>K9m 2014, TOM 145, bmii. 4, cTp. 671 676
© 2014
ON THE EINSTEIN-CARTAN COSMOLOGY VS. PLANCK DATA
D. Palle *
Zavod za teorijsku fiziku, Institut Rugjer BoShovic Bijenicka cesta 54, 10000 Zagreb. Croatia
Received November 11, 2013
The first comprehensive analyses of Planck data reveal that the cosmological model with dark energy and cold dark matter can satisfactorily explain the essential physical features of the expanding Universe. However, the inability to simultaneously fit the large and small scale TT power spectrum, the scalar power index smaller than unity, and the observations of the violation of the isotropy found by few statistical indicators of the CMB urge theorists to search for explanations. We show that the model of the Einstein-Cartan cosmology with clustered dark matter halos and their corresponding clustered angular momenta coupled to torsion can account for small-scale-large-scale discrepancy and larger peculiar velocities (bulk flows) for galaxy clusters. The nonvanishing total angular momentum (torsion) of the Universe enters as a negative effective density term in the Einstein-Cartan equations causing partial cancellation of the mass density. The integrated Sachs-Wolfe contribution of the Einstein-Cartan model is negative, and it can therefore provide partial cancellation of the large-scale power of the TT CMB spectrum. The observed violation of the isotropy appears as a natural ingredient of the Einstein-Cartan model caused by the spin densities of light Majorana neutrinos in the early stage of the evolution of the Universe and bound to the lepton CP violation and matter-antimatter asymmetry.
DOI: 10.7868/S0044451014040090
1. INTRODUCTION AND MOTIVATION
Although the presence of dark matter and dark energy is justified by all cosmological observations, their identification and properties are still far from being established. The measurements of the CMB fluctuations are in this respect especially valuable because of the wealth and accurate information that can be extracted from them.
The most recent disclosed results of the Planck mission contain issues like the temperature power spectrum, gravitational lensing or the integrated Sachs Wolfe (ISW) effect, up to the Sunyaev Zeldovich cluster counts, and isotropy, and non-Gaussianity of the cosmic infrared background. It seems that the old, unexpected features, beyond the ACDM — inflation model persist in data and are even more highlighted: 1. the large-scale temperature power spectrum much lower than the ACDM prediction, limited not only to the low quadrupole fl] but also to almost all multipole moments / < 30 (see Fig. 37 in Ref. [2]), 2. the scalar power spectrum index less than 1 (see Table 8 in
* E-mail: palle'öirb.hr
Ref. [2]), 3. violation of isotropy observed as hemispherical asymmetry, parity asymmetry, quadrupole octo-pole alignment, cold spots, and dipolar asymmetry [3].
If the violation of isotropy will be confirmed by-other complementary cosmic observations of radio galaxies [4], spiral galaxies [5], bulk flows of clusters [6], or quasars [7], it will challenge cosmological principles and call for new theoretical insights.
Assuming that the observed anomalies are real phenomena, we try to understand and elucidate the measured physical features by the Einstein Cart an (EC) cosmology. Incorporating rotating degrees of freedom of matter (spin and angular momentum) and spacetime (torsion) into the relativistic framework, the EC cosmology appears as a nonsingular theory [8, 9]; the cosmic mass density can be fixed [9], the scalar power index can acquire a negative tilt [10], and spin densities trigger density fluctuations [11] and the right-handed vorticity (rotation) of the Universe [12] resulting at later stages of the evolution in the nonvanishing total angular momentum of the Universe [13]. The nonsingular EC cosmology is in conformity with the nonsingular theory of gauge interactions in particle physics [14] that contains light and heavy Majorana neutrinos as hot and cold dark matter particles [15], including other important implications of the perturbative and nonper-
turbativo aspects of strong and electroweak interactions phononionology f 16].
In this paper, we investigate and compare EC and ACDM cosmologies solving evolution equations for the scale-dependent density contrasts, mass fluctuations, peculiar velocities, and the integrated Sachs Wolfe effect. In the next section, we describe the evolution equations and definitions and introducto our simple clustering model. The concluding section deals with the numerical results of the computations, comparisons of the EC and ACDM cosmologies, and final remarks and hints for future research.
2. DEFINITIONS, EQUATIONS, AND THE CLUSTERING MODEL
Because any deviation from cosmic homogeneity and isotropy is very small, we limit our considerations to the homogeneous and isotropic geometry. We start the evolution in the radiation era when the clustering of dark and baryonic matter is negligible. The evolution equations for matter density contrasts in Fourier space are derived in [17]:
d2h 2 da dh dt'2 a dt dt dSm 1 dh (16-, ~df ~ 2 ~dt
= 8irG±\r(2prór kv
dt
"I" Pm$m
1 dh\ 2 dt J
dv ~dt
(1)
= -k
4 u'
du Y 8 ri 2/
_ =rGNu(Pr
Pm + PA)-
We here use the notation
3
a=l
density contrasts are
f]a¡j = -U2 [ôa/û - ha¡j].
Si =
Spi Pi
k is the comoving wave number, R 1
u =
Ro 1 + = '
subscripts m, r, and A denote matter, radiation and the cosmological constant quantities, and v is a velocity. All the quantités are functions of t and k.
These equations can be cast into a more suitable form by eliminating
and by changing the evolution variable to y = In u: (P&m 1 dô„
;9 - „ , -ilm(HrU 1+nm+üA(/) dyl 2 dy
3
+ -(2Í!,.à,. + ttmu6m){ttr + nmu + ÎÎa«4)-1.
dór 4 /dSm kv\ dv Sr k
dy 3 \ dy à J ' dy 4 u
Our notation includes
Pr — - -)■ , Pm — QmPcU PA = ill Am Pc = ÍH°
(2)
8TTGN '
H0 = lOO/i km • s"1 • Mpc 1
and
u = 3.2409 • 10-18/i [Sir«-2+Sim«-1+iiA«2]1/2 s"1.
The evolution equations for the EC cosmology, neglecting small vorticity and acceleration
u> = m = 0, A = 0, Q = Qou~3^2 = torsion,
are derived in [13] (Eq. (14)):
2%
a
2—¿2 u
2—¿3
a
2Q¿2 + (^-gKA - -Q2 -
1 (i\ . 1 ù _ . _ ^ « J 4 « =
2 QSí
(3)
1 (I \ c 1(1 „ .
-KP+- d2---Qdi = 0, 3 at 4 a
1 U I c
-zKp + - d3 = 0,
S = №
sir2
We assume that after the redshift za = 10, the nonlinear bound structures are formed in the form of stars, galaxies, and clusters. The clustering of particles forming halos is described by a model with only two parameters ka and a a- This is applied to both mass and angular momentum clustering:
a=l
Q(a) = (2tt)-3 J d3kQ(u,k) =
= Qo«-3/20(=C; - =), Q(a, k) = Qou~3^2 oxp( —Ik — ka\/(Ta) x
x Q(za ^ z) ^ Qo = Qo(2tt)-3 x
x J d3 k exp ( — I k — ka | /(tq ).
p(u) = (2?r)-3 J d3kp(u,k) = pou~3Q(ta - z), p(u, k) = p0u~3 oxp(-|k - ka\/(Ta)®(za ~ z) (5) Po = Po(27T)-3 j d3k oxp(-|fr - A-g|/<tg).
Fourier transformations of evolution equations (3) take the following form:
(PAit2 u f dil u\ <lA\ß
a J d t/2 a \ du a J dy
=F
a . dd-2 1, =F2-{Q—^ FT a dy
1
1 . « , . --kA+ - 1 A 3 a
1,2
~K{pm6l,2) FT — -{Q"jÖ"I;2)FT ±
±7-{Qd-2;1)FT = 0, 4 a
a\J (PA3 a /du u\ dA*
(6)
a y iij/2 a \ du u J dy
1 . ¿i A .. 1 . . . --K.A + - A3 - -K{pmd3)FT -3 a J 3
- \{Q2&Z)FT = 0.
The Einstein Cartan field equations define the cosmic clocks (see Eq. (15) in Ref. [13]) as follows:
u = H0
fimu 1 + Qa U2 — tu2Q2
o
1/2
du 1
i1mu 1 + Qa U2 — t;u2Q2
O
-1/2
(7)
Almu 2 + 2Qau + -uQ2
O
a 1, i 2_,
- = okA - + oQ"-u 3 6 3
The following definitions and convolutions are used in Eq. (6):
Ai(y.k) = j d3xcxp(ik ■ x)6i(y,x),
{QSi)pT(y, k) = j d3x exp(ik ■ x)Q(y, (y, x) =
= (2n)-3 J d3k'Q(y,k^k')Ai(y,k'),
(Q26t)FT(y<k) = j d3xcx\)(tk ■ x)Q'2(y,x)6i(y,x) =
= (2tt)-6 j d3k'd3k"Ai(y,kl)Q(y,k") x x Q(y, k — k' — k"). Having all the evolution equations for the EC and
ACDM cosmologies, we define initial conditions in the radiation era and choose the parameters of the models:
«¿ = 10-8, 6,,(ui) = k1/2ul 6,n(ui) =
/ \ ozl/2 2
— (ui) = 2 kl^u-, dy
d6m 3 7 1 /2 2 / \ n
— (Ui) = -k ' Ui, v(Ui) = 0,
ACDM:
i!m = 0.34, nA = 0.66, h = 0.67, Q = 0,
EC:
ilm = 2, i)A = 0, h = 0.67,
Q =
0, ; > 10, -2.3«-3/2, 1 < ; < 10, -x/3«-3/2, 0 < c < 1.
Our choice of the torsion (angular momentum) parameters is guided by the condition that at the zero redshift, ilq « — 1 [9, 13] (at the redshifts 1 > i > 0, the galaxy clusters emerge, changing the total angular momentum contribution of the era i > 1), while at the earlier epoch 10 > i > 1, our choice is guided by the condition to roughly match the correct cosmic clocks and the age of the Universe:
10 f du
tu Gyr) = — —
h J u io-3
Qm(j
-1/2
rry(ACDM) = 13.89 Gyr, rr7(EC) = 13.29 Gyr, kwin = 10-3 Mpc-\ kmax = 102 Mpc-1, ka = 1 Mpc-1, aG = 0.25 Mpc-1.
We integrate the above evolution equations to the relative accuracy C(10-4) by lowering the integration steps until the required accuracy is reached. Equations (2) are solved for the evolution from «, = 10-8 to uq = 1/(1 + za) and Eqs. (6) are then solved from uc = 1/(1 + za) to a = 1. The Adams Bashforth Moult on predictor corrector method is used for differential equation integrations (code of L. F. Shampine and M. Iv. Gordon, Sandia Laboratories, Albuquerque, New Mexico) and CUBA Library for Multidimensional Integrations [18]. The next section is devoted to the detailed exposure of the numerical results and comparison between the EC and ACDM models. The relevance of the results for the Planck data are also given here.
7 >K9T®, Bbiii. 4
673
Density contrasts 4500
4000 2500 3000 2500 2000 1500 1000 500
~1-1-1-1-1-r
vxXX
o
-8 -6 -4 -2 0
2 4 6 \n[k, Mpc"1]
Fig. 1. Density contrasts at z = 0 as functions of the wave number fc normalized at kmax: 8m(kmax) = 1. EC — +, ACDM — x
3. RESULTS, DISCUSSION, AND CONCLUSIONS
Because the best fit to the Planck temperature pow
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