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The cosmological constant and the minimum length-scale

In a previous post, we showed how, if the ultimate fate of our Universe is space empty of matter...but not quite....of energy (a de Sitter space), then this future cosmic event horizon (CEH) radius of our current, quantum, Universe set a natural maximum length-scale. Amazingly, the future CEH radius also defines the cosmological constant $\Lambda$.

  

For an observer at O inside the cosmic event horizon (CEH) with radius $l_{\Lambda}$, the universe can be divided into two sub-vacuums, $(A)$ inside the CEH, and $(B)$, outside. The horizon surface $\Sigma$ has entanglement entropy $S_{dS}$ and rest energy $E_H$

What about a Universal minimum length-scale? Might that also be related to a de Sitter space? After all, in a semi-classical analysis, the Planck length $L_p$ plays the role of a minimal length. 

For a long time, it has been observed that the evolution of our Universe can be considered as two asymptotic de Sitter epochs connected by a transition phase parameterized by the brief moment of matter-radiation equality.  We are presently living in the second epoch - our accelerating Universe is quasi-de Sitter. 

Perhaps then, the most natural state of our Universe is dS space, as Sean Carroll posited. The past dS state is during and up to the end of inflation. Although then, by logical extension (i.e. we exist!) it also implies that dS space is highly unstable, giving common ground with Swampland, i.e. in string theory it is basically impossible to construct a meta-stable de Sitter theory. 

This implies that the Universe before inflation would be the actual initial past boundary condition, with the dS-like start of inflation being a later phase change. A key takeaway here is that during and up during and up to the end of inflation, the proper distance $l_{\Lambda}$ to the CEH stays constant (i.e. a dS-like state), while the proper distance between two points increases exponentially.  I know, this is a little difficult to get your head around. What it means is that inflation is not superluminal expansion.

As Lineweaver explains, during inflation (a period of around 60 e-folds) all the energy is in the inflaton which has very few degrees of freedom and low entropy. Inflation ends with a period of reheating, during which the inflation's energy is transferred into a relativistic fluid. This is also known as the Hot Big Bang. After reheating (in $\Lambda$CDM) during radiation domination, the CEH is approximately constant (the CEH proper radius increases as $l_{\Lambda} \propto a$ ),  and in the DE dominated future, the CEH is a constant proper radius. Here $a$ the cosmic scale factor (refer Figure below, also from Lineweaver).  

Alright then, but what about the minimal length-scale? Well, there is an argument (derived from the holographic principle), that the entire universe must be contained within the past horizon of a so-called eternal observer. 

Now, there are multiple lines of evidence that at Planckian scales spacetime behaves effectively two‑dimensional, so the natural “cell” for information is a 2‑sphere of radius $L_p$ (linear size $l_\Lambda=2L_p$, i.e., its diameter). The unit of information is then the area of that sphere in Planck units:\begin{equation}
    I_c \;=\; \frac{A(L_{P})}{L_{P}^{2}} \;=\; \frac{4\pi L_{P}^{2}}{L_{P}^{2}} \;=\; 4\pi.
\end{equation} This means that accessible cosmic information $I_c=4\pi$ is the (properly normalised) number of modes that cross the Hubble radius during inflation.  

So, why $l_\Lambda = 2 L_P$? Well, its the Gauss-Bonnet theorem for a 2-sphere:    \begin{equation}
    \int_{M} K \, dA = 4\pi \chi(S^{2}) = 4\pi
\end{equation} where $\chi(S^{2})=2$ is the Euler characteristic. This means physically, the minimum information content $I_c$ is topologically protected against deformation, it can't be continuously varied to zero.   

We can see this in six different ways!

1. Compton–Schwarzschild Minimal Black Hole

A particle of mass (m) has a Compton wavelength
$
\lambda_C = \frac{\hbar}{m c},
$
while a Schwarzschild black hole of the same mass has radius
$
r_s = \frac{2 G m}{c^2}.
$

If we demand that a black hole’s horizon cannot be smaller than the Planck scale,
$
r_s \geq 2 L_P,
$
then the minimal mass is
$
m_{\rm min} = \frac{c^2 L_P}{G} = M_P,
$
where $(M_P)$ is the Planck mass.

The minimal Schwarzschild radius is therefore
$
r_s = 2 L_P,
$
and the corresponding Bekenstein–Hawking entropy is
$
S_{\rm BH}^{\rm min} = \frac{A}{4 L_P^2} = 4 \pi.
$

This route is a first-principles derivation consistent with both quantum mechanics and general relativity.

2. Generalized Uncertainty Principle (GUP)

String-theory-inspired GUP models modify the Heisenberg relation, leading to a minimal length
$
\Delta x_{\rm min} = 2 \sqrt{\alpha'} = 2 L_P.
$

So, the minimal UV cutoff is $(l_\Lambda = 2 L_P)$.

3. Gauss–Bonnet / Black Hole Crossover

  • The topological (pre-geometric) entropy of a 2-sphere is
    $
    S_{\rm GB}(S^2) = 4 \pi
    $
    —completely metric-independent.

  • Black hole geometric entropy is
    $
    S_{\rm BH}(r) = \frac{\pi r^2}{L_P^2}.
    $

Setting them equal gives
$
4 \pi = \frac{\pi r^2}{L_P^2} \quad \Rightarrow \quad r = 2 L_P.
$

Another derivation, based on emergent geometry.

4. Zero-Point Length / (q)-Metric

Pre-geometric approaches often introduce a modified distance measure:
$
\tilde \sigma = \sigma + 4 L_P^2 \quad \Rightarrow \quad \tilde \sigma_{\rm min}^{1/2} = 2 L_P.
$

Confirms $(l_\Lambda = 2 L_P)$  from zero-point considerations.

5. Covariant Entropy Bound (CEB)

For the minimal causal patch, the covariant entropy bound is saturated:
$
S_{\rm CEB} = 4 \pi \quad \Rightarrow \quad A = 16 \pi L_P^2 \quad \Rightarrow \quad r = 2 L_P.
$

Once again!

6. Kerr / Schwarzschild Symmetry Argument

  • A maximally rotating Planck-mass Kerr black hole has
    $
    r_\pm = r_s/2 = L_P, \quad S = 2\pi, \quad T_H = 0.
    $

  • A non-rotating Schwarzschild black hole has
    $
    r_s = 2 L_P, \quad S = 4 \pi, \quad T_H = \frac{1}{8 \pi} M_P.
    $

Physical argument: minimal pre-geometric patches are spherically symmetric, favoring the Schwarzschild radius:
$
l_\Lambda = 2 L_P.
$

However, this is not quite all. We must ask how these minimal informational bounds dictate the physical limits of mass, acceleration, and force.

Boehmer and Harko proved that in classical GR, the presence of a positive cosmological constant $\Lambda$ sets a minimum (local) mass bound for a stable, gravitationally bound system of radius $r$:

$$m(r) \geqslant \frac{\Lambda c^{2}}{12G} r^{3}$$

Let us define the macroscopic Cosmic Event Horizon (CEH) radius as $R_{IR} = \sqrt{3/\Lambda}$. If we evaluate this mass bound at the macroscopic limit ($r = R_{IR}$), we find the fundamental mass bound of the observable universe:

$$M_{IR} = \frac{3 c^2}{12 G R_{IR}^2} R_{IR}^3 = \frac{c^{2}}{4G} R_{IR}$$

Notice that $M_{IR}$ is exactly half the total mass of the de Sitter horizon ($M_{CEH} = c^2 R_{IR} / 2G$).

But what happens at the local quantum scale? We cannot simply plug the Planck length into an equation driven by the macroscopic $\Lambda$ without breaking classical GR. Instead, we must invoke the UV/IR mixing inherent to the holographic principle. The physics of the macroscopic boundary ($R_{IR}$) is dual to the microscopic minimal cell ($l_{UV} = 2L_P$). They share the same geometric scaling duality, $m/r \propto c^2/G$.

By mapping the Boehmer-Harko macroscopic scaling down to our minimal topological cell $l_{UV} = 2L_P$, we find the semi-classical minimum local mass $m_s$:

$$m_{s} = \frac{c^{2}}{4G} l_{UV} = \frac{c^{2}}{4G} (2L_{P}) = \frac{M_{P}}{2}$$

This perfectly aligns with our earlier constraint. The energy of reheating must have escaped its initial self-gravity, meaning a black hole horizon could not have been present. Writing the Schwarzschild radius $R \geqslant 2GM/c^2$ and substituting our minimal mass $M = M_P/2$, we get $R = L_P$. The semi-classical minimum length scale fundamentally requires the Planck length.

Now that we have established the minimal mass $m_s$, we can determine the maximum local Universal acceleration $a_{UV}$ at this minimal scale. Just as the macroscopic horizon has an acceleration $H = c/R_{IR}$, the microscopic horizon has an acceleration bounded by the minimal length $l_{UV}$:

$$a_{UV} = \frac{c^2}{l_{UV}} = \frac{c^2}{2L_P} = \frac{a_{Planck}}{2}$$

This maximum local acceleration is of the exact same form as the critical acceleration of an electron subject to the Schwinger field in QED, hinting at a deep connection between spacetime geometry and quantum vacuum stability.

Finally, how do these bounds interact? To find the maximum force between two bodies in General Relativity, we multiply the minimal invariant mass $m_s$ by the maximum local acceleration $a_{UV}$. Watch how the length scales cancel entirely:

$$F_{max} = m_{s} \cdot a_{UV} = \left( \frac{c^2}{4G} l_{UV} \right) \left( \frac{c^2}{l_{UV}} \right) = \frac{c^4}{4G}$$

This is the exact value Barrow and Gibbons conjectured for maximum tension in GR. Because the length scale cancels out completely, this proves that the Barrow-Gibbons limit is a scale-invariant topological bound. It is the invariant local limit on observable force, valid identically at the boundary of the observable universe and at the boundary of a single Planckian cell.

 

 


 


 

 

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