# On the first crossing distributions in fractional Brownian motion and the mass function of dark matter haloes

## Abstract

We construct an integral equation for the first crossing distributions for fractional Brownian motion in the case of a constant barrier and we present an exact analytical solution. Additionally we present first crossing distributions derived by simulating paths from fractional Brownian motion. We compare the results of the analytical solutions with both those of simulations and those of some approximated solutions which have been used in the literature. Finally, we present multiplicity functions for dark matter structures resulting from our analytical approach and we compare with those resulting from N-body simulations. We show that the results of analytical solutions are in good agreement with those of path simulations but differ significantly from those derived from approximated solutions. Additionally, multiplicity functions derived from fractional Brownian motion are poor fits of the those which result from N-body simulations. We also present comparisons with other models which are exist in the literature and we discuss different ways of improving the agreement between analytical results and N-body simulations.

- Authors:

- 1st Lyceum of Athens, Ipitou 15, Plaka, 10557, Athens (Greece)
- Dipartimento di Fisica e Astronomia, University Of Catania, Viale Andrea Doria 6, 95125, Catania (Italy)

- Publication Date:

- OSTI Identifier:
- 22679987

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Cosmology and Astroparticle Physics; Journal Volume: 2017; Journal Issue: 03; Other Information: Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 79 ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; ANALYTICAL SOLUTION; APPROXIMATIONS; BROWNIAN MOVEMENT; COMPARATIVE EVALUATIONS; DIFFUSION BARRIERS; DISTRIBUTION; MANY-BODY PROBLEM; MASS; MULTIPLICITY; NONLUMINOUS MATTER; SIMULATION

### Citation Formats

```
Hiotelis, Nicos, and Popolo, Antonino Del, E-mail: adelpopolo@oact.inaf.it, E-mail: hiotelis@ipta.demokritos.gr.
```*On the first crossing distributions in fractional Brownian motion and the mass function of dark matter haloes*. United States: N. p., 2017.
Web. doi:10.1088/1475-7516/2017/03/017.

```
Hiotelis, Nicos, & Popolo, Antonino Del, E-mail: adelpopolo@oact.inaf.it, E-mail: hiotelis@ipta.demokritos.gr.
```*On the first crossing distributions in fractional Brownian motion and the mass function of dark matter haloes*. United States. doi:10.1088/1475-7516/2017/03/017.

```
Hiotelis, Nicos, and Popolo, Antonino Del, E-mail: adelpopolo@oact.inaf.it, E-mail: hiotelis@ipta.demokritos.gr. Wed .
"On the first crossing distributions in fractional Brownian motion and the mass function of dark matter haloes". United States.
doi:10.1088/1475-7516/2017/03/017.
```

```
@article{osti_22679987,
```

title = {On the first crossing distributions in fractional Brownian motion and the mass function of dark matter haloes},

author = {Hiotelis, Nicos and Popolo, Antonino Del, E-mail: adelpopolo@oact.inaf.it, E-mail: hiotelis@ipta.demokritos.gr},

abstractNote = {We construct an integral equation for the first crossing distributions for fractional Brownian motion in the case of a constant barrier and we present an exact analytical solution. Additionally we present first crossing distributions derived by simulating paths from fractional Brownian motion. We compare the results of the analytical solutions with both those of simulations and those of some approximated solutions which have been used in the literature. Finally, we present multiplicity functions for dark matter structures resulting from our analytical approach and we compare with those resulting from N-body simulations. We show that the results of analytical solutions are in good agreement with those of path simulations but differ significantly from those derived from approximated solutions. Additionally, multiplicity functions derived from fractional Brownian motion are poor fits of the those which result from N-body simulations. We also present comparisons with other models which are exist in the literature and we discuss different ways of improving the agreement between analytical results and N-body simulations.},

doi = {10.1088/1475-7516/2017/03/017},

journal = {Journal of Cosmology and Astroparticle Physics},

number = 03,

volume = 2017,

place = {United States},

year = {Wed Mar 01 00:00:00 EST 2017},

month = {Wed Mar 01 00:00:00 EST 2017}

}