We find a simple exact solution of 6-dimensional braneworld which captures some essential features of warped flux compactification, including a warped geometry, compactification, a magnetic flux, and one or two 3-brane(s). In this setup we analyze how the Hubble expansion rate on each brane changes when the brane tension changes. It is shown that effective Newton's constant resulting from this analysis agrees with that inferred by simply integrating extra dimensions out. Based on the result, a general formula for effective Newton's constant is conjectured and its application to cosmology with type IIB warped string compactification is discussed.

We present numerical simulations on the propagation of ultra-high-frequency protons with energies of 10 19.5 -10 22 eV in extragalactic magnetic fields over 1 Gpc. We use the Optical Redshift Survey (ORS) galaxy sample, which allows us to accurately quantify the contribution of nearby sources to the energy spectrum and the arrival distribution, as a source model. The sample is corrected, taking the selection effect and absence of galaxies in the zone of avoidance (jbj < 20 ) into account. We calculate three observable quantities-cosmicray spectrum, harmonic amplitude, and two-point correlation function-from our data of numerical simulations. With these quantities, we compare the results of our numerical calculations with the observation. We find that the arrival distribution of ultra-high-energy cosmic rays (UHECRs) becomes most isotropic as sources are restricted to to luminous galaxies ðM lim ¼ À20:5Þ. However, it is not isotropic enough to be consistent with the Akeno Giant Air Shower Array (AGASA) observation, even for M lim ¼ À20:5. In order to obtain a sufficiently isotropic arrival distribution, we randomly select sources more luminous than À20.5 mag from the ORS sample, which contribute to the observed cosmic-ray flux, and investigate the dependence of the results on their number. We show that three observable quantities, including the GreisenZatsepin-Kuz'min (GZK) cutoff of the energy spectrum, can be reproduced in the case that the number fraction $10 À1.7 of the ORS galaxies more luminous than À20.5 mag is selected as UHECR sources. In terms of the source number density, this constraint corresponds to $10 À6 Mpc À3 . However, since the mean number of sources within the GZK sphere is only $0.5 in this case, the eight AGASA events above 10 20.0 eV, which do not together constitute clustered events, cannot be reproduced. On the other hand, if the cosmic-ray flux measured by the High Resolution Fly's Eye EHE Cosmic-Ray Detector (HiRes), which is consistent with the GZK cutoff, is correct and observational features about the arrival distribution of UHECRs are same as the AGASA, our source model can explain both the arrival distribution and the flux at the same time. Thus, we conclude that a large fraction of the eight AGASA events above 10 20 eV might originate in the top-down scenarios or that the cosmic-ray flux measured by the HiRes experiment might be better. We also discuss the origin of UHECRs below 10 20.0 eV through comparisons between the number density of astrophysical source candidates and our result ($10 À6 Mpc À3 ).

We present numerical simulations of the propagation of ultra-high-energy cosmic rays (UHECRs) above 10 19 eV in a structured extragalactic magnetic field ( EGMF) and simulate their arrival distributions at the Earth. We use the IRAS PSCz catalog in order to construct a model of the EGMF and source models of UHECRs, both of which reproduce the local structures observed around the Milky Way. We also consider modifications of UHECR arrival directions by the Galactic magnetic field. We follow an inverse process of their propagation from the Earth and record the trajectories. This enables us to calculate only trajectories of UHECRs arriving at the Earth, which saves CPU time. From these trajectories and our source models, we construct arrival distributions of UHECRs and calculate their harmonic amplitudes and two-point correlation functions. We estimate the number density of sources that best reproduces the Akeno Ground Air Shower Array (AGASA) observation. As a result, we find that the most appropriate number density of the sources is $5 ; 10 À6 Mpc À3 . This constrains the source candidates of UHECRs. We also demonstrate sky maps of their arrival distribution with the event number expected by future experiments and examine how the EGMF affects their arrival distribution. A main result is the diffusion of clustering events, which are obtained from calculations in the absence of the EGMF. This tendency allows us to reproduce the observed two-point correlation function better. Subject headingg s: cosmic rays -galaxies: general -intergalactic medium -large-scale structure of universemagnetic fields -methods: numerical