MOND is unnecessary

TL;DR

This paper demonstrates that within the standard LambdaCDM model, the holographic principle can explain phenomena typically attributed to MOND, rendering the modified gravity hypothesis unnecessary.

Contribution

The paper shows that the HLSS model, using the holographic principle, accounts for MOND-related phenomena within the LambdaCDM paradigm, challenging the need for modified gravity theories.

Findings

HLSS model explains MOND acceleration

Baryonic Tully-Fisher relation is accounted for

Mass discrepancy-acceleration relation is explained

Abstract

Dark matter seems to account for flat velocity curves in spiral galaxies, with further evidence for dark matter from observations of the colliding "bullet cluster" galaxies 1E0657-56. However, the baryonic Tully-Fisher relation and the mass discrepancy-acceleration relation have been cited (arXiv:1112.3960) as "challenges for the LambdaCDM model." MOND (MOdified Newtonian Dynamics), a modified law of gravity,is invoked in arXiv:1112.3960 to account for those relations. This note shows that the HLSS model in arXiv:1301.0304, employing the holographic principle within the standard LambdaCDM paradigm, readily accounts for the MOND acceleration, the baryonic Tully-Fisher relation, and the mass discrepancy-acceleration relation. After first posting this note, I learned that Man Ho Chan (arXiv:1310.6801) previously reached the same conclusion using a dark matter based analysis independent of the holographic approach used in this paper. These results indicate that the MOND hypothesis is unnecessary.

Equations

$$\frac{G}{R_{\gamma }^2}\left(\frac{R_{\gamma }}{R_S}\right)M_{DM}=\frac{G{M}_B}{R_{\gamma }^2}.$$

$$a_S=\frac{G{M}_{DM}}{R_S^2}+\frac{G{M}_B}{R_S^2}=\frac{G}{R_S^2}(M_{DM}+M_B).$$

$${\left(\frac{V_{observed}}{V_{Newtonian}}\right)}^2\approx \frac{R(a_{DM}+a_B)}{R{a}_b}$$

$$\frac{V_{observed}}{V_{Newtonian}}=\sqrt{1+\frac{a_{DM}}{a_B}}=\sqrt{1+\frac{R{M}_{DM}}{R_S{M}_B}}.$$

$$\frac{a_0}{a_B}=\frac{M_{DM}}{0.19M_B}(\frac{R}{R_S}{)}^2$$

$$\frac{R}{R_S}=\sqrt{\frac{0.19M_B}{M_{DM}}}\sqrt{\frac{a_0}{a_B}}$$

$$\frac{V_{observed}}{V_{Newtonian}}=\sqrt{1+\frac{R{M}_{DM}}{R_S{M}_B}}=\sqrt{1+\sqrt{\frac{0.19M_{DM}}{M_B}}\sqrt{\frac{a_o}{a_B}}}.$$

$$\frac{V_{observed}}{V_{Newtonian}}=\sqrt{1+0.998\sqrt{\frac{a_o}{a_B}}}\approx {\left(\frac{a_0}{a_B}\right)}^{\frac{1}{4}}$$

$$V_{observed}^4={\left(\frac{G{M}_B}{R^2}\right)}^2\left(\frac{R^2}{G{M}_B}\right)a_0=G{M}_B{a}_0,$$

Full text

This note shows that the HLSS model in arXiv:1301.0304, employing the holographic principle within the standard $\Lambda$CDM paradigm, readily accounts for both the MOND acceleration and* the (Vobserved*/VNewtonian) relation. And, after first posting this note, I learned that Man Ho Chan (arXiv:1310.6801) previously reached the same conclusion using a dark matter based analysis independent of the holographic approach used in this paper. These results indicate that the MOND hypothesis is unnecessary.

T.R. Mongan

84 Marin Avenue, Sausalito, California 94965, USA; tmongan@gmail.com

Dark matter seems to account for flat velocity curves in spiral galaxies, and observations of the colliding “bullet cluster” galaxies 1E0657-56 provide further evidence for dark matter. However, the baryonic Tully-Fisher relation and the mass discrepancy-acceleration (or Vobserved/VNewtonian) relation have been cited [1] as “challenges for the $\Lambda CDM$ model.” Ref. 1 invokes MOND (MOdified Newtonian Dynamics), a modified law of gravity involving an acceleration threshold $a_{0}\approx 1.2\times 10^{-8}$cm/sec2, to account for* those relations. In contrast, the results below show that the holographic large scale structure (HLSS) model [2] developed within the $\Lambda$CDM paradigm and employing the holographic principle based on thermodynamics and general relativity [3], accounts for the MOND acceleration threshold, the baryonic Tully-Fisher relation, and the mass discrepancy-acceleration (Vobseerved*/VNewtonian) relation.

In the holographic large scale structure (HLSS) model [2], galaxies with total mass $M_{g}$ inhabit spherical holographic screens with radius $R_{S}=\sqrt{\frac{M_{g}}{0.183g/cm^{2}}}$ if the Hubble constant $H_{0}=$ 67.8 km sec*-1*Mpc${}^{-1}.$ The HLSS model considers galactic matter density distributions $\rho(r)=\frac{M_{g}}{4\pi R_{S}r^{2}}$, where $r$ is the distance from the galactic center. The spherical isothermal halo of dark matter, with radius $R_{S}$ and mass $M_{DM}=0.84M_{g}$, has density distribution $\rho_{DM}(r)=\frac{M_{DM}}{4\pi R_{S}r^{2}}$ so the dark matter mass within radius $R$ is $\frac{R}{R_{S}}M_{DM}.$ There is no singularity in the galactic matter density distribution $\rho(r)=\frac{M_{g}}{4\pi R_{S}r^{2}}$ because mass inside a core volume of radius $R_{c}$ at the galactic center is concentrated in a central black hole with mass $M_{CBH}=\frac{R_{c}}{R_{S}}M_{g}$ [2]. Radial acceleration at radius $R$ due to dark matter is then a_{DM}=\frac{G}{R^{2}}($$\frac{R}{R_{s}})$M_{DM}$. At radii $R$ sufficiently distant from the galactic center that total baryonic mass of the galaxy $M_{B}=0.16M_{g}$ can be treated as concentrated at the galactic center, Newtonian radial acceleration resulting from baryonic matter is $a_{B}=\frac{GM_{B}}{R^{2}}.$ The radius $R_{\gamma}$ where $a_{DM}=a_{B}$ is found from

[TABLE]

Since $M_{DM}=0.84M_{g}$ and $M_{B}=0.16M_{g}$, $R_{\gamma}=0.19R_{S}$, and at that radius $a_{DM}=a_{B}=a_{0}=5.4\times 10^{-8}$cm/sec2, consistent with the MOND estimate $a_{0}\approx 1.2\times 10^{-8}$cm/sec2.

Another indication that the MOND acceleration $a_{0}\approx 1.2\times 10^{-8}$cm/sec2 is a natural scale in the dark matter based HLSS model involves the situation at the radius $R_{S}$ of the spherical holographic screen. Then the Newtonian assumption that total galactic mass can be treated as concentrated at the galactic center is certainly justified. There, the sum of radial acceleration from dark matter and from baryonic matter is

[TABLE]

But $M_{DM}+M_{B}=M_{g}=0.183R_{S}^{2}$, so $a_{S}=0.183G=1.2\times 10^{-8}$cm/sec2, equal to the estimated MOND acceleration $a_{0}\approx 1.2\times 10^{-8}$cm/sec2.

Turning to the (Vobserved/VNewtonian) relation, tangential velocity $V$ at radius $R$ is related to radial acceleration $a_{r}$ by $V^{2}=Ra_{r}$. So, the ratio (Vobserved/VNewtonian) is approximately

[TABLE]

resulting in

[TABLE]

Then, when $R\ll 0.19R_{s}$, $\frac{V_{observed}}{V_{Newtonian}}\approx 1$. Next, using

[TABLE]

and

[TABLE]

yields

[TABLE]

When $a_{B}\ll a_{0},$ $\frac{a_{o}}{a_{B}}\gg 1$ and

[TABLE]

Since $\left(\frac{V_{observed}}{V_{Newtonian}}\right)^{4}=\frac{a_{0}}{a_{B}}$ when $a_{B}\ll a_{0}$, using $a_{B}=\frac{GM_{B}}{R^{2}}$ and $V_{Newtonian}=\sqrt{\frac{GM_{B}}{R^{2}}}$ gives

[TABLE]

also known as the baryonic Tully-Fisher relation.

Finally, with Hubble constant $H_{0}=$ 67.8 km sec*-1Mpc-1*, the cosmological constant $\Lambda=1.12\times 10^{-56}$cm*-2*, and the accelerations $cH_{0}=6.6\times 10^{-8}$cm/sec2 and $c^{2}\sqrt{\frac{\Lambda}{3}}=5.5\times 10^{-8}$cm/sec2 are both consistent with the acceleration $a_{0}=5.4\times 10^{-8}$cm/sec2 estimated above.

In conclusion, MOND is not needed to account for the baryonic Tully-Fisher relation and the mass discrepancy-acceleration (or Vobserved/VNewtonian) relation discussed in Ref. 1. After first posting this note, I learned that Man Ho Chan previously reached the same conclusion [4] using a dark matter based analysis independent of the holographic approach used in this paper.