# On Warm Natural Inflation and Planck 2018 Constraints

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## Abstract

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## 1. Introduction

## 2. Warm Inflation Setup

#### 2.1. Arena

#### 2.2. Power Spectrum

**Strong Limit $Q\gg 1$**:Here, using Equation (27), one can show that$$\begin{array}{ccc}\hfill T& =& {\left(\frac{{Z}^{2}{\left({U}_{\varphi}^{\prime}\right)}^{2}}{4H{C}_{\gamma}{\Gamma}_{0}}\right)}^{1/5}\hfill \end{array}$$We have via Equation (34):$$\begin{array}{ccc}\hfill {\omega}_{k}& =& T\sqrt{\frac{\pi \Gamma}{{H}^{3}}}=\frac{T}{H}\sqrt{3\pi Q}\hfill \end{array}$$Thus, $1+{\nu}_{k}\approx \frac{T}{H}\ll {\omega}_{k}$, and one gets:$$\begin{array}{ccc}\hfill {\Delta}_{R}=\Delta {R}_{s}\phantom{\rule{0.277778em}{0ex}}G& :& \Delta {R}_{s}=\frac{\sqrt{3}TH}{8\sqrt{\pi}}{\u03f5}^{2}{Q}^{\frac{5}{2}}\hfill \end{array}$$Thus, we get$$\begin{array}{ccc}\hfill {n}_{s}-1& =& \frac{1}{H\Delta {R}_{s}}\frac{d\Delta {R}_{s}}{dt}+\frac{\dot{Q}}{H}\frac{{G}_{Q}^{\prime}}{G}\hfill \end{array}$$The first term will give, after lengthy calculations look at [5]:$$\begin{array}{ccc}\hfill \frac{1}{H\Delta {R}_{s}}\frac{d\Delta {R}_{s}}{dt}& =& \frac{1}{Q}\left(-\frac{9}{4}\u03f5+\frac{3}{2}\eta -\frac{9}{4}\beta \right)\hfill \end{array}$$$$\begin{array}{ccc}\hfill \frac{\dot{Q}}{H}\frac{{G}_{Q}^{\prime}}{G}& =& \frac{2.315}{Q}(\u03f5-\beta )\hfill \end{array}$$$$\begin{array}{ccc}\hfill \frac{\dot{Q}}{HQ}=\frac{\dot{\Gamma}}{H\Gamma}-\frac{\dot{H}}{{H}^{2}}& =& \frac{-1}{1+Q}(\beta -\u03f5)\hfill \end{array}$$Thus, we get:$$\begin{array}{ccc}\hfill {n}_{s}-1& =& \frac{1}{Q}\left(-\frac{9}{4}\u03f5+\frac{3}{2}\eta -\frac{9}{4}\beta +2.3\u03f5-2.3\beta \right)\hfill \end{array}$$Note that ${n}_{s}$ involves the temperature T through the expression of $Q=\frac{{\Gamma}_{0}T}{3H}$. Additionally, T plays a role in determining the “end of inflation” field, ${\varphi}_{f}$ being the argument of the slow-roll parameter ($\u03f5,\eta ,\beta $) when it equals $1+Q=1+\frac{{\Gamma}_{0}T}{3H}$, whichever among the three meets the equality first. Determining ${\varphi}_{f}$ allows one to compute the e-folding number by:$$\begin{array}{c}\hfill {N}_{e}\equiv log\frac{{a}_{end}}{{a}_{k}}={\int}_{t}^{{t}_{f}}Hdt={\int}_{{\chi}_{k}}^{{\chi}_{f}}H\frac{d\chi}{\dot{\chi}}\approx {\int}_{{\chi}_{f}}^{{\chi}_{k}}\frac{U}{{U}_{\chi}^{\prime}}(1+Q)d\chi ={\int}_{{\varphi}_{f}}^{{\varphi}_{k}}\frac{U}{{U}_{\varphi}^{\prime}}(1+Q){Z}^{2}d\varphi \end{array}$$The initial time when the inflation started is taken to correspond to the horizon crossing when the dominant quantum fluctuations freeze, transforming into classical perturbations with observed power spectrum.As for the tensor-to-scalar ratio, we get$$\begin{array}{ccc}\hfill r& =& \frac{H}{T}\frac{16\u03f5}{{Q}^{5/2}}{G}^{-1}=\frac{H}{T}\frac{16\u03f5}{0.0185{Q}^{4.815}}\hfill \end{array}$$**Weak Limit $Q\ll 1$**Using Equation (27), one can show that$$\begin{array}{ccc}\hfill T& =& {\left(\frac{{Z}^{2}{\left({U}_{\varphi}^{\prime}\right)}^{2}{\Gamma}_{0}}{36{C}_{\gamma}{H}^{3}}\right)}^{1/3}\hfill \end{array}$$From Equation (34), we have$$\begin{array}{ccc}\hfill {\omega}_{k}& =& \frac{2\pi \Gamma T}{3{H}^{2}}=\frac{2\pi TQ}{H}\hfill \end{array}$$Thus, $1+{\nu}_{k}\approx \frac{T}{H}\gg {\omega}_{k}$, and one gets:$$\begin{array}{ccc}\hfill {\Delta}_{R}=\Delta {R}_{w}\phantom{\rule{0.277778em}{0ex}}G& :& \Delta {R}_{w}=\frac{4TH}{\pi}{\u03f5}^{2}\hfill \end{array}$$Thus, we get$$\begin{array}{ccc}\hfill {n}_{s}-1& =& \frac{1}{H\Delta {R}_{w}}\frac{d\Delta {R}_{w}}{dt}+\frac{\dot{Q}}{H}\frac{{G}_{Q}^{\prime}}{G}\hfill \end{array}$$The first term will give, after lengthy calculations look at [5]:$$\begin{array}{ccc}\hfill \frac{1}{H\Delta {R}_{w}}\frac{d\Delta {R}_{w}}{dt}& =& 1-6\u03f5+2\eta +\frac{{\omega}_{k}}{1+{\omega}_{k}}\left(\frac{15\u03f5-2\eta -9\beta}{4}\right)\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\omega}_{k}=\frac{2\pi TQ}{H}& \ll & 1,\hfill \end{array}$$$$\begin{array}{ccc}\hfill \frac{1}{H\Delta {R}_{w}}\frac{d\Delta {R}_{w}}{dt}& =& 1-6\u03f5+2\eta +\frac{2\pi {\Gamma}_{0}{T}^{2}}{3{H}^{2}}\left(\frac{15\u03f5-2\eta -9\beta}{4}\right)\hfill \end{array}$$As to the second term, we get using Equation (47)$$\begin{array}{ccc}\hfill \frac{\dot{Q}}{H}\frac{{G}_{Q}^{\prime}}{G}& =& 0.456{Q}^{1.364}(\u03f5-\beta )\hfill \end{array}$$Thus, we get:$$\begin{array}{ccc}\hfill {n}_{s}-1& =& 1-6\u03f5+2\eta +\frac{2\pi {\Gamma}_{0}{T}^{2}}{12{H}^{2}}(15\u03f5-2\eta -9\beta )+0.456{Q}^{1.364}(\u03f5-\beta )\hfill \end{array}$$As for the tensor-to-scalar ratio, we get, using $G\approx 1$, the following:$$\begin{array}{ccc}\hfill r& =& \frac{16\u03f5}{{(1+Q)}^{2}}{\mathcal{F}}^{-1}=\frac{8H\u03f5}{T}=\frac{8H{Z}^{2}{\u03f5}^{\varphi}}{T}\hfill \end{array}$$

## 3. Natural Inflation

- Quadratic NMC:$$\begin{array}{ccc}\hfill {\Omega}^{2}\left(\varphi \right)& =& 1+\xi {\varphi}^{2}\hfill \end{array}$$
- Periodic NMC$$\begin{array}{ccc}\hfill {\Omega}^{2}\left(\varphi \right)& =& 1+\lambda \left(1+cos\left(\frac{\varphi}{f}\right)\right)\hfill \end{array}$$

## 4. Comparison to Data: Strong Case

- Quadratic NMCFigure 1 shows the results of scanning the parameters space in the case of strong-limit warm NI with quadratic NMC to gravity. One could accommodate (${n}_{s},r$) but with too little ${N}_{e}$. In the figures, the two colors dots correspond to two choices of the coupling $\xi =-20,-40$.Looking to meet the e-folds constraint, we imposed (${N}_{e}=40$) with ($\xi =-20$), and fixed the values of (${\Gamma}_{0},f,{V}_{0}$) as before, while scanning over ${\varphi}_{*}$. We found the “bench mark”: (${\varphi}_{*}=0.0029$) giving the required e-folds with $r=1.03\times {10}^{-14}$ and Q in the order of $1.3\times {10}^{3}$. However, the scalar spectral index ${n}_{s}$ was large (${n}_{s}=0.98$) outside the acceptable contours.
- Periodic NMCFigure 2 shows the results of scanning the parameters space in the case of strong-limit warm NI with periodic NMC to gravity. As in the case of quadratic NMC, one could accommodate (${n}_{s},r$) but with too little ${N}_{e}$. In the figures, the three colors dots correspond to three choices of the coupling $\lambda (\times {10}^{-6})=5,6,8$.Again, one could meet the acceptable value (${N}_{e}=40$) with ($\lambda =5\times {10}^{-6}$) and the values of (${\Gamma}_{0},f,{V}_{0}$) as before, through scanning over ${\varphi}_{*}$, and finding a “bench mark”: (${\varphi}_{*}=5\times {10}^{-4}$) giving the required e-folds (${N}_{e}=40.14$) with $r=4.2\times {10}^{-20}$ and Q in the order of $1.3\times {10}^{4}$. However, the scalar spectral index ${n}_{s}$ was again large (${n}_{s}=0.98$) outside the acceptable contours.

## 5. Comparison to Data: Weak Case

## 6. Cubic Dissipative Term

- Non-periodic NMC:Scanning over ($\xi \in [19,22]$) with$$\begin{array}{c}{\varphi}_{*}=0.063,f=5,{\Gamma}_{0}=3\times {10}^{-9},{V}_{0}=2\times {10}^{-21},\end{array}$$$$\begin{array}{cc}{\varphi}_{end}\in [8.10,8.34],{A}_{s}\in [2.085,2.128]\times {10}^{-9},& \hfill \\ {n}_{s}\in [0.9433,0.9864],r\in [1.86,1.93]\times {10}^{-13},& \hfill \\ {N}_{e}\in [73.47,73.50],Q\in [28.48,28.73],& \hfill \\ T\in [9.8818,9.8895]\times {10}^{-7},H\in [3.3582,3.3954]\times {10}^{-11}\end{array}$$
- Periodic NMC: Scanning over ($\lambda \in [25,73]$) with$$\begin{array}{c}{\varphi}_{*}=9,f=5,{\Gamma}_{0}={10}^{10},{V}_{0}=1\times {10}^{-18},\end{array}$$$$\begin{array}{cc}{\varphi}_{end}\in [0.1267,0.3601],{A}_{s}\in [0.095,1.5]\times {10}^{-9},& \hfill \\ {n}_{s}\in [0.9519,0.9691],r\in [1.4,2.86]\times {10}^{-13},& \hfill \\ {N}_{e}\in [41,77],Q\in [20.47,23.72],& \hfill \\ T\in [3.7,5.6]\times {10}^{-7},H\in [0.884,2.49]\times {10}^{-11}\end{array}$$

## 7. Summary and Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Note

1 | Note however, that the “measurable” Hubble parameter in the Einstein frame will be the one corresponding to dropping the inhomogeneous term [34]. |

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**Figure 1.**Predictions of warm natural inflation with quadratic NMC to gravity in the strong limit. We took the values in units where Planck mass is unity (${\Gamma}_{0}=7000,f=5,{V}_{0}=5\times {10}^{-6}$). For the black (red) dots, we have $\xi =-20(-40),{\varphi}_{*}\in [3\times {10}^{-4},0.0015]$ ($\in [3\times {10}^{-4},0.0029]$) corresponding to ${N}_{e}\in [14.8,30.4](\in [9.3,26.4])$. Q in both cases is in the order of ${10}^{3}$.

**Figure 2.**Predictions of warm natural inflation with periodic NMC to gravity in the strong limit. We took the values in units where Planck mass is unity (${\Gamma}_{0}=1000,f=100,{V}_{0}=1\times {10}^{-6}$). For the red (black, pink) dots, we have $\lambda =5\times {10}^{6}(6\times {10}^{6},8\times {10}^{6}),{\varphi}_{*}\in [1\times {10}^{-4},1.5\times {10}^{-4}]$ ($\in [1\times {10}^{-4},1.8\times {10}^{-4}],\in [1\times {10}^{-4},2\times {10}^{-4}]$) corresponding to ${N}_{e}\in [24.5,29.3](\in [22.6,29.3],\in [19.9,21.6])$. Q in all cases is in the order of $6\times {10}^{3}$.

**Figure 3.**Predictions of warm natural inflation with NMC to gravity in the weak limit. For the quadratic (periodic) NMC in red (black) dots, we took (the values are given in units where Planck mass is unity): ${\Gamma}_{0}=7.14\times {10}^{-7},f=2,{V}_{0}=2.25\times {10}^{-15}$. We fixed ${\varphi}_{*}=2\left(6.9\right)$ and scanned over $\xi \left(\lambda \right)$ $\in [1.99,2.00]\left(\right[1.04,1.06\left]\right)$. We found ${n}_{s}\in [0.95,0.97]\left([0.95,0.97]\right)$, $r\in [0.015,0.016]\left(\right[0.0385,0.0386\left]\right)$, and we got ${N}_{e}\approx 0.96\left(0.27\right)$. In both cases of NMC, we had Q in the order of ${10}^{-4}$.

**Figure 4.**Above: Cubic dissipation case for both quadratic and periodic NMC, showing consistency with observational data regarding (${n}_{s},r$). Bottom: The quadratic NMC (

**a**) accommodates (${A}_{s},{N}_{e}$), whereas the periodic (

**b**) NMC accommodates ${N}_{e}$ and the magnitude order of ${A}_{s}$.

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**MDPI and ACS Style**

AlHallak, M.; Al-Said, K.K.; Chamoun, N.; El-Daher, M.S.
On Warm Natural Inflation and Planck 2018 Constraints. *Universe* **2023**, *9*, 80.
https://doi.org/10.3390/universe9020080

**AMA Style**

AlHallak M, Al-Said KK, Chamoun N, El-Daher MS.
On Warm Natural Inflation and Planck 2018 Constraints. *Universe*. 2023; 9(2):80.
https://doi.org/10.3390/universe9020080

**Chicago/Turabian Style**

AlHallak, Mahmoud, Khalil Kalid Al-Said, Nidal Chamoun, and Moustafa Sayem El-Daher.
2023. "On Warm Natural Inflation and Planck 2018 Constraints" *Universe* 9, no. 2: 80.
https://doi.org/10.3390/universe9020080