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Problems 11-14

Problem 1.11

a. Bound vs. unbound universes

From Eq. (1.50) we know that

\[ \dot{a}^2 - \frac{8\pi G \rho}{3}a^2 = 2E. \]

At present day \(a_0 = 1\) and this becomes (Neglecting any contribution from radiation)

\[ 2E = H_0^2(1 - \Omega_0) = (1 - \Omega_{\Lambda 0} - \Omega_{m0})H_0^2 = -\frac{kc^2}{x_u^2} \]

The boundary between a bound and unbound universe is then given by \(k=0\) or \(\Omega_{\Lambda 0} + \Omega_{m0} = 1\).

b. Accelerating vs. decelerating universes

Now, Eq. (1.49) can be rewritten as

\[ \frac{\ddot{a}}{a} = \frac{H_0^2}{2}(2\Omega_{\Lambda 0} - \Omega_{m0}a^{-3}). \]

The transition between a decelerating and accelerating universe occurs when \(\ddot{a} = 0\), for all values of \(a\), in particular at present day \(a_0 = 1\), this happens when

\[ 2\Omega_{\Lambda 0} - \Omega_{m0} = 0 \]

c. Recollpapse vs. expand forever

Let's write the Friedmann equation as

\[ \frac{H^2}{H_0^2} = \Omega_{m0}a^{-3} + \Omega_{\Lambda 0} + (1 - \Omega_{m0} - \Omega_{\Lambda 0})a^{-2}. \]

A condition for recollapse is that there exists a time \(t_{\textrm{coll}}\) when \(H(t_{\textrm{coll}}) = 0\), or equivalently a scale factor \(a_{\textrm{coll}} > 1\) such that

\[ \Omega_{m0}a^{-3} + \Omega_{\Lambda 0} + (1 - \Omega_{m0} - \Omega_{\Lambda 0})a^{-2} = 0. \]

Define the function

\[ f(a) = \Omega_{\Lambda 0}a^{3} + (1 - \Omega_{m0} - \Omega_{\Lambda 0})a + \Omega_{m0}. \]

The separatrix in the parameter space between recollapsing and ever-expanding universes is given by the condition that \(f(a)\) has a double root at some \(a = a_{\textrm{coll}} > 1\). This requires that both \(f(a_{\textrm{coll}}) = 0\) and \(f'(a_{\textrm{coll}}) = 0\).

\[ 0 = 3\Omega_{\Lambda 0}a^2 + (1 - \Omega_{m0} - \Omega_{\Lambda 0}) \]

A parametric solution of the separatrix is given by

\[ \Omega_{m0} = \frac{2 a^3}{1 - 3 a^2 + 2 a^3}, \quad \Omega_{\Lambda 0} = \frac{1}{1 - 3 a^2 + 2 a^3}, \quad a > 1. \]

A plot of the different regions in the \(\Omega_{m0}\)-\(\Omega_{\Lambda 0}\) plane is shown below, generated with the following code:

>>> from galactic_dynamics_bt.chapter01.frw_model import plot_frw_model
>>> plot_frw_model()

FRW Model Phase Diagram

Figure P1.11: Phase diagram of FRW cosmological models in the \(\Omega_{m0}\)-\(\Omega_{\Lambda 0}\) plane. The solid line indicates the boundary between bound and unbound models, while the dashed line indicates the boundary between decelerating and accelerating models.

Problem 1.12

The age of the universe at redshift \(z\) can be computed as

>>> from galactic_dynamics_bt.chapter01.universe_age import find_universe_age
>>> z = 0
>>> h7 = 1.05
>>> find_universe_age(
...    z,
...    omega_m0=0.237,
...    omega_lambda0=0.763,
...    omega_gamma0=8.84e-5 / h7**2,
...    H0=70.0 * h7,
...)
14.26742...

For different redshifts we have

Redshift Age of the universe
0 14.267 Gyr
1 6.3286 Gyr
1000 0.4605 Myr

The next figure shows the age of the universe as a function of redshift.

>>> from galactic_dynamics_bt.chapter01.universe_age import plot_universe_age
>>> plot_universe_age()

Age of the Universe vs Redshift Figure P1.12: Age of the universe as a function of redshift for flat cosmologies. Solid line represents the cosmology of Eq (1.73), dashed line are the results of the Planck 2018 cosmology (Aghanim et al., 2020)1.

Problem 1.13

a. Scale factor at matter-radiation equality

Matter density scales as \(\rho_m \propto a^{-3}\), while radiation density scales as \(\rho_\gamma \propto a^{-4}\). The equality between matter and radiation densities occurs when

\[ \rho_m = \rho_\gamma \Rightarrow \Omega_{m0} a^{-3} = \Omega_{\gamma 0} a^{-4} \Rightarrow \Omega_{m0} a = \Omega_{\gamma 0}. \]

This implies that the equality occurs at a scale factor

\[ 1 + z_{\gamma m} = \frac{1}{a_{\gamma m}} = \frac{\Omega_{m0}}{\Omega_{\gamma 0}} = 1.18\times 10^4 h_7^2 \Omega_{m0}. \]

b. Age of the universe at matter-radiation equality

Using the subsitution \(u = \Omega_{\gamma 0} + a \Omega_{m 0}\) we can show that (For \(\Omega_\Lambda = 0\))

\[ H_0t = \int_0^{\Omega_{\gamma 0}/\Omega_{m0}} da \frac{a}{\sqrt{\Omega_{m0} a + \Omega_{\gamma 0}}} = 2(2 - 2^{1/2})\frac{\Omega_{\gamma 0}^{1/2}(1 - \Omega_{m0})}{3\Omega_{m0}^2}. \]

Since \(\Omega_{m0} + \Omega_{\gamma 0} = 1\), we have that the age of the universe at matter-radiation equality is

\[ t_{\gamma m} = \frac{2(2 - 2^{1/2})}{3H_0}\frac{\Omega_{\gamma 0}^{3/2}}{\Omega_{m0}^2}. \]

c. Comoving horizon at matter-radiation equality

The comoving horizon at matter-radiation equality is given by

\[ x_{\gamma m} = c\int_0^{t_{\gamma m}} \frac{dt}{a(t)} = c\int_0^{a_{\gamma m}} \frac{da}{a^2 H(a)} = c\int_0^{a_{\gamma m}} \frac{da}{H_0\sqrt{\Omega_{m0} a + \Omega_{\gamma 0}}}. \]

Similarly as before, using the substitution \(u = \Omega_{\gamma 0} + a \Omega_{m 0}\) we can show that

\[ x_{\gamma m} = (2^{1/2} - 1)\frac{c}{H_0} \frac{\Omega_{\gamma 0}^{1/2}}{\Omega_{m0}}. \]

Problem 1.14

The universe is not opaque for \(z \lesssim 6\) even though it's reionized because the mean electron density is now extremely low due to cosmic expansion, so the Thomson scattering optical depth is tiny, and photons can propagate freely.

References

  1. Aghanim, N., Akrami, Y., Ashdown, M., Aumont, J., Baccigalupi, C., Ballardini, M., Banday, A. J., Barreiro, R. B., Bartolo, N., Basak, S., Battye, R., Benabed, K., Bernard, J.-P., Bersanelli, M., Bielewicz, P., Bock, J. J., Bond, J. R., Borrill, J., Bouchet, F. R., ... Zonca, A. (2020). Planck2018 results: VI. Cosmological parameters. Astronomy &Amp; Astrophysics, 641, A6. https://doi.org/10.1051/0004-6361/201833910 ↩