A new inflaton model beginning near the Planck epoch

Abstract

The Starobinsky model predicts a primordial inflation period without the presence of an inflaton field. The modified version of this model predicts a simple time dependence for the Hubble parameter H(t), which decreases slowly between the Planck epoch and the end of the inflation, H(t) = M Pl -betaM²Pl t, where beta is a dimensionless constant to be adjusted from observations. We investigate an inflaton model which has the same time dependence for H(t). A reverse engineered inflaton potential for the time dependence of H is derived. Normalization of the derived inflaton potential is determined by the condition that the observed density fluctuations, <FONT FACE=Symbol>dr</FONT>/<FONT FACE=Symbol>r »</FONT> 10-5, are created at ~ 60 e-folds before the end of inflation. The derived potential indicates an energy (mass) scale, Mend ~ 10(13) GeV, at the end of inflation. Using the slow roll parameters, which are obtained from this potential, we calculate the spectral index for the scalar modes nS and the relative amplitude of the tensor to scalar modes r. A tensor contribution, r ~ 0.13, and an approximately Harrison-Zeldovich density perturbation spectrum, nS ~ 0.95, are predicted.

Inflation model; Planck epoch; Inflaton model

REGULAR ARTICLES

A new inflaton model beginning near the Planck epoch

Reuven Opher; Ana Pelinson

Departamento de Astronomia, IAG, Universidade de São Paulo, 05508-900, São Paulo, SP, Brazil

ABSTRACT

The Starobinsky model predicts a primordial inflation period without the presence of an inflaton field. The modified version of this model predicts a simple time dependence for the Hubble parameter H(t), which decreases slowly between the Planck epoch and the end of the inflation, H(t) = M_Pl –bM²_Plt, where b is a dimensionless constant to be adjusted from observations. We investigate an inflaton model which has the same time dependence for H(t). A reverse engineered inflaton potential for the time dependence of H is derived. Normalization of the derived inflaton potential is determined by the condition that the observed density fluctuations, dr/r » 10^—5, are created at ~ 60 e-folds before the end of inflation. The derived potential indicates an energy (mass) scale, M_end ~ 10¹³ GeV, at the end of inflation. Using the slow roll parameters, which are obtained from this potential, we calculate the spectral index for the scalar modes n_S and the relative amplitude of the tensor to scalar modes r. A tensor contribution, r ~ 0.13, and an approximately Harrison-Zeldovich density perturbation spectrum, n_S ~ 0.95, are predicted.

Keywords: Inflation model; Planck epoch; Inflaton model

I. INTRODUCTION

The two major problems in cosmology are the origins of the primordial inflation period and the present "inflation" period of the universe. It is possible that both origins are linked. Primordial inflation could have been created by a non-zero vacuum energy. Subsequently, the vacuum energy could have decayed, creating the present period of acceleration. However, strong limits were recently placed on the possible decay of the vacuum energy into cold dark matter (CDM) or cosmic microwave background (CMB) photons [1].

The most popular model for the origin of primordial inflation remains the inflaton (scalar field) model. We investigate here an inflaton model based on the simple time dependence of the Hubble parameter, H(t) [Eq.(3)], that was predicted by the modified Starobinsky model [2],[3]. (See [4] for the original Starobinsky model.) The Starobinsky model suggests that quantum fluctuations created a non-zero vacuum energy that induced the primordial inflation period.

Instead of assuming an ad hoc inflaton potential, as in the standard inflation model, we use the reverse engineering method of Ellis, Murugan and Tsagas [5] to derive the inflaton potential from the H(t) of Eq.(3). The derived potential becomes negligible at the end of inflation, creating the observed density fluctuations, dr/r » 10^—5. These fluctuations are determined by the value of the potential and its first derivative at 60 e-folds before the end of inflation. This condition, together with the time dependence of the potential, determine a mass (energy) scale, M_end~ 10¹³GeV ~ 10^—6M_Pl, at the end of inflation. From the slow roll parameters obtained from the derived potential, we calculate the spectral index of the scalar modes n_S and the relative amplitude of the tensor to scalar modes r. The derived spectral index n_S is in agreement with the WMAP data [6, 7]. The ratio of tensor to scalar modes obtained, r ~ 0.13, is similar to that of most inflation models, which predict r ~ 10 - 30%.

We can compare our scale M_end at the end of inflation with the results of Vilenkin [8] and Starobinsky [9]. Vilenkin noted that, in the Starobinsky model, the Hubble parameter defines a mass (energy) scale with a limiting value, M_end 10¹⁶GeV, at the end of inflation. Starobinsky predicted that M_end 10¹⁴GeV by requiring that the dr/r, resulting from inflation, is sufficiently small. Our derived value, M_end ~ 10¹³GeV, is consistent with the upper limits of both Vilenkin and Starobinsky for M_end.

Although the potential that we obtain [Eq.(6)] is superficially similar to a standard inflation potential that depends on the square of the massive scalar field (see, for example, [10] for a recent review), our inflation model is very much different from the standard model for the following reasons:

1) The standard massive scalar inflation potential has two free parameters: the magnitude of the potential and its first derivative at ~ 60 e-folds before the end of inflation. However, our potential in Eq.(6) is completely determined by a single parameter b, which is derived from the simple time dependence of the Hubble parameter in Eq.(3);

2) In the standard inflation model, there are many possible forms that the massive scalar potential can take. However, the form of our potential, a quadratic dependence on the field, is determined uniquely by Eq.(3);

3) The origin of the potential in the standard model is completely unknown. Moreover, there is no clear justification for its form; and

4) In the standard model, the inflation period begins when there is a displacement of the massive scalar field from the minimum of its potential. The origin of this displacement is left unexplained and the epoch in which it occurs is not specified. However, in our model, the beginning of inflation is specified to occur at the Planck epoch (i.e., at the beginning of the universe). The origin of the inflation is a direct result of the simple time dependence of the Hubble parameter in Eq.(3). Moreover, there is no initial displacement of the field that is left explained.

We present the algorithm for constructing the potential from the time dependence of the Hubble parameter in § 2. In § 3, we use this algorithm to obtain the effective potential from the Hubble parameter, H(t) = M_Pl . From the potential, we calculate the spectral index of the dr/r and the intensity of primordial gravitational waves. The mass (energy) scale at the end of inflation, M_end ~ 10¹³GeV, is determined from the requirement that the potential creates observed dr/r ~ 10^-5 at ~ 60 e-folds before the end of inflation. Finally, our conclusions are presented in § 4.

II. THE FRAMEWORK OF THE SINGLE SCALAR MODEL

Let us assume that there exists an inflaton field, f = f(t), where t is the usual time function, in accordance with the Roberston-Walker symmetry [11]. The Lagrangian containing a minimally coupled scalar field is

where = df/dt. The scalar stress tensor takes the perfect fluid form,

with the following energy density and pressure of the scalar inflaton field:

The classical equation of motion for f(t), which follows from the variation of the action S = ò d⁴x

L, is

where H = (t). The field equations for the Robertson Walker model, with k = 0, are

Following Ellis, Murugan and Tsagas [5], we combine these two independent equations to obtain a more convenient set of equations,

From H(t), the above equations have been used to construct the effective potential in the following manner:

i) Eq.(9) is integrated to obtain f(t) ;

ii) t as a function of f is found;

iii) t(f) is substituted in H(t) to obtain H(f) ; and

iv) the potential V(f) is obtained, using Eq.(8).

III. THE EFFECTIVE INFLATON POTENTIAL

Assuming the simple Hubble parameter time dependence,

we solved Eq.(9) for f(t), obtaining t as a function of f,

where |f₀| > | f|. Choosing the positive sign in Eq.(11), we have -¥ < f < 0, as in [5]. From Eqs.(10) and (11),

Following the algorithm of the previous section to obtain V(f), we substitute Eq.(12) into Eq.(8) to obtain

or, in terms of the time,

A realistic potential V(f) describing inflation should:

1) become negligibly small at the end of the inflationary period, so that there is no important "cosmological constant" entering the FRW era; and

2) produce the density fluctuations at ~ 60 e-folds before the end of inflation (see e.g. [12]),

which are observed to be ~ 10^—5.

For as long as the first term in Eq.(10) dominates, we have the inflationary expansion a(t) = expM_Plt. The second term in Eq.(10) decreases the expansion rate and is important near the maximum value of s(t) = lna(t). Following Vilenkin [8], we characterize the end of inflation by

where H(t_end) = (t = t_end) = M_end and

is a dimensionless parameter (we should expect M_end < M_Pl ~ 10¹⁹ GeV).

The time as a function of µ at the end of inflation is

The number of e-folds of inflation before t_end is

We are interested in N ~ 60, the approximate time t₆₀, when the observed dr/r (scalar) and the primordial gravitational (tensor) fluctuations were created. Substituting Eq.(18) into Eq.(10), we find

where we have used the customary normalization for a(t = 0) = 1. From Eq.(19), we have

Using this result to solve Eq.(10) for t₆₀, we obtain

The slow roll parameters e and h in terms of H(f) are [12]

To first order, the slow roll parameters are related to the ratio r of the tensor to scalar fluctuations, by the relation

and to the spectral index of the scalar dr/r by

[13]. The value for µ that characterizes the end of inflation, is constrained by the condition that e = 1. From this condition and Eq.(12), we obtain

Substituting the time at 60 e-folds before the end of inflation from Eq.(22) and b from Eq.(27) into Eq.(15), we obtain

Using the above result, together with Eq.(17) and the observational evidence that the dr/r produced at ~ 60 e-folds before the end of inflation is ~ 10^—5, we obtain the predicted value of M_end, the mass (energy) scale at the end of inflation,

This value is less than the GUT scale (~ 10¹⁴ — 10¹⁶GeV), but is consistent with the upper limits for the mass (energy) scale at the end of inflation given by Vilenkin [8] and Starobinsky [9].

Evaluating the spectral index of the scalar dr/r from Eqs.(24) and (12), we observe that the parameter h is zero and that e is very small, e ~ 8.3 × 10^—3. From Eq.(26), we have n_S ~ 0.95, an approximately Harrison-Zeldovich spectrum n_S = 1, in agreement with the WMAP data [6,7]. These results do not depend on the exact value of f₀.

From e in Eq.(24) and Eq.(12), we obtain

This value is similar to those predicted by frequently discussed inflation models with r ~ 10% - 30% (e.g., [13]).

IV. CONCLUSIONS

We investigated a model in which the Hubble parameter is decreasing slowly in time, as predicted by the modified Starobinsky model [2],[3], H(t) = M_Pl , and constructed an inflaton potential for H(t). The derived potential, normalized at ~ 60 e-folds before the end of inflation, creates the observed level of dr/r ~ 10^—5 and indicates an energy (mass) scale, M_end ~ 10¹³ GeV, at the end of inflation.

This energy scale at the end of inflation can be compared with those predicted by Vilenkin and Starobinsky. Vilenkin gave a limiting value M_end 10¹⁶GeV for the scale at the end of inflation [8], while Starobinsky predicted M_end 10¹⁴GeV [9].

From the inflaton potential, we calculated the spectral index of the scalar modes. The result, n_S ~ 0.95, is compatible with the WMAP data. The potential also predicts a tensor contribution, r ~ 0.13, in accordance with most inflation models, which predict r ~ 0.10—0.30 and is in agreement with existing observational data.

Acknowledgments

R.O. thanks the Brazilian agencies FAPESP (00/06770-2) and CNPq (300414/82-0) for partial support. A.P. thanks FAPESP for financial support (03/04516-0 and 00/06770-2).

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Received on 18 January, 2006

Publication Dates

• Publication in this collection
  31 July 2006
• Date of issue
  June 2006