Variable gravity: A suitable framework for quintessential inflation

Variable gravity: A suitable framework for quintessential inflation

Md. Wali Hossain wali@ctp-jamia.res.in Centre for Theoretical Physics, Jamia Millia Islamia, New Delhi-110025, India    R. Myrzakulov rmyrzakulov@gmail.com Eurasian International Center for Theoretical Physics, Eurasian National University, Astana 010008, Kazakhstan    M. Sami sami@iucaa.ernet.in Centre for Theoretical Physics, Jamia Millia Islamia, New Delhi-110025, India    Emmanuel N. Saridakis Emmanuel_Saridakis@baylor.edu Physics Division, National Technical University of Athens, 15780 Zografou Campus, Athens, Greece Instituto de Física, Pontificia Universidad de Católica de Valparaíso, Casilla 4950, Valparaíso, Chile
Abstract

In this paper, we investigate a scenario of variable gravity and apply it to the unified description of inflation and late time cosmic acceleration dubbed quintessential inflation. The scalar field called “cosmon” which in this model unifies both the concepts reduces to inflaton at early epochs. We calculate the slow-roll parameters, the Hubble parameter at the end of inflation, the reheating temperature,the tensor-to-scalar ration and demonstrate the agreement of the model with observations and the Planck data. As for the post inflationary dynamics, cosmon tracks the background before it exits the scaling regime at late times. The scenario gives rise to correct epoch sequence of standard cosmology, namely, radiative regime, matter phase and dark-energy. We show that the long kinetic regime after inflation gives rise to enhancement of relic gravity wave amplitude resulting into violation of nucleosynthesis constraint at the commencement of radiative regime in case of an inefficient reheating mechanism such as gravitational particle production. Instant preheating is implemented to successfully circumvent the problem. As a generic feature, the scenario gives rise to a ‘blue’ spectrum for gravity waves on scales smaller than the comoving horizon scale at the commencement of the radiative regime.

pacs:
98.80.-k, 95.36.+x, 04.50.Kd

I Introduction

Theoretical and observational consistencies demand that the standard model of Universe be complemented by early phase of accelerated expansion, dubbed inflation Starobinsky:1980te (); Starobinsky:1982ee (); Guth:1980zm (); Linde:1983gd (); Linde:1981mu (); Liddle:1999mq (); Langlois:2004de (); Lyth:1998xn (); Guth:2000ka (); Lidsey:1995np (); Bassett:2005xm (); Mazumdar:2010sa (); Wang:2013hva (); Mazumdar:2013gya (), and late time cosmic acceleration Copeland:2006wr (); Sahni:1999gb (); Frieman:2008sn (); Padmanabhan:2002ji (); Padmanabhan:2006ag (); Sahni:2006pa (); Peebles:2002gy (); Perivolaropoulos:2006ce (); Sami:2009dk (); Sami:2009jx (); Sami:2013ssa (). Inflation is a remarkable paradigm, a single simple idea which addresses logical consistencies of hot Big Bang and provides a mechanism for primordial perturbations needed to seed the structures in the Universe. As for late time cosmic acceleration, it is now accepted as an observed phenomenon though its underlying cause still remains to be obscure, whereas similar confirmation of inflation is still awaited. Thus, the hot Big Bang and the two phases of accelerated expansion is a theoretically accepted framework for the description of our Universe.

No doubt that inflation is a great idea, the phenomenon should therefore live for ever such that the late time cosmic acceleration is nothing but its reincarnation à la quintessential inflation Peebles:1999fz (); Sahni:2001qp (); Sami:2004xk (); Copeland:2000hn (); Huey:2001ae (); Majumdar:2001mm (); Dimopoulos:2000md (); Sami:2003my (); Dimopoulos:2002hm (); Dias:2010rg (); BasteroGil:2009eb (); Chun:2009yu (); Bento:2008yx (); Matsuda:2007ax (); da Silva:2007vt (); Neupane:2007mu (); Dimopoulos:2007bp (); Gardner:2007ib (); Rosenfeld:2006hs (); Bueno Sanchez:2006ah (); Membiela:2006rj (); Bueno Sanchez:2006eq (); Cardenas:2006py (); Zhai:2005ub (); Rosenfeld:2005mt (); Giovannini:2003jw (); Dimopoulos:2002ug (); Nunes:2002wz (); Dimopoulos:2001qu (); Dimopoulos:2001ix (); Yahiro:2001uh (); Kaganovich:2000fc (); Peloso:1999dm (); Baccigalupi:1998mn (); Hinterbichler:2013we (). The idea was first proposed by Peebles and Vilenkin in 1999 Peebles:1999fz () which was later implemented in the framework of braneworld cosmology Sahni:2001qp (); Sami:2004xk (); Copeland:2000hn (); Huey:2001ae (); Majumdar:2001mm (). At the theoretical level, it sounds pretty simple to implement such a proposal in the language of a single scalar field. The field potential should be shallow at early times, facilitating slow roll, followed by steep behavior thereafter and turning shallow again at late times. The steep potential is needed for radiative regime to commence, such that the field is sub-dominant during radiation era and does not interfere with nucleosynthesis. It should continue to remain in hiding during matter phase, till its late phases, in order not to obstruct structure formation. It is then desirable to have scaling regime, in which the field mimics the background being invisible, allowing the dynamics to be free from initial conditions, which in turn require a particular steep behavior of the potential. At late times the field should overtake the background, giving rise to late-time cosmic acceleration, which is the case if slow roll is ensured or if the potential mimics shallow behavior effectively.

There are several obstacles in implementing the above unification scheme. First, since inflation survives in this scenario until late times, the potential is typically of a run-away type and one therefore requires an alternative mechanism of reheating in this case. One could invoke reheating due to gravitational particle production after inflation Kofman:1994rk (); Dolgov:1982th (); Abbott:1982hn (); Ford:1986sy (); Spokoiny:1993kt (); Kofman:1997yn (); Shtanov:1994ce (); Campos:2002yk (), which is a universal phenomenon. However, the latter is an inefficient process and it might take very long for radiative regime to commence. Clearly, in this case, the scalar field spends long time in the kinetic regime such that the field energy density redshifts with the scale factor as corresponding to equation of state of stiff matter. It is known that the amplitude of gravitational waves produced at the end of inflation enhances during kinetic regime, and if the latter is long, the relic gravitational waves Sahni:2001qp (); Sami:2004xk (); Grishchuk:1974ny (); Grishchuk:1977zz (); Starobinsky:1979ty (); Sahni:1990tx (); Souradeep:1992sm (); Giovannini:1998bp (); Giovannini:1999bh (); Langlois:2000ns (); Kobayashi:2003cn (); Hiramatsu:2003iz (); Easther:2003re (); Brustein:1995ah (); Gasperini:1992dp (); Giovannini:1999qj (); Giovannini:1997km (); Gasperini:1992pa (); Giovannini:2009kg (); Giovannini:2008zg (); Giovannini:2010yy (); Tashiro:2003qp () might come into conflict with nucleosynthesis constraint at the commencement of radiative regime Sahni:2001qp (); Sami:2004xk (); Tashiro:2003qp (). Hence, one should look yet for another alternative reheating mechanism, such as instant preheating Felder:1998vq (); Felder:1999pv (); Campos:2004nc (), to circumvent the said problem.

A second obstacle to the unification is that if we want the scalar field to mimic the background for most of the thermal history, the field potential should behave like a steep exponential potential at least approximately such as the inverse power-law potentials. Since scaling regime is an attractor in such cases, an exit mechanism from scaling regime to late time acceleration should be in place in the scenario.

Let us examine as how to build the unified picture. The single scalar field models aiming for quintessential inflation can be broadly put into two classes: (1) Models in which the field potential has a required steep behavior for most of the history of universe but turn shallow at late times, for instance, the inverse power-law potentials Sahni:2001qp (); Sami:2004xk (); Copeland:2000hn (); Sami:2003my (). (2) Models in which the field potential is shallow at early epochs giving rise to inflation, followed by the required steep behavior.

In the first class of potentials, we can not implement inflation in the standard framework, since slow roll needs to be assisted in this case. For example, one could invoke Randall-Sundrum (RS) braneworld Randall:1999ee (); Randall:1999vf () corrections Sahni:2001qp (); Sami:2004xk (); Copeland:2000hn () to facilitate inflation with steep potential at early epochs. In this case, as the field rolls down to low energy regime, the braneworld corrections disappear, giving rise to a graceful exit from inflation and thereafter the scalar field has the required behavior. However, gravitational particle production Ford:1986sy (); Spokoiny:1993kt () is extremely inefficient in the braneworld inflation Sahni:2001qp () and one could in principle introduce the instant preheating to tackle the relic gravitational waves problem Sami:2004xk (). Unfortunately, the steep braneworld inflation is inconsistent with observations, namely, the tensor-to-scalar ratio of perturbations is too high in this case. Thus, the scenario fails in the early phase, although the late-time evolution is compatible with theoretical consistency and observational requirements Sahni:2001qp (); Sami:2004xk ().

In the second class of potentials, that is shallow at early epochs followed by steep behavior, we need a mechanism to exit from scaling regime. A possible way out is provided by introducing neutrino matter, such that neutrino masses are field-dependent Wetterich:2013aca (); Wetterich:2013jsa (); Wetterich:2013wza (). Such a scenario can be motivated from Brans-Dicke framework, with an additional assumption on the matter Lagrangian in the Jordan frame, namely treating massive neutrinos differently from other forms of matter in a way that the field is minimally coupled to cold dark matter/baryon matter in Einstein frame whereas the neutrino masses grow with the field Wetterich:2013jsa (). In such a scenario neutrinos do not show up in radiation era; their energy density tracks radiation being sub-dominant. However, in the subsequent matter phase at late times, as they become non-relativistic, their masses begin to grow and their direct coupling to scalar field builds up such that the effective potential acquires a minimum at late times giving rise to late time acceleration, provided the field rolls slowly around the effective minimum. At this point, a question arises, namely whether we could do without neutrino matter and the extra assumption in which case the field would couple to matter directly in the Einstein frame and the effective potential would also acquire a minimum. For simplicity let us assume that we are dealing with a constant coupling à la coupled quintessence Amendola:1999er (). In that case it is possible to achieve slow roll around the minimum of the effective potential, provided that is much larger than the slope of the potential, such that the effective equation-of-state parameter has a desired negative value ( where is the slope of the potential). The scaling solution (which is accelerating thanks to non-minimal coupling), an attractor of the dynamics, is approached soon after the Universe enters into matter-dominated regime and consequently we cannot have a viable matter phase in this case. It is therefore necessary that the matter regime is left intact and the transition to accelerated expansion takes place only at late times. The latter can be triggered by massive neutrino matter with field-dependent masses Amendola:2007yx (); Wetterich:2007kr (); Pettorino:2010bv ().

In this paper we consider a scenario of quintessential inflation in the framework of variable gravity model Wetterich:2013jsa (); Wetterich:2013aca (); Wetterich:2013wza (); Wetterich:2014eaa (); Wetterich:2014bma (). We first revisit the model in Jordan frame (Sec. II) and then we transit to the Einstein frame (Sec. III) for detailed investigations of cosmological dynamics by considering the canonical form of the action (Sub Sec. III.1). Behavior of the canonical field with respect to the non canonical field is also examined (Sub Sec. III.2). In the Einstein frame we examine the inflationary phase (Sec. IV), kinetic regime and late time transition to dark energy (Sec. V). Ref.Wetterich:2013jsa () provides broad out line of inflation and late time acceleration in the framework of model under consideration. In this paper, we present complete evolution history by invoking suitable preheating mechanism. We investigate issues related to the spectrum of relic gravity waves (Sub Sec. IV.1) as a generic observational features of quintessential inflation. The relic gravity wave amplitude is defined by the inflationary Hubble parameter whereas the spectrum of the wave crucially depends upon the post inflationary evolution. We investigate the problems related to the long kinetic regime in the scenario and discuss the instant preheating (Sub Sec. IV.2) to tackle the problem. Post inflationary evolution (Sub Sec. V.1) is investigated with canonical action and the epoch sequences (Sub Sec. V.2) are achieved with viable matter phase. Detailed dynamical analysis is performed to check the nature of stability of all fixed points (Sub Sec. V.3). Finally in Sec. VI we summarize the results.

Ii Variable Gravity in Jordan Frame

In this section we revisit and analyze the variable gravity modelWetterich:2013jsa (); Wetterich:2013aca () to be used for our investigations. The scenario of variable gravity is characterized by the following action in the Jordan-frame, :

(1)

where tildes represent the quantities in the Jordan frame. In the above action is the cosmon field with , and apart from the coupling we have considered an effective Planck mass driven by the field. Additionally, and are the matter and radiation actions respectively and is the action for neutrino matter, which we have considered separately since massive neutrinos play an important role in this model during late times. During the radiation era or earlier, neutrinos are ultra relativistic or relativistic, which implies that neutrinos behave as radiation during and before radiation era, with their mass being constant. After the radiation era neutrinos start losing their energy and become non-relativistic, behaving like ordinary matter with zero pressure. During late times neutrino mass starts growing with the field, and along with the cosmon field they give rise to late-time de Sitter solution Fardon:2003eh (); Bi:2003yr (); Hung:2003jb (); Peccei:2004sz (); Bi:2004ns (); Brookfield:2005td (); Brookfield:2005bz (); Amendola:2007yx (); Bjaelde:2007ki (); Afshordi:2005ym (); Wetterich:2007kr (); Mota:2008nj (); Pettorino:2010bv (); LaVacca:2012ir (); Collodel:2012bp ().

In this construction the variation in the particles masses comes from the non-minimal coupling of the field with matter. For radiation this non-minimal coupling does not affect its continuity equation, since the energy-momentum tensor for radiation is traceless. We consider different couplings of the field with matter, radiation, and neutrinos, that is we consider the non-minimal coupling between the cosmon field and matter, and the non-minimal coupling between the cosmon field and the neutrinos. Without loss of generality we consider the non-minimal coupling between the field and radiation to be too. To sum up, we shall use the following actions,

(2)
(3)
(4)

Variation of the action (1) with respect to the metric leads to the Einstein field equation

(5)

where includes the contributions from matter, radiation and neutrinos, that is .

Variation of action (1) with respect to the cosmon field provides its equation of motion of the field, namely

(6)

where and

(7)
(8)
(9)

Here the primes represent derivatives with respect to and the energy-momentum tensors are defined as

(10)

One can easily see that Wetterich:2013jsa ()

(11)
(12)

where and are the masses of matter-particles and neutrino and and are the number densities of the matter-particles and the neutrinos respectively. In the above expression we have followed Wetterich:2013jsa () and for convenience we have defined through

(13)

Comparing (11) and (12) with (7) and (8) respectively, we can see that and . Thus, choosing suitable and we can match our considerations with those of Wetterich:2013jsa (). In particular, according to Wetterich:2013jsa () particles masses vary linearly with the cosmon field, apart from the neutrinos. That is , which leads to . Neutrino mass varies slightly differently than the other particles. In particular, , which gives , with a constant.

We shall consider four matter components in the universe,namely, radiation, baryonic+cold dark matter (CDM), neutrinos and the contribution of the cosmon field. Furthermore, we stress that the late-time dark energy is attributed to two contributions, namely to both the cosmon and the neutrino fields. Thus, the total energy-momentum tensor, which can be calculated from action (1), reads

(14)

where

(15)

with the present value of 111Eq. (15) is calculated by writing Eq. (5) as the standard one, that is where gives the present value of the Newton’s constant. .

The evolution equations of the various sectors in the model at hand read:

(16)
(17)
(18)

which follow from the equations

(19)
(20)
(21)

From Eqs. (16), (17) and (18) we can extract the continuity equation for the cosmon field, which writes as

(22)

Finally, the consistency check of the Eqs. (16), (17), (18) and (22) follows from the conservation equation of the total energy :

(23)

We close this section by mentioning that, although the above model looks similar to extended quintessence Chiba:1999wt (); Uzan:1999ch (); Baccigalupi:2000je (); Amendola:1999er (), or as a special case of the generalized Galileon models , there is a crucial difference, namely that the particle masses depend on , that is the matter energy density and pressure depend on too. This has an important phenomenological consequence the appearance of an effective interaction between the scalar field and matter and neutrinos, described by relations (16), (17) and (22). In the discussion to follow, it would be convenient to work in the Einstein frame.

Iii Variable Gravity in Einstein Frame

In this section we examine the variable gravity model in the Einstein frame and analyze the aspects related to early phase,thermal history and late time evolution. Let us consider the following conformal transformation,

(24)

where is the conformal factor and is the Einstein-frame metric.

Using the conformal transformation (24) one can easily show that

(25)
(26)

Therefore, under the conformal transformation (24) the Jordan-frame action (1) becomes

(27)

where,

(28)
(29)

In this work following Wetterich:2013jsa (), we consider the choice,

(30)
(31)

where and are constants (the tilde in has nothing to do with the frame choice). The parameter is an intrinsic mass scale which plays a crucial role in inflation, when , but can be neglected during and after radiation era when grows to a higher value such that . Hence, for the late time behavior of the model we can use the approximation , which gives approximately a constant :

(32)

One can easily see that in the Einstein frame, neutrino matter is non-minimally coupled to cosmon field whereas matter and radiation are minimally coupled. Indeed, we have

(33)
(34)
(35)

Thus, from (33), (34) and (35) we deduce that only the neutrino mass is field-dependent in the Einstein frame, while the other particles masses remain constant as it should be Wetterich:2013jsa (); Wetterich:2013aca (), that is

(36)
(37)
(38)

Eqs. (36) and (37) imply that and , as usual. However, interestingly enough the neutrino behavior changes from era to era. During radiation epoch or earlier, neutrinos behave as radiation, that is the r.h.s. of Eq. (38) becomes zero and thus . On the other hand, after the radiation epoch neutrinos start becoming non-relativistic and behaving like non-relativistic matter, that is during and after matter era. However, note that Eq. (38) implies that the neutrino mass depends on the field when the r.h.s. of Eq. (38) is non zero, and therefore we deduce that during or after the matter era, the neutrino density does not evolve as .

In order to proceed further, we consider a quadratic potential in the Einstein frame Wetterich:2013jsa (); Wetterich:2013aca ()

(39)

It proves convenient to redefine the field in terms of a new field as

(40)

In this case action (27) becomes

(41)

with

(42)
(43)

where we have defined , which according to Wetterich:2013jsa () . Let us note that the action (41)is a particular case of Horndeski class with higher derivative terms absent and the coefficient of kinetic term having dependence on the field alone. Secondly, the system is free of ghosts as is positive definite in our choice.

Variation of the action (41) with respect to the metric gives

(44)

where

(45)

and

(46)

Moreover, variation of (41) with respect to the re-scaled cosmon field gives its equation of motion, namely

(47)

Finally, note that in terms of the field Eq. (38) becomes,

(48)

which can then be re-expressed in terms of the neutrino mass as Brookfield:2005td (); Brookfield:2005bz ()

(49)

Thus, comparing Eq. (48) and Eq. (49) we deduce that

(50)

where . Since at the present time we can write,

(51)

where is the redshift and is the present value of the neutrino mass.

Finally, from the r.h.s. of Eq. (47) we can define the effective potential

(52)

where and are independent of . This effective potential has a minimum at

(53)

which is the key feature in the scenario under consideration. By setting the model parameters, it is possible to achieve minimum at late times such that the field rolls slowly around the minimum of the effective potential. The role of neutrino matter is solely related the transition to stable de Sitter around the present epoch. Fig. 1 shows the nature of the effective potential (52) and the inset shows the minimum of the effective potential.

Figure 1: Effective potential (52) is plotted against the non-canonical field . are the chosen values of different parameters. Here we should note that if we change the values of the parameters and then the nature of the effective potential does not change and only the position of the minimum shifts along the horizontal axis. The inset shows the minimum of the effective potential.

Using Eq. (53) we get the minimum value of the effective potential (52) for ,

(54)

where .

Eq. (54) can be represented in terms of the neutrino mass by using Eq. (50) and Eq. (51),

(55)

where .

Now Wetterich:2013jsa () where is the present value of the Hubble parameter. So in Eq. (55) the term . If we take then we have in eV, . So will be of the order of only when,

(56)

and since to get late time cosmic acceleration field has to settle down at the minimum of the effective potential during the present time we can safely take , which implies .

iii.1 Canonical form of the action

Let us now transform the scalar-field part of the action (41) to its canonical form through the transformation

(57)
(58)

where is given by (42). Thus, (41) becomes

(59)

where is the conformal coupling in the Einstein frame between the canonical field and neutrinos. As we can see, the scalar field has now the canonical kinetic term.

The dependence of the canonical field can be calculated from Eq. (58), and writes as

(60)

where is an integration constant. We consider 222The choice of also gives as , similar to the field. Therefore, the value of we are getting here can also be obtained from Eq. (60) by putting and considering ., which gives

(61)

Additionally, if we consider very small (according to Wetterich:2013jsa () ) and large comparing to and Wetterich:2013jsa (), we can approximate it as

(62)

Finally, note that in order to write the explicit form of in (59), we need to invert (60) in order to obtain the explicit form of , and then substitute into in (46). However, (60) is a transcendental equation and thus it cannot be inverted. Fortunately, in the following elaboration will appear only through its derivative , which using (57), (58) acquires the simple form

(63)

In order to check whether the behavior of the field can comply with requirements spelled out in the aforesaid discussion, it would be convenient to check for the asymptotic behavior of the potential.

iii.2 Asymptotic behavior

In the previous subsection we extracted the expressions for , and , where is the redefined scalar field, in terms of which the action takes the canonical form. Since the involved expressions are quite complicated, it would be useful to obtain their asymptotic approximations. In particular, we are interested in the two limiting regimes, that is for small ( or equivalently ) and large ( or equivalently ) respectively.

For small from (31),(42) we have

(64)

and then Eq. (60) gives

(65)

Although, as we discussed in the end of the previous subsection, the explicit form of cannot be obtained, since it requires the inversion of the transcendental equation (60) of , its asymptotic form can be easily extracted, since now takes the simple form (65) which can be trivially inverted. In particular, for small the potential becomes

(66)

which for small slope can facilitate slow roll which can continue for large values of . Similarly, for very large values of (), Eqs. (31),(42) lead to

(67)

and then Eq. (60) gives

(68)

Thus, for large the potential reads

(69)

which gives rise to scaling solution for , we shall take to satisfy the nucleosynthesis constraint.

From the above asymptotic expressions, we deduce that the behavior of the canonical field with respect to the non-canonical field , changes from a straight line with slope (for small ) to a straight line with slope 1, and axis intercepts at (for large ). This behavior is always true as long as and . In Fig. 2 we present the change in -field behavior, in terms of the -field.

Figure 2: Blue (solid) line represents the behavior of field (Eq. (60)). Red (dotted) line represents the Eq. (65) and the Green (dashed) line represents the Eq. (68). The figure clearly shows the transition of field from Eq. (65) to Eq. (68). To plot this figure we have taken , and . If one changes the value of and maintaining then only the transition point changes but the behavior remains the same. This plot can be extrapolated for small and large values of field and nature remains the same. If we take values then also the nature remains the same but the slopes of the straight lines get changed.

We next investigate the dynamics of unification in detail which includes inflationary phase,thermal history and late time cosmic acceleration. We shall also examine the issues related to relic gravity waves, a generic feature of the scenario under consideration. To this effect, we shall invoke the instant preheating to circumvent the excessive production of gravity waves.

Iv Inflation

Having presented the scenario of variable gravity Wetterich:2013jsa () in the Jordan and Einstein frames, in this section we proceed to a detailed investigation of the inflationary stage. As discussed earlier, at early times or equivalently for negative values/small positive values of the field, the potential given by Eq. (46) reduces to the canonical potential of (66), which facilitates slow roll for small values of , where consistency with observations demands that . On the other hand, for very large values of , where and the potential is given by (69), we obtain the required scaling behavior in radiation and matter era, for .

The -field slow roll parameters, can be easily expressed in terms of as

(70)
(71)

where we have made use of (63). Since , the slow-roll regime lasts for large values of (since ) such that , and thereafter the field crosses to the kinetic regime where .

Clearly, the large-field slow-roll regime is of great physical interest. In this case the slow-roll parameters are simplified to

(72)

and the kinetic function is given by

(73)

We mention that interpolates between and , as the field evolves from early epochs to late times. At the end of inflation , and then it quickly relaxes to marking the beginning of the kinetic regime. This transition takes place very fast, since the kinetic function decreases exponentially with the field.

It is convenient to express the physical quantities in terms of the non-canonical field too. It is then straightforward to write down the Friedman equation in slow-roll regime as

(74)

which we shall use in the following discussion.

The Number of efoldings are given by,

(75)

Where is the value of field at the end of inflation. Now from Eq. (72) we have which approximates the Eq. (75) by neglecting term with respect to ,

(76)

Fo given efoldings from Eq. (76) we can calculate the value of when inflation started.

The number of e-foldings in the large- approximation are given by

(77)

where designates the field-value where inflation commences. The COBE normalized value of density perturbations Bunn:1996py (); Bunn:1996da ()