From definition to evolution: the kinematics of Ψ(z)
Interpretational layer · no new models · direct consequence of definitions
This note shows that the evolution of Ψ(z) follows directly from its definition and the standard kinematics of FLRW cosmology. No additional assumptions are introduced.
1. Starting point
Define the diagnostic quantity:
\[\Psi(z) = \frac{H(z)}{H_\Lambda(z)} - 1,\]where:
- (H(z)) is the observed expansion rate,
- (H_\Lambda(z)) is the reference ΛCDM expansion history.
This definition is purely operational.
2. Change of variable
It is convenient to use:
\[x = \ln a,\]so that derivatives describe evolution per logarithmic change of scale factor.
3. Evolution of H
In FLRW cosmology, the deceleration parameter is:
\[q = -\frac{\ddot a}{a H^2}.\]From this definition, one obtains the identity:
\[\frac{dH}{dx} = -H(1+q).\]The same holds for the reference model:
\[\frac{dH_\Lambda}{dx} = -H_\Lambda(1+q_\Lambda).\]These relations are purely kinematical and follow from GR.
4. Differentiating Ψ
Differentiate:
\[\Psi = \frac{H}{H_\Lambda} - 1.\]Using the quotient rule:
\[\frac{d\Psi}{dx} = \frac{1}{H_\Lambda}\frac{dH}{dx} - \frac{H}{H_\Lambda^2}\frac{dH_\Lambda}{dx}.\]Substitute the expressions from Section 3:
\[\frac{d\Psi}{dx} = \frac{-H(1+q)}{H_\Lambda} - \frac{H}{H_\Lambda^2}\left[-H_\Lambda(1+q_\Lambda)\right].\]5. Simplification
Simplify step by step:
\[\frac{d\Psi}{dx} = -\frac{H}{H_\Lambda}(1+q) + \frac{H}{H_\Lambda}(1+q_\Lambda).\]Factor:
\[\frac{d\Psi}{dx} = \frac{H}{H_\Lambda}(q_\Lambda - q).\]Using:
\[\frac{H}{H_\Lambda} = 1+\Psi,\]we obtain:
\[\boxed{ \frac{d\Psi}{dx} = (1+\Psi)(q_\Lambda - q). }\]6. Interpretation
This equation is not a model.
It is an identity that follows from:
- the definition of Ψ,
- the standard FLRW relations.
It shows that:
- Ψ evolves only if the real and reference deceleration differ,
- Ψ = 0 corresponds to exact agreement with ΛCDM,
- deviations accumulate through differences in (q).
7. Geometric form
Using:
\[K = -3H,\]the same relation can be written as:
\[\Psi = \frac{K}{K_\Lambda} - 1,\]which shows that Ψ measures relative deformation of the background geometry.
8. Minimal conclusion
The evolution of Ψ(z):
- does not require new physics,
- does not introduce new parameters,
- follows directly from how we compare two expansion histories.
9. Diagnostic meaning
The equation
\[\frac{d\Psi}{d\ln a} = (1+\Psi)(q_\Lambda - q)\]provides a closed, observable-level description of deviation.
It shifts the question from:
what causes acceleration?
to:
how does the observed trajectory differ from the reference one?
10. Scope
This result:
- is exact at the background level,
- is independent of any interpretation (relaxation, memory, etc.),
- serves as a stable foundation for further conceptual layers.
Conclusion
Ψ(z) is not an assumed field.
It is a derived coordinate whose evolution is fully determined by observable kinematics within standard GR.
No additional structure is required to obtain its dynamics.