1 Introduction
Inflation is a successful scenario of the early universe[1-4] However, it is well-known that inflation suffered from the singularity problem.[5-6] This suggests that our understanding about the gravity and the early universe is incomplete. Instead of going to the quantum regime and studying the physics of the "singularity", one might construct classical nonsingular cosmological models alternative or complementary to the inflation scenario. Bouncing cosmology is a class of such models with different applications, see e.g. Refs. [7-9] for earlier studies, Refs. [10-12] for recent reviews.
Building nonsingular cosmological models in the scalar field theories has been still one of the endeavors. It has been observed that the spatially flat bounce models constructed in Horndeski theories[13] inevitably encounter instabilities (or else the singularity in Lagrangian),[14-15] the so-called No-go Theorem, see also Refs. [16-18] for the attempts in the Horndeski theory. Recently, based on the effective field theory (EFT) of cosmological perturbations, it has been found that the solutions of fully stable (without ghosts, gradient instabilities, etc., throughout the whole evolution) cosmological bounce do exist if one goes beyond Horndeski,[19-20] see Refs. [21-23] for the corresponding bounce models performed in full covariant Lagrangians (of the beyond Horndeski theory[24]). The progress caused by "No-go" have also stimulated lots of studies, e.g. Refs. [25-32].
Beyond Horndeski theories are a subclass of the degenerate higher-order scalar-tensor (DHOST) theory.[33-36] Unlike in general relativity (GR), the propagating speed $c_{T}$ of gravitational waves (GW) in the DHOST theory might deviate considerably from the speed of light. Recently, the detection of GW170817[37] and its electromagnetic counterpart has provided a precise measurement for the speed of GWs: it coincides with the speed of light with deviations smaller than a few $\times 10^{-15}$, i.e.\ $c_{T}=1$. This measurement strictly constrained the scalar-tensor theories responsible for the acceleration of the current universe.[38-43] Though the physics implied by GW170817 seems not straightly related with that for the primordial universe, undeniably, such potential relevance will be interesting.
In this paper, inspired by the implication of GW 170817, we will construct the stable cosmological bounce models with $c_{T}=1$. Using the ADM metric, we replace the covariant $c_{T}=1$ DHOST Lagrangian with its ADM form (Sec. 2), and perform the perturbation calculations with it. We construct fully stable bounce models in the beyond Horndeski theory with $c_{T}=1$ (SubSec. 3.1), which is a special subclass of the DHOST theory, and in the full $c_{T}=1$ DHOST theory (SubSec. 3.2), respectively.
2 DHOST Theory with $c_{T}=1$
2.1 The Lagrangians
We begin with the covariant Lagrangian of the beyond Horndeski theory with $c_{T}=1$[38]
$ \mathcal{L}_{c_{T}=1}^{bH}=\sqrt{-g}L_{c_{T}=1}^{bH}=\sqrt{-g}[G_2(\phi,X)+G_3(\phi,X)\square\phi \\ +B_4(\phi,X){R}-\frac{4}{X}B_{4,X}(\phi,X) \\ \times (\phi^\mu\phi^\nu\phi_{\mu\nu}\square\phi -\phi^\mu\phi_{\mu\nu}\phi_\lambda\phi^{\lambda\nu})]\,, $
where $\nabla_\mu\phi\equiv \phi_\mu$, $\nabla_\nu\nabla_\mu\phi\equiv\phi_{\mu\nu}$, and $X\equiv\phi_\mu\phi^\mu$.
We adopt the ADM metric
$ d s^2=-N^2d t^2+h_{ij}(d x^i+N^id t)(d x^j+N^jd t)\,, $
where $N$ is the lapse, $N_i$ is the shift, $h_{ij}$ is the spatial metric. We will use $\eta =\phi$ as the time coordinate in the FRW metric, $d s^2=-N(\eta )^2d\eta ^2+a^2|d\vec{x}|^2$. Dynamics of $\phi$ has been absorbed into $N(\eta )$, since $\phi'\equiv{d\phi}/{d\eta }=1$.
In the unitary gauge $\delta\phi=0$, the covariant Lagrangian $L_{c_{T}=1}^{bH}$ (1) may be rewritten in the ADM form[24]
$ L^{bH}_{c_{T}=1}=P(N,\eta )+Q(N,\eta )K+A(N,\eta )(R-K_2)\,, $
where $R\equiv h^{ij}R_{ij}$ is the Ricci scalar on the spacelike hypersurface, $K\equiv h^{ij}K_{ij}$ is the extrinsic curvature on the spacelike hypersurface and $K_2\equiv K^2-K_{ij}K^{ij}$. The coefficients $P(N,\eta )$, $Q(N,\eta )$ and $A(N,\eta )$ are related with $G_2$, $G_3$, and $B_4$ in Eq. (1) by
$ P(X,\phi)=G_2-\sqrt{-X}\int\frac{G_{3,\phi}}{2\sqrt{-X}}d X\,, \\ Q(X,\phi)=-\int G_{3,X}\sqrt{-X}d X \\ \hphantom{ Q(X,\phi)=}+2(-X)^{3/2}\int\frac{XB_{4,X\phi}-B_{4,\phi}}{X^2}d X\,, \\ A(X,\phi)=B_4\,. $
The covariant Lagrangian $L^{\rm DHOST}_{c_{T}=1}$ of the DHOST theory with $c_{T}=1$ has been identified in Ref. [43]. As pointed out in Ref. [38], $L^{\rm DHOST}_{c_{T}=1}$ may be obtained by performing a conformal rescaling $g_{\mu\nu}\to C(\phi,X)g_{\mu\nu}$ to $L^{bH}_{c_{T}=1}$. Since the light cone is not altered, the corresponding DHOST theory will maintain $c_{T}=1$. Therefore, the ADM Lagrangian of DHOST theories with $c_{T}=1$ may be straightly calculated by rescaling $N\to\sqrt{C}N$, $h_{ij}\to Ch_{ij}$, $h^{ij}\to C^{-1}h^{ij}$, where $C=C(N,\eta)$. Here, without loss of generality, we will set $N_i=0$ in the calculation. Considering $K_{ij}=({1}/{2N})(h_{ij}'-\nabla_i N_j-\nabla_j N_i)$, after some integrations by parts and the redefinition of coefficients, we have
$ \mathcal{L}^{\rm DHOST}_{c_{T}=1}=N\sqrt{h}\,{L}^{\rm DHOST}_{c_{T}=1} \\ = N\sqrt{h}\Big[P+QK+A(R-K_2) \\ -\frac{3AB^2}{2N^2}N^{\prime2}-\frac{2AB}{N}N'K \\ +\frac{B}{a^2}\Big(2\frac{A}{N}+2A_N-\frac{AB}{2}\Big)(\partial N)^2\Big]\,, $
where $B(N,\eta)=\partial_N(\log C)$. It can be checked that this ADM Lagrangian is equivalent to the covariant $L^{\rm DHOST}_{c_{T}=1}$ showed in Ref. [43]. When $B=0$ (or $C=$ const.), ${L}^{\rm DHOST}_{c_{T}=1}$ reduces to the beyond Horndeski Lagrangian $L^{bH}_{c_{T}=1}$ in Eq. (3). In order to write out Eq. (5), we have absorbed the term linear in $N'$ into $P+QK$ by
$$ \frac{f(N,\eta)}{N}N'=n^{\mu}\nabla_\mu F-\frac{1}{N} \partial_\eta F \\ =-F K-\frac{1}{N}\partial_\eta F\,, $$
where $F\equiv\int f d N$.
2.2 The EFT of Scalar Perturbation
The quadratic order EFT of the DHOST theory is[35]
$ S^{\rm quad} = \int d^3x d\eta a^3 \frac{M^2}2\Big\{ \delta K_{ij }\delta K^{ij}- \Big(1+\frac23\alpha_{L}\Big)\delta K^2 +(1+\alpha_{T}) \Big( R \frac{\delta \sqrt{h}}{a^3} + \delta_2 R \Big) \\ + h^2\alpha_{K} \delta N^2+4 h \alpha_{B} \delta K \delta N+ ({1+\alpha_{H}}) R \delta N + 4 \beta_1 \delta K \delta N' + \beta_2 \delta N^{\prime2} + \frac{\beta_3}{a^2}(\partial_i \delta N )^2 \Big\}\,. $
Contracting Eq. (5) with Eq. (6), we can directly read off the effective coefficients in EFT (6),
$ \frac{M^2}{2}= NA\,,\quad \alpha_{L} =0\,,\quad \alpha_{T}=0\,, \\ \frac{M^2}{2}h^2\alpha_{ K}=L_N+\frac{1}{2}NL_{NN}\,, \\ \frac{M^2}{2}4h\alpha_{ B}=NL_{NK}+2h L_{Ns}\,, \\ \frac{M^2}{2}({1+\alpha_{ H}})=A+NA_N\,, \\ \frac{M^2}{2}4\beta_1=-2AB\,,\quad\frac{M^2}{2}\beta_2=-\frac{3AB^2}{2N}\,, \\ \frac{M^2}{2}\beta_3=NB\Big(2\frac{A}{N}+2A_N-\frac{AB}{2}\Big)\,, $
where $\mathcal{H}/N\equiv ({d a/d\eta})/{aN}=H$, and ${L}^{\rm DHOST}_{c_{T}=1}=L$ is set for simplicity. Degenerate conditions have been checked
$ \alpha_{ L}= 0\,,\qquad\beta_2=-6\beta_1^2\,, $
$ \\ \beta_3= -2N\beta_1[2(1+\alpha_{ H})+N\beta_1(1+\alpha_{T})]\,. $
Compared with that in Ref. [35], the condition (9) has been slightly modified, since we have not necessarily $N(\eta)=1$ here.
Use the scalar perturbation
$ N_i\equiv\partial_i\psi\,,\qquad h_{ij}\equiv a^2e^{2\zeta}\delta_{ij}\,, $
to expand (5) or EFT (6). When $N_i\ne0$, $N'$ in $\mathcal{L}^{\rm DHOST}_{c_{T}=1}$ (5) should be promoted to $N'-N^i\partial_i N$. In the corresponding result, $\delta N'\zeta'$ is absorbed into $\tilde{\zeta}^{\prime2}$ by replacing $\zeta$ with a new variable $\tilde{\zeta}=\tilde{\zeta}(\zeta,\delta N)$. Using $\delta L/\delta\psi=0$, and after some integrations by parts, we get the quadratic order Lagrangian of $\zeta$,
$ \mathcal{L}_2=a^3\frac{M^2}{2}\Big[U\zeta^{\prime2}-V\frac{(\partial\zeta)^2}{a^2}\Big]\,, $
with coefficients
$ V=\frac{\Sigma}{\gamma^2}+\frac{6}{N^2}\,, $
$ V=2\Big[\frac{N}{a M^2 }\frac{d}{d\eta}(a\mathcal{M})- 1\Big]\,, $
where
$ \gamma\equiv \frac{\mathcal{H}}{N}+N\mathcal{H}\alpha_{ B}-(N\beta_1)'\,. $
The $\gamma$ in Eq. (14) is related to the $\gamma$ in Refs. [16, 44] by $2A\gamma\to \gamma$.
$ \Sigma\equiv\mathcal{H}^2\Big[\alpha_{ K}+6\Big(\alpha_{ B}^2-\frac{\gamma^2}{\mathcal{H}^2N^2}\Big) \\ \hphantom{\Sigma\equiv} -18\alpha_{ B}\beta_1-\frac{6(\mathcal{H}M^2\alpha_{ B}\beta_1)'}{\mathcal{H}^2M^2}\Big]\,, $
$ \mathcal{M}\equiv\frac{M^2}{\gamma}[(1+\alpha_{ H})/N+\beta_1]\,. $
The absence of ghost suggests $U>0$. The sound speed of scalar perturbation is $c_{ S}^2=V/U$. Gradient stability suggests $c^2_{ S}>0$.
3 Stable Bounce Models
We first set the evolutions of background (the Hubble parameter $H$ and $N$). In our model, $H={\mathcal{H}}/{N}$ follows
$ {\mathcal{H}}/{N}=\frac{\eta}{p(\eta)(1+\eta^2)}\,, $
with
$ p(\eta)=p_i+\frac{1+\tanh(({\eta-\eta_p})/{\tau_p})}{2}(p_f-p_i)\,, $
where $p_f$, $p_i$, $\eta_p$, and $\tau_p=$ const. Initially $\eta\ll -1$, ${\cal H}<0$, the universe contracts with $p(\eta)=p_i$ ($p_i\gg 1$ corresponds to the ekpyrotic contraction[7]). Cosmological bounce occurs at $\eta=0$. Hereafter, the universe expands, and ${\cal H}>0$ has the desired asymptotic form $\sim 1/(p_f\eta)$, see Fig. 1. Meanwhile $N$ follows
$ x(\eta)\equiv \frac{1}{N}=x_i+\frac{1+\tanh(({\eta-\eta_x})/{\tau_x})}{2}(x_f-x_i)\,, $
Fig. 1 The evolution of $H$ with $p_i=8$ and $p_f=3$. |
3.1 In Beyond Horndeski Theory
We set $M_p^2=(8\pi G)^{-1}=1$, and write $P(N,\eta)$, $Q(N,\eta)$ and $A(N,\eta)$ in Eq. (3) as
$ P(N,\eta)=g_1(\eta)\frac{1}{2N^2}+g_2(\eta)\frac{1}{N^4}+g_3(\eta)\,, \\\ noa2 Q(N,\eta)=0\,, \quad A(N,\eta)=\frac{1}{2}+f_1(\eta)\frac{1}{N^2}\,, $
where the $N$-dependent part of $A(N,\eta)$ sets the coefficient $\sim B_{4,X}(\phi, X)\neq 0$ in $L_{c_{T}=1}^{bH}$ (1), and is required for the fully stable bounce.[19-22] $Q(N,\eta)$ is related with the cubic Galileon $G_3(\phi,X)\Box\phi$ in $L_{c_{T}=1}^{bH}$ (1), see Refs. [45-48] for the so-called G-bounce and Ref. [49] for super-bounce. However, $G_3(\phi,X)\Box\phi$ only moves the period of $c_{ S}^2<0$ to the outside of the bounce phase, but cannot dispel it completely, as pointed out in Refs. [16, 46]. Thus we set $Q(N,\eta)=0$ for simplicity.
Since $Q(N,\eta)=0$, Eq. (14) is simplified as
$ \gamma=\frac{\mathcal{H}}{AN}(A-NA_N)\,, $
$ \Sigma=c_1(\eta)\gamma^2\,. $
According to Eq. (3), we have the equations of ${\cal H}$ and $N$ as follows,
$\frac{h^2}{N^2}(A-NA_N)=-P-NP_N-3\frac{h}{N} (NQ_N)\,, \frac{4}{aN}\Big(\frac{a'}{N}A\Big)'=-P-2\frac{h^2}{N^2}A+\frac{1}{N}(Q_{\eta}+Q_NN'). $
One can solve out $g_1(\eta)$, $g_2(\eta)$, and $g_3(\eta)$ in $P(N,\eta)$ algebraically by considering Eqs. (22) and (23), which are showed in Appendix A.
Substituting the corresponding solutions into Eq. (16), we have
$$ \mathcal{M}=\frac{1-4f_1^2x^4}{2\mathcal{H}(1+6f_1x^2)}\,. $$
We choose $f_1(\eta)$ as
$ f_1(\eta)=c_2(\eta)\frac{c_3(\eta)\mathcal{H}(\eta)+1}{2x^2(\eta)}\,,\quad c_2(0)=1\,, $
to make $\mathcal{M}$ not divergent at ${\cal H}=0$. $V>0$ (avoiding the gradient instability) can be insured by adjusting $c_2(\eta)$ and $c_3(\eta)$, noting $1-4f_1^2x^4=0$ at ${\cal H}=0$.
Therefor, with $c_1(\eta)$, $c_2(\eta)$, and $c_3(\eta)$ satisfying certain conditions, we will have a fully stable bounce model. As a concrete example, setting
$ c_1(\eta)=k_1\Big[1-\tanh\Big(\frac{\eta}{\tau_1}\Big)\Big]\,, \\\ c_2(t)=\exp\Big(-\frac{\eta^2}{\tau^2_2}\Big)\,, \quad c_3(\eta)\equiv k_2\,, $
we plot Figs. 2 adn 3 with the parameters $p_i=8$, $p_f=3$, $\tau_p=1$, $\eta_p=0.7$, and $-\eta_x=\tau_x=3$ in Eqs. (17) and (19), as well as $k_1=0.06$, $k_2=2$, $\tau_1=2$, and $\tau^2_2=0.6$ in Eq. (25). Figure 2 shows that the coefficients $g_1(\eta)$, $g_2(\eta)$, $g_3(\eta),$ and $f_1(\eta)$ in Eq. (20) have been fixed. Figure 3 shows that the model is indeed gradient-stable and ghost-free.
That the gravity should asymptotically approach GR requires $f_1\to 0$ in the asymptotic future. The asymptotic behavior of $f_1$ is controlled by $c_2(\eta)$. As a result, the sound speed squared $c^2_{ S}(+\infty)$ is (assume $c_1(+\infty)$ vanishes)
$$ c^2_{ S}(+\infty)=\frac{-xH'+Hx'}{3H^2x^2}\,. $$
Require $c_{ S}^2(+\infty)=1$ and insert background (19), one finds
$ x_f=\frac{p_f}{3}\,. $
Similarly, $x_i$ is related to $p_i$ by requiring $c_{ S}^2(-\infty)=1$.
3.2 In DHOST Theory
The procedure is similar to that in SubSec. 3.1. We write $P(N,\eta)$, $Q(N,\eta)$, $A(N,\eta)$, and $B(N,\eta)$ in Eq. (5) as
$ P(N,t)=g_1(\eta)\frac{1}{2N^2}+g_2(\eta)\frac{1}{N^4}+g_3(\eta)\,, \ \ Q(N,t)=0\,, \\\ A(N,t)=\frac{1}{2}+\frac{g_4(\eta)}{N^2}\,, \quad B(N,t)=b_0\,, $
with $b_0\neq 0$ constant. So $\beta_1$, $\beta_2$, $\beta_3\neq 0$ in the quadratic order EFT of the DHOST theory (6).
Substituting Eq. (27) into Eq. (14), we have $\gamma\sim 2\mathcal{H}+b_0N'$. Considering (19), we have $N'(\pm\infty)=0$. This suggests $\gamma(-\infty)\sim {\cal H}<0$ and $\gamma(+\infty)>0$. In other words, the existence of $b_0$ only shifts the $\gamma$-crossing point to $\eta_0\neq 0$ instead of eliminating it. Therefore, the condition (22) is still needed. To make $\mathcal{M}$ not divergent at $\eta_0$, we might choose $g_4(\eta)$ as
$ g_4=c_2(\eta)\frac{ c_3(\eta) (b_0 N'+2 \mathcal{H})+N^2 (2 N-b_0)}{2 (b_0+2 N)}\,, \\\ c_2(\eta_0)=1\,. $
Thus with $c_1(\eta)$ required in Eq. (22), $c_2(\eta)$ and $c_3(\eta)$ in Eq. (28), we could have a fully stable bounce model based on the DHOST theory (5).
As a concrete example, setting
$ c_1(\eta)\!=\!k_1e^{-\eta^2/\tau_1^2}, \ \ c_2(\eta)\!=\!e^{-(\eta-\eta_0)^2/\tau^2_2}, \ \ c_3(\eta) \!\equiv \! k_2, $
4 Discussion
We have constructed the spatially flat stable cosmological bounce models with GR asymptotics in the $c_{T}=1$ beyond Horndeski theory and in the full $c_{T}=1$ DHOST theory, respectively. In Ref. [23], the stable bouncing solution with $c_{T}=1$ has also been built in the beyond Horndeski theory (but not in the full DHOST theory). Here, since we start straightly from the Lagrangians with the constraint $c_{T}=1$, the procedure of building models (even in full DHOST theory) is simpler.
It is well-known that the solutions of fully stable cosmological bounce do exist in theories beyond Horndeski. Though the simplest implementing is to work in the beyond Horndeski theory,[21-22] the stable bounce in a full DHOST theory is still interesting for study, which might bring unexpected results. In our implementing, we set the parameter $B(\phi,X)=$ const. in the full $c_{T}=1$ DHOST theory (5), see Eq. (27). Generally, it is not this case. The relevant issue will be studied elsewhere.
The singularity of inflation implies that a bounce preceding inflation might occur,[8] see also Refs. [50-54]. Recently, it has been showed in Ref. [55] that the bounce inflation scenario can explain the power deficit of CMB TT-spectrum at low multipoles, specially the dip at multipole $l\sim 20$. Thus it is interesting to embed the bounce models built here into the corresponding scenario, which might bring distinct imprint of DHOST terms in the CMB spectrum.
Appendix A On $g_1$, $g_2$, $g_3$
We give the explicit algebraic solutions of $g_1$, $g_2$, $g_3$ here.
(i) The Beyond Horndeski Model
Recall that $x\equiv1/N$ and $H=h/N$.
$$ g_1=\frac{1}{2 x^2}\big(288 c_1 f_1^2 H^2 x^6+96 c_1 f_1 H^2 x^4+8 c_1 H^2 x^2+12 H x^3 f_1' \\ +36 f_1 H^2 x^2+12 f_1 x^3 H'+24 f_1 H x^2 x'+6 H^2+6 x H'\big)\,, \\ g_2=\frac{1}{8 x^4}\big(-288 c_1 f_1^2 H^2 x^6-96 c_1 f_1 H^2 x^4-8 c_1 H^2 x^2-4 H x^3 f_1' \\ -60 f_1 H^2 x^2-4 f_1 x^3 H'-8 f_1 H x^2 x'-6 H^2-2 x H'\big)\,, \\ g_3=\frac{1}{8} (-18 H^2 + 8 c_1 H^2 x^2 - 36 f_1 H^2 x^2 + 96 c_1 f_1 H^2 x^4 + 288 c_1 f_1^2 H^2 x^6 \\ - 12 H x^3 f_1' - 6 x H' - 12 f_1 x^3 H' - 24 f_1 H x^2 x')\,. $$
(ii) The DHOST Model
$$ g_1= \frac{1}{8 N^3 (2 g_4+N^2)} \big(-36 b_0^2 c_1 g_4^2 N^3 (N')^2-12 b_0^2 c_1 g_4 N^5 (N')^2-b_0^2 c_1 N^7 (N')^2+432 b_0^2 g_4^2 H N^2 N' \\ +144 b_0^2 g_4^2 N^2 N^{\prime\prime}-504 b_0^2 g_4^2 N (N')^2+72 b_0^2 N^4 g_4' N'+144 b_0^2 g_4 N^2 g_4' N' \\ +360 b_0^2 g_4 H N^4 N'+120 b_0^2 g_4 N^4 N^{\prime\prime}-288 b_0^2 g_4 N^3 (N')^2+72 b_0^2 H N^6 N' \\ +24 b_0^2 N^6 N^{\prime\prime}-18 b_0^2 N^5 (N')^2-144 b_0 c_1 g_4^2 H N^3 N'-48 b_0 c_1 g_4 H N^5 N' \\ -4 b_0 c_1 H N^7 N'+720 b_0 g_4^2 H^2 N^2+96 b_0 g_4^2 N^2 H'-816 b_0 g_4^2 H N N' \\ -48 b_0 g_4^2 N N^{\prime\prime}+144 b_0 g_4^2 (N')^2+120 b_0 H N^4 g_4'+240 b_0 g_4 H N^2 g_4' \\ -48 b_0 g_4 N g_4' N'-24 b_0 N^3 g_4' N'+576 b_0 g_4 H^2 N^4+120 b_0 g_4 N^4 H'-456 b_0 g_4 H N^3 N' \\ -48 b_0 g_4 N^3 N^{\prime\prime}+96 b_0 g_4 N^2 (N')^2+108 b_0 H^2 N^6+36 b_0 N^6 H'-24 b_0 H N^5 N' \\ -12 b_0 N^5 N^{\prime\prime}+12 b_0 N^4 (N')^2-144 c_1 g_4^2 H^2 N^3-48 c_1 g_4 H^2 N^5-4 c_1 H^2 N^7 \\ -288 g_4^2 H^2 N-96 g_4^2 N H'+288 g_4^2 H N'-48 H N^3 g_4'-96 g_4 H N g_4'-192 g_4 H^2 N^3 \\ -96 g_4 N^3 H'+192 g_4 H N^2 N'-24 H^2 N^5-24 N^5 H'+24 H N^4 N'\big)\,, \\\ g_2=\frac{1}{32 N (2 g_4+N^2)}\big(36 b_0^2 c_1 g_4^2 N^3 (N')^2+12 b_0^2 c_1 g_4 N^5 (N')^2+b_0^2 c_1 N^7 (N')^2-288 b_0^2 g_4^2 H N^2 N'-96 b_0^2 g_4^2 N^2 N^{\prime\prime} \\ +456 b_0^2 g_4^2 N (N')^2-48 b_0^2 N^4 g_4' N'-96 b_0^2 g_4 N^2 g_4' N'-216 b_0^2 g_4 H N^4 N'-72 b_0^2 g_4 N^4 N^{\prime\prime} \\ +264 b_0^2 g_4 N^3 (N')^2-36 b_0^2 H N^6 N'-12 b_0^2 N^6 N^{\prime\prime}+18 b_0^2 N^5 (N')^2+144 b_0 c_1 g_4^2 H N^3 N' \\ +48 b_0 c_1 g_4 H N^5 N'+4 b_0 c_1 H N^7 N'-432 b_0 g_4^2 H^2 N^2+816 b_0 g_4^2 H N N'+16 b_0 g_4^2 N N^{\prime\prime} \\ -48 b_0 g_4^2 (N')^2-72 b_0 H N^4 g_4'-144 b_0 g_4 H N^2 g_4'+16 b_0 g_4 N g_4' N'+8 b_0 N^3 g_4' N' \\ -288 b_0 g_4 H^2 N^4-24 b_0 g_4 N^4 H'+456 b_0 g_4 H N^3 N'+16 b_0 g_4 N^3 N^{\prime\prime}-32 b_0 g_4 N^2 (N')^2 \\ -36 b_0 H^2 N^6-12 b_0 N^6 H'+24 b_0 H N^5 N'+4 b_0 N^5 N^{\prime\prime}-4 b_0 N^4 (N')^2+144 c_1 g_4^2 H^2 N^3 \\ +48 c_1 g_4 H^2 N^5+4 c_1 H^2 N^7+480 g_4^2 H^2 N+32 g_4^2 N H'-96 g_4^2 H N'+16 H N^3 g_4' \\ +32 g_4 H N g_4'+288 g_4 H^2 N^3+32 g_4 N^3 H'-64 g_4 H N^2 N'+24 H^2 N^5+8 N^5 H' -8 H N^4 N'\big) \,, \\\ g_3= \frac{1}{32 N^5 (2 g_4+N^2)}\big(36 b_0^2 c_1 g_4^2 N^3 (N')^2+12 b_0^2 c_1 g_4 N^5 (N')^2+b_0^2 c_1 N^7 (N')^2-576 b_0^2 g_4^2 H N^2 N'-192 b_0^2 g_4^2 N^2 N^{\prime\prime} \\ +648 b_0^2 g_4^2 N (N')^2-96 b_0^2 N^4 g_4' N'-192 b_0^2 g_4 N^2 g_4' N'-504 b_0^2 g_4 H N^4 N'-168 b_0^2 g_4 N^4 N^{\prime\prime} \\ +408 b_0^2 g_4 N^3 (N')^2-108 b_0^2 H N^6 N'-36 b_0^2 N^6 N^{\prime\prime}+42 b_0^2 N^5 (N')^2+144 b_0 c_1 g_4^2 H N^3 N' \\ +48 b_0 c_1 g_4 H N^5 N'+4 b_0 c_1 H N^7 N'-1008 b_0 g_4^2 H^2 N^2-192 b_0 g_4^2 N^2 H'+816 b_0 g_4^2 H N N' \\ -48 b_0 g_4^2 N N^{\prime\prime}+144 b_0 g_4^2 (N')^2-168 b_0 H N^4 g_4'-336 b_0 g_4 H N^2 g_4'-48 b_0 g_4 N g_4' N' \\ -24 b_0 N^3 g_4' N'-864 b_0 g_4 H^2 N^4-216 b_0 g_4 N^4 H'+456 b_0 g_4 H N^3 N'-48 b_0 g_4 N^3 N^{\prime\prime} \\ +96 b_0 g_4 N^2 (N')^2-180 b_0 H^2 N^6-60 b_0 N^6 H'+24 b_0 H N^5 N'-12 b_0 N^5 N^{\prime\prime}+12 b_0 N^4 (N')^2 \\ +144 c_1 g_4^2 H^2 N^3+48 c_1 g_4 H^2 N^5+4 c_1 H^2 N^7-288 g_4^2 H^2 N-96 g_4^2 N H' +288 g_4^2 H N'-48 H N^3 g_4' \\ -96 g_4 H N g_4'-288 g_4 H^2 N^3-96 g_4 N^3 H'+192 g_4 H N^2 N' -72 H^2 N^5-24 N^5 H'+24 H N^4 N'\big) \,. $$
Acknowledgments
We thank Yong Cai for helpful discussions.