Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 41 / 3 / iv

Table of contents
- Solution iv

Solution

Apply the isotropic cosmological correlation-power-spectrum relation and put

K = k_{m a x}

,

q = Kr

. The band-limited linear density correlation is

ξ (r) = \frac{A}{2 π ^{2} r} \int_{0}^{K} k^{2} sin (k r) d k .

Two integrations by parts give the primitive

- k^{2} cos (k r) / r + 2 k sin (k r) / r^{2} + 2 cos (k r) / r^{3}

. Evaluating both endpoints yields

ξ (r) = \frac{A}{2 π ^{2} r ^{4}} [- q^{2} cos q + 2 q sin q + 2 (cos q - 1)], r > 0.

The apparent singularity at the origin is removable. Either the Taylor series or the original integral gives

ξ (0) = \frac{A K ^{4}}{8 π ^{2}} .

ξ (r) = ξ (0) [1 - q^{2} /9 + q^{4} /240 + O (q^{6})]

. The sharp spectral cutoff produces oscillations and negative correlations at some nonzero separations, even though the matter power spectrum is everywhere nonnegative. A covariance function need not be pointwise nonnegative; its required positivity is that of the covariance quadratic form.

Figure 1.
An abrupt cutoff in a nonnegative density power spectrum produces an oscillating correlation function
. Details.

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