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A cosmological deduction of the order of magnitude of an elementary-particle mass and of the cosmological coincidences
Volume 53A, number 2
PHYSICS LETTERS
2 June 1975
A COSMOLOGICAL DEDUCTION OF THE ORDER OF MAGNITUDE OF AN ELEMENTARY-PARTICLE MASS AND OF THE COSMOLOGICAL COINCIDENCES P.T. LANDSBERG and N.T. BISHOP Department of Mathematics, University, Southampton S09 5NH, England Received 12 April 1975 The assumptions adopted here include a time-dependent gravitational constant G, with H/G time-independent, H being Hubble’s parameter. In terms of h, c, H, G, which are assumed given, we deduce the order of magnitude of an elementary particle mass and the cosmological coincidences, using dimensional considerations.
A number of distinct interpretations of Dirac’s large number hypothesis [1] have recently been offered, based on a rotating cosmological model [2] lepton creation [3] and a statistical spread of elementary masses [4]. In this note a new approach is offered to this problem by using dimensional analysis and three qualitative assumptions which include as a crucial element an (unspecified) time dependence of Newton’s
Table 1 Some masses given by eq. (1)
,
gravitational constant G. The assumptions are: (i) Theories are considered which involve the fundamental parameters h, c, G and H, where H is Hubble’s “constant”; their numerical values are assumed known; (ii) The value of H/G is rn dependent of time; (iii) There exist one or more partides whose bare rest mass, m, is independent of time. These assumptions suffice to deduce in order of magnitude (a) Eddington’s number for the number of particles rn the visible universe, (b) The value of the mass m, and (c) the cosmological coincidences. The first assumption implies that any mass occurring in the theory has the form h’~H~G”c6 and one fmds, if b is an unidentified parameter, that all masses are of the form (see table 1)
b
93/2—1 3/GH
—6
(hc/G)112
(h2H/Gc)113
hH/c2
m(b)/k(b) ____________________________________________ c
It will now be shown that —6b~9. (3) The lower limit is obtained from the smallest mass, ~m, which is the least mass difference which can be determined in the period T H~ since the big bang. It is subject to the uncertainty relation Tc2Sm ‘h, whence ‘~
h/c2T = m(—6)/k(—6), (4) giving the left-hand side of (3). To obtain the righthand side of eq. (3), note that [5] M ~ pR3 M=
c3p~/H3 m(9)
= m(9) (5) 2 GH 2k(9) where M is the mass of the visible universe of density ~-~—-~—
—~-~—
Pc = 3H2/8ir G(t). For a consistent Newtonian model on which eq. (5) can be based see ref. [6]. One can show that b = —1 for the bare rest mass m, by noting from assumptions (ii) and (iii), and eq. (1) that, as far as time-dependence is concerned, P
m(b)
=
k(b)(h3H/G2)h15(c5/PLH2GY)/15
(1)
where the k(b) are unidentified dimensionless constants. This implies fri k(bi)...k(br) (r_~2b 1)]T (2) ~ m(b 1) = [k(r’~bj)V [m For rough estimates the k’s may be replaced by unity in all that follows. The Planck mass is m(3/2)/k(3/2)
5. (6) m(b)~G(~’~)/ It follows that the time-independent particle mass is m
=
m(1).
(7) 109
PHYSICS LE~TTERS
Volume 53A, number 2
Let n be the number of equivalent non-interacting particles of mass m in the visible universe. Then by (5), (7) and (2)
m(9)
m(--l)
k(9)[k(—6)] —
2
rm(1)12
[k(--1)] 3
2 June 1971
stant in the theories envisaged under assumption U). then m(b) 2 involves also2/hc, an unidentified power of the and all masses could confine structure constant e tam such a factor. The argument leading torn m ( 1) for the mass m envisaged under assumption (iv) would, however, remain valid. The effect of interactions
(8) =
k(9)[k(-—1)] [k(3/2)]
~[m(3/2)]4 Lm(—1)
‘~
among the particles of bare mass m( --I) could be to
=
This completes the calculation. For numerical evaluation replace the k’s by unity, to find (H~ 5.7 X 1017 sec)
(c~/GH~h)~I324 X 1080
ii
electric charge is found to be time-independent, as adequately discussed in the literature [8—lOl. One
m —~(h2H/Gc)1I3 10-25 gm. The value of n exceeds Eddington’s original estimate by almost exactly a factor of 100. The mass m ms 0.4 times the rest mass of the charged pion. We now use this approach to re-derive some known results. For example, one might consider Harrison’s [7]
N”
2 and N ~/rn~c’ 2 Grn where m~and m~are neutron and pion rest masses. Ignoring the difference between them for the present purpose, and identifying them both with m(-—l), ~—~-~—
.~IL~L
1
Nj’~],
yield spectrum of masses lie inproton the range a—1.5
also finds e ~ (hc)1/2, whence the present approach yields a e2/hc as time-independent and of order unity, in agreement with what is normally assumed in discussing the cosmological coincidences. This also shows that the omission of the powers of the finestructure constant a in eq. (1) was a consistent proceclure. (Were one to include G and a mass in the expresasion power of the fine-structure as a factor). for the charge, one wouldconstant again arrive at (l)with We are grateful to Dr. A. Pimpale of this Department for helpful discussions. N.T.B. is indebted to the Science Research Council for a research studentship.
N
m(—l) 2 The coincidence N~’‘—‘N
2 is thus contained in (8). Alternatively, taking the view point of Lawrence and Szamosi [4] , the “radius” of the visible universe divided by the Compton wavelength is by (5), table I, (2) and again (5) and (8), taken in that order, R h/mc
2 GM/c ~i/mc
Mm Mm [m(3/2)] 2 m(9)m(—6)
m
m(—6)
=“~‘
Similarly the gravitational fine structure constant is by table 1 and eq. (8) he
Em(312)12
~
References [1] PAM. Dirac, Nature 139 0937) 323; Proc. Roy. Soc. (1974) 439. 245 (1973) 313. [2] A338 G. Cavallo, Nature 13] J.V. Narlikar, Nature 247 (1974) 99. 141 J.K. Lawrence and G. Szamosi, Nature 252 (1974) 538.
151 S. Weinberg, and Cosmology (Wiley, New York) p.476,Gravitation 1972. 16] P.T. Landsberg and N.T. Bishop. M.N.R. Astron. Soc. 171 (1975). 17] E.R. Harrison, Physics Today, December 1972 p. 30.
181 Fl.Peres, Dyson, Phys. Rev. Lett. (1967) 129. 191 A. Phys. Rev. Lett. 19 19 (1967) 1293. 110] T.N. Bacall and M. Schmidt, Phys. Rev. Lett. 19 (1967)
These are the results (7) and (9) of ref. [41. If one allows the electronic charge e to be a con-
110
1294.
PHYSICS LETTERS
2 June 1975
A COSMOLOGICAL DEDUCTION OF THE ORDER OF MAGNITUDE OF AN ELEMENTARY-PARTICLE MASS AND OF THE COSMOLOGICAL COINCIDENCES P.T. LANDSBERG and N.T. BISHOP Department of Mathematics, University, Southampton S09 5NH, England Received 12 April 1975 The assumptions adopted here include a time-dependent gravitational constant G, with H/G time-independent, H being Hubble’s parameter. In terms of h, c, H, G, which are assumed given, we deduce the order of magnitude of an elementary particle mass and the cosmological coincidences, using dimensional considerations.
A number of distinct interpretations of Dirac’s large number hypothesis [1] have recently been offered, based on a rotating cosmological model [2] lepton creation [3] and a statistical spread of elementary masses [4]. In this note a new approach is offered to this problem by using dimensional analysis and three qualitative assumptions which include as a crucial element an (unspecified) time dependence of Newton’s
Table 1 Some masses given by eq. (1)
,
gravitational constant G. The assumptions are: (i) Theories are considered which involve the fundamental parameters h, c, G and H, where H is Hubble’s “constant”; their numerical values are assumed known; (ii) The value of H/G is rn dependent of time; (iii) There exist one or more partides whose bare rest mass, m, is independent of time. These assumptions suffice to deduce in order of magnitude (a) Eddington’s number for the number of particles rn the visible universe, (b) The value of the mass m, and (c) the cosmological coincidences. The first assumption implies that any mass occurring in the theory has the form h’~H~G”c6 and one fmds, if b is an unidentified parameter, that all masses are of the form (see table 1)
b
93/2—1 3/GH
—6
(hc/G)112
(h2H/Gc)113
hH/c2
m(b)/k(b) ____________________________________________ c
It will now be shown that —6b~9. (3) The lower limit is obtained from the smallest mass, ~m, which is the least mass difference which can be determined in the period T H~ since the big bang. It is subject to the uncertainty relation Tc2Sm ‘h, whence ‘~
h/c2T = m(—6)/k(—6), (4) giving the left-hand side of (3). To obtain the righthand side of eq. (3), note that [5] M ~ pR3 M=
c3p~/H3 m(9)
= m(9) (5) 2 GH 2k(9) where M is the mass of the visible universe of density ~-~—-~—
—~-~—
Pc = 3H2/8ir G(t). For a consistent Newtonian model on which eq. (5) can be based see ref. [6]. One can show that b = —1 for the bare rest mass m, by noting from assumptions (ii) and (iii), and eq. (1) that, as far as time-dependence is concerned, P
m(b)
=
k(b)(h3H/G2)h15(c5/PLH2GY)/15
(1)
where the k(b) are unidentified dimensionless constants. This implies fri k(bi)...k(br) (r_~2b 1)]T (2) ~ m(b 1) = [k(r’~bj)V [m For rough estimates the k’s may be replaced by unity in all that follows. The Planck mass is m(3/2)/k(3/2)
5. (6) m(b)~G(~’~)/ It follows that the time-independent particle mass is m
=
m(1).
(7) 109
PHYSICS LE~TTERS
Volume 53A, number 2
Let n be the number of equivalent non-interacting particles of mass m in the visible universe. Then by (5), (7) and (2)
m(9)
m(--l)
k(9)[k(—6)] —
2
rm(1)12
[k(--1)] 3
2 June 1971
stant in the theories envisaged under assumption U). then m(b) 2 involves also2/hc, an unidentified power of the and all masses could confine structure constant e tam such a factor. The argument leading torn m ( 1) for the mass m envisaged under assumption (iv) would, however, remain valid. The effect of interactions
(8) =
k(9)[k(-—1)] [k(3/2)]
~[m(3/2)]4 Lm(—1)
‘~
among the particles of bare mass m( --I) could be to
=
This completes the calculation. For numerical evaluation replace the k’s by unity, to find (H~ 5.7 X 1017 sec)
(c~/GH~h)~I324 X 1080
ii
electric charge is found to be time-independent, as adequately discussed in the literature [8—lOl. One
m —~(h2H/Gc)1I3 10-25 gm. The value of n exceeds Eddington’s original estimate by almost exactly a factor of 100. The mass m ms 0.4 times the rest mass of the charged pion. We now use this approach to re-derive some known results. For example, one might consider Harrison’s [7]
N”
2 and N ~/rn~c’ 2 Grn where m~and m~are neutron and pion rest masses. Ignoring the difference between them for the present purpose, and identifying them both with m(-—l), ~—~-~—
.~IL~L
1
Nj’~],
yield spectrum of masses lie inproton the range a—1.5
also finds e ~ (hc)1/2, whence the present approach yields a e2/hc as time-independent and of order unity, in agreement with what is normally assumed in discussing the cosmological coincidences. This also shows that the omission of the powers of the finestructure constant a in eq. (1) was a consistent proceclure. (Were one to include G and a mass in the expresasion power of the fine-structure as a factor). for the charge, one wouldconstant again arrive at (l)with We are grateful to Dr. A. Pimpale of this Department for helpful discussions. N.T.B. is indebted to the Science Research Council for a research studentship.
N
m(—l) 2 The coincidence N~’‘—‘N
2 is thus contained in (8). Alternatively, taking the view point of Lawrence and Szamosi [4] , the “radius” of the visible universe divided by the Compton wavelength is by (5), table I, (2) and again (5) and (8), taken in that order, R h/mc
2 GM/c ~i/mc
Mm Mm [m(3/2)] 2 m(9)m(—6)
m
m(—6)
=“~‘
Similarly the gravitational fine structure constant is by table 1 and eq. (8) he
Em(312)12
~
References [1] PAM. Dirac, Nature 139 0937) 323; Proc. Roy. Soc. (1974) 439. 245 (1973) 313. [2] A338 G. Cavallo, Nature 13] J.V. Narlikar, Nature 247 (1974) 99. 141 J.K. Lawrence and G. Szamosi, Nature 252 (1974) 538.
151 S. Weinberg, and Cosmology (Wiley, New York) p.476,Gravitation 1972. 16] P.T. Landsberg and N.T. Bishop. M.N.R. Astron. Soc. 171 (1975). 17] E.R. Harrison, Physics Today, December 1972 p. 30.
181 Fl.Peres, Dyson, Phys. Rev. Lett. (1967) 129. 191 A. Phys. Rev. Lett. 19 19 (1967) 1293. 110] T.N. Bacall and M. Schmidt, Phys. Rev. Lett. 19 (1967)
These are the results (7) and (9) of ref. [41. If one allows the electronic charge e to be a con-
110
1294.