We consider a test of the Copernican Principle through observations of the large-scale structures, and for this purpose we study the self-gravitating system in a relativistic huge void universe model which does not invoke the Copernican Principle. If
we focus on the the weakly self-gravitating and slowly evolving system whose spatial extent is much smaller than the scale of the cosmological horizon in the homogeneous and isotropic background universe model, the cosmological Newtonian approximation is available. Also in the huge void universe model, the same kind of approximation as the cosmological Newtonian approximation is available for the analysis of the perturbations contained in a region whose spatial size is much smaller than the scale of the huge void: the effects of the huge void are taken into account in a perturbative manner by using the Fermi-normal coordinates. By using this approximation, we derive the equations of motion for the weakly self-gravitating perturbations whose elements have relative velocities much smaller than the speed of light, and show the derived equations can be significantly different from those in the homogeneous and isotropic universe model, due to the anisotropic volume expansion in the huge void. We linearize the derived equations of motion and solve them. The solutions show that the behaviors of linear density perturbations are very different from those in the homogeneous and isotropic universe model.
We study the two-point correlation function of density perturbations in a spherically symmetric void universe model which does not employ the Copernican principle. First we solve perturbation equations in the inhomogeneous universe model and obtain d
ensity fluctuations by using a method of non-linear perturbation theory which was adopted in our previous paper. From the obtained solutions, we calculate the two-point correlation function and show that it has a local anisotropy at the off-center position differently from those in homogeneous and isotropic universes. This anisotropy is caused by the tidal force in the off-center region of the spherical void. Since no tidal force exists in homogeneous and isotropic universes, we may test the inhomogeneous universe by observing statistical distortion of the two-point galaxy correlation function.
We study the evolution of linear density perturbations in a large spherical void universe which accounts for the acceleration of the cosmic volume expansion without introducing dark energy. The density contrast of this void is not large within the li
ght cone of an observer at the center of the void. Therefore, we describe the void structure as a perturbation with a dimensionless small parameter $kappa$ in a homogeneous and isotropic universe within the region observable for the observer. We introduce additional anisotropic perturbations with a dimensionless small parameter $epsilon$, whose evolution is of interest. Then, we solve perturbation equations up to order $kappa epsilon$ by applying second-order perturbation theory in the homogeneous and isotropic universe model. By this method, we can know the evolution of anisotropic perturbations affected by the void structure. We show that the growth rate of the anisotropic density perturbations in the large void universe is significantly different from that in the homogeneous and isotropic universe. This result suggests that the observation of the distribution of galaxies may give a strong constraint on the large void universe model.
We study gravitational and electromagnetic perturbation around the squashed Kaluza-Klein black holes with charge. Since the black hole spacetime focused on this paper have $SU(2) times U(1) simeq U(2)$ symmetry, we can separate the variables of the e
quations for perturbations by using Wigner function $D^{J}_{KM}$ which is the irreducible representation of the symmetry. In this paper, we mainly treat $J=0$ modes which preserve $SU(2)$ symmetry. We derive the master equations for the $J=0$ modes and discuss the stability of these modes. We show that the modes of $J = 0$ and $ K=0,pm 2$ and the modes of $K = pm (J + 2)$ are stable against small perturbations from the positivity of the effective potential. As for $J = 0, K=pm 1$ modes, since there are domains where the effective potential is negative except for maximally charged case, it is hard to show the stability of these modes in general. To show stability for $J = 0, K=pm 1$ modes in general is open issue. However, we can show the stability for $J = 0, K=pm 1$ modes in maximally charged case where the effective potential are positive out side of the horizon.
