Using the known connection between Schroedinger-like equations and Dirac-like
equations in the supersymmetric context, we discuss an extension of FRW
barotropic cosmologies in which a Dirac mass-like parameter is introduced. New
Hubble cosmological parameters H_K(eta) depending on the Dirac-like parameter
are plotted and compared with the standard Hubble case H_0(eta). The new
H_K(eta) are complex quantities. The imaginary part is a supersymmetric way of
introducing dissipation and instabilities in the barotropic FRW hydrodynamicsComment: 7 pages, 4 figures, accepted at MPL

Lopez-Sandoval, R.

Rosu, H. C.

English

arXiv

H. C. ROSU

R. LÓPEZ-SANDOVAL

Crossref

BAROTROPIC FRW COSMOLOGIES WITH A DIRAC-LIKE PARAMETER

"Using the known connection between Schroedinger-like equations and Dirac-like equations in the supersymmetric context, we discuss an extension of FRW barotropic cosmologies in which a Dirac mass-like parameter is introduced. New Hubble cosmological parameters HK(n) depending on the Dirac-like parameter are plotted and compared with the standard Hubble case H0(n). The new HK(n) are complex quantities. The imaginary part is a supersymmetric way of introducing dissipation and instabilities in the barotropic FRW hydrodynamics.

Rosu Barbus, Haret-Codratian

López Sandoval, Román

Repositorio IPICYT

This is the Author's Pre-print version of the following article: C. ROSU and R. LÓPEZ-SANDOVAL, Mod. Phys. Lett. A 19, 1529 (2004).  Electronic version of an article published as https://doi.org/10.1142/S0217732304013763   © World Scientific Publishing Company http://www.worldscientific.com/worldscinet/mpla       BAROTROPIC FRW COSMOLOGIES WITH A DIRAC-LIKE PARAMETERH.C. ROSU∗ and R. LÓPEZ-SANDOVAL†Potosinian Institute of Science and Technology,IPICyT, Apdo Postal 3-74 Tangamanga, 78231 San Luis Potośı, MexicoUsing the known connection between Schroedinger-like equations and Dirac-like equations in the supersymmetric context, we discuss an extension of FRWbarotropic cosmologies in which a Dirac mass-like parameter is introduced. NewHubble cosmological parameters HK(η) depending on the Dirac-like parameter areplotted and compared with the standard Hubble case H0(η). The new HK(η) arecomplex quantities. The imaginary part is a supersymmetric way of introducingdissipation and instabilities in the barotropic FRW hydrodynamics.1 - IntroductionThe scale factor a(t) of a FRW metric is a function of the comoving time tobeying the Einstein-Friedmann dynamical equations supplemented by the(barotropic) equation of state of the cosmological fluidäa= −4πG3(ρ+ 3p) , (1)(ȧa)2=8πGρ3−κa2, (2)p = (γ − 1)ρ, (3)where ρ and p are the energy density and the pressure, respectively, of theperfect fluid of which a classical universe is usually assumed to be madeof, κ = 0,±1 is the curvature index of the flat, closed, open universe, re-spectively, and γ is the constant adiabatic index of the cosmological fluid.A simple Riccati route of solving the system of eqs. (1)-(3) proposed by∗E-mail: hcr@ipicyt.edu.mx; file romancos.tex†E-mail: sandov@ipicyt.edu.mxFRW barotropic cosmologies with Dirac 2Faraoni,1 has been used by Rosu to develop a factorization (nonrelativisticsupersymmetry) approach of the barotropic FRW cosmologies,2 in which asimple explanation for a currently accelerating universe,3 in terms of ‘Dar-boux fluids’ with effective adiabatic indices is possible.In this work, we first briefly review the supersymmetric factorizationmethods for barotropic FRW cosmologies in section 2. Next, in section 3,we present Dirac-like (first-order) coupled differential equations and theirassociated second-order differential equations. This allows a simple exten-sion of barotropic FRW cosmologies that include a Dirac mass-like constantparameter K.2 - Supersymmetric methodsCombining the equations (1)-(3) and using the conformal time variable ηdefined by dt = a(η)dη one gets the equationa′′a+ (c− 1)(a′a)2+ cκ = 0 . (4)where c = 32γ − 1. The case κ = 0 is directly integrable,1 and will be skippedhenceforth. The logarithmic derivative H0 =a′a is the Hubble parameter ofthe universe which is a fundamental cosmological quantity ever since Hub-ble’s discovery of the expansion of the universe in1929. Interestingly, passingto the function H0 in Eq. (4) the following Riccati equation is obtainedH′0 + cH20 + κc = 0 . (5)Employing now H0 =1cw′κwκone gets the very simple second order differentialequationw′′κ + κc2wκ = 0 . (6)For κ = 1 the solution of the latter is w1 = W1 cos(cη + d) , where d isan arbitrary phase, implying a1(η) = A1[cos(cη + d)]1/c , whereas for κ =−1 one gets w−1 = W−1sinh(cη) and therefore a−1(η) = A−1[sinh(cη)]1/c ,where W±1 and A±1 are amplitude parameters.The point now is that the particular Riccati solutions H+0 = − tan cη andH−0 = cothcη for κ = ±1, respectively, are closely related to the commonfactorizations of the Schrödinger equation. Indeed, Eq. (6) can be writtenw′′− c(−κc)w = 0 (7)and also in factorized form using Eq. (6) one gets(ddη+ cH0)(ddη− cH0)w = w′′− c(H′0 + cH20)w = 0 . (8)FRW barotropic cosmologies with Dirac 3To fix the ideas, we shall call Eq. (8) the bosonic equation. On the otherhand, the supersymmetric partner (or fermionic) equation of Eq. (8) will be(ddη− cH0)(ddη+ cH0)wf = w′′f − c(−H′0 + cH20)wf = w′′f − c · cκ,susywf = 0 .(9)Thus, one can writecκ,susy(η) = −H′0 + cH20 ={c(1 + 2tan2cη) if κ = 1c(−1 + 2coth2cη) if κ = −1for the supersymmetric partner adiabatic index. The solutions are wf =ccos(cη+d) and wf =csinh(cη) for κ = 1 and κ = −1, respectively.3 - Dirac-like formalismThe Dirac equation in the susy nonrelativistic formalism has been discussedby Cooper et al,4 already in 1988. They showed that the Dirac equationwith a Lorentz scalar potential is associated with a susy pair of SchroedingerHamiltonians. This result has been used later by many other authors. Herewe make an application to barotropic FRW cosmologies that we find not tobe a trivial exercise except for the uncoupled ‘zero-mass’ case (subsection3.1).3.1- Let’s introduce now the following two Pauli matricesα = −iσy = −i(0 −ii 0)and β = σx =(0 11 0)and write a cosmological Dirac equationHFRW0 W = [σyDη + σx(icH0)]W = 0 , (10)where W =(w1w2)is a two component ‘zero-mass’ spinor. This is equiva-lent to the following decoupled equationsDηw1 + cH0w1 = 0 (11)−Dηw2 + cH0w2 = 0 . (12)Solving these equations one gets w1 ∝ 1/ cos(cη) and w2 ∝ cos(cη) for κ = 1cosmologies and w1 ∝ 1/sinh(cη) and w2 ∝ sinh(cη) for κ = −1 cosmologies.Thus, we obtainW =(w1w2)=(wfwb).FRW barotropic cosmologies with Dirac 4This shows that the matrix ‘zero-mass’ Dirac equation is equivalent to thetwo Schroedinger equations for the bosonic and fermionic components.3.2- Consider now a “massive” Dirac equationHFRWK W = [σyDη + σx(icHp +K)]W = KW , (13)where K may be considered the mass parameter of the Dirac spinor. Eq. (13)is equivalent to the following system of coupled equationsDηw1 + (icH0 +K)w1 = Kw2 (14)−Dηw2 + (icH0 +K)w2 = Kw1 . (15)These two coupled first-order equations are equivalent to second order dif-ferential equations for the two spinor components:The fermionic spinor component can be found directly as solutions ofD2ηw+1 −[c2(1 + 2 tan2 cη) + 2icK tan cη]w+1 = 0 forκ = 1 (16)andD2ηw−1 −[c2(−1 + 2coth2cη)− 2icKcoth cη]w−1 = 0 forκ = −1 , (17)whereas the bosonic components are solutions ofD2ηw+2 +[c2 − 2icK tan cη]w+2 = 0 for κ = 1 (18)andD2ηw−2 +[− c2 + 2icKcoth cη]w−2 = 0 for κ = −1 . (19)The solutions of the bosonic equations are expressed in terms of the Gausshypergeometric functions 2F1 in the variables y = eicη and y = ecη , respec-tivelyw+2 = Ay−p2F1[−12(p + iq);−12(p− iq), 1 − p;−y2]+Byp 2F1[12(p− iq),12(p + iq), 1 + p;−y2](20)andw−2 = C(−1)−i2ry−ir 2F1[−i2(r + s),−i2(r− s), 1− r; y2]+D(−1)i2ryir 2F1[ i2(r− s),i2(r + s), 1 + r; y2], (21)respectively. The parameters are the following:p = (−1−2Kc)12 , q = (1−2Kc)12 , r = (−1− i2Kc)12 , s = (−1 + i2Kc)12 ,FRW barotropic cosmologies with Dirac 5whereas A, B, C, D are superposition constants.Based on these K- modes, we can introduce bosonic Hubble parametersdepending on the parameter KH+K =1c(DDηlog w+2)(22)andH−K =1c(DDηlog w−2), (23)and similarly for the fermionic components by changing w±2 to w±1 in eqs. (22)and (23), respectively.In the small K limit, K → 0, the ordinary FRW barotropic cosmologiesare obtained. The Hubble parameters corresponding to the K- dependentbosonic modes are plotted in the Figures (1) -(4).4 - ConclusionsWe come now to the interpretation of the mathematical results that wedisplayed in the previous sections. The parameter K introduces an imagi-nary part in the cosmological Hubble parameter H. Since the latter is thelogarithmic derivative of the scale factor of the universe one comes to theconclusion that the supersymmetric techniques presented here are a way toconsider dissipation and instabilities of barotropic FRW cosmologies.References1. V. Faraoni, “Solving for the dynamics of the universe”, Am. J. Phys. 67, 732 (1999)[physics/9901006].2. H.C. Rosu, “Darboux class of cosmological fluids with time-dependent adiabatic in-dices”, Mod. Phys. Lett. A 15, 979 (2000).3. S. Perlmutter et al., “Cosmology from type Ia supernovae”, Bull. Am. Astron. Soc. 29,1351 (1997) [astro-ph/9812473], and “Measurements of Omega and Lambda from 42high-redshift supernovae”, astro-ph/9812133.4. F. Cooper, A. Khare, R. Musto, A. Wipf, “Supersymmetry and the Dirac equation”,Ann. Phys. 187, 1 (1988). See also, R.J. Hughes, V. Alan Kostelecký, M.M. Nieto,“Susy quantum mechanics in a first-order Dirac equation”, Phys. Rev. D 34, 1100(1986); Y. Nogami and F.M. Toyama, “Supersymmetry aspects of the Dirac equationin one dimension with a Lorentz scalar potential”, Phys. Rev. A 47, 1708 (1993).FRW barotropic cosmologies with Dirac 602468100246810-10010Figure 1: The real part of H+K obtained from the logderivative of the bosonicmode w+2 (y;12 ,12 ) for η ∈ [0, 10] and K ∈ [0, 10] in the case of a radiationdominated universe (c = 1) .02468100246810-100102030Figure 2: The imaginary part of H+K for the same case.FRW barotropic cosmologies with Dirac 700.20.40.60.8100.511.52024Figure 3: The real part of H−K calculated from the bosonic mode w−2 (y;12 ,12)for η ∈ [0, 1] and K ∈ [0, 2] and a radiation dominated universe.00.20.40.60.8100.511.52-10-50Figure 4: The imaginary part of H−K for the same case.

Barotropic FRW cosmologies with a Dirac-like parameter

Barotropic FRW cosmologies with a Dirac-like parameter

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