Structure formation in modified gravity models alternative to dark energy

We study structure formation in phenomenological models in which the Friedmann equation receives a correction of the form $H^{\alpha}/r_c^{2-\alpha}$, which realize an accelerated expansion without dark energy. In order to address structure formation in these model, we construct simple covariant gravitational equations which give the modified Friedmann equation with $\alpha=2/n$ where $n$ is an integer. For $n=2$, the underlying theory is known as a 5D braneworld model (the DGP model). Thus the models interpolate between the DGP model ($n=2, \alpha=1$) and the LCDM model in general relativity ($n \to \infty, \alpha \to 0$). Using the covariant equations, cosmological perturbations are analyzed. It is shown that in order to satisfy the Bianchi identity at a perturbative level, we need to introduce a correction term $E_{\mu \nu}$ in the effective equations. In the DGP model, $E_{\mu \nu}$ comes from 5D gravitational fields and correct conditions on $E_{\mu \nu}$ can be derived by solving the 5D perturbations. In the general case $n>2$, we have to assume the structure of a modified theory of gravity to determine $E_{\mu \nu}$. We show that structure formation is different from a dark energy model in general relativity with identical expansion history and that quantitative features of the difference crucially depend on the conditions on $E_{\mu \nu}$, that is, the structure of the underlying theory of modified gravity. This implies that it is essential to identify underlying theories in order to test these phenomenological models against observational data and, once we identify a consistent theory, structure formation tests become essential to distinguish modified gravity models from dark energy models in general relativity.Comment: 12 pages, 3 figure

Are there ghosts in the self-accelerating brane universe?

We study the spectrum of gravitational perturbations about a vacuum de Sitter brane with the induced 4D Einstein-Hilbert term, in a 5D Minkowski spacetime (DGP model). We consider solutions that include a self-accelerating univese, where the accelerating expansion of the universe is realized without introducing a cosmological constant on the brane. The mass of the discrete mode for the spin-2 graviton is calculated for various $Hr_c$, where $H$ is the Hubble parameter and $r_c$ is the cross-over scale determined by the ratio between the 5D Newton constant and the 4D Newton constant. We show that, if we introduce a positive cosmological constant on the brane ($Hr_c >1$), the spin-2 graviton has mass in the range $0 < m^2 < 2H^2$ and there is a normalisable brane fluctuation mode with mass $m^2=2 H^2$. Although the brane fluctuation mode is healthy, the spin-2 graviton has a helicity-0 excitation that is a ghost. If we allow a negative cosmological constant on the brane, the brane bending mode becomes a ghost for $1/2 < Hr_c <1$. This confirms the results obtained by the boundary effective action that there exists a scalar ghost mode for $Hr_c >1/2$. In a self-accelerating universe $Hr_c=1$, the spin-2 graviton has mass $m^2=2H^2$, which is known to be a special case for massive gravitons in de Sitter spacetime where the graviton has no helicity-0 excitation and so no ghost. However, in DGP model, there exists a brane fluctuation mode with the same mass and there arises a mixing between the brane fluctuation mode and the spin-2 graviton. We argue that this mixing presumably gives a ghost in the self-accelerating universe by continuity across $Hr_c=1$, although a careful calculation of the effective action is required to verify this rigorously.Comment: 5 pages, 1 figure, significant revisions, conclusion on the self- accelerating universe change