Measuring distance in a cosmological setting requires some fancy footwork. The goal of this section is to find expressions for D LS , D L D_\text{LS}, D_\text{L} D LS , D L D S D_\text{S} D S
d s 2 = c 2 d t 2 − a 2 ( t ) [ d χ 2 + f k 2 ( χ ) ( d θ 2 + sin 2 θ d ϕ 2 ) ] ds^2 = c^2 dt^2 - a^2(t)
\left[ d\chi^2 + f_k^2(\chi)(d\theta^2 + \sin^2\theta\, d\phi^2) \right] d s 2 = c 2 d t 2 − a 2 ( t ) [ d χ 2 + f k 2 ( χ ) ( d θ 2 + sin 2 θ d ϕ 2 ) ] where a ( t ) a(t) a ( t )
f k ( χ ) = { 1 k sin ( χ k ) ( k > 0 , closed universe ) , χ ( k = 0 , flat universe ) , 1 − k sinh ( χ − k ) ( k < 0 , open universe ) . f_k(\chi) =
\begin{cases}
\dfrac{1}{\sqrt{k}} \sin(\chi \sqrt{k}) & (k>0\quad,\quad\text{closed universe}), \\[1ex]
\chi & (k=0\quad,\quad\text{flat universe}), \\[1ex]
\dfrac{1}{\sqrt{-k}} \sinh(\chi \sqrt{-k}) & (k<0\quad,\quad\text{open universe}).
\end{cases} f k ( χ ) = ⎩ ⎨ ⎧ k 1 sin ( χ k ) χ − k 1 sinh ( χ − k ) ( k > 0 , closed universe ) , ( k = 0 , flat universe ) , ( k < 0 , open universe ) . is a faactor depending on the geometry of the universe. Light follows null-geodets, d s 2 = 0 ds^2=0 d s 2 = 0
d s 2 = 0 ⇒ d χ = c a ( t ) d t . ds^2=0\quad\Rightarrow\quad d\chi=\frac{c}{a(t)}dt. d s 2 = 0 ⇒ d χ = a ( t ) c d t . Comoving distances ¶ The comoving distance d C d_\textrm{C} d C
d C ≡ ∫ t e m t obs d χ d_\textrm{C}\equiv \int_{t_{\rm em}}^{t_\textrm{obs}} d\chi d C ≡ ∫ t em t obs d χ Inserting from the discussion in the previous section we find
d C ≡ ∫ t e m t obs c d t a ( t ) = ∫ obs z c d z ′ H ( z ′ ) , d_\textrm{C}\equiv \int_{t_{\rm em}}^{t_\textrm{obs}} \frac{c\,dt}{a(t)}
= \int_\textrm{obs}^z \frac{c\,dz'}{H(z')}, d C ≡ ∫ t em t obs a ( t ) c d t = ∫ obs z H ( z ′ ) c d z ′ , where H ( z ) H(z) H ( z ) d i j d_{ij} d ij t i t_i t i t j t_j t j t 1 < t 2 < t 3 t_1<t_2<t_3 t 1 < t 2 < t 3 d 13 = d 12 + d 23 d_{13}=d_{12}+d_{23} d 13 = d 12 + d 23 H ( z ) H(z) H ( z )
H 2 ( z ) = H 0 2 ∑ i Ω i , 0 ( 1 + z ) 3 ( 1 + w i ) , H^2(z) = H_0^2 \sum_i \Omega_{i,0}\,(1+z)^{3(1+w_i)}, H 2 ( z ) = H 0 2 i ∑ Ω i , 0 ( 1 + z ) 3 ( 1 + w i ) , where H 0 H_0 H 0
H 2 ( z ) = H 0 2 [ Ω m , 0 ( 1 + z ) 3 + Ω r , 0 ( 1 + z ) 4 + Ω Λ , 0 + Ω k , 0 ( 1 + z ) 2 ] H^2(z) = H_0^2 \left[ \Omega_{m,0}(1+z)^3 + \Omega_{r,0}(1+z)^4 + \Omega_{\Lambda,0} + \Omega_{k,0}(1+z)^2 \right] H 2 ( z ) = H 0 2 [ Ω m , 0 ( 1 + z ) 3 + Ω r , 0 ( 1 + z ) 4 + Ω Λ , 0 + Ω k , 0 ( 1 + z ) 2 ] with current observational values (Planck 2018):
Component Ω i , 0 \Omega_{i,0} Ω i , 0 Matter (dark + baryonic) Ω m , 0 ≈ 0.315 \Omega_{m,0} \approx 0.315 Ω m , 0 ≈ 0.315 Radiation Ω r , 0 ≈ 9.4 × 1 0 − 5 \Omega_{r,0} \approx 9.4\times 10^{-5} Ω r , 0 ≈ 9.4 × 1 0 − 5 Dark energy (Λ \Lambda Λ Ω Λ , 0 ≈ 0.685 \Omega_{\Lambda,0} \approx 0.685 Ω Λ , 0 ≈ 0.685 Curvature Ω k , 0 ≈ 0 \Omega_{k,0} \approx 0 Ω k , 0 ≈ 0
Note that ∑ i Ω i , 0 = 1 \sum_i \Omega_{i,0} = 1 ∑ i Ω i , 0 = 1 Ω k , 0 \Omega_{k,0} Ω k , 0 k = Ω r , 0 = 0. k=\Omega_{r,0}=0. k = Ω r , 0 = 0.
Simpson’s rule to get actual expressions ¶ Define first, for convenience, the dimensionless Hubble rate
E ( z ) ≡ Ω m , 0 ( 1 + z ) 3 + Ω r , 0 ( 1 + z ) 4 + Ω Λ , 0 + Ω k , 0 ( 1 + z ) 2 , E(z) \equiv \sqrt{\Omega_{m,0}(1+z)^3 + \Omega_{r,0}(1+z)^4 + \Omega_{\Lambda,0} + \Omega_{k,0}(1+z)^2}, E ( z ) ≡ Ω m , 0 ( 1 + z ) 3 + Ω r , 0 ( 1 + z ) 4 + Ω Λ , 0 + Ω k , 0 ( 1 + z ) 2 , so that the comoving distance becomes
d ( z ) = c H 0 ∫ 0 z d z ′ E ( z ′ ) d(z) = \dfrac{c}{H_0}\int_0^{z} \dfrac{dz'}{E(z')} d ( z ) = H 0 c ∫ 0 z E ( z ′ ) d z ′ .
Applying Simpson’s rule with n n n h = z / n h = z/n h = z / n w i ∈ { 1 , 4 , 2 , 4 , … , 4 , 1 } w_i \in \{1,4,2,4,\ldots,4,1\} w i ∈ { 1 , 4 , 2 , 4 , … , 4 , 1 }
d ( z ) ≈ c H 0 ⋅ h 3 ∑ i = 0 n w i E ( i h ) . d(z) \approx \frac{c}{H_0} \cdot \frac{h}{3}\sum_{i=0}^{n} \frac{w_i}{E(ih)}. d ( z ) ≈ H 0 c ⋅ 3 h i = 0 ∑ n E ( ih ) w i . Angular-diameter distance$ ¶ The next step is to relate observed angular size to actual, comoving distance.
If an object of physical size Δ w \Delta w Δ w Δ θ \Delta\theta Δ θ
D A ≡ Δ w Δ θ . D_\textrm{A} \equiv \frac{\Delta w}{\Delta\theta}. D A ≡ Δ θ Δ w . In the RW metric, the radius of the spatial sphere is a ( t ) f k ( χ ) a(t)f_k(\chi) a ( t ) f k ( χ ) Δ w = a ( t ) f k ( χ ) Δ θ \Delta w = a(t)f_k(\chi)\Delta\theta Δ w = a ( t ) f k ( χ ) Δ θ
a ( t ) = 1 1 + z a(t)=\frac{1}{1+z} a ( t ) = 1 + z 1 and recadting f ( χ ) → r ( z ) f(\chi)\to r(z) f ( χ ) → r ( z )
D A ( z ) = a ( t ) f k ( χ ) = r ( z ) 1 + z , D_\textrm{A}(z) = a(t)f_k(\chi)=\frac{r(z)}{1+z}, D A ( z ) = a ( t ) f k ( χ ) = 1 + z r ( z ) , where r ( z ) r(z) r ( z )
Thus D A D_\textrm{A} D A apparent size distance : it accounts for the fact that objects at higher redshift were physically closer when the light was emitted, since the scale factor was smaller.
Expressions for D L D_\textrm{L} D L D S D_\textrm{S} D S ¶ We may now write down expressions for D L D_\textrm{L} D L D S D_\textrm{S} D S
D L = r L ( 1 + z L ) and D S = r S ( 1 + z S ) D_\textrm{L}=\frac{r_\textrm{L}}{(1+z_\textrm{L})}\quad\text{and}\quad D_\textrm{S}=\frac{r_\textrm{S}}{(1+z_\textrm{S})} D L = ( 1 + z L ) r L and D S = ( 1 + z S ) r S The ratio is thus
D L D S = ( 1 + z S ) r ( z L ) ( 1 + z L ) r ( z S ) , \frac{D_L}{D_S} = \frac{(1+z_S)\,r(z_\textrm{L})}{(1+z_L)\,r(z_\textrm{S})}, D S D L = ( 1 + z L ) r ( z S ) ( 1 + z S ) r ( z L ) , In the flat case (k = 0 k=0 k = 0
D L D S = ( 1 + z S ) d L ( 1 + z L ) d S . \frac{D_L}{D_S} = \frac{(1+z_S)\,d_\text{L}}{(1+z_L)\,d_\text{S}}. D S D L = ( 1 + z L ) d S ( 1 + z S ) d L . This is as far as we get without actually calculating the comoving distances. Using the results from the comoving-distance section, we obtain the following expressions for D L D_\text{L} D L D L D_\text{L} D L Λ C D M \Lambda CDM Λ C D M
D L = c H 0 ( 1 + z L ) S ( z L ) and D S = c H 0 ( 1 + z S ) S ( z S ) D_\text{L} = \frac{c}{H_0(1+z_\text{L})}S(z_\text{L})\quad\text{and}\quad D_\text{S} = \frac{c}{H_0(1+z_\text{S})}S(z_\text{S}) D L = H 0 ( 1 + z L ) c S ( z L ) and D S = H 0 ( 1 + z S ) c S ( z S ) where we have for simplicity defined the Simpson’s sum
S ( z ) = h 3 ∑ i = 0 n w i E ( i h ) , h = z n . S(z)=\frac{h}{3}\sum_{i=0}^{n}\frac{w_i}{E(i\,h)}, \qquad h = \frac{z}{n}. S ( z ) = 3 h i = 0 ∑ n E ( i h ) w i , h = n z . D S = c H 0 ( 1 + z S ) h S 3 ∑ i = 0 n w i E ( i h S ) , h S = z S n . D_S = \frac{c}{H_0(1+z_S)}\frac{h_S}{3}\sum_{i=0}^{n}\frac{w_i}{E(i\,h_S)}, \qquad h_S = \frac{z_S}{n}. D S = H 0 ( 1 + z S ) c 3 h S i = 0 ∑ n E ( i h S ) w i , h S = n z S . Hence we have
D L D S = ( 1 + z S ) ( 1 + z L ) ⋅ S L S S . \boxed{\frac{D_L}{D_S} = \frac{(1+z_S)}{(1+z_L)}\cdot\frac{S_\text{L}}{S_\text{S}}.} D S D L = ( 1 + z L ) ( 1 + z S ) ⋅ S S S L . CHECK if this gets modified for k ≠ 0 k\neq 0 k = 0
Expression for D LS D_\textrm{LS} D LS ¶ Similarely, one may show that the angular-diameter distance D LS D_\text{LS} D LS
D LS ( z L , z S ) = ( 1 + z S ) − 1 r LS ( z 1 , z 2 ) D_\text{LS}(z_\text{L},z_\text{S}) = (1+z_\text{S})^{-1} r_\text{LS}(z_1,z_2) D LS ( z L , z S ) = ( 1 + z S ) − 1 r LS ( z 1 , z 2 ) where
r LS = r S 1 − k r L 2 − r L 1 − k r S 2 , r_\text{LS} = r_\text{S}\sqrt{1-k r_\text{L}^2} - r_\text{L}\sqrt{1-k r_\text{S}^2}, r LS = r S 1 − k r L 2 − r L 1 − k r S 2 , and k k k z = 0 z=0 z = 0
( 1 + z s ) D LS = ( 1 + z s ) D S 1 − k ( 1 + z L ) 2 D L 2 − ( 1 + z L ) D L 1 − k ( 1 + z S ) 2 D S 2 , (1+z_s)D_\text{LS} = (1+z_s)D_\text{S} \sqrt{1-k(1+z_\text{L})^2 D_\text{L}^2}
-(1+z_\text{L})D_\text{L} \sqrt{1-k(1+z_\text{S})^2 D_\text{S}^2}, ( 1 + z s ) D LS = ( 1 + z s ) D S 1 − k ( 1 + z L ) 2 D L 2 − ( 1 + z L ) D L 1 − k ( 1 + z S ) 2 D S 2 , which in turn gives for a flat universe (k = 0 k=0 k = 0
D LS = D S − 1 + z L 1 + z S D L , \boxed{D_\text{LS} = D_\text{S}
-\frac{1+z_\text{L}}{1+z_\text{S}}D_\text{L},} D LS = D S − 1 + z S 1 + z L D L , Hence, one ought observe that even in the case of a flat universe, D LS ≠ D S − D L D_\text{LS} \neq D_\text{S} - D_\text{L} D LS = D S − D L
The Einstein radius ¶ The Einstein radius (for a circularly symmetric lens) can be expressed as
θ E = M ( θ E ) π Σ cr D L 2 , \theta_E = \sqrt{\frac{M(\theta_E)}{\pi \Sigma_\text{cr} D_L^2}}, θ E = π Σ cr D L 2 M ( θ E ) , where M ( θ E ) M(\theta_E) M ( θ E ) θ E \theta_E θ E
Σ cr = c 2 4 π G D S D L D L S . \Sigma_\text{cr} = \frac{c^2}{4\pi G}\frac{D_S}{D_L D_{LS}}. Σ cr = 4 π G c 2 D L D L S D S . Inserting the above expressions for a flat, homogeneous and isotropic universe, we find for the critical density,
we find
The denominator simplifies as
D L S = D S − 1 + z L 1 + z S D L = c H 0 ( 1 + z S ) ( S S − S L ) , D_{LS} = D_S - \frac{1+z_L}{1+z_S}D_L
= \frac{c}{H_0(1+z_S)}\left(S_S - S_L\right), D L S = D S − 1 + z S 1 + z L D L = H 0 ( 1 + z S ) c ( S S − S L ) , so that
Σ cr = c H 0 ( 1 + z L ) 4 π G ⋅ S S S L ( S S − S L ) . \boxed{\Sigma_\text{cr} = \frac{c\,H_0(1+z_L)}{4\pi G}\cdot\frac{S_S}{S_L(S_S - S_L)}.} Σ cr = 4 π G c H 0 ( 1 + z L ) ⋅ S L ( S S − S L ) S S . The physical Einstein radius in the lens plane is now generally
ξ 0 = 4 G M S L ( S S − S L ) c H 0 ( 1 + z L ) S S . \xi_0 = \sqrt{\frac{4GM\,S_L(S_S - S_L)}{c\,H_0\,(1+z_L)\,S_S}}. ξ 0 = c H 0 ( 1 + z L ) S S 4 GM S L ( S S − S L ) . (where M M M
ξ 0 S I S = 4 π σ v 2 S L ( S S − S L ) c H 0 ( 1 + z L ) S S , \xi_0^{\rm SIS} = \frac{4\pi\sigma_v^2\,S_L(S_S-S_L)}{c\,H_0\,(1+z_L)\,S_S}, ξ 0 SIS = c H 0 ( 1 + z L ) S S 4 π σ v 2 S L ( S S − S L ) ,