Galaxy number counts at second order: an independent approach

The perturbed FLRW spacetime is described in the longitudinal gauge by [26]

$$\begin{equation}\mathrm{d}{s}^{2}={a}^{2}\left[-\left(1+2{{\Phi}}_{1}+{{\Phi}}_{2}\right)\mathrm{d}{\eta }^{2}+\left(1-2{{\Psi}}_{1}-{{\Psi}}_{2}\right){\delta }_{ij}\mathrm{d}{x}^{i}\enspace \mathrm{d}{x}^{j}\right],\end{equation} \tag{ 2.1 }$$

where η is the conformal time, a = a(η) is the scale factor and δ_ij is the flat spatial metric, and we have neglected the vector and tensor modes, we also allow for first and second order anisotropic stresses. From now on we consider perturbations around an FLRW metric up to second-order.

The components of the four-velocity u^μ = dx^μ /dη up to and including second order using the perturbed metric are given by

$$\begin{equation}{u}_{0}=-a\left[1+{{\Phi}}_{1}+\frac{1}{2}{{\Phi}}_{2}-\frac{1}{2}{{{\Phi}}_{1}}^{2}+\frac{1}{2}{v}_{1k}{v}_{1}^{k}\right],\end{equation} \tag{ 2.2 }$$

$$\begin{equation}{u}_{i}=a\left[{v}_{1i}+\frac{1}{2}{v}_{2i}-2{{\Psi}}_{1}{v}_{1k}\right],\end{equation} \tag{ 2.3 }$$

$$\begin{equation}{u}^{0}={a}^{-1}\left[1-{{\Phi}}_{1}-\frac{1}{2}{{\Phi}}_{2}+\frac{3}{2}{{{\Phi}}_{1}}^{2}+\frac{1}{2}{v}_{1k}{v}_{1}^{k}\right],\end{equation} \tag{ 2.4 }$$

$$\begin{equation}{u}^{i}={a}^{-1}\left[{v}_{1}^{i}+\frac{1}{2}{v}_{2}^{i}\right],\end{equation} \tag{ 2.5 }$$

where v_i = ∂_i v, with v the velocity potential.

In a redshift survey galaxy positions are identified by measuring photons produced at the source, denoted by s, and detected by an observer labelled o. In a general spacetime, we consider a lightray with tangent vector k^μ and affine parameter λ, that parametrises the curve the lightray follows. The source, λ_s, and the observer, λ_o, are represented by given values for the affine parameter, as illustrated in figure 1. The components of the photon wavevector can be written as

$$\begin{equation}{\bar{k}}^{\mu }=\frac{\mathrm{d}{x}^{\mu }}{\mathrm{d}\lambda }={a}^{-1}\left[1,{n}^{i}\right],\end{equation} \tag{ 2.6 }$$

where the overbar denotes background quantities, nⁱ is the direction of observation ³ pointing from the observer to the source, and following the normalisation condition: nⁱ n_i = 1.

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The tangent vector is null

$$\begin{equation}{k}_{\mu }{k}^{\mu }=0,\end{equation} \tag{ 2.7 }$$

and geodesic

$$\begin{equation}{k}^{\nu }{\nabla }_{\nu }{k}^{\mu }=0.\end{equation} \tag{ 2.8 }$$

where ∇_ν is the covariant derivative defined by the metric given in equation (2.1). In general, the perturbed wavevector can be written as

$$\begin{equation}{\delta }^{\left(n\right)}{k}^{\mu }={a}^{-1}\left[{\delta }^{\left(n\right)}\nu ,{\delta }^{\left(n\right)}{n}^{i}\right].\end{equation} \tag{ 2.9 }$$

where δ⁽ⁿ⁾ gives the nth order perturbation, and we are following the usual notation for the temporal component, that is k⁰ ≡ ν [24].

The affine parameter of the geodesic equation is also related to the comoving distance (χ) by

$$\begin{equation}\chi ={\lambda }_{\text{o}}-{\lambda }_{\text{s}},\end{equation} \tag{ 2.10 }$$

and in terms of the redshift this is

$$\begin{equation}\chi \left(z\right)={\int }_{0}^{z}\frac{\mathrm{d}\tilde {z}}{\left(1+\tilde {z}\right)\mathcal{H}\left(\tilde {z}\right)}.\end{equation} \tag{ 2.11 }$$

The photon energy measured by an observer with four-velocity u^μ is

$$\begin{equation}\mathcal{E}=-{g}_{\mu \nu }{u}^{\mu }{k}^{\nu }.\end{equation} \tag{ 2.12 }$$

From equation (2.12) the observed redshift of a source (e.g. a galaxy) can be defined as

$$\begin{equation}1+z=\frac{{\mathcal{E}}_{\text{s}}}{{\mathcal{E}}_{\text{o}}},\end{equation} \tag{ 2.13 }$$

From this definition there will be a Doppler effect on the redshift due to the velocities u^μ and the observed redshift is in fact a function of the velocity and the wavevector, i.e. z = z(k^μ , u^μ ).

For a given bundle of lightrays leaving a source, the bundle will invariantly expand and create an area in between the lightrays that conform it, this area can be projected to a screen space, perpendicular to the trajectories of the photons and the four-velocity of the observer, as illustrated in figures 2 and 3.

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The area of a bundle in screen space, $\mathcal{A}$, defines the angular diameter distance d_A , and is directly related to the null expansion θ [27] defined in section 4,

$$\begin{equation}\frac{1}{\sqrt{\mathcal{A}}}\frac{\mathrm{d}\sqrt{\mathcal{A}}}{\mathrm{d}\lambda }=\frac{\mathrm{d}\enspace \mathrm{ln}\enspace {d}_{A}}{\mathrm{d}\lambda }=\frac{1}{2}\theta .\end{equation} \tag{ 2.14 }$$

where λ is the affine parameter defined in equation (2.6). Using equation (2.14) we can compute how the area of the bundle changes along the geodesic trajectory that the photons are following from the source towards the observer.

Number counts relate to the number of sources detected in a bundle of rays, for a small affine parameter displacement λ to λ + dλ at an event P. This corresponds to a physical distance

$$\begin{equation}\mathrm{d}\ell =\left({k}^{\mu }{u}_{\mu }\right)\mathrm{d}\lambda ,\end{equation} \tag{ 2.15 }$$

in the rest frame of a comoving galaxy at said point in space P, if k^μ is a tangent vector to the past directed null geodesics (so that k^μ u_μ > 0).

The cross-sectional area of the bundle is

$$\begin{equation}\mathrm{d}\mathcal{A}={d}_{A}^{2}\left(\lambda \right)\mathrm{d}{\Omega},\end{equation} \tag{ 2.16 }$$

if the geodesics subtend a solid angle dΩ at the observer, this is shown in figure 4.

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From equations (2.15) and (2.16) the corresponding volume element at a point P in space is (see e.g. [25])

$$\begin{equation}\mathrm{d}V=\mathrm{d}\ell \enspace \mathrm{d}\mathcal{A}=\left({k}^{\mu }{u}_{\mu }\right){d}_{A}^{2}\left(\lambda \right)\mathrm{d}\lambda \enspace \mathrm{d}{\Omega}=-\mathcal{E}{d}_{A}^{2}\left(\lambda \right)\mathrm{d}\enspace \lambda \mathrm{d}{\Omega}.\end{equation} \tag{ 2.17 }$$

These covariant definitions lead to the expressions we compute in the following sections at first and second order in CPT.

Let us now look at solutions to the geodesic equation. First, from equation (2.7) and the normalisation of nⁱ we obtain the null condition of the photon wavevector

$$\begin{align}\hfill 0& =2\left[{n}^{i}{{\delta }^{\left(1\right)}n}_{i}-{\delta }^{\left(1\right)}\nu -{{\Phi}}_{1}-{{\Psi}}_{1}\right]\hfill \\ \hfill & \quad +\left[{n}^{i}{{\delta }^{\left(2\right)}n}_{i}-{\delta }^{\left(2\right)}\nu -\frac{1}{2}{{\Phi}}_{2}-\frac{1}{2}{{\Psi}}_{2}-{\left({\delta }^{\left(1\right)}\nu \right)}^{2}+{{\delta }^{\left(1\right)}n}_{i}{{\delta }^{\left(1\right)}n}^{i}\right.\hfill \\ \hfill & \quad -\left.4\left({{\Phi}}_{1}+{{\Psi}}_{1}\right){\delta }^{\left(1\right)}\nu -4{{\Psi}}_{1}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\right].\hfill \end{align} \tag{ 3.1 }$$

For the geodesic equation (2.8), we get the propagation equations for the temporal and spatial perturbations

$$\begin{equation}{k}^{\mu }{\nabla }_{\mu }{\delta }^{\left(n\right)}\nu =\frac{\mathrm{d}{\delta }^{\left(n\right)}\nu }{\mathrm{d}\lambda }+{{\Gamma}}_{\alpha \beta }^{0}{k}^{\alpha }{k}^{\beta }=0,\end{equation} \tag{ 3.2 }$$

$$\begin{equation}{k}^{\mu }{\nabla }_{\mu }{\delta }^{\left(n\right)}{n}^{i}=\frac{\mathrm{d}{\delta }^{\left(n\right)}{n}^{i}}{\mathrm{d}\lambda }+{{\Gamma}}_{\alpha \beta }^{i}{k}^{\alpha }{k}^{\beta }=0,\end{equation} \tag{ 3.3 }$$

where ${{\Gamma}}_{\nu \sigma }^{\mu }$ are the connection coefficients at any given order (n). Their expansion up to second order is provided in appendix A. Substituting and rearranging, we find the geodesic equations in general at first order [28],

$$\begin{equation}\frac{\mathrm{d}{\delta }^{\left(1\right)}\nu }{\mathrm{d}\lambda }=-2\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\lambda }+{{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime },\end{equation} \tag{ 3.4 }$$

$$\begin{equation}\frac{\mathrm{d}{{\delta }^{\left(1\right)}n}^{i}}{\mathrm{d}\lambda }=2\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }{n}^{i}-\left[{{{\Psi}}_{1}}_{,}^{i}+{{{\Phi}}_{1}}_{,}^{i}\right],\end{equation} \tag{ 3.5 }$$

where equation (3.4) there is a term related to the ISW effect defined below. The expressions for the solution at second order are given in appendix C.

We now expand the photon energy $\mathcal{E}=-{g}_{\mu \nu }{u}^{\mu }{k}^{\nu }$ up to second order,

$$\begin{equation}\mathcal{E}=\bar{\mathcal{E}}+{\delta }^{\left(1\right)}\mathcal{E}+\frac{1}{2}{\delta }^{\left(2\right)}\mathcal{E}.\end{equation} \tag{ 3.6 }$$

Using equations (2.1)–(2.3), (3.4), (3.5), (C1) and (C2), we find that $\bar{\mathcal{E}}=1$, and

$$\begin{equation}{\delta }^{\left(1\right)}\mathcal{E}={\delta }^{\left(1\right)}\nu +{{\Phi}}_{1}-\left({v}_{1i}{n}^{i}\right),\end{equation} \tag{ 3.7 }$$

$$\begin{align}\hfill {\delta }^{\left(2\right)}\mathcal{E}& ={\delta }^{\left(2\right)}\nu +{{\Phi}}_{2}-\left({v}_{2i}{n}^{i}\right)+2{{\Phi}}_{1}{\delta }^{\left(1\right)}\nu -2{v}_{1i}{{\delta }^{\left(1\right)}n}^{i}+{{{\Phi}}_{1}}^{2}\hfill \\ \hfill & \quad +\left({v}_{1k}{v}_{1}^{k}\right)+4{{\Psi}}_{1}\left({v}_{1i}{n}^{i}\right).\hfill \end{align} \tag{ 3.8 }$$

The perturbations to the photon energy given in equations (3.7) and (3.8) are written explicitly in terms of the metric potentials in appendix C. Integrating equations (3.4), (3.5), (C1) and (C2), with respect to the affine parameter, λ, from the observer to the source, we find the perturbed photon vector, δk^μ . Note that δk^μ |_o = 0, what can be seen explicitly in equations (C5) and (C6), so that

$$\begin{equation}{\left.{\delta }^{\left(1\right)}\mathcal{E}\right\vert }_{\text{o}}={{\Phi}}_{1}{\vert }_{\text{o}}-{\left({v}_{1i}{n}^{i}\right)}_{\text{o}},\end{equation} \tag{ 3.9 }$$

$$\begin{equation}{\left.{\delta }^{\left(2\right)}\mathcal{E}\right\vert }_{\text{o}}={{\Phi}}_{2}{\vert }_{\text{o}}-{\left({v}_{2i}{n}^{i}\right)}_{\text{o}}+{\left({{\Phi}}_{1}{\vert }_{\text{o}}\right)}^{2}+{\left({v}_{1k}{v}_{1}^{k}\right)}_{\text{o}}+4{{\Psi}}_{1}{\vert }_{\text{o}}{\left({v}_{1i}{n}^{i}\right)}_{\text{o}}.\end{equation} \tag{ 3.10 }$$

The observed redshift is given by equation (2.13), then up to second order that is

$$\begin{equation}1+z=1+\bar{z}+{\delta }^{\left(1\right)}z+\frac{1}{2}{\delta }^{\left(2\right)}z=\frac{{\mathcal{E}}_{\text{s}}}{{\mathcal{E}}_{\text{o}}}=\frac{{\left(\bar{\mathcal{E}}+{\delta }^{\left(1\right)}\mathcal{E}+\frac{1}{2}{\delta }^{\left(2\right)}\mathcal{E}\right)}_{\text{s}}}{{\left(\bar{\mathcal{E}}+{\delta }^{\left(1\right)}\mathcal{E}+\frac{1}{2}{\delta }^{\left(2\right)}\mathcal{E}\right)}_{\text{o}}}.\end{equation} \tag{ 3.11 }$$

Note that in general, the expansion of the perturbed quotient up to second order is

$$\begin{align}\hfill \frac{A+{\delta }^{\left(1\right)}A+\frac{1}{2}{\delta }^{\left(2\right)}A}{B+{\delta }^{\left(1\right)}B+\frac{1}{2}{\delta }^{\left(2\right)}B}& =\frac{A}{B}\left(1+\frac{{\delta }^{\left(1\right)}A}{A}-\frac{{\delta }^{\left(1\right)}B}{B}+\frac{{\delta }^{\left(2\right)}A}{2A}-\frac{{\delta }^{\left(2\right)}B}{2B}\right.\hfill \\ \hfill & \quad -\left.\frac{{\delta }^{\left(1\right)}A}{A}\frac{{\delta }^{\left(1\right)}B}{B}+{\left[\frac{{\delta }^{\left(1\right)}B}{B}\right]}^{2}\right),\hfill \end{align} \tag{ 3.12 }$$

Using equations (3.11) and (3.12), and ignoring the background redshift, we obtain

$$\begin{align}\hfill 1+\hat{z}& =1+\left({\left.{\delta }^{\left(1\right)}\mathcal{E}\right\vert }_{\text{s}}-{\left.{\delta }^{\left(1\right)}\mathcal{E}\right\vert }_{\text{o}}\right)-{\left.{\delta }^{\left(1\right)}\mathcal{E}\right\vert }_{\text{s}}{\left.{\delta }^{\left(1\right)}\mathcal{E}\right\vert }_{\text{o}}\hfill \\ \hfill & \quad +{\left({\left.{\delta }^{\left(1\right)}\mathcal{E}\right\vert }_{\text{o}}\right)}^{2}+\frac{1}{2}\left({\left.{\delta }^{\left(2\right)}\mathcal{E}\right\vert }_{\text{s}}-{\left.{\delta }^{\left(2\right)}\mathcal{E}\right\vert }_{\text{o}}\right).\hfill \end{align} \tag{ 3.13 }$$

From equation (3.13) the redshift of a source s is, at first order,

$$\begin{equation}{\delta }^{\left(1\right)}z={\left.\left({v}_{1i}{n}^{i}+{{\Phi}}_{1}\right)\right\vert }_{\text{s}}^{\text{o}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left[{{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right]\mathrm{d}\lambda ,\end{equation} \tag{ 3.14 }$$

where the integral runs w.r.t. the affine parameter λ along the line of sight. In equation (3.14) we identify the following elements

• (a)  
  The Doppler redshift, which depends on the difference between the peculiar velocities of the source and the observer

  $$\begin{equation}{\delta }^{\left(1\right)}{z}_{\text{Doppler}}={\left({v}_{1i}{n}^{i}\right)}_{\text{o}}-{\left({v}_{1i}{n}^{i}\right)}_{\text{s}}.\end{equation} \tag{ 3.15 }$$
• (b)  
  The Gravitational redshift, which describes the change of energy the photon experiences when it travels from a region with potential Φ₁|_s to a region with potential Φ₁|_o

  $$\begin{equation}{\delta }^{\left(1\right)}{z}_{\text{grav}}={{\Phi}}_{1}{\vert }_{\text{o}}-{{\Phi}}_{1}{\vert }_{\text{s}}.\end{equation} \tag{ 3.16 }$$
• (c)  
  The ISW, which describes the change in energy when a photon travels through a potential well between the source and the observer, this effect is only non-zero when the gravitational potential evolves during and along the photon's trajectory, so that the energy gained by going down the gravitational potential is not cancelled by the energy lost by climbing out the potential at the other end.

  $$\begin{equation}{\delta }^{\left(1\right)}{z}_{\text{ISW}}={\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left[{{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right]\mathrm{d}\lambda .\end{equation} \tag{ 3.17 }$$

At second order the redshift (C8) can also be decomposed as above

$$\begin{equation}{\delta }^{\left(2\right)}z={\delta }^{\left(2\right)}{z}_{\text{Doppler}}+{\delta }^{\left(2\right)}{z}_{\text{grav}}+{\delta }^{\left(2\right)}{z}_{\text{ISW}}+{\delta }^{\left(2\right)}{z}_{\text{NL}},\end{equation} \tag{ 3.18 }$$

where the nonlinear contribution from squared first order quantities denoted by 'NL' is given by

$$\begin{align}\hfill {\delta }^{\left(2\right)}{z}_{\text{NL}}& ={\delta }^{\left(2\right)}{z}_{\text{grav}{\times}\text{grav}}+{\delta }^{\left(2\right)}{z}_{\text{grav}{\times}\text{ISW}}+{\delta }^{\left(2\right)}{z}_{\text{ISW}}\hfill \\ \hfill & \quad +{\delta }^{\left(2\right)}{z}_{\text{grav}{\times}\text{Doppler}}+{\delta }^{\left(2\right)}{z}_{\text{Doppler}{\times}\text{Doppler}}+{\delta }^{\left(2\right)}{z}_{\text{Doppler}{\times}\text{ISW}}.\hfill \end{align} \tag{ 3.19 }$$

Here, just as in the linear case, we have

• (a)  
  Nonlinear Doppler redshift,

  $$\begin{equation*}{\delta }^{\left(2\right)}{z}_{\text{Doppler}}=\frac{1}{2}\left[{\left({v}_{2i}{n}^{i}\right)}_{\text{o}}-{\left({v}_{2i}{n}^{i}\right)}_{\text{s}}\right],\end{equation*}$$
• (b)  
  Nonlinear gravitational redshift,

  $$\begin{equation*}{\delta }^{\left(2\right)}{z}_{\text{grav}}=\frac{1}{2}\left[{{\Phi}}_{2}{\vert }_{\text{o}}-{{\Phi}}_{2}{\vert }_{\text{s}}\right],\end{equation*}$$
• (c)  
  Nonlinear ISW,

  $$\begin{equation*}{\delta }^{\left(2\right)}{z}_{\text{ISW}}=\frac{1}{2}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{2}}^{\prime }+{{{\Psi}}_{2}}^{\prime }\right)\mathrm{d}\lambda .\end{equation*}$$

Here the second order contribution to the effects described above are evident, plus the product of first order contributions:

• (a)  
  Gravitational redshift squared

  $$\begin{equation*}{\delta }^{\left(2\right)}{z}_{\text{grav}{\times}\text{grav}}=-\frac{3}{2}{\left({{\Phi}}_{1}{\vert }_{\text{s}}-{{\Phi}}_{1}{\vert }_{\text{o}}\right)}^{2}+6{{\Phi}}_{1}{\vert }_{\text{s}}{{\Phi}}_{1}{\vert }_{\text{o}},\end{equation*}$$
• (b)  
  Gravitational redshift × ISW

  $$\begin{equation*}{\delta }^{\left(2\right)}{z}_{\text{grav}{\times}\text{ISW}}=2\left({{\Phi}}_{1}{\vert }_{\text{s}}-{{\Phi}}_{1}{\vert }_{\text{o}}\right){\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda \left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right),\end{equation*}$$
• (c)  
  Gravitational redshift × Doppler

  $$\begin{equation*}{\delta }^{\left(2\right)}{z}_{\text{grav}{\times}\text{Doppler}}={{\Phi}}_{1}{\vert }_{\text{o}}{\left({v}_{1i}{n}^{i}\right)}_{\text{s}}-{{\Phi}}_{1}{\vert }_{\text{s}}{\left({v}_{1i}{n}^{i}\right)}_{\text{o}}-2{{\Psi}}_{1}{\vert }_{\text{o}}{\left({v}_{1i}{n}^{i}\right)}_{\text{o}},\end{equation*}$$
• (d)  
  Doppler squared

  $$\begin{align*}\hfill {\delta }^{\left(2\right)}{z}_{\text{Doppler}{\times}\text{Doppler}}& =\frac{1}{2}\left[{\left({v}_{1k}{v}_{1}^{k}\right)}_{\text{s}}-{\left({v}_{1k}{v}_{1}^{k}\right)}_{\text{o}}\right]-{\left({v}_{1i}{n}^{i}\right)}_{\text{o}}\left[{\left({v}_{1i}{n}^{i}\right)}_{\text{s}}-{\left({v}_{1i}{n}^{i}\right)}_{\text{o}}\right],\hfill \end{align*}$$
• (e)  
  Doppler × ISW

  $$\begin{align*}\hfill {\delta }^{\left(2\right)}{z}_{\text{Doppler}{\times}\text{ISW}}& ={\left({v}_{1i}{n}^{i}\right)}_{\text{o}}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda \left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)+2{v}_{1i}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda \left({{{{\Phi}}_{1}}_{,}}^{i}+{{{{\Psi}}_{1}}_{,}}^{i}\right),\hfill \end{align*}$$
• (f)  
  Integrated ISW

  $$\begin{align*}\hfill {\delta }^{\left(2\right)}{z}_{\text{IISW}}& =2{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda \left[{{{\Psi}}_{1}}^{\prime }\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\right]+2{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda \left[{{\Phi}}_{1}\left(\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\lambda }-2\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }\right)\right]\hfill \\ \hfill & \quad -4{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda \left[\left({{\Psi}}_{1}{{{\Phi}}_{1}}_{,i}-{{\Phi}}_{1}{{{\Psi}}_{1}}_{,i}\right){n}^{i}\right]+4{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda \left[{n}^{i}{{{\Phi}}_{1}}_{,i}\left({{\Phi}}_{1}-{{\Psi}}_{1}\right)\right]\hfill \\ \hfill & \quad +{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }\left\{2\left({{{\Phi}}_{1}}^{\prime }-{{{\Psi}}_{1}}^{\prime }\right){\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda \left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\right.\hfill \\ \hfill & \quad -{n}^{i}\left({{{\Phi}}_{1}}_{,i}+{{{\Psi}}_{1}}_{,i}\right){\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda \left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)+\left(\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\lambda }-\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }\right){\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda \left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\hfill \\ \hfill & \quad +2\left({{\Phi}}_{1}-{{\Psi}}_{1}\right){n}^{i}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)}_{,i}\hfill \\ \hfill & \quad +\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\right)\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)}_{,i}\right)\hfill \\ \hfill & \quad -\left.{n}^{i}\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)}_{,i}\right)\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda \left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\right)\right\}.\hfill \end{align*}$$

To measure the angular diameter distance (d_A ) we must define a projector into the screen space perpendicular to the light ray as shown in figure 3. The screen space is orthogonal to the light ray and to the observer four-velocity. In fact the tensor

$$\begin{equation}{\mathcal{P}}_{\mu \nu }={g}_{\mu \nu }+{u}_{\mu }{u}_{\nu }-{n}_{\mu }{n}_{\nu },\end{equation} \tag{ 4.1 }$$

where g_μν is the metric, u_μ is the four-velocity and n_μ is the four-direction of observation ⁴ , projects four-vectors onto screen space, as can be seen in figure 2. The projector tensor satisfies the relations

$$\begin{equation}{{\mathcal{P}}_{\mu }}^{\mu }=2,\qquad {\mathcal{P}}_{\mu \alpha }{{\mathcal{P}}^{\alpha }}_{\nu }={\mathcal{P}}_{\mu \nu },\qquad {\mathcal{P}}_{\mu \nu }{k}^{\nu }={\mathcal{P}}_{\mu \nu }{u}^{\nu }={\mathcal{P}}_{\mu \nu }{n}^{\nu }=0.\end{equation} \tag{ 4.2 }$$

The null expansion, θ, and null shear, Σ_μν , are optical properties given in terms of the tangent vector k^μ by [26, 27, 29]

$$\begin{equation}\theta ={\mathcal{P}}^{\mu \nu }{\nabla }_{\mu }{k}_{\nu },\end{equation} \tag{ 4.3 }$$

$$\begin{equation}{{\Sigma}}_{\mu \nu }={\mathcal{P}}_{\left(\mu ^{\sigma }{\mathcal{P}}_\nu \right)}^{\rho }{\nabla }_{\sigma }{k}_{\rho }-\frac{1}{2}\theta {\mathcal{P}}_{\mu \nu },\end{equation} \tag{ 4.4 }$$

Here θ describes the rate of expansion of the projected area of a bundle of light rays and Σ_μν describes its rate of shear illustrated in figure 3. Note that the wavevector can be obtained from a scalar potential (S), i.e. k_μ = ∇_μ S, and thus there is no null vorticity, that is ω ≡ ∇_[μ k_ν] = 0 [30].

The 'null evolution' is given by the Sachs propagation equations (see e.g. [29] for full derivation)

$$\begin{equation}\frac{\mathrm{d}\theta }{\mathrm{d}\lambda }=-\frac{1}{2}{\theta }^{2}-{{\Sigma}}_{\mu \nu }{{\Sigma}}^{\mu \nu }-{R}_{\mu \nu }{k}^{\mu }{k}^{\nu },\end{equation} \tag{ 4.5 }$$

$$\begin{equation}\frac{\mathrm{d}{{\Sigma}}_{\mu \nu }}{\mathrm{d}\lambda }=-{{\Sigma}}_{\mu \nu }\theta +{C}_{\mu \rho \nu \sigma }{k}^{\rho }{k}^{\sigma },\end{equation} \tag{ 4.6 }$$

where C_μρνσ is the Weyl tensor.

Equations (4.5) and (4.6) allow us to compute the angular diameter distance as a parametric function depending only on the affine parameter, λ, in contrast to previous works where the dependency is on the redshift [13–15] or the conformal time [17]. The advantages of maintaining this dependency are discussed in section 7.

From equations (2.14) and (4.5) we obtain a second order differential equation for the angular diameter distance,

$$\begin{equation}\frac{{\mathrm{d}}^{2}{d}_{A}}{\mathrm{d}{\lambda }^{2}}=-\frac{1}{2}\left({R}_{\mu \nu }{k}^{\mu }{k}^{\nu }+{{\Sigma}}_{\mu \nu }{{\Sigma}}^{\mu \nu }\right){d}_{A}.\end{equation} \tag{ 4.7 }$$

We require appropriate initial conditions to solve (4.7). These can be found from the series expansion of the squared distance given by Kristian and Sachs in [31]:

$$\begin{equation}{d}_{A}^{2}\left({\lambda }_{\text{s}}\right)={\left({u}_{\mu }{k}^{\mu }\right)}_{\text{o}}^{2}\left({\lambda }_{\text{o}}-{\lambda }_{\text{s}}\right)\left[1-\frac{1}{6}{\left({R}_{\mu \nu }{k}^{\mu }{k}^{\nu }\right)}_{\text{o}}{\left({\lambda }_{\text{o}}-{\lambda }_{\text{s}}\right)}^{2}+\cdots \enspace \right],\end{equation} \tag{ 4.8 }$$

from where we obtain the boundary conditions at the observer

$$\begin{equation}{d}_{A}\left({\lambda }_{\text{o}}\right)=0,\quad \text{and},\quad {\left.\frac{\mathrm{d}{d}_{A}}{\mathrm{d}\lambda }\right\vert }_{\text{o}}=-{\mathcal{E}}_{\text{o}}.\end{equation} \tag{ 4.9 }$$

In this section we will define a conformal metric ${g}_{\mu \nu }={a}^{-2}{\hat{g}}_{\mu \nu }$, useful to compute the angular diameter distance. In our notation, a hat ($\hat{\text{\_}}$) denotes quantities on the physical spacetime, while quantities on the conformal spacetime have no hat. The background of the metric g_μν is Minkowski spacetime, which simplifies both the equations and the calculations. Conformal maps preserve both angles and shapes of infinitesimally small figures, but not their overall size [32]. The conformal transformation ${\hat{g}}_{\mu \nu }\to {g}_{\mu \nu }$ maps the null geodesic equation of the perturbed FLRW metric ${\hat{g}}_{\mu \nu }$ to a null geodesic on the perturbed Minkowski metric g_μν [33] and the angular diameter distance transforms as ${\hat{d}}_{A}=a{d}_{A}.$ The affine parameter transforms as $\mathrm{d}\hat{\lambda }={a}^{2}\mathrm{d}\lambda ,$ so that the photon ray vector transforms as ${\hat{k}}^{\mu }={a}^{-2}{k}^{\mu }{\Leftrightarrow}{\hat{k}}_{\mu }={k}_{\mu }$ [34]. For the four-velocity we have ${\hat{u}}_{\mu }=a{u}_{\mu }.$ Finally, the energy transforms as $\hat{\mathcal{E}}=-{\hat{u}}_{\nu }{\hat{k}}^{\nu }=-{a}^{-1}{u}_{\nu }{k}^{\nu }={a}^{-1}\mathcal{E}.$ In Minkowski spacetime we normalize $\mathcal{E}=1$.

Hereafter, and until the end of this section, we will be working in a perturbed Minkowski spacetime, in order to finally conformally transform our result back to an FLRW spacetime.

In the Minkowski background, equation (4.7) simplifies to

$$\begin{equation}\frac{{\mathrm{d}}^{2}{\bar{d}}_{A}}{\mathrm{d}{\lambda }^{2}}=0,\end{equation} \tag{ 4.10 }$$

since ${\bar{R}}_{\mu \nu }$ and the shear vanish in the background. The solution is then

$$\begin{equation}{\bar{d}}_{A}={C}_{1}+\lambda {C}_{2}.\end{equation} \tag{ 4.11 }$$

The initial conditions given in equation (4.9) yield C₁ = 0 and C₂ = −1, so that

$$\begin{equation}{\bar{d}}_{A}\left({\lambda }_{\text{s}}\right)={\lambda }_{\text{o}}-{\lambda }_{\text{s}}.\end{equation} \tag{ 4.12 }$$

Mapping this into the FLRW background we obtain, for the angular diameter distance,

$$\begin{equation}{\hat{d}}_{A}\left({\hat{\lambda }}_{\text{s}}\right)=a\left({\hat{\lambda }}_{\text{s}}\right)\left({\hat{\lambda }}_{\text{o}}-{\hat{\lambda }}_{\text{s}}\right),\end{equation} \tag{ 4.13 }$$

which can be expressed in terms of the comoving distance (2.10) as

$$\begin{equation}{\hat{d}}_{A}\left(\hat{z}\right)=\frac{\chi \left(\hat{z}\right)}{1+\hat{z}}.\end{equation} \tag{ 4.14 }$$

Here we have used the definition of the scale factor $a\left(\hat{z}\right)=1/\left(1+\hat{z}\right)$ and the fact that the comoving distance depends on the redshift as given in equation (2.11).

In general, at first order equation (4.7) takes the form

$$\begin{align}\hfill \frac{{\mathrm{d}}^{2}{\delta }^{\left(1\right)}{d}_{A}}{\mathrm{d}{\lambda }^{2}}& =-\frac{1}{2}\left[2{\bar{R}}_{\mu \nu }{\bar{k}}^{\mu }{\delta }^{\left(1\right)}{k}^{\nu }{\bar{d}}_{A}+{\delta }^{\left(1\right)}{R}_{\mu \nu }{\bar{k}}^{\mu }{\bar{k}}^{\nu }{\bar{d}}_{A}+{\bar{R}}_{\mu \nu }{\bar{k}}^{\mu }{\bar{k}}^{\nu }{\delta }^{\left(1\right)}{d}_{A}\right]\hfill \\ \hfill & \quad -{\bar{d}}_{A}^{\prime }\frac{\mathrm{d}{\delta }^{\left(1\right)}\nu }{\mathrm{d}\lambda }-2{\bar{d}}_{A}^{{\prime\prime}}{\delta }^{\left(1\right)}\nu .\hfill \end{align} \tag{ 4.15 }$$

where we use take the background equivalence between the affine parameter and the conformal time dλ = dη, so that

$$\begin{equation}\frac{\mathrm{d}{\bar{d}}_{A}}{\mathrm{d}\eta }=\frac{\mathrm{d}{\bar{d}}_{A}}{\mathrm{d}\lambda }.\end{equation} \tag{ 4.16 }$$

This relation is only fulfiled in the background. Once perturbations are introduced the relation between the affine parameter and time becomes non-trivial.

In Minkowski spacetime, equation (4.15) simplifies to

$$\begin{align}\hfill \frac{{\mathrm{d}}^{2}{\delta }^{\left(1\right)}{d}_{A}}{\mathrm{d}{\lambda }^{2}}& =-\frac{1}{2}{\bar{d}}_{A}\left[2\left(\frac{{\mathrm{d}}^{2}{{\Psi}}_{1}}{\mathrm{d}{\lambda }^{2}}\right)+{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)-{n}^{i}{n}^{j}{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,ij}-\frac{2}{{\bar{d}}_{A}}\frac{\mathrm{d}{\delta }^{\left(1\right)}\nu }{\mathrm{d}\lambda }\right],\hfill \end{align} \tag{ 4.17 }$$

where we used the background solution for d_A (4.12), the first order perturbation of the Ricci tensor δ⁽¹⁾ R_μν given in appendix B, and equations (1.3) and (1.4).

The solution to (4.17) is, upon several integrations by parts,

$$\begin{align}\hfill \frac{{\delta }^{\left(1\right)}{d}_{A}\left({\lambda }_{\text{s}}\right)}{{\bar{d}}_{A}\left({\lambda }_{\text{s}}\right)}& ={{\Phi}}_{1}{\vert }_{\text{o}}-{{\Psi}}_{1}{\vert }_{\text{o}}-{{\Psi}}_{1}{\vert }_{\text{s}}-{\left({v}_{1i}{n}^{i}\right)}_{\text{o}}-\frac{1}{{\lambda }_{\text{o}}-{\lambda }_{\text{s}}}\left\{2{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {{\Psi}}_{1}\right.\hfill \\ \hfill & \quad +\frac{1}{2}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda \left({\lambda }_{\text{s}}-\lambda \right)\left({\lambda }_{\text{o}}-\lambda \right)\left[{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)-{n}^{i}{n}^{j}\right.\hfill \\ \hfill & \quad {\times}\left.{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,ij}-\left.\frac{2}{{\lambda }_{\text{o}}-{\lambda }_{\text{s}}}\frac{\mathrm{d}{\delta }^{\left(1\right)}\nu }{\mathrm{d}\lambda }\right]\right\}.\hfill \end{align} \tag{ 4.18 }$$

From equation (4.9), in general

$$\begin{equation}{\delta }^{\left(n\right)}{d}_{A}\left({\lambda }_{\text{o}}\right)=0,\quad \text{and},\quad {\left.\frac{\mathrm{d}{\delta }^{\left(n\right)}{d}_{A}}{\mathrm{d}\lambda }\right\vert }_{\text{o}}=-{\delta }^{\left(n\right)}{\mathcal{E}}_{\text{o}}.\end{equation} \tag{ 4.19 }$$

In the absence of anisotropic stress, Φ₁ = Ψ₁ (see, e.g. reference [26]), we recover in equation (4.18) the fully relativistic lensing convergence, usually denoted as κ [9, 17, 30], at first order, which includes Sachs–Wolfe, ISW and Doppler terms in addition to the standard lensing integral

$$\begin{align}\hfill \frac{{\delta }^{\left(1\right)}{d}_{A}\left({\lambda }_{\text{s}}\right)}{{\bar{d}}_{A}\left({\lambda }_{\text{s}}\right)}& =-{{\Phi}}_{1}{\vert }_{\text{s}}-{\left({v}_{1i}{n}^{i}\right)}_{\text{o}}-\frac{1}{{\lambda }_{\text{o}}-{\lambda }_{\text{s}}}\left\{2{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {{\Phi}}_{1}\right.\hfill \\ \hfill & \quad +{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda \left({\lambda }_{\text{s}}-\lambda \right)\left({\lambda }_{\text{o}}-\lambda \right)\hfill \\ \hfill & \quad {\times}\left.\left[{\nabla }^{2}\left({{\Phi}}_{1}\right)-{n}^{i}{n}^{j}{\left({{\Phi}}_{1}\right)}_{,ij}-\frac{1}{{\lambda }_{\text{o}}-{\lambda }_{\text{s}}}\frac{\mathrm{d}{\delta }^{\left(1\right)}\nu }{\mathrm{d}\lambda }\right]\right\}.\hfill \end{align} \tag{ 4.20 }$$

At second order, equation (4.7) takes the form

$$\begin{align}\hfill \frac{{\mathrm{d}}^{2}{\delta }^{\left(2\right)}{d}_{A}}{\mathrm{d}{\lambda }^{2}}& =-\left[2{\bar{k}}^{\mu }{\delta }^{\left(1\right)}{k}^{\nu }{\delta }^{\left(1\right)}{R}_{\mu \nu }+\frac{1}{2}{\bar{k}}^{\mu }{\bar{k}}^{\nu }{\delta }^{\left(2\right)}{R}_{\mu \nu }+{\delta }^{\left(1\right)}{{\Sigma}}_{\mu \nu }{\delta }^{\left(1\right)}{{\Sigma}}^{\mu \nu }\right]{\bar{d}}_{A}\hfill \\ \hfill & \quad -\left[{\bar{k}}^{\mu }{\bar{k}}^{\nu }{\delta }^{\left(1\right)}{R}_{\mu \nu }\right]{\delta }^{\left(1\right)}{d}_{A}-2\left[{\delta }^{\left(1\right)}{k}^{\mu }{\nabla }_{\mu }{\delta }^{\left(1\right)}\nu +3{\delta }^{\left(1\right)}{k}^{\mu }{\bar{k}}^{\alpha }{{\Gamma}}_{\mu \alpha }^{0}\right]{\bar{d}}_{A}^{\prime }\hfill \\ \hfill & \quad -2\left[{\delta }^{\left(2\right)}\nu +{\left({\delta }^{\left(1\right)}\nu \right)}^{2}\right]{\bar{d}}_{A}^{{\prime\prime}}-\left(\frac{\mathrm{d}{\delta }^{\left(2\right)}\nu }{\mathrm{d}\lambda }\right){\bar{d}}_{A}^{\prime }-4\left({\delta }^{\left(1\right)}\nu \right)\left({\delta }^{\left(1\right)}{d}_{A}^{{\prime\prime}}\right)\hfill \\ \hfill & \quad -2\left[{\bar{k}}^{\mu }{\bar{k}}^{\alpha }{{\Gamma}}_{\mu \alpha }+\frac{\mathrm{d}{\delta }^{\left(1\right)}\nu }{\mathrm{d}\lambda }\right]{\delta }^{\left(1\right)}{d}_{A}^{\prime },\hfill \end{align} \tag{ 4.21 }$$

where δ⁽¹⁾Σ_μν is the linear perturbation to the shear. Using equation (4.4) we obtain

$$\begin{equation}\frac{\mathrm{d}{\delta }^{\left(1\right)}{{\Sigma}}_{ij}}{\mathrm{d}\lambda }=\frac{1}{2}{\delta }_{ij}{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)-{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,ij}.\end{equation} \tag{ 4.22 }$$

Without loss of generality, we set the perturbation of the shear at the observer δ⁽¹⁾Σ^μν |_o = 0, and integrating along the line of sight from the observer to the source (λ_o to λ_s), we obtain

$$\begin{equation}{\delta }^{\left(1\right)}{{\Sigma}}_{ij}={\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda \left[\frac{1}{2}{\delta }_{ij}{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)-{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,ij}\right].\end{equation} \tag{ 4.23 }$$

The contraction δ⁽¹⁾Σ_ij δ⁽¹⁾Σ^ij is given by

$$\begin{align}\hfill {\delta }^{\left(1\right)}{{\Sigma}}_{ij}{\delta }^{\left(1\right)}{{\Sigma}}^{ij}& =\left[{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda \left[\frac{1}{2}{\delta }_{ij}{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)-{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,ij}\right]\right]\hfill \\ \hfill & \quad {\times}\left[{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda \left[\frac{1}{2}{\delta }^{ij}{\nabla }^{2}{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)-{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{ij}\right]\right],\hfill \\ \hfill & =\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,ij}\right)\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{ij}\right)\hfill \\ \hfill & \quad -\frac{1}{4}{\left[{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\right]}^{2}.\hfill \end{align} \tag{ 4.24 }$$

In equation (C9) we find the second order part of the angular diameter distance, using the background solution for ${\bar{d}}_{A}$, and the full expression δ⁽²⁾ R_μν , the second order perturbation of the Ricci tensor given in appendix B.

Thus, the total area distance as a function of the affine parameter in a perturbed FLRW spacetime is given by

$$\begin{equation}{\hat{d}}_{A}\left({\lambda }_{\text{s}}\right)=a\left({\lambda }_{\text{s}}\right)\left({\lambda }_{\text{o}}-{\lambda }_{\text{s}}\right)\left[1+\frac{{\delta }^{\left(1\right)}{d}_{A}\left({\lambda }_{\text{s}}\right)}{{\bar{d}}_{A}\left({\lambda }_{\text{s}}\right)}+\frac{1}{2}\frac{{\delta }^{\left(2\right)}{d}_{A}\left({\lambda }_{\text{s}}\right)}{{\bar{d}}_{A}\left({\lambda }_{\text{s}}\right)}\right],\end{equation} \tag{ 4.25 }$$

where the solutions for ${\bar{d}}_{A}\left({\lambda }_{\text{s}}\right)$, δ⁽¹⁾ d_A (λ_s) and δ⁽²⁾ d_A (λ_s) are given in equations (4.12), (4.18) and (C9), respectively. From here onwards, we abandon the conformal Minkowski spacetime and return to FLRW.

In order to compare with previous work done in the literature, we can convert the angular diameter distance in terms of the affine parameter to a function of the observed redshift. To do so, we need to perturbatively invert z(λ) into λ(z) and substitute this into equation (4.25). This means we need d_A on surfaces of constant observed redshift z rather than on surfaces of constant affine parameter λ, which is not observable.

We expand the affine parameter in perturbation theory as

$$\begin{equation}\lambda =\varsigma +{\delta }^{\left(1\right)}\lambda +\frac{1}{2}{\delta }^{\left(2\right)}\lambda ,\end{equation} \tag{ 4.26 }$$

where ς is the affine parameter in redshift space corresponding to the redshift $\hat{z}$, as if there were no perturbations [14]. We define ς using as an anchor the background relation

$$\begin{equation}a\left(\varsigma \right)=\frac{1}{1+\hat{z}},\end{equation} \tag{ 4.27 }$$

with this relation we can fix δ⁽¹⁾ λ and δ⁽²⁾ λ, since it should always hold, and if there are any perturbations, they should cancel since equation (4.27) is only valid in the background. To begin with, we see that at any redshift $\hat{z}$, the derivatives of a with respect to ς are

$$\begin{equation}\frac{1}{a}\frac{\mathrm{d}a}{\mathrm{d}\varsigma }=\mathcal{H}\left(\varsigma \right),\end{equation} \tag{ 4.28 }$$

$$\begin{equation}\frac{1}{a}\frac{{\mathrm{d}}^{2}a}{\mathrm{d}{\varsigma }^{2}}=\left[\frac{\mathrm{d}\mathcal{H}\left(\varsigma \right)}{\mathrm{d}\varsigma }+{\mathcal{H}}^{2}\left(\varsigma \right)\right].\end{equation} \tag{ 4.29 }$$

We now expand the scale factor a about λ up to and including second order perturbations as defined in equation (4.26), we have

$$\begin{align}\hfill a\left(\lambda \right)& =a\left(\varsigma \right)\left[1+\mathcal{H}{\delta }^{\left(1\right)}\lambda +\frac{1}{2}\mathcal{H}{\delta }^{\left(2\right)}\lambda +\frac{1}{2}\left(\frac{\mathrm{d}\mathcal{H}}{\mathrm{d}\lambda }+{\mathcal{H}}^{2}\right)\right.\hfill \\ \hfill & \quad {\times}\left.{\left({\delta }^{\left(1\right)}\lambda \right)}^{2}+\mathcal{O}\left({\delta }^{\left(3\right)}\lambda \right)\right].\hfill \end{align} \tag{ 4.30 }$$

Using equations (3.11) and (4.30) we find that,

$$\begin{align}\hfill \frac{1}{1+\hat{z}}& =\frac{a\left(\varsigma \right)}{a\left({\varsigma }_{\text{o}}\right)}\left[1+\left(\mathcal{H}{\delta }^{\left(1\right)}\lambda -{\delta }^{\left(1\right)}z\right)+\frac{1}{2}\left\{\mathcal{H}{\delta }^{\left(2\right)}\lambda -{\delta }^{\left(2\right)}z+2{\left({\delta }^{\left(1\right)}z\right)}^{2}\right.\right.\hfill \\ \hfill & \quad -\left.\left.2\mathcal{H}{\delta }^{\left(1\right)}\lambda {\delta }^{\left(1\right)}z+\left(\frac{\mathrm{d}\mathcal{H}}{\mathrm{d}\varsigma }+{\mathcal{H}}^{2}\right){\left({\delta }^{\left(1\right)}\lambda \right)}^{2}\right\}\right].\hfill \end{align} \tag{ 4.31 }$$

From the background relation given in equations (4.27) and (4.31) we then find that the perturbations to the affine parameter must follow the following relations:

$$\begin{equation}{\delta }^{\left(1\right)}\lambda =\frac{{\delta }^{\left(1\right)}z}{\mathcal{H}},\end{equation} \tag{ 4.32 }$$

$$\begin{equation}{\delta }^{\left(2\right)}\lambda =\frac{1}{\mathcal{H}}\left[{\delta }^{\left(2\right)}z-{\left({\delta }^{\left(1\right)}z\right)}^{2}\left(1+\frac{1}{{\mathcal{H}}^{2}}\frac{\mathrm{d}\mathcal{H}}{\mathrm{d}\varsigma }\right)\right].\end{equation} \tag{ 4.33 }$$

Finally, using these relations to substitute for $a\left({\lambda }_{\text{s}}\right)\left({\lambda }_{\text{o}}-{\lambda }_{\text{s}}\right)$, we find that the area distance (4.25) becomes

$$\begin{align}\hfill {\hat{d}}_{A}\left(\varsigma \right)& =a\left(\varsigma \right)\left({\varsigma }_{\text{o}}-\varsigma \right)\left\{1+\left[\frac{{\delta }^{\left(1\right)}{d}_{A}}{{\bar{d}}_{A}}\left(\lambda \right)+\left(1-\frac{1}{{\varsigma }_{\text{o}}-\varsigma }{\delta }^{\left(1\right)}z\left(\lambda \right)\right)\right]\right.\hfill \\ \hfill & \quad +\frac{1}{2}\left[\frac{{\delta }^{\left(2\right)}{d}_{A}}{{\bar{d}}_{A}}\left(\lambda \right)+\left(1-\frac{1}{{\varsigma }_{\text{o}}-\varsigma }{\delta }^{\left(2\right)}z\left(\lambda \right)\right)+\frac{{\mathcal{H}}^{\prime }-{\mathcal{H}}^{2}}{{\mathcal{H}}^{3}\left({\varsigma }_{\text{o}}-\varsigma \right)}{\left({\delta }^{\left(1\right)}z\right)}^{2}\right.\hfill \\ \hfill & \quad \left.\left.+2\left(1-\frac{1}{\mathcal{H}\left({\varsigma }_{\text{o}}-\varsigma \right)}\right)\frac{{\delta }^{\left(1\right)}{d}_{A}}{{\bar{d}}_{A}}{\delta }^{\left(1\right)}z\right]\right\}.\hfill \end{align} \tag{ 4.34 }$$

Up until here we have corrected the scale factor from the affine parameter λ to ς. Now we need to convert the first order contributions because they bring additional second order contributions. We introduce that, for a general first order quantity δ⁽¹⁾ X, converting to ς gives

$$\begin{equation}{\delta }^{\left(1\right)}X\left(\lambda \right)={\delta }^{\left(1\right)}X\left(\varsigma \right)+{\left.\frac{\partial {\delta }^{\left(1\right)}X}{\partial \lambda }\right\vert }_{\varsigma }\frac{{\delta }^{\left(1\right)}z\left(\varsigma \right)}{\mathcal{H}},\end{equation} \tag{ 4.35 }$$

where δ⁽¹⁾ X(ς) is to be understood as substituting ς in the expression for δ⁽¹⁾ X(λ), i.e. δ⁽¹⁾ X(λ → ς). The factor ∂_λ δ⁽¹⁾ X|_ς is multiplied by a first order quantity, so the derivative is evaluated in the background. Thus, we can write d_ς δ⁽¹⁾ X. With this, equation (4.34) finally becomes

$$\begin{align}\hfill {\hat{d}}_{A}\left(\hat{z}\right)& =\frac{{\varsigma }_{\text{o}}-\varsigma }{1+\hat{z}}\left\{1+\left[\frac{{\delta }^{\left(1\right)}{d}_{A}}{{\bar{d}}_{A}}+\left(1-\frac{1}{\mathcal{H}\left({\varsigma }_{\text{o}}-\varsigma \right)}\right){\delta }^{\left(1\right)}z\right]\right.\hfill \\ \hfill & \quad +\frac{1}{2}\left[\frac{{\delta }^{\left(2\right)}{d}_{A}}{{\bar{d}}_{A}}+\left(1-\frac{1}{\mathcal{H}\left({\varsigma }_{\text{o}}-\varsigma \right)}\right){\delta }^{\left(2\right)}z\right.\hfill \\ \hfill & \quad +2\left(1-\frac{1}{\mathcal{H}\left({\varsigma }_{\text{o}}-\varsigma \right)}\right)\left(\frac{{\delta }^{\left(1\right)}{d}_{A}}{{\bar{d}}_{A}}+\frac{1}{\mathcal{H}}\frac{\mathrm{d}{\delta }^{\left(1\right)}z}{\mathrm{d}\varsigma }\right){\delta }^{\left(1\right)}z\hfill \\ \hfill & \quad \left.\left.+2\frac{\mathrm{d}}{\mathrm{d}\varsigma }\left(\frac{{\delta }^{\left(1\right)}{d}_{A}}{{\bar{d}}_{A}}\right)\frac{{\delta }^{\left(1\right)}z}{\mathcal{H}}+\frac{{\mathcal{H}}^{\prime }-{\mathcal{H}}^{2}}{{\mathcal{H}}^{3}\left({\varsigma }_{\text{o}}-\varsigma \right)}{\left({\delta }^{\left(1\right)}z\right)}^{2}\right]\right\}.\hfill \end{align} \tag{ 4.36 }$$

We can now write the angular diameter distance as a function of the redshift, although it is written in terms of integrals over the comoving distance χ = ς_o − ς, which depends on the redshift itself by equation (2.11).

Using equation (4.36) written in terms of observable quantities such as the observable redshift and comoving distance, the angular diameter distance will possibly, in principle, be measured with great accuracy by the upcoming surveys, and should complement to the known luminosity distance measurements quite well.

Combining equations (3.14) and (4.20) with (4.36), we have that at linear order the diameter distance as a function of redshift is given by,

$$\begin{align}\hfill {\delta }^{\left(1\right)}{\hat{d}}_{A}\left({\hat{z}}_{\text{s}}\right)& =\frac{{\chi }_{\text{s}}}{1+{\hat{z}}_{\text{s}}}\left\{-{{\Psi}}_{1}{\vert }_{\text{s}}-{{\Psi}}_{1}{\vert }_{\text{o}}-\left(1-\frac{1}{\mathcal{H}{\chi }_{\text{s}}}\right){{\Phi}}_{1}{\vert }_{\text{s}}+\left(2-\frac{2}{\mathcal{H}{\chi }_{\text{s}}}\right){\left({v}_{1i}{n}^{i}\right)}_{\text{o}}\right.\hfill \\ \hfill & \quad +\left(1-\frac{1}{\mathcal{H}{\chi }_{\text{s}}}\right){\left({v}_{1i}{n}^{i}\right)}_{\text{s}}+\left(1-\frac{1}{\mathcal{H}{\chi }_{\text{s}}}\right){\int }_{0}^{{\chi }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\chi \hfill \\ \hfill & \quad -\frac{2}{{\chi }_{\text{s}}}{\int }_{0}^{{\chi }_{\text{s}}}{{\Psi}}_{1}\mathrm{d}\chi -\frac{1}{2{\chi }_{\text{s}}}{\int }_{0}^{{\chi }_{\text{s}}}\mathrm{d}\chi \left(\chi -{\chi }_{\text{s}}\right)\chi \hfill \\ \hfill & \quad {\times}\left.\left[{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)-{n}^{i}{n}^{j}{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,ij}-\frac{2}{\chi }\frac{\mathrm{d}{\delta }^{\left(1\right)}\nu }{\mathrm{d}\varsigma }\right]\right\}.\hfill \end{align} \tag{ 4.37 }$$

The full expression for ${\delta }^{\left(2\right)}{\hat{d}}_{A}\left({\hat{z}}_{\text{s}}\right)$ in terms of the metric potentials is given in appendix C.

The area distance the light-ray bundle creates, changes along the line of sight as seen in figure 2, and we are interested in computing the volume that these hypersurfaces enclose, since therein lie the overdensities we are accounting for.

The volume element (2.17) can be rewritten in terms of the quantities we have computed in the previous sections; the angular diameter distance in equations (4.20) and (C9), and the energy in equations (3.7) and (3.8). It is given up to second order by,

$$\begin{align}\hfill \mathrm{d}V& =-\mathcal{E}{d}_{A}^{2}\left(\lambda \right)\mathrm{d}\lambda \enspace \mathrm{d}{\Omega},\hfill \\ \hfill & =-\bar{\mathcal{E}}{\bar{d}}_{A}^{2}\left[1+2\frac{{\delta }^{\left(1\right)}{d}_{A}}{{\bar{d}}_{A}}+\frac{{\delta }^{\left(1\right)}\mathcal{E}}{\bar{\mathcal{E}}}+{\left(\frac{{\delta }^{\left(1\right)}{d}_{A}}{{\bar{d}}_{A}}\right)}^{2}+\left(\frac{{\delta }^{\left(1\right)}\mathcal{E}}{\bar{\mathcal{E}}}\right)\left(\frac{{\delta }^{\left(1\right)}{d}_{A}}{{\bar{d}}_{A}}\right)\right.+\left.\frac{{\delta }^{\left(2\right)}{d}_{A}}{{\bar{d}}_{A}}+\frac{1}{2}\frac{{\delta }^{\left(2\right)}\mathcal{E}}{\bar{\mathcal{E}}}\right]\hfill \\ \hfill & \quad {\times}\mathrm{d}\lambda \enspace \mathrm{d}{\Omega}.\hfill \end{align} \tag{ 4.38 }$$

The volume element is given in terms of the affine parameter λ, but we need to express our result in terms of the observed redshift z, and so we need to take the volume in bins of dz instead of dλ. To do so we use the fact that we can write the affine parameter as a function of redshift, i.e. λ(z), and using equations (4.26), (4.32) and (4.33), we obtain the relation

$$\begin{align}\hfill \frac{\mathrm{d}\lambda }{\mathrm{d}z}& =-\frac{a}{\mathcal{H}}\left[1+\left(\frac{1}{\mathcal{H}}+\frac{1}{\mathcal{H}\left(1+\bar{z}\right)}\right)\frac{\mathrm{d}{\delta }^{\left(1\right)}z}{\mathrm{d}\lambda }-\frac{{\mathcal{H}}^{\prime }}{{\mathcal{H}}^{2}}{\delta }^{\left(1\right)}z\right]\hfill \\ \hfill & \quad -\frac{a}{2\mathcal{H}}\left[\left(\frac{1}{\mathcal{H}}+\frac{1}{\mathcal{H}\left(1+\bar{z}\right)}\right)\frac{\mathrm{d}{\delta }^{\left(2\right)}z}{\mathrm{d}\lambda }-\frac{{\mathcal{H}}^{\prime }}{{\mathcal{H}}^{2}}{\delta }^{\left(2\right)}z-\frac{{\mathcal{H}}^{\prime }}{{\mathcal{H}}^{3}\left(1+\bar{z}\right)}\left(\frac{\mathrm{d}{\delta }^{\left(1\right)}z}{\mathrm{d}\lambda }\right){\delta }^{\left(1\right)}z\right.\hfill \\ \hfill & \quad +\frac{1}{{\mathcal{H}}^{2}\left(1+\bar{z}\right)}\left(2+\frac{1}{1+\bar{z}}\right){\left(\frac{\mathrm{d}{\delta }^{\left(1\right)}z}{\mathrm{d}\lambda }\right)}^{2}+\left(\frac{{\mathcal{H}}^{\prime }}{{\mathcal{H}}^{2}}\right)\left(1+\frac{{\mathcal{H}}^{\prime }}{{\mathcal{H}}^{2}}\right){\left({\delta }^{\left(1\right)}z\right)}^{2}\hfill \\ \hfill & \quad -\left.\frac{1}{\mathcal{H}}\left(2\frac{{\left({\mathcal{H}}^{\prime }\right)}^{2}}{{\mathcal{H}}^{2}}-\frac{{\mathcal{H}}^{{\prime\prime}}}{\mathcal{H}}\right){\left({\delta }^{\left(1\right)}z\right)}^{2}-\frac{1}{\mathcal{H}}\left(1+\frac{{\mathcal{H}}^{\prime }}{{\mathcal{H}}^{2}}\right)\frac{\mathrm{d}}{\mathrm{d}\lambda }\left[{\left({\delta }^{\left(1\right)}z\right)}^{2}\right]\right],\hfill \end{align} \tag{ 4.39 }$$

modifying equation (4.38) into

$$\begin{align}\hfill \mathrm{d}V\left(z\right)& =-\mathcal{E}\left(z\right){d}_{A}^{2}\left(z\right)\left(\frac{\mathrm{d}\lambda }{\mathrm{d}z}\right)\mathrm{d}z\enspace \mathrm{d}{\Omega},\hfill \\ \hfill & =\frac{\bar{\mathcal{E}}{\bar{d}}_{A}^{2}}{\mathcal{H}\left(1+z\right)}\left[1+\frac{2}{\mathcal{H}}\frac{\mathrm{d}{\delta }^{\left(1\right)}z}{\mathrm{d}\lambda }-\frac{{\mathcal{H}}^{\prime }}{{\mathcal{H}}^{2}}{\delta }^{\left(1\right)}z+2\frac{{\delta }^{\left(1\right)}{d}_{A}}{{\bar{d}}_{A}}+\frac{{\delta }^{\left(1\right)}\mathcal{E}}{\bar{\mathcal{E}}}\right.\hfill \\ \hfill & \quad -\frac{1}{\mathcal{H}}\frac{\mathrm{d}{\delta }^{\left(2\right)}z}{\mathrm{d}\lambda }+\frac{1}{2}\frac{{\mathcal{H}}^{\prime }}{{\mathcal{H}}^{2}}{\delta }^{\left(2\right)}z-\frac{3}{2{\mathcal{H}}^{2}}{\left(\frac{\mathrm{d}{\delta }^{\left(1\right)}z}{\mathrm{d}\lambda }\right)}^{2}-\frac{1}{2}\left(\frac{{\mathcal{H}}^{\prime }}{{\mathcal{H}}^{2}}\right)\left(1+\frac{{\mathcal{H}}^{\prime }}{{\mathcal{H}}^{2}}\right){\left({\delta }^{\left(1\right)}z\right)}^{2}\hfill \\ \hfill & \quad +\frac{1}{2{\mathcal{H}}^{2}}\left(2\frac{{\left({\mathcal{H}}^{\prime }\right)}^{2}}{{\mathcal{H}}^{2}}-\frac{{\mathcal{H}}^{{\prime\prime}}}{\mathcal{H}}\right){\left({\delta }^{\left(1\right)}z\right)}^{2}+\frac{1}{2\mathcal{H}}\left(1+2\frac{{\mathcal{H}}^{\prime }}{{\mathcal{H}}^{2}}\right)\left(\frac{\mathrm{d}{\delta }^{\left(1\right)}z}{\mathrm{d}\lambda }\right){\delta }^{\left(1\right)}z\hfill \\ \hfill & \quad +\left.{\left(\frac{{\delta }^{\left(1\right)}{d}_{A}}{{\bar{d}}_{A}}\right)}^{2}+\left(\frac{{\delta }^{\left(1\right)}\mathcal{E}}{\bar{\mathcal{E}}}\right)\left(\frac{{\delta }^{\left(1\right)}{d}_{A}}{{\bar{d}}_{A}}\right)+\frac{{\delta }^{\left(2\right)}{d}_{A}}{{\bar{d}}_{A}}+\frac{1}{2}\frac{{\delta }^{\left(2\right)}\mathcal{E}}{\bar{\mathcal{E}}}\right]\mathrm{d}z\enspace \mathrm{d}{\Omega}.\hfill \end{align} \tag{ 4.40 }$$

We now give an expression for the volume element order by order. In the background we have

$$\begin{equation}\mathrm{d}\bar{V}=-\bar{\mathcal{E}}{\bar{d}}_{A}^{2}\mathrm{d}\lambda \enspace \mathrm{d}{\Omega}={a}^{2}\left({\lambda }_{\text{s}}\right){\left[{\lambda }_{\text{s}}-{\lambda }_{\text{o}}\right]}^{2}\mathrm{d}\lambda \enspace \mathrm{d}{\Omega},\end{equation} \tag{ 4.41 }$$

$$\begin{equation}=\frac{\bar{\mathcal{E}}{\bar{d}}_{A}^{2}}{\mathcal{H}\left(1+z\right)}\mathrm{d}z\enspace \mathrm{d}{\Omega}=\frac{{\chi }^{2}}{\mathcal{H}\left(1+z\right)}.\end{equation} \tag{ 4.42 }$$

From equation (4.40) and using equations (C5) and (4.37), we have that the first order perturbation to the physical volume is

$$\begin{align}\hfill \mathrm{d}{\delta }^{\left(1\right)}V& =\frac{\bar{\mathcal{E}}{\bar{d}}_{A}^{2}}{\mathcal{H}\left(1+z\right)}\left[\frac{2}{\mathcal{H}}\frac{\mathrm{d}{\delta }^{\left(1\right)}z}{\mathrm{d}\varsigma }-\frac{{\mathcal{H}}^{\prime }}{{\mathcal{H}}^{2}}{\delta }^{\left(1\right)}z+2\frac{{\delta }^{\left(1\right)}{d}_{A}}{{\bar{d}}_{A}}+\frac{{\delta }^{\left(1\right)}\mathcal{E}}{\bar{\mathcal{E}}}\right]\mathrm{d}z\enspace \mathrm{d}{\Omega},\hfill \\ \hfill & =\frac{{\chi }^{2}}{\mathcal{H}\left(1+z\right)}\left[\frac{2}{\mathcal{H}}\left({{{\Psi}}_{1}}^{\prime }-{\partial }_{\chi }{{\Phi}}_{1}+\frac{\mathrm{d}\left({v}_{1i}{n}^{i}\right)}{\mathrm{d}\varsigma }\right)-2\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)-3\left({v}_{1i}{n}^{i}\right)\right.\hfill \\ \hfill & \quad +\left(\frac{{\mathcal{H}}^{\prime }}{{\mathcal{H}}^{2}}+\frac{2}{\mathcal{H}\chi }\right)\left({{\Phi}}_{1}-\left({v}_{1i}{n}^{i}\right)+{\int }_{0}^{\chi }\mathrm{d}\tilde {\chi }\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\right)-\frac{4}{\chi }{\int }_{0}^{\chi }\mathrm{d}\tilde {\chi }{{\Psi}}_{1}\hfill \\ \hfill & \quad -\frac{1}{\chi }{\int }_{0}^{\chi }\mathrm{d}\tilde {\chi }\left(\tilde {\chi }-\chi \right)\tilde {\chi }\left\{{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)-{n}^{i}{n}^{j}{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,ij}-\frac{2}{\tilde {\chi }}\frac{\mathrm{d}{\delta }^{\left(1\right)}\nu }{\mathrm{d}\varsigma }\right\}\hfill \\ \hfill & \quad +\left.3{\int }_{0}^{\chi }\mathrm{d}\tilde {\chi }\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\right],\hfill \end{align} \tag{ 4.43 }$$

and using equations (C6) and (C9) we find that the second order perturbation to the physical volume is

$$\begin{align}\hfill \mathrm{d}{\delta }^{\left(2\right)}V& =\frac{\bar{\mathcal{E}}{\bar{d}}_{A}^{2}}{\mathcal{H}\left(1+z\right)}\left[-\frac{1}{\mathcal{H}}\frac{\mathrm{d}{\delta }^{\left(2\right)}z}{\mathrm{d}\lambda }+\frac{1}{2}\frac{{\mathcal{H}}^{\prime }}{{\mathcal{H}}^{2}}{\delta }^{\left(2\right)}z-\frac{3}{2{\mathcal{H}}^{2}}{\left(\frac{\mathrm{d}{\delta }^{\left(1\right)}z}{\mathrm{d}\lambda }\right)}^{2}\right.\hfill \\ \hfill & \quad -\frac{1}{2}\left(\frac{{\mathcal{H}}^{\prime }}{{\mathcal{H}}^{2}}\right)\left(1+\frac{{\mathcal{H}}^{\prime }}{{\mathcal{H}}^{2}}\right){\left({\delta }^{\left(1\right)}z\right)}^{2}+\frac{1}{2{\mathcal{H}}^{2}}\left(2\frac{{\left({\mathcal{H}}^{\prime }\right)}^{2}}{{\mathcal{H}}^{2}}-\frac{{\mathcal{H}}^{{\prime\prime}}}{\mathcal{H}}\right){\left({\delta }^{\left(1\right)}z\right)}^{2}\hfill \\ \hfill & \quad +\frac{1}{2\mathcal{H}}\left(1+2\frac{{\mathcal{H}}^{\prime }}{{\mathcal{H}}^{2}}\right)\left(\frac{\mathrm{d}{\delta }^{\left(1\right)}z}{\mathrm{d}\lambda }\right){\delta }^{\left(1\right)}z+{\left(\frac{{\delta }^{\left(1\right)}{d}_{A}}{{\bar{d}}_{A}}\right)}^{2}\hfill \\ \hfill & \quad +\left.\left(\frac{{\delta }^{\left(1\right)}\mathcal{E}}{\bar{\mathcal{E}}}\right)\left(\frac{{\delta }^{\left(1\right)}{d}_{A}}{{\bar{d}}_{A}}\right)+\frac{{\delta }^{\left(2\right)}{d}_{A}}{{\bar{d}}_{A}}+\frac{1}{2}\frac{{\delta }^{\left(2\right)}\mathcal{E}}{\bar{\mathcal{E}}}\right]\mathrm{d}z\enspace \mathrm{d}{\Omega}.\hfill \end{align} \tag{ 4.44 }$$

The equivalent expression in terms of the metric potentials is given in appendix C. With the above expansion at hand, we have all the necessary quantities to compute, in the next section, the galaxy number density up to second order, our main result.

Rewriting the result from reference [15], in Poisson gauge, allowing for anisotropic stress. At first order, reference [15] have

$$\begin{align}\hfill {{\Delta}}_{\text{Dio}}^{\left(1\right)}& ={\delta }_{\rho }^{\left(1\right)}+\left(\frac{2}{\mathcal{H}r}+\frac{{\mathcal{H}}^{\prime }}{{\mathcal{H}}^{2}}\right)\left[\left({v}_{1i}{n}^{i}\right)+{\psi }^{I}-{\psi }^{A}+2{\int }_{{\eta }_{\text{s}}}^{{\eta }_{\text{o}}}\mathrm{d}{\eta }^{\prime }{\partial }_{{\eta }^{\prime }}{\psi }^{I}\right]-{\psi }^{I}\hfill \\ \hfill & \quad +\frac{4}{r}{\int }_{{\eta }_{\text{s}}}^{{\eta }_{\text{o}}}\mathrm{d}{\eta }^{\prime }{\psi }^{I}-\frac{2}{r}{\int }_{{\eta }_{\text{s}}}^{{\eta }_{\text{o}}}\mathrm{d}{\eta }^{\prime }\frac{{\eta }^{\prime }-{\eta }_{\text{s}}}{{\eta }_{\text{o}}-{\eta }^{\prime }}{{\Delta}}_{2}{\psi }^{I}\hfill \\ \hfill & \quad +\frac{1}{\mathcal{H}}\left[{\partial }_{\eta }{\psi }^{I}+{\partial }_{r}\left({v}_{1i}{n}^{i}\right)\right]-3{\psi }^{A}+\frac{1}{\mathcal{H}}{\partial }_{\eta }{\psi }^{A},\hfill \end{align} \tag{ 6.2 }$$

where $\mathcal{H}={a}^{\prime }\left(\eta \right)/a\left(\eta \right)$ is the Hubble parameter, the 's' denotes source and the 'o' denotes observer, and

$$\begin{equation}{\psi }^{I}=\frac{\psi +\phi }{2},\qquad \text{and},\qquad {\psi }^{A}=\frac{\psi -\phi }{2},\end{equation} \tag{ 6.3 }$$

which rewriting in our notation is

$$\begin{align}\hfill {{\Delta}}_{\text{Dio}}^{\left(1\right)}& ={\delta }_{\rho }^{\left(1\right)}+\left(\frac{2}{\mathcal{H}r}+\frac{{\mathcal{H}}^{\prime }}{{\mathcal{H}}^{2}}\right)\left[\left({v}_{1i}{n}^{i}\right)+{{\Phi}}_{1}+{\int }_{{\eta }_{\text{s}}}^{{\eta }_{\text{o}}}\mathrm{d}{\eta }^{\prime }\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\right]\hfill \\ \hfill & \quad +\left(2{{\Psi}}_{1}-{{\Phi}}_{1}\right)+\frac{2}{r}{\int }_{{\eta }_{\text{s}}}^{{\eta }_{\text{o}}}\mathrm{d}{\eta }^{\prime }\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\hfill \\ \hfill & \quad -\frac{1}{r}{\int }_{{\eta }_{\text{s}}}^{{\eta }_{\text{o}}}\mathrm{d}{\eta }^{\prime }\frac{{\eta }^{\prime }-{\eta }_{\text{s}}}{{\eta }_{\text{o}}-{\eta }^{\prime }}{{\Delta}}_{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)+\frac{1}{\mathcal{H}}\left[{{{\Psi}}_{1}}^{\prime }+{\partial }_{r}\left({v}_{1i}{n}^{i}\right)\right].\hfill \end{align} \tag{ 6.4 }$$

Computing the difference between equations (6.2) and (5.14), we have

$$\begin{align}\hfill {{\Delta}}_{\text{g}}^{\left(1\right)}-{{\Delta}}_{\text{Dio}}^{\left(1\right)}& \approx \left[\frac{{\delta }^{\left(1\right)}\rho \left({n}^{i},z\right)}{\bar{\rho }\left(z\right)}+3{{\Phi}}_{1}\right]-\left[{\delta }_{\rho }^{\left(1\right)}\right]\hfill \\ \hfill & \quad -\frac{1}{\chi }{\int }_{0}^{\chi }\mathrm{d}\tilde {\chi }\left(\tilde {\chi }-\chi \right)\tilde {\chi }\left[{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)-{n}^{i}{n}^{j}{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,ij}\right]\hfill \\ \hfill & \quad +\left[\frac{1}{r}{\int }_{{\eta }_{\text{s}}}^{{\eta }_{\text{o}}}\mathrm{d}{\eta }^{\prime }\frac{{\eta }^{\prime }-{\eta }_{\text{s}}}{{\eta }_{\text{o}}-{\eta }^{\prime }}{{\Delta}}_{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\right]\hfill \end{align} \tag{ 6.5 }$$

where the first line cancels out from the definition of the comoving density perturbation (${\delta }_{\rho }^{\left(1\right)}$), and the last integral cancels out from the definition of the angular operator Δ₂, both given in reference [15], and we find,

$$\begin{equation}{{\Delta}}_{\text{g}}^{\left(1\right)}-{{\Delta}}_{\text{Dio}}^{\left(1\right)}=0.\end{equation} \tag{ 6.6 }$$

In references [13, 14], their result is written in terms of 'cosmic rulers' and it is given by

$$\begin{align}\hfill {{\Delta}}_{\text{Bertacca}}^{\left(1\right)}& ={\delta }_{\text{g}}^{\left(1\right)}+\frac{1}{2}{\hat{g}}_{\mu }^{\mu \left(1\right)}+{b}_{\text{e}}{\Delta}\mathrm{ln}\enspace {a}^{\left(1\right)}+{\partial }_{{\Vert}}{\Delta}{x}_{{\Vert}}^{\left(1\right)}+\frac{2}{\bar{\chi }}{\Delta}{x}_{{\Vert}}^{\left(1\right)}\hfill \\ \hfill & \quad -2{\kappa }^{\left(1\right)}+{E}_{\hat{0}}^{0\left(1\right)}+{E}_{\hat{0}}^{{\Vert}\left(1\right)},\hfill \end{align} \tag{ 6.7 }$$

which in Poisson gauge, translates into

$$\begin{align}\hfill {{\Delta}}_{\text{Bertacca}}^{\left(1\right)}& =\frac{{\delta }^{\left(1\right)}\rho \left({n}^{i},z\right)}{\bar{\rho }\left({n}^{i},z\right)}-\left(\frac{{\mathcal{H}}^{\prime }}{{\mathcal{H}}^{2}}+\frac{2}{\bar{\chi }\mathcal{H}}\right)\left[{\left({v}_{1i}{n}^{i}-{{\Phi}}_{1}\right)}_{\text{o}}^{\text{s}}-2{\int }_{0}^{\bar{\chi }}{{{\Phi}}_{1}}^{\prime }\mathrm{d}\tilde {\chi }\right]\hfill \\ \hfill & \quad -{{\Phi}}_{1}+\frac{{{{\Phi}}_{1}}^{\prime }}{\mathcal{H}}+\frac{4}{\bar{\chi }}{\int }_{0}^{\bar{\chi }}{{\Phi}}_{1}\mathrm{d}\tilde {\chi }-\frac{1}{\mathcal{H}}\frac{\mathrm{d}}{\mathrm{d}\chi }\left({v}_{1i}{n}^{i}\right)-\frac{1}{\mathcal{H}}\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\chi }\hfill \\ \hfill & \quad -2{\int }_{0}^{\bar{\chi }}\mathrm{d}\tilde {\chi }\left(\bar{\chi }-\tilde {\chi }\right)\frac{\tilde {\chi }}{\bar{\chi }}\left[{\nabla }^{2}{{\Phi}}_{1}+\frac{{\mathrm{d}}^{2}{{\Phi}}_{1}}{\mathrm{d}{\tilde {\chi }}^{2}}+{{{\Phi}}_{1}}^{{\prime\prime}}-2\frac{\mathrm{d}{{{\Phi}}_{1}}^{\prime }}{\mathrm{d}\tilde {\chi }}\right.-\left.\frac{2}{\bar{\chi }}\left(\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\tilde {\chi }}-{{{\Phi}}_{1}}^{\prime }\right)\right],\hfill \end{align} \tag{ 6.8 }$$

where we omitted the terms with the evolution bias b_e. We must rewrite our own result taking Ψ₁ = Φ₁ in equation (5.14) to make the comparison, so we have that

$$\begin{align}\hfill {{\Delta}}_{\text{g}}^{\left(1\right)}\left({n}^{i},z\right)& =\left[\frac{{\delta }^{\left(1\right)}\rho \left({n}^{i},z\right)}{\bar{\rho }\left(z\right)}+3{{\Phi}}_{1}\right]-4{{\Phi}}_{1}+\frac{1}{\mathcal{H}}\left({{{\Phi}}_{1}}^{\prime }-{\partial }_{\chi }{{\Phi}}_{1}+\frac{\mathrm{d}\left({v}_{1i}{n}^{i}\right)}{\mathrm{d}\varsigma }\right)\hfill \\ \hfill & \quad +\left(\frac{{\mathcal{H}}^{\prime }}{{\mathcal{H}}^{2}}+\frac{2}{\mathcal{H}\chi }\right)\left[{{\Phi}}_{1}-\left({v}_{1i}{n}^{i}\right)+2{\int }_{0}^{\chi }\mathrm{d}\tilde {\chi }{{{\Phi}}_{1}}^{\prime }\right]-\frac{4}{\chi }{\int }_{0}^{\chi }\mathrm{d}\tilde {\chi }{{\Phi}}_{1}\hfill \\ \hfill & \quad -\frac{2}{\chi }{\int }_{0}^{\chi }\mathrm{d}\tilde {\chi }\left(\tilde {\chi }-\chi \right)\tilde {\chi }\left[{\nabla }^{2}{{\Phi}}_{1}-{n}^{i}{n}^{j}{{{\Phi}}_{1}}_{,ij}-\frac{1}{\tilde {\chi }}\frac{\mathrm{d}{\delta }^{\left(1\right)}\nu }{\mathrm{d}\varsigma }\right].\hfill \end{align} \tag{ 6.9 }$$

Computing the difference between equations (6.8) and (5.14), we have

$$\begin{equation}{{\Delta}}_{\text{g}}^{\left(1\right)}-{{\Delta}}_{\text{Bertacca}}^{\left(1\right)}\approx \frac{1}{\mathcal{H}}\left(\frac{\mathrm{d}\left({v}_{1i}{n}^{i}\right)}{\mathrm{d}\varsigma }\right)+\frac{1}{\mathcal{H}}\frac{\mathrm{d}}{\mathrm{d}\chi }\left({v}_{1i}{n}^{i}\right),\end{equation} \tag{ 6.10 }$$

where both are derivatives of first order terms with respect to background quantities, and in the background dχ = dς. Note that the direction in the sky, nⁱ , is $\left(-{n}^{i}\right)$ from references [13, 14], so

$$\begin{equation}{{\Delta}}_{\text{g}}^{\left(1\right)}-{{\Delta}}_{\text{Bertacca}}^{\left(1\right)}=0.\end{equation} \tag{ 6.11 }$$

The galaxy overdensity in [17], is given by

$$\begin{equation}{\delta }_{\text{g}}^{\text{obs}\left(1\right)}={\delta }_{\text{g}}^{\text{int}\left(1\right)}+3\delta z+\delta g+2\frac{\delta r}{{\bar{r}}_{z}}-2\kappa +{H}_{z}\frac{\partial \delta r}{\partial z}+\delta {u}^{0}+{V}_{{\Vert}}-{e}_{1}\delta {z}_{{t}_{p}}-{t}_{1}\delta {\mathcal{D}}_{L},\end{equation} \tag{ 6.12 }$$

which in Poisson gauge, allowing for anisotropic stress takes the form

$$\begin{align}\hfill {{\Delta}}_{\text{Yoo}}^{\left(1\right)}& ={\delta }_{\text{g}}^{\text{int}\left(1\right)}+3{\mathcal{H}}_{\text{o}}\delta {\tau }_{\text{o}}-3{{\Phi}}_{1}{\vert }_{\text{o}}^{z}-3{\int }_{0}^{{\bar{r}}_{z}}\mathrm{d}\bar{r}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)+3{{\Phi}}_{1}+3{v}_{1i}{n}^{i}\hfill \\ \hfill & \quad +{{\Phi}}_{1}+3{{\Psi}}_{1}+\frac{2}{{r}_{z}}\left[\delta {\tau }_{\text{o}}-\frac{1}{{\mathcal{H}}_{z}}\left({\mathcal{H}}_{\text{o}}\delta {\tau }_{\text{o}}-{{\Phi}}_{1}{\vert }_{\text{o}}^{z}-{\int }_{0}^{{\bar{r}}_{z}}\mathrm{d}\bar{r}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\right.\right.\hfill \\ \hfill & \quad \left.+\left.{\left({v}_{1i}{n}^{i}\right)}_{\text{o}}^{z}\right)+{\int }_{0}^{{\bar{r}}_{z}}\left({{\Phi}}_{1}-{{\Psi}}_{1}\right)\mathrm{d}\bar{r}-2\kappa +{H}_{z}\frac{\partial }{\partial z}\right.\hfill \\ \hfill & \quad {\times}\left(\delta {\tau }_{\text{o}}-\frac{1}{{\mathcal{H}}_{z}}\left\{{\mathcal{H}}_{\text{o}}\delta {\tau }_{\text{o}}-{{\Phi}}_{1}{\vert }_{\text{o}}^{z}-{\int }_{0}^{{\bar{r}}_{z}}\mathrm{d}\bar{r}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\right.\right.\hfill \\ \hfill & \quad +\left.\left.\left.{\left({v}_{1i}{n}^{i}\right)}_{\text{o}}^{z}\right\}+{\int }_{0}^{{\bar{r}}_{z}}\mathrm{d}\bar{r}\left({{\Phi}}_{1}-{{\Psi}}_{1}\right)\right)-{{\Phi}}_{1}+{v}_{1i}{n}^{i}\right],\hfill \end{align} \tag{ 6.13 }$$

where we did not use the evolution bias or the running and slope of the luminosity.

The difference between equations (6.13) and (5.14) is then,

$$\begin{align}\hfill {{\Delta}}_{\text{g}}^{\left(1\right)}-{{\Delta}}_{\text{Yoo}}^{\left(1\right)}& \approx \left[\frac{{\delta }^{\left(1\right)}\rho \left({n}^{i},z\right)}{\bar{\rho }\left(z\right)}+3{{\Phi}}_{1}\right]-{\delta }_{\text{g}}^{\text{int}\left(1\right)}-3{\mathcal{H}}_{\text{o}}\delta {\tau }_{\text{o}}-\frac{2}{{r}_{z}}\delta {\tau }_{\text{o}}\hfill \\ \hfill & \quad -\frac{1}{\chi }{\int }_{0}^{\chi }\mathrm{d}\tilde {\chi }\left(\tilde {\chi }-\chi \right)\tilde {\chi }\left[{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)-{n}^{i}{n}^{j}{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,ij}\right]+\frac{4}{{r}_{z}}\kappa ,\hfill \end{align} \tag{ 6.14 }$$

where the first line is zero from the definition of ${\delta }_{\text{g}}^{\text{int}\left(1\right)}$, and without loss of generality we take δτ_o = 0, and the integrals in the second line cancel from the definition of κ in reference [17], so

$$\begin{equation}{{\Delta}}_{\text{g}}^{\left(1\right)}-{{\Delta}}_{\text{Yoo}}^{\left(1\right)}=0.\end{equation} \tag{ 6.15 }$$

All data that support the findings of this study are included within the article (and any supplementary files).

Solving equation (2.8) at second order gives

$$\begin{align}\hfill \frac{\mathrm{d}{\delta }^{\left(2\right)}\nu }{\mathrm{d}\lambda }& =-\frac{1}{2}\left[\frac{\mathrm{d}{{\Phi}}_{2}}{\mathrm{d}\lambda }+\frac{\mathrm{d}{{\Psi}}_{2}}{\mathrm{d}\lambda }\right]+2\frac{\mathrm{d}{{\delta }^{\left(1\right)}n}^{i}}{\mathrm{d}\lambda }{{\delta }^{\left(1\right)}n}_{i}-2\frac{\mathrm{d}{\delta }^{\left(1\right)}\nu }{\mathrm{d}\lambda }{\delta }^{\left(1\right)}\nu -\frac{\mathrm{d}{{\delta }^{\left(2\right)}n}^{i}}{\mathrm{d}\lambda }{n}_{i}\hfill \\ \hfill & \quad -4\frac{\mathrm{d}{\delta }^{\left(1\right)}\nu }{\mathrm{d}\lambda }\left[{{\Phi}}_{1}+{{\Psi}}_{1}\right]-4{\delta }^{\left(1\right)}\nu \left[\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\lambda }+\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }\right]-4{{\Phi}}_{1}\left[\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\lambda }+\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }\right]\hfill \\ \hfill & \quad -4\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\lambda }\left[{{\Phi}}_{1}+{{\Psi}}_{1}\right],\hfill \end{align} \tag{ C1 }$$

$$\begin{align}\hfill \frac{\mathrm{d}{{\delta }^{\left(2\right)}n}^{i}}{\mathrm{d}\lambda }& =2\frac{\mathrm{d}{{\Psi}}_{2}}{\mathrm{d}\lambda }{n}^{i}-\left[{{{\Phi}}_{2}}_{,}^{i}+{{{\Psi}}_{2}}_{,}^{i}\right]-4{{\Phi}}_{1}{{{\Psi}}_{1}}^{\prime }{n}^{i}+4{{\delta }^{\left(1\right)}n}^{i}{{{\Psi}}_{1}}^{\prime }+4{{\Phi}}_{1}{{{\Phi}}_{1}}_{,}^{i}\hfill \\ \hfill & \quad -2{{\delta }^{\left(1\right)}n}^{j}{{\delta }^{\left(1\right)}n}_{,j}^{i}-2{{\delta }^{\left(1\right)}n}^{i}{\delta }^{\left(1\right)}\nu +4\left[{n}^{i}{{\delta }^{\left(1\right)}n}^{j}+{n}^{j}{{\delta }^{\left(1\right)}n}^{i}\right.\hfill \\ \hfill & \quad +\left.2{n}^{j}{n}^{i}{{\Psi}}_{1}\right]{{{\Psi}}_{1}}_{,j}-4{{\Psi}}_{1}\left[{{{\Psi}}_{1}}_{,}^{i}-2{n}^{i}{{{\Psi}}_{1}}^{\prime }\right]\hfill \\ \hfill & \quad -4{n}^{j}{{\delta }^{\left(1\right)}n}_{j}\left[\left({{{\Phi}}_{1}}_{,}^{i}+{{{\Psi}}_{1}}_{,}^{i}\right)-{n}^{i}{{{\Psi}}_{1}}^{\prime }\right],\hfill \end{align} \tag{ C2 }$$

Using equation (3.1), and the integrated version of equations (3.4) and (3.5), we rewrite equations (C1) and (C2) purely in terms of the metric potentials,

$$\begin{align}\hfill \frac{\mathrm{d}{\delta }^{\left(2\right)}\nu }{\mathrm{d}\lambda }& =-2\frac{\mathrm{d}{{\Phi}}_{2}}{\mathrm{d}\lambda }+{{{\Phi}}_{2}}^{\prime }+{{{\Psi}}_{2}}^{\prime }+\frac{1}{2}\left[\frac{\mathrm{d}{{\Phi}}_{2}}{\mathrm{d}\lambda }+\frac{\mathrm{d}{{\Psi}}_{2}}{\mathrm{d}\lambda }\right]+4{{\Phi}}_{1}\left[3\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\lambda }-\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }\right]\hfill \\ \hfill & \qquad -4{{{\Phi}}_{1}}^{\prime }\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)+4{{\Phi}}_{1}\left(3{{\Phi}}_{1}+{{\Psi}}_{1}\right)+4{{\Psi}}_{1}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)\mathrm{d}\lambda \hfill \\ \hfill & \qquad -{\left(4{{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,i}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\mathrm{d}\lambda +4\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \hfill \\ \hfill & \qquad -2\left[3{{\Phi}}_{1}+{{\Phi}}_{1}+{{{\Phi}}_{1}}^{\prime }+3{{{\Psi}}_{1}}^{\prime }\right]{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \hfill \\ \hfill & \qquad +6\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,i}\mathrm{d}\lambda \right)\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\mathrm{d}\lambda \right)\hfill \\ \hfill & \qquad +2\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right)\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right),\hfill \end{align} \tag{ C3 }$$

$$\begin{align}\hfill \frac{\mathrm{d}{{\delta }^{\left(2\right)}n}^{i}}{\mathrm{d}\lambda }& =2\frac{\mathrm{d}{{\Psi}}_{2}}{\mathrm{d}\lambda }{n}^{i}-\left[{{{\Phi}}_{2}}_{,}^{i}+{{{\Psi}}_{2}}_{,}^{i}\right]+24{{\Psi}}_{1}{{{\Psi}}_{1}}^{\prime }{n}^{i}+8{{\Phi}}_{1}{{{\Phi}}_{1}}_{,}^{i}-8{n}^{i}{{\Phi}}_{1}{{{\Psi}}_{1}}^{\prime }\hfill \\ \hfill & \quad -4{{{\Psi}}_{1}}^{\prime }{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}+4{{\Phi}}_{1}{{{\Psi}}_{1}}_{,}^{i}-4{{\Psi}}_{1}{{{{\Phi}}_{1}}_{,}}^{i}-8{{\Psi}}_{1}{{{{\Psi}}_{1}}_{,}}^{i}+4{n}^{i}{{\Phi}}_{1}{{{\Phi}}_{1}}^{\prime }\hfill \\ \hfill & \quad -8{{\Psi}}_{1}\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }{n}^{i}+4{{\Psi}}_{1}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}-4{{\Psi}}_{1}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\mathrm{d}\lambda \hfill \\ \hfill & \quad +4{n}^{i}{{{\Psi}}_{1}}_{,j}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{j}\mathrm{d}\lambda -2\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{j}\mathrm{d}\lambda \right)\hfill \\ \hfill & \quad {\times}\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}}_{j}\mathrm{d}\lambda \right)+8{{\Phi}}_{1}{{\Psi}}_{1}{n}^{i}-4{{\Psi}}_{1}{n}^{i}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \hfill \\ \hfill & \quad -4{{\Phi}}_{1}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\mathrm{d}\lambda +2\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right)\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\mathrm{d}\lambda \right)\hfill \\ \hfill & \quad +12{n}^{i}{{\Psi}}_{1}\left[\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }-{{{\Psi}}_{1}}^{\prime }\right]-4{n}^{i}{{{\Psi}}_{1}}_{,j}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{j}\mathrm{d}\lambda \hfill \\ \hfill & \quad -\left[\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }-{{{\Psi}}_{1}}^{\prime }\right]{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\mathrm{d}\lambda -4\left[\left({{{\Phi}}_{1}}_{,}^{i}+{{{\Psi}}_{1}}_{,}^{i}\right)-{n}^{i}{{{\Psi}}_{1}}^{\prime }\right]{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda ,\hfill \end{align} \tag{ C4 }$$

where we integrate along the line of sight from λ_o to λ_s.

The perturbed energy in terms of the metric potentials is given by

$$\begin{equation}{\delta }^{\left(1\right)}\mathcal{E}=\left(-2{\left.{{\Phi}}_{1}\right\vert }_{\text{o}}^{\text{s}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }^{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right)+{{\Phi}}_{1}-{v}_{1i}{n}^{i},\end{equation} \tag{ C5 }$$

$$\begin{align}\hfill {\delta }^{\left(2\right)}\mathcal{E}& =\frac{1}{2}\left[2\left\{\left(-2{\left.{{\Phi}}_{1}\right\vert }_{\text{o}}^{\text{s}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }^{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right)+{{\Phi}}_{1}-{v}_{1i}{n}^{i}\right\}{{\Phi}}_{1}\right.\hfill \\ \hfill & \quad -{{{\Phi}}_{1}}^{2}-5{{{\Psi}}_{1}}^{2}-6{{\Phi}}_{1}{{\Psi}}_{1}+\frac{1}{2}\left({{\Phi}}_{2}-{{\Psi}}_{2}\right)-{n}^{i}{v}_{2i}\hfill \\ \hfill & \quad -4\left({{\Phi}}_{1}+{{\Psi}}_{1}\right){\left.{{\Phi}}_{1}\right\vert }_{\text{o}}^{\text{s}}+2\left({{\Phi}}_{1}+{{\Psi}}_{1}\right){\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \hfill \\ \hfill & \quad -4{n}^{i}{v}_{1i}{\left.{{\Psi}}_{1}\right\vert }_{\text{o}}^{\text{s}}+2{v}_{1i}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\mathrm{d}\lambda \hfill \\ \hfill & \quad +{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left\{-2\frac{\mathrm{d}{{\Phi}}_{2}}{\mathrm{d}\lambda }+{{{\Phi}}_{2}}^{\prime }+{{{\Psi}}_{2}}^{\prime }+\frac{1}{2}\left[\frac{\mathrm{d}{{\Phi}}_{2}}{\mathrm{d}\lambda }+\frac{\mathrm{d}{{\Psi}}_{2}}{\mathrm{d}\lambda }\right]+4{{\Phi}}_{1}\left[3\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\lambda }-\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }\right]\right.\hfill \\ \hfill & \quad -4{{{\Phi}}_{1}}^{\prime }\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)+4{{\Phi}}_{1}\left(3{{\Phi}}_{1}+{{\Psi}}_{1}\right)+4{{\Psi}}_{1}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)\mathrm{d}\lambda \hfill \\ \hfill & \quad -{\left(4{{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,i}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\mathrm{d}\lambda +4\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \hfill \\ \hfill & \quad -2\left[3{{\Phi}}_{1}+{{\Phi}}_{1}+{{{\Phi}}_{1}}^{\prime }+3{{{\Psi}}_{1}}^{\prime }\right]{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \hfill \\ \hfill & \quad +6\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,i}\mathrm{d}\lambda \right)\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\mathrm{d}\lambda \right)\hfill \\ \hfill & \quad +\left.\left.2\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right)\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right)\right\}\mathrm{d}\tilde {\lambda }\right].\hfill \end{align} \tag{ C6 }$$

At second order the redshift is

$$\begin{align}\hfill {\delta }^{\left(2\right)}z& ={\left[{{{\Phi}}_{1}}^{2}+{\left({v}_{1i}{n}^{i}\right)}^{2}-2{{\Phi}}_{1}\left({v}_{1i}{n}^{i}\right)\right]}_{\text{o}}-{{\Phi}}_{1}{\vert }_{\text{o}}{\delta }^{\left(1\right)}\nu {\vert }_{\text{s}}+{\delta }^{\left(1\right)}\nu {\vert }_{\text{s}}{\left({v}_{1i}{n}^{i}\right)}_{\text{o}}-{{\Phi}}_{1}{\vert }_{\text{s}}{{\Phi}}_{1}{\vert }_{\text{o}}\hfill \\ \hfill & \quad +{{\Phi}}_{1}{\vert }_{\text{s}}{\left({v}_{1i}{n}^{i}\right)}_{\text{o}}+{\left({v}_{1i}{n}^{i}\right)}_{\text{s}}{{\Phi}}_{1}{\vert }_{\text{o}}-{\left({v}_{1i}{n}^{i}\right)}_{\text{s}}{\left({v}_{1i}{n}^{i}\right)}_{\text{o}}+\frac{1}{2}\left[2{\delta }^{\left(1\right)}\mathcal{E}{{\Phi}}_{1}-{{{\Phi}}_{1}}^{2}-5{{{\Psi}}_{1}}^{2}\right.\hfill \\ \hfill & \quad -{\left.6{{\Phi}}_{1}{{\Psi}}_{1}+{\delta }^{\left(2\right)}\nu +\frac{1}{2}\left({{\Phi}}_{2}-{{\Psi}}_{2}\right)+2{\delta }^{\left(1\right)}\nu \left({{\Phi}}_{1}+{{\Psi}}_{1}\right)-2{{\delta }^{\left(1\right)}n}^{i}{v}_{1i}-{n}^{i}{v}_{2i}\right]}_{\text{s}}\hfill \\ \hfill & \quad -\frac{1}{2}{\left[{{{\Phi}}_{1}}^{2}-5{{{\Psi}}_{1}}^{2}-6{{\Phi}}_{1}{{\Psi}}_{1}+\frac{1}{2}\left({{\Phi}}_{2}-{{\Psi}}_{2}\right)-\left({v}_{1i}{n}^{i}\right){{\Phi}}_{1}-{n}^{i}{v}_{2i}\right]}_{\text{o}},\hfill \end{align} \tag{ C7 }$$

and using equations (3.4), (3.5), (C1) and (C5), in terms of the metric potentials is

$$\begin{align}\hfill {\delta }^{\left(2\right)}z& ={\left[{{{\Phi}}_{1}}^{2}+{\left({v}_{1i}{n}^{i}\right)}^{2}-2{{\Phi}}_{1}\left({v}_{1i}{n}^{i}\right)\right]}_{\text{o}}-{{\Phi}}_{1}{\vert }_{\text{o}}{\left[-2{{\Phi}}_{1}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right]}_{\text{s}}\hfill \\ \hfill & \quad +{\left[-2{{\Phi}}_{1}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right]}_{\text{s}}{\left({v}_{1i}{n}^{i}\right)}_{\text{o}}-{{\Phi}}_{1}{\vert }_{\text{s}}{{\Phi}}_{1}{\vert }_{\text{o}}\hfill \\ \hfill & \quad +{{\Phi}}_{1}{\vert }_{\text{s}}{\left({v}_{1i}{n}^{i}\right)}_{\text{o}}+{\left({v}_{1i}{n}^{i}\right)}_{\text{s}}{{\Phi}}_{1}{\vert }_{\text{o}}-{\left({v}_{1i}{n}^{i}\right)}_{\text{s}}{\left({v}_{1i}{n}^{i}\right)}_{\text{o}}\hfill \\ \hfill & \quad +\frac{1}{2}\left[2\left\{\left(-2{\left.{{\Phi}}_{1}\right\vert }_{\text{o}}^{\text{s}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }^{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right)+{{\Phi}}_{1}-{v}_{1i}{n}^{i}\right\}{{\Phi}}_{1}\right.\hfill \\ \hfill & \quad -{{{\Phi}}_{1}}^{2}-5{{{\Psi}}_{1}}^{2}-6{{\Phi}}_{1}{{\Psi}}_{1}+\frac{1}{2}\left({{\Phi}}_{2}-{{\Psi}}_{2}\right)-{n}^{i}{v}_{2i}-4\left({{\Phi}}_{1}+{{\Psi}}_{1}\right){\left.{{\Phi}}_{1}\right\vert }_{\text{o}}^{\text{s}}\hfill \\ \hfill & \quad +2\left({{\Phi}}_{1}+{{\Psi}}_{1}\right){\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda -4{n}^{i}{v}_{1i}{\left.{{\Psi}}_{1}\right\vert }_{\text{o}}^{\text{s}}+2{v}_{1i}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\mathrm{d}\lambda \hfill \\ \hfill & \quad +{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left\{-2\frac{\mathrm{d}{{\Phi}}_{2}}{\mathrm{d}\lambda }+{{{\Phi}}_{2}}^{\prime }+{{{\Psi}}_{2}}^{\prime }+\frac{1}{2}\left[\frac{\mathrm{d}{{\Phi}}_{2}}{\mathrm{d}\lambda }+\frac{\mathrm{d}{{\Psi}}_{2}}{\mathrm{d}\lambda }\right]+4{{\Phi}}_{1}\left[3\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\lambda }-\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }\right]\right.\hfill \\ \hfill & \quad -4{{{\Phi}}_{1}}^{\prime }\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)+4{{\Phi}}_{1}\left(3{{\Phi}}_{1}+{{\Psi}}_{1}\right)+4{{\Psi}}_{1}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)\mathrm{d}\lambda \hfill \\ \hfill & \quad -{\left(4{{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,i}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\mathrm{d}\lambda +4\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \hfill \\ \hfill & \quad -2\left[3{{\Phi}}_{1}+{{\Phi}}_{1}+{{{\Phi}}_{1}}^{\prime }+3{{{\Psi}}_{1}}^{\prime }\right]{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \hfill \\ \hfill & \quad +6\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,i}\mathrm{d}\lambda \right)\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\mathrm{d}\lambda \right)\hfill \\ \hfill & \quad \left.\left. +2\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right)\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right)\right\}\mathrm{d}\tilde {\lambda }\right]_{\text{s}}\hfill \\ \hfill & \quad -\frac{1}{2}{\left[{{{\Phi}}_{1}}^{2}-5{{{\Psi}}_{1}}^{2}-6{{\Phi}}_{1}{{\Psi}}_{1}+\frac{1}{2}\left({{\Phi}}_{2}-{{\Psi}}_{2}\right)-\left({v}_{1i}{n}^{i}\right){{\Phi}}_{1}-{n}^{i}{v}_{2i}\right]}_{\text{o}}.\hfill \end{align} \tag{ C8 }$$

Using equations (3.4), (3.5), (C3), (C4), (C6), (4.18), (B1)–(B3) and (4.24) we find that the second order perturbation to the angular diameter distance becomes

$$\begin{align*}\hfill \frac{{\delta }^{\left(2\right)}{d}_{A}\left({\lambda }_{\text{s}}\right)}{{\bar{d}}_{A}\left({\lambda }_{\text{s}}\right)}& =\frac{1}{2}{\left[{{{\Phi}}_{1}}^{2}-5{{{\Psi}}_{1}}^{2}-6{{\Phi}}_{1}{{\Psi}}_{1}+\frac{1}{2}\left({{\Phi}}_{2}-{{\Psi}}_{2}\right)-\left({v}_{1i}{n}^{i}\right){{\Phi}}_{1}-{n}^{i}{v}_{2i}\right]}_{\text{o}}\hfill \\ \hfill & \quad -\frac{1}{{\lambda }_{\text{o}}-{\lambda }_{\text{s}}}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }\left\{4\left(-2{\left.{{\Phi}}_{1}\right\vert }_{\text{s}}^{\text{s}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right)\right.\hfill \\ \hfill & \quad {\times}\left( {{\Phi}}_{1}{\vert }_{\text{o}}-{\left({v}_{1i}{n}^{i}\right)}_{\text{o}}-{{\Phi}}_{1}{\vert }_{\text{o}}^{\text{s}}-\frac{3}{2}\left({{\Psi}}_{1}-{{\Phi}}_{1}\right){\vert }_{\text{o}}^{\text{s}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }\left\{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}^{{\prime\prime}}\right.\right.\hfill \\ \hfill & \quad -\left.{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\right\}-\frac{1}{{\bar{d}}_{A}}\left[2{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}^{\prime }-{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }\right.\hfill \\ \hfill & \quad {\left.{\times}\left.{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\breve {\lambda }\left\{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}^{{\prime\prime}}-{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\right\}\right]\right)}^{{\prime\prime}}+\left({{\Phi}}_{1}{\vert }_{\text{o}}-{\left({v}_{1i}{n}^{i}\right)}_{\text{o}}-{{\Phi}}_{1}{\vert }_{\text{o}}^{\text{s}}\right.\hfill \\ \hfill & \quad -\left.{\left.\frac{3}{2}\left({{\Psi}}_{1}-{{\Phi}}_{1}\right)\right\vert }_{\text{o}}^{\text{s}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }\left\{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}^{{\prime\prime}}-{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\right\}\right.\hfill \\ \hfill & \quad -\left.\frac{1}{{\bar{d}}_{A}}\left[2{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}^{\prime }-{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }\right.\right.\hfill \\ \hfill & \quad {\left.{\times}\left.{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\breve {\lambda }\left\{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}^{{\prime\prime}}-{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\right\}\right]\right)}^{\prime }\left(-2\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\lambda }+{{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\hfill \\ \hfill & \quad +\frac{1}{2}\left({{\Phi}}_{1}{\vert }_{\text{o}}-{\left({v}_{1i}{n}^{i}\right)}_{\text{o}}-{{\Phi}}_{1}{\vert }_{\text{o}}^{\text{s}}-{\left.\frac{3}{2}\left({{\Psi}}_{1}-{{\Phi}}_{1}\right)\right\vert }_{\text{o}}^{\text{s}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda \right.\hfill \end{align*}$$

$$\begin{align*}\hfill & \qquad {\times}{\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }\left\{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}^{{\prime\prime}}-{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\right\}-\left.\frac{1}{{\lambda }_{\text{o}}-{\lambda }_{\text{s}}}\left[2{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda \right.\right.\hfill \\ \hfill & \qquad \left.\left.{\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}^{\prime }-{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\breve {\lambda }\left\{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}^{{\prime\prime}}\right.\right.\right.\hfill \\ \hfill & \qquad \left.\left.\left.-{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\right\}\right]\right)\left(\left[\frac{{\mathrm{d}}^{2}{{\Psi}}_{1}}{\mathrm{d}{\lambda }^{2}}-\frac{{\mathrm{d}}^{2}{{\Phi}}_{1}}{\mathrm{d}{\lambda }^{2}}\right]+2\left[\frac{\mathrm{d}{{{\Phi}}_{1}}^{\prime }}{\mathrm{d}\lambda }+\frac{\mathrm{d}{{{\Psi}}_{1}}^{\prime }}{\mathrm{d}\lambda }\right]\right.\hfill \\ \hfill & \qquad \left.+\left.{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)-\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)\right)\right\}+\frac{1}{{\lambda }_{\text{o}}-{\lambda }_{\text{s}}}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda \hfill \\ \hfill & \qquad {\times}{\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }\left\{\frac{1}{2}\left[-2\frac{\mathrm{d}{{\Phi}}_{2}}{\mathrm{d}\lambda }+{{{\Phi}}_{2}}^{\prime }+{{{\Psi}}_{2}}^{\prime }+\frac{1}{2}\left[\frac{\mathrm{d}{{\Phi}}_{2}}{\mathrm{d}\lambda }+\frac{\mathrm{d}{{\Psi}}_{2}}{\mathrm{d}\lambda }\right]\right.\right.\hfill \\ \hfill & \qquad \left.+\left.4{{\Phi}}_{1}\left[3\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\lambda }-\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }\right]-4{{{\Phi}}_{1}}^{\prime }\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)+4{{\Phi}}_{1}\left(3{{\Phi}}_{1}+{{\Psi}}_{1}\right)\right.\right.\hfill \\ \hfill & \qquad +4{{\Psi}}_{1}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)\mathrm{d}\lambda -{\left(4{{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,i}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\mathrm{d}\lambda \hfill \end{align*}$$

$$\begin{align*}\hfill & \quad +4\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda -2\left[3{{\Phi}}_{1}+{{\Phi}}_{1}+{{{\Phi}}_{1}}^{\prime }+3{{{\Psi}}_{1}}^{\prime }\right]\hfill \\ \hfill & \quad {\times}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda +6\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,i}\mathrm{d}\lambda \right)\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\mathrm{d}\lambda \right)\hfill \\ \hfill & \quad +\left.2\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right)\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right)\right]\hfill \\ \hfill & \quad -\left(-2{\left.{{\Phi}}_{1}\right\vert }_{\text{s}}^{\text{s}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right){\left(-2{\left.{{\Phi}}_{1}\right\vert }_{\text{s}}^{\text{s}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right)}^{\prime }\hfill \\ \hfill & \quad +\left.\left(-2{n}^{i}{\left.{{\Psi}}_{1}\right\vert }_{\text{s}}^{\text{s}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\mathrm{d}\lambda \right)\right.\left.{\left(-2{\left.{{\Phi}}_{1}\right\vert }_{\text{s}}^{\text{s}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right)}_{,i}\right\}\hfill \end{align*}$$

$$\begin{align*}\hfill & \qquad -\frac{1}{2}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }\left\{\frac{1}{2}\left[\frac{{\mathrm{d}}^{2}{{\Psi}}_{1}}{\mathrm{d}{\lambda }^{2}}-\frac{{\mathrm{d}}^{2}{{\Phi}}_{1}}{\mathrm{d}{\lambda }^{2}}\right]+\left[\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\lambda }+\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }\right]\right.\hfill \\ \hfill & \qquad -\left.\frac{1}{2}\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)+\frac{1}{2}{\nabla }^{2}\left({{\Phi}}_{2}+{{\Psi}}_{2}\right)+{\left({{{\Phi}}_{1}}^{\prime }\right)}^{2}-{\left({{{\Psi}}_{1}}^{\prime }\right)}^{2}+2{{\Phi}}_{1}\right.\hfill \\ \hfill & \qquad {\times}\left[ {{{\Phi}}_{1}}^{{\prime\prime}}-{\nabla }^{2}{{\Phi}}_{1}-\frac{{\mathrm{d}}^{2}{{\Psi}}_{1}}{\mathrm{d}{\lambda }^{2}}-3\frac{\mathrm{d}{{{\Phi}}_{1}}^{\prime }}{\mathrm{d}\lambda } \right]-2{{\Psi}}_{1}\left[ {{{\Psi}}_{1}}^{{\prime\prime}}-{\nabla }^{2}{{\Psi}}_{1}-\frac{{\mathrm{d}}^{2}{{\Phi}}_{1}}{\mathrm{d}{\lambda }^{2}}-3\frac{\mathrm{d}{{{\Psi}}_{1}}^{\prime }}{\mathrm{d}\lambda }\right]\hfill \\ \hfill & \qquad -{{{\Phi}}_{1}}_{,i}{{{{\Phi}}_{1}}_{,}}^{i}+{{{\Psi}}_{1}}_{,i}{{{{\Psi}}_{1}}_{,}}^{i}+4{{\Psi}}_{1}\left[\frac{{\mathrm{d}}^{2}{{\Psi}}_{1}}{\mathrm{d}{\lambda }^{2}}-\frac{{\mathrm{d}}^{2}{{\Phi}}_{1}}{\mathrm{d}{\lambda }^{2}}+\frac{\mathrm{d}{{{\Phi}}_{1}}^{\prime }}{\mathrm{d}\lambda }-\frac{\mathrm{d}{{{\Psi}}_{1}}^{\prime }}{\mathrm{d}\lambda }+\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\lambda }\right.\hfill \\ \hfill & \qquad +\left.\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }-{{{\Phi}}_{1}}^{\prime }-{{{\Psi}}_{1}}^{\prime }\right]-2{\left[\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }-\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\lambda }+{{\Phi}}_{1}+{{\Psi}}_{1}\right]}_{,i}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\hfill \\ \hfill & \qquad +4{{\Phi}}_{1}\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }+4{{\Psi}}_{1}\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }+2\left({{\Phi}}_{1}+{{\Psi}}_{1}\right){\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\hfill \\ \hfill & \qquad +4\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }\left[-2{\left.{{\Phi}}_{1}\right\vert }_{\text{o}}^{\text{s}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right]+2{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\hfill \\ \hfill & \qquad {\times}\left[-2{\left.{{\Phi}}_{1}\right\vert }_{\text{o}}^{\text{s}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right]-\frac{3}{2}\left[\frac{\mathrm{d}}{\mathrm{d}\lambda }\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)-2\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\right.\hfill \end{align*}$$

$$\begin{align*}\hfill & \qquad +{\left.{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)\mathrm{d}\lambda \right]}^{2}+8{\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)}^{2}+2{\left[{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)\mathrm{d}\lambda \right]}^{2}\hfill \\ \hfill & \qquad +2{\left[\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\lambda }+\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }\right]}^{2}-\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\left[8\frac{\mathrm{d}}{\mathrm{d}\lambda }\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)+4\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)\right.\hfill \\ \hfill & \qquad +\left.{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\right]-\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right){\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left[8\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)-2{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\right]\mathrm{d}\lambda \hfill \\ \hfill & \qquad +\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,ij}\mathrm{d}\lambda \right)\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{ij}\mathrm{d}\lambda \right)\hfill \\ \hfill & \qquad +\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\mathrm{d}\lambda \right)\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)\mathrm{d}\lambda \right)\hfill \\ \hfill & \qquad -\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\mathrm{d}\lambda \right)\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\mathrm{d}\lambda \right)\hfill \\ \hfill & \qquad -2{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,i}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}+4{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\left[{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,i}\mathrm{d}\lambda \right]\hfill \\ \hfill & \qquad -\left.2\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,i}\mathrm{d}\lambda \right)\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\mathrm{d}\lambda \right)\right\}\hfill \end{align*}$$

$$\begin{align*}\hfill & \quad +\frac{2}{{\lambda }_{\text{o}}-{\lambda }_{\text{s}}}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\breve {\lambda }\left\{\frac{1}{2}\left[\frac{{\mathrm{d}}^{2}{{\Psi}}_{1}}{\mathrm{d}{\lambda }^{2}}-\frac{{\mathrm{d}}^{2}{{\Phi}}_{1}}{\mathrm{d}{\lambda }^{2}}\right]+\left[\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\lambda }+\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }\right]\right.\hfill \\ \hfill & \quad -\left.\frac{1}{2}\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)+\frac{1}{2}{\nabla }^{2}\left({{\Phi}}_{2}+{{\Psi}}_{2}\right)+{\left({{{\Phi}}_{1}}^{\prime }\right)}^{2}-{\left({{{\Psi}}_{1}}^{\prime }\right)}^{2}\right.\hfill \\ \hfill & \quad +2{{\Phi}}_{1}\left[{{{\Phi}}_{1}}^{{\prime\prime}}-{\nabla }^{2}{{\Phi}}_{1}-\frac{{\mathrm{d}}^{2}{{\Psi}}_{1}}{\mathrm{d}{\lambda }^{2}}-3\frac{\mathrm{d}{{{\Phi}}_{1}}^{\prime }}{\mathrm{d}\lambda }\right]\hfill \\ \hfill & \quad -2{{\Psi}}_{1}\left[{{{\Psi}}_{1}}^{{\prime\prime}}-{\nabla }^{2}{{\Psi}}_{1}-\frac{{\mathrm{d}}^{2}{{\Phi}}_{1}}{\mathrm{d}{\lambda }^{2}}-3\frac{\mathrm{d}{{{\Psi}}_{1}}^{\prime }}{\mathrm{d}\lambda }\right]-{{{\Phi}}_{1}}_{,i}{{{{\Phi}}_{1}}_{,}}^{i}+{{{\Psi}}_{1}}_{,i}{{{{\Psi}}_{1}}_{,}}^{i}\hfill \\ \hfill & \quad +4{{\Psi}}_{1}\left[\frac{{\mathrm{d}}^{2}{{\Psi}}_{1}}{\mathrm{d}{\lambda }^{2}}-\frac{{\mathrm{d}}^{2}{{\Phi}}_{1}}{\mathrm{d}{\lambda }^{2}}+\frac{\mathrm{d}{{{\Phi}}_{1}}^{\prime }}{\mathrm{d}\lambda }-\frac{\mathrm{d}{{{\Psi}}_{1}}^{\prime }}{\mathrm{d}\lambda }+\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\lambda }+\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }-{{{\Phi}}_{1}}^{\prime }-{{{\Psi}}_{1}}^{\prime }\right]\hfill \\ \hfill & \quad -2{\left[\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }-\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\lambda }+{{\Phi}}_{1}+{{\Psi}}_{1}\right]}_{,i}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}+4{{\Phi}}_{1}\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }+4{{\Psi}}_{1}\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }\hfill \\ \hfill & \quad +2\left({{\Phi}}_{1}+{{\Psi}}_{1}\right){\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)+4\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }\left[-2{\left.{{\Phi}}_{1}\right\vert }_{\text{o}}^{\text{s}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right]\hfill \\ \hfill & \quad +2{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\left[-2{\left.{{\Phi}}_{1}\right\vert }_{\text{o}}^{\text{s}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right]\hfill \end{align*}$$

$$\begin{align}\hfill & \quad -\frac{3}{2}{\left[\frac{\mathrm{d}}{\mathrm{d}\lambda }\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)-2\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)\mathrm{d}\lambda \right]}^{2}\hfill \\ \hfill & \quad +8{\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)}^{2}+2{\left[{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)\mathrm{d}\lambda \right]}^{2}+2{\left[\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\lambda }+\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }\right]}^{2}\hfill \\ \hfill & \quad -\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\left[8\frac{\mathrm{d}}{\mathrm{d}\lambda }\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)+4\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)+{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\right]\hfill \\ \hfill & \quad -\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right){\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left[8\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)-2{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\right]\mathrm{d}\lambda \hfill \\ \hfill & \quad +\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,ij}\mathrm{d}\lambda \right)\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{ij}\mathrm{d}\lambda \right)\hfill \\ \hfill & \quad +\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\mathrm{d}\lambda \right)\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)\mathrm{d}\lambda \right)\hfill \\ \hfill & \quad -\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\mathrm{d}\lambda \right)\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\mathrm{d}\lambda \right)\hfill \\ \hfill & \quad -2{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,i}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}+4{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\left[{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,i}\mathrm{d}\lambda \right]\hfill \\ \hfill & \quad -\left.2\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,i}\mathrm{d}\lambda \right)\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\mathrm{d}\lambda \right)\right\}.\hfill \end{align} \tag{ C9 }$$

The second order perturbation to the physical volume is

$$\begin{align*}\hfill \mathrm{d}{\delta }^{\left(2\right)}V& =-\bar{\mathcal{E}}{\bar{d}}_{A}^{2}\left[{\left(\frac{{\delta }^{\left(1\right)}{d}_{A}}{{\bar{d}}_{A}}\right)}^{2}+\left(\frac{{\delta }^{\left(1\right)}\mathcal{E}}{\bar{\mathcal{E}}}\right)\left(\frac{{\delta }^{\left(1\right)}{d}_{A}}{{\bar{d}}_{A}}\right)+\frac{{\delta }^{\left(2\right)}{d}_{A}}{{\bar{d}}_{A}}+\frac{1}{2}\frac{{\delta }^{\left(2\right)}\mathcal{E}}{\bar{\mathcal{E}}}\right]\mathrm{d}\lambda \enspace \mathrm{d}{\Omega}\hfill \\ \hfill & ={a}^{2}\left({\lambda }_{\text{s}}\right){\left({\lambda }_{\text{s}}-{\lambda }_{\text{o}}\right)}^{2}\left\{\left({{\Phi}}_{1}{\vert }_{\text{o}}-{\left({v}_{1i}{n}^{i}\right)}_{\text{o}}-{{\Phi}}_{1}{\vert }_{\text{o}}^{\text{s}}-{\left.\frac{3}{2}\left({{\Psi}}_{1}-{{\Phi}}_{1}\right)\right\vert }_{\text{o}}^{\text{s}}\right.\right.\hfill \\ \hfill & \quad +{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }\left\{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}^{{\prime\prime}}-{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\right\}-\left.\frac{1}{{\lambda }_{\text{o}}-{\lambda }_{\text{s}}}\right.\hfill \\ \hfill & \quad {\times}\left.\left[2{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}^{\prime }-{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\breve {\lambda }\left\{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}^{{\prime\prime}}\right.\right.\right.\hfill \\ \hfill & \quad -{\left.\left.\left.{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\right\}\right]\right)}^{2}\left({{\Phi}}_{1}{\vert }_{\text{o}}-{\left({v}_{1i}{n}^{i}\right)}_{\text{o}}-{{\Phi}}_{1}{\vert }_{\text{o}}^{\text{s}}-{\left.\frac{3}{2}\left({{\Psi}}_{1}-{{\Phi}}_{1}\right)\right\vert }_{\text{o}}^{\text{s}}\right.\hfill \\ \hfill & \quad +\left.{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }\left\{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}^{{\prime\prime}}-{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\right\}-\frac{1}{{\lambda }_{\text{o}}-\lambda \text{s}}\right.\hfill \\ \hfill & \quad {\times}\left.\left[2{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}^{\prime }-{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\breve {\lambda }\left\{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}^{{\prime\prime}}\right.\right.\right.\hfill \\ \hfill & \quad \left.\left.-\left.{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\right\}\right]\right)\left(-2{\left.{{\Phi}}_{1}\right\vert }_{\text{o}}^{\text{s}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }^{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda +{{\Phi}}_{1}-{v}_{1i}{n}^{i}\right)\hfill \\ \hfill & \quad +\frac{1}{2}{\left[{{{\Phi}}_{1}}^{2}-5{{{\Psi}}_{1}}^{2}-6{{\Phi}}_{1}{{\Psi}}_{1}+\frac{1}{2}\left({{\Phi}}_{2}-{{\Psi}}_{2}\right)-\left({v}_{1i}{n}^{i}\right){{\Phi}}_{1}-{n}^{i}{v}_{2i}\right]}_{\text{o}}\hfill \end{align*}$$

$$\begin{align*}\hfill & \quad -\frac{1}{{\lambda }_{\text{o}}-{\lambda }_{\text{s}}}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }\left\{4\left(-2{\left.{{\Phi}}_{1}\right\vert }_{\text{s}}^{\text{s}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right)\right.\hfill \\ \hfill & \quad {\times}\left.\left({{\Phi}}_{1}{\vert }_{\text{o}}-{\left({v}_{1i}{n}^{i}\right)}_{\text{o}}-{{\Phi}}_{1}{\vert }_{\text{o}}^{\text{s}}-{\left.\frac{3}{2}\left({{\Psi}}_{1}-{{\Phi}}_{1}\right)\right\vert }_{\text{o}}^{\text{s}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda \right.\right.\hfill \\ \hfill & \quad {\times}{\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }\left\{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}^{{\prime\prime}}-{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\right\}-\left.\frac{1}{{\bar{d}}_{A}}\left[2{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}^{\prime }\right.\right.\hfill \\ \hfill & \quad {\left.-\left.{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\breve {\lambda }\left\{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}^{{\prime\prime}}-{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\right\}\right]\right)}^{{\prime\prime}}\hfill \\ \hfill & \quad +\left({{\Phi}}_{1}{\vert }_{\text{o}}-{\left({v}_{1i}{n}^{i}\right)}_{\text{o}}-{{\Phi}}_{1}{\vert }_{\text{o}}^{\text{s}}-{\left.\frac{3}{2}\left({{\Psi}}_{1}-{{\Phi}}_{1}\right)\right\vert }_{\text{o}}^{\text{s}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }\right.\hfill \\ \hfill & \quad {\times}\left.\left\{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}^{{\prime\prime}}-{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\right\}-\frac{1}{{\bar{d}}_{A}}\left[2{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}^{\prime }\right.\right.\hfill \\ \hfill & \quad {\left.-\left.{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\breve {\lambda }\left\{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}^{{\prime\prime}}-{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\right\}\right]\right)}^{\prime }\hfill \end{align*}$$

$$\begin{align*}\hfill & \quad {\times}\left(-2\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\lambda }+{{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)+\frac{1}{2}\left({{\Phi}}_{1}{\vert }_{\text{o}}-{\left({v}_{1i}{n}^{i}\right)}_{\text{o}}-{{\Phi}}_{1}{\vert }_{\text{o}}^{\text{s}}-{\left.\frac{3}{2}\left({{\Psi}}_{1}-{{\Phi}}_{1}\right)\right\vert }_{\text{o}}^{\text{s}}\right.\hfill \\ \hfill & \quad +{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }\left\{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}^{{\prime\prime}}-{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\right\}\hfill \\ \hfill & \quad -\left.\frac{1}{{\lambda }_{\text{o}}-{\lambda }_{\text{s}}}\left[2{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}^{\prime }-{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }\right.\right.\hfill \\ \hfill & \quad \left.{\times}\left.{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\breve {\lambda }\left\{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}^{{\prime\prime}}-{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\right\}\right]\right)\left(\left[\frac{{\mathrm{d}}^{2}{{\Psi}}_{1}}{\mathrm{d}{\lambda }^{2}}-\frac{{\mathrm{d}}^{2}{{\Phi}}_{1}}{\mathrm{d}{\lambda }^{2}}\right]\right.\hfill \\ \hfill & \quad \left.+\left.2\left[\frac{\mathrm{d}{{{\Phi}}_{1}}^{\prime }}{\mathrm{d}\lambda }+\frac{\mathrm{d}{{{\Psi}}_{1}}^{\prime }}{\mathrm{d}\lambda }\right]+{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)-\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)\right)\right\}+\frac{1}{{\lambda }_{\text{o}}-{\lambda }_{\text{s}}}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda \hfill \\ \hfill & \quad {\times}{\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }\left\{\frac{1}{2}\left[-2\frac{\mathrm{d}{{\Phi}}_{2}}{\mathrm{d}\lambda }+{{{\Phi}}_{2}}^{\prime }+{{{\Psi}}_{2}}^{\prime }+\frac{1}{2}\left[\frac{\mathrm{d}{{\Phi}}_{2}}{\mathrm{d}\lambda }+\frac{\mathrm{d}{{\Psi}}_{2}}{\mathrm{d}\lambda }\right]+4{{\Phi}}_{1}\left[3\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\lambda }-\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }\right]\right.\right.\hfill \\ \hfill & \quad -4{{{\Phi}}_{1}}^{\prime }\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)+4{{\Phi}}_{1}\left(3{{\Phi}}_{1}+{{\Psi}}_{1}\right)+4{{\Psi}}_{1}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)\mathrm{d}\lambda \hfill \end{align*}$$

$$\begin{align*}\hfill & \quad -{\left(4{{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,i}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\mathrm{d}\lambda +4\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \hfill \\ \hfill & \quad -2\left[3{{\Phi}}_{1}+{{\Phi}}_{1}+{{{\Phi}}_{1}}^{\prime }+3{{{\Psi}}_{1}}^{\prime }\right]{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda +6\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,i}\mathrm{d}\lambda \right)\hfill \\ \hfill & \quad {\times}\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\mathrm{d}\lambda \right)+\left.2\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right)\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right)\right]\hfill \\ \hfill & \quad -\left(-2{\left.{{\Phi}}_{1}\right\vert }_{\text{s}}^{\text{s}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right){\left(-2{\left.{{\Phi}}_{1}\right\vert }_{\text{s}}^{\text{s}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right)}^{\prime }\hfill \\ \hfill & \quad +\left.\left(-2{n}^{i}{\left.{{\Psi}}_{1}\right\vert }_{\text{s}}^{\text{s}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\mathrm{d}\lambda \right){\left(-2{\left.{{\Phi}}_{1}\right\vert }_{\text{s}}^{\text{s}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right)}_{,i}\right\}\hfill \\ \hfill & \quad -\frac{1}{2}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }\left\{\frac{1}{2}\left[\frac{{\mathrm{d}}^{2}{{\Psi}}_{1}}{\mathrm{d}{\lambda }^{2}}-\frac{{\mathrm{d}}^{2}{{\Phi}}_{1}}{\mathrm{d}{\lambda }^{2}}\right]+\left[\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\lambda }+\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }\right]-\frac{1}{2}\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)\right.\hfill \\ \hfill & \quad +\left.\frac{1}{2}{\nabla }^{2}\left({{\Phi}}_{2}+{{\Psi}}_{2}\right)+{\left({{{\Phi}}_{1}}^{\prime }\right)}^{2}-{\left({{{\Psi}}_{1}}^{\prime }\right)}^{2}+2{{\Phi}}_{1}\left[{{{\Phi}}_{1}}^{{\prime\prime}}-{\nabla }^{2}{{\Phi}}_{1}-\frac{{\mathrm{d}}^{2}{{\Psi}}_{1}}{\mathrm{d}{\lambda }^{2}}-3\frac{\mathrm{d}{{{\Phi}}_{1}}^{\prime }}{\mathrm{d}\lambda }\right]\right.\hfill \\ \hfill & \quad -2{{\Psi}}_{1}\left[{{{\Psi}}_{1}}^{{\prime\prime}}-{\nabla }^{2}{{\Psi}}_{1}-\frac{{\mathrm{d}}^{2}{{\Phi}}_{1}}{\mathrm{d}{\lambda }^{2}}-3\frac{\mathrm{d}{{{\Psi}}_{1}}^{\prime }}{\mathrm{d}\lambda }\right]-{{{\Phi}}_{1}}_{,i}{{{{\Phi}}_{1}}_{,}}^{i}+{{{\Psi}}_{1}}_{,i}{{{{\Psi}}_{1}}_{,}}^{i}\hfill \end{align*}$$

$$\begin{align*}\hfill & +4{{\Psi}}_{1}\left[\frac{{\mathrm{d}}^{2}{{\Psi}}_{1}}{\mathrm{d}{\lambda }^{2}}-\frac{{\mathrm{d}}^{2}{{\Phi}}_{1}}{\mathrm{d}{\lambda }^{2}}+\frac{\mathrm{d}{{{\Phi}}_{1}}^{\prime }}{\mathrm{d}\lambda }-\frac{\mathrm{d}{{{\Psi}}_{1}}^{\prime }}{\mathrm{d}\lambda }+\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\lambda }+\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }-{{{\Phi}}_{1}}^{\prime }-{{{\Psi}}_{1}}^{\prime }\right]\hfill \\ \hfill & -2{\left[\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }-\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\lambda }+{{\Phi}}_{1}+{{\Psi}}_{1}\right]}_{,i}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}+4{{\Phi}}_{1}\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }+4{{\Psi}}_{1}\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }\hfill \\ \hfill & +2\left({{\Phi}}_{1}+{{\Psi}}_{1}\right){\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)+4\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }\left[-2{\left.{{\Phi}}_{1}\right\vert }_{\text{o}}^{\text{s}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right]\hfill \\ \hfill & +2{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\left[-2{\left.{{\Phi}}_{1}\right\vert }_{\text{o}}^{\text{s}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right]\hfill \\ \hfill & -\frac{3}{2}{\left[\frac{\mathrm{d}}{\mathrm{d}\lambda }\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)-2\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)\mathrm{d}\lambda \right]}^{2}\hfill \\ \hfill & +8{\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)}^{2}+2{\left[{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)\mathrm{d}\lambda \right]}^{2}+2{\left[\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\lambda }+\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }\right]}^{2}\hfill \\ \hfill & -\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\left[8\frac{\mathrm{d}}{\mathrm{d}\lambda }\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)+4\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)+{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\right]\hfill \\ \hfill & -\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right){\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left[8\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)-2{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\right]\mathrm{d}\lambda \hfill \end{align*}$$

$$\begin{align*}\hfill & \quad +\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,ij}\mathrm{d}\lambda \right)\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{ij}\mathrm{d}\lambda \right)+\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\mathrm{d}\lambda \right)\hfill \\ \hfill & \quad {\times}\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)\mathrm{d}\lambda \right)-\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\mathrm{d}\lambda \right)\hfill \\ \hfill & \quad {\times}\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\mathrm{d}\lambda \right)-2{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,i}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}+4{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\hfill \\ \hfill & \quad {\times}\left[{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,i}\mathrm{d}\lambda \right]-\left.2\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,i}\mathrm{d}\lambda \right)\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\mathrm{d}\lambda \right)\right\}\hfill \\ \hfill & \quad +\frac{2}{{\lambda }_{\text{o}}-{\lambda }_{\text{s}}}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda {\int }_{{\lambda }_{\text{s}}}^{{\lambda }_{\text{s}}}\mathrm{d}\tilde {\lambda }{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\breve {\lambda }\left\{\frac{1}{2}\left[\frac{{\mathrm{d}}^{2}{{\Psi}}_{1}}{\mathrm{d}{\lambda }^{2}}-\frac{{\mathrm{d}}^{2}{{\Phi}}_{1}}{\mathrm{d}{\lambda }^{2}}\right]+\left[\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\lambda }+\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }\right]\right.\hfill \\ \hfill & \quad -\left.\frac{1}{2}\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)+\frac{1}{2}{\nabla }^{2}\left({{\Phi}}_{2}+{{\Psi}}_{2}\right)+{\left({{{\Phi}}_{1}}^{\prime }\right)}^{2}-{\left({{{\Psi}}_{1}}^{\prime }\right)}^{2}+2{{\Phi}}_{1}\right.\hfill \\ \hfill & \quad {\times}\left[{{{\Phi}}_{1}}^{{\prime\prime}}-{\nabla }^{2}{{\Phi}}_{1}-\frac{{\mathrm{d}}^{2}{{\Psi}}_{1}}{\mathrm{d}{\lambda }^{2}}-3\frac{\mathrm{d}{{{\Phi}}_{1}}^{\prime }}{\mathrm{d}\lambda }\right]-2{{\Psi}}_{1}\left[{{{\Psi}}_{1}}^{{\prime\prime}}-{\nabla }^{2}{{\Psi}}_{1}-\frac{{\mathrm{d}}^{2}{{\Phi}}_{1}}{\mathrm{d}{\lambda }^{2}}-3\frac{\mathrm{d}{{{\Psi}}_{1}}^{\prime }}{\mathrm{d}\lambda }\right]\hfill \\ \hfill & \quad -{{{\Phi}}_{1}}_{,i}{{{{\Phi}}_{1}}_{,}}^{i}+{{{\Psi}}_{1}}_{,i}{{{{\Psi}}_{1}}_{,}}^{i}+4{{\Psi}}_{1}\left[\frac{{\mathrm{d}}^{2}{{\Psi}}_{1}}{\mathrm{d}{\lambda }^{2}}-\frac{{\mathrm{d}}^{2}{{\Phi}}_{1}}{\mathrm{d}{\lambda }^{2}}+\frac{\mathrm{d}{{{\Phi}}_{1}}^{\prime }}{\mathrm{d}\lambda }-\frac{\mathrm{d}{{{\Psi}}_{1}}^{\prime }}{\mathrm{d}\lambda }+\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\lambda }\right.\hfill \end{align*}$$

$$\begin{align*}\hfill & \quad +\left.\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }-{{{\Phi}}_{1}}^{\prime }-{{{\Psi}}_{1}}^{\prime }\right]-2{\left[\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }-\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\lambda }+{{\Phi}}_{1}+{{\Psi}}_{1}\right]}_{,i}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\hfill \\ \hfill & \quad +4{{\Phi}}_{1}\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }+4{{\Psi}}_{1}\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }+2\left({{\Phi}}_{1}+{{\Psi}}_{1}\right){\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)+4\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }\hfill \\ \hfill & \quad {\times}\left[-2{\left.{{\Phi}}_{1}\right\vert }_{\text{o}}^{\text{s}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right]+2{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\hfill \\ \hfill & \quad {\times}\left[-2{\left.{{\Phi}}_{1}\right\vert }_{\text{o}}^{\text{s}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right]-\frac{3}{2}\left[\frac{\mathrm{d}}{\mathrm{d}\lambda }\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)-2\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\right.\hfill \\ \hfill & \quad +{\left.{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)\mathrm{d}\lambda \right]}^{2}+8{\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)}^{2}+2{\left[{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)\mathrm{d}\lambda \right]}^{2}\hfill \\ \hfill & \quad +2{\left[\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\lambda }+\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }\right]}^{2}-\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\left[8\frac{\mathrm{d}}{\mathrm{d}\lambda }\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)+4\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)\right.\hfill \\ \hfill & \quad +\left.{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\right]-\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right){\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left[8\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)-2{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\right]\mathrm{d}\lambda \hfill \\ \hfill & \quad +\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,ij}\mathrm{d}\lambda \right)\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{ij}\mathrm{d}\lambda \right)+\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\mathrm{d}\lambda \right)\hfill \\ \hfill & \quad {\times}\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)\mathrm{d}\lambda \right)-\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\mathrm{d}\lambda \right)\hfill \\ \hfill & \quad {\times}\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{\nabla }^{2}\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\mathrm{d}\lambda \right)-2{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,i}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}+4{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\hfill \end{align*}$$

$$\begin{align}\hfill & \quad {\times}\left[{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,i}\mathrm{d}\lambda \right]-\left.2\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,i}\mathrm{d}\lambda \right)\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\mathrm{d}\lambda \right)\right\}\hfill \\ \hfill & \quad +\frac{1}{4}\left[2\left\{\left(-2{\left.{{\Phi}}_{1}\right\vert }_{\text{o}}^{\text{s}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }^{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right)+{{\Phi}}_{1}-{v}_{1i}{n}^{i}\right\}{{\Phi}}_{1}\right.-{{{\Phi}}_{1}}^{2}-5{{{\Psi}}_{1}}^{2}\hfill \\ \hfill & \quad -6{{\Phi}}_{1}{{\Psi}}_{1}+\frac{1}{2}\left({{\Phi}}_{2}-{{\Psi}}_{2}\right)-{n}^{i}{v}_{2i}-4\left({{\Phi}}_{1}+{{\Psi}}_{1}\right){\left.{{\Phi}}_{1}\right\vert }_{\text{o}}^{\text{s}}+2\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)\hfill \\ \hfill & \quad {\times}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda -4{n}^{i}{v}_{1i}{\left.{{\Psi}}_{1}\right\vert }_{\text{o}}^{\text{s}}+2{v}_{1i}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\mathrm{d}\lambda \hfill \\ \hfill & \quad +{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left\{-2\frac{\mathrm{d}{{\Phi}}_{2}}{\mathrm{d}\lambda }+{{{\Phi}}_{2}}^{\prime }+{{{\Psi}}_{2}}^{\prime }+\frac{1}{2}\left[\frac{\mathrm{d}{{\Phi}}_{2}}{\mathrm{d}\lambda }+\frac{\mathrm{d}{{\Psi}}_{2}}{\mathrm{d}\lambda }\right]+4{{\Phi}}_{1}\left[3\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\lambda }-\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }\right]\right.\hfill \\ \hfill & \quad -4{{{\Phi}}_{1}}^{\prime }\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)+4{{\Phi}}_{1}\left(3{{\Phi}}_{1}+{{\Psi}}_{1}\right)+4{{\Psi}}_{1}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)\mathrm{d}\lambda \hfill \\ \hfill & \quad -{\left(4{{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,i}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\mathrm{d}\lambda +4\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \hfill \\ \hfill & \quad -2\left[3{{\Phi}}_{1}+{{\Phi}}_{1}+{{{\Phi}}_{1}}^{\prime }+3{{{\Psi}}_{1}}^{\prime }\right]{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \hfill \\ \hfill & \quad +6\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,i}\mathrm{d}\lambda \right)\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\mathrm{d}\lambda \right)\hfill \\ \hfill & \quad +\left.\left.\left.2\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right)\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right)\right\}\mathrm{d}\tilde {\lambda }\right]\right\}\mathrm{d}\lambda \mathrm{d}{\Omega}.\hfill \end{align} \tag{ C10 }$$

Using equations (3.14), (C7) and (5.15) in equation (5.10) we find that the redshift density perturbation at second order is given by

$$\begin{align*}\hfill {\delta }_{z}^{\left(2\right)}\left({n}^{i},z\right)& =\frac{1}{2}\frac{{\delta }^{\left(2\right)}\rho \left({n}^{i},z\right)}{\bar{\rho \left(z\right)}}+\frac{3}{2\left(1+\bar{z}\right)}{\delta }^{\left(2\right)}z\left({n}^{i},z\right)+\frac{3}{{\left(1+z\right)}^{2}}{\left[{\delta }^{\left(1\right)}z\left({n}^{i},z\right)\right]}^{2}\hfill \\ \hfill & =\frac{1}{2}\frac{{\delta }^{\left(2\right)}\rho \left({n}^{i},z\right)}{\bar{\rho \left(z\right)}}+\frac{3}{2\left(1+\bar{z}\right)}\left({\left[{{{\Phi}}_{1}}^{2}+{\left({v}_{1i}{n}^{i}\right)}^{2}-2{{\Phi}}_{1}\left({v}_{1i}{n}^{i}\right)\right]}_{\text{o}}\right.\hfill \\ \hfill & \quad -{{\Phi}}_{1}{\vert }_{\text{o}}{\left[-2{{\Phi}}_{1}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right]}_{\text{s}}+{\left[-2{{\Phi}}_{1}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right]}_{\text{s}}\hfill \\ \hfill & \quad {\times}{\left({v}_{1i}{n}^{i}\right)}_{\text{o}}-{{\Phi}}_{1}{\vert }_{\text{s}}{{\Phi}}_{1}{\vert }_{\text{o}}+{{\Phi}}_{1}{\vert }_{\text{s}}{\left({v}_{1i}{n}^{i}\right)}_{\text{o}}+{\left({v}_{1i}{n}^{i}\right)}_{\text{s}}{{\Phi}}_{1}{\vert }_{\text{o}}-{\left({v}_{1i}{n}^{i}\right)}_{\text{s}}{\left({v}_{1i}{n}^{i}\right)}_{\text{o}}\hfill \\ \hfill & \quad +\frac{1}{2}\left[2\left\{\left(-2{\left.{{\Phi}}_{1}\right\vert }_{\text{o}}^{\text{s}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }^{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right)+{{\Phi}}_{1}-{v}_{1i}{n}^{i}\right\}{{\Phi}}_{1}\right.\hfill \\ \hfill & \quad -{{{\Phi}}_{1}}^{2}-5{{{\Psi}}_{1}}^{2}-6{{\Phi}}_{1}{{\Psi}}_{1}+\frac{1}{2}\left({{\Phi}}_{2}-{{\Psi}}_{2}\right)-{n}^{i}{v}_{2i}-4\left({{\Phi}}_{1}+{{\Psi}}_{1}\right){\left.{{\Phi}}_{1}\right\vert }_{\text{o}}^{\text{s}}\hfill \\ \hfill & \quad +2\left({{\Phi}}_{1}+{{\Psi}}_{1}\right){\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda -4{n}^{i}{v}_{1i}{\left.{{\Psi}}_{1}\right\vert }_{\text{o}}^{\text{s}}+2{v}_{1i}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\mathrm{d}\lambda \hfill \\ \hfill & \quad +{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left\{-2\frac{\mathrm{d}{{\Phi}}_{2}}{\mathrm{d}\lambda }+{{{\Phi}}_{2}}^{\prime }+{{{\Psi}}_{2}}^{\prime }+\frac{1}{2}\left[\frac{\mathrm{d}{{\Phi}}_{2}}{\mathrm{d}\lambda }+\frac{\mathrm{d}{{\Psi}}_{2}}{\mathrm{d}\lambda }\right]+4{{\Phi}}_{1}\left[3\frac{\mathrm{d}{{\Phi}}_{1}}{\mathrm{d}\lambda }-\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }\right]\right.\hfill \\ \hfill & \quad -4{{{\Phi}}_{1}}^{\prime }\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)+4{{\Phi}}_{1}\left(3{{\Phi}}_{1}+{{\Psi}}_{1}\right)+4{{\Psi}}_{1}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{{\prime\prime}}+{{{\Psi}}_{1}}^{{\prime\prime}}\right)\mathrm{d}\lambda \hfill \end{align*}$$

$$\begin{align}\hfill & \qquad -{\left(4{{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,i}{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\mathrm{d}\lambda +4\frac{\mathrm{d}{{\Psi}}_{1}}{\mathrm{d}\lambda }{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \hfill \\ \hfill & \qquad -2\left[3{{\Phi}}_{1}+{{\Phi}}_{1}+{{{\Phi}}_{1}}^{\prime }+3{{{\Psi}}_{1}}^{\prime }\right]{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda +6\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,i}\mathrm{d}\lambda \right)\hfill \\ \hfill & \qquad {\times}\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}{{\left({{\Phi}}_{1}+{{\Psi}}_{1}\right)}_{,}}^{i}\mathrm{d}\lambda \right)+2\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right)\hfill \\ \hfill & \qquad \left.\left.{\times}\left({\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\left({{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right)\mathrm{d}\lambda \right)\right\}\mathrm{d}\tilde {\lambda }\right]_{\text{s}}-\frac{1}{2}\left[{{{\Phi}}_{1}}^{2}-5{{{\Psi}}_{1}}^{2}-6{{\Phi}}_{1}{{\Psi}}_{1}\right.\hfill \\ \hfill & \qquad \left.+{\left.\frac{1}{2}\left({{\Phi}}_{2}-{{\Psi}}_{2}\right)-\left({v}_{1i}{n}^{i}\right){{\Phi}}_{1}-{n}^{i}{v}_{2i}\right]}_{\text{o}}\right)+\frac{3}{{\left(1+z\right)}^{2}}\hfill \\ \hfill & \qquad {\times}{\left\{{\left.\left({v}_{1i}{n}^{i}-{{\Phi}}_{1}\right)\right\vert }_{\text{o}}^{\text{s}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda \left[{{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right]\right\}}^{2}+\frac{1}{\mathcal{H}\left(1+\bar{z}\right)}\hfill \\ \hfill & \qquad {\times}\left(\frac{\mathrm{d}}{\mathrm{d}\eta }\left({v}_{1i}{n}^{i}\right)-\left[{{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right]\right)\left.{\left(\left({v}_{1i}{n}^{i}-{{\Phi}}_{1}\right)\right\vert }_{\text{o}}^{\text{s}}+{\int }_{{\lambda }_{\text{o}}}^{{\lambda }_{\text{s}}}\mathrm{d}\lambda \left[{{{\Phi}}_{1}}^{\prime }+{{{\Psi}}_{1}}^{\prime }\right]\right),\hfill \end{align} \tag{ C11 }$$