PoS(GMC8)062 http://pos.sissa.it/ ∗ [email protected] Speaker. The light propagation is reexamined, allowing forthe various dependence models of of the modied gravity. time We delayinvestigate clarify how (and to induced distinguish those frequency models, shift) when on lightspacetimes propagates modied in parameterized gravity static to spherically models symmetric express and modiedmentioned. gravity. Implications to gravitational lensing are ∗ Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c

The Manchester Microlensing Conference: The 12thWorkshop International Conference and ANGLES Microlensing January 21-25 2008 Manchester, UK Hideki Asada Faculty of Science and Technology, Hirosaki University,E-mail: Hirosaki 036-8561, Japan Gravitational lensing along multiple light pathsprobe as of a physics beyond Einstein gravity PoS(GMC8)062 (2.1) (2.2) Hideki Asada are expected in 7 − is one of parameters 10 g − 6 − ]. Here, , 2 , [ ) 1. 2 r 2 ( 5 r A W = − d c 2 − 10 r 1 ) = 0 × 2 0 r + r ( ) G 2 A 3 1%. . . 2 dr √ ) ) ) r  ( r r 1 ( ( . B B A 2 2 + 2 √ = ( dt b 1 dr ) r r 0 − ( r g ∫ A − ) = = 0 r 2 ]. Therefore, it is important to investigate the Shapiro time delay , r 4 ds ( , t depend on gravity theories. 3 ) r ]; many references are therein). Introducing a new energy or length scale ( 1 B and ) denote the circumference radius and the metric of the unit 2-sphere, respectively. r ( 2 A W d and r The time lapse along a photon path is obtained as We shall consider a static spherically symmetric spacetime, in which light propagates, ex- Assumption: the electromagnetic elds propagate in four-dimensional spacetimes (even if the The theory of general relativity has passed “classical” tests, such as the deection of light, A certain modication, in whatever form, in the standard cosmological model, is strongly The gravitational time delay effect along a light path has successfully tested the Einstein's the- where The functions pressed as whole spacetime is higher dimensional).rical Thus optics photon approximation paths of follow Maxwell null equation). geodesics (as the geomet- 2. Shapiro Time Delay in the parameterized post-Newtonian (PPN)experiment formulation approaches of gravity. the The level sensitivity at in which, the theoretically, Cassini deviations 10 with such a high accuracy.quency shift) We on shall modied discuss gravity models thethe and dependence Shapiro investigate how time of to delay the distinguish ([ those time models(e.g. delay by extra (and using dimension induced scale) fre- may makeit changes is in worthwhile functional forms to of investigatepropagation. the how Throughout gravitational eld. this to paper, Thus probe we such take the a units modied of functional form, by using the light the perihelion shift ofremarkable Mercury and binary the pulsar Shapiro “PSREinstein's time 1913+16”. theory delay, is and correct In with also the a a similar twentieth accuracy systematic century, of test these 0 using tests the proved that the suggested by recent observations(SNIa) such and as the the cosmic magnitude-redshiftadd microwave relation a background of new (CMB) type component anisotropy Ia intotheory by the supernovae of WMAP. energy-momentum general We tensor are relativity intensor itself. forced theories, the to string Indeed, Einstein theories, equation plenty higher dimensional orof of scenarios great modify models and importance the have quantum to gravity. observationally been test Therefore, proposed, these it models. is such as scalar Gravitational lensing by modied gravity 1. Introduction some cosmological models [ ory, where the gravitational timeprinciple. delay is A supplementary signicant to improvement the wasspacecraft gravitational on reported its lensing in way via to 2003 the Fermat's from Saturn, with Doppler tracking of the Cassini PoS(GMC8)062 n 2, B / 3 (2.5) (2.6) (2.3) (2.4) + n = A n for DGP , the extra 2 c E r r Hideki Asada / , > M ) ]. R 1 r √ dt 2 E / v . − y db d = , n for one of massive gravity )( ) B g

1 2 b r m 1 − − ( + 2 n 1 = R R − n n √ . For a receiver at ) A n , , dt B m n / 2, R r r ) + r / m n 1 T 10AU A B R would appear especially in the tail parts of D ( ( = y , the longer the delay + + + 2 d n d ) 1 / n dR 3 M M r r − + n

) n 2 2 r ) 3 1 ) R = R n R − + 2 10 13 with graviton mass − y 1 B / 1 1 − 2 ∫ − as a possible candidate for the dark energy. R + 3 M 10 ( + ≈ ≈ n 2 + L E n ) ) A R ( r r √ R 1 ( ( 3 for the Schwarzschild-de Sitter spacetime, that is, general 2 g n ∫ ( / A B m A n ( L dt db − − 1 ) − + ( R b = n 0 r × r =

n n R r r n B ) = 10AU B n . The larger the index of , where the subscripts from 1 to 3 denote each light path. We assume that t 3 . B ( − 3 0 d R r are approximated as 10 17 + r = / ) − n r 13 and n × r A / , from observations using receivers at very different distances from Sun, as ( A 2 ( 10 ≡ and n B M 2 B ∼ R 2, 2 ∼ ∼ R . denote the impact parameter and the closest point, respectively. Their relation is , r 1

n that is the extra scale within which gravity becomes ve dimensional. (2) = y 0 , and . √ r shows that an extra distortion due to r A 1 ) 2 g c d / n ) 0 r R 1 , r m r r ( n ) ( denotes the mass of the central body. 3 A and A / / b M 2 2 0 are separately determined by using at least three very different light paths [ curves. One can distinguish modied gravity models, which are characterized by various r It is convenient to use the relative change in the frequency, which is caused by the gravitational Figure We consider three light paths, for which the impact parameters of the photon paths are almost For practical calculations, we keep only the leading term at a few AU in the corrections. Examples of modied gravity theories: (1) Up to the linear order, the extra contribution to time delay due to modied gravity is n t = ( is constant in time for simplicity. It is a straightforward task to take account of the eccentricity = − n 2 E and r of the Earth orbit and a difference between the impact parameters. It can be shown that where we dene time delay. This frequency shift isfrequency shift dened is as where 10AU the same (several times of the solardenoted as radius) for convenience sake. The locations of the receivers are y values of shown by Fig. Namely, Gravitational lensing by modied gravity where b where model with A models. (3) relativity with the cosmological constant PoS(GMC8)062 ) = R (3.1) r , n 90 deg.) ( , ) ∼ . Therefore, Hideki Asada 6 10AU , 2 ) = ( R . The long dashed, short r n . , n 11 ( − , ) 10 × 3 10AU , = , is quite large, say 10 2 10 AU is overlapped with the solid curve / and the index n

N 3 r = . ) R 7 r n R 3 r − B ) = ( 10 + R r n , , 064009 (2003). ∼ A , 532 (1994); 2 and n 2 ( ( / 67 , 046007 (2002). N 3 4 1 423 √ 66 = n 1 × as m 0 1 1mas days t 1 - , 374 (2003). 2 425 , 393 (1972). - 39 3 - 0 10 10 10 10 10 10 ------** ** ** ** ** ** Dependence of the frequency shift on the distance 10 10 10 10 10 10 , respectively. The long dashed curve for * * * * * * ) 6 2 4 2 6 4

T. Damour, I. I. Kogan and A. Papazoglou, PRD Statistical investigations of gravitational lensing with future micro-arcsec. astrometry missions The present result suggests that the tail part of gravitational lensing (and time delay) seems

- - - y 1AU , [1] H. Asada, accepted for publication in[2] Phys. Lett. B. B. Bertotti (2008) et (arXiv:0710.0477 al. [gr-qc]). Nature, [3] T. Damour and A. M. Polyakov, Nucl.[4] Phys. B T. Damour, F. Piazza and G. Veneziano,[5] PRD A. I. Vainshtein, PLB 2 denoting the general relativistic case. Here, we assume ( dashed and dotted curves denote the frequency shift for from Sun. The numberone of can source make stars a for ratheraccuracy those optimistic as missions, estimate of the possible constraint on modied gravity with the More detailed investigations on gravitational lensing includinggravity microlensing would as be a probe important. of modied References such as SIM, GAIA and JASMINE willthe bending probably angle become by a the probe solar gravity of is gravity of theories. the order For of instance, 1mas even at a different direction ( sensitive to certain modications to general relativity. 3. Conclusion – Implications to gravitational lensing Figure 1: Gravitational lensing by modied gravity