Wilfrid Laurier University Scholars Commons @ Laurier Physics and Computer Science Faculty Pulications Physics and Computer Science 11-1 Radar Signal Delay in the Dvali-Gaadadze-Porrati Gravity in the Vicinity of the Sun Ioannis Haranas Wilfrid Laurier University, iharanas@wlu.ca Omiros Ragos University of Patras, ragos@math.upatras.gr Ioannis Gkigkitzis East Carolina University, gkigkitzisi@ecu.edu Follow this and additional works at: http://scholars.wlu.ca/phys_faculty Part of the Computer Sciences Commons, Mathematics Commons, and the Physics Commons Recommended Citation Haranas, I., Ragos, O. & Gkigkitzis, I. Radar Signal Delay in the Dvali-Gaadadze-Porrati Gravity in the Vicinity of the Sun. Astrophys Space Sci (1) 4: 81. DOI: 1.17/s159-1-1165-8 This Article is rought to you for free and open access y the Physics and Computer Science at Scholars Commons @ Laurier. It has een accepted for inclusion in Physics and Computer Science Faculty Pulications y an authorized administrator of Scholars Commons @ Laurier. For more information, please contact scholarscommons@wlu.ca.
1 Radar Signal Delay in the Dvali-Gaadadze-Porrati Gravity in the Vicinity of the Sun Ioannis Haranas 1, Omiros Ragos Ioannis Gkigkitzis 1 Department of Physics and Astronomy, York University, 47 Keele Street, Toronto, Ontario, MJ 1P, Canada e:mail: yiannis.haranas@gmail.com Dept. of Mathematics, University of Patras, GR-65 Patras, Greece e-mail: ragos@math.upatras.gr Department of Mathematics, East Carolina University, 14 Austin Building, East Fifth Street Greenville, NC 7858-45, USA e-mail: gkigkitzisi@ecu.edu Astract In this paper we examine the recently introduced Dvali-Gaadadze-Porrati (DGP) gravity model. We use the space time metric in which the local gravitation source dominates the metric over the contriutions from the cosmological flow. Anticipating ideal possile solar system effects we derive expressions for the signal time delays in the vicinity of the sun, and for various angles of the signal approach. We use the corresponding numerical value for the parameter r to e equal to 5 Mpc, and from that we calculate that the time contriution due to DGP correction to the metric is proportional / to / c r. In the vicinity of the Sun and with in the range -/ -/, Δ t is equal to.1 ps. A time signal delay extremely small to measure y today s technology could e proaly measurale in the future years to come, y various future experiments. Key Words: Dvali-Gaadadze-Porrati gravity, radar signal delays, de Sitter ackground, Friedman-Lemaitre-Roertson-Walker phase, accelerating phase. 1. Introduction There is recent attention for a five dimensional gravity model the so called Dvali-Gaadadze-Porrati (DGP). This model explains the oserved acceleration of the expansion of the Universe. Furthermore, it predicts minor post-einstein effects, testale at local scales resulting to information on the Universe s gloal properties in relation to the ongoing cosmological expansion (Iorio, 5). So far, two-ody scenarios have een investigated in which the time rates of change for the longitude of pericenter and the mean anomaly of the secondary have een carried out (Lue and Starkman,, Iorio, ). These effects are functions of eccentricity up to O(e ) ut they do not
depend on the semimajor axis a of the secondary. Following Iorio (, c) one might say that the ideal test-ed for such tests is the inner planets of the solar system. Measurements of such precessions lie in the limit of precision of today s planetary data. For a more detailed and complete overview on the DGP gravity see Lue, (). The DGP model is ased on an extra flat dimension, and a free crossover parameter r which defines a radius eyond which the four-dimensional gravitational theory transitions into a fivedimensional regime, and is defined as follows r k / μ. The constants and k define the energy scales of the theories of gravity: one is Newton s constant, =8G, the other represents the energy scale of the ulk gravity (Sawicki, et al., 7). The crossover parameter is also fixed from oservations of IA type supernova to a value approximately equal to 5 Gpc (Lue and Starkman ). For distances greater than that Newtonian gravity needs to e modified, resulting thus to different explanations from that of dark matter in order to interpret the accelerations oserved in the Universe. In this short contriution we will calculate the signal time delay in the vicinity of a massive ody of mass M, i.e. the Sun when the Newtonian-Einstein gravity is modified due to the Dvali- Gaadadze-Porrati raneworld model. We proceed with in a way similar to Haranas and Ragos (11) and Haranas, et al. (11). We ase our analysis on the angle defined y the distance of the closest approach, and any other point on the traveling radar signal. We apply our results in the vicinity of the Sun and for various angular ranges.. The Dvali-Gaadadze-Porrati Metric Following Lue and Starkman () the metric of a spherical source in a cosmological de Sitter ackground can e descried y the following line element: ds r, wdt A r, wdr B r, wd θ sin dφ dw. (1) c N In the region where the local gravitation dominates the metric over the contriutions from the cosmological flow, the de Sitter solution of the five-dimensional field equations can e expressed y (iid, ): and N GM GMr () rc r r, z 1 nr, z 1
A GM GMr () rc r r, z 1 αr, z 1 r, z r1 r z B, (4) where M is gravitating mass, G is the constant of universal gravitation, the w is the fourth special coordinate, (r, z) are functions of coordinates and time, that can e calculated using the derived field equations. For distances scales much smaller than r r Newton-Einstein gravity is otained with few exceptions that include minor corrections. With reference to Lue and Starkman () wee can write the Dvali-Gaadadze-Porrati (DGP) line element in the following way: ds GM GMr GM GMr 1 r, z dω dw c 1 dt 1 dr r rc r rc r. (5) To deal with the Dvali-Gaadadze-Porrati (DGP) effect on the propagation of electromagnetic signals in the vicinity of the Sun, we incorporate an additional DGP correction term in the Schwarzschild space-time metric coefficients of the line element. We analogously modify the Schwarzschild metric used in the solar system when general relativistic effects are taken into account. Therefore, if r,, are the polar coordinates of any point along the signal s path, and is the corresponding solid angle, the line element takes the form. Next, the photon transmission time can e written as follows: and ds dt, (6) c GM GMr GM GMr ds dt 1 dt 1 rc r rc r, (7) c where the plus sign is related to a the Friedman-Lemaitre-Roertson-Walker phase of the universe, and where the minus sign is related to a self accelerating phase (Iorio, 5a). In this paper we first consider the plus sign or Friedman-Lemaitre-Roertson-Walker phase first and second the minus sign representing the accelerating phase. Next, using r / cos, where is the distance of the closest signal approach, and therefore we have that:
4 sin θ r dθ, cos θ d (8) Also, dr dr r dθ and after sustitution and simplification, Eq (5) ecomes: GM G M 1 cosθ cos θ 4 c c d t c GM GM GM secθ secθ c r c r c sec GMsecθ c r θdθ. (9) Equation (9) contains classical, general relativistic, DGP time delays. Since we are interested in the DGP delay only we may neglect the terms following expression: GM G M 1 cos cos c c we then have to integrate the 4 d t GM GM GM GMsecθ θ θ c sec sec sec c r c r c c r 4 Omitting order O c and c θdθ, (1) O terms for eing to small, we integrate over various angular suintervals of the range ( /, /) to avoid the singularities at /. For any such interval [ a, ] the corresponding radar signal time delay will e: π / 6 GM Δ t secθ GMsecθ sec θdθ c c r cr, (11) π / 6 We start at [ /6, /6] and integrating we otain that: GM GM ln 7 8 GM 8 GM π Δ t F, 7/ 4 (1) c r 6c r c r c r 1 where, F is the elliptic integral function of the first kind (Spiegel, 1968). Similarly, integrating over the range [ /4, /4], we otain that: 1/ 4 GM 8 GM 8 GM π GM Δ t F, c r c r c r 8 c r Next in the range [-/,/] we otain that: cos ln cos π /8 sinπ /8 π /8 sinπ /8 (1).
5 GM 8 GM 8 GM π GM Δ t F, c r c r c r 6 c r 1 ln 1, (14) finally in the range [-4/1, 4/1]or 7 o we otain the following expression GM 5 5 8 c r GM c 157 r 5 8 GM π GM F, c r 5 c r Δ t 1 ln 1 5 5 1 1 5. 5 (15) Next, in the same way, we proceed to calculate the signal delays when the DGP correction to the metric appears with a negative sign. For any interval [ a, ] ( /, / ) the corresponding radar signal time delay will e: Δ t π / 6 GM secθ GMsecθ sec θdθ c c r cr. (16) π / 6 For [ /6, /6], we otain: GM 8 GM 8 GM π GM t F, ln7 7/ 4, (17) c r c r c r 1 6c r Δ where F is the elliptic function of the first kind. Similarly, and for the same angular limits as efore for [ /4, /4], we otain: 1/ 4 GM 8 GM 8 GM π GM Δ t F, c r c r c r 8 c r next, for [ /, /], we otain: cos ln cos π /8 sinπ /8 π /8 sinπ /8, (18) GM 8 GM 8 GM π GM 1 t F, ln (19) c r c r c r 6 c r 1 Δ and finally, for -4/1< < 4/1 we otain that:
6 5 5GM c r 8 15 7 5 c r GM 8 GM π GM F, c r 5 c r Δ t 1 ln 1 5 5 1 1 () 5. 5 In the numerical results section, the evaluation of the elliptic function F will e necessary. This can e accomplished y using the following up to 4 th order series expansion of F given elow: F θ, k sin θ 1 sin θ sin θ sin 4 1 5sin sin θ sin θ sin θ 56 5cos θ sin θ 5cos θ sin 84 96 1 15sin sin θ cos θ sin θ 15cos θ sin 1684 4576 5 45cos θ sin θ 5 7 cos θ sin θ 6144 14 1 9 sin sin θ cos θ 64 k cos θ sin θ 5 k θ θ sin θ θ k 4 k O k 5.., (1) whose numerical values are going to e sustituted for the corresponding functions in Eqs. (1) to () in the section of the numerical results. Therefore we find that, the evaluation to O(k 4 ) of the elliptic integrals of the first time we otain: π 585 9 179π F,.6887, () 1 51 1684 496 π 41 85 417π F,.4154, () 8 14 18 819 π 14491 179π F,.5849, (4) 6 1684 48 π 98681 659964985 417π F,.74414. (5) 5 768 51
7. Numerical Results To apply the aove analysis in the case of the Sun, we have used that we use the numerical values for the mass of the Sun M S = 1.991 kg, and taking R S 6.961 8 m, and using r 5 Gpc = 1.54 1 m (Lue and Starkman ), we taulate in tale 1 the following time signal delays in the vicinity of the Sun for the angular ranges indicated. First, the signal time delay in the vicinity of the Sun and in the range π / 6 θ π / 6 for a Friedman-Lemaitre-Roertson-Walker phase first, we have: / / 6 7 7 Δt 5.9771.1.1, (6) r r r similarly, in the range π / 4 θ π / 4 we otain: / / 5 7 7 Δt 1.181 5.4771 5.4771. (7) r r r Next, in the range π / θ π / we otain / / 5 6 6 Δt.51 1.51 1.51. (8) r r r Finally, in the range 4π /1 θ 4π / 1 we otain: / / 5 6 6 Δt 5.81.1441.1441, (8) r r r where the time is given in picoseconds [ps], is taulated in tale 1 elow: Tale 1. Radar signal time delays due to the DGP gravity in the vicinity of the Sun. Results related to the Friedman-Lemaitre-Roertson-Walker phase of the universe. Angular Range of Closest Approach [rad] Signal Time Delays [ps] π / 6 θ π / 6.61 π / 4 θ π / 4.65 π / θ π /.1 4π /1 θ 4π /1.55
8 Similarly, for the accelerating phase of the universe and for the same order of angles we otain that: / / 6 7 7 Δt 5.9771.1.1, (9) r r r Δt 1.181 5 r 5.4771 7 r / 5.4771 7 r /, () / / 5 6 6 Δt.51 1.51 1.51, (1) r r r Δt 5.81 5 r.1441 6 r /.1441 6 r /, () where the time is given in picoseconds [ps], is taulated in tale elow Tale Radar signal time delays due to the DGP gravity in the vicinity of the Sun. Results related to the accelerating phase of the universe. Angular Range of Closest Approach [rad] Signal Time Delays [ps] π / 6 θ π / 6 -.61 π / 4 θ π / 4 -.65 π / θ π / -.1 4π /1 θ 4π /1 -.55 With reference to Haranas and Ragos (11) we say that to get an idea of today s radar systems, someody could talk aout the sensitivity of a radar, a property that is related to the power of the transmitting radar. Since we are interested in signal time delays and in order to sustantiate our finding we will referrer to today s radar resolution instead something that is related to the detectale times. Quoting Shapiro (1968, 1999), we say that fractional system errors of echo time delays in solar system experiments can e up to 1 part in 1 1 or smaller. Given today s technological progress it might e possile that such effects will e detected in the years to come.
9 Fig. 1 Signal time delay in the vicinity of the Sun as a function of the DGP parameter r and distance of closest approach in Friedman-Lemaitre-Roertson-Walker phase of the universe, in the range [ 4 /1,4 /1]. Fig. 1 Signal time delay in the vicinity of the Sun as a function of the DGP parameter r and distance of closest approach in related to the accelerating phase of the universe in the range [ 4 /1,4 /1].
1 4. Conclusions The signal time delay in the vicinity of the sun for the Dvali-Gaadadze-Porrati metric (DGP) has een calculated, for various angles in the range θ 7. Both algeraic signs have een considered in the metric element. In particular the plus sign is related to the Friedman-Lemaitre-Roertson-Walker phase of the universe, and results to a positive time delay or an addition in the total traveling time due to Dvali-Gaadadze-Porrati, where the negative one is related to a self accelerating phase of the universe, results to a negative time delay or reduction in the total time due the contriution of DGP gravity correction. The signal delays are calculated in fractions of picoseconds. Signal delays of this magnitude might e in the orderline of time detection of today s technology and therefore there might e difficult to detect. Future technologies might e ale to push for such a delectaility limit, and therefore delays attriute to (DGP) gravity might e measured, in solar system experiments. References Haranas I., and Ragos, O., Calculation of Radar Signal Delays in the Vicinity of the Sun due to the Contriution of a Yukawa Correction Term in the Gravitational Potential, Astrophys. Space Sci, vol. 4, No. 1, pp. 71-74, 11. Iorio, L. A comment on the possiility of testing the Dvali-Gaadadze-Porrati gravity model with the outer planets of the Solar System, gr-qc/5115, 5c. Iorio, L. On the effects of the Dvali-Gaadadze-Porrati raneworld gravity on the orital motion of a test particle, Class. Quantum Grav., at press, gr-qc/545, 5a. Iorio, L. On the possiility of testing the Dvali-Gaadadze-Porrati rane-world scenario with orital motions in the Solar System, J. Cosmol. Astropart. Phys., 7, 8, 5. Lue, A. The Phenomenology of Dvali-Gaadadze-Porrati Cosmologies, Phys. Rep., at press, astro-ph/5168, 5. Lue, A., and Starkman, G. Gravitational Leakage into Extra Dimensions Proing Dark Energy Using Local Gravity, Phys. Rev. D, 67 64,. Shapiro I.I., Pettengill G.H., Ash, M.E., Stone M.L., Smith W.B., Ingalls R.P., Brockelman R.A.: Phy. Rev. Letters, 165, (1968). Shapiro I.I.: Reviews of Mod. Physics 71(), S 41, (1999). Sawicki, I., Song, Y. S., and Hul W., Near-horizon solution for Dvali-Gaadadze-Porrati perturations, Phys. Rev. D, 75, 64, 7.