# One-Point Statistics Matter in Extended Cosmologies

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. The Matter PDF in Spheres from Large Deviation Theory

- (i)
- The time- and scale-dependence of the linear variance ${\sigma}_{\mathrm{L}}^{2}(r,z)$.
- (ii)
- The non-linear variance of the log-density ${\sigma}_{ln\rho ,\mathrm{NL}}^{2}(R,z)$.
- (iii)
- The mapping between linear and final densities in spheres, which is taken to be spherical collapse ${\delta}_{\mathrm{L}}\mapsto {\rho}_{\mathrm{SC}}({\delta}_{\mathrm{L}})$ (or its inverse ${\delta}_{\mathrm{L}}^{\mathrm{SC}}(\rho ))$.

#### 2.2. Extended Cosmologies

#### 2.3. Simulations and Model Validation

`pyLDT`(https://github.com/mcataneo/pyLDT-cosmo, accessed on 4 December 2021) a modularised and user-friendly Python code that takes advantage of the PyJulia interface for computationally intensive tasks. The PDFs used in this paper were generated using version 0.4.9 of

`pyLDT`.

## 3. Results

#### 3.1. Matter PDFs in Extended Cosmologies

#### 3.2. Forecasting Constraining Power with the Fisher Formalism

#### 3.3. Response of the PDF to Changes in Cosmological Parameters

#### 3.4. Fisher Forecasts for Modified Gravity Detection and Dark Energy Constraints

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

$\mathsf{\Lambda}$CDM | Lambda Cold Dark Matter (model of cosmology) |

CMB | Cosmic Microwave Background |

DE | Dark Energy |

DGP | Dvali-Gabadadze-Porrati (model of gravity) |

FoM | (Dark Energy) Figure of Merit |

GR | General Relativity |

LDT | Large Deviations Theory |

LSS | Large-Scale Structure (of the universe) |

MG | Modified Gravity |

Probability Distribution Function |

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**Figure 1.**The LDT model for the matter PDF (solid lines) compared to measured PDFs from the Quijote simulations (points). The LDT model remains more accurate than a log-normal approximation with the measured variance (dashed lines) on small scales and at late times.

**Figure 2.**Comparison of the matter PDF in $10\phantom{\rule{4pt}{0ex}}{\mathrm{h}}^{-1}\phantom{\rule{4pt}{0ex}}\mathrm{Mpc}$ spheres in different theories of gravity. The cosmological parameters are chosen such that the clustering amplitude ${\sigma}_{8}$ is the same in all cases at redshift 0. This normalises the overall width of the PDF. The resulting difference in tilt and redshift dependence is due to the change to gravity.

**Figure 3.**Derivatives of the matter PDF in an $f(R)$ modified gravity scenario. The $f(R)$ gravity parameter, $|{f}_{R0}|$, can be distinguished from ${\sigma}_{8}$ by both its redshift dependence and its additional skewness. ${\mathsf{\Omega}}_{m}$ can be disentangled from both by its different effect on skewness, and its effect on the linear growth factor.

**Figure 4.**Derivatives of the matter PDF in an evolving dark energy universe. The dependence of the matter PDF on ${\mathsf{\Omega}}_{m}$ is easily distinguished from the others by its distinct skewness (see Figure 3) and hence not shown here. The ${\sigma}_{8}$, ${w}_{0}$, and ${w}_{a}$ derivatives are similar in shape, but have different redshift evolutions, which allows for degeneracy breaking.

**Figure 5.**Forecast constraints on $f(R)$ gravity using a Euclid-like volume. These are marginalised over all other $\mathsf{\Lambda}$CDM parameters, and include a prior on ${\mathsf{\Omega}}_{b}$ and ${n}_{s}$ described in [15].

**Figure 6.**Forecast constraints on ${w}_{0}{w}_{a}$CDM dark energy using a Euclid-like volume. These are marginalised over all other $\mathsf{\Lambda}$CDM parameters, and include a prior on ${\mathsf{\Omega}}_{b}$ and ${n}_{s}$ described in [15].

**Table 1.**Detection significance for a fiducial $f(R)$ with $|{f}_{R0}|={10}^{-6}$ and DGP model with ${\mathsf{\Omega}}_{\mathrm{rc}}=0.0625$. The stronger $f(R)$ constraints are expected from the additional skewness in the PDF response to $|{f}_{R0}|$ as seen in Figure 3.

$\mathit{f}(\mathit{R})$ Detection | DGP Detection | |
---|---|---|

PDF, 3 scales + prior | $5.15\sigma $ | $1.17\sigma $ |

$P(k)$, ${k}_{\mathrm{max}}=0.2\phantom{\rule{4pt}{0ex}}\mathrm{h}/\mathrm{Mpc}$ + prior | $2.01\sigma $ | $2.42\sigma $ |

PDF + $P(k)$ + prior | $13.40\sigma $ | $5.19\sigma $ |

**Table 2.**Constraints from mildly non-linear scales on ${\sigma}_{8}$, ${w}_{0}$, and ${w}_{a}$ as well as the dark energy Figure of Merit (FoM) coming from the matter PDF, power spectrum, and their combination.

$\frac{\mathit{\sigma}\left[{\mathit{\sigma}}_{8}\right]}{{\mathit{\sigma}}_{8}^{\mathbf{fid}}}$ | $\mathit{\sigma}\left[{\mathit{w}}_{0}\right]$ | $\mathit{\sigma}\left[{\mathit{w}}_{\mathit{a}}\right]$ | FoM | |
---|---|---|---|---|

PDF, 3 scales + prior | 0.18% | 0.37 | 1.25 | 27 |

$P(k),{k}_{\mathrm{max}}=0.2\phantom{\rule{4pt}{0ex}}\mathrm{h}/$Mpc + prior | 0.45% | 0.24 | 1.03 | 50 |

PDF + $P(k)$ + prior | 0.17% | 0.09 | 0.40 | 243 |

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Gough, A.; Uhlemann, C. One-Point Statistics Matter in Extended Cosmologies. *Universe* **2022**, *8*, 55.
https://doi.org/10.3390/universe8010055

**AMA Style**

Gough A, Uhlemann C. One-Point Statistics Matter in Extended Cosmologies. *Universe*. 2022; 8(1):55.
https://doi.org/10.3390/universe8010055

**Chicago/Turabian Style**

Gough, Alex, and Cora Uhlemann. 2022. "One-Point Statistics Matter in Extended Cosmologies" *Universe* 8, no. 1: 55.
https://doi.org/10.3390/universe8010055