2.7 Linear growth rate of the density fluctuation

Most likely our Universe is dominated by collisionless dark matter, and thus λJ is negligibly small. Thus, at most scales of cosmological interest, Equation (54View Equation) is well approximated as
˙a ¨δk + 2-δ˙k − 4πG ρ¯δk = 0. (55 ) a
For a given set of cosmological parameters, one can solve the above equation by substituting the expansion law for a(t) as described in Section 2.3. Since Equation (55View Equation) is the second-order differential equation with respect to t, there are two independent solutions; a decaying mode and a growing mode which monotonically decreases and increases as t, respectively. The former mode becomes negligibly small as the Universe expands, and thus one is usually interested in the growing mode alone.

More specifically those solutions are explicitly obtained as follows. First note that the l.h.s. of Equation (18View Equation) is the Hubble parameter at t, H (t) = a˙∕a:

( Ωm 1 − Ωm − Ω Λ ) H2 = H20 --3-+ -------2----- + Ω Λ (56 ) [ a a ] = H20 Ωm (1 + z)3 + (1 − Ωm − Ω Λ)(1 + z)2 + Ω Λ . (57 )
The first and second differentiation of Equation (56View Equation) with respect to t yields
( ) 2 Ωm-- 1-−-Ωm--−-Ω-Λ 2H H˙ = H 0 − 3a3 − 2 a2 H (58 )
and
( ) ¨ 2 Ωm-- 1 −-Ωm-−-Ω-Λ- H = H 0 92a3 + 2 a2 H, (59 )
respectively. Thus the differential equation for H reduces to
3 Ω H¨ + 2H H˙ = H20H ---m-= 4πG ¯ρH. (60 ) 2a3
This coincides with the linear perturbation equation for δk, Equation (55View Equation). Since H (t) is a decreasing function of t, this implies that H (t) is the decaying solution for Equation (55View Equation). Then the corresponding growing solution D (t) can be obtained according to the standard procedure: Subtracting Equation (55View Equation) from Equation (60View Equation) yields
d da2 d [ d ( D ) ] a2--(D˙H − D ˙H ) + ---(D˙H − D ˙H ) = -- a2H2 -- --- = 0, (61 ) dt dt dt dt H
and therefore the formal expression for the growing solution in linear theory is
∫ t dt′ D (t) ∝ H (t) -----------. (62 ) 0 a2(t′)H2 (t′)
It is often more useful to rewrite D (t) in terms of the redshift z as follows:
5Ω H2 ∫ ∞ 1 + z′ D (z) = ---m--0-H (z ) -------dz′, (63 ) 2 z H3 (z′)
where the proportional factor is chosen so as to reproduce D (z) → 1∕(1 + z) for z → ∞. Linear growth rates for the models described in Section 2.3 are summarized below:

For most purposes, the following fitting formulae [67Jump To The Next Citation Point] provide sufficiently accurate approximations:

g(z) D (z) = ------, (67 ) 1 + z 5Ω-(z)---------------------1-------------------- g(z) = 2 Ω4 ∕7(z ) − λ (z ) + [1 + Ω(z)∕2][1 + λ(z)∕70], (68 )
where
[ ]2 3 Ω (z) = Ωm (1 + z)3 -H0-- = ---------------Ωm-(1 +-z)----------------, (69 ) H (z) Ωm (1 + z)3 + (1 − Ωm − ΩΛ )(1 + z)2 + Ω Λ [ H ]2 Ω λ(z) = ΩΛ ---0- = ----------3----------Λ------------2------. (70 ) H (z) Ωm (1 + z) + (1 − Ωm − Ω Λ)(1 + z ) + Ω Λ
Note that Ωm and ΩΛ refer to the present values of the density parameter and the dimensionless cosmological constant, respectively, which will be frequently used in the rest of the review.

Figure 2View Image shows the comparison of the numerically computed growth rate (thick lines) against the above fitting formulae (thin lines), which are practically indistinguishable.

View Image

Figure 2: Linear growth rate of density fluctuations.


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