In this framework, the following assumptions are made:

• The scalar field is characterized by local fluctuations over a finite volume $V$.

• The average value of the field is zero, i.e. $\langle \varphi \rangle =0$; the field oscillates such that in some regions it is positive while in others it is negative.

• The field interacts weakly with matter and light, permitting the use of linear approximations for its fluctuations.

• The field exhibits an exponential decay profile, $\varphi \left(r\right)={\varphi }_0{\text{e}}^{-mr}$, on macroscopic scales.

• Quantum fluctuations $\xi \left(x,t\right)$ are modeled as Gaussian noise with zero mean.

These assumptions facilitate the analytical derivation of key parameters while capturing the local, oscillatory behavior of the field.

The Planck length ${\ell }_p$ represents the smallest physically meaningful scale [1] :

${\ell }_p=\sqrt{\frac{\hslash G}{c^3}}$(1)

Numerically, this is approximately

${\ell }_p\approx 1.616\times {10}^{-35}\text{m}.$

The mass parameter $m$ sets the scale of the field’s spatial variation. Initially defined as

$m=\frac{1}{r_{\text{scale}}},$

we now confine this to a finite volume by writing

$m=\frac{1}{\lambda {\ell }_p},$

with $\lambda$ being a dimensionless parameter that adjusts the effective extent of the field.

The self-interaction parameter $\alpha$ quantifies the field’s nonlinearity:

$\alpha =\frac{m^2}{{\varphi }_0^2}.$

The scalar field modifies the effective speed of light via local interactions. Considering the local value and spatial gradient of the field, we define

$c_{\text{eff}}\left(r\right)=c_0\left(1+\kappa \left(r\right){\varphi }^2\left(r\right)\right).$

To reflect local fluctuations, the interaction parameter is given by

$\kappa \left(r\right)=\frac{c_{\text{eff}}\left(r\right)-c_0}{c_0\left[{\varphi }^2\left(r\right)+\frac{{\left(\nabla \varphi \right)}^2}{m^2}\right]}.$

Instead of assuming a uniform field, the energy is now computed by integrating the energy density over a finite volume $V$:

$E=\underset{V}{{\displaystyle \int }}\left(\frac{1}{2}{\left(\nabla \delta \varphi \right)}^2+\frac{1}{2}{m}^2\langle {\left(\delta \varphi \right)}^2\rangle +\frac{\alpha }{4}\langle {\left(\delta \varphi \right)}^4\rangle \right)\text{d}V,$(2)

where $\delta \varphi \left(x,t\right)$ represents the local fluctuation of the field, subject to $\langle \varphi \rangle =0$.

The characteristic amplitude ${\varphi }_0$ is initially derived via

${\varphi }_0=\frac{\hslash }{\lambda {\ell }_{p}c},$

but with the inclusion of local fluctuations (and $\langle \varphi \rangle =0$), ${\varphi }_0$ represents the scale of the oscillatory deviations rather than a constant background value.

Recalling from earlier (and from Planck’s original work [1] ),

${\ell }_p=\sqrt{\frac{\hslash G}{c^3}}\approx 1.616\times {10}^{-35}\text{m}.$

The mass parameter is

$m=\frac{1}{\lambda {\ell }_p}.$

For $\lambda ={10}^{10}$,

$m\approx 6.187\times {10}^{24}{\text{m}}^{-1}.$

The scalar field amplitude ${\varphi }_0$ is calculated as:

${\varphi }_0=\frac{\hslash }{\lambda {\ell }_{p}c}.$

Substituting known values:

• $\hslash =1.054\times {10}^{-34}\text{J}\cdot \text{s}$

• $c=3.0\times {10}^8\text{m}/\text{s}$

• ${\ell }_p=1.616\times {10}^{-35}\text{m}$

• $\lambda ={10}^{10}$

We get:

${\varphi }_0\approx 2.177\times {10}^{-18}\left(\text{dimensionless}\right).$

Table 1 summarizes the derived parameters and their numerical values.