The rotation problem

Abstract

Any reasonable form of quantum gravity can explain (by phase interference) why, on a large scale, inertial frames seem not to rotate relative to the average matter distribution in the universe without the need for absolute space, finely tuned initial conditions, or without giving up independent degrees of freedom for the gravitational field. A simple saddlepoint approximation to a path-integral calculation for a perfect fluid cosmology shows that only cosmologies with an average present relative rotation rate smaller than about $T^*H^2 \approx 10^{-71}$ radians per year could contribute significantly to a measurement of relative rotation rate in our universe, where $T^*\approx 10^{-51}$ years is the Planck time and $H \approx 10^{-10}$$\hbox {yr}^{-1}$ is the present value of the Hubble parameter. A more detailed calculation (taking into account that with vorticity flow lines are not normal to surfaces of constant global time, and approximating the action to second order in the mean square vorticity) shows that the saddlepoint at zero vorticity is isolated and that only cosmologies with an average present relative rotation rate smaller than about $T^*H^2 a_1^{1/2} \approx 10^{-73}$ radians per year could contribute significantly to a measurement of relative rotation rate in our universe, where $a_1 \approx 10^{-4}$ is the value of the cosmological scale factor at the time when matter became more significant than radiation in the cosmological expansion. Including inflation with 60 e-foldings in the calculation of the action further restricts relative rotation rate to be smaller than $\approx 10^{-74}$ radians per year. These calculations are consistent with measurements indicating a present relative rotation rate less than about $10^{-20}$ radians per year. The observed lack of relative rotation may be evidence for the existence of quantum gravity.

Appendices

Appendix A: Amplitude for measuring a rotation of the universe

The amplitude for measuring a particular value for some quantity is equal to the amplitude for measuring that value given a particular 4-geometry times the amplitude for that 4-geometry, and then we sum over all 4-geometries.

For example, following [45], the amplitude for the 3-geometry and matter field to be fixed at specified values on two spacelike hypersurfaces is

$$\begin{aligned} \langle ^{(3)}{\mathcal {G}}_f,\phi _f|^{(3)}{\mathcal {G}}_i,\phi _i\rangle = \int \psi [^{(4)}{\mathcal {G}},\phi ] \, {\mathcal {D}} {^{(4)}{\mathcal {G}}} \, {\mathcal {D}} \phi \,, \end{aligned}$$

(8)

where the integral is over all 4-geometries and field configurations that match the given values on the two spacelike hypersurfaces, and

$$\begin{aligned} \psi [^{(4)}{\mathcal {G}},\phi ] \equiv \exp {(i I[^{(4)}{\mathcal {G}},\phi ]/\hbar )} \end{aligned}$$

(9)

is the contribution of the 4-geometry $^{(4)}{\mathcal {G}}$ and matter field $\phi$ on that 4-geometry to the path integral, where $I[^{(4)}{\mathcal {G}},\phi ]$ is the action. The proper time between the two hypersurfaces is not specified. A correct theory of quantum gravity would be necessary to specify the measures ${\mathcal {D}} {^{(4)}{\mathcal {G}}}$ and ${\mathcal {D}} \phi$, but that will not be necessary for the purposes here. Hartle and Hawking [45] restricted the integration in (8) to compact (closed) 4-geometries, but (8) can be applied to open 4-geometries if that is done carefully.

Equation (8) is a path integral. In this case, the “path” is the sequence of 3-geometries that form the 4-geometry ${^{(4)}{\mathcal {G}}}$. Thus, each 4-geometry is one “path.” The space in which these paths exist is often referred to as superspace, e.g. [65]. As pointed out by Hajicek [28], there are two kinds of path integrals: those in which the time is specified at the endpoints, and those in which the time is not specified. The path integral in (8) is the latter. References [28, 53] consider refinements to the path integral in (8), but such refinements are not necessary here.

Because of diffeomorphisms, a given 4-geometry can be specified by different metrics that are connected by coordinate transformations. This makes it difficult to avoid duplications when making path integral calculations. We avoid that difficulty here by considering only simple models.

Let $\psi _i(^{(3)}{\mathcal {G}}_i, \phi _i)$ be the amplitude that the 3-geometry was $^{(3)}{\mathcal {G}}_i$ on some initial space-like hypersurface and that the matter fields on that 3-geometry were $\phi _i$. Let $\psi _f(^{(3)}{\mathcal {G}}_f, \phi _f)$ be the amplitude that the 3-geometry is $^{(3)}{\mathcal {G}}_f$ on some final space-like hypersurface and that the matter fields on that 3-geometry are $\phi _f$. Then, we have

$$\begin{aligned} \psi _f(^{(3)}{\mathcal {G}}_f, \phi _f) = \int \langle ^{(3)}{\mathcal {G}}_f,\phi _f|^{(3)}{\mathcal {G}}_i,\phi _i\rangle \psi _i(^{(3)}{\mathcal {G}}_i, \phi _i) \, {\mathcal {D}} {^{(3)}{\mathcal {G}}_i} \, {\mathcal {D}} \phi _i \,. \end{aligned}$$

(10)

The condition that there are not finely tuned initial conditions is equivalent to $\psi _i(^{(3)}{\mathcal {G}}_i, \phi _i)$ being a broad wave function.

Substituting (8) into (10) gives

$$\begin{aligned} \psi _f(^{(3)}{\mathcal {G}}_f, \phi _f) = \int \int \psi [^{(4)}{\mathcal {G}},\phi ] \, {\mathcal {D}} {^{(4)}{\mathcal {G}}} \, {\mathcal {D}} \phi \, \psi _i(^{(3)}{\mathcal {G}}_i, \phi _i) \, {\mathcal {D}} {^{(3)}{\mathcal {G}}_i} \, {\mathcal {D}} \phi _i \,. \end{aligned}$$

(11)

Although in (11), the integration is over all possible 4-geometries, not just classical 4-geometries, the main contribution to the integral (in most cases) comes from classical 4-geometries, e.g. [38, 53]. Thus, we shall now restrict (11) to be an integration over classical 4-geometries. This is appropriate for our purposes, in any case, since we are trying to explain why we do not measure relative rotation of matter and inertial frames in what appears to be a classical universe.

In principle, the idea is very simple. Any measurement to determine the inertial frame will give a result that depends on the 4-geometry. If several 4-geometries contribute significantly to an amplitude, such as in (11), then any measurement to determine an inertial frame might give the inertial frame corresponding to any one of those 4-geometries. However, the probability for the result being a particular inertial frame will depend on the contribution of the corresponding 4-geometry to calculations such as that in (11).

In this calculation, we consider 4-geometries characterized by a parameter ${\langle \omega _3\rangle }$ which we take to be the rms vorticity on the final space-like hypersurface. Thus, we can rewrite (11) for our purposes as

$$\begin{aligned} \psi _f(^{(3)}{\mathcal {G}}_f, \phi _f) = \int \int _{-\infty }^{\infty } \psi _i(^{(3)}{\mathcal {G}}_i, \phi _i) \psi [^{(4)}{\mathcal {G}},\phi ;\langle \omega _3\rangle ] \, \mathrm {d}{\langle \omega _3\rangle } \, {\mathcal {D}} {^{(3)}{\mathcal {G}}_i} \, {\mathcal {D}} \phi _i \,. \end{aligned}$$

(12)

The integral in (12) is still a path integral. In this case, each value of $\langle \omega _3\rangle$ specifies one “path”, in that it specifies one 4-geometry, and that specifies one sequence of 3-geometries. The space of “paths” in this case is often referred to as a mini-superspace because it is restricted to a much smaller space of 4-geometries. The parameter ${\langle \omega _3\rangle }$, classically determined by initial conditions on the 4-geometry, represents an independent degree-of-freedom of the gravitational field.

Actually, taking the ${\langle \omega _3\rangle }$ integration from $-\infty$ to $\infty$ in (12) is not physically realistic, and might lead to problems if the infinite endpoints contribute significantly to the integration. The largest relative rotation that could possibly be considered without having a theory of quantum gravity would be one rotation of the universe in a Planck time. This would correspond to taking the maximum value of ${\langle \omega _3\rangle }$ to be the reciprocal of the Planck time, $T^*$, or ${\omega }_{\text{ max }} \approx 10^{44} \text{ sec }^{-1}$. Thus, we can rewrite (12) as

$$\begin{aligned} \psi _f(^{(3)}{\mathcal {G}}_f, \phi _f) \approx \int \int _{-{\omega }_{\text{ max }}}^{{\omega }_{\text{ max }}} \psi _i(^{(3)}{\mathcal {G}}_i, \phi _i) \psi [^{(4)}{\mathcal {G}},\phi ;\langle \omega _3\rangle ] \, \mathrm {d}{\langle \omega _3\rangle } \, {\mathcal {D}} {^{(3)}{\mathcal {G}}_i} \, {\mathcal {D}} \phi _i \,. \end{aligned}$$

(13)

We anticipate that the properties of $\psi [^{(4)}{\mathcal {G}},\phi ;\langle \omega _3\rangle ]$ will dominate the integral in (13), so we shall start with

$$\begin{aligned} \psi [^{(4)}{\mathcal {G}},\phi ;\langle \omega _3\rangle ] \approx e^{i I(\langle \omega _3\rangle )/\hbar }, \end{aligned}$$

(14)

where $I(\langle \omega _3\rangle )$ is the action.

Either a stationary-phase path or a steepest-descent path could be used when making the saddlepoint approximation [3, 19, 39], but here, we use a stationary-phase path. Halliwell [29] gives an example of a more detailed path-integral calculation of quantum gravity.

Appendix B: Action for a perfect fluid

We can take the action in (14) to be

$$\begin{aligned} I = \int (-g^{(4)})^{1/2} L \mathrm {d}^4x + \frac{1}{8\pi } \int (g^{(3)})^{1/2} K \mathrm {d}^3x, \end{aligned}$$

(15)

where

$$\begin{aligned} L= L_{\text{ geom }} + L_{\text{ matter }} \end{aligned}$$

(16)

is the Lagrangian, and the surface term is necessary to insure consistency if the action integral is broken into parts [47, 84]. The quantity

$$\begin{aligned} K = g^{(3)ij} K_{ij} = - \frac{1}{2} g^{(3)ij} \frac{\partial g_{ij}^{(3)}}{\partial t} \end{aligned}$$

(17)

is the trace of the extrinsic curvature, where $g_{ij}^{(3)}$ is the 3-metric. In this example, we take the Lagrangian for the geometry as

$$\begin{aligned} L_{\text{ geom }} = \frac{R^{(4)}-2\varLambda }{16\pi }, \end{aligned}$$

(18)

where $R^{(4)}$ is the four-dimensional scalar curvature and $\varLambda$ is the cosmological constant.

For a perfect fluid, the energy momentum tensor is

$$\begin{aligned} T^{\mu \nu } = (\rho + p) u^{\mu } u^{\nu } + p g^{\mu \nu }, \end{aligned}$$

(19)

where p is the pressure, $\rho$ is the density, and u is the 4-velocity. For solutions to Einstein’s field equations⁶

$$\begin{aligned} R^{\mu \nu } = 8\pi \left( T^{\mu \nu } - \frac{1}{2} g^{\mu \nu } T\right) + \varLambda g^{\mu \nu } \end{aligned}$$

(20)

for a perfect fluid, (18) becomes

$$\begin{aligned} L_{\text{ geom }} = \frac{1}{2} \rho - \frac{3}{2} p + \frac{\varLambda }{8\pi }, \end{aligned}$$

(21)

and we can take the Lagrangian for the matter as

$$\begin{aligned} L_{\text{ matter }} = \rho + \alpha (p-\rho ), \end{aligned}$$

(22)

where $\alpha$ is a constant, and we can take $\alpha = 0$ [76], $\alpha = 1$ [60, 77], or $\alpha = \frac{3}{2}$ (from combining (19) with [65, eq. 21.33a]). However, as we shall see, the final result is insensitive to the exact form of the Lagrangian. Substituting (21) and (22) into (16) gives

$$\begin{aligned} {L} = \left( \alpha - \frac{3}{2})(p-\rho \right) + \frac{\varLambda }{8\pi }, \end{aligned}$$

(23)

For some cosmological models, it is possible to represent K as a time-integral of some quantity. For example, for the exact cosmological models considered by [16], the effect of K on the action can be represented as an integral over a 4-volume, allowing us to combine the two terms in (15). In that case, the surface term adds the following term to an effective Lagrangian:

$$\begin{aligned} \frac{3}{2} (p-\rho ) - \frac{3\varLambda }{8\pi } - \frac{3 a_0^2 + q_0^2}{4\pi X^2}, \end{aligned}$$

(24)

where $a_0$, $q_0$, and X are parameters of their model, and $a_0$ in (24) has nothing to do with the cosmological scale factor $a_0$ used here. If so, then we would have

$$\begin{aligned} I = \int (-g^{(4)})^{1/2} {\tilde{L}} \mathrm {d}^4x, \end{aligned}$$

(25)

where ${\tilde{L}}$ can be considered to be an effective Lagrangian, which in the case of (23) and (24) gives

$$\begin{aligned} {\tilde{L}} = \alpha (p-\rho ) - \frac{\varLambda }{4\pi }. \end{aligned}$$

(26)

More generally, we can assume the effective Lagrangian to take the form

$$\begin{aligned} {\tilde{L}} = \alpha _1 p + \alpha _2 \rho + \alpha _3 \varLambda \text{, } \end{aligned}$$

(27)

where $\alpha _1$, $\alpha _2$, and $\alpha _3$ are dimensionless constants of order unity.

Table 2 shows that with vorticity, the density $\rho$ depends on the scale factor along a flow line rather than on the cosmological scale factor a. To take this into account, we can write

$$\begin{aligned} {\tilde{L}}(\ell ) \approx {\tilde{L}}_0 \left[ 1+ f_L(a) \delta + f_{LL}(a) \delta ^2 \right] = H_3^2 F (a) \left[ 1+ f_L(a) \delta + f_{LL}(a) \delta ^2 \right] , \end{aligned}$$

(28)

where $\delta$ [defined in (41)] is proportional to the square of the present local vorticity,

$$\begin{aligned} F(a)= & {} \frac{3}{8\pi } (\alpha _1 w + \alpha _2) \frac{a_1}{a_2} \left( \frac{a_2}{a}\right) ^4 \qquad \text{ for } a \le a_1, \nonumber \\ F(a)= & {} \frac{3}{8\pi } \alpha _2 \left( \frac{a_2}{a}\right) ^3 \,\,\qquad \qquad \qquad \qquad \text{ for } a_1 \le a \le a_2, \nonumber \\ F(a)= & {} 3\alpha _3 \,\,\quad \qquad \qquad \qquad \qquad \qquad \qquad \text{ for } a_2 \le a \le a_3, \end{aligned}$$

(29)

$$\begin{aligned} f_{L}(a)= & {} f_\ell (a_1) + 3 f_\ell (a_2) - 4 f_\ell (a) \quad \text{ for } a \le a_1, \nonumber \\ f_L(a)= & {} 3 f_\ell (a_2) - 3 f_\ell (a) \,\,\,\,\,\quad \qquad \qquad \text{ for } a_1 \le a \le a_2, \nonumber \\ f_L(a)= & {} 0 \,\,\,\qquad \qquad \qquad \qquad \qquad \qquad \qquad \text{ for } a_2 \le a \le a_3, \end{aligned}$$

(30)

$f_\ell (a)$ is defined in (61),

$$\begin{aligned} f_{LL}(a)= & {} f_{\ell \ell }(a_1) + 3 f_\ell (a_2)^2+ 3 f_{\ell \ell }(a_2) + 10 f_\ell (a)^2- 4 f_{\ell \ell }(a) \,\, \nonumber \\ \qquad+ & {} \,3 f_\ell (a_1)f_\ell (a_2) -4 f_\ell (a_1)f_\ell (a) -12 f_\ell (a_2)f_\ell (a) \quad \qquad \qquad \qquad \text{ for } a \le a_1, \nonumber \\ f_{LL}(a)= & {} -9 f_\ell (a_2) f_\ell (a) + 3 f_\ell (a_2)^2+ 3 f_{\ell \ell }(a_2) + 6 f_\ell (a)^2- 3 f_{\ell \ell }(a) \quad \text{ for } a_1 \le a \le a_2, \nonumber \\ f_{LL}(a)= & {} 0 \quad \text{ for } a_2 \le a \le a_3,\nonumber \\ \end{aligned}$$

(31)

and $f_{\ell \ell }(a)$ is defined in (62). Using (64) in (30) gives

$$\begin{aligned} f_{L}(a)= & {} 1 + 3 f_\ell (a_2) - 4 \left( \frac{a}{a_1}\right) ^{2} \text{ for } a \le a_1, \nonumber \\ f_L(a)= & {} -216\sqrt{\frac{a_1}{a_2}} +138\frac{a_1}{a_2} +36\frac{a_1}{a_2} \ln {\frac{a_1}{a_2}} \nonumber \\&\quad +\,216\sqrt{\frac{a_1}{a}} -138\frac{a_1}{a} -36 \frac{a_1}{a} \ln {\frac{a_1}{a}} \text{ for } a_1 \le a \le a_2, \nonumber \\ f_L(a)= & {} 0 \text{ for } a_2 \le a \le a_3. \end{aligned}$$

(32)

Appendix C: Inflation, radiation, matter, and dark-energy eras

There are four cosmological eras to consider. In the very early universe is the inflation era, which ran from roughly $t_i=10^{-36}$ seconds to about $t_0=10^{-34}$ seconds [59, p. 109] after the initial singularity. During the inflation era the universe expanded by N e-foldings. We have that $a_0/a_i = e^N$ and $N=H_{0 \text{ inflation }} (t_0-t_i)\approx H_{0 \text{ inflation }} t_0$. The Planck Collaboration [68] estimates that $50\le N\le 60$. That the cosmological scale factor a varied as $t^{1/2}$ during the radiation era gives the value at the beginning of the radiation era and the end of inflation as about $a_0 \approx 4 \times 10^{-27}$. The main effect of inflation on the calculation of the action is through the parameter $\beta = H_3 t_2 (a_0 /a_1) e^N/N \approx 1.5\times 10^{-23} e^N/N$, where $t_2\approx 9 \times 10^9$ years is the time of matter and dark energy equality, and the other parameters are defined below.

After inflation, radiation dominates over matter to determine the density $\rho$ in the radical in (83). During the radiation era, a varies as $t^{1/2}$. When the cosmological scale factor a reaches a certain size (which we define as $a_1$), matter begins to dominate over radiation to determine the density $\rho$. During the matter era, a varies as $t^{2/3}$. When the cosmological scale factor a gets even larger (to a size we define as $a_2$), the density of matter has fallen low enough that the cosmological constant $\varLambda$ begins to dominate over the density term in (83).

For an equation of state, we take

$$\begin{aligned} p=w\rho , \end{aligned}$$

(33)

where $w=1/3$ in the radiation-dominated era, and $w=0$ in the matter-dominated era. The variation of density $\rho$ with cosmological scale factor a is given by [15, Table 6.1]

$$\begin{aligned} \rho = \rho _{1} (a/a_{1})^{-3(1+w)}, \end{aligned}$$

(34)

where $\rho _{1}$ is the value of $\rho$ at the boundary between the radiation era and the matter era where $a=a_{1}$. Table 2 summarizes some of this.

We can take for the present fraction of radiation density $\varOmega _{\text{ rad }}=5.4\times 10^{-5}$ [59, p. 78]. Otherwise, we take [68]

$$\begin{aligned} a_1 = \frac{\varOmega _{\text{ rad }}}{\varOmega _{\text{ mat }}} \approx \frac{5.4\times 10^{-5}}{.3089} \approx 1.7 \times 10^{-4} \text{, } \end{aligned}$$

(35)

and

$$\begin{aligned} a_2 = \left( \frac{\varOmega _{\text{ mat }}}{\varOmega _\varLambda }\right) ^{1/3} \approx \left( \frac{.3089}{.6911}\right) ^{1/3} \approx 0.76, \end{aligned}$$

(36)

where $\varOmega _{\text{ mat }}$ is the present fraction of matter density (including dark matter), and $\varOmega _\varLambda$ is the present fraction of dark energy. From the present value of the Hubble parameter [68]

$$\begin{aligned} H_3{=}67.74 \text{ km } \hbox {s}^{{-}1} \hbox {Mpc}^{-1} {=}2.195 \times 10^{-18} {\hbox {sec}^{-1}} {=}7.323 \times 10^{-29} {\hbox {cm}^{{-}1}} {=}6.928 \times 10^{{-}11} {\hbox {yr}^{-1}}, \end{aligned}$$

(37)

we can calculate the critical density, and combining with $\varOmega _\varLambda$, we get the value of the density of matter and dark energy when they were equal, which is

$$\begin{aligned} \rho _2 = \frac{\varLambda }{8\pi } \approx 4.4\times 10^{-58} \text{ cm }^{-2}, \end{aligned}$$

(38)

which gives

$$\begin{aligned} \varLambda \approx 1.1 \times 10^{-56} \text{ cm }^{-2}. \end{aligned}$$

(39)

We also have

$$\begin{aligned} \hbar \rightarrow \frac{\hbar G}{c^2} = L^{*2} \approx 2.616 \times 10^{-66} \text{ cm }^{2}. \end{aligned}$$

(40)

Appedix D: Relative rotation

It is necessary to estimate the effect of the vorticity $\omega$ on the total rotation angle $\theta$. Because we expect only very small values of vorticity to contribute significantly to a path integral calculation of vorticity, it is useful to choose a non-dimensional parameter that is small whenever the vorticity is small. In addition, because the action is an even function of the vorticity, and therefore is a function of the square of vorticity, it is useful for that parameter to be proportional to the square of the vorticity. It turns out to be useful to take that parameter as

$$\begin{aligned} \delta \equiv \frac{1}{6} \left( \frac{\omega _3}{H_3}\right) ^2 \left( \frac{a_3}{a_1}\right) \left( \frac{a_3}{a_2}\right) ^{3} \text{, } \end{aligned}$$

(41)

where $\omega _3$ is the present value of the local vorticity, $H_3$ is the present value of the Hubble parameter, $a_1$ is the value of the cosmological scale factor at the time when the matter density became equal to the radiation density, $a_2$ is the value of cosmological scale factor at the time when the cosmological constant surpassed the matter density in the Friedmann equation, and $a_3=1$ is the present value of the cosmological scale factor.

We start with

$$\begin{aligned} \theta = \int _0^{t} \omega \, \mathrm {d} t = \int _{a_i}^{a} \frac{\omega }{\dot{a}} \, \mathrm {d} a = \int _{a_i}^{a} \frac{\omega }{aH_0} \left[ 1+f_H(a) \delta +f_{HH}(a) \delta ^2\right] \, \mathrm {d} a, \end{aligned}$$

(42)

where we have used (91) for $1/\dot{a}$. Using table 2 for $\omega$ gives

$$\begin{aligned} \omega= & {} H_3\sqrt{6\delta } a_1^{-1/2} a_2^{3/2} (\ell _1/a_1)^{-1}(\ell _3/a_3)^{2}a^{-1}(\ell /a)^{-1} \text{ for } a \le a_1, \text{ and } \nonumber \\ \omega= & {} H_3\sqrt{6\delta } a_1^{1/2} a_2^{3/2} (\ell _3/a_3)^{2}a^{-2}(\ell /a)^{-2} \text{ for } a \ge a_1, \end{aligned}$$

(43)

where $\delta$ is defined in (41). Using (60) to expand $\ell _1/a_1$, $\ell _3/a_3$, and $\ell /a$ to second order in $\delta$ allows us to write (43) as

$$\begin{aligned} \omega= & {} \omega _3 \frac{a_3}{a_1}\frac{a_3}{a}[1+g_\omega (a) \delta + g_{\omega \omega }(a) \delta ^2] \text{ for } a \le a_1, \text{ and } \nonumber \\ \omega= & {} \omega _3 \left( \frac{a_3}{a}\right) ^2 [1+g_\omega (a) \delta + g_{\omega \omega }(a) \delta ^2] \text{ for } a \ge a_1, \end{aligned}$$

(44)

where

$$\begin{aligned} g_{\omega }(a)= & {} 2f_\ell (a_3)-f_\ell (a_1)-f_\ell (a) \, \text{ for } a \le a_1 \nonumber \\ g_{\omega }(a)= & {} 2f_\ell (a_3)-2f_\ell (a) \, \text{ for } a\ge a_1, \end{aligned}$$

(45)

$f_\ell (a)$ is defined in (61),

$$\begin{aligned} g_{\omega \omega }(a)= & {} \left[ 2 f_{\ell \ell }(a_3) - f_{\ell \ell }(a_1) - f_{\ell \ell }(a) + f_\ell (a_3)^2+ f_\ell (a_1)^2+ f_\ell (a)^2 \right. \nonumber \\&\left. + f_\ell (a_1)f_\ell (a)-2f_\ell (a_1)f_\ell (a_3)-2f_\ell (a)f_\ell (a_3) \right] \, \text{ for } a \le a_1 \nonumber \\ g_{\omega \omega }(a)= & {} \left[ 2 f_{\ell \ell }(a_3) -2 f_{\ell \ell }(a) + f_\ell (a_3)^2+ 3f_\ell (a)^2 -4f_\ell (a)f_\ell (a_3) \right] \, \text{ for } a\ge a_1,\nonumber \\ \end{aligned}$$

(46)

and $f_{\ell \ell }(a)$ is defined in (62).

Using (43) for $\omega$, (92) for $f_H(a)$, and (60) to expand $\ell _1/a_1$, $\ell _3/a_3$, and $\ell /a$ to second order in $\delta$ allows us to write (42) as

$$\begin{aligned} \theta \approx \sqrt{2 \delta } \left[ \sqrt{f_\theta (a)} - \frac{1}{2} f_{\theta \theta } (a)\delta \right] , \end{aligned}$$

(47)

where

$$\begin{aligned} \sqrt{f_\theta (a)}= & {} H_3\sqrt{3} a_1^{-1/2} a_2^{3/2} \int _{a_i}^{a} \frac{a^{-2}}{H_0} \, \mathrm {d} a \text{ for } a \le a_1, \text{ and } \nonumber \\ \sqrt{f_\theta (a)}= & {} \sqrt{f_\theta (a_1)}+ H_3\sqrt{3} a_1^{1/2} a_2^{3/2} \int _{a_1}^{a} \frac{a^{-3}}{H_0} \, \mathrm {d} a \text{ for } a \ge a_1, \end{aligned}$$

(48)

and

$$\begin{aligned} f_{\theta \theta }(a)= & {} -2\left( 2f_\ell (a_3) - f_\ell (a_1)\right) \sqrt{f_\theta (a)} \nonumber \\&+2 H_3\sqrt{3} a_1^{-1/2} a_2^{3/2} \int _{a_0}^{a} \frac{a^{-2}}{H_0} (f_\ell (a) - f_H(a)) \, \mathrm {d} a \text{ for } a \le a_1, \text{ and } \nonumber \\ f_{\theta \theta }(a)= & {} f_{\theta \theta }(a_1) -4f_\ell (a_3)\left( \sqrt{f_\theta (a)}-\sqrt{f_\theta (a_1)}\right) \nonumber \\&+2 H_3\sqrt{3} a_1^{1/2} a_2^{3/2} \int _{a_1}^{a} \frac{a^{-3}}{H_0} (2f_\ell (a) - f_H(a)) \, \mathrm {d} a \text{ for } a \ge a_1. \end{aligned}$$

(49)

Evaluating (48) gives

$$\begin{aligned} f_\theta (a)= & {} 3 \beta ^2 \left( 1 - \frac{a_0}{a} e^{-N}\right) ^2 \text{ for } a_i \le a \le a_0, \nonumber \\ f_\theta (a)= & {} 3\left[ \beta + \left( 1 - \frac{a_0}{a} \right) \left( \frac{a}{a_1}\right) \right] ^{2} \text{ for } a_0 \le a \le a_1, \nonumber \\ f_\theta (a)= & {} 3 \left[ \beta +3 -2 \left( \frac{a_1}{a}\right) ^{1/2} - \frac{a_0}{a_1} \right] ^2 \text{ for } a_1 \le a \le a_2, \text{ and } \nonumber \\ f_\theta (a)= & {} 3 \left[ \beta +3 - \frac{a_0}{a_1} -\left( \frac{a_1}{a_2}\right) ^{1/2} \left( \frac{3}{2} +\frac{1}{2} \left( \frac{a_2}{a}\right) ^{2} \right) \right] ^2 \text{ for } a_2 \le a \le a_3,\nonumber \\ \end{aligned}$$

(50)

where

$$\begin{aligned} \beta= & {} \frac{H_3}{H_0} \big (\frac{a_1}{a_i}\big ) \left( \frac{a_2}{a_1}\right) ^{3/2} = H_3 \frac{t_0 - t_i}{N} {a_1} \frac{e^N}{a_0} \left( \frac{a_2}{a_1}\right) ^{3/2} \approx H_3 t_0 \left( \frac{a_2}{a_1}\right) ^{3/2} \left( \frac{a_1}{a_0}\right) \frac{e^N}{N} \nonumber \\= & {} H_3 t_1 \left( \frac{a_2}{a_1}\right) ^{3/2} (\frac{a_0}{a_1}) \frac{e^N}{N} = H_3 t_2 (\frac{a_0}{a_1}) \frac{e^N}{N} \approx 1.5 \times 10^{-23} \frac{e^N}{N} ~ \end{aligned}$$

(51)

gives the effect of inflation. We can neglect $a_0$ in some places in (50) to give

$$\begin{aligned} f_\theta (a)= & {} 3 \beta ^2 \left( 1 - \frac{a_0}{a} e^{-N}\right) ^2 \text{ for } a_i \le a \le a_0, \nonumber \\ f_\theta (a)= & {} 3 \left( \beta +\frac{a}{a_1}\right) ^{2} \text{ for } a_0 \le a \le a_1, \nonumber \\ f_\theta (a)= & {} 3 \left[ \beta +3 -2 \left( \frac{a_1}{a}\right) ^{1/2} \right] ^2 \text{ for } a_1 \le a \le a_2, \text{ and } \nonumber \\ f_\theta (a)= & {} 3 \left[ \beta +3 -\left( \frac{a_1}{a_2}\right) ^{1/2} \left( \frac{3}{2} +\frac{1}{2} \left( \frac{a_2}{a}\right) ^{2} \right) \right] ^2 \text{ for } a_2 \le a \le a_3. \end{aligned}$$

(52)

We also have

$$\begin{aligned} \theta ^2 \approx 2 \delta \left[ f_\theta (a) - \sqrt{f_\theta (a)} f_{\theta \theta } (a)\delta \right] , \end{aligned}$$

(53)

and

$$\begin{aligned} \cos {\theta } \approx (1- \frac{1}{2} \theta ^2 + \frac{1}{24} \theta ^4) \approx \left[ 1-f_\theta (a) \delta +\left( \frac{1}{6}f_\theta ^2(a)+\sqrt{f_\theta (a)}f_{\theta \theta } (a)\right) \delta ^2\right] . \end{aligned}$$

(54)

and

$$\begin{aligned} \frac{1}{ \cos {\theta } } \approx (1+ \frac{1}{2} \theta ^2+ \frac{5}{24} \theta ^4) \approx \left[ 1+f_\theta (a) \delta +\left( \frac{5}{6}f_\theta ^2(a)-\sqrt{f_\theta (a)}f_{\theta \theta }(a)\right) \delta ^2\right] . \end{aligned}$$

(55)

Taking the approximate mean of (53), using (52), putting in the appropriate values, including $a_3=1$, gives for the total rms rotation

$$\begin{aligned} \langle \theta \rangle \approx 347 \, \langle \omega _3\rangle /H_3 \approx 347 \, \langle \omega _3\rangle \sqrt{{3}/{\varLambda }}. \end{aligned}$$

(56)

Finally, using the value for $\varLambda$ from (39) and using $\omega _m$ from (103) for $\langle \omega _3\rangle$ gives

$$\begin{aligned} \theta \approx 6 \times 10^{-59} \text{ radians } \approx \times 10^{-58} \text{ radians } \end{aligned}$$

(57)

for the total rotation of the universe from the initial singularity to the present, showing that it should be valid to assume flow lines are normal to surfaces of constant time for the calculations.

To find the limits for the validity of the calculation of $f_I(\langle \omega _3\rangle )$, we use (56) to find the value of $\langle \omega _3\rangle$ for which $\theta =1$ to give

$$\begin{aligned} \langle \omega _3\rangle \approx H_3/347 \approx 0.003 H_3. \end{aligned}$$

(58)

Thus, the approximate calculation of $f_I(\langle \omega _3\rangle )$ is not valid all the way to $\langle \omega _3\rangle =H_3$, but it is valid for $\langle \omega _3\rangle \gg \omega _m$, which is good enough. Of course this is only the local total rotation. If we allow for an additional few orders of magnitude increase because of spatial variation of the vorticity, we still have a small total rotation.

The point is, that it was a good approximation to consider that surfaces of constant global time are normal to flow lines. In addition, there are a few more orders of magnitude to use to allow the approximation to be valid for $\langle \omega _3\rangle$ to be much larger than $\omega _m$ and still be a good approximation.

It also means that the saddlepoint at $\langle \omega _3\rangle =0$ is isolated from any other saddlepoints or other non-analytic points.

Appendix E: Approximate global cosmological scale factor

The flow lines are not normal to surfaces of constant global time, so we have to change variables from $\ell$ to a, where a is the cosmological scale factor. We can use the angle calculations in appendix D to make that conversion. Because the angles are small, we can make approximations. The formulas below apply to each location in the flow because the vorticity can be a function of location.

We use (55) to approximate $1/\cos {\theta }$ to give

$$\begin{aligned} \ell\approx & {} \int \frac{ \, \mathrm {d} a}{\cos {\theta }} \approx \int (1+\frac{1}{2} \theta ^2+\frac{5}{24} \theta ^4) { \, \mathrm {d} a} \nonumber \\\approx & {} \int \left[ 1+f_\theta (a) \delta +\left( \frac{5}{6}f_\theta ^2(a)-\sqrt{f_\theta (a)}f_{\theta \theta }(a)\right) \delta ^2\right] { \, \mathrm {d} a}, \end{aligned}$$

(59)

where a is the cosmological scale factor, which is normal to surfaces of constant global time, and $\delta$ is given by (41).

We can write

$$\begin{aligned} \ell \approx a\left[ 1 + f_\ell (a) \delta + f_{\ell \ell } (a) \delta ^2 \right] , \end{aligned}$$

(60)

where

$$\begin{aligned} f_{\ell } (a) = \frac{1}{a} \int _{a_i}^a f_\theta (a) \, \mathrm {d} a ~ \end{aligned}$$

(61)

and

$$\begin{aligned} f_{\ell \ell } (a) = \frac{1}{a} \int _{a_0}^a \left[ \frac{5}{6} f_\theta ^2(a) - \sqrt{f_\theta (a)}f_{\theta \theta }(a) \right] \, \mathrm {d} a. \end{aligned}$$

(62)

We also have

$$\begin{aligned} \frac{1}{\ell } \approx \frac{1}{a}\left[ 1 - f_\ell (a) \delta + \left( f_\ell ^2 (a)-f_{\ell \ell } (a)\right) \delta ^2 \right] . \end{aligned}$$

(63)

We can use (52) in (61) and neglect $a_0$ in some places to give

$$\begin{aligned} f_\ell (a)= & {} 3\beta ^2 \left( 1 - 2 \frac{a_0}{a} N e^{-N} - 2 \frac{a_0}{a} e^{-N}\ln {\frac{a}{a_0}} - \frac{a_0^2}{a^2} e^{-2N} \right) \text{ for } a_i \le a \le a_0, \nonumber \\ f_\ell (a)= & {} 3\beta ^2 + \left( \frac{a}{a_1}\right) ^{2} \text{ for } a_0 \le a \le a_1, \nonumber \\ f_\ell (a)= & {} 3\beta ^2 + \left( 27 -72 \sqrt{\frac{a_1}{a}} +46 \frac{a_1}{a} -12 \frac{a_1}{a} \ln {\frac{a_1}{a}} \right) \text{ for } a_1 \le a \le a_2, \text{ and } \nonumber \\ f_\ell (a)= & {} f_\ell (a_2) \frac{a_2}{a} + 3 \frac{a_1}{a_2} \left[ \left( 3 \sqrt{\frac{a_2}{a_1}} -\frac{3}{2}\right) ^2 \left( 1 - \frac{a_2}{a} \right) \right. \nonumber \\&\left. +\left( 3 \sqrt{\frac{a_2}{a_1}} -\frac{3}{2}\right) \left( \frac{a_2^2}{a^2} - \frac{a_2}{a} \right) -\frac{1}{12} \left( \frac{a_2^4}{a^4} - \frac{a_2}{a} \right) \right] \text{ for } a_2 \le a \le a_3.\nonumber \\ \end{aligned}$$

(64)

From (64), we have

$$\begin{aligned} f_\ell (a_1) = 3\beta ^2 + 1. \end{aligned}$$

(65)

Using that from (35) $a_1$ is so small, we can write

$$\begin{aligned} f_\ell (a_2) \approx 3\beta ^2 + 27. \end{aligned}$$

(66)

We can write (62) as

$$\begin{aligned} f_{\ell \ell } (a)= & {} \frac{1}{a} \int _{a_0}^a \left[ \frac{5}{6} f_\theta ^2(a) - \sqrt{f_\theta (a)}f_{\theta \theta }(a) \right] \, \mathrm {d} a \text{ for } a_0 \le a \le a_1, \nonumber \\ f_{\ell \ell } (a)= & {} f_{\ell \ell } (a_1) \frac{a_1}{a} + \frac{1}{a} \int _{a_1}^a \left[ \frac{5}{6} f_\theta ^2(a) - \sqrt{f_\theta (a)}f_{\theta \theta }(a) \right] \, \mathrm {d} a \text{ for } a_1 \le a \le a_2, \nonumber \\ f_{\ell \ell } (a)= & {} f_{\ell \ell } (a_2) \frac{a_2}{a} + \frac{1}{a} \int _{a_2}^a \left[ \frac{5}{6} f_\theta ^2(a) - \sqrt{f_\theta (a)}f_{\theta \theta }(a) \right] \, \mathrm {d} a \text{ for } a_2 \le a \le a_3.\nonumber \\ \end{aligned}$$

(67)

Or,

$$\begin{aligned} f_{\ell \ell } (a)= & {} \frac{5}{6} \frac{1}{a} \int _{a_0}^a \left( f_\theta (a)\right) ^2 \, \mathrm {d} a - \frac{1}{a} \int _{a_0}^a f_{\theta \theta }(a) \sqrt{f_\theta (a)} \, \mathrm {d} a \text{ for } a_0 \le a \le a_1, \nonumber \\ f_{\ell \ell } (a)= & {} f_{\ell \ell } (a_1) \frac{a_1}{a} + \frac{5}{6}\frac{1}{a} \int _{a_1}^a \left( f_\theta (a)\right) ^2 \, \mathrm {d} a \nonumber \\&- \frac{1}{a} \int _{a_1}^a f_{\theta \theta }(a) \sqrt{f_\theta (a)} \, \mathrm {d} a \text{ for } a_1 \le a \le a_2, \nonumber \\ f_{\ell \ell } (a)= & {} f_{\ell \ell } (a_2) \frac{a_2}{a} + \frac{5}{6} \frac{1}{a} \int _{a_2}^a \left( f_\theta (a)\right) ^2 \, \mathrm {d} a \nonumber \\&- \frac{1}{a} \int _{a_2}^a f_{\theta \theta }(a) \sqrt{f_\theta (a)} \, \mathrm {d} a \text{ for } a_2 \le a \le a_3. \end{aligned}$$

(68)

Appendix F: Approximate generalized Friedmann equation for small Vorticity

It is possible to find an approximate solution to the field equations that is a perturbation from the standard cosmological model for small vorticity. The difficulty with finding an approximate solution that includes vorticity is that the Raychaudhuri equation is based on a coordinate system that follows flow lines, but surfaces of constant global time cannot be found that are normal to the flow lines. However, in the limit of small vorticity, we can estimate the effect that the scale factor along flow lines $\ell$ is not quite the same as the global cosmological scale factor a.

Derivation of a generalization of the Friedmann equation that includes relative rotation of matter and inertial frames (specifically, shear and vorticity), starts with the Raychaudhuri equation [72, 73, 9, eq. 1.3.4], [10, eq. (36)], [12, 14, eq. 4.12], [17, 24, 25, 57, 58, 81], or the Raychaudhuri–Ehlers equation [15, eq. 6.4].

We start with the Raychaudhuri–Ehlers equation [15, eq. 6.5]

$$\begin{aligned} 3 \frac{\ddot{\ell }}{\ell } = -2\left( \sigma ^2 - \omega ^2\right) + {\nabla }_a\dot{u}^a + \dot{u}_a\dot{u}^a - 4\pi G (\rho +3p) + \varLambda , \end{aligned}$$

(69)

where $\ell$ is a scale factor that follows flow lines, and $\cdot$ is a derivative with respect to a variable $\tau$ that increases along the flow line u.

Multiplying (69) by $\ell {\dot{\ell }}$ and integrating gives

$$\begin{aligned} {{\dot{\ell }}}/{\ell } = \sqrt{ H^2 + H_{\omega }^2 + H_{\sigma }^2 + H_{a}^2 }, \end{aligned}$$

(70)

where

$$\begin{aligned} H \equiv \sqrt{\frac{\varLambda }{3} + \frac{8\pi \rho }{3} - \frac{k}{{\ell }^2} } \end{aligned}$$

(71)

is the Hubble parameter without vorticity, shear, or acceleration,

$$\begin{aligned} H_{\omega }^2 \equiv \frac{4}{3\ell ^2} \int \ell \omega ^2 \, \mathrm {d} \ell ~ \end{aligned}$$

(72)

is the vorticity term, $\omega$ is vorticity,

$$\begin{aligned} H_{\sigma }^2 \equiv -\frac{4}{3\ell ^2} \int \ell \sigma ^2 \, \mathrm {d} \ell ~ \end{aligned}$$

(73)

is the shear term, $\sigma$ is shear, and

$$\begin{aligned} H_{a}^2 \equiv - \frac{2}{3\ell ^2} \int \ell {\dot{u}^a}_{;a} \, \mathrm {d} \ell ~ \end{aligned}$$

(74)

is the acceleration term.

We can take vorticity to depend on the distance along flow lines $\ell$ as

$$\begin{aligned} \omega \propto {\ell }^{-m}, \end{aligned}$$

(75)

and, for small vorticity, we can take $m=1$ in the radiation era and $m=2$ in the matter era [15, Table 6.1].⁷ If we put the $\ell$-variation of vorticity into (72), then we get

$$\begin{aligned} H_{\omega }^2 \approx H_3^2 f_{\omega }(a) \delta , \end{aligned}$$

(76)

where $\delta$ is given by (41),

$$\begin{aligned} H_{3} \approx \sqrt{\varLambda /3} \end{aligned}$$

(77)

is the present value of the Hubble parameter, and

$$\begin{aligned} f_{\omega }(a)= & {} 4 \left( \frac{a_2}{a_1}\right) ^3 \left( \frac{a_1}{a}\right) ^2 \left[ 2 \ln {\frac{a}{a_1}} + \alpha _4 -1 \right] \, \text{ for } a \le a_1 \nonumber \\ f_{\omega }(a)= & {} 4 \left( \frac{a_2}{a_1}\right) ^3 \left( \frac{a_1}{a}\right) ^2 \left[ \alpha _4 - \left( \frac{a_1}{ a}\right) ^2 \right] \, \text{ for } a\ge a_1, \end{aligned}$$

(78)

where $\alpha _4$ is an arbitrary constant of integration in (72).

We can generalize (76) to second order by writing

$$\begin{aligned} H_{\omega }^2 = H_3^2 \left[ f_{\omega }(a) \delta + f_{\omega \omega }(a) \delta ^2 \right] , \end{aligned}$$

(79)

where

$$\begin{aligned} f_{\omega \omega }(a)= & {} 8 \left( \frac{a_2}{a_1}\right) ^3 \left( \frac{a_1}{a}\right) ^2 \left[ \left( 2 f_\ell (a_3) - f_\ell (a_1) - f_\ell (a)\right) \left( 2\ln {\frac{a}{a_1}} \right. \right. \nonumber \\&\left. \left. +\,\alpha _4-1\right) + f_\ell (a) - f_\ell (a_1) \right] \nonumber \, \text{ for } a \le a_1 \nonumber \\ f_{\omega \omega }(a)= & {} 8 \left( \frac{a_2}{a_1}\right) ^3 \left( \frac{a_1}{a}\right) ^2 \left. \bigg [\left( 2f_\ell (a_3) - f_\ell (a_1)- f_\ell (a)\right) \alpha _4 \right. \nonumber \\&\left. -\,2 \left( f_\ell (a_3) - f_\ell (a)\right) \left( \frac{a_1}{ a}\right) ^2\right. \bigg ] \, \text{ for } a\ge a_1. \,\,\,\,\,\,\,\,\,\, \end{aligned}$$

(80)

We can use (64) for $f_\ell (a)$ in (80) to give

$$\begin{aligned} f_{\omega \omega }(a)= & {} 8 \left( \frac{a_2}{a_1}\right) ^3 \left[ 2 \left( 2 f_\ell (a_3) - 1 \right) \left( \frac{a_1}{a}\right) ^2 \ln {\frac{a}{a_1}} -2\ln {\frac{a}{a_1}} \right. \nonumber \\&\left. +\,(2 f_\ell (a_3) - 1) (\alpha _4 -1)\left( \frac{a_1}{a}\right) ^{2} - \left( \frac{a_1}{a}\right) ^{2} +2 -\alpha _4 \right] \, \text{ for } a_0 \le a \le a_1 \nonumber \\ f_{\omega \omega }(a)= & {} 8 \left( \frac{a_2}{a_1}\right) ^3 \left( \frac{a_1}{a}\right) ^{2} \left[ (2f_\ell (a_3) - f_\ell (a_1)-27)\alpha _4 \right. \nonumber \\&\left. -\,\alpha _4 \left( -72 \sqrt{\frac{a_1}{a}} +46 \frac{a_1}{a} +12 \frac{a_1}{a} \ln {\frac{a}{a_1}} \right) \right. \nonumber \\&\left. +\,2 \left( 27 -f_\ell (a_3) -72 \sqrt{\frac{a_1}{a}} +46 \frac{a_1}{a} +12 \frac{a_1}{a} \ln {\frac{a}{a_1}} \right) \left( \frac{a_1}{ a}\right) ^2\right] \nonumber \\&\quad \, \text{ for } a_1 \le a \le a_2 \nonumber \\ f_{\omega \omega }(a)= & {} 8 \left( \frac{a_2}{a_1}\right) ^3 \left( \frac{a_1}{ a}\right) ^2 \left\{ \left( 2f_\ell (a_3) -f_\ell (a_1) -f_\ell (a_2)\frac{a_2}{ a} \right. \right. \nonumber \\&\left. -\,3 \frac{a_1}{a_2} \left[ \left( 3 \sqrt{\frac{a_2}{a_1}} -\frac{3}{2}\right) ^2 \left( 1 - \frac{a_2}{a} \right) \right. \right. \nonumber \\&\left. \left. +\,\left( 3 \sqrt{\frac{a_2}{a_1}} -\frac{3}{2}\right) \left( \frac{a_2^2}{a^2} - \frac{a_2}{a} \right) -\frac{1}{12} \left( \frac{a_2^4}{a^4} - \frac{a_2}{a} \right) \right] \right) \alpha _4 \nonumber \\&-\,2\left( f_\ell (a_3) -f_\ell (a_2)\frac{a_2}{ a} \right. \nonumber \\&\left. \left. -\,3 \frac{a_1}{a_2} \left[ \left( 3 \sqrt{\frac{a_2}{a_1}} -\frac{3}{2}\right) ^2 \left( 1 - \frac{a_2}{a} \right) \right. \right. \right. \nonumber \\&\left. \left. \left. +\,\left( 3 \sqrt{\frac{a_2}{a_1}} -\frac{3}{2}\right) \left( \frac{a_2^2}{a^2} - \frac{a_2}{a} \right) -\frac{1}{12} \left( \frac{a_2^4}{a^4} - \frac{a_2}{a} \right) \right] \right) \left( \frac{a_1}{ a}\right) ^2 \right\} \nonumber \\&\quad \, \text{ for } a_2 \le a \le a_3. \,\,\,\,\,\,\,\,\,\, \end{aligned}$$

(81)

If the vorticity is small enough, then we can find a coordinate system with a cosmological scale factor a normal to surfaces of constant global time, that differ from flow lines by a very small amount. In that case, we can replace the scale factor $\ell$ along flow lines by the cosmological scale factor a, and we calculate the effect of that difference to first and second order. In making this calculation, we replace $\tau$ by the global time t.

We start by using ${\dot{\ell }} \approx \dot{a} / \cos {\theta }$ in (70) and neglect shear and acceleration to give

$$\begin{aligned} \frac{\dot{a}}{a} =\frac{\ell }{a} \cos {\theta } \sqrt{ H^2 + H_{\omega }^2 + H_{\sigma }^2 + H_{a}^2 } ~ \end{aligned}$$

(82)

for the generalized Friedmann equation, where

$$\begin{aligned} H \equiv \sqrt{\frac{\varLambda }{3} + \frac{8\pi \rho }{3} - \frac{k}{{a}^2} } \rightarrow \sqrt{\frac{\varLambda }{3} + \frac{8\pi \rho }{3} } ~ \end{aligned}$$

(83)

is the Hubble parameter without vorticity, shear, or acceleration, and we neglect the spatial curvature term from here on because measurements show it to be nearly zero.

However, because the density $\rho$ in (83) depends on $\ell$ rather than a, we have in analogy with (28) for the same effect on the Lagrangian

$$\begin{aligned} H^2 \approx H_0^2 \left[ 1+ f_L(a) \delta + f_{LL}(a) \delta ^2 \right] , \end{aligned}$$

(84)

where

$$\begin{aligned} H_0 = \sqrt{\frac{\varLambda }{3} + \frac{8\pi \rho _0}{3} } ~ \end{aligned}$$

(85)

is the Hubble parameter without the $\ell$ dependence and $\rho _0$ is the density without the $\ell$ dependence. We also have

$$\begin{aligned} \frac{1}{H^2} \approx \frac{1}{H_0^2} \left[ 1 - f_L(a) \delta + \left( f_L(a)^2-f_{LL}(a)\right) \delta ^2 \right] ~ \end{aligned}$$

(86)

and

$$\begin{aligned} \frac{1}{H} \approx \frac{1}{H_0} \left[ 1 - \frac{1}{2}f_L(a) \delta + \left( \frac{3}{8}f_L(a)^2-\frac{1}{2}f_{LL}(a)\right) \delta ^2 \right] . \end{aligned}$$

(87)

Although in general, the vorticity propagation equation and the shear propagation equation are coupled [11], the coupling terms are second-order, so that for very small vorticity and very small shear, we can neglect the coupling. Therefore, we can neglect the shear term in (82). In addition, we neglect the acceleration term. Using (84) for H, (79) for $H_\omega ^2$, (60) for $\ell$, and (54) for $\cos {\theta }$ gives

$$\begin{aligned}&\frac{\dot{a}}{a} \approx H_0 \left[ 1 +\frac{1}{2}\left( f_L(a)+\frac{H_3^2}{H_0^2}f_\omega (a)\right) \delta \right. \nonumber \\&\left. \quad +\frac{1}{2}\left( f_{LL}(a)+\frac{H_3^2}{H_0^2}f_{\omega \omega }(a)-\frac{1}{4}\left( f_L(a)+\frac{H_3^2}{H_0^2}f_\omega (a)\right) ^2\right) \delta ^2 \right] \times \nonumber \\&\qquad \left[ 1 + f_\ell (a) \delta + f_{\ell \ell } (a) \delta ^2 \right] \left[ 1-f_\theta (a) \delta +\left( \frac{1}{6}f_\theta ^2(a)+\sqrt{f_\theta (a)}f_{\theta \theta } (a)\right) \delta ^2 \right] .\nonumber \\ \end{aligned}$$

(88)

Multiplying out and keeping terms through second order in $\delta$ gives

$$\begin{aligned}&\frac{\dot{a}}{a} \approx H_0 \left\{ 1 +\left[ \frac{1}{2}f_L(a)+ f_\ell (a)-f_\theta (a)+\frac{1}{2}\frac{H_3^2}{H_0^2}f_\omega (a)\right] \delta \right. \nonumber \\&\qquad +\left[ \frac{1}{2}f_{LL}(a)+ f_{\ell \ell } (a)+\frac{1}{6}f_\theta ^2(a)+\sqrt{f_\theta (a)}f_{\theta \theta } (a)-\frac{1}{8}f_L(a)^2 \right. \nonumber \\&\left. \qquad +\frac{1}{2}f_L(a)f_\ell (a) -\frac{1}{2}f_L(a)f_\theta (a) -f_\ell (a)f_\theta (a) \right. \nonumber \\&\qquad \left. \left. +\left( -\frac{1}{4}f_L(a)f_\omega (a)+\frac{1}{2}f_\ell (a)f_\omega (a)-\frac{1}{2}f_\theta (a)f_\omega (a) +\frac{1}{2}f_{\omega \omega }(a) \right. \right. \right. \nonumber \\&\left. \left. \left. \qquad -\frac{1}{8}\frac{H_3^2}{H_0^2}f_\omega (a)^2 \right) \frac{H_3^2}{H_0^2} \right] \delta ^2 \right\} . \end{aligned}$$

(89)

To the same degree of approximation, we have

$$\begin{aligned}&\frac{1}{\dot{a}} \approx \frac{1}{a H_0} \left[ 1 -\frac{1}{2}\left( f_L(a)+\frac{H_3^2}{H_0^2}f_\omega (a)\right) \delta \right. \nonumber \\&\left. \qquad -\frac{1}{2}\left( f_{LL}(a)+\frac{H_3^2}{H_0^2}f_{\omega \omega }(a) -\frac{3}{4}\left( f_L(a)+\frac{H_3^2}{H_0^2}f_\omega (a)\right) ^2\right) \delta ^2 \right] \nonumber \\&\qquad \times \left[ 1 - f_\ell (a) \delta + \left( f_\ell ^2 (a)-f_{\ell \ell } (a)\right) \delta ^2 \right] \left[ 1+f_\theta (a) \delta \right. \nonumber \\&\left. \qquad +\left( \frac{5}{6}f_\theta ^2(a)-\sqrt{f_\theta (a)}f_{\theta \theta } (a)\right) \delta ^2 \right] . \end{aligned}$$

(90)

Multiplying out and keeping terms through second order in $\delta$ gives

$$\begin{aligned} \frac{1}{\dot{a}} \approx \frac{1}{a H_0} \left[ 1+f_H(a) \delta +f_{HH}(a) \delta ^2\right] , \end{aligned}$$

(91)

where

$$\begin{aligned} f_H(a) = -\frac{1}{2}f_L(a)- f_\ell (a)+f_\theta (a)-\frac{1}{2}\frac{H_3^2}{H_0^2}f_\omega (a) ~ \end{aligned}$$

(92)

and

$$\begin{aligned} f_{HH}(a)= & {} -\frac{1}{2}f_{LL}(a) + f_{\ell }^2 (a) - f_{\ell \ell } (a) +\frac{5}{6}f_\theta ^2(a) -\sqrt{f_\theta (a)}f_{\theta \theta } (a) +\frac{3}{8}f_L(a)^2 \nonumber \\&+\frac{1}{2}f_L(a)f_\ell (a) -\frac{1}{2}f_L(a)f_\theta (a) -f_\ell (a)f_\theta (a) \nonumber \\&+\left( \frac{3}{4}f_L(a)f_\omega (a)+\frac{1}{2}f_\ell (a)f_\omega (a)-\frac{1}{2}f_\theta (a)f_\omega (a) \right. \nonumber \\&\left. -\frac{1}{2}f_{\omega \omega }(a) +\frac{3}{8}\frac{H_3^2}{H_0^2}f_\omega (a)^2 \right) \frac{H_3^2}{H_0^2}. \end{aligned}$$

(93)

We can consider that (89) gives the effective generalized Friedmann equation, including to second order the effect that flow lines are not normal to surfaces of constant global time.

Appendix G: Approximate action for small vorticity

The main effect of vorticity and shear on the action is through the generalized Friedmann equation (70). Starting with (25), change the t integration to an integration over a using (91) and use (28) for ${\tilde{L}}$ to give

$$\begin{aligned} I= & {} \int \frac{ V {\tilde{L}} \, \mathrm {d} \, a }{\dot{a}} \approx \int _{a_i}^{a_3} \frac{ V (a) H_3^2 F(a) }{ a H_0} \left[ 1+ f_L(a) {\overline{\delta }} + f_{LL}(a) \overline{\delta ^2} \right] \nonumber \\&\quad \left[ 1+f_H(a) {\overline{\delta }} +f_{HH}(a) \overline{\delta ^2}\right] \, \mathrm {d} \, a, \end{aligned}$$

(94)

where

$$\begin{aligned} V(a) = \frac{4}{3} \pi a^3 r_3^3 ~ \end{aligned}$$

(95)

is the approximate spatial volume,⁸a is the cosmological scale factor, $r_3$ is the present radius of the cosmological horizon,

$$\begin{aligned} {\overline{\delta }} = \frac{1}{6} \overline{\left( \frac{\omega _3}{H_3}\right) ^2} \left( \frac{a_3}{a_1}\right) \left( \frac{a_3}{a_2}\right) ^{3} = \frac{1}{6} \left( \frac{\langle \omega _3\rangle }{H_3}\right) ^2 \left( \frac{a_3}{a_1}\right) \left( \frac{a_3}{a_2}\right) ^{3} \end{aligned}$$

(96)

is the spatial average of $\delta$ given by (41),

$$\begin{aligned} \overline{\delta ^2} = {\overline{\delta }}^2 + \left[ \frac{1}{6} \left( \frac{1}{H_3}\right) ^2 \left( \frac{a_3}{a_1}\right) \left( \frac{a_3}{a_2}\right) ^{3} \right] ^2 \sigma _\omega ^2 \end{aligned}$$

(97)

is the spatial average of the square of $\delta$, $\sigma _\omega ^2$ is the variance of $\omega _3^2$, and we have neglected shear and acceleration, keeping only vorticity. Multiplying out, and keeping terms only up to second order in ${\overline{\delta }}$ gives

$$\begin{aligned}&I \approx \int _{a_i}^{a_3} \frac{ V (a) H_3^2 F(a) }{ a H_0} \left. \bigg \{1+ \left[ f_L(a)+f_H(a) \right] {\overline{\delta }} +\left[ f_L(a)f_H(a) \right. \right. \nonumber \\&\quad \left. \left. \quad + f_{LL}(a) +f_{HH}(a)\right] \overline{\delta ^2} \right. \bigg \} \, \mathrm {d} \, a. \end{aligned}$$

(98)

We can write (98) as

$$\begin{aligned} I \approx I_0 + I_1 {\overline{\delta }} + I_2 \overline{\delta ^2} \text{, } \end{aligned}$$

(99)

where $I_0$ is the action for zero vorticity and zero shear,

$$\begin{aligned} I_1 = \int _{a_i}^{a_3} \frac{ V (a) H_3^2 F(a) }{ a H_0} \left[ f_L(a)+f_H(a) \right] \, \mathrm {d} \, a. \end{aligned}$$

(100)

and

$$\begin{aligned} I_2 = \int _{a_0}^{a_3} \frac{ V (a) H_3^2 F(a) }{ a H_0} \left[ f_L(a)f_H(a) + f_{LL}(a) +f_{HH}(a)\right] \, \mathrm {d} \, a \text{. } \end{aligned}$$

(101)

We can use (95) for V(a), and (96) for ${\overline{\delta }}$ and (97) for $\overline{\delta ^2}$ to write (99) as

$$\begin{aligned} I \approx I_0 + \hbar \left( \frac{\langle \omega _3\rangle }{\omega _m}\right) ^2 \left( \frac{a_3}{a_1}\right) \left[ C_I + \frac{\langle \omega _3\rangle ^2+\sigma _\omega ^2/\langle \omega _3\rangle ^2}{H_3^2} \left( \frac{a_3}{a_1}\right) C_{II}\right] , \end{aligned}$$

(102)

where

$$\begin{aligned} \omega _m = \left( \frac{ \hbar H_3 }{ r_3^3 } \right) ^{1/2} = \frac{T^*}{r_3}\sqrt{\frac{H_3}{r_3}} \approx T^* H_3^2 \approx 10^{-89} \text{ cm }^{-1} \approx 10^{-71} \text{ rad } \hbox {yr}^{-1}, \end{aligned}$$

(103)

$H_3$ is the present value of the Hubble parameter, $r_3$ is the present radius of the universe (which we approximate by the inverse of the Hubble parameter), and $T^*$ is the Planck time,

$$\begin{aligned} C_I = \frac{1}{6} \left( \frac{a_3}{a_2}\right) ^{3} \frac{4}{3} \pi \int _{a_i}^{a_3} a^2 \frac{ H_3 F (a)}{ H_0 } \left[ f_L(a)+f_H(a) \right] \, \mathrm {d} \, a, \end{aligned}$$

(104)

and

$$\begin{aligned} C_{II} = \frac{1}{36} \left( \frac{a_3}{a_2}\right) ^{6} \frac{4}{3} \pi \int _{a_0}^{a_3} a^2\frac{ H_3 F (a)}{ H_0 } \left[ f_L(a)f_H(a) + f_{LL}(a) +f_{HH}(a)\right] \, \mathrm {d} \, a \text{. } \nonumber \\ \end{aligned}$$

(105)

We can use (92) for $f_H(a)$ in (104) to give

$$\begin{aligned} C_I = \frac{1}{6} \left( \frac{a_3}{a_2}\right) ^{3} \frac{4}{3} \pi \int _{a_i}^{a_3} a^2 \frac{ H_3 F (a)}{ H_0 } \left[ f_\theta (a) - f_\ell (a)+\frac{1}{2}f_L(a) - \frac{1}{2} \frac{H_3^2}{H_0^2}f_\omega (a)\right] \, \mathrm {d} \, a.\nonumber \\ \end{aligned}$$

(106)

We can use (92) for $f_H(a)$ and (93) for $f_{HH}(a)$ in (105) to give

$$\begin{aligned} C_{II}= & {} \frac{1}{36} \left( \frac{a_3}{a_2}\right) ^{6} \frac{4}{3} \pi \int _{a_0}^{a_3} a^2\frac{ H_3 F (a)}{ H_0 } \left\{ f_\ell (a)^2- f_{\ell \ell } (a) -f_\theta (a) f_\ell (a) \right. \nonumber \\&+\frac{5}{6} f_\theta (a)^2 - \sqrt{f_\theta (a)} f_{\theta \theta }(a) + \frac{1}{2}f_{LL}(a) + \frac{1}{2}f_{L}(a)f_\theta (a) \nonumber \\&- \frac{1}{2}f_{L}(a)f_\ell (a) - \frac{1}{8}f_{L}^2(a) \nonumber \\&\left. - \frac{1}{2} \frac{H_3^2}{H_0^2} \left[ f_\omega (a)f_\theta (a) - f_\omega (a)f_\ell (a) -\frac{1}{2}f_L(a) f_\omega (a) + f_{\omega \omega }(a) \right. \right. \nonumber \\&\left. \left. - \frac{3}{4} f_\omega (a)^2 \frac{H_3^2}{H_0^2} \right] \right\} \, \mathrm {d} \, a. \end{aligned}$$

(107)

We can write (106) as

$$\begin{aligned} C_I= & {} \left( \frac{a_3}{a_2}\right) ^{3} \frac{2\pi }{9} \left\{ \int _{a_i}^{a_0} a^2 \frac{ H_3 F (a) }{ H_0 } \left[ f_\theta (a) - f_\ell (a) +\frac{1}{2}f_L(a) - \frac{1}{2} \frac{H_3^2}{H_0^2} f_\omega (a) \right] \, \mathrm {d} \, a \right. \nonumber \\&+ \int _{a_0}^{a_1} a^2 \frac{ H_3 F (a) }{ H_0 } \left[ f_\theta (a) - f_\ell (a) +\frac{1}{2}f_L(a) - \frac{1}{2} \frac{H_3^2}{H_0^2} f_\omega (a) \right] \, \mathrm {d} \, a \nonumber \\&+ \int _{a_1}^{a_2} a^2 \frac{ H_3 F (a) }{ H_0 } \left[ f_\theta (a) - f_\ell (a) +\frac{1}{2}f_L(a) - \frac{1}{2} \frac{H_3^2}{H_0^2} f_\omega (a) \right] \, \mathrm {d} \, a \nonumber \\&\left. + \int _{a_2}^{a_3} a^2 \frac{ H_3 F (a) }{ H_0 } \left[ f_\theta (a) - f_\ell (a) +\frac{1}{2}f_L(a) - \frac{1}{2} \frac{H_3^2}{H_0^2} f_\omega (a) \right] \, \mathrm {d} \, a \right\} . \end{aligned}$$

(108)

We can use (85) for $H_0$, (29) for F(a), (52) for $f_\theta (a)$, (64) for $f_\ell (a)$, (32) for $f_L(a)$, and (78) for $f_\omega (a)$, in (108). Keeping only the dominant term in the Hubble factor in each era, using the appropriate dependence on a of $\rho _0$ in each era, performing the integrations, neglecting $a_0$ when appropriate, using (35) and (36) to allow us to neglect $a_1$ in some places, gives

$$\begin{aligned} C_I= & {} \frac{w}{12} \left( \frac{a_1}{a_2}\right) ^{3/2} {a_3^3} \left[ \left( \frac{103}{18} + \frac{4}{3} N -\frac{2}{3} \alpha _4 \right) \beta ^2 +3\beta + {41} - \frac{2}{3} \alpha _4 \right] \alpha _1 \nonumber \\&+\,{a_3^3} \left[ \frac{1}{12} \left( \frac{a_1}{a_2}\right) ^{3/2} \left( \left( \frac{103}{18} + \frac{4}{3} N -\frac{2}{3} \alpha _4 \right) \beta ^2 +3\beta \right) + \left( 1- \left( \frac{a_1}{a_2}\right) ^{1/2} \right) \beta \right. \nonumber \\&\left. +\,6 \left( \frac{a_1}{a_2}\right) ^{1/2} - \frac{1}{15} \frac{a_2}{a_1} \alpha _4 \right] \alpha _2 \nonumber \\&+\,{2\pi } {a_3^3} \left( \frac{a_3}{a_2}- 1\right) \left[ \left( 2\left( \frac{a_3^2}{a_2^2} + \frac{a_3}{a_2}+1\right) - \left( \frac{a_1}{a_2}\right) ^{1/2} \left( \frac{a_3}{a_2}\right) \left( \frac{a_3}{a_2}+1\right) \right) \beta \right. \nonumber \\&\left. +\,\frac{9}{2} \left( \frac{a_3}{a_2}+1\right) - \frac{2 }{3} \frac{a_2}{a_1} \alpha _4 \right] \alpha _3, \end{aligned}$$

(109)

where $\beta$ is given by (51), and gives the effect of inflation on the action.

Putting in values for w, $a_1$, $a_2$, and $a_3=1$ from “Appendix C” gives

$$\begin{aligned} C_I\approx & {} \left( 4\times 10^{-6} -6\times 10^{-8} \alpha _4 \right) \alpha _1 + \left( \beta + 9\times 10^{-2} - 298 \alpha _4 \right) \alpha _2 \nonumber \\&+ 2\pi \left( 2.5 \beta + 3.3 - 941 \alpha _4 \right) \alpha _3. \end{aligned}$$

(110)

We can write (107) as

$$\begin{aligned} C_{II}= & {} \frac{1}{36} \left( \frac{a_3}{a_2}\right) ^{6} \frac{4}{3} \pi \int _{a_0}^{a_1} a^2\frac{ H_3 F (a)}{ H_0 } \left\{ f_\ell (a)^2- f_{\ell \ell } (a) -f_\theta (a) f_\ell (a) \right. \nonumber \\&+\,\frac{5}{6} f_\theta (a)^2 - \sqrt{f_\theta (a)} f_{\theta \theta }(a) + \frac{1}{2}f_{LL}(a) + \frac{1}{2}f_{L}(a)f_\theta (a) \nonumber \\&-\,\frac{1}{2}f_{L}(a)f_\ell (a) - \frac{1}{8}f_{L}^2(a) \nonumber \\&\left. -\,\frac{1}{2} \frac{H_3^2}{H_0^2} \left[ f_\omega (a)f_\theta (a) - f_\omega (a)f_\ell (a) -\frac{1}{2}f_L(a) f_\omega (a) + f_{\omega \omega }(a) \right. \right. \nonumber \\&\left. \left. -\,\frac{3}{4} f_\omega (a)^2 \frac{H_3^2}{H_0^2} \right] \right\} \, \mathrm {d} \, a \nonumber \\&+\,\frac{1}{36} \left( \frac{a_3}{a_2}\right) ^{6} \frac{4}{3} \pi \int _{a_1}^{a_2} a^2\frac{ H_3 F (a)}{ H_0 } \left\{ f_\ell (a)^2- f_{\ell \ell } (a) -f_\theta (a) f_\ell (a) \right. \nonumber \\&+\,\frac{5}{6} f_\theta (a)^2 - \sqrt{f_\theta (a)} f_{\theta \theta }(a) + \frac{1}{2}f_{LL}(a) + \frac{1}{2}f_{L}(a)f_\theta (a) \nonumber \\&-\, \frac{1}{2}f_{L}(a)f_\ell (a) - \frac{1}{8}f_{L}^2(a) \nonumber \\&\left. -\,\frac{1}{2} \frac{H_3^2}{H_0^2} \left[ f_\omega (a)f_\theta (a) - f_\omega (a)f_\ell (a) -\frac{1}{2}f_L(a) f_\omega (a)\right. \right. \nonumber \\&\left. \left. +\,f_{\omega \omega }(a) - \frac{3}{4} f_\omega (a)^2 \frac{H_3^2}{H_0^2} \right] \right\} \, \mathrm {d} \, a \nonumber \\&+\,\frac{1}{36} \left( \frac{a_3}{a_2}\right) ^{6} \frac{4}{3} \pi \int _{a_2}^{a_3} a^2\frac{ H_3 F (a)}{ H_0 } \left\{ f_\ell (a)^2- f_{\ell \ell } (a) -f_\theta (a) f_\ell (a) \right. \nonumber \\&+\,\frac{5}{6} f_\theta (a)^2 - \sqrt{f_\theta (a)} f_{\theta \theta }(a) + \frac{1}{2}f_{LL}(a) + \frac{1}{2}f_{L}(a)f_\theta (a) \nonumber \\&-\, \frac{1}{2}f_{L}(a)f_\ell (a) - \frac{1}{8}f_{L}^2(a) \nonumber \\&\left. -\,\frac{1}{2} \frac{H_3^2}{H_0^2} \left[ f_\omega (a)f_\theta (a) - f_\omega (a)f_\ell (a) -\frac{1}{2}f_L(a) f_\omega (a) \right. \right. \nonumber \\&\left. \left. +\,f_{\omega \omega }(a) - \frac{3}{4} f_\omega (a)^2 \frac{H_3^2}{H_0^2} \right] \right\} \, \mathrm {d} \, a \text{. } \end{aligned}$$

(111)

We can use (85) for $H_0$, keep only the dominant term in the Hubble factor in each era, use (77) for $H_3$, use the appropriate dependence on a of $\rho _0$ in each era, and use (29) for F(a) to give

$$\begin{aligned} C_{II}= & {} \frac{\alpha _1 w+\alpha _2}{72} a_1^{1/2}a_2^{-9/2} a_3^6 \int _{a_0}^{a_1} \left\{ f_\ell (a)^2- f_{\ell \ell } (a) -f_\theta (a) f_\ell (a) \right. \nonumber \\&+\,\frac{5}{6} f_\theta (a)^2 - \sqrt{f_\theta (a)} f_{\theta \theta }(a) + \frac{1}{2}f_{LL}(a) \nonumber \\&+\, \frac{1}{2}f_{L}(a)f_\theta (a) - \frac{1}{2}f_{L}(a)f_\ell (a) - \frac{1}{8}f_{L}^2(a) \nonumber \\&\left. -\,\frac{1}{2} \frac{H_3^2}{H_0^2} \left[ f_\omega (a)f_\theta (a) - f_\omega (a)f_\ell (a) -\frac{1}{2}f_L(a) f_\omega (a) \right. \right. \nonumber \\&\left. \left. +\,f_{\omega \omega }(a) - \frac{3}{4} f_\omega (a)^2 \frac{H_3^2}{H_0^2} \right] \right\} \, \mathrm {d} \, a \nonumber \\&+\, \frac{\alpha _2}{72} a_2^{-9/2} a_3^6 \int _{a_1}^{a_2} a^{1/2} \left\{ f_\ell (a)^2- f_{\ell \ell } (a) -f_\theta (a) f_\ell (a) \right. \nonumber \\&+\,\frac{5}{6} f_\theta (a)^2 - \sqrt{f_\theta (a)} f_{\theta \theta }(a) + \frac{1}{2}f_{LL}(a) \nonumber \\&\quad +\, \frac{1}{2}f_{L}(a)f_\theta (a) - \frac{1}{2}f_{L}(a)f_\ell (a) - \frac{1}{8}f_{L}^2(a) \nonumber \\&\left. -\,\frac{1}{2} \frac{H_3^2}{H_0^2} \left[ f_\omega (a)f_\theta (a) - f_\omega (a)f_\ell (a) -\frac{1}{2}f_L(a) f_\omega (a) \right. \right. \nonumber \\&\left. \left. +\, f_{\omega \omega }(a) - \frac{3}{4} f_\omega (a)^2 \frac{H_3^2}{H_0^2} \right] \right\} \, \mathrm {d} \, a \nonumber \\&+\, \frac{\pi \alpha _3}{9} \left( \frac{a_3}{a_2}\right) ^{6} \int _{a_2}^{a_3} a^2 \left\{ f_\ell (a)^2- f_{\ell \ell } (a) -f_\theta (a) f_\ell (a) \right. \nonumber \\&+\,\frac{5}{6} f_\theta (a)^2 - \sqrt{f_\theta (a)} f_{\theta \theta }(a) \nonumber \\&\quad +\, \frac{1}{2}f_{LL}(a) + \frac{1}{2}f_{L}(a)f_\theta (a) - \frac{1}{2}f_{L}(a)f_\ell (a) - \frac{1}{8}f_{L}^2(a) \nonumber \\&\left. -\,\frac{1}{2} \frac{H_3^2}{H_0^2} \left. \bigg [ f_\omega (a)f_\theta (a) - f_\omega (a)f_\ell (a) \right. \right. \nonumber \\&\left. \left. -\,\frac{1}{2}f_L(a) f_\omega (a) + f_{\omega \omega }(a) - \frac{3}{4} f_\omega (a)^2 \frac{H_3^2}{H_0^2} \right. \bigg ]\right\} \, \mathrm {d} \, a. \end{aligned}$$

(112)

We can use (52) for $f_\theta (a)$, (64) for $f_\ell (a)$, (32) for $f_L(a)$, (78) for $f_\omega (a)$, (68) for $f_{\ell \ell } (a)$, (49) for $f_{\theta \theta }(a)$, (31) for $f_{LL}(a)$, (81) for $f_{\omega \omega }(a)$, neglect $a_0$, and use the fact that $a_1/a_2 \approx 10^{-4}$ to approximate (112) as

$$\begin{aligned} C_{II}\approx & {} \frac{1}{72} \left( \frac{a_1}{a_2}\right) ^{3/2}\left( \frac{a_3}{a_2}\right) ^{3}a_3^{3} \left[ \frac{52007}{9000} -\frac{5}{12}f_\ell (a_2)+\frac{40}{9}f_\ell (a_3) \right. \nonumber \\&\left. +\,\frac{1}{2}f_{\ell \ell }(a_1)+\frac{3}{2}f_{\ell \ell }(a_2) \right. \nonumber \\&\left. +\,\frac{3}{8}f_\ell ^2(a_2)+\frac{3}{2}f_\ell (a_1)f_\ell (a_2)+\left( -\frac{211}{75}+f_\ell (a_2) \right. \right. \nonumber \\&\left. \left. -\,\frac{8}{3}f_\ell (a_3)\right) \alpha _4+\frac{6}{5}\alpha _4^2 \right] (\alpha _1 w + \alpha _2) \nonumber \\&+\,\frac{1}{72} \left( \frac{a_2}{a_1}\right) ^{1/2}\left( \frac{a_3}{a_2}\right) ^{3}a_3^{3} \left[ -144 -\frac{208}{5}\sqrt{\frac{a_2}{a_1}}\alpha _4 +\frac{12}{7} \frac{a_2}{a_1} \alpha _4^2 \right] \alpha _2 \nonumber \\&+\,\frac{ 2\pi }{3} \left( \frac{a_3}{a_2}\right) ^{3} a_3^{3} \left( 1-\frac{a_2}{a_3}\right) \left[ \frac{3}{2} \sqrt{\frac{a_2}{a_1}} -\frac{1}{3} \frac{a_3}{a_1} \alpha _4 + \left( \frac{a_2}{a_1}\right) ^2 \alpha _4^2 \right] \alpha _3.\nonumber \\ \end{aligned}$$

(113)

Or,

$$\begin{aligned} C_{II}\approx & {} \frac{1}{72} \left( \frac{a_1}{a_2}\right) ^{3/2}\left( \frac{a_3}{a_2}\right) ^{3}a_3^{3} \left[ 1066 -72 \sqrt{\frac{a_2}{a_1}} \alpha _4+\frac{6}{5}\alpha _4^2 \right] (\alpha _1 w + \alpha _2) \nonumber \\&+\,\frac{1}{72} \left( \frac{a_2}{a_1}\right) ^{1/2}\left( \frac{a_3}{a_2}\right) ^{3}a_3^{3} \left[ -144 -\frac{208}{5}\sqrt{\frac{a_2}{a_1}}\alpha _4 +\frac{12}{7} \frac{a_2}{a_1} \alpha _4^2 \right] \alpha _2 \nonumber \\&+\,\frac{ 2\pi }{3} \left( \frac{a_3}{a_2}\right) ^{3} a_3^{3} \left( 1-\frac{a_2}{a_3}\right) \left[ \frac{3}{2} \sqrt{\frac{a_2}{a_1}} -\frac{1}{3} \frac{a_3}{a_1} \alpha _4 + \left( \frac{a_2}{a_1}\right) ^2 \alpha _4^2 \right] \alpha _3.\nonumber \\ \end{aligned}$$

(114)

Putting in values for w, $a_1$, $a_2$, and $a_3=1$ from appendix  C gives

$$\begin{aligned} C_{II}\approx & {} \left( 4 \times 10^{-5} -2 \times 10^{-4} {\alpha _4 } +4 \times 10^{-8} {\alpha _4 ^2}\right) \alpha _1 \nonumber \\&+\,\left( -3\times 10^{2} -6\times 10^{3} \alpha _4 +2 \times 10^{4} {\alpha _4 ^2} \right) \alpha _2 \nonumber \\&+\, 8\pi \left( 5 -10^2 \alpha _4 +10^6 \alpha _4^2 \right) \alpha _3. \end{aligned}$$

(115)