Cosmology · The cosmological constant and other dark-energy models

Why the vacuum-energy interpretation remains difficult

Use a deliberately limited cutoff calculation to distinguish regulating a mode sum, choosing a finite energy and predicting a covariant vacuum stress.

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

Recognize the dependence of a regulated zero-point sum on its prescription.
Distinguish a finite energy adjustment from a full stress-tensor calculation.
Explain why fitting a constant contribution does not explain its microscopic origin.

A mathematical mode sum is not an observed density

Quantum field calculations can associate zero-point contributions with field modes. Summing these contributions requires a prescription for handling arbitrarily large wavevectors, called regularization.

Consider only one free massless scalar degree of freedom in flat spacetime. Impose a hard cutoff on spatial wavevector magnitude, |k| < K, where K has units of inverse metres. This is an intentionally limited calculation.

For this prescription, ε_reg = ℏcK⁴/(16π²). If K doubles, this regulated energy density increases by sixteen. Changing a mathematical cutoff is not observing a time-varying cosmic energy density.

Choose an arbitrary reference K_* and U_* = ℏcK_*⁴/(16π²). With q = K/K_*, the regulated density is ε_reg/U_* = q⁴. Neither K_* nor the values below are measured physical cutoff scales.

Holding one energy value fixed is a choice

Supply a formal adjustable energy term B/U_* = 1/10−q⁴. Adding it to ε_reg leaves the total energy density exactly U_*/10 for every q in the table.

At q = 1, the two contributions are 1 and −0.9 in U_* units. At q = 2, they are 16 and −15.9. The changing separate terms cancel by construction.

This arithmetic illustrates the difference between a regulator-dependent term and a chosen finite total. B is a bookkeeping parameter here; it is not a claim that negative-density material has been observed.

The finite value 1/10 was stipulated. Keeping it fixed does not explain why a physical theory should select that value, and does not implement a complete renormalized calculation.

Deliberately incomplete flat-space bookkeeping; q is a regulator ratio, not cosmic time or scale factor
q	ε_reg/U_*	B/U_*	(ε_reg+B)/U_*	P if P_B = −B, in U_* units	Required vacuum P/U_* for this chosen energy
1	1	−0.9	0.1	1.233333	−0.1
2	16	−15.9	0.1	21.233333	−0.1
3	81	−80.9	0.1	107.9	−0.1

The stress tensor exposes a missing step

A Lorentz-invariant vacuum in flat spacetime has a stress tensor proportional to the metric, giving P = −ε. Energy alone is therefore not the whole vacuum calculation.

The spatial-only cutoff used above breaks Lorentz invariance. Its regulated mode stress instead gives P_reg = ε_reg/3. This radiation-like regulated relation is a warning about the prescription; it is not the equation of state of a physical cosmological vacuum fluid.

Even if the energy adjustment B is assigned a Λ-form pressure P_B = −B, the resulting total pressure is P_total/U_* = 4q⁴/3−1/10. It is not the −1/10 required for the selected vacuum-like energy.

The mismatch is 4q⁴U_*/3: matching the energy alone has not restored the required stress symmetry. This failure is intentional in the example, rather than a proposed solution of the cosmological constant problem.

This original deliberately incomplete calculation contains one free massless scalar degree of freedom in flat Minkowski spacetime. A hard cutoff restricts spatial wavevector magnitude to |k| < K, in m⁻¹. q = K/K_* is a mathematical regulator ratio on the finite displayed domain 1 to 3. It is not cosmic time, redshift, the scale factor or an observed physical UV scale. The arbitrary positive reference U_* = ℏcK_*⁴/(16π²) has energy-density units J/m³. Panel A uses linear q and signed energy-density terms divided by U_* from −85 to 85. Solid blue circles show the regulated mode sum ε_reg/U_* = q⁴, giving 1, 16, 81 at q = 1, 2, 3. Dashed orange squares show formal adjustable B/U_* = 1/10−q⁴, giving −0.9, −15.9, −80.9. Dotted green diamonds show their chosen total energy density 0.1 at each q. That small positive total lies close to the explicitly drawn zero reference on this shared signed scale; it is written as 0.1 and is not zero. This formal B is not claimed to be observed negative physical material. Panel B has its own zero-based linear pressure-mismatch ordinate from 0 to 110, using the same q. A dash-dot violet square series shows [P_total−(−ε_total)]/U_* = 4q⁴/3, giving 4/3, 64/3 and 108. The mismatch is not a standard deviation, measurement uncertainty or real vacuum discrepancy. The spatial-only regulator breaks Lorentz invariance; its regulated stress has P_reg = ε_reg/3. A Lorentz-invariant vacuum would instead need P = −ε. Even assigning the bookkeeping term only a Lambda-form pressure P_B = −B leaves P_total/U_* = 4q⁴/3−1/10, not the required −1/10 for the chosen total energy. At q = 2 the total pressure is 637/30, approximately 21.233333 in U_* units. Matching one energy entry has intentionally not restored the full stress symmetry. Distinct signs, direct formulas, shapes and dash patterns distinguish all series without color. The one-half zero-point factor, angular mode density and radial integral yield the quartic coefficient, but this illustration is not covariant renormalization, Wilsonian matching or an interacting/curved-space calculation. No fitted Lambda, numerical physical vacuum discrepancy, microscopic solution or cosmological fluid evolution is inferred. Original Ark teaching model.

Optional mathematical path: count the modes

For one massless degree of freedom, each wavevector contributes zero-point energy ℏc|k|/2. In three dimensions the mode density is d³k/(2π)³, so integrating over a sphere supplies the angular factor 4π.

The remaining radial integral is ∫ from 0 to K of k³ dk = K⁴/4. Combining the one-half zero-point factor and the mode-counting factors gives ε_reg = ℏcK⁴/(16π²).

An isotropic spatial mode stress gives one-third of that regulated energy density as pressure. The simultaneous energy-plus-pressure combination is ε_reg+P_reg = 4ε_reg/3.

A Λ-form counterterm has B+P_B = 0 and therefore cannot cancel that combination by itself. A symmetry-consistent treatment must address the whole stress tensor, not only the scalar energy entry.

State what a fuller theory must do

Regularization makes intermediate expressions manageable using a specified prescription. Renormalization relates parameters and calculated quantities so that physical predictions do not depend on arbitrary regulator choices.

Fitting a finite parameter is different from predicting its physical value or explaining its sensitivity when an effective description is changed. Those questions require more than a cancellation ledger.

The example does not perform covariant renormalization, include interacting fields or curved-spacetime effects, calculate a real numerical vacuum discrepancy, or select a proposed microscopic solution.

Do not insert this regulated energy-pressure pair into a cosmological conservation equation as an evolving fluid. Its purpose is to show why identifying a fitted Λ-like term with a microscopic vacuum contribution demands a careful, complete theory.

Worked reasoning

Change q from 1 to 2

Has a measured cosmological constant increased by sixteen?

The regulated energy term increases from U_* to 16U_*.
The stipulated B changes from −0.9U_* to −15.9U_*, leaving the chosen total 0.1U_*.
With only P_B = −B, the total pressure becomes about 21.233333U_*, rather than the vacuum value −0.1U_*.

No: the cutoff and bookkeeping changed, and this incomplete calculation does not predict a physical vacuum stress.

Try the reasoning

Does q label cosmic time or the scale factor in this example?
Show the answer and reasoning
No: q = K/K_* labels a mathematical spatial cutoff.
The calculation is in flat spacetime and does not supply cosmic evolution.
Why does P_reg = ε_reg/3 matter?
Show the answer and reasoning
It differs from the Lorentz-invariant vacuum relation P = −ε.
The spatial-only regulator breaks the symmetry needed for that vacuum stress.
Does choosing the finite total 0.1U_* explain that value?
Show the answer and reasoning
No.
The value is imposed as an input; neither a measurement nor a dynamical prediction selected it.

What to carry forward

A regulated mode sum is prescription dependent and is not an observed vacuum density.
A vacuum calculation must treat pressure and energy consistently with symmetry.
A finite parameter adjustment does not by itself explain a microscopic cosmological constant.

References

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