# Density perturbations in f(R) gravity theories in metric and Palatini formalisms

## Abstract

We make a detailed study of matter density perturbations in both metric and Palatini formalisms. Considering general theories whose Lagrangian density is a general function, f(R), of the Ricci scalar R, we derive the equation of matter density perturbations in each case, in a number of gauges, including comoving, longitudinal and uniform density gauges. We show that for viable f(R) models that satisfy cosmological and local gravity constraints (LGC), matter perturbation equations derived under a subhorizon approximation are valid even for super-Hubble scales provided the oscillating mode (scalaron) does not dominate over the matter-induced mode. Such approximate equations are especially reliable in the Palatini formalism because of the absence of scalarons. Using these equations we make a comparative study of the behavior of matter density perturbations as well as gravitational potentials for a number of classes of f(R) theories. In the metric formalism the quantity m=Rf{sub ,RR}/f{sub ,R} that characterizes the deviation from the {lambda}CDM model is constrained to be very small during a matter era in order to ensure compatibility with LGC, but the models in which m grows to the order of 10{sup -1} around the present epoch can be allowed. These models also suffer from an additionalmore »

- Authors:

- Department of Physics, Gunma National College of Technology, Gunma 371-8530 (Japan)
- (United Kingdom)

- Publication Date:

- OSTI Identifier:
- 21039086

- Resource Type:
- Journal Article

- Journal Name:
- Physical Review. D, Particles Fields

- Additional Journal Information:
- Journal Volume: 77; Journal Issue: 4; Other Information: DOI: 10.1103/PhysRevD.77.043007; (c) 2008 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0556-2821

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; APPROXIMATIONS; COSMOLOGICAL MODELS; COSMOLOGY; DENSITY; DISTURBANCES; GRAVITATION; LAGRANGIAN FUNCTION; OSCILLATIONS; PERTURBATION THEORY; POTENTIALS; SCALARS

### Citation Formats

```
Tsujikawa, Shinji, Uddin, Kotub, Tavakol, Reza, and School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS.
```*Density perturbations in f(R) gravity theories in metric and Palatini formalisms*. United States: N. p., 2008.
Web. doi:10.1103/PHYSREVD.77.043007.

```
Tsujikawa, Shinji, Uddin, Kotub, Tavakol, Reza, & School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS.
```*Density perturbations in f(R) gravity theories in metric and Palatini formalisms*. United States. doi:10.1103/PHYSREVD.77.043007.

```
Tsujikawa, Shinji, Uddin, Kotub, Tavakol, Reza, and School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS. Fri .
"Density perturbations in f(R) gravity theories in metric and Palatini formalisms". United States. doi:10.1103/PHYSREVD.77.043007.
```

```
@article{osti_21039086,
```

title = {Density perturbations in f(R) gravity theories in metric and Palatini formalisms},

author = {Tsujikawa, Shinji and Uddin, Kotub and Tavakol, Reza and School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS},

abstractNote = {We make a detailed study of matter density perturbations in both metric and Palatini formalisms. Considering general theories whose Lagrangian density is a general function, f(R), of the Ricci scalar R, we derive the equation of matter density perturbations in each case, in a number of gauges, including comoving, longitudinal and uniform density gauges. We show that for viable f(R) models that satisfy cosmological and local gravity constraints (LGC), matter perturbation equations derived under a subhorizon approximation are valid even for super-Hubble scales provided the oscillating mode (scalaron) does not dominate over the matter-induced mode. Such approximate equations are especially reliable in the Palatini formalism because of the absence of scalarons. Using these equations we make a comparative study of the behavior of matter density perturbations as well as gravitational potentials for a number of classes of f(R) theories. In the metric formalism the quantity m=Rf{sub ,RR}/f{sub ,R} that characterizes the deviation from the {lambda}CDM model is constrained to be very small during a matter era in order to ensure compatibility with LGC, but the models in which m grows to the order of 10{sup -1} around the present epoch can be allowed. These models also suffer from an additional fine-tuning due to the presence of scalaron oscillating modes which are absent in the Palatini case. In Palatini formalism LGC and background cosmological constraints provide only weak bounds on |m| by constraining it to be smaller than {approx}0.1. This is in contrast to matter density perturbations which, on galactic scales, place far more stringent constraints on the present deviation parameter m of the order of |m| < or approx. 10{sup -5}-10{sup -4}. This is due to the peculiar evolution of matter perturbations in the Palatini case, which exhibits a rapid growth or a damped oscillation depending on the sign of m.},

doi = {10.1103/PHYSREVD.77.043007},

journal = {Physical Review. D, Particles Fields},

issn = {0556-2821},

number = 4,

volume = 77,

place = {United States},

year = {2008},

month = {2}

}