THE MILNE UNIVERSE AND THE MELIA ‘COSMIC HORIZON’.
By
Richard Blaber.
We revisit the Milne cosmological model and examine Melia’s argument for a ‘cosmic horizon’ in relation to that model, before dismissing both in favour of the standard CDM view.
Imagine a universe which is conformally ‘flat’ and a Ricci-flat manifold on a large scale, i.e., where C = 0 and R(vacuum solution of the Einstein field equations of the General Theory of Relativity, GTR), k4 = 0 in 4-space and [1] k3= –1 in 3-space . The 4-space curvature parameter and Ricci curvature tensor can be equal iff = 0[2],and the GTR does not apply on the large scale, but only on the local scale[3]. = 0 implies, in turn, that the mass density
∞ = 0, which, given that there is mass in this universe, implies that it is a finite mass confined to a finite portion of an infinite volume of 3-space at any given time period, t = age of the universe, tu. The nature of the confinement has changed; at the earliest epoch, tu = 0, the mass was concentrated in a volume of
[1] Here k3 is the 3-space curvature parameter, which is dimensionless and three-valued: –1, when the total mass density parameter, < 1, and space-time is ‘hyperbolic’; 0, when = 1, and space-time is ‘flat’ or Euclidean; and +1, when > 1, and space-time is closed or Riemannian. (Observation indicates that, in our universe, at the present epoch [signified by subscript ‘0’], 0 = 1, k3 = 0.) For more discussion of k4, see below. C is the Weyl conformal tensor (see: Penrose, R [2010], Cycles of Time. An Extraordinary New View of the Universe, London: Vintage Books, pp.130-5, 224 (he refers to C as C); Weyl, H [1918], ‘Reine Infinitesimalgeometrie [Pure Infinitesimal Geometry],’ Mathematische Zeitschrift st 2(3-4):384-411, 1 September 1918, DOI: 10.1007/BF01199420); R is the Ricci curvature tensor, see: Ricci-Curbastro, MMG & Levi-Civita, T (1900), ‘Méthodes de Calcul Différentielle Absolu et leurs Applications [Methods of Absolute Differential Calculus and their Applications],’ Mathematische Annalen 54(1-2):125-201, 1st March 1900, DOI: 10.1007/BF01454201. [2]I.e., the cosmological constant, which appears in the full tensor form of the Einstein 4 equation: R½Rg = T – g, where = 8G/c , g = the metric tensor (see Ricci- Curbastro & Levi-Civita, op.cit.), R the Ricci scalar (scalar curvature), and T the stress- energy-momentum tensor; see: Schoen, R (1984), ‘Conformal Deformation of a Riemannian Metric to Constant Scalar Curvature,’ Journal of Differential Geometry 20:479-95, http://www.math.jhu.edu/~js/Math646/schoen.yamabe.pdf; Kaya, A & Tarman, M (2011), ‘Stress-Energy Tensor of Adiabatic Vacuum in Friedmann-Robertson-Walker Spacetimes,’ Journal of Cosmology and Astroparticle Physics, 2011 Volume, April Issue, DOI: 10.1088/1475-7516/2011/04/040, http://arxiv.org/pdf/1104.5562.pdf. When T = = 0, R = R= 0. If ≠ 0, R = 0, R = –g. [3]The Special Theory of Relativity (STR) would apply at all length scales.
1 space V = 0, a space-time singularity[4], so that then = ∞[5]. As soon as the mass was released from this prison, however, it was liberated into an infinite Euclidean 3-space, with V = ∞, so its density at infinity promptly dropped to ∞ = 0. Its local density at different times since t = 0 has been non-zero positive and finite; in other words, 0 < L ≪ ∞. (However, L → 0 as t → ∞.) Given that it is open, at any given moment t < ∞, the 3-space of this cosmos has negative curvature, and so k3 = –1, as we have seen. The universe we have just described was first adumbrated by EA Milne in 1933[6]. In imaginary Cartesian coordinates, this has the metric[7]:
( ) ( )
(Eq.1a) which is derivable from the Minkowski[8] metric in imaginary Cartesian coordinates, but Milne space only occupies a quarter of Minkowski space[9]:
[4]This was not a curvature singularity, but a Newtonian gravitational singularity, produced by ∇2 = 4G (Poisson’s equation for gravity), where is the gravitational potential in J kg-1 and = –GM/r, where r is distance. When = ∞, r = 0, = –∞. [5]We assume that space-time is continuous and infinitely divisible, not discrete. Thus space- time is not quantised in this universe, and, generally, it is classical rather than quantum mechanical (e.g., there is no Casimir or vacuum energy, see Kaya & Tarman, op.cit.). [6]See: Milne, EA (1933), ‘World-Structure and the Expansion of the Universe,’ Zeitschrift für Astrophysik 6:1-95, BibCode: 1933ZA...... 6....1M, http://articles.adsabs.harvard.edu/cgi- bin/nph- iarticle_query?db_key=AST&bibcode=1933ZA...... 6....1M&letter=0&classic=YES&defaultp rint=YES&whole_paper=YES&page=1&epage=1&send=Send+PDF&filetype=.pdf.. Another feature to note of this universe is that the ratio of the pressure, p (in Pa) to the energy density in J m-3, w = p/c2 = 0/0 = 0. [7]See: Charmousis, C, ‘Introduction to Anti de Sitter Black Holes,’ in Papantonopoulos, E, ed. (2011), From Gravity to Thermal Gauge Theories: The AdS/CFT Correspondence, Part I, Berlin: Springer Verlag, pp.3-26, http://www.physics.ntua.gr/cosmo09/Milos2009/Milos%20Talks%202009/1st%20day/Charm ousis%20Paper.pdf, p.3; here and in Eq.1b, represents proper ‘time’ (in units of imaginary length; not to be confused with the Hubble time, = H-1); and H is the Hubble parameter. ‘AdS/CFT correspondence’, also referred to in the literature as ‘Maldacena duality’ or ‘gauge/gravity duality’ is an abbreviation of ‘Anti-de Sitter space/Conformal Field Theories’, and refers to the fact that the conformal boundary of anti-de Sitter 3-space ( < 0) can be treated as the space-time of a conformal field theory (quantum field theory). See: Maldacena, J (1998), ‘The Large N Limit of Superconformal Field Theories and Supergravity,’ Advances in Theoretical and Mathematical Physics 2:231-52, http://arxiv.org/pdf/hep- th/9711200v3.pdf. [8]Minkowski, H (1909), ‘Raum und Zeit [Space and Time],’ Jahresbericht der Deutschen Mathematiker-Vereinigung 18:75-88, English trans. at: http://en.wikisource.org/wiki/Translation:Space_and_Time, pp.5-9. Strictly speaking, expressing Minkowski space in terms of imaginary or complex dimensions gives it a metric
2
( ) ( )
(Eq.1b)[10]
Instead of considering the local universe, UL (defined below), as a finite portion of an infinite volume of Euclidean 4-space containing a finite quantity of mass- energy, so that k4 = 0, but insufficient for closure of the local 3-space, so that k3 = –1, and that local 3-space is consequently hyperbolic (or ‘Bolyai- Lobachevskian’), let us instead begin by accepting Milne’s basic premise that cosmic observers, wherever they may be located, are equivalent (he calls this ‘The Principle of Extended Relativity’[11]) and adopt an ‘observer-centric’[12] view of the universe, which, in the case of an Earth-based observer, would be a geocentric one. One might, with a high degree of accuracy, describe this as subjective Ptolemaic (or neo-Ptolemaic) cosmology. Next, however, in place of the hyperbolic 3-space, let us substitute a flat (k3 = 0; = 1; = 0) local 3-space, with pressure p = 0 (and thus w = 0). Such a space would constitute a finite hyperplane of a static infinite 4-space co- extensive with ℂ4[13], and thus, if our local volume of 3-space is the only part of this 4-space to contain any mass-energy, ∞ = 0 and k4 = 0. The metric of the 4- space is the Minkowski metric (Eq.1b, see n.8); the metric of the 3-space is for a
signature of +, +, +, + (as opposed to –, +, +, + or +, –, –, –), which converts it into a complex Euclidean 4-space, or complex Hilbert 4-space (see: Bierens, HJ [2007], ‘Introduction to Hilbert Spaces,’ http://econ.la.psu.edu/~hbierens/HILBERT.PDF, pp.1-5, 17). [9]See: Peter, P & Uzan, J-P (2009), Primordial Cosmology, Oxford: OUP, p.144. [10]See: Tegmark, M (2005-6), ‘General Relativity. MIT Course 8.033, Fall 2005, last revised November 7 2006,’ http://ocw.mit.edu/courses/physics/8-033-relativity-fall- 2006/readings/gr.pdf, p.2. [11]See: Robertson, HP (1933), ‘On E.A. Milne’s Theory of World Structure,’ Z für Astrophys 7:153-66, BibCode: 1933ZA...... 7..153R, http://articles.adsabs.harvard.edu/cgi- bin/nph- iarticle_query?db_key=AST&bibcode=1933ZA...... 7..153R&letter=0&classic=YES&default print=YES&whole_paper=YES&page=153&epage=153&send=Send+PDF&filetype=.pdf, pp.154-65. (It is, incidentally, quite wrong for Robertson to refer to Milne’s observers as ‘privileged’: they are anything but privileged, given their equivalence. Milne merely treats each observer’s viewpoint as if it was the centre of the universe; which, from the observer’s perspective, it is.) [12]‘Aisthetescentric’ will probably not have the elegance to thrive as a word. [13]Co-extensive with ℂ4, the set of all imaginary or complex numbers extended in 4 coordinate axes or dimensions from some origin point (Cartesian coordinate system), as opposed to ℝ4, the set of all real numbers extended over four dimensions. This is because we are including time, but measuring time (ict) using imaginary length.
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[14] complex Hilbert 3-space . The deceleration parameter, q0, at the present epoch, is then given by:
(Eq.2)
We find that R = c/H = c = RH, the Hubble radius. Furthermore, the Schwarzschild radius of the mass of the local universe, ML, is given by:
(Eq.3)[15]
If the above prevails, then our universe constitutes the interior of a large black [16] hole , and the distance RS = RH an event horizon, as defined by Rindler (1956)[17]:
‘An event horizon, for a given fundamental observer A, is a (hyper-) surface in space-time which divides all events into two non-empty classes: those that have been, are, or will be observable by A, and those that are forever outside A’s possible powers of observation. ... A
[14]See n.8 above. The metric reads: ds2 = (dx2 + dy2 + dz2). The space is a Euclidean subspace or subset of ℝ3, as well as of ℂ4, and is a subspace of ‘Minkowski’ space, see Eq.1b. [15] 3 Thus ML = c /2G = constant, given constant values for c, G and , the last also implying constant H (= -1). If H = 67.4 km s-1 Mpc-1 = 2.184 × 10-18 s-1 (see: Ade, PAR, et al [2013], ‘Planck 2013 results. XVI. Cosmological parameters,’ http://arxiv.org/pdf/1303.5076v2.pdf, 52 p.12), then ML = 9.2428 × 10 kg. If ≠ constant, M can still be a constant, provided that -2 ≠ 0; = constant (see below) and k = 0, m ∝ . [16]This black hole’s metric is not the Schwarzschild metric, because that applies to the space- time in the immediate vicinity of a non-rotating, electrically neutral black hole, rather than inside it. See: Schwarzschild, K (1916), ‘Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie [On the gravitational field of a mass point according to Einstein’s theory],’ Sitzungberichte der Königlich Preussischen Akademie der Wissenschaften 7:189-96, BibCode: 1916AbhKP1916..189S, http://articles.adsabs.harvard.edu/cgi-bin/nph- iarticle_query?1916AbhKP1916..189S&data_type=PDF_HIGH&whole_paper=YE S&type=PRINTER&filetype=.pdf; Eng trans by Antoci, S & Loinger, A (1999), http://arxiv.org/pdf/physics/9905030v1.pdf. The universe’s actual radius would be given by Ru = ctu < RH = RS. [17]Rindler, W (1956), ‘Visual horizons in world models,’ Monthly Notices of the Royal Astronomical Society 116(6):662-7, BibCode: 1956MNRAS.116..662R, http://adsabs.harvard.edu/full/1956MNRAS.116..662R, p.663.
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particle horizon, for any given fundamental observer A and cosmic instant t0 is a surface in the instantaneous 3-space t = t0, which divides all fundamental particles into two non-empty classes: those that have already been observable by A at time t0 and those that have not’ (ibid., my emphases).
We know from the work of Vesto Slipher[18], Georges Lemaître[19] and Edwin Hubble[20] that galaxies beyond the local group of galaxies[21] are receding from us with a velocity that increases in direct proportion to their distance from us, vr = Hd. At the Hubble distance, vr ≡ c, by definition. In the universe we are describing, and given that, by the STR, no object with non-zero rest mass may travel with v ≥ c, and a black hole has an escape velocity, vesc > c, then, as Melia, Melia & Abdelquader and Melia & Shevchuk[22] argue, the surface of the Hubble sphere (or Hubble volume) constitutes an event horizon as defined by Rindler (op.cit.), or what they term a ‘cosmic horizon’.
[18]Slipher, VM (1913), ‘The Radial Velocity of the Andromeda Nebula,’ Lowell Observatory Bulletin II(8):56-7, http://www.roe.ac.uk/~jap/slipher/slipher_1913.pdf; idem (1915), ‘Spectrographic Observations of Nebulae,’ Report of the 17th Meeting of the American Astronomical Society, Popular Astronomy 23:21-24, http://www.roe.ac.uk/~jap/slipher/slipher_1915.pdf; idem (1917), ‘Nebulae,’ Proceedings of the American Philosophical Society 56:403-9, http://www.roe.ac.uk/~jap/slipher/slipher_1917.pdf. [19]Lemaître, G (1927), ‘Un univers homogène de masse constante et de rayon croissant, rendant compte de la vitesse radiale de nébuleuses extra-galactiques [A homogeneous universe of constant mass and increasing radius accounting for the radial velocity of extra- galactic nebulae],’ Annales de la Société Scientifique de Bruxelles, Sèrie A, 47, BibCode: 1927ASSB...47...49L, http://articles.adsabs.harvard.edu/cgi-bin/nph- iarticle_query?1927ASSB...47...49L&data_type=PDF_HIGH&whole_paper=YES &type=PRINTER&filetype=.pdf. [20]Hubble, EP (1929), A Relation Between Distance and Radial Velocity Among Extra- Galactic Nebulae,’ Proceedings of the National Academy of Sciences of the United States of America 15(3):168-73, 15th March 1929, http://www.pnas.org/content/15/3/168.full.pdf+html. [21]This includes, besides the Milky Way Galaxy, its satellite system, the Andromeda Galaxy and its satellite system, the Triangulum Galaxy, Pisces Dwarf, and some others. See: http://en.wikipedia.org/wiki/Local_Group. [22]Melia, F (2007), ‘The cosmic horizon,’ MNRAS 382(4):1917-21, 16th Nov 2007, DOI: 10.1111/j.1365-2966.2007.12499.x, http://mnras.oxfordjournals.org/content/382/4/1917.full.pdf; idem (2009), ‘Constraints on Dark Energy from the Observed Expansion of Our Cosmic Horizon,’ International Journal of Modern Physics D 18(7):1113, July 2009, DOI: 10.1142/S0218271809014984, http://arxiv.org/pdf/0812.4778.pdf, 15 pp.; idem & Abdelquader, M (2009), ‘The Cosmological Spacetime,’ Int J Mod Phys D 18(12):1889, 30th November 2009, DOI: 10.1142/S0218271809015746, http://arxiv.org/pdf/0907.5394.pdf, 13 pp.; idem & Shevchuk, ASH (2012), ‘The Rh = ct universe,’ MNRAS 419(3):2579-86, Jan 2012, DOI: 10.1111/j.1365-2966.2011.19066.x, http://arxiv.org/pdf/1109.5189.pdf.
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It must be borne in mind that, in this cosmos, space itself is not expanding: matter and energy are merely occupying more of the space available as time progresses. Thus the comoving coordinates and distances of the Friedmann-Lemaître-Robertson-Walker (FLRW)[23] metric do not apply. Distances are measured using the travel times (tltt) of electromagnetic radiation from objects emitting such radiation = tu – tem, where tu is the present age of the universe and tem is the time the electromagnetic radiation was emitted. For distant extragalactic objects, the Lemaître-Hubble formula yields:
≪ √( )
(Eqs.4a-c)
At recessional velocities vr ≪ c, z ≅ ≡ vr/c. Thus, z = 0.008 implies a vr ≅ -1 0.008c = 2,398.339664 km s and a light travel time distance of Dltt = ctltt = d ≅ zc = 1.098 × 1021 km ≅ 116 million light years. This is the approximate distance of galaxies such as the ‘Siamese Twin Galaxies’, in the Virgo Cluster (VCC 1673 and VCC 1676), NGC 4567 and NGC 4568[24]. On the other hand, if vr = 99.9% c, = 0.999 and z = 43.71, zc would give 43.71 c, so we must use vr ≡ c, and derive the value of from:
[23]See: Friedmann, A (1922), ‘Über die Krümmung des Raumes [On the Curvature of Space],’ Zeitschrift für Physik 10(1):377-86, BibCode: 1922ZPhy...10..377F; idem (1924), ‘Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes [On the Possibility of a World with Constant Negative Space Curvature],’ Z für Phys 21(1):326-32, BibCode: 1924ZPhy...21..326F; Lemaître, op.cit.; idem (1931), ‘The Beginning of the World from the Point of View of Quantum Theory,’ Nature 127(3210):706, 9th May 1931, DOI: 10.1038/127706b0, http://www.nature.com/nature/journal/v127/n3210/full/127706b0.html; idem (1933), ‘L’Univers en Expansion [The Expanding Universe],’ ASSB A 53:51, BibCode: 1933ASSB...53...51L; Robertson, HP (1933), ‘Relativistic Cosmology,’ Reviews of Modern Physics 5(1):62-90, 1st Jan 1933, BibCode: 1933RvMP....5...62R, DOI: 10.1103/RevModPhys.5.62; Walker, AG (1933), ‘Distance in an expanding universe,’ MNRAS 94:159-67, BibCode: 1933MNRAS..94..159W, http://articles.adsabs.harvard.edu/cgi-bin/nph- iarticle_query?db_key=AST&bibcode=1933MNRAS..94..159W&letter=0&classic=YES&de faultprint=YES&whole_paper=YES&page=159&epage=159&send=Send+PDF&filetype=.p df. [24]See photograph and information at: http://www.spotastro.com/NGC_4567_Wide.html. NGC 4567 and NGC 4568 were discovered in March 1784 by William Herschel, see: http://cseligman.com/text/atlas/ngc45a.htm#4567. According to the SIMBAD database, -1 NGC 4567’s vr = zc = 2,278.4 km s , its redshift z = 0.0076, and its maximum distance = 19.96 Mpc = 115.7868 Mly, http://simbad.u-strasbg.fr/simbad/sim- id?Ident=%401961459&Name=NGC++4567&submit=display+all+measurements#lab_meas.
6
( )
(Eq.5) finding that = 0.999. While z can exceed 1, cannot; indeed, as vr < c, < 1. Thus, the Hubble radius, RH = c, is an absolute limit, which can be approached, [25] but never reached or crossed. As the Hubble time is ~14.51 Gy , RH ≅ 14.51 Gly. The current age of the universe is estimated to be 13.813 Gy (see: Ade, et al, op.cit.). This means that no object in the universe can be older than 13.813 Gy, or more distant than 13.813 Gly. The most distant objects, on the strength of the argument we have presented, must have recessional velocities of vr = 285,407.46 km s-1, = 0.952 and redshifts, z = 5.377. However, redshifts of z = 1100 have been measured, and z = 90,000,000 is theoretically possible[26]! Consequently, we are left with no alternative but to accept that the Milne picture is wrong[27], and that space is expanding, in accordance with the (modified) FLRW metric (when k = 0, and again using imaginary length for proper time and time):
( ) ( ) ( )( )
(Eq.6) where R(tu0), the scale factor, can be set at 1 at the present epoch. The distances measured by (x, y, z) are then comoving distances as opposed to light travel time distances[28]. However, what matters is the rate at which space is expanding; if
[25]1Gy = 1 billion years. [26]This is the redshift of electromagnetic radiation emitted ~49.293 min after the Big Bang; see: Wright, EL (2006), ‘A Cosmology Calculator for the World Wide Web,’ The Publications of the Astronomical Society of the Pacific 118(850):1711-15, BibCode: 2006PASP..118.1711W, DOI: 10.1086/510102, http://www.astro.ucla.edu/~wright/CosmoCalc.html. In practice, we would have difficulty detecting this, as it originates well before the de-ionisation, or ‘recombination’, era, some 300,000 years after the Big Bang (see: Ryden, B [2003], ‘Cosmic Microwave Background,’ March 2003, http://www.astronomy.ohio-state.edu/~ryden/ast162_9/notes39.html). [27]This does not mean, however, that his ‘Principle of Extended Relativity’ is wrong. The observable universe is still centred on the locus of each hypothetical observer, and the locus of the only observers we know of, as opposed to speculate about, is Earth. [28]A comoving distance is a cosmological distance that does not change over time due to the metric expansion of space, in contrast to a proper distance, which does (although the ratio of the two at the present epoch, with the scale factor equal to 1, is 1). The comoving distance (both radial or line-of-sight and transverse, if k = 0) of an extragalactic object is given by
∫ , where DH is the Hubble distance and E(z) is the dimensionless Hubble ( )
7 the 3-space volume of the Hubble sphere is expanding (i.e., if the Hubble ‘constant’ is not a genuine constant, but a parameter), then if vr = c:
( )
(Eqs.7a-d) the only difference being that vr can exceed c, and at the Hubble distance RH ≅ 14.51 Gly (theoretical light travel time distance), z = 1.479, comoving radial [29] distance, DC ≅ 14.51 Gly and actual light travel time, tlt = 9.492 Gy . This 52 local universe (Hubble volume) still has a mass given by ML = 9.2428 × 10 kg, but its cosmological constant, ≠ 0, a genuine constant, given by
(Eq.8) where = 0.686 is the current mass density parameter generated by ‘dark energy’ or the cosmological constant (see Planck satellite results in Ade, et al, op.cit., p.12.), meaning that 68.6% of all the mass of the universe is in the form 2 of dark energy, and ∝ , the square of the Hubble time. It is this dark energy that ensures that we can be living inside an enormous black hole, and yet not have to worry – for it is more than sufficient to overcome the gravitational force that would otherwise eventually crush the universe back into a space-time singularity[30]. Not only is space expanding: the expansion is accelerating. (How fast depends on the precise nature of the ‘dark energy’ and the value of w[31].) The thesis proposed by Melia and his colleagues (opp.cit.)[32] that the
3 parameter ≡ √[m(1 + z) + ] (see: Hogg, DW [2000], ‘Distance Measures in Cosmology,’ http://arxiv.org/pdf/astro-ph/9905116v4.pdf, pp.1-4). [29]See Wright, op.cit. [30] In a universe where = 0, if RH < RS, > 1, k = 1, and space-time is closed/Riemannian. It expands to a maximum extent, then shrinks to zero volume, infinite density. [31]If dark energy is some form of ‘quintessence’, then, in the case of ‘phantom energy’, with w < –1, the ‘Big Rip’ scenario prevails, see: Caldwell, RR, Kamionkowski, M & Weinberg, NN (2003), Phantom Energy: Dark Energy with w < –1 Causes a Cosmic Doomsday,’ Physical Review Letters 91:071301, 13th August 2003, DOI: 10.1103/PhysRevLett.91.071301, http://arxiv.org/pdf/Astro-ph/0302506.pdf; the ‘Little Rip’ does if w < –1 now but w → –1 asymptotically; the ‘Pseudo-Rip’ if w → constant < –1; and the ‘Quasi-Rip’ if w < –1 now and w > –1 much later. See: Frampton, PH, Ludwick, KJ & Scherrer, RJ (2011), ‘The little rip,’ Physical Review D 84:063003, 6th Sept 2011, DOI: 10.1103/PhysRevD.84.063003, http://arxiv.org/pdf/1106.4996.pdf?origin=publication_detail; idem (2012), ‘The Pseudo-rip: Cosmological models intermediate between the cosmological
8 surface of the Hubble sphere constitutes an event horizon, as defined by Rindler (op.cit.), which has been refuted by Van Oirschot, Kwan & Lewis (2010)[33], is therefore insupportable; it is not even a Rindlerian particle horizon. This honour is reserved for the surface of the sphere defined by the observable universe’s current radius, Robs = 46.156 Gly (tlt = 13.813 Gy = tu, see Wright, op.cit.34). Beyond that lies an unobservable infinity, containing an infinity of ‘parallel universes’[35], all contained within a quasi-Euclidean 4-space[36]. constant and the little rip,’ Physical Review D 85:083001, 3rd April 2012, DOI: 10.1103/PhysRevD.85.083001, http://arxiv.org/pdf/1112.2964.pdf?origin=publication_detail; Brevik, I, Obukhov, VV & Timoshkin, AV (2013), ‘Quasi-Rip and Pseudo-Rip universes induced by the fluid inhomogeneous equation of state,’ Astrophysics and Space Science 344(1):275-9, March 2013, DOI: 10.1007/s10509-012-1328-7, http://arxiv.org/pdf/1212.0391v1.pdf, p.275=p.1. A value for w = –1 is consistent with a ‘heat death’ scenario for the end of the universe; see: Krauss, LM & Starkman, GD (2000), ‘Life, the Universe, and Nothing: Life and Death in an Ever-expanding Universe,’ The Astrophysical Journal 531(1), 1st March 2000, DOI: 10.1086/308434, http://arxiv.org/pdf/astro-ph/9902189.pdf?origin=publication_detail (arxiv copy has 22 pp.). [32]Melia and his colleagues try to support their argument by urging that w = –⅓; however, as Ade, et al, op.cit., p.39, point out, the empirical value of w, from various combined sources of data, including the Planck and Wilkinson Microwave Anisotropy Probe (WMAP) satellites, Baryon Acoustic Oscillation (BAO) observations, and so on, is ≅ –1.13, which, as the authors point out, is consistent with the CDM model’s predicted w = –1. [33]Van Oirschot, P, Kwan, J & Lewis, GF (2010), ‘Through the looking glass: why the “cosmic horizon” is not a horizon,’ MNRAS 404(4):1633-8, DOI: 10.1111/j.1365- 2966.2010.16398.x, http://mnras.oxfordjournals.org/content/404/4/1633.full.pdf. [34] -1 -1 Using a value of 67.356 km s Mpc for H0, slightly different from that in Ade, et al, op.cit., p.12. [35]The ‘Level I’ Multiverse; see: Tegmark, M, ‘The Multiverse Hierarchy,’ in Carr, B, ed (2007), Universe or Multiverse? Cambridge:CUP, http://arxiv.org/pdf/0905.1283v1.pdf, pp.1-4. Tegmark’s use of the term ‘Hubble volume’ is mistaken; this should only refer to the volume of space whose radius is given by RH, the Hubble radius. See: Seshavatharam, UVS & Lakshminarayana, S (2012), ‘Hubble Volume and the Fundamental Interactions,’ International Journal of Astronomy 1(5):87-100, DOI: 10.5923/j.astronomy.201201015.03, http://article.sapub.org/pdf/10.5923.j.astronomy.20120105.03.pdf, p.88. [36] 2 2 2 2 2 The metric of this quasi-Euclidean space is ds = dx1 + dx2 + dx3 + dx4 , with x1 = x, x2 = y, x3 = z and x4 = ict (Minkowski metric). It is reasonable to assume that this space does not expand, but is already as infinite as it can be, with all (or most) of the universes of the Level I Multiverse expanding outwards into it. That there are smaller and greater infinities can be demonstrated from the cardinalities of the sets ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ, with ℵ0 < ℵ1, ℕ being the set of all natural numbers, ℤ that of all integers, ℚ the set of all rational numbers, and ℝ and ℂ as previously defined. ℵ0 (‘aleph-null’) is the cardinality of ℕ, ℤ, and ℚ, which are n n countably, or denumerably, infinite. ‘Aleph-one’, ℵ1 = 했 = |ℝ| = |ℝ | = |ℂ| = |ℂ | > ℵ0 is the cardinality of ℝ and ℂ, which are both uncountably infinite. The cardinality of the quasi- 4 4 Euclidean space ℂ is thus ℵ1 = 했 = |ℂ |. See: http://www.encyclopediaofmath.org/index.php/Continuum_hypothesis; http://mathworld.wolfram.com/Aleph-0.html; http://mathworld.wolfram.com/Aleph-1.html; http://en.wikipedia.org/wiki/Cardinality_of_the_continuum.
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