The Renormalizable Three-Term Polynomial Inflation with Large Tensor-to-Scalar Ratio

The Renormalizable Three-Term Polynomial Inflation with Large Tensor-to-Scalar Ratio

Abstract

We systematically study the renormalizable three-term polynomial inflation in the supersymmetric and non-supersymmetric models. The supersymmetric inflaton potentials can be realized in supergravity theory, and only have two independent parameters. We show that the general renormalizable supergravity model is equivalent to one kind of our supersymmetric models. We find that the spectral index and tensor-to-scalar ratio can be consistent with the Planck and BICEP2 results, but the running of spectral index is always out of the range. If we do not consider the BICEP2 experiment, these inflationary models can be highly consistent with the Planck observations and saturate its upper bound on the tensor-to-scalar ratio (). Thus, our models can be tested at the future Planck and QUBIC experiments.

pacs:
98.80.Cq, 98.80.Es

I Introduction

It is well-known that the standard big bang cosmology has some problems, for instance, the flatness, horizon, and monopole problems, etc, which can be solved naturally by inflation (1); (2); (3); (4). Also, the observed temperature fluctuations in the cosmic microwave background radiation (CMB) strongly suggests an accelerated expansion at a very early stage of our Universe evolution, i.e., inflation. Moreover, the inflationary models predict the cosmological perturbations in the matter density and spatial curvature from the vacuum fluctuations of the inflaton, which can explain the primordial power spectrum elegantly. Besides the scalar perturbation, the tensor perturbation is produced as well, which has special features in the B-mode of the CMB polarization data as a signature of the primordial inflation.

The Planck satellite measured the CMB temperature anisotropy with an unprecedented accuracy. From its first-year observational data (5) in combination with the nine years of Wilkinson Microwave Anisotropy Probe (WMAP) polarization low-multipole likelihood data (6) and the high-multipole spectra data from the Atacama Cosmology Telescope (ACT) (7) and the South Pole Telescope (SPT) (8) (Planck+WP+highL), the scalar spectral index , the running of the scalar spectral index , the tensor-to-scalar ratio , and the scalar amplitude for the power spectrum of the curvature perturbations are respectively constrained to be (9); (10)

(1)

As given by the Planck Collaboration, we also quote 68% errors on the measured parameters and 95% upper limits on the other parameters.

Recently, the BICEP2 experiment announced the discovery of the gravitational waves or primordial tensor perturbations in the B-mode power spectrum around  (11). If confirmed by future experiments, it will definitely be a huge progress in fundamental physics. The measured tensor-to-scalar ratio is

(2)

Subtracting the various dust models and re-deriving the constraint still results in high significance of detection, we have

(3)

Thus, the BICEP2 results are in tension with the Planck results. To be consistent with both experiments, one can consider the running of the spectral index. With it, we have the following results from the Planck+WP+highL data (9)

(4)

And the combined Planck+WP+highL+BICEP2 data give

(5)

Therefore, we must at least require the running of the spectral index to be smaller than 0.0004 at 3 level for any viable inflationary model. However, there might exist the foreground subtleties in the BICEP2 experiment such as dust effects, etc. As we know, the recent observations from the Planck and BICEP2/Keck Array Collaborations provided strong constraints on the primordial tensor fluctuations (12); (13); (14), ( from BICEP2/Keck Array) at 95% Confidence Level (C.L.). Because these results were announced seven months after we submitted our paper to arXiv, we will not consider them here.

Obviously, such a large tensor-to-scalar ratio from the BICEP2 measurement does impose a strong constraint on the inflationary models. For example, most inflationary models from string theory predict small far below and then contradict with the BICEP2 results (15). With or 0.20, we obtain that the Hubble scale during inflation is about  GeV, and the inflaton potential is around the Grand Unified Theory (GUT) scale  GeV which might have some connections with GUTs. From the naive analysis of Lyth bound (16), we will have large field inflation, and then the effective field theory might not be valid since the high-dimensional operators are suppressed by the reduced Planck scale. The inflationary models, which can realize and , have been studied extensively (17); (18); (19); (20); (21); (22); (23); (24); (25); (26); (27); (28); (29); (30); (31); (32); (33); (34); (35); (36); (37); (38); (39); (40); (41); (42); (43); (44); (45); (46); (47); (48); (49); (50). Especially, the simple chaotic and natural inflation models are favoured.

From the particle physics point of view, supersymmetry is the most promising new physics beyond the Standard Model (SM). Especially, it can stabilize the scalar masses, and has a non-renormalized superpotential. Moreover, gravity is very important in the early Universe. Thus, a natural framework for inflationary model building is supergravity theory (51). However, supersymmetry breaking scalar masses in a generic supergravity theory are of the same order as gravitino mass, giving rise to the so-called problem (52), where all the scalar masses are at the order of the Hubble scale due to the large vacuum energy density during inflation (53). Two elegant solutions were proposed: no-scale supergravity (54); (55); (56); (57); (58); (59); (60), and shift symmetry in the Kähler potential (61); (62); (63); (64); (65); (66); (67); (68); (69); (70).

The Planck satellite experiment might measure the tensor-to-scalar ratio down to 0.03-0.05 in one or two years. And the target of future QUBIC experiment is to constrain the tensor-to-scalar ratio of 0.01 at the 90% Confidence Level (C.L.) with one year of data taking from the Concordia Station at C, Antarctica (71). Thus, even if the BICEP2 results on tensor-to-scalar ratio were too large, as long as is not smaller than 0.01, for example, or , how to construct the inflationary models which highly agree with the Planck results and have large tensor-to-scalar ratio is still a very important question since these models can be tested in the near future.

The simple inflationary models have one parameter, for example, the monomial inflaton potentials. So the next to the simple inflationary models have two parameters. In the supergravity models with two parameters, we will generically have three terms due to the square of the F-term. In particular, we show that the general renormalizable supergravity model is equivalent to one kind of our supersymmetric models. Thus, in this paper, we will classify the renormalizable three-term polynomial inflationary models for both supersymmetric and non-supersymmetric models. The supersymmetric inflaton potentials can be obtained from supergravity theory. We find that their spectral indices and tensor-to-scalar ratios can be consistent with the Planck and BICEP2 experiments. However, is always out of the range. In addition, even if we do not consider the BICEP2 results, we find that the three-term polynomial inflationary models can be consistent with the Planck observations. Especially, the tensor-to-scalar ratio can not only be larger than in the region, above the well-known Lyth bound (16), but also saturate the Planck upper bound in the region. Thus, these models produce the typical large field inflation, and can be tested at the future Planck and QUBIC experiments.

This paper is organized as follows. In Section II, we briefly review the slow-roll inflation. In Section III, we construct the supersymmetric models from the supergravity theory. In Section IV, we systematically study the three-term polynomial inflation. Our conclusion is given in Section V.

Ii Brief Review of Slow-Roll Inflation

In the inflation, the slow-roll parameters are defined as

(6)
(7)
(8)

where is the reduced Planck scale, , , and . Also, the scalar power spectrum in the single field inflation is

(9)

where the subscript “*” means the value at the horizon crossing, the scalar amplitude is

(10)

and the scalar spectral index as well as its running at the second order are (72); (73)

(11)
(12)

where with the Euler-Mascheroni constant. Moreover, the tensor power spectrum is

(13)

where the tensor spectral index is (72); (73)

(14)

Thus, the tensor-to-scalar ratio is given by (72); (73)

(15)

Because , we can safely neglect the term at the next leading order in the above equation. Thus, we will take the next leading order approximation for simplicity. Therefore, with the BICEP2 result , we obtain the inflation scale about GeV and the Hubble scale around GeV.

The number of e-folding before the end of inflation is

(16)

where the value of the inflaton at the beginning of the inflation is the value at the horizon crossing, and the value of the inflaton at the end of inflation is defined by either or . From the above equation, we get the Lyth bound (16)

(17)

where is the minimal during inflation. If is a monotonic function of , we have . Thus, for , 0.05, 0.1, 0.16, and 0.21, we obtain the large field inflation due to , , , , and for , respectively. Moreover, to violate the Lyth bound and have the magnitude of smaller than the reduced Planck scale during inflation, we require that be not a monotonic function and have a minimum between and .

In this paper, we will consider the renormalizable three-term polynomial inflation with large tensor-to-scalar ratio. With slow-roll condition, each term in the polynomial potential will be around or smaller. However, without slow-roll condition and with fine-tuning, each term could be much larger than and there exist large cancellations among three terms. Thus, the quantum corrections can be very large and then out of control during large field inflation.

Iii Supergravity Model Building

In this paper, to simplify the discussions, we take . In the non-supersymmetric inflationary models, we will consider the following polynomial potentials at the renormalizable level

(18)

where is the inflaton, and are couplings. In the supersymmetric inflationary models from the supergravity theory, there are some relations among . Before we construct the concrete models, let us briefly review the supergravity model building.

In the supergravity theory with a Kähler potential and a superpotential , the scalar potential is

(19)

where is the inverse of the Kähler metric , and . Moreover, the kinetic term for a scalar field is

(20)

We first briefly review the generic model building. Introducing two superfields and , we consider the Kähler potential and superpotential as below

(21)
(22)

Thus, the above Kähler potential is invariant under the following shift symmetry (61); (62); (63); (64); (65); (66); (67); (68); (69); (70)

(23)

with a dimensionless real parameter. In general, the Kähler potential is a function of and independent on the real part of . Before further discussions, we shall present a few comments on the Kähler potential and superpotential

  • If shift symmetry is a global symmetry, it will be violated by quantum gravity effects, i.e., one might add high-dimensional operators suppressed by the reduced Planck scale. To solve this problem, one can consider gauged discrete symmetry from anomalous gauge symmetry inspired from string models, and then quantum gravity violating effects can be forbidden.

  • Shift symmetry is violated by the superpotential in Eq. (25). In principle, we can break the shift symmetry spontaneuously by introducing a spurion field and extending the shift symmetry as follows (74)

    (24)

    And we consider the following superpotential

    (25)

    which is clearly invariant under the extended shift symmetry. After obtains a non-zero vacuum expectation values, we obtain the superpotential in Eq. (25). The effects from spontaneous shift symmetry breaking have been studied in Ref. (75).

  • In a supersymmetric theory, the superpotential is non-renormalized, while there indeed exist quantum corrections to the Kähler potential in general. In the renormalizable three-term polynomial inflation which we shall study in the following, the inflaton value is about , and each term in the scalar potential is about or smaller during inflation. The Kähler potential for in Eq. (21) is about , and the quantum corrections will be around from the naive dimensional annalyses with loop factor. Thus, such quantum corrections are under control and negligible.

    In addition, supersymmetry is violated during inflation. Thus, the masses for the scalar and fermionic components of any superfield may be splitted. And then we might have additional one-loop effective scalar potential, which may affect the inflation and is beyond the scope of our current paper.

From the above Kähler potential and superpotential, the scalar potential is given by

(26)

Because there is no real component of in the Kähler potential due to the shift symmetry, this scalar potential along is very flat and then is a natural inflaton candidate. From the previous studies (65); (66); (70), we can stabilize the imaginary component of and at the origin during inflation, i.e., and . Therefore, with , we get the inflaton potential

(27)

For a renormalizable superpotential, we have

(28)

where we choose as real numbers. And then the polynomial inflaton potential is

(29)

The polynomial inflations from supergravity model building have been considered before. At the renormalizable level, only the case with and has been studied in the literatures (43); (67); (68). In this paper, we also consider the following three cases with : (1) and ; (2) and ; (3) The most general case with , , and . Moreover, we study the three-term polynomial inflations whose coefficients for the lowest and highest order terms in the inflaton potential can be negative. These inflations cannot be realized in supergravity model building where the coefficients for the lowest and highest order terms must be positive.

Iv The Renormalizable Three-Term Polynomial Inflation

To classify the three-term polynomial inflation at renormalizable level, we consider the following inflaton potential

(30)

where . With , we will study all the renormalizable non-supersymmetric and supersymmetric three-term polynomial inflation with large tensor-to-scalar ratio , which can be consistent with the Planck and/or BICEP2 experiments. For simplicity, we denote the maximum and minimum of the inflaton potential as and , respectively. Because we shall consider the super-Planckian inflation, our inflation around the maxima and minima of inflaton potentials is similar to the inflection point inflation (76); (77); (78); (79).

iv.1 Inflaton Potential with

First, we consider the non-supersymmetric models with the inflaton potential . For , there exists a maximum at . No matter the slow-roll inflation occurs at the right or left of this maximum (which is the same because of symmetry), we cannot find any within the range of the BICEP2 data. And the numerical results for versus is given in Fig. 1. When is within the range , the range of is .

Figure 1: versus for the inflaton potential with and . The inner and outer circles are and regions, respectively.

Moreover, for and , we have a minimum at . We present the numerical results for versus in Fig. 2, where the inner and outer circles are and regions, respectively. For in the range , the range of is , which can be consistent with the BICEP2 experiment. In addition, for the number of e-folding , and are within and regions of the BICEP2 experiment for and and for and , respectively. Also, for , and are within region for and , but no viable parameter space for region. In particular, the best fit point with and for the BICEP2 data can be obtained for , and . For example, , and , and the corresponding , and respectively are , and . Thus, we obtain , which satisfies the Lyth bound. In the following discussions, we will not comment on since the Lyth bound is always satisfied in our models.

Figure 2: versus for the inflaton potential with , , and .

Second, we consider the supersymmetric model with inflaton potential , which has a minimum at . We obtain that for , both and are equal to 1, and then the slow-roll inflation ends. Also, we find that no matter the slow-roll inflation occurs at the left or right of the minimum, and can be written as functions of the e-folding number

(31)

Thus, for , we get and . And for , we get and . In fact, this is similar to the chaotic inflation with inflaton potential .

iv.2 Inflaton Potential with

The inflaton potential is . First, we consider and . Because there is a minimum at and a maximum at , we have three inflationary trajectories, and let us discuss them one by one. When the slow-roll inflation occurs at the left of the minimum, the numerical results for versus is given in Fig. 3. The range of is about for within its range , which is consistent with the BICEP2 results. In the viable parameter space, we generically have . For the number of e-folding , and are within and regions of the BICEP2 experiment for and , respectively. For the number of e-folding , and are within and regions of the BICEP2 experiment for and , respectively. To be concrete, we will present two best fit points for the BICEP2 data. The best fit point with and can be realized for , , for example, and , and the corresponding , and are respectively , and . Another best fit point with and can be obtained for , , for example, , and , and the corresponding , and are , and , respectively. In addition, when slow-roll inflation occurs at the right of the minimum, we also present the numerical results for versus in Fig. 3. The range of is about [0.0337, 0.0669] for within its range . Although we can not fit the BICEP2 data, we still have large enough tensor-to-scalar ratio, which can be tested at the future Planck and QUBIT experiments.

Figure 3: versus for the inflaton potential with , , and , where the inflationary trajectories are at the left and right of the minimum.

Furthermore, for the slow-roll inflation at the right of the maximum, the numerical results for versus is given in Fig. 4. For in the range , the range of is , which is within the reach of the future Planck and QUBIT experiments.

Figure 4: versus for the inflaton potential with , , and , where the inflationary trajectory is at the right of the maximum.

Second, we consider and , the potential will decrease monotonically, and the curves for versus are given in Fig. 5. The range of is about for within its range , which is consistent with the BICEP2 results. In the viable parameter space, we generically have . For the number of e-folding , and are within and regions of the BICEP2 experiment for and , respectively. For the number of e-folding , and are within and regions of the BICEP2 experiment for and , respectively. The best fit point with and can be realized for , and , for example, and , and the corresponding and are respectively and .

Figure 5: versus for the inflaton potential with , , and .

iv.3 Inflaton Potential with

Figure 6: versus for the inflaton potential with , , and for inflation at the left of the maximum.

We consider the non-supersymmetric inflation models with . First, we consider and . There is a maximum at . When slow-roll inflation occurs at the left and right of the maximum, we present the numerical results for versus in Figs. 6 and 7, respectively. For in the range , the corresponding ranges of are and , respectively, which is large enough to be tested at the future Planck and QUBIT experiments.

Figure 7: versus for the inflaton potential with , , and for inflation at the right of the maximum.

Second, we consider and . There exists a minimum at . If the slow-roll inflation occurs at the left of the minimum, we obtain and , which is not consistent with the Planck and BICEP2 data. When the slow-roll inflation occurs at the right of the minimum, the numerical results for versus is given in Fig. 8. With in the range , the range of is about , which agrees with the BICEP2 experiment. Moreover, for the number of e-folding , and are within and regions of the BICEP2 experiment for and , respectively. And for , and are within and regions of the BICEP2 experiment for and , respectively. To be concrete, we will present the best fit point for the BICEP2 data. The best fit point with and can be realized for , , and , for example, , and , and the corresponding , and are respectively , and .

Figure 8: versus for the inflaton potential with , , and .

iv.4 Inflaton Potential with

We consider the inflationary model with potential . First, for and , there exist a minimum at and a maximum at . So we have three inflationary trajectories, and let us discuss them one by one. When the slow-roll inflation occurs at the left of the minimum, we present the numerical results for versus in Fig. 9. For within its range , the range of is about , which agree with the BICEP2 results. For the number of e-folding , and are within and regions of the BICEP2 experiment for and and for , respectively. Also, for , and are within and regions of the BICEP2 experiment for . To be concrete, we will present two best fit points for the BICEP2 data. The best fit point with and can be realized for , and , for instance, , and , and the corresponding , and are respectively , and . Another best fit point with and can be obtained for , , and , for example, and , and the corresponding , and are respectively , and .

In addition, when slow-roll inflation occurs at the right of the minimum, the numerical results for versus are given in Fig. 9 as well. The range of is about for within its range . In the viable parameter space, we have in general. For the number of e-folding , and are within and regions of the BICEP2 experiment for and , respectively. And for the number of e-folding , and are out of the region of the BICEP2 experiment and are within region for . The best fit point with and for the BICEP2 data can be realized for , , and , for instance, , and , and the corresponding , and are respectively , and .

Figure 9: versus for the inflaton potential with where the inflationary trajectories are at the left and right of the minimum.

Furthermore, for the slow-roll inflation at the right of the maximum, the numerical results for versus are given in Fig. 10. For in the range , the range of is , which can be tested at the future Planck and QUBIT experiments.

Figure 10: versus for the inflaton potential with where the inflationary trajectory is at the right of the maximum.

Second, for and , there exist a minimum at and a maximum at . Similar to the above discussions, there exist three inflationary trajectories, and we will discuss them one by one. When the slow-roll inflation occurs at the left of the minimum, we present the numerical results for versus in Fig. 11. For within its range , the range of is about , which can be consistent with the BICEP2 experiment. Generically, we have . For the number of e-folding , and are within and regions of the BICEP2 experiment for and , respectively. Also, for , and are within and regions of the BICEP2 experiment respectively for and . To be concrete, we will present two best fit points for the BICEP2 data. The best fit point with and can be realized for , , and , for instance, , and , and the corresponding , and are respectively , and .

Figure 11: versus for the inflaton potential with where the inflationary trajectory is at the left of the minimum.

In addition, when the slow-roll inflations occur at the right of the minimum and maximum, we present the numerical results for versus in Figs. 12 and 13. For within its range , the corresponding ranges of are respectively and , which are within the reach of the future Planck and BICEP2 experiments.

Figure 12: versus for the inflaton potential with where the inflationary trajectory is at the right of the minimum.
Figure 13: versus for the inflaton potential with where the inflationary trajectory is at the right of the maximum.

Third, for and , there exist a maximum at and a minimum at . Similarly, we have three inflationary trajectories, and will discuss them one by one as well. When the slow-roll inflations occur at the left and right of the maximum, we present the numerical results for versus in Figs. 14 and 15, respectively. For within its range , the corresponding ranges of are and , which can be tested at the future Planck and BICEP2 experiments.

Figure 14: versus for the inflaton potential with where the inflationary trajectory is at the left of the maximum.
Figure 15: versus for the inflaton potential with where the inflationary trajectory is at the right of the maximum.

Furthermore, for the slow-roll inflation at the right of the minimum, the numerical results for versus are given in Fig. 16. For in the range , the range of is , which can be consistent with the BICEP2 experiment. In general, we can take . For the number of e-folding , and are within and regions of the BICEP2 experiment for and respectively. Also, for , and are within and regions of the BICEP2 experiment respectively for and . The best fit point with and can be realized for , and , for instance, , and , and the corresponding , and are respectively , and .

Figure 16: versus for the inflaton potential with where the inflationary trajectory is at the right of the minimum.

iv.5 Inflaton Potential with

First, we consider the non-supersymmetric inflation models with potential . For simplicity, we only study the hill-top scenario with , , and . Thus, there is a maximum at . For the slow-roll inflation occurs at the left of the maximum with , to achieve a proper , we require to get a relatively large , and thus, the term dominates the potential. We present the numerical results for versus in Fig. 17. For in the range , the range of is , which can be consistent with the BICEP2 experiment. In the viable parameter space, we always have . Moreover, for the number of e-folding , and are within and regions of the BICEP2 experiment for and , respectively. Also, for , and are within region for , but no viable parameter space for region. The best fit point with and for the BICEP2 data can be obtained for , , and . For example, , , and , and the corresponding , , and are respectively , and .

Figure 17: versus for the non-supersymmetric inflaton potential with , , , and , where the inflationary trajectory is at the left of the maximum.

In addition, when slow-roll inflation occurs at the right of the maximum, i.e., , the numerical results for versus are given in Fig. 18. For within its range , the range of is about , which is within the reach of the future Planck and QUBIT experiments.

Figure 18: versus for the non-supersymmetric inflaton potential with , , , and , where the inflationary trajectory is at the right of the maximum.

Second, we consider the supersymmetric inflationary model with potential . For simplicity, we assume and . So the potential has two minima at . Without loss of generality, we only consider the positive branch of the filed . The inflationary process can occur at either the left or right of the minimum. When the slow-roll inflation occurs at the left of the minimum, i.e., , we present the numerical results for versus in Fig. 19. For in its range