Varying the Horndeski Lagrangian within the Palatini approach

TL;DR

This paper investigates the effects of varying the Horndeski Lagrangian within the Palatini approach, revealing conditions under which the resulting metric-affine theories are ghost-free and classifying their cosmological solutions.

Contribution

It provides a detailed analysis of metric-affine Horndeski theories, identifying ghost-free subclasses and exploring their cosmological implications.

Findings

Ghost-free subclass identified for linear-in-connection theories.

Classified homogeneous and isotropic cosmologies within these theories.

Non-linear connection terms generally introduce ghosts, but some can be mitigated.

Abstract

We analyse what happens when the Horndeski Lagrangian is varied within the Palatini approach by considering the metric and connection as independent variables. Assuming the connection to be torsionless, there can be infinitely many metric-affine versions $L_{\rm P}$ of the original Lagrangian which differ from each other by terms proportional to the non-metricity tensor. After integrating out the connection, each $L_{\rm P}$ defines a metric theory, which can either belong to the original Horndeski family, or it can be of a more general DHOST type, or it shows the Ostrogradsky ghost. We analyse in detail the subclass of the theory for which the equations are linear in the connection and find that its metric-affine version is ghost-free. We present a detailed classifications of homogeneous and isotropic cosmologies in these theories. Taking into consideration other pieces of the Horndeski Lagrangian which are non-linear in the connection leads to more complex metric-affine theories which generically show the ghost. In some special cases the ghost can be removed by carefully adjusting the non-metricity contribution, but it is unclear if this is always possible. Therefore, the metric-affine generalisations of the Horndeski theory can be ghost-free, but not all of them are ghost-free, neither are they the only metric-affine theories for a gravity-coupled scalar field which can be ghost-free.

Equations

$$S_{\mathrm{H}}[g_{\mu \nu },\varphi ]=\int L_{\mathrm{H}}{d}^{4}x,$$

$$L_{\mathrm{H}}$$

$${\mathcal{L}}_2$$

$${\mathcal{L}}_3$$

$${\mathcal{L}}_4$$

$${\mathcal{L}}_5$$

$$G_{\text{leavevmode}\nu }^{\mu }=R_{\text{leavevmode}\nu }^{\mu }-\frac{1}{2}R{\delta }_{\text{leavevmode}\nu }^{\mu },\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}{\Phi }_{\text{leavevmode}\beta }^{\alpha }=g^{\alpha \sigma }{\nabla }_{\sigma }{\nabla }_{\beta }\varphi ,$$

$$G_4=G_4(\varphi ),\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}{G}_5=0,$$

$$R_{\mu \nu }\to {\stackrel{(\Gamma )}{R}}_{\mu \nu }\equiv {\partial }_{\alpha }{\Gamma }_{\mu \nu }^{\alpha }-{\partial }_{\nu }{\Gamma }_{\mu \alpha }^{\alpha }+{\Gamma }_{\sigma \alpha }^{\alpha }{\Gamma }_{\mu \nu }^{\sigma }-{\Gamma }_{\mu \sigma }^{\alpha }{\Gamma }_{\nu \alpha }^{\sigma },$$

$$R\to \stackrel{(\Gamma )}{R}\equiv g^{\mu \nu }{\stackrel{(\Gamma )}{R}}_{\mu \nu },\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}{G}_{\text{leavevmode}\nu }^{\mu }\to g^{\mu \sigma }{\stackrel{(\Gamma )}{R}}_{\sigma \nu }-\frac{1}{2}\stackrel{(\Gamma )}{R}{\delta }_{\text{leavevmode}\nu }^{\mu },$$

$${\Phi }_{\text{leavevmode}\beta }^{\alpha }\to g^{\alpha \sigma }{\stackrel{(\Gamma )}{\nabla }}_{\sigma }{\stackrel{(\Gamma )}{\nabla }}_{\beta }\varphi .$$

$$L_{\mathrm{H}}\to L_{\mathrm{P}},$$

$$S_{\mathrm{P}}[{\Gamma }_{\alpha \beta }^{\sigma },g_{\mu \nu },\varphi ]=\int L_{\mathrm{P}}{d}^{4}x.$$

$$S_{\mathrm{P}}[\left\{{}_{\mu \nu }^{\alpha }\right\},g_{\mu \nu },\varphi ]=S_{\mathrm{H}}[g_{\mu \nu },\varphi ].$$

$${\Phi }_{\text{leavevmode}\beta }^{\alpha }\to {\stackrel{(\Gamma )}{\nabla }}_{\sigma }(g^{\alpha \sigma }{\stackrel{(\Gamma )}{\nabla }}_{\beta }\varphi )$$

$$Q_{\alpha \mu \nu }\equiv {\stackrel{(\Gamma )}{\nabla }}_{\alpha }{g}_{\mu \nu },$$

$$L_{\mathrm{P}}\to {\tilde{L}}_{\mathrm{P}}=L_{\mathrm{P}}+\Delta L_{\mathrm{P}},$$

$$\Delta L_{\mathrm{P}}=(c_1{Q}_{\text{leavevmode}\text{leavevmode}\alpha }^{\mu \alpha }{\nabla }_{\mu }\varphi +c_2{g}_{\mu \nu }{Q}_{\text{leavevmode}\text{leavevmode}\alpha }^{\mu \nu }{\nabla }^{\alpha }\varphi +c_3{Q}_{\text{leavevmode}\text{leavevmode}\alpha }^{\mu \nu }{\nabla }_{\mu }\varphi {\nabla }_{\nu }\varphi {\nabla }^{\alpha }\varphi +\dots )\sqrt{-g}.$$

$$\Delta L_{\mathrm{P}}=0.$$

$$G_4(X,\varphi )\ne 0,\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}{G}_2=G_3=G_5=0,$$

$$G_5(X,\varphi )\ne 0$$

$$S_{\mathrm{P}}[{\Gamma }_{\mu \nu }^{\alpha },g_{\mu \nu },\varphi ]=M_{\mathrm{Pl}}^2\int \left(G_4(\varphi )\stackrel{(\Gamma )}{R}+K(\varphi ,X)+G_3(\varphi ,X)\stackrel{(\Gamma )}{\square }\varphi \right)\sqrt{-g}{d}^{4}x,$$

$$X=\frac{1}{2}{g}^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi \equiv g^{\mu \nu }{X}_{\mu \nu },\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\stackrel{(\Gamma )}{\square }\varphi =g^{\mu \nu }{\stackrel{(\Gamma )}{\nabla }}_{\mu }{\stackrel{(\Gamma )}{\nabla }}_{\nu }\varphi =g^{\mu \nu }\left({\partial }_{\mu \nu }\varphi -{\Gamma }_{\mu \nu }^{\alpha }{\partial }_{\alpha }\varphi \right).$$

$$\delta {\stackrel{(\Gamma )}{R}}_{\mu \nu }={\stackrel{(\Gamma )}{\nabla }}_{\alpha }\left(\delta {\Gamma }_{\mu \nu }^{\alpha }\right)-{\stackrel{(\Gamma )}{\nabla }}_{\nu }\left(\delta {\Gamma }_{\mu \alpha }^{\alpha }\right),\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\delta \stackrel{(\Gamma )}{\square }\varphi =-g^{\mu \nu }{\partial }_{\alpha }\varphi \delta {\Gamma }_{\mu \nu }^{\alpha }.$$

$$\delta S_{\mathrm{P}}=\int {\Delta }_{\text{leavevmode}\text{leavevmode}\alpha }^{\mu \nu }\delta {\Gamma }_{\mu \nu }^{\alpha }\sqrt{-g}{d}^{4}x,$$

$${\Delta }_{\text{leavevmode}\text{leavevmode}\alpha }^{\mu \nu }=\frac{1}{\sqrt{-g}}{\stackrel{(\Gamma )}{\nabla }}_{\sigma }(\sqrt{-g}{G}_4({\delta }_{\alpha }^{\mu }{g}^{\nu \sigma }-{\delta }_{\alpha }^{\sigma }{g}^{\mu \nu }))-G_3{g}^{\mu \nu }{\partial }_{\alpha }\varphi .$$

$${\Delta }_{\alpha }^{(\mu \nu )}=0.$$

$$\frac{1}{\sqrt{-g}}{\stackrel{(\Gamma )}{\nabla }}_{\mu }\left(\sqrt{-g}{G}_4{g}^{\mu \nu }\right)=\frac{2}{3}{G}_3{\partial }^{\nu }\varphi .$$

$$\frac{1}{\sqrt{-g}}{\stackrel{(\Gamma )}{\nabla }}_{\alpha }\left(\sqrt{-g}{G}_4{g}^{\mu \nu }\right)=G_3\left(\frac{2}{3}{\delta }_{\alpha }^{(\mu }{\partial }^{\nu )}\varphi -g^{\mu \nu }{\partial }_{\alpha }\varphi \right).$$

$$\frac{1}{\sqrt{-g}}{\stackrel{(\Gamma )}{\nabla }}_{\alpha }\sqrt{-g}=-\frac{1}{2}{g}_{\mu \nu }{\stackrel{(\Gamma )}{\nabla }}_{\alpha }{g}^{\mu \nu },$$

$$G_4{\stackrel{(\Gamma )}{\nabla }}_{\alpha }{g}^{\mu \nu }=g^{\mu \nu }{\partial }_{\alpha }{G}_4+\frac{2}{3}{G}_3\left(g^{\mu \nu }{\partial }_{\alpha }\varphi +{\delta }_{\alpha }^{(\mu }{\varphi }^{\nu )}\right).$$

$${\Gamma }_{\mu \nu }^{\alpha }=\left\{{}_{\mu \nu }^{\alpha }\right\}+\frac{1}{2}\left({\delta }_{\mu }^{\alpha }{\partial }_{\nu }\omega +{\delta }_{\nu }^{\alpha }{\partial }_{\mu }\omega -g_{\mu \nu }{\partial }^{\alpha }\omega \right)+\frac{1}{3}\gamma \left({\delta }_{\mu }^{\alpha }{\partial }_{\nu }\varphi +{\delta }_{\nu }^{\alpha }{\partial }_{\mu }\varphi \right).$$

Full text

1 Introduction

The discovery of the cosmic acceleration [1, 2] has invoked a large number of field-theory models of the Dark Energy. Most of them introduce a scalar field, as in the Brans-Dicke, quintessense, $k$-essence, etc. theories (see [3, 4] for reviews), while the others, as for example the $F(R)$ gravity [5, 6], although looking different, are equivalent to the theory with a scalar field. Some of these models were actually introduced long ago in the context of the inflation theory [7]. In view of this interest towards theories with a gravitating scalar field one may ask: what is the most general theory of this type described by second order equations of motion? The answer was obtained already in 1974 by Horndeski [8] (and more recently rediscovered in [9, 10]): this theory is determined by the action

[TABLE]

where, using the parameterization of Ref.[11],

[TABLE]

Here $X=\frac{1}{2}\,\partial_{\mu}\phi\partial^{\mu}\phi$ and $\hat{G}$, $\hat{\Phi}$ denote matrices with components

[TABLE]

while the brackets denote the trace, so that for example $[\hat{\Phi}]=\Box\phi$. This theory incorporates all previously studied models with a single gravity-coupled real scalar field. The coefficient functions $G_{k}(X,\Phi)$ in (1) can be arbitrary, and depending on their choice the properties of the theory can be different.

There is a special subset of the theory, sometimes call Kinetic Gravity Brading (KGB) theory [12], [13], [14] defined by the following choice of the coefficient functions:

[TABLE]

while $G_{2}(\phi,X)$ and $G_{3}(\phi,X)$ can be arbitrary. The speciality of this choice is that it defines theories in which the gravitational waves (GW) propagate with the speed of light around all backgrounds, as demonstrated in [12]. If the property (1.4) is not respected then the GW speed is not constant and the corresponding theories are disfavoured [15, 16, 17, 18] since the recent GW170817 event shows that the GW speed is equal to the speed of light with very high precision [19].

The Horndeski theory can be generalised to the so-called DHOST models containing higher order derivatives in the equations in such a way that the number of propagating degrees of freedom is still three [20, 21, 22, 23, 24]. However, if one restricts only to theories with second order equations of motion, then the Horndeski Lagrangian (1) is the most general one to produce such theories within the metric formulation, that is assuming the connection to be determined by the metric and the covariant derivative of the latter to vanish.

In what follows we shall study theories obtained from the Horndeski Lagrangian (1) without imposing the metricity condition. Specifically, we adopt the Palatini approach and vary the Lagrangian independently with respect to the metric $g_{\mu\nu}$, the scalar field $\phi$, and the connection that we assume to be symmetric, $\Gamma^{\mu}_{\alpha\beta}=\Gamma^{\mu}_{\beta\alpha}$. The equations for $\Gamma^{\mu}_{\alpha\beta}$ are algebraic111Unless $G_{5}(X,\phi)\neq 0$; see the remark after Eq.(1.17). hence the connection is non-dynamical, therefore the number of propagating degrees of freedom is the same as in the original Horndeski theory, unless the ghost emerges.

In fact, we require the connection to be symmetric just for simplicity, in order that the metric-affine theory be maximally close to the original Horndeski theory. Relaxing this condition would give torsion-full generalisations of the theory, and torsion should certainly be taken into account within the most-general metric-affine setting [25, 26, 27]. At the same time, the vector part of the torsion can be gauged-away in projectively-invariant theories [28], whereas requiring the theory to be projectively invariant seems to be necessary for removing the ghost [29]. In this work we shall mainly focus on the ${\cal L}_{3}$ part of the Horndeski theory, whose metric-affine version contains the torsion in the vectorial form that can be gauged away. As a result, choosing the connection to be torsionless from the very beginning does not actually restrict the generality of our analysis. This issue will be further discussed in [30] (see also [31]).

The Ricci tensor in (1) is viewed as function of $\Gamma^{\mu}_{\alpha\beta}$,

[TABLE]

the Ricci scalar and the Einstein tensor in (1) are understood as

[TABLE]

while the covariant derivatives should be computed with respect to $\Gamma^{\mu}_{\alpha\beta}$,

[TABLE]

Making these replacements in (1),(1.3) gives us the metric-affine version of the original Horndeski Lagrangian,

[TABLE]

and this defines the Palatini action

[TABLE]

Of course, this reduces back to the original Horndeski action if the connection is set to be Levi-Civita,

[TABLE]

Any additional matter can be included into the theory by adding to the action the matter term $S_{\rm m}[g_{\mu\nu},{\cal X}]$, where ${\cal X}$ collectively denotes all matter fields. If the matter couples also to the connection and hence the matter term depends on $\Gamma^{\sigma}_{\alpha\beta}$ as well, this may generate a non-zero torsion.

One should say that this metric-affine version of the original theory is not unique. For example, adopting instead of (1.7) the definition

[TABLE]

would give a different Lagrangian $\tilde{L}_{\rm P}$ and a different action $\tilde{S}_{\rm P}$ that also reduce back to $L_{\rm H}$ and $S_{\rm H}$ when $\Gamma^{\mu}_{\alpha\beta}=\left\{{}^{\alpha}_{\mu\nu}\right\}$. However, varying $\tilde{S}_{\rm P}$ and ${S}_{\rm P}$ would not give the same equations. It is clear that the two definitions (1.7) and (1.11) differ from each other by the term containing the covariant derivative of the metric – the non-metricity tensor

[TABLE]

which does not vanish in general. Using this tensor one can construct generalisations of the Lagrangian:

[TABLE]

where

[TABLE]

Here $c_{1}$, $c_{2}$, $c_{3}$ can depend on $X,\phi$ and the dots stand for all possible terms containing higher powers of $Q^{\mu\nu}_{\leavevmode\nobreak\ \leavevmode\nobreak\ \alpha}$ and higher derivatives of $\phi$. As a result, there can be infinitely many different versions $\tilde{L}_{\rm P}$ of the Palatini Lagrangian. All of them coincide when the non-metricity vanishes, but otherwise they lead to different theories. This ambiguity in defining the theory may actually be important for removing the ghost. However, below we shall be considering only the simplest version of the theory for which it is sufficient to choose

[TABLE]

More complex cases will be reported separately [30] (see also [31]).

Therefore, in what follows we shall vary the Palatini action $S_{\rm P}[\Gamma^{\sigma}_{\alpha\beta},g_{\mu\nu},\phi]$ defined by (1.9). We do not expect to get the same equations as those obtained from the metric action $S_{\rm H}[g_{\mu\nu},\phi]$, since already for the $f(R)$ theory the metric formulation and Palatini formulation give different results [5]. The same is expected to happen also for the Horndeski theory.

We find that the resulting theory obtained from $S_{\rm P}[\Gamma^{\sigma}_{\alpha\beta},g_{\mu\nu},\phi]$ can show quite different properties, depending on whether the Lagrangian $L_{\rm P}$ respects or not the KGB condition (1.4). If this condition is respected then $Q_{\mu\nu\alpha}\neq 0$ but the non-metricity contributions can be grouped into additional terms in the effective energy-momentum tensor of the scalar field, and the field equations can be represented in the form containing only the ordinary metric covariant derivatives. Remarkably, these equations turn out to be identically the same as those for a metric KGB theory corresponding to a specific choice of the coefficients $G_{2},G_{3},G_{4}$ in the Lagrangian. Therefore, the Palatini approach yields in this case a theory which is still in the same Horndeski class.

This is a non-trivial result. It can be traced to the fact that the connection in our theory is non-dynamical and can be integrated out, which transforms the metric-affine theory to a metric theory containing additional interactions [33, 32]. When the connection is integrated out, we obtain a metric theory that is again of Horndeski type but with additional scalar-tensor interactions whose structure, however, may in general be not the same as that in the Horndeski theory. As a result, this metric theory may be outside the Hordeski family and may contain ghost. It is therefore rather non-trivial that, when the procedure is applied to the KGB theory, the ghost does not appear and the additional interactions again have the KGB structure. One could say that the KGB family is “stable” under the metric-affine treatment. This, together with the fact that the GW speed is constant, confirms again the special status of these theories. More general subsets of the Horndeski family are not stable in the same sense – their metric-affine versions contain higher derivatives.

If the Lagrangian $L_{\rm P}$ does not respect the condition (1.4) then the equations contain higher derivatives. In some cases they are not dangerous as the theory turns out to be of the DHOST type, however this is not always the case. For example, choosing

[TABLE]

and $\Delta L_{\rm P}=0$ in (1.13), one finds that the ghost is absent if $G_{4}=f(\phi)X$. In some other cases it can be removed via adjusting $\Delta L_{\rm P}\neq 0$ in (1.13) but it remains unclear if such a procedure always works.

The metric-affine versions of theories with

[TABLE]

remain by far almost totally unexplored because the connection then becomes dynamical. Specifically, the connection enters the action algebraically, apart from the terms with $\overset{(\Gamma)}{R}_{\mu\nu}$. Varying the latter with respect to the connection produces result of the form $\Sigma^{\alpha\mu\nu}_{\beta}\overset{(\Gamma)}{\nabla}_{\alpha}\left(\delta\Gamma^{\beta}_{\mu\nu}\right)$ with $\Sigma^{\alpha\mu\nu}_{\beta}$ constructed from $g_{\mu\nu}$ and derivatives of $\phi,X$. After integrating by parts, this gives rise to term $\overset{(\Gamma)}{\nabla}_{\alpha}\Sigma^{\alpha\mu\nu}_{\beta}$ in the equations. If $G_{5}=0$ then $\Sigma^{\beta\mu\nu}_{\alpha}$ does not depend on the connection and the resulting equations are algebraic in $\Gamma^{\beta}_{\mu\nu}$. If $G_{5}\neq 0$ then $\Sigma^{\beta\mu\nu}_{\alpha}$ contains $\overset{(\Gamma)}{\nabla}_{\mu}\overset{(\Gamma)}{\nabla}_{\nu}\phi$ and therefore depends on $\Gamma^{\beta}_{\mu\nu}$ hence $\overset{(\Gamma)}{\nabla}_{\alpha}\Sigma^{\alpha\mu\nu}_{\beta}$ contains derivatives $\Gamma^{\beta}_{\mu\nu}$. As a result, the equations for $\Gamma^{\beta}_{\mu\nu}$ are algebraic if $G_{5}=0$ but they become differential if $G_{5}\neq 0$. The connection starts propagating in the latter case, which should considerably change the physical contents of the theory.

The rest of this text is organised as follows. In Section 2 we perform the Palatini variation of the piece of the Lagrangian respecting the KGB condition (1.4), and in Section 3 we show that the resulting equations actually correspond to one of the metric Horndeski theories. Therefore, varying the same KGB action in the metric approach and in the Palatini approach gives two different theories from the same metric KGB class. In Sections 4,5 we study their solutions to see how much these two theories differ from each other. One should stress that, although both of these models are in the same KGB class and hence their tensor modes propagate with the speed of light, the properties of the scalar mode are not necessarily the same. Cosmologies in these two models are not the same and their stability properties are different. In Section 4 we consider small perturbations of homogeneous and isotropic backgrounds, which gives us conditions for the absence of ghosts and tachyons in the scalar sector. In Section 5 we specify the subclass of theories invariant under shifts $\phi\to\phi+\phi_{0}$ and describe all homogeneous and isotropic cosmologies in these theories. The spectrum of these solutions is surprizingly rich, and we make a comparison with the solutions previously described in the literature [12], [13], [14]. In Section 6 we briefly describe what happens if the condition (1.4) is not respected – a more detailed analysis will be reported separately [30]. We make some concluding remarks in Section 7. The Appendix contains the lengthy expression for the connection arising in the $G_{4}(X,\phi)$-subset of the theory.

The metric-affine formulation for the scalar-tensor theories was recently studied also in [34], [35], [36], [37], [38]. However, to the best of our knowledge, the entire Horndeski family has not been systematically analysed from this viewpoint.

2 Varying the KGB part of the Palatini action

Imposing the KGB condition (1.4), the action (1.9) reduces to

[TABLE]

with the Ricci scalar $\overset{(\Gamma)}{R}$ defined according to (1.5),(1.6); our signature is $(-,+,+,+)$. We assume the connection to be symmetric, $\Gamma^{\alpha}_{\mu\nu}=\Gamma^{\alpha}_{\nu\mu}$, but the Ricci tensor $\overset{(\Gamma)}{R}_{\mu\nu}$ will not in general be symmetric, unless $\Gamma^{\alpha}_{\mu\nu}$ is a Levi-Civita connection. The other quantities in the action are the squared gradient of the scalar field and covariant d’Alembertian,

[TABLE]

Let us vary the action (2.1) independently with respect to $\Gamma^{\alpha}_{\mu\nu}$, $g_{\mu\nu}$, and $\phi$. To vary with respect to $\Gamma^{\alpha}_{\mu\nu}$, we notice that the only connection-dependent terms in the action are $\overset{(\Gamma)}{R}$ and $\overset{(\Gamma)}{\Box}\phi$. The variation $\delta\Gamma^{\alpha}_{\mu\nu}$ is a tensor that induces the variations,

[TABLE]

Injecting this to (2.1), integrating by parts and remembering that the metric is not necessarily covariantly constant with respect to $\overset{(\Gamma)}{\nabla}$, we obtain

[TABLE]

with

[TABLE]

The variation of the action will vanish if

[TABLE]

It follows that $\Delta^{(\mu\nu)}_{\mu}=0$, which yields

[TABLE]

Taking this condition into account, Eq.(2.6) reduces to

[TABLE]

Since one has

[TABLE]

one obtains after simple manipulations the following expression for the covariant derivative of the metric,

[TABLE]

This can be resolved to obtain the connection,

[TABLE]

Here and in what follows we use the functions $\omega,\gamma,\kappa$ related to $G_{4},G_{3},K$ in the action via

[TABLE]

It is worth noting that the first and second terms on the right in (2.11) correspond to the Kristoffel symbols for the conformally related metric $\bar{g}_{\mu\nu}=e^{\omega}g_{\mu\nu}$. However, the last term in (2.11) does not have the Levi-Civita structure.

Injecting the expression for $\Gamma^{\alpha}_{\mu\nu}$ to (1.5) gives the Ricci tensor,

[TABLE]

where $R_{\mu\nu}$ and $\nabla_{\mu}$ are the standard Ricci tensor and covariant derivative constructed from $\left\{{}^{\alpha}_{\mu\nu}\right\}$ while $\Box=\nabla^{\mu}\nabla_{\mu}$. We note that the last term on the right in (2.13) is antisymmetric under $\mu\leftrightarrow\nu$.

Let us now vary the action with respect to $\phi$. One has

[TABLE]

Injecting this to the action and integrating by parts yields

[TABLE]

where

[TABLE]

To compute the expression in the third line we set $G_{3}=\gamma G_{4}$, inject to (2.7), and use (2.9) to obtain

[TABLE]

Since for any vector $I^{\mu}$ one has

[TABLE]

it follows that

[TABLE]

Collecting everything together, the variation of the action with respect to the scalar field is

[TABLE]

with

[TABLE]

Here and below the following two functions are used,

[TABLE]

where the prime denotes differentiation with respect to $\phi$.

Varying the action with respect to the metric is straightforward and yields

[TABLE]

where

[TABLE]

with $\kappa$ defined in (2.12) and \gamma_{\mbox{\tinyX}}=\partial_{X}\gamma, \kappa_{\mbox{\tinyX}}=\partial_{X}\kappa. The Ricci tensor $\overset{(\Gamma)}{R}_{\mu\nu}$ is given by (2.13) and tracing it yields $\overset{(\Gamma)}{R}$. One has

[TABLE]

with $X_{\mu\nu}$ defined in (2.2), hence

[TABLE]

Summarizing the above discussion, the action will be stationary if $E_{(\mu\nu)}=0$ and $E_{\phi}=0$. This yields the field equations which can be rewritten solely in terms of the ordinary metric covariant derivatives. The $E_{(\mu\nu)}=0$ conditions reduce to

[TABLE]

where $G_{\mu\nu}$ is the Einstein tensor for $g_{\mu\nu}$ while the effective energy-momentum tensor

[TABLE]

with $A,B$ defined in (2.22). The condition $E_{\phi}=0$ yields the equation for the scalar field,

[TABLE]

where $J^{\mu},\Sigma$ are defined by (2) with $\overset{(\Gamma)}{\Box}\phi$ given by (2.26) and $\overset{(\Gamma)}{R}$ obtained by tracing $\overset{(\Gamma)}{R}_{\mu\nu}$ in (2.13). A direct verification shows that the differential consequence of (2.27), the covariant conservation condition,

[TABLE]

indeed follows from Eqs.(2.27)–(2.29).

3 Relation to the metric version of the theory

Let us now return to the action (2.1) and assume that $\Gamma^{\alpha}_{\mu\nu}=\left\{{}^{\alpha}_{\mu\nu}\right\}$ is the Levi-Civita connection determined by $g_{\mu\nu}$. The action then reduces to the Horndeski action $S_{\rm H}[g_{\mu\nu},\phi]$. Varying it with respect to $g_{\mu\nu}$ and $\phi$ yields the equations

[TABLE]

where

[TABLE]

and also

[TABLE]

Surprisingly, a direct verification shows that equations (2.27), (2), (2.29), (2) of the metric-affine version can be obtained from equations (3.1)–(3) of the metric version by simply replacing in the latter

[TABLE]

with $A$ given by (2.22). Therefore, the Palatini theory derived from the action (2.1) is actually equivalent to the metric theory derived from the action

[TABLE]

with the new k-essence term

[TABLE]

The explanation of this is as follows. Let us return to the Palatini action (2.1) and inject into it the on-shell value of the connection,

[TABLE]

given by (2.11). Using $\overset{(\Gamma)}{R}_{\mu\nu}$ and $\overset{(\Gamma)}{\Box}\phi$ expressed by (2.13) and (2.26) then yields

[TABLE]

so that the metric action (3.5) is indeed recovered. This implies that the equations derived from both actions should coincide. Indeed, let us vary the scalar field, $\phi\to\phi+\delta\phi$. This induces the variations

[TABLE]

Since the connection is assumed to have the on-shell value, one has

[TABLE]

therefore

[TABLE]

hence the scalar field equation derived from the Palatini action $S_{\rm P}$ coincides with the one obtained from the metric action $\tilde{S}_{\rm H}$. The same applies for equations obtained by varying the metric, hence theories derived from the Palatini action (2.1) and from the metric action (3.5) are equivalent. A similar equivalence holds for all other Horndeski models with $G_{5}=0$ as well, because a non-dynamical connection can always be integrated out and the metric-affine theory reduces to a metric theory.

Summarizing, varying the same action (2.1) within the metric approach and within the Palatini approach yields two different theories from the same metric KGB class. In the former case one obtains the theory with coefficient functions $G_{3}$, $G_{4}$, $K$ while in the latter case one obtains theory with coefficients $G_{3},G_{4},\tilde{K}$, with $\tilde{K}$ defined by (5.14). Both theories are ghost-free and the GW speed is equal to one. Below we shall study solutions of these two theories to see how much they differ from each other.

The change $K\to\tilde{K}$ expressed by (3.6) can be viewed as a “duality relation” between different theories, and one can look for its interpretation, for example within the effective hydrodynamical description developed in Ref.[13]. If the theory is invariant under shifts $\phi\to\phi+\phi_{0}$ then the current in (3) is conserved, which can be expressed as

[TABLE]

with the “fluid 4-velocity” $u_{\mu}=\partial_{\mu}\phi/m$, “acceleration” $a_{\mu}=u^{\alpha}\nabla_{\alpha}u_{\mu}$, “chemical potential” $m=\sqrt{2|X|}$, “density” $n=m\partial_{X}K+{\bm{\kappa}}\theta$ where the “expansion” $\theta=\nabla_{\alpha}u^{\alpha}$, and with the “diffusivity” ${\bm{\kappa}}=2X\partial_{X}G_{3}$ (not to be confused with our $\kappa=K/G_{4}$). The right-hand-side in (3.12) can be interpreted as the “diffusion term” [13].

Now, (3.6) does not change $G_{3}$ and the diffusivity ${\bm{\kappa}}$, but it changes $K$ (equation of state) and the density $n$. If repeated many times, it changes $K$ more and more, but it becomes identity if the theory is chosen such that $A=0$ and hence $\tilde{K}=K$. Unfortunately, this condition does not have non-trivial solutions in the shift-invariant theory222The relation $\tilde{K}=K$ can be realized in a non-trivial way if the theory includes non-metricity terms ${\Delta L}_{\rm P}$ of the type (1.14) [31]. . The duality does not change the right-hand-side of (3.12), but since $n$ changes, the whole diffusion is affected. Eq.(3.12) assumes the standard diffusion form in the limit of small $n$ and $m$, with the diffusion coefficient ${\cal D}=-{\bm{\kappa}}/(m\partial_{m}n)$ [13], but since the duality only affects higher powers of $m$, ${\cal D}$ remains invariant in this limit. The impact of the duality on the diffusion away from the weak field limit should probably be analyzed separately.

4 Stability of the scalar sector

In Section 5 we shall study cosmologies in the Palatini-derived KGB theory and compare them with the those in the metric-derived theory. Even though these two theories belong to the same metric KGB class, hence they are free from Ostrogradsky ghost and contain three propagating degrees of freedom, their scalar sector can be unstable. We therefore study in this Section small perturbations of cosmological backgrounds. Decomposing the perturbations into tensor and scalar parts, one finds, as expected, that tensor modes always propagate with the speed of light. However, depending on the background, the scalar sector may contain ghost and tachyon instabilities. Their absence requires positivity of the kinetic coefficient ${\rm K}$ and sound speed squared $c_{s}^{2}$ derived in Eqs.(4.14),(4). These conditions will be analysed in Section VI.

Let us assume the spacetime metric to be homogeneous and isotropic,

[TABLE]

while the scalar field to depend only on time,

[TABLE]

The Einstein equations (2.27) of the Palatini version of the theory reduce to

[TABLE]

whose consequence is the scalar field equation (2.29). Here $H=\dot{a}/a$ and the prime denotes differentiation with respect to $\phi$.

Suppose one finds a solution of Eqs.(4) (examples will be given below) describing a homogeneous and isotropic background (4.1),(4.2). Consider small perturbations of this background,

[TABLE]

In the linear approximation, the perturbations fulfill the equations obtained by perturbing the background equations,

[TABLE]

The metric perturbations can be decomposed into the scalar, vector, and tensor parts via

[TABLE]

where

[TABLE]

The spatial dependence is given by the plane waves where the wave vector can be oriented along the z-axis, so that the scalar modes are

[TABLE]

the vector amplitudes are chosen as

[TABLE]

while for the tensor modes the only non-trivial components of ${\rm D}_{ik}$ are

[TABLE]

Inserting everything into the perturbation equations (4.5) splits them into three independent groups for the scalar, vector, and tensor modes. These equations determine the effective action which also spits into three independent terms,

[TABLE]

where the bar denotes complex conjugation. One obtains in the tensor sector

[TABLE]

where the kinetic coefficient ${\rm K}=G_{4}=e^{\omega}$ (not to be confused with $K$) is always positive while the sound speed $c_{s}=1$. Therefore, the gravity waves propagate with the speed of light as expected.

The analysis in the vector sector shows that the vector modes have no kinetic term and $I_{\rm V}=0$, hence vector modes do not propagate.

The analysis in the scalar sector is more involved but facilitated by the fact that one can impose the gauge where $\delta\phi=0$ (unless ${\psi}=0$). The equations then imply that the scalar amplitudes $S_{2}$, $S_{3}$, $S_{4}$ can be expressed in terms of $S_{1}$ and the effective action reduces to

[TABLE]

with

[TABLE]

where

[TABLE]

Both the kinetic term ${\rm K}$ and sound speed squared $c_{s}^{2}$ should be positive for the system to be stable. Summarizing, the theory shows two propagating modes in the tensor sector and one scalar mode. The tensor modes propagate with the speed of light, as expected, while properties of the scalar mode depend on the background. The above formulas apply for the Palatini-derived theory described by (2.27), (2), (2.29), (2). The corresponding formulas in the metric theory described by (3.1)–(3) are obtained by making in (4), (4) the inverse to (3.4) replacement : $\kappa\to\kappa-2XA$.

5 Cosmologies

In order to study concrete solutions, we must specify the functions $G_{4}(\phi)$, $G_{3}(\phi,X)$, $K(\phi,X)$. We assume them to be independent of $\phi$,

[TABLE]

in which case the theory is invariant under shifts

[TABLE]

As the simplest option, we assume $G_{3}$ and $K$ to be linear in $X$, hence

[TABLE]

where $\alpha,\beta,\Lambda$ are constant parameters, so that

[TABLE]

Eqs.(2.27) then become

[TABLE]

with the energy-momentum tensor

[TABLE]

while the scalar field equation (2.29) becomes total derivative,

[TABLE]

with the current

[TABLE]

The equations of the corresponding metric version of the theory are obtained by simply omitting in (5.6),(5.8) the terms proportional to $\alpha^{2}$.

We shall now study homogeneous and isotropic cosmologies of this shift-invariant theory, first within its Palatini version and then within its metric version. Even though these two theories belong to the same KGB class, their solutions are different since the equations are not the same and the stability conditions derived in Section 4 are also not the same, since they are background-dependent.

It turns out that the problem reduces to a single algebraic equation (5.24) which determines algebraic curves whose position and critical points determine the properties of the cosmologies. One finds in this way many different solutions: self-accelerating cosmologies, recollapsing cosmologies, and bounces. We give below their detailed classification, analyse their stability conditions, identify stable branches, and make comparison with the previously known results [12], [14].

5.1 Master equations

Assuming the homogeneous and isotropic ansatz (4.1),(4.2) for the fields, the Einstein equations (5.5) reduce to

[TABLE]

with ${\psi}=\dot{\phi}$. These equations can also be obtained by injecting (5.3) to (4). The only non-trivial component of the scalar current (5.8) is

[TABLE]

and the scalar field equation (5.7) reads

[TABLE]

which implies that

[TABLE]

where $C$ is the integration constant – the scalar charge.

The simplest solution of these equations is $C={\psi}=0$ and $3H^{2}=\Lambda$. This solution is stable, although the general stability analysis carried out above does not apply in this particular case since the gauge $\delta\phi=0$ cannot be imposed if ${\psi}=0$. One should repeat the analysis keeping $\delta\phi\neq 0$ and then one finds in the scalar sector ${\rm K}=c_{s}^{2}=1$.

For solutions with ${\psi}\neq 0$ one can use the general formulas (4.14) for the kinetic term and the sound speed, which now reduce to

[TABLE]

If ${\psi}$ does not vanish then (5.13) can be resolved with respect to the Hubble parameter,

[TABLE]

Injecting this to (5.9) yields the algebraic relation between ${\psi}$ and $a$,

[TABLE]

while injecting $H$ to (5.10) determines the derivative of ${\psi}$,

[TABLE]

Eqs.(5.15),(5.16),(5.17) are invariant under

[TABLE]

which provides the one-to-one correspondence between solutions of two theories which differ by the sign of $\alpha$. Therefore, one can assume without loss of generality that $\alpha>0$. The equations are also invariant under the time reversal $t\to-t$, which changes the sign of the first derivatives and of the current, but not of the second derivatives, hence

[TABLE]

This swaps the expanding solutions and contracting solutions.

It follows from (5.15) that if ${\psi}$ approaches zero then either the Hubble rate $H$ should diverge, or, if it remains finite, then the scale factor should $a$ diverge. This corresponds either to the initial singularity or to future infinity. Therefore, between these two extremities ${\psi}$ cannot vanish and should be sign definite, either everywhere positive or everywhere negative.

One can absorb the parameters $\alpha$ and $\beta$ by expressing $a,{\psi},H,\Lambda$ in terms of dimensionless333Assuming the spacetime coordinates to have the dimension of length, $x^{\mu}\sim l$, our normalisation of the action (2.1) implies that $\beta,\phi,a,G_{3},G_{4}$ are dimensionless while $\alpha^{-1}\sim X\sim K\sim\Lambda\sim l^{-2}$. quantities $x,y,h,\lambda$ via

[TABLE]

where the Hubble scale is determined by the length scale $\sqrt{\alpha}$,

[TABLE]

Here ${\bm{\beta}}=\beta$ if $\beta\neq 0$, while if $\beta=0$ then ${\bm{\beta}}$ is an arbitrary dimensionless parameter. The variable $x$ in (5.20) must be non-negative while $y,h,\lambda$ can be positive or negative.

Injecting (5.20) to (5.14)–(5.17) yields,

[TABLE]

where the sign of $C$ should be chosen such that $\pm C/y(x)>0$, hence different values of $C$ correspond to different solutions whose scale factors are “homothetic” to each other. As a result, if $C\neq 0$ then one can assume without loss of generality that either $C=1$ or $C=-1$, depending on sign of $y$. One obtains also

[TABLE]

while Eq.(5.16) reduces to

[TABLE]

with

[TABLE]

Eq.(5.17) yields

[TABLE]

The kinetic term and the sound speed in (5.14) become

[TABLE]

Eqs.(5.23)–(5.1) determine the solutions and their stability.

5.2 Currentless solutions

Let us first consider solutions with vanishing scalar charge, $C=0$, in which case, according to (5.13), the current is zero. If $C=0$ then, according to (5.20), one has either $x={\psi}=0$ hence the system is in vacuum, or $y=0$ and then Eq.(5.24) reduces to

[TABLE]

hence $x=x(\lambda)$ is constant, ${\psi}$ and $H$ are constant as well, and the geometry is de Sitter. If $\epsilon=0$ then $x(\lambda)=(\lambda/3)^{1/3}$. If $\epsilon=\pm 1$ then $f(x)$ diverges for $x\to 0,\infty$ and has a minimum in between, hence for $\lambda$ exceeding some minimal value there are two different solutions of (5.27), $x=x_{+}(\lambda)$ and $x=x_{-}(\lambda)$ (see Fig.1).

Let us assume first that $\beta>0$ hence $\epsilon=1$. Then for $\lambda=0$, for example, one finds two solutions of (5.27) with the following properties:

[TABLE]

Notice that $c_{s}^{2}<0$ for the first of these solutions hence it is unstable.

If $\lambda$ decreases to the negative region then the two values $x=x_{+}(\lambda)$ and $x=x_{-}(\lambda)$ approach each other and merge for $\lambda=-4\sqrt{2}$, in which case

[TABLE]

The Hubble rate vanishes for this solution and the geometry is flat, even though ${\psi}\neq 0$. This solution exists only for $\lambda=-4\sqrt{2}$ but it is unstable since $c_{s}^{2}<0$. This does not mean that flat space is always unstable in the theory, because the flat space solution can also be obtained in a different way: by setting $\lambda=0$ and ${\psi}=0$, in which case it is stable (as was mentioned above, the formulas (5.1) do not apply if ${\psi}=0$).

Let us determine the stability region. If $y=0$ then ${\rm K}$ and $c_{s}^{2}$ defined by (5.1) reduce to

[TABLE]

It follows that ${\rm K}c_{s}^{2}<0$ if $\epsilon=0,-1$ hence all solutions with $\beta\leq 0$ show either ghost or gradient instability. If $\epsilon=1$ then ${\rm K}$ is always positive while $c_{s}^{2}$ will be non-negative if $4-5x^{2}>0$, hence if (see Fig.1)

[TABLE]

Solutions with $x=x_{+}(\lambda)$ always violate this condition hence they are all unstable. Solutions with $x=x_{-}(\lambda)$ fulfill this condition if

[TABLE]

To recapitulate, the currentless solutions are characterised by a constant value of the scalar field gradient and by a constant Hubble rate; their geometry is de Sitter. For $\beta>0$ they exist if only $\lambda\geq-4\sqrt{2}=-5.65$ and they are stable for $\lambda\geq-1.43$. All of such solutions for $\beta=0$ or $\beta<0$ are unstable (we shall later see that if the current does not vanish then stable solutions exist for any $\beta$).

5.3 Solutions with a non-zero current

If $C\neq 0$ then the current is $J^{0}=C/a^{3}\propto y$ hence the amplitude $y$ defined in (5.20) does not vanish. However, since $J^{0}\to 0$ for $a\to\infty$, it follows that $y$ approaches zero at late times and the solutions then approach the described above configurations with constant $x$ and de Sitter geometry. It follows from the above analysis that if $y$ approaches zero then $x$ must approach either $x_{+}(\lambda)$, in which case the product ${\rm K}c_{s}^{2}$ becomes negative and the solution becomes unstable, or $x$ approaches $x_{-}(\lambda)$ and then the solution is stable if $\lambda\geq-\frac{16}{5\sqrt{5}}$.

For $y\neq 0$ Eq.(5.24) can be resolved yielding two different solutions, $y=y_{+}(x)$ or $y=y_{-}(x)$. From now on and till the end of the next sub-section we set $\epsilon=\beta/|\beta|=1$, then

[TABLE]

These functions are defined only in the region where $19\,x^{4}-12\,x^{2}+2\,\lambda\,x\geq 0$. This region must contain a zero of $y(x)$ since we want the solution to approach for $a\to\infty$ one of the de Sitter backgrounds described above. If $\lambda\geq-\frac{16}{5\sqrt{5}}$ then $y_{+}(x)$ vanishes at $x=x_{+}(\lambda)$ which point is known to be unstable, whereas $y_{-}(x)$ vanishes at $x=x_{-}(\lambda)$, and we know that this point is stable. Therefore, we choose

[TABLE]

assuming that $x\to x_{-}(\lambda)$ and hence $y\to 0$ for $a\to\infty$. For finite values of the universe size, when $a<\infty$, one chooses $x\geq x_{-}(\lambda)$, in which case one has $y(x)<0$. According to (5.22), the scale factor is proportional to

[TABLE]

where the sign of $C$ should be chosen such that $\pm C/y(x)>0$. This implies that a$(x)\to\infty$ for $x\to x_{-}$ and a$(x)<\infty$ for $x>x_{-}$. Injecting (5.35) to (5.23) yields the Hubble parameter,

[TABLE]

and similarly injecting to (5.1) yields ${\rm K}(x)$ and $c_{s}^{2}(x)$.

As a result, Eqs.(5.35),(5.36),(5.37) provide the solution in the parametric form, with $x$ being the parameter. Inverting a$(x)$ in (5.36) to obtain $x=x({\rm a})$, the solution can be expressed in terms of the scale factor as shown in Fig.2.

One can see that, as the scale factor a varies from zero to infinity, the gradient squared of the scalar field, $x\sim{\psi}^{2}\sim X$, decreases from infinity to the asymptotic value $x_{-}(\lambda)$. The Hubble function $h$ decreases from infinity to the constant asymptotic value defined by (5.23) with $x=x_{-}(\lambda)$ and $y=0$. The important point is that the kinetic term ${\rm K}$ and the sound speed $c_{s}^{2}$ remain positive for all values of a, as seen in Fig.2, hence the solutions are always stable.

To determine the behaviour near the initial singularity, we notice that when $x$ is large and a is small, then one has from (5.35),(5.36),(5.37)

[TABLE]

hence

[TABLE]

As a result, the system behaves as a perfect fluid with the effective equation of state $w=1/5$, which is somewhere in between the dust $(w=0)$ and radiation $(w=1/3)$.

To recapitulate, the system admits cosmological solutions with a non-zero scalar current. Close to the initial singularity, the squared gradient of the scalar field is $X\propto a^{-6/5}$, which mimics a perfect fluid with the equation of state $w=1/5$. As the size of the universe grows, $X$ and the Hubble rate approach constant values. These solutions are stable.

It is worth noting that these solutions can describe both the expansion and contraction of the universe, according to the choice of sign in Eq.(5.20). Choosing the plus sign yields $H>0$, hence the expansion, in which case one should choose $C<0$ since $y<0$. Choosing the minus sign gives the contraction with $H<0$, and then one should choose $C>0$. The two cases are related by the symmetry (5.19).

5.4 More general solutions

These are defined by the algebraic curve $y(x)$ subject to (5.24). To study this curve, we plot together the functions $y_{+}(x)$ and $y_{-}(x)$ defined by (5.34), which allows us to distinguish different solution types. Depending on value of $\lambda$, these solutions can be classified as follows.

5.4.1 $\lambda<-4\sqrt{2}=-5.65$

Type I. An example of such solutions is shown in Fig.3 for $\lambda=-10$. Both $y_{+}(x)$ and $y_{-}(x)$ are everywhere negative and defined only in the region $x\geq x_{\rm min}(\lambda)$ where the argument of the square root in (5.34) is positive.

In the $x\to x_{\rm min}(\lambda)$ limit the square root vanishes and the $y_{+}(x)$ and $y_{-}(x)$ branches merge at the point $s$ marked on the left panel in Fig.3. Nothing special happens at this point: the solution simply passes from the lower $y_{-}(x)$ branch to the upper $y_{+}(x)$ branch in the direction indicated by the arrow in Fig.3. The direction is determined by the fact that at the lower branch the derivative $\dot{x}\propto p_{-}<0$, as shown in Fig.3, hence $x$ decreases towards the minimal value $x_{\rm min}(\lambda)$, while at the upper branch the derivative $\dot{x}\propto p_{+}>0$ and $x$ increases.

The scale factor ${\rm a}(x)$ obtained from (5.36) increases along the lower branch and the corresponding Hubble parameter is positive, $h_{-}>0$, as shown on the right panel in Fig.3. After passing to the upper branch, the scale factor first continues to increase up to a maximal value, then the Hubble parameter $h_{+}$ changes sign and the universe starts shrinking.

Therefore, the universe starts from zero size at $x=\infty$ and $y=y_{-}=-\infty$, then it expands first along the $y_{-}$ branch and next along the $y_{+}$ branch, then the scale factor reaches a maximal finite value, after which the universe shrinks back to zero size along the $y_{+}$ branch. The sound speed $c_{s}^{2}$ becomes negative at the $y_{+}$ branch, hence the solution is unstable.

The solution remains qualitatively the same for any $\lambda<-4\sqrt{2}=-5.65$, when both $y_{+}$ and $y_{-}$ remain negative, but the maximal value of $y_{+}$ approaches zero from below when $\lambda$ increases.

5.4.2 $\lambda=-4\sqrt{2}$

Type II. The curve $y(x)=y_{-}(x)\cup y_{+}(x)$ remains qualitatively the same as before but the $y_{+}(x)$ branch touches zero from below at the point O indicated on the left panel in Fig.4. The position of this point is described by Eq.(5.30) above. Since $y$ vanishes at this point, the universe size (5.36) becomes infinite. Therefore, there are actually two different solutions in this case. The part of the $\lambda_{1}$-curve on the left panel in Fig.4 which is on the left from the point O describes the universe expanding from zero size to infinity. The part of the curve on the right from O describes the universe shrinking from infinite size to zero.

Since at the point O the sound speed squared becomes negative (see Eq.(5.30)), both of these solutions show a gradient instability.

5.4.3 $-4\sqrt{2}<\lambda\leq 0$

The $y(x)=y_{-}(x)\cup y_{+}(x)$ curve remains qualitatively the same as before but shifts upwards and crosses zero twice at $x=x_{+}$ and $x=x_{-}$ with $x_{\pm}(\lambda)$ defined by Eq.(5.27). This is the case for the $\lambda_{2}$, $\lambda_{3}$, and $\lambda_{4}$ curves on the left panel in Fig.4. Since $y(x_{\pm})=0$, it follows that $a_{\pm}(x_{\pm})=\infty$ (see (5.36)), hence the single curve $y(x)$ determines three different solutions of types III, IV, V described below.

Type III. This solution corresponds to the left part of the $y(x)$ curve where $y(x)<0$ (the $\lambda_{2}$, $\lambda_{3}$, and $\lambda_{4}$ curves in Fig.4). This determines the universe expanding from zero to infinity. This solution can be stable or unstable, depending on the position of the point $s$ of the merging of the $y_{+}(x)$ and $y_{-}(x)$. If the merging point is below the $x$-axis (as for the $\lambda_{2}$-curve in Fig.4) then the solution is unstable. The solution becomes stable for $\lambda=-16/(5\sqrt{5})=-1.43$ when the merging point $s$ is at the $x$-axis (the $\lambda_{3}$-curve in Fig.4), and it remains stable when $s$ moves further up (the $\lambda_{4}$-curve in Fig.4). The profiles of the solution in the latter two cases are similar to those shown in Fig.2.

Type IV. This solution corresponds to the part of the $y(x)$ curve interpolating between $x=x_{-}$ and $x=x_{+}$ (the $\lambda_{2}$, $\lambda_{3}$, and $\lambda_{4}$ curves in Fig.4). This describes a bounce – the universe shrinking to a finite size and then expanding back to infinity. This solution is always unstable since it contains the point $y=0$, $x=x_{+}(\lambda)$ which is known to be unstable.

Type V. This solution corresponds to the right part of the $y(x)$ curve where $y<0$, (the $\lambda_{2}$, $\lambda_{3}$, and $\lambda_{4}$ curves in Fig.4). This also corresponds to the universe expanding from zero to infinity (or contracting, depending on the sign choice in (5.20)), and it is always unstable because it contains the unstable point $y=0$, $x=x_{+}(\lambda)$.

5.4.4 $0<\lambda<8/\sqrt{19}=1.83$

The $y(x)=y_{-}(x)\cup y_{+}(x)$ curve moves further upwards and develops a disjoint part – a small loop, as illustrated by the $\lambda_{5}$-curve shown on the right panel in Fig.4. The curve therefore splits into two disconnected subsets – the compact part (the loop) and the non-compact part. The non-compact part corresponds to three different solutions of Types III–V described above; Type III solution always being stable. The compact part corresponds to a new solution type with the following properties.

Type VI. The small loop shown in Fig.4 touches the vertical axis at the point $(x,y)=(0,1)$ where the universe size $a$ diverges. In the vicinity of this point one has

[TABLE]

where ${H}=\sqrt{\lambda}H_{0}/(6\sqrt{2})$. The evolution along the loop corresponds to the universe starting from an infinite size $a\sim e^{-{H}t}$ in the past, then shrinking to a finite size, bouncing back and expanding again to an infinite size $a\sim e^{+{H}t}$. These solutions show ghost.

5.4.5 $8/\sqrt{19}<\lambda$

If $\lambda$ exceeds the value $8/\sqrt{19}$ then the two disjoint pars of the $y(x)$ curve interconnect to form one connected manifold, as illustrated by the $\lambda_{6}$ curve in Fig.4. This corresponds to four different solutions. The two parts of the curve where $y(x)\leq 0$ correspond to solutions of Types III and V; Type III always being stable. The parts of the curve where $y(x)\geq 0$ correspond to two different solutions of the following new type.

Type VII. The two parts of the $y(x)$-curve which interpolate between points $(x,y)=(0,1)$ and $(x_{-},0)$ or between $(0,1)$ and $(x_{+},0)$ correspond to bounces – the universe starts from and ends up with an infinite size. These solutions are unstable.

Summarizing, the only stable solutions in the above classification are those of Type III; they exist only for $\lambda\geq-16/(5\sqrt{5})=-1.43$. They are qualitatively the same as those previously described in sub-section 5.3.

5.5 Solutions with $\beta\leq 0$

Solving Eq.(5.24) for $\epsilon=0$ or $\epsilon=-1$ yields

[TABLE]

which should be injected to (5.36), (5.37) and to (5.1) to determine the solutions. The behaviour of $y(x)=y_{+}(x)\cup y_{-}(x)$ is shown in Fig.5, and this time one finds only two qualitatively different solution types. First, if $\lambda<0$ then $y(x)$ is illustrated by the $\lambda<0$ curve shown on the left panel in Fig.5. This gives rise to Type I solutions described above, they are always unstable.

If $\lambda\geq 0$ then both $y_{+}(x)$ and $y_{-}(x)$ curves touch the vertical axis at the point with coordinates $(0,\epsilon)$, which corresponds to $a=\infty$ (see Fig.5), hence the solution splits in two. One solution is generated by curves $y_{-}(x)$ which emanate from $(0,\epsilon)$ downwards. These solutions are stable. The other solutions are generated by curves $y_{+}(x)$ emanating from $(0,\epsilon)$ towards increasing values of $y$, all of them are unstable.

Let us describe the stable solutions. If $\epsilon=-1$ and $\lambda\neq 0$ then the Hubble parameter at the point $(0,\epsilon)$ becomes $h_{-}=\sqrt{\frac{\lambda}{2}}$ (see (5.23)), which corresponds to the behaviour (5.40). Therefore, the $y_{-}$ curves for $\lambda\geq 0$ shown in the lower left corner on the left panel in Fig.5 describe the universe starting from zero size in the past and expanding in the future as $a\sim e^{{H}t}$ with ${H}=\sqrt{\lambda}H_{0}/(6\sqrt{2})$. These solutions are stable.

If $\epsilon=-1$ and $\lambda=0$ then one has for small $x$

[TABLE]

Therefore, the $y_{-}$ curve for $\lambda=0$ in the left lower corner on the left panel in Fig.5 describes the universe starting from zero size in the past and entering in the future the $a\sim t^{1/3}$ regime corresponding to the $w=1$ equation of state.

If $\epsilon=0$ but $\lambda\neq 0$ then at small $x$ one has

[TABLE]

where ${H}=\sqrt{\lambda}H_{0}/(6\sqrt{2})$. Therefore, the $y_{-}$ curves for $\lambda>0$ shown in the lower left corner on the right panel in Fig.5 correspond to the universe starting from a zero size in the past and approaching asymptotically the de Sitter phase. The squared gradient of the scalar field $X\sim{\psi}^{2}\sim x$ asymptotically approaches zero. These solutions are stable.

If $\epsilon=\lambda=0$ then for any $x$ one has

[TABLE]

therefore at all times the universe exactly follows the $w=1/5$ equation of state

[TABLE]

This type of behaviour we have already seen in (5.39) close to the singularity, but this time it holds everywhere. This solution is stable.

Summarizing, stable for $\beta\leq 0$ solutions exist for $\lambda\geq 0$ and are generated by the $y_{-}(x)$ curves residing in the lower left corners in the diagrams in Fig.5. They describe universes expanding from zero size to infinity. If $\beta<0$ then the universe approaches in the future the de Sitter phase with the Hubble rate ${H}=\sqrt{\lambda}H_{0}/(6\sqrt{2})$ if $\lambda\neq 0$, while for $\lambda=0$ it expands at late times according to the $w=1$ equation of state. For $\beta=0$ and $\lambda>0$ the universe approaches the de Sitter phase with the same Hubble rate ${H}=\sqrt{\lambda}H_{0}/(6\sqrt{2})$ if $\lambda\neq 0$, whereas for $\beta=0$ and $\lambda=0$ it expands at all times according to the $w=1/5$ equation of state.

5.6 Solutions in the metric version of the theory

Let us now compare studied above solutions in the Palatini version of the theory with those arising in the metric version of the theory. As was mentioned, the equations of the metric version can be obtained from (5.9), (5.10), (5.11), (5.13) by omitting terms proportional to $\alpha^{2}$. Applying the rescaling (5.20) then yields the modified version of Eqs.(5.23), (5.24), (5.25), (5.1). The equation for $y$ becomes

[TABLE]

and one has

[TABLE]

The properties of perturbations are read-off from (4.14),(4), after replacing in these formulas $\kappa\to\kappa-2XA$. This yields the kinetic term and sound speed:

[TABLE]

The procedure is then the same as before: first one solves (5.46) to obtain $y(x)=y_{+}(x)\cup y_{-}(x)$ with

[TABLE]

This determine algebraic curves shown in Figs.6,7 (in the online version of Figs.4–7 the $y_{+}(x)$ and $y_{-}(x)$ amplitudes are shown, respectively, in dark-blue and dark-red). The interpretation of these curves is obtained by injecting $y_{\pm}(x)$ to (5.47) and (5.48): for example, points where $y(x)$ either crosses the horizontal axis or touches the vertical axis correspond to the infinite size of the universe. As a result, the curves in Figs.6,7 corresponds either to universes expanding from zero to infinite size, or to universes expanding only up to a finite size and then shrinking, or to bounces.

The stable solutions are again only those generated by the parts of the $y_{-}(x)$ curves located under the horizontal axis in the left lower corner of the diagrams in Figs.6,7. Such solutions exist for any $\epsilon=0,\pm 1$ but only for $\lambda\geq 0$. Their profiles are qualitatively similar to those shown in Fig.2. The overall conclusion is that, despite their surprising variety, solutions in the Palatini-derived theory and those of the metric theory are qualitatively similar to each other.

5.7 Summary and comparison with the previously studied case

We presented above all homogeneous and isotropic cosmologies in two special shift-symmetric KGB models, one of which is the standard metric theory with the coefficient functions

[TABLE]

The other one is its Palatini version, which, according to the discussion in Section 3, can also be viewed as the standard metric theory with

[TABLE]

As we have seen, the classification of solutions in these two theories is rather complicated, but the solutions are sensitive only to $\Lambda$ and to the sign of $\beta$, while $\alpha$ only changes the scale. Despite some differences between the two theories seen in the figures above, the properties of the solutions turn out to be essentially the same in both cases and the solutions are always found to be either expanding or recollapsing universes, or bounces.

The most interesting physically are the stable expanding cosmologies. They exhibit the effective equation of state $w=1/5$ near singularity and at late times they approach the currentless de Sitter phase with a constant Hubble rate. If $\beta<0$ or $\beta=0$ then the Hubble rate is expressed in terms of $\Lambda$ in the usual way, $H=\sqrt{\Lambda/3}$, but for $\beta>0$ it is not directly related to $\Lambda$, and if $\Lambda=0$ then

[TABLE]

where the numerical coefficient $c=1.64\times 10^{-1}$ in model (5.50) and $c=3.65\times 10^{-3}$ in model (5.51). The theories also show recollapses and bounces, but these solutions are always unstable.

To the best of our knowledge, a similar complete classification of KGB cosmologies has never been reported in the literature. At the same time, it was known before that the equations are completely integrable in the shift-symmetric case, and cosmologies were much studied from a more physical perspective already in the original KGB paper [12].

The specific model considered in Ref.[12] corresponds to (5.50) with $\beta>0$444See Eq.(5.1) in [12] where one should replace $X\to-X$ since the metric signature used in that paper is opposite to ours. but with $\Lambda$ replaced by an external matter energy density, $\Lambda\to\rho_{ext}\sim a^{-3(1+w_{ext})}$. The case of a constant $\rho_{ext}$ was also considered there, and it seems that what is shown in Fig.3 in Ref.[12] corresponds to our solutions with $\epsilon=1$ and $\lambda=3$ on the left panel of our Fig.6. The $\lambda=3$ curve shown there describes three different solutions: a stable cosmology, an unstable cosmology, and a bounce, the two latter showing the ghost. Now, on the right panel in Fig.6 we plotted profiles of these solutions against the dimensionless variable $\chi\sim\psi$ defined as $\chi=\sqrt{x}$ in the $\chi>0$ region and $\chi=-\sqrt{x}$ in the $\chi<0$ region. We used the symmetry (5.19) to relate the values of the solutions for opposite signs of ${\psi}$. The dimensionless Hubble parameter shown in Fig.6 is defined as $h(\chi)=h_{-}$ for $\chi>0$ and $h(\chi)=-h_{+}$ for $\chi<0$, while the dimensionless current is $j(\chi)=-\sqrt{x}y_{-}$ for $\chi>0$ and $j(\chi)=\sqrt{x}y_{+}$ for $\chi<0$. The vertical lines delimit the instability region where $K<0$. The resulting diagram is very similar to Fig.3 in [12] and describes three different cosmologies, since zeros of $j(\chi)\sim\sqrt{x}y\sim{a}^{-1/3}$ correspond to the infinite universe size. The rightmost part of the diagram where $j(\chi)\geq 0$ describes the stable branch.

The main accent Ref.[12] was made on the analysis of the currentless solutions (called fantom attractors) in the presence of a generic external matter. Although this goes beyond the scope of our program, we can recover the same results. Replacing in (5.9) $\Lambda\to\rho_{ext}$ and in (5.10) $\Lambda\to-p_{ext}$ with $p_{ext}=w_{ext}\,\rho_{ext}$, yields

[TABLE]

where $\eta$ distinguishes between theories (5.50) and (5.51) by taking values [math] and $1$, respectively. This defines $\Omega_{X}={\rho_{X}}/{3H^{2}}$, $\Omega_{e}={\rho}/{3H^{2}}$ and $w_{X}={p_{X}}/{\rho_{X}}$, where $H$ is obtained by setting the current to zero (as in (5.15)):

[TABLE]

Injecting this to (5.53) and (5.54) yields algebraic relations which determine $\psi$ and $\dot{\psi}$ in terms of $\rho_{ext}$ and $p_{ext}$. Computing then $\Omega_{X}$ and $w_{X}$ we recover the results of Ref.[12]. For example, we see that at early times the external matter dominates and $\Omega_{X}\ll 1$ while the effective equation of state of the scalar is then determined by $1+w_{X}=-(1+w_{ext})$, whereas at late times $\Omega_{X}\to 1$ and $w_{X}\to-1$.

Bounces in the presence of a “hot matter” were studied in Ref.[14]. Interestingly, solutions whose evolution at the turnaround point is stable were detected, although they still show instability somewhere [14]. Our bounces are also unstable, in addition we do not find solutions whose evolution at the turnaround point would be stable. The difference must be due to the fact that our bounces are “cold” and not “hot”, and also because the model considered in [14] corresponds to555After the replacement $X\to-X$ due to the signature change. $K=\pm X+X^{3}$, which is different from our model (5.51) with the negative coefficient in front of $X^{3}$.

6 More general Horndeski models

We have studied up to now the Palatini version of the Horndeski models respecting the condition (1.4). These theories are described by second order equations and are therefore free of the Ostrogradsky ghost. It turns out that relaxing the condition (1.4) invariably produces higher derivatives within the Palatini approach. However, the ghost does not always arise. To illustrate this, let us consider a simple example obtained by setting in (1)

[TABLE]

with constant $\sigma,\xi$. The Horndeski Lagrangian reduces to

[TABLE]

where the dots denote total derivatives. The term $\nabla^{\mu}(G_{\mu\nu}\sqrt{-g})$ in the second line vanishes, but it would be proportional to the non-metricity within the Palatini approach, hence dropping this term is equivalent to choosing a non-zero $\Delta L_{\rm P}$ in (1.13)666Keeping the $\nabla^{\mu}(G_{\mu\nu}\sqrt{-g})$ term in the metric-affine approach would render the torsion dynamical, as explained after Eq.(1.17).. Consider the metric-affine version of the third line in (6.2),

[TABLE]

where $\overset{(\Gamma)}{R}=g^{\mu\nu}\overset{(\Gamma)}{R}_{\mu\nu}$ and $G_{\mu\nu}=\overset{(\Gamma)}{R}_{\mu\nu}-\frac{1}{2}\overset{(\Gamma)}{R}g_{\mu\nu}$. Varying this with respect to $\phi$ and using (2.18) yields

[TABLE]

In the metric case one has $\overset{(\Gamma)}{\nabla}_{\sigma}g_{\mu\nu}=0$ and $\nabla^{\mu}G_{\mu\nu}=0$ hence the equation reduces to $G_{\mu\nu}\nabla^{\mu}\nabla^{\nu}\phi=0$ which contains only second derivatives. However, if $\overset{(\Gamma)}{\nabla}_{\sigma}g_{\mu\nu}\neq 0$ then $\nabla^{\mu}G_{\mu\nu}\neq 0$ and the equation contains higher derivatives, which can be seen as follows. The Lagrangian can be represented as

[TABLE]

with

[TABLE]

where as usual $X=\frac{1}{2}(\partial\phi)^{2}$. Introducing $h_{\mu\nu}$ defined by the relation

[TABLE]

hence

[TABLE]

the Lagrangian becomes

[TABLE]

It is well-known that varying this Lagrangian with respect to the connection yields

[TABLE]

hence $\Gamma^{\mu}_{\alpha\beta}$ is the Levi-Civita connection for the effective metric $h_{\mu\nu}$. Since the latter contains derivatives in (6.8), it follows that $\Gamma^{\mu}_{\alpha\beta}$ contains second derivatives hence both $G_{\mu\nu}$ and the equation contains third derivatives of $\phi$.

At the same time, the relation (6.8) between $g_{\mu\nu}$ and $h_{\mu\nu}$ is an invertible disformal transformation, hence one can consider $h_{\mu\nu}$, $\phi$ as independent variables instead of $g_{\mu\nu}$, $\phi$. Varying the Lagrangian with respect to $h_{\mu\nu}$ yields

[TABLE]

which are the vacuum Einstein equation for the Ricci tensor constructed from the metric $h_{\mu\nu}$ in the standard way. They imply that $G_{\mu\nu}=0$, hence the scalar field equation (6.4) is fulfilled as well. Therefore, the theory (6.3) is simply the vacuum General Relativity for the effective metric $h_{\mu\nu}$ so that the ghost is absent.

The original metric $g_{\mu\nu}$ is obtained from $h_{\mu\nu}$ by inverting the relation (6.8), and since the latter contains the scalar field $\phi$ remaining undefined, there are infinitely many metrics $g_{\mu\nu}$ for a given Ricci-flat $h_{\mu\nu}$. This ambiguity can be removed by adding $K(X,\phi)\sqrt{-g}$ to the Lagrangian to produce a non-trivial condition for $\phi$. The equations will still contain higher derivatives when expressed in terms of $g_{\mu\nu},\phi$, but they become second order equations when expressed in $h_{\mu\nu},\phi$ variables.

Summarizing, the theory (6.3) contains higher derivatives when parameterized in terms of $g_{\mu\nu}$ and $\phi$ hence it is outside the Horndeski family. At the same time, it is ghost-free since the disformal transformation (6.8) removes the higher derivatives, hence it must belong to the DHOST family (similar examples were considered in [36]).

However, in the generic case the theory turns out to be outside the DHOST family and shows ghost. Consider, for example, the Palatini version of the entire piece of the Horndeski Lagrangian (1) generated by $G_{4}(X,\phi)$,

[TABLE]

Solving the equation for the connection gives

[TABLE]

where $D^{\alpha}_{\mu\nu}$ is displayed in the Appendix. Injecting this back to $L_{\rm P}$ yields for a generic $G_{4}(X,\phi)$ a metric Lagrangian that belongs neither to the Horndeski nor to DHOST family. Therefore the theory contains ghost. For the particular choice $G_{4}(X,\phi)=f(\phi)X$ the theory can be shown to be of the DHOST type, but it is unclear if the ghost can be removed in other cases, for example by adding to the Lagrangian a non-trivial $\Delta L_{\rm P}$ as in (1.13).

The Palatini versions of the parts of the Lagrangian (1) containing $G_{5}(X,\phi)$ remains totally unexplored since the torsion is then dynamical.

7 Concluding remarks

Summarizing the above discussion, we have studied what happens if the Horndeski theory is treated within the Palatini approach. It turns out that there are infinitely many metric-affine versions $L_{\rm P}$ of the original Horndeski Lagrangian which differ from each other by terms proportional to the non-metricity tensor, as expressed by (1.13). Each $L_{\rm P}$ defines a theory which is equivalent to a certain metric theory with the Lagrangian obtained by injecting the algebraic solution for the connection back to $L_{\rm P}$. Therefore, the metric-affine generalisations of the Horndeski theory reduce again to metric theories for a gravity-coupled scalar field.

Every such a metric theory can either belong to the original Horndeski family, or it can be of a more general DHOST type, or it can be something else, in which case it has the Ostrogradsky ghost. Therefore, the metric-affine generalisations of the Horndeski theory can be ghost-free but not all of them are ghost-free.

It is interesting to know when these theories are ghost-free. We were able to give the answer for the KGB subset of the Horndeski theory defined by the condition (1.4): it turns out that its metric-affine version defined by (1.6)–(1.9) is ghost-free because it yields a theory which is again in the metric KGB class. We have also checked that its generalisation defined by (1.13), where $\Delta L_{\rm P}$ contains only the linear in the non-metricity terms shown in (1.14) remains ghost-free [31]. We classified all homogeneous and isotropic cosmologies in these theories.

The situation with more general Horndeski models is more complicated and will be reported separately [30]. It is possible that the metric-affine versions of the parts of the Horndeski Lagrangian containing $G_{4}(X,\phi)$ could be made ghost-free by carefully adjusting $\Delta L_{\rm P}$ but is unclear if this procedure works for generic $G_{4}(X,\phi)$. The situation is totally unexplored if the Lagrangian contains $G_{5}(X,\phi)$.

One should also say that the Horndeski theory is not the only one whose metric-affine versions can be ghost-free. For example, the theory described

[TABLE]

has second order equations but does not reduce to Horndeski theory when the non-metricity vanishes.

Another example is provided by the Lagrangian [35]

[TABLE]

where the terms in the first line are the same as in the Horndeski theory, whereas those in the second line do not have the Horndeski structure. Adding to this a suitably chosen $\Delta L_{\rm P}$ made of the non-metricity and varying yields a particular member of the DHOST family [35] (we were able to confirm this [30]), hence the theory is ghost-free.

An example of a completely different type is provided by the Born-Infeld theory,

[TABLE]

which has second order equation [32]. It follows that the Horndeski Lagrangian is not the most general one that leads to second order field equations within the Palatini approach. An interesting problem would be to find the most general ghost-free metric-affine theory.

Acknowledgments.– It is a pleasure to acknowledge discussions with Evgeny Babichev, Thibault Damour, Alexander Vikman, and especially with Dr. Katsuki Aoki who explained to us his work and helped to clarify a number of important issues. M.S.V. thanks for hospitality the YITP in Kyoto, where a part of this work was completed. His work was also partly supported by the PRC CNRS/RFBR project grant, as well as by the Russian Government Program of Competitive Growth of the Kazan Federal University.

Appendix

Here is the explicit form of the non-metric part of the connection in (6.13):

[TABLE]

with $G_{4X}=\partial_{X}G_{4}(X,\phi)$ and the functions $Y,A,B,C$ defined as

[TABLE]

The function $A,B$ here should not be confused with those used in the main text.