Nonminimal Higgs inflation in the context of warm scenario in the light of Planck data
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Abstract
We investigate the nonminimally Higgs inflaton (HI) model in the context of warm inflation scenario. Warm little inflaton (WLI) model considers the little Higgs boson as inflaton. The concerns of warm inflation model can be eliminated in WLI model. There is a special case of dissipation parameter in WLI model \(\varGamma =\varGamma _0 T\). Using this parameter, we study the potential of HI in Einstein frame . Finally we will constrain the parameters of our model using current Planck observational data.
1 Set up and motivation
Important perturbation parameters in the strong dissipative regime that can be compared with observational data. In these relations we have fixed some parameters as: \(\xi ^2=10^{8}\lambda \) and \(C_{\gamma }=70\)
\(Q\gg 1\)  Theoretical amount  Constant parameter 

\(\varDelta _R\)  \(\varDelta _{0R}\exp (\frac{0.635}{5\sqrt{6}}\frac{\phi }{M_p})\)  \(\frac{\varDelta _{0R}}{\varGamma _0^{3.51}}=1.1\times 10^{2.1}\) 
\(n_s1\)  \(n_{0s}\exp (\frac{4}{5\sqrt{6}}\frac{\phi }{M_p})\)  \(\frac{n_{0s}}{\varGamma _0^{1.6}}=7.62\times 10^{2.6}\) 
r  \(r_0(1n_s)^{\frac{0.635}{4}}\)  \(\frac{r_0}{\varGamma _0^{5.54}}=2.2\times 10^{10.08}\) 
\(n_{run}\)  \(\frac{4}{0.635}(1n_s)^2n_{0r}(1n_s)^{\frac{37}{5}}\)  \(\frac{n_{r0}}{\varGamma _0^{\frac{37}{5}}}=7\) 
2 Important theoretical parameters of warm Higgs inflation
Important perturbation parameters in the weak dissipative regime that can be compared with observational data. In comparison with observation we will fix some parameters as: \(\xi ^2=10^{8}\lambda \) and \(C_{\gamma }=70\)
\(Q\ll 1\)  Theoretical amount  Constant parameter 

\(\varDelta _R\)  \(\varDelta _{0R}\exp (\frac{4}{3\sqrt{6}}\frac{\phi }{M_p})\)  \(P_{0R}\simeq 3.1\times 10^{3}(\frac{\lambda ^2\varGamma _0}{C_{\gamma }\xi ^4})^{\frac{1}{3}}\) 
\(n_s1\)  \(n_{0s}\exp (\frac{2}{\sqrt{6}}\frac{\phi }{M_p})\)  \(n_{0s}=1.8\) 
r  \(r_0(1n_s)^{\frac{4}{3}}\)  \(r_0=4.6\times 10^{1}(\frac{C_{\gamma }\lambda }{6\varGamma _0 \xi ^2})^{\frac{1}{3}}\) 
\(n_{run}\)  \(n_{r0}(1n_s)^2\)  \(n_{r0}=\frac{3}{4}\) 
3 Comparison with observation

The Starobinsky or \(R^{2}\) inflation model [24]: In Starobinsky inflation model the asymptotic behavior of the effective potential is presented as \(V(\phi )\propto [12\mathrm {e} ^{B\phi /M_{pl}}+\mathscr {O}(\mathrm {e}^{2B\phi /M_{pl}})]\) which provides the following predictions in the slowroll limit [25, 26, 27]: \(r\approx 8/B^{2}N^{2}\) and ,\(n_{s}\approx 12/N\) where \(B^{2}=2/3\). Therefore, if we select \(N=50\) then we obtain \((n_{s},r)\approx (0.96,0.0048)\). For \(N=60\) we find \((n_{s},r)\approx (0.967,0.0033)\). It has been found that the Planck data [1, 2] favors the Starobinsky inflation. Obviously, our results (see Figs. 1 and 2) are consistent with those of \(R^{2}\) inflation.

The Standard Higgs boson as the inflaton [5]: In Higgs inflation model the behavior of the effective potential is exponentially flat \(U(\phi )=\frac{\lambda M_p^4}{4\xi ^2}(1+\exp (\frac{2\phi }{\sqrt{6}M_p}))^{2}\) where the \(1\ll \xi \ll \ll 10^{17}\). This form of potential provides the following perturbation parameters in the slowroll limit [5]: \(r\approx 192/(4N+3)^2\) and ,\(n_{s}\approx 18(4N+9)/(4N+3)^2\) Therefore, if we select \(N=60\) we find \((n_{s},r)\approx (0.97,0.0033)\). It has been found that the Planck data [1, 2] favors the nonminimal Higgs inflation. Obviously, our results (see Figs. 1 and 3) are consistent with those of nonminimal Higgs inflation.

The chaotic inflation [28]: In this important model of inflation the form of the potential is presented by \(V(\phi ) \propto \phi ^{k}\). The slowroll parameters for this model are presented as \(\varepsilon =k/4N\), \(\eta =(k1)/2N\) which leads to main perturbation parameters \(n_{s}=1(k+2)/2N\) and \(r=4k/N\). Using special case \(k=2\) and \(N=50\), we present \(n_{s}\simeq 0.96\) and \(r\simeq 0.16\). For \(N=60\) we find \(n_{s}\simeq 0.967\) and \(r\simeq 0.133\). It has been found that the monomial potentials with \(k\ge 2\) are not in agreement with the Planck data [1, 2].
 Hyperbolic model of inflation [29]: In hyperbolic inflation the potential is presented by \(V(\phi ) \propto \mathrm {sinh}^{b}(\phi /f_{1})\). Initially, this form of the potential was proposed for dark energy at the late time [30]. This potential of scalar field has been investigated back in the inflationary era [29] . The slowroll parameters are presented byand$$\begin{aligned} \varepsilon= & {} \frac{b^{2}M_{pl}^{2}}{2f_{1}^{2}}\mathrm {coth}^{2}(\phi /f_1),\\ \eta= & {} \frac{bM_{pl}^{2}}{f_{1}^{2}}\left[ (b1)\mathrm {coth}^{2}(\phi /f_1)+1\right] , \end{aligned}$$where \(\phi _{end}\simeq \frac{f}{2}\mathrm {ln}\left( \frac{\theta +1}{\theta 1}\right) \). Using observational data, it has been constrained the parameters of this model. \(n_{s}\simeq 0.968\), \(r\simeq 0.075\), \(1<b \le 1.5\) and \(f_1\ge 11.7M_{pl}\) [29]. It has been found in our study that warm Higgs inflation model is in agreement with the Planck data for some amounts of dissipation coefficient \(\varGamma _0\) although using observational data the Starobinsky model of inflation is the winner in comparison [1, 2].$$\begin{aligned} \phi =f_1\;\mathrm {cosh}^{1}\left[ e^{NbM_{pl}^{2}/f^{2}}\mathrm {cosh} (\phi _{end}/f_1)\right] . \end{aligned}$$
4 Conclusions
In this work, we studied the observational signatures of warm inflation in the Cosmic Microwave Background data given by Planck 2015. We utilized the paradigm of warm inflation with a Higgs scalar field which is nonminimally coupled to gravity. Within this framework at first, we provided the slowroll parameters and the power spectrum of scalar and tensor fluctuations respectively. Second, we checked the performance of warm Higgs inflationary model against the data provided by Planck2015 data and we found a class of patterns which are consistent with the observations. Finally we compare our model with current predictions with those of viable literature potentials.
Notes
Acknowledgements
I want to thank the Physics School of IPM, which I am a Resident Researcher there, and my colleagues from IPM: Mohammad Mehdi SheikJabbar, Nima Khosravi, Amjad Ashoorioon, Ali Akbar Abolhasani and Seyed Mohammad Sadegh Movahed for some valuable discussions. I want also thanks to my colleagues in BASU: Mohammad Malekjani, Ahmad Mehrabi for some discussions on cosmological perturbation.
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