Cosmology in cubic and $f(P)$ gravity
Cristian Erices Eleftherios Papantonopoulos Emmanuel N. Saridakis





























(αβγωΩσλϵℋℱ∇𝒮𝒜ℬ𝒞𝒩ρ̅p̅δυhttp://doi.orghttp://arxiv.org/abser̊ghmts
ℒ_mαβ̱γδ̣μνłληρ̊σκ̨√(-g)ØΩΘ∇±p_mρ_mḢḦ⃛H
    𝒫
    
    αβ̱γδ̣μνłλϵηζρ̊σκ̨∇□β_1β_2β_3β_4β_5β_6β_7β_8crerices@central.ntua.grDepartment of Physics, National Technical University of Athens, Zografou
    Campus GR 157
    73, Athens, Greece Universidad Católica del Maule, Av. San Miguel 3605, Talca, Chilelpapa@central.ntua.grDepartment of Physics, National Technical University of Athens, Zografou
    Campus GR 157
    73, Athens, Greecemsaridak@phys.uoa.grDepartment of Physics, National Technical University of Athens, Zografou
    Campus GR 157 73, Athens, GreeceDepartment of Astronomy, School of Physical Sciences, University of Science
    and Technology of China, Hefei 230026, China98.80.-k, 95.36.+x, 04.50.Kd
    
    We construct cubic gravity and its f(P) extension and we investigate their early- and 
    late-time cosmological applications. Cubic gravity is based on
    a particular invariant P, constructed from cubic contractions of the
    Riemann tensor, under three requirements: (i)  the resulting theory possesses a
    spectrum identical to that of general relativity, (ii) it is neither topological nor
    trivial in four dimensions, and (iii) it is defined such that it is independent of the 
    dimensions. Relaxing the last condition and restricting the parameters of cubic gravity we 
    can obtain second-order field equations in a cosmological background.
    We show that at early times one can obtain inflationary, de Sitter solutions,
    which are  driven by an effective cosmological constant constructed purely from
    the cubic terms of the simple cubic or f(P) gravity.
    Concerning late-time evolution, the new terms constitute an effective dark-energy 
    sector and we show 
    that the Universe experiences the usual thermal history and the onset  of late-time 
    acceleration. In the case of f(P) gravity, depending on the choice of parameters, we 
    find that the dark-energy equation-of-state parameter can be quintessencelike, 
    phantomlike or it can experience the phantom-divide crossing during the evolution, even if an 
    explicit cosmological constant is absent.
    Cosmology in cubic and f(P) gravity
        Emmanuel N. Saridakis
        December 30, 2023
    ===================================
    
    
    
    § INTRODUCTION
    
    
    Higher-order gravities have been introduced in the general framework of modified theories 
    of gravity, with the aim to describe in a uniform way the history of the Universe; to 
    account for the early-time inflation, the late-time acceleration, and the presence of dark matter; 
    and to be consistent with observations 
    <cit.>. A more 
    theoretical motivation in studying  higher-order corrections to the Einstein-Hilbert term 
    is that these theories arise naturally in the gravitational effective action of a complete 
    string theory <cit.> and they result in a renormalizable and thus quantizable 
    gravitational theory <cit.>.  Additionally,  in such considerations one can 
    construct theories which possess general relativity as a
    particular limit <cit.>. Moreover, certain higher-order gravities  are 
    equivalent to Einstein gravity at the linearized level in the vacuum, and the only 
    physical mode propagated by 
    the metric perturbation is a transverse and massless graviton. Such theories are certain  
    f(Lovelock) theories <cit.>.
    
    Higher-order gravities and more generally modified theories of gravity provide a deeper 
    understanding of Einstein gravity theory itself.
    The construction of modified theories of gravity starts
    from the Einstein-Hilbert Lagrangian and includes extra terms, such as in f(R) gravity 
    <cit.>, f(G)
    gravity <cit.>, Lovelock
    gravity <cit.>, Weyl gravity
    <cit.>, Galileon theory
    <cit.>, etc.
    More radical modifications of Einstein gravity theory are provided by the introduction of 
    torsion terms in f(T) gravity
    <cit.> or f(T,T_G) gravity
    <cit.>. These modified theories of gravity allow us to 
    unveil what features of a gravitational theory are generic and which are specific.
    
    A potential disadvantage of the curvature-based corrections to the Einstein-Hilbert
    action is that the couplings of the different curvature invariants depend on the 
    spacetime dimension D. Hence, they are actually different theories in different 
    dimensions. Furthermore, the higher-order 
    terms may lead to the appearance of higher-than-second-order derivatives in the field equations. Although this does not necessarily mean that
    the theory presents instabilities and pathologies, since the higher-order derivatives may
    be just a reflection of healthy extra degrees of freedom (this can be shown through a
    Hamiltonian analysis), keeping the field equations up to second order is a desirable
    feature since it does ensure that the theory is pathology free.
    
    One interesting class of modified gravity based on higher-order curvature invariants was
    recently constructed in <cit.>, and it uses cubic contractions of the
    Riemann tensor. This theory, called Einsteinian cubic gravity (ECG), possesses   
    basic health conditions coming from nontopological terms. Namely, the coupling 
    parameters, which are dimension independent, allow to propagate a massless and transverse 
    graviton on a maximally symmetric background in the same way as the standard general 
    relativity (GR). In general, such terms contribute with fourth-order derivatives of the 
    metric in the field equations. However, as it was 
    shown in <cit.>, the original form of the theory is sufficient to 
    admit 
    spherically symmetric black hole solutions with a second-order differential equation for the 
    metric 
    function. An extra cubic correction, which is trivial for a spherically symmetric black 
    hole ansatz, 
    allows us additionally to accommodate a Friedmann-Lemaître-Robertson-Walker (FLRW) 
    solution with 
    second-order field equations for the scale factor, leading to a purely geometric 
    inflationary 
    period <cit.>[Black holes solutions were also obtained in 
    <cit.>, alongside the examination of a well-defined cosmology 
    for higher order terms such as the quartic, quintic and  infinite tower of 
    higher-curvature corrections to the Einstein-Hilbert action. ]. All these features make 
    this theory physically interesting and hence worthy of 
    further 
    investigation.
    
    In this work we  study the cosmological applications of cubic gravity, as well as investigate its modifications, namely f(P) gravity. In particular, we are 
    interested in examining whether the cubic terms P can drive the Universe's acceleration at 
    early and late times, and whether we can obtain the usual thermal history of the Universe. 
    We show that since we are considering four-dimensional spacetimes, the condition of 
    dimension-independent coupling parameter can be relaxed,[This can be achieved by 
    considering the dimension-dependent parameters of ECG 
    in <cit.>, namely considering the expressions for the parameters before 
    the dimension-independence condition is imposed, and evaluating in four dimensions.] 
    allowing us to tune the parameters in order to get second-order field equations for the 
    scale factor.
    
    Inspired by ECG, we obtain a modified theory that shares all the good properties of cubic 
    gravity  in four dimensions. In particular, we consider an action  with corrections to the 
    Einstein-Hilbert one, constituted by an arbitrary f(P) function of the cubic term. This 
    provides a remarkable advantage over the cosmological models studied in ECG, since it 
    generates inflationary, de Sitter solutions at early times and reproduces the onset of 
    late-time acceleration and the evolution of the densities in agreement with observations, 
    even when the cosmological constant is zero.
    
    
    The plan of the work is
    as follows. In Sec. <ref> we construct cubic gravity and we extend it to
    the f(P) case. Additionally, we apply these theories in a cosmological framework and we
    extract the Friedmann equations. In Sec. <ref> we perform a detailed
    investigation of the Universe's behavior at both early and late times, examining various
    observables such as the effective dark-energy density and equation-of-state
    parameters. Finally, in Sec. <ref>
     are our conclusions. 
    
    
    
    
    
    § CUBIC AND F(P) GRAVITY
    
    
    In this section we present cubic gravity, as well as its f(P) extension, and we apply
    them in a cosmological framework. The starting point of cubic gravity is to include
    additional terms in the Einstein-Hilbert action that are constructed by contractions of
    three Riemann tensors
    <cit.>. Nevertheless, one desires combinations that lead to a theory
    with the following features:
    (i) it possesses a spectrum identical to that of
    general relativity, i.e. the metric perturbation on a maximally symmetric background
    propagates only a transverse and massless graviton, and (ii) it is neither topological nor trivial in four dimensions. Additionally, we desire
    that (iii) the theory leads to second-order field equations. In ECG 
    an additional condition  is required, namely that the theory should be
    constructed in a dimension-independent way. However, since in the present work we are
    interested in the usual four-dimensional cosmological evolution, we relax such a 
    requirement. This freedom allows us to tune the parameters to satisfy condition (iii) 
    without the inclusion of additional cubic terms. 
    
    Let us focus on four spacetime dimensions. In this case, a general 
    nontopological [thus
    condition (iii)   is satisfied] cubic term P  would be
    
        P=_̱1
        R_^_^R_^_^R_^_^+_̱2
        R_^R_^R^_+_̱3
        R^R_R^_+_̱4
        R R_R^
                    
        +_̱5 R_R^R^+_̱6 R^_R^_R^_+_̱7
        R_R^R+_̱8 R^3,
    
    where   β_i are parameters. One can show that condition (i) above  is satisfied
    if we impose the parameter relations <cit.>
        _̱7=1/12(3_̱1-24_̱2-16_̱3-48_̱4-5_̱5-9_̱6),
              _̱8=1/72(-6_̱1+36_̱2+22_̱3+64_̱4+3_̱5+9_̱6).
    
    
    
    One can now use P as a correction term in  the Einstein-Hilbert action, and construct
    the cubic gravity as
    
        S=∫√(-g)d^4x[1/2κ(R-2Λ)+
        P],
    
    with α a coupling parameter, κ=8π G the Newton's constant, and where we
    have also allowed for an explicit cosmological constant Λ for completeness. We
    mention here that in principle one should include also quadratic corrections to the above
    action, such as the Gauss-Bonnet combination
    G=R^2-R_R^+R_R^. However, in this work we
    focus on four dimensions and such a term can be neglected since it is topological.
    
    
    Variation of the action S+S_m, where S_m accounts for the matter Lagrangian ,
    leads
    to the field equations
    
        G_μν+Λ g_μν=κ (T_+ H_),
    
    with
    
        T_μν=-2/√(-g)δ (√(-g))/δ
        g^μν,
    
    representing the matter energy-momentum tensor and[We use a normalized symmetrization A_(μν):=1/2(A_μν+A_νμ) and antisymmetrization A_[μν]:=1/2(A_μν-A_νμ).]
        H_   =-2/√(-g)δ (√(-g)P)/δ g^μν
           =g_μνP+R^̱̊_(μK_ν)+2∇^∇^K_(),
    
    where  ∇_μ is the covariant derivative with respect to the spacetime metric g_μν. The tensor K_αβμν is defined by
    
        K_αβμν=   ∂ P/∂ R^αβμν
        
        =   12(12R_^R_+6R_^_[^R_]̱̊+2g_[̱R_]R^
           -2g_[R_]̱̊R^-4R_[̊g_][R_]̱^-2R_[R_]).
    
    The tensor H_μν represents the contribution from the cubic term P. Observing the form of  H_  we
    mention that the complication and length of the above general field equations does not easily allow us to perform the Hamiltonian analysis and check whether they present unhealthy high derivatives, in the
    sense that they do not correspond to healthy extra degrees of freedom. Nevertheless, this
    necessary check, i.e. ensuring the condition (iii) above, can be performed around 
    the cosmological
    background on which we focus in this work, as we will shortly see.
    
    In order to proceed to the cosmological application of cubic gravity we consider a   flat
    homogeneous and isotropic FLRW geometry with
      metric
    
        ds^2=-dt^2+a^2(t)δ_ijdx^idx^j ,
    
    where a(t) is the scale factor. Additionally, as usual we consider the matter
    Lagrangian  to correspond to a  perfect fluid with energy-momentum tensor
    T_=(ρ_m+p_m)u_ u_+p_m g_,
    where ρ_m and p_m are   the energy density and pressure respectively and u_
    is the fluid four-velocity. Under these considerations the general field equations
    (<ref>) give rise to the Friedmann equations. These equations can now be easily proven 
    to be second order if we make the additional parameter reduction
      
        _̱6=4_̱2+2_̱3+8_̱4+_̱5.
    
     In particular, they are written as
    
        3H^2   =   ( _̊m+6  H^6)+Λ,
        
        
        3H^2+2Ḣ   =   -[ p_m-6  H^4( H^2+2  )]+Λ,
    
    where H=ȧ/a is the Hubble parameter, with dots
    denoting derivatives with respect to t, and where we have defined the parameter
      
        ≡ -_̱1+4_̱2+2_̱3+8_̱4.
    
    Finally, note that in FLRW geometry, and under the parameter relations
    (<ref>) and (<ref>), the cubic invariant P acquires the simple form
      
        P=6β̃ H^4(2H^2+3Ḣ),
    
    which includes only up to first-order derivatives and hence the Friedmann equations are
    up to second order, as indeed they are constructed to be.
    
    Observing the form of the two Friedmann equations we deduce that they can be rewritten
    in the usual way
    
        3H^2   =   ( _̊m+ _̊cub),
        
        
        3H^2+2Ḣ   =   -( p_m +p_cub),
    
    where
    
        _̊cub≡    6H̱^6+Λ/,
              
        p_cub≡  -6  H̱^4( H^2+2  )-Λ/,
    
    and where we have redefined the sole parameter as ≡̱α.
    Hence, in cubic gravity we obtain an effective sector that incorporates the effects of
    the cubic modification of the action. Finally, one can immediately see that this
    effective sector is conserved, namely
    
        _cub+3H(_̊cub+p_cub)=0,
    
    while the matter sector is conserved too, namely
    
        _m+3H(_̊m+p_m)=0.
    
    
     We close this section by constructing f(P) gravity. In the above we described how one
    can construct a subclass of cubic gravity which in four dimensions is nontopological
    and in a cosmological spacetime it leads to second-order field equations. The theory is
    based on the cubic invariant P of (<ref>), under the parameter choices
    (<ref>) and (<ref>) which ensure the aforementioned requirements. 
    Nevertheless,
    having such an invariant we may generalize cubic gravity to f(P) gravity, characterized
    by the action
    
        S=∫√(-g)d^4x[R/2κ+f(P)],
    
    where  f(P) is an
    arbitrary function of   P.
    Variation of the action S+S_m leads
    to the field equations
    
        G_μν=κ (T_+H̃_),
    
    with  T_ still given by (<ref>) and
    
        H̃_   =-2/√(-g)δ (√(-g)f(P))/δ
        g^μν,
           =g_μνf(P)+R^̱̊_(μK̃_ν)+2∇^∇^K̃_().
    
    Here, tensor K̃_ is given in terms of tensor (<ref>) as K̃_=f'(P)K_ with primes denoting the derivative of a function with respect to its argument.
    
    In the case of FLRW geometry, the field
    equations (<ref>) provide the two Friedmann equations, namely
     
        3H^2   =   (̨_̊m+ _̊f_P),
        
        3H^2+2Ḣ   =   -(̨p_m+ p_f_P),
    
    where
    
        _̊f_P≡-f(P)-18 H^4(H∂_t-H^2-Ḣ)f'(P),
              p_f_P≡ f(P)+6
        H^3[H∂_t^2+2(H^2+2)∂_t-3H^3-5H]f'(P),
    
    and ∂_t≡∂/∂ t and ∂^2_t≡∂^2/∂ t^2. Additionally, note that both the matter and the
    effective f(P) sector are conserved, namely
    
        _m+3H(_̊m+p_m)   =   0,
        _f_P+3H(_̊f_P+p_f_P)   =   0.
    
    Lastly, in the case where f(P)=α P-Λ/, f(P) gravity gives the
    simple cubic gravity of (<ref>).
    
    The action (<ref>) can be recast into a (classically) dynamically equivalent action where the system of equations (<ref>) and (<ref>) can be put explicitly as a second-order system by introducing two scalar variables related by φ=f(ϕ). The details for this procedure can be found in the Appendix.
    
    
    
    § COSMOLOGICAL APPLICATIONS
    
    
    In this section we investigate the cosmological applications of cubic and f(P) gravity
    at both early and late times. 
    Concerning the early-time application as usual we may neglect the matter sector. We are
    interested in obtaining the de Sitter solution, which is the basis of the inflation
    realization. Let us consider first the simple cubic gravity, namely the first
    Friedmann equation (<ref>) without the matter term. In this case we can easily see
    that one obtains the de Sitter solution
    
        H^2=6^1/3β̨+ζ^2/3/
        -6^2/3β̨ζ^
        1/3=const.,
    
    where ζ=3^̨2^̱2Λ+√(3)√(^̨3^̱3(3̨̱Λ^2-2)), provided <̱0. 
    When the 
    cosmological constant is absent we get that H^2=(2κβ)^-1/2 for >̱0. In 
    similar
    lines, in the case of f(P) gravity, i.e. of the Friedmann equation (<ref>),
    without the matter sector, we can see that in general we can obtain the de Sitter
    solution 
    
        H^2=κ/3[18H^6f'(P)-f(P)]=const.
    
    and Ḣ=0. As we observe, in both theories the de Sitter
    solution is driven by an effective cosmological constant that is constructed purely from
    the cubic terms, even if an explicit cosmological constant is absent.
    
    Let us now focus on the late-time evolution.
    We first examine the basic cubic gravity,
    namely (<ref>), where the correction in the general relativity action is just
    P. In this case the cubic correction constitutes the effective dark-energy sector,
    whose energy density and pressure are
    
        ρ_DE   ≡   _̊cub
        
        p_DE   ≡    p_cub,
    
    with _̊cub and p_cub given respectively by (<ref>) and
    (<ref>),
    while the effective dark-energy equation-of-state parameter is
    
        w_DE≡p_DE/ρ_DE.
    
    In order to proceed we define the density parameters Ω_i≡κρ_i/(3H^2), with i standing for matter and dark energy. Lastly,
    it proves convenient to introduce the deceleration
    parameter q, which reads as
      
        q≡-1-Ḣ/H^2=1/2+3/2(w_mΩ_m+w_DEΩ_DE).
    
    
    We elaborate the Friedmann equations  (<ref>) and (<ref>) numerically, using the
    redshift z as the independent variable, defined  as  1+z=a_0/a with the present scale factor set to a_0=1. Concerning the initial 
    conditions, we set
    Ω_DE(z=0)≡Ω_DE0≈0.69  and thus
    Ω_m(z=0)≡Ω_m0≈0.31  as required by observations
    <cit.>. Additionally, we consider the matter sector to be dust, namely we set
     w_m=0.  In the upper graph of Fig. <ref> we depict
    Ω_DE(z) and Ω_m(z)=1-Ω_DE(z). In the middle graph we present the
    corresponding behavior of w_DE(z) given by  (<ref>). Finally, in the
    lower graph we show the deceleration parameter as it arises from  (<ref>).
    
    
    
    From the upper graph of Fig. <ref> we observe that we can obtain the usual
    thermal history, that is the sequence of matter and dark-energy eras, while in the future
    (z→-1) the Universe asymptotically results in a complete dark-energy
    dominated, de Sitter phase. Moreover,  from
    the third graph of Fig. <ref> we see that the transition
    from deceleration to acceleration is realized at z≈ 0.5  in agreement with the
    observed behavior. Additionally, from the middle graph of Fig. <ref> we observe
    that the effective dark-energy equation-of-state parameter w_DE quickly acquires the value -1.
    
    Nevertheless, we must mention here that in the simple cubic gravity  (<ref>) the
    late-time acceleration is triggered mainly by the explicit cosmological constant term
    Λ,
    while the cubic term, quantified by the parameter β in
    (<ref>) and (<ref>), plays a secondary role (a similar result was found 
    in a different context in <cit.>). The reason is
    that
    since the cubic correction in (<ref>) is proportional to H^6, if this term is
    to play the main role in the late-time acceleration then at earlier times it will be
    large enough in order to spoil the desired thermal history of the Universe and lead to an
    “early dark energy”<cit.> behavior. However, even in the case where
    the cubic term plays a
    secondary role in the late-time acceleration of the Universe, its effect can still be
    significant at intermediate times (see the middle graph of  Fig. <ref> where the
    deviation from ΛCDM scenario is clear) as well as at the perturbation level
    reflected in the matter clustering and the large scale structure (LSS). The detailed
    analysis of the perturbations and the comparison with LSS data such as the fσ_8
    ones will be handled in a separate project.
    
    Let us now investigate the extended cubic gravity, namely the f(P) gravity
    (<ref>). In this case, using (<ref>) and (<ref>), the effective dark-energy density and pressure are given by
     
        ρ_DE   ≡   _̊f_P
        
        p_DE   ≡   p_f_P,
    
    and the effective dark-energy equation-of-state parameter as usual is
    w_DE≡  p_DE/ ρ_DE. As a first example we consider the case where
     
        f(P)=α√(P),
    
    without considering an explicit cosmological constant. Additionally, we merger the
    coupling parameter α with the parameter β̃ of P in (<ref>)
    through ≡̱α.
    
    
     
    
    
    We elaborate the Friedmann equations
    (<ref>) and (<ref>) numerically, considering the matter sector to be dust. In
    the
    upper graph of Fig. <ref> we present
    Ω_DE(z) and Ω_m(z), in the middle graph we draw w_DE(z), and in the
    lower graph we depict the deceleration parameter q(z). As we observe, once again we
    obtain  the sequence of matter and dark-energy eras and the onset of late-time
    acceleration. However, the significant advantage compared to the simple cubic gravity of
    Fig. <ref> is that now the above behavior is obtained although we have not
    considered an explicit cosmological constant; namely it arises purely from the f(P)
    modification. This can also be seen in the evolution of w_DE, which exhibits a
    dynamical behavior different than ΛCDM at all times, with an asymptotic value
    slightly into the phantom regime.
    
    In order to examine the effect of f(P) forms on  w_DE, in  Fig. <ref> we
    present the evolution of  w_DE(z) for various f(P) choices. In particular, apart
    from the simple cubic gravity of  Fig. <ref>, which is obtained for  f(P)=β
    P-Λ/, and for the square-root f(P) gravity of (<ref>) of Fig.
    <ref>, we consider the cubic-root case f(P)=α P^1/3, as well as the
    combined cases  f(P)=α√(P)+γ P and  f(P)=α P^1/3+γ P.
    
    As we observe, one can fix the model parameters in order to obtain a w_DE  at
    present (z=0) close to the cosmological constant value, in agreement with observations,
    while the past and future behavior can be different for different scenarios. As we
    mentioned earlier, we note that apart from the linear case, such behavior is achieved
    without the need for an explicit cosmological constant, which is a significant advantage.
    Furthermore, note that w_DE can be  lying in the quintessence
    regime or in the phantom regime or it can experience the phantom-divide crossing during the
    cosmological evolution, which reveals the capabilities of the theory at hand, since it is
    known that the phantom regime cannot be easily obtained <cit.>.
    
    
    
    
    
    
    
    § CONCLUSIONS
    
    
    In this work we constructed cubic gravity and its extension and we investigated their
    cosmological applications for both the early- and late-time Universe. Cubic gravity is based on
    the invariant P that is constructed using particular cubic contractions of the
    Riemann tensor, such that the theory (i) possesses a spectrum identical to that of
    Einstein gravity, i.e. the metric perturbation on a maximally symmetric background
    propagate only a transverse and massless graviton;
    (ii) it is neither topological nor trivial in four dimensions, and  (iii) it is 
    defined such 
    that it is independent of the dimensions.
    By relaxing this last condition, and tuning the coupling parameters, the theory admits 
    second-order 
    field equations for a four-dimensional cosmological ansatz without the inclusion of any 
    additional 
    term. These features make  cubic gravity a
    well-behaved candidate for modified gravity. Additionally, apart from simple cubic
    gravity, which uses P in the Lagrangian, we constructed extensions, namely f(P)
    gravity.
    
    Applying both simple cubic and f(P) gravity in a cosmological framework,  we saw that
    at early times one can easily obtain inflationary, de Sitter solutions,
    which are  driven by an effective cosmological constant that is constructed purely from
    the cubic terms, even if an explicit cosmological constant is absent.
    
    Concerning the late-time evolution, we showed that the cubic terms construct an
    effective dark-energy sector of gravitational origin. In the case of simple cubic gravity
    we obtained the usual thermal history, i.e. the sequence of matter and dark-energy eras,
    with the onset of late-time acceleration at around z≈ 0.5  in agreement with
    observations, while in the future the Universe asymptotically results in a completely
    dark-energy dominated, de Sitter phase. Nevertheless, if we desire not to spoil the
    Universe's evolution at early times then we find that  the background
    late-time acceleration is triggered mainly by the explicit cosmological constant, with
    the cubic term playing a secondary role (which could however be significant at the
    perturbation level).
    
    However, when we proceeded to f(P) gravity we found that we can obtain the    usual
    thermal history and the onset of late-time acceleration even without considering an
    explicit cosmological constant, namely as a result of  the f(P) corrections purely.
    Moreover, examining the evolution of the effective dark-energy equation-of-state
    parameter w_DE we showed that its behavior is determined by the f(P) form as well
    as the parameter choices, and it can be quintessencelike or phantomlike or it can experience
    the phantom-divide crossing during the evolution. Thlikeese features reveal the capabilities
    of f(P) gravity and offer a good motivation for its further investigation.
    
    Finally, we mention  that there are additional studies that
    need to be  performed before  cubic and f(P) gravity can be
    considered as a successful candidate for the description of nature. Firstly, one should
    use data from type Ia supernovae (SNIa),
    cosmic microwave background (CMB) shift parameter, baryon acoustic oscillations (BAO), and
    direct Hubble constant observations, in order to extract constraints on the involved
    forms and parameters. Furthermore, one needs to perform a detailed perturbation
    analysis and confront the theory with data from   CMB temperature and polarization as
    well as from LSS (such as the σ_8 ones). Moreover, one
    could  perform a phase-space analysis in order to reveal the
    global behavior of f(P) cosmology. Lastly, going beyond the cosmological framework, as 
    we mentioned above  the original ECG theory can be corrected in order to admit black hole 
    and cosmological solutions; however these solutions coexist in a disjoint set of values 
    for the cubic coupling parameter. It would be interesting to investigate whether the above 
    f(P) modification provides enough 
    freedom to obtain black hole solutions in a common range of parameters.
    These necessary investigations lie beyond the scope
    of this work and are left for future projects.
    
    
    
    
    
    
    § ACKNOWLEDGMENTS
    
    This article is based upon work from COST Action “Cosmology and Astrophysics Network for
    Theoretical Advances and Training Actions,” supported by the European Cooperation in
    Science and Technology (COST). C.E. acknowledges financial support given by Becas Chile, 
    Comisión Nacional de Investigación Científica y Tecnológica.
    
    
    
    
    *
    
    
    § EQUATIONS OF MOTION IN F(P) GRAVITY AS SECOND-ORDER FIELD EQUATIONS
    
    
    The f(P) gravity can be recast into a (classically) dynamically equivalent action where the Friedmann equations (<ref>) and (<ref>) can be put explicitly as a second-order system. For this let us consider the following action endowed with scalar variables φ and ϕ:
    
        S=∫√(-g)d^4x[R/2κ+f(ϕ)+φ(P-ϕ)].
    
    Variation with respect to φ yields ϕ=P. Replacing this algebraic constraint back into action (<ref>) yields the action (<ref>). Considering the variation with respect to ϕ, it is obtained that φ=f'(ϕ). Variation with respect to the metric g_μν leads to
    
        G_μν   =κ(T_μν+H̃_)
        H̃_   ≡ g_μνf(ϕ)+φR^̱̊_(μK_ν)+2∇^∇^[φ K_()],
    
    which retrieves equations equivalent to (<ref>) when the algebraic constraints are considered. Then, the Friedmann equations (<ref>) and (<ref>), take the same form but with an effective sector given by
    
        _̊f_P≡-f(ϕ)-18 H^4(H∂_t-H^2-Ḣ)φ,
              p_f_P≡ f(ϕ)+6
        H^3[H∂_t^2+2(H^2+2)∂_t-3H^3-5H]φ,
    
    which clearly possesses second-order field equations.
    
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