quantity

scalar spectral index

scalar spectral index — from formula anatomy
16 related articles · 3 formulas

Description

The scalar spectral index (often denoted n_s) is a number that characterizes how the density fluctuations in the early universe vary with scale. In cosmology, it appears in the power spectrum of primordial perturbations, which describes the seeds of cosmic structure. A value of n_s = 1 corresponds to a perfectly scale-invariant spectrum (Harrison–Zel'dovich), while deviations from 1 indicate a slight tilt.

History

The concept was introduced in the context of inflationary cosmology in the early 1980s, with key contributions by Mukhanov, Chibisov, and others. The Harrison–Zel'dovich spectrum was proposed earlier, in the late 1960s and early 1970s.

How it works

The power spectrum of density fluctuations is often approximated as P(k) ∝ k^(n_s - 1), where k is the wavenumber (inverse of scale). If n_s < 1, there is slightly more power on large scales; if n_s > 1, more on small scales. Measurements from the cosmic microwave background and large-scale structure constrain n_s to about 0.96, meaning the universe is nearly but not exactly scale-invariant.

In practice

The scalar spectral index is a key observable for testing models of cosmic inflation and distinguishing between competing theories of the early universe.

Fun fact

The measured value n_s ≈ 0.96 is one of the strongest pieces of evidence that the early universe underwent a period of accelerated expansion (inflation), as simple inflation models naturally produce a slight red tilt.

Concept graph →

Formulas

$$n_s = 1 - \frac{2}{N}$$
Spectral index deviation from scale invariance depends on e-folds.
What is in the formula
n_svariableScalar spectral index — scalar spectral index
NvariableNumber of e-folds of inflation — e folds
$$n_s - 1 = \frac{d\ln\mathcal{P}_\mathcal{R}}{d\ln k}$$
The scalar spectral index measures the deviation from scale invariance of primordial perturbations.
What is in the formula
n_svariablescalar spectral index — scalar spectral index
P_Rvariablepower spectrum of curvature perturbations — power spectrum
kvariablewavenumber — wavenumber reciprocal metre
d/d ln koperatorderivative with respect to natural logarithm of wavenumber — logarithmic derivative
$$n_s - 1 = -2\epsilon - \eta, \quad r = 16\epsilon$$
Relate slow-roll parameters to observable CMB power spectra.
What is in the formula
n_svariableScalar spectral index — scalar spectral index
\epsilonvariableFirst slow-roll parameter — slow roll parameter
\etavariableSecond slow-roll parameter — slow roll parameter second
rvariableTensor-to-scalar ratio — tensor to scalar ratio

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