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Prog. Theor. Phys. Vol. 41 No. 3 (1969) pp. 686-694

The Lagrangian Approach to the Gravitational Instability in an Expanding Universe

Hidekazu Nariai

Research Institute for Theoretical Physics, Hiroshima University, Takehara, Hiroshima-ken

(Received October 21, 1968)

Abstract:

In order to remedy a defect found by Field and Shepley of Lifshitz's relativistic theory for the gravitational instability in an expanding universe, a new approach to the density perturbation mode is proposed. The defect lies in the fact that one solution of the third order differential equation derived by us for the density contrast K = δε/ε₀ has no physical meaning, where ε₀ is the unperturbed density of matter and radiation. The essence of our approach is to adopt the Lagrangian coordinate condition which assures the comoving nature of coordinates even in the perturbed state, in place of Lifshitz's condition which becomes the former condition only when p₀ (the unperturbed pressure of matter and radiation) vanishes. It is shown that the differential equation for K is of the second order even when p₀ = ε₀/3 and is reduced to Bonnor's Newtonian equation when p₀ = 0. It is shown further that, under some favorable condition, the growth rate of K is the highest at the stage p₀ = ε₀/3, i.e. K ∝t which does not necessarily hold in Lifshitz's original theory, where t is the cosmic time. We have also examined how the density perturbation depends sensitively on our selection of coordinate conditions by picking up another coordinate condition.

URL : http://ptp.ipap.jp/link?PTP/41/686/
DOI : 10.1143/PTP.41.686

[ Full Text PDF : FREE ACCESS (546K) ] Citation: Plain Text BiBTeX EndNote(RIS) RSS

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References:

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Citing Article(s) :

1. Progress of Theoretical Physics Vol. 41 No. 5 (1969) pp. 1379-1381 :

   The Gravitational Instability in the Lemaître Universe

   Hidekazu Nariai

2. Progress of Theoretical Physics Vol. 41 No. 6 (1969) pp. 1461-1469 :

   Density Perturbation and Preferential Coordinate Systems in an Expanding Universe

   Kazuo Sakai

3. Progress of Theoretical Physics Vol. 42 No. 1 (1969) pp. 9-23 :

   Formation of Gravitationally Bound Primordial Gas Clouds

   Kenji Tomita

4. Progress of Theoretical Physics Vol. 42 No. 3 (1969) pp. 544-554 :

   Gravitational Instability in the Brans-Dicke Cosmology

   Hidekazu Nariai

5. Progress of Theoretical Physics Vol. 48 No. 5 (1972) pp. 1503-1516 :

   Primordial Irregularities in the Early Universe

   Kenji Tomita

6. Progress of Theoretical Physics Vol. 67 No. 4 (1982) pp. 1076-1092 :

   Primordial Irregularities in an Early Universe with Vacuum Phase Transitions

   Kenji Tomita

7. Progress of Theoretical Physics Vol. 77 No. 3 (1987) pp. 653-670 :

   Gauge-Invariant Perturbations in a Spatially Flat Anisotropic Universe

   Mitsue Den

8. Progress of Theoretical Physics Supplement No.78 (1984) pp. 1-166 :

   Cosmological Perturbation Theory

   Hideo Kodama and Misao Sasaki

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