈ 1 − (C + 1) − E 1 2 π 2 2 2 8 MpcS 1 = ⎧ k a H ⎪ H 2 ⎪ [1 − (C + 1)¯ − C (3¯ −¯η )]2 , for q = 1/2, ⎪ E V E V V π 2 2 ¯ ⎪ 8 MpcS V ⎨⎪ k=a H = ⎪ (5.113) ⎪ ⎪ 2 2 ⎪ ¯V CE qH ⎪ − (C + ) − √ ( ¯ −¯η ) , q, ⎩ 1 E 1 3 V V 2 2 for any 2q 2q 4π M cS¯V p k=a H

where CE is given by

CE =−2 + ln 2 + γ ≈−0.72. (5.114)

4. Using the slow-roll approximations one can further approximate the expression for the sound speed as ⎧ ⎪ − 2¯ + O(¯2 ) +··· , = / , ⎨ 1 V V for q 1 2 2 = 3 cS ( − ) (5.115) ⎪ 1 q 2 ⎩ 1 − ¯V + O(¯ ) +··· , for any q. 3q2 V

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5. Hence using the result in Eq. (5.169) we get the following simpliﬁed expression for the primordial scalar power spectrum: ⎧ ⎪ 5 2 H 2 ⎪ 1 − C + ¯ − C (3¯ −¯η ) , for q = 1/2, ⎨⎪ E V E V V π 2 2¯ 6 8 Mp V = ≈ k a H ζ, ⎪ (5.116) ⎪ ¯ C 2 qH2 ⎪ − (C + − ) V − √ E ( ¯ −¯η ) , q, ⎩ 1 E 1 3 V V 2 2 for any 2q 2q 4π M ¯V p k=a H

where the factor is deﬁned as ⎧ ⎪ 1 ⎨⎪ , for q = 1/2 6 = (5.117) ⎪ 1 − q ⎩ , for any q. 6q

6. Next one can compute the scalar spectral tilt (nS) of the primordial scalar power spectrum as ⎧ ⎪ − − − 2 − C + 8 − C +··· ⎪ 2 1 2 2 1 2 E 1 2 E 2 3 ⎪ 3 ⎪ ⎪ ⎪ = η¯ − ¯ − ¯2 − C + 8 ¯ ( ¯ −¯η ) ⎪ 2 V 8 V 2 V 2 2 E V 3 V V ⎪ 3 ⎪ ⎨⎪ −C ξ¯ 2 − η¯ ¯ + ¯2 +··· , = / , E 2 V 5 V V 36 V for q 1 2 nζ, − 1 ≈ 2 ⎪ −21 − 2 − 2 − (2CE + 3 − 2) 12 − CE 23 +··· ⎪ %1 & ⎪ ⎪ 2 1 2 ¯2 2 ⎪ = η¯ − + 3 ¯ − V − (2C + 3 − 2) ¯ (3¯ −¯η ) ⎪ V V 2 ( )3/2 E V V V ⎪ q q q 2q 2q ⎪ ⎪ C ⎩⎪ E ¯ 2 2 −√ 2ξ − 5η¯V ¯V + 36¯ +··· , for any q. 2q V V (5.118)

7. Next one can compute the running of the scalar spectral tilt (αS) of the primordial scalar power spectrum as dnζ (k) dnζ (k) αζ, = = dlnk dN ⎧ k=a H k=a H ⎪ − ¯ ( +¯ )(η¯ − ¯ ) + ¯ η¯ − ¯2 − ξ¯ 2 ⎪ 4 V 1 V V 3 V 2 10 V V 18 V V ⎪ ⎪ ⎪ −C σ¯ 3 − ¯3 + ξ¯ 2 η¯ − ξ¯ 2 ¯ + ¯2 η¯ − η¯ ¯ ⎪ E 2 V 216 V 2 V V 7 V V 194 V V 10 V V ⎪ ⎪ / ⎪ 8 2 1 4 ⎪ − 2C + 2¯ 10¯ η¯ − 18¯2 − ξ¯ 2 − 4¯ (3¯ −¯η )2 1 − ¯ +··· , for q = 1/2, ⎪ E V V V V V V V V V ⎪ 3 3 ⎪ ¯ ¯ ⎨⎪ 2 V V 2 2 ¯ 2 − 1 + (η¯V − 3¯V ) + 10¯V η¯V − 18¯ − ξ ≈ q q 2q q V V ⎪ ⎪ ⎪ C ⎪ −√ E 2σ¯ 3 − 216¯3 + 2ξ¯ 2 η¯ − 7ξ¯ 2 ¯ + 194¯2 η¯ − 10η¯ ¯ ⎪ V V V V V V V V V V ⎪ 2q ⎪ ⎪ ⎪ 8 2 2 ¯ 2 4 2 ⎪ − 2CE + ¯V 10¯V η¯V − 18¯ − ξ − ¯V (3¯V −¯ηV ) ⎪ 3 q V V (2q)3/2 ⎪ ⎪ 1/4 ⎪ 1 ⎩ × 1 − ¯V +··· , for any q. 3q (5.119)

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8. Finally, one can also compute the running of the running where the helicity index γ is summed over in the Fourier of scalar spectral tilt (κS) of the primordial scalar power modes for the tensor contribution uk and spectrum as d2nζ (k) d2nζ (k) κζ, = = 2 2 dlnk = dN = ⎧ k a H k a H ⎪ − ¯ ( +¯ )(η¯ − ¯ )2 + ¯2 (η¯ − ¯ )2 + ¯ ( +¯ ) ξ¯ 2 − ¯ η¯ + ¯2 ⎪ 8 V 1 V V 3 V 8 V V 3 V 8 V 1 V V 10 V V 18 V ⎪ ⎪ ⎪ + ¯ η¯ (η¯ − ¯ ) + ¯ ξ¯ 2 − ¯ η¯ ⎪ 2 20 V V V 3 V 10 V V 4 V V ⎪ ⎪ / ⎪ 2 1 4 ⎪ − ¯2 (η¯ − ¯ ) − σ¯ 3 + ξ¯ 2 η¯ − ξ¯ 2 ¯ − ¯ +··· , = / , ⎪ 72 V V 3 V V V V V V 1 V for q 1 2 ⎪ 3 ⎨ 3/2 ¯ ≈ 2 V 2 2 2 2 2 ⎪ − ¯V 1 + (η¯V − 3¯V ) + ¯ (η¯V − 3¯V ) ⎪ q 2q q q2 V ⎪ ⎪ ⎪ ¯ ⎪ 2 2 V ¯ 2 2 2 10 ¯ 2 ⎪ + ¯V 1 + ξ − 10¯V η¯V + 18¯ + ¯V η¯V (η¯V − 3¯V ) + 10¯V ξ − 4¯V η¯V ⎪ q q 2q V V q q V ⎪ ⎪ ⎪ 1/4 ⎪ 36 2 3 ¯ 2 ¯ 2 1 ⎩ − ¯ (η¯V − 3¯V ) − σ¯ + ξ η¯V − ξ ¯V 1 − ¯V +··· , for any q. q V V V V 3q (5.120)

⎧ 5.1.5 Computation of tensor power spectrum ⎨⎪ a2 H 2 [2 −¯ ] +··· , for q = 1/2, 2 V 1 d a = η2 ⎪ 2 2 1 a d ⎩ a H 2 − ¯V +··· , for any q. In this subsection our prime objective is to compute the 2q primordial power spectra of tensor quantum fluctuations (5.124) from tachyonic inflation and study the cosmological consequences from the previously mentioned string theory orig- Further replacing the factor aH in Eq. (5.89), we finally get inating tachyonic potentials in the light of Planck 2015 data. the following simplified expression: ⎧ To serve this purpose let us start with the following canon- ⎪ 1 3 2 1 ⎪ +¯ − +···, for q = 1/2, ical variable uk, which can be quantized with the standard ⎨⎪ η2 V 1 d2a 2 4 techniques: ≈ η2 ⎪ a d ⎪ 1 3 1 2 1 γ a γ ⎪ + ¯ − +···, q. ⎩ 2 V for any u ≡ √ Mp h , (5.121) η 2 2q 4 k 2 k (5.125) γ where h is the curvature perturbation in the momentum k Now for further simplification in the computation of tensor space, which can be expressed in terms of the curvature power spectrum we introduce a new factor μ which is defined perturbation in position space through the following Fourier as transformation: ⎧ ⎪ 3 ⎨⎪ +¯V +··· , for q = 1/2, 2 γ ( , ) = 3 γ ( ) ( . ). μ ≈ h t x d khk t exp ik x (5.122) ⎪ (5.126) ⎩⎪ 3 1 + ¯V +··· , for any q. 2 2q Here the superscript γ stands for the helicity index for the transverse and traceless spin-2 graviton degrees of freedom. Hence using Eq. (5.126)inEq.(5.123), we get the following simplified form of the equation of motion: In general, the tensor modes can be written in terms of the % & two orthogonal polarization basis vectors. 2 μ2 − 1 d 2 4 + k − uk(η) = 0, (5.127) Further we derive the equation of motion of the tensor dη2 η2 fluctuation by extremizing the tachyonic model action as and the most general solution of Eq. (5.127) is given by 2 2 d 2 1 d a √ ( ) ( ) + k − uk = 0, (5.123) u (η) = −η D H 1 (−kη) + D H 2 (−kη) , (5.128) dη2 a dη2 k 1 μ 2 μ

123 Eur. Phys. J. C (2016) 76 :278 Page 47 of 130 278 where D1 and D2 are two arbitrary integration constants, On the other hand, the solution stated in Eq. (5.128) deter- which can be fixed from the appropriate choice of the bound- mines the future evolution of the mode including its super- (1) (2) ary conditions. Additionally Hμ and Hμ represent the horizon dynamics at k << aH or |kη| << 1orkη → 0 and Hankel function of the first and second kind with rank μ.Now this is due to to impose the well-known Bunch–Davies boundary condi- − η −μ (1) i k tion at early times we have used lim Hμ (−kη) = (μ) . (5.134) kη→0 π 2 (1) lim Hμ (−kη) kη→−∞ Consequently the solution of the mode function for tensor 1 fluctuations takes the following form: 2 1 π 1 1 = √ exp (ikη) exp i μ + , (5.129) ηπ − η −μ π −η 2 2 i k uk(η) = − (μ) . (5.135) (2) 2 π 2 lim Hμ (−kη) kη→−∞ 1 Finally combining the results obtained in Eqs. (5.94), (5.97) 2 1 π 1 and (5.128) we get = √ exp (−ikη) exp −i μ + . (5.130) π −η 2 2

⎧ √ ⎪ −η (1) (− η) + (2) (− η) , , ⎪ D1 Hμ k D2 Hμ k for AV ⎪ 1 ⎨⎪ ηπ − H (1) (−kη) , +|kη| >> , uk(η) = μ for BD 1 (5.136) ⎪ 2 ⎪ 1 −μ ⎪ ηπ i −kη ⎩⎪ − (μ) , for BD +|kη| << 1, 2 π 2

where AV and BD signify an arbitrary vacuum and the As a result the previously mentioned integration constants Bunch–Davies vacuum respectively. Finally the two point are fixed at the following values: function from the tensor fluctuations for both AV and BD 1 can be expressed as π % & = , D1 (5.131) 4 2 2 h h = δψ δψ = u u k k 2 k k 2 2 k k D2 = 0. (5.132) Mp a Mp π 2 Consequently the solution of the mode function for tensor 3 3 2 = (2π) δ (k + k ) h(k), (5.137) fluctuations takes the following form: k3 1 ηπ where the primordial power spectrum for tensor modes at (1) uk(η) = − Hμ (−kη) . (5.133) any arbitrary momentum scale k can be written for both AV 2 and BD with q = 1/2as

Due to graviton helicity 4523 3 3 2 k Ph(k) k |uk| h(k) ≡ 2 × = 2π 2 π 2a2 M2 ⎧ ⎧ 8 8 p ⎪ ⎪ μ− − μ 8 82 ⎪ ⎪ 22 2 (−kη)3 2 H 2 8 (μ) 8 ⎪ ⎪ 8 8 , for |kη| << 1 ⎪ ⎪ (1 +¯ )2π 2 M2 8 3 8 ⎪ ⎪ V p 2 ⎪ ⎪ 8 8 ⎪ ⎨ 8 82 ⎪ 22μ−2 (1 +¯ )1−2μ H 2 8 (μ) 8 ⎪ V 8 8 , for |kη|=1 for BD, ⎨ ⎪ π 2 2 8 3 8 ⎪ Mp = ⎪ 2 (5.138) ⎪ ⎪ (1) ⎪ ⎪ (−kη)3 H 2|Hμ (−kη) |2 ⎪ ⎩⎪ , for |kη| >> 1, ⎪ ( +¯ )2π 2 ⎪ 1 8V Mp 8 ⎪ 8 82 ⎪ (− η)3 2 8 (1) (− η) + (2) (− η)8 ⎪ 2 k H D1 Hμ k D2 Hμ k ⎩⎪ , for AV, ( +¯ )2π 2 2 1 V Mp

123 278 Page 48 of 130 Eur. Phys. J. C (2016) 76 :278 and similarly the primordial power spectrum for tensor modes at any arbitrary momentum scale k can be written for both AV and BD with any arbitrary q as Due to graviton helicity 4523 3 3 2 k Ph(k) k |uk| h(k) ≡ 2 × = 2π 2 π 2a2 M2 ⎧ ⎧ 8 8 p 2 ⎪ ⎪ 2μ−1 3−2μ 2 8 8 ⎪ ⎪ 2 q (−kη) H 8 (μ) 8 ⎪ ⎪ 8 8 , for |kη| << 1 ⎪ ⎪ 2 8 3 8 ⎪ ⎪ + 1 ¯ π 2 2 2 ⎪ ⎪ 1 2q V Mp ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨⎪ 1−2μ 8 8 ⎪ 2μ−1 + 1 ¯ 2 8 82 ⎪ 2 q 1 q V H 8 (μ) 8 ⎪ 2 , | η|= for BD ⎪ ⎪ 8 8 for k 1 ⎨ ⎪ π 2 M2 8 3 8 ⎪ p 2 = ⎪ (5.139) ⎪ ⎪ ⎪ ⎪ (1) ⎪ ⎪ 2q (−kη)3 H 2|Hμ (−kη) |2 ⎪ ⎪ , for |kη| >> 1. ⎪ ⎩⎪ 2 ⎪ ¯ 1 + 1 ¯ π M2 ⎪ V 2q V p ⎪ 8 8 ⎪ 8 82 ⎪ (− η)3 2 8 (1) (− η) + (2) (− η)8 ⎪ 4q k H C1 Hμ k C2 Hμ k ⎪ , for AV. ⎩⎪ 2 + 1 ¯ π 2 2 1 2q V Mp Now starting from the expression for the primordial power spectrum for the tensor modes one can compute the spectral tilt at any arbitrary momentum scale k for both AV and BD with q = 1/2as

dlnh(k) dlnh(k) nh(k) ≡ = dlnk dN ⎧ ⎧ / ⎪ ⎪ 2 2 1 4 ⎪ ⎪ (3 − 2μ) 1 − ¯ (η¯ − 3¯ ) 1 − ¯ +··· , for |kη| << 1, ⎪ ⎪ V V V V ⎪ ⎪ 3 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1/4 ⎪ ⎪ 2 2 ⎪ ⎪ (3 − 2μ) 1 − ¯ (η¯ − 3¯ ) 1 − ¯ +··· , for |kη|=1, ⎪ ⎪ V V V V ⎪ ⎨ 3 3 ⎪ ⎪ for BD, ⎪ ⎪ ⎪ ⎪ 1/4 1/4 ⎪ ⎪ ¯ − 2¯ (η¯ − ¯ ) − 2¯ − (η¯ − ¯ ) − 2¯ ⎪ ⎪ V V V 3 V 1 V 2 V 3 V 1 V ⎪ ⎪ 3 3 3 ⎨ ⎪ ⎪ ≈ ⎪ (5.140) ⎪ ⎪ (−η) (1) (− η) − (1) (− η) ⎪ ⎪ Hμ−1 k Hμ+1 k ⎪ ⎩⎪ +··· , for |kη| >> 1, ⎪ (1) ⎪ Hμ (−kη) ⎪ ⎪ ⎪ ⎪ 1/4 1/4 ⎪ ¯ − 2¯ (η¯ − ¯ ) − 2¯ − (η¯ − ¯ ) − 2¯ ⎪ V V V 3 V 1 V 2 V 3 V 1 V ⎪ 3 3 3 ⎪ ⎪ ⎪ ⎪ (−η) (1) (− η) − (1) (− η) (−η) (2) (− η) − (2) (− η) ⎪ C1 Hμ−1 k Hμ+1 k C1 Hμ−1 k Hμ+1 k ⎪ +··· , for AV, ⎩⎪ (1) (2) (1) (2) C1 Hμ (−kη) + C2 Hμ (−kη) C1 Hμ (−kη) + C2 Hμ (−kη)

123 Eur. Phys. J. C (2016) 76 :278 Page 49 of 130 278 and for both AV and BD with any arbitrary q as

dlnh(k) dlnh(k) nh(k) ≡ = ⎧ ⎧dlnk dN / ⎪ ⎪ 1 1 1 4 ⎪ ⎪ (3 − 2μ) 1 − ¯ (η¯ − 3¯ ) 1 − ¯ +··· , for |kη| << 1, ⎪ ⎪ q V V V q V ⎪ ⎪ 3 3 ⎪ ⎪ ⎪ ⎪ 1/4 ⎪ ⎪ 1 1 ⎪ ⎪ (3 − 2μ) 1 − ¯V (η¯V − 3¯V ) 1 − ¯V +··· , for |kη|=1, ⎪ ⎨ 3q 3q ⎪ ⎪ for BD, ⎪ ⎪ ¯ 1/4 1/4 ⎪ ⎪ V 1 1 1 2 ⎪ ⎪ − ¯V (η¯V − 3¯V ) 1 − ¯V − (η¯V − 3¯V ) 1 − ¯V ⎪ ⎪ 2q 3q 3q q 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (−η) (1) (− η) − (1) (− η) ⎨⎪ ⎪ Hμ−1 k Hμ+1 k ⎩⎪ +··· , for |kη| >> 1, = (1) ⎪ Hμ (−kη) (5.141) ⎪ ⎪ 1/4 1/4 ⎪ ¯V 1 1 1 2 ⎪ − ¯V (η¯V − 3¯V ) 1 − ¯V − (η¯V − 3¯V ) 1 − ¯V ⎪ 2q 3q 3q q 3 ⎪ ⎪ ( ) ( ) ⎪ (−η) C H 1 (−kη) − H 1 (−kη) ⎪ 1 μ−1 μ+1 ⎪ ⎪ (1) (2) ⎪ C1 Hμ (−kη) + C2 Hμ (−kη) ⎪ ⎪ ⎪ (−η) (2) (− η) − (2) (− η) ⎪ C1 Hμ−1 k Hμ+1 k ⎪ +··· , for AV. ⎩⎪ (1) (2) C1 Hμ (−kη) + C2 Hμ (−kη) One can also consider the following approximations to simplify the ﬁnal derived form of the primordial scalar power spectrum for the BD vacuum with |kcSη|=1 case:

1. We start with the Laurent expansion of the Gamma function: 1 1 π 2 1 γπ2 (μ) = − γ + γ 2 + μ − γ 3 + + 2ζ(3) μ2 + O(μ3). (5.142) μ 2 6 6 2

where γ being the Euler–Mascheroni constant and ζ(3) characterizing the Riemann zeta function of order 3 originating in the expansion of the gamma function. 2. Hence using the result of Eq. (5.142)forq = 1/2 and for arbitrary q we can write: ⎧ π 2 ⎪ 1 1 2 3 ⎪ − γ + γ + +¯V ⎪ 3 +¯ 2 6 2 ⎪ 2 V ⎪ ⎪ γπ2 2 ⎪ 1 3 3 ⎪ − γ + + 2ζ(3) +¯V +··· , for q = 1/2, ⎨ 6 2 2 (μ) = ⎪ π 2 (5.143) ⎪ 1 1 2 3 1 ⎪ ! − γ + γ + + ¯V ⎪ 3 1 2 6 2 2q ⎪ + ¯V ⎪ 2 2q ⎪ ⎪ γπ2 2 ⎪ 1 3 3 1 ⎩ − γ + + 2ζ(3) + ¯V +··· , for any q. 6 2 2 2q

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3. In the slow-roll regime of inﬂation all the slow-roll parameters satisfy the following constraint:

¯V << 1, (5.144)

|¯ηV | << 1, (5.145) |ξ¯ 2 | << , V 1 (5.146) |¯σ 3 | << . V 1 (5.147)

Using these approximations the primordial scalar power spectrum can be expressed as 2 2 CE H ζ, ≈ 1 − (C + 1) − E 1 2 π 2 2 2 8 MpcS 1 = ⎧ k a H ⎪ H 2 ⎪ [1 − (C + 1)¯ − C (3¯ −¯η )]2 , for q = 1/2, ⎨ E V E V V π 2 2 ¯ 8 MpcS V = k= a H (5.148) ⎪ ¯ C 2 qH2 ⎪ − (C + ) V − √ E ( ¯ −¯η ) , q, ⎩ 1 E 1 3 V V 2 2 for any 2q 2q 4π M cS¯V p k=a H

where CE is given by

CE =−2 + ln 2 + γ ≈−0.72. (5.149)

4. Next one can compute the scalar spectral tilt (nS) of the primordial scalar power spectrum as ⎧ ⎪ − + + (C + ) +··· ⎪ 2 1 [1 1 E 1 2] ⎪ ⎪ =− ¯ +¯ + (C + )( ¯ −¯η ) +··· , = / , ⎨⎪ 2 V [1 V 2 E 1 3 V V ] for q 1 2 ≈ − + + (C + ) +··· nh, ⎪ 2 1 [1 1 E 1 2] (5.150) ⎪ ⎪ ¯ ¯ ⎪ V V 2 ⎩⎪ =− 1 + + (CE + 1)(3¯V −¯ηV ) +··· , for any q. q 2q q

5. Next one can compute the running of the tensor spectral tilt (αh) of the primordial scalar power spectrum as dnh(k) dnh(k) αh, = = (5.151) dlnk = dN = ⎧ k a H k a H ⎪ − ¯ ( +¯ )(η¯ − ¯ ) + ¯2 (η¯ − ¯ ) − (C + ) ¯ (η¯ − ¯ )2 ⎪ 4 V 1 V V 3 V 4 V V 3 V 8 E 1 V V 3 V ⎪ ⎪ / ⎪ 2 1 4 ⎪ − ¯ ¯ η¯ − ¯2 − ξ¯ 2 − ¯ +··· , = / , ⎪ 2 V 10 V V 18 V V 1 V for q 1 2 ⎨⎪ 3 ≈ ¯ ¯ (5.152) ⎪ − V + V (η¯ − ¯ ) + 1 ¯2 (η¯ − ¯ ) − 8 (C + ) ¯ (η¯ − ¯ )2 ⎪ 2 1 V 3 V 2 V V 3 V 5/2 E 1 V V 3 V ⎪ q 2q q (2q) ⎪ ⎪ ⎪ 1/4 ⎪ − 2 ¯ ¯ η¯ − ¯2 − ξ¯ 2 − 1 ¯ +··· , . ⎩ V 10 V V 18 V V 1 V for any q q 3q

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6. Finally, one can also compute the running of the running of scalar spectral tilt (κS) of the primordial scalar power spectrum as 2 2 d nh(k) d nh(k) κ , = = h 2 2 dlnk k=a H dN k=a H ⎧ ⎪ − 8¯ (1 +¯ )(η¯ − 3¯ )2 + 8¯2 (η¯ − 3¯ )2 ⎪ V V 6V V V V V 7 ⎪ − (C + ) ¯ (η¯ − ¯ )3 − (η¯ − ¯ ) ¯ η¯ − ¯2 − ξ¯ 2 ⎪ 16 E 1 V V 3 V V 3 V 10 V V 18 V V ⎪ ⎪ 1/4 ⎨⎪ − ¯ ( +¯ ) ¯ η¯ − ¯2 − ξ¯ 2 − 2 ¯ +··· , = / , 4 V 1 V 10 V V 18 V V 1 V for q 1 2 ≈ 3 ⎪ ¯ ¯ ⎪ − 4 ¯ + V (η¯ − ¯ )2 + 2 ¯2 (η¯ − ¯ )2 + 2 ¯ + V ξ¯ 2 − ¯ η¯ + ¯2 ⎪ V 1 V 3 V V V 3 V V 1 V 10 V V 18 V ⎪ q 2q q2 q 2q ⎪ ⎪ 8 6 7 1/4 ⎩⎪ − (C + ) ¯ (η¯ − ¯ )3 − (η¯ − ¯ ) ¯ η¯ − ¯2 − ξ¯ 2 − 1 ¯ +···, . E 1 V V 3 V V 3 V 10 V V 18 V V 1 3q V for any q q (5.153)

5.1.6 Modiﬁed consistency relations

In this subsection we derive the new (modified) consistency relations for single tachyonic field inflation:

Similarly for arbitrary q one can write the following expression for the tensor-to-scalar ratio r at any arbitrary momentum scale: (k) P (k) |u |2 z 2 r(k) ≡ h = 2 h = 2 k ζ (k) Pζ (k) |v |2 a ⎧ ⎧ k 8 8 ⎪ ⎪ 8(μ)82 ⎪ ⎪ 2(μ−ν) 2(ν−μ) 2μ−2 8 8 ⎪ ⎪ 16 × 2 ¯V (−kηcS) c 8 8 , for |kcSη| << 1 ⎪ ⎪ S (ν) ⎪ ⎪ 8 8 ⎪ ⎨ 2(ν−μ) 8 82 ⎪ (μ−ν) ¯V ν− (μ) ⎪ 16 × 22 ¯ 1 + c2 2 8 8 , for |kc η|=1 ⎪ ⎪ V S 8 (ν)8 S for BD ⎨⎪ ⎪ 2q ⎪ (1) = ⎪ 8¯ |Hμ (−kc η) |2 (5.155) ⎪ ⎩⎪ V × S , for |kc η| >> 1. ⎪ 2 (1) S ⎪ c |Hν (−kc η) |2 ⎪ S 8 S 8 ⎪ 8 (1) (2) 82 ⎪ 8D Hμ (−kc η) + D Hμ (−kc η)8 ⎪ 8¯V 1 S 2 S ⎪ × 8 8 , for AV. ⎩⎪ c2 8 (1) (2) 82 S 8C1 Hν (−kcSη) + C2 Hν (−kcSη)8

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2. Next for the BD vacuum with |kcSη|=1 case within the slow-roll regime we can approximately write the following expression for the tensor-to-scalar ratio:

2 h(k) Ph(k) |uk | z 2 r = = 2 = 2 (5.156) ζ (k) Pζ (k) |v |2 a ⎡ k ⎤ 2 ⎢ [1 − (CE + 1)1] ⎥ = ⎣161cS ⎦ C 2 − (C + ) − E 1 E 1 1 2 2 k=a H ⎧ ⎪ − (C + )¯ 2 ⎪ ¯ [1 E 1 V ] ⎪ 16 V cS 2 ⎪ [1 − (CE + 1)¯V − CE (3¯V −¯ηV )] ⎪ k=a H ⎪ ⎡ ⎤ ⎪ ⎪ ⎢ [1 − (C + 1)¯ ]2 ⎥ ⎪ = ⎣ ¯ E V ⎦ , = / , ⎪ 16 V 2 for q 1 2 ⎪ 5 ⎪ 1 − CE + ¯V − CE (3¯V −¯ηV ) ⎨⎪ 6 = ⎡ ⎤ k a H ≈ ¯ 2 (5.157) ⎪ − (C + ) V ⎪ ⎢ 1 E 1 q ⎥ ⎪ 8 ¯ 2 ⎪ ⎣ V cS ⎦ ⎪ q ¯ C 2 ⎪ − (C + ) V − √ E ( ¯ −¯η ) ⎪ 1 E 1 2q 3 V V ⎪ 2q k=a H ⎪ ⎡ ⎤ ⎪ ¯ 2 ⎪ 1 − (C + 1) V ⎪ ⎢ 8 E 2q ⎥ ⎪ = ⎣ ¯V ⎦ , for any q. ⎪ ¯ C 2 ⎩ q − (C + − ) V − √ E ( ¯ −¯η ) 1 E 1 2q 3 V V 2q k=a H

3. Hence the consistency relation between the tensor-to-scalar ratio r and spectral tilt nh for tensor modes for the BD vacuum with the |kcSη|=1 case can be written as ⎡ ⎤ 2 ⎢ [1 − (CE + 1)1] ⎥ =− , × ⎣ ⎦ r 8nh cS 2 − (C + ) − CE + + (C + ) 1 E 1 1 2 2 [1 1 E 1 2] k=a H

⎧ ⎪ [1 − (C + 1)¯ ]2 ⎪ c E V ⎪ S − (C + )¯ − C ( ¯ −¯η ) 2 +¯ + (C + )( ¯ −¯η ) ⎪ [1 E 1 V E 3 V V ] [1 V 2 E 1 3 V V ] = ⎪ k a H ⎪ ⎪ ⎡ ⎤ ⎪ ⎪ ⎪ ⎢ [1 − (C + 1)¯ ]2 ⎥ ⎪ = ⎣ E V ⎦ , for q = 1/2, ⎪ 2 ⎪ − C + 5 ¯ − C ( ¯ −¯η ) +¯ + (C + )( ¯ −¯η ) ⎪ 1 E 6 V E 3 V V [1 V 2 E 1 3 V V ] ⎪ k=a H ⎨⎪ ≈−8nh, × ⎡ ⎤ (5.158) ⎪ ¯ 2 ⎪ 1 − (C + 1) V ⎪ ⎢ E 2q ⎥ ⎪ ⎣cS ⎦ ⎪ ¯ C 2 ¯ ⎪ 1 − (C + 1) V − √ E (3¯ −¯η ) 1 + V + 2 (C + 1)(3¯ −¯η ) ⎪ E 2q 2q V V 2q q E V V ⎪ k=a H ⎪ ⎪ ⎪ ⎡ ⎤ ⎪ ¯ 2 ⎪ 1 − (C + 1) V ⎪ ⎢ E 2q ⎥ ⎪ = ⎣ ⎦ , for any q. ⎪ ¯ C 2 ¯ ⎩ 1 − (C + 1 − ) V − √ E (3¯ −¯η ) 1 + V + 2 (C + 1)(3¯ −¯η ) E 2q 2q V V 2q q E V V 2 34 k=a H 5 Correction factor

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4. Next one can express the ﬁrst two slow-roll parameters ¯V and η¯V in terms of the inﬂationary observables as ⎧ ⎪ nh, r ⎨ ≈− +···≈ +··· , for q = 1/2, 1 2 16 ¯V ≈ ⎩⎪ qr 2q ≈−qnh, +···≈ +··· , for any q, 1 8 ⎧ ⎪ 2 1 r 1 ⎪ 3 − ≈ nζ, − 1 + +···≈ nζ, − 1 − 4n , +··· , for q = 1/2, ⎪ 1 h ⎪ 2 2 % 2 % 2 & & ⎪ 1 1 ⎨ q q 1 2 qr 6q1 − 2 ≈ nζ, − 1 + + 3 +··· η¯V ≈ (5.159) ⎪ 2 2 q q 8 ⎪ 1 % % & & ⎪ ⎪ q 1 2 ⎩⎪ ≈ nζ, − 1 − q + 3 nh, +··· , for any q. 2 q q

5. Then the connecting consistency relation between tensor and scalar spectral tilt and tensor-to-scalar ratio can be expressed as ⎧ ! ⎪ r r r ⎪ − 1 − + 1 − nζ, − CE + nζ, − 1 +··· , for q = 1/2, ⎪ 8c 16 8 ⎪ S % % & & ⎪ ⎨ r 3q 2 2 1 1 − nζ, − 1 + − + 5 r + √ nh, ≈ 8c 8 q q 16 (5.160) ⎪ S 2q ⎪ % % & & ⎪ ⎪ 2 3qr 1 1 2 qr ⎩⎪ + CE − nζ, − 1 + + 3 +··· , for any q. q 8 2 q q 8

Finally, using the approximated version of the expression for cS in terms of slow-roll parameters, one can recast this consistency condition as ⎧ ! ⎪ −r − r + − − C r + − +··· , = / , ⎪ 1 1 nζ, E nζ, 1 for q 1 2 ⎪ 8 24 % % &8 & ⎪ ⎨⎪ r 3q 2 2 1 1 − nζ, − 1 + − + 5 + r + √ nh, ≈ 8 8 q q 16 8 2q (5.161) ⎪ ⎪ % % & & ⎪ ⎪ 2 3qr 1 1 2 qr ⎩⎪ + CE − nζ, − 1 + + 3 +··· , for any q. q 8 2 q q 8

6. Next the running of the sound speed cS can be written in terms of slow-roll parameters as ⎧ 1/4 ⎪ 2 2 ⎨⎪ − ¯V (η¯V − 3¯V ) 1 − ¯V +··· , for q = 1/2, c˙ dlnc dlnc 3 3 S = S = S = S = (5.162) ⎪ 1/4 HcS dN dlnk ⎪ (1 − q) 1 ⎩ − ¯V (η¯V − 3¯V ) 1 − ¯V +··· , for any q, 3q2 3q

which can be treated as another slow-roll parameter in the present context. One can also recast the slow-roll parameter S in terms of the inﬂationary observables as ⎧ ⎪ r r r 1/4 ⎨⎪ − nζ, − 1 + 1 − +··· , for q = 1/2, c˙ dlnc dlnc 48 8 24 S = S = S = S = 1 (5.163) ( − ) 1/4 HcS dN dlnk ⎪ 1 q q r r ⎩ − r nζ, − 1 + 1 − +··· , for any q. 24q2 2 8 24

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7. Further the running of the tensor spectral tilt can be written in terms of the inﬂationary observables as dnh(k) dnh(k) αh, = = (5.164) dlnk k=a H dN k=a H ⎧ ⎪ r r r r r 2 ⎪ − + nζ, − + − nζ, − + ⎪ 1 1 1 ⎪ 8 8 8 8 8 ⎪ ⎪ 2 1/4 ⎪ r r r ⎪ −CE nζ, − 1 + 1 − +··· , for q = 1/2, ⎨ 8 8 24 ≈ 1 (5.165) ⎪ ⎪ r q r r ⎪ − 1 + nζ, − 1 + ⎪ 4 2 8 8 ⎪ ⎪ ⎪ 1 r r 2 r 1/4 ⎩ − (CE + 1) nζ, − 1 + 1 − +··· , for any q. (2q)1/2 8 8 24

8. Next the scalar power spectrum can be expressed in terms of the other inﬂationary observables as ⎧ 2 2 ⎪ 5 r CE r 2H ⎪ 1 − C + + nζ, − 1 + +··· , for q = 1/2, ⎨⎪ E π 2 2 6 16 2 8 Mpr ζ, ≈ (5.166) ⎪ 2 2 ⎪ r CE r 2H ⎩ 1 − (C + 1 − ) + nζ, − 1 + +··· , for any q. E π 2 2 16 2 8 Mpr

9. Further the tensor power spectrum can be expressed in terms of the other inﬂationary observables as ⎧ 2 ⎪ r 2 2H ⎪ 1 − (C + 1) +··· , for q = 1/2, ⎨⎪ E π 2 2 16 Mp h, = (5.167) ⎪ 2 ⎪ r 2 2H ⎩⎪ 1 − (C + 1) +··· , for any q. E π 2 2 16 Mp

10. Next the running of the tensor-to-scalar ratio can be expressed in terms of the inﬂationary observables as dr dr αr, = = =−8αh, +··· dlnk dN ⎧ ⎪ r r r 2 ⎪ r + nζ, − + − r nζ, − + ⎪ 1 1 1 ⎪ 8 8 8 ⎪ ⎪ 2 1/4 ⎪ r r ⎪ −CEr nζ, − 1 + 1 − +··· , for q = 1/2, ⎨ 8 24 ≈ 1 (5.168) ⎪ ⎪ q r r ⎪ 2r 1 + nζ, − 1 + ⎪ 2 8 8 ⎪ ⎪ ⎪ 1 r 2 r 1/4 ⎩ − (CE + 1) r nζ, − 1 + 1 − +··· , for any q. (2q)1/2 8 24

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11. Finally the scale of single field tachyonic inflation can be expressed in terms of the Hubble parameter and the other inflationary observables as ⎧ ⎪ ζ,r π ⎪ Mp ⎪ 2 +··· , for q = 1/2, ⎪ C ⎨⎪ − C + 5 r + E − + r 1 E 6 16 2 nζ, 1 8 = ≈ Hinf H ⎪ (5.169) ⎪ ζ,r π ⎪ Mp ⎪ 2 +··· , for any q. ⎪ C ⎩ − (C + − ) r + E − + r 1 E 1 16 2 nζ, 1 8

One can recast this statement in terms of the inﬂationary potential as ⎧ ⎪ ζ, √ ⎪ 4 3 r π M ⎪ 2 p ⎪ 1 +··· , for q = 1/2, ⎪ C ⎪ − C + 5 r + E − + r ⎨ 1 E 6 16 2 nζ, 1 8 4 = 4 ≈ Vinf V √ (5.170) ⎪ 4 3ζ,r ⎪ π M ⎪ 1 2 p +··· , . ⎪ for any q ⎪ C ⎩⎪ − (C + − ) r + E − + r 1 E 1 16 2 nζ, 1 8

5.1.7 Field excursion for tachyon

In this subsection we explicitly derive the expression for the field excursion4 for tachyonic inflation defined as

|T |=|Tcmb − Tend|=|T − Tend| (5.171) where Tcmb, Tend and T signify the tachyon field value at the time of horizon exit, at end of inflation and at pivot scale respectively. Here we perform the computation for both AV and BD vacuum. For for the sake of simplicity the pivot scale is fixed at the horizon exit scale. To compute the expression for the field excursion we perform the following steps:

1. We start with the operator identity for single field tachyon using which one can write expression for the tachyon field variation with respect to the momentum scale (k) or number of e-foldings (N) in terms of the inflationary observables as ⎧ 1 ⎪ r r 1/4 ⎨⎪ Mp 1 − +··· , for q = 1/2, 1 dT dT dT 8V (T )α 24 = = ≈ 1 / (5.172) H dt dN dlnk ⎪ qr Mp r 1 4 ⎩ 1 − +··· , for any arbitrary q, 4V (T )α 2q 24

where the tensor-to-scalar ratio r is a function of k or N. 2. Next using Eq. (5.172) we can write the following integral equation: ⎧ 1 k ⎪ r r 1/4 ⎪ dlnk Mp 1 − +··· ⎪ 8α 24 ⎪ kend 1 ⎪ N ⎪ r r 1/4 ⎪ = dN M 1 − +··· , for q = 1/2, T ⎨ p N 8α 24 dT V (T ) ≈ end 1 (5.173) ⎪ k / T ⎪ qr M r 1 4 end ⎪ k p − +··· ⎪ dln α 1 ⎪ kend 4 2q 24 ⎪ 1 ⎪ N qr M r 1/4 ⎩⎪ = N p − +··· , q. d α 1 for any arbitrary Nend 4 2q 24

4 In the context of effective field theory, with a minimally coupled scalar field with Einstein gravity we compute the field excursion formula in Refs. [44,129–132]. 123 278 Page 56 of 130 Eur. Phys. J. C (2016) 76 :278

3. Next we parametrize the form of the tensor-to-scalar ratio for q = 1/2 and for any arbitrary q at any arbitrary scale as ⎧ ⎧ , ⎪ ⎪r for Case I ⎪ ⎪ ⎪ ⎪ − + ⎪ ⎪ nh, nζ, 1 ⎪ ⎨ k , ⎪ r for Case II ⎪ k for BD, ⎪ ⎪ ⎪ ⎪ α −α ⎨ ⎪ h, ζ, k ⎪ n ,−nζ,+1+ ln +··· ( ) = ⎪ k h 2! k r k ⎪ ⎩r for Case III, (5.174) ⎪ k∗ ⎪ ⎪ 8 8 ⎪ 8 (1) (2) 82 ⎪ 8D1 Hμ (−kcSη) + D2 Hμ (−kcSη)8 ⎪ qr × , ⎪ 8 8 for AV ⎩ 2c2 8 (1) (2) 82 S 8C1 Hν (−kcSη) + C2 Hν (−kcSη)8

where k∗ is the pivot scale of momentum. One can also express Eq. (5.174) in terms of the number of e-foldings (N)as ⎧ ⎧ ⎪ ⎪r for Case I, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪r exp (N − N)(nh, − nζ, + 1) for Case II, ⎪ ⎨⎪ ⎪ ⎪ for BD, ⎪ ⎪r exp (N − N) (nh, − nζ, + 1) ⎨⎪ ⎪ ⎪ ( ) = ⎪ α − α r N ⎪ ⎪ h, ζ, (5.175) ⎪ ⎩ + (N − N) +··· for Case III, ⎪ 2! ⎪ ⎪ 8 8 ⎪ 8 (1) (2) 82 ⎪ 8D Hμ (−kc η exp [N − N]) + D Hμ (−kc η exp [N − N])8 ⎪ qr 1 S 2 S ⎪ × 8 8 for AV ⎩⎪ 2c2 8 (1) (2) 82 S 8C1 Hν (−kcSη exp [N − N]) + C2 Hν (−kcSη exp [N − N])8

k = ( − ) . where k and N are connected through the following expression: k exp [ N N ] Here the three possibilities for the BD vacuum are: Case I stands for a situation where the spectrum is scale invariant. This is a similar situation to the one considered in the case of the Lyth bound. This possibility also surmounts to the Harrison & Zeldovich spectrum, which is completely ruled out by Planck 2015+WMAP9 data within 5σ C.L. Case II stands for a situation where the spectrum shows a power-law feature through the spectral tilt (nζ , nh).This possibility is also tightly constrained by the WMAP9 and Planck 2015+WMAP9 data within 2σ C.L. Case III signiﬁes a situation where the spectrum shows deviation from power low in the presence of running of the spectral tilt (αζ ,αh) along with logarithmic correction in the momentum scale as appearing in the exponent. This possibility is favored by WMAP9 data and tightly constrained within 2σ window by Planck+WMAP9 data.

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4. For any value of q including q = 1/2 we need to compute the following integral: 1 k M 1/4 qr p − r dlnk 1 k 4α 2q 24 end⎧ 1 / ⎪ qr Mp r 1 4 k ⎪ 1 − ln for Case I, ⎪ 4α 2q 24 k ⎪ end ⎪ ⎪ qr Mp ⎪ 4α q 1 3 3 r r 1/4 ⎪ F , ; ; + − ⎪ ( − + ) 2 1 2 1 ⎪ 3 nh, nζ, 1 2 4 2 24 24 ⎪ ⎧ ⎪ − + ⎪ nh, nζ, 1 ⎪ ⎨ nh,−nζ,+1 ⎪ kend 2 1 3 3 r kend ⎪ − 2 F1 , ; ; ⎪ k ⎩ 2 4 2 24 k ⎪ ⎪ ⎫⎤ ⎪ % & / ⎪ n ,−nζ,+1 1 4⎬ ⎪ r k h ⎪ + − end ⎦ , ⎪ 2 1 ⎭ for Case II ⎪ 24 k ⎨⎪ (5.176) 1 ( − + )2 ≈ 3 nh, nζ, 1 for BD, πqr Mp − (α −α ) ⎪ 4 h, ζ, ⎪ e ⎪ α (α , − αζ,) 48q ⎪ h ⎪ ⎪ (n ,−nζ,+1)2 ⎪ h , − ζ, + ⎪ 2(α ,−αζ,) nh n 1 ⎪ 12e h erfi √ ⎪ 2 α , − αζ, ⎪ h ⎪ √ ⎪ n , − nζ, + 1 α , − αζ, k ⎪ − h√ + h end ⎪ erfi α − α ln ⎪ 2 h, ζ, 2 k ⎪ √ %√ & ⎪ ⎪ 3r 3(nh, − nζ, + 1) ⎪ − erfi √ ⎪ 24 2 α , − αζ, ⎪ h ⎪ %√ & ⎪ (α − α ) ⎪ 3(nh, − nζ, + 1) 3 h, ζ, kend ⎩⎪ −erfi √ + ln for Case III, 2 αh, − αζ, 2 k

Similarly for AV we get the following result: 1 k M 1/4 qr p − r dlnk 1 k 4α 2q 24 end⎧ 1 % & / ⎪ 1 4 ⎪ r Mp qr k ⎪ 1 − ln for Case I, ⎨ α 2 (5.177) 8 2cS 48cS kend ≈ 1 % & / for AV. ⎪ 2 1 4 ⎪ r Mp |D| qr |D| k ⎩⎪ 1 − ln for Case II, α | | 2 | |2 8 2cS C 48cS C kend

In terms of the number of e-foldings N one can re-express Eqs. (5.176) and (5.177)as

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1 k M 1/4 qr p − r dlnk 1 k 4α 2q 24 end⎧ 1 / ⎪ qr Mp r 1 4 ⎪ 1 − (N − Nend) for Case I, ⎪ 4α 2q 24 ⎪ ⎪ ⎪ qr Mp ⎪ α 1/4 ⎪ 4 q 1, 3; 3; r + − r ⎪ 2 F1 2 1 ⎪ 3(nh, − nζ, + 1) 2 4 2 24 24 ⎪ ⎪ − + ⎪ nh, nζ, 1 ⎪ (N −N) 1 3 3 r (n ,−nζ,+1)(N −N) ⎪ −e 2 end F , ; ; e h end ⎪ 2 1 ⎪ 2 4 2 24 ⎪ ⎪ r 1/4 ⎪ (nh,−nζ,+1)(Nend−N) ⎪ +2 1 − e for Case II, ⎪ 24 ⎪ ⎪ 1 ( − + )2 (5.178) ⎨ 3 nh, nζ, 1 πqr Mp − (α −α ) ≈ e 4 h, ζ, , ⎪ α(α − α ) for BD ⎪ h, ζ, 48q ⎪ ( − + )2 ⎪ nh, nζ, 1 − + ⎪ (α −α ) nh, nζ, 1 ⎪ 2 h, ζ, √ ⎪ 12e erfi ⎪ 2 αh, − αζ, ⎪ ⎪ √ ⎪ nh, − nζ, + 1 αh, − αζ, ⎪ −erfi √ + (Nend − N) ⎪ 2 α , − αζ, 2 ⎪ h ⎪ √ %√ & ⎪ ( − + ) ⎪ 3r 3 nh, nζ, 1 ⎪ − erfi √ ⎪ 24 2 αh, − αζ, ⎪ ⎪ %√ & ⎪ ( − + ) (α − α ) ⎪ 3 nh, nζ, 1 3 h, ζ, ⎩ −erfi √ + (Nend − N) for Case III, 2 αh, − αζ, 2

1 k qr M r 1/4 k p − dln α 1 kend 4 2q 24 ⎧ 1 % & ⎪ 1/4 ⎪ r Mp qr ⎪ 1 − (N − N ) for Case I, ⎨⎪ α 2 end (5.179) 8 2cS 48cS ≈ for AV. ⎪ 1 % & / ⎪ 2 1 4 ⎪ r Mp |D| qr |D| ⎩⎪ 1 − (N − N ) for Case II, α | | 2 | |2 end 8 2cS C 48cS C

Here the two possibilities for AV vacuum are: Case I stands for a situation where the spectrum is characterized by the constraint i) D1 = D2 = C1 = C2 = 0, ii) D1 = D2, C1 = C2 = 0, iii) D1 = D2 = 0, C1 = C2. Case II stands for a situation where the spectrum is characterized by the constraint i) μ ≈ ν, D1 = D2 = D = 0 and C1 = C2 = C = 0, ii) μ ≈ ν, D1 = D = 0, D2 = 0 and C1 = C = 0, C2 = 0, iii) μ ≈ ν, D2 = D = 0, D1 = 0 and C2 = C = 0, C1 = 0. 5. Next we assume that the generic tachyonic potential V (T ) can be expressed as

∞ n V (T ) ( ) = + 1 d ( − )n , V T V0 n T T0 (5.180) n! dT = n=1 T T0

where the contribution from V0 ﬁx the scale of potential and the higher order Taylor expansion co-efﬁcients characterize the shape of potential.

123 Eur. Phys. J. C (2016) 76 :278 Page 59 of 130 278

6. Further we need to compute the following integral: ∞ T T 1 dn V (T ) dT V (T ) ≈ dT V + (T − T )n 0 n! dT n 0 Tend Tend = T =T0 n 1 ∞ T 1 dn V (T ) ≈ dT V 1 + (T − T )n +··· 0 ! n 0 T 2n V0 dT T =T end ⎧ n=1 0 ⎫ ⎪ dn V (T ) ⎪ ⎨ ∞ n+ n+ ⎬ dT n ( − ) 1 − ( − ) 1 T =T0 T T0 Tend T0 ≈ V0T 1 + +··· (5.181) ⎩⎪ 2(n + 1)n!V T ⎭⎪ n=1 0

7. Next we assume that n n+1 n+1 1 d V (T ) (T − T0) − (Tend − T0) << 1. (5.182) V dT n T 0 T =T0

Consequently from Eq. (5.181) we get T dT V (T ) ≈ V0T (5.183) Tend

8. Finally using Eqs. (5.178), (5.179), (5.183) and (5.173) we get ⎧ 1 qr r 1/4 ⎪ 1 − ( − ) , ⎪ 1 N Nend for Case I ⎪ 4α V0 2q 24 ⎪ ⎪ qr 1 ⎪ α / ⎪ 4 V0 q 1 3 3 r r 1 4 ⎪ 2 F1 , ; ; + 2 1 − ⎪ 3(n , − nζ, + 1) 2 4 2 24 24 ⎪ h ⎪ n −n +1 ⎪ h, ζ, ( − ) 1 3 3 r ( − + )( − ) ⎪ − 2 Nend N , ; ; nh, nζ, 1 Nend N ⎪ e 2 F1 e ⎪ 2 4 2 24 ⎪ / ⎪ r ( − + )( − ) 1 4 ⎪ +2 1 − e nh, nζ, 1 Nend N for Case II, ⎪ 24 ⎪ ⎪ 1 3(n −n +1)2 ⎨ π − h, ζ, qr 1 (α −α ) T e 4 h, ζ, ≈ α(α − α ) for BD, (5.184) M ⎪ h, ζ, V0 48q p ⎪ ⎪ (n ,−nζ,+1)2 ⎪ h n , − nζ, + 1 ⎪ 2(α ,−αζ,) h ⎪ 12e h erfi √ ⎪ 2 αh, − αζ, ⎪ √ ⎪ − + α − α ⎪ nh, nζ, 1 h, ζ, ⎪ −erfi √ + (Nend − N) ⎪ 2 α , − αζ, 2 ⎪ √ h%√ & ⎪ ⎪ 3r 3(nh, − nζ, + 1) ⎪ − erfi √ ⎪ α − α ⎪ 24 2 h, ζ, ⎪ %√ & ⎪ ( − + ) (α − α ) ⎪ 3 nh, nζ, 1 3 h, ζ, ⎩ −erfi √ + (Nend − N) for Case III, 2 αh, − αζ, 2 ⎧ % & ⎪ 1 1/4 ⎪ r 1 qr ⎪ 1 − (N − N ) for Case I, ⎨ α 2 end T 8 V0 2cS 48cS ≈ % & for AV. (5.185) M ⎪ 1 1/4 p ⎪ r |D| qr |D|2 ⎩⎪ 1 − (N − N ) for Case II, α | | 2 | |2 end 8 V0 2cS C 48cS C

123 278 Page 60 of 130 Eur. Phys. J. C (2016) 76 :278

9. Next using Eq. (5.170)inEqs.(5.184) and (5.185) we get ⎧ 1 ⎪ qr r 1/4 ⎪ 1 − ( − ) ⎪ 1 N Nend ⎪ 4α 3ζ,r 24 ⎪ 2q π M2 ⎪ 2 p ⎪ ⎪ r CE ⎪ 1 − (CE + 1 − ) + √ nζ, − 1 ⎪ 16 2 2q ⎪ ⎪ ⎪ r ⎪ + 1 + 3 2q − 6q for Case I, ⎪ ⎪ 8 ⎪ ⎪ qr 1 ⎪ ⎪ 4α 3ζ,r ⎪ q π M2 1 3 3 r r 1/4 ⎪ 2 p , ; ; + − ⎪ 2 F1 2 1 ⎪ 3(n , − nζ, + 1) 2 4 2 24 24 ⎪ h ⎪ − + ⎪ nh, nζ, 1 ⎪ (N −N) 1 3 3 r (n ,−nζ,+1)(N −N) ⎪ −e 2 end F , ; ; e h end ⎪ 2 1 2 4 2 24 ⎪ ⎪ / ⎪ r ( − + )( − ) 1 4 ⎪ +2 1 − e nh, nζ, 1 Nend N ⎪ 24 ⎪ ⎪ ⎪ r CE ⎪ 1 − (CE + 1 − ) + √ nζ, − 1 ⎪ 16 2 2q ⎪ ⎪ ⎨ r T ≈ + 1 + 3 2q − 6q for Case II, , ⎪ 8 for BD (5.186) Mp ⎪ ⎪ π ⎪ qr ( − + )2 ⎪ α(α −α ) 3 nh, nζ, 1 ⎪ h, ζ, − (α −α ) ⎪ e 4 h, ζ, ⎪ ⎪ 3 ζ,r π 2 ⎪ 48q 2 Mp ⎪ ⎪ (n ,−nζ,+1)2 ⎪ h , − ζ, + ⎪ 2(α ,−αζ,) nh n 1 ⎪ 12e h erfi √ ⎪ 2 αh, − αζ, ⎪ √ ⎪ − + α − α ⎪ nh, nζ, 1 h, ζ, ⎪ − √ + (N − N) ⎪ erfi α − α end ⎪ 2 h%, ζ, 2 & ⎪ √ √ ⎪ 3r 3(n , − nζ, + 1) ⎪ − √h ⎪ erfi α − α ⎪ 24 2 h, ζ, ⎪ %√ & ⎪ (α − α ) ⎪ 3(nh, − nζ, + 1) 3 h, ζ, ⎪ −erfi √ + (Nend − N) ⎪ 2 α , − αζ, 2 ⎪ h ⎪ ⎪ r C ⎪ − (C + − ) + √E − ⎪ 1 E 1 nζ, 1 ⎪ 16 2 2q ⎪ ⎪ r ⎩⎪ + 1 + 3 2q − 6q for Case III, 8 ⎧ / ⎪ 1 4 ⎪ 1 − qr ( − ) ⎪ 1 2 N Nend ⎪ r 48cS ⎪ ⎪ α ⎪ 8 3 ζ,r 2 ⎪ 2cS π M ⎪ 2 p ⎪ ⎪ r CE r ⎨⎪ 1 − (CE + 1 − ) + √ nζ, − 1 + 1 + 3 2q − 6q for Case I, T 16 2 2q 8 ≈ for AV. (5.187) ⎪ 1/4 Mp ⎪ 2 ⎪ 1 | | − qr |D| ( − ) ⎪ D 1 2 2 N Nend ⎪ r 48c |C| ⎪ S ⎪ α ⎪ 8 3 ζ,r 2 ⎪ 2cS |C| π M ⎪ 2 p ⎪ C ⎪ r E r ⎩ 1 − (CE + 1 − ) + √ nζ, − 1 + 1 + 3 2q − 6q for Case II, 16 2 2q 8

123 Eur. Phys. J. C (2016) 76 :278 Page 61 of 130 278 T Further using the approximated form of the sound speed cS sinh2 1 T0 the expression for the ﬁeld excursion for AVcan be rewritten ¯V = , (5.190) 2g cosh T as T0 ⎧ % & / ⎪ 1 4 ⎪ qr ⎪ 1 1 − (N − N ) ⎪ − (1−q) r end ⎪ 48 1 3q 8 ⎪ r ⎪ ⎪ 8α ( − ) 3ζ,r ⎪ − 1 q r π M2 ⎪ 2 1 3q 8 2 p ⎪ C ⎪ r E r ⎨ − (C + − ) + √ nζ, − + + q − q , T 1 E 1 1 1 3 2 6 for Case I ≈ % 16 2 2q & 8 . 1/4 for AV (5.188) Mp ⎪ ⎪ qr |D|2 ⎪ 1 |D| 1 − (N − N ) ⎪ − (1−q) r |C|2 end ⎪ r 48 1 3q 8 ⎪ ⎪ ⎪ 8α ( − ) 3ζ,r ⎪ − 1 q r |C| π M2 ⎪ 2 1 3q 8 2 p ⎪ ⎪ r CE r ⎩ 1 − (CE + 1 − ) + √ nζ, − 1 + 1 + 3 2q − 6q for Case II, 16 2 2q 8

⎡ ⎤ sinh2 T 5.1.8 Semi-analytical study and cosmological parameter 1 ⎣ T0 T ⎦ η¯V = − sech , (5.191) estimation g T T0 cosh T 0 ⎡ ⎤ In this subsection our prime objective are: sinh2 T sinh2 T ξ¯ 2 = 1 T0 ⎣ T0 − T ⎦ , V 2 5 sech g cosh T cosh T T0 • To compute various inflationary observables from vari- T0 T0 ants of tachyonic single field potentials as mentioned ear- (5.192) ⎡ lier in this paper. sinh2 T sinh2 T • To estimate the relevant cosmological parameters from σ¯ 3 = 1 T0 ⎣ T0 V the proposed models. g3 T T cosh T cosh T • Next to compare the effectiveness of all of these models ⎧ 0 0 ⎫ ⎤ in the light of recent Planck 2015 data alongwith other ⎨sinh2 T ⎬ × T0 − T + 2 T ⎦ , combined constraints. ⎩ 18 sech ⎭ 5 sech cosh T T0 T0 • Finally to check the compatibility of all of these models T0 with the CMB TT, TE and EE angular power spectra as (5.193) observed by Planck 2015. where the factor g is defined as

Model I: inverse cosh potential αλT 2 M4 αT 2 g = 0 = s 0 . (5.194) 2 ( π)3 2 For the single field case the first model of the tachyonic poten- Mp 2 gs Mp tial is given by λ V (T ) = , (5.189) cosh T T0 where λ characterizes the scale of inflation and T0 is the parameter of the model. In Fig. 1 we have depicted the symmetric behavior of the inverse cosh potential with respect to scaled field coordinate T/T0 in dimensionless units around theoriginfixedatT/T0 = 0. In this case the tachyon field started rolling down from the top height of the potential and takes part in the inflationary dynamics.

Next using speciﬁed form of the potential the potential Fig. 1 Variation of the inverse cosh potential V (T )/λ with ﬁeld T/T0 dependent slow-roll parameters are computed as in dimensionless units 123 278 Page 62 of 130 Eur. Phys. J. C (2016) 76 :278

Next we compute the number of e-foldings from this model: ⎧ ⎡ ⎤ ⎪ tanh Tend ⎪ 2T0 ⎪ g ln ⎣ ⎦ , for q = 1/2, ⎪ ⎨ tanh T 2T0 N(T ) = ⎡ ⎤ (5.195) ⎪ T ⎪ tanh end ⎪ ⎣ 2T0 ⎦ , . ⎪ 2qgln for any arbitrary q ⎩ tanh T 2T0 Further using the condition to end inﬂation:

¯V (Tend) = 1, (5.196)

|¯ηV (Tend)|=1, (5.197) we get the following ﬁeld value at the end of inﬂation: −1 Tend = T0 sech (g). (5.198)

Next using N = Ncmb = N and T = Tcmb = T at the horizon crossing we get ⎧ ⎪ − N 1 − ⎨⎪ tanh 1 exp − tanh sech 1(g) , for q = 1/2 g 2 = × T 2T0 ⎪ (5.199) ⎪ − N 1 − ⎩⎪ tanh 1 exp −√ tanh sech 1(g) , for any arbitrary q. 2qg 2 Consequently the field excursion can be computed as ⎧ 8 8 8 8 ⎪ 8 − N 1 − − 8 ⎨⎪ 82 tanh 1 exp − tanh sech 1(g) − sech 1(g)8 , for q = 1/2, g 2 | |= × 8 8 T T0 ⎪ 8 8 (5.200) ⎪ 8 −1 N 1 −1 −1 8 ⎩ 82 tanh exp −√ tanh sech (g) − sech (g)8 , for any q. 2qg 2 In the slow-roll regime of inflation the following approximations hold good: ⎧ ⎪ 1 ⎪ , for q = 1/2, ⎨⎪ N tanh g T ≈ cosh ⎪ (5.201) T0 ⎪ 1 ⎪ , for any arbitrary q, ⎩ tanh √N ⎧ 2qg ⎪ 1 ⎪ , for q = 1/2, ⎨⎪ N sinh g T ≈ sinh ⎪ (5.202) T0 ⎪ 1 ⎪ , for any arbitrary q. ⎩ sinh √N 2qg Using Eqs. (5.201) and (5.202) in the definition of potential dependent slow-roll parameter finally we compute the following inflationary observables: ⎧ ⎪ N ⎨⎪ sinh2 , for q = 1/2, gλ g ζ, ≈ × (5.203) 12π 2 M4 ⎪ N p ⎩⎪ 2q sinh2 √ , for any arbitrary q, 2qg

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⎧ 2 ⎪ 1 αζ, ≈ × nζ, − 1 ⎪ , for q = 1/2, ⎨⎪ N g ⎧ tanh g − ≈−2 × ⎪ 1 nζ, 1 ⎪ ⎪ , for q = 1/2, g ⎪ 1 ⎪ ⎪ √ , for any arbitrary q, ⎨ sinh 2N ⎩⎪ g 2q tanh √N 2qg × ⎪ 1 ⎪ , , ⎧ (5.204) ⎪ √ for any arbitrary q ⎩ 2q sinh √2N ⎪ 1 2qg ⎪ , for q = 1/2, ⎨⎪ sinh2 N (5.211) 2 g αζ, ≈− × g2 ⎪ 1 ⎪ , for any arbitrary q, 2 ⎩⎪ κζ, ≈ × nζ, − 1 2q sinh2 √N 2 2qg g ⎧ ⎪ 1 ⎧ (5.205) ⎪ , for q = 1/2, ⎪ ⎪ N ⎨ sinh2 N ⎪ cosh g g ⎪ , for q = 1/2, × ⎪ ⎪ ⎨⎪ sinh3 N ⎪ 1 4 g ⎪ , for any arbitrary q. κζ, ≈− × ⎩⎪ N 3 2 √ g ⎪ N 2q sinh ⎪ cosh 2qg ⎪ 2qg , , ⎪ for any arbitrary q (5.212) ⎩ (2q)3/2 sinh3 √N 2qg (5.206) Let us now discuss the general constraints on the parame- ⎧ ters of tachyonic string theory including the factor q and on ⎪ 1 ⎪ , for q = 1/2, the parameters appearing in the expression for the inverse ⎨⎪ sinh 2N 16 g cosh potential. In Fig. 2a, b, we have shown the behavior of r ≈ × (5.207) g ⎪ 1 the tensor-to-scalar ratio r with respect to the scalar spec- ⎪ , for any arbitrary q. ⎩⎪ 2q sinh √2N tral index nζ and the model parameter g for the inverse cosh 2qg potential respectively. In both the figures the purple and blue For the inverse cosh potential we get the following con- colored line represent the upper bound of the tensor-to-scalar sistency relations: ratio allowed by Planck+ BICEP2+Keck Array joint constraint and only Planck 2015 data respectively. For both the r ≈ 4(1 − nζ,) figures the red, green, brown, orange colored curves repre- ⎧ sent q = 1/2, q = 1, q = 3/2 and q = 2, respectively. The ⎪ 1 , = / , ⎪ for q 1 2 cyan color shaded region bounded by two vertical black col- ⎨⎪ cosh2 N g ored lines in Fig. 2a represent the Planck 2σ allowed region × √ (5.208) ⎪ 2q and the rest of the light gray shaded region shows the 1σ ⎪ , , ⎩⎪ for any arbitrary q allowed range, which is at present disfavored by the Planck cosh2 N 2qg 2015 data and Planck+ BICEP2+Keck Array joint constraint. λ The rest of the region is completely ruled out by the present ζ, ≈ 12π 2 M4(1 − nζ,) observational constraints. From Fig. 2a, b, it is also observed ⎧ p that, within 50 < N < 70, the inverse cosh potential is ⎪ 2N ⎨⎪ sinh , for q = 1/2, favored only for the characteristic index 1/2 < q < 2, by g × Planck 2015 data and Planck+ BICEP2+Keck Array joint ⎪ 2N = / = = / ⎩⎪ 2q sinh √ , for any arbitrary q, analysis. Also in Fig. 2aforq 1 2, q 1, q 3 2 2qg and q = 2wefixN/g ∼ 0.8. This implies that for (5.209) 50 < N < 70, the prescribed window for g from the r–nζ plot is given by 63 < g < 88. In Fig. 2b, we have explic- 2λ itly shown that the in r–g plane the observationally favored ζ, ≈ 3π 2 M4r lower bound for the characteristic index is q ≥ 1/2. It is ⎧ p additionally important to note that, for q >> 2, the tensor- ⎪ N ⎨⎪ tanh , for q = 1/2, to-scalar ratio computed from the model is negligibly small g × for the inverse cosh potential. This implies that if the infla- ⎪ N ⎩⎪ tanh √ , for any arbitrary q, tionary tensor mode is detected close to its present upper 2qg bound on tensor-to-scalar ratio then all q >> 2 possibilities (5.210) for tachyonic inflation can be discarded for the inverse cosh 123 278 Page 64 of 130 Eur. Phys. J. C (2016) 76 :278

constraint on scalar spectral tilt is satisfied within the window of the tensor-to-scalar ratio, 0.090 < r < 0.12 for 50 < N < 70. Finally, for the q = 2 situation, the value for the tensor-to-scalar ratio is r < 0.12, which is tightly constrained from the upper bound of the spectral tilt from Planck 2015 observational data. In Fig. 3a–c, we have depicted the behavior of the scalar power spectrum ζ vs. the stringy parameter g, scalar spectral tilt nζ vs. the stringy parameter g and scalar power spectrum ζ vs. scalar spectral index nζ for the inverse cosh (a) vs . potential respectively. It is important to note that, for all of the figures the red, green, brown, orange colored curves represent q = 1/2, q = 1, q = 3/2 and q = 2 respectively. The purple and blue colored line represent the upper and lower bound allowed by WMAP+Planck 2015 data respectively. The cyan color shaded region bounded by two vertical black colored lines represent the Planck 2σ allowed region and the rest of the light gray shaded region is the 1σ region, which is presently disfavored by the joint Planck+WMAP constraints. The rest of the region is completely ruled out by the present observational constraints. From Fig. 3aitis clearly observed that the observational constraints on the (b) vs . amplitude of the scalar mode fluctuations satisfy within the Fig. 2 Behavior of the tensor-to-scalar ratio r with respect to a the window 80 < g < 100. For g > 100 the corresponding scalar spectral index nζ and b the parameter g for the inverse cosh amplitude falls down in a non-trivial fashion by following potential. The purple and blue colored line represent the upper bound the exact functional form as stated in Eq. (5.203). Next using of tensor-to-scalar ratio allowed by Planck+ BICEP2+Keck Array joint the behavior as shown in Fig. 3b, the lower bound on the constraint and only Planck 2015 data respectively. For both the figures the red, green, brown, orange colored curves represent q = 1/2, q = 1, stringy parameter g is constrained by g > 50 by using the q = 3/2andq = 2 respectively. The cyan color shaded region bounded non-trivial relationship as stated in Eq. (5.204). But at this by two vertical black colored lines in a represents the Planck 2σ lower bound of the parameter g the amplitude of the scalar light gray shaded σ allowed region and the rest of the region shows the 1 power spectrum is larger compared to the present observa- allowed range, which is at present disfavored by the Planck data and Planck+ BICEP2+Keck Array joint constraint. From a and b,itisalso tional constraints. This implies that, to satisfy both the con- observed that, within 50 < N < 70, the inverse cosh potential is straints from the amplitude of the scalar power spectrum favored only for the characteristic index 1/2 < q < 2, by Planck 2015 and its spectral tilt, within 2σ CL the constrained numeri- b data and Planck+ BICEP2+Keck Array joint analysis. In ,wehave cal value of the stringy parameter is lying within the window explicitly shown that in the r–g plane the observationally favored lower bound for the characteristic index is q ≥ 1/2 80 < g < 100.AlsoinFig.3cforq = 1/2, q = 1, q = 3/2 and q = 2wefixN/g ∼ 0.8, which further implies that for 50 < N < 70, the prescribed window for g from ζ –nζ potential. On the contrary, if inflationary tensor modes are plot is given by 63 < g < 88. If we additionally impose never detected by any of the future observational probes then the constraint from the upper bound on tensor-to-scalar ratio the q >> 2 possibilities for tachyonic inflation in the case then also the allowed parameter range is lying within a sim- of the inverse cosh potential are highly prominent. Also it ilar window, i.e., 88 < g < 100. is important to mention that, in Fig. 2b within the window In Fig. 4a, b, we have shown the behavior of the running 0 < g < 100, if we take a smaller value of g, then the of the scalar spectral tilt αζ and running of the running of the inflationary tensor-to-scalar ratio also gradually decreases. scalar spectral tilt κζ with respect to the scalar spectral index To analyze the results more clearly let us describe the cos- nζ for the inverse cosh potential with g = 88, respectively. mological features from Fig. 2a in detail. Let us first start For both the figures the red, green, brown, orange colored with the q = 1/2 situation, in which the 2σ constraint on the curves represent q = 1/2,q = 1, q = 3/2 and q = 2, respec- scalar spectral tilt is satisfied within the window of tensor- tively. The cyan color shaded region bounded by two vertical to-scalar ratio, 0.015 < r < 0.025 for 50 < N < 70. Next black colored lines in both the plots represent the Planck 2σ for the q = 1 case, the same constraint is satisfied within allowed region and the rest of the light gray shaded region the window of the tensor-to-scalar ratio, 0.050 < r < 0.075 shows the 1σ allowed range, which is at present disfavored for 50 < N < 70. Further for the q = 3/2 case, the same by the Planck data and Planck+ BICEP2+Keck Array joint 123 Eur. Phys. J. C (2016) 76 :278 Page 65 of 130 278

Fig. 3 Variation of the a scalar power spectrum ζ vs. scalar spectral index nζ , b scalar power spectrum ζ vs. the stringy parameter g and c scalar spectral tilt nζ vs. the stringy parameter g.Thepurple and blue colored line represent the upper and lower bound allowed by WMAP+Planck 2015 data respectively. The green dotted region bounded by two vertical black colored lines represent the Planck 2σ allowed region and the rest of the light gray shaded region is disfavored by the Planck+WMAP constraint

(a) Δζ vs g. (b) nζ vs g.

constraint. From both of these figures, it is also observed where λ characterize the scale of inflation and T0 is the param- that, within 50 < N < 70, the inverse cosh potential is eter of the model. In Fig. 5 we have depicted the behavior favored for the characteristic index 1/2 < q < 2, by Planck of the logarithmic potential with respect to scaled field coor- 2015 data and Planck+ BICEP2+Keck Array joint analysis. dinate T/T0 in dimensionless units. In this case the tachyon From Fig. 4a, b, it is observed that within the 2σ observed field started rolling down from the top height of the potential range of the scalar spectral tilt nζ , as the value of the char- either from the left or right hand side and takes part in the acteristic parameter q increases, the value of the running αζ inflationary dynamics. and running of the running κζ decreases for the inverse cosh Next using specified form of the potential the potential potential. It is also important to note that for 1/2 < q < 2, dependent slow-roll parameters are computed as − the numerical value of the running αζ ∼ O(−10 4) and − κζ ∼ O(− 6) 2 2 running of the running 10 , which are perfectly T 2 T T . σ 4 ln 1 + ln consistent with the 1 5 constraints on running and running ¯ = 1 T0 T0 T0 , V (5.214) of the running as obtained from Planck 2015 data. 2g 2 3 + T 2 T 1 T ln T Model II: logarithmic potential 0 0 For single field case the second model of tachyonic potential is given by 2 2 ln T + 1 + ln T 1 T0 T0 η¯ = , 2 2 V 2 (5.215) ( ) = λ T T + , g 2 V T ln 1 (5.213) 1 + T ln2 T T0 T0 T0 T0 123 278 Page 66 of 130 Eur. Phys. J. C (2016) 76 :278 4ln T 1 + ln T 3 + 2ln T ξ¯ 2 = 1 T0 T0 T0 , V (5.216) g2 2 4 1 + T ln2 T T0 T0 2 4ln2 T 1 + ln T 1 + 2ln T σ¯ 3 =−1 T0 T0 T0 , V g3 2 6 1 + T ln2 T T0 T0 (5.217) where the factor g is defined as (a) αζ VS nζ . αλT 2 M4 αT 2 g = 0 = s 0 . (5.218) 2 ( π)3 2 Mp 2 gs Mp Next we compute the number of e-foldings from this model: ⎧ ⎧ ⎪ ⎨ 2Ei 2 ln T +1 Ei 4 ln T +1 ⎪ g T0 T0 ⎪ − − ⎪ 2 ⎩ e2 e4 ⎪ ⎪ ⎪ T 4 11 3 T 1 T ⎪ + − + 2 ⎪ ln ln ⎪ T0 32⎛ 8 T0 ⎞⎫4 T0 ⎪ T ⎪ 2 T ⎬ ⎪ T ln T ⎪ + + ⎝ 0 ⎠ , = / , ⎪ ln ⎭ for q 1 2 ⎪ T0 1+ln T ⎪ T0 T ⎨⎪ ⎧ end ⎨ 2Ei 2 ln T +1 N(T )= g T0 ⎪ 2q − ⎪ ⎩ 2 ⎪ 2 e ⎪ ⎪ T κζ VS nζ . ⎪ Ei 4 ln +1 (b) ⎪ − T0 ⎪ 4 ⎪ e ⎪ T 4 T T Fig. 4 Behavior of the a running of the scalar spectral tilt αζ and b ⎪ + 11 − 3 + 1 2 ⎪ ln ln κ ⎪ T0 32 8 T0 4 T0 running of the running of the scalar spectral tilt ζ with respect to the ⎪ ⎛ ⎞⎫T ⎪ T ⎬ scalar spectral index nζ for the inverse cosh potential with g = 88. For ⎪ 2 ln ⎪ + T + ⎝ T0 ⎠ , . ⎪ ln ⎭ for any arbitrary q both figures the red, green, brown, orange colored curves represent ⎩ T0 1+ln T T0 q = 1/2, q = 1, q = 3/2andq = 2 respectively. The cyan color Tend shaded region bounded by two vertical black colored lines in a represent (5.219) the Planck 2σ allowed region and the rest of the light gray shaded region shows the 1σ allowed range, which is at present disfavored by the Planck Further using the condition to end inflation: data and Planck+ BICEP2+Keck Array joint constraint. From a and b, it is also observed that, within 50 < N < 70, the inverse cosh potential ¯V (Tend) = 1, (5.220) / < < is favored for the characteristic index 1 2 q 2, by Planck 2015 |¯η (T )|=1, (5.221) data and Planck+ BICEP2+Keck Array joint analysis V end we get the following transcendental equation: T 2 T 2 T 1 + ln end end ln2 end − 1 T0 T0 T0 T = ln end , (5.222) T0 from which we get the following sets of real solutions for the field value:

Tend = (0.07 T0, 0.69 T0, 1.83 T0) . (5.223) Then using this result we need to numerically solve the transcendental equation of T which involves N explicitly. How- ever, in the slow-roll regime of inflation we get the following simplified expression for the field value T in terms of N, Fig. 5 Variation of the logarithmic potential V (T )/λ with field T/T0 in dimensionless units Tend and T0:

123 Eur. Phys. J. C (2016) 76 :278 Page 67 of 130 278 ⎧ ⎪ 2 ⎪ 2N Tend ⎨⎪ + , for q = 1/2, g T0 T ≈ T0 × (5.224) ⎪ 2 ⎪ 2N Tend ⎩ √ + , for any arbitrary q, 2qg T0 where we have explicitly used the fact that in the slow-roll regime the quadratic term gives the dominant contribution in N. Also the field excursion can be computed as ⎧ 8 8 ⎪ 8 8 ⎪ 8 2N 8 ⎨⎪ 8 + c2 − c8 , for q = 1/2, 8 g 8 | |= × 8 8 T T0 ⎪ 8 8 (5.225) ⎪ 8 2N 8 ⎩⎪ 8 √ + c2 − c8 , for any arbitrary q, 8 2qg 8 where c = 0.07, 0.69, 1.83. During the numerical estimation we have taken c = 0.07 as it is compatible with the observational constraints from Planck 2015 data. Finally using the previously mentioned definition of potential dependent slow-roll parameter we compute the following inflationary observables: ⎧ ⎪ 2 2 4 ⎪ + 1 2N + Tend 2 2N + Tend ⎪ 1 4 g T ln g T ⎪ 0 0 ⎪ , for q = 1/2, ⎪ 2 2 2 2 ⎨⎪ 2N + Tend ln2 2N + Tend 1 + 1 ln 2N + Tend gλ g T0 g T0 2 g T0 ζ, ≈ × (5.226) 48π 2 M4 ⎪ 2 2 4 p ⎪ 2q 1 + 1 √2N + Tend ln2 √2N + Tend ⎪ 4 2qg T0 2qg T0 ⎪ . ⎪ for any q ⎪ 2 2 2 2 ⎩ √2N + Tend ln2 √2N + Tend 1 + 1 ln √2N + Tend 2qg T0 2qg T0 2 2qg T0 ⎧ ϑ ⎪ ⎪ , for q = 1/2, ⎪ 2 2 3 ⎨⎪ 1 + 1 2N + Tend ln2 2N + Tend 4 4 g T0 g T0 nζ, − 1 ≈ × ϑ (5.227) g ⎪ ⎪ . for any q ⎪ 2 2 3 ⎩⎪ 1 + 1 √2N + Tend ln2 √2N + Tend 4 2qg T0 2qg T0

⎧ ⎪ 2 2 2 2 ⎪ 2N + Tend 2 2N + Tend + 1 2N + Tend ⎪ g T ln g T 1 2 ln g T ⎪ 0 0 0 ⎪ , for q = 1/2, ⎪ 2 2 3 ⎨⎪ 1 + 1 2N + Tend ln2 2N + Tend 32 4 g T0 g T0 r ≈ × (5.228) g ⎪ 2 2 2 2 ⎪ √2N + Tend ln2 √2N + Tend 1 + 1 ln √2N + Tend ⎪ 2qg T0 2qg T0 2 2qg T0 ⎪ 1 , , ⎪ for any arbitrary q ⎪ 2q 2 2 3 ⎩ 1 + 1 √2N + Tend ln2 √2N + Tend 4 2qg T0 2qg T0 where

123 278 Page 68 of 130 Eur. Phys. J. C (2016) 76 :278 ⎧ % & % & % & ⎪ 2 2 2 2 ⎪ 3 2N T 2N T 1 2N T ⎪ 1 − + end ln2 + end 1 + ln + end ⎪ ⎪ 4 g T0 g T0 2 g T0 ⎪ % & % & % & ⎪ 2 2 2 ⎪ 1 2N Tend 1 2N Tend 2 2N Tend ⎨⎪ + ln + 1 + + ln + , for q = 1/2, 2 g T0 4 g T0 g T0 ϑ ≈ % & % & (5.229) ⎪ 2 2 2 ⎪ G N N T N T ⎪ − 2 √2 + end + 1 √2 + end ⎪ 1 ln 1 ln ⎪ 4 qg 2qg T0 2 2qg T0 ⎪ % & % & % & ⎪ 2 2 2 ⎪ P 2N Tend 1 2N Tend 2N Tend ⎩⎪ + ln √ + 1 + √ + ln2 √ + , for any arbitrary q. 2 2qg T0 4 2qg T0 2qg T0

Here the constants G and P are defined as 1 G = 1 + 2 2q , (5.230) 2q 1 P = √ . (5.231) 2q For the inverse cosh potential we get the following consistency relations: ⎧ ⎪ 2 2 2 2 ⎪ 2N + Tend ln2 2N + Tend 1 + 1 ln 2N + Tend ⎪ g T0 g T0 2 g T0 ⎪ , q = / , ⎨ |ϑ | for 1 2 ≈ ( − ) × r 8 1 nζ, (5.232) ⎪ 2 2 2 2 ⎪ 2N Tend 2 2N Tend 1 2N Tend ⎪ √ + ln √ + 1 + ln √ + ⎪ 2qg T0 2qg T0 2 2qg T0 ⎩ , for any q 2q|ϑ| ⎧ |ϑ | ⎪ ⎪ , for q =1/2, ⎪ 2 2 2 2 ⎪ 2N T 2N T 1 2N T ⎨ (nζ, − 1) + end ln2 + end 1 + ln + end λ g T0 g T0 2 g T0 ζ, ≈ × |ϑ | 12π 2 M4 ⎪ 2q p ⎪ . for any q ⎪ 2 2 2 2 ⎪ 2N T 2N T 1 2N T ⎩ (nζ, − 1) √ + end ln2 √ + end 1 + ln √ + end 2qg T0 2qg T0 2 2qg T0 (5.233) ⎧ % & % & 2 2 ⎪ 1 2N T 2N T ⎪ + + end 2 + end , = / , ⎨⎪ 1 ln for q 1 2 2λ 4 g T0 g T0 ζ, ≈ × % & % & (5.234) π 2 4 ⎪ 3 Mpr ⎪ 2 2 ⎪ 1 2N Tend 2 2N Tend ⎩⎪ 1 + √ + ln √ + , for any arbitrary q. 4 2qg T0 2qg T0 Let us now discuss the general constraints on the parameters of tachyonic string theory including the factor q and on the paramters appearing in the expression for the logarithmic potential. In Fig. 6a, b, we have shown the behavior of the tensor-to-scalar ratio r with respect to the scalar spectral index nζ and the model parameter g for the logarithmic potential respectively. In both the figures the purple and blue colored line represent the upper bound of the tensor-to-scalar ratio allowed by Planck+ BICEP2+Keck Array joint constraint and only Planck 2015 data respectively. For both the figures the red, green, brown, orange colored curves represent q = 1/2, q = 1, q = 3/2 and q = 2, respectively. The cyan color shaded region bounded by two vertical black colored lines in Fig. 6a represent the Planck 2σ allowed region and the rest of the light gray shaded region shows the 1σ allowed range, which is at present disfavored by the Planck 2015 data and Planck+ BICEP2+Keck Array joint constraint. The rest of the region is completely ruled out by the present observational constraints. From Fig. 6a, b, it is also observed that, within 50 < N < 70, the logarithmic potential is favored for the characteristic index 1/2 < q < 2, by Planck 2015 data and Planck+ BICEP2+Keck Array joint analysis. Also in Fig. 6aforq = 1/2, q = 1, q = 3/2 and q = 2wefixN/g ∼ 0.7. This implies that for 50 < N < 70, the prescribed window for g from r–nζ plot is given by 71.4 < g < 100. In Fig. 6b, we have explicitly shown that the in r − g plane the observationally favored lower bound for 123 Eur. Phys. J. C (2016) 76 :278 Page 69 of 130 278

also gradually decreases. After that within 71 < g < 300 the value of the tensor-to-scalar ratio slightly increases and again it falls down to lower value within the the interval 120 < g < 1000. In Fig. 7a–c, we have depicted the behavior of the scalar power spectrum ζ vs. the stringy parameter g, scalar spectral tilt nζ vs. the stringy parameter g and scalar power spectrum ζ vs. scalar spectral index nζ for the logarithmic potential respectively. It is important to note that, for all of the figures the red, green, brown, orange colored curves represent q = 1/2, q = 1, q = 3/2 and q = 2 respectively. (a) vs . The purple and blue colored line represent the upper and lower bound allowed by WMAP+Planck 2015 data respectively. The cyan color shaded region bounded by two vertical black colored lines represent the Planck 2σ allowed region and the rest of the light gray shaded region is the 1σ region, which is presently disfavored by the joint Planck+WMAP constraints. The rest of the region is completely ruled out by the present observational constraints. From Fig. 7aitis clearly observed that the observational constraints on the amplitude of the scalar mode fluctuations satisfy within the window g = (26, 90) for q = 1/2, g = (20, 60) for q = 1, g = (18, 48) for q = 3/2 and g = (16, 38) for q = 2. For all the cases outside the mentioned window for the param- (b) vs . eter g the corresponding amplitude gradually increases in a non-trivial fashion by following the exact functional form as Fig. 6 Behavior of the tensor-to-scalar ratio r with respect to a the stated in Eq. (5.226). Next using the behavior as shown in scalar spectral index nζ and b the parameter g for the logarithmic poten- Fig. 7b, the lower bound on the stringy parameter g is con- tial. The purple and blue colored line represent the upper bound of > tensor-to-scalar ratio allowed by Planck+ BICEP2+Keck Array joint strained by g 45 by using the non-trivial relationship as constraint and only Planck 2015 data respectively. For both figures stated in Eq. (5.227)forq = 2. Similarly by observing the the red, green, brown, orange colored curves represent q = 1/2, Fig. 7b one can find the other lower bounds on g from differ- = = / = q 1, q 3 2andq 2, respectively. The cyan color shaded region ent value of q. But at this lower bound of the parameter g the bounded by two vertical black colored lines in a represent the Planck 2σ allowed region and the rest of the light gray shaded region shows amplitude of the scalar power spectrum is very larger com- the 1σ allowed range, which is at present disfavored by the Planck data pared to the present observational constraints. This implies and Planck+ BICEP2+Keck Array joint constraint. From a and b,itis that, to satisfy both the constraints from the amplitude of the < < also observed that, within 50 N 70, the logarithmic potential is scalar power spectrum and its spectral tilt within 2σ CL the favored for the characteristic index 1/2 < q < 2, by Planck 2015 data and Planck+ BICEP2+Keck Array joint analysis. In a, we have explic- constrained numerical value of the stringy parameter is lying itly shown that in the r–nζ plane the observationally favored lower within the window 48 < g < 90 in which q = 2 case is bound for the characteristic index is q ≥ 1/2 slightly disfavored compared to the other lower values of q studied in the present context. Also in Fig. 7cforq = 1/2, the characteristic index is q ≥ 1/2. It is additionally impor- and q = 1wefixN/g ∼ 0.7, which further implies that tant to note that, for q >> 2, the tensor-to-scalar ratio com- for 50 < N < 70, the prescribed window for g from the puted from the model is negligibly small for the logarithmic ζ –nζ plot is given by 71.4 < g < 100. If we additionally potential. This implies that if the inflationary tensor mode is impose the constraint from the upper bound on tensor-to- detected close to its present upper bound on tensor-to-scalar scalar ratio then also the allowed parameter range is lying ratio then all q >> 2 possibilities for tachyonic inflation can within a similar window, i.e., 71.4 < g < 90 and combin- be discarded for the logarithmic potential. On the contrary, ing all the constraints the allowed value of the characteristic if inflationary tensor modes are never detected by any of the index is lying within 1/2 < q < 1. However, q = 1/2is future observational probes then the q >> 2 possibilities tightly constrained as depicted in Fig. 7c. for tachyonic inflation in the case of a logarithmic potential Model III: exponential potential Type I are highly prominent. Also it is important to mention that, in For the single field case the third model of the tachyonic < < Fig. 6b within the window 40 g 150, if we increase potential is given by the value of g, then the inflationary tensor-to-scalar ratio 123 278 Page 70 of 130 Eur. Phys. J. C (2016) 76 :278

Fig. 7 Variation of the a scalar power spectrum ζ vs. scalar spectral index nζ , b scalar power spectrum ζ vs. the stringy parameter g and c scalar spectral tilt nζ vs. the stringy parameter g.Thepurple and blue colored line represent the upper and lower bound allowed by WMAP+Planck 2015 data respectively. The green dotted region bounded by two vertical black colored lines represent the Planck 2σ allowed region and the rest of the light gray shaded region is disfavored by the Planck+WMAP constraint (a) Δζ vs g. (b) nζ vs g.

T V (T ) = λ exp − , (5.235) T0 where λ characterizes the scale of inflation and T0 is the parameter of the model. In Fig. 8 we have depicted the behavior of the exponential potential Type-I with respect to the scaled field coordinate T/T0 in dimensionless units. In this case the tachyon field started rolling down from the top height of the potential from the left hand side and takes part in the inflationary dynamics. Next using specified form of the potential the potential dependent slow-roll parameters are computed as 1 T ( )/λ ¯ = exp , (5.236) Fig. 8 Variation of the exponential potential Type-I V T with field V T/T in dimensionless units 2g T0 0 1 T η¯V = exp , (5.237) g T0 where the factor g is defined as ξ¯ 2 = 1 2T , V 2 exp (5.238) g T0 2 2 1 3T α λT M4 α T σ¯ 3 = , g = 0 = s 0 . (5.240) V 3 exp (5.239) 2 ( π)3 2 g T0 Mp 2 gs Mp 123 Eur. Phys. J. C (2016) 76 :278 Page 71 of 130 278

⎧ Next we compute the number of e-foldings from this model: ⎪ 4 ⎪ − , for q = 1/2, ⎧ ⎨⎪ 1 3 ⎪ T T N + ⎪ g − − − end , q = / , 2 ⎨ exp exp for 1 2 κζ, ≈ T0 T0 ⎪ 4 N(T ) = ⎪ − , for any arbitrary q, ⎪ ⎪ 3 ⎪ T T ⎩⎪ / N ⎩⎪ − − − end , . (2q)3 2 √ + 1 2qg exp exp for any arbitrary q q 2 T0 T0 2 (5.241) (5.249) ⎧ ⎪ 8 Further using the condition to end inflation: ⎪ , for q = 1/2, ⎨ + 1 N 2 ¯V (Tend) = 1, (5.242) r ≈ 8 (5.250) ⎪ , q. ⎩⎪ for any arbitrary 2q √N + 1 we get the following field value at the end of inflation: 2q 2

Tend = T0 ln(2g). (5.243) For the inverse cosh potential we get the following consistency relations: Next using N = Ncmb = N and T = Tcmb = T at the ⎧ horizon crossing we get ⎨ 1, for q = 1/2, ⎧ r ≈ 4(1 − nζ,) × 1 ⎩ √ , for any arbitrary q, ⎪ g ⎪ , = / , 2q ⎪ ln 1 for q 1 2 ⎨⎪ + N (5.251) ⎡ 2 ⎤ T = T0 × (5.244) ⎪ ⎪ ⎣ g ⎦ λ 1, for q = 1/2, ⎪ ln , for any arbitrary q. ζ, ≈ × ⎩ 1 + √N 3gπ 2 M4(1 − nζ,)2 1, for any arbitrary q, 2 2q p (5.252) Also the ﬁeld excursion can be computed as 8 8 ⎧ ⎧ 8 8 ⎪ 1 ⎪ 1 ⎪ N + , for q = 1/2, ⎪ 8ln 8 , for q = 1/2, ⎨ ⎪ 8 + 8 2λ 2 ⎨ 1 2N ζ, ≈ × π 2 4 ⎪ 8 ⎡ ⎤8 3g Mpr ⎪ N 1 |T |=T0 × 8 8 (5.245) ⎩⎪ √ + , , ⎪ 8 8 for any arbitrary q ⎪ 8 ⎣ 1 ⎦8 , . 2q 2 ⎩⎪ 8ln 8 for any arbitrary q 8 1 + √2N 8 (5.253) 2q

Finally we compute the following inflationary observ- 1 2 1, for q = 1/2, ables: αζ, ≈− nζ, − 1 × 2 1, for any arbitrary q, ⎧ % & ⎪ + 1 2 ⎪ N (5.254) ⎪ 2 , for q = 1/2, ⎨ g gλ ⎛ ⎞ ζ, ≈ × 2 , = / , 12π 2 M4 ⎪ √N + 1 1 3 1 for q 1 2 p ⎪ 2q 2 κζ, ≈ nζ, − 1 × (5.255) ⎪ 2q ⎝ ⎠ , for any arbitrary q, 2 1, for any arbitrary q. ⎩ g (5.246) Let us now discuss the general constraints on the parameters of tachyonic string theory including the factor q and ⎧ on the parameters appearing in the expression for expo- ⎪ 2 ⎪ − , for q = 1/2, nential potential Type-I. In Fig. 9a, we have shown the ⎨⎪ + 1 N 2 − ≈ behavior of the tensor-to-scalar ratio r with respect to the nζ, 1 ⎪ 2 ⎪ − , , scalar spectral index nζ for exponential potential Type-I ⎩⎪ √ for any arbitrary q 2q √N + 1 respectively. In both the figures the purple and blue colored 2q 2 line represent the upper bound of the tensor-to-scalar ratio (5.247) allowed by Planck+ BICEP2+Keck Array joint constraint ⎧ and only Planck 2015 data respectively. For both the fig- ⎪ 2 ⎪ − , for q = 1/2, ures the red, green, brown, orange colored curve represent ⎪ 1 2 ⎨ N + q = 1/2, q = 1, q = 3/2 and q = 2, respectively. The cyan α ≈ 2 ζ, ⎪ 2 color shaded region bounded by two vertical black colored ⎪ − , for any arbitrary q, ⎩⎪ 2 lines in Fig. 9a represent the Planck 2σ allowed region and 2q √N + 1 2q 2 the rest of the light gray shaded region shows the 1σ allowed (5.248) range, which is at present disfavored by the Planck 2015 data 123 278 Page 72 of 130 Eur. Phys. J. C (2016) 76 :278

(a) vs . (b) Δζ vs nζ .

Fig. 9 In a the behavior of the tensor-to-scalar ratio r with respect shown that in r − nζ plane the observationally favored lower bound to the scalar spectral index nζ and b the parameter g for exponential for the characteristic index is q ≥ 1/2. Variation of the scalar power potential Type-I. The purple and blue colored line represent the upper spectrum ζ vs. scalar spectral index nζ is shown in b.Thepurple bound of the tensor-to-scalar ratio allowed by Planck+ BICEP2+Keck and blue colored line represent the upper and lower bounds allowed Array joint constraint and only Planck 2015 data respectively. For both by WMAP+Planck 2015 data respectively. The green dotted region figures the red, green, brown, orange colored curves represent q = 1/2, bounded by two vertical black colored lines represents the Planck 2σ q = 1, q = 3/2andq = 2, respectively. The cyan color shaded region allowed region and the rest of the light gray shaded region is disfavored bounded by two vertical black colored lines in a represent the Planck by the Planck+WMAP constraint. From a and b, it is also observed that, 2σ allowed region and the rest of the light gray shaded region shows the within 50 < N < 70, the exponential potential Type-I is favored for 1σ allowed range, which is at present disfavored by the Planck data and the characteristic index 1 < q < 2, by Planck 2015 data and Planck+ Planck+ BICEP2+Keck Array joint constraint. In a,wehaveexplicitly BICEP2+Keck Array joint analysis and Planck+ BICEP2+Keck Array joint constraint. The rest all of the figures the red, green, brown, orange colored curves of the region is completely ruled out by the present obser- represent q = 1/2, q = 1, q = 3/2 and q = 2 respectively. vational constraints. From Fig. 9a, it is also observed that, The purple and blue colored line represent the upper and within 50 < N < 70, the exponential potential Type-I is lower bound allowed by WMAP+Planck 2015 data respec- favored for the characteristic index 1/2 < q < 2, by Planck tively. The cyan color shaded region bounded by two vertical 2015 data and Planck+ BICEP2+Keck Array joint analysis. black colored lines represent the Planck 2σ allowed region Also in Fig. 9aforq = 1/2, q = 1, q = 3/2 and q = 2 and the rest of the light gray shaded region is the 1σ region, we fix N/g ∼ 0.8. This implies that for 50 < N < 70, which is presently disfavored by the joint Planck+WMAP the prescribed window for g from r–nζ plot is given by constraints. The rest of the region is completely ruled out 63 < g < 88. To analyze the results more clearly let us by the present observational constraints. Also in Fig. 9bfor describe the cosmological features from Fig. 9a in detail. q = 1/2, q = 1, q = 3/2 and q = 2wefixN ∼ 70. But Let us first start with the q = 1/2 situation, in which the the plots can be reproduced for 50 < N < 70 also consid- 2σ constraint on the scalar spectral tilt is satisfied within ering the present observational constraints from Planck 2015 the window of the tensor-to-scalar ratio, 0.10 < r < 0.12 data. It is important to mention here that if we combine the for 50 < N < 70. Next for the q = 1 case, the same constraints obtained from Fig. 9a, b, then by comparing the constraint is satisfied within the window of the tensor-to- behavior of the GTachyon in the r–nζ and ζ –nζ planes we scalar ratio, 0.07 < r < 0.11 for 50 < N < 70. Further clearly observe that the q = 1/2 case is almost discarded as for the q = 3/2 case, the same constraint on scalar spec- it is not consistent with both of the 2σ constraints simulta- tral tilt is satisfied within the window of the tensor-to-scalar neously. ratio, 0.06 < r < 0.085 for 50 < N < 70. Finally, for In Fig. 10a, b, we have shown the behavior of the run- q = 2 situation, the value for the tensor-to-scalar ratio is ning of the scalar spectral tilt αζ and running of the run- 0.05 < r < 0.075. ning of the scalar spectral tilt κζ with respect to the scalar In Fig. 9b, we have depicted the behavior of the scalar spectral index nζ for exponential potential Type I respec- power spectrum ζ vs. scalar spectral tilt nζ for exponential tively. For both the figures the red colored curve represent potential Type-I respectively. It is important to note that, for the behavior for any arbitrary values of q. The cyan color

123 Eur. Phys. J. C (2016) 76 :278 Page 73 of 130 278

(a) αζ VS nζ .

Fig. 11 Variation of the exponential potential Type-II V (T )/λ with ﬁeld T/T0 in dimensionless units

power spectrum in this case. To produce the correct value of the amplitude of the scalar power spectra we fix the parameter 360 < g < 400. Model IV: exponential potential-type II For the single field case the first model of the tachyonic potential is given by (b) κζ VS nζ . 2 Fig. 10 a αζ b Behavior of the running of the scalar spectral tilt and ( ) = λ − T , running of the running of the scalar spectral tilt κζ with respect to the V T exp (5.256) T0 scalar spectral index nζ for exponential potential Type-I. For both the figures red colored curves represent for any value of q, respectively. The cyan color shaded region bounded by two vertical black colored lines where λ characterize the scale of inflation and T0 is the param- in a represent the Planck 2σ allowedregionandtherestofthelight eter of the model. In Fig. 11 we have depicted the symmetric σ gray shaded region shows the 1 allowed range, which is at present behavior of the exponential potential Type-I with respect to disfavored by the Planck data and Planck+ BICEP2+Keck Array joint / constraint. From a and b, it is also observed that, within 50 < N < 70, the scaled field coordinate T T0 in dimensionless units. In the exponential potential Type-I is favored for the characteristic index this case the tachyon field started rolling down from the top 1 < q < 2, by Planck 2015 data and Planck+ BICEP2+Keck Array height of the potential situated at the origin and takes part in joint analysis the inflationary dynamics. Next using specified form of the potential the potential dependent slow-roll parameters are computed as shaded region bounded by two vertical black colored lines σ in both the plots represent the Planck 2 allowed region and 1 T 2 T 2 the rest of the light gray shaded region shows the 1σ allowed ¯V = 4 exp , (5.257) 2g T0 T0 range, which is at present disfavored by the Planck data and Planck+ BICEP2+Keck Array joint constraint. From both of 2 2 η¯ = 1 T − T , these figures, it is also observed that, within 50 < N < 70, V 2 1 exp (5.258) g T0 T0 the exponential potential Type-I is favored for the characteristic index 1 < q < 2, by Planck 2015 data and 2 2 2 ξ¯ 2 = 1 T T − T , Planck+ BICEP2+Keck Array joint analysis. It is also impor- V 2 8 2 3 exp 2 g T0 T0 T0 tant to note that for any values of q, the numerical value − (5.259) of the running αζ ∼ O(−10 4) and running of the run- κ ∼ O(− −5) 2 4 2 ning ζ 10 , which are perfectly consistent with σ¯ 3 = 1 T T − T + V 3 16 4 12 3 the 1.5σ constraints on running and running of the running g T0 T0 T0 as obtained from Planck 2015 data. For q = 1/2, q = 1, 2 q = 3/2 and q = 2 g is not explicitly appearing in the var- T × exp 3 , (5.260) ious inflationary observables except the amplitude of scalar T0 123 278 Page 74 of 130 Eur. Phys. J. C (2016) 76 :278 where the factor g is defined as Next using N = Ncmb = N and T = Tcmb = T at the horizon crossing we get α λT 2 M4 α T 2 g = 0 = s 0 . (5.261) ⎧ 2 3 2 N T M (2π) gs M ⎪ + end , = / , p p ⎨⎪ exp ln for q 1 2 g T0 5 = × Next we compute the number of e-foldings from this model: T T0 ⎪ ⎪ N T ⎧ ⎩⎪ √ + end , . T T exp ln for any arbitrary q ⎪ − end , = / , 2qg T0 ⎨⎪ g ln ln for q 1 2 T0 T0 ( ) ≈ (5.267) N T ⎪ ⎪ T T ⎩ 2qg ln − ln end , for any arbitrary q. Also the field excursion can be computed as T0 T0 8 8 ⎧8 8 (5.263) ⎪8 N 8 ⎪8exp + ln p − p8 , for q = 1/2, ⎨ g Further using the condition to end inflation: |T |=T × 8 8 0 ⎪8 8 ⎪8 N 8 ¯ (T ) = 1. (5.264) ⎩8exp √ +ln p − p8 , for any arbitrary q V end 2qg we get the following transcendental equation to determine (5.268) the field value at the end of inflation: g where p = 1, . But during numerical estimation we only 2 2 2 Tend Tend = g exp (5.265) = g = T0 T0 2 take p 2 because p 1 is disfavored by the Planck 2015 data. which we need to solve numerically for a given value of g. Finally we compute the following inflationary observ- For the sake of simplicity in the further computation one can ables: consider the following possibilities: ⎧ ⎨ , g ∼ 1T0 for 2 e Tend ≈ g (5.266) ⎩ T , for g << 1 2 0 2

⎧ ⎪ − 2N + Tend ⎪ exp 2exp g 2ln T ⎪ 0 , = / , ⎪ for q 1 2 ⎨⎪ exp 2N + 2ln Tend gλ g T0 ζ, ≈ × (5.269) π 2 M4 ⎪ 48 p ⎪ 2N Tend ⎪ exp −2exp √ + 2ln ⎪ 2qg T0 , , ⎩⎪ 2q for any arbitrary q exp √2N + 2ln Tend 2qg T0

⎧ 2 ⎪ 2 T 2N ⎪ − + end , = / , ⎨⎪ 1 2 exp for q 1 2 g T0 g nζ, − 1 ≈ (5.270) ⎪ 2 ⎪ 2 Tend 2N ⎩⎪ −√ 1 + 2 exp √ , for any arbitrary q. 2qg T0 2qg

5 It is important to note that here we have used the following approximation: T 2 T Ei − ≈ γ + 2ln +··· (5.262) T0 T0 where all the terms represented via ··· are negligibly small in the series expansion. Here γ represents the Euler constant, with the numerical value γ 0.577216. 123 Eur. Phys. J. C (2016) 76 :278 Page 75 of 130 278

⎧ ⎪ 2 ⎪ Tend 2N ⎨⎪ exp , for q = 1/2, 8 T0 g αζ, ≈ × (5.271) g2 ⎪ 2 ⎪ 1 T 2N ⎩⎪ end exp √ , for any arbitrary q, 2q T0 2qg ⎧ ⎪ 2 ⎪ Tend 2N ⎨⎪ exp , for q = 1/2, 16 T0 g κζ, ≈− × (5.272) g3 ⎪ 2 ⎪ 1 T 2N ⎩⎪ end √ , q, 3/2 exp for any arbitrary (2q) T0 2qg ⎧ 2 2 ⎪ T 2N T 2N ⎪ end + end , = / , ⎨⎪ exp exp for q 1 2 32 T0 g T0 g r ≈ × (5.273) g ⎪ 2 2 ⎪ 1 Tend 2N Tend 2N ⎩⎪ exp √ + exp √ , for any arbitrary q. 2q T0 2qg T0 2qg For the inverse cosh potential we get the following consistency relations: ⎧ ⎪ g 1 g ⎪ 1 − nζ, − 1 exp 1 − nζ, − 1 , for q = 1/2, ⎨ 2 2 2 ≈ 16 × r ⎪ √ √ (5.274) g ⎪ 1 2qg 1 2qg ⎩ 1 − nζ, − 1 exp 1 − nζ, − 1 , for any arbitrary q, 2q 2 2 2 ⎧ g ⎪ exp − 1 − nζ, − 1 ⎪ 2 , = / , ⎪ g for q 1 2 ⎪ 1 − nζ, − 1 λ ⎨ 2 g √ ζ, ≈ × 2qg (5.275) 24π 2 M4 ⎪ exp − 1 − nζ, − 1 p ⎪ 2 ⎪ 2q √ , for any arbitrary q, ⎩ 2qg − − 2 1 nζ, 1 ⎧ 2 ⎪ T 2N ⎪ − end , = / , ⎨⎪ exp exp for q 1 2 2λ T0 g ζ, ≈ × (5.276) 3π 2 M4r ⎪ 2 p ⎪ Tend 2N ⎩⎪ exp − exp √ , for any arbitrary q, T0 2qg ⎧ g ⎪ − − , = / , ⎨⎪ 1 nζ, 1 for q 1 2 4 2 αζ, ≈ × √ (5.277) 2 ⎪ g ⎪ 1 2qg ⎩ 1 − nζ, − 1 , for any arbitrary q, 2q 2 ⎧ g ⎪ − − , = / , ⎨⎪ 1 nζ, 1 for q 1 2 8 2 κζ, ≈− × √ (5.278) 3 ⎪ g ⎪ 1 2qg ⎩ 1 − nζ, − 1 , for any arbitrary q. (2q)3/2 2

123 278 Page 76 of 130 Eur. Phys. J. C (2016) 76 :278

resent q = 1/2, q = 1, q = 3/2 and q = 2, respectively. The cyan color shaded region bounded by two vertical black colored lines in Fig. 12a represent the Planck 2σ allowed region and the rest of the light gray shaded region shows the 1σ allowed range, which is at present disfavored by the Planck 2015 data and Planck+ BICEP2+Keck Array joint constraint. The rest of the region is completely ruled out by the present observational constraints. From Fig. 12a, b, it is also observed that, within 50 < N < 70, the exponential potential Type-II is favored only for the characteristic index 1/2 < q < 2, by Planck 2015 data and Planck+ BICEP2+Keck Array joint analysis. Also in Fig. 12afor q = 1/2, q = 1, q = 3/2 and q = 2wefixN/g ∼ 0.85. (a) vs . This implies that for 50 < N < 70, the prescribed window for g from r–nζ plot is given by 59 < g < 82.3 considering 1/2 < q < 2. It is additionally important to note that, for q << 1/2, the tensor-to-scalar ratio computed from the model is negligibly small for exponential potential Type-II. This implies that if the inflationary tensor mode is detected near its present upper bound on the tensor-to-scalar ratio then all q << 1/2 possibilities for tachyonic inflation can be discarded for exponential potential Type-II. On the contrary, if inflationary tensor modes are never detected by any of the future observational probes then q << 1/2 possibilities for tachyonic inflation in the case of an exponential potential Type-II is highly prominent. Also it is important to mention that, in Fig. 12b within the window 30 < g < 93, if we take a (b) vs . smaller value of g, then the inflationary tensor-to-scalar ratio also gradually decreases. To analyze the results more clearly Fig. 12 Behavior of the tensor-to-scalar ratio r with respect to a the let us describe the cosmological features from Fig. 12ain scalar spectral index nζ and b the parameter g for exponential potential detail. Let us first start with the q = 1/2 situation, in which purple blue colored line upper Type-II. The and represent the bound σ of the tensor-to-scalar ratio allowed by Planck+ BICEP2+Keck Array the 2 constraint on the scalar spectral tilt is satisfied within joint constraint and only Planck 2015 data respectively. For both figures the window of the tensor-to-scalar ratio, 0 < r < 0.065 for = / = the red, green, brown, orange colored curves represent q 1 2, q 1, 50 < N < 70. Next for the q = 1 case, the same con- = / = q 3 2andq 2, respectively. The cyan color shaded region bounded straint is satisfied within the window of the tensor-to-scalar vertical black colored lines a σ by two in represent the Planck 2 allowed . < < . < < region and the rest of the light gray shaded region shows the 1σ allowed ratio, 0 030 r 0 12 for 50 N 70. Further for range, which is at present disfavored by the Planck data and Planck+ the q = 3/2 case, the same constraint on scalar spectral tilt BICEP2+Keck Array joint constraint. From a and b, it is also observed is satisfied within the window of the tensor-to-scalar ratio, that, within 50 < N < 70, the logarithmic potential is favored for the 0.045 < r < 0.12 for 50 < N < 70. Finally, for the / < q < characteristic index 1 2 2, by Planck 2015 data and Planck+ = BICEP2+Keck Array joint analysis q 2 situation, the value for the tensor-to-scalar ratio is 0.060 < r < 0.12, which is tightly constrained from the upper bound of spectral tilt from Planck 2015 observational Let us now discuss the general constraints on the parame- data. ters of tachyonic string theory including the factor q and on In Fig. 13a–c, we have depicted the behavior of the scalar the parameters appearing in the expression for exponential power spectrum ζ vs. the stringy parameter g, scalar spec- potential Type-II. In Fig. 12a, b, we have shown the behav- tral tilt nζ vs. the stringy parameter g and scalar power spec- ior of the tensor-to-scalar ratio r with respect to the scalar trum ζ vs. scalar spectral index nζ for exponential potential, spectral index nζ and the model parameter g for exponential Type-II respectively. It is important to note that, for all of the potential Type-II respectively. In both the figures the purple figures the red, green, brown, orange colored curves rep- and blue colored line represent the upper bound of tensor-to- resent q = 1/2, q = 1, q = 3/2 and q = 2 respectively. scalar ratio allowed by Planck+ BICEP2+Keck Array joint The purple and blue colored line represent the upper and constraint and only Planck 2015 data respectively. For both lower bound allowed by WMAP+Planck 2015 data respec- the figures the red, green, brown, orange colored curve rep- tively. The cyan color shaded region bounded by two vertical 123 Eur. Phys. J. C (2016) 76 :278 Page 77 of 130 278

Fig. 13 Variation of the a scalar power spectrum ζ vs. scalar spectral index nζ , b scalar power spectrum ζ vs. the stringy parameter g and c scalar spectral tilt nζ vs. the stringy parameter g.Thepurple and blue colored line represent the upper and lower bound allowed by WMAP+Planck 2015 data respectively. The green dotted region bounded by two vertical black colored lines represent the Planck 2σ allowed region and the rest of the light gray shaded region is disfavored by the Planck+WMAP constraint (a) Δζ vs g. (b) nζ vs g.

black colored lines represent the Planck 2σ allowed region which further implies that for 50 < N < 70, the prescribed and the rest of the light gray shaded region is the 1σ region, window for g from ζ –nζ plot is given by 59 < g < 82.3. If which is presently disfavored by the joint Planck+WMAP we additionally impose the constraint from the upper bound constraints. The rest of the region is completely ruled out on the tensor-to-scalar ratio then also the allowed parameter by the present observational constraints. From Fig. 13ait range is lying within the window, i.e., 73 < g < 82.3. is clearly observed that the observational constraints on the In Fig. 14a, b, we have shown the behavior of the running amplitude of the scalar mode fluctuations satisfy within the of the scalar spectral tilt αζ and running of the running of the window 55 < g < 93 considering 1/2 < q < 2. For scalar spectral tilt κζ with respect to the scalar spectral index g > 93 the corresponding amplitude falls down in a non- nζ for exponential potential Type II with g = 76, respec- trivial fashion by following the exact functional form as tively. For both the figures the red, green, brown, orange stated in Eq. (5.269). Next using the behavior as shown in colored curves represent q = 1/2, q = 1, q = 3/2 and Fig. 13b, the lower bound on the stringy parameter g is con- q = 2, respectively. The cyan color shaded region bounded strained by g > 73 by using the non-trivial relationship as by two vertical black colored lines in both plots represent the stated in Eq. (5.270). But at this lower bound of the param- Planck 2σ allowed region and the rest of the light gray shaded eter g the amplitude of the scalar power spectrum is slightly region shows the 1σ allowed range, which is at present disfa- outside to the present observational constraints. This implies vored by the Planck data and Planck+ BICEP2+Keck Array that, to satisfy both the constraints from the amplitude of joint constraint. From both of these figures, it is also observed the scalar power spectrum and its spectral tilt, within 2σ CL that, within 50 < N < 70, the exponential potential Type-II the constrained numerical value of the stringy parameter is is favored for the characteristic index 1/2 < q < 2, by the lying within the window 73 < g < 93. Also in Fig. 13cfor Planck 2015 data and Planck+ BICEP2+Keck Array joint q = 1/2, q = 1, q = 3/2 and q = 2wefixN/g ∼ 0.85, analysis. From Fig. 14a, b, it is observed that within the 2σ

123 278 Page 78 of 130 Eur. Phys. J. C (2016) 76 :278

(a) αζ vs nζ.

Fig. 15 Variation of the inverse power-law potential V (T )/λ with the ﬁeld T/T0 in dimensionless units

where λ characterize the scale of inflation and T0 is the parameter of the model. In Fig. 15 we have depicted the symmetric behavior of the inverse power-law potential with respect to the scaled field coordinate T/T0 in dimensionless units. In this case the tachyon field started rolling down from the top height of the potential situated at the origin and takes part in the inflationary dynamics. Next using specified form of the potential the potential (b) κζ vs nζ. dependent slow-roll parameters are computed as Fig. 14 Behavior of the a running of the scalar spectral tilt αζ and b running of the running of the scalar spectral tilt κζ with respect to the 6 T scalar spectral index nζ for the inverse cosh potential with g = 88. For 16 1 T0 both figures the red, green, brown, orange colored curves represent ¯V = , (5.280) = / = = / = 2g 4 q 1 2, q 1, q 3 2andq 2 respectively. The cyan color 1 + T shaded region bounded by two vertical black colored lines in a represent T0 the Planck 2σ allowed region and the rest of the light gray shaded region shows the 1σ allowed range, which is at present disfavored by the Planck 4 data and Planck+ BICEP2+Keck Array joint constraint. From a and b, 4 5 T − 3 it is also observed that, within 50 < N < 70, the inverse cosh potential 1 T0 / < < η¯ = , (5.281) is favored for the characteristic index 1 2 q 2, by Planck 2015 V g 4 data and Planck+ BICEP2+Keck Array joint analysis 1 + T T0

observed range of the scalar spectral tilt nζ ,asthevalueof 2 8 4 96 T 5 T − 10 T + 1 the characteristic parameter q increases, the value of the run- T0 T0 T0 ξ¯ 2 = 1 , α κ V (5.282) ning ζ decreases and running of the running ζ increases g2 4 2 for exponential potential Type-II. It is also important to note 1 + T T0 that for 1/2 < q < 2, the numerical value of the running −4 −6 αζ ∼ O(10 ) and running of the running κζ ∼ O(−10 ), 6 4 8 4 which are perfectly consistent with the 1.5σ constraints on 384 T 5 T 7 T −31 T +13 −1 T0 T0 T0 T0 σ¯ 3 = 1 , running and running of the running as obtained from the V g3 4 3 1+ T Planck 2015 data. T0 Model V: inverse power-law potential (5.283) For single field case the first model of tachyonic potential is given by where the factor g is defined as λ V (T ) = , (5.279) 2 4 2 4 α λT M α T + T g = 0 = s 0 . (5.284) 1 T 2 ( π)3 2 0 Mp 2 gs Mp 123 Eur. Phys. J. C (2016) 76 :278 Page 79 of 130 278

⎧ Next we compute the number of e-foldings from this model: ⎪ 3 ⎧ ⎡ ⎤ ⎪ − , for q = 1/2, ⎨ (N + 1) ⎪ ⎪ g ⎢ 1 1 ⎥ nζ, − 1 ≈ (5.291) ⎪ ⎣ − ⎦ , q = / , ⎪ 3 ⎪ 2 2 for 1 2 ⎪ − , q. ⎪ 8 T ⎪ for any arbitrary ⎨ T end ⎩ √N T0 T0 + 1 N(T )= ⎡ ⎤ ⎧ 2q ⎪ ⎪ ⎪ 3 ⎪ g ⎢ 1 1 ⎥ ⎪ − , for q = 1/2, ⎪ 2q ⎣ − ⎦ , for any arbitrary q. ⎨⎪ ( + )2 ⎩⎪ 8 2 2 N 1 T Tend α ≈ T0 T0 ζ, 3 ⎪ − , q, (5.285) ⎪ √ 2 for any arbitrary ⎩⎪ 2q √N + 1 2q Further using the condition to end inflation: ⎧ (5.292) ⎪ − 6 , = / , ¯V (Tend) = 1, (5.286) ⎪ for q 1 2 ⎨ (N + 1)3 κζ, ≈ 6 (5.293) we get the following field values at the end of inflation: ⎪ − , , ⎪ 3 for any arbitrary q 1 ⎩⎪ 2q √N + 1 g 2q Tend ∼ T0. (5.287) ⎧ 8 ⎪ 16 ⎪ , for q = 1/2, ⎨ (N + 1) Next using N = Ncmb = N and T = Tcmb = T at the r ≈ (5.294) horizon crossing we get ⎪ 16 , . ⎩⎪ for any arbitrary q ⎧ 2q √N + 1 2q ⎪ 1 , = / , ⎪ for q 1 2 ⎪ 8 ( + ) For the inverse cosh potential we get the following consis- ⎨ g N 1 tency relations: T = T0 × ⎪ 1 ⎧ ⎪ 1 , for any arbitrary q, ⎪ , = / , ⎪ ⎨ 1 for q 1 2 ⎩ 8 √N + 1 16 g 2q r ≈ (1 − nζ,) × 1 (5.295) 3 ⎩⎪ , for any arbitrary q, (5.288) 2q ⎧ 1 ⎨⎪ , for q = 1/2, Also the field excursion can be computed as g4λ(1 − nζ,)3 221184 ζ, ≈ × π 2 4 ⎪ | |= 12 Mp ⎩ 2q T T0 8 8 , for any arbitrary q, ⎧ 8 1 8 221184 ⎪ 8 8 ⎪ 8 1 g 8 (5.296) ⎪ 8 − 8 , for q = 1/2, ⎧ ⎪ 8 ⎪ 1 ⎪ 8 ( + ) 8 8 ⎪ , = / , ⎨⎪ g N 1 4 3 ⎨ for q 1 2 8 8 g λr 33554432 × 8 8 ζ, ≈ × ⎪ 8 1 8 12π 2 M4 ⎪ (2q)3 ⎪ 8 8 p ⎩ , , ⎪ 8 1 g 8 for any arbitrary q ⎪ 81 − 8 , for any arbitrary q. 33554432 ⎪ 8 8 8 (5.297) ⎩ 8 8 √N + 1 8 ⎧ g 2q ⎪ , = / , ⎨ 1 for q 1 2 (5.289) 1 2 αζ, ≈− nζ, − 1 × 1 3 ⎩⎪ √ , for any arbitrary q, Finally we compute the following inflationary observ- 2q ables: ⎧ (5.298) ⎪ , = / , ⎨ 1 for q 1 2 gλ 2 3 ζ, ≈ κζ, ≈ nζ, − × π 2 4 1 ⎪ 1 12 Mp 9 ⎩ , for any arbitrary q. ⎧ 2q 1 ⎪ , for q = 1/2, ⎪ 3 (5.299) ⎨⎪ 8 ( + ) 16 g N 1 × Let us now discuss the general constraints on the param- ⎪ 2q eters of tachyonic string theory including the factor q and ⎪ , for any arbitrary q, ⎩⎪ 3 on the parameters appearing in the expression for inverse 16 8 √N + 1 g 2q power-law potential. In Fig. 16a, we have shown the behav- (5.290) ior of the tensor-to-scalar ratio r with respect to the scalar 123 278 Page 80 of 130 Eur. Phys. J. C (2016) 76 :278

(a) rnvs ζ (b) Δζ vs nζ

Fig. 16 In a the behavior of the tensor-to-scalar ratio r with respect to teristic index is q ≥ 1/2. Variation of the scalar power spectrum ζ the scalar spectral index nζ and b the parameter g for inverse power-law vs. scalar spectral index nζ is shown in b.Thepurple and blue colored potential. The purple and blue colored line represent the upper bound line represent the upper and lower bound allowed by WMAP+Planck of the tensor-to-scalar ratio allowed by Planck+ BICEP2+Keck Array 2015 data respectively. The green dotted region bounded by two verti- joint constraint and only Planck 2015 data respectively. For both figures cal black colored lines represent the Planck 2σ allowed region and the the red, green, brown, orange colored curves represent q = 1/2, q = 1, rest of the light gray shaded region is disfavored by the Planck+WMAP q = 3/2andq = 2, respectively. The cyan color shaded region bounded constraint. In b, we have explicitly shown that in ζ − nζ plane the by two vertical black colored lines in a represent the Planck 2σ allowed observationally favored value of the characteristic index is q = 1and region and the rest of the light gray shaded region shows the 1σ allowed q = 3/2. From a and b, it is also observed that, within 50 < N < 70, range, which is at present disfavored by the Planck data and Planck+ the inverse power-law potential is favored for the characteristic index BICEP2+Keck Array joint constraint. In a, we have explicitly shown 3/2 < q < 1, by Planck 2015 data and Planck+ BICEP2+Keck Array that in r–nζ plane the observationally disfavored value of the charac- joint analysis spectral index nζ for inverse power-law potential respec- 0.07 < r < 0.10 for 50 < N < 70. Further for the tively. In both the figures the purple and blue colored q = 3/2 case, the same constraint on scalar spectral tilt line represent the upper bound of the tensor-to-scalar ratio is satisfied within the window of the tensor-to-scalar ratio, allowed by Planck+ BICEP2+Keck Array joint constraint 0.05 < r < 0.07 for 50 < N < 70. Finally, for the and only Planck 2015 data respectively. For both the fig- q = 2 situation, the value for the tensor-to-scalar ratio is ures the red, green, brown, orange colored curve represent 0.035 < r < 0.05. q = 1/2, q = 1, q = 3/2 and q = 2, respectively. The In Fig. 16b, we have depicted the behavior of the scalar cyan color shaded region bounded by two vertical black power spectrum ζ vs. scalar spectral tilt nζ for inverse colored lines in Fig. 16a represent the Planck 2σ allowed power-law potential respectively. It is important to note that, region and the rest of the light gray shaded region shows for all of the figures the red, green, brown, orange colored the 1σ allowed range, which is at present disfavored by the curves represent q = 1/2, q = 1, q = 3/2 and q = 2 Planck 2015 data and Planck+ BICEP2+Keck Array joint respectively. The purple and blue colored line represent the constraint. The rest of the region is completely ruled out upper and lower bound allowed by WMAP+Planck 2015 by the present observational constraints. From Fig. 16a, it data respectively. The cyan color shaded region bounded is also observed that, within 50 < N < 70, the inverse by two vertical black colored lines represent the Planck 2σ power-law potential is favored for the characteristic index allowed region and the rest of the light gray shaded region 1 < q < 2, by Planck 2015 data and Planck+ BICEP2+Keck is the 1σ region, which is presently disfavored by the joint Array joint analysis. Also in Fig. 16aforq = 1/2, q = 1, Planck+WMAP constraints. The rest of the region is com- q = 3/2 and q = 2wefixN ∼ 70. One can draw pletely ruled out by the present observational constraints. similar characteristic curves for any value of the number Also in Fig. 16bforq = 1/2, q = 1, q = 3/2 and of e-foldings lying within the window 50 < N < 70 q = 2wefixN ∼ 70. But the plots can be reproduced also. To analyze the results more clearly let us describe the for 50 < N < 70 also considering the present observational cosmological features from Fig. 16a in detail. Let us first constraints from Planck 2015 data. It is clearly observed from start with the q = 1/2 situation, in which the 2σ con- Fig. 16b that the Planck 2015 constraint on amplitude of the straint on the scalar spectral tilt is disfavored for 50 < scalar power spectrum ζ and the scalar spectral tilt nζ dis- N < 70.Nextfortheq = 1 case, the same constraint favor q = 1/2 and q = 2 values for inverse power-law is satisfied within the window of the tensor-to-scalar ratio, potential. 123 Eur. Phys. J. C (2016) 76 :278 Page 81 of 130 278

ζ –nζ plane then the stringent window on the characteristic index of GTachyon is lying within 1 < q < 3/2. It is also important to note that for any values of q, the numerical − value of the running αζ ∼ O(−10 4) and running of the − running κζ ∼ O(−10 5), which are perfectly consistent with the 1.5σ constraints on running and running of the running as obtained from Planck 2015 data. For q = 1/2, q = 1, q = 3/2 and q = 2 g does not explicitly appear in the various inﬂationary observables except the amplitude of scalar power spectrum in this case. To produce the correct value of the amplitude of the scalar power spectra we ﬁx the parameter αζ vs nζ. (a) 600 < g < 700.

5.1.9 Analyzing CMB power spectrum

In this section we explicitly study the cosmological consequences from the CMB TT, TE, EE, BB angular power spectrum computed from all the proposed models of inflation.6 The angular power spectra or equivalently the two point correlator of the X and Y fields are defined as [10] l XY 1 ∗ C ≡ a , aY,m, X, Y = T, E, B. 2 + 1 X m m=−l

(b) κζ vs nζ. (5.300) Here l characterizes the CMB multipole and m signifies Fig. 17 Behavior of the a running of the scalar spectral tilt αζ and b = running of the running of the scalar spectral tilt κζ with respect to the the magnetic quantum number, which runs from m scalar spectral index nζ for the inverse cosh potential with g = 88. For −l,...,+l. In general, the field X(nˆ) and Y (nˆ) can be both figures the red, green, brown, orange colored curves represent q = expressed in terms of the following harmonic expansion: 1/2, q = 1, q = 3/2andq = 2 respectively. The cyan color shaded ⎧ region bounded by two vertical black colored lines in a and a represent ∞ +l ⎪ the Planck 2σ allowed region and the rest of the light gray shaded region ⎪ (ˆ), = , ⎪ aT,lmVlm n for X T shows the 1σ allowed range, which is at present disfavored by the Planck ⎪ ⎪ l=0 m=−l data and Planck+ BICEP2+Keck Array joint constraint. From a and b, ⎪ ⎨⎪ ∞ +l it is also observed that, within 50 < N < 70, the inverse cosh potential is favored for the characteristic index 1/2 < q < 2, by Planck 2015 X(nˆ), Y (nˆ) = aE,lmVlm(nˆ), for X = E, ⎪ data and Planck+ BICEP2+Keck Array joint analysis ⎪ l=0 m=−l ⎪ ⎪ ∞ +l ⎪ ⎩⎪ aB,lmVlm(nˆ), for X = B, In Fig. 17a, b, we show the behavior of the running of the l=0 m=−l scalar spectral tilt αζ and running of the running of the scalar (5.301) spectral tilt κζ with respect to the scalar spectral index nζ for where nˆ is the arbitrary directional unit vector in CMB map. inverse power-law potential respectively. For both the figures Here V (nˆ) are the spherical harmonics which are chosen the red, green, brown, and orange colored curves represent lm to be the basis of the harmonic expansion of the temperature q = 1/2, q = 1, q = 3/2, and q = 2, respectively. The cyan anisotropy and the E and B polarization. Further using the color shaded region bounded by two vertical black colored inflationary power spectra at any momentum scale k: lines in both plots represent the Planck 2σ allowed region σ and the rest of the light gray shaded region shows the 1 (k) ≡{ζ (k), h(k)} (5.302) allowed range, which is at present disfavored by the Planck data and Planck+ BICEP2+Keck Array joint constraint. From 6 In this work we have not considered the possibility of other cross both of these figures, it is also observed that, within 50 < correlators, i.e., TB, EB as there is no observational evidence of such N < 70, the inverse power-law potential is favored for the contributions in the CMB map. Also till date there is no observational characteristic index 1/2 < q < 2, by Planck 2015 data evidence for inflationary origin of BB angular power spectrum except from CMB lensing [133]. However, for the sake of completeness in and a Planck+ BICEP2+Keck Array joint analysis. If we this paper we show the theoretical BB angular power spectra from the additionally impose the constraints from the r–nζ and the various models of the tachyonic potential. 123 278 Page 82 of 130 Eur. Phys. J. C (2016) 76 :278 the angular power spectra for CMB temperature fluctuations h(k) = r(k)ζ (k) α κ h, k h, k and polarization can be expressed as [10] n ,+ ln + ln2 +··· k h 2 k 6 k = h, . (5.313) XY 2 2 C = k k (k) (k) (k), k π d 2345 2 X 34 Y 5 (5.303) Inflation Anisotropies It is important to note that the cosmological significance of where the anisotropic integral kernel can be written as [10] the E and B decomposition of CMB polarization carries the η0 following significant features: ( ) = η ( ,η) ( [η − η]) . X k d 2SX 34k 5 2PX k 340 5 (5.304) 0 Sources Projection • Curvature (density) perturbations create only polarizing The integral (5.303) relates the inhomogeneities predicted E-modes. • by inflation, (k), to the anisotropies observed in the CMB, Vector (vorticity) perturbations create mainly B-modes. XY However, it is important to note that here the contribu- C . The correlations between the different X and Y modes tions of vector modes decay with the expansion of the are related by the transfer functions X(k) and Y (k). The transfer functions may be written as the line-of-sight universe and are therefore sub-dominant at the epoch of integral Eq. (5.304) which factorizes into physical source recombination. For this reason we have neglected such terms S (k,η)and geometric projection factors P (k[η − sub-dominant effects from our rest of the analysis. X X 0 • η]) through combinations of Bessel functions. Tensor (gravitational wave) perturbations create both E- Further expressing Eq. (5.303) in terms of TT, TE, EE, modes and B-modes. But to quantify the exact contri- BB correlation we get bution of the primordial gravitational waves, specifically in the inflationary B modes, one needs to separate all For curvature perturbation the other significant contributions in the B modes, i.e., the primordial magnetic field, gravitational lensing, non- TT 2 2 C = k k (k) (k) (k), π d S T T (5.305) Gaussianity, etc., But at present in the cosmological literature no such sophisticated techniques or algorithms TE = ( π)2 2 ( ) ( ) ( ), C 4 k dk S k T k E k (5.306) are available using which one can separate all of these significant contributions completely. EE 2 2 C = (4π) k dk S(k) E(k) E(k), (5.307)

For tensor perturbation To compute the momentum integrals numerically and to analyze the various features of CMB angular power spec- BB 2 2 C = (4π) k dk h(k) B(k) B(k) (5.308) tra from all of the inflationary models mentioned in the last section, here we use a semi-analytical code “CAMB”. TT 2 2 C = k k (k) (k) (k), For the numerical analysis we use here the best fit model π d h T T (5.309) parameters corresponding to all of the five tachyonic mod- TE 2 2 els, which are compatible with Planck 2015 data. Addition- C = ( π) k k (k) (k) (k), 4 d h T E (5.310) ally we take the CDM background. Further we put all these EE 2 2 inputs to “CAMB” and modify the inbuilt parameterization C = (4π) k dk (k) (k) (k), (5.311) h E E of the power spectrum for scalar and tensor modes accord- ingly. After performing all the numerical computations via where the inflationary power spectra {ζ (k), (k)} are h “CAMB” finally from our analysis we have generated all parametrized at any arbitrary momentum scale k as7 the theoretical CMB angular power spectra from which we α κ ζ, k ζ, k nζ,−1+ ln + ln2 +··· observed the following significant features: k 2 cSk 6 cSk ζ (k) = ζ, , cSk (5.312) • In Fig. 18,atlow region (2 < l < 49) the contributions from the running (αζ ,αh), and running of running (κζ ,κ ) are very small. Their additional contribution to 7 However, it is important to mention here that if we know the mode h scalar and tensor mode functions, which are obtained from the exact the CMB power spectrum for scalar and tensor modes solution of Mukhanov–Sasaki equation for exactly Bunch–Davies vac- becomes unity (O(1)) within low-l region and the origi- uum or for any arbitrary vacuum then one can implement the exact nal power spectrum becomes unchanged. As a result the structural form of the primordial power spectra. But within the slow- tachyonic models will be well fitted with the CMB TT roll regime of inflation the presented version of the parametrization of the power spectra is exactly compatible with the exact power spectrum spectrum at low-l region within high cosmic variance as from the solution of the Mukhanov–Sasaki equation. observed by Planck except for a few outliers according 123 Eur. Phys. J. C (2016) 76 :278 Page 83 of 130 278

( +1) TT 2 vs (scalar) TT (a) l l Cl / πl (b) l(l +1)Cl /2πlvs (scalar)

( +1) TT 2 vs (scalar) TT (c) l l Cl / πl (d) l(l +1)Cl /2πlvs (scalar)

TT (e) l(l +1)Cl /2πlvs (scalar)

Fig. 18 We show the variation of CMB TT angular power spectrum with respect to the multipole, l, for scalar modes for all ﬁve tachyonic models

123 278 Page 84 of 130 Eur. Phys. J. C (2016) 76 :278

to the Planck 2013 data release.8 But to update our anal- But till date only the contribution from the lensing B- ysis with the latest Planck 2015 data set we have not modes are detected via South Pole Telescope and Planck shown such high cosmic variance explicitly. In case of 2015 +BICEP2/Keck Array joint mission. But confirm- WMAP9 data for CMB TT spectrum the cosmic vari- ing the sole inflationary origin from the detection of the ance in the low-l region is not very large compared to de-lensed version of the signal is not sufficient to draw the Planck low-l data. But our tachyonic models are also any final conclusion.9 There are other possibilities as well fitted with WMAP9 low-l data, which we have not well through which it is possible to generate CMB B- shown explicitly in the plot to update the analysis using modes—those components are the primordial magnetic Planck 2015 data. It is also important to mention that if field, gravitational lensing, CP asymmetry in the lepton we incorporate the uncertainties in the scanning multi- sector of particle physics, etc. pole l for the measurement of CMB TT spectrum, then also our prescribed analysis is fairly consistent with the 5.2 Computation for assisted inflation Planck 2015 data. Further if we move toward the high regime (47 < l < 2500) the contributions of running In the case of assisted inflation [134–137] all the tachyons and running of running become stronger, and this will would follow a similar trajectory with a unique late time enhance the power spectrum to a permissible value such attractor.10 In more technical language one can state that that it will accurately fit high-l data within very a small ∼ ∼··· ≡ = ∀ = , ,..., . cosmic variance as observed by Planck. In this way one T1 T2 TM Ti T i 1 2 M (5.314) can easily scan over all the multipoles starting from low-l Most importantly the detailed computation of the density and to high-l using the same momentum dependent parame- tensor perturbation responsible for the anisotropy of CMB terization of the tensor-to-scalar ratio. depends on this late time attractor behavior of the fields. • From Fig. 18, we see that the Sachs–Wolfe plateau obtained from our proposed tachyonic models is non- 5.2.1 Condition for inflation flat, confirming the appearance of running, and running of the running in the spectrum observed for low For assisted tachyonic inflation, the prime condition for infla- l region (l < 47). For larger values of the multipole tion is given by (47 < l < 2500), the CMB anisotropy spectrum is dominated by the baryon acoustic oscillations (BAO), giving ¨ M (ρ + ) (ρ + ) ˙ 2 a i 3pi 3p rise to several ups and downs in the CMB TT spectrum. In H + H = =− = > 0 a 6M2 6M2 the low l region due to the presence of very large cosmic i=1 p p variance there may be other pre-inflationary scenarios (5.315) which might be able to describe the TT-power spectrum which can be re-expressed in terms of the following con- better. In our study we have considered only the pos- straint condition in the context of assisted tachyonic infla- sibility for which the behavior of tachyonic models is tion: analyzed for both low and high l regions. • From Figs. 18, 19, 20, we observe that if we include M ( ) V Ti − 3α ˙ 2 the uncertainties in multipole l as well in the observed 1 Ti ˙ 2 2 i=1 3M2 1 − α T CMB angular power spectra then the proposed tachyonic p i models is fairly consistent with the CMB TT, TE, EE for ( ) = MV T − 3α ˙ 2 > . scalar mode from Planck 2015 data. 1 T 0 (5.316) 3M2 1 − αT˙ 2 2 • In Figs. 21, 22, 23 and 24 we have explicitly shown the p theoretical CMB BB, TT, TE and EE angular power spec- Here Eq. (5.316) implies that to satisfy inflationary con- tra from the tensor mode. Most importantly, if the infla- straints in the slow-roll regime the following constraint tionary paradigm is responsible for the nearly de Sitter always holds good: expansion of the early universe then the CMB BB spec- 1 2 tra for tensor modes is one of the prime components T˙ < ∀ i = , ,...,M, α 1 2 (5.317) through which one can detect the contribution for pri- 3 1 mordial gravitational waves via the tensor-to-scalar ratio. 6 T¨ < H T˙ < H ∀ i = , ,...,M. 3 α 1 2 (5.318)

8 From Fig. 18 it is observed that for the inverse cosh and logarithmic 9 model q = 1/2, exponential Type-I and Type-II model q = 1, inverse In this respect one may consider the alternative frameworks of inﬂa- power-law model q = 1andq = 3/2 generalized characteristic indices tionary paradigm as well. are well ﬁtted with Planck 2015 data. 10 We also suggest the reader to read Ref. [138] for completeness. 123 Eur. Phys. J. C (2016) 76 :278 Page 85 of 130 278

( +1) TE 2 vs (scalar) TE (a) l l Cl / πl (b) l(l +1)Cl /2πlvs (scalar)

TE ( +1) TE 2 vs (scalar) (c) l(l +1)Cl /2πlvs (scalar) (d) l l Cl / πl

TE (e) l(l +1)Cl /2πlvs (scalar)

Fig. 19 We show the variation of CMB TE angular power spectrum with respect to the multipole, l, for scalar modes for all ﬁve tachyonic models

123 278 Page 86 of 130 Eur. Phys. J. C (2016) 76 :278

EE ( +1) EE 2 vs (scalar) (a) l(l +1)Cl /2πlvs (scalar) (b) l l Cl / πl

EE ( +1) EE 2 vs (scalar) (c) l(l +1)Cl /2πlvs (scalar) (d) l l Cl / πl

EE (e) l(l +1)Cl /2πlvs (scalar)

Fig. 20 We show the variation of CMB EE angular power spectrum with respect to the multipole, l, for scalar modes for all ﬁve tachyonic models

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BB ( +1) BB 2 vs (tensor) (a) l(l +1)Cl /2πlvs (tensor) (b) l l Cl / πl

BB ( +1) BB 2 vs (tensor) (c) l(l +1)Cl /2πlvs (tensor) (d) l l Cl / πl

BB (e) l(l +1)Cl /2πlvs (tensor)

Fig. 21 We show the variation of CMB BB angular power spectrum with respect to the multipole, l, for tensor modes for all ﬁve tachyonic models

123 278 Page 88 of 130 Eur. Phys. J. C (2016) 76 :278

TT (a) l(l +1)Cl /2πlvs (tensor) TT vs (b) l(l +1)Cl /2πl(tensor)

TT (d) ( +1) TT 2 vs (tensor) (c) l(l +1)Cl /2πlvs (tensor) l l Cl / πl

TT (e) l(l +1)Cl /2πlvs (tensor)

Fig. 22 We show the variation of CMB TT angular power spectrum with respect to the multipole, l, for tensor modes for all ﬁve tachyonic models 123 Eur. Phys. J. C (2016) 76 :278 Page 89 of 130 278

TE (b) ( +1) TE 2 vs (tensor) (a) l(l +1)Cl /2πlvs (tensor) l l Cl / πl

TE ( +1) TE 2 vs (tensor) (c) l(l +1)Cl /2πlvs (tensor) (d) l l Cl / πl

TE (e) l(l +1)Cl /2πlvs (tensor)

Fig. 23 We show the variation of CMB TE angular power spectrum with respect to the multipole, l, for tensor modes for all ﬁve tachyonic models

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EE vs EE (a) l(l +1)Cl /2πl(tensor) (b) l(l +1)Cl /2πlvs (tensor)

( +1) EE 2 vs (tensor) EE (c) l l Cl / πl (d) l(l +1)Cl /2πlvs (tensor)

EE (e) l(l +1)Cl /2πlvs (tensor)

Fig. 24 We show the variation of CMB EE angular power spectrum with respect to the multipole, l, for tensor modes for all ﬁve tachyonic models

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⎧ √ ⎪ 1 αβ ⎪ √ ∂μ −gV (T ,χ ) 1 + g ∂αχ ∂β χ ⎪ −g ef f 1 i i i ⎪ ⎪ α χ¨ dV (T ,χ ) ⎨⎪ = i + α χ˙ + ef f 1 i , = / , 2 3H i for q 1 2 1 − α χ˙ Vef f (T1,χi )dχi 0 = √ i (5.328) ⎪ 1 αβ q ⎪ √ ∂μ −gVef f (T1,χi ) 1 + g ∂αχi ∂β χi ⎪ −g ⎪ ⎪ 2qα χ¨ χ˙ dVef f (T ,χi ) ⎩⎪ = i + 6qα H i + 1 , for any q, − αχ˙ 2 − α( − )χ˙ 2 ( ,χ ) χ 1 i 1 1 2q i Vef f T1 i d i

Consequently the field equations are approximated by ( ,χ ) ( ) where Vef f T1 i represents the effective potential. The ˙ dV Ti 3Hα Ti + ≈ 0 ∀ i = 1, 2,...,M, (5.319) minimum in the χi direction is always appearing at χi = 0, V (Ti )dTi regardless of the explicit behavior of the tachyon field T1. Similarly, in the most generalized case, Consequently the late time solution has all the Ti equal. From M ( ) this analysis it is also observed that the length of the time V Ti − ( + )α ˙ 2 − 1 1 q T interval to reach this attractor behavior will depend on the ˙ 2 1 q i = 3M2 1 − α T i 1 p i initial separation, i.e., the tachyon field value χi and the extent ( ) = MV T − ( + )α ˙ 2 > . of the frictional contribution coming from the expansion rate − 1 1 q T 0 (5.320) 2 − α ˙ 2 1 q H. 3Mp 1 T It is important to mention that, for assisted inflation, the Here Eq. (5.320) implies that to satisfy inflationary con- equation of motion for each tachyon fields follows the fol- straints in the slow-roll regime the following constraint lowing simple relationship: always holds good: dlnTi dlnT ˙ 1 ∀ i = 1, 2,...,M. (5.329) T < ∀ i = 1, 2,...,M, (5.321) dt dt α(1 + q) Further substituting Eq. (5.324)inEqs.(5.319) and (5.323) 9 T¨ < H T˙ < H ∀ i = , ,...,M. we get 3 α( + ) 1 2 (5.322) 1 q √ 3MV(T ) dV (T ) Consequently the field equations are approximated by α T˙ + i ≈ 0, (5.330) ( ) i ( ) ˙ dV Ti Mp V Ti dTi 6qα H Ti + ≈ 0 ∀ i = 1, 2,...,M. (5.323) √ V (Ti )dTi MV(T ) dV (T ) q √ αT˙ + i ≈ . 6 i ( ) 0 (5.331) Also for both cases in the slow-roll regime the Friedmann 3Mp V Ti dTi equation is modified as M Finally the general solution for both cases can be expressed V (Ti ) MV(T ) ( ) H 2 ≈ = . in terms of the single field tachyonic potential V T as 2 2 (5.324) = 3Mp 3Mp √ √ i 1 T 3Mα i V (T )V (Ti ) The equation of motions can be mapped to the equations of t − t ≈− dT , (5.332) in M i V (T ) a model with a single tachyonic field using the following √ p Tin √ i 6q Mα Ti V (T )V (T ) redefinitions:√ t − t ≈− √ dT i . (5.333) ˜ in i V (T ) T = MT, (5.325) 3Mp Tin i M ˜ ˜ Further using Eqs. (5.15), (5.16) and (5.324) we get the fol- V = V (T ) = MV(T ) = V (Ti ). (5.326) lowing solution for the scale factor in terms of the tachyonic i=1 field for the usual q = 1/2 and for a generalized value of q: Now to show the late time attractor behavior of the solution ⎧ of scalar fields we keep T1, but we replace the rest with the ⎪ α M Ti V (T )V (T ) redefined fields: ⎪ exp − dT i , for q = 1/2, ⎨⎪ M2 i V (T ) χ = − ∀ = ,..., . p Tin i i Ti T1 i 2 M (5.327) a = ain × ⎪ ⎪ 2qα M Ti V (T )V (T ) ⎩⎪ exp − dT i , for any arbitrary q. Using Eq. (5.327), at late times the equation of motion can 2 i ( ) Mp Tin V Ti be recast as (5.334)

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5.2.2 Analysis using slow-roll formalism

Here our prime objective is to define slow-roll parameters for tachyon inflation in terms of the Hubble parameter and the assisted tachyonic inflationary potential. Using the slow-roll approximation one can expand various cosmological observables in terms of small dynamical quantities derived from the appropriate derivatives of the Hubble parameter and of the inflationary potential. To start with here we use the horizon-flow parameters based on the derivatives of the Hubble parameter with respect to the number of e-foldings N, defined as tend N(t) = H(t) dt, (5.335) t where tend signifies the end of inflation. Further, using Eqs. (5.5), (5.9), (5.10) and (5.335), we get ⎧ ⎪ 2H ⎪ − , q = / , ˙ ⎨ 3 for 1 2 dTi Ti 3α H % & = = ˙ ⎪ 2H 1 − α (1 − 2q)T 2 (5.336) dN H ⎩⎪ − i , for any arbitrary q, 3α H 3 2q ˙ where H > 0, which makes always Ti > 0 during inflationary phase. Further using Eq. (5.336) we get the following differential operator identity for tachyonic inflation: ⎧ ⎪ 2H d ⎪ − , q = / , ⎨ 3 for 1 2 1 d d d 3α H d%Ti & = = = ⎪ 2H 1 − α (1 − 2q)T˙ 2 d (5.337) H dt dN dlnk ⎪ − i , q. ⎩ 3 for any arbitrary 3α H 2q dTi Further using the differential operator identity as mentioned in Eq. (5.337) we get the following Hubble flow equation for tachyonic inflation for j ≥ 0: ⎧ ⎪ 2H d j ⎪ − , q = / , ⎨ 3 for 1 2 1 di d j 3α H d%Ti & = = = j+1 j ⎪ 2H 1 − α (1 − 2q)T˙ 2 d (5.338) H dt dN ⎪ − i j , q. ⎩ 3 for any arbitrary 3α H 2q dTi For a realistic estimate from the single field tachyonic inflationary model substituting the free index j to j = 0, 1, 2in Eqs. (5.23) and (5.338) we get the contributions from the first three Hubble slow-roll parameters, which can be represented as ⎧ 2 ⎪ 2 H 3 ⎪ = α T˙ 2, for q = 1/2, ⎪ 3α H 2 2 i ⎪ % & ˙ ⎨ 2 − α( − ) ˙ 2 dln| | H 2 H 1 1 2q Ti = 0 =− = (5.339) 1 2 ⎪ 3α H 2 2q dN H ⎪ % & ⎪ ⎪ 3 2q ⎩⎪ = α T˙ 2 , for any arbitrary q, i − α( − ) ˙ 2 2 1 1 2q Ti ⎧ ⎪ 2 T¨ ⎪ 1 = i , q = / , ⎪ α 2 ˙ for 1 2 ⎪ 3 1 H H Ti ⎨⎪ % & | | ¨ ˙ dln 1 H 2 1 − α (1 − 2q)T 2 2 = = + 21 = i 1 (5.340) dN H H˙ ⎪ α ⎪ 3 1 2q H ⎪ ⎪ 2T¨ ⎩⎪ = i , for any arbitrary q, ˙ − α( − ) ˙ 2 H T 1 1 2q Ti

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... | | ¨ ¨ 2 ˙ 2 = dln 2 = 1 H − H − H + H 3 ˙ 3 3 ˙ 4 4 dN 2 H 2 H H H 2 H 2 H

⎧ 1 ... ⎪ 21 2T i 2 ⎪ 2 = + − , for q = 1/2, ⎪ 3α H 2 ˙ 1 2 ⎪ H Ti 2 ⎪ ⎪ % & ⎨ − α( − ) ˙ 2 2 1 1 1 2q Ti 2 = (5.341) ⎪ 3α 2q H ⎪ ⎪ ... ⎪ 2 T α( − ) ¨ 2 ⎪ i + − 2 4 1 2q Ti ⎪ 2 ˙ 1 ⎪ H Ti 2 2 ⎩⎪ = + H , for any arbitrary q. 1 − α(1 − 2q)T˙ 2 − α( − ) ˙ 2 2 i 1 1 2q Ti Further using Eqs. (5.339), (5.340), (5.343), (5.34) and (5.35) one can re-express the Hubble slow-roll parameters in terms of the potential dependent slow-roll parameter as ⎧ ⎪ M2 V 2(T ) (T ) ⎪ p i = V i ≡¯ (T ), q = / , ⎨ 2 V i for 1 2 2α MV(T )V (Ti ) MV(T )α 1 ≈ (5.342) ⎪ 2 2 ⎪ Mp V (Ti ) V (Ti ) ¯V (Ti ) ⎩ = ≡ , q, 2 for any 4qα MV(T )V (Ti ) 2qMV(T )α 2q ⎧ % & 2 2 ⎪ Mp V (Ti ) V (Ti ) ⎪ 3 − 2 ⎪ α ( ) 2( ) ( ) ( ) ⎪ M V T V Ti V T V Ti ⎪ ⎪ ( ( ) − η ( )) ⎪ 2 3 V Ti V Ti ⎪ = = 2 (3¯V (Ti ) −¯ηV (Ti )) , for q = 1/2, ⎨⎪ MV(T )α % & 2 ≈ ⎪ M2 V 2(T ) V (T ) ⎪ √ p i − i ⎪ 3 2 2 ⎪ 2qα M V (T )V (Ti ) V (T )V (Ti ) ⎪ ⎪ ⎪ 2 ( ( ) − η ( )) ⎪ q 3 V Ti V Ti 2 ⎩⎪ = = (3¯ (T ) −¯η (T )) , for any q MV(T )α q V i V i ⎧ % & 4 2 4 ⎪ Mp V (T )V (T ) V (T )V (T ) V (T ) ⎪ 2 i i − 10 i i + 9 i ⎪ 2 2 2 2 3 4 ⎪ M V (T )α V (Ti ) V (Ti ) V (Ti ) ⎪ ⎪ ⎪ 2ξ 2 (T ) − 5η (T ) (T ) + 362 (T ) ⎪ = V i V i V i V i ⎪ 2 2( )α2 ⎪ M V T ⎪ ⎪ ⎪ = 2ξ¯ 2 (T ) − 5η¯ (T )¯ (T ) + 36¯2 (T ) , for q = 1/2, ⎨⎪ V i V i V i V i 32 ≈ % & (5.343) ⎪ 4 2 4 ⎪ Mp V (T )V (T ) V (T )V (T ) V (T ) ⎪ √ 2 i i − 10 i i + 9 i ⎪ 2 2 2 2 3 4 ⎪ M 2qV (T )α V (Ti ) V (Ti ) V (Ti ) ⎪ ⎪ ⎪ 2ξ 2 (T ) − 5η (T ) (T ) + 362 (T ) ⎪ = V i √V i V i V i ⎪ 2 2 2 ⎪ M 2qV (T )α ⎪ ⎪ ¯ 2 2 ⎪ 2ξ (Ti ) − 5η¯V (Ti )¯V (Ti ) + 36¯ (Ti ) ⎩ = V √ V , for any q, 2q

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= / ,η ,ξ2 ,σ3 where d dTi and the potential dependent slow-roll parameters V V V V are defined as 2 2 Mp V (Ti ) V (Ti ) = , (5.344) 2 V (Ti ) ( ) η ( ) = 2 V Ti , V Ti Mp (5.345) V (Ti ) V (T )V (T ) ξ 2 (T ) = M4 i i , V i p 2 (5.346) V (Ti ) % & V 2(T )V (T ) σ 3 (T ) = M6 i i , V i p 3 (5.347) V (Ti ) which is exactly similar to the expression for the slow-roll parameter as appearing in the context of single field slow-roll inflationary models. However, for the sake of clarity here we introduce new sets of potential dependent slow-roll parameters for tachyonic inflation by rescaling with the appropriate powers of αV (T ): ( ) M2 ( ) 2 ¯ ¯ ( ) = V Ti = p V Ti = V , V Ti Mα V (T ) 2Mα V (T ) V (Ti ) M (5.348) η ( ) M2 ( ) η¯ η¯ ( ) = V Ti = p V Ti = V , V Ti Mα V (T ) Mα V (T ) V (Ti ) M (5.349) 2 4 ¯ 2 ξ (Ti ) Mp V (T )V (T ) ξ ξ¯ 2 (T ) = V = i i = V , (5.350) V i 2 2 2 2 2 2 2 2 M α V (T ) M α V (T ) V (Ti ) M % & 3 6 2 3 σ (Ti ) Mp V (T )V (T ) σ¯ σ¯ 3 (T ) = V = i i = V . (5.351) V i 3 3 3 2 3 3 3 3 M α V (T ) M α V (T ) V (Ti ) M ¯ η¯ ξ¯ 2 σ¯ 3 where V , V , V and V are the single field tachyonic slow-roll parameters. Further using Eqs. (5.344)–(5.470) we get the following operator identity for tachyonic inflation: ⎧ ⎪ 1/4 ⎪ 2¯V (Ti ) 2 d ⎪ M − ¯ (T ) , q = / , ⎨⎪ ( )α p 1 V i for 1 2 1 d d d MV T 3 dTi = = ≈ (5.352) H dt dN dlnk ⎪ / ⎪ ¯ ( ) M 1 4 ⎪ 2 V Ti p − 1 ¯ ( ) d , . ⎩ 1 V Ti for any arbitrary q MV(T )α 2q 3q dTi Finally using Eq. (5.352) we get the following sets of flow equations in the context of tachyonic inflation: ⎧ / ⎪ d¯ (T ) 2 2 ¯ 1 4 ⎪ V i = ¯ (η¯ − ¯ ) − V , q = / , ⎨ 2 V V 3 V 1 for 1 2 d1 dN M 3 M = (5.353) dN ⎪ 1/4 ⎪ 1 d¯V (Ti ) ¯V 1 ¯V ⎩ = (η¯V − 3¯V ) 1 − , for any q, 2q dN qM2 3q M

⎧ / ⎪ 2 2 ¯ 1 4 ⎪ ¯ η¯ − ¯2 − ξ¯ 2 − V , q = / , ⎨ 2 10 V V 18 V V 1 for 1 2 d2 M 3 M = (5.354) dN ⎪ 2 ¯ 1/4 ⎩⎪ q 2 ¯ 2 1 V 10¯V η¯V − 18¯ − ξ 1 − , for any q, M2 V V 3q M

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⎧ ⎪ 1 σ¯ 3 − ¯3 + ξ¯ 2 η¯ − ξ¯ 2 ¯ ⎪ 2 V 216 V 2 V V 7 V V ⎪ M3 ⎪ ¯ 1/4 ⎪ 2 2 V ⎨ + 194¯ η¯V − 10η¯V ¯V 1 − ¯V , for q = 1/2, d( ) V 3 M 2 3 = (5.355) dN ⎪ √1 σ¯ 3 − ¯3 + ξ¯ 2 η¯ − ξ¯ 2 ¯ ⎪ 2 V 216 V 2 V V 7 V V ⎪ M3 2q ⎪ ¯ 1/4 ⎪ 2 1 V ⎩ + 194¯ η¯V − 10η¯V ¯V 1 − , for any q, V 3q M where we use the following consistency conditions for the rescaled potential dependent slow-roll parameters: ⎧ / ⎪ 2 2 ¯ 1 4 ⎨⎪ ¯ (η¯ − ¯ ) − V , q = / , ¯ 2 V V 3 V 1 for 1 2 d V = M 3 M ⎪ ¯ 1/4 (5.356) dN ⎩⎪ 2 1 V ¯V (η¯V − 3¯V ) 1 − , for any q, M2 3q M

⎧ / ⎪ 1 2 ¯ 1 4 ⎨⎪ ξ¯ 2 − ¯ η¯ − V , q = / , η¯ 2 V 4 V V 1 for 1 2 d V = M 3 M ⎪ ¯ 1/4 (5.357) dN ⎩⎪ 1 ¯ 2 1 V ξ − 4¯V η¯V 1 − , for any q, M2 V 3q M ⎧ ⎪ ¯ 1/4 ⎪ 1 σ¯ 3 + ξ¯ 2 η¯ − ξ¯ 2 ¯ − 2 V , = / , ¯ 2 ⎨ V V 1 for q 1 2 dξ M3 V V V 3 M V = (5.358) dN ⎪ ¯ 1/4 ⎪ 1 3 ¯ 2 ¯ 2 1 V ⎩ σ¯ + ξ η¯V − ξ ¯V 1 − , for any q, M3 V V V 3q M

⎧ / ⎪ 1 2 ¯ 1 4 ⎨⎪ σ¯ 3 (η¯ − 12¯ ) 1 − V , for q = 1/2, dσ¯ 3 4 V V V V = M 3 M ⎪ ¯ 1/4 (5.359) dN ⎩⎪ 1 3 1 V σ¯ (η¯V − 12¯V ) 1 − , for any q. M4 V 3q M In terms of the slow-roll parameters, the number of e-foldings can be re-expressed as ⎧ 1 1 ⎪ 3α Ti,end H(T ) 3α Tend H(T ) ⎪ √ dT = √ dT ⎪ i ¯ ( ) ⎪ 2 Ti 1 2 Ti V Ti ⎪ ⎪ T ⎪ α M i V (T )V (Ti ) ⎪ ≈ dT , for q = 1/2, ⎨ 2 ( ) i M T , V Ti = p i end N ⎪ 1 ⎪ 3α Ti,end H(T ) Ti,end H(T ) ⎪ 2q √ dT = 3αq √ dT ⎪ i ¯ ( ) i ⎪ 2 Ti 1 Ti V Ti ⎪ √ ⎪ T ⎪ 2qα M i V (T )V (Ti ) ⎩ ≈ dT , for any arbitrary q, 2 ( ) i Mp Ti,end V Ti where Ti,end characterizes the tachyonic field value at the end of inflation t = tend for all i fields participating in assisted inflation. As in the case of assisted inflation all the M fields are identical; then one can re-express the number of e-foldings as ⎧ T 2 T 2 ⎪ α M i V (Ti ) α M V (T ) ⎪ dT = dT, for q = 1/2, ⎨ 2 ( ) i 2 ( ) Mp Ti,end V Ti Mp Tend V T N = √ √ (5.360) ⎪ ⎪ 2qα M Ti V 2(T ) 2qα M T V 2(T ) ⎩⎪ i dT = dT, for any arbitrary q. 2 ( ) i 2 ( ) Mp Ti,end V Ti Mp Tend V T

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˙ 5.2.3 Basics of tachyonic perturbations α V (Ti ) Ti (˙ t, k) + H(t, k) =− δT , 2 i (5.364) 1 − αT˙ 2 Mp In this subsection we explicitly discuss the cosmological i ( , ) − ( , ) = . linear perturbation theory within the framework of assisted t k t k 0 (5.365) tachyonic inflation. Let us clearly mention that here we have Similarly, for any arbitrary q the perturbed Einstein field various ways of characterizing cosmological perturbations in equations can be expressed as the context of inflation, which finally depend on the choice of k2 1 gauge. Let us do the computation in the longitudinal gauge, 3H H(t, k) + (˙ t, k) + =− δρ , (5.366) 2( ) 2 i where the scalar metric perturbations of the FLRW back- a t 2Mp ground are given by the following infinitesimal line element: (¨ t, k) + 3H H(t, k) + (˙ t, k) + H(˙ t, k) 2 2 2 i j ds =−(1 + 2(t, x)) dt + a (t) (1 − 2(t, x)) δijdx dx , 2 (5.361) ˙ k 1 +2H(t, k) + ((t, k) − (t, k)) = δpi , 3a2(t) 2M2 where a(t) is the scale factor, (t, x) and (t, x) charac- p terizes the gauge invariant metric perturbations. Specifically, (5.367) the perturbation of the FLRW metric leads to the perturba- (˙ t, k) + H(t, k) tion in the energy-momentum stress tensor via the Einstein field equation or equivalently through the Friedmann equa- α ( ) − α( − ) ˙ 2 ˙ V Ti 1 1 2q Ti Ti tions. For the perturbed metric as mentioned in Eq. (5.361), =− δTi , (5.368) − α ˙ 2 1−q M2 the perturbed Einstein field equations can be expressed for 1 Ti p the q = 1/2 case of the tachyonic inflationary set-up as ( , ) − ( , ) = . k2 1 t k t k 0 (5.369) 3H H(t, k) + (˙ t, k) + =− δρ , 2( ) 2 i ( , ) ( , ) a t 2Mp Here t k and t k are the two gauge invariant (5.362) metric perturbations in the Fourier space. Additionally, it is important to note that in Eq. (5.65), the two gauge invari- (¨ t, k) + 3H H(t, k) + (˙ t, k) + H(˙ t, k) ant metric perturbations (t, k) and (t, k) are equal in the k2 1 + 2H˙ (t, k) + ((t, k) − (t, k)) = δp , context of a minimally coupled tachyonic string field theo- 2( ) 2 i 3a t 2Mp retic model with an Einstein gravity sector. In Eqs. (5.62) and (5.363) (5.63) the perturbed energy density δρ and pressure δp are given by

⎧ α ( ) ˙ δ ˙ + ˙ 2( , ) ⎪ V (Ti )δTi V Ti Ti Ti T t k ⎪ + i , = / , ⎪ 3/2 for q 1 2 ⎪ − α ˙ 2 1 − αT˙ 2 ⎪ 1 Ti i ⎪ ⎨⎪ 6 7 V (T ) 1 − α(1 − 2q)T˙ 2 δT − 4α(1 − 2q)V (T )T˙ δT˙ δρ = i i i i i i (5.370) i ⎪ − ⎪ − α ˙ 2 1 q ⎪ 1 Ti ⎪ ⎪ α( − ) ( ) − α( − ) ˙ 2 ˙ δ ˙ + ˙ 2( , ) ⎪ 2 1 q V Ti 1 1 2q Ti Ti Ti Ti t k ⎩⎪ + , for any arbitrary q. − α ˙ 2 2−q 1 Ti ⎧ α ( ) ˙ δ ˙ + ˙ 2( , ) ⎪ ˙ 2 V Ti Ti Ti T t k ⎪ −V (Ti ) 1 − α T δTi + , for q = 1/2, ⎨ i ˙ 1 − α T 2 δp = i ⎪ q 2qα V (T ) T˙ δT˙ + T˙ 2(t, k) ⎪ ˙ 2 i i i i ⎩ −V (Ti ) 1 − α T δTi + , for any arbitrary q. i − α ˙ 2 1−q 1 Ti

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Similarly after the variation of the tachyonic field we • Further, the time evolution of the curvature perturbation get the following expressions for the perturbed equation of can be expressed as motion: ⎧ ¨ ˙ ˙ ˙ 2 ⎪ 2α Ti Ti δTi + T (t, k) ⎪ δT¨ + 3HδT˙ + i ⎪ i i − α ˙ 2 ⎪ 1 Ti ⎪ ⎪ − α ˙ 2 2 2 ⎪ Mp 1 Ti k 2k ⎪ + − H˙ (t, k) − (t, k) ⎪ 2 3 2 ⎪ α V (Ti ) a a ⎪ ⎪ −3 (¨ t, k) + 4H(˙ t, k) + H(˙ t, k) + H˙ (t, k) + 4H 2(t, k) ⎪ ⎪ ( ) ⎪ − α ˙ 3 − 2V Ti − α ˙ 2 ( , ) − (˙ , ) + (˙ , ) ˙ ⎪ 6H Ti 1 Ti t k t k 3 t k Ti ⎪ α V%(Ti ) & ⎪ ⎪ ˙ 2 2 ⎪ Mp 1 − α T V (T ) V (T ) ⎪ − i i − i , for q = 1/2, ⎨⎪ α ( ) 2( ) V Ti V Ti 0 ≈ 2αT¨ T˙ δT˙ + T˙ 2(t, k) (5.371) ⎪ δ ¨ + δ ˙ + i i i i ⎪ Ti 3H Ti ( − ) ⎪ ˙ 2 2 1 q ⎪ 1 − α T ⎪ − i ⎪ ˙ 2 1 q 2 2 ⎪ Mp 1 − α T k 2k ⎪ + i − 3H˙ (t, k) − (t, k) ⎪ α ( ) − α( − ) ˙ 2 2 2 ⎪ V Ti 1 1 2q T a a ⎪ i ⎪ −3 (¨ t, k) + 4H(˙ t, k) + H(˙ t, k) + H˙ (t, k) + 4H 2(t, k) ⎪ ⎪ ⎪ 2 V (T ) 2(1−q) ⎪ − HαT˙ 3 − i − αT˙ 2 (t, k) − (˙ t, k) + (˙ t, k) T˙ ⎪ 6 i α ( ) 1 i 3 i ⎪ q V %Ti & ⎪ ( − ) ⎪ − α ˙ 2 2 1 q 2 ⎪ Mp 1 T V (Ti ) V (Ti ) ⎪ − √ i − , q. ⎩ 2 for any 2qα V (Ti ) V (Ti ) Further we will perform the following steps throughout M δ ¯ ˙ pi the next part of the computation: ζ = H , (5.375) ρi + pi i=1 • First of all we decompose the scalar perturbations into δ ¯ two components-(1) entropic or isocurvature perturba- where pi characterizes the non-adiabatic or entropic tions which can be usually treated as the orthogonal pro- contribution in the first order linearized cosmological per- δ ¯ jective part to the trajectory and (2) adiabatic or curva- turbation. In the present context pi can be expressed as ture perturbations which can be usually treated as the ∂p parallel projective part to the trajectory. δ p¯ = p˙ = i δS, i i ∂ (5.376) • Within the framework of first order cosmological per- S S turbation theory we define a gauge invariant primordial curvature perturbation on the scales outside the horizon: where characterizes the relative displacement between hypersurfaces of uniform pressure and density. In the case M M of the assisted or multi-component fluid dynamical sys- H ζ = ζ + ζ = − δρ . tem, there are two contributions to the δ p¯ appearing in r T˜ i (5.372) i ρ˙i i=1 i=1 the computation:

• Next we consider the uniform density hypersurface in δ p¯i = δ p¯ , ˜ + δ p¯int, (5.377) which rel Ti

where δ p¯rel,i and δ p¯int characterize the relative and δρ = 0 ∀ i = 1, 2,...,M. (5.373) i intrinsic contributions to the multi-component ﬂuid system given by Consequently the curvature perturbation is governed by 1 M δ ¯ = ρ˙ ρ˙ 2 − 2 S ∝˙ρ , prel,T˜ r T˜ cr c , ˜ rT˜ r i 3Hρ˙ i S Ti i ζ = ζ + ζ ˜ = . (5.374) r Ti i=1 (5.378) 123 278 Page 98 of 130 Eur. Phys. J. C (2016) 76 :278 2 δ p¯ = δ p¯ ˜ + δ p¯ , = δpα − cαδρα . This clearly implies that int int,Ti int r ˜ α=r,Ti δ ¯ = δ − 2δρ = (5.379) pint,r pr cr r 0 (5.385)

2 2 Here c ˜ and cr are the effective adiabatic sound speed i.e., the intrinsic non-adiabatic pressure vanishes identi- S,Ti and sound speed due to radiation. Also the relative cally. However, in terms of the effective tachyonic field ˜ entropic perturbation is defined as T this situation is not very simple, as the equation of state parameter and the intrinsic sound speed changes with the % & number of component fields M. During the inflationary δρ δρ ˜ S = ζ − ζ =− r − Ti epoch all M fields obey the following equation of state: rT˜ 3 r T˜ 3H i i ρ˙ ρ˙ ˜ r Ti δ p ˜ δr T˜ Ti 2 = − i , w ˜ = −1 = c ˜ = Constant (5.380) Ti ρ S,Ti 1 + w 1 + w ˜ T˜ r Ti i ∀ i = 1, 2,...,M. (5.386)

where w and w ˜ signify the equation of state parameters r Ti due to radiation and tachyonic matter, δ and δ ˜ charac- For all M effective tachyon fields the fluctuation in the r Ti terize the energy density contrasts due to radiation and pressure can be expressed as tachyonic matter; M is the number of tachyonic matter 2 contents, defined by δp ˜ w ˜ δρ ˜ = c ˜ δρ ˜ −δρ ˜ . (5.387) Ti Ti Ti S,Ti Ti Ti

δρα ˜ Consequently the intrinsic non-adiabatic pressure within δα = ∀ α = r, Ti (∀i = 1, 2,...,M). (5.381) ρα each tachyonic field amongst M fields can be expressed as In a generalized prescription the pressure perturbation in 2 arbitrary gauge can be decomposed into the following δp ˜ = δp ˜ − c δρ ˜ 0. (5.388) int,Ti Ti S,T˜ Ti contributions: i Now in the present context the relative contributions to M M δ = δ = 2 δρ + 1 ρ˙ ρ˙ 2 − 2 S the non-adiabatic pressure of M tachyonic fields in this p pi cS r T˜ cr c , ˜ rT˜ 3Hρ˙ i S Ti i context can be expressed as i=1 i=1 M 2 2 2 + δpα − c δρα . (5.382) δp , ˜ ˜ ∝ c ˜ − c ˜ 0, (5.389) α rel Ti Tj S,Ti S,Tj = ˜ i 1 α=r,Ti since As δ p¯ ˜ ∝˙ρ , the relative non-adiabatic pressure per- rel,Ti r turbation is heavily suppressed in the computation and c2 c2 −1 ∀ i, j = 1, 2,...,M. (5.390) S,T˜ S,T˜ one can safely ignore such contributions during the epoch i j of inflation. Here the relative entropic perturbation S ˜ rTi This implies that the total non-adiabatic pressure is neg- remains small and finite in the limit of small ρ˙ due to r ligible during inflation, i.e., the smallness of the curvature perturbations ζ and ζ ˜ . r Ti • Now let us consider a situation where the equation of 2 δ p¯ = δ p¯ ˜ + δ p¯ , = δpα − cαδρα = 0. state parameter corresponding to radiation is int int,Ti int r ˜ α=r,Ti w = 2 = . (5.391) r cr Constant (5.383)

and it is completely justified to write the fluctuation in Consequently the fluctuation in pressure due to radiation the pressure for M individual tachyonic field contents as can be expressed in terms of the fluctuation of the density: δ ¯ = δ ¯ = 1 ρ˙ ρ˙ 2 − 2 S ∝˙ρ . δ = w δρ = 2δρ = × δρ ∝ δρ . pi prel,T˜ r T˜ cr c , ˜ rT˜ r pr r r cr r Constant r r i 3Hρ˙ i S Ti i (5.384) (5.392) 123 Eur. Phys. J. C (2016) 76 :278 Page 99 of 130 278

Using these results the total fluctuation in the pressure perturbation rapidly becomes adiabatic perturbation of can be re-expressed as the same amplitude, as local pressure differences, due to the local fluctuations in the equation of state, re-distribute M M the energy density. However, this change is slightly less δ = δ = 2 δρ + δ ¯ = 2 δρ + ρ˙ p pi cS pi cS AM r efficient during the epoch of radiation compared to the i=1 i=1 tachyonic matter dominated epoch and can only occur 2 δρ, cS (5.393) after the decoupling between the photons and baryons, in the specific case of baryonic isocurvature perturbation. where A is the proportionality constant and we have Causality precludes this re-distribution on scales bigger safely ignored the contribution from the isocurvatrure than the Hubble radius, and thus any entropy perturba- modes due to its smallness in the case of assisted tachy- tion on these scales remains with a constant amplitude. onic inflation. Consequently we get Also it is important to note that the entropic or isocurvature perturbations are not affected by Silk damping, δ p¯i = p˙i = AMρ˙r 0 which is exactly contrary to the curvature or adiabatic ⇒ ρr = Constant ⇒ ζ = Constant, (5.394) perturbations.

which is exactly similar with the usual single field slow- 5.2.4 Computation of scalar power spectrum roll conditions in the context of the tachyonic inflationary set-up discussed earlier. Finally, in the uniform density In this subsection we will not derive the results for assisted hypersurfaces, the curvature perturbation can be written inflation as it is exactly the same as derived in the case of in terms of the tachyonic field fluctuations on spatially single tachyonic inflation. But we will state the results for flat hypersurfaces as the BD vacuum where the changes will appear due to the presence of M identical copies of the tachyon field. One can M δ δ similarly write down the detailed expressions for AVas well. ζ =− Ti =− T H ˙ HM ˙ The changes will appear in the expressions for the following Ti T i=1% & inflationary observables at the horizon crossing: δ ˜ =− T . HM ˙ (5.395) • In the present context amplitude of scalar power spectrum T˜ can be computed as

⎧ 2 2 ⎪ ¯V CE MH ⎪ − (C + ) − ( ¯ −¯η ) , q = / , ⎪ 1 E 1 3 V V 2 2 for 1 2 ⎪ M M 8π M cS¯V ⎨ p k=a H ζ, = (5.396) ⎪ ⎪ ¯ C 2 qMH2 ⎪ − (C + ) V − √ E ( ¯ −¯η ) , q, ⎩ 1 E 1 3 V V 2 2 for any 2qM 2qM 4π M cS¯V p k=a H

2 = / 2 C =− + + γ ≈ • But in the case of multi-tachyonic inflation where the M where H MV 3Mp and E 2 ln 2 number of fields are not identical with each other the sit- −0.72. Using the slow-roll approximations one can fur- uation is not simpler like assisted inflation. In that case ther approximate the expression for the sound speed as during the short time intervals when fields decay in a ⎧ % & ⎪ 2 ¯ ¯2 very much faster rate and its corresponding equation of ⎪ 1 − V + O V +···, for q = 1/2, ⎨⎪ 2 state changes rapidly then it would be really interesting to 3 M M 2 = cS % & investigate the production of isocurvature perturbations. ⎪ ( − ) ¯2 ⎪ 1 q V On the scale smaller than the Hubble radius any entropy ⎩⎪ 1 − ¯V + O +···, for any q. 3q2 M M2 (5.397)

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Hence using the result in Eq. (5.408) we get the following simpliﬁed expression for the primordial scalar power spectrum: ⎧ ⎪ 5 ¯ C 2 MH2 ⎪ 1 − C + V − E (3¯ −¯η ) , for q = 1/2, ⎨⎪ E V V π 2 2¯ 6 M M 8 Mp V ≈ k=a H ζ, ⎪ (5.398) ⎪ ¯ C 2 qMH2 ⎪ − (C + − ) V − √ E ( ¯ −¯η ) , q. ⎩ 1 E 1 3 V V 2 2 for any 2qM 2qM 4π M ¯V p k=a H

• Next one can compute the scalar spectral tilt (nS) of the primordial scalar power spectrum as ⎧ ⎪ 1 2 2 8 ⎪ (2η¯ − 8¯ ) − ¯2 − 2C + ¯ (3¯ −¯η ) ⎪ V V 2 V 2 E V V V ⎪ M M M 3 ⎪ ⎪ C ⎪ − E ξ¯ 2 − η¯ ¯ + ¯2 +··· , = / , ⎪ 2 V 5 V V 36 V for q 1 2 ⎪ M2 ⎪ ⎨⎪ % & 1 2 1 1 2 ¯2 nζ, − 1 ≈ η¯ − + 3 ¯ − V (5.399) ⎪ V V 2 2 ⎪ M q M q q 2q M ⎪ ⎪ ⎪ − 2 ( C + − ) ¯ ( ¯ −¯η ) ⎪ / 2 E 3 2 V 3 V V ⎪ (2q)3 2 M2 ⎪ ⎪ C ⎪ E ¯ 2 2 ⎩ −√ 2ξ − 5η¯V ¯V + 36¯ +··· , for any q. 2qM2 V V

• Next one can compute the running of the scalar spectral tilt (αS) of the primordial scalar power spectrum as ⎧ ⎪ 4 ¯V 2 ⎪ − ¯ 1 + (η¯ − 3¯ ) + 10¯ η¯ − 18¯2 − ξ¯ 2 ⎪ M2 V M V V M2 V V V V ⎪ ⎪ ⎪ C ⎪ − E 2σ¯ 3 − 216¯3 + 2ξ¯ 2 η¯ − 7ξ¯ 2 ¯ + 194¯2 η¯ − 10η¯ ¯ ⎪ 3 V V V V V V V V V V ⎪ M ⎪ ⎪ ⎪ 1 8 2 ¯ 2 ⎪ − 2CE + 2¯V 10¯V η¯V − 18¯ − ξ ⎪ M3 3 V V ⎪ ⎪ ⎪ ¯ 1/4 ⎪ − ¯ ( ¯ −¯η )2 − 2 V +··· , = / , ⎪ 4 V 3 V V 1 for q 1 2 ⎨⎪ 3 M αζ, ≈ ⎪ 2 ¯ ¯ 1 2 ⎪ − V 1 + V (η¯ − 3¯ ) + 10¯ η¯ − 18¯2 − ξ¯ 2 ⎪ 2 V V 2 V V V V ⎪ q qM 2qM M q ⎪ ⎪ C ⎪ −√ E σ¯ 3 − ¯3 + ξ¯ 2 η¯ − ξ¯ 2 ¯ + ¯2 η¯ − η¯ ¯ ⎪ 2 V 216 V 2 V V 7 V V 194 V V 10 V V ⎪ 2qM3 ⎪ ⎪ ⎪ ⎪ − 1 C + 8 2 ¯ ¯ η¯ − ¯2 − ξ¯ 2 ⎪ 2 E V 10 V V 18 V V ⎪ M3 3 q ⎪ ⎪ ⎪ ¯ 1/4 ⎪ 4 2 1 V ⎩ − ¯V (3¯V −¯ηV ) 1 − +··· , for any q. (2q)3/2 3q M

• Finally, one can also compute the running of the running of scalar spectral tilt (κS) of the primordial scalar power spectrum as

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⎧ ⎪ − 1 ¯ ( +¯ )(η¯ − ¯ )2 + ¯2 (η¯ − ¯ )2 ⎪ 8 V 1 V V 3 V c V V 3 V ⎪ M3 ⎪ ⎪ ⎪ + ¯ ( +¯ ) ξ¯ 2 − ¯ η¯ + ¯2 ⎪ 8 V 1 V V 10 V V 18 V ⎪ ⎪ ⎪ ⎪ + ¯ η¯ (η¯ − ¯ ) + ¯ ξ¯ 2 − ¯ η¯ ⎪ 2 20 V V V 3 V 10 V V 4 V V ⎪ ⎪ ⎪ / ⎪ 2 ¯ 1 4 ⎪ −72¯2 (η¯ − 3¯ ) − σ¯ 3 + ξ¯ 2 η¯ − ξ¯ 2 ¯ 1 − V +··· , for q = 1/2, ⎪ V V V V V V V V ⎪ 3 M ⎨ / κζ, ≈ 1 2 3 2 ¯ 2 2 (5.400) ⎪ − ¯ 1 + V (η¯ − 3¯ )2 + ¯2 (η¯ − 3¯ )2 ⎪ 3 V V V 2 V V V ⎪ M q 2qM q q M ⎪ ⎪ ⎪ ¯ ⎪ + 2 2 ¯ + V ξ¯ 2 − ¯ η¯ + ¯2 ⎪ V 1 V 10 V V 18 V ⎪ q q 2q ⎪ ⎪ ⎪ ⎪ + 2 10¯ η¯ (η¯ − ¯ ) + ¯ ξ¯ 2 − ¯ η¯ ⎪ V V V 3 V 10 V V 4 V V ⎪ q q ⎪ ⎪ ⎪ ¯ 1/4 ⎪ 36 2 3 ¯ 2 ¯ 2 1 V ⎩⎪ − ¯ (η¯V − 3¯V ) − σ¯ + ξ η¯V − ξ ¯V 1 − +··· , for any q. q V V V V 3q M

5.2.5 Computation of tensor power spectrum

In this subsection we will not derive the results for assisted inflation as it is exactly the same as derived in the case of single tachyonic inflation. But we will state the results for the BD vacuum where the changes will appear due to the presence of M identical copies of the tachyon field. One can similarly write down the detailed expressions for AV as well. The changes will appear in the expressions for the following inflationary observables at the horizon crossing:

• In the present context amplitude of tensor power spectrum can be computed as ⎧ ⎪ ¯ 2 2 ⎪ V 2H ⎪ 1 − (CE + 1) , for q = 1/2, ⎨ M π 2 M2 p k=a H h, = (5.401) ⎪ ¯ 2 2 ⎪ V 2H ⎪ 1 − (CE + 1) , for any q, ⎩ 2qM π 2 M2 p k=a H

where CE =−2 + ln 2 + γ ≈−0.72. • Next one can compute the scalar spectral tilt (nS) of the primordial scalar power spectrum as ⎧ ⎪ 2 ¯V 2 ⎪ − ¯ 1 + + (C + 1)(3¯ −¯η ) +··· , for q = 1/2, ⎨⎪ M V M M E V V nh, ≈ (5.402) ⎪ ¯ ¯ ⎪ V V 1 2 ⎩ − 1 + + (CE + 1)(3¯V −¯ηV ) +··· , for any q. qM 2qM M q

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• Next one can compute the running of the tensor spectral tilt (αh) of the primordial scalar power spectrum as ⎧ ⎪ 4 ¯ 4 ⎪ − ¯ 1 + V (η¯ − 3¯ ) + ¯2 (η¯ − 3¯ ) ⎪ 2 V V V 3 V V V ⎪ M M M ⎪ ⎪ ⎪ − 8 (C + ) ¯ (η¯ − ¯ )2 ⎪ E 1 V V 3 V ⎪ M3 ⎪ ⎪ / ⎪ 2 2 ¯ 1 4 ⎪ − ¯ ¯ η¯ − ¯2 − ξ¯ 2 − V +··· , q = / , ⎪ 3 V 10 V V 18 V V 1 for 1 2 ⎨⎪ M 3 M α , ≈ (5.403) h ⎪ ¯ ¯ ⎪ V V 1 2 ⎪ − 2 1 + (η¯V − 3¯V ) + ¯ (η¯V − 3¯V ) ⎪ qM2 2qM q2 M3 V ⎪ ⎪ ⎪ 8 ⎪ − (C + 1) ¯ (η¯ − 3¯ )2 ⎪ ( )5/2 3 E V V V ⎪ 2q M ⎪ ⎪ / ⎪ 1 2 1 ¯ 1 4 ⎪ − ¯ ¯ η¯ − ¯2 − ξ¯ 2 − V +··· , q. ⎩ 3 V 10 V V 18 V V 1 for any M q 3q M

• Finally, one can also compute the running of the running of scalar spectral tilt (κS) of the primordial scalar power spectrum as ⎧ ⎪ 8 8 ⎪ − ¯ (1 +¯ )(η¯ − 3¯ )2 + ¯2 (η¯ − 3¯ )2 ⎪ 3 V V V V 4 V V V ⎪ M M ⎪ ⎪ ! ⎪ − 16 (C + ) ¯ (η¯ − ¯ )3 − (η¯ − ¯ ) ¯ η¯ − ¯2 − ξ¯ 2 ⎪ E 1 V V 3 V V 3 V 10 V V 18 V V ⎪ M4 ⎪ ⎪ / ⎪ 4 ¯ 2 ¯ 1 4 ⎪ − ¯ 1 + V 10¯ η¯ − 18¯2 − ξ¯ 2 1 − V +··· , for q = 1/2, ⎪ 3 V V V V V ⎪ M M 3 M ⎪ ⎨ ¯ 4 V 2 2 2 2 κh, ≈ − ¯V 1 + (η¯V − 3¯V ) + ¯ (η¯V − 3¯V ) (5.404) ⎪ qM3 2qM q2 M4 V ⎪ ⎪ ⎪ 2 ¯ ⎪ + ¯ 1 + V ξ¯ 2 − 10¯ η¯ + 18¯2 ⎪ 3 V V V V V ⎪ qM 2q ⎪ ⎪ ! ⎪ − 8 (C + ) ¯ (η¯ − ¯ )3 − (η¯ − ¯ ) ¯ η¯ − ¯2 − ξ¯ 2 ⎪ 4 E 1 V V 3 V V 3 V 10 V V 18 V V ⎪ qM ⎪ ⎪ / ⎪ 1 ¯ 1 4 ⎩⎪ × 1 − V +··· , for any q. 3q M

5.2.6 Modiﬁed consistency relations

In this subsection we derive the new (modified) consistency relations for single tachyonic field inflation:

1. Next for the BD vacuum with |kcSη|=1 case within slow-roll regime we can approximately write the following expression for the tensor-to-scalar ratio:

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⎧ ⎡ ⎤ ⎪ ¯ 2 ⎪ 1 − (C + 1) V ⎪ ⎢16 E M ⎥ ⎪ ⎣ ¯V cS ⎦ ⎪ ¯ C 2 ⎪ M − (C + ) V − E ( ¯ −¯η ) ⎪ 1 E 1 M M 3 V V ⎪ k=a H ⎪ ⎡ ⎤ ⎪ ¯ 2 ⎪ 1 − (C + 1) V ⎪ ⎢16 E M ⎥ ⎪ = ⎣ ¯V ⎦ , for q = 1/2, ⎪ ¯ C 2 ⎨⎪ M − C + 5 V − E ( ¯ −¯η ) 1 E 6 M M 3 V V k=a H r ≈ ⎡ ⎤ (5.405) ⎪ ¯ 2 ⎪ 1 − (C + 1) V ⎪ ⎢ 8 E 2qM ⎥ ⎪ ⎣ ¯V cS ⎦ ⎪ ¯ C 2 ⎪ qM − (C + ) V − √E ( ¯ −¯η ) ⎪ 1 E 1 2qM 3 V V ⎪ M 2q k=a H ⎪ ⎡ ⎤ ⎪ ¯ 2 ⎪ 1 − (C + 1) V ⎪ ⎢ 8 E 2qM ⎥ ⎪ = ⎣ ¯V ⎦ , for any q. ⎪ ¯ C 2 ⎩ qM − (C + − ) V − √E ( ¯ −¯η ) 1 E 1 2qM 3 V V M 2q k=a H

2. Hence the consistency relation between the tensor-to-scalar ratio r and spectral tilt nT for tensor modes for BD vacuum with |kcSη|=1 case can be written as

⎧ ⎡ ⎤ ⎪ ¯ 2 ⎪ 1 − (C + 1) V ⎪ ⎢ E M ⎥ ⎪ ⎣cS ⎦ ⎪ ¯ 2 ¯ ⎪ − (C + ) V − CE ( ¯ −¯η ) + V + 2 (C + )( ¯ −¯η ) ⎪ 1 E 1 M M 3 V V 1 M M E 1 3 V V ⎪ k=a H ⎪ ⎡ ⎤ ⎪ ¯ 2 ⎪ 1 − (C + 1) V ⎪ ⎢ E M ⎥ ⎪ = ⎣ ⎦ , for q = 1/2, ⎪ ¯ 2 ¯ ⎪ − C + 5 V − CE ( ¯ −¯η ) + V + 2 (C + )( ¯ −¯η ) ⎨ 1 E 6 M M 3 V V 1 M M E 1 3 V V k r ≈−8nh, × ⎡ ⎤ (5.406) ⎪ ¯ 2 ⎪ 1 − (C + 1) V ⎪ ⎢ E 2qM ⎥ ⎪ ⎣cS ⎦ ⎪ ¯ C 2 ¯ ⎪ − (C + ) V − √E ( ¯ −¯η ) + V + 1 2 (C + )( ¯ −¯η ) ⎪ 1 E 1 2qM 3 V V 1 2qM M q E 1 3 V V ⎪ M 2q k=a H ⎪ ⎡ ⎤ ⎪ ¯ 2 ⎪ 1 − (C + 1) V ⎪ ⎢ E 2qM ⎥ ⎪ = ⎣ ⎦ , for any q. ⎪ ¯ C 2 ¯ ⎩ 1 − (C + 1 − ) V − √E (3¯ −¯η ) 1 + V + 1 2 (C + 1)(3¯ −¯η ) E 2qM M 2q V V 2qM M q E V V 2 34 k 5 Correction factor

3. Next one can express the ﬁrst two slow-roll parameters ¯V and η¯V in terms of the inﬂationary observables as ⎧ ⎨⎪ nh, M r M 1 ≈− +···≈ +··· , for q = 1/2, ¯V ≈ 2 16 (5.407) ⎩⎪ qr M 2q ≈−qnh, M +···≈ +··· , for any q, 1 8

⎧ ⎪ M r M ⎪ − 2 ≈ − + +···≈ − − +··· , = / , ⎪ 3 1 nζ, 1 nζ, 1 4nh, for q 1 2 ⎪ 21 2 1 % 2 % 2 & & ⎨⎪ − q ≈ q − + 1 + 2 qr +··· η¯ ≈ 6q 1 2 M nζ, 1 3 V ⎪ 2 2 q q 8 (5.408) ⎪ 1 % % & & ⎪ ⎪ q 1 2 ⎩⎪ ≈ M nζ, − 1 − q + 3 nh, +··· , for any q. 2 q q

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4. Then the connecting consistency relation between tensor and scalar spectral tilt and tensor-to-scalar ratio can be expressed as ⎧ ! ⎪ − r − r + − − C r + − +··· , = / , ⎪ 1 1 nζ, E nζ, 1 for q 1 2 ⎪ 8cS 16 % % &8 & ⎪ ⎨ r 3q 2 2 1 1 − nζ, − 1 + − + 5 r + √ n , ≈ h ⎪ 8cS 8 q q 16 2q (5.409) ⎪ % % & & ⎪ ⎪ 2 3qr 1 1 2 qr ⎩ + CE − nζ, − 1 + + 3 +··· , for any q. q 8 2 q q 8

Finally using the approximated version of the expression for cS in terms of slow-roll parameters one can recast this consistency condition as ⎧ ! r r r ⎪ − 1 − + 1 − nζ, − C + nζ, − 1 +··· , for q = 1/2, ⎪ % % E & & ⎪ 8 24 8 ⎨⎪ r q 1 − nζ, − + 3 2 − 2 + 1 + + √ ≈ 1 5 r nh, ⎪ 8 8 q q 16 8 2q (5.410) ⎪ % % & & ⎪ ⎪ 2 3qr 1 1 2 qr ⎩ + CE − nζ, − 1 + + 3 +··· , for any q. q 8 2 q q 8

5. Next the running of the sound speed cS can be written in terms of slow-roll parameters as

c˙ dlnc dlnc S = S = S = S Hc dN dlnk ⎧ S 1/4 ⎪ 2 2 ¯V ⎨ − ¯V (η¯V − 3¯V ) 1 − +··· , for q = 1/2, = 3M2 3 M ⎪ ( − ) ¯ 1/4 (5.411) ⎩⎪ 1 q 1 V − ¯V (η¯V − 3¯V ) 1 − +··· , for any q, 3q2 M2 3q M

which can be treated as another slow-roll parameter in the present context. One can also recast the slow-roll parameter S in terms of the inﬂationary observables as ⎧ ⎪ r r r 1/4 ⎨⎪ − nζ, − 1 + 1 − +··· , for q = 1/2, 48 8 24 S = 1 (5.412) ⎪ (1 − q) q r r 1/4 ⎩ − r nζ, − 1 + 1 − +··· , for any q. 24q2 2 8 24

6. Further the running of the tensor spectral tilt can be written in terms of the inﬂationary observables as ⎧ 2 ⎪ − r + r − + r − r − + r ⎪ 1 nζ, 1 nζ, 1 ⎪ 8 8 8 8 8 ⎪ ⎪ r r 2 r 1/4 ⎨⎪ −CE nζ, − 1 + 1 − +··· , for q = 1/2, 81 8 24 αh, = (5.413) ⎪ r q r r ⎪ − 1 + nζ, − 1 + ⎪ ⎪ 4 2 8 8 ⎪ / ⎩⎪ 1 r r 2 r 1 4 − (CE + 1) nζ, − 1 + 1 − +··· , for any q. (2q)1/2 8 8 24

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7. Next the scalar power spectrum can be expressed in terms of the other inﬂationary observables as ⎧ ⎪ 2 2 ⎪ 5 r CE r 2H ⎪ 1 − C + + nζ, − 1 + +··· , for q = 1/2, ⎨ E π 2 2 6 16 2 8 Mpr ζ, ≈ (5.414) ⎪ 2 2 ⎪ r CE r 2H ⎩⎪ 1 − (C + 1 − ) + nζ, − 1 + +··· , for any q. E π 2 2 16 2 8 Mpr

8. Further the tensor power spectrum can be expressed in terms of the other inﬂationary observables as ⎧ 2 ⎪ r 2 2H ⎨⎪ 1 − (C + 1) +··· , for q = 1/2, E π 2 2 = 16 Mp h, 2 (5.415) ⎪ r 2 2H ⎩⎪ 1 − (C + 1) +··· , for any q. E π 2 2 16 Mp

9. Next the running of the tensor-to-scalar ratio can be expressed in terms of the inﬂationary observables as

αr, =−8αh, +··· ⎧ ⎪ r r r 2 ⎪ r 1 + nζ, − 1 + − r nζ, − 1 + ⎪ 8 8 8 ⎪ / ⎪ r 2 r 1 4 ⎨ −CEr nζ, − 1 + 1 − +··· , for q = 1/2, 8 24 ≈ 1 (5.416) ⎪ q r r ⎪ 2r 1 + nζ, − 1 + ⎪ 2 8 8 ⎪ ⎪ 1 r 2 r 1/4 ⎩ − (CE + 1) r nζ, − 1 + 1 − +··· , for any q. (2q)1/2 8 24

10. Finally the scale of single field tachyonic inflation can be expressed in terms of the Hubble parameter and the other inflationary observables as ⎧ ⎪ ζ,r π ⎪ 2 Mp ⎪ +··· , for q = 1/2, ⎨⎪ 5 r C r 1 − C + + E nζ, − 1 + E 6 16 2 8 Hinf = H ≈ (5.417) ⎪ ζ,r π ⎪ Mp ⎪ 2 +··· , for any q. ⎪ C ⎩ − (C + − ) r + E − + r 1 E 1 16 2 nζ, 1 8

One can recast this statement in terms of the inﬂationary potential as ⎧ √ ⎪ 4 3ζ,r ⎪ π M ⎪ 1 2 p +··· , = / , ⎪ for q 1 2 ⎪ C ⎨⎪ − C + 5 r + E − + r 1 E 6 16 2 nζ, 1 8 4 4 V = V ≈ inf ⎪ √ (5.418) ⎪ 4 3 ζ,r π M ⎪ 2 p ⎪ 1 +··· , for any q. ⎪ C ⎩ − (C + − ) r + E − + r 1 E 1 16 2 nζ, 1 8

5.2.7 Field excursion for tachyon

In this subsection we explicitly derive the expression for the field excursion for tachyonic inflation defined as

|T |=|Tcmb − Tend|=|T − Tend| (5.419) 123 278 Page 106 of 130 Eur. Phys. J. C (2016) 76 :278 where Tcmb, Tend and T signify the tachyon field value at the time of horizon exit, at end of inflation and at pivot scale respectively. Here we perform the computation for both AV and BD vacuum. For for the sake of simplicity the pivot scale is fixed at the horizon exit scale. To compute the expression for the field excursion we perform the following steps: 1. We start with the operator identity for single field tachyon using which one can write expression for the tachyon field variation with respect to the momentum scale (k) or number of e-foldings (N) in terms of the inflationary observables as ⎧ 1 ⎪ 1/4 ⎪ r − r +··· , = / , ⎨ Mp 1 for q 1 2 1 dT dT dT 18MV(T )α 24 = = ≈ / (5.420) H dt dN dlnk ⎪ qr Mp r 1 4 ⎩ 1 − +··· , for any arbitrary q, 4MV(T )α 2q 24

where the tensor-to-scalar ratio r is a function of k or N. 2. Next using Eq. (5.172) we can write the following integral equation: ⎧ 1 ⎪ k r r 1/4 ⎪ k M − +··· ⎪ dln α p 1 ⎪ kend 8M 24 ⎪ 1 ⎪ N r r 1/4 ⎪ = − +··· , = / , T ⎨ dN Mp 1 for q 1 2 N 8Mα 24 dT V (T ) ≈ end 1 (5.421) ⎪ k qr M r 1/4 Tend ⎪ p − +··· ⎪ dlnk 1 ⎪ k 4Mα 2q 24 ⎪ end 1 ⎪ N qr M r 1/4 ⎩⎪ = N p − +··· , q. d α 1 for any arbitrary Nend 4M 2q 24

3. Next we use the same parametrization of the tensor-to-scalar ratio for q = 1/2 and for any arbitrary q at any arbitrary scale as mentioned earlier for the single tachyonic field case. 4. For any value of q including q = 1/2 we need to compute the following integral: 1 k qr M r 1/4 k p − dln α 1 kend 4M 2q 24 ⎧ 1 ⎪ qr M r 1/4 k ⎪ p − , ⎪ α 1 ln for Case I ⎪ 4M 2q 24 kend ⎪ ⎡⎧ ⎫ ⎪ qr Mp ⎨ ⎬ ⎪ α 1/4 ⎪ 4M q ⎣ 1, 3; 3; r + − r ⎪ ⎩2 F1 2 1 ⎭ ⎪ 3(nh, − nζ, + 1) 2 4 2 24 24 ⎪ ⎧ ⎪ − + ⎪ nh, nζ, 1 ⎪ ⎨ nh,−nζ,+1 ⎪ kend 2 1 3 3 r kend ⎪ − 2 F1 , ; ; ⎪ k ⎩ 2 4 2 24 k ⎪ ⎪ % & ⎫⎤ ⎪ − + 1/4⎬ ⎪ nh, nζ, 1 ⎪ + − r kend ⎦ , ⎪ 2 1 for Case II ⎨⎪ 24 k ⎭ (5.422) 1 ( − + )2 ≈ 3 nh, nζ, 1 for BD. ⎪ πqr Mp − (α −α ) ⎪ e 4 h, ζ, ⎪ α(α − α ) ⎪ M h, ζ, 48q ⎪ ( − + )2 ⎪ nh, nζ, 1 − + ⎪ (α −α ) nh, nζ, 1 ⎪ e 2 h, ζ, √ ⎪ 12 erfi α − α ⎪ 2 h, ζ, ⎪ √ ⎪ nh, − nζ, + 1 αh, − αζ, kend ⎪ −erfi √ + ln ⎪ 2 α , − αζ, 2 k ⎪ √ h%√ & ⎪ ⎪ 3r 3(nh, − nζ, + 1) ⎪ − erfi √ ⎪ 24 2 α , − αζ, ⎪ %√ h & ⎪ ⎪ 3(n , − nζ, + 1) 3(αh, − αζ,) k ⎩⎪ −erfi √h + ln end for Case III, 2 αh, − αζ, 2 k 123 Eur. Phys. J. C (2016) 76 :278 Page 107 of 130 278

Similarly for AV we get the following result: ⎧ % & 1 1/4 ⎪ r M qr k 1 ⎪ p − , ⎨⎪ 1 ln for Case I k 1/4 α 2 qr Mp r 8M 2cS 48cS kend dlnk 1 − ≈ 1 % & / for AV. Mα q ⎪ 2 1 4 kend 4 2 24 ⎪ r Mp |D| qr |D| k ⎩⎪ 1 − ln for Case II, α | | 2 | |2 8M 2cS C 48cS C kend (5.423) In terms of the number of e-foldings N one can re-express Eqs. (5.422) and (5.423)as 1 k qr M r 1/4 k p − dln α 1 kend 4M 2q 24 ⎧ 1 / ⎪ qr Mp r 1 4 ⎪ 1 − (N − N ) for Case I, ⎪ 4Mα 2q 24 end ⎪ ⎪ qr Mp ⎪ 4Mα q 1 3 3 r r 1/4 ⎪ F , ; ; + − ⎪ ( − + ) 2 1 2 1 ⎪ 3 nh, nζ, 1 2 4 2 24 24 ⎪ ⎪ − + ⎪ nh, nζ, 1 ⎪ (N −N) 1 3 3 r (n ,−nζ,+1)(N −N) ⎪ −e 2 end F , ; ; e h end ⎪ 2 1 ⎪ 2 4 2 24 ⎪ ⎪ / ⎪ r ( − + )( − ) 1 4 ⎪ +2 1 − e nh, nζ, 1 Nend N for Case II, ⎪ ⎪ 24 ⎪ ⎪ 1 ( − + )2 ⎨⎪ 3 nh, nζ, 1 (5.424) πqr Mp − (α −α ) 4 h, ζ, ≈ e for BD, ⎪ Mα (αh, − αζ,) 48q ⎪ ⎪ ⎪ (n ,−nζ,+1)2 ⎪ h n , − nζ, + 1 ⎪ 2(α ,−αζ,) h ⎪ 12e h erfi √ ⎪ 2 αh, − αζ, ⎪ ⎪ √ ⎪ nh, − nζ, + 1 αh, − αζ, ⎪ −erfi √ + (N − N) ⎪ α − α end ⎪ 2 h, ζ, 2 ⎪ √ %√ & ⎪ ⎪ 3r 3(nh, − nζ, + 1) ⎪ − erfi √ ⎪ 24 2 α , − αζ, ⎪ h ⎪ %√ & ⎪ (α − α ) ⎪ 3(nh, − nζ, + 1) 3 h, ζ, ⎩⎪ −erfi √ + (Nend − N) for Case III, 2 αh, − αζ, 2 ⎧ % & 1 1/4 ⎪ r M qr 1 ⎪ p − ( − ) , ⎨⎪ 1 N Nend for Case I k 1/4 α 2 qr Mp r 8M 2cS 48cS dlnk 1 − ≈ 1 % & / for AV. α q ⎪ 2 1 4 kend 4 2 24 ⎪ r Mp |D| qr |D| ⎩⎪ 1 − (N − N ) for Case II, α | | 2 | |2 end 8M 2cS C 48cS C (5.425) Here the two possibilities for AV vacuum as appearing for the assisted inflationary framework are: Case I stands for a situation where the spectrum is characterized by the constraint i) D1 = D2 = C1 = C2 = 0, ii) D1 = D2, C1 = C2 = 0, iii) D1 = D2 = 0, C1 = C2. Case II stands for a situation where the spectrum is characterized by the constraint i) μ ≈ ν, D1 = D2 = D = 0 and C1 = C2 = C = 0, ii) μ ≈ ν, D1 = D = 0, D2 = 0 and C1 = C = 0, C2 = 0, iii) μ ≈ ν, D2 = D = 0, D1 = 0 and C2 = C = 0, C1 = 0.

123 278 Page 108 of 130 Eur. Phys. J. C (2016) 76 :278

5. Next following exactly the same steps for single field inflation and also using Eqs. (5.424), (5.425) and (5.421) we get ⎧ 1 ⎪ qr 1 r 1/4 ⎪ − (N − N ) , ⎪ α 1 end for Case I ⎪ 4M V0 2q 24 ⎪ ⎪ ⎪ ⎪ qr 1 ⎪ α 1/4 ⎪ 4M V0 q 1, 3; 3; r + − r ⎪ 2 F1 2 1 ⎪ 3(nh, − nζ, + 1) 2 4 2 24 24 ⎪ ⎪ ⎪ − + ⎪ nh, nζ, 1 ( − ) 1 3 3 r ( − + )( − ) ⎪ − 2 Nend N , ; ; nh, nζ, 1 Nend N ⎪ e 2 F1 e ⎪ 2 4 2 24 ⎪ ⎪ ⎪ / ⎪ r ( − + )( − ) 1 4 ⎪ + − nh, nζ, 1 Nend N , ⎪ 2 1 e for Case II ⎪ 24 ⎪ ⎪ ⎪ 1 ( − + )2 ⎨ 3 nh, nζ, 1 T πqr 1 − (α −α ) ≈ 4 h, ζ, , e for BD (5.426) M ⎪ Mα (α , − αζ,)V 48q p ⎪ h 0 ⎪ ⎪ ⎪ (n ,−nζ,+1)2 ⎪ h , − ζ, + ⎪ 2(α ,−αζ,) nh n 1 ⎪ 12e h erfi √ ⎪ 2 α , − αζ, ⎪ h ⎪ ⎪ √ ⎪ nh, − nζ, + 1 αh, − αζ, ⎪ −erfi √ + (N − N) ⎪ α − α end ⎪ 2 h, ζ, 2 ⎪ ⎪ √ %√ & ⎪ ⎪ 3r 3(nh, − nζ, + 1) ⎪ − erfi √ ⎪ 24 2 α , − αζ, ⎪ h ⎪ ⎪ %√ & ⎪ (α − α ) ⎪ 3(nh, − nζ, + 1) 3 h, ζ, ⎩⎪ −erfi √ + (Nend − N) for Case III, 2 αh, − αζ, 2

⎧ 1 % & / ⎪ 1 4 ⎪ r 1 qr ⎪ 1 − (N − N ) for Case I, ⎨⎪ 8MαV 2c 2 end 0 S 48cS T ≈ . ⎪ % & for AV (5.427) Mp ⎪ 1 1/4 ⎪ r |D| qr |D|2 ⎩⎪ 1 − (N − N ) for Case II, α | | 2 | |2 end 8M V0 2cS C 48cS C

123 Eur. Phys. J. C (2016) 76 :278 Page 109 of 130 278

6. Next using Eq. (5.418)inEqs.(5.426) and (5.427) we get

⎧ 1 ⎪ qr 1 r 1/4 ⎪ 1 − (N − Nend) ⎪ α ⎪ 4M 3 ζ,r π 2 24 ⎪ 2q Mp ⎪ 2 ⎪ r C r ⎪ − (C + − ) + √E − + , ⎪ 1 E 1 nζ, 1 for Case I ⎪ 16 2 2q 8 ⎪ ⎪ ⎪ qr 1 ⎪ ⎪ 4Mα 3ζ,r ⎪ q π M2 1 3 3 r r 1/4 ⎪ 2 p F , ; ; + 2 1 − ⎪ ( − + ) 2 1 ⎪ 3 nh, nζ, 1 2 4 2 24 24 ⎪ ⎪ − + ⎪ nh, nζ, 1 ⎪ (N −N) 1 3 3 r (n ,−nζ,+1)(N −N) ⎪ −e 2 end F , ; ; e h end ⎪ 2 1 ⎪ 2 4 2 24 ⎪ ⎪ / ⎪ r ( − + )( − ) 1 4 ⎪ +2 1 − e nh, nζ, 1 Nend N ⎪ 24 ⎪ ⎪ ⎪ r C r ⎪ − (C + − ) + E − + , ⎨⎪ 1 E 1 nζ, 1 for Case II T 16 2 8 ≈ for BD, (5.428) ⎪ π Mp ⎪ qr ( − + )2 ⎪ 3 nh, nζ, 1 ⎪ Mα (αh,−αζ,) − ⎪ 4(α ,−αζ,) ⎪ e h ⎪ 3ζ,r ⎪ 48q π M2 ⎪ 2 p ⎪ (n ,−nζ,+1)2 ⎪ h n , − nζ, + ⎪ 2(α ,−αζ,) h 1 ⎪ 12e h erfi √ ⎪ 2 αh, − αζ, ⎪ ⎪ √ ⎪ nh, − nζ, + 1 αh, − αζ, ⎪ −erfi √ + (N − N) ⎪ α − α end ⎪ 2 h, ζ, 2 ⎪ √ %√ & ⎪ ⎪ 3r 3(nh, − nζ, + 1) ⎪ − erfi √ ⎪ 24 2 α , − αζ, ⎪ h ⎪ %√ & ⎪ (α − α ) ⎪ 3(nh, − nζ, + 1) 3 h, ζ, ⎪ − √ + (N − N) ⎪ erfi α − α end ⎪ 2 h, ζ, 2 ⎪ ⎪ ⎪ r CE r ⎩ 1 − (CE + 1 − ) + nζ, − 1 + for Case III, 16 2 8

⎧ 1/4 ⎪ qr ⎪ 1 1 − (N − N ) ⎪ 48c2 end ⎪ r S ⎪ ⎪ 8Mα 3ζ,r ⎪ 2c π M2 ⎪ S 2 p ⎪ C ⎪ − (C + − ) r + E − + r , ⎨⎪ 1 E 1 nζ, 1 for Case I T 16 2 8 ≈ for AV. (5.429) Mp ⎪ 1/4 ⎪ qr |D|2 ⎪ 1 |D| 1 − (N − N ) ⎪ 48c2 |C|2 end ⎪ r S ⎪ ⎪ 8Mα 3ζ,r ⎪ 2c |C| π M2 ⎪ S 2 p ⎪ C ⎩⎪ r E r 1 − (CE + 1 − ) + nζ, − 1 + for Case II, 16 2 8

Further using the approximated form of the sound speed cS the expression for the ﬁeld excursion for AV can be rewritten as 123 278 Page 110 of 130 Eur. Phys. J. C (2016) 76 :278

⎧ % & ⎪ 1/4 ⎪ ⎪ − qr ( − ) ⎪ 1 1 ( − ) N Nend ⎪ 48 1− 1 q r ⎪ r 3q 8 ⎪ ⎪ α ⎪ 8M − (1−q) r 3 ζ,r π 2 ⎪ 2 1 3q 8 2 Mp ⎪ ⎪ r C r ⎪ − (C + − ) + E − + , ⎨⎪ 1 E 1 nζ, 1 for Case I T 16 2 8 ≈ for AV. (5.430) M ⎪ % & / p ⎪ 1 4 ⎪ qr |D|2 ⎪ 1 |D| 1 − (N − N ) ⎪ − (1−q) r |C|2 end ⎪ r 48 1 3q 8 ⎪ ⎪ ⎪ 8Mα ( − ) 3ζ,r ⎪ 2 1 − 1 q r |C| π M2 ⎪ 3q 8 2 p ⎪ C ⎪ r E r ⎩ 1 − (CE + 1 − ) + nζ, − 1 + for Case II, 16 2 8

This implies that for BD and AV we get roughly the fol- sake of simplicity here we define a new stringy parameter g˜, lowing result from this analysis: which is given by 8 8 8 8 8T 8 1 8T 8 α λT 2 M4 α T 2 8 8 = √ × 8 8 , g˜ = gM = 0 ×M = s 0 ×M, 8 8 8 8 (5.431) 2 3 2 (5.432) M M M (2π) gs M p Assisted M p Single 2 34p 5 2 34 p5 which means that if the number of tachyonic fields partici- where the terms pointed by the 2345 symbol signify the exact pating in the assisted inflation gradually increases, then the contribution from the single tachyonic field. Now from the tachyonic field excursion for assisted inflation becomes more observational constraints instead of constraining the param- and more sub-Planckian compared to the single field result. eter g here we need to constrain the value of g˜ for all five tachyonic potentials mentioned earlier. Let us mention all 5.2.8 Semi-analytical study and cosmological parameter the constraints on the stringy parameter g˜ for the assisted estimation inflationary framework:

In this subsection our prime objective are: • Model I: inverse cosh potential For q = 1/2, q = 1, q = 3/2 and q = 2wefixN/g˜ ∼ • To compute various inflationary observables from vari- 0.8, which further implies that for 50 < N < 70, the ants of tachyonic potentials in the presence of M identical prescribed window for g˜ from the ζ + nζ plot is given degrees of freedom. by 63 < g˜ < 88. If we additionally impose the constraint • To estimate the relevant cosmological parameters from from the upper bound on the tensor-to-scalar ratio then the proposed models. also the allowed parameter range is lying within a similar • To compare the effectiveness of all of these models in window, i.e., 88 < g˜ < 100. the light of recent Planck 2015 data along with other combined constraints. Model II: logarithmic potential • Finally to check the compatibility of all of these models For q = 1/2, and q = 1wefixN/g˜ ∼ 0.7, which with the CMB TT, TE and EE angular power spectra as further implies that for 50 < N < 70, the prescribed observed by Planck 2015. window for g˜ from the ζ + nζ plot is given by 71.4 < g < 100. If we additionally impose the constraint from However, instead of computing everything in detail we will the upper bound on the tensor-to-scalar ratio then also the not do any further computation in the context of assisted allowed parameter range is lying within a similar window, inflation using all the individual five potentials for that we i.e., 71.4 < g˜ < 90. have already done the analysis in the context of single tachy- Model III: exponential potential-type I onic field earlier in this paper. In this context the results are exactly the same for all potentials that have already done for a For q = 1/2, q = 1, q = 3/2 and q = 2 case g˜ is not single field, provided for all the models the stringy parameter explicitly appearing in the various inflationary observ- g, appearing almost everywhere, is rescaled by the number ables except the amplitude of scalar power spectrum in of identical scalar fields in the present context, i.e., for the this case. To produce the correct value of the ampli- 123 Eur. Phys. J. C (2016) 76 :278 Page 111 of 130 278

Table 1 Comparison between the field excursion obtained from all value for large M by a large amount, compared to the value obtained the tachyonic potentials from the single field and assisted inflation- from single inflationary set-up. Technically this implies the doing effec- ary frameworks. Here we define y0 = T0/Mp,whereMp = 2.43 × tive field theory (EFT) with assisted framework is safer compared to 1018 GeV. The numerics clearly implies that the assisted tachyonic infla- single field case, as it involves an additional parameter M tionary framework pushes the field excursion value to a sub-Planckian Model Parameters Single field Assisted (2σ bound) X S := (|T |/Mp)Single X A := (|T |/Mp)Assisted

. . Inverse cosh potential 80 < g, g˜ < 100 0.34y < X < 1.00y 0√34y0 < X < 1√00y0 0 S 0 M A M 1/2 < q < 2 EFT For y0 < 1.00 EFT For I. y0 < 1.00 II. M ≥ 2 . . Logarithmic potential 71.4 < g, g˜ < 100 0.93y < X < 1.12y 0√93y0 < X < 1√12y0 0 S 0 M A M 1/2 < q < 1 EFT For y0 < 0.892 EFT For I. y0 < 0.892 c = 0.07 II. M ≥ 2 . . Exponential potential Type-I 360 < g, g˜ < 400 4.26y < X < 4.27y 4√26y0 < X < 4√27y0 0 S 0 M A M 1 < q < 2 EFT For y0 < 0.234 EFT For I. y0 < 0.234 II. M ≥ 2 . . Exponential potential Type-II 73 < g, g˜ < 82.33.18y < X < 6.27y 3√18y0 < X < 6√27y0 0 S 0 M A M 1/2 < q < 2 EFT For y < 0.160 EFT For I. y < 0.160 0 0 < = g < . ≥ 6 p 2 6 4 II. M 2 . . Inverse power-law potential 600 < g, g˜ < 700 7.2y < X < 7.9y 7√2y0 < X < 7√9y0 0 S 0 M A M 1 < q < 3/2 EFT For y0 < 0.127 EFT For I. y0 < 0.127 II. M ≥ 2

tude of the scalar power spectra we fix the parameter value for large M with large amount, compared to the value 360 < g˜ < 400. obtained from the single inflationary set-up. Technically this implies that effective field theory (EFT) with assisted frame- Model IV: exponential potential-type II work is safer compared to the single field case, as it involves For q = 1/2, q = 1, q = 3/2 and q = 2wefixN/g ∼ an additional parameter M. 0.85, which further implies that for 50 < N < 70, the Additionally it is important to mention here that the other prescribed window for g from ζ + nζ constraints is conclusions and the rest of the constraints are exactly the givenby59< g˜ < 82.3. If we additionally impose the same as analyzed in the single field case. Similarly the CMB constraint from the upper bound on the tensor-to-scalar TT, TE, EE spectra for the scalar modes are exactly the same ratio then also the allowed parameter range is lying within as obtained in the context of single tachyonic inflation and the window, i.e., 73 < g˜ < 82.3. compatible with the observed Planck 2015 data. Model V: inverse power-law potential For the q = 1/2, q = 1, q = 3/2 and q = 2 cases 5.3 Computation for the multi-field inflation g˜ is not explicitly appearing in the various inflationary observables except the amplitude of scalar power spec- In case of multi-tachyonic inflation all the tachyons are not trum in this case. To produce the correct value of the identical. In more technical language for the most generalized amplitude of the scalar power spectra we fix the param- prescription one can state that eter 600 < g˜ < 700. T1 = T2 =··· = TM . (5.433)

In Table 1 we have shown the comparison between the field In the next subsections we will explore the detailed features of excursion obtained from all the tachyonic potentials from sin- multi-tachyonic inflation by computing the curvature, isocur- gle field and assisted inflationary framework. Here we define vature and tensor perturbations and then we discuss the obser- 18 y0 = T0/Mp, where Mp = 2.43 × 10 GeV. The numerics vational constraints and cosmological consequences from the clearly implies that the assisted tachyonic inflationary frame- set-up. We will give all the analytical results for M non- work pushes the field excursion value to a sub-Planckian identical tachyonic fields and for completeness also give the 123 278 Page 112 of 130 Eur. Phys. J. C (2016) 76 :278 results for M number of non-identical inverse cosh separable ¨ ˙ 9 Ti < 3H Ti < H ∀ i = 1, 2,...,M. (5.441) potential. α(1 + q)

5.3.1 Condition for inflation Consequently the field equations are approximated by ⎛ ⎞ ⎛ ⎞ −1 For assisted tachyonic inflation, the prime condition for infla- M ∂ M ˙ ⎝ ⎠ ⎝ ⎠ tion is given by 6qα H Ti + V (Tj ) V (Tj ) ≈ 0 ∂Ti j=1 j=1 M (5.442) a¨ (ρ + 3p ) H˙ + H 2 = =− i i > 2 0 (5.434) a 6Mp i=1 Also for both cases in the slow-roll regime the Friedmann equation is modified as which can be re-expressed in terms of the following constraint condition in the context of assisted tachyonic infla- M V (T ) tion: H 2 ≈ i . (5.443) 3M2 i=1 p M ( ) V Ti − 3α ˙ 2 > . 1 Ti 0 (5.435) Further substituting Eq. (5.443)inEqs.(5.438) and (5.442) = 2 − α ˙ 2 2 i 1 3Mp 1 Ti we get

⎛ ⎞ ⎛ ⎞ √ −1 Here Eq. (5.316) implies that to satisfy inﬂationary con- M M M 3α ˙ ⎝ ⎠ ∂ ⎝ ⎠ straints in the slow-roll regime the following constraint V (Tj ) Ti + V (Tj ) V (Tj ) ≈ 0 Mp ∂Ti always holds good: j=1 j=1 j=1 ∀i = 1, 2,...,M, (5.444) ⎛ ⎞ ⎛ ⎞ 1 −1 2 q M M ∂ M ˙ < ∀ = , ,..., , √6 ( )α ˙ + ⎝ ( )⎠ ⎝ ( )⎠≈ Ti i 1 2 M (5.436) V Tj Ti V Tj V Tj 0 α M ∂Ti 3 1 3 p j=1 j=1 j=1 6 ∀ = , ,..., . T¨ < H T˙ < H ∀ i = , ,...,M. i 1 2 M (5.445) i 3 i α 1 2 (5.437)

Consequently the field equations are approximated by Finally the general solution for both cases can be expressed in terms of the single field tachyonic potential V (T ) as ⎛ ⎞ ⎛ ⎞ −1 M M √ ∂ α ˙ + ⎝ ( )⎠ ⎝ ( )⎠ ≈ , 3α 3H Ti V Tj V Tj 0 t − tin,i ≈− ∂Ti Mp j=1 j=1 ⎡⎛ ⎞ ⎛ ⎞⎤− −1 1 (5.438) T M M M i ⎢⎝ ⎠ ∂ ⎝ ⎠⎥ dTi V (Tj ) ⎣ V (Tj ) V (Tj ) ⎦ , ∂Ti Tin,i j=1 j=1 j=1 Similarly, in the most generalized case, (5.446) qα − ≈−√6 M t tin,i V (Ti ) 3Mp − ( + )α ˙ 2 > . ⎡ ⎤ − 1 1 q Ti 0 (5.439) ⎛ ⎞ ⎛ ⎞ −1 2 ˙ 2 1 q −1 = 3M 1 − α T T M M M i 1 p i i ⎢⎝ ⎠ ∂ ⎝ ⎠⎥ dTi V (Tj ) ⎣ V (Tj ) V (Tj ) ⎦ . ∂Ti Tin,i j=1 j=1 j=1 Here Eq. (5.439) implies that to satisfy inflationary con- (5.447) straints in the slow-roll regime the following constraint always holds good: Further using Eqs. (5.446), (5.447) and (5.443) we get the following solution for the scale factor in terms of the tachy- ˙ < 1 ∀ = , ,..., , Ti i 1 2 M (5.440) q = / α (1 + q) onic field for the usual 1 2 and for a generalized value of q: 123 Eur. Phys. J. C (2016) 76 :278 Page 113 of 130 278

⎧ ⎡ ⎡⎛ ⎞ ⎛ ⎞⎤− ⎤ ⎪ −1 1 ⎪ M M M ⎪ ⎢ α M Ti ⎢ ∂ ⎥ ⎥ ⎪ ⎣− T V (T ) ⎣⎝ V (T )⎠ ⎝ V (T )⎠⎦ ⎦ , q = / , ⎪ exp 2 d i j j j for 1 2 ⎨⎪ M ∂Ti p Tin j=1 j=1 j=1 a = ain,i × ⎡ ⎡⎛ ⎞ ⎛ ⎞⎤− ⎤ (5.448) ⎪ −1 1 ⎪ M M M ⎪ ⎢ 2qα M Ti ⎢ ∂ ⎥ ⎥ ⎪ ⎣− T V (T ) ⎣⎝ V (T )⎠ ⎝ V (T )⎠⎦ ⎦ . q. ⎪ exp 2 d i j j j for any ⎩ M ∂Ti p Tin j=1 j=1 j=1

2 ∂ ( )∂ ( ) Sometimes it is convenient to identify the inflaton field Mp Tj V Ti Tk V Ti ; (T ) = , V Tj Tk i 2 (5.455) direction as the direction in field space corresponding to the 2 V (Ti ) evolution of the background spatially homogeneous tachy- ∂ ( )∂ ∂ ∂ ( ) 2 4 Tj V Ti Tj Tj Tj V Ti onic fields during inflation. To serve this purpose one can ξ ; (Ti ) = M , V Tj Tj Tj Tj p 2( ) write V Ti (5.456) M ∂ ( )∂ ∂ ∂ ( ) σ = σˆ ˙ , 2 4 Tj V Ti Tj Tk Tk V Ti i Ti dt (5.449) ϑ ; (Ti ) = M , V Tj Tj Tk Tk p 2( ) i=1 V Ti (5.457) where the inflaton direction is defined as ⎡ ⎤ ∂T V (Ti )∂T ∂T ∂T V (Ti ) −1/2 ϑ2 (T ) = M4 k j k k , M V ;Tk Tj Tk Tk i p 2 V (Ti ) σˆ ≡ T˙ × ⎣ T˙ 2⎦ . (5.450) i j (5.458) j=1 ∂ ( )∂ ∂ ∂ ( ) 2 4 Tj V Ti Tk Tk Tk V Ti The M evolution equations for the homogeneous scalar fields ϑ ; (Ti ) = M , V Tj Tk Tk Tk p V 2(T ) can be written as the evolution of single scalar field in the i slow-roll regime as (5.459) ⎛ ⎞ ⎛ ⎞ σ 3 (T ) −1 V ;Tj Tj Tj Tj Tj Tj i M M ⎝ ⎠ ∂ ⎝ ⎠ 3Hσ˙ + V (Tj ) V (Tj ) ∂T V (Ti )∂T V (Ti )∂T ∂T ∂T ∂T V (Ti ) ∂σ = M6 j j j j j j , (5.460) j=1 j=1 p V 3(T ) ⎛ ⎞ ⎛ ⎞ i −1 M M ∂ M ϒ3 (T ) ⎝ ⎠ ⎝ ⎠ V ;Tj Tj Tk Tk Tk Tk i = 3Hσ˙ + σî V (Tj ) V (Tj ) = 0. ∂Ti i=1 j=1 j=1 ∂T V (Ti )∂T V (Ti )∂T ∂T ∂T ∂T V (Ti ) = M6 j j k k k k , p 3 (5.461) (5.451) V (Ti )

3 ϒ ; (Ti ) 5.3.2 Analysis using slow-roll formalism V Tj Tk Tk Tk Tk Tk ∂T V (Ti )∂T V (Ti )∂T ∂T ∂T ∂T V (Ti ) = M6 j k k k k k , p 3 (5.462) Here our prime objective is to define slow-roll parameters for V (Ti ) tachyon inflation in terms of the Hubble parameter and the 3 ϒ ; (Ti ) multi-tachyonic inflationary potential, where the M tachyon V Tj Tj Tj Tk Tj Tk fields are not identical. Using the slow-roll approximation ∂T V (Ti )∂T V (Ti )∂T ∂T ∂T ∂T V (Ti ) = M6 j j j k k k , one can expand various cosmological observables in terms p 3 (5.463) V (Ti ) of small dynamical quantities derived from the appropriate 3 derivatives of the Hubble parameter and of the inflationary ϒ ; (Ti ) V Tj Tj Tj Tj Tk Tk potential. To start with, in the present context the potential ∂T V (Ti )∂T V (Ti )∂T ∂T ∂T ∂T V (Ti ) = M6 j j j j k k , dependent slow-roll parameters are defined as p 3 (5.464) V (Ti ) 2 2 ϒ3 ( ) M ∂ V (T ) V ;T T T T T T Ti ( ) = p Tj i , j j j j j k V ;Tj Tj Ti (5.452) ∂ ( )∂ ( )∂ ∂ ∂ ∂ ( ) 2 V (T ) Tj V Ti Tj V Ti Tj Tj Tj Tk V Ti i = M6 , p 3 (5.465) ∂ ∂ V (T ) V (Ti ) η ( ) = 2 Tj Tj i , V ;Tj Tj Ti Mp (5.453) 3 V (T ) ϒ ; (Ti ) i V Tj Tk Tj Tj Tj Tj ∂ ∂ V (T ) ∂ ( )∂ ( )∂ ∂ ∂ ∂ ( ) 2 Tj Tk i Tj V Ti Tk V Ti Tj Tj Tj Tj V Ti ; (T ) = M , (5.454) = M6 , V Tj Tk i p p 3 (5.466) V (Ti ) V (Ti ) 123 278 Page 114 of 130 Eur. Phys. J. C (2016) 76 :278

3 ϒ3 ( ) ϒ ; (Ti ) ; Ti V Tj Tk Tj Tj Tj Tk ϒ¯ 3 ( ) = V Tj Tj Tj Tk Tj Tk , V ;T T T T T T Ti (5.472) ∂ ( )∂ ( )∂ ∂ ∂ ∂ ( ) j j j k j k 9 3 Tj V Ti Tk V Ti Tj Tj Tj Tk V Ti 3 M = M6 , α = V (Tj ) p 3 (5.467) j 1 V (Ti ) 3 3 ϒ ; (Ti ) ϒ ; (Ti ) 3 V Tj Tj Tj Tj Tk Tk V Tj Tk Tj Tj Tk Tk ϒ¯ (T ) = , V ;Tj Tj Tj Tj Tk Tk i 9 3 ∂ ( )∂ ( )∂ ∂ ∂ ∂ ( ) 3 M Tj V Ti Tk V Ti Tj Tj Tk Tk V Ti α V (T ) = M6 , (5.468) j=1 j p 3( ) V Ti 3 ϒ ; (Ti ) ϒ3 (T ) ϒ¯ 3 ( ) = V Tj Tj Tj Tj Tj Tk , V ;Tj Tk Tj Tk Tk Tk i ; Ti (5.473) V Tj Tj Tj Tj Tj Tk 9 3 ∂ ( )∂ ( )∂ ∂ ∂ ∂ ( ) α3 M ( ) Tj V Ti Tk V Ti Tj Tk Tk Tk V Ti j= V Tj = M6 . 1 p 3 (5.469) V (Ti ) ϒ3 ( ) V ;T T T T T T Ti ϒ¯ 3 (T ) = j k j j j j , V ;Tj Tk Tj Tj Tj Tj i 9 3 3 M α = V (Tj ) However, for the sake of clarity here we introduce new j 1 3 sets of potential dependent slow-roll parameters for multi- ϒ ; (Ti ) ϒ¯ 3 ( ) = V Tj Tk Tj Tj Tj Tk , V ;T T T T T T Ti (5.474) tachyonic inflation by rescaling with the appropriate powers j k j j j k 9 3 9 α 3 M ( ) α M ( ) j=1 V Tj of k=1 V Tk : 3 ϒ ; (Ti ) ϒ¯ 3 ( ) = V Tj Tk Tj Tj Tk Tk , V ;T T T T T T Ti j k j j k k 9 3 ; (Ti ) α 3 M ( ) V Tj Tj j=1 V Tj ¯V ;T T (Ti ) = 9 , j j M α V (T ) 3 k=1 k ϒ ; (Ti ) ¯ 3 V Tj Tk Tj Tk Tk Tk ϒ ; (Ti ) = . (5.475) ; (Ti ) V Tj Tk Tj Tk Tk Tk 9 3 ¯ V Tj Tk 3 M V ;T T (Ti ) = 9 , α = V (Tj ) j k α M ( ) j 1 j=1 V Tj where in all cases i, j, k = 1, 2,...,M and j = k.Forthe η ; (T ) V Tj Tj i sake of simplicity let us parametrize the flow-functions in the η¯V ;T T (Ti ) = 9 , j j α M ( ) j=1 V Tj slow-roll regime via two angular parameters, by making use of the following transformation equations: ; (T ) ¯ V Tj Tk i V ;T T (Ti ) = 9 , j k M ¯ ; ( ) α V (T ) V Tj Tj Ti j=1 j cos α ; := , (5.476) V Tj Tj ¯ ξ 2 ( ) V V ;T T T T Ti ξ¯ 2 (T ) = j j j j , ¯ ( ) V ;Tj Tj Tj Tj i 9 2 V ;T T Ti M α ; := k k , α 2 V (T ) sin V Tk Tk (5.477) j=1 j ¯V ϑ2 ( ) β := ¯ , ; Ti cos V ;Tk Mp V ;Tk (5.478) ¯ 2 V Tj Tj Tk Tk ϑ ; (Ti ) = , V Tj Tj Tk Tk 9 2 sin β ; := M ¯ ; . (5.479) α2 M ( ) V Tj p V Tk j=1 V Tj For the two-field set-up this can be visualized in a better way. 2 ϑ ; (Ti ) ¯ 2 V Tk Tj Tk Tk In that case we need to fix j = 1, k = 2or j = 2, k = 1. and ϑ ; (T ) = , V Tk Tj Tk Tk i 9 2 = , α2 M ( ) i is the free index which can take values i 1 2 depend- j=1 V Tj ∂ ∂ ing on the field derivative, T1 or T2 acting on it. In the ϑ2 ( ) V ;T T T T Ti present context these two sets of angular parameters physi- ϑ¯ 2 (T ) = k k j k , (5.470) V ;Tj Tk Tk Tk i 9 2 cally represent the angle between the adiabatic perturbation, α2 M ( ) j=1 V Tj the tangent of the first slow-roll parameter and the field con- σ 3 ( ) tents. Here also we define the following reduced parameters V ;T T T T T T Ti σ¯ 3 (T ) = j j j j j j , for multi-tachyonic inflation: V ;Tj Tj Tj Tj Tj Tj i 9 3 α3 M ( ) j=1 V Tj M M ¯ = ¯ ( ) 3 V V ;Tj Tj Ti ϒ ; (Ti ) ϒ3 ( ) = V Tj Tj Tk Tk Tk Tk , i=1 j=1 V ;T T T T T T Ti (5.471) j j k k k k 9 3 α 3 M V (T ) M M M j=1 j ¯ ¯ + V ;T T (Ti ) + V ;T T (Ti ) , 3 j k k j ϒ ; (Ti ) = = = ϒ¯ 3 ( ) = V Tj Tk Tk Tk Tk Tk , i 1 j 1 k 1 V ;T T T T T T Ti j k k k k k 9 3 (5.480) α 3 M ( ) j=1 V Tj 123 Eur. Phys. J. C (2016) 76 :278 Page 115 of 130 278

M M +ϒ¯ 3 (T ) + ϒ¯ 3 (T ) η¯ = η¯ ( ) V ;Tj Tk Tk Tj Tk Tk i V ;Tj Tk Tk Tk Tj Tk i V V ;Tj Tj Ti i=1 j=1 +ϒ¯ 3 (T ) + ϒ¯ 3 (T ) V ;Tj Tk Tk Tk Tk Tj i V ;Tk Tj Tj Tk Tk Tk i M M M ¯ ¯ +ϒ¯ 3 (T ) + ϒ¯ 3 (T ) + ; (Ti )+ ; (Ti ) , (5.481) V ;Tk Tj Tk Tj Tk Tk i V ;Tk Tj Tk Tk Tj Tk i V Tj Tk V Tk Tj i=1 j=1 k=1 ¯ 3 ¯ 3 +ϒ ; (Ti ) + ϒ ; (Ti ) . M M V Tk Tj Tk Tk Tk Tj V Tj Tj Tj Tk Tk Tk ξ¯ 2 = ξ¯ 2 (T ) (5.483) V V ;Tj Tj Tj Tj i i=1 j=1 M M M ¯ 2 ¯ 2 + ϑ ; (Ti ) + ϑ ; (Ti ) V Tj Tj Tk Tk V Tk Tj Tk Tk Now simplify the results let us mention the symmetries = = = i 1 j 1 k 1 appearing due to various permutation of the indices: +ϑ¯ 2 (T ) + ϑ¯ 2 (T ) V ;Tj Tk Tk Tk i V ;Tj Tk Tj Tk i +ϑ¯ 2 (T ) + ϑ¯ 2 (T ) ¯ ¯ V ;Tj Tk Tk Tj i V ;Tk Tk Tj Tk i V ;T T (Ti ) = V ;T T (Ti ), (5.484) j k k j ¯ 2 ¯ ; ( ) = ¯ ; ( ), +ϑ (T ) , (5.482) V Tj Tk Ti V Tk Tj Ti (5.485) V ;Tk Tk Tk Tj i ¯ 2 ¯ 2 ¯ 2 ϑ ; (T ) = ϑ ; (T ) = ϑ ; (T ), M M V Tj Tj Tk Tk i V Tj Tk Tj Tk i V Tj Tk Tk Tj i σ¯ 3 = σ¯ 3 (T ) (5.486) V V ;Tj Tj Tj Tj Tj Tj i i=1 j=1 ϑ¯ 2 (T ) = ϑ¯ 2 (T ) = ϑ¯ 2 (T ), V ;Tk Tj Tk Tk i V ;Tk Tk Tj Tk i V ;Tk Tk Tk Tj i M M M (5.487) + ϒ3 ( ) V ;T T T T T T Ti ϒ¯ 3 ( ) = ϒ¯ 3 ( ) j j k k k k V ;T T T T T T Ti V ;T T T T T T Ti i=1 j=1 k=1 j j j k k k j j k j k k = ϒ¯ 3 (T ) = ϒ¯ 3 (T ), ¯ 3 ¯ 3 V ;Tj Tj Tk Tk Tj Tk i V ;Tj Tj Tk Tk Tk Tj i + ϒ ; (Ti ) + ϒ ; (Ti ) V Tj Tk Tk Tk Tk Tk V Tj Tj Tj Tj Tk Tk (5.488) +ϒ¯ 3 ( ) + ϒ¯ 3 ( ) ¯ 3 ¯ 3 V ;T T T T T T Ti V ;T T T T T T Ti ϒ (T ) = ϒ (T ) j j j j j k j k j j j j V ;Tj Tj Tj Tj Tk Tk i V ;Tj Tj Tj Tk Tj Tk i ¯ 3 ¯ 3 ¯ 3 ¯ 3 +ϒ (T ) + ϒ (T ) = ϒ ; (T ) = ϒ ; (T ) V ;Tj Tk Tj Tj Tj Tk i V ;Tj Tk Tj Tj Tk Tk i V Tj Tj Tj Tk Tk Tj i V Tj Tj Tk Tj Tk Tj i ¯ 3 ¯ 3 = ϒ¯ 3 (T ), (5.489) +ϒ (T ) + ϒ (T ) V ;Tj Tj Tk Tj Tj Tk i V ;Tj Tk Tj Tk Tk Tk i V ;Tj Tj Tk Tj Tk Tk i ϒ¯ 3 (T ) = ϒ¯ 3 (T ) +ϒ¯ 3 ( ) + ϒ¯ 3 ( ) V ;Tj Tj Tj Tj Tj Tk i V ;Tj Tj Tj Tj Tk Tj i V ;T T T T T T Ti V ;T T T T T T Ti j j k k j k j j k k k j ¯ 3 ¯ 3 = ϒ ; (Ti ) = ϒ ; (Ti ), ¯ 3 ¯ 3 V Tj Tj Tj Tk Tj Tj V Tj Tj Tk Tj Tj Tj +ϒ ; (Ti ) + ϒ ; (Ti ) V Tj Tj Tj Tk Tj Tk V Tj Tj Tj Tk Tk Tj (5.490) +ϒ¯ 3 (T ) + ϒ¯ 3 (T ) ϒ¯ 3 (T ) = ϒ¯ 3 (T ), (5.491) V ;Tj Tj Tk Tj Tk Tj i V ;Tj Tj Tk Tj Tj Tk i V ;Tj Tk Tj Tj Tj Tj i V ;Tk Tj Tj Tj Tj Tj i ¯ 3 ¯ 3 ϒ¯ 3 ( ) = ϒ¯ 3 ( ) +ϒ (T ) + ϒ (T ) V ;T T T T T T Ti V ;T T T T T T Ti V ;Tj Tj Tj Tj Tk Tj i V ;Tj Tj Tj Tk Tj Tj i j k j j j k j k j j k j ¯ 3 ¯ 3 = ϒ ; (Ti ) = ϒ ; (Ti ), ¯ 3 ¯ 3 V Tj Tk Tj Tk Tj Tj V Tj Tk Tk Tj Tj Tj +ϒ ; (Ti ) + ϒ ; (Ti ) V Tj Tj Tk Tj Tj Tj V Tk Tj Tj Tj Tj Tj (5.492) +ϒ¯ 3 (T ) + ϒ¯ 3 (T ) ϒ¯ 3 (T ) = ϒ¯ 3 (T ) V ;Tj Tk Tj Tj Tk Tj i V ;Tj Tk Tj Tk Tj Tj i V ;Tk Tj Tj Tj Tj Tk i V ;Tk Tj Tj Tj Tk Tj i ¯ 3 ¯ 3 = ϒ¯ 3 (T ) = ϒ¯ 3 (T ), +ϒ (T ) + ϒ (T ) V ;Tk Tj Tj Tk Tj Tj i V ;Tk Tj Tk Tj Tj Tj i V ;Tj Tk Tk Tj Tj Tj i V ;Tk Tj Tj Tj Tj Tk i (5.493) ¯ 3 ¯ 3 +ϒ ; (Ti ) + ϒ ; (Ti ) V Tk Tj Tj Tj Tk Tj V Tk Tj Tj Tk Tj Tj ϒ¯ 3 (T ) = ϒ¯ 3 (T ) V ;Tj Tk Tj Tj Tk Tk i V ;Tj Tk Tj Tk Tj Tk i +ϒ¯ 3 (T ) + ϒ¯ 3 (T ) ¯ 3 ¯ 3 V ;Tk Tj Tk Tj Tj Tj i V ;Tj Tk Tj Tk Tj Tk i = ϒ (T ) = ϒ (T ) V ;Tj Tk Tk Tj Tj Tk i V ;Tj Tk Tj Tk Tk Tj i +ϒ¯ 3 (T ) + ϒ¯ 3 (T ) = ϒ¯ 3 (T ), (5.494) V ;Tj Tk Tk Tj Tj Tk i V ;Tj Tk Tj Tk Tk Tj i V ;Tj Tk Tk Tj Tk Tj i ¯ 3 ¯ 3 ϒ¯ 3 ( ) = ϒ¯ 3 ( ) +ϒ (T ) + ϒ (T ) V ;T T T T T T Ti V ;T T T T T T Ti V ;Tj Tk Tk Tj Tk Tj i V ;Tk Tj Tj Tj Tk Tk i k j j j k k k j j k j k ¯ 3 ¯ 3 ¯ 3 ¯ 3 = ϒ ; (Ti ) = ϒ ; (Ti ) +ϒ (T ) + ϒ (T ) V Tk Tj Tk Tj Tj Tk V Tk Tj Tj Tk Tk Tj V ;Tk Tj Tj Tk Tj Tk i V ;Tk Tj Tk Tj Tj Tk i ¯ 3 = ϒ ; (Ti ), (5.495) +ϒ¯ 3 (T ) + ϒ¯ 3 (T ) V Tk Tj Tk Tj Tk Tj V ;Tk Tj Tj Tk Tk Tj i V ;Tk Tj Tk Tj Tk Tj i 123 278 Page 116 of 130 Eur. Phys. J. C (2016) 76 :278

¯ 3 ¯ 3 M M ϒ ; (Ti ) = ϒ ; (Ti ) V Tk Tj Tj Tk Tk Tk V Tk Tj Tk Tj Tk Tk η¯ = η¯ ( ), V V ;Tj Tj Ti (5.507) = ϒ¯ 3 (T ) = ϒ¯ 3 (T ), V ;Tk Tj Tk Tk Tj Tk i V ;Tk Tj Tk Tk Tk Tj i i=1 j=1 (5.496) M M ¯ 3 ¯ 3 ξ¯ 2 = ξ¯ 2 ( ), ϒ (T ) = ϒ (T ) V V ;T T T T Ti (5.508) V ;Tj Tk Tj Tk Tk Tk i V ;Tj Tk Tk Tj Tk Tk i j j j j i=1 j=1 ¯ 3 ¯ 3 = ϒ ; (Ti ) = ϒ ; (Ti ). V Tj Tk Tk Tk Tj Tk V Tj Tk Tk Tk Tk Tj M M (5.497) σ¯ 3 = σ¯ 3 (T ). (5.509) V V ;Tj Tj Tj Tj Tj Tj i = = Using these results one can finally re-express the reduced i 1 j 1 slow-roll parameters as The cross terms only appear when the structural form of M M M M M the M number of tachyons are different. For example, if we ¯ = ¯ ( ) + ¯ ( ), choose V V ;Tj Tj Ti 2 V ;Tj Tk Ti i=1 j=1 i=1 j=1 k=1 V (T1) = A1 exp(−T1/T01), (5.510) (5.498) V (T ) = B cosh(T /T ), (5.511) M M M M M 2 2 2 02 η¯ = η¯ ( ) + ¯ ( ), ········· ·················· (5.512) V V ;Tj Tj Ti 2 V ;Tj Tk Ti i=1 j=1 i=1 j=1 k=1 then all the cross terms in slow-roll vanish. Also for non- (5.499) separable potentials this explanation works. M M ¯ 2 ¯ 2 ξ = ξ ; (Ti ) V V Tj Tj Tj Tj 5.3.3 The δN formalism for Multi tachyons i=1 j=1 M M M ¯ 2 In this section we have used the δN formalism to compute + 3ϑ ; (Ti ) V Tj Tj Tk Tk the inflationary observables, for the multi-tachyonic field set- i=1 j=1 k=1 up. Here N signifies the number of e-foldings, which can be + 3ϑ¯ 2 (T ) + ϑ¯ 2 (T ) , (5.500) V ;Tk Tj Tk Tk i V ;Tj Tk Tk Tk i expressed in the multi-tachyonic set-up as ⎧ M M M ⎪ α Ti V 2(T ) ⎪ i dT , for q = 1/2, σ¯ 3 = σ¯ 3 ( ) ⎨⎪ 2 ( ) i V V ;T T T T T T Ti : Mp = Ti,end V Ti j j j j j j = tend = i 1 = = N t Hdt √ i 1 j 1 ⎪ M T 2 ⎪ 2qα i V (Ti ) ⎪ T , q. M M M ⎩ 2 d i for any arbitrary M V (Ti ) ¯ 3 3 p i=1 Ti,end +ϒ ; (Ti ) + ϒ ; (Ti ) V Tj Tk Tk Tk Tk Tk V Tj Tj Tk Tk Tk Tk (5.513) i=1 j=1 k=1 + 5ϒ¯ 3 (T ) + 4ϒ¯ 3 (T ) V ;Tj Tj Tj Tj Tk Tk i V ;Tj Tj Tj Tj Tj Tk i In the non-attractor regime, the δN formalism shows vari- + ϒ¯ 3 ( ) + ϒ¯ 3 ( ) ous non-trivial features which have to be taken into account 2 V ;T T T T T T Ti 8 V ;T T T T T T Ti j k j j j j j k j j j k during explicit calculations. Once the solution reaches the ¯ 3 ¯ 3 + 10ϒ ; (Ti ) + 8ϒ ; (Ti ) V Tj Tk Tj Tj Tk Tk V Tj Tk Tj Tk Tk Tk attractor behavior, the dominant contribution comes from ¯ 3 only on the perturbations of the scalar-field trajectories with +4ϒ ; (Ti ) . (5.501) V Tj Tj Tj Tk Tk Tk respect to the tachyon field value at the initial hypersurface, ˙ Now in our computation we take separable potentials hav- Ti , as the velocity, Ti , is uniquely determined by Ti where ing same structural form for all M number of non-identical i = 1, 2,...,M. However, in the non-attractor regime of tachyons. In such case, for an example if take: solution, both the information from the field value Ti and ˙ also Ti are required to determine the trajectory. This can be V (T ) = A exp(−T /T ), (5.502) ˙ 1 1 1 01 understood by providing two initial conditions on Ti and Ti V (T2) = A2 exp(−T2/T02), (5.503) on the initial hypersurface. During the computation of the ········· ·················· (5.504) trajectories let us assume here that the universe has already arrived at the adiabatic limit via attractor phase by this epoch, V (T ) = A exp(−T /T ), (5.505) M M M 0M or equivalently it can be stated that a typical phase transition which implies the structural form, we get the following sim- phenomenon appears to an attractor phase at the time t = t∗. plified expression: More specifically, in the present context, we have assumed M M that the evolution of the universe is unique after the value ¯ = ¯ ( ), of the scalar field arrives at T = T∗ where it is mimicking V V ;Tj Tj Ti (5.506) i=1 j=1 the role of a standard clock, irrespective of the value of its 123 Eur. Phys. J. C (2016) 76 :278 Page 117 of 130 278

M M M i 1 i j = N,i δT + N,ijδT δT 2! i=1 i=1 j=1 M M M 1 i j k + N,ijkδT δT δT +··· , (5.514) 3! i=1 j=1 k=1

where we use the following short hand notations:

= ∂ , N,i Ti N (5.515) = ∂ ∂ , N,ij Ti Tj N (5.516) = ∂ ∂ ∂ . N,ijk Ti Tj Tk N (5.517) Fig. 25 Diagrammatic representation of δN formalism. In this δ ∀ = , ,..., schematic picture (ti ) and (t f ) represent the initial and final hyper- More precisely here Ti i 1 2 M represent the surface where time arrow flows from ti → t f deviations of the fields from their unperturbed values in some specified region of the universe. When the potential is sum-separable, the derivatives of N can be simplified to the following expressions:

( ) + c 1 V Ti Zi N,i = 9 , (5.518) 2¯V (T ) M ( ) i j=1 V T ⎡ j ⎤ η¯ (T ) V (T ) + Z c ⎣ V i i i ⎦ N,ij = δij 1 − 9 2¯V (T ) M ( ) i j=1 V Tj

1 c + 9 Z , , (5.519) ¯ ( ) M ( ) j i 2 V Tj m=1 V Tm

c c where Zi and Zi, j is defined as % & M c ¯V (T ) Z c = V (T c ) i − V (T c), (5.520) Fig. 26 Diagrammatic representation of trajectories and perturbation i m ¯c i m= V decompositions in the field space. Here it is explicitly shown that any 1 9 2 perturbation including field perturbations can be decomposed into a M ( c ) curvature component “curve” and an isocurvature component “iso” m=1 V Tm 2 Z c =− 9 i, j M ¯ ( ) = V (T ) V Tj p 1 p M ˙ ¯ (T ) ¯ (T ) velocity T∗. Let us mention that only in this case δN is equal × ¯ (T ) V i − δ V j − δ V k ¯ ik ¯ jk to the final value of the comoving curvature perturbation ζ = V V k 1 which is conserved at t ≥ t∗.InFigs.25 and 26,wehave η¯ (T ) shown the schematic picture of the δN formalism and the tra- × − V k 1 ¯ jectories and perturbation decompositions in the field space V % c & for the multi-field tachyon inflation. M = 2 ( ) K , On sufficiently large scales for given suitable assumptions V Tm ij (5.521) ¯V (Tj ) as regards the dynamical behavior which permit us to ignore m=1 time derivatives of the perturbations, we expect that each horizon volume will evolve in such a manner that it behaves where indicates the horizon crossing and c denotes the as a self contained universe, and consequently the curvature constant density surface. In this context additionally we get perturbation can be written beyond linear order in cosmolog- M V (T ) M ∂T c ( ) ical perturbation theory as [18,139] = j − i V Ti , dN dTj ∂ j V (Ti ) ∂T ∂i V (Tk) j=1 i=1 j ζ = δN (5.522) 123 278 Page 118 of 130 Eur. Phys. J. C (2016) 76 :278 where Gram–Schmidt orthogonalization technique one can deter- % & 9 mine s . ∂T c M V (T c) ¯ (T c) ¯ (T c) p i =−9 k=1 k V i 9 V i − δ . On the other hand for the multi-tachyonic scenario the ∂T M ( ) ¯ (T ) M ¯ ( ) ij j m=1 V Tm V j p=1 V Tp isocurvature perturbations Sp is the orthogonal projection (5.523) along the sK directions:

Further using this, one can write the following differential M M H jm operator identity which is very useful to compute various Sp =− spjG δTm. (5.530) T˙ inﬂationary observables mentioned in the next section: 0 j=1 m=1

−1 M V (T ) M ∂T c ( ) Using this one can write down the expression for the normal- d = j − i V Ti d . ized isocurvature perturbation: dN ∂ j V (Ti ) ∂T ∂i V (Tk) dT j=1 i=1 j j (5.524) M−1 M−1 SkSk = SpkSqk In the next section we will write all the inﬂationary observ- p=1 q=1 δ ables in terms of the components of N by including all the π 2 3 3 2 ¯ effects of non-linearities in cosmological perturbations for = (2π) δ (k + k ) S (k), (5.531) k3 multi-tachyons. where 5.3.4 Computation of scalar power spectrum M−1 M−1 M M 3 ( ) 1 p q jm k Pζ k ¯ S (k) = s s G , j m 2 In this subsection we start with two point function for multi- 2¯V 2π p=1 q=1 j=1 m=1 tachyonic scalar modes. Implementing the δN procedure we (5.532) get:

M M π 2 where we have explicitly shown all the summations to i j 3 3 2 ζkζ = ζ ζ =(2π) δ (k + k ) ζ (k), indicate that the isocurvature basis vectors are (M − 1)- k k k k3 i=1 j=1 dimensional. For the sake of simplicity applying the Gram– (5.525) Schmidt orthogonalization technique one can choose a basis where Gij is diagonal and given by Gij = δij. where the primordial power spectrum for the scalar modes Similarly, for completeness one can deﬁne the adiabatic- at any arbitrary momentum scale k can be expressed as isocurvature cross spectra in the following fashion: M M 3 ( ) ij k Pζ k M−1 ζ (k) = N,i N j G . (5.526) 2π 2 ζ¯ S = ζ¯ S i=1 j=1 k k k pk p=1 Sometimes it is convenient to express everything in terms π 2 3 3 2 ¯ of the normalized adiabatic curvature power spectrum in the = (2π) δ (k + k ) ζS (k), (5.533) k3 following way: π 2 where 3 3 2 ζ¯ ζ¯ =( π) δ (k + k ) ¯ ζ (k), k k 2 3 (5.527) k M−1 M M 3 ( ) ¯ 1 p jm mj k Pζ k ζS (k) = s G + G . where j m 2 2¯V 2π p=1 j=1 m=1 M M ¯ H ij (5.534) ζ =− i δTj G , (5.528) T˙ 0 i=1 j=1 Cross correlations are generically expected if the background M M 3 ( ) 1 ij k Pζ k trajectory is curved as the modes of interest leave the horizon. ¯ ζ (k) = G . i j 2 (5.529) 2¯V 2π Now we write down the expressions for all the inﬂationary i=1 j=1 observables computed from the multi-tachyonic set-up at a ˙ Ti horizon crossing: Here i = ˙ is a basis vector that projects δTi along the T0 direction of classical background trajectory. The vector and a complementary set of (M − 1) mutually orthonor- • In the present context the amplitude of scalar power spec- mal basis vectors sp form the kinematic basis. Applying the trum can be computed as 123 Eur. Phys. J. C (2016) 76 :278 Page 119 of 130 278

⎧ ⎪ M M 2 ⎪ ij 2 H ⎪ N, N, G − (C + )¯ − C ( ¯ −¯η ) , q = / , ⎪ i j [1 E 1 V E 3 V V ] 2 for 1 2 ⎪ 4π cS ⎨ i=1 j=1 = ζ, ⎪ (5.535) ⎪ M M ¯ C 2 2 ⎪ ij V E qH ⎪ N, N, G − (C + ) − √ ( ¯ −¯η ) , q, ⎩ i j 1 E 1 3 V V 2 for any 2q 2q 2π cS i=1 j=1 9 2 = / 2 = M ( )/ 2 C =− + + γ ≈− . . where H V 3Mp k=1 V Tk 3Mp and E 2 ln 2 0 72 Using the slow-roll approximations one can further approximate the expression for the sound speed as ⎧ ⎪ 2 2 ⎪ 1 − ¯V + O ¯ +··· , for q = 1/2, ⎨⎪ 3 V 2 = cS ⎪ (5.536) ⎪ ( − ) ⎪ 1 q 2 ⎩ 1 − ¯V + O ¯ +··· , for any q, 3q2 V where ¯V and η¯V are the cumulative or total contribution to the slow-roll parameter as deﬁned earlier. Hence using the result in Eq. (5.535) we get the following simpliﬁed expression for the primordial scalar power spectrum: ⎧ ⎪ M M 2 2 ⎪ ij 5 H ⎪ N,i N, j G 1 − CE + ¯V − CE (3¯V −¯ηV ) , for q = 1/2 ⎪ 6 4π 2 ⎨ i=1 j=1 = ζ, ⎪ (5.537) ⎪ M M ¯ C 2 2 ⎪ ij V E qH ⎩⎪ N,i N, j G 1 − (CE + 1 − ) − √ (3¯V −¯ηV ) , for any q. 2q 2q 2π 2 i=1 j=1 Similarly, using the normalized adiabatic curvature power spectrum, we get ⎧ M M ⎪ 5 2 H 2 ⎪ Gij 1 − C + ¯ − C (3¯ −¯η ) , for q = 1/2, ⎪ i j E V E V V π 2¯ ⎨ = = 6 8 V ¯ i 1 j 1 ζ, = (5.538) ⎪ ⎪ M M ¯ C 2 qH2 ⎪ Gij − (C + − ) V − √ E ( ¯ −¯η ) , q. ⎩ i j 1 E 1 3 V V 2 for any 2q 2q 4π ¯V i=1 j=1

• In the present context the normalized amplitude of isocurvature power spectrum can be computed as ⎧ M−1 M ⎪ H 2 ⎪ s psq G jm [1 − (C + 1)¯ − C (3¯ −¯η )]2 , for q = 1/2, ⎪ j m E V E V V π 2 ¯ ⎨ , = , = 8 cS V ¯ p q 1 j m 1 S, = (5.539) ⎪ − ⎪ M1 M ¯ C 2 qH2 ⎪ s psq G jm − (C + ) V − √ E ( ¯ −¯η ) , q, ⎩ j m 1 E 1 3 V V 2 for any 2q 2q 4π cS¯V p,q=1 j,m=1 9 2 = / 2 = M ( )/ 2 C =− + +γ ≈− . . where H V 3Mp k=1 V Tk 3Mp and E 2 ln 2 0 72 Further using an approximated expression for cS in the slow-roll regime and also using the result in Eq. (5.539) we get the following simpliﬁed expression for the power spectrum: ⎧ M−1 M ⎪ 5 2 H 2 ⎪ s psq G jm 1 − C + ¯ − C (3¯ −¯η ) , for q = 1 ⎪ j m E V E V V π 2¯ 2 ⎨ , = , = 6 8 V ¯ p q 1 j m 1 S, = (5.540) ⎪ − ⎪ M1 M ¯ C 2 qH2 ⎪ s psq G jm − (C + − ) V − √ E ( ¯ −¯η ) , q. ⎩ j m 1 E 1 3 V V 2 for any 2q 2q 4π ¯V p,q=1 j,m=1

123 278 Page 120 of 130 Eur. Phys. J. C (2016) 76 :278

• Similarly the normalized amplitude of the adiabatic-isocurvature cross power spectrum can be computed as ⎧ M−1 M ⎪ H 2 ⎪ s p G jm + Gmj − (C + )¯ − C ( ¯ −¯η ) 2 , q = / , ⎪ j m [1 E 1 V E 3 V V ] 2 for 1 2 ⎨ 8π cS¯V p=1 j,m=1 ¯ ζS, = ⎪ M− M ⎪ 1 ¯ C 2 qH2 ⎪ s p G jm + Gmj − (C + ) V − √ E ( ¯ −¯η ) , q, ⎩ j m 1 E 1 3 V V 2 for any 2q 2q 4π cS¯V p=1 j,m=1 (5.541) 9 2 = / 2 = M ( )/ 2 C =− + +γ ≈− . . where H V 3Mp k=1 V Tk 3Mp and E 2 ln 2 0 72 Further using an approximated expression for cS in the slow-roll regime and also using the result in Eq. (5.541) we get the following simpliﬁed expression for the power spectrum: ⎧ M−1 M ⎪ 5 2 H 2 ⎪ s p G jm + Gmj − C + ¯ − C ( ¯ −¯η ) , q = 1 ⎪ j m 1 E V E 3 V V 2 for 2 ⎨ 6 8π ¯V p=1 j,m=1 ¯ ζS, = ⎪ M− M ⎪ 1 ¯ C 2 qH2 ⎪ s p G jm + Gmj − (C + − ) V − √ E ( ¯ −¯η ) q. ⎩ j m 1 E 1 3 V V 2 for any 2q 2q 4π ¯V p=1 j,m=1 (5.542)

• Next one can compute the scalar spectral tilt (nζ , nS , nζS ) of the primordial adiabatic, isocurvature and cross power spectrum as ⎧ ⎪ 2 ⎪ ( η¯ − ¯ ) − 9 9 ⎪ 2 V 8 V M M ⎪ N, N, Gij ⎪ i=1 j=1 i j ⎪ 9 9 9 9 9 ⎪ M M M M M ⎪ V, (T )N, N, GilG jk ⎪ + 2 n=1 i=1 j=1 l=1 k=1 ij n l k +··· , = / , ⎪ 9 9 9 for q 1 2 ⎪ M ( ) M M i j ⎨ m=1 V Tm i=1 j=1 N,i N, j G − ≈ % & nζ, 1 ⎪ (5.543) ⎪ 2 1 2 1 ⎪ η¯ − + ¯ − 9 9 ⎪ V 3 V M M ⎪ q q q q N, N, Gij ⎪ i=1 j=1 i j ⎪ 9 9 9 9 9 ⎪ M M M M M il jk ⎪ = = = = = V,ij(Tn)N,l N,k G G ⎪ + 1 n 1 i 1 j 1 l 1 k 1 +··· , , ⎩⎪ 9 9 9 for any q M ( ) M M i j q m=1 V Tm i=1 j=1 N,i N, j G

and additionally we have

nS, − 1 ≈ nζ, − 1 ≈ nζS, − 1. (5.544)

• One can also compute the expression for the running and running of the running by following the same procedure as mentioned above.

For completeness let us mention the behavior of the cosmological perturbation at later times for multi Gtachyonic inflation. To start with it is important to mention here that in a more generalized physical prescription the time dependence of adiabatic and entropy perturbations in the large cosmological scale limit can always be written in the following simplified form for multi Gtachyonic inflationary paradigm as dζ = ζ˙ = ϒ HS, (5.545) dt dS = S˙ = HS, (5.546) dt

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where ϒ and physically represent the generalized time- power spectra explicitly computed in the previous subsec- dependent dimensionless functions in cosmological pertur- tion. After doing the detailed analysis one can finally com- bation theory of multi Gtachyonic fields. Now to extract pute the resulting curvature and entropic primordial power more informations form Eqs. (5.545) and (5.546), we fur- spectra at the beginning point of the radiation dominated ther integrate both equations over the specified cosmological epoch: time scale and finally following this prescription one can ¯ ˆ 2 ¯ easily compute the expression for the generalized form of ζ = 1 + TζS ζ,, (5.552) the transfer matrix which relates the curvature and entropic ¯ ˆ 2 ¯ S = T ζ,, (5.553) perturbations generated during the situation where the fluctu- SS ¯ ˆ ˆ ¯ ating mode is stretched outside the expansion (Hubble) scale ζS = TζS TSSζ,. (5.554) during the epoch of inflation to the curvature and entropic perturbations at later time via the following simplified form For the sake of simplicity let us define a dimensionless cos- of the matrix equation: mological measure of the correlation function in terms of a correlation angle θ in the following simplified way: % & ζ( ) ˆ ζ( ) ¯ ˆ t 1 TζS t ζS TζS = ˆ , (5.547) θ ≡ = , S(t) 0 TSS S(t) cos (5.555) ¯ ¯ ˆ 2 ζ S 1 + TζS ⎛ ⎞ where the transfer functions are represented by the following 2 ¯ equations in the context of multi-tachyonic inflation from the ζS 1 sin θ ≡ 1 − ⎝ ⎠ = , (5.556) effective GTachyon set-up: ¯ ˆ 2 ζ S 1 + TζS t −1 ˆ ˆ ˆ θ = cot (TζS ), (5.557) TζS (t, t) = ϒ(t )TSS(t, t )H(t )dt , (5.548) t t which are surely very useful for further computation in the ˆ TSS(t, t) = exp (t )H(t )dt . (5.549) present context. Further using Eq. (5.552)inEq.(5.555), t finally we get the following expression for the scalar metric perturbation at the horizon crossing which is expressed in It is also important to note that the evolution in the large- terms of the observed curvature perturbation at later times scale limit is independent of the cosmological scale under in the cosmic time scale and also in terms of the cross- consideration in the present context and consequently the ˆ correlation angle: derived generalized form of the transfer functions TζS and ˆ ¯ ¯ 2 TSS are implicit functions of the cosmological scale due to ζ, ζ sin θ. (5.558) their dependence upon the cosmic time scale t(k) as appearing in the argument of the transfer functions. The scale depen- Next using Eqs. (5.552–5.554), the spectral indices of the pri- dence of the transfer functions are governed by the following mordial power spectrum at later stage of times in cosmolog- sets of evolution equations: ical time scale can be expressed by the following simplified expressions: −1∂ + ˆ + ϒ = , H t TζS 0 (5.550) − = − + −1 ∂ ˆ θ, nζ 1 nζ, 1 H t TζS sin 2 (5.559) −1 ˆ H ∂t + TSS = 0. (5.551) −1 ˆ nS − 1 = nζ, − 1 + 2H ∂t TSS , (5.560) ! −1 ˆ ˆ nC − 1 = nζ, − 1 + H ∂ TζS tan θ + ∂ TSS , where we introduce the notation ∂t = ∂/∂t to define the t t partial differentiation in a simplified way. In the present con- (5.561) text, the cosmological (momentum) scale dependence of the generalized transfer functions for the multi-tachyonic infla- which are very useful to study the scale dependent behav- tionary paradigm is explicitly determined by the two fac- ior of the primordial power spectra in the present context. tors, ϒ and , which physically represent the cosmic time Following the same procedure one can also compute the scale evolution of the curvature and entropic fluctuations at expressions for the running and running of the running of the horizon crossing during the epoch of multi GTachyonic the spectral indices. Additionally, it is important to mention inflation. here that the overall amplitude of the generalized transfer ˆ ˆ Now one can apply the above mentioned generalized functions TζS and TSS are dependent on the time scale evo- transfer matrix as stated in Eq. (5.547) to the primordial scalar lution after horizon crossing via the reheating phenomenon 123 278 Page 122 of 130 Eur. Phys. J. C (2016) 76 :278 and into the radiation dominated epoch. But in spite of this • Finally the tensor-to-scalar ratio for the multi-tachyonic important fact, the spectral tilts of the resulting primordial set-up can be expressed as perturbation spectra can be finally expressed in terms of the ⎧ slow-roll parameters at the horizon crossing during the epoch ⎪ 8 ⎪ 9 9 +··· , for q = 1/2, of inflation and also in terms of the cross-correlation angle ⎨ M M ij i j=1 N,i N, j G θ, which we have already introduced earlier in Eq. (5.557). r ≈ ⎪ 4 ⎩⎪ 9 9 +··· , for any q. M M ij q i j=1 N,i N, j G 5.3.5 Computation of tensor power spectrum (5.566) In this subsection we will not derive the crucial results for On the contrary, compared to the scalar (curvature and multi-Gtachyonic inflation. But we will state the results for entropic) part of the cosmological perturbations, the ten- BD vacuum where the changes will appear due to the pres- sor perturbations remain frozen in on the large cosmolog- ence of M number of different tachyon field. One can simi- ical scales and finally decoupled from the scalar (curvature larly write down the detailed expressions for AV (including and entropic) part of the cosmological perturbations at the α vacuum) as well. linear order of the cosmological perturbation theory. Conse- M M π 2 quently, in the present context, the primordial cosmological i j 3 3 2 h h = h h =(2π) δ (k + k ) (k), perturbation spectrum for gravitational waves is given by the k k k k k3 h i=1 j=1 following simplified expression: (5.562) h = h,, (5.567) = . where the primordial power spectrum for the scalar modes nh nh, (5.568) at any arbitrary momentum scale k can be expressed as Finally, it also important to note that the consistency con- 9 9 M M dition for the tensor-to-scalar amplitudes of the primordial N, N, Gmn k3 Pζ (k) 4k3 Pζ (k) ( ) = 9m=1 9n=1 m n = . power spectrum at the Hubble-crossing can be rewritten, h k 8 M M ij 2π 2 π 2 i=1 j=1 N,i N j G using Eqs. (5.558) and (5.567), as a model independent con- (5.563) sistency relation between the tensor-to-scalar amplitudes of the primordial power spectrum at late times in cosmological The changes will appear in the expressions for the following scale for GTachyon as inflationary observables at the horizon crossing: h 2 r = −8nh sin θ +··· ζ • In the present context amplitude of tensor power spec- 2 2 =−8nh, sin θ +···=r sin θ +··· . (5.569) trum can be computed as In the present context, the scale dependence of the final scalar ⎧ (curvature and entropic) power spectra depends on both on ⎪ 2H 2 the cosmological scale dependence of the initial spectral ⎪ [1 − (C + 1)¯ ]2 , for q = 1/2, ⎨⎪ E V π 2 2 index (nζ,) and on the explicit form of the generalized trans- Mp = h, ⎪ fer functions TζS and TSS. Here we have not explicitly dis- ⎪ ¯ 2 2H 2 ⎪ 1 − (C + 1) V , for any q, cussed the cosmological parameter estimation and numerical ⎩ E π 2 2 2q Mp estimations from a specific class of multi-tachyonic poten- (5.564) tials. But to understand this more clearly, one can carry forward the results obtained in this section to test various models of the multi-tachyonic potential. where CE =−2 + ln 2 + γ ≈−0.72. • Next one can compute the tensor spectral tilt (nh)ofthe primordial scalar power spectrum as

⎧ ⎪ −2¯V [1 +¯V + 2 (CE + 1)(3¯V −¯ηV )] +··· , for q = 1/2, ⎨⎪ nh, ≈ (5.565) ⎪ ¯V ¯V 2 ⎩⎪ − 1 + + (CE + 1)(3¯V −¯ηV ) +··· , for any q. q 2q q

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5.3.6 Analytical study for the multi-field model Next we compute the number of e-foldings from this model: ⎧ ⎡ ⎤ Here we compute the expression for the inverse cosh poten- ⎪ Tend, j ⎪ M tanh tial. One can repeat the computation for the other proposed ⎪ ⎣ 2T0 j ⎦ ⎪ g j ln , for q = 1/2, ⎪ Tj models of multi-field inflation as well. For the multi-field ⎪ = tanh ⎨ j 1 2T0 j case the inverse cosh potential is given by = N ⎪ ⎡ ⎤ ⎪ Tend, j λ ⎪ M tanh ( ) = j ∀ = , ,..., , ⎪ ⎣ 2T0 j ⎦ V Tj j 1 2 M (5.570) ⎪ 2qgj ln , for any arbitrary q. Tj ⎩⎪ Tj cosh = tanh T0 j j 1 2T0 j and the total effective potential is given by (5.577) M M Further using the condition to end inflation: λ j V = V (Tj ) = , (5.571) Tj ¯ ( ) = , = = cosh V Tend,i 1 (5.578) j 1 j 1 T0 j |¯η ( )|= , where λ j characterize the scale of inflation in each branch and V Tend,i 1 (5.579) T0 j are j number of different parameter of the model. Next we get the following field value at the end of inflation: using specified form of the potential the potential dependent −1 slow-roll parameters for i th species are computed as Tend,i = T0i sech (gi ). (5.580) ⎡ ⎤ 2 Ti Next using N = N , = N, and T = T , = T at λ sinh i cmb i i i cmb i i 1 ⎣ i T0i ⎦ ¯V (Ti ) = 9 , the horizon crossing we get M Tm 2g Ti = λmsech i cosh m 1 T0m T0i ⎧ ⎪ − N,i ⎪ tanh 1 exp − , for q = 1/2, (5.572) ⎨ g i 1 ≈ , × η¯ ( ) = Ti 2T0 i ⎪ V Ti 9 ⎪ − N,i M λ Tm ⎩ tanh 1 exp − √ . for any arbitrary q. m=1 msech T0m 2qgi λ T T (5.581) × i tanh2 i − sech2 i . g T T i 0i 0i Using these results one can compute (5.573) 9 ¯ (T c) ( ) − ( c) + M ( c ) V i Also the total contribution in the slow-roll is expressed V T V T m=1 V Tm ¯c 1 i i V N,i = 9 through the reduced slow-roll parameters: ¯ ( ) M 2 V T V (T ) ⎡ ⎤ i j=1 j T M λ sinh2 i 1 ⎣ i T0i ⎦ ¯ = , T T c 9 c ¯ (T c) V 9 λ i − λ i + M λ Tm V i M λ Tm 2gi Ti i sech T i sech T m=1 m sech T ¯c m=1 msech T i=1 cosh T = 0i 0i 0m V 0m 0i 2 Ti λ sinh 9 T (5.574) i T0i M λ j T j=1 j sech gi cosh i T0 j 1 T0i η¯V = 9 M Tm = λ sech (5.582) m 1 m T0m M λ which is very useful to further compute the inflationary i 2 Ti 2 Ti × tanh − sech . observables from the multi-tachyonic set-up. Here indi- gi T0i T0i i=1 cates the horizon crossing and c denotes the constant density (5.575) surface. Using these results finally we compute the following infla- Additionally here we introduce a new factor g j for each non- identical j number of multi-tachyonic fields, defined as tionary observables: αλ 2 4 α 2 j T j Msj T j g = 0 = 0 . (5.576) j 2 ( π)3 2 Mp 2 gsj Mp

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⎛ ⎞ 2 ⎜ ⎟ ⎜ T T c 9 c ¯ (T c) ⎟ M λ i − λ i + M λ Tm V i ⎜ i sech T i sech T m=1 msech T ¯c ⎟ = ⎜ 0i 0i 0m V ⎟ ζ, ⎜ ⎟ ⎜ 2 Ti ⎟ i=1 ⎝ λ sinh 9 T ⎠ i T0i M λ j T j=1 j sech gi cosh i T0 j T0i

⎧ ⎧ ⎫ ⎪ 9 T ⎪ ⎪ M j ⎪ ⎪ ⎨ = λ j sech ⎬ ⎪ j 1 T0 j ⎪ ⎪ , for q = 1/2, ⎪ ⎪ 12π 2 M2 ⎪ ⎨⎪ ⎩ p ⎭ × ⎧ ⎫ (5.583) ⎪ 9 T ⎪ ⎪ M j ⎪ ⎪ ⎨q = λ j sech ⎬ ⎪ j 1 T0 j ⎪ , for any q, ⎪ ⎪ π 2 2 ⎪ ⎩⎪ ⎩⎪ 6 Mp ⎭⎪ ⎧ ⎧ ⎫ 2 ⎪ ⎪ 9 T ⎪ ⎪ ⎪ M j ⎪ ⎪ ⎨ = λ j sech ⎬ ⎪ j 1 T0 j ⎪ , for q = 1/2, ⎪ ⎪ π 2 2 ⎪ ⎪ ⎪ 12 Mp ⎪ ⎡ ⎤−1 ⎪ ⎩ ⎭ 2 Ti ⎨⎪ M λ sinh c = ⎣ i T0i ⎦ × ζ,c ⎪ ⎧ ⎫ (5.584) 2gi Ti ⎪ 2 i=1 cosh ⎪ ⎪ 9 T ⎪ T0i c ⎪ ⎪ M j ⎪ ⎪ ⎨q = λ j sech ⎬ ⎪ j 1 T0 j ⎪ , for any q, ⎪ ⎪ 6π 2 M2 ⎪ ⎩⎪ ⎩⎪ p ⎭⎪ c ⎧ 2 ⎪ (2η¯ − 8¯ ) − ⎛ ⎞ +··· , for q = 1/2, ⎪ V V 2 ⎪ ⎪ ⎪ ⎜ c 9 c ¯ ( c) ⎟ ⎪ ⎜ Ti Ti M Tm V Ti ⎟ ⎪ 9 λi sech −λi sech + = λm sech c ⎪ M ⎜ T0i T0i m 1 T0m ¯ ⎟ ⎪ ⎜ V ⎟ ⎪ i=1 ⎡ ⎤ ⎪ ⎜ 2 Ti ⎟ ⎪ ⎝ λ sinh T 9 T ⎠ ⎪ ⎣ i 0i ⎦ M λ j ⎪ j=1 j sech ⎪ gi Ti T0 j ⎪ cosh T ⎪ 0i ⎪ ⎨⎪ % & nζ, − 1 ≈ 2 1 2 (5.585) ⎪ η¯V − + 3 ¯V ⎪ q q q ⎪ ⎪ ⎪ 1 ⎪ − ⎛ ⎞ +··· , for any q, ⎪ 2 ⎪ ⎪ ⎪ ⎜ c 9 c ¯ ( c) ⎟ ⎪ ⎜ Ti Ti M Tm V Ti ⎟ ⎪ 9 λi sech −λi sech + = λm sech c ⎪ M ⎜ T0i T0i m 1 T0m ¯ ⎟ ⎪ ⎜ V ⎟ ⎪ q i=1 ⎡ ⎤ ⎪ ⎜ 2 Ti ⎟ ⎪ ⎝ λ sinh T 9 T ⎠ ⎪ ⎣ i 0i ⎦ M λ sech j ⎩ g T j=1 j T i cosh i 0 j T0i ⎧ 9 ⎪ M ¯ ( )η¯ ( ) ⎪ i=1 V Ti V Ti ⎪ −6¯V + 2 +··· , for q = 1/2, ⎨ ¯V nζ,c − 1 ≈ 9 (5.586) ⎪ M ¯ ( )η¯ ( ) ⎪ 2 i=1 V Ti V Ti ⎩⎪ −3 ¯V + ··· , for any q, q q¯V

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⎧ ⎪ 8 ⎪ ⎛ ⎞ +··· , for q = 1/2, ⎪ 2 ⎪ ⎪ ⎪ ⎜ c 9 c ¯ ( c) ⎟ ⎪ ⎜ Ti Ti M Tm V Ti ⎟ ⎪ 9 λi sech −λi sech + = λm sech c ⎪ M ⎜ T0i T0i m 1 T0m ¯ ⎟ ⎪ ⎜ V ⎟ ⎪ i=1 ⎡ ⎤ ⎪ ⎜ 2 Ti ⎟ ⎪ ⎝ λ sinh T 9 T ⎠ ⎪ ⎣ i 0i ⎦ M λ sech j ⎨⎪ g T j=1 j T i cosh i 0 j ≈ T0i r ⎪ 4 (5.587) ⎪ ⎛ ⎞ +··· , for any q. ⎪ 2 ⎪ ⎪ ⎪ ⎜ c 9 c ¯ ( c) ⎟ ⎪ ⎜ Ti Ti M Tm V Ti ⎟ ⎪ 9 λi sech −λi sech + = λm sech c ⎪ M ⎜ T0i T0i m 1 T0m ¯ ⎟ ⎪ ⎜ V ⎟ ⎪ q i=1 ⎡ ⎤ ⎪ ⎜ 2 Ti ⎟ ⎪ ⎝ λ sinh T 9 T ⎠ ⎪ ⎣ i 0i ⎦ M λ sech j ⎩⎪ g T j=1 j T i cosh i 0 j T0i

For the inverse cosh potential we get the following con- assisted ﬁeld cases we have derived the results in the quasi-de sistency relations: Sitter background in which we have utilized the details of: ⎧ ( η¯ − ¯ ) +··· , = / , ⎨⎪ 2 V 8 %V & for q 1 2 r nζ, − 1 + ≈ 2 1 2 (5.588) 4 ⎩⎪ η¯ − + 3 ¯ +··· , for any q. q V q q V

6 Conclusion (1) cosmological perturbations and quantum fluctuations for scalar and tensor modes, (2) slow-roll prescription up to all In this paper, we have explored various cosmological conse- orders. In this context we have derived the expressions for quences from the GTachyonic field. We start with the basic all inflationary observables using any arbitrary vacuum and introduction of tachyons in the context of non-BPS string also for the Bunch–Davies vacuum by exactly solving the theory, where we also introduce the GTachyon field, in the Mukhanov–Sasaki equation as obtained from the fluctuation presence of which the tachyon action is modified, and one of scalar and tensor modes. For the single field and assisted can quantify the amount of the modification via a super- field cases in the presence of a GTachyon we have derived script q instead of 1/2. This modification exactly mimics the inflationary Hubble flow and potential dependent flow the role of effective field theory operators and, studying the equations, new sets of consistency relations, which are valid various cosmological features from this theory, one of the in the slow-roll regime and also derived the expression for final objectives is to constrain the index q and a specific the field excursion formula for the tachyon in terms of infla- ∝ α 4/ α combination ( Ms gs) of the Regge slope parameter , tionary observables from both of the solutions obtained from the string coupling constant gs and the mass scale of tachyon arbitrary and the Bunch–Davies initial conditions for infla- Ms, from the recent Planck 2015 and Planck+BICEP2/Keck tion. Particularly the derived formula for the field excursion Array joint data. To serve this purpose, we introduce various for the GTachyon can be treated as one of the probes through types of tachyonic potentials: the inverse cosh, logarithmic, which one can: (1) test the validity and the applicability of exponential and inverse polynomials, using which we con- the effective field theory prescription within the present set- strain the index q. To explore this issue in detail, we start with up, (2) distinguish between various classes of models from the detailed characteristic features of each of the potentials. an effective field theory point of view. Also we have shown Next we discuss the dynamics of GTachyon as well as usual that in the case of assisted tachyon inflation the validity of the tachyon for single, assisted and multi-field scenario. We also effective field theory prescription is much better compared to derive the dynamical solutions for various phases of the uni- the single field case. This is because of the fact that the field << >> verse, including two situations, T T0 and T T0, excursion for assisted inflation is expressed in terms of the where T0 is interpreted as the minimum or the mass scale field excursion for the√ single field, provided the multiplicative of the tachyon. Next we have explicitly studied the infla- scaling factor is 1/ M, where M is the number of identical tionary paradigm from single field, assisted field and multi- tachyons participating in assisted inflation. This derived for- field tachyon set-ups. Specifically for the single field and mula also suggests that if M is a very large number then it

123 278 Page 126 of 130 Eur. Phys. J. C (2016) 76 :278 is very easy to validate effective theory techniques, as in that the primordial non-Gaussianity in the presence of a non- case |T |Assisted << Mp. Next using the explicit form of canonical GTachyonic sector is completely unknown. We the tachyonic potentials we have studied the inflationary con- have also plans to extend this project in that direction. straints and quantified the allowed range of the generalized • The generation of the seed (primordial) magnetic field index q for each of the potentials. Hence using each of the from the inflationary sector [129,147] is a long stand- specific forms of the tachyonic potentials in the context of the ing issue in primordial cosmology. To address this well- single field scenario, we have studied the features of CMB known issue one can also study the cosmological con- angular power spectrum from TT, TE and EE correlations sequences from the effective field theory of inflationary from scalar fluctuations within the allowed range of q for magnetogenesis from the GTachyon within the frame- each of the potentials. We also put in the constraints from the work of Type-IIA/IIB string theory. Planck temperature anisotropy and polarization data, which • Also, our present analysis has been performed for five dif- shows that our analysis matches well with the data. We have ferent potentials motivated from tachyonic string theory, additionally studied the features of the tensor contribution in specifically in the context of single field and assisted field the CMB angular power spectrum from TT, BB, TE and EE inflation. For the multi-field we have quoted the results correlations, which will give more interesting information only for a specific kind of separable potential. It would be in the near future once one can detect the signature of pri- really very interesting to check whether we can study the mordial B-modes. Further, using the δN formalism we have behavior of non-separable potentials in the present con- derived the expressions for the inflationary observables in the text [148,149]. Also it is interesting to reconstruct any context of multi-field tachyons. We have also demonstrated general form of a tachyonic inflationary potential, using the results for the two-field tachyonic case to understand the which one can study the applicability of the effective field cosmological implications of the results in a better way. theory framework within this present set-up. The future prospects of our work are appended below: • One can also carry forward our analysis in the context of non-minimal set-up where the usual Einstein–Hilbert • Throughout this analysis one finds each an every detail, term in the gravity sector is coupled with the GTachy- but one can further study the features of primordial non- onic field. By applying the conformal transformation in Gaussianity from the single field, assisted field and multi- the metric one can construct an equivalent representa- field cases [96,97,140–144]. Due to the presence of a tion of the gravity sector11 in which the gravity sector is generalized index q one would expect that the consis- decoupled from the tachyonic sector and the usual tachy- tency relations, which connect non-Gaussian parameters onic matter sector modified in the presence of an extra with the inflationary observables, are getting modified conformal factor. This clearly implies that the tachyonic in the slow-roll regime of inflation and consequently one potentials that we have studied in this paper all get mod- can generate large amount of primordial non-Gaussianity ified via the additional conformal factor. Consequently from the GTachyonic set-up. Also our aim is general- it is expected that the cosmological dynamics will be ize the results as well as the consistency relations in all modified in this context. Also one can check the appli- order of cosmological perturbations by incorporating the cability of the slow-roll prescription, which helps further effect of sound speed cS and the generalized parameter to understand the exact behavior of cosmological pertur- q. Using this set-up one can also compute the cosmo- bations in the presence of conformally rescaled modified logical Ward identities for inflation [143–145], through tachyonic potentials. which one can write down the recursion relation between • The detailed study of the generation of the dark matter correlation functions. This will help to derive the mod- using effective field theory framework,12 other crucial ified consistency relations in the present context. Addi- particle physics issues i.e., the reheating phenomenon, tionally, one can also explore the possibility of devising leptogenesis and baryogenesis from the GTachyonic set- cosmological observables which violate Bell’s inequali- up is also an unexplored issue till date. ties recently pointed out in Ref. [146]. Such observables could be used to argue that cosmic scale features were produced by quantum mechanical effects in the very early universe, specifically in the context of inflation. We are 11 In the cosmological literature this frame is known as the Einstein planning to report on these issues explicitly very soon. frame. • One would also like to ask what happened when we 12 For completeness, it is important to note that, very recently in add additional higher derivatives in the gravity sector Ref. [150], we have explicitly studied the effective field theory frame- within tachyonic set-up, which are important around the work from membrane inflationary paradigm with Randall Sundrum single brane set-up. We also suggest the reader to study Refs. [151,152], string scale Ms. How the higher derivative gravity sec- in which the effective theory of dark matter has already been studied tor will change the cosmological dynamics and details of very well. 123 Eur. Phys. J. C (2016) 76 :278 Page 127 of 130 278

Acknowledgments SC would like to thank Department of Theoretical 9. D. Baumann, L. McAllister, Inflation and string theory. Physics, Tata Institute of Fundamental Research, Mumbai for providing arXiv:1404.2601 [hep-th] me Visiting (Post-Doctoral) Research Fellowship. SC takes this oppor- 10. D. Baumann, TASI lectures in inflation (2009). arXiv:0907.5424 tunity to thank sincerely to Sandip P. Trivedi, Shiraz Minwalla, Soumi- [hep-th] tra SenGupta, Varun Sahni, Sayan Kar and Supratik Pal for their con- 11. R. Durrer, The theory of CMB anisotropies. J. Phys. Stud. 5, 177 stant support and inspiration. SC takes this opportunity to thank all the (2001). arXiv:astro-ph/0109522 active members and the regular participants of weekly student discus- 12. W. Hu, S. Dodelson, Cosmic microwave background sion meet “COSMOMEET” from Department of Theoretical Physics anisotropies. Ann. Rev. Astron. Astrophys. 40, 171 (2002). and Department of Astronomy and Astrophysics, Tata Institute of Fun- arXiv:astro-ph/0110414 damental Research for their strong support. SC also thanks Sandip 13. M. Kamionkowski, A. Kosowsky, The cosmic microwave back- Trivedi and Shiraz Minwalla for giving the opportunity to be part of ground and particle physics. Ann. Rev. Nucl. Part. Sci. 49,77 the String Theory and Mathematical Physics Group. SC also thanks the (1999). arXiv:astro-ph/9904108 other post-docs and doctoral students from String Theory and Mathe- 14. C. Cheung, P.Creminelli, A.L. Fitzpatrick, J. Kaplan, L. Senatore, matical Physics Group for providing an excellent academic ambiance The effective field theory of inflation. JHEP 0803, 014 (2008). during the research work. SC also thanks Institute of Physics (IOP), arXiv:0709.0293 [hep-th] Bhubaneswar and specially Prof. Sudhakar Panda, for arranging the 15. S. Weinberg, Effective field theory for inflation. Phys. Rev. D 77, official visit where the problems have been formulated and the initial 123541 (2008). arXiv:0804.4291 [hep-th] part have been done. SC also thanks the organizers of COSMOASTRO, 16. D. Lopez Nacir, R. A. Porto, L. Senatore, M. Zaldarriaga, Dissi- 2015, Institute of Physics (IOP), Bhubaneswar, for giving the oppor- pative effects in the effective field theory of inflation. JHEP 1201, tunity to give an invited talk on related issues of effective field theory 075 (2012). arXiv:1109.4192 [hep-th] and primordial non-Gaussianity. SC additionally thanks Inter University 17. S. Choudhury, A. Mazumdar, E. Pukartas, Constraining N = Centre for Astronomy and Astrophysics (IUCAA), Pune and specially 1 supergravity inflationary framework with non-minimal Kähler Varun Sahni for extending hospitality during the work. Additionally operators. JHEP 1404, 077 (2014). arXiv:1402.1227 [hep-th] SC takes this opportunity to thank the organizers of STRINGS, 2015, 18. S. 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