Abstract
In this paper, we investigate the possible parameter space of Palatini–Horndeski theory with gravitational waves in a spatially flat Universe. We develop a general method for obtaining the speed of gravitational waves in the Palatini formalism in the cosmological background and we find that if the theory satisfies the following condition: in any spatially flat cosmological background, the tensor gravitational wave speed is the speed of light c, then only \(S = \int d^4x \sqrt{-g} \big [K(\phi ,X)-G_{3}(\phi ,X){{\tilde{\Box }}}\phi +G_{4}(\phi ){\tilde{R}}\big ]\) is left as the possible action in Palatini–Horndeski theory. We also find that when \(G_{5}(\phi ,X)\ne 0\), the tensor part of the connection will propagate and there are two different tensor gravitational wave speeds.
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1 Introduction
The successful detection of gravitational waves made up the last puzzle missing in the experimental verification of general relativity [1,2,3,4,5]. Therefore, general relativity has become the most successful theory of gravity so far.
However, there are still many theoretical problems that can not be explained by general relativity, such as how to explain the hierarchy between the Planck scale and the electroweak scale [6,7,8] and how to quantize gravity [9]. In addition, the phenomena observed in experiments, such as the accelerated expansion of the Universe [10] and the flat rotation curves of galaxies [11], can not be explained by general relativity. For these reasons, many modified theories of gravity were considered [6,7,8, 12,13,14,15,16] in the hope of answering the problems that general relativity could not answer.
Adding additional scalar field is one way to modify gravity. Theories obtained in this way are called scalar–tensor theories. In Ref. [17], a tentative indication for scalar transverse gravitational waves was reported. If this is further confirmed in the future, it will strongly suggest that the gravity theory describing our world should have a scalar degree of freedom. In order to avoid the Ostrogradsky instability [18,19,20,21], we expect to give priority to those theories that can derive second-order field equations. In the metric formalism, the most general scalar–tensor theory that can derive second-order field equations is Horndeski theory [13].
However, Refs. [22,23,24] pointed out that the observation of the speed of tensor gravitational waves in the Universe by the gravitational wave event GW170817 together with the gamma ray burst GRB170817A would severely constrain the possible parameter space of metric Horndeski theory. Specifically, GW170817 and GRB170817A require the tensor gravitational wave speed \(c_g\) to meet [25, 26]
This shows that in a very high precision, we can say that the tensor gravitational wave speed in the Universe is equal to the basic constant c (speed of light). Considering that the cosmic background is also evolving during gravitational wave propagation, the most economic and natural assumption made by this observation result for the theory seems to be that: in any cosmological background, tensor gravitational waves always propagate at the speed of light. However, the possible subclasses of metric Horndeski theory satisfying this assumption only remain [22, 27]
This constraint limits the application of scalar–tensor theories. Therefore, We expect to find scalar–tensor theories beyond the metric Horndeski framework. Reference [28] studied the orbital evolution of eccentric binary systems in metric Horndeski gravity. There are also many studies using GW170817 to constrain modify gravity theories [29,30,31,32,33,34,35,36,37,38,39,40].
Further analysis shows that not all higher derivative theories have the Ostrogradsky instability. The higher derivative theory without the Ostrogradsky instability is required to satisfy the degeneracy condition [21, 41, 42]. In the metric formalism, the scalar–tensor theory with higher derivative but without the Ostrogradsky instability is called degenerate higher-order scalar–tensor (DHOST) theory [43,44,45,46,47]. In addition to DHOST theory, considering the teleparallel framework is another way to go beyond the metric Horndeski framework. In teleparallel Horndeski theory established by Bahamonde et al., metric Horndeski theory is included in the teleparallel framework as one of many subclasses [48, 49].
Considering the scalar–tensor theory in the Palatini formalism may be another way to go beyond metric Horndeski framework. There have been some works on scalar–tensor gravity in the Palatini formalism [50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67]. Cosmology in Palatini–Horndeski theory is different from that in metric Horndeski theory and their stability properties are different [60]. Different from metric Horndeski theory, under some parameter spaces, the connection of Palatini–Horndeski theory will introduce some new degrees of freedom [60]. In addition, the polarization modes of gravitational waves in Palatini–Horndeski theory are different from that in metric Horndeski theory [63]. Thus, it seems that Palatini–Horndeski theory may be different from metric Horndeski. However, it is necessary to further investigate the possible parameter space of Palatini–Horndeski theory.
In this paper, we will find possible subclasses of Palatini-Horndeski theory that satisfied the following condition: the speed of tensor gravitational waves is the speed of light in any spatially flat cosmological background. In Sect. 2, we review Palatini–Horndeski theory. In Sect. 3, we develop a general method in the Palatini formalism to obtain the speed of tensor gravitational waves in the spatially flat cosmological background and constrain the parameter space. The conclusion is given in Sect. 4.
We use the natural system of units in this paper. Greek alphabet indices \((\mu ,\nu ,\lambda ,\rho )\) and Latin alphabet indices (i, j, k, l) range over spacetime indices (0, 1, 2, 3) and space indices (1, 2, 3), respectively.
2 Palatini–Horndeski theory
In the Palatini formalism, the connection is independent of the metric. Therefore, it is necessary to take the variations of the action with respect to the metric and the connection independently. The Riemann tensor \({\tilde{R}}^{\mu }_{\ \nu \rho \sigma }\) and Ricci tensor \({\tilde{R}}_{\mu \nu }\) in the Palatini formalism are defined as
Furthermore, we assume that the connection is nontorsion: \(\varGamma ^{\lambda }_{\mu \nu }=\varGamma ^{\lambda }_{\nu \mu }\).
The action of Palatini–Horndeski theory is defined as follows:
where
Here, \({\tilde{\Box }}={\tilde{\nabla }}^{\mu }{\tilde{\nabla }}_{\mu }\), \(X=-\frac{1}{2}\partial _{\mu }\phi \partial ^{\mu }\phi \), and \( K, G_{3}, G_{4}\) and \(G_{5}\) are real analytic functions of the variables \(\phi \) and X. To distinguish the quantities in the metric formalism, we add tilde to represent the corresponding quantities defined in the Palatini formalism. A comma in subscript means partial derivative, e.g., \(G_{4,X}\equiv \partial G_4 /\partial X\).
In the Palatini formalism, the compatibility condition \({\tilde{\nabla }}_{\lambda }g_{\mu \nu }=0\) is generally no longer valid. Therefore, the definition of \({\tilde{\Box }}\phi , {\tilde{\nabla }}^{\mu }{\tilde{\nabla }}^{\nu }\phi \) and \( {\tilde{\nabla }}^{\mu }{\tilde{\nabla }}_{\nu }\phi \) in the action (5) in the Palatini formalism is not unique [60, 62]. In this paper, we take the definition in Ref. [63]:
3 The speed of tensor gravitational waves
In this section, we will calculate the speed of tensor gravitational waves propagating in a spatially flat cosmological background and find possible subclasses of Palatini–Horndeski theory that satisfy the following condition: the speed of tensor gravitational waves is the speed of light in any spatially flat cosmological background.
We first consider the background evolution of a spatially flat Universe in the Palatini–Horndeski theory. For a spatially flat Universe, the metric \(g_{\mu \nu }\) is the spatially flat Friedmann-Robertson-Walker (FRW) metric, and the connection \(\varGamma ^{\lambda }_{\mu \nu }\) and scalar field \(\phi \) are only functions of time:
We consider that the connection has spatial isotropy, that is, under the spatial rotation transformation, the components of the connection \(\varGamma ^{\lambda }_{\mu \nu }\) are invariant. This condition further limits the value of the connection. Specifically, under the spatial rotation transformation, the transformation law of the components of the connection is the same as that of the third-order tensor, which requires that the connection \(\varGamma ^{\lambda }_{\mu \nu }\) satisfies [68, 69]
Here, \(\delta _{ij}\) is the Kronecker delta, and \(\varepsilon _{ijk}\) is the Levi-Civita tensor. It can be seen that for the components of the connection, only \(\varGamma ^{0}_{00}, \varGamma ^{0}_{11}=\varGamma ^{0}_{22}=\varGamma ^{0}_{33}\) and \(\varGamma ^{1}_{01}=\varGamma ^{2}_{02}=\varGamma ^{3}_{03}\) may not be zero.
By substituting Eqs. (11) and (12) into the action (5), we obtain the action that describes the evolution of a spatially flat Universe:
where,
Here and below, the dot on the letter represents the derivative of the corresponding quantity with respect to time, \(K,G_3,G_4,G_5,G_{4,X}\), and \(G_{5,X}\) are functions of \((\phi ,\frac{{\dot{\phi }}^2}{2N^2})\).
One may want to set \(N(t)=1\) at the level of action, and then obtain the evolution equations by varying the variables \(\left( a,\phi ,\varGamma ^{0}_{00},\varGamma ^{0}_{11},\varGamma ^{1}_{01}\right) \). However, this will miss one equation [70]. Thus, in order to obtain complete evolution equations, N should be kept in the action.
In addition to the gravitational field, the ideal fluid material field is also distributed in the spatially flat Universe. Therefore, in addition to the gravitational field action (13), we should also add an action \(S_m\) that describes the ideal fluid into the the total action
Here, S is defined by (13). In the Palatini formalism, \(S_m\) is only a function of the metric and the material field, and it is independent of the connection. Varying the action \(S_m\) with respect to \(g_{\mu \nu }\), we obtain
Here, \(T^{\mu \nu }\) is the energy-momentum tensor of the ideal fluid:
where \(\epsilon \) is the matter density, P is the matter pressure. The four-velocity \(u^{\mu }\) satisfies \(u^{0}=\frac{1}{N}, u^{i}=0\).
By varying the action (15) with respect to \(N,\phi ,a,\varGamma ^{0}_{00},\varGamma ^{0}_{11}\) and \(\varGamma ^{1}_{01}\), we obtain the background equations:
Here, L is defined by (14). Because the specific expressions of the background equations (18) are very lengthy and easy to obtain, they will not be listed in this paper. In the following, we take \(N(t)=1\).
In order to study the tensor gravitational waves, we need to obtain the linear perturbation equations of the tensor perturbations on the spatially flat cosmological background.
Since the metric and connection are independent in the Palatini formalism, they should be perturbed independently:
We take the part describing tensor gravitational waves in perturbations:
Here, \(H_{ij}, A_{ij}, B_{ij}, C_{ij}\) and \(D_{ij}\) are symmetric transverse traceless tensors. They satisfy
Only in this paragraph, we use \(\delta ^{ij}\) (\(\delta _{ij}\)) to raise and lower the index. In Appendix 1, we give the decomposition of the connection and explain why the perturbations describing the tensor gravitational waves are given by Eq. (20).
Without losing generality, we consider the propagation direction of gravitational waves as \(+z\) direction. At this time, it can be seen from (20) that the components of the perturbations \(h_{\mu \nu }\) and \(\varSigma ^{\lambda }_{\mu \nu }\) that may not be zero are
By expanding the second-order terms of the perturbations (22) in the action (15) and varying the action with respect to the perturbations, we can obtain the linear perturbation equations describing the tensor gravitational waves. These equations are very lengthy and easy to obtain, so they are not listed here.
Now we have obtained the linear perturbation equations describing the tensor gravitational waves. Next, we will use the equations to obtain the speed of the tensor gravitational waves.
Before that, we will take metric Horndeski theory as an example to demonstrate how to obtain the speed of tensor gravitational waves from the linear perturbation equation. In metric Horndeski theory, the linear perturbation equation describing the tensor gravitational waves is given by [27]:
where h is the component \(h_{11}\) or \(h_{12}\), b and \(c_t\) are functions of time, and \(\varDelta \) is the Laplace operator. For h(t, z) propagating along the \(+z\) direction, we make a Fourier transform:
By substituting Eq. (24) into Eq. (23), and using the linearity of Eq. (23), we obtain the following equation:
This allows us to consider only the case with a single spatial wave vector \(k_3\):
where f(t) can always be expressed as
Here, F is the norm of f(t) and \({k_0}(t) t\) is the argument. Therefore, F and \({k_0}\) are real numbers.
Considering that the gravitational wave is observed near time \(t_0\) and the observation duration is \(\varDelta T\), that is, the observation time \(t \in [t_0-\frac{\varDelta T}{2},t_0+\frac{\varDelta T}{2}]\). The duration \(\varDelta T\) is about the same order of magnitude as the period of the gravitational wave, and during this time, the amplitude and phase of the gravitational wave change very little:
Thus, \(h=F(t)e^{i[{k_0(t) t-k_3 z}]}\) can be approximated as a plane gravitational wave near \(t_0\):
For the evolution of the cosmic background, the changes of a, b and \(c_t\) in Eq. (23) during this period are also small:
So Eq. (23) near \(t_0\) can be approximated as
By substituting Eq. (29) into Eq. (31), we can obtain
The gravitational waves we can observe have large \(k_0\) and \(k_3\), which makes the linear term of wave vector component (uniformly recorded as k) in the above equation negligible compared with the quadratic term of k. Thus, by Eq. (32), the relationship between \(k_0\) and \(k_3\) will satisfy
Just write (29) as
and using Eq. (33), we can see that the tensor gravitational wave speed \(c_g\) at time \(t_0\) is
The speed (1) obtained by this method is the same as that of Refs. [27, 71].
Similar to the above analysis, for Palatini–Horndeski theory near a certain time, we also approximate the coefficients of perturbations in the linear perturbation equations to constants which are independent of time. In addition, we also approximate the perturbations (22) to the form of plane gravitational waves:
Here, \({\bar{h}}_{\mu \nu }\) and \({\bar{\varSigma }}^{\lambda }_{\mu \nu } \) are amplitudes. Similar to the above example of metric Horndeski theory, by substituting (36) into the approximated linear perturbation equations, we can obtain the linear equations with amplitudes \({\bar{h}}_{\mu \nu }\) and \({\bar{\varSigma }}^{\lambda }_{\mu \nu }\) as the variable. This equations can be written in matrix form:
where A is a \(10 \times 10\) matrix and it depends on variables \(k_0\) and \(k_3\). X is a column vector composed of the components of the amplitudes \({\bar{h}}_{\mu \nu }\) and \({\bar{\varSigma }}^{\lambda }_{\mu \nu }\). The specific expression of A is very lengthy and easy to obtain, so it is not listed in this paper.
Equation (37) has a gravitational wave solution if and only if
As in the above example, we consider \(k_0\) and \(k_3\) to be large. Thus, the lower power terms of k in \(\det (A)\) are ignored, and only the highest power terms of k are retained. We mark the remaining quantity in \(\det (A)\) as \({\mathcal {A}}\). Therefore, according to the equation
we can know the relationship between \(k_0\) and \(k_3\), so as to solve the speed of tensor gravitational waves.
Now, we will calculate the speed of tensor gravitational waves propagating in a spatially flat cosmological background.
We divide the parameter space of Palatini–Horndeski theory into two classes.
Class I: \(G_{5}(\phi ,X)=0\). In this class, by solving Eq. (39), we find that the tensor gravitational wave speed \(c_g\) is given by
Thus, the condition that the tensor gravitational wave speed is always the speed of light in any spatially flat cosmological background requires
for any spatially flat cosmological background.
In this class, by solving the background equation (18), we find that \(\big (\ddot{a},\dddot{\phi },\varGamma ^{0}_{00},{\varGamma }^{1}_{01},{\varGamma }^{0}_{11}\big )\) can be expressed as functions of \(\big (a,{\dot{a}},\phi ,{\dot{\phi }},\ddot{\phi },P,\epsilon \big )\). Further considering the equation of state and the energy conservation equation of the ideal fluid, we can also use \(\big (P,a,{\dot{a}}\big )\) to express \(\big ({\dot{P}},{\dot{\epsilon }}\big )\). Therefore, as long as we know \(\big (a,{\dot{a}},\phi ,{\dot{\phi }},\ddot{\phi },P,\epsilon \big )\), we can obtain \(\big (\ddot{a},\dddot{\phi },\varGamma ^{0}_{00},{\varGamma }^{1}_{01},{\varGamma }^{0}_{11},{\dot{P}},{\dot{\epsilon }}\big )\). This determines the initial value condition of the background equation (18). The specific expressions of these variables are very lengthy and easy to obtain, so we have not listed them.
Considering that there are different equations of state for different types of matters, P and \(\epsilon \) can be considered as independent variables. Therefore, the condition that the tensor gravitational wave speed is the speed of light in any spatially flat cosmological background is equivalent to the following condition: at any values of the variables \(\big (a,{\dot{a}},\phi ,{\dot{\phi }},\ddot{\phi },P,\epsilon \big )\), condition (41) is always true. This requires that \(G_{4,X}\) is always vanishing.
In this way, we find that in Class I, only subclass
satisfies the condition that the tensor gravitational wave speed is the speed of light in any spatially flat cosmological background.
Class II: \(G_{5}(\phi ,X)\ne 0\). In this class, by solving the background equation (18), we can see that \(\big ({\dot{a}},\dddot{\phi },{\dot{\varGamma }}^{0}_{00},{\dot{\varGamma }}^{0}_{11},{\dot{\varGamma }}^{1}_{01}\big )\) can be expressed as the functions of \(\big (a,\phi ,{\dot{\phi }},\ddot{\phi },{\varGamma }^{0}_{00},{\varGamma }^{0}_{11},{\varGamma }^{1}_{01},P,\epsilon \big )\). Further considering the state equation and the energy conservation equation of the ideal fluid, we can also use \(\big (P,a,{\dot{a}}\big )\) to express \(\big ({\dot{P}},{\dot{\epsilon }}\big )\). Therefore, as long as we know \(\big (a,\phi ,{\dot{\phi }},\ddot{\phi },{\varGamma }^{0}_{00},{\varGamma }^{0}_{11},{\varGamma }^{1}_{01},P,\epsilon \big )\), we can solve \(\big ({\dot{a}},\dddot{\phi },{\dot{\varGamma }}^{0}_{00}, {\dot{\varGamma }}^{0}_{11},{\dot{\varGamma }}^{1}_{01},{\dot{P}},{\dot{\epsilon }}\big )\). This determines the initial value condition of the background equation (18). The specific expressions of these variables are very lengthy, so we do not list them. By substituting these expressions into Eq. (39) and solving it, we can obtain the tensor gravitational wave speed expressed by the variables \(\big (a,\phi ,{\dot{\phi }},\ddot{\phi },{\varGamma }^{0}_{00},{\varGamma }^{0}_{11},{\varGamma }^{1}_{01},P,\epsilon \big )\).
In fact, the tensor gravitational wave speed we solved is not unique in this class, and it has two possible solutions \(c_{g1}\) and \(c_{g2}\). These two speeds are generally different. However, when the matter pressure \(P=0\), we have \(c_{g1}=c_{g2}\). The specific expressions of \(c_{g1}\) and \(c_{g2}\) are very lengthy, so we do not list them.
The first speed \(c_{g1}^2\) can be expressed as a fraction
It can be seen that \(c_{g1}=1\) is equivalent to the numerator part \({\mathcal {N}}\) on the right side of Eq. (43) minus the denominator part \({\mathcal {D}}\) equal to 0:
By expanding the brackets, M can be expressed as a polynomial about the variables \(\big (a,\phi ,{\dot{\phi }},\ddot{\phi },{\varGamma }^{0}_{00},{\varGamma }^{0}_{11},{\varGamma }^{1}_{01},P,\epsilon \big )\). If we require the tensor gravitational wave speed \(c_{g1}\) to be the speed of light under any spatially flat cosmological background, then for any values of the variables \(\big (a,\phi ,{\dot{\phi }},\ddot{\phi },{\varGamma }^{0}_{00},{\varGamma }^{0}_{11},{\varGamma }^{1}_{01},P,\epsilon \big )\), this polynomial should be 0. We notice that in this polynomial, the term where \((\varGamma ^{0}_{00})^4\epsilon \) appears is
Therefore, the above condition requires
If substituting the condition \(G_5=\frac{5}{2}{{\dot{\phi }}}^2G_{5,X}\) into Eq. (44), we again notice that in this polynomial, the term where \(({\varGamma ^{1}_{01}})^4\epsilon \) appears is
Therefore, the above condition further requires \(G_{5,X}=0\). Combining with the condition (46) we have \(G_5=0\). However, this is inconsistent with the assumption \(G_5 \ne 0\) in this class.
For the second solution \(c_{g2}\), using the same analysis method as that used to analyze the first solution \(c_{g1}\), we find that the condition of \(c_{g2}=1\) also requires \(G_5=0\).
To sum up, for Palatini–Horndeski theory, the parameter space satisfying the condition that the tensor gravitational wave speed is the speed of light under any spatially flat cosmological background is only \(G_{4,X}=0\) and \(G_5=0\).
4 Conclusion
In this paper, we calculated the speed of tensor gravitational waves in the spatially flat cosmological background. Unlike the metric formalism, in the Palatini formalism, the background perturbation describing tensor gravitational waves is no longer just the tensor perturbation of metric \(h^{TT}_{\mu \nu }\), but there are also several tensor perturbations of connection (see Eq. (22)). Therefore, linear perturbation equations describing tensor gravitational waves in the Palatini formalism are often a system of equations. For such a system of equations, we developed a general method in the Palatini formalism to obtain the speed of tensor gravitational waves in the spatially flat cosmological background in Sect. 3. This method is generally applicable to any theory in the Palatini formalism.
It is worth noting that we found that there are two possible speeds of tensor gravitational waves in Class II. This is due to the additional degrees of freedom introduced by the tensor perturbations of the connection. It seems to imply that if we observe two tensor gravitational waves with different speeds in the future, the theory of gravitation describing our world may be described by the Palatini formalism.
However, if we further require the tensor gravitational wave speed to be the speed of light c in any spatially flat cosmological background, then only
is left as the possible action in the above two subclasses of Palatini–Horndeski theory. Reference [60] pointed out that the action (48) in the Palatini formalism is actually equivalent to the following action in the metric formalism:
Here,
It can be seen that the action (48) in the Palatini formalism actually still belongs to metric Horndeski theory. Therefore, Palatini–Horndeski theory described by action (48) does not have the Ostrogradsky instability. It should be noted that the action (49) is the only subclass of metric Horndeski theory that is compatible with the condition that the tensor gravitational wave speed is the speed of light c in any spatially flat cosmological background.
Despite some progress, the maximum parameter space without the Ostrogradsky instability in Palatini–Horndeski theory has not yet been fully found. Specifically, when \(G_{5}(\phi ,X)\ne 0\), the connection will introduce additional degrees of freedom. It is completely unknown whether the theory has the Ostrogradsky instability in this case [60]. When only considering the case of the evolution of a spatially flat Universe, conducting the Hamiltonian analysis of the action (13), one can find that there is at least one constraint in the theory. It can be expected that by carefully analyzing the constraint, all parameter spaces without the Ostrogradsky instability can be found in the considered case. Providing such an analysis will help us deepen our understanding of the Ostrogradsky instability in the Palatini-Horndeski theory. This requires future research.
Finally, it should be pointed out that this does not mean that the scalar–tensor gravity in the Palatini formalism must not beyond the framework of metric Horndeski theory. This is because in this paper, we have taken the simplest and natural assumption for the speed of tensor gravitational waves, that is, in any spatially flat cosmological background, the tensor gravitational wave speed is the speed of light c. By weakening this assumption, it is possible to find other parameter spaces of Palatini–Horndeski theory compatible with GW170817. In addition, Palatini–Horndeski theory considered in this paper is not the most general theory of scalar–tensor gravity in the Palatini formalism. A more general discussion needs to study more general action. These all need to be studied in future work.
Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors’ comment: The results of this paper are all derived from analytical calculations. Therefore, there is no associated data.]
References
B.P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), Observation of gravitational waves from a binary black hole merger. Phys. Rev. Lett. 116, 061102 (2016)
B.P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), GW151226: observation of gravitational waves from a 22-solar-mass binary black hole coalescence. Phys. Rev. Lett. 116, 241103 (2016)
B.P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), GW170104: observation of a 50-solar-mass binary black hole coalescence at Redshift 0.2. Phys. Rev. Lett. 118, 221101 (2017)
B.P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), GW170814: a three-detector observation of gravitational waves from a binary black hole coalescence. Phys. Rev. Lett. 119, 141101 (2017)
B.P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), GW170817: observation of gravitational waves from a binary neutron star inspiral. Phys. Rev. Lett. 119, 161101 (2017)
N. Arkani-Hamed, S. Dimopoulos, G. Dvali, The hierarchy problem and new dimensions at a millimeter. Phys. Lett. B 429, 263 (1998)
L. Randall, R. Sundrum, A large mass hierarchy from a small extra dimension. Phys. Rev. Lett. 83, 3370 (1999)
L. Randall, R. Sundrum, An alternative to compactification. Phys. Rev. Lett. 83, 4690 (1999)
A. Shomer, A pedagogical explanation for the non-renormalizability of gravity. (2007). arXiv:0709.3555
A.G. Riess, A.V. Filippenko, P. Challis et al., Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron. J. 116, 1009 (1998)
V.C. Rubin, N. Thonnard, W.K. Ford, Rotational properties of 21 SC galaxies with a large range of luminosities and radii, from NGC 4605 (R=4kpc) to UGC 2885 (R=122kpc). Astrophys. J. 238, 471 (1980)
C. Brans, R.H. Dicke, Mach’s principle and a relativistic theory of gravitation. Phys. Rev. 124, 925 (1961)
G.W. Horndeski, Second-order scalar–tensor field equations in a four-dimensional space. Int. J. Theor. Phys. 10, 363 (1974)
H.A. Buchdahl, Non-linear Lagrangians and cosmological theory. Mon. Not. Roy. Astron. Soc. 150, 1 (1970)
C. Skordis, T. Zlosnik, New relativistic theory for modified Newtonian dynamics. Phys. Rev. Lett. 127, 161302 (2021)
T. Cliftona, P.G. Ferreira, A. Padilla, C. Skordis, Modified gravity and cosmology. Phys. Rep. 513, 1 (2012)
Z. Chen, C. Yuan, Q. Huang, Non-tensorial gravitational wave background in NANOGrav 12.5-year data set. Sci. China Phys. Mech. Astron. 64, 120412 (2021)
M. Ostrogradsky, Memoires sur les equations differentielle relatives au probleme des isoperimetres. Mem. Ac. St. Petersbourg VI 4, 385 (1850)
R.P. Woodard, Avoiding dark energy with \(1/R\) modifications of gravity. Lect. Notes. Phys. 720, 403 (2007)
H. Motohashi, T. Suyama, Quantum Ostrogradsky theorem. J. High Energy Phys. 09, 32 (2020)
A. Ganz, K. Noui, Reconsidering the Ostrogradsky theorem: higher-derivatives Lagrangians, ghosts and degeneracy. Class. Quantum Gravity 38, 075005 (2021)
P. Creminelli, F. Vernizzi, Dark energy after GW170817 and GRB170817A. Phys. Rev. Lett. 119, 251302 (2017)
C.D. Kreisch, E. Komatsu, Cosmological constraints on Horndeski gravity in light of GW170817. J. Cosmol. Astropart. Phys. 12, 030 (2018)
Y. Gong, E. Papantonopoulos, Z. Yi, Constraints on scalar–tensor theory of gravity by the recent observational results on gravitational waves. Eur. Phys. J. C 78, 738 (2018)
B.P. Abbottet et al. (LIGO Scientific Collaboration and Virgo Collaboration), Gravitational waves and gamma-rays from a binary neutron star merger: GW170817 and GRB 170817A. Astrophys. J. 848, L13 (2017)
B.P. Abbottet et al. (LIGO Scientific Collaboration and Virgo Collaboration), Tests of general relativity with GW170817. Phys. Rev. Lett. 123, 011102 (2019)
R. Kase, S. Tsujikawa, Dark energy in Horndeski theories after GW170817: a review. Int. J. Mod. Phys. D 28, 1942005 (2019)
A. Chowdhuri, A. Bhattacharyya, Study of eccentric binaries in Horndeski theory. Phys. Rev. D 106, 064046 (2022)
Y. Gong, E. Papantonopoulos, Z. Yi, Constraints on scalar–tensor theory of gravity by the recent observational results on gravitational waves. Eur. Phys. J. C 78, 738 (2018)
A. Emir Gumrukcuoglu, M. Saravani, T.P. Sotiriou, Hořava gravity after GW170817. Phys. Rev. D 97, 024032 (2018)
Y. Gong, S. Hou, D. Liang, E. Papantonopoulos, Gravitational waves in Einstein–æther and generalized TeVeS theory after GW170817. Phys. Rev. D 97, 084040 (2018)
Y.F. Cai, C. Li, E.N. Saridakis, L.Q. Xue, f(T) gravity after GW170817 and GRB170817A. Phys. Rev. D 97, 103513 (2018)
J. Oost, S. Mukohyama, A. Wang, Constraints on Einstein–æether theory after GW170817. Phys. Rev. D 97, 124023 (2018)
Y. Gong, S. Hou, E. Papantonopoulos, D. Tzortzis, Gravitational waves and the polarizations in Hořava gravity after GW170817. Phys. Rev. D 98, 104017 (2018)
D. Liang, R. Xu, X. Lu, L. Shao, Polarizations of gravitational waves in the Bumblebee Gravity Model. arXiv: 2207.14423
V. Oikonomou, F. Fronimos, Reviving non-minimal Horndeski-like theories after GW170817: kinetic coupling corrected Einstein–Gauss–Bonnet inflation. Class. Quantum Gravity 38, 035013 (2021)
V. Oikonomou, F. Fronimos, Generalized Horndeski-like Einstein Gauss–Bonnet inflation with massless primordial gravitons. Nucl. Phys. B 971, 115522 (2021)
L. Visinelli, N. Bolis, S. Vagnozzi, Brane-world extra dimensions in light of GW170817. Phys. Rev. D 97, 064039 (2018)
A. Casalino, M. Rinaldi, L. Sebastiani, S. Vagnozzi, Mimicking dark matter and dark energy in a mimetic model compatible with GW170817. Phys. Dark Univ. 22, 108 (2018)
A. Casalino, M. Rinaldi, L. Sebastiani, S. Vagnozzi, Alive and well: mimetic gravity and a higher-order extension in light of GW170817. Class. Quantum Gravity 36, 017001 (2019)
H. Motohashi, K. Noui, T. Suyama, M. Yamaguchi, D. Langlois, Healthy degenerate theories with higher derivatives. J. Cosmol. Astropart. Phys. 07, 033 (2016)
D. Langlois, K. Noui, Degenerate higher derivative theories beyond Horndeski: evading the Ostrogradski instability. J. Cosmol. Astropart. Phys. 02, 034 (2016)
M. Crisostomi, K. Koyama, G. Tasinato, Extended scalar–tensor theories of gravity. J. Cosmol. Astropart. Phys. 04, 044 (2016)
J. Achour, M. Crisostomi, K. Koyama, D. Langlois, K. Noui, G. Tasinato, Degenerate higher order scalar-tensor theories beyond Horndeski up to cubic order. J. High Energy Phys. 12, 100 (2016)
D. Langlois, Dark energy and modified gravity in degenerate higher-order scalar-tensor (DHOST) theories: a review. Int. J. Mod. Phys. D 28, 1942006 (2019)
T. Kobayashi, Horndeski theory and beyond: a review. Rep. Prog. Phys. 82, 086901 (2019)
J. Gleyzes, D. Langlois, F. Piazza, F. Vernizzi, New class of consistent scalar–tensor theories. Phys. Rev. Lett. 114, 211101 (2015)
S. Bahamonde, K.F. Dialektopoulos, J.L. Said, Can Horndeski theory be recast using teleparallel gravity? Phys. Rev. D 100, 064018 (2019)
S. Bahamonde, K.F. Dialektopoulos, V. Gakis, J.L. Said, Reviving Horndeski theory using teleparallel gravity after GW170817. Phys. Rev. D 101, 084060 (2020)
U. Lindstrom, The Palatini variational principle and a class of scalar–tensor theories. Nuovo Cim. B 35, 130 (1976)
F. Bauer, D.A. Demir, Inflation with non-minimal coupling: metric versus Palatini formulations. Phys. Lett. B 665, 222 (2008)
M. Li, X. Wang, Metric-affine formalism of higher derivative scalar fields in cosmology. J. Cosmol. Astropart. Phys. 07, 010 (2012)
T. Markkanen, T. Tenkanen, V. Vaskonen, H. Veermäe, Quantum corrections to quartic inflation with a non-minimal coupling: metric vs. Palatini. J. Cosmol. Astropart. Phys. 03, 029 (2018)
L. Jarv, A. Racioppi, T. Tenkanen, Palatini side of inflationary attractors. Phys. Rev. D 97, 083513 (2018)
K. Aoki, K. Shimada, Galileon and generalized Galileon with projective invariance in a metric-affine formalism. Phys. Rev. D 98, 044038 (2018)
A. Kozak, A. Borowiec, Palatini frames in scalar–tensor theories of gravity. Eur. Phys. J. C 79, 335 (2019)
K. Shimada, K. Aoki, K.I. Maeda, Metric-affine gravity and inflation. Phys. Rev. D 99, 104020 (2019)
R. Jinno, K. Kaneta, K. Oda, S.C. Park, Hillclimbing inflation in metric and Palatini formulations. Phys. Lett. B 791, 396 (2019)
K. Aoki, K. Shimada, Scalar-metric-affine theories: can we get ghost-free theories from symmetry? Phys. Rev. D 100, 044037 (2019)
T. Helpin, M. Volkov, Varying the Horndeski Lagrangian within the Palatini approach. J. Cosmol. Astropart. Phys. 01, 044 (2020)
T. Helpin, M. Volkov, A metric-affine version of the Horndeski theory. Int J Mod. Phys. A 35, 2040010 (2020)
M. Kubota, K. Oda, K. Shimada, M. Yamaguchi, Cosmological perturbations in Palatini formalism. J. Cosmol. Astropart. Phys. 3, 006 (2021)
Y.Q. Dong, Y.X. Liu, Polarization modes of gravitational waves in Palatini–Horndeski theory. Phys. Rev. D 105, 064035 (2022)
I.D. Gialamas, A.B. Lahanas, Reheating in \(R^2\) Palatini inflationary models. Phys. Rev. D 101, 084007 (2020)
I.D. Gialamas, A. Karam, A. Racioppi, Dynamically induced Planck scale and inflation in the Palatini formulation. J. Cosmol. Astropart. Phys. 11, 014 (2020)
I.D. Gialamas, A. Karam, A. Lykkas, T.D. Pappas, Palatini-Higgs inflation with non-minimal derivative coupling. Phys. Rev. D 102, 063522 (2020)
I.D. Gialamas, A. Karam, T.D. Pappas, V.C. Spanos, Scale-invariant quadratic gravity and inflation in the Palatini formalism. Phys. Rev. D 104, 023521 (2021)
A. Minkevich, A. Garkun, Isotropic cosmology in metric-affine gauge theory of gravity. (1998). arXiv:gr-qc/9805007
D. Iosifidis, Cosmological hyperfluids, torsion and non-metricity. Eur. Phys. J. C 80, 1042 (2020)
H. Motohashi, T. Suyama, K. Takahashi, Fundamental theorem on gauge fixing at the action level. Phys. Rev. D 94, 124021 (2016)
T. Kobayashi, M. Yamaguchi, J. Yokoyama, Generalized G-inflation: inflation with the most general second-order field equations. Prog. Theor. Phys. 126, 511 (2011)
Acknowledgements
We would like to thank Yu-Peng Zhang for useful discussion. This work is supported in part by the National Key Research and Development Program of China (Grant no. 2020YFC2201503), the National Natural Science Foundation of China (Grants no. 11875151, no. 12247101, and no. 12047501), the 111 Project (Grant no. B20063), the Department of education of Gansu Province: Outstanding Graduate “Innovation Star” Project (Grant no. 2022CXZX-059) and Lanzhou City’s scientific research funding subsidy to Lanzhou University.
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Appendix A: Decomposition of connection
Appendix A: Decomposition of connection
In this appendix, we use \(\delta ^{ij}\) (\(\delta _{ij}\)) to raise and lower the index. Therefore, it will not cause ambiguity if the upper and lower indices of a tensor are not distinguished.
We can decompose the perturbation of the connection \(\varSigma ^{\lambda }_{\mu \nu }\) into the following forms:
Here,
The notations used in the appendix are not related to those in the text and should not be confused. The method to prove that the perturbation of the connection \(\varSigma ^{\lambda }_{\mu \nu }\) can always be decomposed into the above form is the same as the method for the perturbation of the metric \(h_{\mu \nu }\).
Similarly, we can decompose the linear perturbation equations into several sets of coupled equations. The perturbation describing the tensor gravitational waves is the transverse traceless tensor part of the metric perturbation \(h^{TT}_{\mu \nu }\). Consider that each term in the linear perturbation equations is a combination of \(\delta _{ij}\), \(\partial _{i}\), a time dependent function and a perturbation, and since \(h^{TT}_{\mu \nu }\) is a transverse traceless symmetric tensor, only \(S_{ij}\), \(U_{ij}\), \(C_{ij}\) and \(D_{ij}\) are coupled with \(h^{TT}_{\mu \nu }\) in the perturbations of the connection (A1–A4). It is the reason why we take the perturbations (20) in the text.
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Dong, YQ., Liu, YQ. & Liu, YX. Constraining Palatini–Horndeski theory with gravitational waves after GW170817. Eur. Phys. J. C 83, 702 (2023). https://doi.org/10.1140/epjc/s10052-023-11861-9
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DOI: https://doi.org/10.1140/epjc/s10052-023-11861-9