Scalar–tensor–vector gravity

<h2 id="overview">Overview</h2>
Scalar–tensor–vector gravity theory,<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> also known as MOdified Gravity (MOG), is based on an <a href="/facts/Action_principle/NhWS2Mts">action principle</a> and postulates the existence of a <a href="/facts/Vector_field/ccFNuPPp">vector field</a>, while elevating the three constants of the theory to <a href="/facts/Scalar_fields/8MtPTGob">scalar fields</a>. In the <a href="/facts/Weak-field_approximation/mbwlg5GW">weak-field approximation</a>, STVG produces a <a href="/facts/Yukawa_potential/TmPzXYpx">Yukawa</a>-like modification of the gravitational force due to a point source. Intuitively, this result can be described as follows: far from a source gravity is stronger than the Newtonian prediction, but at shorter distances, it is counteracted by a repulsive <a href="/facts/Fifth_force/SsPYPtTw">fifth force</a> due to the vector field.
STVG has been used successfully to explain <a href="/facts/Galaxy_rotation_curve/YJyb1Jy3">galaxy rotation curves</a>,<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> the mass profiles of galaxy clusters,<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> gravitational lensing in the <a href="/facts/Bullet_Cluster/40yuPD4C">Bullet Cluster</a>,<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> and cosmological observations<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> without the need for <a href="/facts/Dark_matter/y5HgX55N">dark matter</a>. On a smaller scale, in the Solar System, STVG predicts no observable deviation from general relativity.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> The theory may also offer an explanation for the origin of <a href="/facts/Inertia/bzK9uzI2">inertia</a>.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>

<h2 id="mathematical-details">Mathematical details</h2>
STVG is formulated using the action principle. In the following discussion, a <a href="/facts/Metric_(general_relativity)/lJoCMdve">metric</a> signature of 
 
 
 
 [
 +
 ,
 −
 ,
 −
 ,
 −
 ]
 
 
 {\displaystyle [+,-,-,-]}
 
 will be used; the speed of light is set to 
 
 
 
 c
 =
 1
 
 
 {\displaystyle c=1}
 
, and we are using the following definition for the <a href="/facts/Ricci_tensor/qujou7QS">Ricci tensor</a>:

R
          
            α
            β
          
        
        =
        
          ∂
          
            γ
          
        
        
          Γ
          
            α
            β
          
          
            γ
          
        
        −
        
          ∂
          
            β
          
        
        
          Γ
          
            α
            γ
          
          
            γ
          
        
        +
        
          Γ
          
            α
            β
          
          
            γ
          
        
        
          Γ
          
            γ
            δ
          
          
            δ
          
        
        −
        
          Γ
          
            α
            δ
          
          
            γ
          
        
        
          Γ
          
            γ
            β
          
          
            δ
          
        
        .
      
    
    {\displaystyle R_{\alpha \beta }=\partial _{\gamma }\Gamma _{\alpha \beta }^{\gamma }-\partial _{\beta }\Gamma _{\alpha \gamma }^{\gamma }+\Gamma _{\alpha \beta }^{\gamma }\Gamma _{\gamma \delta }^{\delta }-\Gamma _{\alpha \delta }^{\gamma }\Gamma _{\gamma \beta }^{\delta }.}

We begin with the <a href="/facts/Einstein%25E2%2580%2593Hilbert_action/DGbTkCe9">Einstein–Hilbert Lagrangian</a>:

L
            
          
          
            G
          
        
        =
        −
        
          
            1
            
              16
              π
              G
            
          
        
        (
        R
        +
        2
        Λ
        )
        
          
            −
            g
          
        
        ,
      
    
    {\displaystyle {\mathcal {L}}_{G}=-{\frac {1}{16\pi G}}(R+2\Lambda ){\sqrt {-g}},}

where 
 
 
 
 R
 
 
 {\displaystyle R}
 
 is the trace of the Ricci tensor, 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is the gravitational constant, 
 
 
 
 g
 
 
 {\displaystyle g}
 
 is the determinant of the metric tensor 
 
 
 
 
 g
 
 α
 β
 
 
 
 
 {\displaystyle g_{\alpha \beta }}
 
, while 
 
 
 
 Λ
 
 
 {\displaystyle \Lambda }
 
 is the cosmological constant.
We introduce the <a href="/facts/Proca_action/avxS9IyC">Maxwell-Proca</a> <a href="/facts/Lagrangian_(field_theory)/p9BCUz57">Lagrangian</a> for the STVG covector field 
 
 
 
 
 ϕ
 
 α
 
 
 
 
 {\displaystyle \phi _{\alpha }}
 
:

L
            
          
          
            ϕ
          
        
        =
        −
        
          
            1
            
              4
              π
            
          
        
        ω
        
          [
          
            
              
                1
                4
              
            
            
              B
              
                α
                β
              
            
            
              B
              
                α
                β
              
            
            −
            
              
                1
                2
              
            
            
              μ
              
                2
              
            
            
              ϕ
              
                α
              
            
            
              ϕ
              
                α
              
            
            +
            
              V
              
                ϕ
              
            
            (
            ϕ
            )
          
          ]
        
        
          
            −
            g
          
        
        ,
      
    
    {\displaystyle {\mathcal {L}}_{\phi }=-{\frac {1}{4\pi }}\omega \left[{\frac {1}{4}}B^{\alpha \beta }B_{\alpha \beta }-{\frac {1}{2}}\mu ^{2}\phi _{\alpha }\phi ^{\alpha }+V_{\phi }(\phi )\right]{\sqrt {-g}},}

where 
 
 
 
 
 B
 
 α
 β
 
 
 =
 
 ∂
 
 α
 
 
 
 ϕ
 
 β
 
 
 −
 
 ∂
 
 β
 
 
 
 ϕ
 
 α
 
 
 =
 (
 
 d
 
 ϕ
 
 )
 
 α
 β
 
 
 
 
 {\displaystyle B_{\alpha \beta }=\partial _{\alpha }\phi _{\beta }-\partial _{\beta }\phi _{\alpha }=(\mathrm {d} \phi )_{\alpha \beta }}
 
 is the field strength of 
 
 
 
 
 ϕ
 
 α
 
 
 
 
 {\displaystyle \phi _{\alpha }}
 
 (given by the <a href="/facts/Exterior_derivative/p7HA0wze">exterior derivative</a>), 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
 is the mass of the vector field, 
 
 
 
 ω
 
 
 {\displaystyle \omega }
 
 characterizes the strength of the coupling between the fifth force and matter, and 
 
 
 
 
 V
 
 ϕ
 
 
 
 
 {\displaystyle V_{\phi }}
 
 is a self-interaction potential.
The three constants of the theory, 
 
 
 
 G
 ,
 μ
 ,
 
 
 {\displaystyle G,\mu ,}
 
 and 
 
 
 
 ω
 ,
 
 
 {\displaystyle \omega ,}
 
 are promoted to scalar fields by introducing associated kinetic and potential terms in the Lagrangian density:

L
            
          
          
            S
          
        
        =
        −
        
          
            1
            G
          
        
        
          [
          
            
              
                1
                2
              
            
            
              g
              
                α
                β
              
            
            
              (
              
                
                  
                    
                      
                        ∂
                        
                          α
                        
                      
                      G
                      
                        ∂
                        
                          β
                        
                      
                      G
                    
                    
                      G
                      
                        2
                      
                    
                  
                
                +
                
                  
                    
                      
                        ∂
                        
                          α
                        
                      
                      μ
                      
                        ∂
                        
                          β
                        
                      
                      μ
                    
                    
                      μ
                      
                        2
                      
                    
                  
                
                −
                
                  ∂
                  
                    α
                  
                
                ω
                
                  ∂
                  
                    β
                  
                
                ω
              
              )
            
            +
            
              
                
                  
                    V
                    
                      G
                    
                  
                  (
                  G
                  )
                
                
                  G
                  
                    2
                  
                
              
            
            +
            
              
                
                  
                    V
                    
                      μ
                    
                  
                  (
                  μ
                  )
                
                
                  μ
                  
                    2
                  
                
              
            
            +
            
              V
              
                ω
              
            
            (
            ω
            )
          
          ]
        
        
          
            −
            g
          
        
        ,
      
    
    {\displaystyle {\mathcal {L}}_{S}=-{\frac {1}{G}}\left[{\frac {1}{2}}g^{\alpha \beta }\left({\frac {\partial _{\alpha }G\partial _{\beta }G}{G^{2}}}+{\frac {\partial _{\alpha }\mu \partial _{\beta }\mu }{\mu ^{2}}}-\partial _{\alpha }\omega \partial _{\beta }\omega \right)+{\frac {V_{G}(G)}{G^{2}}}+{\frac {V_{\mu }(\mu )}{\mu ^{2}}}+V_{\omega }(\omega )\right]{\sqrt {-g}},}

where 
 
 
 
 
 V
 
 G
 
 
 ,
 
 V
 
 μ
 
 
 ,
 
 
 {\displaystyle V_{G},V_{\mu },}
 
 and 
 
 
 
 
 V
 
 ω
 
 
 
 
 {\displaystyle V_{\omega }}
 
 are the self-interaction potentials associated with the scalar fields.
The STVG action integral takes the form

S
        =
        ∫
        
          (
          
            
              
                L
              
            
            
              G
            
          
          +
          
            
              
                L
              
            
            
              ϕ
            
          
          +
          
            
              
                L
              
            
            
              S
            
          
          +
          
            
              
                L
              
            
            
              M
            
          
          )
        
         
        
          
            d
            
              4
            
          
        
        x
        ,
      
    
    {\displaystyle S=\int {({\mathcal {L}}_{G}+{\mathcal {L}}_{\phi }+{\mathcal {L}}_{S}+{\mathcal {L}}_{M})}~\mathrm {d^{4}} x,}

where 
 
 
 
 
 
 
 L
 
 
 
 M
 
 
 
 
 {\displaystyle {\mathcal {L}}_{M}}
 
 is the ordinary matter Lagrangian density.

<h2 id="spherically-symmetric-static-vacuum-solution">Spherically symmetric, static vacuum solution</h2>
The <a href="/facts/Field_equations/7HuKY1qu">field equations</a> of STVG can be developed from the action integral using the <a href="/facts/Variational_principle/a5UkKEBj">variational principle</a>. First a test particle Lagrangian is postulated in the form

L
            
          
          
            
              T
              P
            
          
        
        =
        −
        m
        +
        α
        ω
        
          q
          
            5
          
        
        
          ϕ
          
            μ
          
        
        
          u
          
            μ
          
        
        ,
      
    
    {\displaystyle {\mathcal {L}}_{\mathrm {TP} }=-m+\alpha \omega q_{5}\phi _{\mu }u^{\mu },}

where 
 
 
 
 m
 
 
 {\displaystyle m}
 
 is the test particle mass, 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 is a factor representing the nonlinearity of the theory, 
 
 
 
 
 q
 
 5
 
 
 
 
 {\displaystyle q_{5}}
 
 is the test particle's fifth-force charge, and 
 
 
 
 
 u
 
 μ
 
 
 =
 d
 
 x
 
 μ
 
 
 
 /
 
 d
 s
 
 
 {\displaystyle u^{\mu }=dx^{\mu }/ds}
 
 is its four-velocity. Assuming that the fifth-force charge is proportional to mass, i.e., 
 
 
 
 
 q
 
 5
 
 
 =
 κ
 m
 ,
 
 
 {\displaystyle q_{5}=\kappa m,}
 
 the value of 
 
 
 
 κ
 =
 
 
 
 G
 
 N
 
 
 
 /
 
 ω
 
 
 
 
 {\displaystyle \kappa ={\sqrt {G_{N}/\omega }}}
 
 is determined and the following equation of motion is obtained in the spherically symmetric, static gravitational field of a point mass of mass 
 
 
 
 M
 
 
 {\displaystyle M}
 
:

r
              ¨
            
          
        
        =
        −
        
          
            
              
                G
                
                  N
                
              
              M
            
            
              r
              
                2
              
            
          
        
        
          [
          
            1
            +
            α
            −
            α
            (
            1
            +
            μ
            r
            )
            
              e
              
                −
                μ
                r
              
            
          
          ]
        
        ,
      
    
    {\displaystyle {\ddot {r}}=-{\frac {G_{N}M}{r^{2}}}\left[1+\alpha -\alpha (1+\mu r)e^{-\mu r}\right],}

where 
 
 
 
 
 G
 
 N
 
 
 
 
 {\displaystyle G_{N}}
 
 is <a href="/facts/Newton%2527s_constant/GdQ8tcBD">Newton's constant</a> of gravitation. Further study of the field equations allows a determination of 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 and 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
 for a point gravitational source of mass 
 
 
 
 M
 
 
 {\displaystyle M}
 
 in the form<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>

μ
        =
        
          
            D
            
              M
            
          
        
        ,
      
    
    {\displaystyle \mu ={\frac {D}{\sqrt {M}}},}

α
        =
        
          
            
              
                G
                
                  ∞
                
              
              −
              
                G
                
                  N
                
              
            
            
              G
              
                N
              
            
          
        
        
          
            M
            
              (
              
                
                  M
                
              
              +
              E
              
                )
                
                  2
                
              
            
          
        
        ,
      
    
    {\displaystyle \alpha ={\frac {G_{\infty }-G_{N}}{G_{N}}}{\frac {M}{({\sqrt {M}}+E)^{2}}},}

where 
 
 
 
 
 G
 
 ∞
 
 
 ≃
 20
 
 G
 
 N
 
 
 
 
 {\displaystyle G_{\infty }\simeq 20G_{N}}
 
 is determined from cosmological observations, while for the constants 
 
 
 
 D
 
 
 {\displaystyle D}
 
 and 
 
 
 
 E
 
 
 {\displaystyle E}
 
 galaxy rotation curves yield the following values:

D
        ≃
        
          25
          
            2
          
        
        ⋅
        
        10
        
          M
          
            ⊙
          
          
            1
            
              /
            
            2
          
        
        
          
            k
            p
            c
          
          
            −
            1
          
        
        ,
      
    
    {\displaystyle D\simeq 25^{2}\cdot \,10M_{\odot }^{1/2}\mathrm {kpc} ^{-1},}

E
        ≃
        
          50
          
            2
          
        
        ⋅
        
        10
        
          M
          
            ⊙
          
          
            1
            
              /
            
            2
          
        
        ,
      
    
    {\displaystyle E\simeq 50^{2}\cdot \,10M_{\odot }^{1/2},}

where 
 
 
 
 
 M
 
 ⊙
 
 
 
 
 {\displaystyle M_{\odot }}
 
 is the mass of the <a href="/facts/Sun/CadPCVHh">Sun</a>. These results form the basis of a series of calculations that are used to confront the theory with observation.

<h2 id="agreement-with-observations">Agreement with observations</h2>
STVG/MOG has been applied successfully to a range of astronomical, astrophysical, and cosmological phenomena.
On the scale of the Solar System, the theory predicts no deviation<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> from the results of Newton and Einstein. This is also true for star clusters containing no more than a few million solar masses.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>
The theory accounts for the rotation curves of spiral galaxies,<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> correctly reproducing the <a href="/facts/Tully%25E2%2580%2593Fisher_relation/gP6WRYIc">Tully–Fisher law</a>.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a>
STVG is in good agreement with the mass profiles of galaxy clusters.<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>
STVG can also account for key cosmological observations, including:<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>

<ul><li>The acoustic peaks in the <a href="/facts/Cosmic_microwave_background/ThB2HkLS">cosmic microwave background</a> radiation;</li>
<li>The <a href="/facts/Accelerating_universe/F7X1PAMq">accelerating expansion of the universe</a> that is apparent from <a href="/facts/Type_Ia_supernova/W4rEFuDK">type Ia supernova</a> observations;</li>
<li>The <a href="/facts/Matter_power_spectrum/kM46Rlq9">matter power spectrum</a> of the universe that is observed in the form of galaxy-galaxy correlations.</li></ul>
<h2 id="problems-and-criticism">Problems and criticism</h2>
A 2017 article on Forbes by <a href="/facts/Ethan_Siegel/qm9R9mIe">Ethan Siegel</a> states that the Bullet Cluster still "proves dark matter exists, but not for the reason most physicists think". There he argues in favor of dark matter over non-local gravity theories, such as STVG/MOG. Observations show that in "undisturbed" <a href="/facts/Galaxy_cluster/BhIEgbx8">galaxy clusters</a> the reconstructed mass from <a href="/facts/Gravitational_lens/LmxzvLs8">gravitational lensing</a> is located where matter is distributed, and a separation of matter from gravitation only seems to appear after a collision or interaction has taken place. According to Ethan Siegel: "Adding dark matter makes this work, but non-local gravity would make differing before-and-after predictions that can't both match up, simultaneously, with what we observe."<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Modified_Newtonian_dynamics/3kenEm7h">Modified Newtonian dynamics</a></li>
<li><a href="/facts/Nonsymmetric_gravitational_theory/49EEeEwM">Nonsymmetric gravitational theory</a></li>
<li><a href="/facts/Tensor%25E2%2580%2593vector%25E2%2580%2593scalar_gravity/Eu9wnpb3">Tensor–vector–scalar gravity</a></li>
<li><a href="/facts/Reinventing_Gravity/kqvLNJuY">Reinventing Gravity</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">McKee, M. (25 January 2006). "Gravity theory dispenses with dark matter". New Scientist. Retrieved 2008-07-26. <a href="https://www.newscientist.com/article.ns?id=dn8631" target="_blank">https://www.newscientist.com/article.ns?id=dn8631</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Moffat, J. W. (2006). "Scalar–Tensor–Vector Gravity Theory". Journal of Cosmology and Astroparticle Physics. 2006 (3): 4. arXiv:gr-qc/0506021. Bibcode:2006JCAP...03..004M. doi:10.1088/1475-7516/2006/03/004. <a href="/wiki/Journal_of_Cosmology_and_Astroparticle_Physics" target="_blank">/wiki/Journal_of_Cosmology_and_Astroparticle_Physics</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Brownstein, J. R.; Moffat, J. W. (2006). "Galaxy Rotation Curves Without Non-Baryonic Dark Matter". Astrophysical Journal. 636 (2): 721–741. arXiv:astro-ph/0506370. Bibcode:2006ApJ...636..721B. doi:10.1086/498208. <a href="/wiki/Astrophysical_Journal" target="_blank">/wiki/Astrophysical_Journal</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Brownstein, J. R.; Moffat, J. W. (2006). "Galaxy Cluster Masses Without Non-Baryonic Dark Matter". Monthly Notices of the Royal Astronomical Society. 367 (2): 527–540. arXiv:astro-ph/0507222. Bibcode:2006MNRAS.367..527B. doi:10.1111/j.1365-2966.2006.09996.x. <a href="https://doi.org/10.1111%2Fj.1365-2966.2006.09996.x" target="_blank">https://doi.org/10.1111%2Fj.1365-2966.2006.09996.x</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Brownstein, J. R.; Moffat, J. W. (2007). "The Bullet Cluster 1E0657-558 evidence shows Modified Gravity in the absence of Dark Matter". Monthly Notices of the Royal Astronomical Society. 382 (1): 29–47. arXiv:astro-ph/0702146. Bibcode:2007MNRAS.382...29B. doi:10.1111/j.1365-2966.2007.12275.x. <a href="https://doi.org/10.1111%2Fj.1365-2966.2007.12275.x" target="_blank">https://doi.org/10.1111%2Fj.1365-2966.2007.12275.x</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Moffat, J. W.; Toth, V. T. (2007). "Modified Gravity: Cosmology without dark matter or Einstein's cosmological constant". arXiv:0710.0364 [astro-ph]. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Moffat, J. W.; Toth, V. T. (2008). "Testing modified gravity with globular cluster velocity dispersions". Astrophysical Journal. 680 (2): 1158–1161. arXiv:0708.1935. Bibcode:2008ApJ...680.1158M. doi:10.1086/587926. <a href="/wiki/Astrophysical_Journal" target="_blank">/wiki/Astrophysical_Journal</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Moffat, J. W.; Toth, V. T. (2009). "Modified gravity and the origin of inertia". Monthly Notices of the Royal Astronomical Society Letters. 395 (1): L25. arXiv:0710.3415. Bibcode:2009MNRAS.395L..25M. doi:10.1111/j.1745-3933.2009.00633.x. <a href="https://doi.org/10.1111%2Fj.1745-3933.2009.00633.x" target="_blank">https://doi.org/10.1111%2Fj.1745-3933.2009.00633.x</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Moffat, J. W.; Toth, V. T. (2009). "Fundamental parameter-free solutions in Modified Gravity". Classical and Quantum Gravity. 26 (8): 085002. arXiv:0712.1796. Bibcode:2009CQGra..26h5002M. doi:10.1088/0264-9381/26/8/085002. <a href="/wiki/Classical_and_Quantum_Gravity" target="_blank">/wiki/Classical_and_Quantum_Gravity</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Moffat, J. W.; Toth, V. T. (2008). "Testing modified gravity with globular cluster velocity dispersions". Astrophysical Journal. 680 (2): 1158–1161. arXiv:0708.1935. Bibcode:2008ApJ...680.1158M. doi:10.1086/587926. <a href="/wiki/Astrophysical_Journal" target="_blank">/wiki/Astrophysical_Journal</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Moffat, J. W.; Toth, V. T. (2008). "Testing modified gravity with globular cluster velocity dispersions". Astrophysical Journal. 680 (2): 1158–1161. arXiv:0708.1935. Bibcode:2008ApJ...680.1158M. doi:10.1086/587926. <a href="/wiki/Astrophysical_Journal" target="_blank">/wiki/Astrophysical_Journal</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Brownstein, J. R.; Moffat, J. W. (2006). "Galaxy Rotation Curves Without Non-Baryonic Dark Matter". Astrophysical Journal. 636 (2): 721–741. arXiv:astro-ph/0506370. Bibcode:2006ApJ...636..721B. doi:10.1086/498208. <a href="/wiki/Astrophysical_Journal" target="_blank">/wiki/Astrophysical_Journal</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Moffat, J. W.; Toth, V. T. (2009). "Fundamental parameter-free solutions in Modified Gravity". Classical and Quantum Gravity. 26 (8): 085002. arXiv:0712.1796. Bibcode:2009CQGra..26h5002M. doi:10.1088/0264-9381/26/8/085002. <a href="/wiki/Classical_and_Quantum_Gravity" target="_blank">/wiki/Classical_and_Quantum_Gravity</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Brownstein, J. R.; Moffat, J. W. (2006). "Galaxy Cluster Masses Without Non-Baryonic Dark Matter". Monthly Notices of the Royal Astronomical Society. 367 (2): 527–540. arXiv:astro-ph/0507222. Bibcode:2006MNRAS.367..527B. doi:10.1111/j.1365-2966.2006.09996.x. <a href="https://doi.org/10.1111%2Fj.1365-2966.2006.09996.x" target="_blank">https://doi.org/10.1111%2Fj.1365-2966.2006.09996.x</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Moffat, J. W.; Toth, V. T. (2007). "Modified Gravity: Cosmology without dark matter or Einstein's cosmological constant". arXiv:0710.0364 [astro-ph]. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Siegel, Ethan (9 November 2017). "The Bullet Cluster proves dark matter exists, but not for the reason most physicists think". Forbes. <a href="/wiki/Ethan_Siegel" target="_blank">/wiki/Ethan_Siegel</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
</ol>

Scalar–tensor–vector gravity open-in-new

Scalar–tensor–vector gravity