SpectralIndex

The term "spectral index" originated in the field of astronomy and astrophysics and is used to describe how the intensity of a signal (like radiation) changes with frequency or wavelength. The term is borrowed from spectroscopy where "spectral" refers to the breakdown of light or other electromagnetic radiation into its component frequencies. An "index" is a number that quantifies some property, in this case, the dependence of the signal on frequency.

In the context of the cosmic microwave background (CMB) and cosmology, the spectral index is a measure of the primordial power spectrum's dependence on the scale of fluctuations. In the simplest models of cosmic inflation, quantum fluctuations in the early universe are stretched to cosmological scales, resulting in a nearly scale-invariant distribution of primordial density perturbations.

This distribution can be characterized by a power spectrum, which is typically a power-law function of the wavenumber $k$, representing different scales of the universe. The spectral index $n_s$ is the exponent in this power-law relationship. Mathematically, this is expressed as:

$$P(k) \sim A_s \left(\frac{k}{k_{\text{pivot}}}\right)^{n_s - 1}$$

where:

• $P(k)$ is the power spectrum as a function of the wavenumber $k$.
• $A_s$ is the amplitude of the fluctuations.
• $k$ is the wavenumber corresponding to different scales in the universe.
• $k_{\text{pivot}}$ is a pivot scale chosen for reference.
• $n_s$ is the spectral index.

A spectral index of $n_s = 1$ corresponds to a scale-invariant spectrum, meaning the amplitude of the fluctuations is the same at all scales. Observations of the CMB by the Planck satellite and other experiments have found that $n_s$ is slightly less than 1, indicating a slight preference for larger scales (smaller $k$), which is a key prediction of cosmic inflation.

The term "spectral index" is also used in other areas of astrophysics with similar meanings. For instance, in radio astronomy, the spectral index defines how the flux density of an astronomical source changes with frequency.