Лямбда-член как вторая фундаментальная константа

в гравитационной физике

Ю. В. Д у м и н

Теоретический отделИЗМИРАН

г.Троицк Московской обл.142190 Россия

Max Planck Institute(MPIPKS)

Nöethnitzer Strasse 38,01187 Dresden, Germany

Всероссийское совещание по прецизионной физике и фундаментальным физическим константамWorkshop on Precision Physics and Fundamental Physical Constants

ОИЯИ, Дубна, 5 – 9 декабря 2011

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

ABSTRACTIntroduction - 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

ABSTRACTIntroduction - 2

Gravitational field of a point-like mass (Kottler, 1918):

In the quasi-Newtonian limit:

“anti-gravity”

ABSTRACTHistory 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).

ABSTRACTHistory 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”.

supernovae Ia

CMB fluctuations

Lyman-alpha forest

Estimating the Lambda-Term from the Large-Scale Cosmological Data

ABSTRACTEstimating 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.

For example, the following arguments do not work in the case of the Lambda-dominated 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 - 2

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 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).

;

Estimating the Lambda-Term from the Local Planetary Dynamics - 3

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 - 4

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 - 5

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 - 6

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).

rg = 0.01

R0 = 1

r0 = 1000

r0 = 2000

r0 = 5000

r0 =

t

r

Note: The curves are wavy because the initial (unperturbed) planetary orbit was taken to be slightly elliptical.

Estimating the Lambda-Term from the Local Planetary Dynamics - 7

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 - 8

The difference 2.2 cm/yr may be attributed to the local Hubble expansion with rate

H0(loc) = 56 ± 8 (km/s)/Mpc .

Immediatemeasurementby LLR

Indirect derivationfrom the Earth’srotation 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

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.

Rate of the lunar orbital increase

Estimating the Lambda-Term from the Local Planetary Dynamics - 9

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.