By T Padmanabhan

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**Extra resources for Cosmological constant - the weight of the vacuum**

**Example text**

In both the cases, −1 < w < 1, though it is possible to construct more complicated scalar field Lagrangians with even w < −1. ) Since the quintessence field (or the tachyonic field) has an undetermined free function V (φ), it is possible to choose this function in order to produce a given H(a). ) and ρφ (a) is due to a scalar field. When w(a) is given, one can determine the V (φ) using either (47) or (52). For quintessence, (47) along with (43) gives 3H 2(a) φ˙ 2 (a) = ρ(1 + w) = (1 − Q)(1 + w); 8πG 41 2V (a) = ρ(1 − w) = 3H 2 (a) (1 − Q)(1 − w) 8πG (53) For tachyonic scalar field, (52) along with (43) gives φ˙ 2 (a) = (1 + w); V (a) = ρ(−w)1/2 = 3H 2 (a) (1 − Q)(−w)1/2 8πG (54) Given Q(a), w(a) these equations implicitly determine V (φ).

This means that clusters will be comparatively rare at high redshifts in a ΩNR = 1 universe compared to models with cosmological constant. 5 in a universe with cosmological constant[212,213,214,215,75]. ) Hence the evolution of cluster population as a function of redshift can be used to discriminate between these models. An indirect way of measuring this abundance is through the lensing effect of a cluster of galaxy on extended background sources. Typically, the foreground clusters shears the light distribution of the background object and leads to giant arcs.

The resulting equation is valid in the linear regime and hence will be satisfied by each of the Fourier modes δk (t) obtained by Fourier transforming δ(t, x) with respect to x. Taking δ(t, x) = D(t)f (x), the D(t) satisfies the equation ¨ + 2 a˙ D˙ = 4πGρb D D a (66) The power spectra P (k, t) =< |δk (t)|2 > at two different redshifts in the linear regime are related by P (k, zf ) = T 2 (k, zf , zi , bg)P (k, zi) (67) where T (called transfer function) depends only on the parameters of the background universe (denoted by ‘bg’) but not on the initial power spectrum and can be computed by solving (66).