# Lagrangian bias in the local bias model

## Abstract

It is often assumed that the halo-patch fluctuation field can be written as a Taylor series in the initial Lagrangian dark matter density fluctuation field. We show that if this Lagrangian bias is local, and the initial conditions are Gaussian, then the two-point cross-correlation between halos and mass should be linearly proportional to the mass-mass auto-correlation function. This statement is exact and valid on all scales; there are no higher order contributions, e.g., from terms proportional to products or convolutions of two-point functions, which one might have thought would appear upon truncating the Taylor series of the halo bias function. In addition, the auto-correlation function of locally biased tracers can be written as a Taylor series in the auto-correlation function of the mass; there are no terms involving, e.g., derivatives or convolutions. Moreover, although the leading order coefficient, the linear bias factor of the auto-correlation function is just the square of that for the cross-correlation, it is the same as that obtained from expanding the mean number of halos as a function of the local density only in the large-scale limit. In principle, these relations allow simple tests of whether or not halo bias is indeed local in Lagrangian space.more »

- Authors:

- SISSA, International School for Advanced Studies, Via Bonomea 265, 34136, Trieste (Italy)
- Center for Particle Cosmology, University of Pennsylvania, 209 S 33rd Street, Philadelphia, PA 19104 (United States)

- Publication Date:

- OSTI Identifier:
- 22279685

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Cosmology and Astroparticle Physics

- Additional Journal Information:
- Journal Volume: 2012; Journal Issue: 11; Other Information: Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1475-7516

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 79 ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; ASTROPHYSICS; CORRELATION FUNCTIONS; CORRELATIONS; COSMOLOGY; DENSITY; FLUCTUATIONS; LAGRANGIAN FUNCTION; MASS; NONLUMINOUS MATTER; RENORMALIZATION; SPACE; VISIBLE RADIATION

### Citation Formats

```
Frusciante, Noemi, and Sheth, Ravi K., E-mail: nfruscia@sissa.it, E-mail: sheth@ictp.it.
```*Lagrangian bias in the local bias model*. United States: N. p., 2012.
Web. doi:10.1088/1475-7516/2012/11/016.

```
Frusciante, Noemi, & Sheth, Ravi K., E-mail: nfruscia@sissa.it, E-mail: sheth@ictp.it.
```*Lagrangian bias in the local bias model*. United States. doi:10.1088/1475-7516/2012/11/016.

```
Frusciante, Noemi, and Sheth, Ravi K., E-mail: nfruscia@sissa.it, E-mail: sheth@ictp.it. Thu .
"Lagrangian bias in the local bias model". United States. doi:10.1088/1475-7516/2012/11/016.
```

```
@article{osti_22279685,
```

title = {Lagrangian bias in the local bias model},

author = {Frusciante, Noemi and Sheth, Ravi K., E-mail: nfruscia@sissa.it, E-mail: sheth@ictp.it},

abstractNote = {It is often assumed that the halo-patch fluctuation field can be written as a Taylor series in the initial Lagrangian dark matter density fluctuation field. We show that if this Lagrangian bias is local, and the initial conditions are Gaussian, then the two-point cross-correlation between halos and mass should be linearly proportional to the mass-mass auto-correlation function. This statement is exact and valid on all scales; there are no higher order contributions, e.g., from terms proportional to products or convolutions of two-point functions, which one might have thought would appear upon truncating the Taylor series of the halo bias function. In addition, the auto-correlation function of locally biased tracers can be written as a Taylor series in the auto-correlation function of the mass; there are no terms involving, e.g., derivatives or convolutions. Moreover, although the leading order coefficient, the linear bias factor of the auto-correlation function is just the square of that for the cross-correlation, it is the same as that obtained from expanding the mean number of halos as a function of the local density only in the large-scale limit. In principle, these relations allow simple tests of whether or not halo bias is indeed local in Lagrangian space. We discuss why things are more complicated in practice. We also discuss our results in light of recent work on the renormalizability of halo bias, demonstrating that it is better to renormalize than not. We use the Lognormal model to illustrate many of our findings.},

doi = {10.1088/1475-7516/2012/11/016},

journal = {Journal of Cosmology and Astroparticle Physics},

issn = {1475-7516},

number = 11,

volume = 2012,

place = {United States},

year = {2012},

month = {11}

}