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How does one input move both results?

In the stated early universe scenario, N measures an interval of expansion. Move it to see the pattern tilt rise while the relative gravitational wave strength falls.

Step 1 of 7

Start with a stated early universe scenario.

The leading R + R² branch uses N★ for an amount of expansion. This illustration starts at 55; the registered range is 50 to 60. Reheating and the branch assumptions supply that choice.

1. The stated inflation branch and external N★. Choose N★ = 55 within the declared range. One shared expansion input. 2. N★ = 55. Compute 1 − 2/N★. nₛ ≈ 0.963636. 3. The same N★ = 55. Compute 12/N★². r ≈ 0.003967. 4. The two equations with the same N★. Substitute 1 − nₛ = 2/N★. r = 3(1 − nₛ)². 5. The separate mass branch and c₃. Compute c₃ to the power 7/2. Mscal/M̄ ≈ 1.256494e-5. 6. The separate vacuum branch, alpha and reference scale. Evaluate the exponential and prefactor. ρΛ/M̄⁴ ≈ 7.12533e-121. 7. The mass ratio and vacuum density ratio. Take the density’s fourth root; compare dimensionless energies. Mass/reference and vacuum energy/reference.
Arrows carry the named inputs into later calculations. Added inputs stay marked. Reduced motion is on. Use Next step or choose a stage below.
Carry forward
The stated inflation branch and external N★
Added at this step
R + R² branch and the external expansion input
Operation
Choose N★ = 55 within the declared range
Result
One shared expansion input
50≤N⋆≤60,N⋆=55 in this example\begin{gathered}50\le N_\star\le60,\\[.4em]N_\star=55\ \text{in this example}\end{gathered}

In words: Choose N★ = 55 within the declared range

Why this step works and what it assumes

This control does not derive the reheating history or select the external expansion interval. It follows the published leading branch.

Read the original calculation
See where this fits in the whole theory
Try the individual controls

Try this. Move N star from fifty to sixty. Compare how the two displayed results change together.

One choice changes two results

The inflation branch uses a supplied value called N. Increase it and the pattern tilt rises while the relative wave strength falls.

Pattern tilt

0.9636360.96000.9667

nₛ = 1 − 2 / N

Relative wave strength

0.0039670.00330.0048

r = 12 / N²

The same input appears in both formulas. The wave readout r is wave power divided by density-fluctuation power. N depends on the stated inflation and reheating history; this control does not derive that external choice. Animation shows formula dependence, rather than time passing in the early universe.

Choose the expansion input

Set N star between fifty and sixty. It describes the amount of expansion used by the stated early Universe branch.

Read the density result

The first number describes how the power of early density fluctuations changes with size.

Read the wave result

The second compares gravitational wave power with density fluctuation power. Both numbers must use the same N star.

Explore the full interactive calculation

One input moves two observables.

Move along the leading R² branch. Each point has one tilt and one tensor ratio.

The leading relations are nₛ = 1 − 2/N★ and r = 12/N★². The source branches also specify a scalaron mass and a vacuum energy density relative to the same reference scale.
Spectral tilt nₛ0.963636
Tensor ratio r0.003967

The frozen band uses N★ from 50 to 60. Reheating selects its physical location; the slider explores that external choice.

ns=1−2N⋆,r=12N⋆2n_s=1-\frac{2}{N_\star},\qquad r=\frac{12}{N_\star^2}

In words: Spectral tilt is one minus two over N star; the tensor ratio is twelve over N star squared.

r=3(1−ns)2r=3(1-n_s)^2

In words: Eliminating N star puts both observables on the same curve: r equals three times one minus n s squared.

How this fits into the whole theory

Both results use the same amount of expansion. One calculated curve fixes their positions together.

What this experiment establishes

These readouts belong to the stated R + R² and vacuum branches. N star and reheating remain external inputs, while physical gravity and dimensional scales retain their source assumptions and calibration.