ЯДЕРНАЯ ФИЗИКА, 2009, том 72, № 5, с. 788-795
= ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
IS IT POSSIBLE TO ESTIMATE THE HIGGS MASS FROM THE CMB POWER SPECTRUM?
© 2009 A. B. Arbuzov1)*, B. M. Barbashov1), A. Borowiec2), V. N. Pervushin1)**, S. A. Shuvalov3), A. F. Zakharov4)
Received October 14, 2008
General Relativity and Standard Model are considered as a theory of dynamical scale symmetry with definite initial data compatible with the accepted Higgs mechanism. In this theory the Early Universe behaves like a factory of electroweak bosons and Higgs scalars, and it gives a possibility to identify three peaks in the Cosmic Microwave Background power spectrum with the contributions of photonic decays and annihilation processes of primordial Higgs, W and Z bosons in agreement with the QED coupling constant, Weinberg's angle, and Higgs' particle mass of about 118 GeV.
PACS:95.30.Sf, 98.80.-k, 98.80.Es
1. INTRODUCTION
The observational data [1] on the Cosmic Microwave Background (CMB) power spectrum show several clear peaks at the orbital momenta i\ — 220, t2 — 546, t3 — 800. These phenomena are explained in the ACDM model [2] by acoustic inhomogeneities of the scalar metric component treated as a dynamical variable. By adjusting parameters of the equations for the acoustic excitations one can provide a good fit of the observed peaks and predict other peaks with higher t values, which can be found in future observations. Recall that the ACDM model requires the acoustic explanation of the CMB power spectrum by introduction of a dynamical scalar metric component that is absent in the Wigner classification of relativistic states [3]. The dynamical scalar metric component is introduced by the ACDM model without any substantial motivation and clear discussion of the reasons for introducing new concepts. Moreover, this ACDM explanation contradicts the vacuum postulate. Since the CMB power spectrum is one of the highlights of the present-day Cosmology with far-reaching implications and more precise observations are planned for near future [1], the detailed investigation of any possible flaw of the standard theory deserves attention and a public discussion.
'Joint Institute for Nuclear Research, Dubna, Russia.
2)Institute of Theoretical Physics, University of Wroclaw, Poland.
3)Russian Peoples Friendship University, Moscow, Russia.
4)Institute of Theoretical and Experimental Physics, Moscow, Russia.
E-mail: arbuzov@theor.jinr.ru E-mail: pervush@theor.jinr.ru
In this paper we try to describe the CMB power spectrum in accord with the well-established Wigner's theory of the relativistic state classification, where any relativistic particle in quantum field theory can be associated with a unitary irreducible representation of the Poincare group given in a definite frame with a positive energy.
The cosmological scale factor, its local excitations used for description of the CMB power spectrum, and Poincare group transformations can be naturally included in the Wigner classification, if General Relativity is considered as the theory of the joint nonlinear realization of the affine and conformal symmetries with the Poincare group of the vacuum stability [4], where the scale invariance of laws of Nature [5, 6] is realized dynamically by means of the dilaton Goldstone field.
The dynamical scale symmetry plays a role of the principle of a choice of variables in the accepted General Relativity (GR) and the Standard Model (SM) [7]. The dilaton Goldstone field compensates all scale transformations of fields including the cosmo-logical scale factor describing expansion of the Universe lengths in the Standard Cosmology [2]. Nevertheless, the cosmological dynamics can be introduced by help of Einstein's cosmological principle [8] that means averaging all scalar characteristics including the dilaton field over a constant Universe volume. This cosmological dynamics of the zeroth dilaton mode explains the redshift by a permanent increase of all masses in the Universe and leads to the Conformal Cosmology [9—16], where all measurable quantities are identified with the conformal ones (conformal time, coordinate distance, and constant conformal temperature).
General Relativity considered as the theory of dynamical scale symmetry [4, 6] changes the numerical analysis of supernovae type Ia data [17, 18] and shows the dominance of the scalar field kinetic energy in all epochs of the Universe evolution including the chemical evolution, recombination, and SN explosions.
In the paper we try to describe the CMB power spectrum [19] in GR as the theory of dynamical scale symmetry in accord with the classification of relativistic states [3].
2. DILATONIC VARIABLES IN GENERAL RELATIVITY
Let us consider the accepted GR supplemented by the SM and an additional scalar field Q governing the Universe evolution
Su[g,F\ =
= /
R(g)
+ Lsm(F ) + o^ Qdrq
phenomenological applications, one can identify this choice with the CMB co-moving reference frame. In terms of the dilaton variables, the GR action takes the form
Sqr = - J éx^f^g
= d x
R(g)
\ab)
(4)
Nd 24Nd
_4D R(3)(e) + 8eD/2Ae-D/2
(1)
where units K = c = Mplmc]í^/3/(S^т) = 1 are used throughout the paper. This action depends on a set of scalar, spinor, vector, and tensor fields F(n) = = ,g^v with their conformal weights n =
= —1, -3/2,0,2, respectively.
Following the foundation of the GR as a dynamical scale symmetry [4, 6] we define all observable fields F(n) as scale-invariant quantities using the following scale transformations of these fields F(n) in action (1) including the metric components g^v:
F(n) = exp{nD}F(n), Fnv = exp[2D]g^v, (2)
where D is the dilaton compensating scale transformations of all these fields. Any concrete choice of the dilaton as a metric functional D[g] means a gauge fixing. In [12, 20, 21] this functional is chosen in the form of D[g] = — log \g(3) \/6 in accordence with the accepted definition of transverse traceless graviton physical variables given in a definite frame
distinguishing the spatial metric components gj. Therefore, one can remove any scale factor from the spatial metric components in the Dirac—ADM parametrization [20] in terms of the simplex components Fo), Fb) in the Minkowskian tangent space— time
- NDe
where R(3)(e) is a curvature, A = d^e^)eja)dj\ is the Laplace operator, and vD = [d0D + dlNl/3\, v(ab) = = e(a)iv(b) + e(b)iv(a), v(a)i = [doe(a)i + e(a)idiNl —
— e(a)ldiNl/3\, are velocities of the metric components, d0 = (d0 — Nldl).
Simplex (3) as an object of frame transformations from the Earth frame to the CMB one moving to Leo with the measurable velocity 368 km/s separates the latter from the unmeasurable diffeomorphisms x0 —
— xS0 = xS0(x0), xk — xk = xP(x°,xk). The principle of diffeo(d)-invariance of observables, D(x°) = = D(x°), is at heart of GR. One can see that variables (2) and interval (3) define a d-invariant finite coordinate volume fVo d3x = V0 < to, d-invariant evolution parameter in the field space of events, and a d-invariant time interval N0dx° = dr by Einstein's cosmological principle [8] as averaging of the dilaton D and the inverse Dirac lapse function ND-1 over this volume:
V0-1 J d3xD(r,xk) = (D)(t),
(5)
Vo
V0-1 J d3xN-1 = (N-1) = N0
J-1-
Vo
ds = uo — tib, iF(o) = e- NDdx , (3)
iF(b) = e(b)j (dxj + Nj dx°),
where e^ are the triads [21] with the unit spatial metric determinant le^ \ = 1, ND is the Dirac lapse function, and Nj are the shift vector components. In
The scale-invariant variables and d-invariant evolution parameter as the dilaton zeroth mode (D)(t) are compatible with a definite d-invariant cosmological dynamics known as the Conformal Cosmology [9— 16] that strongly differs from the heuristic phenomenology of the accepted Standard Cosmology [2]. Principles of the conformal symmetry, relativistic (frame) symmetry, and d-invariance of observables and the Dirac Hamiltonian approach to GR completely determine the finite volume generalization of Einstein theory [13—16]
Su[D,F\
D=(D)+D
= Sz[(D)\+Su[D,F\, (6)
6
2
2
6
6
where
Stirn _
(7)
'U
= Vo J dT[-dr(D)2 + dr(0)2 + 3T(Q)2]
dr=NudxU
t=0
is the zeroth-mode action and the second term Su repeats actions (1) and (4) for nonzero harmonics associated with local excitations.
3. COSMOLOGICAL DYNAMICS OF THE ZEROTH DILATON MODE
Let us consider the Early Universe when one can neglect all these local excitations SU ~ 0 (complete expressions of action (6) see in Appendix A). In this case, the cosmological evolution of the Empty Universe arises in the form of a conformal mechanics of zeroth harmonics of all scalar fields F = D,fi,Q with equations d2(F) = 0 and the initial data
{</>)i = Mw/(gV2), <9r ((/>)/ = 0; (Q)i _ Qo, dr (Q)i _ Ho
(8)
defined so that the mechanism of spontaneous electroweak symmetry breaking does not differ from the accepted one in SM. Recall, that the accepted spontaneous symmetry breaking mechanism is based on the Coleman—Weinberg potential equation dV((4>)I)/d(4>)i = 0 in the perturbation theory restricted by the constraint dT ($>) = 0. In our perturbation theory (8), loop diagrams also lead to the effective potential with the same equation dV((4>)I)/d(4>)I = 0 treated as a constraint that keeps the vacuum equation d'22(^) = 0.
Using as an example this potential free model of the Empty Universe, one can see that the Standard Cosmology observable quantities are connected with the conformal ones by relation (2):
F
(n)SC
= e-n(D)F(
(n)CC-
This relation determines the scale factor
-(D) _
a(z) _ (1 + z)
-l
conformai masses and time
m _ a(z)m0, dn _ dTa2(T),
(9)
(10)
(ii:
and the horizon H = H0a-2. In this case, the dilaton solution of the motion equation d2 (D) = 0 takes the form
(D) = (D)o + Ho(r - to). (12)
In terms of the effective cosmological factor (10) and conformal time (11) this solution becomes
(13)
i(r]) = ao\/l + 2H0(r) - T]o).
The cosmological dynamics of the Conformal Cosmolog
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