# Is it no Longer Necessary to Test Cosmologies with Type Ia Supernovae?

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. A Brief Overview of SNeIa Cosmology

^{−2}s

^{−1}, $\mathcal{L}={10}^{-2M/5}\times 3.02\times {10}^{35}$ erg s

^{−1}[7], one obtains

## 3. A Non-Standard Approach to SNeIa Data

**(a)**About a decade ago, a new approach that does not respect the standard procedure described above was adopted to analyze the SNeIa data. Initiated by the SuperNova Legacy Survey (SNLS) [15] in 2006, this approach simply assumes, rather than examines, that the standard cosmology is consistent with the SNeIa observations and limits itself to calculating confidence intervals (ellipses) of parameters. Under this approach, ${\chi}^{2}$ is calculated from

**(b)**In passing, it would also be worthwhile to bring to the notice of the reader another approach that also does not seem perfectly consistent with Equation (16). Some authors (for example, [8,12,13]) perform the SNeIa data-fitting by considering the ${\chi}^{2}$-statistic in the form

^{−1}within its likely range 200 Km ${\mathrm{s}}^{-1}\le {\sigma}_{v}\le $ 500 Km s

^{−1}. They further add 2500 km s

^{−1}in the quadrature to ${\sigma}_{v}$ for high-redshift SNeIa whose redshifts are determined from the broad features in the SN spectrum. Let us note that $m(z)$ is as non-linear as

**A misunderstanding about the Milne model:**It would also be worthwhile to clear a misunderstanding related to Milne’s model, which persists in the literature. In Table 1, we notice that, besides the ΛCDM and other dark energy models, the Milne model also fares well with the data (see also, [21]). The remarkable fact is that this coasting model does so without requiring any dark energy and accelerated expansion.

## 4. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Appendix A: QSSC

## Appendix B: The Milne Model

## References

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1. | Here $H\equiv \dot{S}/S$ is the Hubble parameter with S being the scale factor of the homogeneous-isotropic universe (the “over-dot” represents derivative with respect to the cosmic time t), $q\equiv -\ddot{S}/(S{H}^{2})$ is the deceleration parameter, $k=\pm 1,\phantom{\rule{3.33333pt}{0ex}}0$ is the curvature parameter of the R-W spacetime, $\rho ,\phantom{\rule{3.33333pt}{0ex}}p$ are respectively the density and pressure of the cosmic matter, and $c,\phantom{\rule{3.33333pt}{0ex}}G$ are respectively the speed of light in vacuum and the Newtonian constant of gravitation. |

2. | The present value of the scale factor ${S}_{0}$ can be calculated, for different values of k, from Equation (1) in terms of ${\rho}_{0}$ and ${H}_{0}$ giving ${S}_{0}=c{H}_{0}^{-1}\sqrt{k/({\mathsf{\Omega}}_{\mathrm{m}}-1)}$, where ${\mathsf{\Omega}}_{\mathrm{m}}\equiv {\rho}_{0}/{\rho}_{c}$ is the present density of the universe in the unit of critical density ${\rho}_{c}\equiv 3{H}_{0}^{2}/(8\pi G)$. |

3. | The value of this constant depends on the chosen units in which ${d}_{\mathrm{L}}$ and ${H}_{0}$ are measured. For example, if ${d}_{\mathrm{L}}$ is measured in Mpc and ${H}_{0}$ in km s ^{−1} Mpc^{−1}, then this constant comes out as ≈25. |

4. | Perhaps $\mathcal{M}$ is called so because it serves as the absolute magnitude corresponding to the “Hubble constant-free luminosity distance” in Equation (10). It is easy to check that ${H}_{0}{d}_{\mathrm{L}}$ is Hubble constant-free. |

**Table 1.**Best-fit parameters of some selected cosmological models fitted to different SNeIa data sets.

Models | ${\mathsf{\Omega}}_{\mathbf{m}}$ | ${\mathsf{\Omega}}_{\mathbf{DE}}$ | ${\mathit{\omega}}_{\mathbf{DE}}$ (Constant) | $\mathcal{M}$ | ${\mathit{q}}_{\mathbf{0}}$ | ${\mathit{\chi}}^{\mathbf{2}}$ | DoF | P (%) |
---|---|---|---|---|---|---|---|---|

(9 high z + 27 low z) MLCS SNeIa from Riess et al. [8] (1998) | ||||||||

ΛCDM | $0.15\pm 1.28$ | $0.60\pm 1.47$ | $-1$ | 43.31 | $-0.53$ | 44.0 | 33 | 9.5 |

ΛCDM (${\mathsf{\Omega}}_{\mathrm{total}}$ = 1) | $0.26\pm 0.10$ | $1-{\mathsf{\Omega}}_{\mathrm{m}}$ | $-1$ | 43.30 | $-0.60$ | 44.0 | 34 | 11.7 |

${\rho}_{\mathrm{DE}}$CDM (${\mathsf{\Omega}}_{\mathrm{total}}$ = 1) | $0.14\pm 1.34$ | $1-{\mathsf{\Omega}}_{\mathrm{m}}$ | $-0.79\pm 1.79$ | 43.31 | $-0.52$ | 44.0 | 33 | 9.5 |

Milne model | 43.35 | 0 | 47.1 | 35 | 8.3 | |||

54 SNeIa from Perlmutter et al. [10] (1999) | ||||||||

ΛCDM | $0.79\pm 0.47$ | $1.40\pm 0.65$ | $-1$ | $23.91$ | $-1.01$ | 56.9 | 51 | 26.6 |

ΛCDM (${\mathsf{\Omega}}_{\mathrm{total}}$ = 1) | $0.28\pm 0.08$ | $1-{\mathsf{\Omega}}_{\mathrm{m}}$ | $-1$ | 23.94 | $-0.58$ | 57.7 | 52 | 27.3 |

${\rho}_{\mathrm{DE}}$CDM (${\mathsf{\Omega}}_{\mathrm{total}}$ = 1) | $0.48\pm 0.15$ | $1-{\mathsf{\Omega}}_{\mathrm{m}}$ | $-2.10\pm 1.83$ | $23.91$ | $-1.14$ | 57.2 | 51 | 25.7 |

Milne model | 24.03 | 0 | 61.5 | 53 | 19.8 | |||

“Gold Sample” of 157 SNeIa from Riess et al. [12] (2004) | ||||||||

ΛCDM | $0.46\pm 0.10$ | $0.98\pm 0.19$ | $-1$ | $43.32$ | $-0.75$ | 175.0 | 154 | 11.8 |

ΛCDM (${\mathsf{\Omega}}_{\mathrm{total}}$ = 1) | $0.31\pm 0.04$ | $1-{\mathsf{\Omega}}_{\mathrm{m}}$ | $-1$ | $43.34$ | $-0.54$ | 177.1 | 155 | 10.8 |

${\rho}_{\mathrm{DE}}$CDM (${\mathsf{\Omega}}_{\mathrm{total}}$ = 1) | $0.49\pm 0.06$ | $1-{\mathsf{\Omega}}_{\mathrm{m}}$ | $-2.33\pm 1.07$ | $43.30$ | $-1.28$ | 173.7 | 154 | 13.2 |

Milne model | 43.40 | 0 | 191.7 | 156 | 2.7 | |||

“New Gold Sample” of 182 SNeIa from Riess et al. [13] (2007) | ||||||||

ΛCDM | $0.48\pm 0.09$ | $0.96\pm 0.18$ | $-1$ | 43.36 | $-0.72$ | 156.4 | 179 | 88.7 |

ΛCDM (${\mathsf{\Omega}}_{\mathrm{total}}$ = 1) | $0.34\pm 0.04$ | $1-{\mathsf{\Omega}}_{\mathrm{m}}$ | $-1$ | $43.40$ | $-0.49$ | 158.7 | 180 | 87.1 |

${\rho}_{\mathrm{DE}}$CDM (${\mathsf{\Omega}}_{\mathrm{total}}$ = 1) | $0.46\pm 0.06$ | $1-{\mathsf{\Omega}}_{\mathrm{m}}$ | $-1.75\pm 0.63$ | 43.35 | $-0.92$ | 156.6 | 179 | 88.5 |

Milne model | $43.45$ | 0 | 174.3 | 181 | 62.6 | |||

New Gold Sample + the most distant SN UDS10Wil of z = 1.914 [14] (2013) | ||||||||

ΛCDM | $0.50\pm 0.09$ | $0.99\pm 0.17$ | $-1$ | 43.36 | $-0.74$ | 157.0 | 180 | 89.1 |

ΛCDM (${\mathsf{\Omega}}_{\mathrm{total}}$ = 1) | $0.35\pm 0.04$ | $1-{\mathsf{\Omega}}_{\mathrm{m}}$ | $-1$ | $43.40$ | $-0.48$ | 160.1 | 181 | 86.6 |

${\rho}_{\mathrm{DE}}$CDM (${\mathsf{\Omega}}_{\mathrm{total}}$ = 1) | $0.47\pm 0.06$ | $1-{\mathsf{\Omega}}_{\mathrm{m}}$ | $-1.80\pm 0.62$ | $43.35$ | $-0.94$ | 157.5 | 180 | 88.6 |

Milne model | $43.45$ | 0 | 178.3 | 182 | 56.4 |

**Table 2.**Different possible ΛCDM models which provide ${\chi}^{2}$/DoF $\approx 1$ for suitably chosen ${\sigma}_{\mathrm{int}}$ to fit 115 SNeIa from Astier et al. [15].

${\mathit{\sigma}}_{\mathbf{int}}$ | ${\mathsf{\Omega}}_{\mathbf{m}}$ | ${\mathsf{\Omega}}_{\mathsf{\Lambda}}$ | $\mathcal{M}$ | ${\mathit{\chi}}^{\mathbf{2}}$/DoF | Varied in ${\mathit{\chi}}^{\mathbf{2}}$-Minimization |
---|---|---|---|---|---|

0.131 | 0.26 | $1-{\mathsf{\Omega}}_{\mathrm{m}}$ | 43.16 | 112.97/113 | ${\mathsf{\Omega}}_{\mathrm{m}}$, $\mathcal{M}$ |

0.131 | 0.31 | 0.81 | 43.15 | 112.09/112 | ${\mathsf{\Omega}}_{\mathrm{m}}$, ${\mathsf{\Omega}}_{\mathsf{\Lambda}}$, $\mathcal{M}$ |

0.143 | 0 | 0 | 43.26 | 114.40/114 | $\mathcal{M}$ |

0.172 | 0.3 | 0 | 43.32 | 113.94/114 | $\mathcal{M}$ |

0.132 | 0 | 0.38 | 43.17 | 112.97/113 | ${\mathsf{\Omega}}_{\mathsf{\Lambda}}$, $\mathcal{M}$ |

0.135 | $-0.24$ | 0 | 43.20 | 112.89/113 | ${\mathsf{\Omega}}_{\mathrm{m}}$, $\mathcal{M}$ |

**Table 3.**Different plausible models in QSSC which provide ${\chi}^{2}$/DoF $\approx 1$ for suitably chosen ${\sigma}_{\mathrm{int}}$ to fit 115 SNeIa from Astier et al. [15]. The parameters ${z}_{\mathrm{max}}$, ${\mathsf{\Omega}}_{\mathsf{\Lambda}}$ ($<0$), $\kappa $ (measured in the units of ${10}^{5}$ cm

^{2}/g), ${\rho}_{\mathrm{g}}$ (in ${10}^{-34}$ g/cm

^{3}), and ${H}_{0}$ (in 100 Km s

^{−1}Mpc

^{−1}) characterize a typical model in QSSC [16].

${\mathit{\sigma}}_{\mathbf{int}}$ | ${\mathsf{\Omega}}_{\mathsf{\Lambda}}$ | $\mathit{\kappa}{\mathit{\rho}}_{\mathbf{g}}{\mathit{H}}_{\mathbf{0}}^{\mathbf{-}\mathbf{1}}$ | ${\mathit{z}}_{\mathbf{max}}$ | $\mathcal{M}$ | ${\mathit{\chi}}^{\mathbf{2}}$/DoF | Varied in ${\mathit{\chi}}^{\mathbf{2}}$-Minimization |
---|---|---|---|---|---|---|

0.14 | $-0.3$ | 8.49 | 10 | 43.22 | 113.62/113 | $\kappa {\rho}_{\mathrm{g}}{H}_{0}^{-1}$, $\mathcal{M}$ |

0.15 | $-0.1$ | 5 | 10 | 43.28 | 113.22/114 | $\mathcal{M}$ |

0.16 | $-0.2$ | 5 | 10 | 43.31 | 114.3/114 | $\mathcal{M}$ |

0.173 | $-0.3$ | 5 | 10 | 43.34 | 113.80/114 | $\mathcal{M}$ |

0.147 | $-0.1$ | 5 | 8 | 43.27 | 114.63/114 | $\mathcal{M}$ |

0.16 | $-0.2$ | 5 | 8 | 43.30 | 111.94/114 | $\mathcal{M}$ |

0.17 | $-0.3$ | 5 | 8 | 43.33 | 114.02/114 | $\mathcal{M}$ |

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Vishwakarma, R.G.; Narlikar, J.V. Is it no Longer Necessary to Test Cosmologies with Type Ia Supernovae? *Universe* **2018**, *4*, 73.
https://doi.org/10.3390/universe4060073

**AMA Style**

Vishwakarma RG, Narlikar JV. Is it no Longer Necessary to Test Cosmologies with Type Ia Supernovae? *Universe*. 2018; 4(6):73.
https://doi.org/10.3390/universe4060073

**Chicago/Turabian Style**

Vishwakarma, Ram Gopal, and Jayant V. Narlikar. 2018. "Is it no Longer Necessary to Test Cosmologies with Type Ia Supernovae?" *Universe* 4, no. 6: 73.
https://doi.org/10.3390/universe4060073