Fine Structure Constant

arXiv:gr-qc/0112011v1 7 Dec 2001 Possible cosmological implications of a time varying fine structure constant. Marcelo.S.Berman(1) and Luis.A.Trevisan(2) (1) Tecpar-Instituto de Tecnologia do Paraná. Grupo de Projetos Especiais. R.Prof Algacyr M. Mader 3775-CIC-CEP 81350-010 Curitiba-PR-Brazil Email: marsambe@tecpar.br (2) Universidade Estadual de Ponta Grossa, Demat, CEP 84010-330, Ponta Grossa,Pr, Brazil email: latrevis@uepg.br October 24, 2018 Abstract Webb et al. [2] experimental results on the fine strucutre “constant ” variation with the age of the Universe, is here analized. By using the experimental data on the fine-structure “constant ”’s variation with the age of the Universe, and Dirac’s LNH (Large Number Hypothesis), we find how should vary the total number of nucleons in the Universe, the speed of light, Newton’s gravitational “constant ” and the energy density, and we make an estimate on the deceleration parameter, finding that the Universe would be accelerating, just as Supernovae observations have concluded. PACS 98.80 Hw 1 POSSIBLE COSMOLOGICAL IMPLICATIONS OF A TIME VARYING FINE STRUCTURE CONSTANT. MARCELO S. BERMAN and LUIS A. TREVISAN In the two landmarking letters, Webb et al. [1] and Webb et al.[2] have provided experimental data on quasars that span 23% to 87% of the age of the Universe, finding a variation in the fine structure constant, given by ∆α/α ∼ = −0.72x10−5 . We shall show that this result is coherent with other cosmological facts. One can ask wether this variation could be caused by a variyng speed of light. Gomide [18] has long ago studied cosmological models with varying c, and/or varying ε0 . Dirac’s LNH (Large Numbers Hypothesis)[9][10][11] was included in his framework. Barrow[6] and Barrow and Magueijo [19] have studied how a time varying c would explain both, the α̇ 6= 0 and the Supernovae observations. We refer to their papers for further information. Our framework will presently be different from those references. If we perceive the present Universe as having constant deceleration parameter, q, like, for instance, Einstein-de Sitter’s Universe, where q = 1/2=const., we may use Berman’s[3] [4]formula for the Hubble’s parameter, H= 1 1 −1 = t mt 1+q (1) H = Ṙ/R (2) where and R̈R (3) Ṙ2 Overdots are for time-derivatives, and R is the scale-factor in Robertson Walker’s metric, q=− ds2 = dt2 − R2 (t) (dx2 + dy 2 + dz 2 ) 2 (1 + kr4 )2 where k = 0, ±1 is the tricurvature. We may now express Webb’s et al’s[2] experimental result as:   α̇ ≃ −1.1 × 10−5H(1 + q) α exp We define 2 (4) (5) e2 (6) ~c and we shall suppose that the time-variation of α is caused by a varying speed of light, as in Barrow’s paper’s[5][6][7]. From (6) we find, α≡ ċ α̇ =− . (7) α c If we suppose that the speed of light varies with a power-law of time, say: c = Atn (8) (A = const.); we find, from the above experimental values, n = 1.1 × 10−5 . (9) We see that the speed of light varies slowly with the age of the Universe, as Berman and Trevisan [8] have shown elsewhere, for another model. Now, let us consider Dirac’s large number’s hypothesis[9][10][11]. The number of nucleons in the Universe is called N, and we find roughly that both the ratio of the Hubble’s length and the classical electronic radius and the ratio of electrostatic√and gravitational interactions between a proton and a electron are of order N , with N ∼ 1080 ,i. e. cH −1 e2 me c2 ≈ √ N (10) √ e2 ≈ N Gmp me (11) ρ(cH −1)3 ≈N mp (12) Dirac’s LNH can be presented by the assumption that N varies with the age of the Universe, so that a present “large ” number N only means that the Universe is “old ”. When we plug law (1) into (10)-(11)-(12), we find that, √ ρ ∝ t−2 N (13) G ∝ (N)−1 (14) 3 1 1 c ∝ (N) 6 t− 3 (15) In order to accommodate (15) with (8) and (9), we assume that N ∝ t2.0001 (16) G ∝ t−1.00005 (17) ρ ∝ t−0.99995 (18) and, then, This “solves ” Dirac’s LNH, for the experimental found fine-structure “constant ’s ” time-variation. We find Ġ/G ∼ = 1.0H(1 + q) while it has = 1.0t−1 ∼ been found under Lunar laser ranging and Viking radar measurements by Williams et al. [12] and Reasenberg [13], that Ġ/G = σH with |σ| < 0.6. Will [14][15] comments that these two kinds of measurements give the best limits on σ[20]. This means that – 0.4>q > −1.6. A negative q is a good result because of Supernovae observations [16]. We have thus shown that ∆α/α should really be negative, for a positive result could mean a positive deceleration parameter. As a bonus, we found how N, G, ρ, and the speed of light may vary with the age of the Universe, in order to be in accordance with LNH and Webb et al’s results. In the model presented here, α = α(t), because c = c(t). It is important to remark that the electric permittivity of the vacuum, as well as its magnetic permeability, and Planck’s constant, are thought really constants; other possibility was devised by Berman and Trevisan [17]. It is important to elaborate different models and make some predictions in order to decide among them. In fact a Superunification theory will only survive in case that such evolution of the constants values with the age of the Universe will be explained by this theory. Acknowledgments Both authors thank support by Prof. Ramiro Wahrhaftig, Secretary of Science, Technology , and Higher Education of the State of Paraná, and by our Institutions, especially to Jorge L.Valgas, Roberto Merhy, Mauro K. Nagashima, Carlos Fior, C.R. Kloss, J.L.Buso, and Roberto Almeida. References [1] Webb, J.K.; et al. –Phys. Rev. Lett. 82, 884 (1999) 4 [2] Webb,J.K.; et al. –Phys. Rev. Lett. 87, 091301 (2001). [3] Berman, M.S. –Nuovo Cimento 74B, 182 (1983) [4] Berman, M.S.;Gomide,F.M.–General Relat. and Grav. 20, 191 (1988) [5] Barrow, J.D. –arXiv: astro-ph./9811022-1 [6] Barrow,J.D. –Astrophys.J. 532, L87 (1999) [7] Barrow,J.D–arXiv:gr-qc/9711084-1 [8] Berman, M.S.;Trevisan, L.A. –Submitted to Phys.Rev.D15 (2001) [9] Dirac, P.A.M.–Proc.R.Soc. A165, 199 (1938) [10] Dirac, P.A.M–Proc.R.Soc. A338, 439 (1974) [11] Raychaudhuri, A.K. “Theoretical Cosmology ”, Oxford U.P., Clarendon, pages 157-8. [12] Williams, J.G.; et al. .;–Phys. Rev. Lett, 36, 551 (1976) [13] Reasenberg R.D.; –Phil. Trans. Roy.Soc (London) 310,22 (1983) [14] Will, C.;–Experimental Gravitation from Newton´s Princípia to Einstein´s General Relativity, in “300 years of Gravitation ”-Edited by Werner Israel and Stephen Hawking–Cambridge U.P.(1987) [15] Will, C.;–The Confrontation between General Relativity and Experiment: a 1995 UPDATE, in “General Relativity-Prooceedings of the Forty Sixth Scotish Universities Summer School in Physics ”, Edited by Graham Hall and J.R.Pulham –Institute of Physics Publishing and SUSSP publications,pg 239 (1995) [16] Perlmutter S. et al, Ap. J. 483, 565 (1997); Perlmutter et al (The Supernovae Cosmology Project), Nature 391 51 (1998); Garnavich et al.– Ap. J. Lett. 493 L53-57 (1998);–Schmidt, B.P.;–Ap. J. 507, 46-63 (1998); Riess, A.G.; et al.;–Ap.J. 116, 1009 (1998). [17] Berman, M.S.; Trevisan, L.A.;–to be submitted. [18] Gomide, F.M.;–Lett. Nuovo Cimento 15, 595 (1976). [19] Barrow, J.D; Magueijo, J.;–Astrophysical Journal, 532, L87, (1999). [20] Dickey, G.O.; et al. ;–Science 265, 482 (1994). 5