# Testing Predictions of the Quantum Landscape Multiverse 3: The Hilltop Inflationary Potential

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## Abstract

**:**

## 1. Introduction

## 2. The Modified Hilltop Potential

## 3. Analysis Method

`cosmomc`[26], with a convergence diagnostic based on the Gelman and Rubin statistic, where we modified the CAMB code [27], to include the primordial power spectrum of our model. It implements an efficient sampling of the posterior distribution using the fast/slow parameter decorrelations [28], and it includes the support for the Planck data release 2015 Likelihood Code [20] (see http://cosmologist.info/cosmomc/).

## 4. Results $\mathit{p}=\mathbf{4}$

## 5. Results $\mathit{p}=\mathbf{6}$

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**$68\%$ c.l. constraints on cosmological parameters considering the minimal standard cosmological $\Lambda $CDM model and its extensions, for different combinations of datasets.

Planck TT | Planck TTTEEE | Planck TT | Planck TTTEEE | Planck TT | Planck TTTEEE | |

+ lowP | + lowP | + lowP | + lowP | + lowP | + lowP | |

${\Omega}_{\mathrm{b}}{h}^{2}$ | $0.02224\phantom{\rule{0.166667em}{0ex}}\pm 0.00023$ | $0.02225\phantom{\rule{0.166667em}{0ex}}\pm 0.00016$ | $0.02230\phantom{\rule{0.166667em}{0ex}}\pm 0.00037$ | $0.02220\phantom{\rule{0.166667em}{0ex}}\pm 0.00024$ | $0.02228\phantom{\rule{0.166667em}{0ex}}\pm 0.00023$ | $0.02229\phantom{\rule{0.166667em}{0ex}}\pm 0.00016$ |

${\Omega}_{\mathrm{c}}{h}^{2}$ | $0.1195\phantom{\rule{0.166667em}{0ex}}\pm 0.0022$ | $0.1197\phantom{\rule{0.166667em}{0ex}}\pm 0.0014$ | $0.1205\phantom{\rule{0.166667em}{0ex}}\pm 0.0041$ | $0.1191\phantom{\rule{0.166667em}{0ex}}\pm 0.0031$ | $0.1195\phantom{\rule{0.166667em}{0ex}}\pm 0.0022$ | $0.1196\phantom{\rule{0.166667em}{0ex}}\pm 0.0015$ |

$\tau $ | $0.077\phantom{\rule{0.166667em}{0ex}}\pm 0.019$ | $0.078\phantom{\rule{0.166667em}{0ex}}\pm 0.017$ | $0.080\phantom{\rule{0.166667em}{0ex}}\pm 0.022$ | $0.077\phantom{\rule{0.166667em}{0ex}}\pm 0.018$ | $0.076\phantom{\rule{0.166667em}{0ex}}\pm 0.020$ | $0.075\phantom{\rule{0.166667em}{0ex}}\pm 0.017$ |

$log\left({10}^{10}{A}_{S}\right)$ | $3.087\phantom{\rule{0.166667em}{0ex}}\pm 0.036$ | $3.092\phantom{\rule{0.166667em}{0ex}}\pm 0.033$ | $3.096\phantom{\rule{0.166667em}{0ex}}\pm 0.047$ | $3.088\phantom{\rule{0.166667em}{0ex}}\pm 0.038$ | $3.085\phantom{\rule{0.166667em}{0ex}}\pm 0.037$ | $3.085\phantom{\rule{0.166667em}{0ex}}\pm 0.033$ |

${n}_{S}$ | $0.9666\phantom{\rule{0.166667em}{0ex}}\pm 0.0062$ | $0.9652\phantom{\rule{0.166667em}{0ex}}\pm 0.0047$ | $0.969\phantom{\rule{0.166667em}{0ex}}\pm 0.016$ | $0.9620\phantom{\rule{0.166667em}{0ex}}\pm 0.0097$ | $0.9660\phantom{\rule{0.166667em}{0ex}}\pm 0.0061$ | $0.9649\phantom{\rule{0.166667em}{0ex}}\pm 0.0048$ |

r | $<0.0472$ | $<0.0463$ | $\left(0\right)$ | $\left(0\right)$ | $\left(0\right)$ | $\left(0\right)$ |

${N}_{\mathrm{eff}}$ | $\left(3.046\right)$ | $\left(3.046\right)$ | $3.13{\phantom{\rule{0.166667em}{0ex}}}_{-0.34}^{+0.30}$ | $2.99\phantom{\rule{0.166667em}{0ex}}\pm 0.20$ | $\left(3.046\right)$ | $\left(3.046\right)$ |

w | $(-1)$ | $(-1)$ | $(-1)$ | $(-1)$ | $-1.54{\phantom{\rule{0.166667em}{0ex}}}_{-0.40}^{+0.20}$ | $-1.55{\phantom{\rule{0.166667em}{0ex}}}_{-0.38}^{+0.19}$ |

${H}_{0}$ | $67.42\phantom{\rule{0.166667em}{0ex}}\pm 0.99$ | $67.31\phantom{\rule{0.166667em}{0ex}}\pm 0.64$ | $68.0{\phantom{\rule{0.166667em}{0ex}}}_{-3.0}^{+2.6}$ | $66.8\phantom{\rule{0.166667em}{0ex}}\pm 1.6$ | $>80.9$ | $>81.3$ |

${\sigma}_{8}$ | $0.828\phantom{\rule{0.166667em}{0ex}}\pm 0.014$ | $0.830\phantom{\rule{0.166667em}{0ex}}\pm 0.013$ | $0.834{\phantom{\rule{0.166667em}{0ex}}}_{-0.025}^{+0.022}$ | $0.828\phantom{\rule{0.166667em}{0ex}}\pm 0.018$ | $0.98{\phantom{\rule{0.166667em}{0ex}}}_{-0.06}^{+0.11}$ | $0.98{\phantom{\rule{0.166667em}{0ex}}}_{-0.06}^{+0.10}$ |

Best Fit | Best Fit | Best Fit | Best Fit | Best Fit | Best Fit | |

${\Omega}_{\mathrm{b}}{h}^{2}$ | $0.02224$ | $0.02228$ | $0.02224$ | $0.02217$ | $0.02233$ | $0.02230$ |

${\Omega}_{\mathrm{c}}{h}^{2}$ | $0.1196$ | $0.1198$ | $0.1196$ | $0.1183$ | $0.1191$ | $0.1195$ |

$\tau $ | $0.080$ | $0.083$ | $0.078$ | $0.078$ | $0.078$ | $0.0747$ |

$log\left({10}^{10}{A}_{S}\right)$ | $3.093$ | $3.101$ | $3.089$ | $3.087$ | $3.088$ | $3.082$ |

${n}_{S}$ | $0.9663$ | $0.9659$ | $0.969$ | $0.961$ | $0.967$ | $0.9654$ |

r | $0.0000$ | $0.0001$ | $\left(0\right)$ | $\left(0\right)$ | $\left(0\right)$ | $\left(0\right)$ |

${N}_{\mathrm{eff}}$ | $\left(3.046\right)$ | $\left(3.046\right)$ | $3.04$ | $2.94$ | $\left(3.046\right)$ | $\left(3.046\right)$ |

w | $(-1)$ | $(-1)$ | $(-1)$ | $(-1)$ | $-1.94$ | $-1.95$ |

${H}_{0}$ | $67.38$ | $67.32$ | $67.34$ | $66.52$ | 100 | $99.9$ |

${\sigma}_{8}$ | $0.831$ | $0.834$ | $0.829$ | $0.826$ | $1.09$ | $1.09$ |

${\chi}^{2}$ | $11261.9$ | $12935.6$ | $11261.9$ | $12935.2$ | $11258.9$ | $12932.3$ |

**Table A2.**$95\%$ c.l. constraints on cosmological parameters considering the unmodified Hilltop inflationary model with $p=4$ for different combinations of datasets.

Planck TT | Planck TTTEEE | Planck TT | Planck TTTEEE | Planck TT | Planck TTTEEE | |

+ lowP | + lowP | + lowP | + lowP | + lowP | + lowP | |

${\Omega}_{\mathrm{b}}{h}^{2}$ | $0.02224{\phantom{\rule{0.166667em}{0ex}}}_{-0.00044}^{+0.00045}$ | $0.02223{\phantom{\rule{0.166667em}{0ex}}}_{-0.00030}^{+0.00031}$ | $0.02234{\phantom{\rule{0.166667em}{0ex}}}_{-0.00070}^{+0.00066}$ | $0.02220{\phantom{\rule{0.166667em}{0ex}}}_{-0.00046}^{+0.00047}$ | $0.02229\phantom{\rule{0.166667em}{0ex}}\pm 0.00045$ | $0.02226\phantom{\rule{0.166667em}{0ex}}\pm 0.00031$ |

${\Omega}_{\mathrm{c}}{h}^{2}$ | $0.1196{\phantom{\rule{0.166667em}{0ex}}}_{-0.0043}^{+0.0044}$ | $0.1198\phantom{\rule{0.166667em}{0ex}}\pm 0.0029$ | $0.1208{\phantom{\rule{0.166667em}{0ex}}}_{-0.0076}^{+0.0079}$ | $0.1192{\phantom{\rule{0.166667em}{0ex}}}_{-0.0060}^{+0.0062}$ | $0.1192{\phantom{\rule{0.166667em}{0ex}}}_{-0.0043}^{+0.0044}$ | $0.1195\phantom{\rule{0.166667em}{0ex}}\pm 0.0029$ |

$\tau $ | $0.077{\phantom{\rule{0.166667em}{0ex}}}_{-0.037}^{+0.038}$ | $0.079{\phantom{\rule{0.166667em}{0ex}}}_{-0.33}^{+0.032}$ | $0.081{\phantom{\rule{0.166667em}{0ex}}}_{-0.039}^{+0.041}$ | $0.077{\phantom{\rule{0.166667em}{0ex}}}_{-0.034}^{+0.035}$ | $0.076{\phantom{\rule{0.166667em}{0ex}}}_{-0.037}^{+0.036}$ | $0.074{\phantom{\rule{0.166667em}{0ex}}}_{-0.033}^{+0.034}$ |

${10}^{12}{V}_{0}/{M}^{4}$ | $<14.7$ | $<13.7$ | $<14.5$ | $<13.4$ | $<14.9$ | $<16.8$ |

${10}^{11}{\lambda}_{hill}$ | $0.307{\phantom{\rule{0.166667em}{0ex}}}_{-0.055}^{+0.076}$ | $0.311{\phantom{\rule{0.166667em}{0ex}}}_{-0.050}^{+0.069}$ | $0.308{\phantom{\rule{0.166667em}{0ex}}}_{-0.052}^{+0.072}$ | $0.309{\phantom{\rule{0.166667em}{0ex}}}_{-0.051}^{+0.069}$ | $0.311{\phantom{\rule{0.166667em}{0ex}}}_{-0.056}^{+0.074}$ | $0.314{\phantom{\rule{0.166667em}{0ex}}}_{-0.059}^{+0.084}$ |

${c}_{hill}$ | $0.0031\phantom{\rule{0.166667em}{0ex}}\pm 0.0013$ | $0.00318\phantom{\rule{0.166667em}{0ex}}\pm 0.00095$ | $<0.00499$ | $0.0034\phantom{\rule{0.166667em}{0ex}}\pm 0.0018$ | $0.0030\phantom{\rule{0.166667em}{0ex}}\pm 0.0012$ | $0.0031\phantom{\rule{0.166667em}{0ex}}\pm 0.0010$ |

r | $<0.116$ | $<0.108$ | $<0.116$ | $<0.105$ | $<0.118$ | $<0.131$ |

${N}_{\mathrm{eff}}$ | $\left(3.046\right)$ | $\left(3.046\right)$ | $3.17{\phantom{\rule{0.166667em}{0ex}}}_{-0.59}^{+0.55}$ | $3.00{\phantom{\rule{0.166667em}{0ex}}}_{-0.39}^{+0.41}$ | $\left(3.046\right)$ | $\left(3.046\right)$ |

w | $(-1)$ | $(-1)$ | $(-1)$ | $(-1)$ | $-1.54{\phantom{\rule{0.166667em}{0ex}}}_{-0.49}^{+0.59}$ | $-1.56{\phantom{\rule{0.166667em}{0ex}}}_{-0.46}^{+0.55}$ |

${H}_{0}$ | $67.4\phantom{\rule{0.166667em}{0ex}}\pm 1.9$ | $67.3\phantom{\rule{0.166667em}{0ex}}\pm 1.3$ | $68.4\phantom{\rule{0.166667em}{0ex}}\pm 5.0$ | $66.9\phantom{\rule{0.166667em}{0ex}}\pm 3.1$ | $>65$ | $>66$ |

${\sigma}_{8}$ | $0.828\phantom{\rule{0.166667em}{0ex}}\pm 0.028$ | $0.831\phantom{\rule{0.166667em}{0ex}}\pm 0.026$ | $0.835{\phantom{\rule{0.166667em}{0ex}}}_{-0.041}^{+0.042}$ | $0.828{\phantom{\rule{0.166667em}{0ex}}}_{-0.034}^{+0.035}$ | $0.98{\phantom{\rule{0.166667em}{0ex}}}_{-0.17}^{+0.14}$ | $0.98{\phantom{\rule{0.166667em}{0ex}}}_{-0.16}^{+0.13}$ |

Best Fit | Best Fit | Best Fit | Best Fit | Best Fit | Best Fit | |

${\Omega}_{\mathrm{b}}{h}^{2}$ | $0.02216$ | $0.02238$ | $0.02210$ | $0.02224$ | $0.02229$ | $0.02238$ |

${\Omega}_{\mathrm{c}}{h}^{2}$ | $0.1222$ | $0.1183$ | $0.1188$ | $0.1175$ | $0.1193$ | $0.1185$ |

$\tau $ | $0.074$ | $0.082$ | $0.073$ | $0.103$ | $0.088$ | $0.093$ |

${10}^{12}{V}_{0}/{M}^{4}$ | $0.1$ | $0.1$ | $2.8$ | $0.7$ | $3.0$ | $15.5$ |

${10}^{11}{\lambda}_{hill}$ | $0.277$ | $0.269$ | $0.296$ | $0.289$ | $0.299$ | $0.389$ |

${c}_{hill}$ | $0.0039$ | $0.0030$ | $0.0039$ | $0.0034$ | $0.0030$ | $0.0023$ |

r | $0.001$ | $0.001$ | $0.024$ | $0.006$ | $0.025$ | $0.119$ |

${N}_{\mathrm{eff}}$ | $\left(3.046\right)$ | $\left(3.046\right)$ | $2.91$ | $3.00$ | $\left(3.046\right)$ | $\left(3.046\right)$ |

w | $(-1)$ | $(-1)$ | $(-1)$ | $(-1)$ | $-0.89$ | $-1.11$ |

${H}_{0}$ | $66.3$ | $67.9$ | $66.0$ | $66.9$ | $64.3$ | $71.4$ |

${\sigma}_{8}$ | $0.838$ | $0.826$ | $0.822$ | $0.843$ | $0.805$ | $0.870$ |

${\chi}^{2}$ | $11264.4$ | $12944.8$ | $11263.9$ | $12943.9$ | $11263.7$ | $12942.6$ |

**Table A3.**$95\%$ c.l. constraints on cosmological parameters considering the unmodified Hilltop inflationary model with $p=6$ for different combinations of datasets.

Planck TT | Planck TTTEEE | Planck TT | Planck TTTEEE | Planck TT | Planck TTTEEE | |

+ lowP | + lowP | + lowP | + lowP | + lowP | + lowP | |

${\Omega}_{\mathrm{b}}{h}^{2}$ | $0.02233{\phantom{\rule{0.166667em}{0ex}}}_{-0.00044}^{+0.00045}$ | $0.02226\phantom{\rule{0.166667em}{0ex}}\pm 0.00031$ | $0.02264{\phantom{\rule{0.166667em}{0ex}}}_{-0.00044}^{+0.00021}$ | $0.02258{\phantom{\rule{0.166667em}{0ex}}}_{-0.00041}^{+0.00037}$ | $0.02239\phantom{\rule{0.166667em}{0ex}}\pm 0.00045$ | $0.02230{\phantom{\rule{0.166667em}{0ex}}}_{-0.00030}^{+0.00031}$ |

${\Omega}_{\mathrm{c}}{h}^{2}$ | $0.1158{\phantom{\rule{0.166667em}{0ex}}}_{-0.0038}^{+0.0039}$ | $0.1179\phantom{\rule{0.166667em}{0ex}}\pm 0.0027$ | $0.1234{\phantom{\rule{0.166667em}{0ex}}}_{-0.0065}^{+0.0068}$ | $0.1232\phantom{\rule{0.166667em}{0ex}}\pm 0.0057$ | $0.1156{\phantom{\rule{0.166667em}{0ex}}}_{-0.0038}^{+0.0039}$ | $0.1176{\phantom{\rule{0.166667em}{0ex}}}_{-0.0028}^{+0.0027}$ |

$\tau $ | $0.049\phantom{\rule{0.166667em}{0ex}}\pm 0.030$ | $0.044\phantom{\rule{0.166667em}{0ex}}\pm 0.027$ | $0.055{\phantom{\rule{0.166667em}{0ex}}}_{-0.029}^{+0.026}$ | $0.055{\phantom{\rule{0.166667em}{0ex}}}_{-0.029}^{+0.027}$ | $0.049\phantom{\rule{0.166667em}{0ex}}\pm 0.029$ | $0.043{\phantom{\rule{0.166667em}{0ex}}}_{-0.026}^{+0.027}$ |

${10}^{12}{V}_{0}/{M}^{4}$ | $48\phantom{\rule{0.166667em}{0ex}}\pm 10$ | $50\phantom{\rule{0.166667em}{0ex}}\pm 10$ | $46\phantom{\rule{0.166667em}{0ex}}\pm 10$ | $48{\phantom{\rule{0.166667em}{0ex}}}_{-9}^{+10}$ | $48\phantom{\rule{0.166667em}{0ex}}\pm 10$ | $49{\phantom{\rule{0.166667em}{0ex}}}_{-9}^{+10}$ |

${10}^{11}{\lambda}_{hill}$ | $>0.953$ | $>0.962$ | $>0.941$ | $>0.957$ | $>0.948$ | $>0.961$ |

${c}_{hill}$ | $0.00177{\phantom{\rule{0.166667em}{0ex}}}_{-0.00097}^{+0.00093}$ | ${0.00220}_{-0.00077}^{+0.00075}$ | $<0.00127$ | $<0.00212$ | $0.00174{\phantom{\rule{0.166667em}{0ex}}}_{-0.00097}^{+0.00091}$ | $0.00216{\phantom{\rule{0.166667em}{0ex}}}_{-0.00077}^{+0.00075}$ |

r | $0.398{\phantom{\rule{0.166667em}{0ex}}}_{-0.057}^{+0.058}$ | $0.405{\phantom{\rule{0.166667em}{0ex}}}_{-0.056}^{+0.059}$ | $0.386{\phantom{\rule{0.166667em}{0ex}}}_{-0.057}^{+0.061}$ | $0.399{\phantom{\rule{0.166667em}{0ex}}}_{-0.055}^{+0.057}$ | $0.398{\phantom{\rule{0.166667em}{0ex}}}_{-0.057}^{+0.059}$ | $0.402{\phantom{\rule{0.166667em}{0ex}}}_{-0.055}^{+0.060}$ |

${N}_{\mathrm{eff}}$ | $\left(3.046\right)$ | $\left(3.046\right)$ | $3.56{\phantom{\rule{0.166667em}{0ex}}}_{-0.32}^{+0.30}$ | $3.42{\phantom{\rule{0.166667em}{0ex}}}_{-0.35}^{+0.31}$ | $\left(3.046\right)$ | $\left(3.046\right)$ |

w | $(-1)$ | $(-1)$ | $(-1)$ | $(-1)$ | $-1.61{\phantom{\rule{0.166667em}{0ex}}}_{-0.32}^{+0.40}$ | $-1.66{\phantom{\rule{0.166667em}{0ex}}}_{-0.31}^{+0.41}$ |

${H}_{0}$ | $69.0\phantom{\rule{0.166667em}{0ex}}\pm 1.8$ | $68.0{\phantom{\rule{0.166667em}{0ex}}}_{-1.2}^{+1.3}$ | $72.1{\phantom{\rule{0.166667em}{0ex}}}_{-2.2}^{+1.9}$ | $70.6{\phantom{\rule{0.166667em}{0ex}}}_{-2.5}^{+2.3}$ | $>81$ | $>81$ |

${\sigma}_{8}$ | $0.792{\phantom{\rule{0.166667em}{0ex}}}_{-0.021}^{+0.022}$ | $0.796\phantom{\rule{0.166667em}{0ex}}\pm 0.020$ | $0.818{\phantom{\rule{0.166667em}{0ex}}}_{-0.028}^{+0.027}$ | $0.819{\phantom{\rule{0.166667em}{0ex}}}_{-0.030}^{+0.027}$ | $0.97{\phantom{\rule{0.166667em}{0ex}}}_{-0.12}^{+0.09}$ | $0.98{\phantom{\rule{0.166667em}{0ex}}}_{-0.11}^{+0.09}$ |

Best Fit | Best Fit | Best Fit | Best Fit | Best Fit | Best Fit | |

${\Omega}_{\mathrm{b}}{h}^{2}$ | $0.02223$ | $0.02223$ | $0.02249$ | $0.02246$ | $0.02236$ | $0.02223$ |

${\Omega}_{\mathrm{c}}{h}^{2}$ | $0.1170$ | $0.1179$ | $0.1183$ | $0.1224$ | $0.1151$ | $0.1181$ |

$\tau $ | $0.041$ | $0.034$ | $0.031$ | $0.041$ | $0.024$ | $0.033$ |

${10}^{12}{V}_{0}/{M}^{4}$ | 50 | 54 | 51 | 48 | 52 | 54 |

${10}^{11}{\lambda}_{hill}$ | $0.965$ | $0.933$ | $0.880$ | $0.957$ | $0.885$ | $0.938$ |

${c}_{hill}$ | $0.0020$ | $0.0021$ | $0.0007$ | $0.0013$ | $0.0015$ | $0.0022$ |

r | $0.41$ | $0.45$ | $0.45$ | $0.41$ | $0.45$ | $0.44$ |

${N}_{\mathrm{eff}}$ | $\left(3.046\right)$ | $\left(3.046\right)$ | $3.32$ | $3.35$ | $\left(3.046\right)$ | $\left(3.046\right)$ |

w | $(-1)$ | $(-1)$ | $(-1)$ | $(-1)$ | $-1.20$ | $-1.15$ |

${H}_{0}$ | $68.4$ | $68.0$ | $71.3$ | $69.9$ | $75.9$ | $72.7$ |

${\sigma}_{8}$ | $0.790$ | $0.788$ | $0.782$ | $0.806$ | $0.826$ | $0.831$ |

${\chi}^{2}$ | $11296.6$ | $12979.9$ | $11288.7$ | $12979.0$ | $11288.3$ | $12973.5$ |

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**Figure 1.**The temperature CMB angular power spectrum by varying the SUSY-breaking scale b associated with the landscape effects, for the modified Hilltop model with $p=4$ (

**upper panels**) and $p=6$ (

**bottom panels**).

**Figure 2.**The polarization CMB angular power spectra by varying the SUSY-breaking scale b associated with the landscape effects, for the modified Hilltop model with $p=4$ (

**upper panels**) and $p=6$ (

**bottom panels**).

**Figure 3.**The matter power spectrum by varying the SUSY-breaking scale b associated with the landscape effects, for the modified Hilltop model with $p=4$ (

**left panel**) and $p=6$ (

**right panel**).

**Figure 4.**Constraints at $68\%$ and $95\%$ confidence levels on the ${12}^{10}{V}_{0}/{M}^{4}$ vs. $log\left(b\right[GeV\left]\right)$ plane, in our modified $\Lambda $CDM+r Hilltop inflation with $p=4$. Looking at the Table 2, we can see that the best fits for these parameters are ${12}^{10}{V}_{0}/{M}^{4}=0.70$ and $log\left(b\right[GeV\left]\right)=11.8$ for PlanckTT+lowP, while they are ${12}^{10}{V}_{0}/{M}^{4}=0.18$ and $log\left(b\right[GeV\left]\right)=15.7$ for PlanckTTTEEE+lowP.

**Figure 5.**Constraints at $68\%$ and $95\%$ confidence levels in our modified $\Lambda $CDM+r+${N}_{\mathrm{eff}}$ Hilltop scenario with $p=4$. Looking at the Table 3, we can see that the best fits for these parameters are ${12}^{10}{V}_{0}/{M}^{4}=0.49$, $log\left(b\right[GeV\left]\right)=15.3$, ${N}_{\mathrm{eff}}=3.00$ and ${H}_{0}=66.9$ for PlanckTT+lowP, while they are ${12}^{10}{V}_{0}/{M}^{4}=0.25$, $log\left(b\right[GeV\left]\right)=8.4$, ${N}_{\mathrm{eff}}=3.01$ and ${H}_{0}=66.9$ for PlanckTTTEEE+lowP.

**Figure 6.**Comparison of the temperature CMB angular power spectrum computed for the best fit of our modified Hilltop model with $p=6$ (magenta), the best fit of our modified Hilltop model with $p=4$ (red), and the best fit obtained with a minimal standard cosmological model $\Lambda $CDM+r (cyan), with Planck 2015 TT+lowP data (points with error bars). The main differences between the two models are at lower-ℓ and on the amplitude of the peaks that the Hilltop model with $p=6$, modified for the entanglement, prefers slightly lower.

**Figure 7.**Comparison of the polarization CMB angular power spectra computed for the best fit of our modified Hilltop model with $p=6$ (magenta), the best fit of our modified Hilltop model with $p=4$ (red), and the best fit obtained with a minimal standard cosmological model $\Lambda $CDM+r (cyan), with Planck 2015 TT+lowP data (points with error bars).

**Figure 8.**Constraints at $68\%$ and $95\%$ confidence levels on the ${\sigma}_{8}$ vs. $\tau $ plane, in our modified $\Lambda $CDM+r Hilltop inflation with $p=6$. Looking at the Table 5, we can see that the best fits for these parameters are ${\sigma}_{8}=0.793$ and $\tau =0.043$ for PlanckTT+lowP, while they are ${\sigma}_{8}=0.798$ and $\tau =0.044$ for PlanckTTTEEE+lowP.

**Figure 9.**1D posteriors for our modified $\Lambda $CDM+r Hilltop inflation model with $p=6$ (black solid line) and the standard $\Lambda $CDM model (red solid line). The region between the grey dashed lines is the $1\sigma $ constraint obtained by KiDS-450.

**Figure 10.**Constraints at $68\%$ and $95\%$ confidence levels on the ${12}^{10}{V}_{0}/{M}^{4}$ vs. $Log\left(b\right[GeV\left]\right)$ plane, in our modified $\Lambda $CDM+r Hilltop inflation with $p=6$. Looking at the Table 5, we can see that the best fits for these parameters are ${12}^{10}{V}_{0}/{M}^{4}=46$ and $log\left(b\right[GeV\left]\right)=11.7$ for PlanckTT+lowP, while they are ${12}^{10}{V}_{0}/{M}^{4}=48$ and $log\left(b\right[GeV\left]\right)=18.8$ for PlanckTTTEEE+lowP.

**Figure 11.**Constraints at $68\%$ and $95\%$ confidence levels in our modified $\Lambda $CDM+r+${N}_{\mathrm{eff}}$ Hilltop scenario with $p=6$. Looking at the Table 6, we can see that the best fits for these parameters are ${12}^{10}{V}_{0}/{M}^{4}=40$, $log\left(b\right[GeV\left]\right)=11$, ${N}_{\mathrm{eff}}=3.66$ and ${H}_{0}=73.1$ for PlanckTT+lowP, while they are ${12}^{10}{V}_{0}/{M}^{4}=48$, $log\left(b\right[GeV\left]\right)=18.9$, ${N}_{\mathrm{eff}}=3.61$ and ${H}_{0}=71.8$ for PlanckTTTEEE+lowP.

Parameter | $\mathit{p}=4$ | $\mathit{p}=6$ |
---|---|---|

${\Omega}_{\mathrm{b}}{h}^{2}$ | $[0.005,0.1]$ | $[0.005,0.1]$ |

${\Omega}_{\mathrm{cdm}}{h}^{2}$ | $[0.001,0.99]$ | $[0.001,0.99]$ |

${\Theta}_{\mathrm{s}}$ | $[0.5,10]$ | $[0.5,10]$ |

$\tau $ | $[0.01,0.8]$ | $[0.01,0.8]$ |

$log\left(b\right[GeV\left]\right)$ | $[1,19]$ | $[1,19]$ |

${10}^{12}{V}_{0}/{M}^{4}$ | $[0.02,40]$ | $[10,80]$ |

${10}^{11}{\lambda}_{hill}$ | $[0.2,0.6]$ | $[0.2,1.0]$ |

${c}_{hill}$ | $[0,1]$ | $[0,0.05]$ |

${N}_{\mathrm{eff}}$ | $[0.05,10]$ | $[0.05,10]$ |

w | $[-3.0,0.3]$ | $[-3.0,0.3]$ |

**Table 2.**$95\%$ c.l. constraints on cosmological parameters in our baseline $\Lambda $CDM+r scenario from different combinations of datasets with a modified Hilltop inflation with $p=4$.

Planck TT | Best Fit | Best Fit | Best Fit | |||

+ lowP | + lowP | + lowP + BAO | + lowP + BAO | + tau055 | + tau055 | |

${\Omega}_{\mathrm{b}}{h}^{2}$ | ${0.02225}_{-0.00044}^{+0.00047}$ | $0.02225$ | $0.02227{\phantom{\rule{0.166667em}{0ex}}}^{+0.00040}-0.00039$ | $0.02224$ | ${0.02210}_{-0.00044}^{+0.00042}$ | $0.02193$ |

${\Omega}_{\mathrm{c}}{h}^{2}$ | ${0.1195}_{-0.0042}^{+0.0044}$ | $0.1199$ | $0.1189{\phantom{\rule{0.166667em}{0ex}}}_{-0.0026}^{+0.0025}$ | $0.1195$ | ${0.1213}_{-0.0042}^{+0.0044}$ | $0.1228$ |

$\tau $ | ${0.078}_{-0.037}^{+0.037}$ | $0.076$ | $0.080{\phantom{\rule{0.166667em}{0ex}}}_{-0.034}^{+0.036}$ | $0.084$ | $0.059{\phantom{\rule{0.166667em}{0ex}}}_{-0.018}^{+0.017}$ | $0.059$ |

${10}^{12}{V}_{0}/{M}^{4}$ | $<11.7$ | $0.70$ | $<12.5$ | $0.53$ | $<29.2$ | $5.6$ |

$log\left(b\right[GeV\left]\right)$ | $>6.75$ | $11.8$ | $>6.86$ | $13.3$ | $>7.44$ | $10.2$ |

${10}^{11}{\lambda}_{hill}$ | ${0.304}_{-0.048}^{+0.059}$ | $0.276$ | ${0.304}_{-0.048}^{+0.063}$ | $0.280$ | ${0.36}_{-0.10}^{+0.12}$ | $0.31$ |

${c}_{hill}$ | $0.0031\phantom{\rule{0.166667em}{0ex}}\pm 0.0012$ | $0.0033$ | ${0.00295}_{-0.00089}^{+0.00094}$ | $0.0034$ | ${0.0033}_{-0.0012}^{+0.0013}$ | $0.0037$ |

r | $<0.0941$ | $0.0061$ | $<0.101$ | $0.0046$ | $<0.219$ | $0.047$ |

${H}_{0}$ | $67.4\pm 1.9$ | $67.2$ | ${67.7}_{-1.1}^{+1.2}$ | $67.5$ | ${66.6}_{-1.8}^{+1.8}$ | $66.0$ |

${\sigma}_{8}$ | ${0.829}_{-0.028}^{+0.029}$ | $0.830$ | $0.828{\phantom{\rule{0.166667em}{0ex}}}_{-0.028}^{+0.029}$ | $0.836$ | ${0.820}_{-0.020}^{+0.021}$ | $0.826$ |

${\chi}^{2}$ | $11266.6$ | $11271.1$ | $771.4$ | |||

Planck TTTEEE | Best Fit | Best Fit | Best Fit | |||

+ lowP | + lowP | + lowP + BAO | + lowP + BAO | + tau055 | + tau055 | |

${\Omega}_{\mathrm{b}}{h}^{2}$ | ${0.02224}_{-0.00032}^{+0.00032}$ | $0.02227$ | ${0.02228}_{-0.00026}^{+0.00028}$ | $0.02234$ | ${0.02216}_{-0.00028}^{+0.00030}$ | $0.02226$ |

${\Omega}_{\mathrm{c}}{h}^{2}$ | ${0.1198}_{-0.0030}^{+0.0028}$ | $0.1193$ | ${0.1192}_{-0.0021}^{+0.0021}$ | $0.1181$ | $0.1208\phantom{\rule{0.166667em}{0ex}}\pm 0.0027$ | $0.1205$ |

$\tau $ | ${0.078}_{-0.033}^{+0.034}$ | $0.091$ | ${0.082}_{-0.032}^{+0.032}$ | $0.091$ | ${0.061}_{-0.016}^{+0.017}$ | $0.073$ |

${10}^{12}{V}_{0}/{M}^{4}$ | $<13.4$ | $0.18$ | $<12.4$ | $0.17$ | $<26.6$ | $11.6$ |

$log\left(b\right[GeV\left]\right)$ | $>6.90$ | $15.7$ | $>6.80$ | $14.8$ | $>7.24$ | $16.8$ |

${10}^{11}{\lambda}_{hill}$ | ${0.310}_{-0.049}^{+0.068}$ | $0.279$ | ${0.307}_{-0.047}^{+0.063}$ | $0.274$ | ${0.35}_{-0.09}^{+0.11}$ | $0.362$ |

${c}_{hill}$ | ${0.00318}_{-0.00099}^{+0.00094}$ | $0.0033$ | ${0.00306}_{-0.00079}^{+0.00085}$ | $0.0030$ | $0.0033\pm 0.0011$ | $0.0033$ |

r | $<0.106$ | $0.0016$ | $<0.0983$ | $0.0014$ | $<0.199$ | $0.091$ |

${H}_{0}$ | ${67.3}_{-1.2}^{+1.4}$ | $67.4$ | ${67.54}_{-0.94}^{+0.93}$ | $68.0$ | $66.8\phantom{\rule{0.166667em}{0ex}}\pm 1.2$ | $67.1$ |

${\sigma}_{8}$ | ${0.830}^{+0.026}-0.025$ | $0.839$ | $0.831\phantom{\rule{0.166667em}{0ex}}\pm 0.026$ | $0.833$ | $0.820{\phantom{\rule{0.166667em}{0ex}}}_{-0.015}^{+0.016}$ | $0.827$ |

${\chi}^{2}$ | $12943.7$ | $12949.1$ | $2451.7$ |

**Table 3.**$95\%$ c.l. constraints on cosmological parameters in our baseline $\Lambda $CDM+r+${N}_{\mathrm{eff}}$ scenario from different combinations of datasets with a modified Hilltop inflation with $p=4$.

Planck TT | Best Fit | Best Fit | Best Fit | |||

+ lowP | + lowP | + lowP + BAO | + lowP + BAO | + tau055 | + tau055 | |

${\Omega}_{\mathrm{b}}{h}^{2}$ | ${0.02236}_{-0.00067}^{+0.00065}$ | $0.02204$ | ${0.02236}_{-0.00047}^{+0.00048}$ | $0.02221$ | ${0.02183}_{-0.00072}^{+0.00069}$ | $0.2154$ |

${\Omega}_{\mathrm{c}}{h}^{2}$ | ${0.1210}_{-0.0077}^{+0.0075}$ | $0.1203$ | ${0.1210}_{-0.0075}^{+0.0078}$ | $0.1191$ | ${0.1178}_{-0.0078}^{+0.0077}$ | $0.1151$ |

$\tau $ | ${0.082}_{-0.042}^{+0.041}$ | $0.078$ | ${0.082}_{-0.036}^{+0.035}$ | $0.075$ | $0.058{\phantom{\rule{0.166667em}{0ex}}}_{-0.018}^{+0.017}$ | $0.062$ |

${10}^{12}{V}_{0}/{M}^{4}$ | $<11.7$ | $0.49$ | $<11.7$ | $1.8$ | $<27.1$ | $7.3$ |

$log\left(b\right[GeV\left]\right)$ | $>6.42$ | $15.3$ | $>6.42$ | $18.2$ | $>7.11$ | $11.7$ |

${10}^{11}{\lambda}_{hill}$ | ${0.302}_{-0.047}^{+0.061}$ | $0.278$ | ${0.302}_{-0.049}^{+0.063}$ | $0.288$ | ${0.35}_{-0.09}^{+0.12}$ | $0.342$ |

${c}_{hill}$ | $<0.00482$ | $0.0036$ | ${0.0025}_{-0.0017}^{+0.0016}$ | $0.0036$ | ${0.0046}_{-0.0031}^{+0.0032}$ | $0.0055$ |

r | $<0.0928$ | $0.016$ | $<0.0941$ | $0.0013$ | $<0.197$ | $0.057$ |

${N}_{\mathrm{eff}}$ | ${3.18}_{-0.57}^{+0.58}$ | $3.00$ | ${3.18}_{-0.44}^{+0.47}$ | $3.19$ | ${2.74}_{-0.60}^{+0.64}$ | $2.52$ |

${H}_{0}$ | ${68.5}_{-4.9}^{+5.0}$ | $66.9$ | ${68.5}_{-2.9}^{+3.0}$ | $68.3$ | ${64.1}_{-5.4}^{+5.5}$ | $62.5$ |

${\sigma}_{8}$ | ${0.836}_{-0.042}^{+0.042}$ | $0.826$ | ${0.836}_{-0.038}^{+0.040}$ | $0.839$ | ${0.809}_{-0.026}^{+0.026}$ | $0.808$ |

${\chi}^{2}$ | $11268.0$ | $11271.7$ | $770.4$ | |||

Planck TTTEEE | Best Fit | Best Fit | Best Fit | |||

+ lowP | + lowP | + lowP + BAO | + lowP + BAO | + tau055 | + tau055 | |

${\Omega}_{\mathrm{b}}{h}^{2}$ | ${0.02219}_{-0.00046}^{+0.00049}$ | $0.02216$ | ${0.02230}_{-0.00037}^{+0.00038}$ | $0.02238$ | ${0.02195}_{-0.00046}^{+0.00045}$ | $0.02192$ |

${\Omega}_{\mathrm{c}}{h}^{2}$ | ${0.1192}_{-0.0059}^{+0.0062}$ | $0.1194$ | ${0.1196}_{-0.0061}^{+0.0061}$ | $0.1189$ | ${0.1177}_{-0.0060}^{+0.0063}$ | $0.1154$ |

$\tau $ | $0.077\pm 0.035$ | $0.067$ | ${0.082}_{-0.031}^{+0.030}$ | $0.087$ | $0.059\pm 0.017$ | $0.054$ |

${10}^{12}{V}_{0}/{M}^{4}$ | $<10.5$ | $0.25$ | $<12.0$ | $1.5$ | $<24.4$ | $5.7$ |

$log\left(b\right[GeV\left]\right)$ | $>6.85$ | $8.4$ | $>6.77$ | $9.0$ | $>7.27$ | $18.1$ |

${10}^{11}{\lambda}_{hill}$ | ${0.303}_{-0.044}^{+0.055}$ | $0.269$ | ${0.306}_{-0.047}^{+0.061}$ | $0.288$ | ${0.34}_{-0.08}^{+0.11}$ | $0.316$ |

${c}_{hill}$ | ${0.0035}_{-0.0018}^{+0.0018}$ | $0.0035$ | ${0.0030}_{-0.0014}^{+0.0015}$ | $0.0033$ | ${0.0042}_{-0.0019}^{+0.0020}$ | $0.0046$ |

r | $<0.0841$ | $0.0022$ | $<0.0957$ | $0.013$ | $<0.182$ | $0.047$ |

${N}_{\mathrm{eff}}$ | ${2.99}_{-0.39}^{+0.41}$ | $3.01$ | ${3.07}_{-0.36}^{+0.37}$ | $3.02$ | ${2.81}_{-0.38}^{+0.42}$ | $2.68$ |

${H}_{0}$ | ${66.9}_{-3.0}^{+3.2}$ | $66.9$ | $67.7\pm 2.4$ | $67.4$ | ${65.1}_{-3.0}^{+3.2}$ | $64.4$ |

${\sigma}_{8}$ | ${0.827}_{-0.034}^{+0.036}$ | $0.822$ | $0.832\phantom{\rule{0.166667em}{0ex}}\pm 0.032$ | $0.834$ | $0.810\phantom{\rule{0.166667em}{0ex}}\pm 0.024$ | $0.799$ |

${\chi}^{2}$ | $12946.0$ | $12948.3$ | $2447.9$ |

**Table 4.**$95\%$ c.l. constraints on cosmological parameters in our baseline wCDM+r scenario from different combinations of datasets with a modified Hilltop inflation with $p=4$.

Planck TT | Best Fit | Best Fit | Best Fit | |||

+ lowP | + lowP | + lowP + BAO | + lowP + BAO | + tau055 | + tau055 | |

${\Omega}_{\mathrm{b}}{h}^{2}$ | ${0.02228}_{-0.00044}^{+0.00048}$ | $0.02236$ | ${0.02225}_{-0.00041}^{+0.00043}$ | $0.02249$ | $0.02215\pm 0.00042$ | $0.02222$ |

${\Omega}_{\mathrm{c}}{h}^{2}$ | ${0.1193}_{-0.0045}^{+0.0043}$ | $0.1193$ | $0.1192\pm 0.0038$ | $0.1176$ | ${0.1210}_{-0.0042}^{+0.0041}$ | $0.1211$ |

$\tau $ | ${0.076}_{-0.037}^{+0.039}$ | $0.086$ | ${0.078}_{-0.037}^{+0.037}$ | $0.078$ | $0.058{\phantom{\rule{0.166667em}{0ex}}}_{-0.016}^{+0.017}$ | $0.054$ |

${10}^{12}{V}_{0}/{M}^{4}$ | $<12.5$ | $1.6$ | $<12.0$ | $3.8$ | $<28.3$ | $1.2$ |

$log\left(b\right[GeV\left]\right)$ | $>6.91$ | $7.4$ | $>6.70$ | $17.0$ | $>7.12$ | $14.4$ |

${10}^{11}{\lambda}_{hill}$ | ${0.306}_{-0.050}^{+0.063}$ | $0.284$ | ${0.303}_{-0.049}^{+0.062}$ | $0.294$ | ${0.35}_{-0.10}^{+0.12}$ | $0.273$ |

${c}_{hill}$ | ${0.0030}_{-0.0012}^{+0.0012}$ | $0.0030$ | $0.0030\pm 0.0011$ | $0.0024$ | ${0.0032}_{-0.0012}^{+0.0012}$ | $0.0038$ |

r | $<0.100$ | $0.013$ | $<0.0972$ | $0.033$ | $<0.210$ | $0.010$ |

w | ${-1.53}_{-0.49}^{+0.60}$ | $-1.78$ | ${-1.02}_{-0.15}^{+0.15}$ | $-0.97$ | ${-1.48}_{-0.60}^{+0.69}$ | $-1.68$ |

${H}_{0}$ | $>65$ | $94.1$ | ${68.1}_{-3.3}^{+3.4}$ | $67.3$ | $>62$ | $88.3$ |

${\sigma}_{8}$ | ${0.98}_{-0.17}^{+0.14}$ | $1.06$ | ${0.834}_{-0.050}^{+0.053}$ | $0.812$ | ${0.95}_{-0.19}^{+0.16}$ | $1.00$ |

${\chi}^{2}$ | $11262.7$ | $11272.6$ | $769.9$ | |||

Planck TTTEEE | Best Fit | Best Fit | Best Fit | |||

+ lowP | + lowP | + lowP + BAO | + lowP + BAO | + tau055 | + tau055 | |

${\Omega}_{\mathrm{b}}{h}^{2}$ | ${0.02227}_{-0.00030}^{+0.00031}$ | $0.02231$ | ${0.02225}_{-0.00029}^{+0.00030}$ | $0.02230$ | ${0.02218}_{-0.00028}^{+0.00029}$ | $0.02231$ |

${\Omega}_{\mathrm{c}}{h}^{2}$ | ${0.1196}_{-0.0028}^{+0.0029}$ | $0.1195$ | ${0.1197}_{-0.0027}^{+0.0029}$ | $0.1195$ | ${0.1206}_{-0.0028}^{+0.0028}$ | $0.1201$ |

$\tau $ | ${0.074}_{-0.034}^{+0.032}$ | $0.082$ | $0.078\pm 0.033$ | $0.090$ | ${0.060}_{-0.017}^{+0.017}$ | $0.055$ |

${10}^{12}{V}_{0}/{M}^{4}$ | $<13.5$ | $0.12$ | $<11.9$ | $0.62$ | $<23.4$ | $5.5$ |

$log\left(b\right[GeV\left]\right)$ | $>6.96$ | $13.5$ | $>6.93$ | $14.6$ | $>7.40$ | $14.5$ |

${10}^{11}{\lambda}_{hill}$ | ${0.308}_{-0.051}^{+0.066}$ | $0.275$ | ${0.306}_{-0.048}^{+0.061}$ | $0.281$ | ${0.343}_{-0.087}^{+0.097}$ | $0.307$ |

${c}_{hill}$ | ${0.0031}_{-0.0009}^{+0.0010}$ | $0.0034$ | ${0.00320}_{-0.00096}^{+0.00096}$ | $0.0032$ | $0.0032\pm 0.0010$ | $0.0034$ |

r | $<0.107$ | $0.0010$ | $<0.0947$ | $0.0052$ | $<0.178$ | $0.046$ |

w | ${-1.56}_{-0.47}^{+0.59}$ | $-1.36$ | ${-1.03}_{-0.13}^{+0.11}$ | $-1.04$ | ${-1.54}_{-0.52}^{+0.65}$ | $-1.48$ |

${H}_{0}$ | $>66$ | $78.5$ | ${68.3}_{-2.9}^{+3.2}$ | $68.8$ | $>64$ | $82.4$ |

${\sigma}_{8}$ | ${0.99}_{-0.17}^{+0.13}$ | $0.934$ | ${0.839}_{-0.040}^{+0.044}$ | $0.852$ | ${0.97}_{-0.18}^{+0.14}$ | $0.946$ |

${\chi}^{2}$ | $12941.8$ | $12950.7$ | $2447.8$ |

**Table 5.**$95\%$ c.l. constraints on cosmological parameters in our baseline $\Lambda $CDM+r scenario from different combinations of datasets with a modified Hilltop inflation with $p=6$.

Planck TT | Best Fit | Best Fit | Best Fit | |||

+ lowP | + lowP | + lowP + BAO | + lowP + BAO | + tau055 | + tau055 | |

${\Omega}_{\mathrm{b}}{h}^{2}$ | ${0.02232}_{-0.00045}^{+0.00046}$ | $0.02229$ | $0.02222\phantom{\rule{0.166667em}{0ex}}\pm 0.00039$ | $0.02222$ | ${0.02222}_{-0.00043}^{+0.00044}$ | $0.02209$ |

${\Omega}_{\mathrm{c}}{h}^{2}$ | ${0.1158}_{-0.0038}^{+0.0040}$ | $0.1172$ | $0.1175{\phantom{\rule{0.166667em}{0ex}}}_{-0.0025}^{+0.0024}$ | $0.1179$ | ${0.1181}_{-0.0036}^{+0.0038}$ | $0.1191$ |

$\tau $ | ${0.048}_{-0.031}^{+0.030}$ | $0.043$ | $0.042{\phantom{\rule{0.166667em}{0ex}}}_{-0.029}^{+0.026}$ | $0.051$ | $0.055\phantom{\rule{0.166667em}{0ex}}\pm 0.017$ | $0.057$ |

${10}^{12}{V}_{0}/{M}^{4}$ | $48\phantom{\rule{0.166667em}{0ex}}\pm 10$ | 46 | $49\phantom{\rule{0.166667em}{0ex}}\pm 10$ | 48 | $60\phantom{\rule{0.166667em}{0ex}}\pm 10$ | 56 |

$log\left(b\right[GeV\left]\right)$ | $>8.10$ | $11.7$ | $>8.01$ | $16.8$ | $>8.30$ | $8.0$ |

${10}^{11}{\lambda}_{hill}$ | $>0.951$ | $0.999$ | $>0.954$ | $0.999$ | $>0.936$ | $0.998$ |

${c}_{hill}$ | $0.00178{\phantom{\rule{0.166667em}{0ex}}}_{-0.00098}^{+0.00095}$ | $0.0021$ | ${0.00211}_{-0.00074}^{+0.00070}$ | $0.0020$ | ${0.00234}_{-0.00083}^{+0.00080}$ | $0.0025$ |

r | $0.398\phantom{\rule{0.166667em}{0ex}}\pm 0.065$ | $0.38$ | $0.401{\phantom{\rule{0.166667em}{0ex}}}_{-0.064}^{+0.065}$ | $0.39$ | $0.469{\phantom{\rule{0.166667em}{0ex}}}_{-0.077}^{+0.087}$ | $0.443$ |

${H}_{0}$ | $68.9\pm 1.8$ | $68.3$ | $68.2\pm 1.1$ | $68.1$ | $67.9\phantom{\rule{0.166667em}{0ex}}\pm 1.7$ | $67.4$ |

${\sigma}_{8}$ | $0.791{\phantom{\rule{0.166667em}{0ex}}}_{-0.023}^{+0.022}$ | $0.793$ | $0.793\phantom{\rule{0.166667em}{0ex}}\pm 0.021$ | $0.800$ | ${0.806}_{-0.017}^{+0.016}$ | $0.810$ |

${\chi}^{2}$ | $11296.3$ | $11302.4$ | $781.8$ | |||

Planck TTTEEE | Best Fit | Best Fit | Best Fit | |||

+ lowP | + lowP | + lowP + BAO | + lowP + BAO | + tau055 | + tau055 | |

${\Omega}_{\mathrm{b}}{h}^{2}$ | ${0.02226}_{-0.00032}^{+0.00031}$ | $0.02213$ | ${0.02224}_{-0.00028}^{+0.00027}$ | $0.02228$ | $0.02222\phantom{\rule{0.166667em}{0ex}}\pm 0.00029$ | $0.02216$ |

${\Omega}_{\mathrm{c}}{h}^{2}$ | $0.1179{\phantom{\rule{0.166667em}{0ex}}}_{-0.0027}^{+0.0028}$ | $0.1183$ | ${0.1182}_{-0.0021}^{+0.0020}$ | $0.1182$ | $0.1189\phantom{\rule{0.166667em}{0ex}}\pm 0.0026$ | $0.1196$ |

$\tau $ | $0.044{\phantom{\rule{0.166667em}{0ex}}}_{-0.027}^{+0.026}$ | $0.044$ | $0.043\phantom{\rule{0.166667em}{0ex}}\pm 0.026$ | $0.048$ | $0.054\phantom{\rule{0.166667em}{0ex}}\pm 0.016$ | $0.054$ |

${10}^{12}{V}_{0}/{M}^{4}$ | $49\phantom{\rule{0.166667em}{0ex}}\pm 10$ | 48 | $50\phantom{\rule{0.166667em}{0ex}}\pm 10$ | 50 | $60\phantom{\rule{0.166667em}{0ex}}\pm 10$ | 59 |

$log\left(b\right[GeV\left]\right)$ | $>8.16$ | $18.8$ | $>8.24$ | $16.5$ | $>7.93$ | $10.6$ |

${10}^{11}{\lambda}_{hill}$ | $>0.964$ | $0.996$ | $>0.962$ | $0.995$ | $>0.957$ | $0.999$ |

${c}_{hill}$ | ${0.00219}_{-0.00072}^{+0.00074}$ | $0.0023$ | ${0.00224}_{-0.00066}^{+0.00065}$ | $0.0021$ | ${0.00249}_{-0.00066}^{+0.00065}$ | $0.0027$ |

r | $0.405{\phantom{\rule{0.166667em}{0ex}}}_{-0.056}^{+0.057}$ | $0.397$ | $0.406{\phantom{\rule{0.166667em}{0ex}}}_{-0.058}^{+0.059}$ | $0.40$ | $0.463{\phantom{\rule{0.166667em}{0ex}}}_{-0.064}^{+0.068}$ | $0.45$ |

${H}_{0}$ | ${68.0}_{-1.3}^{+1.2}$ | $67.8$ | ${67.88}_{-0.91}^{+0.95}$ | $67.9$ | $67.6\phantom{\rule{0.166667em}{0ex}}\pm 1.2$ | $67.3$ |

${\sigma}_{8}$ | $0.796{\phantom{\rule{0.166667em}{0ex}}}_{-0.020}^{+0.019}$ | $0.798$ | $0.796{\phantom{\rule{0.166667em}{0ex}}}_{-0.019}^{+0.020}$ | $0.802$ | $0.808\phantom{\rule{0.166667em}{0ex}}\pm 0.015$ | $0.811$ |

${\chi}^{2}$ | $12981.0$ | $12986.4$ | $2466.4$ |

**Table 6.**$95\%$ c.l. constraints on cosmological parameters in our baseline $\Lambda $CDM+r+${N}_{\mathrm{eff}}$ scenario from different combinations of datasets with a modified Hilltop inflation with $p=6$.

Planck TT | Best Fit | Best Fit | Best Fit | |||

+ lowP | + lowP | + lowP + BAO | + lowP + BAO | + tau055 | + tau055 | |

${\Omega}_{\mathrm{b}}{h}^{2}$ | ${0.02236}_{-0.00044}^{+0.00041}$ | $0.02278$ | ${0.02248}_{-0.00044}^{+0.00043}$ | $0.02262$ | ${0.02248}_{-0.00052}^{+0.00049}$ | $0.02234$ |

${\Omega}_{\mathrm{c}}{h}^{2}$ | ${0.1236}_{-0.0067}^{+0.0068}$ | $0.1245$ | ${0.1256}_{-0.0073}^{+0.0071}$ | $0.1283$ | ${0.1231}_{-0.0073}^{+0.0074}$ | $0.1212$ |

$\tau $ | ${0.055}_{-0.030}^{+0.026}$ | $0.051$ | ${0.049}_{-0.029}^{+0.028}$ | $0.067$ | $0.058{\phantom{\rule{0.166667em}{0ex}}}_{-0.017}^{+0.016}$ | $0.056$ |

${10}^{12}{V}_{0}/{M}^{4}$ | $46\phantom{\rule{0.166667em}{0ex}}\pm 10$ | 40 | $47\phantom{\rule{0.166667em}{0ex}}\pm 10$ | 46 | $55{\phantom{\rule{0.166667em}{0ex}}}_{-10}^{+20}$ | 51 |

$log\left(b\right[GeV\left]\right)$ | $>7.83$ | $11.0$ | $>7.75$ | $9.5$ | $>7.55$ | $16.8$ |

${10}^{11}{\lambda}_{hill}$ | $>0.937$ | $0.995$ | $>0.940$ | $0.998$ | $>0.916$ | $0.993$ |

${c}_{hill}$ | $<0.00127$ | $0.0000$ | $<0.00185$ | $0.0000$ | $<0.00256$ | $0.0017$ |

r | $0.388{\phantom{\rule{0.166667em}{0ex}}}_{-0.061}^{+0.068}$ | $0.35$ | $0.392{\phantom{\rule{0.166667em}{0ex}}}_{-0.069}^{+0.071}$ | $0.38$ | $0.446{\phantom{\rule{0.166667em}{0ex}}}_{-0.086}^{+0.097}$ | $0.41$ |

${N}_{\mathrm{eff}}$ | $3.56{\phantom{\rule{0.166667em}{0ex}}}_{-0.34}^{+0.30}$ | $3.66$ | ${3.54}_{-0.41}^{+0.36}$ | $3.73$ | ${3.42}_{-0.46}^{+0.44}$ | $3.28$ |

${H}_{0}$ | ${72.1}_{-2.3}^{+2.0}$ | $73.1$ | $71.0{\phantom{\rule{0.166667em}{0ex}}}_{-2.4}^{+2.3}$ | $72.2$ | ${70.6}_{-3.5}^{+3.2}$ | $69.5$ |

${\sigma}_{8}$ | ${0.819}_{-0.028}^{+0.027}$ | $0.817$ | ${0.821}_{-0.033}^{+0.032}$ | $0.846$ | ${0.821}_{-0.027}^{+0.026}$ | $0.815$ |

${\chi}^{2}$ | $11287.9$ | $11298.3$ | $781.3$ | |||

Planck TTTEEE | Best Fit | Best Fit | Best Fit | |||

+ lowP | + lowP | + lowP + BAO | + lowP + BAO | + tau055 | + tau055 | |

${\Omega}_{\mathrm{b}}{h}^{2}$ | $0.02258{\phantom{\rule{0.166667em}{0ex}}}_{-0.00042}^{+0.00040}$ | $0.02275$ | $0.02248{\phantom{\rule{0.166667em}{0ex}}}_{-0.00037}^{+0.00036}$ | $0.02246$ | $0.02236\phantom{\rule{0.166667em}{0ex}}\pm 0.00042$ | $0.02216$ |

${\Omega}_{\mathrm{c}}{h}^{2}$ | ${0.1233}_{-0.0058}^{+0.0057}$ | $0.1257$ | ${0.1235}_{-0.0057}^{+0.0059}$ | $0.1210$ | ${0.1213}_{-0.0058}^{+0.0061}$ | $0.1180$ |

$\tau $ | $0.054{\phantom{\rule{0.166667em}{0ex}}}_{-0.030}^{+0.028}$ | $0.070$ | $0.050{\phantom{\rule{0.166667em}{0ex}}}_{-0.029}^{+0.026}$ | $0.062$ | $0.056{\phantom{\rule{0.166667em}{0ex}}}_{-0.016}^{+0.017}$ | $0.053$ |

${10}^{12}{V}_{0}/{M}^{4}$ | $48\phantom{\rule{0.166667em}{0ex}}\pm 10$ | 48 | $48\phantom{\rule{0.166667em}{0ex}}\pm 10$ | 52 | $58\phantom{\rule{0.166667em}{0ex}}\pm 10$ | 56 |

$log\left(b\right[GeV\left]\right)$ | $>7.99$ | $18.9$ | $>8.02$ | $8.4$ | $>8.00$ | $8.5$ |

${10}^{11}{\lambda}_{hill}$ | $>0.955$ | $0.997$ | $>0.956$ | $0.994$ | $>0.950$ | $0.994$ |

${c}_{hill}$ | $<0.00212$ | $0.0002$ | ${0.0013}_{-0.0012}^{+0.0011}$ | $0.0015$ | ${0.0020}_{-0.0013}^{+0.0012}$ | $0.0026$ |

r | $0.398{\phantom{\rule{0.166667em}{0ex}}}_{-0.060}^{+0.063}$ | $0.39$ | $0.400{\phantom{\rule{0.166667em}{0ex}}}_{-0.055}^{+0.058}$ | $0.42$ | $0.453{\phantom{\rule{0.166667em}{0ex}}}_{-0.071}^{+0.076}$ | $0.44$ |

${N}_{\mathrm{eff}}$ | $3.42{\phantom{\rule{0.166667em}{0ex}}}_{-0.35}^{+0.33}$ | $3.61$ | $3.38\pm 0.34$ | $3.26$ | ${3.21}_{-0.35}^{+0.38}$ | $3.00$ |

${H}_{0}$ | $70.6{\phantom{\rule{0.166667em}{0ex}}}_{-2.6}^{+2.4}$ | $71.8$ | $69.9{\phantom{\rule{0.166667em}{0ex}}}_{-2.2}^{+2.1}$ | $69.4$ | ${68.7}_{-2.6}^{+2.8}$ | $67.4$ |

${\sigma}_{8}$ | ${0.819}_{-0.030}^{+0.028}$ | $83.9$ | $0.817\phantom{\rule{0.166667em}{0ex}}\pm 0.030$ | $0.819$ | $0.816{\phantom{\rule{0.166667em}{0ex}}}_{-0.023}^{+0.024}$ | $0.803$ |

${\chi}^{2}$ | $12979.9$ | $12984.3$ | $2466.7$ |

**Table 7.**$95\%$ c.l. constraints on cosmological parameters in our baseline wCDM+r scenario from different combinations of datasets with a modified Hilltop inflation with $p=6$.

Planck TT | Best Fit | Best Fit | Best Fit | |||

+ lowP | + lowP | + lowP + BAO | + lowP + BAO | + tau055 | + tau055 | |

${\Omega}_{\mathrm{b}}{h}^{2}$ | $0.02239{\phantom{\rule{0.166667em}{0ex}}}_{-0.00044}^{+0.00045}$ | $0.02246$ | $0.02227\phantom{\rule{0.166667em}{0ex}}\pm 0.00042$ | $0.02207$ | $0.02228{\phantom{\rule{0.166667em}{0ex}}}_{-0.00042}^{+0.00043}$ | $0.02230$ |

${\Omega}_{\mathrm{c}}{h}^{2}$ | $0.1156{\phantom{\rule{0.166667em}{0ex}}}_{-0.0038}^{+0.0039}$ | $0.1148$ | $0.1165\pm 0.0035$ | $0.1169$ | $0.1177\phantom{\rule{0.166667em}{0ex}}\pm 0.0037$ | $0.1182$ |

$\tau $ | $0.050\phantom{\rule{0.166667em}{0ex}}\pm 0.029$ | $0.043$ | ${0.045}_{-0.031}^{+0.028}$ | $0.044$ | $0.055\phantom{\rule{0.166667em}{0ex}}\pm 0.017$ | $0.052$ |

${10}^{12}{V}_{0}/{M}^{4}$ | $48\phantom{\rule{0.166667em}{0ex}}\pm 10$ | 44 | $48\phantom{\rule{0.166667em}{0ex}}\pm 10$ | 46 | $60\phantom{\rule{0.166667em}{0ex}}\pm 10$ | 55 |

$log\left(b\right[GeV\left]\right)$ | $>7.85$ | $8.6$ | $>8.03$ | $14.3$ | $>8.08$ | $11.1$ |

${10}^{11}{\lambda}_{hill}$ | $>0.947$ | $0.988$ | $>0.952$ | $0.999$ | $>0.932$ | $0.998$ |

${c}_{hill}$ | ${0.00172}_{-0.00096}^{+0.00094}$ | $0.0016$ | ${0.00193}_{-0.00093}^{+0.00091}$ | $0.0022$ | ${0.00288}_{-0.00084}^{+0.00079}$ | $0.0026$ |

r | $0.400{\phantom{\rule{0.166667em}{0ex}}}_{-0.061}^{+0.064}$ | $0.38$ | $0.398{\phantom{\rule{0.166667em}{0ex}}}_{-0.064}^{+0.065}$ | $0.39$ | $0.466{\phantom{\rule{0.166667em}{0ex}}}_{-0.077}^{+0.087}$ | $0.43$ |

w | ${-1.61}_{-0.31}^{+0.40}$ | $-1.74$ | $-{0.96}_{-0.13}^{+0.12}$ | $-1.00$ | ${-1.56}_{-0.42}^{+0.53}$ | $-1.68$ |

${H}_{0}$ | $>81$ | $96.8$ | ${67.2}_{-2.9}^{+3.0}$ | $68.2$ | $87{\phantom{\rule{0.166667em}{0ex}}}_{-20}^{+10}$ | $91.2$ |

${\sigma}_{8}$ | ${0.97}_{-0.12}^{+0.09}$ | $0.997$ | ${0.779}_{-0.043}^{+0.045}$ | $0.792$ | ${0.96}_{-0.15}^{+0.12}$ | $0.994$ |

${\chi}^{2}$ | $11289.9$ | $11302.8$ | $778.5$ | |||

Planck TTTEEE | Best Fit | Best Fit | Best Fit | |||

+ lowP | + lowP | + lowP + BAO | + lowP + BAO | + tau055 | + tau055 | |

${\Omega}_{\mathrm{b}}{h}^{2}$ | ${0.02230}_{-0.00030}^{+0.00031}$ | $0.02234$ | $0.02224\phantom{\rule{0.166667em}{0ex}}\pm 0.00030$ | $0.02215$ | $0.02226\phantom{\rule{0.166667em}{0ex}}\pm 0.00029$ | $0.02234$ |

${\Omega}_{\mathrm{c}}{h}^{2}$ | $0.1176\pm 0.0027$ | $0.1173$ | $0.1182\pm 0.0025$ | $0.1183$ | $0.1186\phantom{\rule{0.166667em}{0ex}}\pm 0.0026$ | $0.1184$ |

$\tau $ | $0.043\phantom{\rule{0.166667em}{0ex}}\pm 0.027$ | $0.43$ | $0.043\pm 0.026$ | $0.047$ | $0.054\phantom{\rule{0.166667em}{0ex}}\pm 0.016$ | $0.054$ |

${10}^{12}{V}_{0}/{M}^{4}$ | $49\phantom{\rule{0.166667em}{0ex}}\pm 10$ | 47 | $50{\phantom{\rule{0.166667em}{0ex}}}_{-9}^{+10}$ | 49 | $59\phantom{\rule{0.166667em}{0ex}}\pm 10$ | 53 |

$log\left(b\right[GeV\left]\right)$ | $>7.95$ | $17.3$ | $>8.05$ | $14.4$ | $>7.88$ | $11.0$ |

${10}^{11}{\lambda}_{hill}$ | $>0.960$ | $0.991$ | $>0.963$ | $0.988$ | $>0.951$ | $0.997$ |

${c}_{hill}$ | $0.00215{\phantom{\rule{0.166667em}{0ex}}}_{-0.00078}^{+0.00074}$ | $0.0022$ | ${0.00224}_{-0.00073}^{+0.00072}$ | $0.0021$ | ${0.00245}_{-0.00066}^{+0.00064}$ | $0.0022$ |

r | $0.403{\phantom{\rule{0.166667em}{0ex}}}_{-0.058}^{+0.061}$ | $0.39$ | $0.407{\phantom{\rule{0.166667em}{0ex}}}_{-0.055}^{+0.057}$ | $0.40$ | $0.460{\phantom{\rule{0.166667em}{0ex}}}_{-0.066}^{+0.070}$ | $0.42$ |

w | ${-1.66}_{-0.32}^{+0.42}$ | $-1.77$ | ${-1.01}_{-0.12}^{+0.11}$ | $-1.05$ | ${-1.60}_{-0.41}^{+0.51}$ | $-1.59$ |

${H}_{0}$ | $>81$ | $95.1$ | $68.0{\phantom{\rule{0.166667em}{0ex}}}_{-2.8}^{+3.1}$ | $69.1$ | $88{\phantom{\rule{0.166667em}{0ex}}}_{-20}^{+10}$ | $87.4$ |

${\sigma}_{8}$ | ${0.98}_{-0.12}^{+0.09}$ | $1.01$ | ${0.797}_{-0.036}^{+0.037}$ | $0.812$ | ${0.97}_{-0.14}^{+0.11}$ | $0.972$ |

${\chi}^{2}$ | $12973.7$ | $12986.9$ | $2459.9$ |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Di Valentino, E.; Mersini-Houghton, L.
Testing Predictions of the Quantum Landscape Multiverse 3: The Hilltop Inflationary Potential. *Symmetry* **2019**, *11*, 520.
https://doi.org/10.3390/sym11040520

**AMA Style**

Di Valentino E, Mersini-Houghton L.
Testing Predictions of the Quantum Landscape Multiverse 3: The Hilltop Inflationary Potential. *Symmetry*. 2019; 11(4):520.
https://doi.org/10.3390/sym11040520

**Chicago/Turabian Style**

Di Valentino, Eleonora, and Laura Mersini-Houghton.
2019. "Testing Predictions of the Quantum Landscape Multiverse 3: The Hilltop Inflationary Potential" *Symmetry* 11, no. 4: 520.
https://doi.org/10.3390/sym11040520