# The Line-of-Sight Analysis of Spatial Distribution of Galaxies in the COSMOS2015 Catalogue

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## Abstract

**:**

## 1. Introduction

## 2. The COSMOS2015 Catalogue

#### 2.1. Description

#### 2.2. Photometric Redshifts

#### 2.3. Selection Effects

## 3. The Method for Line-of-Sight Analysis of the LSSU in Spatial Bins

#### 3.1. Spatial Distribution of Galaxies

#### 3.2. Radial Histogram of the Number of Galaxies

#### 3.3. The Approximations of Homogeneous Distribution

#### 3.4. The Fluctuation Amplitudes

#### 3.5. Comparison with the SCM Predictions

#### 3.6. Alternative Approach

#### 3.7. Calculating the Fluctuation Scales

#### 3.8. Applications of the Method

## 4. Results

#### 4.1. Entire Sample of the COSMOS2015 Catalogue

#### 4.2. The w-Sampling

- any w (no sampling), i.e., the entire sample;
- $w>0.7$ (weak sampling), i.e., galaxies with relative error $\sigma \left(z\right)<30\%$ at significance level of $1\sigma $ or $\sigma \left(z\right)<90\%$ at significance level of $3\sigma $;
- with $w>0.9$ (medium sampling), i.e., galaxies with relative error $\sigma \left(z\right)<10\%$ at significance level of $1\sigma $ or $\sigma \left(z\right)<30\%$ at significance level of $3\sigma $;
- with $w>0.97$ (strong sampling), i.e., galaxies with relative error $\sigma \left(z\right)<3\%$ at significance level of $1\sigma $ or $\sigma \left(z\right)<9\%$ at significance level of $3\sigma $.

## 5. Discussion

## 6. Conclusions

- The method for analyzing radial fluctuations of the number of galaxies along the line of sight (see the last work Shirokov et al. [9]) now takes into account the integral values within each bin for all quantities. That increases the mathematical rigor and certainty of the results. The target variables became more robust for comparative analysis of different samples of galaxies with the developed algorithms for error estimation. The use of logarithmic redshift bins can better take into account the photometric errors. Moreover, the metric bin size in logarithmic scale slightly depends on the redshift. Instead of analytical bias functions, numerical estimates of galaxy biases can be obtained from N-body simulations of the Universe (in a grid of model parameters) or by using the concept of model fractal catalogs, as in Shirokov et al. [30].
- For the case of the theoretical form of approximation of homogeneity in the $\Lambda $CDM frameworks, the average standard deviation of detected structures from homogeneity is ${\sigma}_{\mathrm{mean}}^{\Lambda \mathrm{CDM}}=0.09\pm 0.02$, and the average characteristic size of structures is ${R}_{\mathrm{mean}}^{\Lambda \mathrm{CDM}}=790\pm 150$ Mpc. The maximum size of the detected structure is ${R}_{j}=1,754\pm 40$ Mpc, and the minimum one is ${R}_{j}=147\pm 24$ Mpc.
- For the case of the empirical approximation of homogeneity, the average standard deviation of detected structures from homogeneity is ${\sigma}_{\mathrm{mean}}^{\mathrm{empiric}}=0.08\pm 0.01$, and the average characteristic size of structures is ${R}_{\mathrm{mean}}^{\mathrm{empiric}}=640\pm 140$ Mpc. The maximum size of the detected structure is ${R}_{j}=2,000\pm 80$ Mpc, and the minimum one is ${R}_{j}=145\pm 36$ Mpc.
- We have introduced the selection parameter w to take into account different uncertainty of redshifts $\sigma \left(z\right)$. At different values of the parameter w, we have obtained similar results.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

LSM | least-squares method |

LSSU | large-scale structure of Universe |

SCM | standard cosmological model ($\Lambda $CDM) |

PLCF | power-law correlation function |

## Appendix A

**Figure A1.**Histograms of the radial distribution of the COSMOS2015 photometric redshifts and corresponding fluctuation patterns for the disjoint 162,318 COSMOS galaxies (

**top**) and 29,470 UltraVISTA galaxies (

**bottom**) with a bin $\Delta z=0.1$ and ${z}_{\mathrm{max}}=3$ without sampling by $\sigma \left(z\right)$. The dashed lines indicate the least squares fit (

**left**) and Poisson noise 5$\sigma $ (

**right**).

**Figure A2.**Histograms of radial distribution of the COSMOS2015 photometric redshifts and the corresponding fluctuation patterns for the disjoint COSMOS galaxies (

**top**) and UltraVISTA galaxies (

**bottom**) with a bin $\Delta z=0.1$ and ${z}_{\mathrm{max}}=6$ without sampling by $\sigma \left(z\right)$. The dashed lines indicate the least squares fits (

**left**) and Poisson noise 5$\sigma $ (

**right**).

**Table A1.**Tables of structures for disjoint the COSMOS (top) and UltraVISTA (bottom) samples from the COSMOS2015 catalogue for a bin $\Delta z=0.1$ with ${z}_{\mathrm{max}}=3$ and without ${\sigma}_{z}$ selection. The approximation is by empirical Formula (6). In the last string there are the means of corresponding values (by all structures): mean of redshifts ${z}_{\mathrm{mean}}$, mean of structure sizes in Mpc ${R}_{\mathrm{mean}}$, mean of Poisson noise level $1\overline{{\sigma}_{\mathrm{P}}}$, mean of observed cosmic variance ${\sigma}_{\mathrm{mean}}$, mean of dark matter variance $\overline{{\sigma}_{\mathrm{dm}}}$, mean of dark matter bias $\overline{{b}_{\mathrm{dm}}}$, mean of Peebles correlation function variance $\overline{{\sigma}_{\mathrm{pl}}}$ and its bias $\overline{{b}_{\mathrm{pl}}}$ (see details in the text).

Sample | j | n | $\overline{\mathit{z}}$ | $\mathsf{\Delta}\mathit{r}\left(\mathit{z}\right)$, Mpc | ${\mathit{\sigma}}_{\mathbf{p}}$ | ${\mathit{\sigma}}_{\mathbf{obs}}$ | ${\mathit{\sigma}}_{\mathbf{dm}}$ | ${\mathit{b}}_{\mathbf{dm}}$ | ${\mathit{\sigma}}_{\mathbf{pl}}$ | ${\mathit{b}}_{\mathbf{pl}}$ |
---|---|---|---|---|---|---|---|---|---|---|

Only COSMOS | 1 | 3 | 0.15 ± 0.10 | ${774}_{-183}^{+203}$ | 0.008 | 0.100 ± 0.063 | 0.188 | 0.5 ± 0.3 | 0.199 | 0.5 ± 0.3 |

${z}_{\mathrm{max}}=3$ | 2 | 4 | 0.40 ± 0.15 | ${1012}_{-154}^{+183}$ | 0.006 | 0.063 ± 0.078 | 0.140 | 0.5 ± 0.6 | 0.198 | 0.3 ± 0.4 |

w = all | 3 | 4 | 0.80 ± 0.15 | ${799}_{-121}^{+145}$ | 0.006 | 0.025 ± 0.009 | 0.094 | 0.3 ± 0.1 | 0.178 | 0.1± 0.1 |

4 | 3 | 1.05 ± 0.10 | ${459}_{-108}^{+121}$ | 0.007 | 0.020 ± 0.057 | 0.070 | 0.3 ± 0.8 | 0.151 | 0.1 ± 0.4 | |

5 | 7 | 1.45 ± 0.30 | ${1111}_{-79}^{+108}$ | 0.005 | 0.105 ± 0.034 | 0.073 | 1.4 ± 0.5 | 0.184 | 0.6± 0.2 | |

6 | 6 | 2.00 ± 0.25 | ${704}_{-62}^{+79}$ | 0.006 | 0.185 ± 0.081 | 0.056 | 3.3 ± 1.4 | 0.168 | 1.1 ± 0.5 | |

7 | 5 | 2.45 ± 0.20 | ${462}_{-53}^{+62}$ | 0.008 | 0.058 ± 0.030 | 0.046 | 1.3 ± 0.6 | 0.154 | 0.4 ± 0.2 | |

8 | 3 | 2.75 ± 0.10 | ${204}_{-49}^{+53}$ | 0.011 | 0.014 ± 0.026 | 0.035 | 0.3 ± 0.4 | 0.125 | 0.1 ± 0.1 | |

means: | 1.38 ± 0.17 | ${691}_{-152}^{+166}$ | 0.007 | 0.071 ± 0.021 | 0.088 | 1.0 ± 0.4 | 0.170 | 0.4 ± 0.1 | ||

Only UltraVISTA | 1 | 2 | 0.10 ± 0.05 | ${397}_{-193}^{+203}$ | 0.028 | 0.202 ± 0.438 | 0.157 | 1.3 ± 2.8 | 0.155 | 1.3 ± 2.8 |

${z}_{\mathrm{max}}=3$ | 2 | 3 | 0.25 ± 0.10 | ${734}_{-173}^{+193}$ | 0.018 | 0.109 ± 0.071 | 0.155 | 0.7 ± 0.5 | 0.187 | 0.6 ± 0.4 |

w = all | 3 | 4 | 0.50 ± 0.15 | ${955}_{-145}^{+173}$ | 0.014 | 0.004 ± 0.044 | 0.124 | 0.0 ± 0.0 | 0.192 | 0.0 ± 0.0 |

4 | 4 | 0.90 ± 0.15 | ${753}_{-114}^{+137}$ | 0.014 | 0.077 ± 0.082 | 0.087 | 0.9 ± 0.9 | 0.174 | 0.4 ± 0.5 | |

5 | 5 | 1.25 ± 0.20 | ${823}_{-92}^{+114}$ | 0.013 | 0.071 ± 0.066 | 0.075 | 0.9 ± 0.9 | 0.176 | 0.4 ± 0.4 | |

6 | 5 | 1.65 ± 0.20 | ${666}_{-75}^{+92}$ | 0.014 | 0.111 ± 0.028 | 0.062 | 1.8 ± 0.4 | 0.167 | 0.7 ± 0.2 | |

7 | 9 | 2.25 ± 0.40 | ${1012}_{-53}^{+75}$ | 0.012 | 0.104 ± 0.031 | 0.055 | 1.9 ± 0.6 | 0.175 | 0.6 ± 0.2 | |

means: | 0.99 ± 0.18 | ${763}_{-151}^{+170}$ | 0.016 | 0.097 ± 0.023 | 0.102 | 1.2 ± 0.3 | 0.175 | 0.7 ± 0.2 |

**Table A2.**Tables of structures for disjoint the COSMOS (top) and UltraVISTA (bottom) samples from the COSMOS2015 catalogue for a bin $\Delta z=0.1$ with ${z}_{\mathrm{max}}=6$ and without ${\sigma}_{z}$ selection. The approximation is by empirical Formula (6). In the last string there are the means of corresponding values (by all structures): mean of redshifts ${z}_{\mathrm{mean}}$, mean of structure sizes in Mpc ${R}_{\mathrm{mean}}$, mean of Poisson noise level $1\overline{{\sigma}_{\mathrm{P}}}$, mean of observed cosmic variance ${\sigma}_{\mathrm{mean}}$, mean of dark matter variance $\overline{{\sigma}_{\mathrm{dm}}}$, mean of dark matter bias $\overline{{b}_{\mathrm{dm}}}$, mean of Peebles correlation function variance $\overline{{\sigma}_{\mathrm{pl}}}$ and its bias $\overline{{b}_{\mathrm{pl}}}$ (see details in the text).

Sample | j | n | $\overline{\mathit{z}}$ | $\mathsf{\Delta}\mathit{r}\left(\mathit{z}\right)$, Mpc | ${\mathit{\sigma}}_{\mathbf{p}}$ | ${\mathit{\sigma}}_{\mathbf{obs}}$ | ${\mathit{\sigma}}_{\mathbf{dm}}$ | ${\mathit{b}}_{\mathbf{dm}}$ | ${\mathit{\sigma}}_{\mathbf{pl}}$ | ${\mathit{b}}_{\mathbf{pl}}$ |
---|---|---|---|---|---|---|---|---|---|---|

Only COSMOS | 1 | 3 | 0.15 ± 0.10 | ${774}_{-183}^{+203}$ | 0.008 | 0.098 ± 0.064 | 0.188 | 0.5 ± 0.3 | 0.199 | 0.5 ± 0.3 |

${z}_{\mathrm{max}}=6$ | 2 | 4 | 0.40 ± 0.15 | ${1012}_{-154}^{+183}$ | 0.006 | 0.063 ± 0.078 | 0.140 | 0.5 ± 0.6 | 0.198 | 0.3 ± 0.4 |

w = all | 3 | 4 | 0.80 ± 0.15 | ${799}_{-121}^{+145}$ | 0.006 | 0.024 ± 0.009 | 0.094 | 0.3 ± 0.1 | 0.178 | 0.1 ± 0.1 |

4 | 3 | 1.05 ± 0.10 | ${459}_{-108}^{+121}$ | 0.007 | 0.021 ± 0.057 | 0.070 | 0.3 ± 0.8 | 0.151 | 0.1 ± 0.4 | |

5 | 7 | 1.45 ± 0.30 | ${1111}_{-79}^{+108}$ | 0.005 | 0.106 ± 0.034 | 0.073 | 1.5 ± 0.5 | 0.184 | 0.6± 0.2 | |

6 | 6 | 2.00 ± 0.25 | ${704}_{-62}^{+79}$ | 0.006 | 0.185 ± 0.081 | 0.056 | 3.3 ± 1.4 | 0.168 | 1.1 ± 0.5 | |

7 | 5 | 2.45 ± 0.20 | ${462}_{-53}^{+62}$ | 0.008 | 0.058 ± 0.030 | 0.046 | 1.2 ± 0.6 | 0.154 | 0.4 ± 0.2 | |

8 | 3 | 2.75 ± 0.10 | ${204}_{-49}^{+53}$ | 0.012 | 0.016 ± 0.025 | 0.035 | 0.3 ± 0.5 | 0.125 | 0.1 ± 0.1 | |

9 | 7 | 3.35 ± 0.30 | ${497}_{-37}^{+45}$ | 0.010 | 0.066 ± 0.032 | 0.039 | 1.7 ± 0.8 | 0.155 | 0.4 ± 0.2 | |

10 | 3 | 3.75 ± 0.10 | ${145}_{-35}^{+37}$ | 0.017 | 0.014 ± 0.015 | 0.028 | 0.0 ± 0.0 | 0.118 | 0.0 ± 0.0 | |

11 | 13 | 4.45 ± 0.60 | ${717}_{-25}^{+35}$ | 0.011 | 0.128 ± 0.028 | 0.034 | 3.7 ± 0.8 | 0.159 | 0.8 ± 0.2 | |

12 | 3 | 5.25 ± 0.10 | ${96}_{-23}^{+24}$ | 0.033 | 0.041 ± 0.036 | 0.022 | 1.2 ± 1.0 | 0.112 | 0.2 ± 0.2 | |

means: | 2.32 ± 0.20 | ${582}_{-124}^{+134}$ | 0.011 | 0.068 ± 0.015 | 0.069 | 1.3 ± 0.4 | 0.158 | 0.4 ± 0.1 | ||

Only UltraVISTA | 1 | 3 | 0.25 ± 0.10 | ${734}_{-173}^{+193}$ | 0.018 | 0.132 ± 0.095 | 0.155 | 0.9 ± 0.6 | 0.187 | 0.7 ± 0.5 |

${z}_{\mathrm{max}}=6$ | 2 | 4 | 0.50 ± 0.15 | ${955}_{-145}^{+173}$ | 0.014 | 0.045 ± 0.045 | 0.124 | 0.3 ± 0.4 | 0.192 | 0.2 ± 0.2 |

w = all | 3 | 4 | 0.90 ± 0.15 | ${753}_{-114}^{+137}$ | 0.014 | 0.130 ± 0.087 | 0.087 | 1.5 ± 1.0 | 0.174 | 0.7 ± 0.5 |

4 | 5 | 1.25 ± 0.20 | ${823}_{-92}^{+114}$ | 0.013 | 0.056 ± 0.065 | 0.075 | 0.7 ± 0.8 | 0.176 | 0.3 ± 0.4 | |

5 | 4 | 1.60 ± 0.15 | ${511}_{-79}^{+92}$ | 0.015 | 0.101 ± 0.028 | 0.059 | 1.7 ± 0.5 | 0.158 | 0.6 ± 0.2 | |

6 | 11 | 2.25 ± 0.50 | ${1271}_{-51}^{+79}$ | 0.010 | 0.175 ± 0.039 | 0.057 | 3.1 ± 0.7 | 0.179 | 1.0 ± 0.2 | |

7 | 3 | 2.85 ± 0.10 | ${197}_{-47}^{+51}$ | 0.022 | 0.002 ± 0.023 | 0.035 | 0.0 ± 0.0 | 0.124 | 0.0 ± 0.0 | |

8 | 15 | 3.65 ± 0.70 | ${1063}_{-30}^{+47}$ | 0.012 | 0.249 ± 0.041 | 0.040 | 6.2 ± 1.0 | 0.166 | 1.5 ± 0.3 | |

9 | 6 | 4.60 ± 0.25 | ${285}_{-26}^{+30}$ | 0.024 | 0.043 ± 0.029 | 0.030 | 1.2 ± 0.8 | 0.143 | 0.3 ± 0.2 | |

10 | 4 | 5.00 ± 0.15 | ${154}_{-24}^{+26}$ | 0.033 | 0.061 ± 0.032 | 0.025 | 2.0 ± 1.0 | 0.127 | 0.4 ± 0.2 | |

11 | 4 | 5.30 ± 0.15 | ${143}_{-23}^{+24}$ | 0.036 | 0.044 ± 0.045 | 0.024 | 1.0 ± 1.1 | 0.126 | 0.2 ± 0.2 | |

means: | 2.56 ± 0.24 | ${626}_{-141}^{+150}$ | 0.019 | 0.094 ± 0.023 | 0.065 | 1.9 ± 0.5 | 0.159 | 0.6 ± 0.1 |

## Appendix B

**Figure A3.**Histogram of radial distribution of the Deep-UltraVISTA [6] photometric redshifts recalculated to metric distances in comoving space with a bin $\Delta R=200$ Mpc without sampling by $\sigma \left(z\right)$. The dashed line indicates the least-squares fit by the theoretical approximation of homogeneity (according to the SCM) by Equation (3).

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**Figure 1.**The projection on the sky of 25,750 galaxies from the COSMOS2015 catalogue in the range $1.0<z<1.2$ with the error $\sigma \left(z\right)<10$%. The left angular map shows galaxies as points. The right angular map is gaussian interpolation in 36 × 42 pixels (each of which is equal to 2 squared minutes) plotted by using the matplotlib.pyplot library. The color palette shows the galaxy count in one pixel.

**Figure 2.**(

**left**) Histogram of the radial distribution of photometric redshifts from the COSMOS2015 catalogue in the range $0<z<6$ for 518,404 galaxies within bins $\Delta z=0.1$, and the approximations. Dotted lines mark the least-squares fittings. (

**right**) The fluctuation pattern. Dotted lines mark Poisson noise 5$\sigma $.

**Figure 3.**High resolution differential histograms of radial distribution of galaxies from the COSMOS2015 catalogue in redshift space (

**left**) with $\Delta z=0.03$, and metric space (

**right**) with $\Delta r\left(z\right)=50$ Mpc for the various values of the sampling parameter w.

**Figure 4.**Histograms of radial distribution of the photometric redshifts and corresponding fluctuation patterns for samples with the parameter $w=0.7$ for 391,098 galaxies (

**top**), and $w=0.9$ for 248,183 galaxies (

**bottom**) with a bin $\Delta z=0.1$. The dashed lines indicate the least squares fit (

**left**) and Poisson noise 5$\sigma $ (

**right**).

**Table 1.**The metric size of bins $\Delta z=0.1$ in Mpc at various redshifts. $\Delta r\left(z\right)$ is the size in the moment $t=t\left(z\right)$ (comoving distances), and $\Delta {d}_{\mathrm{L}}$ is the size in the moment $t=t\left(0\right)$ (proper distances).

$\overline{\mathit{z}}$ | 0.05 | 0.95 | 1.45 | 1.55 | 1.65 | 1.95 | 2.95 | 3.95 | 4.95 | 5.95 |
---|---|---|---|---|---|---|---|---|---|---|

$\Delta r(z,\Delta z)$, Mpc | 407 | 244 | 184 | 175 | 166 | 144 | 95 | 68 | 52 | 41 |

$\Delta {d}_{\mathrm{L}}(z,\Delta z)$, Mpc | 448 | 784 | 866 | 879 | 890 | 920 | 989 | 1032 | 1063 | 1087 |

**Table 2.**Tables of structures for w-samples from the COSMOS2015 catalogue for a bin $\Delta z=0.1$ with ${z}_{\mathrm{max}}=6$ and the various ${\sigma}_{z}$ selection. The approximation is by empirical Formula (6). In the last string there are the means of corresponding values (by all structures): mean of redshifts ${z}_{\mathrm{mean}}$, mean of structure sizes in Mpc ${R}_{\mathrm{mean}}$, mean of Poisson noise level $1\overline{{\sigma}_{\mathrm{P}}}$, mean of observed cosmic variance ${\sigma}_{\mathrm{mean}}$, mean of dark matter variance $\overline{{\sigma}_{\mathrm{dm}}}$, mean of dark matter bias $\overline{{b}_{\mathrm{dm}}}$, mean of Peebles correlation function variance $\overline{{\sigma}_{\mathrm{pl}}}$ and its bias $\overline{{b}_{\mathrm{pl}}}$ (see details in the text).

Sample | j | n | $\overline{\mathit{z}}$ | $\mathsf{\Delta}\mathit{r}\left(\mathit{z}\right)$, Mpc | ${\mathit{\sigma}}_{\mathbf{p}}$ | ${\mathit{\sigma}}_{\mathbf{obs}}$ | ${\mathit{\sigma}}_{\mathbf{dm}}$ | ${\mathit{b}}_{\mathbf{dm}}$ | ${\mathit{\sigma}}_{\mathbf{pl}}$ | ${\mathit{b}}_{\mathbf{pl}}$ |
---|---|---|---|---|---|---|---|---|---|---|

COSMOS2015 | 1 | 3 | 0.15 ± 0.10 | ${774}_{-183}^{+203}$ | 0.005 | 0.131 ± 0.064 | 0.188 | 0.7 ± 0.3 | 0.199 | 0.7 ± 0.3 |

${z}_{\mathrm{max}}=6$ | 2 | 3 | 0.35 ± 0.10 | ${694}_{-163}^{+183}$ | 0.004 | 0.065 ± 0.077 | 0.132 | 0.5 ± 0.6 | 0.179 | 0.4 ± 0.4 |

w = all | 3 | 3 | 0.55 ± 0.10 | ${618}_{-145}^{+163}$ | 0.004 | 0.044 ± 0.050 | 0.104 | 0.4 ± 0.5 | 0.168 | 0.3 ± 0.3 |

4 | 5 | 0.85 ± 0.20 | ${1035}_{-114}^{+145}$ | 0.003 | 0.058 ± 0.049 | 0.097 | 0.6 ± 0.5 | 0.188 | 0.3 ± 0.3 | |

5 | 8 | 1.50 ± 0.35 | ${1265}_{-75}^{+108}$ | 0.003 | 0.072 ± 0.028 | 0.073 | 1.0 ± 0.4 | 0.187 | 0.4 ± 0.2 | |

6 | 4 | 2.00 ± 0.15 | ${421}_{-65}^{+75}$ | 0.004 | 0.072 ± 0.050 | 0.050 | 1.4 ± 1.0 | 0.151 | 0.5 ± 0.3 | |

7 | 5 | 2.35 ± 0.20 | ${482}_{-55}^{+65}$ | 0.005 | 0.058 ± 0.022 | 0.048 | 1.2 ± 0.5 | 0.156 | 0.4 ± 0.1 | |

8 | 12 | 3.10 ± 0.55 | ${1000}_{-37}^{+55}$ | 0.004 | 0.162 ± 0.033 | 0.045 | 3.6 ± 0.7 | 0.168 | 1.0 ± 0.2 | |

9 | 5 | 3.95 ± 0.20 | ${273}_{-32}^{+36}$ | 0.010 | 0.059 ± 0.039 | 0.032 | 1.8 ± 1.2 | 0.140 | 0.4 ± 0.3 | |

10 | 4 | 4.40 ± 0.15 | ${180}_{-28}^{+31}$ | 0.014 | 0.050 ± 0.055 | 0.028 | 1.7 ± 1.9 | 0.129 | 0.4 ± 0.4 | |

11 | 7 | 4.85 ± 0.30 | ${321}_{-24}^{+28}$ | 0.013 | 0.126 ± 0.046 | 0.030 | 4.2 ± 1.5 | 0.145 | 0.9 ± 0.3 | |

means: | 2.19 ± 0.22 | ${642}_{-137}^{+148}$ | 0.006 | 0.082 ± 0.012 | 0.075 | 1.6 ± 0.4 | 0.164 | 0.5 ± 0.1 | ||

COSMOS2015 | 1 | 3 | 0.15 ± 0.10 | ${774}_{-183}^{+203}$ | 0.008 | 0.195 ± 0.035 | 0.188 | 1.0 ± 0.2 | 0.199 | 1.0 ± 0.2 |

${z}_{\mathrm{max}}=6$ | 2 | 3 | 0.35 ± 0.10 | ${694}_{-163}^{+183}$ | 0.005 | 0.039 ± 0.116 | 0.132 | 0.3 ± 0.9 | 0.179 | 0.2 ± 0.6 |

w = 0.7 | 3 | 3 | 0.55 ± 0.10 | ${618}_{-145}^{+163}$ | 0.005 | 0.101 ± 0.080 | 0.104 | 1.0 ± 0.8 | 0.168 | 0.6 ± 0.5 |

4 | 5 | 0.85 ± 0.20 | ${1035}_{-114}^{+145}$ | 0.003 | 0.135 ± 0.069 | 0.097 | 1.4 ± 0.7 | 0.188 | 0.7 ± 0.4 | |

5 | 8 | 1.50 ± 0.35 | ${1265}_{-75}^{+108}$ | 0.003 | 0.110 ± 0.036 | 0.073 | 1.5 ± 0.5 | 0.186 | 0.6 ± 0.2 | |

6 | 4 | 2.00 ± 0.15 | ${421}_{-65}^{+75}$ | 0.005 | 0.078 ± 0.055 | 0.050 | 1.5 ± 1.1 | 0.151 | 0.5 ± 0.4 | |

7 | 5 | 2.35 ± 0.20 | ${482}_{-55}^{+65}$ | 0.005 | 0.042 ± 0.022 | 0.048 | 0.9 ± 0.5 | 0.156 | 0.3 ± 0.1 | |

8 | 12 | 3.10 ± 0.55 | ${1000}_{-37}^{+55}$ | 0.005 | 0.189 ± 0.035 | 0.045 | 4.2 ± 0.8 | 0.168 | 1.1 ± 0.2 | |

9 | 3 | 3.75 ± 0.10 | ${145}_{-35}^{+37}$ | 0.013 | 0.010 ± 0.017 | 0.028 | 0.0 ± 0.0 | 0.118 | 0.0 ± 0.0 | |

10 | 4 | 4.00 ± 0.15 | ${202}_{-32}^{+35}$ | 0.013 | 0.092 ± 0.059 | 0.030 | 3.0 ± 1.9 | 0.131 | 0.7 ± 0.4 | |

11 | 7 | 4.55 ± 0.30 | ${347}_{-26}^{+31}$ | 0.014 | 0.116 ± 0.044 | 0.031 | 3.7 ± 1.4 | 0.147 | 0.8 ± 0.3 | |

12 | 5 | 5.25 ± 0.20 | ${193}_{-23}^{+25}$ | 0.025 | 0.269 ± 0.164 | 0.026 | 10.3 ± 6.3 | 0.134 | 2.0 ± 1.2 | |

means: | 2.37 ± 0.21 | ${598}_{-134}^{+144}$ | 0.009 | 0.115 ± 0.022 | 0.071 | 2.6 ± 0.8 | 0.160 | 0.8 ± 0.2 | ||

COSMOS2015 | 1 | 3 | 0.15 ± 0.10 | ${774}_{-183}^{+203}$ | 0.010 | 0.260 ± 0.108 | 0.188 | 1.4 ± 0.6 | 0.199 | 1.3 ± 0.5 |

${z}_{\mathrm{max}}=6$ | 2 | 4 | 0.50 ± 0.15 | ${955}_{-145}^{+173}$ | 0.005 | 0.054 ± 0.086 | 0.124 | 0.4 ± 0.7 | 0.192 | 0.3 ± 0.5 |

w = 0.9 | 3 | 5 | 0.85 ± 0.20 | ${1035}_{-114}^{+145}$ | 0.004 | 0.118 ± 0.080 | 0.097 | 1.2 ± 0.8 | 0.188 | 0.6 ± 0.4 |

4 | 3 | 1.15 ± 0.10 | ${434}_{-102}^{+114}$ | 0.005 | 0.011 ± 0.073 | 0.066 | 0.2 ± 1.0 | 0.149 | 0.1 ± 0.4 | |

5 | 14 | 1.90 ± 0.65 | ${1958}_{-55}^{+102}$ | 0.003 | 0.196 ± 0.043 | 0.066 | 3.0 ± 0.7 | 0.189 | 1.0 ± 0.2 | |

means: | 0.91 ± 0.24 | ${1031}_{-287}^{+302}$ | 0.005 | 0.128 ± 0.046 | 0.108 | 1.2 ± 0.5 | 0.183 | 0.7 ± 0.2 | ||

COSMOS2015 | 1 | 3 | 0.35 ± 0.10 | ${694}_{-163}^{+183}$ | 0.011 | 0.290 ± 0.173 | 0.132 | 2.2 ± 1.3 | 0.179 | 1.6 ± 1.0 |

${z}_{\mathrm{max}}=6$ | 2 | 5 | 0.65 ± 0.20 | ${1166}_{-129}^{+163}$ | 0.006 | 0.089 ± 0.055 | 0.114 | 0.8 ± 0.5 | 0.196 | 0.5 ± 0.3 |

w = 0.97 | 3 | 3 | 0.95 ± 0.10 | ${487}_{-114}^{+129}$ | 0.007 | 0.120 ± 0.229 | 0.075 | 1.6 ± 3.1 | 0.153 | 0.8 ± 1.5 |

4 | 3 | 1.35 ± 0.10 | ${388}_{92}^{+102}$ | 0.010 | 0.041 ± 0.135 | 0.059 | 0.7 ± 2.2 | 0.144 | 0.3 ± 0.9 | |

5 | 9 | 1.85 ± 0.40 | ${1216}_{-62}^{+92}$ | 0.010 | 0.231 ± 0.074 | 0.064 | 3.6 ± 1.2 | 0.182 | 1.3 ± 0.4 | |

means: | 1.03 ± 0.18 | ${790}_{-212}^{+227}$ | 0.009 | 0.154 ± 0.046 | 0.089 | 1.8 ± 0.5 | 0.171 | 0.9 ± 0.3 |

**Table 3.**Tables of structures for w-samples from the COSMOS2015 catalogue for a bin $\Delta z=0.1$ with ${z}_{\mathrm{max}}=6$ and the various ${\sigma}_{z}$ selection. The approximation is by theoretical Formula (3). In the last string there are the means of corresponding values (by all structures): mean of redshifts ${z}_{\mathrm{mean}}$, mean of structure sizes in Mpc ${R}_{\mathrm{mean}}$, mean of Poisson noise level $1\overline{{\sigma}_{\mathrm{P}}}$, mean of observed cosmic variance ${\sigma}_{\mathrm{mean}}$, mean of dark matter variance $\overline{{\sigma}_{\mathrm{dm}}}$, mean of dark matter bias $\overline{{b}_{\mathrm{dm}}}$, mean of Peebles correlation function variance $\overline{{\sigma}_{\mathrm{pl}}}$ and its bias $\overline{{b}_{\mathrm{pl}}}$ (see details in the text).

Sample | j | n | $\overline{\mathit{z}}$ | $\mathsf{\Delta}\mathit{r}\left(\mathit{z}\right)$, Mpc | ${\mathit{\sigma}}_{\mathbf{p}}$ | ${\mathit{\sigma}}_{\mathbf{obs}}$ | ${\mathit{\sigma}}_{\mathbf{dm}}$ | ${\mathit{b}}_{\mathbf{dm}}$ | ${\mathit{\sigma}}_{\mathbf{pl}}$ | ${\mathit{b}}_{\mathbf{pl}}$ |
---|---|---|---|---|---|---|---|---|---|---|

COSMOS2015 | 1 | 3 | 0.15 ± 0.10 | ${774}_{-183}^{+203}$ | 0.005 | 0.169 ± 0.133 | 0.188 | 0.9 ± 0.7 | 0.199 | 0.9 ± 0.7 |

${z}_{\mathrm{max}}=6$ | 2 | 3 | 0.35 ± 0.10 | ${694}_{-163}^{+183}$ | 0.004 | 0.110 ± 0.084 | 0.132 | 0.8 ± 0.6 | 0.179 | 0.6 ± 0.5 |

w = all | 3 | 4 | 0.60 ± 0.15 | ${900}_{-137}^{+163}$ | 0.003 | 0.038 ± 0.038 | 0.112 | 0.3 ± 0.3 | 0.186 | 0.2 ± 0.2 |

4 | 4 | 0.90 ± 0.15 | ${753}_{-114}^{+137}$ | 0.003 | 0.023 ± 0.057 | 0.087 | 0.3 ± 0.6 | 0.174 | 0.1 ± 0.3 | |

5 | 7 | 1.35 ± 0.30 | ${1172}_{-83}^{+114}$ | 0.003 | 0.076 ± 0.027 | 0.077 | 1.0 ± 0.4 | 0.186 | 0.4 ± 0.1 | |

6 | 8 | 2.00 ± 0.35 | ${989}_{-60}^{+83}$ | 0.003 | 0.114 ± 0.041 | 0.059 | 1.9 ± 0.7 | 0.176 | 0.6 ± 0.2 | |

7 | 7 | 2.75 ± 0.30 | ${616}_{-45}^{+57}$ | 0.005 | 0.134 ± 0.048 | 0.046 | 2.9 ± 1.1 | 0.161 | 0.8 ± 0.3 | |

8 | 6 | 3.30 ± 0.25 | ${420}_{-38}^{+45}$ | 0.006 | 0.082 ± 0.026 | 0.039 | 2.1 ± 0.7 | 0.151 | 0.5 ± 0.2 | |

means: | 1.42 ± 0.21 | ${790}_{-137}^{+155}$ | 0.004 | 0.093 ± 0.017 | 0.092 | 1.3 ± 0.3 | 0.177 | 0.5 ± 0.1 | ||

COSMOS2015 | 1 | 3 | 0.15 ± 0.10 | ${774}_{-183}^{+203}$ | 0.007 | 0.367 ± 0.129 | 0.188 | 2.0 ± 0.7 | 0.199 | 1.8 ± 0.7 |

${z}_{\mathrm{max}}=6$ | 2 | 3 | 0.35 ± 0.10 | ${694}_{-163}^{+183}$ | 0.005 | 0.015 ± 0.134 | 0.132 | 0.1 ± 1.0 | 0.179 | 0.1 ± 0.7 |

w = 0.7 | 3 | 3 | 0.55 ± 0.10 | ${618}_{-145}^{+163}$ | 0.005 | 0.083 ± 0.066 | 0.104 | 0.8 ± 0.6 | 0.168 | 0.5 ± 0.4 |

4 | 5 | 0.85 ± 0.20 | ${1035}_{-114}^{+145}$ | 0.003 | 0.067 ± 0.060 | 0.097 | 0.7 ± 0.6 | 0.188 | 0.4 ± 0.3 | |

5 | 8 | 1.40 ± 0.35 | ${1335}_{-79}^{+114}$ | 0.003 | 0.097 ± 0.032 | 0.076 | 1.3 ± 0.4 | 0.189 | 0.5 ± 0.2 | |

6 | 14 | 2.40 ± 0.65 | ${1561}_{-45}^{+79}$ | 0.003 | 0.196 ± 0.039 | 0.055 | 3.5 ± 0.7 | 0.180 | 1.1 ± 0.2 | |

7 | 6 | 3.30 ± 0.25 | ${420}_{-38}^{+45}$ | 0.007 | 0.044 ± 0.024 | 0.039 | 1.1 ± 0.6 | 0.151 | 0.3 ± 0.2 | |

means: | 1.29 ± 0.25 | ${920}_{-195}^{+211}$ | 0.005 | 0.124 ± 0.046 | 0.099 | 1.4 ± 0.4 | 0.179 | 0.7 ± 0.2 | ||

COSMOS2015 | 1 | 3 | 0.15 ± 0.10 | ${774}_{-183}^{+203}$ | 0.010 | 0.360 ± 0.203 | 0.188 | 1.9 ± 1.1 | 0.199 | 1.8 ± 1.0 |

${z}_{\mathrm{max}}=6$ | 2 | 3 | 0.35 ± 0.10 | ${694}_{-163}^{+183}$ | 0.007 | 0.066 ± 0.109 | 0.132 | 0.5 ± 0.8 | 0.179 | 0.4 ± 0.6 |

w = 0.9 | 3 | 3 | 0.55 ± 0.10 | ${618}_{-145}^{+163}$ | 0.006 | 0.029 ± 0.071 | 0.104 | 0.3 ± 0.7 | 0.168 | 0.2 ± 0.4 |

4 | 3 | 1.05 ± 0.10 | ${459}_{-108}^{+121}$ | 0.005 | 0.001 ± 0.109 | 0.070 | 0.0 ± 0.0 | 0.151 | 0.0 ± 0.0 | |

5 | 4 | 1.50 ± 0.15 | ${538}_{-83}^{+97}$ | 0.006 | 0.059 ± 0.053 | 0.062 | 1.0 ± 0.9 | 0.160 | 0.4 ± 0.3 | |

6 | 4 | 1.80 ± 0.15 | ${463}_{-71}^{+83}$ | 0.007 | 0.024 ± 0.043 | 0.054 | 0.4 ± 0.8 | 0.154 | 0.2 ± 0.3 | |

7 | 7 | 2.25 ± 0.30 | ${756}_{-55}^{+71}$ | 0.006 | 0.205 ± 0.065 | 0.053 | 3.9 ± 1.2 | 0.168 | 1.2 ± 0.4 | |

8 | 12 | 3.10 ± 0.55 | ${1000}_{-374}^{+55}$ | 0.006 | 0.483 ± 0.076 | 0.045 | 10.8 ± 1.7 | 0.168 | 2.9 ± 0.5 | |

means: | 1.34 ± 0.19 | ${663}_{-130}^{+146}$ | 0.006 | 0.153 ± 0.064 | 0.089 | 2.7 ± 1.3 | 0.168 | 1.0 ± 0.4 | ||

COSMOS2015 | 1 | 3 | 0.55 ± 0.10 | ${618}_{-145}^{+163}$ | 0.008 | 0.005 ± 0.066 | 0.104 | 0.0 ± 0.0 | 0.168 | 0.0 ± 0.0 |

${z}_{\mathrm{max}}=6$ | 2 | 3 | 0.75 ± 0.10 | ${548}_{-129}^{+145}$ | 0.007 | 0.057 ± 0.067 | 0.087 | 0.7 ± 0.8 | 0.160 | 0.4 ± 0.4 |

w = 0.97 | 3 | 4 | 1.20 ± 0.15 | ${633}_{-97}^{+114}$ | 0.008 | 0.063 ± 0.139 | 0.072 | 0.9 ± 1.9 | 0.166 | 0.4 ± 0.8 |

4 | 13 | 2.05 ± 0.60 | ${1678}_{-53}^{+92}$ | 0.008 | 0.338 ± 0.080 | 0.062 | 5.5 ± 1.3 | 0.185 | 1.8 ± 0.4 | |

5 | 25 | 3.85 ± 1.20 | ${1754}_{-25}^{+53}$ | 0.014 | 1.104 ± 0.175 | 0.040 | 27.3 ± 4.3 | 0.170 | 6.5 ± 1.0 | |

6 | 4 | 5.20 ± 0.15 | ${147}_{-23}^{+25}$ | 0.054 | 0.074 ± 0.087 | 0.025 | 2.1 ± 2.4 | 0.126 | 0.4 ± 0.5 | |

means: | 2.27 ± 0.38 | ${896}_{-282}^{+290}$ | 0.017 | 0.273 ± 0.173 | 0.065 | 7.3 ± 4.4 | 0.162 | 1.9 ± 1.0 |

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**MDPI and ACS Style**

Nikonov, M.; Chekal, M.; Shirokov, S.; Baryshev, A.; Gorokhov, V.
The Line-of-Sight Analysis of Spatial Distribution of Galaxies in the COSMOS2015 Catalogue. *Universe* **2020**, *6*, 215.
https://doi.org/10.3390/universe6110215

**AMA Style**

Nikonov M, Chekal M, Shirokov S, Baryshev A, Gorokhov V.
The Line-of-Sight Analysis of Spatial Distribution of Galaxies in the COSMOS2015 Catalogue. *Universe*. 2020; 6(11):215.
https://doi.org/10.3390/universe6110215

**Chicago/Turabian Style**

Nikonov, Maxim, Mikhail Chekal, Stanislav Shirokov, Andrey Baryshev, and Vladimir Gorokhov.
2020. "The Line-of-Sight Analysis of Spatial Distribution of Galaxies in the COSMOS2015 Catalogue" *Universe* 6, no. 11: 215.
https://doi.org/10.3390/universe6110215