The Talmud Yerushalmi teaches, "Anyone who sleeps on the Rosh Hashanah – his fortune will sleep." On the basis of this passage, it has become customary not to sleep during the day on Rosh Hashanah, as this could lead one’s good fortune to "sleep," Heaven forbid, during the coming year.
It would appear at first glance that in order to avoid this undesirable consequence of sleeping on Rosh Hashanah, we need to wake up right at the crack of dawn, which is usually sometime after 5am. After all, we need to avoid sleeping during the day, and the day starts at dawn, and it should thus be necessary to wake up already at dawn. Indeed, this is the view taken by several authorities, including the Ben Ish Hai (Parashat Nisavim, 11) and Kaf Ha’haim Sofer (584:37).
Hacham Ovadia Yosef, however, disagreed, and maintained that especially if somebody is up late on the night of Rosh Hashanah studying Torah or reciting Tehillim, he does not have to awaken at dawn. This was also the view taken by Rav Shlomo Zalman Auerbach (Jerusalem, 1910-1995), who explained that going to sleep during the day is different from remaining asleep after the day begins. When one goes to sleep during the day of Rosh Hashanah, he shows his disregard for Rosh Hashanah and that he is not concerned about the judgment on this day. But if he was sleeping during the night and remained asleep past dawn, as he normally does, this does not reflect any disregard or lack of concern. Therefore, it is not necessary to wake up at the crack of dawn on Rosh Hashanah.
The question arises, however, as to whether one may go back to sleep if he happened to wake up at dawn. Those who have been waking up early for Selihot during the weeks before Rosh Hashanah might wake up at or shortly after dawn on Rosh Hashanah morning, as they had been accustomed to doing, and then wish to go back to sleep. Seemingly, this should be forbidden, as suggested by a parallel case relevant to the laws of Sukkot. When rain falls on Sukkot, one is permitted to sleep indoors and does not have to sleep in the Sukka. If the rain stops during the night, he nevertheless does not have to go back outside into the Sukka, but if he wakes up after dawn and sees that the rain has stopped, and he wishes to go back to sleep, then he must sleep in the Sukka. Seemingly, we should apply this rule to Rosh Hashanah, as well, and if one wakes up at or after dawn, he should not be allowed to go back to sleep.
In truth, however, Hacham Ovadia maintains that even on Sukkot, one does not have to sleep in the Sukka after dawn if the rain has stopped. In his work Hazon Ovadia – Sukkot (p. 207), he writes that the Halacha requiring one to go out into the Sukka in such a case applied only in times when people would normally arise at dawn. Nowadays, however, when people normally sleep past dawn, one who wakes up after dawn on Sukkot and wishes to go back to sleep may sleep inside, even though the rain has stopped. Similarly, one may go back to sleep on Rosh Hashanah morning after dawn, if it is still earlier than the time when people normally wake up.
Needless to say, there is a general rule of "Zerizin Makdimin La’misvot," which means that we should always try to perform Misvot as early as possible. Thus, it would certainly be praiseworthy for one who can to wake up already at dawn on Rosh Hashanah. Strictly speaking, however, this is not necessary, and as long as one wakes up at a normal time in the morning, he does not have to fear that his good fortune will "sleep" during the coming year.
Summary: Although it is customary not to sleep during the day on Rosh Hashanah, one does not have to wake up already at dawn, and may sleep until the time when people normally wake up.