sented an explanation of its anomalous evolution in terms of time dependence of G$_{eff}$ . Since γ has characteristic time and wavenumber dependence, future detailed observations may yield useful information on the validity of f ( R ) gravity through this quantity, although current constraints have been obtained assuming that it is constant both in time and in wavenumber[4, 39]. Another related observational test of this model is supplied by the large-scale structure of the Universe which should be different from that in the ΛCDM model. In particular, voids are expected to be more pronounced since the effective gravi- tational constant is bigger inside them compared to large matter overdensities where it is practically equal to that measured in laboratory. 

## Acknowledgments 

HM and JY are grateful to T. Narikawa and K. Yamamoto for useful communications. We thank T. Kobayashi for poiting out a typo in Ref. [15] and in the first version of this manuscript. AS acknowledges RESCEU hospitality as a visiting professor. He was also partially supported by the grant RFBR 08-02-00923 and by the Scientific Programme “As- tronomy” of the Russian Academy of Sciences. This work was supported in part by JSPS Grant-in-Aid for Scientific Research No. 19340054(JY), JSPS Core-to-Core program “In- ternational Research Network on Dark Energy”, and Global COE Program “the Physical Sciences Frontier”, MEXT, Japan. 

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