# Weak Lensing Data and Condensed Neutrino Objects

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Neutrino Magnetic Moment Power Loss

## 3. Condensed Neutrinos is an Idea Many Decades Old

^{2}[11].5 Mass mixing is only possible if the neutrinos have very close spacing of their masses:

## 4. Neutrino Equation of State

_{1}, ν

_{2}, ν

_{3}, the three neutrino flavors may initially have three different Fermi momenta, Figure 2. However, mixing causes the higher Fermi levels to vacate and fill unfilled other flavor neutrino levels, resulting in the same Fermi momentum for all three neutrino flavors. The same is true for the three flavor anti-neutrinos. Thus, the most general Condensed Neutrino Object will be described by two Fermi momenta. However, spectroscopically, no measurement can be made to differentiate these different types and, since we are only interested in the mass and radius of these large objects, we describe them by setting the neutrino Fermi momentum equal to the anti-neutrino Fermi momentum. Furthermore, since the neutrino masses are nearly identical, we use one neutrino mass m

_{ν}for all six flavors and call it the “neutrino degenerate mass”. Thus, the equation of state will be a Fermion fluid with common mass m

_{ν}and 6 flavors (3 neutrino and 3 anti-neutrino), each having 2 degrees of freedom from their spin vector direction, and each flavor contributing to the pressure. The CNO EOS [12] is

_{f}/m

_{ν}c; p

_{f}= Fermi momentum; and with c = speed of light (as before). The equation of hydro-statistic equilibrium is

_{ν}is determined. Observationally: x(0) << 1. The value of m

_{ν}has to be determined by observational astronomical data, and be consistent with any future direct measurement of any neutrino or anti-neutrino flavor.

## 5. CNO are Stable Objects

_{total}is the total mass in solar masses (M

_{Θ}), ly (light years), Ω (gravitational potential energy of formation), R

_{S}(Schwarzschild radius).

_{ν}. By fitting the Dark Matter observational data, an approximate value of m

_{ν}can be predicted. It is this predicted value that the KATRIN experiment [17] should use, since its team is directly measuring the mass of the electron anti-neutrino, which should be nearly the value of the common degenerate neutrino mass.

## 6. Materials and Methods

^{®}from Wolfram Research was used to solve numerically the hydrostatic equation for Condensed Neutrino Objects. These equations and details were sufficient for other researchers to replicate, and were originally provided in [12]. The Dark Matter signature for CNO with the methods and arguments used to identify and constrain the applicable data were provided in [20]. A solution for our Local Group CNO, with sufficient detail for replication, were first obtained using the quite familiar Monte Carlo method and was described in [21].

^{®}to provide non-linear fits of our theoretical density profiles to the Einasto density profile for a wide range of potential degenerate neutrino mass and x(0) solution pairs. Mathematica

^{®}was also used to provide new results for a number of commonly used goodness-of-fit tests for our functional form (shown below—the fit derivation is found in the appendix of [20]). We now propose a comparison of our initial Monte Carlo (FORTRAN) results with our results from a Latin Hypercube implementation in C++, providing a comparison of these results for the center solutions of the CNO embedding the Local Group of galaxies on a starfield night sky. We also include a revised plot of potential/applicable CNO solutions for the CNO_LG with the identified weak lensing data.

## 7. Updated and New Results

#### 7.1. Einasto Density Profile Fits

^{2}the fit to the data overlap the theoretical values extremely well up to a CNO radius ~1.7 Mpc. Beyond those values, the theoretical density/radius values were generally less than (i.e., fall short of) the corresponding Einasto profile density fit’s value.

^{2}) to theoretical spherical CNOs for various x(0) boundary conditions. R

_{max}is the spherical CNO’s outer boundary for a given degenerate neutrino mass and x(0) is the (internal) boundary condition. Also of note in Table 3 is the nearly constant shape parameter with values of ~2.20 across a range of degenerate neutrino masses and x(0) constraint parameters. This is an important finding that directly contradicts Einasto fit values from N-body modeling studies and will be revisited in more detail in the discussion section.

#### 7.2. Dark Matter Galaxy Cluster Data

^{2}and a non-linear fit to the data revealing a solution of 0.826907 eV/c

^{2}± 0.05862 eV/c

^{2}at the 95% confidence interval. The data sources, as noted in the table’s type column, use different methods for estimating the mass of the Dark Matter contained in the various clusters. Figure 4 shows the data points from the various utilized mass estimate methods6.

^{®}we ran 5000 Monte Carlo iterations from uniformly distributed random draws of radii in the same range as the data (i.e., from 1.19 to 3 Mega parsecs). The results from this test for the five-common goodness-of-fit measures provided by Mathematica

^{®}are listed in Table 5. We see that, since the p-values tend to easily exceed the 95% confidence rejection level (i.e., p-values are greater than 0.05) we cannot reject our null hypothesis that Equation (6) describes the available data.

#### 7.3. Monte Carlo and Latin Hypercube Local Group CNO Center Locations

_{1}) and the mass at the center (R

_{C}) were computed by integrating our degenerate neutrino density solutions for various potential degenerate neutrino masses with CNO central Fermi momentum boundary conditions from the CNO center in Equation (8) out to these specific (R

_{1}and R

_{C}) distances11.

^{2}) for various central boundary condition values for the Fermi momentum ratio x(0) = $\frac{Pf}{{m}_{\nu}c}$ (x-axis). Galactic speed constraints from the scientific literature for the Milky Way at 8 and 20 Kilo-parsec (kpc) were used to define the greyed-out area of acceptable Fermi momentum ratio solutions.

## 8. Discussion

^{2}range. This value can be tested in the upcoming KATRIN measurement [17] of the electron anti-neutrino in beta-decay.

## Author Contributions

## Conflicts of Interest

## References

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1 | The authors performed a rough order of magnitude calculation on the number of expected CNOs in [11]. |

2 | v is the neutrino speed, c is the speed of light. |

3 | MRI (magnetic resonance imaging) uses this radiation physics. |

4 | |

5 | eV/c ^{2} is electron Volts divided by the speed of light squared. |

6 | The authors are aware of efforts to use lensing of the Cosmic Microwave Background for mapping Dark Matter distributions (https://arxiv.org/pdf/1707.09369.pdf). However, we are awaiting appropriate data to include in future analysis. |

7 | For a definition of M _{200}, R_{200}, M_{vir} and R_{vir.} please see https://en.wikipedia.org/wiki/Virial_mass. |

8 | Coordinates obtained from Wolfram Research’s Mathematica ^{®} 11.2. |

9 | See for example https://en.wikipedia.org/wiki/Latin_hypercube_sampling for a description of Latin hypercube sampling. |

10 | |

11 | The rotation speed of a spiral arm is relative to the spiral galaxy’s center. |

**Figure 1.**Majorana neutrinoless double beta decay4, not seen experimentally.

**Figure 3.**The figure shows the mass density (ρ) per solar mass per Kilo-parsec (kpc) cubed against the CNO radius in Mega-parsecs (Mpc) for a degenerate neutrino mass of 0.8 eV/c

^{2}for central x(0) boundary conditions ranging from 0.005 to 0.01 with overlaid Einasto density model fits as: (

**a**) linear-linear plot; (

**b**) log-log plot.

**Figure 5.**The log-log plotted virial mass and radii from identified weak lensing data with best fit to the data (0.827 eV/c

^{2}) line and the 95% Confidence Interval lines of ±0.059 eV/c

^{2}from the best fit line, respectively.

**Figure 6.**Our Local Group CNO spiral galaxy orientation. Two of the three embedded spiral galaxies have spin axis which are radially aligned to the CNO center, while the Milky Way has a canted spin axis from the center.

**Figure 7.**The local group’s CNO center plotted using The Aerospace Corporation’s SOAP version 14.0.7 as determined by independent implementations in C++ and FORTRAN.

**Figure 8.**The geometry for the CNO’s center to distances from the center of the galaxies was used for computing the density differences for: (

**a**) the canted Milky Way’s center to 8 kilo parsecs out from the center using the law of cosines; (

**b**) the simpler geometry for M31 from the center to 15 kilo parsecs out with no cant angle.

**Figure 9.**Plots potential local group spiral galactic speed solutions (y-axis) for a CNO for a degenerate neutrino mass of 0.826907 eV/c

^{2}against various central boundary condition values for the Fermi momentum ratio (x-axis).

**Figure 10.**The plotted virial mass and radii from identified weak lensing data now also displaying acceptable Local Group CNO solutions, including the CNO mass and radii values used in Figure 11.

**Figure 11.**Plots the FORTRAN 3D solution bounding CNO with the locations of M31, M33 and the Milky Way.

x(0) (for m_{ν}c^{2} = 1 eV) | M_{total} (M_{Θ}) | R_{0} (ly) | Ω (joules) | R_{0} (R_{S}) |
---|---|---|---|---|

0.001 | 3.100 × 10^{13} | 13.01 × 10^{6} | −1.764 × 10^{54} | 1.344 × 10^{6} |

0.01 | 9.809 × 10^{14} | 4.12 × 10^{6} | −5.569 × 10^{57} | 1.345 × 10^{4} |

0.1 | 3.077 × 10^{16} | 1.30 × 10^{6} | −1.739 × 10^{61} | 135.3 |

1.0 | 5.27 × 10^{17} | 3.72 × 10^{5} | −1.969 × 10^{64} | 2.254 |

_{Θ}is the solar mass, R

_{0}is the radius, ly is light years, Rs is the Schwarzschild radius.

Interesting Property | Consequence |
---|---|

CNO obey Pauli Exclusion Principle | CNO cannot grow from fusion (like Black Holes)—CNO repel each other |

Embedded matter (galaxies and gas) undergo simple harmonic motion | Embedded gas is not in thermodynamic equilibrium, contrary to assumptions in x-ray data analysis |

CNO can be internally excited (lowest excited state is a quadrupole oscillation) | CNO do not necessarily have a spherical shape |

CNO are primarily composed of the original cosmological neutrinos and anti-neutrinos formed in the Big Bang | CNO are common objects (not rare) |

Spherical CNO Properties | Einasto Density Parameters | |||||
---|---|---|---|---|---|---|

m_{ν} (eV/c^{2}) | x(0) | M(R_{max}) (10^{14} Mo) | R_{max} (Kpc) | ρ_{−2} (per Mo/Kpc^{3}) | r_{−2} (Kpc) | α |

0.7 | 0.005 | 7.07563 | 3646.02 | 8336.41 | 1853.96 | 2.2081 |

0.006 | 9.30115 | 3329 | 14,409.6 | 1692.09 | 2.20887 | |

0.007 | 11.7211 | 3082.04 | 22,878.5 | 1566.69 | 2.20832 | |

0.008 | 14.3206 | 2883.2 | 34,148.4 | 1465.54 | 2.20801 | |

0.009 | 17.0877 | 2718.57 | 48,638.3 | 1381.42 | 2.20893 | |

0.01 | 20.0131 | 2579.16 | 66,700.5 | 1310.7 | 2.20801 | |

0.015 | 37.4921 | 2107 | 22,5235 | 1069.76 | 2.20918 | |

0.02 | 56.5911 | 1823.84 | 533,578 | 926.671 | 2.20704 | |

0.8 | 0.005 | 5.41728 | 2794 | 14,219.9 | 1419.48 | 2.20743 |

0.006 | 7.12107 | 2550 | 24,564.9 | 1295.97 | 2.20644 | |

0.007 | 8.97393 | 2360 | 39,046.8 | 1199.14 | 2.20924 | |

0.008 | 10.9657 | 2208 | 58,287.6 | 1121.65 | 2.20922 | |

0.009 | 13.0828 | 2081.46 | 82,932 | 1057.9 | 2.20699 | |

0.01 | 15.3225 | 1974.72 | 113,784 | 1003.47 | 2.20745 | |

0.015 | 28.1461 | 1612.37 | 384,147 | 819.097 | 2.20792 | |

0.02 | 73.6874 | 1397 | 909,020 | 709.983 | 2.20246 | |

0.95 | 0.005 | 3.84162 | 1983 | 28,278.5 | 1006.5 | 2.20706 |

0.006 | 5.04993 | 1808 | 48,899.8 | 918.405 | 2.20894 | |

0.007 | 7.27676 | 1674 | 77,601.6 | 850.565 | 2.20691 | |

0.008 | 7.77518 | 1565.37 | 115,867 | 795.492 | 2.20753 | |

0.009 | 9.42287 | 1477 | 164,901 | 750.167 | 2.20606 | |

0.01 | 10.9897 | 1401 | 226,441 | 711.225 | 2.209 | |

0.015 | 19.9595 | 1143.38 | 762,388 | 581.445 | 2.2013 | |

0.02 | 31.1658 | 990.5 | 1,807,020 | 503.513 | 2.20053 |

**Table 4.**Identified weak lensing data and two data points from weak lensing and strong lensing data which used an NFW density fit.

Cluster | z | M_{vir} (10^{14} Mo) | R_{vir} (Mpc) | Type | Reference |
---|---|---|---|---|---|

MS2137-23 * | 0.310 | ${7.72}_{-0.42}^{+0.47}$ | 1.89 ± 0.04 | SL+WL (NFW fit) | [24] |

Coma (Abell 1656) ** | 0.024 | ${6.1}_{-3.5}^{+12.1}$ | ${2.5}_{-0.5}^{+0.8}$ | WL NFW | [25] |

A914 | 0.193 | 11 ± 6 | ${1.23}_{-0.12}^{+0.13}$ | WL | [26] |

A1351 | 0.328 | 33 ± 14 | ${1.68}_{-0.13}^{+0.18}$ | WL | [26] |

A1576 | 0.299 | 40 ± 14 | ${1.52}_{-0.21}^{+0.10}$ | WL | [26] |

A1722 | 0.326 | 12 ± 6 | ${1.86}_{-0.19}^{+0.22}$ | WL | [26] |

A1995 | 0.321 | 11 ± 4 | ${1.19}_{-0.20}^{+0.15}$ | WL | [26] |

A2261 | 0.225 | 22 ± 2 | ~3 | SL+WL (NFW fit) | [27] |

A1689 | 0.183 | 13 ± 2.05 | 2.011 ± 0.113 | 3D WL | [28] |

A1703 | 0.281 | 13.25 ± 2.21 | 1.915 ± 0.148 | 3D WL | [28] |

A370 | 0.375 | 23.99 ± 2.49 | 2.215 ± 0.079 | 3D WL | [28] |

Cl0024 + 17 | 0.395 | 13.29 ± 2.24 | 1.799 ± 0.105 | 3D WL | [28] |

RXJ1347 − 11 | 0.451 | 11.5 ± 2.50 | 1.663 ± 0.115 | 3D WL | [28] |

_{200}and R

_{200}, thus may be an underestimate of M

_{vir}and R

_{vir.}; ** Uses R

_{vir}at ${h}_{70}^{-1}$.7

**Table 5.**Goodness-of-fit statistics from 5000 Monte Carlo iterations for the five common tests provided by Mathematica

^{®}. CI is Confidence Interval; IQR is Interquartile Range.

Goodness-of-Fit statistics m_{ν} = 0.8269 (5000 Monte Carlo iterations) | Mean p-value | Mean CI | Variance | Median p-value | IQR |
---|---|---|---|---|---|

Anderson-Darling [29] | 0.464 | ±0.006 | 0.049 | 0.415 | 0.397 |

Cramér-von Mises [30] | 0.446 | ±0.006 | 0.048 | 0.379 | 0.384 |

Kolmogorov-Smirnov [31] | 0.513 | ±0.007 | 0.063 | 0.379 | 0.403 |

Pearson χ^{2} [32] | 0.564 | ±0.005 | 0.031 | 0.53 | 0.304 |

Watson U^{2} [30] | 0.33 | ±0.006 | 0.041 | 0.261 | 0.313 |

**Table 6.**Coordinates of the Local Group spiral galaxies.8

Name | Distance (kpc) | Dec (°) | RA (°) |
---|---|---|---|

M31 | 788.333 | 41.2689 | 10.685 |

M33 | 862.417 | 30.6581 | 23.466 |

Milky Way | 7.61113 | −29.0078 | 266.417 |

Quantity | Predicted Value |
---|---|

Center CNO-LG distance from MW | 673.422 kpc |

Center CNO-LG distance from M33 | 740.423 kpc |

Center CNO-LG distance from M31 | 656.015 kpc |

Algorithm error center CNO-LG | 0.000706162 kpc |

Right ascension of CNO-LG center | −26.6588° |

Declination of CNO-LG center | 0.91773° |

Galactic longitude of CNO-LG center | 62.83928785° |

Galactic latitude of CNO-LG center | −42.77834848° |

Milky way cant angle | 47.221° |

Local Group Galaxy | Distance from Center (kpc) | V_{DM} Low (km/s) | V_{DM} High (km/s) | Reference |
---|---|---|---|---|

M33 | 15 | ~65 | ~110 | [35] |

M31 | 30 | ~50 | ~110 | [36] |

Milky Way | 8 | ~80 | ~260 | [37] |

Milky Way | 20 | ~250 | ~400 | [37] |

**Table 9.**Local Group CNO-induced approximate galactic rotational speeds for the degenerate neutrino masses determined from the weak lensing data fits.

0.768290 eV/c^{2} | 0.826907 eV/c^{2} | 0.88552 eV/c^{2} | |||||||
---|---|---|---|---|---|---|---|---|---|

x(0) boundary values → | 0.014 | 0.017 | 0.021 | 0.013 | 0.017 | 0.022 | 0.013 | 0.019 | 0.025 |

M33 (@ 15 kpc) (km/s) | 26 | 31 | 38 | 24 | 30 | 35 | 23 | 27 | 16 |

M31 (@ 30 kpc) (km/s) | 56 | 70 | 88 | 54 | 71 | 88 | 54 | 73 | 76 |

MW (@ 8 kpc) (km/s) | 162 | 203 | 253 | 155 | 204 | 250 | 156 | 203 | 201 |

MW (@ 20 kpc) (km/s) | 257 | 321 | 399 | 246 | 322 | 393 | 246 | 319 | 310 |

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Morley, P.; Buettner, D.
Weak Lensing Data and Condensed Neutrino Objects. *Universe* **2017**, *3*, 81.
https://doi.org/10.3390/universe3040081

**AMA Style**

Morley P, Buettner D.
Weak Lensing Data and Condensed Neutrino Objects. *Universe*. 2017; 3(4):81.
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**Chicago/Turabian Style**

Morley, Peter, and Douglas Buettner.
2017. "Weak Lensing Data and Condensed Neutrino Objects" *Universe* 3, no. 4: 81.
https://doi.org/10.3390/universe3040081