The age of the Universe as published by NASA is 13.77 billion years old plus or minus errors of a few million years. While Genesis recorded the time from the creation of the Universe until Adam and Eve to be six days. The problem exists in reading the literal day in English as 24 hours. The original Hebrew word "yom" translated into English as "day" can be use to mean a long period of time. This is actually the position held by most Hebrew scholar, it is perfectly fine as it is with six "yom", equivalent to 13+ billion years.
There is actually no need to bring in the expanding space-time to fit in the 24-hour day interpretation. I will not even attempt to translate the different rates of expansion to fit in a 24-hour day.
However, what i found interesting is that the "yom" in Genesis 1 are marked by evening and morning, and if you allow for the passage of time to be different based on the observer reference point, we can actually see a pattern of light and darkness rather easily across the first 3 days. This should not be interpreted as contradiction as the passage of time can be different for different observers. These are all provided for by Einstein Special-Relativity. There are text book written about Special Relativity, but a good article to read online is wikipedia's Introduction to Special Relativity
On the other hand, to claim that the Universe and Earth is only 6000 years old is not necessary. Most older annotated Bible will place the creation date to about 4000BC, these additional "notes" are usually printed on the side column, or as reading guides at the beginning or introduction to the Bible, and are not actually part of the original Bible, these are extra information that the publishers had added, thinking it would be helpful at that time, but created more confusion now.