Reconstruction of inflationary potentials

The reconstruction of inflationary potentials

Jianmang Lin, Qing Gao, and Yungui Gong,
School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China
School of Physical Science and Technology, Southwest University, Chongqing 400715, China
563939360@qq.comgaoqing01good@163.comyggong@mail.hust.edu.cn
Abstract

The observational data on the anisotropy of the cosmic microwave background constraints the scalar spectral tilt and the tensor to scalar ratio which depend on the first and second derivatives of the inflaton potential. The information can be used to reconstruct the inflaton potential in the polynomial form up to some orders. However, for some classes of potentials, and behave as and universally in terms of the number of e-folds . The universal behaviour of can be used to reconstruct a class of inflaton potentials. By parametrizing one of the parameters , and , and fitting the parameters in the models to the observational data, we obtain the constraints on the parameters and reconstruct the classes of the inflationary models which include the chaotic inflation, T-model, hilltop inflation, s-dual inflation, natural inflation and inflation.

keywords:
cosmology: inflation.

1 Introduction

The quantum fluctuation during inflation seeds the large-scale structure and imprints the information of early Universe in the anisotropies of the cosmic microwave background (CMBR). For the slow-roll inflation with a single scalar field, the scalar (Mukhanov & Chibisov, 1981) and tensor (Starobinsky, 1979) power spectra can be parametrized by the slow-roll parameters (Stewart & Lyth, 1993). In particular, the scalar spectral tilt , the amplitude of the scalar spectrum and the tensor to scalar ratio can be constrained by the observational data on CMBR. For an inflationary model, we can calculate the slow-roll parameters and the observables and , and compare the results with the observations. However, there are lots of inflationary models with different potentials (Martin et al., 2014a) and it is not an easy task to compare all the models and constrain the model parameters in the models, although we may use Bayesian evidence to select best models (Martin et al., 2014b). Recently, the temperature and polarization measurements on the CMBR by the Planck survey gave the results (68% CL), (68% CL) and (95% CL) (Planck Collaboration I, 2015; Planck Collaboration XX, 2015). If we take the number of e-folds before the end of inflation at the horizon exit , then the measured scalar spectral tilt can be approximated as , and can be described as with . The small value of alleviates the problem imposed by the Lyth bound (Choudhury & Mazumdar, 2014b; Gao et al., 2014; Choudhury, 2015; Gao et al., 2015). Since the running of the scalar spectral index is a second-order effect, it is expected to be in the order of for slow-roll inflationary models (Gao & Gong, 2014). The running of the scalar spectral index is constrained to be (68% CL) (Planck Collaboration XX, 2015).

For chaotic inflation with the power-law potential (Linde, 1983), we get with being the number of e-folds before the end of inflation at the horizon exit. For the hilltop models with the potential (Boubekeur & Lyth, 2005), we get and . For the Starobinsky model (Starobinsky, 1980), we get and . For a class of inflationary models with non-minimal coupling to gravity (Kallosh et al., 2014), it was found that and . For natural inflation (Freese et al., 1990), we get and . Therefore, there are some universal behaviours for and which are consistent with the observations for a large class of inflationary models. The suggested simple relation between and tells us that it will be more easier to compare the models with the observational data if the slow-roll parameters and the observable can be expressed in terms of . By parametrizing the slow-roll parameter as , Mukhanov derived the corresponding inflaton potential (Mukhanov, 2013). For the simple inverse power-law form , Roset divided some inflationary models into two universal classes with the same behaviour for and different power-law behaviour for (Roest, 2014). More complicated forms of were also proposed by Garcia-Bellido and Roset (Garcia-Bellido & Roest, 2014). The parameters and in the parametrization were fitted to the observational data in (Barranco et al., 2014). The consequences of the two fixed points on and in the parametrization were discussed in (Boubekeur et al., 2015). In addition to the parametrization of , the reconstruction of the inflaton potential from the parametrization of with , and from the assumption between the amplitude of the power spectrum and were also considered (Chiba, 2015; Creminelli et al., 2015; Gobbetti et al., 2015). On the other hand, the reconstruction of the inflaton potential from the power spectra by the method of functional reconstruction were usually applied (Hodges & Blumenthal, 1990; Copeland et al., 1993; Liddle & Turner, 1994; Lidsey et al., 1997; Peiris & Easther, 2006; Norena et al., 2012; Choudhury & Mazumdar, 2014a; Ma & Wang, 2014; Myrzakulov et al., 2015; Choudhury, 2016).

In this paper, we discuss the reconstruction of the inflaton potential from the parametrizations of , and . The parameters in the models are fitted to the Planck temperature and polarization data. The paper is organized as follows. In the Section 2, we review the general relations between , , and by applying the slow-roll formula. The reconstruction of the inflaton potential from is presented in Section 3. The reconstruction of the inflaton potential from is presented in Section 4. The reconstruction of the inflaton potential from is presented in Section 5. The conclusions are drawn in Section 6.

2 General Relations

For the single slow-roll inflation, to the first order of approximation, we have

(1)

Since

(2)

so

(3)

If we parametrize the slow-roll parameter as a function of the number of e-folds before the end of inflation, then we can derive the parametrization and . Conversely, if we parametrize , we can solve equation (3) to get , and then .

From the energy conservation of the scalar field, we have

(4)

Since , , and

(5)

so

(6)

and the inflton always rolls down the potential during inflation. Substituting the above result into equation (3), we get (Chiba, 2015)

(7)

If we have any one of the functions , and , we can derive the other functions by using equations. (3), (6) and (7).

To derive the form of the potential , we need to find the functional relationship for the scalar field. Note that

(8)

where the sign depend on the sign of the first derivative of the potential and the scalar field is normalized by the Planck mass , so

(9)

Once one of the functions , , and is known, in principle we can derive , and the potential by using the relations (3), (6), (7) and (8).

Before we present particular parametrizations, we briefly discuss the effect of second-order corrections. To the second-order of approximation, we have (Stewart & Lyth, 1993; Schwarz et al., 2001)

(10)
(11)
(12)

where the Euler constant , and

(13)

The observational data requires , so and are at most in the order of , their derivatives with respect to have at most the order of , so is at most in the order of . Therefore, the running of the scalar spectral index and the second-order corrections will be in the order of , we may neglect the second-order corrections.

3 The parametrization of the spectral tilt

We approximate as

(14)

where the constants and are both positive, and the constant accounts for the contribution from the scalar field at the end of inflation. With this approximation, then is well behaved at the end of inflation when . Note that the functional form of the potential is not affected by . The observational results favour , so we consider only.

For , the solution to Equation (3) is

(15)

where is an integration constant. This is a generalization of the Mukhanov parametrization (Mukhanov, 2013). Since , so . If , then must be a very small number to ensure that , and its contribution is negligible so that we can take it to be zero. Therefore, we consider only. Since

(16)

so requires that .

Combining Equations (2) and (15), we get

(17)

For and , and , so the tensor to scalar ratio is small and only contributes to .

Either solving Equation (7) with the parametrization (14), or solving Equation (6) with the solution (15), we get (Chiba, 2015)

(18)

where is an integration constant.

Let us consider the special case first. For , we get

(19)

and

(20)

Therefore, both and contribute to the scalar spectral tilt and

(21)

Substituting Equation (19) into Equation (9), we get

(22)

or

(23)

where is an arbitrary integration constant, and . Since , so and this model corresponds to large field inflation. Combining Equations (23) and (18), we get the power-law potential for chaotic inflation (Linde, 1983)

(24)

For simplicity, we apply the Planck 2015 68% CL constraints on and to the relation (21) (Planck Collaboration XX, 2015), and we find that no satisfies the 68% CL constraints as shown in Fig. 1. At about the 99.8% CL level, and for , and for . From the above analysis, we see that the power-law potential is disfavoured by the observations at the 68% CL, and the filed excursion for the inflaton is super-Planckian. For , is marginally consistent with the observational constraint at the 99.8% CL, it is possible that the field excursion of the inflaton is sub-Planckian and the tensor to scalar ratio is close to zero.

Figure 1: The marginalized 68%, 95% and 99.8% CL contours for and from Planck 2015 data (Planck Collaboration XX, 2015) and the observational constraints on the parametrizations (15) and . The cyan regions are for the parametrization (15) with , the brown regions are for the parametrization (15) with , and the magenta areas are for the parametrization .

Next we consider the special case with . For this case, we get

(25)
(26)

where is an arbitrary integration constant and . Combining Equations (25) and (18), we get the corresponding T-model potential (Kallosh & Linde, 2013),

(27)

where . If or , then the potential reduces to the quadratic potential. From the discussion on the power-law potential, we know that the inflaton is a large field and it is disfavoured by the observations at the 68% CL level. If or , then the potential becomes

(28)

The potential includes the models with -attractors (Kallosh et al., 2013) and the Starobinsky model (Starobinsky, 1980) when or . By fitting Equations (14) and (15) to the Planck 2015 data, for , we get and at the 99.8% CL, for , we get and at the 99.8% CL. The results are shown in Fig. 1. Here we extend the integration constant to the region of , and we verify the conclusion that is very small if as discussed above. The above results also tell us that this model can be either small field inflation or large field inflation depending on the value of . If , then and is small. If is close to , then is close to zero, and is large.

For the general case with and , we get

(29)

The analytical form of the potential is not apparent, so we analyse the asymptotic form of the potential. For , the potential will be the same as the case with , and it is the power-law potential. For , Equation (15) can be approximated as

(30)

and Equation (29) can be approximated as

(31)

and

(32)

Combining Equations (18) and (31), for , the potential is

(33)

If , then for small field , the potential reduces to the hilltop potential (Boubekeur & Lyth, 2005) with (Garcia-Bellido & Roest, 2014; Creminelli et al., 2015). If , then for large field , the potential reduces to the form with (Garcia-Bellido & Roest, 2014; Creminelli et al., 2015). Fitting the model with general and to the observational data (Planck Collaboration XX, 2015), we find the constraints on and for and the results are shown in Fig. 2. The results tell us that the model can accommodate both small and large field inflation. Because , so the parametrization (14) requires that . At the 99.8% CL, , so and if we take .

Figure 2: The marginalized 68%, 95% and 99.8% CL contours for and from Planck 2015 data (Planck Collaboration XX, 2015) and the observational constraints on the parametrization (15). The left-hand panel shows the contours and the right-hand panel shows the constraints on and for . The green, yellow and red regions correspond to 68%, 95% and 99.8% CLs, respectively.

4 The parametrization of

For the parametrization

(34)

we get

(35)

Note that for the parametrization in this section, we take for simplicity, , and the parameter so that we consider the contribution from , i.e., at the end of inflation, . When , we require . Substitute the parametrization (34) into Equation (6), we get

(36)

Using Equation (8), we get

(37)

and

(38)

Combining Equations (36) and (37), we get

(39)

where . For , the potential is

(40)

By fitting Equations (34) and (35) with to the Planck 2015 data (Planck Collaboration XX, 2015), we find that no and satisfies the 99.8% CL constraints.

For , the potential is

(41)

Fitting Equations (34) and (35) with to the Planck 2015 data (Planck Collaboration XX, 2015), we find that no and satisfies the 68% CL constraints, so the model is disfavoured at the 68% CL. The 95% and 99.8% CL constraints on and for are shown in Fig. 3. By using the 99.8% CL constraints, we find that , so the field excursion of the inflaton in this model is super-Planckian.

Figure 3: The marginalized 68%, 95% and 99.8% CL contours for and from Planck 2015 data (Planck Collaboration XX, 2015) and the observational constraints on the parametrizations (34) with . The left-hand panel shows the contours and the right-hand panel shows the constraints on and for . The yellow and red regions correspond to 95% and 99.8% CLs, respectively.

For the parametrization

(42)

we get

(43)

Substitute the parametrization (42) into Equation (6), we get

(44)

From Equation (8), we get

(45)

For , we get

(46)

and the potential

(47)

If , we recover the potential for the s-dual inflation (Anchordoqui et al., 2014). Fitting Equations (42) and (43) with to the Planck 2015 data (Planck Collaboration XX, 2015), we obtain the constraints on the parameters and for and the results are shown in Fig. 4. From Fig. 4, we see that the s-dual inflation is consistent with the observational data. By using the 99.8% CL constraints, we find that , so the model includes both the large field and small field inflation. If is close to zero, then is small.

Figure 4: The marginalized 68%, 95% and 99.8% CL contours for and from Planck 2015 data (Planck Collaboration XX, 2015) and the observational constraints on the parametrizations (42) with . The left-hand panel shows the contours and the right-hand panel shows the constraints on and for . The green, yellow and red regions correspond to 68%, 95% and 99.8% CLs, respectively. The solid line in the right-hand panel corresponds to the s-dual inflation.

For , we get the potential,

(48)

or

(49)

If we take and , then we recover the potential for natural inflation (Freese et al., 1990). Fitting Equations (42) and (43) with to the Planck 2015 data (Planck Collaboration XX, 2015), we obtain the constraints on the parameters and for and the results are shown in Fig. 5. From Fig. 5, we see that natural inflation is disfavoured at the 68% CL. By using the 99.8% CL constraints, we find that , so the model includes both the large field and small field inflation, and the small field inflation is achieved when is close to zero.

Figure 5: The marginalized 68%, 95% and 99.8% CL contours for and from Planck 2015 data (Planck Collaboration XX, 2015) and the observational constraints on the parametrizations (42) with . The left-hand panel shows the contours and the right-hand panel shows the constraints on and for . The green, yellow and red regions correspond to 68%, 95% and 99.8% CLs, respectively. The solid line in the right-hand panel corresponds to the natural inflation.

For the parametrization

(50)

we get

(51)
(52)

and

(53)

For , we get the potential

(54)

If , then the potential reduces to the hilltop potential with for small . If , then the potential reduces to the double well potential for small (Olive, 1990). Fitting Equations (50) and (51) with to the Planck 2015 data (Planck Collaboration XX, 2015), we obtain the constraints on the parameters and for and the results are shown in Fig. 6. From Fig. 6, we see that the double well potential is excluded by the observational data and the hilltop potential with is disfavoured at the 68% CL. By using the 99.8% CL constraints, we find that , so the model includes both the large field and small field inflation. If is close to zero, then is small.

Figure 6: The marginalized 68%, 95% and 99.8% CL contours for and from Planck 2015 data (Planck Collaboration XX, 2015) and the observational constraints on the parametrizations (50) with . The left-hand panel shows the contours and the right-hand panel shows the constraints on and for . The green, yellow and red regions correspond to 68%, 95% and 99.8% CLs, respectively. The black solid line in the right-hand panel corresponds to the hilltop potential with , and the magenta solid line corresponds to the double well potential.

For , we get the potential

(55)

Fitting Equations (50) and (51) with to the Planck 2015 data (Planck Collaboration XX, 2015), we obtain the constraints on the parameters and for and the results are shown in Fig. 7. By using the 99.8% CL constraints, we find that , so the model includes both the large field and small field inflation, and the small field inflation is achieved if is close to zero.

Figure 7: The marginalized 68%, 95% and 99.8% CL contours for and from Planck 2015 data (Planck Collaboration XX, 2015) and the observational constraints on the parametrizations (50) with . The left-hand panel shows the contours and the right-hand panel shows the constraints on and for . The green, yellow and red regions correspond to 68%, 95% and 99.8% CLs, respectively.

5 The parametrization of

Combining Equations (6) and (8), we get

(56)

Once the functional form is known, we can derive the potential form . Let us first consider the power-law parametrization

(57)

For , from Equation (56), we get the power-law potential,

(58)

where is and integration constant. From Equation (8), we get

(59)

so

(60)

and . The results are the same as those discussed in Section 3 with , and .

For , we derive the potential,

(61)

the scalar spectral tilt,

(62)

the tensor to scalar ratio,

(63)

and the parameters , and satisfy the relation . Note that Equations (62) and (63) include the special case which corresponds to the power-law potential. Fitting Equations (62) and (63) for the parametrization (57) to the Planck 2015 data (Planck Collaboration XX, 2015), we obtain the constraints on the parameters and for and the results are shown in Fig. 8. By using the 99.8% CL constraints, we find that . If is near zero, then it is large field inflation. If is near zero, then it is small field inflation. In the left-hand panel of Fig. 8, the lower bounds of the contours are set by .

Figure 8: The marginalized 68%, 95% and 99.8% CL contours for and from Planck 2015 data (Planck Collaboration XX, 2015) and the observational constraints on the parametrizations (57). The left-hand panel shows the contours and the right-hand panel shows the constraints on and for . The green, yellow and red regions correspond to 68%, 95% and 99.8% CLs, respectively.

Next we consider the parametrization . Combining Equations (1), (3), (8) and (56), we get

(64)
(65)
(66)
(67)

From the end of inflation condition , we get , so , and and depend on the parameter only. If is small, then and , and the model will behave like the model (14) with and small , so the two models will cover some common regions in the graph. The fitting results are shown in Fig. 1. In particular, for , the potential can be approximated as

(68)

Therefore, the inflation is also included in this model.

In the last we consider the exponential parametrization with . Following the same procedure as above, we obtain

(69)
(70)
(71)

The model parameters satisfy the relation , so and . Since , the parameter should be negative. From the constraint on , we can get the upper limit on , with this upper limit, we find that the model is not consistent with the observational data.

6 Conclusions

For the double well potential , the potential and the potential , the predicted and are not consistent with the Planck 2015 data. The power-law potential , the potential , the natural inflation and the hilltop potential with are disfavoured by the observational data at the 68% CL. At the 99.8% CL, we find that for the power-law potential if we take the number of e-folds before the end of inflation . For the power-law potential , the T-model potential which includes the -attractors and the Starobinsky model, the hilltop potential with and , the potential with and , and the potential , their spectral tilts have the universal behavior .

For the parametrization , we get and the corresponding potential which includes the s-dual inflation. For the parametrization , we get and the corresponding potential which includes the natural inflation. For the parametrization , we get the corresponding potential which includes the hilltop potential with and the double well potential. The tensor to scalar ratio for these models can easily be small due to the factor in . For the parametrization , the corresponding potential is , both and depend on only and the model has the universal behavior if is small. All these models can achieve both small and large field inflation.

Based on the slow-roll relations (3), (6), (7) and (8), by parameterizing one of the parameters , and , and fitting the parameters in the models to the observational data, we not only obtain the constraints on the parameters, but also easily reconstruct the classes of the inflationary models which include the chaotic inflation, T-model, hilltop inflation, s-dual inflation, natural inflation and inflation, and the reconstructed inflationary models are consistent with the observations. Since the observational data only probes a rather small intervals of scales, the reconstructed potentials approximate the inflationary potential only in the slow-roll regime for the observational scales . Outside the slow-roll regime, the inflationary potential can be rather different, but it does not mean that the reconstructed potential is not applicable in that regime. Once the potential is obtained, we can either apply the slow-roll formulae or work out the exact solutions.

acknowledgements

This research was supported in part by the Natural Science Foundation of China under Grants Nos. 11175270 and 11475065, and the Program for New Century Excellent Talents in University under Grant No. NCET-12-0205.

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