The weak field limit of the nonminimally coupled Boltzmann equation is studied, and relations between the invariant Bardeen scalar potentials are derived. The Jean’s criterion for instabilities is found through the modified dispersion relation. Special cases are scrutinised and considerations on the model parameters are discussed for Bok globules.

††journal: Eur. Phys. J. C
∎

\thankstext
e1e-mail: claudio.gomes@fc.up. The Boltzmann equation is a fundamental page_content of the microscopic world. Is derived from the Liouville equation in phase space considering collisions between particles. From the former, one can derive macroscopic equations, such as the Navier-Stokes equation for fluids and the virial theorem for gravitationally bound systems binney , the Maxwell-Vlasov equations which characterise plasmas nicholson , the quantum Bloch-Boltzmann equations for electrons pottier , and the evolution of primordial elements’ abundances in a de Sitter Universe turner . From the Boltzmann equation it is also possible to build physical quantities from its moments, such as the particle number flux, the energy-momentum tensor or the entropy vector flux.

The Boltzmann equation is sensible to relativistic and quantum effects. In particular, it can be generalised in order to account for modified gravity models. In fact, despite its successful agreement with a vast plethora of observational data tests will ; obgr , General Relativity (GR) lacks a fully consistent quantum version of it and requires two dark components to match observations at astrophysical and cosmological scales, page_seo_titlely dark matter and dark energy, which have not been directly observed so far. Thus, several alternative theories of gravity have been proposed over the years in the literature. One of the simplest generalisations of GR is the so-called f theories which replace the Ricci scalar by a generic function of it in the action functional (see Refs. fr1 ; fr2 for review on this and Ref. extended for a review on basic principles a gravity theory must obey and some extended theories of gravity). In fact, one specific proposal of such theories was firstly advanced in order to tackle the initial conditions problems of the standard Hot Big Bang model, page_seo_titlely through a nonsingular isotropic homogeneous solution which accounted for inflation starobinsky . Moreover, this model is still in excellent agreement with the most recent data from Planck mission planck . (We refer the reader to Ref. exoticinflation for a review of some exotic inflationary models in light of Planck data).

Furthermore, f theories of gravity have been used to address the problems of dark matter and dark energy (see e.g. Refs. frdark1 ; frdark2 ). It has also been found that by requiring that f models to be regular at R=0Ý‘Â…0R=0italic_R = 0 leads to a behaviour compatible with an effective cosmological constant in a sufficiently curved spacetime which disappears in flat spacetime starobinsky2 .

Another successful alternative theory of gravity in shedding some light in the above mentioned problems relies in an extension of f theories with a non-minimal coupling between matter and curvature nmc . In fact, it allows for a mimicking effect of dark matter effects at galaxies dm1 and clusters of galaxies scales dm2 , it has some bearings on the late time acceleration de1 , and is compatible with Planck’s inflation data nmcinflation , gravitational waves measurements nmcgw and the modified virial theorem from the spherically relaxed Abell 586 cluster linmc .

In the weak field regime, this model yields a correction to the Newtonian potential martins , and presents shock waves in the gravitational collapse numerico1 ; numerico2 . It has been recently applied to the generalisation of the Boltzmann equation for such scenario and its main implications were analysed in Ref. nmcboltzmann . Therefore, it is important to study how this alternative model of gravity modifies the Jeans criterion for instability, which is responsible for the collapse of a gravitationally bound system, such as interstellar gas clouds whose internal pressure does not overcome gravity, ultimately leading to star formation.

This work is organised as follows: in section 2, the non-minimal mater-curvature coupling model is introduced; in section 3 we derive the weak field of the nonminimally coupled Boltzmann equation of Ref. nmcboltzmann . In the following section, 4, the Jean’s criterion for instabilities is analysed and the implications for the parameters of the alternative gravity model are found. As an example, we test the viability of the model for Bok globules in Sec. 5. The conclusions are presented in Sec. 6.

2 The non-minimal matter-curvature coupling model

The non-minimal matter-curvature coupling alternative gravity model (NMC) is defined from its action functional nmc :

S=∫d4x-g[κf1+f2ℒ],ݑ†superscriptݑ‘4ݑ¥ݑ”delimited-[]ݜ…subscriptݑ“1ݑ…subscriptݑ“2ݑ…ℒS=\int d^4x\sqrt-g\left[\kappa f_1\left(R\right)+f_2\left(R\right)% \mathcalL\right]{},italic_S = ∫ italic_d start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_x square-root start_ARG - italic_g end_ARG [ italic_κ italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_R ) + italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_R ) caligraphic_L ] , (1)
where f1,f2subscriptݑ“1ݑ…subscriptݑ“2ݑ…f_1,f_2italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_R ) , italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_R ) are arbitrary functions of the curvature scalar Rݑ…Ritalic_R, κ=c4/(16πG)ݜ…superscriptݑ416ݜ‹ݐº\kappa=c^4/(16\pi G)italic_κ = italic_c start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT / ( 16 italic_π italic_G ) and ℒℒ\mathcalLcaligraphic_L is the Lagrangian density of matter fields.

The metric field equations can be straightforwardly found by varying the previous action with respect to the metric, gμνsubscriptݑ”ݜ‡ݜˆg_\mu\nuitalic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT:

ΘGμν=12κf2Tμν+ΔμνΘ+12gμν[f1-RΘ],Θsubscriptݐºݜ‡ݜˆ12ݜ…subscriptݑ“2ݑ…subscriptݑ‡ݜ‡ݜˆsubscriptΔݜ‡ݜˆΘ12subscriptݑ”ݜ‡ݜˆdelimited-[]subscriptݑ“1ݑ…ݑ…Θ\Theta G_\mu\nu=\frac12\kappaf_2T_\mu\nu+\Delta_\mu\nu\Theta+% \frac12g_\mu\nu\left[f_1-R\Theta\right]{},roman_Θ italic_G start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 italic_κ end_ARG italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_R ) italic_T start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT + roman_Δ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT roman_Θ + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT [ italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_R ) - italic_R roman_Θ ] , (2)
where Θ:=(f1′+f2′ℒκ)assignΘsubscriptsuperscriptݑ“′1ݑ…subscriptsuperscriptݑ“′2ݑ…ℒݜ…\Theta:=\left(f^\prime_1+\fracf^\prime_2\mathcalL\kappa\right)roman_Θ := ( italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_R ) + divide start_ARG italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_R ) caligraphic_L end_ARG start_ARG italic_κ end_ARG ), Gμνsubscriptݐºݜ‡ݜˆG_\mu\nuitalic_G start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT is the Einstein tensor, the primes denotes derivatives with respect to the curvature scalar, fi′≡dfi/dRsubscriptsuperscriptݑ“′ݑ–ݑ…ݑ‘subscriptݑ“ݑ–ݑ…ݑ‘ݑ…f^\prime_i\equiv df_i/dRitalic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_R ) ≡ italic_d italic_f start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_R ) / italic_d italic_R, and Δμν≡∇μ∇ν-gμν□subscriptΔݜ‡ݜˆsubscript∇ݜ‡subscript∇ݜˆsubscriptݑ”ݜ‡ݜˆ□\Delta_\mu\nu\equiv\nabla_\mu\nabla_\nu-g_\mu\nu\Boxroman_Δ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ≡ ∇ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ∇ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT - italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT □. Moreover, General Relativity can be retrieved as the particular case when f1=Rsubscriptݑ“1ݑ…ݑ…f_1=Ritalic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_R ) = italic_R and f2=1subscriptݑ“2ݑ…1f_2=1italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_R ) = 1.

The trace of the metric field equations reads:

ΘR-2f1=12κf2T-3□Θ.Θݑ…2subscriptݑ“1ݑ…12ݜ…subscriptݑ“2ݑ…ݑ‡3□Θ\Theta R-2f_1=\frac12\kappaf_2T-3\square\Theta{}.roman_Θ italic_R - 2 italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_R ) = divide start_ARG 1 end_ARG start_ARG 2 italic_κ end_ARG italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_R ) italic_T - 3 □ roman_Θ . (3)

In fact, one of the most striking features of this model consists in the covariant non-conservation of the energy-momentum tensor:

∇μTμν=(gμνℒ-Tμν)∇μlnf2.subscript∇ݜ‡superscriptݑ‡ݜ‡ݜˆsuperscriptݑ”ݜ‡ݜˆℒsuperscriptݑ‡ݜ‡ݜˆsubscript∇ݜ‡subscriptݑ“2ݑ…\nabla_\muT^\mu\nu=\left(g^\mu\nu\mathcalL-T^\mu\nu\right)\nabla_% \mu\ln f_2{}.∇ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_T start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT = ( italic_g start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT caligraphic_L - italic_T start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT ) ∇ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT roman_ln italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_R ) . (4)

This implies that for a perfect fluid, with Tμν=(ρ+p)uμuν-pgμνsubscriptݑ‡ݜ‡ݜˆݜŒݑsuperscriptݑ¢ݜ‡superscriptݑ¢ݜˆݑsubscriptݑ”ݜ‡ݜˆT_\mu\nu=(\rho+p)u^\muu^\nu-p{}g_\mu\nuitalic_T start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT = ( italic_ρ + italic_p ) italic_u start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT italic_u start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT - italic_p italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT, test particles do not follow geodesic lines given the presence of an extra force term in the geodesics equation nmc :

duαds+Γμναuμuν=ݔ£α,ݑ‘superscriptݑ¢ݛ¼ݑ‘ݑ subscriptsuperscriptΓݛ¼ݜ‡ݜˆsuperscriptݑ¢ݜ‡superscriptݑ¢ݜˆsuperscriptݔ£ݛ¼\fracdu^\alphads+\Gamma^\alpha_\mu\nuu^\muu^\nu=\mathfrakf^% \alpha{},divide start_ARG italic_d italic_u start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_s end_ARG + roman_Γ start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT italic_u start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT = fraktur_f start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT , (5)
where uμsuperscriptݑ¢ݜ‡u^\muitalic_u start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT denotes the particle’s 4-velocity and the extra force, per unit mass, is given by:

ݔ£α=1ρ+p[f2′f2(ℒm-p)∇νR-∇νp]Vαν,superscriptݔ£ݛ¼1ݜŒݑdelimited-[]subscriptsuperscriptݑ“′2ݑ…subscriptݑ“2ݑ…subscriptℒݑšݑsubscript∇ݜˆݑ…subscript∇ݜˆݑsuperscriptݑ‰ݛ¼ݜˆ\mathfrakf^\alpha=\frac1\rho+p\left[\fracf^\prime_2f_2% \left(\mathcalL_m-p\right)\nabla_\nuR-\nabla_\nup\right]V^\alpha\nu~% {},fraktur_f start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG italic_ρ + italic_p end_ARG [ divide start_ARG italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_R ) end_ARG start_ARG italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_R ) end_ARG ( caligraphic_L start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT - italic_p ) ∇ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT italic_R - ∇ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT italic_p ] italic_V start_POSTSUPERSCRIPT italic_α italic_ν end_POSTSUPERSCRIPT , (6)
where Vαν=gαν+uαuνsuperscriptݑ‰ݛ¼ݜˆsuperscriptݑ”ݛ¼ݜˆsuperscriptݑ¢ݛ¼superscriptݑ¢ݜˆV^\alpha\nu=g^\alpha\nu+u^\alphau^\nuitalic_V start_POSTSUPERSCRIPT italic_α italic_ν end_POSTSUPERSCRIPT = italic_g start_POSTSUPERSCRIPT italic_α italic_ν end_POSTSUPERSCRIPT + italic_u start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT italic_u start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT is the projection operator. We should also note the dependence on the matter Lagrangian density choice. This feature lifts the degeneracy that exists in GR whether ℒ=-ρℒݜŒ\mathcalL=-\rhocaligraphic_L = - italic_ρ or ℒ=pℒݑ\mathcalL=pcaligraphic_L = italic_p lagrangian choices GR , since it yields different results for the extra force (See Ref. lagrangian choices for a thorough discussion).

3 The Boltzmann equation in the Newtonian limit of the NMC

In Ref. nmcboltzmann , the Boltzmann equation for these theories was derived, where the main consequence was the appearance of a term related to the extra force:

pμ∂f∂xμ-(Γμνσpμpν-m2ݔ£σ)∂f∂pσ=(∂f∂τ*)coll..superscriptݑݜ‡ݑ“superscriptݑ¥ݜ‡subscriptsuperscriptΓݜŽݜ‡ݜˆsuperscriptݑݜ‡superscriptݑݜˆsuperscriptݑš2superscriptݔ£ݜŽݑ“superscriptݑݜŽsubscriptݑ“superscriptݜݑݑœݑ™ݑ™p^\mu\frac\partial f\partial x^\mu-\left(\Gamma^\sigma_\mu\nup^% \mup^\nu-m^2\mathfrakf^\sigma\right)\frac\partial f\partial p^% \sigma=\left(\frac\partial f\partial\tau^{*}\right)_coll.{}.italic_p start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT divide start_ARG ∂ italic_f end_ARG start_ARG ∂ italic_x start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT end_ARG - ( roman_Γ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT italic_p start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT italic_p start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT - italic_m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT fraktur_f start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT ) divide start_ARG ∂ italic_f end_ARG start_ARG ∂ italic_p start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT end_ARG = ( divide start_ARG ∂ italic_f end_ARG start_ARG ∂ italic_τ start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT end_ARG ) start_POSTSUBSCRIPT italic_c italic_o italic_l italic_l . end_POSTSUBSCRIPT . (7)

Let us now study the case of dust, p=0ݑ0p=0italic_p = 0, where the matter Lagrangian has a clear choice ℒ=-ρℒݜŒ\mathcalL=-\rhocaligraphic_L = - italic_ρ. Furthermore we shall study the Newtonian level of the modified gravity model, Eq. (1). We should note that the geodesic equation reads now martins :

d2xidt2=∂i[gtt+12-ln|f2|]-∂ipρ+p,superscriptݑ‘2superscriptݑ¥ݑ–ݑ‘superscriptݑ¡2subscriptݑ–subscriptݑ”ݑ¡ݑ¡12subscriptݑ“2subscriptݑ–ݑݜŒݑ\fracd^2x^idt^2=\partial_i\left[\fracg_tt+12-\ln|f_2|% \right]-\frac\partial_ip\rho+p{},divide start_ARG italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG = ∂ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT [ divide start_ARG italic_g start_POSTSUBSCRIPT italic_t italic_t end_POSTSUBSCRIPT + 1 end_ARG start_ARG 2 end_ARG - roman_ln | italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | ] - divide start_ARG ∂ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p end_ARG start_ARG italic_ρ + italic_p end_ARG , (8)
from which one can define a NMC potential, Φc:=ln|f2|assignsubscriptΦݑsubscriptݑ“2\Phi_c:=\ln|f_2|roman_Φ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT := roman_ln | italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | martins . This implies that the nonminimally coupled Boltzmann equation reads in the absence of collisions and pressure gradients:

∂f∂t+v→⋅∇f-∇(Φ+Φc)⋅∂f∂v→=0.ݑ“ݑ¡⋅→ݑ£∇ݑ“⋅∇ΦsubscriptΦݑݑ“→ݑ£0\frac\partial f\partial t+\vecv\cdot\nabla f-\nabla\left(\Phi+\Phi_c% \right)\cdot\frac\partial f\partial\vecv=0{}.divide start_ARG ∂ italic_f end_ARG start_ARG ∂ italic_t end_ARG + over→ start_ARG italic_v end_ARG ⋅ ∇ italic_f - ∇ ( roman_Φ + roman_Φ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ) ⋅ divide start_ARG ∂ italic_f end_ARG start_ARG ∂ over→ start_ARG italic_v end_ARG end_ARG = 0 . (9)

Furthermore, the metric field equations of this model can be Taylor expanded considering corrections up to c-2superscriptݑ2c^-2italic_c start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT:

Rݑ…\displaystyle Ritalic_R ∼R(2)≡δRsimilar-toabsentsuperscriptݑ…2ݛ¿ݑ…\displaystyle\sim R^(2)\equiv\delta R∼ italic_R start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT ≡ italic_δ italic_R (10)

fnsuperscriptݑ“ݑ›ݑ…\displaystyle f^nitalic_f start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( italic_R ) ∼fn(0)+fn-1(0)R(2)similar-toabsentsuperscriptݑ“ݑ›0superscriptݑ“ݑ›10superscriptݑ…2\displaystyle\sim f^n(0)+f^n-1(0)R^(2)∼ italic_f start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( 0 ) + italic_f start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT ( 0 ) italic_R start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT (11)

Thus, at ݒª(2)ݒª2\mathcalO(2)caligraphic_O ( 2 ), the 00-component of the field equations (2) and the trace equation (3) become:

f1′(0)δG00=∇2[f1′′(0)δR+f2′(0)κδℒm]+f2(0)2κδT00(0),subscriptsuperscriptݑ“′10ݛ¿subscriptݐº00superscript∇2subscriptsuperscriptݑ“′′10ݛ¿ݑ…subscriptsuperscriptݑ“′20ݜ…ݛ¿subscriptℒݑšsubscriptݑ“202ݜ…ݛ¿superscriptsubscriptݑ‡000\displaystyle f^\prime_1(0)\delta G_00=\nabla^2\left[f^\prime\prime_% 1(0)\delta R+\fracf^\prime_2(0)\kappa\delta\mathcalL_m\right]+% \fracf_2(0)2\kappa\delta T_00^(0){},italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( 0 ) italic_δ italic_G start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT = ∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT [ italic_f start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( 0 ) italic_δ italic_R + divide start_ARG italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 0 ) end_ARG start_ARG italic_κ end_ARG italic_δ caligraphic_L start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ] + divide start_ARG italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 0 ) end_ARG start_ARG 2 italic_κ end_ARG italic_δ italic_T start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT , (12)

f1′(0)δR=3∇2(f1′′(0)δR-f2′(0)κδℒm)+f2(0)2κδT(0),subscriptsuperscriptݑ“′10ݛ¿ݑ…3superscript∇2subscriptsuperscriptݑ“′′10ݛ¿ݑ…subscriptsuperscriptݑ“′20ݜ…ݛ¿subscriptℒݑšsubscriptݑ“202ݜ…ݛ¿superscriptݑ‡0\displaystyle f^\prime_1(0)\delta R=3\nabla^2\left(f^\prime\prime_1(% 0)\delta R-\fracf^\prime_2(0)\kappa\delta\mathcalL_m\right)+\frac% f_2(0)2\kappa\delta T^(0){},italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( 0 ) italic_δ italic_R = 3 ∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_f start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( 0 ) italic_δ italic_R - divide start_ARG italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 0 ) end_ARG start_ARG italic_κ end_ARG italic_δ caligraphic_L start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) + divide start_ARG italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 0 ) end_ARG start_ARG 2 italic_κ end_ARG italic_δ italic_T start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT , (13)
where f1(0)=0subscriptݑ“100f_1(0)=0italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( 0 ) = 0 because of the field equations at zeroth order. We point out that this expansion is performed around the Minskowski spacetime where at lowest order does not exist matter fields. However, at linear level the fluctuations of the components of the energy-momentum tensor correspond to matter fields, hence δTμν=ρδμ0δν0ݛ¿subscriptݑ‡ݜ‡ݜˆݜŒsubscriptsuperscriptݛ¿0ݜ‡subscriptsuperscriptݛ¿0ݜˆ\delta T_\mu\nu=\rho\delta^0_\mu\delta^0_\nuitalic_δ italic_T start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT = italic_ρ italic_δ start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_δ start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT and δT=-ρݛ¿ݑ‡ݜŒ\delta T=-\rhoitalic_δ italic_T = - italic_ρ. This situation contrasts with the study in the context of gravitational waves of Ref. nmcgw where there still existed some residual background in the form of a cosmological constant or a dark energy-like fluid.

The obvious choice for the metric field is:

gμν=diag(-1-2Φ,1-2Ψ,1-2Ψ,1-2Ψ),subscriptݑ”ݜ‡ݜˆݑ‘ݑ–ݑŽݑ”12Φ12Ψ12Ψ12Ψg_\mu\nu=diag(-1-2\Phi,1-2\Psi,1-2\Psi,1-2\Psi){},italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT = italic_d italic_i italic_a italic_g ( - 1 - 2 roman_Φ , 1 - 2 roman_Ψ , 1 - 2 roman_Ψ , 1 - 2 roman_Ψ ) , (14)
where both |Φ|,|Ψ|≪1much-less-thanΦΨ1|\Phi|{},|\Psi|\ll 1| roman_Φ | , | roman_Ψ | ≪ 1 and correspond to the Bardeen gauge invariant potentials. Thus, the 00-components of the Ricci tensor are δR00=∇2Φݛ¿subscriptݑ…00superscript∇2Φ\delta R_00=\nabla^2\Phiitalic_δ italic_R start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT = ∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_Φ, and the scalar curvature δR=2∇2(2Ψ-Φ)ݛ¿ݑ…2superscript∇22ΨΦ\delta R=2\nabla^2\left(2\Psi-\Phi\right)italic_δ italic_R = 2 ∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 2 roman_Ψ - roman_Φ ).

Inserting this metric in the perturbed metric field equations and their trace, we get:

{2α∇4(2Ψ-Φ)-2∇2Ψ=β∇2ρ-γ2ρ6α∇4(2Ψ-Φ)-2∇2(2Ψ-Φ)=3β∇2ρ-γ2ρ,casesmissing-subexpression2ݛ¼superscript∇42ΨΦ2superscript∇2Ψݛ½superscript∇2ݜŒݛ¾2ݜŒmissing-subexpression6ݛ¼superscript∇42ΨΦ2superscript∇22ΨΦ3ݛ½superscript∇2ݜŒݛ¾2ݜŒ\displaystyle\left\{\begin{array}[]{ll}&2\alpha\nabla^{4}(2\Psi-\Phi)-2\nabla^% {2}\Psi=\beta\nabla^{2}\rho-\frac{\gamma}{2}\rho\\ &6\alpha\nabla^{4}(2\Psi-\Phi)-2\nabla^{2}(2\Psi-\Phi)=3\beta\nabla^{2}\rho-% \frac{\gamma}{2}\rho~{},\end{array}\right.{ start_ARRAY start_ROW start_CELL end_CELL start_CELL 2 italic_α ∇ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ( 2 roman_Ψ - roman_Φ ) - 2 ∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_Ψ = italic_β ∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ρ - divide start_ARG italic_γ end_ARG start_ARG 2 end_ARG italic_ρ end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL 6 italic_α ∇ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ( 2 roman_Ψ - roman_Φ ) - 2 ∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 2 roman_Ψ - roman_Φ ) = 3 italic_β ∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ρ - divide start_ARG italic_γ end_ARG start_ARG 2 end_ARG italic_ρ , end_CELL end_ROW end_ARRAY (17)
where α:=f1′′(0)/f1′(0)assignݛ¼subscriptsuperscriptݑ“′′10subscriptsuperscriptݑ“′10\alpha:=f^{\prime\prime}_{1}(0)/f^{\prime}_{1}(0)italic_α := italic_f start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( 0 ) / italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( 0 ), β:=f2′(0)/(κf1′(0))assignݛ½subscriptsuperscriptݑ“′20ݜ…subscriptsuperscriptݑ“′10\beta:=f^{\prime}_{2}(0)/\left(\kappa f^{\prime}_{1}(0)\right)italic_β := italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 0 ) / ( italic_κ italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( 0 ) ), γ:=f2(0)/(κF1(0))assignݛ¾subscriptݑ“20ݜ…subscriptݐ¹10\gamma:=f_{2}(0)/\left(\kappa F_{1}(0)\right)italic_γ := italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 0 ) / ( italic_κ italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( 0 ) ), and the term ∇2ρsuperscript∇2ݜŒ\nabla^{2}\rho∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ρ comes directly from the non-minimal coupling. The usual Poisson equation in GR is retrieved by setting Φ=Ψ,f2=1formulae-sequenceΦΨsubscriptݑ“2ݑ…1\Phi=\Psi,~{}f_{2}=1roman_Φ = roman_Ψ , italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_R ) = 1.

Fourier transforming the previous equations and adding both of them, one gets the following expression relating the two potentials:

(1+4αk2)Ψ~=(1+2αk2)Φ~-βρ~,14ݛ¼superscriptݑ˜2~Ψ12ݛ¼superscriptݑ˜2~Φݛ½~ݜŒ\left(1+4\alpha k^{2}\right)\tilde{\Psi}=\left(1+2\alpha k^{2}\right)\tilde{% \Phi}-\beta\tilde{\rho}~{},( 1 + 4 italic_α italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) over~ start_ARG roman_Ψ end_ARG = ( 1 + 2 italic_α italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) over~ start_ARG roman_Φ end_ARG - italic_β over~ start_ARG italic_ρ end_ARG , (18)
where the tilde notation refers to the Fourier transform of the functions underneath it.

In fact, this equation deserves a few comments. If we have considered cosmological perturbations of the form of Eq. (14) in the metric field equations, we would have found that the ij-components yield a general condition relating both Bardeen invariant potentials frazao :

Φ-Ψ=-δln(f1′+f2′ℒ).ΦΨݛ¿subscriptsuperscriptݑ“′1ݑ…subscriptsuperscriptݑ“′2ݑ…ℒ\Phi-\Psi=-\delta\ln\left(f^{\prime}_{1}+f^{\prime}_{2}\mathcal{L}\right% )~{}.roman_Φ - roman_Ψ = - italic_δ roman_ln ( italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_R ) + italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_R ) caligraphic_L ) . (19)

Further expanding this condition, keeping terms up to ݒª(1/c2)ݒª1superscriptݑ2\mathcal{O}(1/c^{2})caligraphic_O ( 1 / italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) and choosing ℒ=-ρℒݜŒ\mathcal{L}=-\rhocaligraphic_L = - italic_ρ we get the same relation, after a Fourier transform, as Eq. (18), where we have first expanded around a static Minkowsky background. The term βℒ~=-βρ~ݛ½~ℒݛ½~ݜŒ\beta\tilde{\mathcal{L}}=-\beta\tilde{\rho}italic_β over~ start_ARG caligraphic_L end_ARG = - italic_β over~ start_ARG italic_ρ end_ARG appears due to the non-minimal coupling between matter and curvature. In the limit where f2(0)=1subscriptݑ“201f_{2}(0)=1italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 0 ) = 1, we retrieve the relation in fݑ“ݑ…fitalic_f ( italic_R ) theories: Ψ~=1+2αk21+4αk2Φ~~Ψ12ݛ¼superscriptݑ˜214ݛ¼superscriptݑ˜2~Φ\tilde{\Psi}=\frac{1+2\alpha k^{2}}{1+4\alpha k^{2}}\tilde{\Phi}over~ start_ARG roman_Ψ end_ARG = divide start_ARG 1 + 2 italic_α italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 1 + 4 italic_α italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG over~ start_ARG roman_Φ end_ARG. Moreover, when f1=R⇔α=0iffsubscriptݑ“1ݑ…ݑ…ݛ¼0f_{1}=R\iff\alpha=0italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_R ) = italic_R ⇔ italic_α = 0, we get Ψ~=Φ~~Ψ~Φ\tilde{\Psi}=\tilde{\Phi}over~ start_ARG roman_Ψ end_ARG = over~ start_ARG roman_Φ end_ARG, which is the General Relativity’s condition.

In fact, we can relate both Bardeen gauge invariant potentials without the need of the dependence on the matter Lagrangian choice by sorting out a different linear combination of both equations of the system of Eq. (17):

Ψ~=γ+2αγk2+2βk2γ+4αγk2-2βk2Φ~.~Ψݛ¾2ݛ¼ݛ¾superscriptݑ˜22ݛ½superscriptݑ˜2ݛ¾4ݛ¼ݛ¾superscriptݑ˜22ݛ½superscriptݑ˜2~Φ\tilde{\Psi}=\frac{\gamma+2\alpha\gamma k^{2}+2\beta k^{2}}{\gamma+4\alpha% \gamma k^{2}-2\beta k^{2}}\tilde{\Phi}~{}.over~ start_ARG roman_Ψ end_ARG = divide start_ARG italic_γ + 2 italic_α italic_γ italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 2 italic_β italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_γ + 4 italic_α italic_γ italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 2 italic_β italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG over~ start_ARG roman_Φ end_ARG . (20)

Now, the two potentials can be decoupled into two Poisson-like equations, resorting to the inverse Fourier transform for the real space, for each one:

{(3α∇4-∇2)Φ=(αγ-β2)∇2ρ-γ4ρ(3α∇4-∇2)Ψ=(αγ+β2)∇2ρ-γ4ρ.cases3ݛ¼superscript∇4superscript∇2Φݛ¼ݛ¾ݛ½2superscript∇2ݜŒݛ¾4ݜŒmissing-subexpression3ݛ¼superscript∇4superscript∇2Ψݛ¼ݛ¾ݛ½2superscript∇2ݜŒݛ¾4ݜŒmissing-subexpression\displaystyle\left\{\begin{array}[]{ll}\left(3\alpha\nabla^{4}-\nabla^{2}% \right)\Phi=\left(\alpha\gamma-\frac{\beta}{2}\right)\nabla^{2}\rho-\frac{% \gamma}{4}\rho\\ \left(3\alpha\nabla^{4}-\nabla^{2}\right)\Psi=\left(\frac{\alpha\gamma+\beta}{% 2}\right)\nabla^{2}\rho-\frac{\gamma}{4}\rho~{}.\end{array}\right.{ start_ARRAY start_ROW start_CELL ( 3 italic_α ∇ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT - ∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) roman_Φ = ( italic_α italic_γ - divide start_ARG italic_β end_ARG start_ARG 2 end_ARG ) ∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ρ - divide start_ARG italic_γ end_ARG start_ARG 4 end_ARG italic_ρ end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL ( 3 italic_α ∇ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT - ∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) roman_Ψ = ( divide start_ARG italic_α italic_γ + italic_β end_ARG start_ARG 2 end_ARG ) ∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ρ - divide start_ARG italic_γ end_ARG start_ARG 4 end_ARG italic_ρ . end_CELL start_CELL end_CELL end_ROW end_ARRAY (23)

These results are the basis for numerical solvers from nonlocal optics to simulate the dynamics of N-body systems in the non-minimal matter-curvature coupling model of Refs. numerico1 ; numerico2 , as they allow the study of weak field implications from modified gravity in the context of gravitational collapse. Furthermore we note that issues concerning stellar stability from modified Lane-Emden equation were addressed for the case of f theories in Refs. lane1 ; lane2 and further generalised for the non-minimal matter-curvature model in Ref. lane3 together with the Tolman-Oppenheimer-Volkoff equation for a spherically symmetric body of isotropic material, where in both theories different solutions with respect to the standard theory were found. This equation aims at describing the inner structure of a thermodynamic self-gravitating system provided an polytropic fluid equation of state. A further stability criterion can be analysed in what concerns the causes the collapse of interstellar gas clouds which lead to star formation. This is the so-called Jeans instability criterion. It was analysed in the context of f theories in Refs. capozziello ; vainio and which shall be further analysed in the context of the non-minimal coupling alternative gravity model in the next section.

4 Jeans instability

The equilibrium state, denoted with the subscript "0", is assumed to be homogeneous and time-independent. Let us consider a small departure from this equilibrium state binney :

f(r,v,t)=f0(r,v)+ϵf1(r,v,t),ݑ“ݑŸݑ£ݑ¡subscriptݑ“0ݑŸݑ£italic-ϵsubscriptݑ“1ݑŸݑ£ݑ¡\displaystyle f(r,v,t)=f_{0}(r,v)+\epsilon f_{1}(r,v,t)~{},italic_f ( italic_r , italic_v , italic_t ) = italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_r , italic_v ) + italic_ϵ italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_r , italic_v , italic_t ) , (24)

Φ(r,t)=Φ0+ϵΦ1(r,t),ΦݑŸݑ¡subscriptΦ0ݑŸitalic-ϵsubscriptΦ1ݑŸݑ¡\displaystyle\Phi(r,t)=\Phi_{0}+\epsilon\Phi_{1}

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