# 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 |

## References

- Planck Collaboration; Aghanim, N.; Akrami, Y.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M.; Banday, A.J.; Barreiro, R.B.; Bartolo, N.; et al. Planck 2018 results. VI. Cosmological parameters. Astron. Astrophys.
**2020**, 641, A6. [Google Scholar] [CrossRef] [Green Version] - Carrasco, J.J.M.; Hertzberg, M.P.; Senatore, L. The effective field theory of cosmological large scale structures. J. High Energy Phys.
**2012**, 2012, 82. [Google Scholar] [CrossRef] [Green Version] - Laureijs, R.; Amiaux, J.; Arduini, S.; Auguères, J.L.; Brinchmann, J.; Cole, R.; Cropper, M.; Dabin, C.; Duvet, L.; Ealet, A.; et al. Euclid Definition Study Report. arXiv
**2011**, arXiv:1110.3193. [Google Scholar] - Ivezić, Ž.; Kahn, S.M.; Tyson, J.A.; Abel, B.; Acosta, E.; Allsman, R.; Alonso, D.; AlSayyad, Y.; Anderson, S.F.; Andrew, J.; et al. LSST: From Science Drivers to Reference Design and Anticipated Data Products. Astrophys. J.
**2019**, 873, 111. [Google Scholar] [CrossRef] - Levi, M.; Bebek, C.; Beers, T.; Blum, R.; Cahn, R.; Eisenstein, D.; Flaugher, B.; Honscheid, K.; Kron, R.; Lahav, O.; et al. The DESI Experiment, a whitepaper for Snowmass 2013. arXiv
**2013**, arXiv:1308.0847. [Google Scholar] - Douspis, M.; Salvati, L.; Aghanim, N. On the tension between Large Scale Structures and Cosmic Microwave Background. arXiv
**2019**, arXiv:1901.05289. [Google Scholar] - Di Valentino, E.; Anchordoqui, L.A.; Akarsu, Ö.; Ali-Haimoud, Y.; Amendola, L.; Arendse, N.; Asgari, M.; Ballardini, M.; Basilakos, S.; Battistelli, E.; et al. Cosmology Intertwined III: Fσ
_{8}and S_{8}. Astropart. Phys.**2021**, 131, 102604. [Google Scholar] [CrossRef] - Perivolaropoulos, L.; Skara, F. Challenges for ΛCDM: An update. arXiv
**2021**, arXiv:2105.05208. [Google Scholar] - Planck Collaboration; Akrami, Y.; Arroja, F.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M.; Banday, A.J.; Barreiro, R.B.; Bartolo, N.; et al. Planck 2018 results. IX. Constraints on primordial non-Gaussianity. Astron. Astrophys.
**2020**, 641, A9. [Google Scholar] [CrossRef] - Bernardeau, F. The nonlinear evolution of rare events. Astrophys. J.
**1994**, 427, 51–71. [Google Scholar] [CrossRef] - Valageas, P. Dynamics of gravitational clustering. II. Steepest-descent method for the quasi-linear regime. Astron. Astrophys.
**2002**, 382, 412–430. [Google Scholar] [CrossRef] [Green Version] - Bernardeau, F.; Pichon, C.; Codis, S. Statistics of cosmic density profiles from perturbation theory. Phys. Rev. D
**2014**, 90, 103519. [Google Scholar] [CrossRef] [Green Version] - Bernardeau, F.; Reimberg, P. Large deviation principle at play in large scale structure cosmology. Phys. Rev. D
**2016**, 94, 063520. [Google Scholar] [CrossRef] [Green Version] - Uhlemann, C.; Codis, S.; Pichon, C.; Bernardeau, F.; Reimberg, P. Back in the saddle: Large-deviation statistics of the cosmic log-density field. Mon. Not. R. Astron. Soc.
**2016**, 460, 1529–1541. [Google Scholar] [CrossRef] [Green Version] - Cataneo, M.; Uhlemann, C.; Arnold, C.; Gough, A.; Li, B.; Heymans, C. The matter density PDF for modified gravity and dark energy with Large Deviations Theory. arXiv
**2021**, arXiv:2109.02636. [Google Scholar] - Villaescusa-Navarro, F.; Hahn, C.; Massara, E.; Banerjee, A.; Delgado, A.M.; Ramanah, D.K.; Charnock, T.; Giusarma, E.; Li, Y.; Allys, E.; et al. The Quijote Simulations. Astrophys. J. Suppl. Ser.
**2020**, 250, 2. [Google Scholar] [CrossRef] - Chevallier, M.; Polarski, D. Accelerating Universes with Scaling Dark Matter. Int. J. Mod. Phys. D
**2001**, 10, 213–223. [Google Scholar] [CrossRef] [Green Version] - Linder, E.V. Exploring the Expansion History of the Universe. Phys. Rev. Lett.
**2003**, 90, 091301. [Google Scholar] [CrossRef] [Green Version] - Hu, W.; Sawicki, I. Models of f(R) cosmic acceleration that evade solar system tests. Phys. Rev. D
**2007**, 76, 064004. [Google Scholar] [CrossRef] [Green Version] - Schmidt, F. Cosmological simulations of normal-branch braneworld gravity. Phys. Rev. D
**2009**, 80, 123003. [Google Scholar] [CrossRef] [Green Version] - Cataneo, M.; Lombriser, L.; Heymans, C.; Mead, A.J.; Barreira, A.; Bose, S.; Li, B. On the road to percent accuracy: Non-linear reaction of the matter power spectrum to dark energy and modified gravity. Mon. Not. R. Astron. Soc.
**2019**, 488, 2121–2142. [Google Scholar] [CrossRef] - Uhlemann, C.; Friedrich, O.; Villaescusa-Navarro, F.; Banerjee, A.; Codis, S. Fisher for complements: Extracting cosmology and neutrino mass from the counts-in-cells PDF. Mon. Not. R. Astron. Soc.
**2020**, 495, 4006–4027. [Google Scholar] [CrossRef] - Planck Collaboration; Blanchard, A.; Camera, S.; Carbone, C.; Cardone, V.F.; Casas, S.; Clesse, S.; Ilić, S.; Kilbinger, M.; Kitching, T.; et al. Euclid preparation. VII. Forecast validation for Euclid cosmological probes. Astron. Astrophys.
**2020**, 642, A191. [Google Scholar] [CrossRef] - Friedrich, O.; Uhlemann, C.; Villaescusa-Navarro, F.; Baldauf, T.; Manera, M.; Nishimichi, T. Primordial non-Gaussianity without tails-how to measure f
_{NL}with the bulk of the density PDF. Mon. Not. R. Astron. Soc.**2020**, 498, 464–483. [Google Scholar] [CrossRef] - Barthelemy, A.; Codis, S.; Uhlemann, C.; Bernardeau, F.; Gavazzi, R. A nulling strategy for modelling lensing convergence in cones with large deviation theory. Mon. Not. R. Astron. Soc.
**2020**, 492, 3420–3439. [Google Scholar] [CrossRef] - Boyle, A.; Uhlemann, C.; Friedrich, O.; Barthelemy, A.; Codis, S.; Bernardeau, F.; Giocoli, C.; Baldi, M. Nuw CDM cosmology from the weak lensing convergence PDF. arXiv
**2020**, arXiv:2012.07771. [Google Scholar] [CrossRef] - Thiele, L.; Hill, J.C.; Smith, K.M. Accurate analytic model for the weak lensing convergence one-point probability distribution function and its autocovariance. Phys. Rev. D
**2020**, 102, 123545. [Google Scholar] [CrossRef] - Repp, A.; Szapudi, I. Galaxy bias and σ
_{8}from counts in cells from the SDSS main sample. Mon. Not. R. Astron. Soc. Lett.**2020**, 498, L125–L129. [Google Scholar] [CrossRef] - Friedrich, O.; Halder, A.; Boyle, A.; Uhlemann, C.; Britt, D.; Codis, S.; Gruen, D.; Hahn, C. The PDF perspective on the tracer-matter connection: Lagrangian bias and non-Poissonian shot noise. arXiv
**2021**, arXiv:2107.02300. [Google Scholar] [CrossRef] - Gruen, D.; Friedrich, O.; Krause, E.; DeRose, J.; Cawthon, R.; Davis, C.; Elvin-Poole, J.; Rykoff, E.S.; Wechsler, R.H.; Alarcon, A.; et al. Density split statistics: Cosmological constraints from counts and lensing in cells in DES Y1 and SDSS data. Phys. Rev. D
**2018**, 98, 023507. [Google Scholar] [CrossRef] [Green Version] - Friedrich, O.; Gruen, D.; DeRose, J.; Kirk, D.; Krause, E.; McClintock, T.; Rykoff, E.; Seitz, S.; Wechsler, R.; Bernstein, G.; et al. Density split statistics: Joint model of counts and lensing in cells. Phys. Rev. D
**2018**, 98. [Google Scholar] [CrossRef] [Green Version] - Hunter, J.D. Matplotlib: A 2D graphics environment. Comput. Sci. Eng.
**2007**, 9, 90–95. [Google Scholar] [CrossRef] - Hinton, S. ChainConsumer. J. Open Source Softw.
**2016**, 1, 45. [Google Scholar] [CrossRef] - Harris, C.R.; Millman, K.J.; van der Walt, S.J.; Gommers, R.; Virtanen, P.; Cournapeau, D.; Wieser, E.; Taylor, J.; Berg, S.; Smith, N.J.; et al. Array programming with NumPy. Nature
**2020**, 585, 357–362. [Google Scholar] [CrossRef] [PubMed] - Virtanen, P.; Gommers, R.; Oliphant, T.E.; Haberland, M.; Reddy, T.; Cournapeau, D.; Burovski, E.; Peterson, P.; Weckesser, W.; Bright, J.; et al. SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python. Nat. Methods
**2020**, 17, 261–272. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**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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**MDPI and ACS Style**

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