# Cosmic massive graviton bounds

In the paper, Relativistic theory of gravitation and the graviton rest mass. Theoretical and Mathematical Physics, 65(1), 971–979. doi:10.1007/bf01028629, the mass bound

$$m_g<6.4\cdot 10^{-66}g=6.4\cdot 10^{-69}kg$$

is obtained. However, putting the numbers with $$\rho=10^{-29}g/cm^3=10^{-26}kg/m^3$$ I get $$m_g< 6.4\cdot 10^{-60}kg=6.4\cdot 10^{-57}g$$, with $$k=8\pi G_N$$. Where am I missing the power counting?

• This? inis.iaea.org/collection/NCLCollectionStore/_Public/17/084/… Is an English language version available? Apr 1 at 14:12
• Yes: link.springer.com/article/10.1007/BF01028629 @JamesK Apr 1 at 15:30
• @JamesK Even if taking into account the values in the russian version, if you see the English variant seems wrong as well...I am puzzled because I am pretty sure I did the conversion factors right...and that is why I ask here...$\hbar=1.05\cdot 10^{-33}J/Hz$ and $c=3\cdot 10^8m/s$, from the English variant of the article you obtain $\sqrt{2\kappa\rho}\hbar/ c$. Oh, wait...I think I can see it now...Let me try a thing... Apr 1 at 15:34
• Python's astropy.units can help you converting between units.
– pela
Apr 3 at 11:33

I think I lost a $$c$$ factor...

I try

$$m<\sqrt{2\kappa\rho}\dfrac{\hbar}{c^2}=10^{-34}\dfrac{\sqrt{16\pi \cdot 6.674\cdot 10^{-11}\cdot 10^{-28}}}{9\cdot 10^{16}}=0.64\cdot 10^{-69}kg=0.64\cdot 10^{-66}g$$

But yet, I am missing a 10 factor...Not a big thing like before...But...As well: The translated into English version seems to be bad in the units of $$\rho_0$$ and the final numbers, but also the Russian version! Or I am doing something wrong... Am I correct now or have I lost a 10 somewhere here?

Remark: the density should be $$\rho_0=10^{-26}kg/m^3=10^{-29}g/cm^3\sim 6protons/m^3$$ to get the result of the article.

Dimensional check:

$$[m]=[\sqrt{2\kappa\rho}\dfrac{\hbar}{c^2}]=M=\dfrac{ML^2T^{-1}\sqrt{L^3T^{-2}M^{-1}ML^{-3}}}{L^2T^{-2}}$$

• This does not seem like an answer, but something you should have edited into your original question. Apr 2 at 1:05
• @StephenG-HelpUkraine I think it IS an answer, since it remarks that the likely missing factor is in the mass/energy density (the number seems wrong by different reasons in both, the Russian and the English papers) Apr 3 at 17:36