#
Cosmological Properties of the Cosmic Web^{ †}

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^{†}

## Abstract

**:**

## 1. Introduction

`Quijote`simulations [4] to study the cosmic web properties, and we show using a deep neural network that they indeed allow for information beyond the matter field only, the paper is organised as follows: in Section 2 we describe our cosmic web segmentation technique, in Section 3 we illustrate the dynamical properties of the cosmic web environments and their dependence on the cosmological model, while, in Section 4, we present the results of the cosmological parameters inference using the cosmic web categories using a deep neural network. At the end, in Section 5, we discuss and explain the results; all the results of our study are presented in details in two forthcoming papers [5,6].

## 2. TWEB Algorithm

## 3. Cosmic Web Properties

#### 3.1. Observables

#### 3.2. Data

`Quijote`suite of simulations [4], in particular the high-resolution fiducial model, with ${1024}^{3}$ particles in a box of 1 (Gpc·h${}^{-1}$)${}^{3}$, and cosmological parameters $[{\Omega}_{m},{\Omega}_{b},h,{\sigma}_{8},{n}_{s}]=[0.3175,0.049,0.6711,0.834,0.9624]$ for redshifts $z=127,3,2,1,0.5,0$.

#### 3.3. Non-Gaussianities

- $\sigma =\sqrt{<{(\delta -<\delta >)}^{2}>}$
- $S=\sqrt{<{(\delta -<\delta >)}^{3}>}/{\sigma}^{3}$
- $K=\sqrt{<{(\delta -<\delta >)}^{4}>}/{\sigma}^{4}-3$

#### 3.4. Cosmological Dependence of CW Environment

## 4. Cosmological Parameters Inference

#### 4.1. Method

#### 4.2. Performance Evaluation and Results

- Visually, by plotting a predicted vs. true value scatter plot: the less scattered around the identity line, the better the performance is (cf. Figure 4).
- Quantitatively by computing the relative squared error $RSE=\frac{{\sum}_{i=1}^{{n}_{test}}{({y}_{pred}^{i}-{y}_{true}^{i})}^{2}}{{\sum}_{i=1}^{{n}_{test}}{({y}_{true}^{i}-<{y}_{true}>)}^{2}}$ where ${y}_{true}$ and ${y}_{pred}$ are, respectively, the true and predicted parameter, and $<{y}_{true}>$ is the average of the true parameters; the RSE allows us to compare different inference results: the lower the RSE, the better the performance is (cf. Table 1).

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. DNN Architecture

- Input layer: working with one field, our input layer has 500 points (number of $P\left(k\right)$ points); when we combine all the fields, we stack all the power spectra creating an input of 2500 points
- Hidden layers: In the case of one field, the hidden layers have 1024 neurones each; in the case of combined fields, we use hidden layers with 2048 neurones. We also use a dropout layer [10] of 0.3 rate between every two hidden layers; the number of hidden layers was individually tuned for every observable and is summarized in the table below.

Observable | Hidden Layers | Learning Rate |
---|---|---|

$P\left(k\right)$—matter | 2 | 5 × 10${}^{-6}$ |

$P\left(k\right)$—voids | 2 | 1 × 10${}^{-6}$ |

$P\left(k\right)$—walls | 3 | 1 × 10${}^{-5}$ |

$P\left(k\right)$—filaments | 3 | 5 × 10${}^{-6}$ |

$P\left(k\right)$—nodes | 3 | 1 × 10${}^{-6}$ |

$P\left(k\right)$—all | 3 | 1 × 10${}^{-6}$ |

## References

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**Figure 1.**Evolution wrt to the cosmic epoch of the probability density function of each cosmic web environment; the top left panel is for the whole matter field, while the bottom right panel is the $P\left(\delta \right)$ of all categories at $z=0$.

**Figure 2.**Evolution of each of the four moments of the density field for each category wrt scale factor $a=\frac{1}{1+z}$; for the mean density, we plot the absolute value on a logarithmic scale, but the mean density for voids and walls are always negative.

**Figure 3.**Scatter plots of some cosmological parameters with colour maps corresponding to the values of the observable written on the top of each panel.

**Figure 4.**Ground truth vs. predicted values scatter plot; using the column, we can compare how the fit changes for one parameter using different categories. Horizontally, we can compare given category predictions for different parameters.

**Table 1.**Relative squared error for each cosmological parameter using different categories, the last line is when we combine all the categories, and we obtain the best results except for ${\Omega}_{b}$, the best results for each parameters are written in bold.

Category | ${\mathit{RSE}}_{{\mathbf{\Omega}}_{\mathit{m}}}$ | ${\mathit{RSE}}_{{\mathbf{\Omega}}_{\mathit{b}}}$ | ${\mathit{RSE}}_{\mathit{h}}$ | ${\mathit{RSE}}_{{\mathit{n}}_{\mathit{s}}}$ | ${\mathit{RSE}}_{{\mathit{\sigma}}_{8}}$ |
---|---|---|---|---|---|

Matter | 0.0098 | 0.4507 | 0.5097 | 0.1085 | 0.0022 |

Voids | 0.0405 | 0.4003 | 0.4322 | 0.0388 | 0.0034 |

Walls | 0.0802 | 0.9419 | 0.8534 | 0.0340 | 0.0027 |

Filament | 0.0105 | 0.3760 | 0.3975 | 0.0569 | 0.0018 |

Nodes | 0.0047 | 0.5115 | 0.5379 | 0.1555 | 0.0055 |

All | 0.0041 | 0.6317 | 0.3565 | 0.0198 | 0.0016 |

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**MDPI and ACS Style**

Shalak, M.; Alimi, J.-M.
Cosmological Properties of the Cosmic Web. *Phys. Sci. Forum* **2023**, *7*, 53.
https://doi.org/10.3390/ECU2023-14046

**AMA Style**

Shalak M, Alimi J-M.
Cosmological Properties of the Cosmic Web. *Physical Sciences Forum*. 2023; 7(1):53.
https://doi.org/10.3390/ECU2023-14046

**Chicago/Turabian Style**

Shalak, Majd, and Jean-Michel Alimi.
2023. "Cosmological Properties of the Cosmic Web" *Physical Sciences Forum* 7, no. 1: 53.
https://doi.org/10.3390/ECU2023-14046