A Study of the Early Cosmic Dynamics in a Multifield Model of Inflation and Curvature Perturbations

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Introduction
Infation as an exponentially growing phase in the history of the very early universe was introduced in the framework of the standard model of cosmology which entails the big bang origin of the observable universe.It is believed to have resolved a number of enigmatic issues encountered in the standard model.It sets the initial conditions for the big bang origin on one hand and explains the growth of cosmic structure on the other hand.Tere is a large number of infationary models proposed with single as well as with multiple scalar felds.Te infationary models with more than one light feld are referred to as multifeld models.Obtaining the infationary era in multifeld models with two or more felds has comparatively more perspectives, but less predictive power of observables [1][2][3][4].A notable diference between single-feld and multifeld models is their sensitivity to respond to the initial conditions.Multifeld models increase the characterization and features of the adiabatic spectrum by creating isocurvature or entropy perturbations that have an impact on anisotropies of the CMB [1,3,[5][6][7][8].Te generation of density perturbations in some multifeld models is treated in such a way as if to decouple it from the dynamics of the infationary era.
At the end of infation in a multifeld scenario, primordial density perturbations can be created on account of the inhomogeneous phase of reheating or modulated hybrid infation if the decaying of dark energy turns to be sensitive to the local values of multifelds except for the infation [9][10][11].Te curvaton infation on the other hand traverses its path averse to it [12][13][14] where, at some point, after infation a weakly coupled feld produces primordial density perturbations during its decay into radiation.Isocurvature density perturbations are also a relict from the curvaton scenario but the abundance of it comes out in many distinct ways in relation to thermal equilibrium abundance at the time when the curvaton sets to decay [15].Te realization of infation can also be availed through multifeld scenarios comprising the double and hybrid models as well.Te infationary theory [16] resolves the fatness, homogeneity, and monopole problems in a very natural way and leads to predict the density perturbations which show approximately the scale invariance and are compatible with the current observational data available.
Cosmic infationary models propound that the early universe underwent an incipient phase of accelerated expansion driven by the dynamics of single, double, or more scalar felds [17,18], while the brief phase of accelerated expansion to be called infation takes the responsibility of seeding all structure formation of the universe we observe today in the form of quantum fuctuations accompanying it in the very tiny fraction of the frst second.Quantum fuctuations [19] are considered to intertwine with the exponential expansion; however, they are supposed to be frozen in Hubble radius while they horizon-cross it.Nonetheless, once the infationary phase comes to an end, they stretch to cosmological scales and grow out to the scale of the present-day universe we do observe it [20].
Infationary dynamics are backed up by the contribution of generic entropic perturbations to the adiabatic one.In single-feld models of infation, adiabatic perturbations are produced when in the models of multifeld both types of perturbations are generated, namely, entropic and adiabatic [21].Infation has thus transformed into the robust paradigmatic theory for understanding the properties and characteristics of our entire observable universe.Nonetheless, infation models consisting of a single feld are characterized by a few fne-tuning problems in general, particularly on the parametric scales.For instance, the masses of the felds and their couplings with one another, in addition to the values of the felds , make it imperative to substantially transform the theories concerning the highenergy regime more realistic.
On taking a number of felds altogether into consideration, it was observed that they can function and operate in coordination with one another in order to give rise to a brief era of infation with the help of the assisted infation mechanism proposed by Liddle et al. [22], although neither of these felds has the ability to sustain the infationary era individually and alone.Te multifeld infationary models diminish the enigmatic problems faced by single-feld infationary models which can be thought of as a fascinating feature to substantiate and materialize the infationary scenario.Te evolution of the universe faces problems when we use a single tachyonic feld to derive infation because, in this case, a larger anisotropy is likely to be generated.Piao studied a model of assisted infation [23] by taking multitachyon felds to derive an infationary period.Te spectrum of curvature perturbations of multifeld infation with a small feld potential was studied [24] by Ahmad et al.Tey put to use the Sasaki-Stewart formalism and reached the results which were obtained with the assumption that isocurvature, i.e., entropy perturbations can, nonetheless, be neglected.Piao investigated that [25] primordial density perturbations can be possibly generated by taking into account the sufcient number of e-folds and by making the use of the decaying speed of sound in a gradually expanding phase of the universe.Some interesting and appealing topics concerning the applications can be studied in [26][27][28][29][30][31].
Cai et al. investigated the entropy perturbations in infation and computed the entropy corrections to the power spectrum of curvature perturbations [32] by fnding out a transfer coefcient analytically.He described a correlation function between entropic and curvature perturbations for this purpose.Te mechanism of relating the power spectrum to the slow-roll parameters is described in [33,34] with a detailed account presented there.Te governing evolutionary equations in the background for the process of driving the primordial power spectrum [35] are given by Tere are appealing related discussions with regard to the application and implementation of these ideas [36][37][38][39][40][41][42].
Avgoustidis et al. investigated [43] the importance of slowroll corrections in multifeld infationary models when the evolution of cosmological perturbations in the form of quantum fuctuations takes place.Tey studied the evolution of curvature and isocurvature perturbations to the next order in the regime of slow-roll infation.Cosmological observables are sensitive during the time of reheating phase in multifeld infationary models.A study was carried out by Hotinli et al. to examine [44] the observables during this phase by devising a method that permits a method, implementing the semianalytic remedying of the efect of perturbative reheating on cosmological perturbations, and using the technique of abrupt decay approximation.Tey further showed that the rate at which the scalar felds decay into radiation afects the tensorto-scalar ratio "r" and scalar spectral index "n s ."A method was presented by Frazer [45] for deriving the analytical expression of the density function of cosmological observables in multifeld models of infation using semiseparable potentials.Frazer found that the sharp peak of the density function is very faintly sensitive to the distribution of initial conditions which means infationary models of multifeld may possess a density function for the observables that peaked sharply.
Te dynamics of the exact multifeld scenarios have been investigated in the classical style in [46] for the case of the hybrid infationary model.Asadi and Nozari investigated a multifeld model with two felds to study its reheat phase in order to have some constraints in the parametric space.Tey found the number of e-folds and the temperature during this era of the reheating phase of their model [47].A class of multifeld models based on those felds that decay or get stabilized in a staggered style during infation was explored by Battefeld and Battefeld [48].Tey observed that felds remain fat before marching towards a steep downfall in assisted infation, and when these felds face such a decrease, their decay rate is measured dynamically and the transfer of 2 Advances in Astronomy energy takes place in the other degrees of freedom.A further decrease in potential energy caused by the decay of the felds contributes to the observables such as spectral index and tensor-to-scalar ratio.Te number of e-folds is bounded for the acceleratedly expanding universe that emerges out from the de Sitter epoch asymptotically [49], and the multifeld model of dark energy is investigated.
In this work, we intend to investigate an infationary model with multifelds in the context of their number of efolds, slow-roll parameters, and spectral indices.Te multifeld infationary models possess some remarkably interesting signatures which the single-feld models digress due to taking into consideration a single feld and have more perspectives for the observational evidence which provides motivation to study these models theoretically.Te study of infationary phase driven combinedly by multifelds usually by axions spaced sparsely is of great interest.Te curvature perturbations are an infationary relic that seeds the structure formation specifcally.Te investigations of the spectrum of these perturbations in multifeld infation are carried out enormously.For the case of equal and unequal masses by considering the suitable initial conditions, these are investigated [50][51][52][53].When we use the power-law potential for multifelds, the spectrum for these perturbations comes out to be redder than it is when a single scalar feld is employed [54][55][56][57][58][59].Spontaneous symmetry breaking naturally gives rise to the small feld models with multifelds where the felds usually begin with unstable equilibrium about the origin and roll down towards a stable minimum.Multifeld infation could also lead to fguring out a mechanism to understand the quantum gravity, an interplay between gravity and quantum feld theory.
Te remaining part of the paper is organized as follows: In Section 2, we perform calculations to determine the expressions for the number of e-folds and spectral index, following the formalism developed by Sasaki and Stewart.Specifcally, we focus on the case where p > 2. Additionally, this section includes an explanatory, albeit brief discussion of the resulting outcomes.In Section 3, we examine the scenario where p � 2. We observe that assigning a negative value to p yields nonsensical results for the number of e-folds and spectral index, while assigning a value of − 2 provides results that are interpretable.Section 3 also includes graphical representations of the results and accompanying remarks that elucidate the relationships among the derived cosmological observables.Te fnal section, Section 4, summarizes the overall fndings of the study and their implications.By considering diferent values of p, we derive predictions regarding the number of e-folds and spectral index.

Driving Multifield Inflation Based on Small
Field Potential and the Spectrum of Curvature Perturbation

Introduction to the Multifeld Potential.
While the universe undergoes an infationary phase, the geometry of spacetime tends to be fat, so while addressing the multifeld model of infation we consider cosmic geometry to be fat.So, after proposing the geometry to be fat, we begin by considering the following potential for investigation in the present work: where the subscript "i" in the potential in question stands for the i th feld in addition to the entities related to it.Furthermore, Λ i is the mass scale and μ i is a parameter which describes the height and tilt of the potential of the i th feld, respectively.Te parameters p and μ i are free variables to choose suitably from the underlying conditions.Te potential given in equation ( 1) is the multifeld version of the brane infationary potential V(ϕ) � Λ(1 − (ϕ/μ) − p ) used in the brane model of infation.In the brane model, the infation is proposed to engender by the motion of branes in the extra dimensions.Te efective Lagrangian for such a system leads to the following expression: , where T 3 is the tension of a light brane and r is related to the distance between the two branes.Other parameters are defned for the system on the same lines.Te efective Lagrangian for the brane infation could be worked out to have the form of the potential expressed in equation (1) for the case i � 1 with an arbitrary value of p [60].Te case i � 1 related to the potential has already been frequently studied in the literature (e.g., see reference [60] and the references furnished therein in the corresponding section).Te potential can be considered a generalized version of the small feld models of infation as discussed in [60][61][62][63][64][65][66] for the negative values of p. Various potentials bearing resemblance to such models in many aspects are also investigated in [67][68][69].
Infationary cosmology by posing an ultrafast phase of cosmic expansion in its early evolution solves many problems, while single-feld infation despite showing extraordinary consistency to the observational data available today leaves room for considering the multifeld models of infation.Being motivated by Lyth bound, in addition, in particle physics, multifeld models come to the scene naturally, especially in the realm of high energy physics beyond the standard models of particle physics such as supersymmetry (SUSY), supergravity (SUGRA), and string theory where generally many felds are considered to be present.In these theories, infation is thought to be driven by the presence of more than one scalar feld where these felds may interact similar to particle interactions.Te presence of more than one feld leads to predictions that could quietly be revolutionary in comparison to single-fled infation.Te choice of the potential in equation ( 1) is motivated by the presence of multiple scalars in the context of the axions of string theory where brane infation is thought to be caused by branes.In the context of superstring theory, by compactifying six dimensions, the model incorporates multiple scalars such as axion, dilaton, and spin two modes of tensor perturbations in its low states of energy.Tus, we see that physically or cosmologically, the choice of the potential in equation ( 1) is well justifed and this model has Advances in Astronomy a correspondence in elementary particle physics.On plotting the potential, we see that it demonstrates an increasing function of the feld, so that the infation feld advances from the right-hand direction to the left.Te feld would disappear for ϕ � μ or ϕ/μ � 1 and it might, therefore work in the domain ϕ/μ > 1. Te study of this model hence should be carried out only in the region lying within the limit ϕ/μ > 1.
In addition, the brane infation conforms to the condition μ/M pl ≪ 1 and occurs following this condition.In Figure 1, the plot of the potential is demonstrated, where the potential and logarithm of the potential are plotted for i � 1 and p � 2.
Te dynamics in the background of a multifeld model of infation can be realized and understood by describing them in terms of dimensionless slow-roll parameters ε, η ‖ , and η ⊥ , which is similar to the situation of a single-feld model, however, the second slow-roll parameter η is required to be modifed in the scenario due to multifeld infation likely to be confronted with the eta problem.Te parameter ε is the frst slow-roll parameter and η ‖ and η ⊥ give the slow-roll rate of the felds along the perpendicular direction of the motion of the felds.Te parameter η ⊥ gives the turn rate of the felds along the perpendicular direction of motion.Te slow roll would last as long as ε ≪ 1 and |η ‖ | ≪ 1, whereas the parameter η ⊥ gives the turn rate of the felds perpendicular to the motion of the felds.A comparatively larger value of η ⊥ may pose as an interesting phenomena to the multifeld scenario, however, it does not imply that it will necessarily violate the slow-roll conditions and will destroy it altogether as is described for multifeld infation.It is also manifested from the slow-roll parameters defned for the multifeld infation that Hubble parameter H and the feld derivative z t ϕ would grow gradually.

Analytical Analysis for the Case Related to p > 2.
Considering the potential used usually for the brane infationary phase concerning p > 2, this potential was found to be relevant and useful in many situations occurring in the viable phenomena ensuing from the real world [70][71][72][73][74][75].It is interesting and signifcant to note that when we assign the value p � 2, then the model under consideration could be thought to be the degenerate version of the infationary model in the small feld realm.Te potential we consider here is with − p. Tis is the profle representing small feld infation and can be regarded as the lowest order of Taylor series expansion of an arbitrary potential about the origin of maxima and minima of it.Te equation of motion of the scalar feld ϕ i while it slow-rolls during the slow-roll phase is where during the slow-roll phase, € ϕ i ≈ 0 and accordingly from equation (2), we are left with the following expression which dominates the phase: and the number of e-folds N during the phase of the infationary scenario can be calculated by the usual formula as Te lower limit ϕ s i in the integral marks the point of time at the beginning of the infationary phase when the corresponding perturbations cross the horizon and the upper limit ϕ e i in the integral corresponds to the point in time when the infationary phase terminates.It is noticeable, however, that ϕ s i < ϕ e i in general and the coming of infation to an end is consistent with the condition ϕ e i ≤ μ i .In addition, the model of small feld potential under consideration satisfes the constraint μ i ≤ M P .For further evaluation of equation ( 4), we have the following from equation (1): . ( 5) Furthermore, we have which simplifes to and by substituting equation ( 7) into (4), we get For the small feld infation, it turns out that the value of μ i is less than the Planck's mass M p, i.e., μ i ≤ M p and the infation comes to an end for ϕ e i ≤ μ i .Tis causes the quadratic terms, i.e., (ϕ s i ) 2 and (ϕ e i ) 2 to disappear due to ϕ s i ≤ ϕ e i ≤ M p .Ten, from equation ( 8), we have or which is further simplifed to If we approximate the expression 1 − (ϕ s i /ϕ e i ) p+2 to unity, then we are left with 4 Advances in Astronomy Curvature perturbations, as well as isocurvature perturbations both, exist in multifeld infation models; however, to keep things simpler, it is considered here that during the slow-roll phase, isocurvature perturbations are suppressed and can be neglected.Te remaining perturbations which are due to the curvature only can be tackled suitably by a mechanism developed by Sasaki and Stewart known as Sasaki-Stewart formalism [7,8,17,[76][77][78][79][80][81].Tus, we see that the magnitude of these curvature perturbations at the end of infation could be worked out on the spatial hypersurfaces of constant density denoted by ζ as where δp na � δp − _ p/ _ ρδρ.We know that curvature perturbation is a gauge invariant quantity and therefore has an arbitrary nature and leads to temperature fuctuations in CMB and spawns the seeds for cosmic structure.
Diferentiating with respect to time and using the energy conservation equation give It reduces to the following on superhorizon scales (k ≪ aH): which on superhorizon scales vanishes in single-feld infation.Tese perturbations, however, in multiple scalars' case are measured by means of the power spectrum and its related parameter bispectrum.
where κ � |k 1 | � |k 2 | � |k 3 | and the power spectrum is worked out to be P ζ (κ) � 2π 2 /κ 3 P ζ (κ) and the spectral index n s − 1 using this formalism is given as In equation (12), we replace μ i /φ s i by ω i, i.e., then equation ( 12) can be reexpressed in the following form: Now, we substitute from equation ( 19) and get then equation ( 19) is written as Te reduced Planck mass can be expressed in terms of the newly defned constant B 1 as Now, we calculate all three terms in the Sasaki-Stewart formalism from equation ( 1) by squaring both sides as Advances in Astronomy Te second and third terms can be neglected as the constraint φ e i ≤ μ i is satisfed for the infation to come to an end.In addition, the model of small feld potential under consideration satisfes the condition μ i ≤ M P and equation ( 23) also has to satisfy φ s i ≤ φ s i ≤ μ i ≤ M P which again motivates to ignore the terms that may result in quadratic form.Te same will be applicable for μ i /φ s i ≃ ω i in equation ( 18) in conjunction with equations ( 44) and ( 49), wherever applicable.
Now, let us consider that then equation ( 24) takes the following form: Now, from equation ( 5), by squaring both sides and employing equation (18), we obtain the following equation: Now, let us take the expression then, equation (27) becomes Now, from equation ( 6), the simplifcation after squaring both sides and by using equation (18) gives By using the defnition from equation (18) in equation (30), we obtain and by absorbing φ 2 i into the defnition of ω i as given in equation (18), and after simplifcation, we obtain Let us now take Now, equation (33) takes the following form by using the above-defned constant: Ten, equation ( 5) by diferentiating once again gives Hence, by using equation ( 18), we have By fnding out the product of equations ( 30) and ( 35) and by using equation (18), we get and then, equation (37) becomes with On substituting from equations ( 22), ( 24)-( 26), ( 29), (34), and (39) in equation ( 17), we determine the following equation as follows: 6 Advances in Astronomy Equation (41) gives the following expression after simplifying it: By multiplying and dividing the 1st and 2nd terms in the abovementioned expression with p + 1 and by simplifying, we get In equation (43), on the right-hand side, the 1st and 2nd terms inside the parenthesis vanish due to ω − np±n i ≃ 0, and the remaining part is represented as We further write down the abovementioned equation in a suitable form by adding and subtracting 1 on the righthand side within the parenthesis as or Let us replace the given equation as then equation ( 46) has the following form: For R(ω i ) ≃ 0, equation (48) serves to calculate the spectral index for a single scalar feld.However, the term R(ω i ) adds in for the case when we are considering multifelds.Terefore, it is important to determine this factor.We will use the defnitions of involved constants to fnd out the approximate value of the R(ω i ) for larger values of p than 2. From equations ( 20), ( 25), ( 28), (33), and ( 38) by substituting for the constants B 1 , B 2 , B 4 , B 5 in equation ( 47) to determine R(ω i ) we get From equation ( 36), we have the expression for ω p i and by considering Ten, by using the value of ω p i in equation ( 49), we get Te expression of R(ω i ) found in equation ( 51) is due to the consideration of multifelds instead of a single feld.Equation (48) now takes the following form: and with R(ω i ) ≃ 0, equation ( 48) comes out to be Te expression in equation ( 52) represents the spectral index corresponding to multifelds, while equation (53) demonstrates the value of the spectral index conforming to the case when a single scalar feld is taken into account.In this case, the masses of all the felds considered are of the same value at the time of horizon-crossing.Tis poses the case when the spectral index of the multifeld is the same and corresponds to the spectral index of the single scalar feld [50].It is also clear that the term R(ω i ) appears due to the consideration of multifelds.It can also be observed in this regard that the value of R(ω i ) will be positive for ω i < ω i+1 and m 2 i < m 2 i+1 .However, it will turn into a negative for Te positive value of R(ω i ) is interpreted to be its spectrum which is redder for multifelds as compared to its corresponding spectrum emerging from a single feld.While the nagative value attached to that implies the corresponding spectrum would be less redder comparatively.However, a stringent condition begs further work to develop.

Analytical Analysis for the Case
Related to p � − 2, +2.Now, we will discuss some specifc cases for the values of p.We will investigate for p � 2, − 2 and will observe what efect it bores upon the expressions of number of e-folds and spectral indices.

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From equation (12), for the number of e-folds with p � 2, we get Equation ( 55) leads to the absurd result which is Similarly, the spectral index n s − 1 from equation ( 52) for p � 2 can be calculated as For N ≃ 0, as we have an absurd result from the abovementioned equation, equation ( 57) leads to the value of the spectral index n s − 1 approaching to ∞ which is again meaningless seemingly.For p � − 2, equation (1) takes the following form: Equations ( 5) and ( 35) are reduced to the following expressions: and the expression for the number of e-folds N is given by Finding the values of the following from equation (58) results in: Similarly, we get and fnally, we have By substituting equations ( 61)-( 64) in equation ( 17), it produces the following expression for the spectral index: where we used the condition that ω − 2 i ≃ 0 and from equation (3), we have Tus, by fnding V i ′ (φ i ) and V j ′ (φ j ) from equation ( 58) and by using them in equation ( 65), we obtain By integrating the abovementioned equation between the limits φ s i andφ e i for the i th feld and between φ s j and φ e i for the corresponding j th feld, we get and after simplifcation, it gives For some fxed value of k, the summation sign  k can be dropped out of the expression as In a more simplifed form, it can be written as Te number of e-folds in equation (60) becomes Tus, from equation ( 72), we can fnd the expression for Planck mass in terms of the number of e-folds as 8 Advances in Astronomy By using the value of Planck mass M 2 pl from equation ( 73), the expression for spectral index in equation ( 65) now reads as where for Λ i � Λ k , all the felds φ i or φ k will possess the same value of Λ i .In this case, the expression for the spectral index in the abovementioned equation reduces to the following form: Te results of equations ( 74) and ( 75) are independent of the choice of the values of k as are considered to the uncompromised level.In equation (75), if the multifelds happen to be such that they can avail the chance of having the same μ i and μ j � μ k , then we would have It can be noted from equation ( 76) that all the terms included in the ln(φ s k /φ e k ) might be equivalent on the basis of equation (72).On the other hand, equation (76) represents the same equation for the corresponding single-feld case.Te value of ln (φ s k /φ e k ) in equation ( 72) will be smaller for the bigger value of μ i , when Λ i is taken as equivalent to μ i .If we consider μ k � Max(μ n ), where n pertains to natural numbers, then it leads to μ i /μ k < 1 implying that the spectrum is redder than its corresponding spectrum which results from equation ( 76) for a single scalar feld φ k .In this case, the value of ln(φ s k /φ e k ) would represent almost the smallest value from all the values of ln(φ s i /φ e i ).It would accordingly indicate that in the context of equation ( 76), the value of k approaches nearer to unity in case of a single scalar feld φ k .On the other hand, if we take into account μ k � Min(μ n ), where n belongs to natural numbers, then this would give rise to μ i /μ k > 1 which leads to the factual result that the spectrum is less red than its corresponding spectrum which results from equation (76) in the case of a single scalar feld φ k .In this case, the value of ln(φ s k /φ e k ) would represent almost the biggest value out of all the values of ln(φ s i /φ e i ) showing that in equation (76) in the case of a single scalar feld φ k , the value of k shifts away from unity.It means that the value of the scalar spectral index falls between that of a single scalar feld in general for the biggest μ k and the smallest accordingly.In Tables 1 and 2, the range for the e-folds N that is cosmologically viable as concerns for the early cosmic evolution corresponding to p and that of spectral index n s corresponding to N is shown, respectively.
Ten, in Figures 2 and 3, the spectral index (n s ) is plotted against the e-folding number N for a range of values.
Te presence of multiple scalars could lead to a somewhat revolutionary infationary paradigm such that the dynamics due to these felds are quite naive compared to the single feld.For example, specifcally, the generation of primordial perturbation spectra and nonadiabatic feld perturbations could afect the evolution of curvature perturbations and the detection of non-Gaussianity.Te infation in the present case of potential occurs long enough so that it is sufciently long that the observable universe comes close to being spatially fat.In order to understand the tensor-to-scalar ratio occurring in the model under consideration, we determine frst the spectrum of scalar curvature perturbations using the relation given beneath as it is fgured out in references [51,80].
which comes out approximately to be Substituting from equations ( 1) and ( 5) into equation (78) and by simplifying, it produces Table 1: Spectral index (n s ) in terms of e-folding number N for a range of values of p.

Values of p
Spectral index (n s ) in terms of e-folds (N) and with the help of equation ( 12), it becomes and where it gives an additional relation that is For tensor perturbations in the case of a general multifeld model, the relation can be used suitably as given in reference [52].
Now, the tensor-to-scalar ratio can be computed as Te recent BICEP results put the following constraint on the tensor-to-scalar ratio as an upper bound r < 0.034(95 % CL) Friedmann evolution equation: for vanishing the curvature term k and the energy density ρ ⟶ V i (φ i ) during the infationary phase, and this further gives Equation ( 85) takes the following form in the light of equations ( 1) and (87): (88) Figure 2: Plot of spectral index (n s ) against the e-folding number (N) for the values of p given in Table 1.(a) the plot is directly drawn between two quantities, (b) the logarithm of the plot is displayed.Figure 3: Plot of spectral index (n s ) against the e-folding number (N) for the values of p given in Table 1.(a) the plot is simply presented between the parameters, (b) it is drawn after taking the logarithm of (n s ). 10

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It can be seen from equation ( 88) that the tensor-toscalar r ratio is dominantly dependent on the number of efolds N as well as on the distribution of μ i .Tis dependence indeed motivates us to stay focused on determining the spectral index corresponding to the model in question.Now, considering that the felds are uncoupled such that the dynamics during slow-roll infation are governed by the following equation: Ten, using equations ( 3), (5), and (86) in equation ( 89) leads to Te quantum fuctuations during this phase occur as where δφ i ∼ H/2π and by making use of equations ( 1) and (86), it gives Te critical feld values can be reached at Δφ ≃ δφ, and for p � 2, we have Now, by using the slow-roll condition ε ≪ 1, from equations ( 1) and ( 5), we get or toward the end of the slow-roll phase, we get Reheating in single-feld infation models could take place by the breakdown of a slow-roll constraint, while in multifeld models, it could be geared by an instability when the felds reach a minimum.Te reheating phase comes at the end of the infationary period when the proposed multifelds lose energy to transform into other relics such as radiation and particles which grow to the present structure formation.At the end of the multifeld infationary phase, the reheating phase can be understood through isocurvature perturbations.In a fat FLRW background, the line element for the linear perturbations reads as where χ and G represent lapse function and shear, respectively.Ten, we have Te adiabatic perturbations for a fuid in general, that is, the adiabatic pressure can be written in expanded form as δP na � δP − c 2 s δρ, where c 2 s � z t P/z t ρ.For the two scalar felds, we have the following: which on comparison gives (99) By reheating when the felds decay into the fuids which transform into radiation and particles, the pressure perturbation of nonadiabatic fuid becomes where c 2 s � (z t ρ σ /z t ρ)c 2 σ .Tus, through efective feld equations coupling the felds by means of their decay products, the reheating period after a multifeld infation occurs, where the nonadiabatic pressure is inside the few orders of magnitude of the pressure perturbations when the infation ends.When diferent mass scales are given, we have to use a robust technique which provides some suitable and viable solution.R. Easter and L. McAllister devised a very powerful technique to work out the mass scales concerning the multifelds [51].Te method is frequently employed in infation or multiple feld scenarios.Tey suggested a new technique known to be as the law regarding the distribution of mass scales in general and is named after the two inventors as Marcenco-Pastur law.In multifeld models of infation, it is customary to make use of random matrix theory which might play a very basic and important role in the distribution of diferent masses related to the spectrum.Tis is accomplished by using some suitable law, the best example is the Mar � c enko-Pastur law.In the beginning, the law was employed frst in the string theory where the problem of the distribution of masses related to the axion feld was being faced.In multifeld models, diferent trajectories of infation occur and therefore, they become susceptible to the initial conditions as the values of the felds lay in the background dynamics.As in most cases, the infationary scenarios are based on the hypothesis taken ad hoc which poses the problem of fnding the initial conditions not to be much reliable.Tus, based on it, infationary parameters, in some specifc scenarios are predicted not to depend largely on priors of initial conditions [81].In this work, we utilize the Mar � c enko-Pastur law for the distribution of mass scales for the factors μ and Λ i .Te Mar � c enko-Pastur law puts to use two parameters μ and β, where β stands for the factor μ expressed as the ratio of rows and columns of the mass.For any mass-scale matrix of order (n + r) × n, we can write it by β � n/n + r.Now, the values that the parameter μ can have, which has to be the smallest value on one hand and the largest value on the other hand, can be determined by the following expressions, respectively: whereas during slow-roll approximation, the feld values can be worked out to be where T(t) � (φ n (t)/φ n (t 0 )) specifes the ratio of relatively larger feld values between t 0 and t where they stand for some initial and later times, respectively.Now, we introduce z � 2 ln[T(t)]/y in equations ( 93) and (95), where φ 2 j is replaced by φ 2 j (t 0 )e zμ 2 i .Straightforwardly, now we can fgure out the values of mass distributions on the average within the respective range regardless of the feld value distributions in the beginning when the correlation relations are evaded and overlooked between them.Ten, by applying the power series expansion, we can fnd out the average value of the term involving exponentiation.

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Advances in Astronomy where In Figure 4, the distribution of mass scales is plotted according to Mar � c enko-Pastur law.
We can have values of the functions f 1 (t, β) and f 2 (t, β) corresponding to the distinct values as adapted by or assigned to the parameter z.However, for comparatively bigger values of it, the functions behave like a constant as the fgures show it.In the case, when values of the felds and mass scales are equivalent, the functions f 1 (t, β) � f 2 (t, β) ≃ 1 and from equation (106) the value of α ≃ φ 2 and m � m, in this case, which leads to regain the values of the concerned felds.

Comments on Conclusions
In this article, we conducted an investigation into the model of infationary phase dynamics by considering multifelds where a small feld potential written in general form is under consideration.It stands for multifelds, however, we fgured out the results for up to two felds and presented the outcomes analytically.Te model is characterized by two free parameters p and μ i which are free to choose as constrained by their predicted range of values.Te variable p is importantly negative in the model we worked out and is arbitrarily chosen.Te potential upon which the model is based represents the small feld infation and can be regarded as Taylor series expansion about the origin of its minima and maxima in its lowest order.In these models of infation, the feld is usually considered to begin with an unstable equilibrium around the origin and then to roll down along its potential about the origin.As the feld expression denotes a generalized potential to stand for multiple scalars connoting the infationary potential, i denotes an ith feld taken into account out of multiple felds.Te parameters Λ i and μ i denote the height and tilt of the ith chosen potential in the multiple felds.Te spectrum of curvature perturbations that give rise to the growth of cosmic structure is an important relic from infation.We investigated this spectrum for the potential under consideration here.In the frst place, we considered the case for a value of p larger than 2. In this case, in general, when the multifelds have the equivalent masses, the equations of motion give rise to those of single-feld infation producing the phase of nonperturbations.Tis occurs due to relative mass diferences in the multifelds and it could be observed that the spectrum comes out to be more or less redder in comparison with the corresponding single-feld model accordingly.Te multifelds under consideration as well as their efective masses play a very signifcant role due to the dependence of the results at the time of horizoncrossing.We noted that the result corresponds to that of a single scalar feld when the efective masses of all the felds are taken to be equivalent.Te spectrum in this case results to be the same and therefore coincides with the spectrum of It can be noted that the law of large numbers of mass scales ensures that the mass distribution of N felds obeys the distribution probability similar to that of a single feld.the (a) plot is simply presented, (b) it is drawn after taking its logarithm.
Advances in Astronomy a single feld.It is concluded that the results for the values of p > 2, p � 2, and p � − 2 are diferent and the behaviors of the feld potentials and the corresponding spectrums are distinct as well as diferent in their nature.
It can be further noted that all the terms included in the factor ln(φ s k /φ e k ) might be equivalent on account of the result determined.With some extra terms, the two expressions represent the same equation for the corresponding single-feld case.When Λ i are taken to be equivalent to μ i , the larger value of μ i corresponds to the minimum value of ln (φ s k /φ e k ).When we consider μ k � Max(μ n ), where n denotes natural numbers, it leads to μ i /μ k < 1 which implies that the spectrum is redder than its corresponding spectrum resulting from the result of a single scalar feld φ k .In this case, the value of ln(φ s k /φ e k ) would represent almost the smallest value from all the values of ln(φ s i /φ e i ) which indicates that in equation ( 76), expressing the case of a single scalar feld φ k , the value of k tends to get nearer to unity.On the other hand, when we take the μ k � Min(μ n ) for n to be a natural number, it gives rise to μ i /μ k > 1, which resultantly leads to the result stating that the spectrum is less red than its corresponding spectrum resulting from the result for a single scalar feld φ k .In this case, the value of ln(φ s k /φ e k ) would represent almost the larger one out of all the values of ln(φ s k /φ e k ) which shows that in the case of a single scalar feld φ k , the value of k shifts away from unity.It means that the value of the scalar spectral index falls between that of a single feld in general for the biggest μ k and the smallest accordingly.
Te results we came across depend on the efective masses and the values of the felds, however, they emerge irrespective of the consideration for the initial conditions.Due to the spectrum being calculated on the time of horizoncrossing, these occur at this time.In order to obtain these results we only require to satisfy the constraints concerning the slow-roll approximation of the felds in the beginning only.Te following condition δφ j / _ φ j � δφ j / _ φ j is required to be imposed so that the isocurvature perturbations can be ignored.By implementing the condition, it seems as though the felds are confned to some specifc trajectories.Although the isocurvature perturbation modes look plausible to be taken into account, for the time being, we evaded them to keep the things simple and to stick to the main theme, however, this is underway in our next investigation.
From the investigations conducted with regard to the observable parameters e.g., slow-roll parameters, e-folding number, and spectral index, we see that they efectively infuence the infationary scenario when a host of a large number of scalar felds is taken into account as the multifeld case demands.Multifeld models might predict a range of values for the spectral index, although the initial values of the multifeld scalars depend upon the coefcient μ.In Figures 2  and 3  of values of the spectral index against e-folds falls in the viable limit for cosmological evolution.Te range of values for spectral index with an increasing number of e-fold is listed in Tables 1 and 2. Te scalar feld infationary models in conjunction with the potential in question such as natural infation, double-well infationary model, and brane infationary model are also of concern.Te recent Planck results put the stringent constraint on the spectral index n s , that is, n s � 0.9649 ± 0.0042(68 % C.L.) which can be used to see as how the model in question contrasts with it.

Figure 1 :
Figure 1: Te plot of the potential for the brane infationary scenario.(a) the potential is plotted in simple scale, whereas on the (b) it is plotted after taking its logarithm as a function of ϕ/μ for p � 2.

Figure 4 :
Figure 4: Te fgure demonstrates the mass distribution according to the Marčhenko-Pastur law as it takes place against the dimensionless mass variables in the case β that takes on diferent values.Te parameter c is along the parallel axis when the functions are along vertical axes.It can be noted that the law of large numbers of mass scales ensures that the mass distribution of N felds obeys the distribution probability similar to that of a single feld.the (a) plot is simply presented, (b) it is drawn after taking its logarithm.
, the spectral index (n s ) is plotted against the e-folding number N for a range of values.It illustrates the behavior and trend of the spectral index against the number of e-folds N where 0.70 < n s < 0.97 corresponds to values N � 20, 30, 40, 50, 60, 70 for plot (a) in Figure 2 and 0.85 < n s < 0.98 corresponds to values N � 20, 30, 40, 50, 60, 70, 80, 90 for the logarithm of plot (b) in Figure 2. Te range

Table 2 :
Tabulation of spectral index (n s ) against the number of e-folds N.