# Characterizing the Lyα forest flux probability distribution function using Legendre polynomials

## Abstract

The Lyman-α forest is a highly non-linear field with considerable information available in the data beyond the power spectrum. The flux probability distribution function (PDF) has been used as a successful probe of small-scale physics. In this paper we argue that measuring coefficients of the Legendre polynomial expansion of the PDF offers several advantages over measuring the binned values as is commonly done. In particular, the *n*-th Legendre coefficient can be expressed as a linear combination of the first n moments, allowing these coefficients to be measured in the presence of noise and allowing a clear route for marginalisation over mean flux. Moreover, in the presence of noise, our numerical work shows that a finite number of coefficients are well measured with a very sharp transition into noise dominance. This compresses the available information into a small number of well-measured quantities. In conclusion, we find that the amount of recoverable information is a very non-linear function of spectral noise that strongly favors fewer quasars measured at better signal to noise.

- Authors:

- Brookhaven National Lab. (BNL), Upton, NY (United States)

- Publication Date:

- Research Org.:
- Brookhaven National Laboratory (BNL), Upton, NY (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)

- OSTI Identifier:
- 1413953

- Report Number(s):
- BNL-114793-2017-JA

Journal ID: ISSN 1475-7516; TRN: US1800602

- Grant/Contract Number:
- SC0012704

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of Cosmology and Astroparticle Physics

- Additional Journal Information:
- Journal Volume: 2017; Journal Issue: 10; Journal ID: ISSN 1475-7516

- Publisher:
- Institute of Physics (IOP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 79 ASTRONOMY AND ASTROPHYSICS; Lyman alpha forest; intergalactic media; cosmological simulations

### Citation Formats

```
Cieplak, Agnieszka M., and Slosar, Anze.
```*Characterizing the Lyα forest flux probability distribution function using Legendre polynomials*. United States: N. p., 2017.
Web. doi:10.1088/1475-7516/2017/10/013.

```
Cieplak, Agnieszka M., & Slosar, Anze.
```*Characterizing the Lyα forest flux probability distribution function using Legendre polynomials*. United States. doi:10.1088/1475-7516/2017/10/013.

```
Cieplak, Agnieszka M., and Slosar, Anze. Thu .
"Characterizing the Lyα forest flux probability distribution function using Legendre polynomials". United States.
doi:10.1088/1475-7516/2017/10/013. https://www.osti.gov/servlets/purl/1413953.
```

```
@article{osti_1413953,
```

title = {Characterizing the Lyα forest flux probability distribution function using Legendre polynomials},

author = {Cieplak, Agnieszka M. and Slosar, Anze},

abstractNote = {The Lyman-α forest is a highly non-linear field with considerable information available in the data beyond the power spectrum. The flux probability distribution function (PDF) has been used as a successful probe of small-scale physics. In this paper we argue that measuring coefficients of the Legendre polynomial expansion of the PDF offers several advantages over measuring the binned values as is commonly done. In particular, the n-th Legendre coefficient can be expressed as a linear combination of the first n moments, allowing these coefficients to be measured in the presence of noise and allowing a clear route for marginalisation over mean flux. Moreover, in the presence of noise, our numerical work shows that a finite number of coefficients are well measured with a very sharp transition into noise dominance. This compresses the available information into a small number of well-measured quantities. In conclusion, we find that the amount of recoverable information is a very non-linear function of spectral noise that strongly favors fewer quasars measured at better signal to noise.},

doi = {10.1088/1475-7516/2017/10/013},

journal = {Journal of Cosmology and Astroparticle Physics},

number = 10,

volume = 2017,

place = {United States},

year = {Thu Oct 12 00:00:00 EDT 2017},

month = {Thu Oct 12 00:00:00 EDT 2017}

}