Beyond the Hubble's law.
In a 2014 paper, Silva  introduced an expression for Universe's scale factor to describe the Universe's expansion,
a(t) = exp ([H.sub.0][T.sub.0]/[beta]([(t/[T.sub.0]).sup.[beta]] - 1)), (1)
[beta] = 1 + [H.sub.0][T.sub.0] (-1/2[[OMEGA].sub.m]([T.sub.0]) + [[OMEGA].sub.[LAMBDA]]([T.sub.0]) - 1), (2)
[H.sub.0] is the Hubble constant, [T.sub.0] is the Universe current age, [[OMEGA].sub.m]([T.sub.0]) is the cosmic matter density parameter (baryonic + non-baryonic matter), [[OMEGA].sub.[LAMBDA]]([T.sub.0]) is the cosmic dark energy density parameter .
In reference  matter and dark energy are treated as perfect fluids and it is shown that it very difficult to distinguish between closed (k = 1), flat (k = 0) and open (k = -1) universes. In this paper we intuitively adopt k = 1 and explore the Universe as being closed.
The spacetime metric for k = 1 according to Friedmann-Lemaitre-Robertson-Walker (FLRW) is [1,3]
d[s.sup.2] = [R.sup.2]([T.sub.0])[a.sup.2](t)(d[[psi].sup.2] + [sin.sup.2] [psi] (d[[theta].sup.2] + [sin.sup.2] [theta]d[[phi].sup.2])) -[c.sup.2]d[t.sup.2] (3)
where [psi], [theta] and [phi] are the the comoving space coordinates (0 [less than or equal to] [psi] [less than or equal to] [pi], 0 [less than or equal to] [phi] and 0 [less than or equal to] [phi] [less than or equal to] 2[pi]); R([T.sub.0]) is the current Universe's radius of curvature. This proper time t is the cosmic time.
It is known that at t = 380,000 yr [equivalent] [10.sup.-4] Gyr, after the Big Bang, the Universe became transparent and the first microwave photons started traveling freely through it. They constitute what is called the Cosmic Microwave Background (CMB).
The observer (Earth) is assumed to occupy position [psi] = 0 for any time t in the comoving reference system. To reach the observer at the Universe age T the CMB photons leave a specific position [[psi].sub.T] (t [equivalent] [10.sup.-4] Gyr). They follow a null geodesic.
It's time to make the following observation: since we will be dealing with large times values (some giga years) we have no loss if we treat t [equivalent] [10.sup.-4] Gyr as t [equivalent] 0 Gyr for practical purposes.
For a null geodesic we have:
-[c dt/R(0)] = d[psi], (4)
[[psi].sub.T] = [c/R](0)] [[integral].sup.T.sub.0] 1/a(t) dt. (5)
We have seen then that CMB photons emitted at [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] should arrive at the observer, [psi] = 0 and [T.sub.0]. Along their trajectory, other emitted photons, at later times, by astronomical objects that lie on the way, join the the photons troop and eventually reach the observer. They form the picture of the sky that the observer "sees". Certainly CMB photons emitted at [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] will reach the observer at times latter than [T.sub.0].
2 The receding velocity
As the Universe expands, the streching distance between the observer and any astronomical object at time t is given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)
The receding velocity of any astronomical object with respect to the observer is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)
where we have used the fact that
[??](t) = a(t)H(t). (8)
According to reference ,
H(t) = [H.sub.0] [(t/[T.sub.0]).sup.[beta]-1] (9)
By performing the integration in equation (8) we have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
where [GAMMA](A, B) and [GAMMA](A, C) are incomplete Gamma Functions .
Taking into account that
[GAMMA](A, B) - [GAMMA](A, C) = [GAMMA](A, B, C), (11)
where [GAMMA](A, B, C) are generalized incomplete Gamma Functions , we have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)
3 Comparison to Hubble's law
By replacing t by [T.sub.0] - [d.sub.lb]/c, where [d.sub.lb] = [ct.sub.lb] is the so called lookback distance, [t.sub.lb] being the lookback time:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)
Figure 1 shows that the receding velocities increase as the lookback distance increases, initially in a linear way. Distant astronomical objects are seen to recede at much faster velocities than the nearest ones.
By expanding expression (13) in power series of [d.sub.lb], and retaining the lowest order term we get
[v.sub.rec] ([d.sub.lb]) = [H.sub.0][V.sub.rec]([d.sub.lb]) + higher order terms. (14)
The Hubble's law,
[v.sub.rec]([d.sub.lb]) = [H.sub.0][v.sub.rec]([d.sub.lb]), (15)
is an approximation to our just obtained expression. According to the present work, Hubble's law holds up to ~ 7 Gly or, equivalently ~ 2.1 x [10.sup.3] Megaparsecs.
For this work, we have used the following experimental data :
[H.sub.0] = 69.32 [kms.sup.-1][Mpc.sup.-1] = 0.0709 [Gyr.sup.-1],
[T.sub.0] = 13.772 Gyr. (16)
As indicated by references [1,6] the present scale factor predicts
that the Universe goes from a matter era to a dark energy era at the age of [T.sub.*] = 3.214 Gyr. Before that matter dominated, and after that dark energy era dominates.
The Universe's scale factor a(t) = exp([[H.sub.0][T.sub.0]/[beta]] (([(t/[T.sub.0]).sup.[beta]] - 1)), with [beta] = 1 + [H.sub.0][T.sub.0] (-1/2[[OMEGA].sub.m]([T.sub.0]) + [[OMEGA].sub.[LAMBDA]]([T.sub.0]) - 1) introduced by Silva  has been used to find an expression for the receding velocities of astronomical objects caused by the expansion of the Universe. The expression found, equation (13), is a generalization of Hubble's law. This later one should be valid up to ~ 2.1 x [10.sup.3] Megaparsecs.
After such very good results we feel very stimulated with the idea that expression (1) is a very good candidate for describing the geometrical evolution of our Universe.
Submitted on October 16, 2016 / Accepted on October 21, 2016
[1.] Silva N.P. A Model for the Expansion of the Universe. Progress in Physics, 2014, v. 10(2), 93-97.
[2.] Silva N.P. The Universe's Expansion Factor and the Hubble's Law. ResearchGate, 2016, https://www.researchgate.net/publication/296060 533.
[3.] Peebles PJ.E. The Large-scale Structure of the Universe. Princeton University Press, 1980.
[4.] Wolfram Research, Inc., Mathematica, Version 10, Champaign, IL, 2014.
[5.] Bennet C.L. et al. Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Final Maps and Results. arXiv: astro-ph.CO. 2013.
[6.] Silva N.P. The Universe's Scale Factor: From Negative to Positive Acceleration of the Expansion. ResearchGate, 2016, https://doi.org/10.13140/RG.2.1.1540.3608.
Nilton Penha Silva
Departmento de Fisica (Retired Professor), Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brazil.
Please note: Some tables or figures were omitted from this article.
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|Author:||Silva, Nilton Penha|
|Publication:||Progress in Physics|
|Date:||Jan 1, 2017|
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