Bounce in GR and higher-order derivative operators

TL;DR

This paper demonstrates that stable, nonsingular cosmologies can be achieved within General Relativity by incorporating higher-order derivative operators of scalar fields, challenging the notion that modifications to GR are necessary for such solutions.

Contribution

It shows that higher-order derivative operators in scalar-tensor theories enable stable nonsingular cosmologies within GR without requiring modifications.

Findings

Stable nonsingular cosmologies are possible within GR using higher-order derivatives.

Higher-order operators do not introduce instabilities at energies below the Planck scale.

This approach challenges the need for modifying GR to violate the null energy condition.

Abstract

Recent progress seems to suggest that one must modify General Relativity (GR) to stably violate the null energy condition and avoid the cosmological singularity. However, with the higher-order derivative operators of scalar field (a subclass of the degenerate higher-order scalar-tensor theory), we show that at energies well below the Planck scale, fully stable nonsingular cosmologies can actually be implemented within GR.

Equations

$$S=\int d^{4}x\sqrt{-g}\left(\frac{M_p^2}{2}R+L_m[g_{\mu \nu }]\right).$$

$$L\sim \frac{M_P^2}{2}{}^{(4)}R+P(\varphi ,X)+\mathcal{O}\left((\square \varphi {)}^2,({\varphi }^{\mu \nu }{)}^2\cdots \right).$$

$$L_{c_T=1}^{DHOST}=$$

$$+B{\varphi }^{\mu }{\varphi }^{\nu }{\varphi }_{\mu \nu }\square \varphi +\frac{B}{A}\left(2A_X+\frac{XB}{2}\right)({\varphi }_{\mu }{\varphi }^{\mu \nu }{\varphi }_{\nu }{)}^2,$$

$$d{s}^2=-N^{2}d{t}^2+h_{ij}(d{x}^i+N^{i}dt)(d{x}^j+N^{j}dt),$$

$$B=-\frac{4}{X}{A}_X+A\tilde{B},$$

$$L_{c_T=1}^{DHOST}=L_{c_T=1}^{bH}+\Delta L,$$

$$\Delta L=$$

$$+\frac{X\tilde{B}}{2}\left(-\frac{4}{X}{A}_X+A\tilde{B}\right)({\varphi }_{\mu }{\varphi }^{\mu \nu }{\varphi }_{\nu }{)}^2.$$

$${\varphi }^{\mu }{\varphi }^{\nu }{\varphi }_{\mu \nu }$$

$$\square \varphi$$

$${\varphi }^{\mu }{\varphi }_{\mu \nu }{\varphi }_{\lambda }{\varphi }^{\lambda \nu }$$

$$\Delta L=-\frac{3A{\tilde{B}}^2}{8N^{12}}{N}^{\prime 2}+\frac{A\tilde{B}}{N^6}{N}^{\prime }K-\frac{\tilde{B}}{N^5}\left(\frac{A}{N}+N{A}_N+\frac{A\tilde{B}}{8N^5}\right)(\partial N{)}^2$$

$$L_{c_T=1}^{DHOST}=$$

$$+\tilde{B}\left(2\frac{A}{N}+2A_N-\frac{A\tilde{B}}{2}\right)(\partial N{)}^2.$$

$$N^i={\delta }^{ij}{\partial }_j\psi ,h_{ij}=a^2(\eta )e^{2\zeta }{\delta }_{ij},$$

$$K_i^j=\frac{1}{N}\left[\left(\mathcal{H}+\dot{\zeta }-\mathcal{H}\frac{\delta N}{N}\right){\delta }_i^j-\frac{{\delta }^{jk}}{a^2}{\partial }_i{\partial }_k\psi \right]+\mathcal{O}(\delta N^2),$$

$$\mathcal{R}=-\frac{2}{a^{2}}\big{[}2\partial^{2}\zeta+(\partial\zeta)^{2}-4\zeta\partial^{2}\zeta\big{]}+\mathcal{O}(\zeta^{3}),$$

$${\mathcal{L}}_{kinetic}^{(2)}=a^3\frac{A}{N}\left(-6{\dot{\zeta }}^2-6\tilde{B}\dot{\zeta }\delta \dot{N}-\frac{3{\tilde{B}}^2}{2}\delta {\dot{N}}^2\right)=-\frac{6a^{3}A}{N}(\dot{\zeta }+\tilde{B}\delta \dot{N}/2{)}^2$$

$${\mathcal{L}}^{(2)}=a^{3}NA\left[U{\dot{\tilde{\zeta }}}^2-V\frac{(\partial \tilde{\zeta }{)}^2}{a^2}\right]$$

$$U=\frac{\Sigma }{{\gamma }^2}+\frac{6}{N^2},$$

$$V=\frac{2}{aA}\frac{d}{d\eta }\left(a\mathcal{M}\right)-2,$$

$$\gamma \equiv \left(\frac{1}{N}+N{\alpha }_B\right)\mathcal{H}+\dot{\tilde{B}}/2,$$

$$\Sigma \equiv {\mathcal{H}}^2\left[{\alpha }_K+6\left({\alpha }_B^2-\frac{{\gamma }^2}{{\mathcal{H}}^2{N}^2}\right)+\frac{9{\alpha }_B\tilde{B}}{N}+\frac{3d({\alpha }_B\mathcal{H}A\tilde{B})/d\eta }{{\mathcal{H}}^{2}NA}\right],$$

$$\mathcal{M}\equiv\frac{1}{\gamma}\left[\big{(}{A\over N}+A_{N}\big{)}-A{\tilde{B}}/2\right].$$

$${\alpha }_B=\frac{1}{4NA\mathcal{H}}(N{L}_{NK}+2\mathcal{H}{L}_{NS}),{\alpha }_K=\frac{1}{NA{\mathcal{H}}^2}\left(L_N+\frac{N}{2}{L}_{NN}\right),$$

$$L_{c_T=1,A=M_P^2/2}^{DHOST}=$$

$$+\tilde{B}\left(\frac{M_P^2}{N}-\frac{M_P^2\tilde{B}}{4}\right)(\partial N{)}^2,$$

$$a\mathcal{M}\big{|}_{f}-a\mathcal{M}\big{|}_{i}>\int_{i}^{f}aAd\eta.$$

$$\mathcal{M}=\frac{M_P^2}{2\gamma }\left(\frac{1}{N}-\tilde{B}/2\right)$$

$$\mathcal{H}=H/N=\frac{\eta }{3(1+{\eta }^2)},$$

$$\tilde{P}(N,\eta )=\frac{g_1(\eta )}{2N^2}+\frac{g_2(\eta )}{N^4}+g_3(\eta ),$$

$$\Sigma =c_1(\eta ){\gamma }^2.$$

Full text

I Introduction

It is well-known that General Relativity (GR) suffers the singularity problem, which indicates that our understanding about gravity and the origin of the universe is incomplete Hawking1970:singularity ; Guth2003:inflationary . It is still an elusive task to look for an ultraviolet (UV)-complete theory to describe what happens at the ”singularity”. However, searching for fully stable nonsingular cosmologies with the effective field theory (EFT), which captures low energy behaviors of the complete theory, might be an alternative approach.

In spatially flat nonsingular cosmologies, the Null Energy Condition (NEC) must be violated for a period. However, it is often accompanied by (ghost, gradient) instabilities Rubakov2016:generalized ; Kobayashi2016:generic , or singularities (strong coupling) in the perturbed action, see also Refs.Easson:2011zy ; Ijjas:2016tpn ; Ijjas2017:fully ; Dobre2018:unbraiding . Recently, it has been found that fully stable nonsingular cosmological solutions do exist in the EFT beyond Horndeski Cai:2016thi ; Creminelli:2016zwa ; Cai:2017tku ; Cai:2017dyi ; Kolevatov:2017voe ; Mironov:2018oec . Degenerate higher-order scalar-tensor (DHOST) theory Langlois:2018dxi actually is a rich pool for such EFTs Ye:2019frg . However, it is noteworthy that in the nonsingular models built, the gravity has been no longer GR-like333By ”GR”, we refer to a theory where matter is minimally coupled to the Einstein-Hilbert action, i.e:

$S=\int d^{4}x\sqrt{-g}\left(\frac{M^{2}_{p}}{2}R+L_{m}[g_{\mu\nu}]\right).$

In particular, the scalar field is minimally coupled to the gravitational metric $g_{\mu\nu}$..

Recently, the LIGO Scientific and Virgo Collaborations have detected the gravitational wave (GW) signals of binary black holes (BH) Abbott:2016blz and binary neutron star mergers TheLIGOScientific:2017qsa , which opened a new window to probe the gravity physics. The results of all tests performed in Refs.TheLIGOScientific:2016src ; Abbott:2018lct showed perfect agreement with GR, particularly in the strong-field regime. Currently, GR is still a well-established effective theory in the low energy regime of the UV-complete theory, though it must break down around the Planck energy.

How to implement the nonsingular bounce with GR? It is well-known that the $P(\phi,X)$ theory can hardly bring a stable NEC violation. To stably realize such a violation, one may include higher-order derivative operators $(\Box\phi)^{2}$, $(\phi^{\mu\nu})^{2}\cdots$ in the $P(\phi,X)$ theory, and set the EFT as, e.g.ArkaniHamed:2003uy ,

[TABLE]

Generally, higher-order corrections ${\cal O}\left((\Box\phi)^{2},(\phi^{\mu\nu})^{2}\cdots\right)$ are generated if one integrates out the massive particles beyond the cutoff scale deRham:2017aoj ; deRham:2018red . One frequently studied case is ${\cal O}\sim(\Box\phi)^{2}$, see e.g.Creminelli:2006xe ; Li:2005fm ; Buchbinder:2007ad . However, the corresponding EFT must beg unknown physics in the sufficiently far past, otherwise the higher-order derivative operator will show itself the Ostrogradski ghost. It’s possible to include such higher-order derivative operators in the so-called DHOST theory Langlois:2015cwa ; Langlois:2017mxy , see also Gleyzes:2014dya , without introducing any Ostrogradski instability Motohashi:2016ftl ; Motohashi:2014opa .

Nevertheless, which operator in ${\cal O}\left((\Box\phi)^{2},(\phi^{\mu\nu})^{2}\cdots\right)$ is indispensable for achieving a pathology-free bounce in GR is still not clear so far. In this paper, we will propose a consistent (1)-like EFT for spatially-flat fully stable nonsingular cosmologies. We, with it, will discuss how to evade the No-go Theorem Rubakov2016:generalized ; Kobayashi2016:generic plaguing the cosmologists, and show a concrete example for the cosmological bounce.

II DHOST theory with $c_{T}=1$

II.1 Reducing to GR

We begin with the DHOST theory with $c_{T}=1$ ($c_{T}$ is the speed of GWs) Langlois:2017dyl

[TABLE]

where $\phi_{\mu}\equiv\nabla_{\mu}\phi$, $\phi_{\mu\nu}\equiv\nabla_{\nu}\nabla_{\mu}\phi$ and $X\equiv\phi_{\mu}\phi^{\mu}$. The coefficients $A$, $Q$ and $B$ only depend on $\phi$ and $X$. According to the classification in Ref.Achour:2016rkg , theory (2) belongs to class Ia DHOST theories. Generally, $B$ and $A$ are independent functions. However, if $B=-\frac{4}{X}A_{X}$, $L^{DHOST}_{c_{T}=1}$ will reduce to the $c_{T}=1$ beyond-Horndeski theory $L^{bH}_{c_{T}=1}$ Creminelli2017:dark .

It is significant to notice that if setting $A=const.$ and $Q=0$, $L^{bH}_{c_{T}=1}$ will reduce to GR, while $L^{DHSOT}_{c_{T}=1}$ will become GR plus extra DHOST operators (higher-order derivative operators). The latter is not covered by the beyond-Horndeski Lagrangian Gleyzes:2014dya but belongs to a subclass of the $c_{T}=1$ DHOST theory. Degenerate conditions required by the DHOST theory guarantee that such a combination of higher-order derivative operators is free of the Ostrodradsky ghost. A (1)-like EFT will be Ostrodradsky ghost-free, only if the degenerate conditions are satisfied.

II.2 Perturbation in DHOST theories with $c_{T}=1$

We adopt the ADM metric,

[TABLE]

where $N$ is the lapse, $N^{i}$ is the shift and $h_{ij}$ is the spatial metric. In the following we will work in the unitary gauge and use $\eta=\phi$ as the time coordinate (assuming $\phi_{\mu}$ is timelike). In particular, in this gauge $\phi_{\mu}=\delta_{\mu}^{0}$ and the dynamics of $\phi$ is absorbed into $N(\eta)$, as $\dot{\phi}\equiv{d\phi}/{d\eta}=1$ (for any operator $\mathcal{O}$, $\dot{\mathcal{O}}$ refers to derivatives with respect to the clock time $\phi$, or equivalently $\eta$, and not as usual to the cosmic time $t$).

Defining

[TABLE]

we have

[TABLE]

where

[TABLE]

In the unitary gauge, one has $L^{bH}_{c_{T}=1}={\tilde{P}}+{\tilde{Q}}K+A(\mathcal{R}-\mathcal{K}_{2})$ Gleyzes:2014dya , where $\mathcal{R}\equiv h^{ij}\mathcal{R}_{ij}$ is the Ricci scalar on the spacelike hypersurface, $K\equiv h^{ij}K_{ij}$ is the extrinsic curvature and $\mathcal{K}_{2}\equiv K^{2}-K_{ij}K^{ij}$. The DHOST operators follow

[TABLE]

Thus we have

[TABLE]

where the equality $X=-1/N^{2}$ is used. Replacing $-{\tilde{B}}/(2N^{5})$ with ${\tilde{B}}$, we get the ADM form of $L^{DHOST}_{c_{T}=1}$ (2)

[TABLE]

We will work with (6). To study the stability of perturbations, we expand $L^{DHOST}_{c_{T}=1}$ in (6) to second order. Defining the metric perturbation

[TABLE]

we have $\mathcal{L}^{(2)}=3a^{3}N\zeta\delta L+a^{3}\delta N\delta L+a^{3}N\delta_{2}L$ at quadratic order, where $\delta_{2}L$ refers to the expansion of $L$ at second order. To proceed, we first expand $K_{i}^{\ j}$ and $\mathcal{R}$,

[TABLE]

[TABLE]

where $\mathcal{H}\equiv{da\over ad\eta}=NH$, and $H$ is the Hubble parameter. The kinetic term in $L^{DHOST}_{c_{T}=1}$ (6) is contributed by $-A\mathcal{K}_{2}-\frac{3A\tilde{B}^{2}}{2N^{2}}N^{\prime 2}-\frac{2A\tilde{B}}{N}N^{\prime}K$. Considering (7) and (8), one finds that

[TABLE]

is diagonal for $\tilde{\zeta}=\zeta+\tilde{B}\delta N/2$. The coefficients of the operators $N^{\prime 2}$ and $N^{\prime}K$ should satisfy a relation in the DHOST theory ($\beta_{2}=-6\beta_{1}^{2}$, see e.g.Ref.Langlois:2017mxy ). As a result, $\mathcal{L}^{(2)}_{kinetic}$ is necessarily diagonal. Confronting ${\tilde{\zeta}}$ with the constraint $\delta L/\delta(\partial^{2}\psi)=0$, we get

[TABLE]

with

[TABLE]

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

Following the notation in Langlois:2017mxy , one sets $\alpha_{B}$ and $\alpha_{K}$ as the coefficients of the operators $\delta K\delta N$ and $\delta N^{2}$ respectively,

[TABLE]

where $\mathcal{S}\equiv K_{ij}K^{ij}$. It can be checked444In Ref.Langlois:2017mxy , a different time parametrization is chosen such that $\bar{N}=1$. that our calculation conform with Ref.Langlois:2017mxy by setting $N=1$.

III Bounce in GR

III.1 Expelling No-go with higher-order derivative operators

In the Horndeski theory, fully stable nonsingular cosmological solutions are prohibited, the so-called No-go Theorem Rubakov2016:generalized ; Kobayashi2016:generic , see also Kolevatov:2016ppi ; Akama:2017jsa ; Ijjas2018:space-time ; Banerjee:2018svi for relevant studies. One way out is going beyond Horndeski, as pointed out in Refs.Cai:2016thi ; Creminelli:2016zwa . In particular, in the beyond-Horndeski subclass of the DHOST theory, solutions of fully stable nonsingular cosmologies have been found Cai:2017dyi ; Kolevatov:2017voe ; Mironov:2018oec ; Ye:2019frg .

Setting $A=M_{P}^{2}/2=const.$ in (6), we have

[TABLE]

which also belongs to a subclass of the DHOST theory. Recall the redefinition (4) and replacement $-{\tilde{B}}/(2N^{5})\to{\tilde{B}}$ in Sect.II, then the coefficient $B$ in the covariant theory (2) is related to the $\tilde{B}$ in (16) by $B=-M_{P}^{2}N^{5}{\tilde{B}}$. It is also noticed that if $A=const.$, ${\tilde{Q}}=0$ in (16) is equivalent to $Q=0$ and $P={\tilde{P}}$ in $L^{DHOST}_{c_{T}=1}$ (2). Thus if ${\tilde{Q}}=0$, (16) is actually a (1)-like EFT.

The essence of the No-go proof is rewriting $V>0$ ($c_{S}^{2}>0$) in (11) as the integral inequality, see Kobayashi:2019hrl for a review,

[TABLE]

In the nonsingular models, the integral $\int_{i}^{f}aAd\eta$ will diverge, thus $\mathcal{M}$ must cross 0 at a certain time. According to (14), we have

[TABLE]

for (16). Thus we might get $\mathcal{M}=0$ by adjusting ${\tilde{B}}(N,\eta)$, or equivalently $B(X,\phi)$ in $L^{DHOST}_{c_{T}=1}$ (2). This suggests that it is possible to build fully stable nonsingular cosmological models with (16) (equivalently, (1)-like EFTs).

III.2 An example

To show that the observation made in Sect.III.1 is correct, we will present a concrete model for the nonsingular bounce, which might have significant applications in early universe scenarios, e.g.Khoury2001:ekpyrotic ; Piao:2003zm ; Piao:2004me ; Qiu:2011cy .

We adopt

[TABLE]

with $N(\eta)=1$ as the background solution. When $\eta<0$, the universe contracted with ${\cal H}\sim 1/\eta<0$. Cosmological bounce happened at $\eta=0$. We might set ${\tilde{P}}(N,\eta)$ and ${\tilde{B}}(N,\eta)$ in $L^{DHOST}_{c_{T}=1,A=const.}$ (16) as

[TABLE]

and ${\tilde{B}}(N,\eta)=g_{4}(\eta)$. Here, since ${\tilde{Q}}=0$, ${\tilde{P}}(N,\eta)$ is actually equivalent to $P(X,\phi)=g_{1}(\phi)X/2+g_{2}(\phi)X^{2}+g_{3}(\phi)$ in $L^{DHOST}_{c_{T}=1}$ (2).

One simple possibility for (13) is, see also Ye:2019frg ,

[TABLE]

According to Eq.(10), we will have $U>0$ for a suitable $c_{1}(\eta)$. Combining Eq.(21) with the background equations (25) and (26) in Appendix A, we get the algebraical solutions of $g_{1}(\eta)$, $g_{2}(\eta)$ and $g_{3}(\eta)$, see Appendix B.

Inserting ${\tilde{B}}(N,\eta)=g_{4}(\eta)$ into Eq.(12), we have $\gamma=\mathcal{H}+{\dot{g}_{4}}/{2}$. Thus

[TABLE]

Requiring that around $\eta=0$, $1-g_{4}/2=0$ and $\dot{g}_{4}\sim{\cal H}$ (so ${\cal M}=0$), we consider such a $g_{4}$,

[TABLE]

with $\lambda$ set by $g_{4}(0)={\mu}e^{1/\lambda^{2}}\Gamma(0,1/\lambda^{2})/3=2$. Fig.1 plots the evolutions of $\dot{g}_{4}$ for $\mu=0.9$ and $\cal H$. When $|\eta|\gg\lambda$, $g_{4}=0$, we will have a $P(X,\phi)$ EFT with GR. Inserting (22) into (11), we have $V(\eta=0)=\frac{2(2\mu-1)}{-\mu+1}$, so $c_{S}^{2}(\eta=0)=V/U>0$ suggests $0.5<\mu<1$.

As a concrete example, we plot Figs.2 and 3 with $c_{1}(\eta)=150e^{-\eta^{2}/500}$. We see that the model is fully stable. As pointed out in Ref.Achour:2016rkg , class Ia DHOST theories can be disformally transformed to Horndeski. It’s proved in Appendix C that such field redefinition is ill-defined in the example considered here.

IV Discussion

Currently, GR is the well-tested effective theory of gravity. Based on the higher-order derivative operators, which might capture the physics of a UV-complete theory, we propose a consistent EFT

[TABLE]

for the spatially-flat fully stable nonsingular cosmologies. It belongs to a subclass ($A=M_{P}^{2}/2$, $Q=0$) of the $c_{T}=1$ DHOST theory (2). It has been speculated that the higher-order derivative operators ${\cal O}\left((\Box\phi)^{2},(\phi^{\mu\nu})^{2}\cdots\right)$ in the EFT (1) might play crucial roles in nonsingular cosmologies. Here, we clearly showed what kind of ${\cal O}\left((\Box\phi)^{2},(\phi^{\mu\nu})^{2}\cdots\right)$ is required for the full stability of nonsingular cosmologies.

We discussed how to evade the No-go Theorem with the EFT (24) (its ADM Langrangian (16)). In Refs.Cai:2016thi ; Creminelli:2016zwa ; Cai:2017dyi ; Kolevatov:2017voe , the operator ${\cal R}\delta g^{00}$ is used to expel the No-go. However, in their implementation, besides higher-order derivative operators, the corresponding covariant EFT also includes the derivative coupling of $\phi$ to gravity $\sim X\,^{(4)}R$. Here, we found that the No-go can be evaded solely by introducing the higher-order derivative operators ${\cal O}\left((\Box\phi)^{2},(\phi^{\mu\nu})^{2}\cdots\right)$ (the DHOST operators) in (24) without modifying GR. A concrete model of the cosmological bounce have been presented in Sect.III.2. Generally, all the operators compatible with the symmetry of the problem are expected to be generated at quantum level. However, only a finite subset of all possible higher-order derivative operators is considered in the example studied. It would thus be interesting to study whether such model is protected against quantum corrections Pirtskhalava:2015nla ; Santoni:2018rrx . It might be also interesting to apply the EFT (24) to regulate the singularity of the BH, e.g.Mironov:2018pjk ; Franciolini:2018aad ; Mironov:2018uou .

Recently, the well-posedness of the initial value problem (IVP) has been promoted in non-perturbative cosmologies Ijjas:2018cdm . An issue worthy of exploring is whether the IVP for (24) is well-posed.

Acknowledgments

We thank Yong Cai for helpful discussions. This work is supported by NSFC, Nos.11575188, 11690021.

Appendix A The background equations

Varying (6) with respect to $N$ and $\mathcal{H}$, respectively, we get

[TABLE]

[TABLE]

Appendix B Solutions of $g_{1}$, $g_{2}$ and $g_{3}$

[TABLE]

[TABLE]

[TABLE]

In Sect.III.2, since $N=1$, (27), (28) and (29) will be simplified.

Appendix C Disformal transformations

In this appendix, we will show that the field redefinition relating the example in Sect.III.2 to a Horndeski theory is ill-defined. According to Ref.Achour:2016rkg , theory (2) can be disformally transformed to a Horndeski theory by the field redefinition $\tilde{g}_{\mu\nu}=\Omega(X,\phi)g_{\mu\nu}+\Gamma(X,\phi)\phi_{\mu}\phi_{\nu}$ where

[TABLE]

A necessary condition for an invertible disformal transformation is Langlois:2017mxy

[TABLE]

For the specific example studied in Sect.III.2, $A=M_{p}^{2}/2,\ B=-M_{p}^{2}N^{5}g_{4},\ X=-1/\bar{N}^{2}=-1$, thus $\Omega-X\Omega_{X}-X^{2}\Gamma_{X}=(1-g_{4}/2)\Omega$. According to (23), the disformal transformation is singular at the bounce point.