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IL NUOVO CIMENTO VOL. 112 B, N. 6 Giugno 1997 NOTE BREVI

On the self-shielding universe

Progress Center, University of Florida - Pharmos, Alachua, FL 32615, USA

(ricevuto il 10 Aprile 1995; approvato il 25 Marzo 1997)

Summary. — Because the mass of the proton is much larger than the mass of the

electron, and the non-baryonic dark matter does not affect the optical depth, the observable universe is not self-shielded, i.e. does not possess an optical depth of unity. PACS 98.80.Dr – Theoretical cosmology.

It was recently suggested [1] that the observable universe could be self-shielded for electromagnetic radiation. This assertion was based on the remark that by postulating a self-shielded, flat universe, containing only one type of particle of charge equal to the elementary charge e, a characteristic mass m04 (He4/Gc3)1 /3 of the order of the electron mass emerges, where H is the Hubble constant, G is the gravitational constant, and c is the speed of light.

The objective of this note is to demonstrate that the universe is not and was not self-shielded for most of its existence. The quantities of interest are: the optical depth

t , the critical density rc, the density parameter V , the number density of particles n (all referring to the observable universe) and the Thompson cross-section s of the corresponding particle. One has

t 4nsc/H , (1) n 4Vrc/m , (2) rc4 ( 3 /8 p)(H2/G) , (3) s 4 (8p/3)(e2/mc2)2. (4)

From these equations it is easy to see that a flat universe (V 41) containing only one type of particle of mass equal to the electron mass me, has an optical depth

t 4127h, where h4H/100 km s21_Mpc21_{. Not only this value is two orders of} magnitude larger than unity, but the deviation is in the wrong direction when compared with the real universe. As known, the real universe is a “two-particle universe” in which one particle, the nucleon, has a mass mnmuch larger than the other particle, the electron. Because of this, and because the Thompson cross-section of the 927

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electron is much larger than the cross-section of the nucleon, the mass term in eq. (2) is

mn, while the mass term in eq. (4) is me.

Another problem is that, as far as we know, the non-baryonic dark matter does not affect the optical depth. Therefore, regardless of the fact that the universe is or is not flat, the V parameter in eq. (2) is the baryon density parameter VBC 0 .0145 h22[2].

When one takes into account all these aspects, one obtains the following expression of the photon mean free path d 4 (ns)21_{in the real universe:}

d 4 (re/VB)(NE/ND) , (5)

where r 4e2_/m

ec2is the classical radius of the electron, NE4 (c/Hre)2is Eddington’s large number [3], and ND4 e2/Gmnmeis Dirac’s large number [4]. It may be remarked that the value of d( 1029m ) is independent of the present uncertainty in Hubble’s parameter. The optical depth of the observable universe is, then,

t 4c/HdC1023_h21_. (6)

Because the density scales as R23_{, the optical depth scales as R}22_{. This means that} the universe ceased to be self-shielding since the time of galaxy formation, 10–15 billion years ago [5].

R E F E R E N C E S

[1] SHAWG. E., Nuovo Cimento B, 109 (1994) 1119.

[2] COPIC. J., SCHRAMMD. N. and TURNERM. S., Science, 267 (1995) 192.

[3] EDDINGTONA. S., Relativity Theory of Protons and Electrons (Cambridge University Press, Cambridge) 1936.

[4] DIRACP. A. M., Proc. R. Soc. London, Ser. A, 165 (1938) 199.

[5] PEEBLES P. J. E., The Large-scale Structure of the Universe (Princeton University Press, Princeton, N.J.) 1980.