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4.1 The role of Lambda for inflation

A decade later when Einstein and others wanted to drop the cosmological constant, because they no longer considered it necessary in an expanding universe, the astrophysicist among cosmologists, Lemaître, offered physical interpretations for Lambda. He suggested:

1. the cosmological constant to be related to the driving mechanism of expansion
2. a physical process to be responsible for the existence of Lambda.

While earlier explanations did not survive time, the concept proposed three years later has won surprising actuality in connection with the inflationary universe.

Lemaître (1934):

``The problem of the universe is essentially an application of the law of gravitation to a region of extremely low density. The mean density of matter up to a distance of some ten millions of light years from us is of the order of 10-30 g cm-3; if all the atoms of the stars were equally distributed through space there would be about one atom per cubic yard, or the total energy would be that of an equilibrium radiation at the temperature of liquid hydrogen [12 K]. The theory of relativity points out the possibility of a modification of the law of gravitation under such extreme conditions. It suggests that, when we identify gravitational mass and energy, we have to introduce a constant. Everything happens as though the energy in vacuo would be different from zero. In order that absolute motion, i.e., motion relative to vacuum, may not be detected, we must associate pressure p = rho c2 to the density of energy rho c2 of vacuum. This is essentially the meaning of the cosmological constant lambda which corresponds to a negative density of vacuum rho0 according to

Equation 37

From the equations given later in the text:

Equation 38

Equation 39

with chi defined in the usual way as varying between 0 and 2pi over the complete map of the universe, he obtains for the location of the observer

Equation 40

with the solution

Equation 41

With the then accepted value for the Hubble constant, H0 = 500 km s-1 Mpc-1, Lemaître obtained the numerical value quoted for the negative vacuum density.

It can also be deduced that Lemaître might have been aware of the fact that Lambda can be obtained from observations and that the data available to him give

Lambda = 10-54 cm-2.

The connection with the Hubble constant appeared indirectly already in de Sitter's paper (1930, Sect. 3.1.4). The relation

R (t) = exp (Lambda t).

is required for the inflationary universe, where in the current models a rapid increase of R (t) occurs because of the large amount of false vacuum energy, proportional to Lambda, suddenly becoming available.

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