Perturbation of a Planetary Orbit by the Dark Energy

1. 15th Lomonosov Conference on Elementary Particle Physics 18 – 24 August 2011 Moscow State University, Moscow, Russia Perturbation of a Planetary Orbit by the Dark Energy Yurii V. Dumin Theoretical Department, Institute of Terrestrial Magnetism, Ionosphere, and Radio Wave Propagation (IZMIRAN), Russian Academy of Sciences 142190 Troitsk, Moscow region, Russia

2. ABSTRACT Introduction: The Problem of Local Cosmological Effects - 1 The question if the planetary orbits (and other local dynamics) can be affected by the cosmological expansion was put forward by G.C. McVittie as early as 1933; and a quite large number of researchers dealt with this problem subsequently: G. Jaernefelt, J.F. Cardona & J.M. Tejeiro, A. Einstein & E.G. Straus, W.B. Bonnor, E. Schuecking, Yu.V. Dumin, R.H. Dicke & P.J.E. Peebles, A. Dominguez & J.Gaite, V.S. Brezhnev, D.D. Ivanenko S.A. Klioner & M.H. Soffel, & B.N. Frolov, L. Iorio, J. Pachner, G.A. Krasinsky & V.A. Brumberg, P.D. Noerdlinger & V. Petrosian, D.F. Mota & C. van de Bruck, R. Gautreau, M. Nowakowski, I. Arraut, C.G. Boehmer F.I. Cooperstock, V. Faraoni & A. Balaguera-Antolinez, & D.N. Vollick, C.H. Gibson & R.E. Schild, M. Mars, M. Sereno & Ph. Jetzer, J.M.M. Senovilla & R. Vera, V. Faraoni & A. Jacques, J.L. Anderson, E. Hackmann & C. Laemmerzahl, ... The most frequent conclusion was that the effect of cosmological (Hubble) expansion at interplanetary scales should be negligible or absent at all. However:  various estimates disagree with each other,  the most of arguments are not applicable to the Lambda-dominated cosmology.

3. Introduction: The Problem of Local Cosmological Effects - 2 For example, the following arguments do not work in the case when the cosmological background is formed by the perfectly-uniform “dark energy”:  Einstein–Straus theorem;  virial criterion of gravitational binding;  Einstein–Infeld–Hoffmann (EIH) surface integral method.

4. ; Theoretical Treatment of the Planetary Perturbation by the Lambda-term - 1 The most straightforward approach to answer the question of local cosmological influences is to consider the two-body motion (e.g. a test particle in the field of a massive central body) embedded in the expanding Universe.  This was done in a number of previous works (listed in the Introduction).  The main problem is a perturbation of the background cosmological matter distributionby the central body.  The situation is substantially simplified for the “dark energy”-dominated Universe(because of the perfectly uniform distribution of the Lambda-term). The starting point of a few recent considerations was Kottler (Schwarzschild – deSitter) solution of the General Relativity equations (e.g. E. Hackmann &C. Laemmerzahl; M. Nowakowski, I. Arraut, C.G. Boehmer & A. Balaguera-Antolinez): and it was found that influence of the Lambda-term in the Solar system should be negligible. Unfortunately, these authors took into account only the “conservative” effects, because the above metric suffers from the lack of the adequate cosmological asymptotics (i.e.does not reproduce the standard Hubble flow at infinity).

5. Theoretical Treatment of the Planetary Perturbation by the Lambda-term - 2 The basic idea of our approach is to perform the entire analysis, just from the beginning, in the coordinate system possessing the correct (Robertson–Walker) cosmological asymptotics at infinity [Yu.V. Dumin, Phys. Rev. Lett., v.98, p.059001 (2007)]: , . , Up to the first non-vanishing terms of rg and 1/r0 , this metric can be reduced to

6. For example, in the Earth–Moon system: rg~ 10–2 m, R0 ~ 109 m, r0~ 1027 m; i.e. the characteristic scales of the problem differ by many orders of magnitude. Theoretical Treatment of the Planetary Perturbation by the Lambda-term - 3 Equations of motion of a test particle (including the first non-vanishing terms of rg and 1/r0):

7. Results of Numerical Integration - 1 To simplify calculations, let us assume that difference between the characteristic scales (Schwarzschild radius, the planetary orbit radius, and de Sitter radius) is not so much as in reality, e.g. rg = 0.01, R0 =1. r0 = r0 =5000 r0 =2000 r0 =1000

8. Results of Numerical Integration - 2 rg = 0.01 R0 = 1 r0 =1000 r0 =2000 r0 =5000 r0 = r Note: The curves are wavy because the initial (unperturbed) planetary orbit was taken to be slightly elliptical. t Dashed lines represent the standard Hubble flow (unperturbed by the central gravitating mass). As follows from these plots, in certain circumstances the perturbation caused by the -term (“dark energy”) becomes substantial (and even can reach the rate of the standard Hubble flow at infinity).

9. An Empirical Evidence in Favor of the Local Cosmological Influences - 1 Disagreement in the rates of secular increase in the lunar semi-major axis derived from the astrometric data and lunar laser ranging (LLR):

10. An Empirical Evidence in Favor of the Local Cosmological Influences - 2 Immediate Indirect derivation Rate of the lunar orbital increase measurement from the Earth’s rotation deceleration by LLR (1) geophysical tides (1) geophysical tides Effects involved (2) local Hubble expansion Numerical value 3.8±0.1 cm/yr 1.6±0.2 cm/yr The difference 2.2 cm/yr may be attributed to the local Hubble expansion with rate H0(loc) = 56±8 (km/s)/Mpc . If the local Hubble expansion is formed only by the uniformly distributed -term (“dark energy”), while the irregularly distributed (aggregated) forms of matter begin to contributeat the larger scales, then So, the ratio of the local to global Hubble expansion rate should be At 0 = 0.75 and D0 = 0.25, we get H0/H0(loc) 1.15 ; so that H0 = 65 ± 9 (km/s)/Mpc, which is in agreement with cosmological data.

11. Discussion and Summary  The problem of cosmological influences on the local (e.g. planetary) systems is studied for almost 80 years but still remains a poorly understood subject (especially, in the case of arbitrary energy–momentum tensor and inhomogeneous background mater distribution).  A theoretical consideration of the two-body problem becomes much simpler and straightforward in the case of the -dominated cosmological background. However, the analytical perturbation theory is still under development; while the numerical calculations encounter the problem of very different spatial scales involved, so that only some toy models were reliably computed by now.  Consideration of a few toy models shows that perturbation of the planetary orbit by the -term can be significant, and the rate of its secular increase in some circumstances can approach the rate of the standard Hubble flow at infinity.  An important empirical evidence for the probable cosmological effects in the Earth–Moon system is the disagreement between LLR data and angular astrometric measurements of the Earth’s rotation deceleration, which can be resolved by the inclusion of the local cosmological effects caused by the -term [Yu.V. Dumin, Adv. Space Res., v.31, p.2461 (2003); in Proc. 11th Marcel Grossmann Meeting on General Relativity (World Sci., Singapore, 2008), p.1752].  Therefore, inclusion of the effects by -term into the high-precision models of planetary motion should enable astronomers either to derive the value of “dark energy” density immediately from the local planetary dynamics or to impose constraints on its existence.  The same effects may be important also in the dynamics of close relativistic systems (e.g. binary pulsars), but this subject was not yet studied in detail.