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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2018

Paper 1

Put $u=\tanh(\lambda t)=e^{\phi/\phi_0}$. The <hyperbolic cotangent> double-angle identity gives
$$
\coth(2\lambda t)=\frac{1+u^2}{2u}
=\cosh\left(\frac{\phi}{\phi_0}\right),
$$
so
$$
\boxed{\
H(\phi)=\lambda\frac{\phi_0^2}{M_{\rm Pl}^2}
\cosh\left(\frac{\phi}{\phi_0}\right)\
}.
$$
Also
$$
\operatorname{csch}^2(2\lambda t)
=\cosh^2\left(\frac{\phi}{\phi_0}\right)-1.
$$
Solving the <Friedmann equation> for the scalar potential,
$$
c^2V=3M_{\rm Pl}^2H^2-\frac12\dot\phi^2,
$$
therefore gives
$$
\boxed{\
V(\phi)=\frac{2\lambda^2\phi_0^2}{c^2}
\left[
\left(\frac{3\phi_0^2}{2M_{\rm Pl}^2}-1\right)
\cosh^2\left(\frac{\phi}{\phi_0}\right)+1
\right]\
}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 15B / c / Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 15B / c / Solution

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Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 15B / c / Solution

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