Cosmology · Origins, primordial tests and the limits of knowledge

Singularities and the limits of classical extrapolation

Compare a curvature-divergent dust geometry with an incomplete expanding de Sitter chart to identify what a past boundary does and does not establish.

Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.

Back to the chapter

After reading, you can

Distinguish curvature divergence from geodesic incompleteness.
Calculate normalized past null-affine intervals in two stated geometries.
Interpret a classical extrapolation without selecting an unsupported origin scenario.

A classical boundary is a question about a model

The hot early history and the ultimate origin of that history are different scientific questions. Extending an effective classical description backward can identify where that description stops supplying a regular continuation.

A spacetime is geodesically incomplete when an inextendible geodesic within it has a finite interval of its natural parameter: proper time for a massive freely falling observer, or an affine parameter for a light ray.

Curvature divergence is a different diagnostic. A finite affine interval does not by itself require a diverging curvature scalar, and a chart or region can admit an extension to a larger regular spacetime.

Two explicitly supplied geometries make these distinctions calculable. Their times and densities are model inputs, not measurements of the age, earliest accessible epoch or absolute beginning of the observed Universe.

A flat dust model develops divergent curvature

First supply a spatially flat classical homogeneous dust solution with positive reference time t_* in seconds, t>0, a(t) = (t/t_*)^(2/3) and pressure P = 0. Normalize a(t_*) = 1.

Its expansion rate is H = 2/(3t). In general relativity, the mass-equivalent density is ρ = 1/(6πGt²) in kg/m³; the energy density is ρc².

For a flat FLRW metric, the curvature invariant K = R_abcd R^abcd is 12[(ä/a)²+H⁴]/c⁴. It has units m⁻⁴ and does not depend on the sign convention chosen for the Riemann tensor.

Here ä/a = −2/(9t²), giving K = 80/(27c⁴t⁴). Thus this exact classical dust extrapolation has divergent curvature as t approaches zero; dust alone is not a complete account of early thermal processes.

Dust quantities in declared reference units; every row belongs to the same t>0 model
t/t_*	a	Ht_*	6πGρt_*²	Kc⁴t_*⁴
1	1	2/3	1	80/27
1/2	0.629961	4/3	4	1280/27
1/4	0.396850	8/3	16	20480/27

Normalize the affine parameter before comparing intervals

For a radial null geodesic in a flat FLRW geometry, an affine parameter can be normalized so that dλ = [a(t)/a(t_f)]dt at a chosen reference time t_f. This convention makes dλ/dt = 1 at that reference.

Choose t_f = t_* for the dust model. Its past affine interval is Δλ = ∫₀ᵗ_* (t/t_*)^(2/3)dt = 3t_*/5, finite under the stated normalization.

A comoving dust observer has proper time equal to cosmic time, so its proper interval from t = 0 to t_* is also finite, equal to t_*. These are different kinds of parameter interval.

An affine parameter has a normalization freedom. The number 3t_*/5 is a result under the chosen convention, not an absolute physical light-travel distance or a universal parameter-independent age.

An expanding de Sitter chart supplies a counterexample

Now separately supply the flat expanding de Sitter chart a(t) = exp(t/t_*) for −∞<t<∞. Its H = 1/t_* is constant; a positive vacuum component has ρ = 3/(8πGt_*²) and P = −ρc².

Both ä/a and H² equal 1/t_*², so K = 24/(c⁴t_*⁴) remains finite and constant. There is no curvature divergence as the chart coordinate t goes to minus infinity.

Normalize at t_f = 0, where a = 1. The past null-affine interval is ∫₋∞⁰ exp(t/t_*)dt = t_*, even though the coordinate-time range is infinite.

This expanding chart is past null-incomplete and can be extended to the larger de Sitter spacetime. Its incomplete null rays do not establish a curvature singularity or the past incompleteness of the full extended geometry.

Different supplied geometries can have finite past null-affine intervals for different reasons
Geometry and reference	Past Δλ/t_*	Curvature K	Interpretation
Dust, t_f = t_*	3/5	80/(27c⁴t⁴)	Diverges at the classical t = 0 endpoint
Expanding de Sitter chart, t_f = 0	1	24/(c⁴t_*⁴)	Finite; chart extends to a larger spacetime

Two separate supplied spatially flat FLRW geometries use arbitrary positive reference time t_* in seconds and dimensionless a. Normalize the null-affine parameter by dλ = [a(t)/a(t_f)]dt and dλ/dt = 1 when a = 1; this chosen λ has time units and no absolute light-travel-distance meaning. Panel A shows dust a = (t/t_*)^(2/3) on its physical domain t > 0. The linear horizontal x = t/t_* axis runs from 0 to 1, but the solid blue curve is drawn only on 1/8 ≤ x ≤ 1; no regular t = 0 endpoint is drawn. The zero-based linear vertical a axis runs from 0 to 1.1. Hollow circles at x = 1/4,1/2,1 give a approximately 0.396850,0.629961,1. Dust H = 2/(3t), mass-equivalent density ρ = 1/(6πGt²) kg/m³ and P = 0 give K = R_abcd R^abcd = 80/(27c⁴t⁴), in metres to the minus four, diverging as t approaches zero. The full analytic past normalized affine interval at t_f = t_* is Δλ/t_* = integral from 0 to 1 of x^(2/3)dx = 3/5, not just the area in the finite plotted window. Panel B is a different geometry, the flat expanding de Sitter chart a = exp(t/t_*), defined for all real t. The dashed orange curve with hollow squares is shown only in the finite window −4 ≤ x ≤ 0; the earlier tail to minus infinity is not drawn. Its linear a ordinate is also 0 to 1.1; marked a at x = −4,−2,0 is approximately 0.018316,0.135335,1. H = 1/t_* and K = 24/(c⁴t_*⁴) are constant. At t_f = 0 the full analytic integral from minus infinity to 0 is Δλ/t_* = 1 despite infinite cosmic-coordinate time range. This expanding chart is null-incomplete and admits extension to larger regular de Sitter spacetime; its chart boundary does not prove divergent curvature or past incompleteness of the full extension. Curvature and normalized affine intervals answer different questions. Suitable averaged expansion along the specified congruence/null or noncomoving timelike trajectory is a conditional theorem assumption, not an unconditional first-event proof. No observed cosmic age, early thermal completion or selected origin model is supplied. Separate named panels, circle/solid versus square/dashed encodings, explicit drawn domains and printed exact intervals provide noncolor access. Original Ark teaching model.

Incompleteness does not choose a physical completion

The Borde–Guth–Vilenkin argument uses a suitable positive average expansion along a past-directed null or noncomoving timelike geodesic. Under its stated kinematical setup, that geodesic reaches a past boundary in a finite affine or proper interval.

The required average, congruence and geodesic matter. The argument is not an unconditional statement about every trajectory in every proposed cosmology, and its conclusion does not identify what physical description lies beyond the boundary.

A curvature-divergent extrapolation motivates studying the limits of classical assumptions. A regular extendible patch instead demonstrates why the region being described must be specified before interpreting incompleteness.

A successful origin proposal must provide its own consistent dynamics, domains and testable consequences. Neither of these calculations selects a quantum state, a bounce, an absolute first event or a unique explanation of the observed hot history.

Worked reasoning

Integrate the dust affine interval

What is the past null-affine interval when dλ/dt = 1 at t = t_*?

Set u = t/t_*, so Δλ/t_* = ∫₀¹ u^(2/3)du.
The antiderivative is (3/5)u^(5/3).
Evaluate at 1 and 0 to obtain Δλ = 3t_*/5.

The interval is finite, 3t_*/5, under the declared affine normalization.

Try the reasoning

How does dust K change when t is reduced from t_* to t_*/2?
Show the answer and reasoning
It increases by a factor of 16.
K is proportional to t⁻⁴ in this supplied geometry.
Does a finite past null-affine interval alone imply divergent curvature?
Show the answer and reasoning
No.
The flat expanding de Sitter chart has a finite past null-affine interval and constant finite K.
Does past incompleteness select a unique physical origin scenario?
Show the answer and reasoning
No.
A continuation or completion requires additional physical assumptions and independently testable predictions.

What to carry forward

Curvature divergence and geodesic incompleteness are distinct diagnostics.
Specify the spacetime region and parameter normalization before interpreting a boundary.
A classical limit identifies a question for further physics without choosing its answer.

References

Sources supporting the explanations, physical distinctions and examples in this lesson.