Our best understanding of the universe is rooted in a cosmological model known as LCDM. The CDM stands for cold dark matter, where most of the matter in the universe isn't stars and planets, but a strange form of matter that is dark and nearly invisible. The L, or lambda, represents dark energy. It is the symbol used in the equations of general relativity to describe the Hubble parameter, or the rate of cosmic expansion. Although the LCDM model matches our observations incredibly well, it isn't perfect. And the more data we gather on the early universe, the less perfect it seems to be.
A central difficulty is the fact that increasingly, our various measures of the Hubble parameter aren't lining up. For example, if we use fluctuations in the cosmic microwave background to calculate the parameter, we get a value of about 68 km/s per megaparsec. If we look at distant supernova to measure it, we get a value of around 73 km/s per megaparsec.
Several studies have looked at this, drawing upon a range of sky surveys. Determining the age of stars andis particularly difficult, so the resulting data is a bit fuzzy. While it's possible to fit the data to the range of Hubble parameters we have from direct measures, the age-distance data suggests the universe is a bit older than the LCDM allows. In other words, IF the age data is truly accurate, there is a discrepancy between cosmic age and stellar ages.
Another approach uses gravitational lensing. If a nearby galaxy happens to be between us and a distant supernova, the light from the supernova can be gravitationally lensed around the galaxy, creating multiple images of the supernova. Since the light of each image travels a different path, each image takes a different amount of time to reach us. When we are lucky we can see the supernova multiple times.
When we look at fluctuations within the CMB, most of them are on a scale predicted by the LCDM model, but there are some larger scale fluctuations that are not, which we call anomalies. Most of these anomalies can be accounted for by the integrated Sachs-Wolfe effect. How this pertains to cosmic inflation is that since most of the ISW happens in the early period of the universe, it puts limits on how much you can tweak dark energy to address the tension problem.
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