arXiv:1501.01919v1[physics.gen-ph]4Jan2015
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Paradoxes of cosmological physics
in the beginning of the 21-st century
Yurij Baryshev∗
Astronomical Institute, Saint Petersburg State University,
Saint Petersburg, 198504, Russia
∗E-mail: yubaryshev@mail.ru
In the history of cosmology physical paradoxes played important role for development
of contemporary world models. Within the modern standard cosmological model
there are both observational and conceptual cosmological paradoxes which stimulate to
search their solution. Confrontation of theoretical predictions of the standard cosmological
model with the latest astrophysical observational data is considered. A review of
conceptual problems of the Friedmann space expending models, which are in the bases
of modern cosmological model, is discussed. The main paradoxes, which are discussed
in modern literature, are the Newtonian character of the exact Friedmann equation, the
violation of the energy conservation within any comoving local volume, violation of the
limiting recession velocity of galaxies for the observed high redshift objects. Possible
observational tests of the nature of the cosmological redshift are discussed.
Keywords: cosmological models; observational tests; standard model; paradoxes.
1. The standard cosmological model
Nowadays the expanding Big Bang cosmological model is generally accepted as the
standard cosmological model (SCM) for description of the structure and evolution
of the physical Universe (Peebles1
, Weinberg2
, Baryshev & Teerikorpi3
). SCM is
based on the geometrical gravity theory (general relativity) and uses the description
of all physical processes in expanding space. The fundamental assumptions of the
SCM are:
• Homogeneous and isotropic matter distribution in the expanding Universe
(ρ = ρ(t); p = p(t); gik
= gik
(t)) a
.
• General relativity is applicable to the whole Universe (gik; ℜiklm; T ik
(m+de)).
• Laboratory physics can be extended into the expanding space.
• Inflation in the early Universe is needed for flatness, isotropy and for initial
conditions of large scale structure formation.
aWe use main definitions and notations similar to Landau & Lifshitz4, so 4-dimensional tensor
indices are denoted by Latin letters i, k, l... which take on the values 0, 1, 2, 3, and the metric has
signature (+, −, −, −).
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1.1. Einstein’s cosmological principle
The fundamental basic element of the SCM is the Einstein’s Cosmological Principle,
which states that the universe is spatially homogeneous and isotropic at
enough ”large scales”. The term ”large scales” relates to the fact that the
universe is obviously inhomogeneous at scales of galaxies and clusters of galaxies.
The hypothesis of homogeneity and isotropy of the matter distribution in space
means that starting from certain scale rhom, for all scales r > rhom we can consider
the total energy density ε = ρc2
and the total pressure p as a function of time
only, i.e. ε(r, t) = ε(t) and p(r, t) = p(t) . Here the total energy density and the
total pressure are the sum of the energy densities for matter and dark energy :
ε = εm + εde, and p = pm + pde.
An ideal fluid equation of state p = γ̺c2
is usually considered for cosmological
fluid, where usual matter and dark energy have following partial equations of state:
pm = βεm with 0 ≤ β ≤ 1, and pde = wεde with −1 ≤ w < 0. Recently values
w < −1 also were considered for description the ”fantom” energy.
1.2. Expanding space paradigm
An important consequence of homogeneity and isotropy is that the line element
ds2
= gikdxi
dxk
may be presented in the Robertson-Walker form:
ds2
= c2
dt2
− S2
(t)dχ2
− S2
(t)I2
k (χ)(dθ2
+ sin2
θdφ2
) , (1)
where χ, θ, φ are the ”spherical” comoving space coordinates, t is synchronous time
coordinate, and Ik(χ) = (sin(χ), χ, sinh(χ)), corresponding to curvature constant
values k = (+1, 0, − 1) respectively. S(t) is the scale factor, which determines the
time dependence of the metric.
The expanding space paradigm states that the proper (internal) metric distance
r to a galaxy with fixed co-moving coordinate χ from the observer is given by
relation r(t) = S(t) · χ and increases with time t as the scale factor S(t).
Note that physical dimension of metric distance [r] = cm , hence, if physical
dimension [S] = cm, then χ is the dimensionless comoving coordinate distance. In
direct mathematical sense χ is the spherical angle and S(t) is the radius of the sphere
(or pseudosphere) embedded in the 4-dimensional Euclidean space. It means that
the ”cm” (the measuring rod) itself is defined as unchangeable unit of length in the
embedding 4-d Euclidean space.
It is important to point out that the hypothesis of homogeneity and isotropy
of space implies that for a given galaxy the recession velocity is proportional to
distance (exact linear velocity-distance relation for all RW metrics Eq. (1)):
Vexp(r) =
dr
dt
=
dS
dt
χ =
dS
dt
·
r
S
= H(t)r = c
r
rH
(2)
where H = ˙S/S is the Hubble constant (also is a function of time) and rH = c/H(t)
is the Hubble distance at the time t.
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1.3. Geometrical gravity theory
The Einstein-Hilbert field equations of the general relativity have the form:
ℜik
−
1
2
gik
ℜ =
8 π G
c4
(T ik
(m) + T ik
(de)) , (3)
where ℜik
is the Ricci tensor, T ik
(m) is the energy-momentum tensor (EMT) of the
matter, which includes all kinds of material substances, such as particles, fields,
radiation, and T ik
(de) is the EMT of dark energy, in particular, the cosmological
vacuum is described by T ik
(vac) = gik
Λ, where Λ is Einstein’s cosmological constant.
Usually T ik
(m) and T ik
(de) are considered as independent quantities, though there are
models with interacting matter and dark energy5
.
It is important to note that T ik
(m) does not contain the energy-momentum tensor
of the gravity field itself, because gravitation in general relativity is a property of
space and is not a material field. This is why there is no such concepts as gravity
force and energy of gravitational field in general relativity.
A mathematical consequence of the field equations (Eq. (3)) is that the covariant
derivative of the left side equals zero (due to Bianchi identity), so for the right side
we also have
(T ik
(m) + T ik
(de)) ; i = 0 . (4)
The continuity equation (Eq. (4)) also gives the equations of motion for the considered
matter.b
1.4. Friedmann’s equations
In comoving coordinates the total EMT has the form T i
k = diag(ε, −p, −p, −p) and
for the case of unbounded homogeneous matter distribution given by metric Eq.
(1), the Einstein’s equations (Eq. (3)) are directly reduced to the Friedmann’s
equations (FLRW model). From the initial set of 16 equations we have only two
independent equations for the (0,0) and (1,1) components, to which we must add
the continuity equation (Eq. (4)) which has the form 3 ˙S/S = − ˙ε/(ε + p).
Using the definition of the Hubble constant H = ˙S/S , the Friedmann’s equations
get the form:
H2
−
8πG
3
̺ = −
kc2
S2
, or 1 − Ω = −Ωk , (5)
and
¨S = −
4πG
3
̺ +
3p
c2
S , or q =
1
2
Ω 1 +
3p
̺c2
, (6)
b As it was emphasized by Landau & Lifshitz4 the Eq.(4) is not a conservation law of the
energy-momentum of the particles plus gravity system, because Tik
(m)
does not contain the energymomentum
of the gravity field itself.
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where Ω = ̺/̺crit, ̺crit = 3H2
/8πG, Ωk = kc2
/S2
H2
q = − ¨SS/ ˙S2
, and Ω, p, ̺ are
the total quantities, i.e. the sum of corresponding components for matter and dark
energy. Solving the Friedmann’s equations one finds the dependence on time the
scale factor S(t) or the metric distance r(t), which is the mathematical presentation
of the space expansion.
1.5. Fundamental conclusions of the SCM
There are many explained astrophysical phenomena in the frame of the SCM, such
as cosmological redshift of distant objects, cosmic microwave background radiation,
large scale structure formation, chemical composition of matter and other. The
main observational conclusions of the SCM are:
• Cosmological redshift (1+z) = λ0/(λ1) = S0/S1 , and the linear velocity-distance
relation Vexp = H × r is the consequence of the space expansion r(t) = S(t) × χ
of the homogeneous Universe.
• Cosmic microwave background radiation is the result of the photon gas cooling
in the expanding space T (z) = T0(1 + z).
• Small anisotropy ∆T/T (θ) of the CMBR is determined by the initial spectrum
of density fluctuations which are the source of the large scale structure of the
Universe.
• The physics of the expanding Universe is described by the LCDM model which
predicts the following matter budget at present epoch: 70% of unobservable in
lab dark energy, 25% unknown nonbaryonic cold dark matter and 5% ordinary
matter . Visible galaxies contribution is less than 0.5%.
2. Observational puzzles of the SCM
In spite of evident successes of the SCM there are also observational facts which
present severe problems for the SCM. We emphasize here several such problems
which were discussed recently in the literature.
(1) Absurd Universe. The visible matter of the Universe, the part which we can
actually observe, is a surprisingly small (about 0.5%) piece of the predicted
matter content and this looks like an ”Absurd Universe”6
. What is more, about
95% of the cosmological matter density, which determine the dynamics of the
whole Universe has unknown physical nature. Turner7
emphasized that modern
SCM predicts with high precision the values for dark energy and nonbaryonic
cold dark matter, but ”we have to make sense to all this”.
(2) The cosmological constant problem. One of the most serious problem of the
LCDM model is that the observed value of the cosmological constant Λ is about
120 orders of magnitude smaller than the expectation from the physical vacuum
(as discussed by Weinberg8
and Clifton et al.9
). In fact the critical density of
the Ω = 1 universe is ̺crit = 0.853 × 10−29
g/cm3
, while the Planck vacuum has
̺vac ≈ 10+94
g/cm3
.
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(3) The cold dark matter crisis on galactic and subgalactic scales. There are number
of problems with predicting behavior of baryonic and nonbaryonic matter
within galaxies. It was discussed by Tasitsiomi10
and Kroupa11
that there
are discrepancies between observed and predicted galaxy density profiles (the
cusp problem), small number of observed satellites galaxies (missing satellites
problem), and observed tight correlation between dark matter and baryons in
galaxies, which is not expected within LCDM galaxy formation theory.
(4) The LCDM crisis at super-large scales. The most recent observational facts
which contradict the LCDM picture of the large scale structure formation,
come from: the SDSS and 2dF galaxy redshift surveys (Sylos Labini12
), problems
with observations of baryon acoustic oscillations (Sylos Labini et al.13
),
existence of structures with sizes ∼ 400 Mpc/h in the local Universe (Gott
et al.14
, Tully et al.15
)and ∼ 1000 Mpc/h radial structures in the very deep
galaxy surveys (Nabokov & Baryshev16
), existence of qso structures with sizes
∼ 500 Mpc/h (Clowes et al.17
, Einasto et al.18
), alternative interpretation of
the shape of the CMBR fluctuations correlation function (Lopez-Corredoira &
Gabrielli19
), lack of CMBR power at angular scales larger 60 degrees and correlation
of CMBR quadrupole and octopole with ecliptic plain ( Copi et al.20
),
see also recent review by Perivolaropoulos21
.
3. Conceptual paradoxes of the SCM
The existence of the mentioned above observational puzzles in the SCM interpretations
of the astrophysical data rises a question: Does the contemporary standard
cosmological model present the ultimate physical picture of the Universe?
Philosophical, methodological and sociological aspects of the development of the
science on the whole Universe was recently analyzed by Lopez-Corredoira22
, who
emphasized the important role of alternative ideas in cosmology, though usually
they have small funding in modern scientific society. The mathematical and physical
basis for the SCM and alternatives was considered by Baryshev & Teerikorpi3
.
As it is natural for progress in physics we should carefully analyze the fundamental
assumptions laying in the basis of the physical theories. In the Sec.1 we
have formulated several fundamental assumptions in the basis of SCM which have
led us to the serious observational puzzles (Sec.2). As was emphasized by Turner7
for making new cosmology one has to answer a new set of questions and the future
world model will reveal deep connection between fundamental physics and cosmology:
”There may even be some big surprises: time variation of the constants or a
new theory of gravity that eliminates the need for dark matter and dark energy” 7
.
Intriguingly, besides the mentioned above observational puzzles there are several
deep conceptual problems in the foundation of the SCM. Their solution could open
the door to construction more firmly established future cosmology. Below we present
several such conceptual difficulties/paradoxes of the SCM, which already have been
discussed in the literature:
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• Gravitation field energy paradox: in the framework of the Einstein’s geometrical
gravity theory (General Relativity) there is no physical concept of the
energy-momentum density of the gravitational field (also there is no physical concept
of the energy quanta of the gravitational field), though field energy exists
for all other fundamental physical interactions.
• 1st Harrison’s paradox: physics of space expansion contains such puzzling
phenomena as continuous creation of vacuum, violation of energy conservation,
violation of limiting velocity by receding galaxies.
• 2nd Harrison’s paradox: the cosmological redshift in expanding space is not
the Doppler effect, but it is a new physical phenomenon which does not tested
in the lab, the global gravitational cosmological redshift should be taken into
account.
• Hubble-de Vaucouleurs’ paradox: in the expanding space the linear Hubble
law is the fundamental consequence of the homogeneity, however modern observations
reveal existence of strongly inhomogeneous fractal large-scale galaxy
distribution at scales at least up to 100 Mpc, while the linear Hubble law starts
from 1 Mpc, i.e. just inside inhomogeneous structure.
3.1. Gravitation theory
Though Einstein’s general relativity solved the old gravitational paradox of the
Newtonian gravity theory, the geometrical gravity leads to the new form of the
paradox at a deep conceptual level - absence of the energy-momentum tensor of the
gravitational field.
According to Landau & Lifshitz4
(paragraph 101 ”The energy-momentum
pseudotensor”) the Einstein’s field equations (Eq.(3)) should contains ”the fourmomentum
of matter plus gravitational field; the latter is not included in the expression
for T ik
”. This is why Landau & Lifshitz4
claimed that the continuity
equation (Eq. (4): (T ik
) ; = 0 ) ” does not generally express any conservation law
whatever”.
The ”pseudo-tensor” character of the gravity field in GR has remarkable history
(see e.g. Baryshev23
) and had been discussed from time to time for a century, causing
surprises for each new generation of physicists. Rejection of the Minkowski space
inevitably leads to deep difficulties with the definition of the energy-momentum for
the gravitational field and its conservation.
However this conceptual problem can be solved within non-metric Feynman’s
Field Gravitation approach24
, which is the theory of the symmetric second rank
tensor field (gravitational potentials) in Minkowski space, and which unite gravity
with the other fundamental forces of nature (consistent development of the FG see
e.g. Sokolov & Baryshev25
and Baryshev23
). As Feynman24
emphasized: ”geometrical
interpretation is not really necessary or essential for physics” (p.113).
The Feynman’s Field Gravitation (FG) theory contains the concept of gravitational
EMT and conservation of the total EMT. The geometrical gravity theory
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may be considered as an approximation of the relativistic quantum field gravity
(like geometrical optics for electrodynamics). There are achievable in near future
experiments and astrophysical observations which can distinguish between GR and
FG, like additional scalar gravitational radiation (Baryshev26
).
3.2. Physics of space expansion
Mathematically space expansion is a continuous increasing with time of the distance
r(t) between galaxies. It is given by relation r(t) = S(t) · χ where S(t) is the scale
factor from Eq. 1. But what does space expansion mean physically?
Cosmological physics of the expanding space is essentially different from the
lab physics and even contains deep paradoxes which should be studied carefully27
.
Physically expansion of the universe means the continues creation of space together
with physical vacuum. Real Universe is not homogeneous, it contains atoms, planets,
stars, galaxies. In fact bounded physical objects like particles, atoms, stars and
galaxies do not expands. So inside these objects there is no space creation. This is
why the creation of space is a new cosmological phenomenon, which has not been
tested yet in physical laboratory.
The first puzzling feature of the space expansion physics is that the Friedmann’s
equations Eq. (6, 5) in terms of the metric distance r(t) = S(t) · χ get the exact
Newtonian form:
¨r = −
GMg(r)
r2
, and
V 2
exp
2
−
GM
r
= const , (7)
where Mg(r) = −4πG
3 ̺ + 3p
c2 r3
is the gravitating mass of a ball with radius r(t).
So according to general relativity the dynamics of the whole universe is determined
by the exact Newtonian acceleration and Newtonian kinetic plus potential energy
conservation (here velocity of light c does not change the Newtonian character of
the equations).
The second puzzling fact of the space expanding universe is that in the case of
the equation of state p = γ̺c2
the mass (energy) of any local ball is changing with
time as:
Mg(r) = −
4πG
3
(1 + 3γ) ̺r3
∝ S−3γ
(t) . (8)
For example for photon gas γ = 1/3 and the mass-energy of the initially hot radiation
is cooling proportional to the scale factor S(t).
In cosmology Eq.(8) gives us a possibility to calculate of how much the energy
increases or decreases inside any finite comoving volume but it does not tell us
where the energy comes from or where it goes. As Harrison emphasized: ”The
conclusion, whether we like it or not, is obvious: energy in the universe is not
conserved” (Harrison28
, p.276).
Another puzzling consequence of the Friedmann’s equations Eq. (7) is that in
exact general relativistic expansion dynamics of the universe there is no relativistic
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effects due to the velocity of a receding galaxy. The expansion velocity is larger
than the velocity of light for distances larger than the Hubble distance: Vexp > c
for r > RH, where RH = c/H (see also Eq. (2)).
3.3. The nature of cosmological redshift
In the Sandage’s list of the ”23 astronomical problems”29
the number fifteen (the
first in the extragalactic section) sounds intriguingly:”Is the expansion real?”.
In fact the literature on the SCM contains acute discussion on the nature of
the cosmological redshift3
, subject which constantly produces ”common big bang
misconceptions” or the ”expanding confusions”. A summary of such discussions was
done by Francis et al.30
who confronts Rees & Weinberg claim: how is it possible for
space, which is utterly empty, to expand? How can nothing expand? The answer is:
space does not expand. Cosmologists sometimes talk about expanding space, but they
should know better, with the state by Harrison28
: expansion redshifts are produced
by the expansion of space between bodies that are stationary in space.
In mathematical language within FLRW space expanding model the cosmological
redshift is a new physical phenomenon where due to the expansion of space
the wave stretching of the traveling photons occurs via the Lemaitre’s equation
(1 + z) = λ0/λ1 = S0/S1, which is different from the familiar in lab the Dopplers
effect. One can also see this if compare relativistic Doppler and cosmological
FLRW velocity-redshift V(z) relations. The relativistic Doppler relation has the
form VDop(z) = c(2z + z2
)/(2 + 2z + z2
) and the velocity always less than c, while
expanding space velocity Vexp can be arbitrary large3
.
It is important to note that on the verge of modern technology there are direct
observational tests of the physical nature of the cosmological redshift. First crucial
test of the reality of the space expansion was suggested by Sandage31
, who noted
that the observed redshift of a distant object (e.g. quasar) in expanding space must
be changing with time according to relation dz/dt = (1 + z)H0 − H(z). In terms of
radial velocity, the predicted change dv/dt ∼ 1 cm s−1
/yr. This may be within the
reach of the future 42m ELT telescope32
.
Even within the Solar System there is a possibility to test the global expansion
of the universe. According to recent papers by Kopeikin33,34
the equations
of light propagation used currently by Space Navigation Centers for fitting range
and Doppler-tracking observations of celestial bodies contain some terms of the
cosmological origin that are proportional to the Hubble constant H0. Such project
as PHARAO may be an excellent candidate for measuring the effect of the global
cosmological expansion within Solar System, which has a well-predicted frequency
drift magnitude ∆ν/ν = 2H0∆t ≈ 4×10−15
(H0/70kms−1
Mpc−1
)(∆t/103
s), where
H0 is the Hubble constant ∆t is the time of observations. In the case of the nonexpanding
Universe the frequency drift equals zero.
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3.4. Fractality of large-scale galaxy distribution
Modern observations of the 3-dimensional galaxy distribution, obtained from huge
redshift surveys (such as 2dF and SDSS), demonstrate that at least for interval
of scales 1 ÷ 100 Mpc/h there is a power law relation between the galaxy number
density n(R) and the radiuses of spheres R, so that n(R) ∝ R−γ
(see reviews by
Sylos Labini12
and Baryshev & Teerikorpi3
). Such power law behavior is known
as the de Vaucouleurs law35
. Note that the power law correlation function
is the characteristic feature of the discrete stochastic fractal structures in physics
(phase transitions, strange attractors, structure growth) and has clear mathematical
presentation (e.g. Gabrielli et al.36
).
At the same time modern observations of the Hubble law in the local Universe
based on Cepheid distances to local galaxies, supernova distances, Tully-Fisher
distances and other distance indicators, demonstrate that the perfect linear Hubble
law is well established within the same distance interval of scales 1 ÷ 100 Mpc/h (e.
g. Sandage37
, Baryshev & Teerikorpi3
).
A puzzling conclusion is that the Hubble law, the strictly linear redshift-distance
relation, is observed just inside strongly inhomogeneous galaxy distribution, i.e.
deeply inside fractal structure at scales 1÷100 Mpc/h. This empirical fact presents
a profound challenge to the standard model where the homogeneity is the basic explanation
of the Hubble law, and ”the connection between homogeneity and Hubble’s
law was the first success of the expanding world model” (Peebles et al.38
).
However, contrary to this expectation, modern data show a good linear Hubble
law even for nearby galaxies. It leads to a new conceptual puzzle that the linear
Hubble law is not a consequence of the homogeneity. A cosmological model which
can unite the Hubble law with fractality of matter distribution and the cosmic
microwave background radiation was suggested by Baryshev39
.
4. Conclusions
The positive role of the physical paradoxes in the science of the Universe is well
known from the history of cosmology3
. Even the known phenomena may have different
interpretations, each corresponding to a specific choice of the basic framework
able to explain key observations. Theoretical physics is a developing subject and new
physics may offer a variety of new cosmological applications. Finally, observations
and theoretical understanding are always limited, hence even a quite credible world
model has its limitations, too (in current cosmology 99.5% of the needed mass has
unknown nature). These emphasize the importance of crucial observational tests as
the only safe way to decide between alternative cosmological ideas.
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