# Conformal GUT inflation, proton lifetime and non-thermal leptogenesis

## Abstract

In this paper, we generalize Coleman–Weinberg (CW) inflation in grand unified theories (GUTs) such as \(\text {SU}(5)\) and \(\text {SO}(10)\) by means of considering two complex singlet fields with conformal invariance. In this framework, inflation emerges from a spontaneously broken conformal symmetry. The GUT symmetry implies a potential with a CW form, as a consequence of radiative corrections. The conformal symmetry flattens the above VEV branch of the CW potential to a Starobinsky plateau. As a result, we obtain \(n_{s}\sim 1-\frac{2}{N}\) and \(r\sim \frac{12}{N^2}\) for \(N\sim \) 50–60 e-foldings. Furthermore, this framework allow us to estimate the proton lifetime as \(\tau _{p}\lesssim 10^{40}\) years, whose decay is mediated by the superheavy gauge bosons. Moreover, we implement a type I seesaw mechanism by weakly coupling the complex singlet, which carries two units of lepton number, to the three generations of singlet right handed neutrinos (RHNs). The spontaneous symmetry breaking of global lepton number amounts to the generation of neutrino masses. We also consider non-thermal leptogenesis in which the inflaton dominantly decays into heavy RHNs that sources the observed baryon asymmetry. We constrain the couplings of the inflaton field to the RHNs, which gives the reheating temperature as \(10^{6}\text { GeV}\lesssim T_{R}<10^{9}\) GeV.

## 1 Introduction

Primordial inflation is a successful paradigm for the description of the early Universe and it is strongly supported by the current observational data [1, 2, 3, 4, 5, 6]. Primordial perturbations, when the scales exiting the horizon \(\left( k\sim aH\right) \), are eventually responsible for the structure formation in the Universe. From *Planck* 2015 [6, 7], the key observables of inflation, namely, the scalar tilt and the ratio of tensor to scalar power spectra, are constrained as \(n_{s}=0.968\pm 0.006\), \(r<0.09\) at \(95\%\) confidence level. The CMB power spectra is observed to be nearly adiabatic, scale invariant and Gaussian [6, 8]. Although the physical nature of the inflaton is still uncertain [9, 10], the models based on *f*(*R*) or a canonical scalar field with a flat potential are favoured with respect to the data. Since the inflationary scale is in general expected to be \(\sim 10^{16}\,\text {GeV}\), it is natural consider the inflaton to be a scalar field associated with grand unified theory (GUT) groups, such as \(\text {SU}(5)\) and \(\text {SO}(10)\). The Shafi-Vilenkin (SV) model [11] is one of the first realistic model of inflation which was based on \(\text {SU}(5)\) GUT [12]. In this framework, inflation is a result of the spontaneous breaking of \(\text {SU}(5)\rightarrow \text {SU}(3)_{c}\times \text {SU}(2)_{L}\times \text {U}(1)_{Y}\) by a GUT field (**24-plet** adjoint Higgs) and an inflaton, which is a SU(5) singlet that rolls down to a vacuum expectation value (VEV). The success of the SV model is that it can lead to a successful baryogenesis after inflation and predicts a proton life time above the current lower bound [13, 14]. In this model, the scalar field potential is of a Coleman–Weinberg (CW) form, according to which primordial gravitational waves are constrained by \(0.02\le r\le 0.1\) [15]. Although the SV model is well within the current bounds of *Planck* 2015, several extensions of this model were studied to get smaller values of *r*. In [16, 17, 18, 19, 20], CW inflation was studied in the context of induced gravity, non-minimal coupling and brane-world scenario, where the tensor to scalar ratio was obtained to be \(r\sim \mathcal {O}\left( 10^{-2}\right) -\mathcal {O}\left( 10^{-3}\right) \). We note that all these modifications necessarily introduce an additional parameter whose value determines the shape of the inflaton potential in the Einstein frame.^{1}

Moreover, extensions of the SV model within particle physics offer rich physics beyond the Standard Model (SM). Therefore, the SV model is embedded in a higher gauge group such as \(\text {SO}\left( 10\right) \), which can be broken to the SM via an intermediate group \(\text {G}_{422}=\text {SU}(4)_{c}\times \text {SU}\left( 2\right) _{L}\times \text {SU}\left( 2\right) _{R}\) [23, 24]. Obtaining successful inflation in \(\text {SO}\left( 10\right) \) is more realistic with additional benefits to explain physics beyond SM, such as neutrino physics, matter anti-matter asymmetry through non-thermal leptogenesis, monopoles and dark matter (DM) [14]. For example, Ref. [25] considered a complex singlet scalar being coupled to right handed neutrinos (RHNs), followed by implementing type I seesaw mechanism. This approach unified inflation with Majorana DM together with the scheme of generating neutrino masses. In [26] an additional \(\text {U}(1)_{B-L}\) symmetry was considered in the SM i.e., \(\text {SU}(3)_{c}\times \text {SU}(2)_{L}\times \text {U}(1)_{Y}\times \text {U}\left( 1\right) _{B-L}\), where \(B-L\) symmetry can be spontaneously broken when the scalar field takes the VEV. In this setup, we can explain the baryon asymmetry of the Universe through non-thermal leptogenesis [24, 27, 28, 29]. Recently, CW inflation was studied in an extension with \(\text {SO}(10)\) and \(\text {E}_{6}\), pointing out the possibility of observing primordial monopoles [30].

*N*is the number of e-foldings before the end of inflation. There has been a growing interest on embedding these models in string theory and supergravity (SUGRA) aiming for a UV completion [33, 34]. Recently, a UV completion of the Starobinsky model was proposed in the context of non-local gravity inspired from string field theory [35, 36]. Starobinsky like models were also developed in \(\mathcal {N}=1\) SUGRA, namely, no scale [37] and \(\alpha \)-attractor models [38] where an additional physical parameter leads to any value of \(r<0.1\). In [39] \(\alpha -\)attractor models were studied in the non-slow-roll context where a new class of potentials were shown to give the same predictions. On the other hand, Higgs inflation is particularly interesting due to the fact that Higgs was the only scalar so far found at LHC. But for it to be an inflaton candidate compatible with CMB data, we require a very large non-minimal coupling \(\left( \xi \gg 1\right) \) to Ricci scalar. It was known that a scalar field with large non-minimal coupling gives rise to a \(R^{2}\) term considering 1-loop quantum corrections. Consequently, renormalization group (RG) analysis shows that Higgs inflation is less preferable compared to Starobinsky model [40, 41]. This result not only applies to Higgs inflation but also to any arbitrary scalar with very large non-minimal coupling. Furthermore, in both \(R^{2}\) and Higgs inflation the inflaton field rolls down to zero after inflation.

^{2}Differently, in GUT theories the inflaton field acquires a VEV due to its interaction with GUT fields.

The main goal of this paper is to generalize the SV model in order to achieve \(r\sim \mathcal {O}\left( 10^{-3}\right) \) without introducing any additional parameters that affect the flatness of the inflaton potential (in Einstein frame), coasting towards a Starobinsky plateau.^{3} In our construction, we introduce conformal symmetry (or local scale invariance) in a GUT model. It was shown by Wetterich [42] that scale symmetries play a crucial role in the construction of realistic cosmological models based on particle physics. Moreover, scale symmetries successfully explain the hierarchy of different scales such as the Planck and the Higgs mass [43, 44, 45, 46]. Therefore, it is natural to consider scale invariance in constructing an inflationary scenario, through which we can obtain a dynamical generation of the Planck mass, inflationary scale and particle physics scales beyond SM. In this regard, we consider two complex singlet fields \(\left( \bar{X},\,\Phi \right) \) of \(\text {SU}(5)\) or \(\text {SO}(10)\) and couple them to the Ricci scalar and adjoint Higgs field \(\left( \Sigma \right) \), such that the total action would be conformally invariant. We obtain inflation as a result of spontaneous breaking the conformal and GUT symmetries. The former occurs due to gauge fixing of one singlet field to a constant for all spacetime and the latter occurs due to \(\Sigma \) field taking its GUT VEV. Here the inflaton is identified with the real part of the second singlet (\(\phi =\sqrt{2}\mathfrak {Re}\left[ \Phi \right] \)), whereas the imaginary part is the corresponding Nambu–Goldstone boson is assumed to pick up a mass due to the presence of small explicit soft lepton number violation terms in the scalar potential [25]. We also assume \(\Phi \) carries two units of lepton number and it is coupled to the RHNs. Near the end of inflation, the inflaton is supposed to reach its VEV and also the global lepton number is violated. Thereafter, we study the dominant decay of inflaton into heavy RHNs producing non-thermal leptogenesis. We compute the corresponding reheating temperatures and also discuss the issue of producing the observed baryon asymmetry. Our study completes with an observationally viable inflationary scenario, predicting proton life time, neutrino masses and producing non-thermal leptogenesis from heavy RHNs.

The paper is briefly organized as follows. In Sect. 2, we describe toy models with conformal and scale invariance. We identify the interesting aspects of spontaneous symmetry breaking of these symmetries leading to viable inflationary scenarios. In Sect. 3, we briefly present the SV model and the computation of the proton life time. In Sect. 4 we propose our generalization of the SV model by introducing an additional conformal symmetry. We report the inflationary predictions of the generalized model together with estimates of proton life time. In Sect. 5 we further explore the nature of inflaton couplings to the SM Higgs and singlet RHNs through type I seesaw mechanism. In the view of the dominant decay of the inflaton into heavy RHNs, we constrain the Yukawa couplings of the inflaton field compatible with the generation of light left handed neutrino masses. In Sect. 6 we implement non-thermal leptogenesis and compute the reheating temperatures corresponding to the dominant decay of inflaton to heavy RHNs. We additionally comment on the necessary requirements for the production of observed baryon asymmetry through CP violation decays of RHNs. In Sect. 7 we summarize our results pointing to future steps. We provide an Appendix A summarizing the effects of geometric destabilization from fields space of inflaton and the presence of heavy fields in our model. In this paper we follow the units \(\hbar =1,\,c=1,\,\) \(m_\mathrm{P}^{2}=\frac{1}{8\pi G}\).

## 2 Conformal vs scale invariance

Models with global and local scale invariance [Weyl invariance (or) conformal invariance] are often very useful to address the issue of hierarchies in both particle physics and cosmology [43, 44, 45, 47, 48, 49]. Models with these symmetries contains no input mass parameters. The spontaneous breaking of those symmetries induced by the VEV’s of the scalar fields present in the theory, generates a hierarchy of mass scales e.g., Planck mass, GUT scale and neutrino masses.^{4} Moreover, it is a generic feature that scale or conformal symmetry breaking induce a flat direction in the scalar field potential [42], which makes these models even more interesting in the context of inflation. Another motivation to consider scale invariance for inflationary model building comes from CMB power spectra which is found to be nearly scale invariant [6].

In this section, we present firstly a toy model (with two fields) that is (global) scale invariant and present the generic form of (scale invariant) potentials and their properties. We review the presence of a massless Goldstone boson that appears as a result of spontaneous breaking of global scale invariance. In the following, we discuss the two field conformally invariant model, in which case the presence of a massless Goldstone boson can be removed by appropriate gauge fixing. The resultant spontaneous breaking of conformal symmetry (SBCS) turns to be very useful to obtain a Starobinsky like inflation.^{5}^{6} We will later explore the role of SBCS in a more realistic inflationary setting based on GUTs.

### 2.1 Scale invariance

Here we discuss a toy model with two scalar fields (in view of Refs. [42, 53, 62, 63]) and point out interesting features that we later utilize in our construction.

*V*, i.e., \(\partial _{\phi }V=\partial _{\chi }V=0\) can also be written as \(f\left( \rho \right) =f^{\prime }\left( \rho \right) =0\). One of the conditions fix the ratio of the VEV’s of the fields, while the other gives a relation between couplings (if \(\langle \phi \rangle \ne 0\) and \(\langle \chi \rangle \ne 0\)). The interesting property here is that if \(\langle \phi \rangle \propto \langle \chi \rangle \) there exists a flat direction for the field \(\phi \) (see [42] for detailed analysis). This will be more useful in the context of local scale invariant model.

### 2.2 Conformal invariance

From the above action we can define an effective Planck mass \(m_{eff}^{2}=\frac{\chi ^{2}-\phi ^{2}}{6}\) which evolves with time. In these theories, we would recover the standard Planck scale \(m_\mathrm{P}\) when the fields reach their VEV. Note that the field \(\chi \) contains a wrong sign for the kinetic term but it is not a problem as we can gauge fix the field at \(\chi =\text {constant}=\sqrt{6}M\) for all spacetime where \(M\sim \mathcal {O}\left( m_\mathrm{P}\right) \). This particular gauge choice is so called \(c-\)gauge^{7} which spontaneously breaks the conformal symmetry. It was argued that the theories in this gauge are of interest especially in cosmological models based on particle physics [46]. We will further see in this paper that fixing the scale *M* sources the hierarchy of mass scales related to inflation and particle physics (e.g., neutrino masses). In the inflationary models based on GUTs it is natural that the field \(\phi \) takes a non-zero VEV, i.e., \(\langle \phi \rangle \ne 0\) in which case it is useful to assume \(6M^{2}-\langle \phi \rangle ^{2}=6m_\mathrm{P}^{2}\) in order to generate Planck mass. Moreover, its also necessary to keep the evolution of the field \(\phi \lesssim \sqrt{6}M\) in order to avoid an anti-gravity regime.

In the next sections, we will study realistic GUT inflationary models where the inflaton rolls down to non-zero VEV and sources interesting implications in particle physics sector.

## 3 Coleman–Weinberg GUT inflation

^{8}\(g^{2}\), therefore the radiative corrections in \(\left( \Sigma ,\,H_{5}\right) \) sector can be neglected. The coefficient \(\gamma \) takes a relatively smaller value and \(0<\lambda _{i}\ll g^{2}\) and \(\lambda _{1}\lesssim \text {max}\left( \lambda _{2}^{2},\,\lambda _{3}^{2}\right) \).

*C*are dimensionfull and dimensionless constants respectively. Substituting (16) in (19) we obtain the effective potential for the field \(\phi \) in the direction of \(\sigma \propto \phi \). We set \(V_{0}=\frac{A\mu ^{4}}{4}\) which is the vacuum energy density i.e., \(V\left( \phi =0\right) \) and the constant

*C*can be chosen such that \(V\left( \phi =\mu \right) =0\). Therefore, the potential (19) can be written as

*Planck*2015 data [6, 15].

## 4 GUT inflation with conformal symmetry

As discussed in Sect. 2, conformal symmetry is useful to generate flat potentials and the hierarchy of mass scales. Therefore, embedding conformal symmetry in GUT inflation is more realistic and helpful to generate simultaneously a Planck scale \(m_\mathrm{P}\) along with the mass scale of X Bosons \(M_{X}\sim 10^{15}\,\text {GeV}\) that sources proton decay. In this section, we extend the previously discussed CW inflation by means of introducing conformal symmetry in SU(5) GUT theory. We then obtain an interesting model of inflation by implementing spontaneous breaking of conformal symmetry together with GUT symmetry.^{9} We start with two complex singlet fields^{10} of \(\text {SU}(5)\) \(\left( \Phi ,\,\bar{X}\right) \) where the real part of \(\Phi \) (\(\phi =\sqrt{2}\mathfrak {Re}\left[ \Phi \right] \)) is identified as the inflaton. Gauge fixing the field \(\bar{X}\) causes SBCS as discussed in Sect. 2. It is worth to note here that the same framework we study here, based on \(\text {SU}(5)\) GUT, can be easily realized in the \(\text {SO}(10)\) GUT. Therefore, the two complex singlets of \(\text {SU}(5)\) considered here are also singlets of \(\text {SO}(10)\) [14, 24].

^{11}\(m_{\text {Im}\Phi }^{2}\gg H_{inf}^{2}\). To arrange this, we can add a new term to the potential (24) as

^{12}

We note here that the CW potential we considered is the standard one obtained from 1-loop correction in Minkowski space-time. In the de Sitter background, 1-loop corrections are in principle different and their significance was discussed in literature [81, 82, 83]. Recently, in Ref. [84], it was argued that during slow-roll inflation we can neglect the contribution of 1-loop corrections in the gravity sector. In addition, the contributions from higher loops can also be neglected by the consideration of the slow-rolling scalar field. Refs. [85, 86] provide quantum corrections calculated for the cases of non-minimally coupled scalar fields.

### 4.1 Inflationary predictions and proton lifetime

*H*is the Hubble parameter and the prime \(^{\prime }\) denotes derivative with respect to e-folding number \(N=\ln a\left( t\right) \) before the end of inflation. The scalar power spectrum is given by

*X*bosons mass and proton life time using (21) and (22). We also show our results for the case when the inflaton field rolls from above VEV (AV) i.e., when \(\phi >\mu \). The predictions of below VEV (BV) branch i.e., when \(\phi <\mu \) are not very interesting as those are nearly same in the original CW inflation without any conformal symmetry [14]. This is evident from Fig. 1 where we can see only the AV branch of the potential significantly different in our case, whereas the BV branch is nearly same as in the SV model. Therefore, our interest in this paper is restricted to AV branch. For this case, from Table 1 we can see that the inflationary predictions of the model almost remains the same for any value of inflaton VEV. Note that even though the inflaton field values are trans-Planckian, the values of \(n_s\),

*r*remain the same. This is due to the fact that when \(\varphi \gg \mu \) the shape of the potential is exponentially flat like in Starobinsky model. Therefore, inflationary predictions only depend on the potential plateau rather than the field values (shift symmetry).

Inflationary predictions of the AV branch solutions for different parameter values

| \(\mu \) | \(\langle \varphi \rangle \) | \(A\) | \(V_{0}^{1/4}\) | \(V\left( \phi _{0}\right) ^{1/4}\) | \(H_{inf}\) | \(N_{e}\) | \(\varphi _{0}\) | \(\varphi _{e}\) | \(n_{s}\) | \(r\) | \(-\alpha _{s}\) | \(-\beta _{s}\) | \(M_{X}\) | \(\tau _{p}\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

\(\left( m_\mathrm{P}\right) \) | \(\left( m_\mathrm{P}\right) \) | \(\left( m_\mathrm{P}\right) \) | \(\left( 10^{-12}\right) \) | \(\left( 10^{16}\,\text {Gev}\right) \) | \(\left( 10^{16}\,\text {Gev}\right) \) | \(\left( 10^{13}\,\text {Gev}\right) \) | \(\left( m_\mathrm{P}\right) \) | \(\left( m_\mathrm{P}\right) \) | \(\left( 10^{-4}\right) \) | \(\left( 10^{-5}\right) \) | \(\left( \sim 10^{16}\,\text {Gev}\right) \) | \(\left( \text {years}\right) \) | |||

1.1 | 1.123 | 1.09 | 4.79 | 0.29 | 0.85 | 1.74 | 50 | 7.24 | 2.10 | 0.960 | 0.0048 | 8.07 | 2.67 | 0.57 | \(5.0\times 10^{34}\) |

3.95 | 0.27 | 0.82 | 1.59 | 55 | 7.35 | 2.10 | 0.963 | 0.0039 | 6.67 | 2.36 | 0.54 | \(4.2\times 10^{34}\) | |||

3.32 | 0.26 | 0.78 | 1.46 | 60 | 7.46 | 2.10 | 0.966 | 0.0033 | 5.61 | 2.90 | 0.52 | \(3.6\times 10^{34}\) | |||

1.5 | 2.738 | 2.36 | 6.87 | 0.76 | 0.85 | 1.71 | 50 | 7.95 | 3.093 | 0.960 | 0.0046 | 7.88 | 3.38 | 1.53 | \(2.6\times 10^{36}\) |

5.69 | 0.72 | 0.81 | 1.56 | 55 | 8.07 | 3.093 | 0.964 | 0.0038 | 6.52 | 2.27 | 1.46 | \(2.1\times 10^{36}\) | |||

4.79 | 0.70 | 0.77 | 1.43 | 60 | 8.17 | 3.093 | 0.967 | 0.0032 | 5.48 | 1.56 | 1.39 | \(1.8\times 10^{36}\) | |||

2 | 4.242 | 3.23 | 7.59 | 1.21 | 0.84 | 1.70 | 50 | 8.63 | 3.897 | 0.960 | 0.0045 | 7.79 | 3.47 | 2.47 | \(1.6\times 10^{37}\) |

6.29 | 1.15 | 0.80 | 1.55 | 55 | 8.75 | 3.897 | 0.964 | 0.0037 | 6.45 | 2.22 | 2.30 | \(1.3\times 10^{37}\) | |||

5.29 | 1.11 | 0.77 | 1.52 | 60 | 8.85 | 3.897 | 0.967 | 0.0032 | 5.42 | 1.48 | 2.21 | \(1.1\times 10^{37}\) | |||

3 | 6.928 | 4.32 | 7.92 | 1.99 | 0.84 | 1.68 | 50 | 9.61 | 5.956 | 0.960 | 0.0044 | 7.73 | 2.44 | 3.99 | \(1.2\times 10^{38}\) |

6.57 | 1.90 | 0.80 | 1.53 | 55 | 9.72 | 5.956 | 0.964 | 0.0037 | 6.40 | 1.99 | 3.81 | \(1\times 10^{38}\) | |||

5.54 | 1.82 | 0.77 | 1.41 | 60 | 9.82 | 5.956 | 0.967 | 0.0031 | 5.39 | 1.24 | 3.65 | \(8.5\times 10^{37}\) | |||

5 | 12 | 5.62 | 8.07 | 3.47 | 0.84 | 1.68 | 50 | 12.5 | 7.95 | 0.960 | 0.0044 | 7.69 | 3.04 | 6.95 | \(7.8\times 10^{38}\) |

6.70 | 3.32 | 0.80 | 1.53 | 55 | 12.7 | 7.95 | 0.964 | 0.0037 | 6.37 | 2.51 | 6.63 | \(9.2\times 10^{38}\) | |||

5.65 | 3.18 | 0.77 | 1.41 | 60 | 12.8 | 7.95 | 0.967 | 0.0031 | 5.35 | 1.97 | 6.35 | \(1.3\times 10^{40}\) | |||

10 | 24.37 | 7.33 | 8.13 | 7.07 | 0.84 | 1.68 | 50 | 12.5 | 7.95 | 0.960 | 0.0044 | 7.68 | 5.20 | 14.1 | \(1.9\times 10^{40}\) |

6.75 | 6.75 | 0.80 | 1.53 | 55 | 12.7 | 7.95 | 0.964 | 0.0037 | 6.35 | 4.04 | 13.5 | \(1.6\times 10^{40}\) | |||

5.69 | 6.47 | 0.77 | 1.41 | 60 | 12.8 | 7.95 | 0.967 | 0.0031 | 5.33 | 3.96 | 12.9 | \(1.3\times 10^{40}\) |

In Fig. 2 we depict the evolution of field \(\phi \) (also for the canonically normalized field \(\varphi \)) and slow-roll parameter \(\epsilon \) for particular parameter values.

## 5 Type I seesaw mechanism and neutrino masses

*l*is the lepton doublet, \(\tau _{2}\) is the second Pauli matrix. Here \(Y_{D}\) is the Yukawa coupling matrix of the SM Higgs coupling to the left handed neutrinos and \(Y_{N}\) is the coupling matrix of the singlet field to the three generations of Majorana RHNs \(\left( \nu _{R}^{i}\right) \). In principle, we can also weakly couple the inflaton with the SM Higgs boson as

^{13}

## 6 Reheating and non-thermal leptogenesis

^{14}\(^{,}\)

^{15}which requires \(m_{\varphi }\gtrsim 2m_{\nu _{R}}\). The mass of the canonically normalized field \(\varphi \) at the minimum of the potential is given by the second derivative of the potential (37)

Hierarchical masses for RHNs \(m_{\nu _{R}^{1}}\ll m_{\nu _{R}^{2}}\sim m_{\nu _{R}^{3}}\). To arrange this we require the coupling constants to be \(Y_{N_{1}}\ll Y_{N_{2}}\sim Y_{N_{3}}\). We assume that the inflaton decays equally into the two heavy RHNs \(\nu _{R}^{2,3}\) and the corresponding reheating temperature can be computed using [26, 27]

^{16}of our case taking \(c_{1}\approx 0\) and \(c_{2}=c_{3}=1\).

The decays of RH Majorana neutrinos \(\nu _{R}^{i}\) break the lepton number conservation and leads to CP violation. There are two decay channels

*H*and

*l*denote the Higgs field and the lepton doublets of the SM. The (lepton asymmetry generated by the CP violation) decay of \(\nu _{R}^{i}\) is measured by the following quantity

*s*indicates the entropy density, \(\text {Br}_{i}\) denotes the branching ratio

The production of RH Majorana neutrinos happens non-thermally and sufficiently late so that the produced lepton asymmetry sources the baryon asymmetry at a later stage. This essentially requires \(m_{\nu _{R}^{1}}\gtrsim T_{R}\) so that the later decay of lightest RH Majorana neutrino \(\nu _{R}^{1}\) does not wash away the produced lepton asymmetry by the heavy ones. We assume there is an accidental \(B-L\) conservation

^{17}such that sphaleron process is active which brings a part of the above lepton asymmetry into the baryon asymmetry (see Refs. [106, 107, 108] for details). As the reheating temperature in our case is \(T_{R}\sim 10^{6}-10^{9}\,\text {GeV}\) (see Fig. 3), we take \(Y_{N}^{1}\sim 10^{-10}-10^{-9}\) such that \(m_{\nu _{R}^{1}}\sim 10^{8}-10^{9}\,\text {GeV}\) . Therefore, with values \(m_{\nu _{R}^{2,3}}\sim 10^{10}-10^{12}\,\text {GeV}\) , \(m_{\nu _{R}^{1}}\sim 10^{8}-10^{9}\,\text {GeV}\) and \(T_{R}\sim 10^{6}-10^{9}\,\text {GeV}\), we have met the conditions for successful leptogenesis which are \(m_{\nu _{R}^{2}}\sim m_{\nu _{R}^{3}}\gg m_{\nu _{R}^{1}}\) and \(m_{\nu _{R}^{1}}\gtrsim T_{R}\).

## 7 Conclusions

Coleman–Weinberg inflation [11] has been a successful and realistic model based on GUT and is consistent with the current Planck data with \(r\gtrsim 0.02\) [15]. In this work, we have further generalized the framework of CW inflation with an additional conformal symmetry. Spontaneous breaking of conformal symmetry is useful to create a hierarchy of mass scales, therefore it is natural to realize this symmetry in GUT models. In this respect, two complex singlet fields of \(\text {SU}(5)\) or *SO*(10) were considered and are coupled to the GUT fields in a suitable manner. We have showed that this setup, upon spontaneous breaking of GUT and conformal symmetry, leads to an interesting inflationary scenario driven by the real part of the singlet field. In our model, the above VEV branch of CW potential gets flattened to a Starobinsky plateau, allowing for \(n_{s}\sim 1-\frac{2}{N}\) and \(r\sim \frac{12}{N^2}\) for \(N\sim 50-60\) number of e-foldings. Therefore, our model is observationally fits with the same predictions of the Starobinsky and Higgs inflation. Moreover, the VEV of the inflaton affects the masses of the superheavy gauge bosons that mediate the proton decay. We calculated the corresponding estimates for the proton life time above the current lower bound from Super-K data as \(\tau _{p}\left( p\rightarrow \pi ^{0}+e^{+}\right) >1.6\times 10^{34}\). In the next step, we introduced a coupling between the complex singlet field with the generation of three singlet RHNs, where the singlet field is assumed to carry two units of lepton number. We implemented a type I seesaw mechanism, where spontaneous symmetry breaking of global lepton number results in generating neutrino masses. We put an upper bound to the inflaton couplings to RHNs, assuming inflation is dominated by inflaton couplings to GUT field. For the non-thermal leptogenesis to happen, we have considered a dominant decay of the inflaton into some of the RHNs and obtained the corresponding reheating temperatures as \(10^{6}\text { GeV}\lesssim T_{R}<10^{9}\) GeV. Furthermore, our proposed extension of CW inflation can be tested within future CMB and particle physics experiments [109].

In this work, we mainly restricted to a non-supersymmetric construction of GUT inflation with conformal symmetry. It would be interesting to consider this model in GUT based SUGRA framework with superconformal symmetries, which we defer for future investigations.

## Footnotes

- 1.
- 2.
Although the SM Higgs field rolls to its electroweak VEV it is negligible compared to the energy scale of inflation.

- 3.
- 4.
For example, single scalar field models with spontaneously broken scale invariance by the 1-loop corrections were studied in [50, 51, 52]. In [53] a two field model with scale invariance was studied to generate the hierarchy of mass scales and the dynamical generation of Planck mass from the VEV’s of the scalar fields. Recently in [54], some constraints were derived on these models from Big Bang Nucleosynthesis (BBN).

- 5.
- 6.
- 7.
- 8.
The field \(\Sigma \) interacts with vector boson

*X*with a coupling constant*g*. - 9.
- 10.
A complex singlet is required to implement type I mechanism which we later explain in Sect. 5.

- 11.
Where \(H_{inf}\) is the Hubble parameter during inflation.

- 12.
A similar scenario happens in the context of hybrid inflationary scenario discussed in [80].

- 13.
The kinetic terms and couplings of SM Higgs and RHNs to the Ricci scalar are irrelevant here and can be neglected in comparison with the inflaton dynamics.

- 14.The inflaton could also decay into Higgs field but we have chosen the coupling of the Higgs field to the inflaton as \(\lambda _h\ll Y_N^i \lesssim \mathcal {O}\left( 10^{-6} \right) \). For these couplings, the decay rate of the inflaton to a pair of Higgs bosons is negligible [93, 94]. However, there can be a period of parametric resonance in the phase of preheating right after the end of inflation, during which the number of Higgs particles can grow exponentially [93, 94]. Around the VEV, the inflaton potential (37) can be approximated asThen we can apply the results of [93, 94] to estimate the effect of parametric resonance. The inflaton field oscillates around the minimum as$$\begin{aligned} V_E\left( \varphi \right) = \frac{1}{2}m_\varphi ^2\left( \varphi -\langle \varphi \rangle \right) ^2 = \frac{1}{2}m^2\hat{\varphi }^2. \end{aligned}$$where \(\hat{\varphi }_A\left( t \right) \approx \frac{m_p}{\sqrt{3\pi }mt}\) is the amplitude of oscillations of the inflaton field. The regime of parametric resonance occurs as far as \(\hat{\varphi }_A > \frac{\lambda _h^2}{8\pi }\langle \varphi \rangle \) and when \(\hat{\varphi }_A\) drops to smaller values then standard perturbation theory dominates. To estimate the effect of parametric resonance in our case we compute the number of oscillations at the end of parametric resonance (\(N_f\)). Following estimates from [93] we especially have \(N_f\approx \frac{mt_f}{2\pi }\) where \(t_f\) is the instant when parametric resonance ends, by means of$$\begin{aligned} \hat{\varphi }(t) \approx \hat{\varphi }_A(t)\sin (mt), \end{aligned}$$As a result, we can further obtain [93]$$\begin{aligned} \lambda _h\hat{\varphi }_A\approx \frac{\lambda _hM_p}{3m_{\varphi }t_f} \approx m. \end{aligned}$$since \(m_\varphi \sim \mathcal {O}\left( 10^{-6} \right) \) from (57) and \(\lambda _h\ll 10^{-6}\) in our case. Therefore, the effects of parametric resonance in our case is negligible for our chosen values of inflaton-Higgs couplings.$$\begin{aligned} N_f \sim \frac{\lambda _hm_P}{6\pi m_{\varphi }} \ll 1, \end{aligned}$$
- 15.
- 16.
- 17.
\(B,\,L\) refers to baryon number and lepton number, respectively.

- 18.
Note that we gauge fixed the conformal field at \(X=\sqrt{3}M\).

- 19.
The Ricci scalar from a fields space metric in the case of \(\alpha -\) attractor models is \(\mathcal {R}_K=-\frac{2}{3\alpha }\) [75].

## Notes

### Acknowledgements

We thank the anonymous referee for very useful comments. We would like to thank Qaisar Shafi for numerous useful discussions and feedback during this project. We would like to thank C. Pallis, N. Okada and D. Raut for useful discussions and comments. K. S. K thanks P. Parada and K. N. Deepthi for useful discussions. This research work was supported by the Grant UID/MAT/00212/2013 and COST Action CA15117 (CANTATA). K. S. K thanks FCT BD Grant SFRH/BD/51980/2012 and Netherlands Organization for Scientific Research (NWO) Grant number 680-91-119. PVM is grateful to DAMTP, University of Cambridge for providing an excellent research environment for his sabbatical and he is also thankful to Clare Hall college, Cambridge for a Visiting Fellowship.

## References

- 1.A.A. Starobinsky, A new type of isotropic cosmological models without singularity. Phys. Lett. B
**91**, 99–102 (1980)ADSzbMATHCrossRefGoogle Scholar - 2.A.H. Guth, The inflationary universe: a possible solution to the horizon and flatness problems. Phys. Rev. D
**23**, 347–356 (1981)ADSzbMATHCrossRefGoogle Scholar - 3.A.H. Guth, The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems. Adv. Ser. Astrophys. Cosmol.
**3**, 139 (1987)Google Scholar - 4.A.D. Linde, A new inflationary universe scenario: a possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems. Phys. Lett.
**108B**, 389–393 (1982)ADSCrossRefGoogle Scholar - 5.A.D. Linde, A New inflationary universe scenario: a possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems. Adv. Ser. Astrophys. Cosmol.
**3**, 149 (1987)Google Scholar - 6.Planck Collaboration, P. A. R. Ade et al., Planck 2015 results. XX. Constraints on inflation. Astron. Astrophys.
**594**, A20 (2016) arXiv:1502.02114 [astro-ph.CO] - 7.BICEP2 Collaboration, Planck Collaboration Collaboration, P. Ade et al., A Joint analysis of BICEP2/Keck Array and Planck data. Phys. Rev. Lett. (1502.00612)Google Scholar
- 8.Planck Collaboration Collaboration, P. Ade et al., Planck 2015 results. XVII. Constraints on primordial non-Gaussianity. arXiv:1502.01592 [astro-ph.CO]
- 9.J. Martin, C. Ringeval, V. Vennin, Encyclopedia inflationaris. Phys. Dark Univ.
**5–6**, 75–235 (2013). arXiv:1303.3787 Google Scholar - 10.J. Martin, The observational status of cosmic inflation after Planck. Astrophys. Sp. Sci. Proc.
**45**, 41–134 (2016). arXiv:1502.05733 [astro-ph.CO]CrossRefGoogle Scholar - 11.Q. Shafi, A. Vilenkin, Inflation with SU(5). Phys. Rev. Lett.
**52**, 691–694 (1984)ADSCrossRefGoogle Scholar - 12.H. Georgi, S.L. Glashow, Unity of all elementary particle forces. Phys. Rev. Lett.
**32**, 438–441 (1974)ADSCrossRefGoogle Scholar - 13.Q. Shafi, V.N. Senoguz, Coleman–Weinberg potential in good agreement with wmap. Phys. Rev. D
**73**, 127301 (2006). arXiv:astro-ph/0603830 [astro-ph]ADSCrossRefGoogle Scholar - 14.M.U. Rehman, Q. Shafi, J.R. Wickman, GUT inflation and proton decay after WMAP5. Phys. Rev. D
**78**, 123516 (2008). arXiv:0810.3625 [hep-ph]ADSCrossRefGoogle Scholar - 15.N. Okada, V. N. Şenoğuz, Q. Shafi. The observational status of simple inflationary models: an update. arXiv:1403.6403 [hep-ph]
- 16.A. Cerioni, F. Finelli, A. Tronconi, G. Venturi, Inflation and reheating in induced gravity. Phys. Lett. B
**681**, 383–386 (2009). arXiv:0906.1902 [astro-ph.CO]ADSCrossRefGoogle Scholar - 17.G. Panotopoulos, Nonminimal GUT inflation after Planck results. Phys. Rev. D
**89**(4), 047301 (2014). arXiv:1403.0931 [hep-ph]ADSCrossRefGoogle Scholar - 18.G. Barenboim, E.J. Chun, H.M. Lee, Coleman–Weinberg inflation in light of Planck. Phys. Lett. B
**730**, 81–88 (2014). arXiv:1309.1695 [hep-ph]ADSCrossRefGoogle Scholar - 19.K. Kannike, A. Racioppi, M. Raidal, Linear inflation from quartic potential. JHEP
**01**, 035 (2016). arXiv:1509.05423 [hep-ph]ADSMathSciNetzbMATHCrossRefGoogle Scholar - 20.A. Racioppi, Coleman–Weinberg linear inflation: metric vs. Palatini formulation. JCAP
**1712**(12), 041 (2017). arXiv:1710.04853 [astro-ph.CO]ADSMathSciNetCrossRefGoogle Scholar - 21.R. Kallosh, A. Linde, D. Roest, Universal attractor for inflation at strong coupling. Phys. Rev. Lett.
**112**(1), 011303 (2014). arXiv:1310.3950 [hep-th]ADSCrossRefGoogle Scholar - 22.B.J. Broy, D. Coone, D. Roest, Plateau inflation from random non-minimal coupling. JCAP
**1606**(06), 036 (2016). arXiv:1604.05326 [hep-th]ADSCrossRefGoogle Scholar - 23.G. Lazarides, Q. Shafi, Extended structures at intermediate scales in an inflationary cosmology. Phys. Lett. B
**148**, 35–38 (1984)ADSCrossRefGoogle Scholar - 24.G. Lazarides, Q. Shafi, Origin of matter in the inflationary cosmology. Phys. Lett. B
**258**, 305–309 (1991)ADSCrossRefGoogle Scholar - 25.S.M. Boucenna, S. Morisi, Q. Shafi, J.W.F. Valle, Inflation and majoron dark matter in the seesaw mechanism. Phys. Rev. D
**90**(5), 055023 (2014). arXiv:1404.3198 [hep-ph]ADSCrossRefGoogle Scholar - 26.N. Okada, Q. Shafi, Observable gravity waves from U(1)\({B-L}\) Higgs and Coleman-Weinberg inflation. arXiv:1311.0921 [hep-ph]
- 27.T. Asaka, H.B. Nielsen, Y. Takanishi, Nonthermal leptogenesis from the heavier Majorana neutrinos. Nucl. Phys. B
**647**, 252–274 (2002). arXiv:hep-ph/0207023 [hep-ph]ADSCrossRefGoogle Scholar - 28.V.N. Senoguz, Q. Shafi, GUT scale inflation, nonthermal leptogenesis, and atmospheric neutrino oscillations. Phys. Lett. B
**582**, 6–14 (2004). arXiv:hep-ph/0309134 [hep-ph]ADSCrossRefGoogle Scholar - 29.V. N. Senoguz, Q. Shafi, \(U(1)(B-L):\) Neutrino physics and inflation. In: Proceedings on 11th International Symposium on particles, strings and cosmology (PASCOS 2005): Gyeongju, Korea, 30 May–4 June 2005. (2005). arXiv:hep-ph/0512170 [hep-ph]
- 30.V.N. Şenoğuz, Q. Shafi, Primordial monopoles, proton decay, gravity waves and GUT inflation. Phys. Lett. B
**752**, 169–174 (2016). arXiv:1510.04442 [hep-ph]ADSCrossRefGoogle Scholar - 31.A.A. Starobinsky, The Perturbation spectrum evolving from a nonsingular initially De-Sitte r Cosmology and the microwave background anisotropy. Sov. Astron. Lett.
**9**, 302 (1983)ADSGoogle Scholar - 32.F.L. Bezrukov, M. Shaposhnikov, The Standard Model Higgs boson as the inflaton. Phys. Lett. B
**659**, 703–706 (2008). arXiv:0710.3755 [hep-th]ADSCrossRefGoogle Scholar - 33.A. Linde. Inflationary cosmology after Planck (2013), arXiv:1402.0526
- 34.S. Ferrara, R. Kallosh, A. Linde, M. Porrati, Minimal supergravity models of inflation. Phys. Rev. D
**88**(8), 085038 (2013). arXiv:1307.7696 [hep-th]ADSCrossRefGoogle Scholar - 35.A.S. Koshelev, L. Modesto, L. Rachwal, A.A. Starobinsky, Occurrence of exact \(R^2\) inflation in non-local UV-complete gravity. JHEP
**11**, 067 (2016). arXiv:1604.03127 [hep-th]ADSzbMATHCrossRefGoogle Scholar - 36.A.S. Koshelev, K. Sravan Kumar, A.A. Starobinsky, \(R^2\) inflation to probe non-perturbative quantum gravity. JHEP
**03**, 071 (2018). arXiv:1711.08864 [hep-th]ADSzbMATHCrossRefGoogle Scholar - 37.J. Ellis, D.V. Nanopoulos, K.A. Olive, Starobinsky-like inflationary models as avatars of no-scale supergravity. JCAP
**1310**, 009 (2013). arXiv:1307.3537 [hep-th]ADSCrossRefGoogle Scholar - 38.R. Kallosh, A. Linde, D. Roest, Superconformal inflationary \(\alpha \)-sttractors. JHEP
**1311**, 198 (2013). arXiv:1311.0472 ADSzbMATHCrossRefGoogle Scholar - 39.K.S. Kumar, J. Marto, P.V. Moniz, S. Das, Non-slow-roll dynamics in \(\alpha -\)attractors. JCAP
**1604**(04), 005 (2016). arXiv:1506.05366 [gr-qc]MathSciNetCrossRefGoogle Scholar - 40.A. Salvio, A. Mazumdar, Classical and quantum initial conditions for Higgs inflation. Phys. Lett. B
**750**, 194–200 (2015). arXiv:1506.07520 [hep-ph]ADSCrossRefGoogle Scholar - 41.X. Calmet, I. Kuntz, Higgs Starobinsky inflation. Eur. Phys. J. C
**76**(5), 289 (2016). arXiv:1605.02236 [hep-th]ADSCrossRefGoogle Scholar - 42.C. Wetterich, Cosmology and the fate of dilatation symmetry. Nucl. Phys. B
**302**, 668–696 (1988)ADSCrossRefGoogle Scholar - 43.G. ’t Hooft, Local conformal symmetry: the missing symmetry component for space and time. arXiv:1410.6675 [gr-qc]
- 44.I. Quiros, Scale invariant theory of gravity and the standard model of particles, arXiv:1401.2643 [gr-qc]
- 45.E. Scholz, Paving the way for transitions—a case for Weyl geometry, arXiv:1206.1559 [gr-qc]
- 46.I. Bars, P. Steinhardt, N. Turok, Local conformal symmetry in physics and cosmology. Phys. Rev. D
**89**(4), 043515 (2014). arXiv:1307.1848 [hep-th]ADSCrossRefGoogle Scholar - 47.F. Englert, C. Truffin, R. Gastmans, Conformal invariance in quantum gravity. Nucl. Phys. B
**117**, 407–432 (1976)ADSCrossRefGoogle Scholar - 48.S. Deser, Scale invariance and gravitational coupling. Ann. Phys.
**59**, 248–253 (1970)ADSMathSciNetCrossRefGoogle Scholar - 49.M. Shaposhnikov, D. Zenhausern, Quantum scale invariance, cosmological constant and hierarchy problem. Phys. Lett. B
**671**, 162–166 (2009). arXiv:0809.3406 [hep-th]ADSCrossRefGoogle Scholar - 50.M. Rinaldi, L. Vanzo, Inflation and reheating in theories with spontaneous scale invariance symmetry breaking. Phys. Rev. D
**94**(2), 024009 (2016). arXiv:1512.07186 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 51.M. Rinaldi, L. Vanzo, S. Zerbini, G. Venturi, Inflationary quasiscale-invariant attractors. Phys. Rev. D
**93**, 024040 (2016). arXiv:1505.03386 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 52.C. Csaki, N. Kaloper, J. Serra, J. Terning, Inflation from Broken Scale Invariance. Phys. Rev. Lett.
**113**, 161302 (2014). arXiv:1406.5192 [hep-th]ADSCrossRefGoogle Scholar - 53.P.G. Ferreira, C.T. Hill, G.G. Ross, Scale-independent inflation and hierarchy generation. Phys. Lett. B
**763**, 174–178 (2016). arXiv:1603.05983 [hep-th]ADSzbMATHCrossRefGoogle Scholar - 54.K. Kannike, M. Raidal, C. Spethmann, H. Veermäe, Evolving Planck mass in classically scale-invariant theories, arXiv:1610.06571 [hep-ph]
- 55.J. Garcia-Bellido, J. Rubio, M. Shaposhnikov, D. Zenhausern, Higgs-Dilaton cosmology: from the early to the late universe. Phys. Rev. D
**84**, 123504 (2011). arXiv:1107.2163 [hep-ph]ADSCrossRefGoogle Scholar - 56.F. Bezrukov, G.K. Karananas, J. Rubio, M. Shaposhnikov, Higgs-Dilaton cosmology: an effective field theory approach. Phys. Rev. D
**87**(9), 096001 (2013). arXiv:1212.4148 [hep-ph]ADSCrossRefGoogle Scholar - 57.G.K. Karananas, J. Rubio, On the geometrical interpretation of scale-invariant models of inflation. Phys. Lett. B
**761**, 223–228 (2016). arXiv:1606.08848 [hep-ph]ADSCrossRefGoogle Scholar - 58.J. Rubio, C. Wetterich, Emergent scale symmetry: connecting inflation and dark energy. Phys. Rev. D
**96**(6), 063509 (2017). arXiv:1705.00552 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 59.R. Kallosh, A. Linde, Superconformal generalizations of the Starobinsky model. JCAP
**1306**, 028 (2013). arXiv:1306.3214 ADSMathSciNetCrossRefGoogle Scholar - 60.R. Kallosh, A. Linde, Multi-field conformal cosmological attractors. JCAP
**1312**, 006 (2013). arXiv:1309.2015 ADSCrossRefGoogle Scholar - 61.A.S. Koshelev, K. Sravan Kumar, P. Vargas Moniz, Effective models of inflation from a nonlocal framework. Phys. Rev. D
**96**(10), 103503 (2017). arXiv:1604.01440 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 62.C. Wetterich, Cosmologies with variable Newton’s ’Constant’. Nucl. Phys. B
**302**, 645–667 (1988)ADSCrossRefGoogle Scholar - 63.D.M. Ghilencea, Manifestly scale-invariant regularization and quantum effective operators. Phys. Rev. D
**93**(10), 105006 (2016). arXiv:1508.00595 [hep-ph]ADSMathSciNetCrossRefGoogle Scholar - 64.W.F. Kao, Scale invariance and inflation. Phys. Lett. A
**154**, 1–4 (1991)ADSCrossRefGoogle Scholar - 65.I. Bars, S.-H. Chen, P.J. Steinhardt, N. Turok, Complete set of homogeneous isotropic analytic solutions in scalar-tensor cosmology with radiation and curvature. Phys. Rev. D
**86**, 083542 (2012). arXiv:1207.1940 [hep-th]ADSCrossRefGoogle Scholar - 66.A.D. Linde, Particle physics and inflationary cosmology. Contemp. Concepts Phys.
**5**, 1–362 (1990). arXiv:hep-th/0503203 [hep-th]Google Scholar - 67.M. Magg, Q. Shafi, Symmetry breaking patterns in SU(5). Z. Phys. C
**4**, 63 (1980)ADSCrossRefGoogle Scholar - 68.Super-Kamiokande Collaboration, H. Nishino et al., Search for proton decay via p —> e+ pi0 and p —> mu+ pi0 in a large water cherenkov detector,
*Phys. Rev. Lett.***102**, 141801 (2009), arXiv:0903.0676 [hep-ex] - 69.Super-Kamiokande Collaboration, K. Abe et al., Search for Proton Decay via \(p \rightarrow e^+\pi ^0\) and \(p \rightarrow \mu ^+\pi ^0\) in 0.31 megaton\(\cdot \)years exposure of the Super-Kamiokande Water Cherenkov Detecto. Phys. Rev. D95(1), 012004 (2017), arXiv:1610.03597 [hep-ex]
- 70.G. Esposito, G. Miele, L. Rosa, Cosmological restrictions on conformally invariant SU(5) GUT models. Class. Quantum Gravit.
**10**, 1285–1298 (1993). arXiv:gr-qc/9506093 [gr-qc]ADSzbMATHCrossRefGoogle Scholar - 71.J.L. Cervantes-Cota, H. Dehnen, Induced gravity inflation in the SU(5) GUT. Phys. Rev. D
**51**, 395–404 (1995). arXiv:astro-ph/9412032 [astro-ph]ADSCrossRefGoogle Scholar - 72.F. Buccella, G. Esposito, G. Miele, Spontaneously broken SU(5) symmetries and one loop effects in the early universe. Class. Quantum Gravit.
**9**, 1499–1509 (1992). arXiv:gr-qc/9506091 [gr-qc]ADSCrossRefGoogle Scholar - 73.R. Kallosh, A. Linde, T. Rube, General inflaton potentials in supergravity. Phys. Rev. D
**83**, 043507 (2011). arXiv:1011.5945 [hep-th]ADSCrossRefGoogle Scholar - 74.D.I. Kaiser, Conformal transformations with multiple scalar fields. Phys. Rev. D
**81**, 084044 (2010). arXiv:1003.1159 [gr-qc]ADSCrossRefGoogle Scholar - 75.S. Renaux-Petel, K. Turzyński, Geometrical destabilization of inflation. Phys. Rev. Lett.
**117**(14), 141301 (2016). arXiv:1510.01281 [astro-ph.CO]ADSCrossRefGoogle Scholar - 76.A.R. Brown, Hyperbolic inflation. Phys. Rev. Lett.
**121**(25), 251601 (2018). arXiv:1705.03023 [hep-th]ADSCrossRefGoogle Scholar - 77.P. Christodoulidis, D. Roest, E. I. Sfakianakis, Angular inflation in multi-field \({\alpha }\)-attractors, arXiv:1803.09841 [hep-th]
- 78.S. Garcia-Saenz, S. Renaux-Petel, J. Ronayne, Primordial fluctuations and non-Gaussianities in sidetracked inflation. JCAP
**1807**(07), 057 (2018). arXiv:1804.11279 [astro-ph.CO]ADSMathSciNetCrossRefGoogle Scholar - 79.O. Iarygina, E.I. Sfakianakis, D.-G. Wang, A. Achucarro, Universality and scaling in multi-field \(\alpha \)-attractor preheating. JCAP
**1906**(06), 027 (2019). arXiv:1810.02804 [astro-ph.CO]ADSCrossRefGoogle Scholar - 80.W. Buchmuller, K. Ishiwata, Grand unification and subcritical hybrid inflation. Phys. Rev. D
**91**(8), 081302 (2015). arXiv:1412.3764 [hep-ph]ADSCrossRefGoogle Scholar - 81.D. Boyanovsky, H.J. de Vega, N.G. Sanchez, Quantum corrections to the inflaton potential and the power spectra from superhorizon modes and trace anomalies. Phys. Rev. D
**72**, 103006 (2005). arXiv:astro-ph/0507596 [astro-ph]ADSCrossRefGoogle Scholar - 82.D. Boyanovsky, C. Destri, H.J. De Vega, N.G. Sanchez, The effective theory of inflation in the standard model of the universe and the CMB+LSS data analysis. Int. J. Mod. Phys. A
**24**, 3669–3864 (2009). arXiv:0901.0549 [astro-ph.CO]ADSzbMATHCrossRefGoogle Scholar - 83.C. Destri, H.J. de Vega, N.G. Sanchez, Higher order terms in the inflaton potential and the lower bound on the tensor to scalar ratio r. Ann. Phys.
**326**, 578–603 (2011). arXiv:0906.4102 [astro-ph.CO]ADSzbMATHCrossRefGoogle Scholar - 84.R.K. Jain, M. Sandora, M.S. Sloth, Radiative Ccorrections from heavy fast-roll fields during inflation. JCAP
**1506**, 016 (2015). arXiv:1501.06919 [hep-th]ADSCrossRefGoogle Scholar - 85.K. Kirsten, G. Cognola, L. Vanzo, Effective Lagrangian for selfinteracting scalar field theories in curved space-time. Phys. Rev. D
**48**, 2813–2822 (1993). arXiv:hep-th/9304092 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 86.T. Markkanen, A. Tranberg, Quantum corrections to inflaton and curvaton dynamics. JCAP
**1211**, 027 (2012). arXiv:1207.2179 [gr-qc]ADSCrossRefGoogle Scholar - 87.F. Bezrukov, M. Shaposhnikov, Standard Model Higgs boson mass from inflation: two loop analysis. JHEP
**07**, 089 (2009). arXiv:0904.1537 [hep-ph]ADSCrossRefGoogle Scholar - 88.D.P. George, S. Mooij, M. Postma, Quantum corrections in Higgs inflation: the Standard Model case. JCAP
**1604**(04), 006 (2016). arXiv:1508.04660 [hep-th]ADSCrossRefGoogle Scholar - 89.J. Fumagalli, M. Postma, UV (in)sensitivity of Higgs inflation. JHEP
**05**, 049 (2016). arXiv:1602.07234 [hep-ph]ADSCrossRefGoogle Scholar - 90.C. Pallis, Q. Shafi, Gravity waves from non-minimal quadratic inflation. JCAP
**1503**(03), 023 (2015). arXiv:1412.3757 [hep-ph]ADSMathSciNetCrossRefGoogle Scholar - 91.C. van de Bruck, C. Longden, Running of the running and entropy perturbations during inflation. Phys. Rev. D
**94**(2), 021301 (2016). arXiv:1606.02176 [astro-ph.CO]ADSCrossRefGoogle Scholar - 92.Planck Collaboration, P. A. R. Ade et al., Planck 2015 results. XIII. Cosmological parameters, Astron. Astrophys.
**594**, A13 (2016), arXiv:1502.01589 [astro-ph.CO] - 93.L. Kofman, A. D. Linde, and A. A. Starobinsky, Towards the theory of reheating after inflation,
*Phys. Rev.*D56 (1997) 3258–3295, arXiv:hep-ph/9704452 [hep-ph]ADSCrossRefGoogle Scholar - 94.P.B. Greene, L. Kofman, A.D. Linde, A.A. Starobinsky, Structure of resonance in preheating after inflation. Phys. Rev. D
**56**, 6175–6192 (1997). arXiv:hep-ph/9705347 [hep-ph]ADSCrossRefGoogle Scholar - 95.Y. Ema, R. Jinno, K. Mukaida, K. Nakayama, Violent preheating in inflation with nonminimal coupling. JCAP
**1702**(02), 045 (2017). arXiv:1609.05209 [hep-ph]ADSCrossRefGoogle Scholar - 96.M.P. DeCross, D.I. Kaiser, A. Prabhu, C. Prescod-Weinstein, E.I. Sfakianakis, Preheating after multifield inflation with nonminimal couplings, II: resonance structure. Phys. Rev. D
**97**(2), 023527 (2018). arXiv:1610.08868 [astro-ph.CO]ADSCrossRefGoogle Scholar - 97.T. Krajewski, K. Turzyński, M. Wieczorek, On preheating in \(\alpha \)-attractor models of inflation, arXiv:1801.01786 [astro-ph.CO]
- 98.M. Fukugita, T. Yanagida, Baryogenesis without grand unification. Phys. Lett. B
**174**, 45–47 (1986)ADSCrossRefGoogle Scholar - 99.M.Yu. Khlopov, A.D. Linde, Is It easy to save the gravitino? Phys. Lett.
**138B**, 265–268 (1984)ADSCrossRefGoogle Scholar - 100.I.V. Falomkin, G.B. Pontecorvo, M.G. Sapozhnikov, MYu. Khlopov, F. Balestra, G. Piragino, Low-energy anti-P HE-4 annihilation and problems of the modern cosmology, gut and susy models. Nuovo Cim.
**A79**, 193–204 (1984). [Yad. Fiz.39,990(1984)]ADSCrossRefGoogle Scholar - 101.MYu. Khlopov, YuL Levitan, E.V. Sedelnikov, I.M. Sobol, Nonequilibrium cosmological nucleosynthesis of light elements: calculations by the Monte Carlo method. Phys. At. Nucl.
**57**, 1393–1397 (1994). [Yad. Fiz.57,1466 (1994)]Google Scholar - 102.MYu. Khlopov, A. Barrau, J. Grain, Gravitino production by primordial black hole evaporation and constraints on the inhomogeneity of the early universe. Class. Quantum Gravit
**23**, 1875–1882 (2006). arXiv:astro-ph/0406621 [astro-ph]ADSMathSciNetzbMATHCrossRefGoogle Scholar - 103.M. Flanz, E.A. Paschos, U. Sarkar, Baryogenesis from a lepton asymmetric universe. Phys. Lett. B
**345**, 248–252 (1995). arXiv:hep-ph/9411366 [hep-ph]. [Erratum: Phys. Lett.B382,447(1996)]ADSCrossRefGoogle Scholar - 104.W. Buchmuller, M. Plumacher, CP asymmetry in Majorana neutrino decays. Phys. Lett. B
**431**, 354–362 (1998). arXiv:hep-ph/9710460 [hep-ph]ADSCrossRefGoogle Scholar - 105.K. Hamaguchi, Cosmological baryon asymmetry and neutrinos: baryogenesis via leptogenesis in supersymmetric theories. PhD thesis, Tokyo U., 2002. arXiv:hep-ph/0212305 [hep-ph]
- 106.S.Yu. Khlebnikov, M.E. Shaposhnikov, The statistical theory of anomalous fermion number nonconservation. Nucl. Phys. B
**308**, 885–912 (1988)ADSCrossRefGoogle Scholar - 107.J.A. Harvey, E.W. Kolb, D.B. Reiss, S. Wolfram, Calculation of cosmological baryon asymmetry in grand unified gauge models. Nucl. Phys. B
**201**, 16–100 (1982)ADSCrossRefGoogle Scholar - 108.J.A. Harvey, M.S. Turner, Cosmological baryon and lepton number in the presence of electroweak fermion number violation. Phys. Rev. D
**42**, 3344–3349 (1990)ADSCrossRefGoogle Scholar - 109.P. Creminelli, D.L. López Nacir, M. Simonović, G. Trevisan, M. Zaldarriaga, Detecting primordial \(B\)-modes after Planck. JCAP
**1511**(11), 031 (2015). arXiv:1502.01983 [astro-ph.CO]ADSCrossRefGoogle Scholar - 110.A. Achúcarro, R. Kallosh, A. Linde, D.-G. Wang, Y. Welling, Universality of multi-field \(\alpha \)-attractors. JCAP
**1804**(04), 028 (2018). arXiv:1711.09478 [hep-th]ADSCrossRefGoogle Scholar - 111.P. Christodoulidis, D. Roest, E. I. Sfakianakis, Attractors, bifurcations and curvature in multi-field inflation, arXiv:1903.03513 [gr-qc]
- 112.M.P. DeCross, D.I. Kaiser, A. Prabhu, C. Prescod-Weinstein, E.I. Sfakianakis, Preheating after multifield inflation with nonminimal couplings, III: dynamical spacetime results. Phys. Rev. D
**97**(2), 023528 (2018). arXiv:1610.08916 [astro-ph.CO]ADSCrossRefGoogle Scholar - 113.M.P. DeCross, D.I. Kaiser, A. Prabhu, C. Prescod-Weinstein, E.I. Sfakianakis, Preheating after Multifield inflation with nonminimal couplings, I: covariant formalism and attractor behavior. Phys. Rev. D
**97**(2), 023526 (2018). arXiv:1510.08553 [astro-ph.CO]ADSCrossRefGoogle Scholar - 114.R. Nguyen, J. van de Vis, E. I. Sfakianakis, J. T. Giblin, and D. I. Kaiser, Nonlinear dynamics of preheating after multifield inflation with nonminimal couplings, arXiv:1905.12562 [hep-ph]

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}