Angular diameter distance

Angular Diameter Distance Derivation

Definition

The angular diameter distance $d_A(z)$ relates an object’s physical transverse size $L$ to its observed angular size $\theta$: $\theta =\frac{L}{d_A(z)}$

General Expression

The angular diameter distance is given by: $d_A(z)=\frac{r(z)}{1+z}$

where $r(z)$ is the comoving distance to redshift $z$.

Comoving Distance Derivation

The comoving distance is: $r(z)=c{\int }_0^z\frac{d{z}^{\prime }}{H(z^{\prime })}$

where the Hubble parameter $H(z)$ in terms of scale factor $a=\frac{1}{1+z}$ is:

$H(z)=H_0\sqrt{{\Omega }_m{a}^{-3}+{\Omega }_r{a}^{-4}+{\Omega }_{\Lambda }{a}^{-3(1+w)}+{\Omega }_k{a}^{-2}}$

For a flat universe (${\Omega }_k=0$) with cosmological constant ($w=-1$): $H(z)=H_0\sqrt{{\Omega }_m{a}^{-3}+{\Omega }_r{a}^{-4}+{\Omega }_{\Lambda }}$

Angular Diameter Distance in Terms of Scale Factor

Substituting $a=\frac{1}{1+z}$ and $dz=-\frac{da}{a^2}$:

$d_A(a)=\frac{c}{H_0}\cdot a{\int }_a^1\frac{d{a}^{\prime }}{a^{\prime 2}E(a^{\prime })}$

where $E(a)=\frac{H(a)}{H_0}=\sqrt{{\Omega }_m{a}^{-3}+{\Omega }_r{a}^{-4}+{\Omega }_{\Lambda }+{\Omega }_k{a}^{-2}}$

Simplified Forms

Matter + Lambda Dominated (ignoring radiation)

For $a\ll 1$ (early universe) or when ${\Omega }_r$ is negligible: $E(a)=\sqrt{{\Omega }_m{a}^{-3}+{\Omega }_{\Lambda }+{\Omega }_k{a}^{-2}}$

Flat Matter + Lambda Universe

$d_A(a)=\frac{c}{H_0}\cdot a{\int }_a^1\frac{d{a}^{\prime }}{a^{\prime 2}\sqrt{{\Omega }_m{a}^{\prime -3}+{\Omega }_{\Lambda }}}$

Luminosity Distance

Definition

The luminosity distance $d_L$ is defined by the inverse-square law relating bolometric luminosity $\mathcal{L}$ (power emitted) to observed bolometric flux $\mathcal{F}$: $\mathcal{F}=\frac{\mathcal{L}}{4\pi d_L^2}$

Relation to Comoving/Transverse Comoving Distance

In an FRW spacetime, photon energies redshift by a factor $(1+z)$ and photon arrival rates are time-dilated by another factor $(1+z)$. Combined with geometric dilution over the transverse comoving distance $D_M$, this yields: $d_L(z)=(1+z)D_M(z)$

For spatial curvature, the transverse comoving distance is $D_M(z)=S_k(\chi (z))$ with $\chi (z)=c{\int }_0^z\frac{d{z}^{\prime }}{H(z^{\prime })}$

and

\begin{cases} \frac{1}{\sqrt{\Omega_k}}\sinh\!\big(\sqrt{\Omega_k}\,\chi H_0/c\big)\,\frac{c}{H_0}, & \Omega_k>0 \\ \chi, & \Omega_k=0 \\ \frac{1}{\sqrt{|\Omega_k|}}\sin\!\big(\sqrt{|\Omega_k|}\,\chi H_0/c\big)\,\frac{c}{H_0}, & \Omega_k<0 \end{cases}$$ (Equivalently: $D_M = \frac{c}{H_0\sqrt{|\Omega_k|}}\,\mathrm{sinn}\!\big(\sqrt{|\Omega_k|}\,\chi H_0/c\big)$.) ### In Terms of Scale Factor Using $a = \frac{1}{1+z}$ and $dz = -\frac{da}{a^2}$, the line-of-sight comoving distance becomes $$\chi(a)= c\int_a^1 \frac{da'}{a'^2 H(a')} = \frac{c}{H_0}\int_a^1 \frac{da'}{a'^2 E(a')}$$ so the luminosity distance is $$d_L(a) = \frac{1}{a}\,D_M(a) = \frac{1}{a}\,S_k\!\big(\chi(a)\big)$$ For a flat universe ($\Omega_k=0$), $D_M=\chi$ and this simplifies to $$d_L(a)=\frac{c}{H_0}\,\frac{1}{a}\int_a^1 \frac{da'}{a'^2 E(a')}$$ ### Distance Duality The Etherington reciprocity relation gives $$d_L(z) = (1+z)^2 d_A(z)$$ which in scale factor form is $$d_L(a) = \frac{1}{a^2}\,d_A(a)$$ ## Age of the Universe and Lookback Time ### Cosmic Time vs Scale Factor For an FRW universe, the Hubble parameter is $$H(a) = \frac{\dot a}{a}$$ so $$dt = \frac{da}{aH(a)}$$ ### Age Today Taking the Big Bang at $a=0$ and today at $a=1$, the age of the universe is $$t_0 = \int_0^1 \frac{da}{aH(a)} = \frac{1}{H_0}\int_0^1 \frac{da}{aE(a)}$$ with $$E(a)=\sqrt{\Omega_r a^{-4}+\Omega_m a^{-3}+\Omega_k a^{-2}+\Omega_{de} a^{-3(1+w)}}$$ and for a cosmological constant ($w=-1$, $\Omega_{de}=\Omega_\Lambda$): $$E(a)=\sqrt{\Omega_r a^{-4}+\Omega_m a^{-3}+\Omega_k a^{-2}+\Omega_\Lambda}$$ ### Age at Scale Factor a The cosmic time at scale factor $a$ is $$t(a) = \int_0^a \frac{da'}{a'H(a')} = \frac{1}{H_0}\int_0^a \frac{da'}{a' E(a')}$$ ### Lookback Time The lookback time to scale factor $a$ (or redshift $z=1/a-1$) is $$t_L(a) = t_0 - t(a) = \int_a^1 \frac{da'}{a'H(a')} = \frac{1}{H_0}\int_a^1 \frac{da'}{a'E(a')}$$ ## Key Properties 1. **Maximum**: $d_A(z)$ reaches a maximum at $z \approx 1.5$ (depending on cosmology) 2. **Duality**: Related to luminosity distance by $d_L = (1+z)^2 d_A$ 3. **Behavior**: - At low $z$: $d_A \approx \frac{cz}{H_0}$ - At high $z$: $d_A \propto \frac{1}{(1+z)H_0}$ ## Numerical Integration For practical calculations, the integral is typically evaluated numerically: $$d_A(a) = \frac{c}{H_0} \cdot a \int_a^1 \frac{da'}{a'^2 \sqrt{\Omega_m a'^{-3} + \Omega_r a'^{-4} + \Omega_\Lambda + \Omega_k a'^{-2}}}$$