CC BY
41
UDC 524.8
Fractal Properties of the Universe A. A. Agapov*, I. K. Rozgacheva^
* Peoples' Friendship University of Russia 6, Miklukho-Maklaya str., Moscow, 117198, Russia
1 VINITI RAS 20, Usievicha str., Moscow, A-190, 125190, Russia Moscow State Pedagogical University, Moscow, Russia
The large-scale structure of the Universe is revealed to be characterized by a range of power-laws. The power-laws are evidences of fractality because they may be interpreted through a conception of the Universe as an assembly of self-similar space-time domains. We accept the hypothesis that the matter of the Universe is described by the scalar charged meson field possessing the rotary symmetry. On basis of the hypothesis, the fractal cosmological model with scale invariant Lagrange's field equation and Einstein's equation permitting physical explanation of these properties is constructed. The field energy densities (which are constant) and the space-time metrics of different domains differ in constant factors only. Therefore, the space-time domains are geometrically similar and evolve similarly. Fractal properties of initial cosmological density perturbations remain and lead to presence of the fractal properties of the Universe's large-scale structure which formed from them. The nonsingular, compacted, pulsating and doubly-connected cosmological model as a partial solution for the homogeneous, isotropic and flat case is constructed. A background radiation power spectrum has been computed. The spectrum is shown to be close to the observable angular power spectrum of the SDSS-quasar distribution on the celestial sphere.
Key words and phrases: quasars, large-scale structure, fractal dimension, complex field, rotary symmetry, fractal properties of the large-scale structure, fractal cosmological model, background radiation.
1. Introduction
The large-scale structure of the Universe is the structure of the galaxy and quasar distribution in whole observable Universe's volume. The structure is sponge-like and composed of regions of a higher galaxy number density (filaments and planes of galaxies and galaxy clusters) bordering huge voids. Scales of these structures are equal to tens and hundreds of megaparsecs (Mpc).
Investigation of geometrical properties of the large-scale structure of the Universe through galaxies, galaxy clusters, quasars and the CMB temperature anisotropy is an important area at present for understanding of galaxy and Universe evolution. Estimation of distances to galaxies is complicated by low quality of spectra, by uncertainty of galaxy peculiar motion and by dependence on a cosmological model determining distance as a function of redshift. Consequently, investigation of three-dimensional galaxy distribution gives not quite reliable results. Therefore, study of statistical and topological properties of the large-scale structure through two-dimensional galaxy distribution is relevant despite obvious mistakes related to overlapping of clustering areas.
The purposes of the present work are:
- the data processing on the quasar distribution on the celestial sphere according to the SDSS DR7 catalogue and on the CMB temperature anisotropy according to WMAP-7;
- revealing of the fractal properties of the Universe's large-scale structure;
— construction of the fractal cosmological model permitting physical explanation of these properties;
— computation of a background radiation power spectrum within the fractal model framework.
Received 11th June, 2012.
The authors thank the FCPK grant 16.740.11.0465 for support of the work.
The SDSS DR7 catalogue [1] containing 105,783 quasars with redshifts 0.0645 < z < 5.4608 was used for investigation of the quasar distribution. The catalogue was compiled by a wide-angle, narrow-angle and pencil beam surveys. For the purpose of the present work the area of the celestial sphere with the equatorial coordinates 9h < a < 16h, 0° < 5 < 55° covered by the wide-angle survey was chosen. The main fractal properties of the large-scale structure are represented in Section 2. These properties may be interpreted through a conception of the Universe as an assembly of self-similar space domains.
In Section 3 it is shown that the hypothesis of Gaussian (thermal) spectrum of initial cosmological density perturbations implies they possessed fractal properties, their correlation function was a power-law. The perturbations lead to the Universe's large-scale structure formation due to gravitational instability. However, the equations of the General Theory of Relativity are not scale invariant. Due to this, the fractal properties may not remain during gravitational fluctuations evolution, generally.
The hypothesis of the rotary symmetry of the charged scalar meson matter field (complex field) ^ which then possesses the form ^ = is considered in Sec-
tion 4. The Universe is composed of space-time domains related by the discrete scaling: ^ o / 7, ^ o 7</5. On basis of this hypothesis the cosmological model with scale invariant (i.e. invariant under the scaling) Lagrange's field equation and Einstein's equation is constructed. The general solution of the equations is derived. The field energy densities E and E (which are constant) and the space-time metrics
gmn and gmn differ in a constant factor only. Therefore, these space-time
domains are geometrically similar and evolve similarly. The fractal properties of the initial density perturbations remain and lead to presence of the fractal properties of the Universe's large-scale structure.
The partial solution of the equations in homogeneous and isotropic case is adduced in Section 5. An equation for the scale factor is derived and solved. The cosmological model is nonsingular, the Universe is found to be compacted, pulsating and doubly-connected.
In Section 6 the anisotropy of a background radiation within framework of this model is considered. When photons pass through the space domains their energies change due to gravitational frequency shift. An observer receives them and detects spots with different brightness on the celestial sphere. The power spectrum of the brightness anisotropy is calculated and revealed to be close to the observed angular power spectrum of the SDSS-quasar distribution on the celestial sphere. Only qualitatively it conforms to the angular power spectrum of CMB.
2. Fractal Properties of the Large-Scle Structure
We performed the SDSS-quasars distribution analysis in framework of the standard cosmological model with parameters: H0 = 70 km, ■ s-1 ■ Mpc-1 is the Hubble constant, Om = 0.3, Oa = 0.7 are the dimensionless density parameters of dust and A-term respectively [2,3]. Cosmological distance to a quasar with redshift z (comoving distance) is determined by the formula [4]:
Distances measured in the Hubble distance c/H0 are used below.
A variation of SDSS-quasars number density in an element of comoving volume of a spatial cone of the SDSS catalogue is shown on fig.la (the number density is normalized to a maximal value). As one can see, there was an epoch of high galaxy activity in the Universe evolution at redshift range 0.35 < z < 2.30 which has been chosen for the further investigation. For decrease of significance of evolution effects
this range has been divided into six layers, possessing qualitatively resembling shape of the luminosity function. This implies that the same common properties of the large-scale structure are displayed in every layer.
o l z 2 3 4 5 6
1a) 1b)
Figure 1. a) Number density of SDSS-quasars in an element of comoving volume; b) The correlation function of SDSS quasars for the layer 1.04 < z < 1.36
1. The correlation dimension computation methods are given in papers [5-7], for example. This value characterizes quasar clumping degree and difference of the quasar distribution from a homogenous and isotropic one. The dependence of a quasar number N(r) in a sphere on its radius r for the chosen redshift range is described by a power-law:
N (<r) - rdc, (1)
where the exponent (correlation dimension) is equal to dc = 2.17 [2,3].
The power-law (1) is usually considered as an indication of fractality of a spatial sources distribution. Similar N (r) dependence is typical for galaxies (dc « 1.15 ^ 2.25) and it is a large-scale structure common law [7,8].
2. An analogous relation arises for two-dimensional quasar distribution on the celestial sphere between a number of quasars with angular distances less than tf and sin (tf/2):
N (<tf) - (sin 0 , (2)
where a = 1.49 ^ 1.56 for different redshift layers [5,6].
3. For each layer, the luminosity function parameters and the angular correlation function w (tf) of the observable quasar distribution as well as the angular power spectrum m for its expansion in spherical functions series have been computed using methods described in [2,3,9,10]. The power spectrum is usually used for comparison of large-scale structure formation models with observations because the power spectrum depends on an adopted cosmological model. This fact enables to estimate cosmological parameters.
Both the angular correlation function and the angular power spectrum are revealed to possess the similar shape in every layer. They may be described by the power laws
u (tf) - tf1-7, (3)
ui - F"-3, (4)
respectively with the layer-averaged parameters (7) = 2.08 and (7^) = 1.92. Here, I is a multipole moment number. Examples of the correlation function and of the power spectrum plot for the layer 1.04 < z < 1.36 are shown on fig.1b and fig.2a respectively [2,3].
4. CMB photons show us the Universe as it was at the recombination epoch. The WMAP experiment proves convincingly that CMB temperature angular fluctuations exist. The angular power spectrum for the expansion of the CMB temperature anisotropy in a spherical functions series is a power-law as well [2,3]:
Q — l-L74. (5)
The power spectrum plot is shown on fig.2b. It qualitatively corresponds to the power spectrum of the quasar distribution (4): u — I-1.08.
5. For estimation of quasar clump sizes the spherical wavelet transform [11] was used. The wavelet analysis permits to single out quasar clump areas: the wavelet coefficients are positive in these areas. Large quasar groups are discovered in papers [2, 3]. For study of fractal properties of the large quasar groups we should determine a number of groups Nc with a certain angular size $c. Their relation has been revealed to be a power-law:
Nc (#c) — (6)
These power-laws are evidences of fractality because they may be interpreted through a conception of the Universe as an assembly of self-similar space domains. These large quasar groups mark the domains. Let each i-th domain with a size n contains mi pairs of objects (galaxies or quasars). Self-similarity implies the sizes of domains and numbers of pairs form geometric progressions:
i—1 i—1 ri = qn-1 = roq , mi = pmt_i = mop ,
where i = 1, 2, 3,... and p > 1, q > 1 are constant numbers. The total number of galaxies in the sample and the radius of the sample volume are equal
N/2 N/2 N/2 _ 1
N = 2^mi = 2mo^2p%_1 = 2mo-—1— ~ 2moP12-1, rN/2 = roq12-1.
i=i i=1 p 1
Then we obtain the relation (1):
ln p
ln q ^ ln p
ln q
N = 2mo(ln' — (rN/2)
3. Fractal Spectrum of Initial Cosmological Density
Perturbations
We accept a hypothesis that these fractal properties are consequences of fractal properties of initial matter density perturbations which further led to star, galaxy and cluster formation due to gravitational instability [12,13]. Now we observe traces of these fractal properties through quasars.
This interpretation follows from the hypothesis of Gaussian (thermal) spectrum of the initial density perturbations. Let's consider a spherical volume V containing mass M in a continuous medium. Probability of appearance of thermal density fluctuation near this mass is defined by the formula [14,15]
W
exp
C-v ~2
(—Y_—(—\ (
\t) - 2kT\dp)TV
dP\ (5V\2_ Rmn dp)t\ V ) kT
where p is medium density, P is pressure, cv is medium heat capacity at constant volume, 5T and 5V are independent fluctuations of temperature and volume respectively, Rmin is minimal work necessary for reversible removal of a mass SM for a distance Sr in the gravity field of the mass M. In case of radial displacement the work is equal to
M -SM 4n
Rmin ~ G-¡5— 5r = —Gpr • 5M • Sr
r 2 3
in Newtonian approximation, where 5M = 4tt pr25r is a mass of a spherical layer. In this case, probability of a thermal fluctuation is equal
W =
1
2tTAt A,
exp
1
'6T\2
2AT2 VTJ
1
2Ar 2
( 7
where the variances are equal to
AT 2 = c,
A. ^ f 12np
\ kT
dP) t
8^Gpr 5
i
to
The root-mean-square relative density fluctuation (fluctuation spectrum) are equal
A,
\
dP\ dp) T
o SnGp 5 r3 + r5 9
-1/2
. (7)
At spatial scales for which pressure gradients are important, i.e. when the first term in the brackets dominates, we have the "white noise" spectrum (Zel'dovich-Harrison
At large scales for which gravity effects are important,
-2,5
spectrum): Ap ~ r 1,s.
i.e. when the second term dominates, one has the following spectrum Ap ~ r The spectrum (7) is scale invariant because the fraction doesn't change under scale transformations. This thermal fluctuations spectrum is an example of fractal spectrum.
The root-mean-square relative density fluctuation is an estimate of the correlation function according to the correlation function definition [10]: £ ~ Ap. For a random fluctuation average number of neighbour fluctuations within distance less then may
be estimated as
r
{N) « 4tï {n) j (1 + 0 f2df,
where n is a mean fluctuations number density. In case of fluctuations clumping and £ ^ 1 there are fractal laws like (1): {N) ~ r1'5 for the white noise and {N) ~ r0'5 at large scales.
The observable correlation dimension value dc ~ 1,15 + 2,25 may follow from the spectrum
\
(( H )}
therefore, it permits not only the white noise spectrum.
Thereby, the fractal laws (1)-(6) are expected to exist in Newtonian approximation. However, it is not quite so in the general theory of relativity because Einstein's tensor is not invariant under scale transformation of the Riemannian space-time [16,17]. Generally, if the large-scale structure evolution is described by Einstein's gravity theory the fractal properties may not conserve, even if the initial fluctuations had the thermal spectrum.
4. General Solution of Lagrange's and Einstein's Equations for the Complex Field with the Rotary Symmetry
We consider a hypothesis that the matter of the Universe is described by the charged scalar meson field (complex field) which possesses the rotary symmetry [12,13]:
^* = = const, (8)
(where the asterisk denotes complex conjugation and ^ is the field amplitude related to the field charge Q ~ ). The dynamic system of gravity and complex ^ fields is described by Einstein-Hilbert action within general relativity framework:
* = -- ^"S.
where G is Newton's gravity constant, R is scalar curvature, g < 0 is determinant of the metric tensor. We use the following form of complex field Lagrangian:
L = L (gik -Uo^*\ (9)
hc \ dx% dxk ) '
where U is a constant field potential parameter, h is Planck's constant, c is light velocity. The field Lagrange's equation is
1 a H)=-(10)
y—j dxk \ dxl J d^
In Einstein's equation
-
Rfc - W = KTk (11)
energy-momentum tensor of the complex field is equal
Tk = d^ dL + dr dL _ 6kL = 2. ki( dSdS* + didr\ _ 6kL
1 dxfaf dxi d(ÇÉl) 1 he \dxl dxl dxl dxl J 1 '
I dxk I I dxk I
where Rk is Ricci tensor, k = 8ttG / c4 is Einstein's gravity constant, 5k is the Kronecker delta.
It's easily seen that a field of the form of
f = ^eiLp, f* = ^e-iLp, (12)
satisfies the condition (8), where field phase p (x*) is a differentiable function. Then the equation (10) is satisfied by the general solution
i ( dxl dxk + dxl k + dxk ^ , ()
r
U0 V dx% dxk dxl
1 dV ( im
kl .
■1 ^ (g+aA ,
U0 dxkdxJ \ dxm J '
dp
where derivative and covariant vector m must satisfy equations ox1
gik ^ dn¡ = Uo, gik( =0,
ax% axK \dx%) ,k
(14)
0
ai;k = 0, aia1 = -3Uo, °—a% = 0.
ox1
Covariant ai and contravariant ak vectors satisfy equations:
°ai = -3 °2,-P °ak = 3ak °2,-P (15)
dxl dx%dxv dxl dxkdxl
This general solution contains isotropic and anisotropic solutions because the derivative may depend on direction.
The Lagrangian (9) is equal to 0. The Ricci tensor and the energy-momentum tensor for this solution are equal to:
fí = d!k_ dTk + r _ =±i d2y da1 d2y dal\ +
xl xk + ik lm il km uAdxidxkdxl dxidxldxkj
1 ( a2v a2v a2v \ ( inav , A™
+ U02 \dxldxk dxmdxl dxldxl dxmdxk ) v dxn +a >a Tik =
he dx% dxk o p
Functions ——, a,i, a1 are derived from equations (11) and (15).
ox1
The energy density is constant and positive:
*=¿ as as*+uo*-)=^ > 0.
Therefore, the solution (12)-(13) corresponds to a stationary field condition.
Let the Universe is composed of space-time domains related by the discrete scaling:
^ o ^ /7, ( o 7(. (16)
The Christoffel symbols and the Ricci tensor Rik don't change under this transformation. Vector ai, metric tensor, enrgy density, mixed components of energy-momentum and Ricci tensors are multiplied by constant factors:
ai o 7 an, gik (ip) o 72pik (fy , (17)
E o 1 ^E, Rk o \^Rk, T? o 1 ^T?.
72 U0 72 U0 72 U0
Therefore, Einstein's and Lagrange's equations don't change. These domains are geometrically similar and evolve similarly. The fractal properties of the initial density perturbations remain and lead to presence of the fractal properties of the Universe's large-scale structure.
The phase path of the fields ip and ip* is a circle (8):
pi* = p12 + p22 = p = h + iip2, p* = h — iip2.
The function ( is a degree of rotation round the circle. Length of a circle arc i.e. an interval of a set {ipi,ip2} is equal
dF2 = (dpi)2 + (diP2)2 = dipdip* = *2^ dxidxk.
The relation between the phase space interval dF and the space-time interval d s is
(18)
d F2 =
4
Uods2 — (a J^ + ak J1) dxidxk
The first equation (15) has the following solution:
d(
ai = — 3—- + di,
axi
(ddi \
where di is a constant vector f —i = 0 j. The expression (18) shows that the vector
di may be chosen so that the phase space interval is proportional to the time interval: d F ~ di. Therefore, the solution (12)-(13) may describe a time-pulsating cosmological model. As energy density of the system is finite this model must be nonsingular [12,13].
5. Partial Solution for Homogeneous and Isotropic Case
Let's consider a partial solution corresponding to a homogeneous, isotropic and
flat case
dip
a (
a xi a x2
Mixed Ricci tensor components and scalar curvature are equal:
H = J1 Ids 2 = <?dt 2 — a2
dx3 dx
(dx1)2 + (dx2)2 + (dx3)2
№ = 1
3( +3( V a ) t \ a /
1 2 3 1
K1 = K2 = K3 =
'01) + 3Î at)
^ a J t U/
2
2
R = —1 c2
61 01) + 12 f—N 2
a /1
where a is a scale factor of the model, index denotes partial derivative with cosmolog-ical time. We may add a total derivative with time of any function to the Lagrangian:
1 f.kdfdi>*TT^„,*\ , dF
1 ( ikd^ d^* A dF
and Einstein's equation (11) with the energy-momentum tensor
T = 2^dF
Tik , ^ r> O U 9ik j,
h c ox 0xK di comes to the following two equations:
3 f - )2 = 2^ ^ ( ^ V -nC2 dF,
Va/ he \&t J di
2
aA ,o/aA21 = (M2 — ^dF (19)
he ^ a2 \dx ) K dt . (19)
Lagrange's equation (10) turns into two equations:
M2 -1 (M2 =Uo,
\c dt ) a2 \dx ) '
ldV 3dV + 3 a, ^ = 0. (20)
c2 dt2 a2 dx2 c2 a dt
Four equations (19)-(20) determine four functions: a, , , ° ^ . Equations (19)
d d x d x d
and the first equation (20) lead to the equation determining scale factor a:
(T), + 2 - = (21)
The hyperbolic solution of this equation corresponds to the case of zero field La-grangian and dF = 0: a = a0 J cosh i-j, where t = . If we choose
d F 1
—— = — U0^2 the equation has the periodic solution d h
a = a0\l cos ( — + f ) (22)
with period 2kt. The scale factor (22) turns into zero at the moment t* when — + < =
T
K ft*
— ± Kn. The solution (22) is not singular within the interval 0 < — < 2k if — > 2k. 2
k
This condition permits to choose the phase: — ±Kn — < > 2k. Therefore the model is not singular if
3 K
<< — — ±Kn. (23)
a
The equation (21) has the integral for the periodic solution (22) that is a map of the phase path (8) in the space-time:
00V- 4r2 (01 )2 = 1.
a )
a )
Derivatives of the phase coordinate h are equal:
(th) "iU°
( ^ ) +1
a
2
1 Tn2 = 1U
a2\dx 12U0
( a0 ) - 3
One can define the metric tensor in the form analogous to the general definition (13) through these expressions. Further, the expressions imply that the parameter a0 is a maximal scale factor value for the solution (22): a ^ ~^a0. The comoving radial coordinate of the horizon is equal:
/^ ■ 2).
where F (^, 2) is an elliptic integral of the first kind possessing recurring values with period 2ttt.
Since the horizon comoving radial coordinate values are repeated, a model with pulsating space-time corresponds to the solution (22). This model is compacted, i.e. the total space volume is finite and the evolution in time is a periodic process of space expansion and contraction. In the presence of the phase restriction (23) the space contracts to minimal nonzero volume. The two-dimensional analogy of such spacetime is a torus with variable thickness where parallels are lines of time (lines of constant space coordinates) and meridians are space coordinate lines. Analogous compacted model has been constructed in the paper [18] and possible astrophysical consequences of space volume finiteness are discussed there. It has been showed there that dynamical entropy of the complex field is increasing during space pulsating [12,13].
6. Anisotropy of a Background Radiation
Light signals transfer along isotropic geodesics of space-time (13). An isotropic vector satisfies equations:
PiP1 = 0, pi;k = 0.
Under solution (13) these equations are satisfied by the vector
Pi = + *. (24)
Under the discrete scaling (16) this vector transforms by the rule:
Pi o 7pi. (25)
Therefore, an isotropic vector remains isotropic under the scaling.
Self-similarity of space-time domains described by the solution (13) permits consideration of an assembly of such domains because transition from any domain to another resolves itself into dilatation or compression of an interval. Directions of isotropic geodesics don't change. Let's consider transfer of photons of a background radiation within such fractal structure. We take into account gravitational influence of every domain only and leave out of account direct interaction with substance (absorption and
scattering). In this case, energy of transferring photon changes due to gravitational frequency shift. Let the background radiation is homogeneous and isotropic, and an observer receives photons which passed through the fractal structure. Brightness differs from one domain to another because energy of photons changes being multiplied by a scaling factor for each domain according to the expression (25). Therefore, the observer notices spots of different brightness in the distribution of the background radiation brightness on the celestial sphere.
An angular distance between a center of the j-th spot and any point of the spot is equal
" = arccos (cos 5j cos 5 cos (a — aj) + sin 5j sin 5),
where (5, a) and (5j ,aj) are declinations and right ascensions of the point and of the spot center respectively. The whole spot is described by Legendre polynomial of degree lj and its representation through spherical harmonics (the addition theorem):
Pi; (cos ") = 2r+T £ YlJm (j)Y^ (a, 5). (26)
j m= — lj
The multipole number lj and the spot's size "j are related by the expression k 180°
dj ~ — = ——. The polynomial (26) is of the bell shape in the range 0 ^ "j ^ dj
j j
with maximum equal to 1 when "dj = 0.
Summarized brightness distribution of the background radiation may be expressed now as
N
F (a, 5) = Y^ljPi U,a, 5), (27)
j=i
where the scaling factor jj takes into account change of photons' energy when leaving the j-th domain. For determination of the power spectrum of the background radiation brightness distribution anisotropy the function (27) should be expanded in a spherical harmonics series:
F (a, 6) = ^2 amiY^n (j, a, S).
The power spectrum is a function Ci = ^—r Ylm=—i \a-m.i|2. Using definitions (26)
2 t 1
and (27) we can determine the serial expansion coefficients ami:
4K N m=l aml = jj ^ Y-m (j).
j=l m= — l
The normalization condition for the spherical harmonics on a whole sphere is used here:
| YlJmY^dn = 5U].
In the simplest case of symmetric spots, the weight factors jj are proportional to
'
the spot's angular size, jj ~ dj & —, and determine the dependence of the expansion
j
N N 1
coefficients on the multipole numbers: ami ^^^jj ~ ^^ -p. In this case, the power
j=i j=i lj
200 Bulletin of PFUR. Series Mathematics. Information Sciences. Physics. No 1, 2013. Pp. 189-201 spectrum may be close to the power-law:
The model power spectrum (28) closely corresponds to the power spectrum of SDSS-quasars (4).
The main results of the present work are following.
- Revealing of fractal properties of the large-scale structure of the Universe which are described by the power-laws (1)-(6). The fractal dimension value for the large quasar group distribution in sizes is compared to that of polygonal path of Brownian particle (length distribution of segments). This analogy indicates that initial density perturbations from which large quasar groups arise, apparently, had a thermal spectrum.
- Interpretation of these properties through a conception of the Universe as an assembly of self-similar space-time domains related by the scaling (16).
- Construction on basis of this hypothesis of the fractal cosmological model with scale invariant (i.e. invariant under the scaling) Lagrange's and Hilbert-Einstein equations permitting physical explanation of these properties.
- Construction of the nonsingular, compacted, pulsating and doubly-connected cosmological model as a partial solution for the homogeneous, isotropic and flat case.
- Computation of a background radiation power spectrum within the fractal cos-mological model. The spectrum is shown to be close to the observable angular power spectrum of the SDSS-quasar distribution on the celestial sphere. It differs from the average power spectrum of the observable CMB anisotropy (WMAP-7). This fact will be a subject of investigation in further works.
Computation of the cosmological density perturbation evolution due to gravitational instability within fractal cosmological model framework will be the next step in this investigation.
1. Shneider D. P., Richards G. T., Hall P. B. et al. The Sloan Digital Sky Survey Quasar Catalog V. Seventh Data Release. — arXiv:1004.1167v1.
2. Rozgacheva I. K., Borisov A. A., Agapov A. A. et al. Fractal Properties of the Large-Scale Structure. — arXiv:1201.5554v2.
3. Fractal Properties of the Large-Scale Structure / I. K. Rozgacheva, A. A. Borisov, A. A. Agapov et al. // Nelineinii mir. — 2012. — Vol. 10, No 5. — Pp. 300-311. — In Russian.
4. Zel'dovich Y. B., Novikov I. D. The Structure and Evolution of the Universe. — Moscow: Nauka, 1975. — In Russian.
5. Agapov A. A., Rozgacheva I. K. Observational Fractal Properties of the Quasar Distribution According to SDSS Catalogue // Nelineinii mir. — 2011. — Vol. 9, No 6. — Pp. 384-390. — In Russian.
6. Rozgacheva I. K., Agapov A. A. Fractal Properties of SDSS Quasars. — arXiv:1101.4280.
7. Baryshev Y., Teerikorpi P. The Fractal Analysis of the Large Scale Galaxy Distribution // Bull. Spec. Astrophys. Obs. — 2006. — Vol. 59. — Pp. 92-154.
8. Jones B. J. T., Mariines V. J., Saar A., Trimble V. Scaling Laws in the Distribution of Galaxies. — arXiv:astro-ph/0406086.
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7. Conclusion
References
9. Peebles P. J. E., Hauser M. G. Statistical Analysis of Catalogs of Extra-galactic Objects. III. The Shane-Wirtanen and Zwicky Catalogs // Astro-phys.J.Suppl.Ser. — 1974. — Vol. 28, No 253. — Pp. 19-36.
10. Peebles P. J. E. The Large-Scale Structure of the Universe. — Princeton: Princeton University Press, 1980.
11. Wavelets on the Sphere: Implementation and Approximations / J.-P. Antoine, L. Demanet, L. Jacques, P. Vandergheynst // Applied and Computational Harmonic Analysis. — 2002. — Vol. 13, No 3. — Pp. 177-200.
12. Rozgacheva I. K., Agapov A. A. The Fractal Cosmological Model. — arXiv:1103.0552.
13. Rozgacheva I. K., Agapov A. A. The Fractal Cosmological Model // Nelineinii mir. — 2011. — Vol. 9, No 10. — Pp. 668-676. — In Russian.
14. Landau L. D., Lifshitz E. M. Statistical Physics. — Oxford: ButterworthHeinemann, 1980. — Vol. 5 of Course of Theoretical Physics.
15. Rozgacheva I. K. Role of Massive Neutrinos in the Gravitational Cosmological Perturbations Evolution // Astronomicheskii Zhurnal. — 1984. — Vol. 61, No 4. — P. 654. — In Russian.
16. Jackiw R. Introducing Scale Symmetry // Phys. Today. — 1972. — Vol. 25, No 1. — P. 23.
17. Scale-Covariant Theory of Gravitation and Astrophysical Application / V. Canuto, P. J. Adams, S.-H. Hsieh, E. Tsiang // Physical Rev. D. — 1977. — Vol. 16, No 6. — Pp. 1643-1663.
18. Rozgacheva I. K. A Doubly-Connected Cosmological Model // Astronomicheskii Zhurnal. — 1997. — Vol. 74, No 2. — P. 165.
УДК 524.8
Фрактальные свойства Вселенной А. А. Агапов*, И. К. Розгачева^
* Кафедра теоретической физики Российский университет дружбы народов ул. Миклухо-Маклая, д. 6, Москва, Россия, 117198
1 ВИНИТИ РАН ул. Усиевича,, д. 20, Москва, А-190, 125190, Россия Московский педагогический государственный университет ул. Малая Пироговская, д.1, стр. 1, Москва, 119991, Россия
Обнаружено, что крупномасштабная структура Вселенной характеризуется рядом степенных зависимостей. Эти степенные законы являются признаками фрактальности, потому что их можно объяснить, если представить Вселенную как совокупность самоподобных пространственно-временных областей. Выдвигается гипотеза, что материя Вселенной описывается скалярным заряженным мезонным полем с вращательной симметрией. На основе этой гипотезы построена фрактальная космологическая модель с масштабно инвариантными уравнениями Лагранжа и Эйнштейна, которая позволяет дать физическую трактовку фрактальных свойств крупномасштабной структуры. Плотности энергии (являющиеся постоянными) и метрические тензоры различных пространственно-временных областей отличаются лишь постоянным множителем. Следовательно, эти области геометрически подобны и эволюционируют одинаково. Фрактальные свойства начальных космологических флуктуаций плотности сохраняются и приводят к наличию фрактальных свойств у крупномасштабной структуры, которая из них образовалась. Построена несингулярная, компактная, пульсирующая и двусвязная космологическая модель как частное решение для однородного, изотропного и плоского случая. Выведен спектр мощности фонового излучения в данной модели. Этот спектр близок к наблюдаемому угловому спектру мощности распределения SDSS-квазаров на небесной сфере.
Ключевые слова: квазары, крупномасштабная структура, фрактальная размерность, комплексное поле, вращательная симметрия, фрактальные свойства крупномасштабной структуры, фрактальная космологическая модель, фоновое излучение.
