Equations of Motion in an Expanding Universe

Abstract

We make use of an effective field-theoretical approach to derive post-Newtonian equations of motion of hydrodynamical inhomogeneities in cosmology. The matter Lagrangian for the perturbed cosmological model includes dark matter, dark energy, and ordinary baryonic matter. The Lagrangian is expanded in an asymptotic Taylor series around a Friedmann-Lemeître-Robertson-Walker background. The small parameter of the decomposition is the magnitude of the metric tensor perturbation. Each term of the expansion series is gauge-invariant and all of them together form a basis for the successive post-Newtonian approximations around the background metric. The approximation scheme is covariant and the asymptotic nature of the Lagrangian decomposition does not require the post-Newtonian perturbations to be small though computationally it works the most effectively when the perturbed metric is close enough to the background metric. Temporal evolution of the background metric is governed by dark matter and dark energy and we associate the large-scale inhomogeneities of matter as those generated by the primordial cosmological perturbations in these two components with $\delta \rho /\rho \le 1$. The small scale inhomogeneities are generated by the baryonic matter which is considered as a bare perturbation of the background gravitational field, dark matter and energy. Mathematically, the large scale structure inhomogeneities are given by the homogeneous solution of the post-Newtonian equations while the small scale inhomogeneities are described by a particular solution of these equations with the stress-energy tensor of the baryonic matter that admits $\delta \rho /\rho \gg 1$. We explicitly work out the field equations of the first post-Newtonian approximation in cosmology and derive the post-Newtonian equations of motion of the large and small scale inhomogeneities which generalize the covariant law of conservation of stress-energy-momentum tensor of matter in asymptotically-flat spacetime.

Appendices

Appendix 1: Hilbert and Einstein Lagrangians

We define the Christoffel symbols of the second kind as usual [46]

$$\begin{aligned} \Gamma ^\alpha {}_{\beta \gamma }\equiv & {} \frac{1}{2} g^{\alpha \delta }\left(g_{\delta \beta ,\gamma }+g_{\delta \gamma ,\beta }-g_{\beta \gamma ,\delta }\right).\end{aligned}$$

(261)

The Christoffel symbols of the first type

$$\begin{aligned} \Gamma _{\alpha \beta \gamma }= & {} g_{\alpha \sigma }\Gamma ^\sigma {}_{\beta \gamma }=\frac{1}{2}\left(g_{\alpha \beta ,\gamma }+g_{\alpha \gamma ,\beta }-g_{\beta \gamma ,\alpha }\right),\end{aligned}$$

(262)

We notice the symmetry with respect to the last two indices $\Gamma _{\alpha \beta \gamma }=\Gamma _{\alpha (\beta \gamma )}$. There is no any symmetry with respect to the first two indices. In general,

$$\begin{aligned} \Gamma _{\alpha \beta \gamma }= & {} \Gamma _{(\alpha \beta )\gamma }+\Gamma _{[\alpha \beta ]\gamma }, \end{aligned}$$

(263)

where

$$\begin{aligned} \Gamma _{(\alpha \beta )\gamma }=\frac{1}{2} g_{\alpha \beta ,\gamma },\qquad \Gamma _{[\alpha \beta ]\gamma }=\frac{1}{2} \left(g_{\gamma \alpha ,\beta }-g_{\gamma \beta ,\alpha }\right), \end{aligned}$$

(264)

There are two, particularly useful symbols that are obtained by contracting indices of the Christoffel symbols of the first kind. They are denoted as

$$\begin{aligned} \mathcal{Y}_\alpha \equiv \Gamma ^\beta {}_{\alpha \beta },\qquad \mathcal{Y}^\alpha =g^{\alpha \beta }\mathcal{Y}_\beta ,\end{aligned}$$

(265)

and

$$\begin{aligned} \Gamma ^\alpha \equiv g^{\beta \gamma }\Gamma ^\alpha {}_{\beta \gamma },\qquad \Gamma _\alpha =g_{\alpha \beta }\Gamma ^\beta ,\end{aligned}$$

(266)

Direct inspection shows that

$$\begin{aligned} \mathcal{Y}_\alpha= & {} =\frac{1}{2} g^{\beta \gamma }g_{\beta \gamma ,\alpha }=\left(\ln {\sqrt{-g}}\right)_{,\alpha }.\end{aligned}$$

(267)

The two symbols are interrelated

$$\begin{aligned} \Gamma _\alpha= & {} -\mathcal{Y}_a+g^{\beta \gamma }g_{\alpha \beta ,\gamma },\end{aligned}$$

(268)

$$\begin{aligned} \Gamma ^\alpha= & {} -\mathcal{Y}^a-g^{\alpha \beta }{}_{,\beta },\end{aligned}$$

(269)

We define the Riemann tensor as follows [46]

$$\begin{aligned} R^\alpha {}_{\mu \beta \nu }= & {} \Gamma ^\alpha {}_{\mu \nu ,\beta }-\Gamma ^\alpha {}_{\mu \beta ,\nu }+\Gamma ^\alpha {}_{\beta \gamma }\Gamma ^\gamma {}_{\mu \nu }-\Gamma ^\alpha {}_{\nu \gamma }\Gamma ^\gamma {}_{\mu \beta }. \end{aligned}$$

(270)

Second covariant derivatives of tensors do not commute due to the curvature of spacetime. For example, denoting the covariant derivatives with a vertical bar we have for a covector field ${\mathcal F}_\alpha$ and a covariant tensor of second rank, ${\mathcal F}_{\alpha \beta }$ the following commutation relations

$$\begin{aligned} {\mathcal F}_{\alpha |\beta \gamma }= & {} {\mathcal F}_{\alpha |\gamma \beta }+R^\mu {}_{\alpha \beta \gamma }{\mathcal F}_\mu ,\end{aligned}$$

(271)

$$\begin{aligned} {\mathcal F}_{\alpha \beta |\gamma \delta }= & {} {\mathcal F}_{\alpha \beta |\delta \gamma }+R^\mu {}_{\alpha \gamma \delta }{\mathcal F}_{\mu \beta }+R^\mu {}_{\beta \gamma \delta }{\mathcal F}_{\alpha \mu }. \end{aligned}$$

(272)

Riemann tensor can be also expressed in terms of the second partial derivatives of the metric tensor and the Christoffel symbols

$$\begin{aligned} R_{\alpha \mu \beta \nu }=\frac{1}{2}\left(g_{\mu \beta ,\alpha \nu }+g_{\nu \alpha ,\beta \mu }-g_{\alpha \beta ,\mu \nu }-g_{\mu \nu ,\alpha \beta } \right)+\Gamma _{\rho \mu \beta }\Gamma ^\rho {}_{\alpha \nu }-\Gamma _{\rho \mu \nu }\Gamma ^\rho {}_{\alpha \beta }. \end{aligned}$$

(273)

Contraction of two indices in the Riemann tensor yields the Ricci tensor

$$\begin{aligned} R_{\mu \nu }= & {} \Gamma ^\alpha {}_{\mu \nu ,\alpha }-\mathcal{Y}_{\mu ,\nu }+\mathcal{Y}_\gamma \Gamma ^\gamma {}_{\mu \nu }-\Gamma ^\alpha {}_{\nu \gamma }\Gamma ^\gamma {}_{\mu \alpha },\end{aligned}$$

(274)

or, in terms of the second derivatives from the metric tensor and the Christoffel symbols,

$$\begin{aligned} R_{\mu \nu }=\frac{1}{2}g^{\kappa \epsilon }\left(g_{\mu \kappa ,\epsilon \nu }+g_{\nu \kappa ,\epsilon \mu }-g_{\kappa \epsilon ,\mu \nu }-g_{\mu \nu ,\kappa \epsilon }\right) +g^{\kappa \epsilon }\Gamma _{\rho \mu \epsilon }\Gamma ^\rho {}_{\kappa \nu }-\Gamma _{\rho \mu \nu }\Gamma ^\rho . \end{aligned}$$

(275)

One more contraction of indices in the Ricci tensor brings about the Ricci scalar which we shall write down in the form suggested by Fock [14, Appendix B]

$$\begin{aligned} R= & {} L+{\mathcal Y}_\alpha \Gamma ^\alpha -{\mathcal Y}_\alpha {\mathcal Y}^\alpha +\Gamma ^\alpha {}_{,\alpha }-{\mathcal Y}^\alpha {}_{,\alpha }, \end{aligned}$$

(276)

where

$$\begin{aligned} L= & {} g^{\mu \nu }\left(\Gamma ^\alpha {}_{\nu \gamma }\Gamma ^\gamma {}_{\mu \alpha }-{\mathcal Y}_\alpha \Gamma ^\alpha {}_{\mu \nu }\right), \end{aligned}$$

(277)

is (up to a constant factor) the gravitational Lagrangian introduced by Einstein [79] as an alternative to the gravitational Lagrangian, R, of Hilbert. The Hilbert Lagrangian is the Ricci scalar which depends on the second derivatives of the metric tensor while the Einstein Lagrangian does not.

The two Lagrangians are interrelated

$$\begin{aligned} R=L+\left(-g\right)^{-1/2}\mathcal{A}^\alpha {}_{,\alpha }, \end{aligned}$$

(278)

where

$$\begin{aligned} \mathcal{A}^\alpha =\sqrt{-g}\left(\Gamma ^\alpha -{\mathcal Y}^\alpha \right), \end{aligned}$$

(279)

is a vector density of weight $+1$. After performing differentiation in (278), and accounting for (265) we can easily prove that (278) reproduces (276).

One more form of relation between R and L will be useful for calculating the variational derivative in Appendix section “Variational Derivative from the Hilbert Lagrangian”. To this end we introduce a new notation

$$\begin{aligned} \Gamma\equiv & {} \Gamma ^\alpha {}_{,\alpha }+{\mathcal Y}_\alpha \Gamma ^\alpha ,\end{aligned}$$

(280)

and notice that

$$\begin{aligned} g^{\alpha \beta }{\mathcal Y}_{\alpha ,\beta }= & {} {\mathcal Y}^\alpha {}_{,\alpha }+{\mathcal Y}_\alpha \Gamma ^\alpha +{\mathcal Y}_\alpha {\mathcal Y}^\alpha , \end{aligned}$$

(281)

Equations (280), (281) allows us to cast (276) to the following form

$$\begin{aligned} R= & {} L+\Gamma +{\mathcal Y}_\alpha \Gamma ^\alpha -g^{\alpha \beta }{\mathcal Y}_{\alpha ,\beta }, \end{aligned}$$

(282)

that was found by Fock [14, Appendix B].

Appendix 2: Variational Derivatives

1.1 Variational Derivative from the Hilbert Lagrangian

The goal of this section is to prove relation (45) being valid on the background manifold ${\bar{{\mathcal M}}}$. We shall omit the bar over the background geometric objects as it does not bring about confusion. We notice that the Hilbert Lagrangian density, ${{\mathcal L}}^\mathrm{\scriptscriptstyle G}=-(16\pi )^{-1}\sqrt{-g}R$, differs from ${{\mathcal L}}^\mathrm{\scriptscriptstyle E}=-(16\pi )^{-1}\sqrt{-g}L$ by a total derivative that is a consequence of (278). Due to relation (9) the Lagrangian derivatives from ${{\mathcal L}}^\mathrm{\scriptscriptstyle G}$ and ${{\mathcal L}}^\mathrm{\scriptscriptstyle E}$ coincides

$$\begin{aligned} \frac{\delta {{\mathcal L}}^\mathrm{\scriptscriptstyle G}}{\delta \mathfrak {g}^{\mu \nu }}=\frac{\delta {{\mathcal L}}^\mathrm{\scriptscriptstyle E}}{\delta \mathfrak {g}^{\mu \nu }}, \end{aligned}$$

(283)

thus, pointing out that we can safely operate with the Einstein Lagrangian density ${{\mathcal L}}^\mathrm{\scriptscriptstyle E}$. Because of (35), we have

$$\begin{aligned} \frac{\delta {{\mathcal L}}^\mathrm{\scriptscriptstyle E}}{\delta \mathfrak {g}^{\mu \nu }}=\frac{1}{\sqrt{-g}}A^{\rho \sigma }_{\mu \nu }\frac{\delta {{\mathcal L}}^\mathrm{\scriptscriptstyle E}}{\delta g^{\rho \sigma }}, \end{aligned}$$

(284)

which suggests that calculation of the variational derivative with respect to the metric tensor is sufficient.

Calculation of the variational derivative $\delta {{\mathcal L}}^\mathrm{\scriptscriptstyle E}/\delta g^{\rho \sigma }$ demands the partial derivatives of the contravariant metric and Christoffel symbols with respect to $g^{\mu \nu }$. The partial derivatives of the metric are calculated with the help of (37), (30). The Christoffel symbols are given in terms of the partial derivatives from covariant metric tensor, $g_{\alpha \beta ,\gamma }$ which are not conjugated with the dynamic variable $g^{\alpha \beta }$. Thus, calculation of the partial derivative with respect to $g^{\mu \nu }$ from the Christoffel symbols demands its transformation to the form where the conjugated variables $g^{\alpha \beta }{}_{,\gamma }$ are used instead. This form of the Christoffel symbols is

$$\begin{aligned} \Gamma ^\alpha {}_{\beta \gamma }=\frac{1}{2} \left(g^{\rho \kappa }{}_{,\sigma }g^{\alpha \sigma }g_{\rho \beta }g_{\kappa \gamma }-g^{\alpha \sigma }{}_{,\beta }g_{\gamma \sigma }-g^{\alpha \sigma }{}_{,\gamma }g_{\beta \sigma }\right). \end{aligned}$$

(285)

Taking the partial derivative of (285) with respect to the contravariant metric yields

$$\begin{aligned} \frac{\partial \Gamma ^\alpha {}_{\beta \gamma }}{\partial g^{\mu \nu }}=-g^{\alpha \sigma }\left\rbrace \Gamma _{[\sigma \beta ](\mu }g_{\nu )\gamma }+\Gamma _{[\sigma \gamma ](\mu }g_{\nu )\beta }+\Gamma _{(\beta \gamma )(\mu }g_{\nu )\sigma }\right\lbrace , \end{aligned}$$

(286)

and

$$\begin{aligned} \frac{\partial {\mathcal Y}_\alpha }{\partial g^{\mu \nu }}=-\Gamma _{(\mu \nu )\alpha }, \end{aligned}$$

(287)

where we have used (263). Contracting (286), (287) with the Christoffel symbols and the metric tensor results in

$$\begin{aligned} g^{\sigma \gamma }\frac{\partial \Gamma ^\alpha {}_{\beta \gamma }}{\partial g^{\mu \nu }}\Gamma ^\beta {}_{\sigma \alpha }= & {} -2\Gamma ^\alpha {}_{\beta \mu }\Gamma ^\beta {}_{\nu \alpha },\end{aligned}$$

(288)

$$\begin{aligned} g^{\sigma \gamma }\frac{\partial {\mathcal Y}_\beta }{\partial g^{\mu \nu }}\Gamma ^\beta {}_{\sigma \gamma }= & {} -\Gamma _{(\mu \nu )\alpha }\Gamma ^\alpha ,\end{aligned}$$

(289)

$$\begin{aligned} g^{\sigma \gamma }\frac{\partial \Gamma ^\beta {}_{\sigma \gamma }}{\partial g^{\mu \nu }}{\mathcal Y}_\beta= & {} \Gamma _{(\mu \nu )\alpha }{\mathcal Y}^\alpha -\Gamma _{\alpha \mu \nu }{\mathcal Y}^\alpha -{\mathcal Y}_\mu {\mathcal Y}_\nu . \end{aligned}$$

(290)

Partial derivatives of the Christoffel symbols with respect to the metric derivatives are calculated from (285) with the help of (38). We get

$$\begin{aligned} \frac{\partial \Gamma ^\alpha {}_{\beta \gamma }}{\partial g^{\mu \nu }{}_{,\rho }}= & {} \frac{1}{2}\left[g^{\rho \alpha }g_{\beta (\mu }g_{\nu )\gamma }-\delta ^\rho _\gamma \delta ^\alpha _{(\mu }g_{\nu )\beta }-\delta ^\rho _\beta \delta ^\alpha _{(\mu }g_{\nu )\gamma }\right],\end{aligned}$$

(291)

$$\begin{aligned} \frac{\partial \mathcal{Y}_\beta }{\partial g^{\mu \nu }{}_{,\rho }}= & {} -\frac{1}{2}g_{\mu \nu }\delta ^\rho _\beta . \end{aligned}$$

(292)

Contracting (291), (292) with the Christoffel symbols and the metric tensor results in

$$\begin{aligned} g^{\sigma \gamma }\frac{\partial \Gamma ^\alpha {}_{\beta \gamma }}{\partial g^{\mu \nu }{}_{,\rho }}\Gamma ^\beta {}_{\sigma \alpha }= & {} -\frac{1}{2}\Gamma ^\rho {}_{\mu \nu },\end{aligned}$$

(293)

$$\begin{aligned} g^{\sigma \gamma }\frac{\partial {\mathcal Y}_\beta }{\partial g^{\mu \nu }{}_{,\rho }}\Gamma ^\beta {}_{\sigma \gamma }= & {} -\frac{1}{2} g_{\mu \nu }\Gamma ^\rho ,\end{aligned}$$

(294)

$$\begin{aligned} g^{\sigma \gamma }\frac{\partial \Gamma ^\beta {}_{\sigma \gamma }}{\partial g^{\mu \nu }{}_{,\rho }}{\mathcal Y}_\beta= & {} \frac{1}{2} g_{\mu \nu }{\mathcal Y}^\rho -\delta ^\rho {}_{(\mu }{\mathcal Y}_{\nu )}. \end{aligned}$$

(295)

Explicit expression for the variational derivative of the Einstein Lagrangian is

$$\begin{aligned} -16\pi \frac{\delta {{\mathcal L}}^\mathrm{\scriptscriptstyle E}}{\delta g^{\mu \nu }}= & {} \left(\frac{\partial \sqrt{-g}}{\partial g^{\mu \nu }}g^{\sigma \gamma }+\sqrt{-g}\frac{\partial g^{\sigma \gamma }}{\partial g^{\mu \nu }}\right)\left(\Gamma ^\alpha {}_{\beta \gamma }\Gamma ^\beta {}_{\sigma \alpha }-{\mathcal Y}_\beta \Gamma ^\beta {}_{\sigma \gamma }\right)\\\nonumber+ & {} \sqrt{-g}g^{\sigma \gamma }\left(2\frac{\partial \Gamma ^\alpha {}_{\beta \gamma }}{\partial g^{\mu \nu }}\Gamma ^\beta {}_{\sigma \alpha }-\frac{\partial {\mathcal Y}_\beta }{\partial g^{\mu \nu }}\Gamma ^\beta {}_{\sigma \gamma }-\frac{\partial \Gamma ^\beta {}_{\sigma \gamma }}{\partial g^{\mu \nu }}{\mathcal Y}_\beta \right)\\\nonumber- & {} \frac{\partial }{\partial x^\rho }\left[\sqrt{-g}g^{\sigma \gamma }\left(2\frac{\partial \Gamma ^\alpha {}_{\beta \gamma }}{\partial g^{\mu \nu }{}_{,\rho }}\Gamma ^\beta {}_{\sigma \alpha }-\frac{\partial {\mathcal Y}_\beta }{\partial g^{\mu \nu }{}_{,\rho }}\Gamma ^\beta {}_{\sigma \gamma }-\frac{\partial \Gamma ^\beta {}_{\sigma \gamma }}{\partial g^{\mu \nu }{}_{,\rho }}{\mathcal Y}_\beta \right)\right] \end{aligned}$$

(296)

Replacing the partial derivatives in (296) with the corresponding right sides of Eqs. (37), (30), (293)–(295) and taking the partial derivative with respect to spatial coordinates, yields

$$\begin{aligned} -16\pi \frac{\delta {{\mathcal L}}^\mathrm{\scriptscriptstyle E}}{\delta g^{\mu \nu }}=\sqrt{-g}\left(R_{\mu \nu }-\frac{1}{2} g_{\mu \nu }R\right), \end{aligned}$$

(297)

where we have used expressions (274), (276) for the Ricci tensor and Ricci scalar respectively. Substituting Eq. (297) to (284) yields

$$\begin{aligned} \frac{\delta {{\mathcal L}}^\mathrm{\scriptscriptstyle E}}{\delta \mathfrak {g}^{\mu \nu }}=-\frac{1}{16\pi }R_{\mu \nu }. \end{aligned}$$

(298)

1.2 Variational Derivatives of Dynamic Variables with Respect to the Metric Tensor

1.2.1 Variational Derivatives of Dark Matter Variables

The primary thermodynamic variable of dark matter is $\mu _\mathrm{m}$ defined in (125). Variational derivative from $\mu _\mathrm{m}$ is calculated directly from its definition and yields

$$\begin{aligned} \frac{\delta {\bar{\mu }}_\mathrm{m}}{\delta \bar{g}_{\mu \nu }}=\frac{1}{2}{\bar{\mu }}_\mathrm{m}\bar{u}^\mu \bar{u}^\nu . \end{aligned}$$

(299)

Variational derivative of pressure $\bar{p}_\mathrm{m}$ is obtained from thermodynamic relation (121a) by making use of the chain differentiation rule along with (299), that is

$$\begin{aligned} \frac{\delta \bar{p}_\mathrm{m}}{\delta \bar{g}_{\mu \nu }}=\frac{1}{2}\bar{\rho }_\mathrm{m}{\bar{\mu }}_\mathrm{m}\bar{u}^\mu \bar{u}^\nu . \end{aligned}$$

(300)

Variational derivative of the rest mass and energy density are obtained by making use of (299) along with equation of state that allows us to express partial derivatives of $\rho _\mathrm{m}$ and $\epsilon _\mathrm{m}$ in terms of the variational derivative for $\mu _\mathrm{m}$. More specifically,

$$\begin{aligned} \frac{\delta \bar{\rho }_\mathrm{m}}{\delta \bar{g}_{\mu \nu }}= & {} \frac{1}{2}\frac{c^2}{c^2_\mathrm{s}}\bar{\rho }_\mathrm{m}\bar{u}^\mu \bar{u}^\nu ,\end{aligned}$$

(301)

$$\begin{aligned} \frac{\delta {\bar{\epsilon }}_\mathrm{m}}{\delta \bar{g}_{\mu \nu }}= & {} \frac{1}{2}\frac{c^2}{c^2_\mathrm{s}}\bar{\rho }_\mathrm{m}{\bar{\mu }}_\mathrm{m}\bar{u}^\mu \bar{u}^\nu \end{aligned}$$

(302)

where the speed of sound appears explicitly. Variational derivatives from products and/or ratios of the thermodynamic quantities are calculated my applying the chain rule of differentiation and the above equations,

$$\begin{aligned} \frac{\delta \left(\bar{\rho }_\mathrm{m}{\bar{\mu }}_\mathrm{m}\right)}{\delta \bar{g}_{\mu \nu }}= & {} \frac{1}{2}\left(1+\frac{c^2}{c^2_\mathrm{s}}\right)\bar{\rho }_\mathrm{m}{\bar{\mu }}_\mathrm{m}\bar{u}^\mu \bar{u}^\nu ,\end{aligned}$$

(303)

$$\begin{aligned} \frac{\delta }{\delta \bar{g}_{\mu \nu }}\left(\frac{\bar{\rho }_\mathrm{m}}{{\bar{\mu }}_\mathrm{m}}\right)= & {} -\frac{1}{2}\left(1-\frac{c^2}{c^2_\mathrm{s}}\right)\frac{\bar{\rho }_\mathrm{m}}{{\bar{\mu }}_\mathrm{m}}\bar{u}^\mu \bar{u}^\nu ,\end{aligned}$$

(304)

$$\begin{aligned} \frac{\delta \left(\bar{p}_\mathrm{m}-{\bar{\epsilon }}_\mathrm{m}\right)}{\delta \bar{g}_{\mu \nu }}= & {} \frac{1}{2}\left(1-\frac{c^2}{c^2_\mathrm{s}}\right)\bar{\rho }_\mathrm{m}{\bar{\mu }}_\mathrm{m}\bar{u}^\mu \bar{u}^\nu . \end{aligned}$$

(305)

1.2.2 Variational Derivatives of Dark Energy Variables

The primary thermodynamic variable of dark energy is ${\bar{\mu }}_\mathrm{q}$ defined in (125). Variational derivative from ${\bar{\mu }}_\mathrm{q}$ is calculated directly from its definition,

$$\begin{aligned} \frac{\delta {\bar{\mu }}_\mathrm{q}}{\delta \bar{g}_{\mu \nu }}=\frac{1}{2}{\bar{\mu }}_\mathrm{q}\bar{u}^\mu \bar{u}^\nu . \end{aligned}$$

(306)

Variational derivative of the mass density $\bar{\rho }_\mathrm{q}$ of the dark energy “fluid” follows directly from $\bar{\rho }_\mathrm{q}={\bar{\mu }}_\mathrm{q}$, and reads

$$\begin{aligned} \frac{\delta \bar{\rho }_\mathrm{q}}{\delta \bar{g}_{\mu \nu }}=\frac{1}{2}\bar{\rho }_\mathrm{q}\bar{u}^\mu \bar{u}^\nu . \end{aligned}$$

(307)

Variational derivative of pressure $\bar{p}_\mathrm{q}$ is obtained from definition (138) along with (306), which yields

$$\begin{aligned} \frac{\delta \bar{p}_\mathrm{q}}{\delta \bar{g}_{\mu \nu }}=\frac{1}{2}\bar{\rho }_\mathrm{q}{\bar{\mu }}_\mathrm{q}\bar{u}^\mu \bar{u}^\nu . \end{aligned}$$

(308)

Variational derivative of energy density ${\bar{\epsilon }}_\mathrm{q}$ is obtained by making use of (306) along with (137). More specifically,

$$\begin{aligned} \frac{\delta {\bar{\epsilon }}_\mathrm{q}}{\delta \bar{g}_{\mu \nu }}=\frac{1}{2}\bar{\rho }_\mathrm{q}{\bar{\mu }}_\mathrm{q}\bar{u}^\mu \bar{u}^\nu . \end{aligned}$$

(309)

Variational derivatives from products and ratios of other quantities are calculated my making use of the chain rule of differentiation and the above equations

$$\begin{aligned} \frac{\delta \left(\bar{\rho }_\mathrm{q}{\bar{\mu }}_\mathrm{q}\right)}{\delta \bar{g}_{\mu \nu }}= & {} \bar{\rho }_\mathrm{q}{\bar{\mu }}_\mathrm{q}\bar{u}^\mu \bar{u}^\nu ,\end{aligned}$$

(310)

$$\begin{aligned} \frac{\delta }{\delta \bar{g}_{\mu \nu }}\left(\frac{\bar{\rho }_\mathrm{q}}{{\bar{\mu }}_\mathrm{q}}\right)= & {} 0,\end{aligned}$$

(311)

$$\begin{aligned} \frac{\delta \left(\bar{p}_\mathrm{q}-{\bar{\epsilon }}_\mathrm{q}\right)}{\delta \bar{g}_{\mu \nu }}= & {} 0. \end{aligned}$$

(312)

1.2.3 Variational Derivatives of Four-Velocity of the Hubble Flow

Variational derivatives from four-velocity of the fluid are derived from the definition (163) of the four-velocity given in terms of the potential $\bar{\Phi }$ or ${\bar{\Psi }}$ which are independent dynamic variables that do not depend on the metric tensor. Taking variational derivative from (163) and making use either (299) or (306) we obtain

$$\begin{aligned} \frac{\delta \bar{u}_\alpha }{\delta \bar{g}_{\mu \nu }}= & {} -\frac{1}{2}\bar{u}_\alpha \bar{u}^\mu \bar{u}^\nu ,\end{aligned}$$

(313)

$$\begin{aligned} \frac{\delta \bar{u}^\alpha }{\delta \bar{g}_{\mu \nu }}= & {} -\frac{1}{2}\bar{u}^\alpha \bar{u}^\mu \bar{u}^\nu -\bar{g}^{\alpha (\mu }\bar{u}^{\nu )},\end{aligned}$$

(314)

$$\begin{aligned} \frac{\delta \left(\bar{u}^\alpha \phi _\alpha \right)}{\delta \bar{g}_{\mu \nu }}= & {} -\phi ^{(\mu }\bar{u}^{\nu )}-\frac{1}{2}\bar{u}^\mu \bar{u}^\nu \left(\bar{u}^\alpha \phi _\alpha \right), \end{aligned}$$

(315)

where Eq. (315) accounts for the fact that $\phi _\alpha$ is an independent variable that does not depend on the metric tensor.

1.2.4 Variational Derivatives of the Metric Tensor Perturbations

Variational derivatives from the metric tensor perturbations $l^{\alpha \beta }$ are determined by taking into account that $l^{\alpha \beta }=\mathfrak {h}^{\alpha \beta }/\sqrt{\bar{g}}$ and $\mathfrak {h}^{\alpha \beta }$ is an independent dynamic variable which does not depend on the metric tensor. Therefore, its variational derivative is nil, and we have

$$\begin{aligned} \frac{\delta l^{\alpha \beta }}{\delta \bar{g}_{\mu \nu }}=\frac{\delta }{\delta \bar{g}_{\mu \nu }}\left( \frac{\mathfrak {h}^{\alpha \beta }}{\sqrt{\bar{g}}}\right) =\mathfrak {h}^{\alpha \beta }\frac{\delta }{\delta \bar{g}_{\mu \nu }}\left( \frac{1}{\sqrt{\bar{g}}}\right) =-\frac{1}{2}l^{\alpha \beta }\bar{g}^{\mu \nu }. \end{aligned}$$

(316)

Other variational derivatives are derived by making use of tensor operations of rising and lowering indices with the help of $\bar{g}_{\alpha \beta }$ and applying from (316). It gives

$$\begin{aligned} \frac{\delta l_{\alpha \beta }}{\delta \bar{g}_{\mu \nu }}= & {} -\frac{1}{2}l_{\alpha \beta }\bar{g}^{\mu \nu }+2l_\alpha {}^{(\mu }\delta ^{\nu )}_\beta ,\end{aligned}$$

(317)

$$\begin{aligned} \frac{\delta l}{\delta \bar{g}_{\mu \nu }}= & {} l^{\mu \nu }-\frac{1}{2}l\bar{g}^{\mu \nu },\end{aligned}$$

(318)

$$\begin{aligned} \frac{\delta \mathfrak {q}}{\delta \bar{g}_{\mu \nu }}= & {} -\mathfrak {q}\left(\bar{u}^\mu \bar{u}^\nu +\frac{1}{2}\bar{g}^{\mu \nu }\right)+\frac{1}{2}\left(l^{\mu \nu }+l\bar{u}^\mu \bar{u}^\nu \right),\end{aligned}$$

(319)

$$\begin{aligned} \frac{\delta }{\delta \bar{g}_{\mu \nu }}\left(l^{\alpha \beta }l_{\alpha \beta }-\frac{l^2}{2}\right)= & {} 2l^{\alpha (\mu }l^{\nu )}{}_\alpha -ll^{\mu \nu }-\bar{g}^{\mu \nu }\left(l^{\alpha \beta }l_{\alpha \beta }-\frac{l^2}{2}\right). \end{aligned}$$

(320)

1.3 Variational Derivatives with Respect to Matter Variables

1.3.1 Variational Derivatives of Dark Matter Variables

The dark matter variables do not depend on the Clebsch potential $\bar{\Phi }$ directly but merely on its first derivative $\bar{\Phi }_\alpha$. Therefore, any variational derivative of dark matter variable, say, $\mathcal{Q}=\mathcal{Q}(\bar{\Phi }_\alpha )$, is reduced to a total divergence

$$\begin{aligned} \frac{\delta \mathcal{Q}}{\delta \bar{\Phi }}=-\frac{\partial }{\partial x^\alpha }\frac{\partial \mathcal{Q}}{\partial \bar{\Phi }_\alpha }. \end{aligned}$$

(321)

We present a short summary of the partial derivatives with respect to $\bar{\Phi }_\alpha$.

$$\begin{aligned} \frac{\partial {\bar{\mu }}_\mathrm{m}}{\partial \bar{\Phi }_{\alpha }}= & {} \bar{u}^\alpha ,\end{aligned}$$

(322)

$$\begin{aligned} \frac{\partial \bar{p}_\mathrm{m}}{\partial \bar{\Phi }_{\alpha }}= & {} \bar{\rho }_\mathrm{m}\bar{u}^\alpha ,\end{aligned}$$

(323)

$$\begin{aligned} \frac{\partial \bar{\rho }_\mathrm{m}}{\partial \bar{\Phi }_{\alpha }}= & {} \frac{c^2}{c^2_\mathrm{s}}\frac{\bar{\rho }_\mathrm{m}}{{\bar{\mu }}_\mathrm{m}}\bar{u}^\alpha ,\end{aligned}$$

(324)

$$\begin{aligned} \frac{\partial {\bar{\epsilon }}_\mathrm{m}}{\partial \bar{\Phi }_{\alpha }}= & {} \frac{c^2}{c^2_\mathrm{s}}\bar{\rho }_\mathrm{m}\bar{u}^\alpha ,\end{aligned}$$

(325)

$$\begin{aligned} \frac{\partial \left(\bar{\rho }_\mathrm{m}{\bar{\mu }}_\mathrm{m}\right)}{\partial \bar{\Phi }_{\alpha }}= & {} \left(1+\frac{c^2}{c^2_\mathrm{s}}\right)\bar{\rho }_\mathrm{m}\bar{u}^\alpha ,\end{aligned}$$

(326)

$$\begin{aligned} \frac{\partial }{\partial \bar{\Phi }_{\alpha }}\left(\frac{\bar{\rho }_\mathrm{m}}{{\bar{\mu }}_\mathrm{m}}\right)= & {} -\left(1-\frac{c^2}{c^2_\mathrm{s}}\right)\frac{\bar{\rho }_\mathrm{m}}{{\bar{\mu }}_\mathrm{m}^2}\bar{u}^\alpha ,\end{aligned}$$

(327)

$$\begin{aligned} \frac{\partial \left(\bar{p}_\mathrm{m}-{\bar{\epsilon }}_\mathrm{m}\right)}{\partial \bar{\Phi }_{\alpha }}= & {} +\left(1-\frac{c^2}{c^2_\mathrm{s}}\right)\bar{\rho }_\mathrm{m}\bar{u}^\alpha . \end{aligned}$$

(328)

Partial derivatives of four velocity

$$\begin{aligned} \frac{\partial \bar{u}_\alpha }{\partial \bar{\Phi }_{\beta }}=-\frac{\bar{P}_\alpha {}^\beta }{{\bar{\mu }}_\mathrm{m}},\qquad \frac{\partial \bar{u}^\alpha }{\partial \bar{\Phi }_{\beta }}=-\frac{\bar{P}^{\alpha \beta }}{{\bar{\mu }}_\mathrm{m}}. \end{aligned}$$

(329)

It allows us to deduce, for example,

$$\begin{aligned} \frac{\partial \left(\bar{u}^\alpha \phi _\alpha \right)}{\partial \bar{\Phi }_{\beta }}= & {} -\frac{1}{{\bar{\mu }}_\mathrm{m}}\bar{P}^{\alpha \beta }\phi _\beta ,\end{aligned}$$

(330)

$$\begin{aligned} \frac{\partial \mathfrak {q}}{\partial \bar{\Phi }_{\alpha }}= & {} -\frac{2}{{\bar{\mu }}_\mathrm{m}}\bar{P}^\alpha {}_\mu l^{\mu \nu }\bar{u}_\nu . \end{aligned}$$

(331)

1.3.2 Variational Derivatives of Dark Energy Variables

The dark energy variables depend on both the scalar potential ${\bar{\Psi }}$ and its first derivative ${\bar{\Psi }}_\alpha$ in the most generic situation. This is because there is a potential of the scalar field $W(\bar{\Phi })$ that is absent in case of the dark matter. Therefore, variational derivative of the dark energy variable, say, $\mathcal{a}=\mathcal{A}({\bar{\Psi }},{\bar{\Psi }}_\alpha )$, is

$$\begin{aligned} \frac{\delta \mathcal{A}}{\delta {\bar{\Psi }}}=\frac{\partial \mathcal{A}}{\partial {\bar{\Psi }}}-\frac{\partial }{\partial x^\alpha }\frac{\partial \mathcal{A}}{\partial {\bar{\Psi }}_\alpha }. \end{aligned}$$

(332)

Partial derivatives $\partial \mathcal{A}/\partial {\bar{\Psi }}=(\partial \mathcal{A}/\partial W)(\partial W/\partial {\bar{\Psi }}$, and their particular form depends on the shape of the potential W. As for the partial derivatives with respect to the derivatives of the field, they can be calculated explicitly for each variable, and we present a short summary of these partial derivatives below. More specifically,

$$\begin{aligned} \frac{\partial {\bar{\mu }}_\mathrm{q}}{\partial {\bar{\Psi }}_{\alpha }}= & {} \bar{u}^\alpha ,\end{aligned}$$

(333)

$$\begin{aligned} \frac{\partial \bar{p}_\mathrm{q}}{\partial {\bar{\Psi }}_{\alpha }}= & {} \bar{\rho }_\mathrm{q}\bar{u}^\alpha ,\end{aligned}$$

(334)

$$\begin{aligned} \frac{\partial \bar{\rho }_\mathrm{q}}{\partial {\bar{\Psi }}_{\alpha }}= & {} \bar{u}^\alpha ,\end{aligned}$$

(335)

$$\begin{aligned} \frac{\partial {\bar{\epsilon }}_\mathrm{q}}{\partial {\bar{\Psi }}_{\alpha }}= & {} \bar{\rho }_\mathrm{q}\bar{u}^\alpha ,\end{aligned}$$

(336)

$$\begin{aligned} \frac{\partial \left(\bar{\rho }_\mathrm{q}{\bar{\mu }}_\mathrm{q}\right)}{\partial {\bar{\Psi }}_{\alpha }}= & {} 2\bar{\rho }_\mathrm{q}\bar{u}^\alpha ,\end{aligned}$$

(337)

$$\begin{aligned} \frac{\partial }{\partial {\bar{\Psi }}_{\alpha }}\left(\frac{\bar{\rho }_\mathrm{q}}{{\bar{\mu }}_\mathrm{q}}\right)= & {} 0,\end{aligned}$$

(338)

$$\begin{aligned} \frac{\partial \left(\bar{p}_\mathrm{q}-{\bar{\epsilon }}_\mathrm{q}\right)}{\partial {\bar{\Psi }}_{\alpha }}= & {} 0. \end{aligned}$$

(339)

Partial derivatives of four velocity

$$\begin{aligned} \frac{\partial \bar{u}_\alpha }{\partial {\bar{\Psi }}_{\beta }}=-\frac{\bar{P}_\alpha {}^\beta }{{\bar{\mu }}_\mathrm{q}},\qquad \frac{\partial \bar{u}^\alpha }{\partial {\bar{\Psi }}_{\beta }}=-\frac{\bar{P}^{\alpha \beta }}{{\bar{\mu }}_\mathrm{q}}. \end{aligned}$$

(340)

It allows us to deduce, for example,

$$\begin{aligned} \frac{\partial \left(\bar{u}^\alpha \psi _\alpha \right)}{\partial {\bar{\Psi }}_{\beta }}= & {} -\frac{1}{{\bar{\mu }}_\mathrm{q}}\bar{P}^{\alpha \beta }\psi _\beta ,\end{aligned}$$

(341)

$$\begin{aligned} \frac{\partial \mathfrak {q}}{\partial {\bar{\Psi }}_{\alpha }}= & {} -\frac{2}{{\bar{\mu }}_\mathrm{q}}\bar{P}^\alpha {}_\mu l^{\mu \nu }\bar{u}_\nu . \end{aligned}$$

(342)