# Dark Matter Sterile Neutrino from Scalar Decays

## Abstract

**:**

## 1. Introduction

## 2. Sterile Neutrino Production by the Scalar Decay (SDP)

## 3. Parameterization and Methods

`camb`1 [43] to allow the calculation of sterile neutrino SDP formalism presented in the previous section.

`CosmoMC`2 [44] to sample from the space of cosmological and sterile neutrino SDP mechanism parameters and generate estimates of their posterior mean and confidence intervals. We first run the modified versions of

`C`osmoMC and

`camb`setting to zero the additional parameters of the SDP model. We find good consistency between our bounds and the existing constraints for the $\Lambda $CDM model [1]. We use the default convergence settings implemented in

`C`osmoMC: $\mathrm{MPI}\text{\_}\mathrm{Limit}\text{\_}\mathrm{Converge}=0.025$ and $\mathrm{MPI}\text{\_}\mathrm{Limit}\text{\_}\mathrm{Converge}\text{\_}\mathrm{Err}=0.2$. With these choices, the

`C`osmoMC run stops when the confidence interval for each parameter at 95% C.L. is accurate at $0.2\phantom{\rule{0.166667em}{0ex}}\sigma $. This error can be reduced, but in this case the computing time increases to reach the convergence limit. This is critical for non-standard models as SDP for which the execution time is larger than in the standard case because of numerical evolution of momentum distributions in

`camb`. The final runs are based on 120 independent channels, reaching the convergence criterion $(R-1)=0.007$, defined as the ratio between the variance of the means and the mean of variances for the second half of chains [44].

## 4. Cosmological Datasets

`Commander`for $2\le l\le 29$,

`CamSpec`$50\le l\le 2500$,

`LowLike`for $2\le l\le 32$ for polarization data and

`Lensing`for $40\le l\le 400$ for lensing data. As sterile neutrino production is expected to affect the redshift–distance relations and the growth of structures, we include in the analysis the Planck power spectrum of the reconstructed lensing potential [2]. We will refer to the combination of these measurements as the Planck+lens dataset.

## 5. Analysis and Results

#### 5.1. Sensitivity of Cosmological Data to ${m}_{{\nu}_{s}}$ and ${f}_{S}$

#### 5.2. Cosmological Constraints

## 6. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notes

1 | http://camb.info, CAMB v.1.1.2 accessed on 31 May 2020. |

2 | http://cosmologist.info, accessed on 31 May 2020. |

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**Figure 1.**The dependence of the sterile neutrino momentum distributions on the co-moving momentum $q=p/T$ for the scalar mass ${M}_{S}$ = 533 GeV and different values of the Higgs coupling, ${\lambda}_{H}$, and Yukawa coupling, ${y}_{k}$. The dotted line indicates the averaged co-moving momentum corresponding to sterile neutrino thermal distribution. Other parameters are fixed to: ${\Omega}_{b}{h}^{2}=0.0226$, ${\Omega}_{c}{h}^{2}=0.112$, ${\Omega}_{\nu}{h}^{2}=0.00064$, ${H}_{0}$ = 70 km s${}^{-1}$Mpc${}^{-1}$.

**Figure 2.**The dependence of the abundances of scalar (continuous lines) and sterile neutrino (dashed lines) on the time parameter $r={m}_{h}/T$ for the same models presented in Figure 1. The vertical blue line indicates the temperature of the electroweak phase transition (EWPT).

**Figure 3.**Redshift dependence of the variation of BAO characteristic parameter, $\Delta ({r}_{s}\left({z}_{d}\right)/{D}_{V}\left(z\right))$, for models sharing the same sterile neutrino mass fraction ${f}_{S}$ in the SDP scenario. The production mechanism parameters are indicated for each case. Other parameters are fixed to: ${\Omega}_{b}{h}^{2}=0.0226$, ${\Omega}_{c}{h}^{2}=0.112$, ${\Omega}_{\nu}{h}^{2}=0.00064$, ${H}_{0}$ = 70 km s${}^{-1}$Mpc${}^{-1}$.

**Figure 4.**The likelihood probability distribution of the estimated free-streaming horizon wave-number, ${k}_{fs}^{min}$, from the fit of SDP models with Planck + lens + BAO + DES datasets.

**Figure 5.**The dependence of the deflection angle power spectra on sterile neutrino mass fraction ${f}_{S}$ in models sharing the same sterile neutrino mass ${m}_{{\nu}_{s}}$, obtained in the SDP scenario. For comparison, we plot the deflection angle power spectrum for the $\Lambda $CDM model. The contributions of the wave-numbers in the range $0.1<k<3$ Mpc${}^{-1}$ are also indicated (dotted lines). Other parameters are fixed to: ${\Omega}_{b}{h}^{2}=0.0226$, ${\Omega}_{c}{h}^{2}=0.112$, ${\Omega}_{\nu}{h}^{2}=0.00064$, ${H}_{0}$ = 70 km s${}^{-1}$Mpc${}^{-1}$.

**Figure 6.**The marginalized likelihood probability distributions and the joint confidence regions (68% and 95% CL) for SDP mechanism parameters color-coded by the scalar mass values ${M}_{S}$. The dominant effect on SDP mechanism is given by the strength of the Higgs coupling, ${\lambda}_{H}$, that sets ${M}_{S}$, and the strength of Yukawa coupling, ${y}_{k}$, that sets ${m}_{{\nu}_{s}}$. The best fit values of the SDP parameters lead to ${f}_{S}$ = 0.86 ± 0.07 (68% C.L.), indicating that SDP is a dominant mechanism.

**Table 1.**Priors and constraints for the $\Lambda $CDM-ext parameters adopted in the analysis. All priors are uniform in the listed intervals. We assume a flat Universe.

Parameter | Prior |
---|---|

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

${\Omega}_{c}{h}^{2}$ | [0.001, 0.5 ] |

$100{\theta}_{s}$ | [0.5, 10] |

$\tau $ | [0.01, 0.9] |

$\mathrm{log}\left({10}^{1}0{A}_{s}\right)$ | [2.5, 5] |

${n}_{s}$ | [0.5, 1.5] |

${m}_{\nu}\left(\mathrm{eV}\right)$ | [0, 6] |

${N}_{eff}$ | [3.046, 8] |

${H}_{0}\left({\mathrm{km}\phantom{\rule{0.166667em}{0ex}}\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}\right)$ | [20, 100] |

**Table 2.**Priors and constraints on the additional parameters for SDP models. All priors are uniform in the listed intervals.

SDP Parameter | Prior |
---|---|

${m}_{{\nu}_{s}}\left(\mathrm{keV}\right)$ | [2, 30] |

${y}_{k}$ | [${10}^{-10}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.166667em}{0ex}}{10}^{-8}]$ |

${\lambda}_{H}$ | [${10}^{-8}\phantom{\rule{0.166667em}{0ex}},{10}^{-4}$] |

${M}_{S}\left(\mathrm{GeV}\right)$ | [10, 1000] |

${\Omega}_{{\nu}_{s}}{h}^{2}$ | [0.001, 0.5] |

**Table 3.**The mean values and the absolute errors of the main parameters obtained from the fit of $\Lambda $CDM-ext and SDP models with the Planck + lensing + BAO + DES datasets. The errors are quoted at 68% C.L. The upper limits are quoted at 95% C.L. The first group of parameters are the base cosmological parameters sampled in the Monte-Carlo Markov Chain analysis with uniform priors. The others are derived parameters.

Parameter | $\Lambda $CDM-Ext | SDP |
---|---|---|

${\Omega}_{b}{h}^{2}$ | 0.0223 ± 0.0002 | 0.0219 ± 0.0003 |

${\Omega}_{c}{h}^{2}$ | 0.122 ± 0.004 | 0.121 ± 0.004 |

100${\theta}_{MC}$ | 1.0412 ± 0.0008 | 1.0413 ± 0.0009 |

$\tau $ | 0.087 ± 0.015 | 0.069 ± 0.012 |

$\sum {m}_{\nu}$ | <0.321 | <0.198 |

${N}_{eff}$ | 3.520 ± 0.259 | 3.380 ± 0.243 |

${f}_{S}$ | 0.860 ± 0.071 | |

M${}_{S}$ (GeV) | 533.60 ± 47.21 | |

${10}^{-6}{\lambda}_{H}$ | 3.780 ± 0.642 | |

${10}^{-9}{y}_{k}$ | 3.451 ± 1.820 | |

${\Omega}_{m}$ | 0.295 ± 0.013 | 0.284 ± 0.011 |

${\sigma}_{8}$ | 0.808 ± 0.021 | 0.832 ± 0.019 |

${\sigma}_{8}{({\Omega}_{m}/0.3)}^{0.5}$ | 0.801 ± 0.004 | 0.809 ± 0.005 |

${\sigma}_{8}{e}^{-\tau}$ | 0.741 ± 0.021 | 0.776 ± 0.018 |

${H}_{0}$ | 70.512 ± 1.556 | 71.210 ± 1.433 |

${m}_{{\nu}_{s}}$(keV) | 7.882 ± 0.731 |

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**MDPI and ACS Style**

Popa, L.A.
Dark Matter Sterile Neutrino from Scalar Decays. *Universe* **2021**, *7*, 309.
https://doi.org/10.3390/universe7080309

**AMA Style**

Popa LA.
Dark Matter Sterile Neutrino from Scalar Decays. *Universe*. 2021; 7(8):309.
https://doi.org/10.3390/universe7080309

**Chicago/Turabian Style**

Popa, Lucia Aurelia.
2021. "Dark Matter Sterile Neutrino from Scalar Decays" *Universe* 7, no. 8: 309.
https://doi.org/10.3390/universe7080309