Transcript Заголовок слайда отсутствует

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
Теоретический отдел
ИЗМИРАН
г.Троицк Московской обл.
142190 Россия
Max Planck Institute
(MPIPKS)
Nöethnitzer Strasse 38,
01187 Dresden, Germany
Contents:
1. Introduction
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
ABSTRACT
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
ABSTRACT
Introduction - 2
Gravitational field of a point-like mass (Kottler, 1918):
In the quasi-Newtonian limit:
“anti-gravity”
ABSTRACT
History of
the Lambda-Term - 1
• Late 17th century: Newton – two alternative forms of the law of gravity:
or
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 General
Relativity 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 apparent
concentration of quasars near the redshift value z = 2 (not confirmed later).
ABSTRACT
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 early
Universe, inspired by the elementary-particle physics.
• 1990s – the Lambda-term began to be widely discussed again in the
cosmology of the late Universe, especially, to explain formation of the
large-scale structure.
• 1998: Reiss, et al., Perlmutter, et al. – existence of the Lambda-term was
strongly supported by the data on the accelerated expansion of the Universe
following 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 CMB
anisotropy data by WMAP satellite and ground-based instruments as well
as by other kinds of cosmological probes, such as 2dF and SDSS galaxy
surveys.
• 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 problem
of variability of the “fundamental constants”.
Estimating the Lambda-Term from the Large-Scale
Cosmological Data
supernovae Ia
CMB fluctuations
Lyman-alpha forest
ABSTRACT
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.
However:
 various estimates disagree with each other,
 the most of arguments are not applicable to the Lambda-dominated
cosmology.
Estimating the Lambda-Term from the Local Planetary Dynamics - 2
For example, the following arguments do not work in the case of the Lambdadominated cosmology:
 Einstein–Straus theorem;
 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 distribution
by 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).
Estimating the Lambda-Term from the Local Planetary Dynamics - 4
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
.
Estimating the Lambda-Term from the Local Planetary Dynamics - 5
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.
Estimating the Lambda-Term from the Local Planetary Dynamics - 6
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
Estimating the Lambda-Term from the Local Planetary Dynamics - 7
rg = 0.01
R0 = 1
r0 = 1000
r0 = 2000
r
r0 = 5000
r0 = 
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 becomes substantial (and even can reach the rate of the standard Hubble
flow at infinity).
Estimating the Lambda-Term from the Local Planetary Dynamics - 8
 A well-known disagreement in the rates of secular increase in the lunar semi-major axis
derived from the astrometric data and lunar laser ranging (LLR):
Estimating the Lambda-Term from the Local Planetary Dynamics - 9
Immediate
measurement
by LLR
Indirect derivation
from the Earth’s
rotation deceleration
Effects involved
(1) geophysical tides
(2) local Hubble expansion
(1) geophysical tides
Numerical value
3.8 ± 0.1 cm/yr
1.6 ± 0.2 cm/yr
Rate of the lunar
orbital increase
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 contribute
at 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.
Discussion and Summary
 Existence of the Lambda-term is established by now quite reliably from a
number of cosmological tests.
 All the available estimates of the value of Lambda-term were obtained from
the large-scale cosmological data.
 Since most of the commonly-used arguments against the local Hubble
expansion are not applicable to the Lambda-dominated cosmology, there
are some perspectives to get the value of Lambda-term also from the
high-precision astrometric measurements of the planetary systems or
compact relativistic objects (e.g. binary pulsars).
 The question of the dynamic or static nature of the Lambda-term remains
open, because the results published by now are very contradictory.