Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 9E / b

Table of contents
- Solution b

Solution

Each photon loses a factor

(1 + z)^{- 1}

in energy, and cosmological time dilation reduces the photon arrival rate by another factor

(1 + z)^{- 1}

. Hence

L_{received} = \frac{L}{( 1 + z ) ^{2}},

so

n = 2

. The physical area of the sphere today is

4 πa (t_{0})^{2} x^{2}

F = \frac{L}{4 πa ( t _{0} ) ^{2} x ^{2} ( 1 + z ) ^{2}} .

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