Reheating and Dark Radiation after Fibre Inflation

Reheating and Dark Radiation after Fibre Inflation

Abstract

We study perturbative reheating at the end of fibre inflation where the inflaton is a closed string modulus with a Starobinsky-like potential. We first derive the spectral index and the tensor-to-scalar ratio as a function of the number of efoldings and the parameter which controls slow-roll breaking corrections. We then compute the inflaton couplings and decay rates into ultra-light bulk axions and visible sector fields on D7-branes wrapping the inflaton divisor. This leads to a reheating temperature of order GeV which requires efoldings. Ultra-light axions contribute to dark radiation even if is almost negligible in the generic case where the visible sector D7-stack supports a non-zero gauge flux. If the parameter is chosen to be small enough, is then in perfect agreement with current observations while turns out to be of order . If instead the flux on the inflaton divisor is turned off, which, when used as a prior for Planck data, requires . After is fixed to obtain such a value of , primordial gravity waves are larger since .

a,b,c]Michele Cicoli, a]Gabriel A. Piovano \affiliation[a]Dipartimento di Fisica e Astronomia, Università di Bologna,
via Irnerio 46, 40126 Bologna, Italy
\affiliation[b]INFN, Sezione di Bologna, viale Berti Pichat 6/2, 40127 Bologna, Italy \affiliation[c]ICTP, Strada Costiera 11, Trieste 34014, Italy \emailAddmichele.cicoli@unibo.it \emailAddgabriel.piovano@studio.unibo.it \keywordsReheating, Inflation, String vacua

1 Introduction

The expansion rate of the Universe is exponential whenever the first slow-roll parameter . Moreover, if the second slow-roll parameter , this accelerated expansion can last for enough e-foldings to solve the flatness and horizon problems. However controls the ratio between the inflaton mass and the Hubble scale during inflation, implying that the inflaton has to be a light scalar. However, due to the generic absence of symmetries protecting scalar masses against any kind of quantum corrections, most would-be inflatons would lead to . This problem becomes even more severe for large field models with trans-Planckian field ranges since one has to make sure that for .

Thus successful inflationary model building can be achieved only in the presence of a symmetry which can just be postulated from the low-energy effective field theory point of view while it can in principle be derived from a consistent UV framework like string theory [1, 2]. Hence the best case scenario for inflation is a framework where the inflaton is the pseudo Nambu-Goldstone boson of a spontaneously broken effective global symmetry. The inflaton is light since its direction is flat up to small effects which break the symmetry explicitly. The two main Abelian symmetries used in string compactifications for inflationary model building are compact shifts for axions [3] and non-compact rescalings for Kähler moduli [4].

Another crucial issue to trust inflationary models is moduli stabilisation. In fact, one needs to control the directions orthogonal to the inflaton to prevent dangerous runaways and to ensure that the inflationary dynamics is indeed single-field. Moreover, moduli stabilisation is fundamental to fix all the mass and energy scales in the model.

The presence of a symmetry and full moduli stabilisation are already considered as two considerable achievements for successful inflationary model building and most of the models in the literature, which are considered to be good working examples, satisfy these two requirements. However in order to check the consistency of each inflationary model, one has to study additional formal and phenomenological features. From the theoretical point of view, one has to provide a concrete Calabi-Yau embedding with an explicit orientifold involution and brane setup which are compatible with tadpole cancellation and the presence of a chiral SM-like visible sector [5, 6, 7]. On the other hand, more phenomenological implications of any model, like the predictions for cosmological observables as the scalar spectral index or the tensor-to-scalar ratio , depend crucially on reheating and the details of the post-inflationary evolution of our Universe. In fact, one can extract trustable predictions from an inflationary model only by developing a thorough understanding of what happens after the end of inflation since the exact number of efoldings , corresponding to horizon exit of CMB modes, depends on the reheating temperature and the fact that the Universe is radiation dominated or not all the way from reheating to Big Bang Nucleosynthesis (BBN). Notice that non-standard post-inflationary cosmologies tend to arise rather naturally from string theory due to the presence of long-lived moduli which can give rise to periods of matter domination before BBN [8, 9, 10]. Given that determines the point where and have to be evaluated, one cannot make a clear prediction without studying reheating.

Moreover, is a quantity which should match the most recent value from cosmological data, while the value of turns out to be an actual prediction for future observations. However the spectral index inferred from CMB data depends crucially on the choice of some fundamental parameters. A crucial quantity which is positively correlated with is the effective number of neutrino-like species [11]. If this quantity is set equal to its SM value, , Planck data give () [12]. On the contrary, if one sets as a prior , Planck data lead to a considerably larger spectral index, () [11]. Hence it is crucial to predict the precise value of in order to know which value of should be matched.

Interestingly, another rather generic feature of 4D string models is the presence of ultra-light axions which tend to be the supersymmetric partners of Kähler moduli [13, 14, 15, 16]. These fields can be produced either directly from the inflaton decay or from the decay of the lightest modulus which triggers reheating [17, 18, 19, 20, 21, 22]. Given that the moduli are SM-singlets, in general the branching ratio for their decay into ultra-light axions is non-negligible, leading to a non-zero contribution to . Notice that the mass of these axions tends to be exponentially smaller than the gravitino mass, and so these pseudo-scalars are so light that they never become non-relativistic. The goal of each string inflationary model which aims at yielding trustable predictions is thus to provide a detailed description of reheating which can allow to compute and , and to follow all the post-inflationary evolution all the way down to BBN. Perturbative reheating after the end of some string inflationary models has already been studied in [23, 24] without however taking into account the crucial issue of dark radiation overproduction due to a non-zero inflaton branching ratio into ultra-light axions.

In this paper we shall study perturbative reheating for a well-known class of string models called Fibre Inflation [25, 26]. These constructions are within the framework of type IIB LVS models [27, 28]. The underlying Calabi-Yau manifold has a fibred structure and its volume is controlled by two Kähler moduli associated with the volumes of the 2D base and the 4D K3 or fibre [29]. The internal space features also additional rigid del Pezzo divisors which support non-perturbative effects. At leading order in an expansion in inverse powers of the exponentially large (in string units) internal volume, contributions to the scalar potential [30] and non-perturbative effects [31] fix all the moduli expect for three directions which remain flat. These correspond to the Kähler modulus parameterising the fibre volume and two axions and associated respectively with the reduction of the 10D bulk 4-form on the fibre divisor and the 4-cycle containing the 2D base. At perturbative level, only develops a potential by either 1-loop open string effects [32, 33, 34] or higher derivative corrections to the effective action [35, 36]. Depending on which of these contributions to the inflationary potential is present or dominates (due to topological properties, details of the brane setup and tuning of flux-dependent coefficients), several slightly different inflationary models can emerge [25, 37, 38]. However all of these models are characterised by a similar shape of the inflaton potential which resembles Starobinsky inflation [39] and supergravity -attractors [40, 41] since it features a constant plus negative exponentials which generate a trans-Planckian inflationary plateau. The inflaton range is around with larger values bounded by the size of the underlying Kähler cone [42]. Hence the final prediction for primordial gravity waves is . The stability of the inflationary potential against further corrections is guaranteed by an effective rescaling shift symmetry [43] and globally consistent Calabi-Yau embeddings with consistent brane setup and a chiral visible sector have been recently constructed in [5, 6].

The actual predictions for and depend just on two quantities: the number of efoldings between CMB horizon exit and the end of inflation, and the naturally small coefficient (for a string coupling in the perturbative regime) controlling the strength of the positive exponentials which dominate the inflationary potential at large field values and destroy its flatness. Thus the first step of our analysis will be the derivation of the relations and . In turn, depends on the reheating temperature and the equation of state parameter for the reheating epoch. In this case, the inflaton oscillates around the minimum after the end of inflation and behaves as a classical condensate which redshifts as matter, setting , before decaying into hidden and visible sector particles.

In this paper, we shall focus only on the perturbative decay of the inflaton since studies of oscillon formation in Fibre Inflation have revealed that there is no efficient particle production due to non-perturbative preheating effects after the end of inflation [44]. However ref. [45] has recently claimed that parametric self-resonance can produce inflaton quanta after the end of Fibre Inflation even it is not efficient enough to convert all the energy of the inflaton condensate into particle production. The disagreement between the two results might be due to the fact that the analysis of [44] is based on a numerical study while the analysis of [45] is based on an analytical approximation. Due to the intrinsically non-linear nature of the preheating phase, we think that numerical results are more trustable. Hence we would tend to conclude that the process through which the inflaton energy gets transferred into SM degrees of freedom is dominated by a purely perturbative dynamics. Let us mention that, even if there is production of inflaton particles and/or the formation of oscillons after the end of inflation, one would still need to study their perturbative decay unless oscillons collapse into black holes before decaying (see [46] for the study of the stability of oscillon solutions for axion-like potentials).

Let us stress also that, given that the inflaton is the lightest Kähler modulus, no period of matter domination can arise between reheating from the inflaton decay and BBN. The two axions and could in principle dominate the energy density after the end of inflation but they receive a potential only via tiny non-perturbative corrections to the superpotential which give them a mass that is exponentially smaller than the gravitino mass. Hence these pseudo-scalars are almost massless and could just behave as relativistic degrees of freedom which belong to the hidden sector and can contribute to . Present upper bounds on this quantity set strong constraints on our brane setup. In fact, if the SM lives on D3-branes at singularities [47, 48], the visible sector is sequestered from the bulk [49, 50], resulting in an effective decoupling of the inflaton from the the visible sector degrees of freedom [17, 18]. Consequently, the main inflaton decay channels are just to and leading to a value of which is way too large [19]. Therefore the SM is forced to live on stacks of D7-branes wrapped around the inflaton cycle. In this case the visible sector is unsequestered and the soft terms turn out to be of order the gravitino mass [51]. Given that the inflaton is much lighter than the gravitino, the inflaton decay to supersymmetric particles is kinematically forbidden and the dominant inflaton decay channels are into Higgses, SM gauge bosons and hidden sector ultra-light bulk axions [20]. Thus in this case the branching ratio of the inflaton decay into and is much smaller than unity, resulting in dark radiation in agreement with observational bounds.

We will find two separate cases depending on the gauge flux on the visible sector stack of D7-branes wrapping the fibre divisor. In the generic case where this flux is non-zero, the inflaton coupling to visible sector gauge bosons is enhanced. Hence turns out to be negligible, in perfect agreement with current observations since Planck+galaxy BAO data give () [12]. On the other hand, when the gauge flux is absent, ultra-light axions generate a considerable contribution to dark radiation of order . Notice that such a large value of is not necessarily ruled out since a combined Planck+galaxy BAO+HST analysis gives () [12]. Moreover, when LyaF BAO measurements are added to Planck+galaxy BAO data, the amount of dark radiation increases to () [52]. Finally in a comprehensive combination of SN and LyaF BAO measurements with Planck+galaxy BAO+HST data, is pulled to similarly high values of order () [53].

Our study of reheating will also allow us to fix the reheating temperature at GeV which implies . Notice that a similar result for the reheating temperature of Fibre Inflation models has already been found in [23] by considering just the inflaton decay into visible sector gauge bosons. Our analysis is however deeper since it includes also the study of the additional leading order inflaton decay channels into Higgses and ultra-light bulk axion and the corresponding model building constraints from the requirement of avoiding the overproduction of axionic dark radiation. The two results for the order of magnitude of the reheating temperature however agree since we will find that the inflaton decay into gauge bosons tends to dominate over the decay into Higgs degrees of freedom. We shall then plug in the inflationary relations and which simplify to and . The microscopic parameter can be fixed from by using the result of our computation of as a precise prior for Planck data. Once is fixed, the relation gives the prediction of Fibre Inflation models for the amplitude of primordial gravity waves. Notice that our analysis is complementary to the study of reheating for Fibre Inflation performed in [54] which worked out as a function of and independently of the microscopic details of the model.

In the generic case where is close to zero, Planck data give centered around [12] which can be easily obtained for a sufficiently small value of the microscopic parameter of order . The tensor-to-scalar ratio then turns out to be of order . If instead since the gauge flux on visible D7-stack is vanishing, we shall consider as the number to be matched to fix from . In fact, Fig. 21 of [11] (see also Fig. 20 of the same paper for values in deviations from the CDM model with ) gives a spectral index in the range at for , and so it seems reasonable to consider for . In this case and the relation leads to larger primordial tensor modes since .

Notice that in both cases, due to the rather large value of , the inflationary scale is high, GeV, and the gravitino mass is around GeV. An interesting implication of this result is that the visible sector cannot be a simple constrained MSSM with universal scalar and gaugino masses since the scale of supersymmetry breaking would be too high to allow for a correct Higgs mass around GeV [55]. Therefore in Fibre Inflation models the visible sector has to have a richer structure, involving non-universal soft terms [56], additional fields like in the NMSSM [57] or a larger gauge group.

The plan of this paper is as follows. In Sec. 2 we briefly describe the dynamics of Fibre Inflation models and we derive the dependence of the spectral index and the tensor-to-scalar ratio on the number of efoldings and the coefficient of the corrections which steepen the potential at large field values. In Sec. 3 we first illustrate the brane setup and derive the inflaton couplings to all visible sector particles and bulk axions, and then compute the reheating temperature, the number of efoldings, the amount of dark radiation and the final predictions for and . We finally discuss our results and present our conclusions in Sec. 4.

2 Fibre inflation

2.1 General idea

Fibre Inflation [25, 26] is a class of type IIB string inflationary models where the inflaton is a Kähler modulus which behaves as the pseudo Nambu-Goldstone boson of a slightly broken non-compact Abelian symmetry [43]. The perturbative Kähler potential of type IIB compactifications looks like:

(1)

where the tree-level part is:

(2)

and leads to the famous no-scale cancellation. Hence at tree-level the scalar potential for all Kähler moduli is flat. On the other hand, the leading order [30] and corrections to [32, 33, 34] in 4D Einstein frame scale as:

(3)

where the ’s are 2-cycle volumes while and are coefficients. Focusing on just one Kähler modulus, the ratio between and effects scales as:

(4)

However due to the extended no-scale structure [34], there is a cancellation in the leading contribution of to the scalar potential, implying that effects are subdominant with respect to effects at the level of the scalar potential even if they are the leading perturbative contributions to . In fact and corrections to the scalar potential scale as:

(5)

where denotes 4-cycle moduli and the tree-level flux superpotential. In the case of just one Kähler modulus , and so , implying that:

(6)

Therefore the flatness of the tree-level potential is broken at leading order by corrections which however depend just on the overall volume , and so lift just one direction in Kähler moduli space. This means that all directions orthogonal to the overall volume remain exactly flat at leading order, and so are very promising inflaton candidates. These remaining flat directions can be lifted by subdominant perturbative effects which can depend on all Kähler moduli, or even by higher derivative corrections to the effective action [35, 36].

Moreover, these directions orthogonal to enjoy an effective approximate shift symmetry since both and depend just on [43]. This shift symmetry is broken by effects since depends on all Kähler moduli but the fact that this is a small effect with respect to the tree-level contribution guarantees that higher dimensional operators are sufficiently suppressed. In fact, these operators are suppressed by both powers of and the small ratio .

2.2 Inflationary potential

Compactification manifold

Fibre Inflation models involve a Calabi-Yau manifold with at least 3 Kähler moduli [25]:

  1. : plays the rôle of the inflaton and parametrises the volume of a K3 or divisor fibred over a base. This field is stabilised at subleading order due to string loop corrections to [28] or higher derivative terms [35]. The field comes from the reduction of the 10D bulk form over and it is fixed due to tiny non-perturbative corrections to the superpotential [58, 59]. This field is much lighter than the inflaton and acquires isocurvature fluctuations during inflation. However present strong bounds on isocurvature fluctuations [60] due not apply to our case since is too light to behave as dark matter.

  2. : parametrises the volume of the base of the fibration and mainly controls the overall volume which is stabilised at leading order via corrections. The volume modulus is a spectator during inflation since it turns out to be heavier than the inflaton. The axion comes from the reduction of over and, similarly to , is fixed due to tiny non-perturbative effects and it is much lighter than the inflaton during inflation.

  3. : controls the size of a blow-up mode which is required to perform a full stabilisation of at leading order. Both and are fixed by non-perturbative corrections to and are heavier than during inflation.

Expanding the Kähler form in a basis of dual 2-forms as:

(7)

the Calabi-Yau volume can expressed in terms of 2-cycle moduli as [29]:

(8)

where is the volume of the base while is the size of the K3 or fibre. Using the expressions of the 4-cycle moduli:

(9)

the volume (8) can be rewritten as:

(10)

where and .

Effective action

The 4D supergravity theory is characterised by the following Kähler potential and superpotential . The tree-level together with the leading and perturbative corrections reads:

(11)

with where is the Calabi-Yau Euler number [30]. The 1-loop open string correction comes from the tree-level propagation of closed Kaluza-Klein strings between non-intersecting stacks of branes, while is due to the tree-level exchange of winding strings between intersecting branes [32]. These corrections have been conjectured to take the form [33]:

(12)

where is the 2-cycle perpendicular to the -th couple of parallel branes, while is the 2-cycle which is the intersection locus of the -th couple of intersecting D7-branes/O7-planes.

The superpotential receives a tree-level contribution from background fluxes which is just a constant after dilaton and complex structure moduli stabilisation. Moreover can depend on all Kähler moduli at non-perturbative level:

(13)

The parameters and are constants. In particular, with in the case of an ED3-instanton while is the rank of the condensing gauge group in the case of gaugino condensation on D7-branes [31].

The F-term scalar potential originating from (11) and (13), can be organised in an expansion in inverse powers of :

(14)

where we did not include the tree-level scalar potential which scales as since the no-scale cancellation implies that is exactly flat for all -moduli at this order of approximation. The potential beyond tree-level can be studied order by order in the inverse volume expansion, showing that at each order it can provide stable minima for some of the moduli. Notice that in each contribution in (14) we have shown explicitly only the dependence on the moduli which are fixed at that order of approximation.

Moduli stabilisation at

Focusing on the large volume limit ,1 the leading contributions to the potential for the -moduli come from corrections to and -dependent non-perturbative effects which give a typical LVS potential [27, 28]:

(15)

Notice that, at this order in the expansion, does not depend either on or on the axions and . This is because the dominant contribution to the potential of arises only via string loops at order [28]. On the other hand, and develop a potential at even more subleading order, with , due to tiny non-perturbative effects [58].

The potential admits a supersymmetry-breaking AdS minimum at exponentially large volume (at first order in an expansion in ):

(16)

Moduli stabilisation at order

Without loss of generality and following [25], we shall lift one of the three remaining flat directions by the inclusion of string loops while we shall ignore contributions from higher derivative corrections since, as studied in [37] and [38], these terms do not modify qualitatively the inflationary dynamics but give rise just to some slight quantitative differences. The effects (12) generate a scalar potential of the form [34]:

(17)

In our case, focusing just on and -dependent loop corrections, the resulting scalar potential becomes [25]:

(18)

where , , are given by:

(19)

The parameters , and depend on the complex structure moduli which are fixed by background fluxes. Thus, from the string landscape point of view, these are tunable coefficients which can be adjusted due to phenomenological requirements. As shown in [26], leads to a prediction for ruled out by observations. Hence from now on we focus only on the case with which, when , yields a minimum for at:

(20)

justifying the scaling of (18). Notice that at the minimum the moduli scale as since:

(21)

Moduli stabilisation at

The leading contribution to the scalar potential from and -dependent non-perturbative corrections to is given by (we neglect mixed terms since they have a double exponential suppression):

(22)

This potential turns out to be more suppressed than the ones analysed before since it scales as but it is still the dominant potential for and which are stabilised at:

(23)

We stress again that this minimum is AdS and so an additional term is needed to uplift the potential to a nearly Minkowski vacuum. As recently reviewed in [59], there are several mechanisms to achieve a dS vacuum via either anti-branes [61], T-branes [62], non-perturbative effects at singularities [63] or non-zero F-terms of the complex structure moduli [64].

Mass spectrum

At leading order, the scalar potential depends just on three moduli: the blow-up mode and its associated axion , and a particular combination of and corresponding to the overall volume . Hence, at this level of approximation, only these three fields acquire masses of order:

(24)

where the gravitino mass scales as . As shown in [23, 65], the kinetic terms for the Kähler moduli can be diagonalised at leading order in giving:

(25)

which implies:

(26)

Hence corresponds to , and so it is the direction fixed at leading order, while corresponds to the combination of and which remains flat. This remaining flat direction is lifted at subleading order by perturbative effects and plays the rôle of the inflaton. The mass of at the minimum then sets the value of the Hubble constant during inflation :

(27)

However during inflation the inflaton, which from (26) we can parametrise as , is displaced from its minimum. Thus its potential (18) becomes exponentially suppressed in terms of the canonically normalised inflaton which turns out to be very light since:

(28)

where denotes the shift of from its minimum. Hence drives a period of slow-roll inflation while the other moduli sit approximately at their minima (since they are heavy). For , the shifts of the minima of the heavy fields are negligible and the dynamics is effectively single-field. The pictorial view of the inflationary process is rather simple. Inflation starts in a region where the fibre modulus is much larger than the base modulus , while during the inflationary evolution the fibre decreases while the base becomes larger keeping approximately constant. At the end of inflation the inflaton reaches its minimum where, according to (21), the Calabi-Yau is anisotropic since the base is much larger than the fibre.

On the other hand the two axions and are almost massless since they are stabilised at order :

(29)

Given that these pseudoscalar fields are in practice massless, their contribution to the total dark matter density is negligible, and so present isocurvature fluctuation bounds [60] can be easily satisfied.

2.3 Single-field inflation

The inflationary dynamics can be studied by expanding in terms of its canonically normalised counterpart using (25), and then writing to denote the shift from the minimum:

(30)

After expressing in terms of the canonically normalised field , the inflationary potential (18) looks like:2

(31)

where:

(32)

In the previous expression we added -independent contributions to the scalar potential needed to uplift the original AdS vacuum to a nearly Minkowski one which can come from several sources of positive energy like anti-branes [61], T-branes [62], non-perturbative effects at singularities [63] or non-vanishing F-terms of the complex structure moduli [64]. Moreover notice that the prefactor of (31) sets both the inflaton mass at the minimum and the Hubble scale during inflation :

(33)

in perfect agreement with (27). Clearly, the inflaton mass quickly becomes exponentially smaller than for . Fig. 1 shows the inflationary potential for different values of .

Figure 1: Inflationary potential for different values of setting . The plot shows also the end point of inflation used in the following analysis.

Inflationary predictions

Starting from the inflationary potential (31), the slow-roll parameters become:

(34)
(35)

The slow-roll parameter vanishes at the two inflection points where the two negative exponentials compete with each other, and for where the positive exponential becomes comparable in size with . The slow-roll parameter at becomes , signaling that inflation ends close to the first inflection point. In fact, around , independently of the microscopic parameters since the term proportional to can be neglected in the vicinity of the minimum. As in [25], there is an inflationary plateau to the right of the first inflection point and inflation takes place for field values in the window since the spectral index is always too blue for .

The number of efoldings between the point of horizon exit and the end of inflation is then computed as:

(36)

with:

(37)

where we have neglected the term in proportional to since slow-roll inflation occurs in the region with .

The main cosmological observables we are interested in are the scalar spectral index and the tensor-to-scalar ratio which have to be evaluated at horizon exit as:

(38)

We would like now to solve the integral (36), and then invert the relation to obtain . Substituting this relation into and , we would end up with and , where depends on the post-inflationary evolution and is a tunable parameter which can be adjusted to reproduce the correct Planck value of the spectral index once the prediction for is determined.

However the integral (36) cannot be solved analytically. We shall therefore consider two simplified cases where an approximated analytical solution can be provided:

  1. : In this case we can set and (37) simplifies to:

    (39)

    and so the integral in (36) can be solved exactly yielding:

    (40)

    where . This equation can be inverted iteratively to give:

    (41)

    In this region where , the slow-roll parameters (34) and (35) can be well approximated as:

    (42)

    which implies an interesting relation between the cosmological observables and :

    (43)

    If we substitute (42) into (43) with given by (41), the spectral index can be expressed as a function of the number of efoldings as:

    (44)

    while the tensor-to-scalar ratio is given from (43) by .

    Figure 2: Scalar spectral index and tensor-to-scalar ratio as a function of the number of e-foldings for the case with (red lines) and the case where the positive exponential in cannot be neglected (blue lines fixing ).
  2. : In this case we need to keep the term proportional to but the integrand (37) can be expanded in power series as:

    (45)

    and so the integral in (36) now admits the following solution:

    We can again invert this equation iteratively, obtaining:

    In this region where , the slow-roll parameters (34) and (35) can be well approximated as:

    (47)

    where:

    (48)

    This gives a modified relation between the cosmological observables and since:

    (49)

    and:

    (50)

    implying that, for fixed , this case gives a larger with respect to the first case. This result is not surprising since in the case where the positive exponential contribution to is not negligible, the inflationary potential turns out to be steeper. If we substitute (48) into (49) and (50) with given by (2), we can obtain the functions