Influence of the Cosmological Lambda-Term on the Dynamics of Compact Objects

1. Actual Problems in Extragalactic Astronomy, Pushchino, 24–27 April 2006 Influence of the Cosmological Lambda-Term on the Dynamics of Compact Objects Yurii V. Dumin IZMIRAN Troitsk, Moscow reg., 142190 Russia dumin@yahoo.com

2. The problem to be addressed: How much the perfectly-uniform component of the energy–momentum tensor, represented by -term, can affect the Hubble expansion at small scales? Historical landmarks: 1918 F. Kottler Über die physikalischen Grundlagen der Einsteinschen Gravitationstheorie.Ann. Phys.,56, 401–462 1933 G.C. McVittie The mass-particle in an expanding universe.MNRAS,93, 325–339…… …………… …………………………………………… Modern mathematical approaches for treating this problem:  Einstein–Straus theorem; virial criterion of gravitational binding; Einstein–Infeld–Hoffmann (EIH) surface integral method

3. Pointlikegravitatingmass Static Schwarz-schild metricinside an emptycavity M Time-dependentFRW metric inthe backgroundmatter Movingcontactboundary Einstein – Straus theorem: There is a continuous spherically-symmetric solution of the General Relativity equations of the following form: So, the gravitational field of a point-like mass is static in its local empty neighborhood and experiences the Hubble expansion only outside the contact boundary. The Einstein–Straus theorem is evidently inapplicable to the-dominated cosmological models, because it is meaningless to consider an empty cavity in the vacuum energy distribution.

4. The problem to be addressed: How much the perfectly-uniform component of the energy–momentum tensor, represented by -term, can affect the Hubble expansion at small scales? Historical landmarks: 1918 F. Kottler Über die physikalischen Grundlagen der Einsteinschen Gravitationstheorie.Ann. Phys.,56, 401–462 1933 G.C. McVittie The mass-particle in an expanding universe.MNRAS,93, 325–339…… …………… …………………………………………… Modern mathematical approaches for treating this problem:  Einstein–Straus theorem; virial criterion of gravitational binding; Einstein–Infeld–Hoffmann (EIH) surface integral method

5. A promising way for a detection of the small-scale Hubble expansion, associated with -term, is utilizing the available data of lunar laser ranging (LLR).

6. Years Accuracy Comments reflection from the natural lunar surface early 1960’s a few kilometers deployment of five reflector arrays on the lunar surface early 1970’s 2030 cm new generation of the ground-based ranging stations late 1980’s 23 cm (under the most favorable atmospheric conditions) present time a few millimeters Lunar Laser Ranging: history and main achievements An expected increase in the lunar semi-major axis due to the “standard” Hubble expansion (i.e. with the same rate as at intergalactic scales) isabout 50 cm for the period of 20 years. On the other hand, the instrumental errors of LLR measurements duringthe last 20 years were about 23 cm. So, the resulting accuracy shouldbe sufficiently good. The main obstacle is to exclude the effect of geophysical tides, whichalso contributes to the secular increase in the Earth–Moon distance.

7. As follows from the law of con-servation of angular momentum , the variation in the Earth–Moon distance R is related to the change of the Earth’s diurnal period TE by the formula . . , where . Because of the relaxation effects,the Earth’s tidal bulge is not perfectly aligned in the direction to Moon. As a result, there is a torque moment and the exchange of angular momentum, leading to increase in the Earth–Moon distance R.

8. Immediate measu-rement by LLR Independent estimate fromthe Earth’s tidal deceleration Method (1) geophysical tides Effects involved (1) geophysical tides (2) local Hubble expansion (1.6±0.2)2.5 cm/yrfrom the Earth’s angularvelocity (2.4±0.2)(3.3±0.2) cm/yrfrom the tidal amplitude(torque moment) Numerical values 3.8±0.1 cm/yr Comparison of the rates of secular increase in the lunar orbit

9. . . Type of measurement–TE R H0(loc) H0(ms/cyr) (cm/yr) (km/s/Mpc) (km/s/Mpc) Angular velocityof the Earth’s rotation: astrometric observations 0.880.10 1.60.2 568 598for ~350 years observations of solar 1.4 2.5 33 35eclipses for ~1500 years Tidal amplitude(torque moment):  Schwiderski 1.350.1 2.40.2 365 385  FES95.2 1.780.1 3.20.2 155 165  CSR3 1.830.1 3.30.2 135 146 Summary and conclusions 1. There is a considerable disagreement between the various types of data on the Earth’s tidal deceleration. 2. The values of H0 derived from the astrometric observations during the last 350 years are in reasonable agreement with the commonly-accepted ones. 3. Even the smallest values of H0 are significantly different from zero.

10. It is reasonable to assume that the local Hubble expansion is formed onlyby the uniformly-distributed “dark energy”, while the irregularly distributed (aggregated) forms of matter begin to contribute at the larger scales and, thereby, increase the rate of Hubble expansion up to the standard value. In the flat Universe filled only by vacuum and dust-like (“cold”) matter . So, the ratio of the local to total Hubble expansion should be . AtV0 = 0.73 andD0 = 0.27 , we getH0(loc)/H0  1 0.05. Relation between the local and global Hubble constants

11. . . Type of measurement–TE R H0(loc) H0(ms/cyr) (cm/yr) (km/s/Mpc) (km/s/Mpc) Angular velocityof the Earth’s rotation: astrometric observations 0.880.10 1.60.2 568 598for ~350 years observations of solar 1.4 2.5 33 35eclipses for ~1500 years Tidal amplitude(torque moment):  Schwiderski 1.350.1 2.40.2 365 385  FES95.2 1.780.1 3.20.2 155 165  CSR3 1.830.1 3.30.2 135 146 Summary and conclusions 1. There is a considerable disagreement between the various types of data on the Earth’s tidal deceleration. 2. The values of H0 derived from the astrometric observations during the last 350 years are in reasonable agreement with the commonly-accepted ones. 3. Even the smallest values of H0 are significantly different from zero.