The decrease in the Sun's size is a creationist belief that is supposed to disprove the age of the universe as stated by science. It started when two astronomers, J. A. Eddy and A. A. Boornazian, published―as a side note―their deduction that the Sun was shrinking at a rate of 0.1- 0.2% per century. Eddy and Boornazian had studied British Royal Observatory data recorded since 1750, in order to announce the finding in 1979.
Full of sound and fury ... Signifying nothing!The timeline of the life of a star is so long that any trend observed during the tiny period of human civilization is much more likely to be fluke rather than a permanent change.
What made creationists pounce on the study was the mistaken assumption that the Sun had always been shrinking at that rate. Like all creationist theories, the creationist interpretation of the flawed study is filled with deliberately incomplete research, leaving wide gaps between the strictly defined margins of the science in it. In fact, vital tenets of creationism itself have also been completely ignored in their desperate attempt to discredit the Darwinian theory of evolution.
To understand the misunderstanding about Eddy and Boornazian's study, we need to look at some basic principles of the theory of the contraction of the Sun, and the development of man's understanding about the Sun.
Before nuclear fusion was discovered, a scientist named Hermann Helmholtz had proposed a theory for the luminescence of the Sun. He stated that the Sun had been larger in the past, and was slowly collapsing into itself due to its own gravity, releasing gravitational energy in the form of light and heat. Under the conditions of the scientific development at the time, this theory was indeed admirably logical. However, when nuclear reactions were discovered, nuclear fusion was estimated to be the source of solar energy, since it provided a much better and more complete explanation of solar phenomena than mere gravitational contraction.
Even before Helmholtz, a popular theory was that the Sun was simply a burning mass of fossil fuels! According to the scientific limitations of the time, it was indeed a believable theory, however ridiculous it may seem today. Creation science has, thankfully, developed enough to see that this theory is clearly ridiculous, but it clearly has some way to go before it understands why Helmholtz's theory had to be discarded.
When Eddy and Boornazian's findings were made public, creationists equated the findings with Helmholtz's theory (which was a big mistake). If the rate of shrinkage observed by Eddy and Boornazian had been constant for the entire history of the Sun, it would have been much larger in the past than it is now. Life on Earth would have had to form inside (or very close to) the Sun according to the evolutionary timescale. Thus, the creationists concurred that the evolutionary timescale had been conclusively disproved, and a much more recent conception of life on Earth had to be the actual eventuality.
Even today, creationists maintain the stance that Helmholtz's estimation about the Sun's energy source "was discarded because it did not provide the vast time span demanded by the theory of organic evolution" (quoted from the website of the Institute of Creation Research). Since Helmholtz's theory (which had countless loopholes) supposedly rendered evolution improbable, it was supposedly discarded by scientists in favor of nuclear fusion, which fit the evolutionary timescale.
The creationists shot themselves in the foot by needlessly mixing up Helmholtz's proposition and the 1979 findings. The rate of shrinkage proposed by Eddy and Boornazian is much higher―more than quadruple―than the rate expected from a Helmholtz contraction. Since the creationists have to claim that the rate remained constant throughout the Sun's history, they simply turn a blind-eye towards this anomaly.
There is no obligation that the rate observed by Eddy and Boornazian had to have remained constant throughout the Sun's stellar evolution; this was a very creative assumption by the creationist camp.
While it is true that the evolutionary timescale would have to be severely reformed if it was proven that the Sun had indeed been shrinking its entire life, the creationists would have to settle for a different version of the creation of the world than the one provided in their scriptures. But since the word of God is infallible, no creation scientist mentions this inconvenient truth.
Their research was later found to have been faulty, and was discredited by scientists. But it did help establish that the Sun goes through a 76-year cycle of fluctuations in size. Most stars in the universe increase or decrease in size regularly. It is a part of a cycle, and should only be viewed as such. All Eddy and Boornazian discovered was that the Sun had shrunk by that rate during 1750 and 1979. No other study done before or after their research found any continuous change in the Sun's size.
The main difference between science and religious pseudo-science is that one constantly updates and corrects itself, and the other hasn't changed ideologically since it tried to silence Copernicus and Galileo. Scientists of Helmholtz's time had no doubt opposed the theory at first, before accepting it for want of a more sophisticated alternative. Even Darwin's theory of evolution was opposed by prominent contemporaneous scientists such as Lord Kelvin, the eponym of the Kelvin scale of temperature. Since the scientific knowledge of the time was severely limited (relative to modern times), scientists in those days were simply doing their job by opposing the new theories until sufficient evidence had been gathered for the same. Once nuclear fusion was discovered, scientists realized that it explained solar phenomena that couldn't be explained by Helmholtz's theory of gravitational contraction. Thus, nuclear fusion replaced Helmholtz's theory in the scientific world, while creationists, resistant to change, have not stopped citing a flawed study and a flawed theory. The shrinking Sun is treated by scientists in nearly the same vein as Noah's Ark and the great global flood, and that's exactly where the theory belongs.