Universality of the spherical collapse with respect to the matter type: the case of a barotropic fluid with linear equation of state

Abstract

We study the spherical collapse of an over-density of a barotropic fluid with linear equation of state in a cosmological background. Fully relativistic simulations are performed by using the Baumgarte–Shibata–Shapiro–Nakamura formalism jointly with the Valencia formulation of the hydrodynamics. This permits us to test the universality of the critical collapse with respect to the matter type by considering the constant equation of state parameter $\omega$ as a control parameter. We exhibit, for a fixed radial profile of the energy-density contrast, the existence of a critical value $\omega ^*$ for the equation of state parameter under which the fluctuation collapses to a black hole and above which it is diluting. It is shown numerically that the mass of the formed black hole, for subcritical solutions, obeys a scaling law $M\propto |\omega - \omega ^*|^\gamma$ with a critical exponent $\gamma$ independent on the matter type, revealing the universality. This universal scaling law is shown to be verified in the empty Minkoswki and de Sitter space-times. For the full matter Einstein-de Sitter universe, the universality is not observed if conformally flat sub-horizon initial conditions are used. But when super-horizon initial conditions computed from the long-wavelength approximation are used, the universality appears to be true.

Appendices

Appendix A: Recovering the primitive variables from the conserved ones

The conserved variables are defined in such a way:

$$\begin{aligned} {\left\{ \begin{array}{ll} D &{}= \rho W\\ S_i &{}= (e+p)W^2v_i\\ \tau &{}= (e+p)W^2-p-D \end{array}\right. } \end{aligned}$$

(97)

where the rest-mass density $\rho$, the energy density e, the pressure p, the velocity $v_i$ and the Lorentz factor $W = \frac{1}{\sqrt{1-v^2}}$ are the primitive variables. In general, the inversion of this relation is not analytical and requires a numerical root-finding procedure (see [37] for more details).

In the case of the barotropic equation of state $p = \omega e$, this inversion can be made analytically. To obtain it, we start by squaring the second equation of (97):

$$\begin{aligned} S^2 := S^iS_i = (e+p)^2W^4v^2. \end{aligned}$$

(98)

Recalling that $v^2 = 1-\frac{1}{W^2}$, (98) becomes

$$\begin{aligned} S^2 = (e+p)^2W^4 - (e+p)^2W^2. \end{aligned}$$

(99)

The third equation of (97) gives

$$\begin{aligned} (e+p)^2W^4= & {} (\tau +D+p)^2\\ (e+p)^2W^2= & {} (e+p)(\tau +D+p). \end{aligned}$$

Reinserting in (99) gives the relation²

$$\begin{aligned} S^2 = (\tau +D)^2 + (\tau + D)(p-e) - pe. \end{aligned}$$

(100)

Using now the equation of state $p=\omega e$, we obtain a quadratic equation in e:

$$\begin{aligned} -\omega e^2 + (\omega -1)(\tau +D)e + (\tau +D)^2 - S^2 = 0. \end{aligned}$$

(101)

Its solutions are

$$\begin{aligned} e = (\tau +D) - \frac{S^2}{\tau +D},\quad \text { if}\,\omega =0, \end{aligned}$$

(102)

and

$$\begin{aligned} e = \tau +D, \quad \text { if}\,\omega =-1, \end{aligned}$$

(103)

and

(104)

The sign we have to consider depends on the value of $\omega$.

If $0 < \omega \le 1$, we have $(\omega - 1)(\tau +D) \le 0$ and thus we have to take the plus to keep a non negative energy density.

If $\omega > 1$, the positivity of the interior of the root and the fact that

$$\begin{aligned} (\omega +1)(\tau +D)^2 - 4\omega S^2 \le (\omega +1)\left[ (\tau +D)^2-S^2\right] \end{aligned}$$

(105)

imply that

$$\begin{aligned} e_+e_- = \frac{(\tau +D)^2-S^2}{-\omega } \le 0, \end{aligned}$$

(106)

where the equality holds if and only if $e = 0$. The latter case is trivial,because it requires $D = 0$, $S_i = 0$ and $\tau = 0$, and in the other cases, this means that the solutions $e_+$ and $e_-$ have opposite signs and that only one is positive.

If $\omega < 0$, the situation is less obvious because both solutions can be positive. For example, if $\omega \in ]-1;0[$, we have

$$\begin{aligned} e_+e_-= & {} \frac{(\tau +D)^2-S^2}{-\omega }\\= & {} \frac{\left( (e+p)W^2-p\right) ^2-(e+p)^2W^4v^2}{-\omega } \\= & {} \frac{\left( (\omega +1)^2W^4 +\omega ^2 - 2 \omega (\omega +1)W^2\right) e^2}{-\omega } \\&-\frac{ (\omega + 1)^2W^4v^2e^2 }{-\omega }\\= & {} \frac{\left[ (\omega +1)^2W^4(1-v^2) + \omega ^2 - 2\omega (\omega +1)W^2\right] e^2 }{-\omega }\\> & {} 0, \end{aligned}$$

because all the terms of the numerator are positive. This means that both solutions have the same sign and thus are positive. The choice must be done thanks to the continuity of the solution with time, which can be difficult numerically.

Ones the energy density e is computed, the other variables follow easily:

$$\begin{aligned} p= & {} \omega e \end{aligned}$$

(107)

$$\begin{aligned} W= & {} \sqrt{\frac{\tau +D +p}{e+p}}\end{aligned}$$

(108)

$$\begin{aligned} v_i= & {} \frac{S_i}{(e+p)W^2}\end{aligned}$$

(109)

$$\begin{aligned} \rho= & {} \frac{D}{W}. \end{aligned}$$

(110)

Appendix B: Splitting for the source terms in the PIRK operators

The evolution equations are written in the form (40) because we are using the PIRK algorithm. The splitting has been chosen to ensure the scheme to be as stable as possible (see [24]). In the first step, the hydrodynamical conserved variables, the cosmological scale factor a, the lapse $\alpha$, the elements of the conformal 3-metric $\hat{a}$ and $\hat{b}$ and $\psi$ are evolved explicitly. These are thus included in the $L_1$ operator. In the second step, the extrinsic curvature is evolved. This means that K and $A_a$ are split into the following $L_2$ and $L_3$ operators³:

$$\begin{aligned} L_{2(A_a)}= & {} -\left( \nabla ^r\nabla _r \alpha - \frac{1}{3}\nabla ^2 \alpha \right) \nonumber \\&+ \alpha \left( R^r_r - \frac{1}{3}R\right) , \end{aligned}$$

(111)

$$\begin{aligned} L_{3(A_a)}= & {} \alpha K A_a - \frac{16\pi }{3}\left( S_a - S_b\right) ,\end{aligned}$$

(112)

$$\begin{aligned} L_{2(K)}= & {} -\nabla ^2\alpha , \end{aligned}$$

(113)

$$\begin{aligned} L_{3(K)}= & {} \alpha \left( A_a^2 + 2A_b^2 + \frac{1}{3} K^2 \right) \nonumber \\&+ 4\pi \alpha \left( E + S_a + 2S_b\right) . \end{aligned}$$

(114)

Finally, the auxiliary variable $\hat{\Delta }^r$ is evolved partially implicitly:

$$\begin{aligned} L_{2\left( \hat{\Delta }^r\right) }= & {} -\frac{2}{\hat{a}}\left( A_a\partial _r \alpha + \alpha \partial _r A_a\right) - \frac{4\alpha }{r\hat{b}}\left( A_a -A_b\right) \nonumber \\&+ \frac{\xi \alpha }{\hat{a}}\Bigg [\partial _r A_a - \frac{2}{3}\partial _r K + 6A_a\frac{\partial _r\psi }{\psi }\nonumber \\&+ \left( A_a-A_b\right) \left( \frac{2}{r}+\frac{\partial _r\hat{b}}{\hat{b}}\right) \Bigg ], \end{aligned}$$

(115)

$$\begin{aligned} L_{3\left( \hat{\Delta }^r\right) }= & {} 2\alpha A_a \hat{\Delta }^r - 8\pi j_r \frac{\xi \alpha }{\hat{a}}. \end{aligned}$$

(116)

Note that general expressions can be found in [31].

Appendix C: Kodama mass and mean energy-density contrast in BSSN variables

The Kodama mass was first defined in [48] but we take [13] and [47] as references.

Recall that, in spherical symmetry, the areal radius is the positive quantity R(t, r) defined by the area A(t, r) of the surface defined by constant t and r coordinates in such a way:

$$\begin{aligned} A = 4\pi R^2. \end{aligned}$$

(117)

In our BSSN metric (2), the areal radius is simply the square root of its $\theta \theta$ component:

$$\begin{aligned} R = \sqrt{g_{\theta \theta }} = \psi ^2a\sqrt{\hat{b}}r. \end{aligned}$$

(118)

Consider now the 2-metric $G_{AB} = \begin{pmatrix} g_{tt} &{} g_{tr}\\ g_{rt} &{} g_{rr} \end{pmatrix}$ with $A,B\in {t,r}$. We define the Kodama vector by

$$\begin{aligned} K^A = \epsilon ^{AB}\partial _BR, \end{aligned}$$

(119)

where $\epsilon _{AB} = \sqrt{-G}\varepsilon _{AB}$ with $\varepsilon _{AB}$ being the Levi-Civita symbol and $\epsilon ^{AB} = G^{AC}G^{BD}\epsilon _{CD}$. Working in the zero shift gauge gives

$$\begin{aligned} K^t= & {} -\frac{\partial _rR}{\alpha \psi ^2a\sqrt{\hat{a}}} \end{aligned}$$

(120)

$$\begin{aligned} K^r= & {} \frac{\partial _tR}{\alpha \psi ^2a\sqrt{\hat{a}}}. \end{aligned}$$

(121)

The tensor $K^A$ is extended to a 4-vector $K^\mu$ by posing $K^\theta = K^\phi = 0$. The quantity $S^\mu = T^\mu _\nu K^\nu$ is thus a conserved current (see [48] and [47] for explanations) and its integral, the Kodama mass, is a conserved quantity. The Kodama mass within a sphere of radius r at time t is thus defined by

$$\begin{aligned} M_K(t,r) := 4\pi \int ^r_0S^t\alpha (t,x) R^2(t,x)dx. \end{aligned}$$

(122)

By developing $S^t$, we find

$$\begin{aligned} M_K(t,r) = 4\pi \int ^r_0\left[ T^t_t\left( -R^2\partial _rR\right) +T^t_r\left( R^2\partial _tR\right) \right] dx. \end{aligned}$$

(123)

The expression (9) gives, in the case of a universe filled with one fluid of matter (other cases do not change much things),

$$\begin{aligned} T^t_t= & {} -(e+p)W^2+p = -E \end{aligned}$$

(124)

$$\begin{aligned} T^t_r= & {} (e+p)\frac{W^2}{\alpha }v_r = \frac{S_r}{\alpha }. \end{aligned}$$

(125)

In terms of BSSN variables, we have

$$\begin{aligned} R^2\partial _rR= & {} \psi ^6a^3r^2 \sqrt{\frac{\hat{b}}{\hat{a}}}\left( 1+2r\frac{\partial _r\psi }{\psi }+\frac{r\partial _r\hat{b}}{2\hat{b}}\right) \end{aligned}$$

(126)

$$\begin{aligned} R^2\partial _tR= & {} \psi ^6a^3r^3\sqrt{\frac{\hat{b}}{\hat{a}}}\left( \frac{\dot{a}}{a}+2\frac{\partial _t\psi }{\psi } + \frac{\partial _t\hat{b}}{2\hat{b}}\right) \nonumber \\= & {} -\alpha \psi ^6a^3r^3\sqrt{\frac{\hat{b}}{\hat{a}}} \left( \frac{K}{3}+A_b\right) , \end{aligned}$$

(127)

where we have use (11) and (121314) for the last equality. In conclusion, the expression for the Kodama mass in BSSN variables is

$$\begin{aligned} M_K(t,r)= & {} 4\pi a^3 \int ^r_0 \psi ^6 x^2\sqrt{\frac{\hat{b}}{\hat{a}}}\Bigg [ E\left( 1+2r\frac{\partial _r\psi }{\psi }+\frac{r\partial _r\hat{b}}{2\hat{b}}\right) \nonumber \\&- xS_r\left( \frac{K}{3}+A_b\right) \Bigg ]dx. \end{aligned}$$

(128)

The corresponding quantity for the Friedmann universe used as background is thus

$$\begin{aligned} \overline{M_K}(t,r) = 4\pi a^3\int ^r_0 \overline{e} x^2dx = \frac{4}{3}\pi (ar)^3\overline{e}. \end{aligned}$$

(129)

Note that in the definition of the compaction function (55) we need to compute it at the same areal radius than the local Kodama mass, that is to say

$$\begin{aligned} \overline{M_K}(t,\psi ^2\sqrt{\hat{b}}r)= & {} \frac{4}{3}\pi a^3\psi ^6\overline{e} \sqrt{\frac{\hat{b}}{\hat{a}}}r^3\\= & {} 4\pi a^3\overline{e}\int ^{\psi ^2\sqrt{\hat{b}}r}_0 x^2dx\\= & {} 4\pi a^3\overline{e} \int ^r_0 \psi ^6y^2 \sqrt{\frac{\hat{b}}{\hat{a}}}\left( 1+2y\frac{\partial _r\psi }{\psi }+\frac{y\partial _r\hat{b}}{2\hat{b}}\right) dy, \end{aligned}$$

where we made the change of variable $x = \psi (t,y)^2\sqrt{\hat{b}(t,y)}y$ for the last equality. The last expression, though less simple, can be useful because of its similarity with the first term of (128). For example, in a comoving gauge it gives the relation

$$\begin{aligned}&M_K(t,r) - \overline{M_K}(t,\psi ^2\sqrt{\hat{b}}r) \nonumber \\&\quad =4\pi a^3 \overline{e}\int ^r_0 \delta \psi ^6 \sqrt{\frac{\hat{b}}{\hat{a}}}x^2\Bigg (1+2x\frac{\partial _r\psi }{\psi }+\frac{x\partial _r\hat{b}}{2\hat{b}}\Bigg )dx, \end{aligned}$$

(130)

only in term of the energy-density contrast $\delta = \frac{e-\overline{e}}{\overline{e}}$.

Concerning the mean energy-density contrast, recall that it is defined in the following way:

$$\begin{aligned} \delta _\text {mean}(t,r) = \frac{\int ^R_0\delta R^2dR}{\int ^R_0R^2dR}. \end{aligned}$$

(131)

By using BSSN equations, it gives

$$\begin{aligned} \delta _\text {mean}(t,r) = \frac{\int ^r_0 \delta \psi ^6a^3 \sqrt{\frac{\hat{b}}{\hat{a}}}x^2\left( 1+2x\frac{\partial _r\psi }{\psi }+\frac{x\partial _r\hat{b}}{2\hat{b}}\right) dx}{\int ^r_0\psi ^6a^3 \sqrt{\frac{\hat{b}}{\hat{a}}}x^2\left( 1+2x\frac{\partial _r\psi }{\psi }+\frac{x\partial _r\hat{b}}{2\hat{b}}\right) dx}, \end{aligned}$$

(132)

where the denominator can be replaced by $\frac{R^3}{3} = \frac{\psi ^6a^3\sqrt{\frac{\hat{b}}{\hat{a}}}r^3}{3}$. We thus see the direct relation between the mean energy-density contrast and the compaction in the comoving gauge by looking at the relation (130).

Appendix D: Derivation of the long-wavelength solution in the BSSN variables

We assume that, at fixed time, the universe becomes locally flat homogeneous and isotropic in the limit $\epsilon \rightarrow 0$. We thus assume, still following [50], that $\hat{\gamma }_{ij} = O(\epsilon ^2)$. As described in details in [47], we then obtain

$$\begin{aligned} \psi= & {} O(\epsilon ^0), \end{aligned}$$

(133)

$$\begin{aligned} v^i= & {} O(\epsilon ),\end{aligned}$$

(134)

$$\begin{aligned} v_i= & {} O(\epsilon ^3),\end{aligned}$$

(135)

$$\begin{aligned} D_i v^i= & {} O(\epsilon ^4),\end{aligned}$$

(136)

$$\begin{aligned} W= & {} 1+O(\epsilon ^6),\end{aligned}$$

(137)

$$\begin{aligned} \delta= & {} O(\epsilon ^2),\end{aligned}$$

(138)

$$\begin{aligned} \hat{A}_{ij}= & {} O(\epsilon ^2),\end{aligned}$$

(139)

$$\begin{aligned} h_{ij}= & {} O(\epsilon ^2),\end{aligned}$$

(140)

$$\begin{aligned} \chi= & {} O(\epsilon ^2),\end{aligned}$$

(141)

$$\begin{aligned} \kappa= & {} O(\epsilon ^2), \end{aligned}$$

(142)

where we used the following notations

$$\begin{aligned} \delta:= & {} \frac{e-\overline{e}}{\overline{e}},\\ h_{ij}:= & {} \hat{\gamma }_{ij} - \overline{\hat{\gamma }}_{ij},\\ \chi:= & {} \alpha - 1,\\ \kappa:= & {} \frac{K-\overline{K}}{\overline{K}}. \end{aligned}$$

Moreover, the conformal factor can be decomposed in such a way:

$$\begin{aligned} \psi (t,x^i) = \Psi (x^i)\left( 1+\xi (t,x^i)\right) , \end{aligned}$$

(143)

where $\Psi = O(\epsilon ^0)$ and $\xi = O(\epsilon ^2)$. In fact, as also shown in [50] and [47], all slicings coincide up to $O(\epsilon )$, which allows to apply the long-wavelength scheme in any slicing. After having done it in the special case where $p = \omega e$, the evolution equations become

$$\begin{aligned}&\dot{\delta } + 6 \dot{\xi } + 3H\omega \left( \chi + \kappa \right) = O(\epsilon ^4), \end{aligned}$$

(144)

$$\begin{aligned}&\frac{1}{1+\omega }\dot{\delta }+6\dot{\xi } = O(\epsilon ^4),\end{aligned}$$

(145)

$$\begin{aligned}&\partial _t\left( a^3(1+\omega )\overline{e}u_i\right) = -a^3\overline{e}\left[ \omega \partial _i \delta + (1+\omega )\partial _i \chi \right] + O(\epsilon ^5),\end{aligned}$$

(146)

$$\begin{aligned}&\overline{\Delta }\Psi = -2\pi \Psi ^5a^2\overline{e}(\delta -2\kappa ) + O(\epsilon ^4),\end{aligned}$$

(147)

$$\begin{aligned}&H^{-1}\dot{\kappa } = \frac{3\omega -1}{2}\kappa - \frac{3(1+\omega )}{2}\chi -\frac{1+3\omega }{2}\delta + O(\epsilon ^4),\end{aligned}$$

(148)

$$\begin{aligned}&\partial _t h_{ij} = -2 \hat{A}_{ij} + O(\epsilon ^4),\end{aligned}$$

(149)

$$\begin{aligned}&\partial _t\hat{A}_{ij} + 3H\hat{A}_{ij} = \frac{1}{a^2\Psi ^4}\Bigg [-\frac{2}{\Psi }\left( \overline{D}_i\overline{D}_j\Psi - \frac{1}{3}\overline{\hat{\gamma }}_{ij}\overline{\Delta }\Psi \right) \nonumber \\&+ \frac{6}{\Psi ^2}\left( \overline{D}_i\Psi \overline{D}_j\Psi - \frac{1}{3}\overline{\hat{\gamma }}_{ij}\overline{D}^k\Psi \overline{D}_k\Psi \right) \Bigg ]+ O(\epsilon ^4),\nonumber \\\end{aligned}$$

(150)

$$\begin{aligned}&\overline{D}_i\left( \Psi ^6\hat{A}^i_j\right) +2H\Psi ^6\overline{D}_j\kappa = 8\pi \Psi ^6(1+\omega )\overline{e}u_j +O(\epsilon ^5), \end{aligned}$$

(151)

where $\overline{D}$ and $\overline{\Delta }$ are the covariant derivative and the Laplacian operators related to the flat metric written in the coordinates $\lbrace x^i\rbrace$. In spherical coordinates, we have that this metric is given by $\overline{\hat{\gamma }}_{ij} = \text {diag}(1,r^2,r^2\sin ^2\theta )$. The resulting equations have been solved in [47] for the constant mean curvature (CMC) slicing (for which $K = \overline{K}$), the comoving slicing, the uniform-density slicing (for which $\delta = 0$) and the geodesic slicing. The paper also gives comparison of the solutions between these four slicings. In our case, we will solve them for a general Bona-Masso slicing (5354). This solution has, to our knowledge, never been derived in the literature.

We start with the computation of the variables $\hat{A}_{ij}$ and $h_{ij}$ which, as explained in [47], do not depend on the slicing to $O(\epsilon ^2)$. If we use the intermediate variable

$$\begin{aligned} p_{ij}:= & {} \frac{1}{\Psi ^4}\Bigg [-\frac{2}{\Psi }\left( \overline{D}_i\overline{D}_j\Psi - \frac{1}{3}\overline{\hat{\gamma }}_{ij}\overline{\Delta }\Psi \right) \nonumber \\&+\frac{6}{\Psi ^2}\left( \overline{D}_i\Psi \overline{D}_j\Psi - \frac{1}{3}\overline{\hat{\gamma }}_{ij}\overline{D}^k\Psi \overline{D}_k\Psi \right) \Bigg ], \end{aligned}$$

(152)

the explicit expressions of $\hat{A}_{ij}$ and $h_{ij}$ are given, solving (149) and (150), by

$$\begin{aligned}&\hat{A}_{ij} = \frac{2}{3\omega +5}p_{ij} H \left( \frac{1}{aH}\right) ^2 + O(\epsilon ^4), \end{aligned}$$

(153)

$$\begin{aligned}&h_{ij} = -\frac{4}{(3\omega +5)(3\omega +1)}p_{ij}\left( \frac{1}{aH}\right) ^2+ O(\epsilon ^4). \end{aligned}$$

(154)

If we want to use spherical symmetry and the BSSN variables presented in Sect. 2, we must use the quantities

$$\begin{aligned} p_a := p^r_r&= \frac{1}{\Psi ^4}\Bigg [-\frac{4}{3\Psi }\left( \partial ^2_r\Psi -\frac{1}{r}\partial _r\Psi \right) \nonumber \\&\quad + \frac{4}{\Psi ^2}\left( \partial _r\Psi \right) ^2\Bigg ], \end{aligned}$$

(155)

$$\begin{aligned} p_b := p^\theta _\theta&= \frac{1}{\Psi ^4}\Bigg [\frac{2}{3\Psi }\left( \partial ^2_r\Psi -\frac{1}{r}\partial _r\Psi \right) \nonumber \\&\quad - \frac{2}{\Psi ^2}\left( \partial _r\Psi \right) ^2\Bigg ]. \end{aligned}$$

(156)

With these definitions, we can write

$$\begin{aligned} A_a= & {} \frac{2}{3\omega +5}p_a H \left( \frac{1}{aH}\right) ^2 + O(\epsilon ^4),\\ A_b= & {} \frac{2}{3\omega +5}p_b H \left( \frac{1}{aH}\right) ^2 + O(\epsilon ^4),\\ \hat{a}= & {} 1 - \frac{4}{(3\omega +5)(3\omega +1)}p_a\left( \frac{1}{aH}\right) ^2 + O(\epsilon ^4),\\ \hat{b}= & {} 1 - \frac{4}{(3\omega +5)(3\omega +1)}p_b\left( \frac{1}{aH}\right) ^2 + O(\epsilon ^4), \end{aligned}$$

where we have $p_a+2p_b = 0$, implying the desired conditions $A_a + 2A_b = 0$ and $\hat{a}\hat{b}^2 = 1 + O(\epsilon ^4)$. The variable $\hat{\Delta }^r$ can be determined thanks to equation (5).

We will now determine the expressions for $\delta$, $\xi$, $\kappa$ and $\chi$ by solving equations (144), (145), (147), (148) and (5354). The combination of the first two ones gives

$$\begin{aligned} H^{-1}\dot{\delta } + 3(1+\omega )(\chi + \kappa ) = O(\epsilon ^4). \end{aligned}$$

(157)

The Bona-Masso equation (5354) reads, after being expanded around $\alpha = 1$,

$$\begin{aligned} H^{-1}\dot{\chi } = 3f(1) \kappa +O(\epsilon ^4). \end{aligned}$$

(158)

With equation (148) and the change of variable $s = \ln a$, we obtain the differential linear system

$$\begin{aligned} \partial _s \begin{pmatrix} \kappa \\ \delta \\ \chi \end{pmatrix} = M(\omega ) \begin{pmatrix} \kappa \\ \delta \\ \chi \end{pmatrix} + O(\epsilon ^4), \end{aligned}$$

(159)

where the matrix $M(\omega )$ is given by

$$\begin{aligned} \displaystyle M(\omega ) = \displaystyle \begin{pmatrix} \displaystyle \frac{3\omega -1}{2} &{} \displaystyle -\frac{1+3\omega }{2} &{} \displaystyle -\frac{3(1+\omega )}{2}\\ \displaystyle -3(1+\omega ) &{} \displaystyle 0 &{} \displaystyle -3(1+\omega )\\ \displaystyle 3f(1) &{} \displaystyle 0 &{} \displaystyle 0 \end{pmatrix}. \end{aligned}$$

(160)

This matrix admits the following eigenvalues:

$$\begin{aligned}&\lambda _* = 1+3\omega , \end{aligned}$$

(161)

$$\begin{aligned}&\lambda _\pm = -\frac{3}{4}\left[ 1+\omega \pm \sqrt{(1+\omega )(1+\omega -8f(1))}\right] . \end{aligned}$$

(162)

The general solution of the system (159) is thus given by

$$\begin{aligned} \begin{pmatrix} \kappa \\ \delta \\ \chi \end{pmatrix} \quad = C_* \mathbf {v}_* a^{\lambda _*} + C_+ \mathbf {v}_+ a^{\lambda _+} + C_- \mathbf {v}_- a^{\lambda _-} + O(\epsilon ^4), \end{aligned}$$

(163)

where $\mathbf {v}_*$, $\mathbf {v}_+$ and $\mathbf {v}_-$ are the eigenvectors associated respectively to $\lambda _*$, $\lambda _+$ and $\lambda _-$ and $C_*$, $C_+$ and $C_-$ are integration constants.

For the geodesic slicing, $\lambda _+ = \displaystyle -\frac{3(1+\omega )}{4} <0$ and $\lambda _- = 0$. In the other Bona–Masso slicings of Sect. 3.1.3, $\lambda _+$ and $\lambda _-$ are complex conjugates with a negative real part. Since we only take pure growing modes, we set $C_+ = C_- = 0$. We have that the eigenvector of $\lambda _*$ is given by

$$\begin{aligned} \begin{pmatrix} v_*^1\\ v_*^2\\ v_*^3 \end{pmatrix} = \begin{pmatrix} (1+3\omega )^2\\ -3(1+\omega )(1+3\omega +3f(1))\\ 3(1+3\omega )f(1) \end{pmatrix}. \end{aligned}$$

(164)

We can then deduce the two useful relations

$$\begin{aligned} \kappa= & {} \frac{v_*^1}{v_*^2}\delta + O(\epsilon ^4), \end{aligned}$$

(165)

$$\begin{aligned} \chi= & {} \frac{v_*^3}{v_*^2}\delta + O(\epsilon ^4). \end{aligned}$$

(166)

By inserting (165) in (147), we finally obtain

$$\begin{aligned} \delta = \frac{v_*^2}{v_*^2-2v_*^1}F \left( \frac{1}{aH}\right) ^2 + O(\epsilon ^4), \end{aligned}$$

(167)

where we have defined

$$\begin{aligned} F := -\frac{4}{3} \frac{\overline{\Delta \Psi }}{\Psi ^5}. \end{aligned}$$

(168)

We immediately deduce the values of $\kappa$, $\chi$ and $\xi$ thanks to (165), (166) and (145):

$$\begin{aligned} \kappa= & {} \frac{v_*^1}{v_*^2-2v_*^1}F \left( \frac{1}{aH}\right) ^2 + O(\epsilon ^4), \end{aligned}$$

(169)

$$\begin{aligned} \chi= & {} \frac{v_*^3}{v_*^2-2v_*^1}F \left( \frac{1}{aH}\right) ^2 + O(\epsilon ^4), \end{aligned}$$

(170)

$$\begin{aligned} \xi= & {} -\frac{1}{6(1+\omega )} \frac{v_*^2}{v_*^2-2v_*^1}F \left( \frac{1}{aH}\right) ^2 + C\nonumber \\&\quad + O(\epsilon ^4), \end{aligned}$$

(171)

where the integration constant C can be absorbed into $\Psi$ (see [47] in a similar case).

The last equation that remains to solve is (146) to find the velocity $v_i$. By using the expressions of $\delta$ and $\chi$, we find

$$\begin{aligned} u_i= & {} -\frac{2}{3\omega +5}\cdot \frac{\omega v_*^2+(1+\omega ) v_*^3}{(1+\omega )(v_*^2-2v_*^1)}\partial _i F \nonumber \\&\cdot \frac{1}{a}\left[ \int ^a_0 \frac{d\tilde{a}}{H(\tilde{a})} +C\right] \left( \frac{1}{aH}\right) ^2 + O(\epsilon ^5), \end{aligned}$$

(172)

where the integration constant C must be set to zero to keep only growing modes (see [47]). We deduce the value for $v_i$:

$$\begin{aligned} v_i = -\frac{2}{3\omega +5}\cdot \frac{\omega v_*^2+(1+\omega ) v_*^3}{(1+\omega )(v_*^2-2v_*^1)}\partial _i F a \left( \frac{1}{aH}\right) ^3 + O(\epsilon ^5). \end{aligned}$$

(173)

To summarize the results in terms of BSSN variables in spherical symmetry, the long-wavelength solution is given by

$$\begin{aligned} \psi= & {} \Psi \left( 1 -\frac{1}{6(1+\omega )} \frac{v_*^2}{v_*^2-2v_*^1}F \left( \frac{1}{aH}\right) ^2\right) \nonumber \\&\quad + O(\epsilon ^4),\\ A_a= & {} \frac{2}{3\omega +5}p_a H \left( \frac{1}{aH}\right) ^2 + O(\epsilon ^4),\\ A_b= & {} \frac{2}{3\omega +5}p_b H \left( \frac{1}{aH}\right) ^2 + O(\epsilon ^4),\\ \hat{a}= & {} 1 - \frac{4}{(3\omega +5)(3\omega +1)}p_a\left( \frac{1}{aH}\right) ^2 + O(\epsilon ^4),\\ \hat{b}= & {} 1 - \frac{4}{(3\omega +5)(3\omega +1)}p_b\left( \frac{1}{aH}\right) ^2 + O(\epsilon ^4),\\ \delta= & {} \frac{v_*^2}{v_*^2-2v_*^1}F \left( \frac{1}{aH}\right) ^2 + O(\epsilon ^4),\\ K= & {} \overline{K}\left( 1+ \frac{v_*^1}{v_*^2-2v_*^1}F \left( \frac{1}{aH}\right) ^2\right) +O(\epsilon ^4),\\ \alpha= & {} 1+\frac{v_*^3}{v_*^2-2v_*^1}F \left( \frac{1}{aH}\right) ^2 + O(\epsilon ^4),\\ v_r= & {} -\frac{2}{3\omega +5}\cdot \frac{\omega v_*^2+(1+\omega ) v_*^3}{(1+\omega )(v_*^2-2v_*^1)}\partial _i F a \left( \frac{1}{aH}\right) ^3\nonumber \\&\quad + O(\epsilon ^5). \end{aligned}$$