Simple single field inflation models and the running of spectral index

# Simple single field inflation models and the running of spectral index

## Abstract

The BICEP2 experiment confirms the existence of primordial gravitational wave with the tensor-to-scalar ratio ruled out at level. The consistency of this large value of with the Planck data requires a large negative running of the scalar spectral index. Herein we propose two types of the single field inflation models with simple potentials to study the possibility of the consistency of the models with the BICEP2 and Planck observations. One type of model suggested herein is realized in the supergravity model building. These models fail to provide the needed even though both can fit the tensor-to-scalar ratio and spectral index.

Single field inflation; BICEP2 results; Supergravity model building
###### pacs:
98.80.Cq, 98.80.Es
1

## I Introduction

The observed temperature fluctuations in the cosmic microwave background radiation (CMB) strongly suggested that our Universe might experience an accelerated expansion, more precisely, inflation (1); (2); (3); (4), at a seminal stage of evolution. In addition to the solution to the problems in the standard big bang cosmology such as the flatness, horizon, and monopole problems, the inflation models predict the cosmological perturbations in the matter density from the inflaton vacuum fluctuations, which describes the primordial power spectrum consistently. Besides the scalar perturbation, the tensor perturbation is generated as well, which gives the B-mode polarisation as a signature of the primordial gravitational wave.

Recently, the BICEP2 experiment has discovered the primordial gravitational wave with the B-mode power spectrum around  (5). If it is confirmed, it will seemingly forward the study in fundamental physics. BICEP2 experiment (5) has measured the tensor-to-scalar ratio to be at the 68% confidence level for the lensed-CDM model, with disfavoured at level. Subtracting the various dust models and re-deriving the constraint still results in high significance of detection, it results in . From the first-year observations, the Planck temperature power spectrum (6) in combination with the nine years of Wilkinson Microwave Anisotropy Probe (WMAP) polarization low-multipole likelihood (7) and the high-multipole spectra from the Atacama Cosmology Telescope (ACT) (8) and the South Pole Telescope (SPT) (9) (Planck+WP+highL) constrained the tensor-to-scalar ratio to be at the 95% confidence level (10); (11). Therefore, the BICEP2 result is in disagreement with the Planck result. To reduce the inconsistency between Planck and BICEP2 experiments, we need to include the running of the spectral index . With the running of the spectral index, the 68% constraints from the Planck+WP+highL data are and with at the 95% confidence level. Thus, the running of the spectral index needs to be smaller than 0.008 at the 3 level for any viable inflation model.

Because different inflationary models predict different magnitudes for the tensor perturbations, such large tensor-to-scalar ratio from the BICEP2 measurement will give a strong constraint on the inflation models. Also, the inflaton potential is around the Grand Unified Theory (GUT) scale  GeV, and Hubble scale is about  GeV. From the naive analysis of Lyth bound (12), a large field inflation will be experienced, and then the validity of effective field theory will be challenged since the high-dimensional operators are suppressed by the reduced Planck scale. The inflation models, which can have and , have been studied extensively (13); (14); (15); (16); (17); (18); (19); (20); (21); (22); (23); (24); ?; ?; ?; (27); (28); (29); (30); (31); ?; ?; ?; (35); (36); (37); ?; ?; (40); (41); (42); (43). Specifically, the simple chaotic and natural inflation models are preferred.

Conversely, supersymmetry is the most promising extension for the particle physics Standard Model (SM). Specifically, it can stabilize the scalar masses, and has a non-renormalized superpotential. Also, gravity is critical in the early Universe, so it seems to us that supergravity theory is a natural framework for inflation model building (44); ?; (46); ?. However, supersymmetry breaking scalar masses in a generic supergravity theory are of the same order as the gravitino mass, inducing the reputed problem (48); ?; ?; ?; ?; ?; (54), where all the scalar masses are of the order of the Hubble parameter because of the large vacuum energy density during inflation (55). There are two elegant solutions: no-scale supergravity (56); ?; ?; ?; ?; (61); (62); (63); (64); (65); (66), and shift-symmetry in the Kähler potential (67); (68); (69); (70); (71); (72); (73); (74); (75); (76).

Thus, three issues need to be addressed regarding the criteria of the inflation model building:

Firstly (C-1), the spectral index is around 0.96, and the tensor-to-scalar ratio is around 0.16/0.20.

Secondly (C-2), to reconcile the Planck and BICEP2 results, we need to have . For simplicity, we do not consider the alternative approach here (21); (22).

Lastly (C-3), we need to violate the Lyth bound and try to realize the sub-Planckian inflation. For simplicity, we will not consider the alternative mechanisms such as two-field inflation models (77); (29), and the models which employ symmetries to control the quantum corrections like the axion monodromy (78).

It seemingly appears that (C-1) can be satisfied by a considerable amount of inflaton potentials, thus, this is not a difficulty to overcome. In this paper, we will propose two types of the simple single field inflation models, and show that their spectral indices and tensor-to-scalar ratios are highly consistent with both the Planck and BICEP2 experiments. We construct one type of inflation models from the supergravity theory with shift symmetry in the Kähler potential. However, in these simple inflation models, we will show that (C-2) and (C-3) can not be satisfied.

## Ii Slow-roll Inflation

The slow-roll parameters are defined as

 ϵ=M2plV2ϕ2V2, (1) η=M2plVϕϕV, (2) ξ2=M4plVϕVϕϕϕV2, (3)

where , , and . For the single field inflation, the scalar power spectrum is

 PR=As(kk∗)ns−1+n′sln(k/k∗)/2, (4)

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

 As≈124π2M4plΛ4ϵ, (5)

the scalar spectral index and the running are given (54); (79) by

 ns≈1+2η−6ϵ+2[13η2+(8C−1)ϵη−(53+12C)ϵ2−(C−13)ξ2], (6) n′s=16ϵη−24ϵ2−2ξ2, (7)

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

 PT=AT(kk∗)nt, (8)

the tensor spectral index and the tensor to scalar ratio (54); (79) are

 nt=−2ϵ[1+(4C+113)ϵ−2(23+C)η]≈−2ϵ, (9) r=ATAs=16ϵ[1+8(C+23)(2ϵ−η)]≈16ϵ. (10)

With the BICEP2 result , the energy scale of inflation is Gev and the slow roll parameter to the first order approximation. If , then the second order correction for the scalar spectral index in Eq. (6) is negligible, we have

 n′s=r(ns−1)2+3r232−2ξ2. (11)

It is clear that with the observational results. Therefore, to get , we need to consider large and the second order correction to the scalar spectral index in Eq. (6). If , then the main contribution to the running of the spectral comes from . For slow roll parameters and , we have and . Note that , we can neglect the term at the next leading order in Eqs. (9) and (10). Thus, we will take the leading order approximation and for simplicity.

The number of e-folds before the end of inflation is given by

 N(ϕ)=∫tetHdt≈1M2pl∫ϕϕeV(ϕ)Vϕ(ϕ)dϕ=1√2Mpl∫ϕϕedϕ√ϵ, (12)

where the value of the inflaton at the end of inflation is defined by . Now let us briefly consider the Lyth bound (12). From the above equation, we have

 Δϕ≡|ϕ∗−ϕe|>√2ϵminN(ϕ)Mpl , (13)

where is the minimal during inflation. If is a monotonous function of , we have . With the BICEP2 result , we can obtain the large field inflation because of . Thus, to violate the Lyth bound and have the magnitude of smaller than the reduced Planck scale during inflation, we require that is not a monotonous function and it has a minimum between and .

## Iii Single Field Inflation Models with Simple Potentials

### iii.1 Inflaton Potentials

Herein, we will describe one type of the single field inflation models with simple potentials. The inflation models with potential  (76) have been studied systematically previously, while such type of potentials may have the unlikeliness problem (80) unless both and are even. Conversely, for the S-dual inflation with the potential  (13), the slow-roll parameters are

 ϵ=M2pl2M2tanh2(ϕ/M), (14) η=4ϵ−M2plM2, (15) ξ2=24ϵ2−10(MplM)2ϵ. (16)

To satisfy slow-roll condition, must be large, then always and inflation will not end. Thus we need another mechanism to end inflation. For the S-dual inflation, we have

 r=16ϵ=8g−2tanh2(ϕ/M)≤8/g2, (17) ns=1+r8−2g2≤1−r8, (18) n′s=r4(1g2−r8)≥0. (19)

The number of e-folds before the end of inflation is given as

 Missing or unrecognized delimiter for \left (20)

If we take , and , we get , , and , which is marginally consistent with the observational constraint at the 95% confidence level.

Let us suggest one type of the inflation models with the hybrid monomial and S-dual potentials , here is an even integer. The slow-roll parameters are

 ϵ=12g2[nϕ/M−tanh(ϕ/M)]2, (21) η=1g2[n(n−1)(ϕ/M)2−2nϕ/Mtanh(ϕ/M)+ 2tanh2(ϕ/M)−1], (22) ξ2=1g4[6tanh4(ϕ/M)−12nϕ/Mtanh3(ϕ/M)+ (9n2−3n(ϕ/M)2−5)tanh2(ϕ/M)+ (8nϕ/M−4n3−6n2+2n(ϕ/M)3)tanh(ϕ/M)+ n2(n−1)(n−2)(ϕ/M)4−3n2(ϕ/M)2]. (23)

So the spectral index also satisfies the bound

 η=2ϵ+g−2[tanh2(ϕ/M)−n(ϕ/M)2−1]≤2ϵ, (24) ns=1+2η−3r8≤1−r8. (25)

For , and , we obtain , , , and . If we choose , and , we obtain , , , and . For , and , we get , , , and . The results for , and are shown in Figs. 1 and 2. We also show the observational constraints. From Figs. 1 and 2, it can be seen that these models are marginally consistent with the observational results at the 95% confidence level. For , if there is an increase of the value of or the value of the energy scale , then both and increase when is retained, then the results are almost unchanged when . For and , if there is an increase in and is fixed, then increases and moves closer to zero. At the 95% confidence level, we find that for , for and for .

### iii.2 Supergravity Model Building

For a given Kähler potential and a superpotential in the supergravity theory, we have the following scalar potential

 V=eK((K−1)i¯jDiWD¯j¯¯¯¯¯¯W−3|W|2) , (26)

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

 L = K¯ji∂μΦi∂μ¯Φ¯j . (27)

Introducing two superfields and , we consider the following Kähler potential and superpotential

 K=−12(Φ+¯Φ)2+X¯X−δ(X¯X)2 , (28)
 W = Xf(Φ) . (29)

Therefore, the Kähler potential is invariant under the following shift symmetry (67); (68); (69); (70); (71); (72); (73); (74); (75); (76) is thus

 Φ→Φ+iCMpl , (30)

with a dimensionless real parameter. In general, the Kähler potential is a function of and independent on the imaginary part of .

We can obtain the scalar potential as follows

 V = eK[|(Φ+¯Φ)Xf(Φ)+X∂f(Φ)∂Φ|2−3|Xf(Φ)|2 (31) +|(¯X−2δX¯X2)Xf(Φ)+f(Φ)|2] .

Because there is no imaginary component of in the Kähler potential because of the shift symmetry, the potential along is considerably flat and then is a natural inflaton candidate. From the previous studies (71); (72); (76), the real component of and can be stabilized at the origin during inflation, i.e., and . Therefore, with , we obtain the inflaton potential

 V = |f(ϕ/√2)|2 . (32)

If we chosse as below

 f(Φ) = √V0(√2Φ/M)me−iΦ/√2M+eiΦ/√2M , (33)

with a positive integer, we realize the potential with . The slow-roll parameters for this type of models are

 ϵ=12g2[nϕ/M−2tanh(ϕ/M)]2, (34) η=1g2[n(n−1)(ϕ/M)2−2sech2(ϕ/M)+4tanh2(ϕ/M) −4nϕ/Mtanh(ϕ/M)], (35) ξ2=1g4[nϕ/M−2tanh(ϕ/M)]×[n(2−3n+n2)(ϕ/M)3− 6n(n−1)(ϕ/M)2tanh(ϕ/M)+12nϕ/Mtanh2(ϕ/M)−8tanh3(ϕ/M)+ 2sech2(ϕ/M)(8tanh(ϕ/M)−3nϕ/M)]. (36)

Thus the spectral index also satisfies the bound

 η=2ϵ−g−2[2sech2(ϕ/M)+n(ϕ/M)2]≤2ϵ, (37) ns=1+2η−3r8≤1−r8. (38)

For , , and , we obtain , , , and . If we take , and , then we obtain , , , and . For , and , we obtain , , , and . The results for , and are shown in Figs. 3 and 4. We also show the observational constraints. From Figs. 3 and 4, it can be seen from the models that they are also marginally consistent with the observational results at the 95% confidence level. For , if is fixed and increase the value of or the value of the energy scale , then both and increase, with the result almost unchanged when . For and , if we increase and retain fixed, then increases and moves closer to zero. At the 95% confidence level, we find that for , for and for .

## Iv Discussion

Herein we have proposed one type of the single field inflation models with the hybrid monomial and S-dual potentials and found that given by the model is around when and are consistent with the BICEP2 constraints. If we increase the model parameter or , for the same value of , then the tensor-to-scalar ratio increases, but the running of the scalar spectral index moves closer to zero except for . Therefore, the model parameters are constrained by the observational results. At the 95% confidence level, we obtained that for , for and for .

Then we used the supergravity model building method to propose another type of models with the potentials . The behavior of this model is similar to the inflation model with the potential and the model is more constrained by the observational data. At the 95% confidence level, we found that for , for and for . The running of the scalar spectral index for both models is at the order of . Both models failed to provide the second order slow-roll parameter as large as the first order slow-roll parameters and . Furthermore, to obtain large , the inflaton will experience a Planck excursion because of the Lyth bound. This is the common problem for single field inflation as suggested by Gong  (17). To violate the Lyth bound and result in sub-Planckian inflaton field, the slow roll parameter needs not be retained by a monotonous function during inflation (43); (81). Recently Ben-Dayan and Brustein  (81) found that , and for a single field inflation with polynomial potential. It remains unclear if a single field inflation model can be constructed which both contain a large and .

###### Acknowledgements.
This research was supported in part by the Natural Science Foundation of China under grant numbers 11075194, 11135003, 11175270 and 11275246, the National Basic Research Program of China (973 Program) under grant number 2010CB833000 (TL), the Program for New Century Excellent Talents in University under grant No. NCET-12-0205 and the Fundamental Research Funds for the Central Universities under grant No. 2013YQ055.

### Footnotes

1. preprint: arXiv: 1404.7214

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