Dark matter gravitational interactions

The non-Riemannian Einstein tensor is not, in general, divergenceless.

“Unlike the Einstein tensor associated with the Levi-Civita connection, the Einstein tensor Ein (defined in (2.3)) is associated with the non-Riemannian connection ∇ and is not, in general, divergenceless [23]:”

• cites [23] Tucker R W, 1996 sourceSupported

The exterior derivative of both sides of equation (4.5) yields (cmα/h)^2 d*α + σ djq = 0.

“while the exterior derivative of both sides of (4.5) yields cm α h 2 d α + σ dj q = 0.”

• AssumptionThe authors state this as an assumption: Proca charged particles are conserved: d(nq Vq)=0, equivalently djq=0.

In the domain Y(x)<4^{1/3} θ^{1/3}, Y(x) has an acceleration phase (∂2Y/∂x2>0, qd<0).

“We also show that in the domain Y (x) < 4 1/3 ϑ 1/3 , Y (x) has an acceleration phase (∂ 2 Y/∂x 2 > 0, q d < 0 obtained by differentiating (7.7)).”

The vanishing of the transverse components yields relativistic Navier-Stokes-type equations for neutral and charged fluids, respectively.

“while the vanishing of the transverse components yields the appropriate relativistic Navier-Stoke's-type equations for neutral and charged fluids, respectively,”

Taking the typical observational value vc≈220 km s^-1, equation (5.12) gives ρq(r)≈6×10^-28 kg m^-3 for r=10 kpc.

“Taking a typical observational value [33] v c ≈ 220 km s -1 equation (5.12) gives a value ρ q (r) ≈ 6 × 10 -28 kg m -3 of the dark matter density for r = 10 kpc.”

• cites [33] Peebles P J E, 1993 sourceNot checked — no full text of the source

The Green function associated with (4.24) is A = -1/(4πr) e^{-(cmα/h)r}, where r = sqrt((x1)^2+(x2)^2+(x3)^2).

“The Green function associated with (4.24) is given by A = -1 4πr e -(cm α /h)r (4.25)where r = (x 1 ) 2 + (x 2 ) 2 + (x 3 ) 2 .”

Since the Proca field couples to a current jq of Proca charged matter, d*F + (cmα/h)^2 α + σ jq = 0.

“Since the Proca field couples to a current j q of Proca charged matter we have d F + cm α h 2 α + σ j q = 0. (4.5)”

• AssumptionThe authors state this as an assumption: The Proca charge current takes the convective form jq = q nq Vq with constant Proca charge q.
• AssumptionThe authors state this as an assumption: Proca charged particles are conserved: d(nq Vq)=0, equivalently djq=0.

From the particle conservation equation (4.8), n0=N0/S^3, where N0 is a dimensionless constant.

“From the particle conservation equation (4.8) it follows that n 0 = N 0 S 3 (7.1) where N 0 is a dimensionless constant.”

Inserting this value in (8.14) yields Smin>10 cm.

“Inserting this value in (8.14) yields the value of the minimal scale factor S min > 10 cm.(8.20)”

In all three cases the Levi-Civita curvature is regular throughout spacetime.

“In all three cases the Levi-Civita curvature is regular throughout spacetime.”

Substituting equations (2.20) and (2.21) into (2.19) yields a Proca-type equation d*dQ + β3 Q = 0.

“Substituting (2.20) and (2.21) into (2.19) one now obtains a Proca-type equation in the form d dQ + β 3 Q = 0 (2.23)”

The asymptotic dark matter pressure distribution is Pq(r)=kB^2 T^2/(2πGμq^2 r^2).

“and the asymptotic dark matter pressure distribution P q (r) = k 2 B T 2 2πGµ 2 q r 2 . (5.10)”

The non-Riemannian Einstein tensor differs from the Levi-Civita Einstein tensor by terms involving the tensor λ and its derivatives.

“It follows from (2.6) that Ein differs from the Levi-Civita Einstein tensor Ein • by terms involving the tensor λ and its derivatives.”

In the theory, the weak field limit admits both massless tensor gravitational quanta (traditional gravitons) and massive vector gravitational quanta.

“In such a theory the weak field limit admits both massless tensor gravitational quanta (traditional gravitons) and massive vector gravitational quanta.”

Comparing equations (4.16) and (4.5) gives Z = σ q nq Vq.

“Comparing (4.16) and (4.5) one has Z = σ qn q V q (4.17)”

• AssumptionThe authors state this as an assumption: The Proca charge current takes the convective form jq = q nq Vq with constant Proca charge q.

Relation (5.11) may be used to eliminate kBT/μq in (5.9), yielding the approximate dark matter density ρq(r)≈vc^2/(4πG r^2).

“This relation may be used to eliminate k B T /µ q in (5.9) yielding the approximate dark matter density ρ q (r) ≈ v 2 c 4πGr 2 .(5.12)”

For a large class of actions containing torsion and metric gradient fields, the terms involving λ and its derivatives can be dramatically simplified.

“However, for a large class of actions containing the torsion and metric gradient fields one finds that these terms can be dramatically simplified.”

If k=0 or -1, the universe contracts to a minimal scale and then expands to infinity.

“If k = 0 or -1 the universe contracts to a minimal scale and then expands to infinity.”

It follows from (6.16), (7.3), (8.9) and (8.15) that τ = S0/Sr (Smin/Sr)^2.

“It follows from (6.16), (7.3), (8.9) and (8.15) that τ = 0 r r i = S 0 S r S min S r 2 .(8.18)”

A gravitational Proca field configuration that gives rise to attractive forces between dark matter charges of like polarity exhibits homogeneous isotropic eternal cosmologies free of cosmological curvature singularities.

“A gravitational Proca field configuration that gives rise to attractive forces between dark matter charges of like polarity exhibits homogeneous isotropic eternal cosmologies that are free of cosmological curvature singularities”

The vanishing of the tangential components of the field equations (4.3) and (4.4) gives rise to the isentropic conditions V0(s0)=0 and Vq(sq)=0.

“Thus the vanishing of the tangential components of the field equations (4.3) and (4.4) gives rise to the isentropic conditions: V 0 (s 0 ) = 0 (4.20) and V q (s q ) = 0 (4.21)”

In the negligible-pressure limit, relation (4.2) yields ρf≈μf nf, where the constant μf is identified as the Newtonian mass of the neutral and Proca charged particle, respectively.

“In the case of negligible pressure (P f /c 2 ρ f , f = 0, q) the relation (4.2) yields ρ f ≈ µ f n f where the constant µ f is identified as the Newtonian mass of the neutral and Proca charged particle, respectively.”

With ∂S/∂t=0 at S=Smin, Λ=1/(8πG) Smin^6 -3k c^2 Smin^4.

“With ∂S/∂t = 0 at S = S min then = 1 8πG S 6 min -3kc 2 S 4 min .(8.6)”

The Einstein field equation is Ein•(Xa,Xb)1 + Eab + Tab = 0.

“Thus the Einstein field equation is Ein • (X a , X b ) 1 + E ab + T ab = 0 where L1 g = -h ab T ab .”

With Nq from (8.10) and q≈lpl, mα≈mpl, (7.6) gives Λ=3×10^24 kg m^-3.

“With the value of N q from (8.10) and q ≈ ℓ pl , m α ≈ m pl , (7.6) gives = 3 × 10 24 kg m 3 .(8.17)”

• AssumptionThe authors state this as an assumption: The necessary parameters are estimated by requiring q≈ℓpl and mα≈mpl.

Such solutions admit dense hot pre-inflationary epochs each with a characteristic scale factor that may be correlated with the dark matter density in the current era of expansion.

“Such solutions do, however, admit dense hot pre-inflationary epochs each with a characteristic scale factor that may be correlated with the dark matter density in the current era of expansion.”

Every component 4-form that gives rise, by variation, to field equations for any dynamical tensor fields in the action excluding the metric also gives rise to a divergenceless second-rank tensor.

“Thus we have shown that every component 4-form r that gives rise, by variation, to a set of field equations {F k = 0} for any of the dynamical tensor fields in the action, excluding the metric g, also gives rise to a divergenceless second-rank tensor T r .”

The mass of the Proca field is determined by the coupling constants in the parent non-Riemannian action.

“The mass of the Proca field is determined by the coupling constants in the parent non-Riemannian action.”

Tab exactly cancels Eab, leaving the Einstein equation in the form Ein• + lg + κτ = 0.

“Remarkably, Tab exactly cancels E ab leaving the Einstein equation in the form Ein • + ℓg + κτ = 0. (2.26)”

For σ=-1, where θ<0, one finds solutions for Y exhibiting zeros where the Levi-Civita geometry is singular as in the standard model of cosmology.

“For σ = -1 where < 0 one finds solutions for Y exhibiting zeros where the Levi-Civita geometry is singular as in the standard model of cosmology.”

The Proca charge conservation equation (4.7) yields the general solution nq=Nq/S^3, where Nq is a dimensionless constant.

“The Proca charge conservation equation (4.7) yields the general solution n q = N q S 3 (6.5) where N q is a dimensionless constant.”

The asymptotic dark matter mass density distribution is ρq(r)=kBT/(2πGμq r^2).

“This gives rise to the asymptotic dark matter mass density distribution ρ q (r) = k B T 2πGµ q r 2 (5.9)”

The Yukawa potential energy between two Proca charged particles with separation r is V(r) = -σ c^4 q^2/(32π^2 G r) e^{-(cmα/h)r}.

“Thus the 'Yukawa potential energy' V (r) between these two Proca charged particles with separation r is V (r) = - σ c 4 32π 2 G q 2 r e -(cm α /h)r . (4.27)”

Under approximately adiabatic conditions, (4.2) yields ρq≈μq nq with constant μq.

“Under approximately adiabatic conditions (4.2) yields ρ q ≈ µ q n q with constant µ q .”

The divergence of the Proca stress tensor is ∇• • = i_Z F + (cmα/h)^2 *d*α α.

“Together with (4.11) the divergence of becomes ∇ • • = i Z F + cm α h 2 -1 d α α (4.15”

• AssumptionThe authors state this as an assumption: The Proca charge current takes the convective form jq = q nq Vq with constant Proca charge q.

For pressure-free dusts with conserved particle currents, ρ0=μ0 n0 and ρq=μq nq, with the constant chemical potentials μq and μ0 interpreted as the masses of the Proca charged and neutral particles.

“It then follows from P 0 = 0, P q = 0 and (6.8) that ρ 0 = µ 0 n 0 ρ q = µ q n q with the constant chemical potentials µ q , µ 0 interpreted as the masses of the Proca charged and neutral particles.”

For u≫1, equation (5.7) has the asymptotic solution N(u)≈2u/η^2, corresponding to M(r)≈2kBT μq r/G, irrespective of the sign of σ.

“For u 1 equation (5.7) has the asymptotic solution N (u) ≈ 2u/η 2 , corresponding to M(r) ≈ 2k B T µ q G r (5.8) irrespective of the sign of σ .”

Eliminating Smin using (8.14) and assuming the desired value τ<1, solving (8.18) for Sr gives Sr>2×10^8 m.

“Eliminating S min using (8.14) and assuming the desired value τ < 1 we may solve (8.18) for S r : S r > 2 × 10 8 m (8.19)”

If the inflationary era ends at time tr with scale factor Sr and ρinflation(tr)=ρFriedmann(tr), then Λ=8πGMq/Sr^3.

“If the inflationary era ends at time t r with a scale factor S r = S(t r ) and ρ inflation (t r ) = ρ Friedmann (t r ) (8.15) one has = 8πGM q S 3 r .(8.16)”

In the low-energy and nonrelativistic limit, (4.23) and (4.25) imply μq ẍk = -σ c^4 q^2/(8πG) ∂A/∂x^k.

“If this particle also has Proca charge q and mass µ q then in the low-energy and nonrelativistic limit (4.23) and (4.25) imply that µ q ẍk = - σ c 4 q 2 8πG ∂ ∂x k A.”

For an action sum of 4-forms, there exists a set of symmetric second-degree tensors {Ts} satisfying ∇• • Ts = 0 in terms of the Levi-Civita connection.

“Then in general there exists a set of symmetric second degree tensors {T s }, each member of which is constructed from an element s belonging to a subset { s } contained in { r }, satisfying ∇ • • T s = 0 in terms of the Levi-Civita connection ∇ •”

The term Lλ in equation (2.11) does not contribute.

“From (2.10) one sees that the term L λ in (2.11) does not contribute.”

If there exists a vector field V=K satisfying the Killing condition, then J_rK defines a genuine conserved current dJ_rK=0.

“For example, if there exists a vector field V = K where K satisfies the Killing condition (L K g = 0) then both terms on the right-hand side of (3.2) are zero and J rK defines a genuine conserved current: dJ rK = 0.”

Solutions to the field equations derived from the action L_EH(g,λ)+F may be generated from solutions to the Levi-Civita Einstein-Proca system (2.23), (2.26) with arbitrary cosmological constant l.

“Thus the solutions to the field equations derived from the action L EH (g, λ) + F where F is given by (2.18) may be generated from solutions to the Levi-Civita Einstein-Proca system (2.23), (2.26) with arbitrary cosmological constant ℓ.”

The model is consistent with the observed behaviour of galactic rotation curves according to Newtonian dynamics.

“The model is also consistent with the observed behaviour of galactic rotation curves according to Newtonian dynamics.”

Depending on the sign adopted for the coupling of the Proca field, the interaction may be responsible for removing the classical curvature singularity associated with traditional Friedmann cosmologies.

“Depending on the sign adopted for the coupling of the Proca field in the fundamental action defining the theory, this interaction may also be responsible for removing the classical curvature singularity associated with the traditional Friedmann cosmologies [10].”

• cites [10] Hawking S W, 1973 sourceNot checked — no full text of the source

Stars composed of ordinary matter experience Newtonian forces due to an all-pervading background of massive gravitationally charged cold dark matter.

“The essential idea is that stars composed of ordinary (as opposed to dark) matter experience Newtonian forces due to the presence of an all-pervading background of massive gravitationally charged cold dark matter.”

• AssumptionThe authors state this as an assumption: Ordinary matter has zero coupling to the Proca field.
• AssumptionThe authors state this as an assumption: Dark matter has a non-zero coupling to the Proca field.

Without a significant contribution from directly unobserved matter it is hard to reconcile Einstein's theory of gravitation and its Newtonian limit with the dynamics of the Universe.

“In particular, without a significant contribution from directly unobserved matter it is hard to reconcile Einstein's theory of gravitation (and its Newtonian limit) with the dynamics of the Universe.”

The interaction mediated by the new Proca component of gravitation is expected to modify the traditional gravitational interaction on small scales.

“The interaction mediated by the new Proca component of gravitation is expected to modify the traditional gravitational interaction on small scales.”

The conjecture that hidden matter exists seems necessary to explain the Newtonian dynamics of both individual galaxies and galactic clusters.

“the conjecture that hidden matter also exists seems necessary to explain the Newtonian 934 R W Tucker and C Wang dynamics of both individual galaxies and galactic clusters.”

The model is based on a theory in which a modification of Einsteinian gravity at very short distances can be expressed in terms of the gradient of the Einstein metric and the torsion of a non-Riemannian connection on the bundle of linear frames over spacetime.

“The model is based on a theory in which a modification of Einsteinian gravity at very short distances can be expressed in terms of the gradient of the Einstein metric and the torsion of a non-Riemannian connection on the bundle of linear frames over spacetime.”

The presence of the Proca field in the Einstein equations with σ=1 can modify the singular nature of the geometry associated with Friedmann cosmologies.

“In the above we have shown that the presence of the Proca field in the Einstein equations with σ = 1 can modify the singular nature of the geometry associated with the Friedmann cosmologies.”

Dark matter in the Universe may be endowed with a new kind of gravitational charge that couples via a short-range gravitational interaction mediated by a massive vector field.

“We argue that the conjectured dark matter in the Universe may be endowed with a new kind of gravitational charge that couples via a short-range gravitational interaction mediated by a massive vector field.”

A model assimilates the proposed dark matter gravitational charge into current inflationary cosmology.

“A model is constructed that assimilates this concept into ideas of current inflationary cosmology.”

If the current Friedmann era is dominated by dark matter with negligible pressure, the current scale factor may be estimated as S(t0)≈3×10^10 light years = 3×10^26 m.

“If the current Friedmann era of the Universe with t = t 0 ≈ 3.5 × 10 17 s is dominated by dark matter with negligible pressure (ρ q (t 0 ) P q (t 0 )/c 2 ), then the current scale factor may be estimated [42] to be S(t 0 ) ≈ 3 × 10 10 light years = 3 × 10 26 m.”

• cites [42] Fowler W A, 1987, Q. J. R. Astron. Soc.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

With σ=-1 and mα=0 the model is analogous to the classical Einstein-Maxwell-fluid system.

“With σ = -1 and m α = 0 the model is analogous to the classical Einstein-Maxwell-fluid system.”

The novel gravitational interactions are predicted to have a significant influence on pre-inflationary cosmology.

“The novel gravitational interactions are predicted to have a significant influence on pre-inflationary cosmology.”

The existence of such singularity-free eternal cosmologies eliminates the horizon problem associated with the standard big-bang scenario.

“thus eliminating the horizon problem associated with the standard big-bang scenario.”

If dark matter dominates over ordinary matter in the later phase, the Proca field mass is of the order of the Planck mass, and the coupling to dark matter is of the same order as the fine-structure constant, then the hypotheses are consistent with both the inflationary scenario and the observed galactic rotation curves according to Newtonian dynamics.

“If one assumes that the amount of dark matter dominates over the ordinary matter in the later phase of evolution of the Universe, that the Proca field mass is of the order of the Planck mass and the appropriate coupling to the dark matter is of the same order as the fine-structure constant then one finds that such hypotheses are consistent with both the inflationary scenario of modern cosmology as”

• AssumptionThe authors state this as an assumption: The amount of dark matter dominates over ordinary matter in the later phase of evolution of the Universe.
• AssumptionThe authors state this as an assumption: The Proca field mass is of the order of the Planck mass.
• AssumptionThe authors state this as an assumption: The appropriate coupling to dark matter is of the same order as the fine-structure constant.

The new physics resides in a component of gravitation mediated by a Proca field.

“The simplicity of the model is a consequence of the structure of a class of non-Riemannian actions whose dynamical consequences imply that the new physics resides in a component of gravitation mediated by a Proca field.”

• AssumptionThe authors state this as an assumption: Dark matter may couple gravitationally through both its mass and a new kind of gravitational charge.