**3.8 A Synthesis of LCDM with MOND in an Universal Lambda Milgröm Deceleration**

[Excerpt from Wikipedia: https://en.wikipedia.org/wiki/Modified_Newtonian_dynamics

Several independent observations point to the fact that the visible mass in galaxies and galaxy clusters is insufficient to account for their dynamics, when analysed using Newton's laws. This discrepancy – known as the "missing mass problem" – was first identified for clusters by Swiss astronomer Fritz Zwicky in 1933 (who studied the Coma cluster),^{[4]}^{[5]} and subsequently extended to include spiral galaxies by the 1939 work of Horace Babcock on Andromeda.^{[6]} These early studies were augmented and brought to the attention of the astronomical community in the 1960s and 1970s by the work of Vera Rubin at the Carnegie Institute in Washington, who mapped in detail the rotation velocities of stars in a large sample of spirals. While Newton's Laws predict that stellar rotation velocities should decrease with distance from the galactic centre, Rubin and collaborators found instead that they remain almost constant^{[7]} – the rotation curves are said to be "flat". This observation necessitates at least one of the following: 1) There exists in galaxies large quantities of unseen matter which boosts the stars' velocities beyond what would be expected on the basis of the visible mass alone, or 2) Newton's Laws do not apply to galaxies. The former leads to the dark matter hypothesis; the latter leads to MOND.

MOND was proposed by Mordehai Milgrom in 1983

The basic premise of MOND is that while Newton's laws have been extensively tested in high-acceleration environments (in the Solar System and on Earth), they have not been verified for objects with extremely low acceleration, such as stars in the outer parts of galaxies. This led Milgrom to postulate a new effective gravitational force law (sometimes referred to as "Milgrom's law") that relates the true acceleration of an object to the acceleration that would be predicted for it on the basis of Newtonian mechanics.^{[1]} This law, the keystone of MOND, is chosen to reduce to the Newtonian result at high acceleration but lead to different ("deep-MOND") behaviour at low acceleration:

........(1)

Here **F _{N}** is the Newtonian force, m is the object's (gravitational) mass,

**a**is its acceleration, μ(x) is an as-yet unspecified function (known as the "interpolating function"), and a

_{0}is a new fundamental constant which marks the transition between the Newtonian and deep-MOND regimes. Agreement with Newtonian mechanics requires μ(x) → 1 for x >> 1, and consistency with astronomical observations requires μ(x) → x for x << 1. Beyond these limits, the interpolating function is not specified by the theory, although it is possible to weakly constrain it empirically.

^{[8]}

^{[9]}Two common choices are:

("Simple interpolating function"), and ("Standard interpolating function").

Thus, in the deep-MOND regime (a << a_{0}):

Applying this to an object of mass m in circular orbit around a point mass M (a crude approximation for a star in the outer regions of a galaxy), we find:

.......(2)

that is, the star's rotation velocity is independent of its distance r from the centre of the galaxy – the rotation curve is flat, as required. By fitting his law to rotation curve data, Milgrom found a_{0} ≈ 1.2 x 10^{−10} m s^{−2} to be optimal. This simple law is sufficient to make predictions for a broad range of galactic phenomena. Milgrom's law can be interpreted in two different ways. One possibility is to treat it as a modification to the classical law of inertia (Newton's second law), so that the force on an object is not proportional to the particle's acceleration **a** but rather to μ(a/a_{0})**a**. In this case, the modified dynamics would apply not only to gravitational phenomena, but also those generated by other forces, for example electromagnetism.^{[10]} Alternatively, Milgrom's law can be viewed as leaving Newton's Second Law intact and instead modifying the inverse-square law of gravity, so that the true gravitational force on an object of mass m due to another of mass M is roughly of the form GMm/(μ(a/a_{0})r^{2}). In this interpretation, Milgrom's modification would apply exclusively to gravitational phenomena. [End of excerpt]

For **LCDM:** acceleration a: a = G{M_{BM}+m_{DM}}/R^{2}

**For MOND:** acceleration a: a+a_{mil} = a{a/a_{o}} = GM_{BM}/R^{2} = v^{4}/a_{o}.R^{2} for v^{4} = GM_{BM}a_{o} a_{mil }= a{a/a_{o}-1} = a{a-a_{o}}/a_{o} = GM_{BM}/R^{2} - a

For Newtonian acceleration a: G{M_{BM}+m_{DM}}/R^{2} = a = GM_{BM}/R^{2} - a_{mil}

a_{mil} = - Gm_{DM}/R^{2} = (a/a_{o})(a-a_{o}) and relating the Dark Matter to the Milgröm constant in interpolation a_{mil}

for the Milgröm deceleration applied to the Dark Matter and incorporating the radial independence of rotation velocities in the galactic structures as an additional acceleration term in the Newtonian gravitation as a function for the total mass of the galaxy and without DM in MOND.

Both, LCDM and MOND consider the Gravitational 'Constant' constant for all accelerations and vary either the mass content in LCDM or the acceleration in MOND in the Newtonian Gravitation formulation respectively. The standard gravitational parameter GM in a varying mass term G(M+m) = M(G+DG) reduces to Gm=DGM for a varying Gravitational parameter G in (G+DG) = f(G).

The Dark Matter term Gm_{DM} can be written as Gm_{DM}/R^{2} = -a_{mil} = a - a^{2}/a_{o} = DGM/R^{2} to identify the Milgröm acceleration constant as an intrinsic and universal deceleration related to the Dark Energy and the negative pressure term of the cosmological constant invoked to accommodate the apparent acceleration of the universal expansion (q_{dS }= -0.5585).

DG = G_{o}-G(n) in a_{mil} = -2cH_{o}/[n+1]^{3} = {G_{o}-G(n)}M/R^{2} for some function G(n) descriptive for the change in f(G).

The Milgröm constant so is not constant, but emerges as the initial boundary condition in the Instanton aka the Quantum Big Bang and is identfied as the parametric deceleration parameter in Friedmann's solutions to Einstein's Field Equations in a_{mil}.a_{o} = a(a-a_{o}) and a_{o}(a_{mil} + a) = a^{2} or a_{o} = a^{2}/(a_{mil}+a).

A(n)= -2cH_{o}/[n+1]^{3} = -2cH_{o}^{2}/R_{H}[n+1]^{3} and calculates as -1.112663583x10^{-9} (m/s^{2})* at the Instanton and as -1.16189184x10^{-10} (m/s^{2})* for the present time coordinate.

The Gravitational Constant G(n)=G_{o}X^{n} in the standard gravitational parameter represents a finestructure in conjunction with a subscale quantum mass evolution for a proto nucleon mass m_{c} = alpha^{9}.m_{Planck} from the gravitational interaction finestructure constant a_{g} = 2pG_{o}m_{c}^{2}/hc = 3.438304..x10^{-39} = alpha^{18} to unify electromagnetic and gravitational quantum interactions.

The proto nucleon mass m_{c}(n) so varies as complementary finestructure to the finestructure for G in m_{c}Y^{n} for a truly constant G_{o} as defined in the interaction unification. G(n)M(n)=G_{o}X^{n}.M_{o}Y^{n} = G_{o}M_{o}(XY)^{n }= G_{o}M_{o} in the macro evolution of baryonic mass seedling M_{o} and G_{o}m_{c} in the micro evolution of the nucleonic seed remain constant to describes a particular finestructure for the timeframe in the cosmogenesis when the nonluminous Dark Matter remains separate from the luminous Baryon mass.

The DM-BM intersection coordinate is calculated for a cycletime n=H_{o}t=1.4142..or at an universal true electromagnetic age of 23.872 billion years. At that time, the {BM-DM-DE} mass density distribution will be {5.536%; 22.005%; 72.459%}, with the G(n)M(n) assuming a constant value in the Hubble cycle. The Dark Energy pressure will be P_{PBM∩DM }= -3.9300x10^{-11} (N/m^{2})* with a corresponding 'quasi cosmological constant' of L_{BM∩DM }= -6.0969x10^{-37} (s^{-2})*.

Within a local inertial frame of measurement; the gravitational constant so becomes a function of the micro evolution of the proto nucleon mass m_{c} from the string epoch preceding the Instanton. A localized measurement of G so engages the value of the mass of a neutron as evolved m_{c} in a coupling to the evolution of the macro mass seedling M_{o} and so the baryonic omega W_{o}=M_{o}/M_{H} = 0.02803 in the critical density r_{critical }= 3H_{o}^{2}/8pG_{o} = 3M_{H}/4pR_{H}^{3} = 3c^{2}/8pG_{o}R_{H}^{2} for the zero curvature and a Minkowski flat cosmology.

The finestructure for G so engages both the micro mass m_{c} and the macro mass M_{o}, the latter being described in the overall Hypermass evolution of the universe as a Black Hole cosmology in a 5/11D AdS 'closed' spacetime encompassing the dS spacetime evolution of the 4/10D 'open' universe. Details are described in a later section of this discourse.

The Milgröm 'constant' so relates an intrinsic Dark Energy cosmology to the macrocosmic hypermass evolution of Black Holes at the cores of galaxies and becomes universally applicable in that context. No modification of Newtonian gravitation is necessary, if the value of a locally derived and measured G is allowed to increase to its string based (Planck-Stoney) value of G_{o}=1/k=4pe_{o} = 1.111..x10^{-10} string unification units [C*=m^{3}/s^{2}] and relating spacial volume to angular acceleration in gravitational parameter GM.

The necessity for Dark Matter to harmonise the hypermass evolution remains however, with the Dark Energy itself assuming the form of the Milgröm deceleration.

a_{mil} = -2cH_{o}/[n+1]^{3} = -{G_{o}-G(n)}M/R^{2} = -G_{o}{1-X^{n}}M/R^{2} for the gravitational parameter GM coupled to the size of a galactic structure harbouring a central Black Hole-White Hole/Quasar power source.

**G**

_{o}M/R^{2}= 2cH_{o}/{(1-X^{n})(n+1)^{3}}For a present n=1.13242 ......{(1-X^{n})(n+1)}^{3} = 4.073722736.. for M/R^{2} = constant = 2.48906

For the Milky Way barred spiral galaxy and a total BM+DM mass of 1.7x10^{42} kg, the mass distribution would infer a diameter of 1.6529x10^{21} m or 174,594 light years, inclusive the Dark Matter halo extension.

For the Andromeda barred spiral galaxy and a total BM+DM mass of 3x10^{42} kg, the galaxy's diameter would increase to 2.1957x10^{21} m or 231,930 light years for a total matter distribution.