# Gravitational time delay of light for various models of modified gravity

###### Abstract

We reexamined the gravitational time delay of light, allowing for various models of modified gravity. We clarify the dependence of the time delay (and induced frequency shift) on modified gravity models and investigate how to distinguish those models, when light propagates in static spherically symmetric spacetimes. Thus experiments by radio signal from spacecrafts at very different distances from Sun and future space-borne laser interferometric detectors could be a probe of modified gravity in the solar system.

###### pacs:

04.80.Cc, 04.50.+h, 95.30.Sf, 95.36.+xThe nature of dark energy and dark matter has become a central issue in modern cosmology. Recent observations such as the magnitude-redshift relation of type Ia supernovae (SNIa) SN and the cosmic microwave background (CMB) anisotropy by WMAP WMAP strongly suggest a certain modification, in whatever form, in the standard cosmological model. We are forced to add a new component into the energy-momentum tensor in the Einstein equation or modify the theory of general relativity itself Kamionkowski . Indeed, there have been a lot of proposals motivated by, for instance, scalar tensor theories, string theories, higher dimensional scenarios and quantum gravity (For recent reviews of modified gravity models inspired by the dark energy observation, e.g., theories ). Therefore, it is of great importance to observationally test these models.

The theory of general relativity has passed “classical” tests, such as the deflection of light, the perihelion shift of Mercury and the Shapiro time delay, and also a systematic test using the remarkable binary pulsar “PSR 1913+16” and several binary pulsars now known Will . In the twentieth century, these tests proved that the Einstein’s theory is correct with a similar accuracy of .

Since the time delay effect along a light path in the gravitational field was first noticed in 1964 by Shapiro Shapiro , this effect has successfully tested the Einstein’s theory Will06 . A significant improvement was reported in 2003 from Doppler tracking of the Cassini spacecraft on its way to the Saturn, with Cassini . Here, is one of parameters in the parameterized post-Newtonian (PPN) formulation of gravity Will . The bending and delay of photons by the curvature of spacetime produced by any mass are proportional to , where is unity in general relativity but zero in the Newtonian theory, and the quantity is thus considered as a measure of a deviation from general relativity. The sensitivity in the Cassini experiment approaches the level at which, theoretically, deviations are expected in some cosmological models DP ; DPV . Therefore, it is important to investigate the Shapiro time delay with such a high accuracy.

In addition to the above theoretical motivation, there are advances in technologies concerning the high precision measurement of time and frequency such as optical lattice clocks Katori and attoseconds ( s) laser technologies Sansone . ASTROD project with three spacecrafts aims at measuring at the level of Ni .

The purpose of this paper is to clarify the dependence of the time delay (and induced frequency shift) on modified gravity models and investigate how to distinguish those models by using the Shapiro time delay. An important point in this paper is that we allow for various modified gravity theories beyond the scope of the PPN formulation. Introducing a new energy or length scale (e.g. extra dimension scale) may make changes in functional forms of the gravitational field. Thus it is worthwhile to investigate how to probe such a modified functional form, by using the light propagation in the solar system. Throughout this paper, we take the units of .

In this paper, we assume that the electromagnetic fields propagate in four-dimensional spacetimes (even if the whole spacetime is higher dimensional). Thus photon paths follow null geodesics (as the geometrical optics approximation of Maxwell equation).

We shall consider a static spherically symmetric spacetime, in which light propagates, expressed as

(1) |

where and denote the circumference radius and the metric of the unit 2-sphere, respectively. The functions and depend on gravity theories.

The time lapse along a photon path is obtained as

(2) |

where and denote the impact parameter and the closest point, respectively. Their relation is .

According to a concordance between solar-system experiments and the theory of general relativity, we can assume that the spacetime is expressed as the Schwarzschild metric (rigorously speaking, its weak field approximation) with a small perturbation induced by modified gravity. For practical calculations, we keep only the leading term at a few AU in the corrections. Namely, and are approximated as

(3) | |||||

(4) |

where denotes the mass of the central body. Here, , , and rely on a theory which we wish to test. For simplicity, we assume , which corresponds to a wide class of theories of gravity.

Examples of modified gravity theories are as follows. (1) , for DGP model with is the extra scale within which gravity becomes five dimensional DGP . (2) , and with graviton mass for one of massive gravity models MG1 ; MG2 . (3) , for the Schwarzschild-de Sitter spacetime, that is, general relativity with the cosmological constant as a possible candidate for the dark energy, though this is not a manifest modification of gravity. The solar system experiments are not sensitive to this model with KKL . Here, it should be noted that the examples (1) and (2) give conformally flat spacetimes (in the weak field approximation) and their conformal factors generate the gravitational time delay (and induced frequency shift), though the null geodesic in any conformally flat spacetime is mapped into that in the Minkowski one.

The Cassini experiment has put the tightest constraint on the solar gravity, especially near the solar surface with the accuracy of Cassini . This implies that deviations in and must be less than , that is, , .

We consider the round-trip time between pulse transmission and echo reception, denoted by . The pulse is emitted from Earth at , and reflected at .

Up to the linear order in , and , is expressed as

(5) | |||||

The extra time delay induced by a correction to general relativity is expressed as

(6) | |||||

where we define nondimensional radial coordinates as , and . For a radar tracking of a spacecraft such as Cassini, and are of the order of 1 AU ( km), and is several times of the solar radius ( km). Equation (6) can be rewritten by using special functions, though it seems less informative. Therefore, we take expansions of Eq. (6) in because of . For , we obtain

(7) |

whereas the second term of R.H.S. becomes for .

It is convenient to use the relative change in the frequency, which is caused by the gravitational time delay Bertotti , because the Doppler shift due to the receiver’s motion has no effect owing to the cancellation at both the receipt and emission of radio signal Bertotti . This frequency shift is defined as . Indeed, the frequency shift was used by the Cassini experiment. For a case of , which is valid for the Cassini experiment, the general relativistic contribution is expressed as Will

(8) |

We pay attention to the extra contribution due to modified gravity. For , the extra frequency shift becomes

(9) |

while we obtain for . Here we used , near the solar conjunction (). The total frequency shift is the sum, . The impact parameter of light path changes with time, because of the motion of the emitter and receiver with respect to Sun. For simplicity, we assume that they move at constant velocity during short-time observations. The impact parameter changes as , where denotes the minimum of the impact parameter near the solar conjunction at , and is the velocity component perpendicular to the line of sight.

Here, we make an order-of-magnitude estimate of the frequency shift. First, we obtain , where the dot denotes the time derivative, and is the orbital velocity of Earth ( 30 km/s). The Cassini experiment reported at the level of by careful processing of the frequency fluctuations largely due to the solar corona and the Earth’s troposphere Cassini . Multi-band measurements are preferred in order to avoid the astrophysical effect of the corona and interplanetary plasma on the delay, which is proportional to the square inverse of the frequency.

For a receiver at , the extra frequency shift is

(10) | |||||

where . The larger the index of , the longer the delay .

Figure 1 shows that an extra distortion due to would appear especially in the tail parts of curves. According to the fact that no deviation from general relativity has been reported by the Cassini experiment Cassini , we can put a constraint as at AU. On the other hand, Eq. gives for and , for instance, which are thus rejected. One can distinguish modified gravity models, which are characterized by various values of , , , from observations using receivers at very different distances from Sun, as shown by Fig. 1.

Figure 2 shows the dependence of on and . Hence, one can put a constraint on and from observed.

Equation (8) shows that the frequency shift depends only on the impact parameter but not the locations of the emitter and receiver. Strictly speaking, still has weak dependence on and as shown by Eq. (5). On the other hand, depends strongly on and . The dependence of and on and plays a crucial role in constraining (or detecting) a correction to general relativity in the solar system.

Let us imagine that time delays (or induced frequency shifts) are measured along two light trajectories, whose impact parameters are denoted as and , respectively. Then, we make a comparison of the two time delays. If they are in good agreement after taking account of a difference in the impact parameters, general relativity can be verified again. Otherwise, a certain modification could be required for the solar gravitational field. At this stage, however, one can say nothing about functional forms of the correction because the parameters of both and , which we wish to determine, enter the frequency shift.

In order to break this degeneracy, therefore, we consider three light paths, for which the impact parameters of the photon paths are almost the same (several times of the solar radius) for convenience sake. The locations of the receivers are denoted as , and , where the subscripts from 1 to 3 denote each light path. We assume that is constant in time for simplicity. It is a straightforward task to take account of the eccentricity of the Earth orbit and a difference between the impact parameters.

We make use of a difference such as and , in order to cancel out general relativistic parts. We find

(11) |

It should be noted that is proportional to . Hence, the following ratio depends only on as

(12) |

Thereby, one can determine the index . Next, one obtains by substituting the determined into Eq. (11).

In summary, we have clarified the dependence of the gravitational time delay on modified gravity models. For neighboring light rays, Eq. (7) gives almost the same value so that one can hardly distinguish models of gravity. This implies that we should prepare receivers at very different distances from Sun. Our result could be used for exterior planets explorers such as New Horizons, which were launched in 2006 and their primary target is Pluto and its moon, Charon at distance from Sun 40 AU NH . In future practical data analyses, however, it would be safer to use the original integral form as Eq. (6), because Eq. (7) is an approximate expression.

Furthermore, becomes the same order of , for future space-borne laser interferometric detectors such as LISA LISA , DECIGO DECIGO and especially ASTROD Ni . These detectors are in motion in our solar system. Namely, and change with time. Therefore, the sophisticated experiments by space-borne laser interferometric detectors, which are originally designed to detect time-dependent part of gravity, i.e. gravitational waves, could probe also a time-independent part of gravity at the relative level of . It would be important to make a feasibility estimate for these detectors. Clearly, a stronger test can be done by not a single experiment but combining several ones. In addition, more precise measurements of the Shapiro time delay with binary pulsars may put a constraint on the modifications discussed in this paper, especially in the strong self-gravitating regime. Further investigations along these lines will be done in the future.

###### Acknowledgements.

The author would like to thank S. Kawamura, N. Mio, M. Sasaki, M. Shibata and T. Tanaka for useful conversations. This work was supported by a Japanese Grant-in-Aid for Scientific Research from the Ministry of Education, No. 19035002.## References

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