How to convert units when calculating a dimensionless quantity?

[Safinaz]

Homework Statement

For instance consider calculating this dimensionless quantity:

$P= \frac{1}{H^4~~ (GeV)^4} \times \left(\frac{1}{k_0 ~~~(Mpc)^{-1}}\right)^{0.2} ~~(1)$

Relevant Equations

Where H and k are constants. How to convert or choose units to make $P$ dimensionless.
where Mpc$^{-1} = 6.6 \times 10^{-39}$ GeV.

The original quantity is given in this paper: [reference][1], equations: (31-33-34), where $a(\eta)= \frac{1}{H\eta}$, so I considered in (1) only the constants which share by dimensions to $P$.

Any help is appreciated!


[1]: https://arxiv.org/pdf/hep-th/0703290

 

[anuttarasammyak]

where Mpc$^{-1} = 6.6 \times 10^{-39}$ GeV.

I am intereted in how do you get this relation ? Mpc has dimension of length L. GeV has dimensitn of energy ML^2T^-2.

 

[RiScJ]

I am intereted in how do you get this relation ? Mpc has dimension of length L. GeV has dimensitn of energy ML^2T^-2.

This is in a system of natural units where $\hbar=c=1$.

 

[Orodruin]

To add to that, in natural units length and time both have dimensions of inverse energy. It is incredibly common to use such units particularly in high-energy physics, relativity, and related fields.

 

[anuttarasammyak]

(30) and (31) seem to suggest that P(k) has same dimension with k^5. Is it OK? I have not found your (1) in the paper. Where is it ?