# Lambda as a unit of length

1. Jul 29, 2008

### stevebd1

Apparently, one way to understand the effects of Λ on space-time is to apply the following equation which provides the invariant cosmological length-scale (while Planck length is the invariant quantum length-scale that replaces zero), length-scale being a measure of the space-time sheet-

$$\Lambda =\frac{1}{L^2}$$

where L is 'the genuine physical meaning of the cosmological constant Λ, a maximal, impassable scale, invariant under dilations, that replaces infinity from the view point of its physical properties'.
Source- 'Scale-Relativistic Cosmology' by Laurent Nottale http://luth2.obspm.fr/~luthier/nottale/NewCosUniv.pdf

The equation can be re-written-

$$L=\sqrt{\frac{1}{\Lambda}}$$

For a cosmological constant of 1.252x10^-52 m^-2, this works out at 8.937x10^25 metres which equals 9.447x10^9 Lys (9.45 billion Lys).

What is this length exactly? Is it suppose to relate to the cosmic light horizon (13.7 billion Lys) and 9.45 billion years is considered acceptable because it's within 1 order or is it a length in it's own right, and if this is considered the maximum length for a space-time sheet, what exactly becomes variable at this distance? If the Planck length is considered the quantum length-scale and anything smaller than this is quantum foam, what is supposed to happen at the cosmic length scale?

regards
Steve

Last edited by a moderator: Apr 23, 2017
2. Aug 1, 2008

### stevebd1

On page six of the above paper, 'Scale-Relativistic Cosmology' by Laurent Nottale, the following statement appears-

'..This limit corresponds to the Planck length-time scale towards very small scales and to the cosmic scale IL toward very large scales. Therefore in the special scale-relativity framework, these two scales can neither be reached nor crossed (i.e. nothing actually exists beyond them), and they replace from the viewpoint of their physical properties respectively the zero and the infinite.' (page 6)

(the paper uses a figure of 1.362x10^-52 m^-2 for Λ giving a corresponding figure of ~9 billion light years for IL)

This idea of a space-time sheet having a maximum length of 9-10 billion light years seems to be at odds with the cosmic light horizon at 13.7 billion years and the corresponding comoving radius of 46.2 billion light years. Is it an alternative theory or something that works in conjunction with what we observe? Is it assumed that if 9-10 billion light years is the cosmic physical limit, light from objects outside this limit can still cross this boundary to reach use eventually? Would the cosmological length-scale be described as a radius or a length/diameter with us being in the middle of the space-time sheet?

Steve

Last edited: Aug 1, 2008
3. Aug 16, 2008

### stevebd1

While there is plenty of evidence of the Planck scale (lP) being used in quantum and astrophysics, is there a significant use for the cosmic scale (IL) in equations other than IL $=\sqrt{1/\Lambda}$ or is it simply just 'a number'? Does curvature become significant at IL and should be taken into account when calculating distances, is there any other evidence that supports this cosmic scale of ~9 billion lightyears (say, in the CMB)?

Steve

Last edited: Aug 16, 2008