Parameters of the Selfvariating Universe (SVU)

Parameters of the Selfvariating Universe (SVU) Kyriaos Kefalas () (2) () Department of Physis, Astrophysis Laboratory, National and Kapodistrian University of Athens, Panepistimioupolis, GR 5 783 Zographos, Athens, Greee (2) Institute for the Advanement of Physis and Mathematis, 3 Pouliou str., 523 Athens, Greee email : refalas@ apminstitute.org Abstrat The selfvariating universe (SVU) is the osmologial model of the Theory of Selfvariations (TSV) proposed by Manousos in (Manousos 2007) and (Manousos 203 b). The Theory of Selfvariations (TSV) relies on the selfvariation priniple and on the zero ar length propagation priniple (Manousos 203 b, 203 d). The selfvariation priniple predits a slight inrease of the rest mass of material partiles and of the absolute eletri harge of partiles of matter. The selfvariating universe (SVU) is a stati and flat universe, where osmologial redshift is explained without the need of the expansion. It predits an exponential law for the osmologial redshift / distane relation (Manousos 203 b, 204 a) and a variation of the fine struture onstant (Manousos 203, 9), whih has been measured by Webb (20). In this paper we investigate the onsequenes of a larger set for the SV parameters than the one used by Manousos in (Manousos 204 a,, 3). The investigation leads to a narrow set of possible values. However to aurately fix all SV parameters, a preise determination of the distane of astronomial objets, independent of any osmologial model and a more aurate measurement of the variation of the fine struture onstant is needed. The disovery of the variation of α, is a strong support for the TSV, as no other mainstream theory an explain it. We have found that in one ase, the SV universe (whih begun with almost zero rest masses) is heading to a reversal proedure where the SV rest mass unit will beome infinite. After that, it enters a reversed phase where SV rest mass unit aquires negative values. The reversal time T R, lies in the future and an be no less than 3 Bill years from now. We also predit an anisotropy of osmologial redshift, for objets of the same distane and peuliar veloity. Keywords : Theory of Selfvariations (TSV), Selfvariating universe (SVU), Cosmologial redshift, SV rest mass unit, SV eletri harge unit, Selfvariation parameters, Non expanding universe, Stati universe, Fine struture onstant. Introdution.. The fundamental laws of the selfvariating universe (SVU) The selfvariating universe (SVU) is the osmologial model of the Theory of Selfvariations (TSV) proposed by Manousos in (Manousos 2007) and desribed in detail in (Manousos 203 b, 204 b). The SVU is based on the following laws of the TSV : Law of rest mass selfvariation Every rest mass m 0, in the universe inreases slightly with time, aording to the equation : t m 0 (t) - b ħ 2 t m 0 (t) m 0 (t) = 0 () where, is the speed of light in vauum, ħ, is the redued Plan onstant and b, is a dimensionless arbitrary onstant (Manousos 203 b, 7.2, 203 d, 3.4). All three onstants,, ħ, b, are onsidered by the TSV to be universal spaetime onstants. Law of eletri harge selfvariation The eletri harge q, of material partiles in the universe varies slightly with time, aording to the equation : t q(t) - b e ħ t q(t) = 0 (2) V e q(t) where, V e, is the selfvariation eletri potential whih aording to (Manousos 203 b, 7.2,), 3, 4, where the potential V e, is denoted as V 0, an vary slightly, taing positive or negative values. The value of the selfvariation potential V e, whose units

2 SVP 0 vd, Parameters of the SVU, De 204.nb are, J C -, depends on the partile involved and on the distribution of harges in the surrounding spaetime. The parameter b e, is a dimensionless arbitrary universal spaetime onstant. The selfvariation law requires that the absolute harge of partiles of matter slightly inreases with time. The SVU model predited by the TSV is an overall flat, stati universe with overall zero energy density, where expansion is not needed to explain osmologial redshift and other osmologial data. It has immense (possibly infinite) spatial and temporal dimensions whih greatly exeed the urrently observed universe and has existed for an indefinite period of time (Manousos 203, 2, 203 d, 3). In the SVU model the universe did not begun with a Big Bang and there is no need for an inflationary phase to explain its large sale features. In ontrast it quietly begun from a vauum (or nearly vauum) state with an impereptible inrease of the rest mass of material partiles. This inrease is ongoing sine then. Hene osmologial redshift is explained by the impliation that, partiles of matter of distant astronomial objets had smaller rest mass (and absolute eletri harge) in the past and therefore emitted photons with larger wavelengths on the first plae. Generally it is assumed that these photons have traveled through spaetime with unaltered wavelength. The TSV also predits that the SV potential is suh that the absolute eletri harge of antimatter partiles dereases with time, leading to neutral partiles. This (partly) explains the matter antimatter asymmetry we observe today. One of the most surprising results of the TSV is that there is a unique way aording to whih the selfvariations an happen (Manousos 203 b, 203 ). The selfvariations are quantified in the SVU model by the laws of selfvariations (), (2), whih are the only possible assuming the priniples of Relativity (inorporated in the TSV as the zero ar length propagation priniple). The TSV predits that the energy inrease due to the inrease of a partile ' s rest mass aused by the selfvariation, is ounterbalaned by the ontinual emission of generalized photons from the partile (Manousos 203 b), suh that total energy is onserved. The result is that the total energy ontent of the SV universe is the same at all times and overall amounts to zero energy density. For more information about matter - antimatter asymmetry see (Manousos 203 d, 3). For more information about the CMBR anisotropy see (Manousos 203 d, 6). For information about generalized photons see (Manousos 203 b). 2. Solutions of the selfvariation equations for the SVU 2.. The SV rest mass and SV eletri harge units In this paper we will use the notation, log x log e x, for the natural logarithm. The general solutions of equations (), (2) whih give the rest mass and eletri harge are : m 0 (t) = b ħ 2 - e (t+μ) (3) q (t) = b e e ħ V e - e e (t+μ e ) where, e, and μ, μ e, are onstants of integration having units s - and s, respetively. Manousos in (Manousos 203 b) investigates the onsequenes of eq. (3), (4). He defines the parameter, A = e (t+μ), and onsiders the ase where, b =, > 0, and, 0 < A <. In (Manousos 203 d) he does a somewhat different investigation for eq. (e4). In this paper we want to enlarge the investigation of the above equations and for this purpose we start with as few assumptions for the parameters as possible. As restritions for the parameters follow from neessary or plausible impliations of the assumptions, we will write them down as parameter onditions (C n). At the end we will sum everything together. To simplify our exposition we define a spaetime oordinate system, x = (r, θ, φ, t) = (r, t), entered on Earth now, hene the urrent osmi time is, t 0 = 0. Let, μ = l + log b, μ e = l e + log b e, where l, l e, have time units. We also define the dimensionless parameters, A = e (t+l), e B e = e e(t+l e ), A0 = e l, and, B 0 = e e l e. Then we have : (4)

2/0/5 SVP 0 vd, Parameters of the SVU, De 204.nb 3 m 0 (t) = b ħ 2 - b e (t+l) = b ħ 2 m 0 = m 0 (0) = b ħ 2 - b A (5) - b A 0 (6) where m 0, is the urrent, the same everywhere in the universe. Also onsidering that the SV eletri potential V e, may generally depend on x, we have : q (t) = b e e ħ V e - b e e e (t+l e ) = b e e ħ V e q 0 = q (0) = b e e ħ V 0 - b e B e (7) - b e B 0 (8) q E = b e e ħ V E - b e B 0 (9) where q 0, V 0, are the urrent loal values of q, V e, at position r, and q E, V E, are the urrent Earth values of q, V e. Notie that the earth value of the SV eletri harge, depends on t : q E (t) = b e e ħ V E (t) - b e B e (0) The parameters b, b e, are arbitrary dimensionless multipliers for the rest mass and eletri harge, so we only onsider the values, b, b e = ±. For these values the above equations represent rest mass and eletri harge units, whih from now on will be alled the SV rest mass unit and the SV eletri harge unit respetively. Eqs. (5-0) give the SV units in various ases. In this paper we only onsider partiles with finite non zero rest mass and eletri harge, so we require that,, e, V e 0. In partiular we onsider partiles with finite non zero urrent and loal SV rest mass and eletri harge units. All these restritions onstitute our first parameter ondition (C0) : b, b e = ±, e, V e 0 If, b = t + l, l 0 If, b e = t + l e, l e 0 (C0) V 0, V E 0 If, b = A, A 0 If, b e = B e, B 0 Notie that the parameters A, B e, e, are not exatly the same as the A, B,, parameters used by Manousos in his papers (Manousos 203 d, 2, 4). 2.2. Derivatives of the SV units We onsider the time derivatives of the SV units onsidering, t A = A, t B e = e B e, and, b 2 = b e 2 =, whih must satisfy the requirements of the selfvariation laws : t m 0 (t) = 2 ħ 2 A ( - b A) 2 = 2 A m 0 (t) 2 > 0, t A, A 0 > 0 () ħ Suffiient (but however not neessary) onditions for () are :, t, l. We also have : t m 0 (t) m 0 (t) = b A - b A If we assume that the SV eletri potential V e, is time independent, we have for partiles of matter : t q(t) = e 2 ħ V e B t q (t) = q (t) V e B e q (t) ħ (sine, q(t) 0) ( - b e B) 2 (2) = b e e B e - b e B e > 0, t (3) As above we only onsider real values of the parameters, e, l e, (whih in this ase are neither suffiient nor generally

4 SVP 0 vd, Parameters of the SVU, De 204.nb only parameters, e, e, ( generally neessary onditions to satisfy eq.(3)). Then : B e, B 0 > 0 q(t) V e > 0 b b e = b e = - e e > 0 e > 0 e < 0 - b e - + B B e e B e All these restritions onstitute the parameter ondition (Ca) for matter partiles :, e, t, l, l e A, A 0 > 0 B e, B 0 > 0 (Ca) Partiles of matter - Time independent V e b e = b e = - e < 0 B e, B 0 > e > 0 e < 0 0 < B e, B 0 < B e, B 0 > 0 (Cb) q(t) V e > 0 q 0 V 0 > 0 q E V E > 0 If more generally we assume that the SV eletri potential V e, is time dependent, we have : t q(t) = q(t) V e B e q(t) ħ - t V e V e Hene the parameter onditions for matter partiles are : = b e e B e - b e B e - t V e V e > 0, t (4) Partiles of matter - Time dependent V e b e = b e = - e B e - t V e > 0 e B e + t V e < 0 -B e V e +B e V e (C) In this ase the parameter ondition C is more ompliated and will not be onsidered here in greater detail (it an be found online at : http : // apminstitute.org/ategory/physis /). In any ase and for the rest of this paper, we assume only the parameter ondition Ca and Cb, unless otherwise notied. As we wor through this paper we will gather more onditions for the parameters, implied by various assumptions and observational data and we will try to estimate possible values for the parameters. 2.3. Introdution of the parameters M, and W We define two time dependent parameters, M, W, with units s -, whih satisfy the differential equations : Hene : t M M = t W = t m 0 m 0 = M(0) W 0 = t q 0 W = (5) (6) q 0 = W(0) (7) (8) M = e t W = W 0 e e t (9) (20) where as above we assume that V e, is time independent. These parameters are arefully hosen and onnet the SVU with the SCM.In setion 4.2 we will see how they are related to the Hubble parameter. From eqs. (7), (8), onsidering eq.(2), (3) we have : = b A 0 - b A 0 = W 0 = M = b e - μ - b (2) b e e e - e μ e - be = b e e B 0 - b e B 0 (22) A = b A = b e t A 0 - b A 0 e - μ - b (23)

2/0/5 SVP 0 vd, Parameters of the SVU, De 204.nb 5 B W = W 0 = b e e B 0 = b e e ee t B 0 - b e B 0 e - (24) e μ e - be thus : W 0 A 0 =, and, B 0 = (25) b( + ) b e ( e + W 0 ) We will all M, the Manousos parameter and, the Manousos onstant. From C0, Ca, we have that, M,, W, W 0 0. Below we analyze the onditions under whih eqs. (2), (22), are valid. > 0 M > 0, and, < 0 M < 0, and, 0 < A 0 < > 0 0 < A 0 < < 0 b = A 0 >, or, - < < 0 or b = A 0 >, or, 0 < < - b = - 0 < A 0 A 0-2 -2 < - b = - 0 < A 0 A 0-2 - < -2 (C2a) W 0 > 0 W > 0, and, W 0 < 0 W < 0, and, 0 < B 0 < e > 0 0 < B 0 < e < 0 b e = B 0 >, or, -W 0 < e < 0 or b e = B 0 >, or, 0 < e < -W 0 b e = - 0 < B 0 B 0 e -2 W 0-2 W 0 e < -W 0 b e = - 0 < B 0 B 0 e -2 W 0 -W 0 < e -2 W 0 (C2b) If we ombine C2b with Cb (where the SV potential is assumed time independent ), we see that all ases were, W 0 < 0, are exluded and we are left with the first part of ondition C2b only : W 0 > 0 W > 0 (C2) This is a remarable restrition of the parameter onditions C2b, valid for partiles of matter. In the other ase where the SV potential is time dependent, the ombination of ondition C with C2b, leads to roughly similar results, in the ase that, t V e 0,. Thus we are almost onfident that the urrent onstant W 0, has a positive value in both ases. 3. The fine struture onstant α 3.. Relative SV eletri potential Manousos states in (Manousos 203 d, 3) that the SV eletri potential only slightly varies with position.we will onfirm this in setion 3.4. In this paper we will tae a referene value to be its urrent Earth value V E. Sine, V E 0. we an define a relative SV eletri potential, V R 0, generally depending on x, as : V R = V e V E (26) 3.2. The SV rest mass and SV eletri harge units observed at distane r, from Earth We annot diretly measure the urrent SV rest mass and SV eletri harge units at any partiular x. This is alled by Manousos (Manousos 203 b, 4.9), the internality of the universe in the measurement proess. However we an find the relation between the urrent SV units and the SV units measured at a osmologial distane r, as observed from Earth. The loo ba

6 SVP 0 vd, Parameters of the SVU, De 204.nb time (in the SVU) from a distanes r, is, Δt = r. From eqs. (5), (6), we find the value of the relative SV rest mass unit at the time of photon emission : m 0 (-Δt) = b ħ 2 - b e (-Δt+l) m 0 (r) = b ħ 2 m 0 (r) m 0 = - b e - r +l - b A 0 - b A 0 e - r = + - e - r (27) where, m 0 (r), is the SV rest mass unit observed from Earth at distane r (in our time epoh), relative to its urrent value. m 0 In the same way we get the orresponding equations for the relative SV eletri harge unit from eqs. (7), (9), (26) : q(r) q E = V R - b e B 0 3.3. Distane in the SCM and SVU - b e B 0 e - e r = V R + W 0 e - e - We have to larify the notion of distane we refer to in the SVU model and how this distane relates to the distanes that are in use in the ontext of the SCM. The flat and stati non expanding SVU model is ompatible with a unique distane notion, whih does not overall hange with time. In the expanding universe of the SCM, and partiular in the ontext of the Friedmann Lemaître Robertson Waler solution, we have two distanes whih onern us here; the light travel (or loo ba) distane given by : r L = z d v H 0 0 ( + v) Ω m ( + v) 3 + Ω ( + v) 2 + Ω Λ and the omoving distane given by : r = z H 0 0 Ω m ( + v) 3 + Ω ( + v) 2 + Ω Λ where Ω m, is total matter density, Ω Λ, is dar energy density and, Ω = - (Ω m + Ω Λ ), represents the urvature. The urrent Λ CDM models sets these values at : Ω Λ = 0.728, Ω m = 0.262, and, Ω = 0.002. However we will use a flat universe (Ω = 0), as is predited by the SVU model. The omoving distane of the Λ CDM model, whih is the proper distane at urrent time epoh, is the one that best mathes the distane notion of the SVU. Thus we set : r r These relations will enable us to assign the observed osmologial data (for ex. redshift) to distanes in the SVU model. Notie also that the light travel (loo ba) distane in the SVU, is generally the same as the distane, sine light is generally assumed to be reeived at the same wavelength as it is emitted. Also notie that we do not restrit ourselves to the urrently used set of osmologial parameters given above, but we will explore the whole interval of values : 0 Ω Λ. In partiular we will use four values : Ω Λ = {0, 0.5, 0.728, 0.99}. The reason is that osmologial distanes are not aurately measured independently of the osmology used. Whereas redshift is a diret observable, the urrent proper distane of an astronomial objet is usually only indiretly inferred, via a partiular hoie of a osmology parameter set. Thus hanging the osmology used, results in onsiderable differenes for the ' proper ' distane. 3.4. Variation of the fine struture parameter The definition of the fine struture onstant is : e 2 α = 4 π ε 0 ħ d v e r (28)

2/0/5 SVP 0 vd, Parameters of the SVU, De 204.nb 7 The fine struture parameter is atually a onstant only if all quantities on the right hand side are onstants. Sine the SVU model predits a variation of the fundamental eletri harge e, it omes to no surprise that J. K. Webb (Webb 20) measured a variation in α, Δα = α(r) - α ±0-5 (29) α α at light travel SCM distanes a, r L 3 Gp. This variation an be regarded as a onfirmation of the priniples of the Theory of Selfvariations (at least of the eletri harge selfvariation), sine it is not predited by the usual physial theories (Manousos 203, 9). In the SVU model this variation is explained by the dependene of the SV eletri harge unit on r, and V e. Setting : α(r) = e(r)2 4 π ε 0 ħ where, e(r), is the fundamental eletri harge observed urrently from Earth at distane r, and e, is the urrent fundamental eletri harge (i.e. the eletron or proton harge) as measured urrently on Earth. From eq. (28) we have : (30) Δα = e(r)2 α e 2 - = q(r)2 - = q2 E V 2 R + W 0 - e - e r e 2 - ±0-5 (3) Two fators influene the value of α. The flutuation of the (relative) SV eletri potential V R, and the selfvariation of the eletri harge, expressed here by the fator, + W 0 - e - e r. If we ignore the eletri harge selfvariation assuming, W 0 e - e - e r 0, we have a variation in α, oupled only with the SV eletri potential : Δα = α V 2 - (32) R If we ignore the relative SV eletri potential assuming, V R, we have a variation in α, oupled only with the eletri harge selfvariation : Δα = α + W 0 - e - e r e 2 - (33) The total effet stems from both fators and we want to estimate their relative strength. The measurements of Webb, King et al. have found that the variation of α, has a large sale dipole form, where at opposite poles, Δα, has opposite signs approximately, Δα ±0-5, at light travel SCM distanes a 3 Gp. Also there seem to be random flutuations at smaller α sales Δα ±0-5, at distanes 3 Gp (Webb 20). Both these fats are ompatible with an underlying flutuation of α V R. Contrariwise the selfvariation itself, requires an isotropi derease of α, in all diretions of the sy. In (Webb 20), (King 202), Webb, King et al., indiate a small differene between the absolute average values of Δα, at the two poles : Δα + + Δα - -0. 0-5 α α Sine they averaged data from a lot of soures, measured with two different telesopes (Ke and VLT), it is possible that this differene, is the true manifestation of the eletri harge selfvariation. In any ase, from these data, we estimate an upper bound for the underlying isotropi derease aused by the eletri harge selfvariation, of about δ -0-6 e

8 SVP 0 vd, Parameters of the SVU, De 204.nb (34) for an SCM light travel distane, r L 3 Gp, whih orresponds, z =.5, and thus to omoving distanes of the Λ CDM model for various values of Ω Λ : Ω Λ r (Gp) 0 3.25 0.5 4.0 0.728 4.6 0.99 6.5 The omoving distane intended by Webb is, 4.6 Gp (Ω Λ = 0.726, was used in (Webb 20)). Sine we onsider that the atual proper SVU distane has to be measured by other means, independent of any osmologial model, we will use in this paper an even wider range of, 3 r 8 Gp, as a possible range of SVU distanes, orresponding to the SCM light travel distane, r L 3 Gp, in Webb ' s paper. The distane unertainty of r, might loo huge at, z =.5, however as the unertainty inreases with z, the orresponding unertainty at for ex., z = 0., is only 0 %, orresponding to distanes between, 400-445 Mp; an unertainty margin whih might indeed seem too narrow (see Fig.3). From eq. (33) we have : 0 > + W 0 - e - e r e 0 < W 0 - e - e r e 2 - δ or -2 + δ 2 W 0 - e - e r < -2 e +δ - - δ 2 where for this and the following equations we use inequalities, sine we onsider that, δ -0-6, is an upper bound for the manifestation of the eletri harge selfvariation. Form ondition Cb (i.e. q(t) V e > 0, q E V E > 0) and eq. (28), we infer that, + W 0 - e - e r e latter ondition and eep only : From C3 we have : 0 < W 0 - e - e r - δ 5 0-7 e 2 (C3) where, r < r, and, 3 r 8 Gp, orresponding to SCM light travel distane, r L 3 Gp 0 < W 0 - δ e 2 - e - e r - > 0. Hene we disard the The parameter e, is very lose to zero and an be either positive or negative, but not exatly zero (ondition C0). The graph below reflets this inequality and plots the W 0 area, against e. (35)

2/0/5 SVP 0 vd, Parameters of the SVU, De 204.nb 9 W 0 ( s - ).4 0-23.2 0-23 W 0 δ < - = 5 0-7 e 2. 0-23 8. 0-24 6. 0-24 4. 0-24 r = 3 Gp Values of W 0, for δ -0-6 W 0 e 0 2. 0-24 r = 8 Gp e ( s - ) -2. 0-7 -. 0-7 0. 0-7 2. 0-7 3. 0-7 Fig. : Possible values of W 0, against e, for, r = 3, 4, 4.6, 8 Gp (urves in desending order) aording to ondition C3. From the graph we see that the unertainty of r, does not affet the order of magnitude of the SV parameters W 0, and e. For, e 0, and various r, we have the values : r (Gp) 0 < W 0 3.0.6 0-24 4.0.2 0-24 4.6.05 0-24 8.0 6. 0-25 A similar value is given by Manousos in (204 a, 3).Theoretially the urve of Fig. extends on the right to muh larger values for both W 0, and e. However the justifiation for the sale hosen in this figure, omes from onditions C6, C7 W 0 H 0 2.2 0-8 in setion 4.2. In partiular : e < 0-8 0 < W 0 < 2 0-24 (C3a) Form eq. (3) we see that the ombined effet of the SV eletri potential and the eletri harge selfvariation on osmologial redshift, for distanes up to a few Gp, is very small : - 0-5 W V 2 0 R + - e - e r e Combining with ondition C3 we have -. 0-5 V R 2 + 0.9 0-5 hene : 2 + 0-5 - 5.5 0-6 V R + 4.5 0-6 (C4) for distanes, 3 r 8 Gp, orresponding to SCM light travel distane, r L 3 Gp 4. Cosmologial redshift 4.. The exponential redshift/ distane relation The TSV predits that in the SVU, the photon emissions of distant astronomial objets happened at a time when the rest mass and absolute harge of partiles of matter were smaller and hene the emisssion / absorption wavelengths were larger. In this

0 SVP 0 vd, Parameters of the SVU, De 204.nb way osmologial redshift is explained in a flat stati (non expanding) universe, where generally we assume that radiation from distant astronomial objets is reieved at the same wavelength as it is emitted. Let, λ(r), be the wavelength reieved today (at urrent time epoh) of photons emitted from atoms at position r. Let λ E, be the orresponding photon wavelength in our laboratory. Cosmologial redshift (measured at urrent time epoh t 0 ) is defined as : z = λ (r) λ E - (36) Aording to the simple Bohr model, whih however is reasonably exat for simple atoms, the photon wavelength is inversely proportional to the rest mass and the 4 th power of the eletri harge of the emitting partile, so we have : z = m 0 m 0 (r) q E q(r) 4 - (37) Cosmologial redshift depends on both the relative SV rest mass and the relative SV eletri harge units. From eq. (27), (28), we have : z = V R 4 + - e - r + W 0 - e - e r e 4 - (38) This exponential redshift / distane relation gives osmologial redshift (for simple atoms) in the flat non expanding universe aording to the SVU model. For small osmologial distanes up to few hundred Mp, it should be nearly oinident with the Hubble law. In the next setion we will see that the derivative of z, (relative to distane) at, r = 0, is atually the Hubble law and this enables us to estimate some SV parameters. From onditions C3, C4, we are motivated to define a mean redshift, whih ompensates for the flutuations of the SV eletri potential : z M = + z V R 4 - (39) z M = + Combining eq. (40) we get ondition C5 : - e - r + W 0 - e - e r e - e - r z =.5 (C5) where, 3 r 8 Gp 4 - (40) where we have assumed that in the SVU model, a redshift of, z =.5, orresponds to distanes, 3 r 8 Gp. Taing the ratio with Cd3 we have : 0 < W 0 - e - e r e - e - r 3 0-7 (4) Notie that the exat value of r, does not affet the order of magnitude of this result, whih shows how muh weaer the eletri harge selfvariation is relative to the rest mass selfvariation. Hene we an write the exponential redshift / distane relation in the SVU, in a muh simpler but nevertheless quite aurate approximation : z V R 4 + z M - e - r - (42) - e - r (43) These are important relations beause they are simple and very aurate for distanes up to a few Gp. From eq. (39) we have :

2/0/5 SVP 0 vd, Parameters of the SVU, De 204.nb z V R 4 ( + zm ) - (44) Sine V R, is slightly flutuating, we expet that for a partiular distane r (where z M, is fixed), the atual measured osmologial redshift z, will show slight flutuations and some anisotropi behavior over the entire sy. This an be revealed by exat measurements, espeially for large redshifts. For small redshifts the peuliar veloities will ause an additional superimposed effet. 4.2. The Hubble law and its role in the SCM and SVU From eq. (38), we have that the derivative of the osmologial redshift over distane is : r z = + z r + e - + + e W 0 4 e e r e - + 4 r V R V R (45) and at the limit, r 0, we have, V R, thus : r z(0) = + 4 W 0 + 4 r V R (0) (46) From ondition C4, we an assume that most probably the SV potential derivative is negligible. Thus : r z(0) r z M (0) = + 4 W 0 (47) Sine mean redshift inreases with distane (in onformity with the Hubble law) we have, + 4 W 0 > 0, and sine W 0 > 0 (from ondition C2), we have, > 0. From eq. (43) we also have : r z M (0) (48) Combining these we get another ondition for the selfvariation parameters, W 0, based on the ondition (C5), where in this paper,, will mean ' smaller by several orders of magnitude '. Hene we have : r V R (0) 0 0 < W 0 (C6) We assume that the Hubble law refers to mean redshift (sine the flutuation of redshift is negligible for small distanes and has not been notied so far). From this law we have : z M (0) = H 0 r r z M (0) = H 0 (49) Combining with eqs. (48-49) with ondition C7, we get an estimate of the value of : H 0 = + 4 W 0 H 0 2.2 0-8 s - 68 m s Mp (C7) Here we learly see the roles of the parameters H 0, and W 0,, in the SCM and SVU models respetively. In summary :. In the SCM, H 0, determines the rate of hange of osmologial redshift with distane in our viinity (via the Hubble law), whereas in the SVU, + 4 W 0, is the linear oeffiient of the mean exponential redshift / distane relation. 2. In the SCM, H, is oupled with the expansion rate of the universe, whereas in the SVU, M, and W, are oupled with the universal rest mass and eletri harge selfvariation respetively. From onditions (C5) we have : - e - r z = z + WL j - r r z where WL j, j = 0, -, are branhes of the Lambert W funtion. From C8 : e - r z

2 SVP 0 vd, Parameters of the SVU, De 204.nb z + WL j - r r z e - r z (50) We set, z =.5, and tae only the real values of, to establish ondition C9 : -4.6 0-8 s - ( 0) 0.47 0-8 s - -2. ( 0) 0.2 (C8) where, 3 Gp r (z =.5) 8 Gp. For, r < 6.62 Gp, we have, < 0, otherwise, > 0. Below we give a graph of..0 0.5 0.0 Values of ( For distane, r = 3-8 Gp, whihorrespondsto, z =.5 ) 2 4 6 6.62 8 3 5 7 r (Gp) -0.5 -.0 -.5-2.0-2.5 Fig. 2 : Values of, for, r = 3-8 Gp, orresponding to, z =.5. 4.3. Cosmologial redshift limit as, r The long range behavior of osmologial redshift as, r +, depends on the signs of the parameters in eq. (40). From ondition C2, C6 we have, W 0, > 0. Thus the long range behavior depends only on the signs of, e : For,, e > 0 z M z max = + + W 0 - e In all other ases z M + (5) We have measured redshifts of astronomial objets in exess of 0. Also the CMBR has a redshift in exess of 0 3, whih is quite ompatible with the SVU model. Hene we onlude that in the ase of finite z max, we have : + + W 0 e > + z max We saw in ondition (C8) that for positive, the ratio, > z max, or, z max W 0 > + z max - M e 0 +, is small. Then ombining ondition C8 we have::, e > 0 either, < z max or, z max where, z max > 0 3 0.2, and, e W 0 < + z max - (C9)

2/0/5 SVP 0 vd, Parameters of the SVU, De 204.nb 3 4.4. Comparison of the inverse redshift/ distane relations in the SVU and the Λ CDM models Inverting equation (43) we get the inverse mean redshift / distane relation in the SVU model : r - log - z M (52) Below we plot this equation for various values of,, in the range given by ondition C8. For omparison we also plot the omoving distane of the SCM model, based on the Robertson Waler metri for an expanding universe as given by the Λ CDM model. The urves reflet various values of dar energy density, 0 < Ω Λ < 0.99, for a flat universe (Ω = 0). Thus : r = z H 0 0 Ω m ( + v) 3 + Ω Λ d v where we equate the independent variable z, with the mean redshift z M, of the SVU model so that, r = r. The most favorite urrent value of dar energy density is, Ω Λ = 0.728. However dar energy density has not been measured diretly and this value reflets the need for an almost flat universe in the ontext of the Λ CDM model. In turn the Λ CDM model is used to give omoving distanes. Hene the r, urves heavily depend on the Λ CDM model, as diret proper distane measurements of astronomial objets are very diffiult to get and have great unertainty, espeially beyond small redshifts. So the interpretation of the r, is only shematial sine the Λ CDM model is not ompatible with the SVU model. What really matters is a orrelation of redshift with proper distane measured independently of any model. From this orrelation we ould infer the exat values of the SV parameters. It must not go unnotied that the proximity of the urves of the SVU and the Λ CDM model is remarable for distanes of many Gp, for the partiular range of the SV onstants. This range has not been wored out in order to produe a partiular shape of the distane / redshift urve, but from the selfvariation laws, the variation of the fine struture parameter, the assumption of the validity of the Hubble law for small distanes and a possible range of distanes for a single redshift value (z =.5). From this proximity it is lear that if we are to rely on the redshift / distane data orrelation in order to hoose between the models, the data have to be partiularly preise.

where, K = 4 SVP 0 vd, Parameters of the SVU, De 204.nb 4 Gp Ω Λ = 0.99 Ω Λ = 0.728 Ω Λ = 0.5 Ω Λ = 0 A B C 3 Distane vs Redshift for the SVU and Λ CDM models Area (A : 0 < K < 0.2) Area (B : -.0 < K < 0) Area (C : -2. < K < -) 2 0.0 0.5.0.5 2.0 2.5 3.0 Fig. 3 : Cosmologial distane vs. redshift in the SVU and the flat SCM (where the Λ CDM are from left to right, Ω Λ = 0.99, 0.728, 0.5, 0.0). The straight line (K = 0) is the naive Hubble law (almost oinident with the urve, Ω Λ = 0.99, whih is slightly to the right). The SVU model predits three possible areas A, B, C, where the atual distane / redshift urve an exist, depending on the value of,. Notie also from onditions C0, C0 (setion 5.) that : 0, -. The urrent Λ CDM model with, Ω Λ = 0.728, seems to plae the omoving distane (whih is simply equivalent to distane in the SVU model) in the area B. The position of this urve orresponds to, -0.6. However area B, as we will see in setion 5., leads to a negative SV rest mass unit. For omparison we give a urve of the Λ CDM model with, Ω Λ = 0.5, whih would marginally plae the urve on the area C (very lose to the urve, K = -.0). However the urrent Λ CDM model assumes the Robertson Waler solution whih explains osmologial redshift in an expanding universe. In the SVU model the ause z M

2/0/5 SVP 0 vd, Parameters of the SVU, De 204.nb 5 of redshift is different and obeys a different relation (eq. 40). Future exat measurements of distane with redshift (independently of any model) will show whih model best fits the data. 4.5. Change of SV rest mass and SV eletri harge units with redshift Inserting eq. (52) into eq.(27) we get the relative SV rest mass unit with respet to mean osmologial redshift : m 0 (z M ) (53) m 0 + z M This is a surprisingly simple and important equation of the SVU model, as it does not depend on any SV parameters. Many physial quantities depend on the SV rest mass unit, whih diretly affets the rest mass of all material partiles in astronomial objets. Our equation is the same as Manousos derived in (Manousos 203, 4), however we arrived at it under more general assumptions for the SV parameters. This equation neglets the eletri harge selfvariation. If we use the eq.(40) instead of (43), we get the exat relation between the relative SV rest mass unit and mean redshift. Unfortunately eq. (40) annot be inverted, so we start with the inversion of eq. (27) and then we insert into eq. (40), to get : + z M = m 0 m 0 (r) + W 0 e - + - m 0 m 0 (r) e / 4 Sine this equation annot be diretly inverted we expand into series and invert. The result is : m 0 (z M ) m 0 = + b z M + b 2 z M 2 + b 3 z M 3 +... where the oeffiients b n, are given by the equations : b = + γ n-2 b n = - ( + γ) 2 n- n - where, γ = 4 W 0, and, P n e least quadrati in, e + n γ + 4 n! n+ P 2 n-2 e (54) (55), n 2 (56), is a polynomial of degree n, in, e, with no onstant and linear terms i.e. the lowest term is at. Manousos in (Manousos 203 d, 4) implies that, e. Although this does not neessarily follow from the parameter onditions gathered so far, it is ompatible with the slow rate of the eletri harge selfvariation and is most probably true. We all this assumption A : e (A) The exat form of the polynomials, P n e assumption (A) we have that, P 2 n-2 e formula : γ n-2 b n = - ( + γ) 2 n- n -, is available online at, http : // apminstitute.org/ ategory/ physis /. Under, n 2. Hene we an alulate the oeffiients b n, with the approximate + n, n 2 To get an idea in this ase we insert some numerial values for the SV parameters : (56 ')

6 SVP 0 vd, Parameters of the SVU, De 204.nb e = 0-5 γ = 0-5 / = 0-3 γ = -0-5 / = 0-3 γ = 0-5 / = - γ = -0-5 / = - b 0.99999.0000 0.99999.0000 b 2 -.000 0-5.000 0-5 -5.000 0-6 5.000 0-6 b 3-5.003 0-9 5.004 0-9.667 0-6 -.667 0-6 b 4-3.336 0-2 3.336 0-2 -8.333 0-7 8.334 0-7 b 5-2.502 0-5 2.502 0-5 5.000 0-7 -5.000 0-7 More oeffiients are available online at, http : // apminstitute.org/ ategory/ physis /. Inserting eq. (52) into eq. (28) we get the SV eletri harge unit with respet to mean redshift : q 0 (z M ) + W 0 - - z M q 0 V R e e / - (57) where always, - z M > 0. This is the analog to eq. (53), only that in this ase the SV parameters do not anel out. Both equations are very useful to investigate physial phenomena in the SVU model, whih depend on the rest mass and eletri harge. Eq. (57) is derived under the same approximation as eq. (53). We an also get the exat equation giving the dependene of redshift from eletri harge, by inverting eq. (28) and inserting into eq. (38) : + z = V R 4 + - + e W 0 - q 0 V R q 0 (r) e / q 0 V R q 0 (r) Setting the relative SV potential, V R, we get an equation analogous with eq. (54), whih an be inverted in a similar way. 4 (58) 5. Evolution of the SV rest mass unit 5.. Evolution of the SV rest mass unit From eqs. (5), (7), we have : m 0 (t) = b ħ 2 + + - e t (59) The beginning and eventual evolution of the SV rest mass unit depends ruially on the sign of. In the table below we give the limit of, m 0 (t), for the various ases : m 0 (t) t - t + > 0 0 < 0 0 (60) where : = b ħ 2 (6) is an extremely small positive or negative universal mass onstant depending on the SV onstants b, and. It is interesting that, onsidering Cd8, eq.(59) alone does not guarantee a positive SV rest mass unit. From onditions (C8) we have :

2/0/5 SVP 0 vd, Parameters of the SVU, De 204.nb 7 0 < < 5.42 0-69 -2. < < - 0 < < 0.2 > 0 (C0) - < < 0 < 0 From eqs.(9), (59), (6), we have : m 0 (t) = + + - M m 0 = + (62) (63) Another interesting feature of eq. (59) is that there are ases where, + - e T R = 0, for some finite time T R. At time, t = T R, the SV rest mass unit beomes infinite, m 0 (T R ) = ±. We an alulate T R, by : T R = log + (64) where from ondition C2a we have, -. At time T R, we have a sign reversal for the SV rest mass unit, where it hanges from positive to negative or vie versa. The reversal time T R, is real only for, > -. In the other ase where, -2. < < -, there is no sign reversal and the evolution of the SV rest mass unit follows a smooth pattern. But what does the sign reversal mean? Assuming that the urrent SV rest mass unit is positive, the universe will enter a reverse phase after time T R, where the SV rest mass unit will be negative. Under the rest mass selfvariation law stated at the beginning, all rest masses in the universe must follow that pattern. The universe will have entered a reverse phase predited by the SVU model. The next figure shows the reversal time T R, with respet to the value of,. Reversal time T R, of the SV rest mass unit 00 Ga 80 K = 60 40 20 T H K -.0-0.8-0.6-0.4-0.2 0.2 Fig. 4 : Sign reversal time T R, of the SV rest mass unit in Ga (where positive is the future).

8 SVP 0 vd, Parameters of the SVU, De 204.nb From Fig.4 we see that the reversal time is positive for all allowed values of, > -, thus a sign reversal of the SV rest mas unit has not happened in the past history of the universe. The sign reversal will happen in the future. However it will not happen for at least 3 Bill years from now. It is interesting and ertainly no oinidene that for, 0, the reversal time is, T R T H = H 0 = 4.4 Ga, thus omparable with the Hubble time, though T R, lies in the future. We will ome ba to this ase in setion 5.2. In the following diagrams we show the time evolution of the SV rest mass unit relative to the SV mass onstant. Current time epoh is set at, t = 0. 5 Relative SV rest mass unit: m 0 (t) where, K =, > 0 0 A 5 K = 0.2 K = -2. C K = - t(ga) 0 T R T H -50-00 -50 50 00 50 200 K = 0.2 A Fig. 5 a : Evolution of SV rest mass unit (eq. 59) with time (Ga), given in, units. The shaded areas A and C are shown. T R, is the reversal time and T H, is the Hubble time. -5 Area A, orresponds to, 0 < < 0.2, and area C, to, -2. < < -. In both ases the rest mass onstant, is positive. In the ase A, the urrent SV rest mass unit is positive, however a finite reversal time is expeted, after whih the universe will enter a reverse phase, with negative rest mass unit. In ase C, no suh reversal will tae plae and the SV rest mass unit was and will remain forever positive. We do not now yet in whih universe we live, aording to the SVU options. To find whih is the ase, we have to determine,, more preisely. The next diagram shows ase B, where, - < < 0, and urrently both the rest mass onstant, and the SV rest mass unit are negative. In this ase also there is a finite reversal time, after whih the SV rest mass unit will enter a positive phase (notie that the diagram shows their ratio). Sine by onvention we assign positive values to rest mass, we probably have to exlude this ase.

2/0/5 SVP 0 vd, Parameters of the SVU, De 204.nb 9 5 Relative SV rest mass unit: m 0 (t) where, K =, < 0 0 B 5 K = -0.7 t(ga) -50-00 -50 T H T R 50 00 50 200 - K = -0.7 B Fig. 5 b : Evolution of SV rest mass unit (eq. 59) with time (Ga), given in, units. The shaded area B is shown. T R, is the reversal time and T H, is the Hubble time. In this diagram the urve with, K = -0.7, is an example urve, as the whole shaded areas are possible. 5.2. The ase of the reversed rest mass selfvariation law -5 For all ases of the SV parameters whih predit a reversal time, the alulated reversal time lies in the future. However extending the parameter onditions to omplex values might bring about negative reversal times lying in the past. Another interesting ase is the reversal of the selfvariation law itself. In this ase, without hanging eq.(), we may postulate that the SV rest mass unit dereases with time. This is not so grave as it seems, beause eq.(), stems from two symmetri equations involving rest mass and rest energy (see (Manousos 203 b, 5.2)). The reversal of the law just means to reverse rest mass with rest energy. What is interesting in this ase, is that urrent SV rest mass unit is positive and the reversal time is roughly the Hubble time, but lying in our past. Hene in this model, the ' beginning ' of the universe is atually the beginning of the urrent phase, starting with a phase reversal, whih has happened roughly 4 Bill years ago. Prior to this, there existed a reversed phase with negative SV rest mass unit. This is a surprising predition of the SVU model, sine it atually opens the possibility of a phase reversal in the universe. If this atually happens or not, is a ase for further study. 5.3. Evolution of the SV eletri harge unit The evolution of the SV eletri harge unit follows a similar pattern, sine it is based on a similar selfvariation law. However matters are ompliated by the SV eletri potential. Also the parameter onditions gathered in this paper, are less strit than for the SV rest mass unit, leaving more options open.certainly in some ases there is also a reversal time for the eletri harge. 6. Derivatives 6.. SV rest mass unit From eq.(5), (62) we have : t m 0 (t) = M m 0 (t) + - M (65)

20 SVP 0 vd, Parameters of the SVU, De 204.nb From eq.(7) we have : r m 0 (r) m 0 (r) r m 0 (r) m 0 = - = - m 0 (r) e - r (66) m 0 m 0 (r) m 0 2 e - r < 0 (67) r m 0 m 0 = - and onsidering eq.(7) : (68) r m 0 + t m 0 = 0 (69) In the same way as we alulated eqs.(27) and (62) we have : 6.2. Redshift m 0 (r, t) m 0 = m 0 (t) m 0 = From eq.(43) we have : + - M e - r + - M (70) (7) r z M e - r > 0 (72) From eq.(70) we alulate redshift at any time, z M (t), in the same way as eq.(43) : z M (r, t) - M e - r (73) t z M (r, t) -M e - r < 0 (74) t z M - e - r < 0 (75) where z M, is the urrent redshift at any distane. Hene ombining with eq.(72) : t z M + r z M 0 (76) From eq.(6) the sign of, t M, depends on the sign of. From this and ondition C7 it follows that the sign of, t H, may be positive or negative. 7. Disussion We have started from very broad assumptions about the SV parameters : b, b e = ±, and,, e \0 μ, μ e We have also introdued the new parameters, M, W \0, as a ombination of the above. Despite the fat that we have started with these broad assumptions, we have onsiderably narrowed their possible values. However the possibilities left open are more than that presented by Manousos in (Manousos 203 b, 7.3, 203, 5, 203 d, 2, 4, 204 a,, 3), where the assumptions were : b = b e =, and,, > 0 μ = μ e 0 < A, B < t + μ < 0 Notie however that the parameters B e, W, e, used in this paper are not exatly oinident with, B, W,, used by Manousos

2/0/5 SVP 0 vd, Parameters of the SVU, De 204.nb 2 sine the latter inorporate the SV eletri potential. We have deliberately done so, in order to separate the effets of the SV eletri potential with that of the eletri harge selfvariation. In setion 3 we have too into aount that Webb (20) desribes the variation of the fine struture onstant α, as having a large dipole form, with only a small isotropi omponent. Finally for the first time we present the different possible ases for the evolution of the SV universe. The hoie of ases depends on the values of the SV parameters. We have also found that the time derivatives, t z, t M, and, t H, may tae positive or negative values. We have onfirmed that the equation whih gives the relative SV rest mass unit, m 0 (z M ) (42) m 0 + z M is almost independent of the SV parameters, whih is the same result as in Manousos (203, 4). We have however alulated the small higher order terms, whih are not independent of the SV parameters. Finally it is also oneivable that the investigation of the onsequenes of the SVU model an proeed by starting with omplex values of the parameters, as long as it produes real valued observables. 8. Conlusion The SVU model of the theory of selfvariations (TSV) predits a flat stati non expanding universe, in whih the rest mass of material partiles and the absolute eletri harge of partiles of matter, slightly inrease with time. The SVU model onforms with all urrent osmologial observations (Manousos 203, 204 ). It partiular it explains the variation, over large distanes, of the fine struture onstant observed by Webb (20), sine at large distane we see the smaller value of the eletri harge in the past. No mainstream osmology an explain this variation. The rest mass of all material partiles follows the evolution of the SV rest mass unit, whih we have studied in this paper. At large osmologial distane we observe a smaller value of the rest mass in the past. This, in ombination with the variation of the eletri harge, explains osmologial redshift without the need of the expansion of the universe and all other assumptions that follow from the expansion and are inorporated in the SCM. The SVU model predits an exponential redshift / distane relation, whih is very losely mathed by the redshift / omoving distane relation of the Λ CDM model. This is expeted, as the omoving distane of the Robertson Waler metri orresponds to the distane in the stati SVU model. We have found that, depending on the exat values of the SV parameters, the SV rest mass unit (and thus the rest mass of all partiles), will either evolve towards a limit mass onstant, in the infinite future, or it will aquire an infinite value at a finite time in the future, alled the reversal time T R. After the reversal time the universe will enter into a reversed phase, where the rest mass unit has a negative sign. We have found that, due to the restritions of the parameters studied in this paper, the reversal time an be no less than 3 Bill years from now. It may however be hundreds of Bill years in the future. Preise measurements of the distane of astronomial objets, independently of any osmologial model, will show the preise values of the SV parameters and thus selet between the different evolutions in the ontext of the SVU model. In the ase where,, e > 0, redshift tends to a limit z max, at infinite distane. In all other ases redshift tends to infinity. We have found also that the SVU model predits a flutuation of osmologial redshift produing an anisotropy over the sy, for astronomial objets lying at the same distane from us and ideally having the same peuliar veloities. This flutuation is in part overed by the peuliar veloities. The flutuation however is proportional to redshift, and inreases with distane, while peuliar veloities probably have an upper bound throughout the observed universe. 9. Referenes King, J.A., Webb, J.K., Murphy, M.T., Flambaum, V.V., Carswell, R.F., Bainbridge et al., 202. Spatial variation in the fine - struture onstant new results from VLT / UVES.Monthly Noties of the Royal Astronomial Soiety, 422 (4), 3370-344. DOI : 0. / j.365-2966.202.20852.x Manousos, E., 204. Selfvariations vs Standard Cosmologial Model Using as Criterion the Cosmologial Data.GJSFR - A, 4 (2) : 35-4 Manousos, E., 204 b. The replaement of the potentials as a onsequene of the limitations set by the law of selfvariations on

22 SVP 0 vd, Parameters of the SVU, De 204.nb the physial laws.ijeit, 3 (0) : 8-85 Manousos, E., 204 a. The numerial values of the fundamental parameters of the model of selfvariations and the age of the universe.phys.int., 5 : 8-4. Manousos, E., 203 d. On the absene of antimatter from the universe, dar matter, the fine struture parameter, the flutuations of the osmi mirowave baground radiation and the temperature differene between the northern and southern hemisphere of the universe.physis International 4.2 : 20-34. DOI : 0.3844 / pisp.203.20.34 Manousos, E., 203. Cosmologial data ould have a mirosopi, not marosopi, ause.amerian Journal of Spae Siene : 0-2. DOI : 0.3844 /ajssp.203.9.2 Manousos, E., 203 b. Mass and Charge Selfvariation : A Common Underlying Cause for Quantum Phenomena and Cosmologial Data.Progress in Physis : 9 (3), 73-4 Manousos, E., 203 a. The Cause of the Inreased Luminosity Distanes of Supernovae Reorded in the Cosmologial Data. Progress in Physis : 9 (3), 5-6 Manousos, E., 2007. The theory of self - variations.a ontinuous slight inrease of the harges and the rest masses of the partiles an explain the osmologial data.nuovo Cimento B, 2007, 359 388.DOI : 0.393 / nb / i2006-0 59-9 Webb, J.K., J.A.King, M.T.Murphy, V.V.Flambaum and R.F.Carswell et al., 20. Indiations of a spatial variation of the fine struture onstant.phys.rev.lett.doi : 0.03 / PhysRevLett 07.90