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Physics LibreTexts

S16. Distance and Magnitude - SOLUTIONS

  • Page ID
    7852
  • Exercise 16.1.1

    Answer

    \(d_L = d_A (1+z)\)

    Exercise 16.2.1

    Answer

    The path is at constant \(\theta\) and \(\phi\) so only \(r\) is varying. Therefore \(\sqrt{ds^2} = a(t) dr/\sqrt{1-kr^2}\). The comoving length is given by integrating this up, while setting the scale factor equal to unity so \(\ell = \int_0^r dr/\sqrt{1-kr^2}\). For \(kr^2 << 1\), \(\ell \simeq \int_0^r dr(1+k r^2/2) = r + kr^3/6\).

    Exercise 16.3.1

    Answer

    Curvature affects the history of the expansion rate, \(H(a)\), via the Friedmann equation. It also affects the time a photon has to travel to come from coordinate distance \(r\) due to the \(dr^2/(1-kr^2)\) term in the invariant distance equation.

    Exercise 16.4.1

    Answer

    I have not produced a written solution for this yet.

    Exercise 16.4.2

    Answer

    I have not produced a written solution for this yet.

    Exercise 16.4.3

    Answer

    I have not produced a written solution for this yet.