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| {{lowercase|title=''f''(''R'') gravity}}
| | Today, there are several other types of web development and blogging software available to design and host your website blogs online and that too in minutes, if not hours. What I advise you do next is save the backup data file to a remote place like a CD-ROM, external disk drive if you have one or a provider such as Dropbox. A pinch of tablet centric strategy can get your Word - Press site miles ahead of your competitors, so here are few strategies that will give your Wordpress websites and blogs an edge over your competitors:. s and intelligently including a substantial amount of key words in the title tags, image links, etc. provided by Word - Press Automatic Upgrade, so whenever you need to update the new version does not, it automatically creates no webmaster. <br><br>These folders as well as files have to copied and the saved. The higher your blog ranks on search engines, the more likely people will find your online marketing site. You are able to set them within your theme options and so they aid the search engine to get a suitable title and description for the pages that get indexed by Google. Furthermore, with the launch of Windows 7 Phone is the smart phone market nascent App. W3C compliant HTML and a good open source powered by Word - Press CMS site is regarded as the prime minister. <br><br>You can down load it here at this link: and utilize your FTP software program to upload it to your Word - Press Plugin folder. It was also the very first year that the category of Martial Arts was included in the Parents - Connect nationwide online poll, allowing parents to vote for their favorite San Antonio Martial Arts Academy. Those who cannot conceive with donor eggs due to some problems can also opt for surrogacy option using the services of surrogate mother. If you loved this article therefore you would like to get more info with regards to [http://roaaad.com/link//backup_plugin_7695079 backup plugin] please visit the internet site. To turn the Word - Press Plugin on, click Activate on the far right side of the list. For any web design and development assignment, this is definitely one of the key concerns, specifically for online retail outlets as well as e-commerce websites. <br><br>A built-in widget which allows you to embed quickly video from popular websites. This plugin allows a webmaster to create complex layouts without having to waste so much time with short codes. re creating a Word - Press design yourself, the good news is there are tons of Word - Press themes to choose from. The most important plugins you will need are All-in-One SEO Pack, some social bookmarking plugin, a Feedburner plugin and an RSS sign up button. If your blog employs the permalink function, This gives your SEO efforts a boost, and your visitors will know firsthand what's in the post when seeing the URL. <br><br>Under Settings —> Reading, determine if posts or a static page will be your home page, and if your home page is a static page, what page will contain blog posts. Here's a list of some exciting Word - Press features that have created waves in the web development industry:. While deciding couple should consider the expertise of the doctor,clinics success rate,the costs of fertility treatment,including fertility tests and IVF costs and overall ones own financial budget. Web developers and newbies alike will have the ability to extend your web site and fit other incredible functions with out having to spend more. Press CTRL and the numbers one to six to choose your option. |
| {{DISPLAYTITLE:''f''(''R'') gravity}}
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| | |
| ''' ''f''(''R'') gravity''' is a type of modified gravity theory which generalizes [[Albert Einstein|Einstein's]] [[General Relativity]]. ''f''(''R'') gravity is actually a family of theories, each one defined by a different function of the [[Scalar curvature|Ricci scalar]]. The simplest case is just the function being equal to the scalar; this is General Relativity. As a consequence of introducing an arbitrary function, there may be freedom to explain the [[Accelerating universe|accelerated expansion]] and [[structure formation]] of the Universe without adding unknown forms of [[dark energy]] or [[dark matter]]. Some functional forms may be inspired by corrections arising from a [[Quantum gravity|quantum theory of gravity]]. ''f''(''R'') gravity was first proposed in 1970 by [[Hans Adolph Buchdahl]]<ref>{{cite journal| title = Non-linear Lagrangians and cosmological theory| last=Buchdahl |first=H. A.| journal = [[Monthly Notices of the Royal Astronomical Society]]| volume = 150| pages = 1–8| year = 1970| url = http://adsabs.harvard.edu/abs/1970MNRAS.150....1B| bibcode = 1970MNRAS.150....1B }}</ref> (although ''φ'' was used rather than ''f'' for the name of the arbitrary function). It has become an active field of research following work by Starobinsky.<ref>{{cite journal| title = A new type of isotropic cosmological models without singularity| last=Starobinsky |first=A. A.| journal = [[Physics Letters B]] | volume = 91| pages = 99–102| year = 1980|doi = 10.1016/0370-2693(80)90670-X| bibcode = 1980PhLB...91...99S }}</ref> A wide range of phenomena can be produced from this theory by adopting different functions; however, many functional forms can now be ruled out on observational grounds, or because of pathological theoretical problems.
| |
| | |
| ==Introduction==
| |
| | |
| In ''f''(''R'') gravity, one seeks to generalise the [[Lagrangian]] of the [[Einstein-Hilbert action]]:
| |
| :<math>S[g]= \int {1 \over 2\kappa} R \sqrt{-g} \, \mathrm{d}^4x </math>
| |
| to | |
| :<math>S[g]= \int {1 \over 2\kappa} f(R) \sqrt{-g} \, \mathrm{d}^4x </math>
| |
| where ''κ'' = 8''πGc''<sup>−4</sup>, ''g'' = |''g<sub>μν</sub>''| is the determinant of the [[metric tensor]] and ''f''(''R'') is some function of the [[scalar curvature|Ricci Curvature]].
| |
| | |
| ==Metric ''f''(''R'') Gravity==
| |
| | |
| ===Derivation of field equations===
| |
| In metric ''f''(''R'') gravity, one arrives at the field equations by varying with respect to the metric and not treating the connection independently. For completeness we will now briefly mention the basic steps of the variation of the action. The main steps are the same as in the case of the variation of the [[Einstein-Hilbert action]] (see the article for more details) but there are also some important differences.
| |
| | |
| The variation of the determinant is as always:
| |
| :<math>\delta \sqrt{-g}= -\frac{1}{2} \sqrt{-g} g_{\mu\nu} \delta g^{\mu\nu}</math>
| |
| The [[Ricci scalar]] is defined as
| |
| :<math> R = g^{\mu\nu} R_{\mu\nu}.\!</math>
| |
| Therefore, its variation with respect to the inverse metric ''g<sup>μν</sup>'' is given by
| |
| :<math>
| |
| \begin{align}
| |
| \delta R &= R_{\mu\nu} \delta g^{\mu\nu} + g^{\mu\nu} \delta R_{\mu\nu}\\
| |
| &= R_{\mu\nu} \delta g^{\mu\nu} + g^{\mu\nu}(\nabla_\rho \delta \Gamma^\rho_{\nu\mu} - \nabla_\nu \delta \Gamma^\rho_{\rho\mu})
| |
| \end{align}
| |
| </math>
| |
| For the second step see the article about the [[Einstein-Hilbert action]]. Since ''δΓ<sup>λ</sup><sub>μν</sub>'' is the difference of two connections, it should transform as a tensor. Therefore, it can be written as
| |
| :<math>\delta \Gamma^\lambda_{\mu\nu}=\frac{1}{2}g^{\lambda a}\left(\nabla_\mu\delta g_{a\nu}+\nabla_\nu\delta g_{a\mu}-\nabla_a\delta g_{\mu\nu} \right).</math>
| |
| Substituting into the equation above:
| |
| :<math>\delta R= R_{\mu\nu} \delta g^{\mu\nu}+g_{\mu\nu}\Box \delta g^{\mu\nu}-\nabla_\mu \nabla_\nu \delta g^{\mu\nu}</math>
| |
| where ∇<sub>''μ''</sub> is the [[covariant derivative]] and □ = ''g<sup>μν</sup>''∇<sub>''μ''</sub>∇<sub>''ν''</sub> is the [[D'Alembertian|D'Alembert]] operator.
| |
| | |
| Now the variation in the action reads:
| |
| :<math>
| |
| \begin{align}
| |
| \delta S[g]&= \int {1 \over 2\kappa} \left(\delta f(R) \sqrt{-g}+f(R) \delta \sqrt{-g} \right)\, \mathrm{d}^4x \\
| |
| &= \int {1 \over 2\kappa} \left(F(R) \delta R \sqrt{-g}-\frac{1}{2} \sqrt{-g} g_{\mu\nu} \delta g^{\mu\nu} f(R)\right) \, \mathrm{d}^4x \\
| |
| &= \int {1 \over 2\kappa} \sqrt{-g}\left(F(R)(R_{\mu\nu} \delta g^{\mu\nu}+g_{\mu\nu}\Box \delta g^{\mu\nu}-\nabla_\mu \nabla_\nu \delta g^{\mu\nu}) -\frac{1}{2} g_{\mu\nu} \delta g^{\mu\nu} f(R) \right)\, \mathrm{d}^4x
| |
| \end{align}
| |
| </math>
| |
| where ''F''(''R'') = ∂''f''(''R'')/∂''R''. Doing integration by parts on the second and third terms we get:
| |
| :<math>
| |
| \begin{align}
| |
| \delta S[g]&= \int {1 \over 2\kappa} \sqrt{-g}\delta g^{\mu\nu} \left(F(R)R_{\mu\nu}-\frac{1}{2}g_{\mu\nu} f(R)+[g_{\mu\nu}\Box -\nabla_\mu \nabla_\nu]F(R) \right)\, \mathrm{d}^4x.
| |
| \end{align}
| |
| </math>
| |
| | |
| By demanding that the action remains invariant under variations of the metric, ''δS''[''g''] = 0, one obtains the field equations:
| |
| :<math>F(R)R_{\mu\nu}-\frac{1}{2}f(R)g_{\mu\nu}+\left[g_{\mu\nu} \Box-\nabla_\mu
| |
| \nabla_\nu \right]F(R) = \kappa T_{\mu\nu},</math>
| |
| | |
| where ''T<sub>μν</sub>'' is the [[energy-momentum tensor]] defined as
| |
| :<math>T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g} L_\mathrm{m})}{\delta g^{\mu\nu}},</math>
| |
| where ''L''<sub>m</sub> is the matter Lagrangian.
| |
| | |
| === The generalized Friedmann equations ===
| |
| Assuming a [[Robertson-Walker metric]] with scale factor ''a''(''t'') we can find the generalized [[Friedmann equations]] to be (in units where ''κ'' = 8''πGc''<sup>−4</sup> = 1):
| |
| | |
| :<math>3F H^{2} = \rho_{{\rm m}}+\rho_{{\rm rad}}+\frac{1}{2}(FR-f)-3H{\dot F}</math>
| |
| :<math>-2F\dot{H} = \rho_{{\rm m}}+\frac{4}{3}\rho_{{\rm rad}}+\ddot{F}-H\dot{F},</math>
| |
| | |
| where
| |
| :<math>H = \frac{\dot{a}}{a},</math>
| |
| the dot is the derivative with respect to the cosmic time ''t'', and the terms ''ρ''<sub>m</sub> and ''ρ''<sub>rad</sub> represent the matter and radiation densities respectively; these satisfy the continuity equations:
| |
| :<math> \dot{\rho}_{{\rm m}}+3H\rho_{{\rm m}}=0;</math>
| |
| :<math> \dot{\rho}_{{\rm rad}}+4H\rho_{{\rm rad}}=0.</math>
| |
| | |
| ===Modified Newton's constant===
| |
| An interesting feature of these theories is the fact that the [[gravitational constant]] is time and scale dependent. To see this, add a small scalar perturbation to the metric (in the [[Newtonian gauge]]):
| |
| :<math>\mathrm{d}s^2 = -(1+2\Phi)\mathrm{d}t^2 +\alpha^2 (1-2\Psi)\delta_{ij}\mathrm{d}x^i \mathrm{d}x^j \,</math>
| |
| | |
| where Φ and Ψ are the Newtonian potentials and use the field equations to first order. After some lengthy calculations, one can define a [[Poisson equation]] in the Fourier space and attribute the extra terms that appear on the right hand side to an effective gravitational constant ''G''<sub>eff</sub>.
| |
| Doing so, we get the gravitational potential (valid on sub-horizon scales ''k''<sup>2</sup> ≫ ''a''<sup>2</sup>''H''<sup>2</sup>):
| |
| :<math> \Phi = -4 \pi G_\mathrm{eff} \frac{a^2}{k^2} \delta\rho_\mathrm{m} </math>
| |
| where ''δρ''<sub>m</sub> is a perturbation in the matter density and ''G''<sub>eff</sub> is:
| |
| :<math> G_\mathrm{eff}=\frac{1}{8\pi F}\frac{1+4\frac{k^2}{a^2R}m}{1+3\frac{k^2}{a^2R}m},</math>
| |
| with
| |
| :<math> m\equiv\frac{RF_{,R}}{F}.</math>
| |
| | |
| === Massive gravitational waves ===
| |
| This class of theories when linearized exhibits three polarization modes for the [[gravitational waves]], of which two correspond to the massless [[graviton]] (helicities ±2) and the third (scalar) is coming from the fact that if we take into account a conformal transformation, the fourth order theory ''f''(''R'') becomes [[general relativity]] plus a [[scalar field]]. To see this, identify
| |
| :<math> \Phi \rightarrow f'(R)~~~~~\textrm{and}~~~~ \frac{dV}{d\Phi}\rightarrow\frac{2f(R)-R f'(R)}{3},</math>
| |
| and use the field equations above to get
| |
| :<math>\Box \Phi=\frac{\mathrm{d}V}{\mathrm{d}\Phi}</math>
| |
| Working to first order of perturbation theory:
| |
| :<math> g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}\, </math> | |
| :<math> \Phi=\Phi_0+\delta \Phi \,</math>
| |
| and after some tedious algebra, one can solve for the metric perturbation, which corresponds to the gravitational waves. A particular frequency component, for a wave propagating in the ''z''-direction, may be written as
| |
| :<math> h_{\mu\nu}(t,z;\omega)=A^{+}(\omega)(t-z)e^{+}_{\mu\nu}+A^{\times}(\omega)(t-z)e^{\times}_{\mu\nu} +h_f(v_\mathrm{g} t-z;\omega) \eta_{\mu\nu} </math>
| |
| where
| |
| :<math> h_f\equiv \frac{\delta \Phi}{\Phi_0},</math>
| |
| and ''v''<sub>g</sub>(ω) = dω/d''k'' is the [[group velocity]] of a [[wave packet]] ''h<sub>f</sub>'' centred on wave-vector ''k''. The first two terms correspond to the usual [[Gravitational waves#Linear approximation|transverse polarizations]] from general relativity, while the third corresponds to the new massive polarization mode of ''f''(''R'') theories. The transverse modes propagate at the [[speed of light]], but the scalar mode moves at a speed ''v''<sub>G</sub> < 1 (in units where ''c'' = 1), this mode is dispersive.
| |
| | |
| == Equivalent formalism ==
| |
| | |
| Under certain additional conditions [See Fiziev (2013) in Further reading, below] we can simplify the analysis of ''f''(''R'') theories by introducing an [[auxiliary field]] Φ. Assuming ''f′′''(''R'') ≠ 0 for all ''R'', let ''V''(Φ) be the [[Legendre transform]] of ''f''(''R'') so that Φ = ''f′''(''R'') and ''R'' = ''V′''(Φ). Then, one obtains the O'Hanlon (1972) action
| |
| :<math>S = \int d^4x \sqrt{-g} \left[ \frac{1}{2\kappa}\left(\Phi R - V(\Phi)\right) + \mathcal{L}_{\text{m}}\right].</math>
| |
| | |
| We have the Euler-Lagrange equations
| |
| :<math>V'(\Phi)=R</math>
| |
| :<math>\Phi \left( R_{\mu\nu} - \frac{1}{2}g_{\mu\nu} R \right) + \left(g_{\mu\nu}\Box -\nabla_\mu \nabla_\nu \right) \Phi + \frac{1}{2} g_{\mu\nu}V(\Phi) = \kappa T_{\mu\nu}</math>
| |
| Eliminating Φ, we obtain exactly the same equations as before. However, the equations are only second order in the derivatives, instead of fourth order.
| |
| | |
| We are currently working with the [[Jordan and Einstein frames|Jordan frame]]. By performing a conformal rescaling
| |
| :<math>\tilde{g}_{\mu\nu}=\Phi g_{\mu\nu},</math>
| |
| we transform to the [[Einstein frame]]:
| |
| :<math>R=\Phi^{-1} \left[ \tilde{R} + \frac{3\tilde{\Box} \Phi}{\Phi} -\frac{9}{2}\left(\frac{\tilde{\nabla} \Phi}{\Phi}\right)^2 \right]</math>
| |
| :<math>S = \int d^4x \sqrt{-\tilde{g}}\frac{1}{2\kappa}\left[ \tilde{R} - \frac{3}{2}\left( \frac{\tilde{\nabla}\Phi}{\Phi} \right)^2 - \frac{V(\Phi)}{\Phi^2} \right]</math>
| |
| after integrating by parts.
| |
| | |
| Defining
| |
| :<math>\tilde{\Phi} = \sqrt{3} \ln{\Phi}</math>,
| |
| and substituting
| |
| :<math>S = \int \mathrm{d}^4x \sqrt{-\tilde{g}}\frac{1}{2\kappa}\left[ \tilde{R} - \frac{1}{2}\left(\tilde{\nabla}\tilde{\Phi}\right)^2 - \tilde{V}(\tilde{\Phi}) \right]</math>
| |
| :<math>\tilde{V}(\tilde{\Phi}) = e^{-2/\sqrt{3}\;\tilde{\Phi}}V(e^{\tilde{\Phi}/\sqrt{3}}).</math>
| |
| This is general relativity coupled to a real scalar field: using ''f''(''R'') theories to describe the accelerating universe is practically equivalent to using [[quintessence (physics)|quintessence]]. | |
| | |
| ==Palatini ''f''(''R'') Gravity==
| |
| | |
| In [[Palatini variation|Palatini]] ''f''(''R'') gravity, one treats the metric and [[Connection (mathematics)|connection]] independently and varies the action with respect to each of them separately. The matter Lagrangian is assumed to be independent of the connection. These theories have been shown to be equivalent to [[Brans–Dicke theory]] with ''ω'' = −3/2.<ref name="flanagan04">{{cite journal| title = The conformal frame freedom in theories of gravitation| last= Flanagan |first=E. E.| journal = [[Classical and Quantum Gravity]] | volume = 21| pages = 3817| year = 2004| doi = 10.1088/0264-9381/21/15/N02 | bibcode = 2004CQGra..21.3817F |arxiv = gr-qc/0403063| issue = 15 }}</ref><ref name="olmo05">{{cite journal| title = The Gravity Lagrangian According to Solar System Experiments| last= Olmo |first=G. J.| journal = [[Physical Review Letters]] | volume = 95| pages = 261102| year = 2005| doi = 10.1103/PhysRevLett.95.261102 | bibcode = 2005PhRvL..95z1102O |arxiv = gr-qc/0505101| issue = 26| pmid = 16486333 }}</ref> Due to the structure of the theory, however, Palatini ''f''(''R'') theories appear to be in conflict with the Standard Model,<ref name="flanagan04"/><ref>{{cite journal| title =How (not) to use the Palatini formulation of scalar-tensor gravity| last1= Iglesias |first1=A. |last2=Kaloper |first2=N. |last3=Padilla |first3=A. |last4=Park |first4=M.| journal = [[Physical Review D]] | volume = 76| pages = 104001| year = 2007| doi = 10.1103/PhysRevD.76.104001 | bibcode = 2007PhRvD..76j4001I |arxiv = 0708.1163| issue =10 }}</ref> may violate Solar system experiments,<ref name="olmo05"/> and seem to create unwanted singularities.<ref>{{cite journal| title =A no-go theorem for polytropic spheres in Palatini ''f''(''R'') gravity| last1=Barausse |first1=E. |last2=Sotiriou |first2=T. P. |last3=Miller |first3=J. C.| journal = [[Classical & Quantum Gravity]] | volume = 25| pages = 062001| year = 2008| doi = 10.1088/0264-9381/25/6/062001 | bibcode = 2008CQGra..25f2001B |arxiv = gr-qc/0703132| issue =6 }}</ref>
| |
| | |
| ==Metric-Affine ''f''(''R'') Gravity==
| |
| | |
| In [[metric-affine gravitation theory|metric-affine]] ''f''(''R'') gravity, one generalizes things even further, treating both the metric and connection independently, and assuming the matter Lagrangian depends on the connection as well.
| |
| | |
| ==Observational tests==
| |
| | |
| As there are many potential forms of ''f''(''R'') gravity, it is difficult to find generic tests. Additionally, since deviations away from General Relativity can be made arbitrarily small in some cases, it is impossible to conclusively exclude some modifications. Some progress can be made, without assuming a concrete form for the function ''f''(''R'') by [[Taylor series|Taylor expanding]]
| |
| :<math>f(R) = a_0 + a_1 R + a_2 R^2 + \ldots</math>
| |
| The first term is like the [[cosmological constant]] and must be small. The next coefficient ''a''<sub>1</sub> can be set to one as in general relativity. For metric ''f''(''R'') gravity (as opposed to Palatini or metric-affine ''f''(''R'') gravity), the quadratic term is best constrained by [[fifth force]] measurements, since it leads to a [[Yukawa potential|Yukawa]] correction to the gravitational potential. The best current bounds are |''a''<sub>2</sub>| < 4 × 10<sup>−9</sup>m<sup>2</sup> or equivalently |''a''<sub>2</sub>| < 2.3 × 10<sup>22</sup> GeV<sup>−2</sup>.<ref name="Berry">
| |
| {{cite journal| title = Linearized ''f''(''R'') gravity: Gravitational radiation and Solar System tests| last1= Berry |first1=C. P. L. |last2= Gair |first2=J. R.| journal = [[Physical Review D]] | volume = 83| pages = 104022| year = 2011| doi = 10.1103/PhysRevD.83.104022| bibcode = 2011PhRvD..83j4022B |arxiv = 1104.0819| issue = 10 }}</ref><ref>{{cite journal| title = Dark Matter from R<sup>2</sup> Gravity| last1=Cembranos |first1=J. A. R.| journal = [[Physical Review Letters]] | volume = 102| pages = 141301| year = 2009| doi = 10.1103/PhysRevLett.102.141301| bibcode = 2009PhRvL.102n1301C |arxiv = 0809.1653| issue = 14| pmid = 19392422 }}
| |
| </ref>
| |
| | |
| The [[parameterized post-Newtonian formalism]] is designed to be able to constrain generic modified theories of gravity. However, ''f''(''R'') gravity shares many of the same values as General Relativity, and is therefore indistinguishable using these tests.<ref>{{cite journal| title = Parametrized post-Newtonian limit of fourth-order theories of gravity| last1= Clifton |first1=T.| journal = [[Physical Review D]] | volume = 77| pages = 024041 | year = 2008| doi = 10.1103/PhysRevD.77.024041| bibcode = 2008PhRvD..77b4041C |arxiv = 0801.0983| issue = 2 }}</ref> In particular light deflection is unchanged, so ''f''(''R'') gravity, like General Relativity, is entirely consistent with the bounds from [[Cassini–Huygens#Tests_of_general_relativity|Cassini tracking]].<ref name="Berry" />
| |
| | |
| == Tensorial generalization ==
| |
| | |
| ''f''(''R'') gravity as presented in the previous sections is a scalar modification of general relativity. More generally, we can have a | |
| :<math>\int \mathrm{d}^Dx \sqrt{-g}\, f(R, R^{\mu\nu}R_{\mu\nu}, R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma})</math> | |
| coupling involving invariants of the [[Ricci tensor]] and the [[Weyl tensor]]. Special cases are ''f''(''R'') gravity, [[conformal gravity]], [[Gauss-Bonnet gravity]] and [[Lovelock gravity]]. It is suggested to consider dependency to the [[covariant derivative]] of the [[Riemann tensor]] in order to resolve more problems.<ref>{{cite journal |last=Exirifard |first=Q. |year=2010|title= Phenomenological covariant approach to gravity|journal=[[General Relativity and Gravitation]] |doi=10.1007/s10714-010-1073-6|bibcode = 2011GReGr..43...93E |volume=43 |pages=93–106 |arxiv = 0808.1962 }}</ref> Notice that with any nontrivial tensorial dependence, we typically have additional massive spin-2 degrees of freedom, in addition to the massless graviton and a massive scalar. An exception is Gauss-Bonnet gravity where the fourth order terms for the spin-2 components cancel out.
| |
| | |
| == See also ==
| |
| | |
| * [[Extended theories of gravity]]
| |
| * [[Gauss-Bonnet gravity]]
| |
| * [[Lovelock gravity]]
| |
| | |
| ==References==
| |
| <references/>
| |
| | |
| ==Further reading==
| |
| | |
| *{{cite journal
| |
| |last1=Carroll |first1=S. M.
| |
| |last2=Duvvuri |first2=V.
| |
| |last3=Trodden |first3=M.
| |
| |last4=Turner |first4=M. S.
| |
| |year=2004
| |
| |title=Is Cosmic Speed-Up Due to New Gravitational Physics?
| |
| |journal=[[Physical Review D]]
| |
| |volume=70 |issue=4 |pages=043528
| |
| |arxiv=astro-ph/0306438
| |
| |bibcode=2004PhRvD..70d3528C
| |
| |doi=10.1103/PhysRevD.70.043528
| |
| }}
| |
| *{{cite journal
| |
| |last1=Capozziello |first1=S.
| |
| |last2=Cardone |first2=V. F.
| |
| |last3=Troisi |first3=A.
| |
| |year=2006
| |
| |title=Dark energy and dark matter as curvature effects
| |
| |journal=[[Journal of Cosmology and Astroparticle Physics]]
| |
| |volume=2006 |issue=8 |pages=1
| |
| |arxiv=astro-ph/0602349
| |
| |bibcode=2006JCAP...08..001C
| |
| |doi=10.1088/1475-7516/2006/08/001
| |
| }}
| |
| *{{cite journal
| |
| |last1=Tsujikawa |first1=Shinji
| |
| |year=2007
| |
| |title=Matter density perturbations and effective gravitational constant in modified gravity models of dark energy
| |
| |journal=[[Physical Review D]]
| |
| |volume=76 |issue=2 |pages=023514
| |
| |arxiv=0705.1032
| |
| |bibcode=2007PhRvD..76b3514T
| |
| |doi=10.1103/PhysRevD.76.023514
| |
| }}
| |
| *{{cite arxiv
| |
| |last1=Faraoni |first1=F.
| |
| |year=2008
| |
| |title=f(R) gravity: Successes and challenges
| |
| |eprint=0810.2602
| |
| |class=gr-qc
| |
| }}
| |
| *{{cite journal
| |
| |last1=Flanagan |first1=E. E.
| |
| |year=2004
| |
| |title=Palatini form of 1/R gravity
| |
| |journal=[[Physical Review Letters]]
| |
| |volume=92 |issue=7 |pages=071101
| |
| |arxiv=astro-ph/0308111
| |
| |bibcode=2004PhRvL..92g1101F
| |
| |doi=10.1103/PhysRevLett.92.071101
| |
| }}
| |
| *{{cite journal
| |
| |last1=Capozziello |first1=S.
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| | |
| ==External links==
| |
| *[http://xstructure.inr.ac.ru/x-bin/theme3.py?level=1&index1=3649 ''f''(''R'') gravity on arxiv.org]
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| *[http://inspirehep.net/record/925916 Extended Theories of Gravity]
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| {{theories of gravitation}}
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| [[Category:Theories of gravitation]]
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