CRI (Color Rendering Index)
CRI (Color Rendering Index) tells us how well a light source does at accurately reproducing colors in an object relative to a natural or black body radiation source (like the sun, incandescent bulb). It really falls flat, though, and a different standard has come out called TM-30. It doesn't actually replace CRI because they are standards from two different organizations, the CIE (International Commission on Illumination) for CRI, and ANSI/IES (American National Standards Institute/Illuminating Engineering Society) for TM-30.
The CIE 1960 UCS Color Space
CRI plays a larger role in lux to PPFD (umol/m2/sec) conversions than color temperature. Higher CRI lighting will have a greater amount of deeper reds, which naturally have a lower luminous flux at the same radiant flux (because it takes into account the sensitivity of our eyes by wavelength). In other words, the deeper reds have a lower luminous efficiency. You can see the differences in my spectroradiometer SPD charts. A white LED that is 100% efficient that draws one watt of power (one joule per second) will output about 320 lumens of light at CRI 80. An LED with a CRI of 80 that outputs 200 lumens per watt will have an efficiency of 200/320= 63%. But an LED with a CRI of 100 that output 200 lumens per watt will have an efficiency of 200/280 = 71%.
Fidelity Index (Rf) is used with TM-30 measurements and is sort of like CRI (0-100 scale with higher being better, but CRI can also have a negative number), but there's 99 color evaluation samples with a wide range of hue (base color), chroma (amount of saturation), and lightness. It is the average amount of 'color smearing' in the 99 color samples, or the average of how far off one is from the color samples. An ultra high CRI bulb can have a TM-30-15 Rf of 94, and around 60 should be the minimum for indoor lighting (higher for living areas). The US Dept of Energy has a great tutorial on TM-30. Gamut Index (Gf) with TM-30 ranges from 80-120 and is basically the amount of saturation with 100 being a neutral saturation. It is the color gamut area. Lower Gf white lights will make objects appear duller with higher Gf having colors more saturated.