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    HLPaper 3
    1.

    Show that the temperature of the universe is inversely proportional to the cosmic scale factor.

    [3]
    Verified
    Solution

    «wavelength of light/CBR» λ∝R\lambda \propto Rλ∝R ✓ reference to Wien's law showing that λ∝1T\lambda \propto \frac{1}{T}λ∝T1​ ✓ combine to get result ✓

    2.

    The present temperature of the cosmic microwave background (CMB) radiation is 3 K. Estimate the size of the universe relative to the present size of the universe when the temperature of the CMB was 300 K.

    [1]
    Verified
    Solution

    RpastRnow=3300=0.01\frac{R_{\text{past}}}{R_{\text{now}}}=\frac{3}{300}=0.01Rnow​Rpast​​=3003​=0.01 ✓

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