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On neutrino absorption in cosmology
Volume 53A, number 1
PHYSICS LETTERS
19 May 1975
ON NEUTRINO ABSORPTION IN COSMOLOGY R. BURMAN* Department of Physics, University of Western Australia, Australia Received 15 October 1974 This lette~ deals with the self-consistency or otherwise, in the absorber theory of radiation, of retarded and advanced neutsino fields in the Robertson-Walker cosmologies. In the Wheeler-Feynman absorber theory of radiation, retarded or advanced solutions of the field equations are self-consistent if absorption is complete. Cosmic absorption of retarded neutrino fields has been investigated by several authors [ 1 - 4 ] for conformally flat cosmological models of zero spatial curvature. In this letter the completeness or otherwise of cosmic neutrino absorption will be investigated using a technique introduced, in the case of electromagnetic absorption, by Davies [5] ; the technique applies to the Robertson-Walker cosmologies of arbitrary spatial curvature. Suppose that all matter in the universe can be represented by a smoothed-out number density N. Let R denote the cosmological scale factor, a function of cosmic time t. Let o e denote an effective cross section equal to ¢ok/Nc where" ¢o is the angular wave frequency, k is the imaginary part of the refractive index and c is the speed of light in free space; % is related to the neutrino cross section o used in refs. [1-4] by o e = (c/t.,o)('tto) 112 Sin ~ where ~ is the phase angle of the complex scattering length. From the work of Davies, absorption will be complete for retarded and advanced fields if
respectively, where f and p denote the distant future and distant past. Consider models of class I, in which there is continuous creation of matter at such a rate as to keep N constant, and models of class II, in which matter is conserved so that N (x R - 3 . Let E denote the neutrino IWesentaddress: Astxonomy Cenlze, University of Sussex, Falmer, Brighton BNI 9QH, UK.
energy; as the wave propagates, E ~ R -1 because of the cosmic expansion. Take R ~ t n with 0 < t < 0% corresponding to expanding "big-bang" models. For retarded neutrino fields write o e cc E'I; in the standard current-current theory of weak interactions 7 = 0; in the photon-neutrino theory 7 = - 2 . In class I universes, (1 a) shows that full absorption occurs for all n in both theories. In class II universes, full absorption occurs for n < 1/3 and for n < 1 in the respective theories. Consider advanced neutrino fields. Assume that oe approaches a constant as E -~ oo. In class I universes, ( l b ) implies that full absorption does not occur. In class II universes with a "hot big bang", as they travel into the past advanced waves that are not absorbed first eventually enter an epoch in which the cosmic matter is in thermal equilibrium with electromagnetic radiation at relativistic temperatures. As this epoch is entered there is a rapid increase in N because of thermal electron-positron pair production [6] ; within this epoch the mass density of the cosmic medium varies as R - 4 , but N still varies as R -3. (This point was incorrectly treated in the discussion of retarded neutrino fields in ref. [4] ). Thus full absorption of advanced waves occurs for n ~ 1/3. Assume, for the sake of discussion, that the absorber theory of radiation is valid for neutrinos, that only retarded neutrino fields can be observed in the actual universe and that o e saturates at high energies. In determining what cosmological models are to be eliminated, under these assumptions, as possible models of the actual universe, the relevant values of n are those applicable for t -+ 0 and t -~ 0% such values will be denoted by n o and no. respectively. No class I universe is eliminated. Class II universes with n o > 1/3 are eliminated. Class II universes with n , > 1/3 or n.. > 1 17
Volume 53A, number 1
PHYSICS LETTERS
are eliminated if the current-current or photon-neutrino theories, respectively, are valid. The interaction o f neutrinos with most o f the matter in the universe is not affected much by the state o f the matter, which can therefore be represented by a smoothed-out density as above. For matter in black holes the interaction can be represented by a constant a e. Suppose that in class I universes there is a constant number density of such objects: they contribute to the absorption o f retarded waves in the same way as does the smoothed-out matter in the current-current theory, while for advanced waves the two types o f absorption behave in the same manner. Suppose that in class II universes the number density o f such objects varies as R - 3 in the distant future: again they contribute to the absorption o f retarded waves in the same way as does the smoothed-out matter in the currentcurrent theory. Thus, allowing for b o t h types o f absorption, the models eliminated are the same as stated above. In the standard "hot big-bang" model of general relativity, n = 1/2 i n the radiation-domina.ted epoch
18
19 May 1975
so that this model has consistent advanced fields (unless these are rejected on thermodynamic grounds [7] ) and is eliminated. The spatially flat form of the BransDicke cosmology, which includes as a special case the Einstein-de Sitter universe, is eliminated if the standard current-current theory is valid since retarded fields are then inconsistent [4]. The Friedmann models with zero cosmological constant and negative spatial curvature have no. = 1 and are hence eliminated if the current-current theory is valid I thank the referee for some helpful remarks.
References [1] [2] [3] 14} [5] [6] [7]
J.V. Narlikar, Proc. Roy. Soc, A270 (1962) 553. P. Ray Chaudhuri, J. Phys. A: Gen. Phys. 3 (1970) L5. R. Burman, Observatory 92 (1972) 128. R. Burman, Observatory 92 (1972) 131. P.C.W. Davies, J. Phys. A: Gen Phys. 5 (1972) 1722. R.J. Gould, Ann. Rev. Astron. Astrophys. 6 (1968) 195. P.C.W. Davies, J. Phys. A 8(1975) 272.
PHYSICS LETTERS
19 May 1975
ON NEUTRINO ABSORPTION IN COSMOLOGY R. BURMAN* Department of Physics, University of Western Australia, Australia Received 15 October 1974 This lette~ deals with the self-consistency or otherwise, in the absorber theory of radiation, of retarded and advanced neutsino fields in the Robertson-Walker cosmologies. In the Wheeler-Feynman absorber theory of radiation, retarded or advanced solutions of the field equations are self-consistent if absorption is complete. Cosmic absorption of retarded neutrino fields has been investigated by several authors [ 1 - 4 ] for conformally flat cosmological models of zero spatial curvature. In this letter the completeness or otherwise of cosmic neutrino absorption will be investigated using a technique introduced, in the case of electromagnetic absorption, by Davies [5] ; the technique applies to the Robertson-Walker cosmologies of arbitrary spatial curvature. Suppose that all matter in the universe can be represented by a smoothed-out number density N. Let R denote the cosmological scale factor, a function of cosmic time t. Let o e denote an effective cross section equal to ¢ok/Nc where" ¢o is the angular wave frequency, k is the imaginary part of the refractive index and c is the speed of light in free space; % is related to the neutrino cross section o used in refs. [1-4] by o e = (c/t.,o)('tto) 112 Sin ~ where ~ is the phase angle of the complex scattering length. From the work of Davies, absorption will be complete for retarded and advanced fields if
respectively, where f and p denote the distant future and distant past. Consider models of class I, in which there is continuous creation of matter at such a rate as to keep N constant, and models of class II, in which matter is conserved so that N (x R - 3 . Let E denote the neutrino IWesentaddress: Astxonomy Cenlze, University of Sussex, Falmer, Brighton BNI 9QH, UK.
energy; as the wave propagates, E ~ R -1 because of the cosmic expansion. Take R ~ t n with 0 < t < 0% corresponding to expanding "big-bang" models. For retarded neutrino fields write o e cc E'I; in the standard current-current theory of weak interactions 7 = 0; in the photon-neutrino theory 7 = - 2 . In class I universes, (1 a) shows that full absorption occurs for all n in both theories. In class II universes, full absorption occurs for n < 1/3 and for n < 1 in the respective theories. Consider advanced neutrino fields. Assume that oe approaches a constant as E -~ oo. In class I universes, ( l b ) implies that full absorption does not occur. In class II universes with a "hot big bang", as they travel into the past advanced waves that are not absorbed first eventually enter an epoch in which the cosmic matter is in thermal equilibrium with electromagnetic radiation at relativistic temperatures. As this epoch is entered there is a rapid increase in N because of thermal electron-positron pair production [6] ; within this epoch the mass density of the cosmic medium varies as R - 4 , but N still varies as R -3. (This point was incorrectly treated in the discussion of retarded neutrino fields in ref. [4] ). Thus full absorption of advanced waves occurs for n ~ 1/3. Assume, for the sake of discussion, that the absorber theory of radiation is valid for neutrinos, that only retarded neutrino fields can be observed in the actual universe and that o e saturates at high energies. In determining what cosmological models are to be eliminated, under these assumptions, as possible models of the actual universe, the relevant values of n are those applicable for t -+ 0 and t -~ 0% such values will be denoted by n o and no. respectively. No class I universe is eliminated. Class II universes with n o > 1/3 are eliminated. Class II universes with n , > 1/3 or n.. > 1 17
Volume 53A, number 1
PHYSICS LETTERS
are eliminated if the current-current or photon-neutrino theories, respectively, are valid. The interaction o f neutrinos with most o f the matter in the universe is not affected much by the state o f the matter, which can therefore be represented by a smoothed-out density as above. For matter in black holes the interaction can be represented by a constant a e. Suppose that in class I universes there is a constant number density of such objects: they contribute to the absorption o f retarded waves in the same way as does the smoothed-out matter in the current-current theory, while for advanced waves the two types o f absorption behave in the same manner. Suppose that in class II universes the number density o f such objects varies as R - 3 in the distant future: again they contribute to the absorption o f retarded waves in the same way as does the smoothed-out matter in the currentcurrent theory. Thus, allowing for b o t h types o f absorption, the models eliminated are the same as stated above. In the standard "hot big-bang" model of general relativity, n = 1/2 i n the radiation-domina.ted epoch
18
19 May 1975
so that this model has consistent advanced fields (unless these are rejected on thermodynamic grounds [7] ) and is eliminated. The spatially flat form of the BransDicke cosmology, which includes as a special case the Einstein-de Sitter universe, is eliminated if the standard current-current theory is valid since retarded fields are then inconsistent [4]. The Friedmann models with zero cosmological constant and negative spatial curvature have no. = 1 and are hence eliminated if the current-current theory is valid I thank the referee for some helpful remarks.
References [1] [2] [3] 14} [5] [6] [7]
J.V. Narlikar, Proc. Roy. Soc, A270 (1962) 553. P. Ray Chaudhuri, J. Phys. A: Gen. Phys. 3 (1970) L5. R. Burman, Observatory 92 (1972) 128. R. Burman, Observatory 92 (1972) 131. P.C.W. Davies, J. Phys. A: Gen Phys. 5 (1972) 1722. R.J. Gould, Ann. Rev. Astron. Astrophys. 6 (1968) 195. P.C.W. Davies, J. Phys. A 8(1975) 272.