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Всероссийское совещание по прецизионной физике и фундаментальным физическим константам Workshop on Precision Physics and Fundamental Physical Constants ОИЯИ, Дубна, 5 – 9 декабря 2011. Лямбда-член как вторая фундаментальная константа в гравитационной физике. Ю. В. Д у м и н.
Всероссийское совещание по прецизионной физике и фундаментальным физическим константам
Workshop on Precision Physics and Fundamental Physical Constants
ОИЯИ, Дубна, 5–9 декабря 2011
Лямбда-член как вторая фундаментальная константа
в гравитационной физике
Ю. В. Думин
г.Троицк Московской обл.
Max Planck Institute
Nöethnitzer Strasse 38,
01187 Dresden, Germany
2. History of the Lambda-term
3. Estimating the Lambda-term from the large-scale cosmological data
4. Estimating the Lambda-term from the local planetary dynamics
5. Discussion and summary
Introduction - 1
The second fundamental constant appears naturally by the least-action principle in a curved space-time:
The resulting dynamical equation is
Two fundamental constants appearing here are
Introduction - 2
Gravitational field of a point-like mass (Kottler, 1918):
In the quasi-Newtonian limit:
History of the Lambda-Term - 1
•Late 17th century: Newton – two alternative forms of the law of gravity:
Unique features of these two laws:
(1) the orbits of the test bodies (planets) are closed curves (conical sections);
(2) the interaction between two extended massive spheres is equivalent to the interaction between the points in their centers with the masses of the spheres.
•Late 18th – early 19th century: Laplace – an attempt to explain the anomalous motion of Mercury’s perihelion by the modified law of gravity:
•1917: Einstein – the Lambda-term was introduced to the equations of GeneralRelativity to provide a stationary state of the Universe.
•1922: Friedmann, 1927: Lemaitre – the Lambda-term became unnecessary after finding the non-stationary cosmological solutions and identifying them with the effect of Hubble expansion.
•1967: Petrosian et al. – Lambda-term is actively discussed again in the framework of non-stationary cosmology to explain the apparentconcentration of quasars near the redshift value z =2 (not confirmed later).
History of the Lambda-Term - 2
•1980: Guth, 1982: Linde, et al. – the “dynamical” Lambda-term (scalar field)began to be actively exploited in the inflationary models of the earlyUniverse, inspired by the elementary-particle physics.
•1990s – the Lambda-term began to be widely discussed again in thecosmology of the late Universe, especially, to explain formation of thelarge-scalestructure.
•1998: Reiss, et al., Perlmutter, et al. – existence of the Lambda-term wasstrongly supported by the data on the accelerated expansion of the Universefollowing from the Type Ia supernovae distribution.
•1999: Turner – the term “dark energy” was introduced to denote the energy density associated with Lambda-term.
•2000s – existence of the Lambda-term was further supported by the CMBanisotropy data by WMAP satellite and ground-based instruments as wellas by other kinds of cosmological probes, such as 2dF and SDSS galaxysurveys.
• The question if the Lambda-term is a genuine constant or a dynamic quantity(scalar field) remains open. It is actually one of items of the general problemof variability of the “fundamental constants”.
Estimating the Lambda-Term from the Large-Scale Cosmological Data
Estimating the Lambda-Term from the Local Planetary Dynamics - 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.
various estimates disagree with each other,
the most of arguments are not applicable to the Lambda-dominated cosmology.
For example, the following arguments do not work in the case of the Lambda-dominated cosmology:
virial criterion of gravitational binding;
Einstein–Infeld–Hoffmann (EIH) surface integral method.
Estimating the Lambda-Term from the Local Planetary Dynamics - 3
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).
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
Equations of motion of a test particle (including the first non-vanishing terms of rg and 1/r0):
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 from each other by many orders of magnitude.
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.
rg = 0.01
R0 = 1
Note: The curves are wavy because the initial (unperturbed) planetary orbit was taken to be slightly elliptical.
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 becomes substantial (and even can reach the rate of the standard Hubble flow at infinity).
A well-known disagreement in the rates of secular increase in the lunar semi-major axisderived from the astrometric data and lunar laser ranging (LLR):
Rate of the lunar orbital increase
from the Earth’s
(1) geophysical tides
(1) geophysical tides
(2) local Hubble expansion
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 reasonable agreement with cosmological data.
Existence of the Lambda-term is established by now quite reliably from anumber of cosmological tests.
All the available estimates of the value of Lambda-term were obtained fromthe large-scale cosmological data.
Since most of the commonly-used arguments against the local Hubbleexpansion are not applicable to the Lambda-dominated cosmology, thereare some perspectives to get the value of Lambda-term also from thehigh-precision astrometric measurements of the planetary systems orcompact relativistic objects (e.g. binary pulsars).
The question of the dynamic or static nature of the Lambda-term remainsopen, because the results published by now are very contradictory.