About inserting function in front of scale factor after solving it from Friedmann equation without cosmological constant

This is just an idea to improve Friedmann equation without cosmological constant. Instead adding a constant to the left sides of the Friedmann equations (or adding cosmological constant into Einstein's field equation) , the scale factor is multiplied by a function L(t) AFTER it is first solved from the Friedmann's first equation without lambda.

$$ (\frac{\dot{a}}{a})^2 = \frac{8 \pi G}{3}(\frac{\rho_{m,0}}{a^3}+\frac{\rho_{r,0}}{a^4}) $$

$$ a_{r}= a \cdot L(t) $$

$$ L(t_0) = 1 $$

$$ a(t_0) = 1 $$

,where the new scale factor a_r is the real,observed scale factor. The function L(t) is unknown but may be exponential, assuming that whatever the phenomenon is responsible of the function L – it is proportionally same in every infinitesimal time interval:

$$ L(t) = e^{k(t-t_0)} $$

$$ L(t_0) = 1 $$

,where k is a constant.

The actual phenomenon that causes equation 2 and function L(t) is unknown. Why it is added in front of scale factor after solving it - It is something that happens to space in every moment ”in all cases” - hence scale factor can be multiplied by L(t)

Some mathematics:

$$ \dot{a_{r}} = \dot{a} \cdot L + a \cdot \dot{L} $$

$$ \frac{\dot{a_{r}}}{a_r} = \frac{\dot{a}}{a} + \frac{\dot{L}}{L} $$

Keeping L exponential the Hubble parameter becomes:

$$ H_{r}(t) = \frac{\dot{a_{r}}}{a_r} = \frac{\dot{a}}{a} + k = H(t) + k $$

The following may not be useful - combining the first and the second equation the combined Friedmann's first equation becomes

$$ (\frac{\dot{a_{r}}}{a_r} - \frac{\dot{L}}{L})^2 = \frac{8 \pi G}{3}[\frac{\rho_{r,0}}{a_{r}^4}L^{4} + \frac{\rho_{m,0}}{a_{r}^3}L^{3}] $$

But it is more convenient to write it in form of two equations:

$$ (\frac{\dot{a}}{a})^2 = \frac{8 \pi G}{3}(\frac{\rho_{m,0}}{a^3}+\frac{\rho_{r,0}}{a^4}) $$

$$ a_{r}= a \cdot L(t) $$

$$ L(t) = e^{k(t-t_0)} $$

$$ L(t_0) = 1 $$

$$ a(t_0) = 1 $$

I could solve this numerically and plot it and compare it to the solution of Lambda-CDM model or even to observations (measured values for scale factor a). I could also try to find best fit function L(t). But i am not very familiar with python plotting. I ddn't find the data either. If i was, i probably wouldn't write here.

The essential problem is of course that is there any phenomenon that could cause this function L(t) ?

Edited November 6Nov 6 by caracal
mistake in formula