Marital relations on Rosh Hashanah:
One is to avoid marital relations on both nights of Rosh Hashanah, in order to not enter into a state of Keri.
Mikveh Night: If the 1st or 2nd night of Rosh Hashanah is the night of Mikvah then one is obligated to have intercourse that night. [One who refrains from doing so is considered as if he is performing a Mitzvah as the result of an Aveira. One is, however, to diminish as much as possible in the duration of the intimacy. If a woman delayed going to Mikveh prior to Rosh Hashanah, some Poskim rule that she is not to immerse on the night of Rosh Hashanah. However, many Poskim rule that she is to immerse.]
Strong desire: Those who feel a strong desire, which may prove difficult to withhold, are rather to have intercourse than come to do a sin, and it is even a Mitzvah to do so in such a case. [The same applies if one has a constant erection, or has gotten the thoughts of women in his mind.]
Peru Urevu: Some Poskim rule that if one has yet to fulfill the Mitzvah of Peru Urevu, and his wife is able to conceive, then he may have intimacy on the above nights. From other Poskim, however, it is implied that even in such a case one is to be stringent.
Immersing in a Mikveh the next morning: A man who had marital relations on the night of Rosh Hashanah, such as in the event of Mikveh night or a strong desire, is to immerse in a Mikveh the next morning prior to Shacharis. [In the event that one knows he will be unable to immerse in a Mikveh the next morning, he is nevertheless obligated in marital relations if his wife went to Mikveh that night.]
Three days before Rosh Hashanah: Three days prior to Rosh Hashanah, the Chazan must remove himself from all matters of impurity including marital relations.
 M”A 581:16; Sha’ar Hakavanos p. 90; M”B 240:7; 581:26; based on Seder Hayom; Sh’lah p. 213 brought in Kaf Hachaim 581:86; Kaf Hachaim 240:5
 Sh’lah p. 213 brought in Kaf Hachaim 581:86; Nevertheless, one is to do so with fear and trepidation and diminish in his physical enjoyment. [ibid; Beir Heiytiv 581:16]
 Chayeh Adam 139:7; Shulchan Hatahor 1:241-2;
 K’neses Hagedola brought in M”A 581:15; Elya Raba 581:38; Chayeh Adam 139:7; M”E 581:5 [in contradiction of 581:53-see Kitzei Hamteh ibid 98]; Sheyikadesh Atzmo 8:6; So rule whenever a woman has pushed off her Mikveh to the night of Yom Tov: Shut Chut Hashani, brought in Toras Hashelamim 197:3
 The reason: As it is not considered a Tevilas Mitzvah [ibid] and therefore due to the holiness of Rosh Hashanah it is not allowed. [So is implied from above Poskim that it is not prohibited due to the prohibition of immersing on Shabbos but rather due to the holiness of Rosh Hashanah.]
 Peri Chadash 581; Birkeiy Yosef 18; Sha’areiy Teshuvah 581:16; Machatzis Hashekel 581:15 [that so is implied from Sh’lah]; M”E 581:53 [in contradiction of 581:5-see Kitzei Hamteh ibid 98]; Kaf Hachaim 581:87
Why is this not forbidden due to the prohibition of a woman immersing on Shabbos if she could have immersed beforehand? The custom to prohibit a woman to immerse on Shabbos if she could have done so beforehand [see Admur 326:8; Rama Yoreh Deah 197:2] does not apply on Yom Tov. [Toras Hashelamim 197:3; Sidrei Taharah 197:2; Chochmas Adam 118:8; Lechem Vesimla 197:10; Levushei Mordechai 93; Sha’areiy Tohar 17:4; Divrei Yatziv 104; Nitei Gavriel 31:13]
 Siddur Sh’lah; M”B 240:7 in name of Chochmas Adam; 581:26; Ben Ish Chaiy Vayeira 2:23; Taharas Yisrael 240:11; Kaf Hachaim 240:5; Piskeiy Teshuvos 240:10; See Seder Hayom brought in Kaf Hachaim 581:86; Machazik Bracha 581:4; Ben Ish Chai Netzavim 9; Chayeh Adam 139:7; Kaf Hachaim 581:88; See Sheyikadesh Atzmo 8:7
 See Chayeh Adam 139:7; Sheyikadesh Atzmo 8:8
 Biur Halacha 240:1 “Mileil Shabbos” in name of Yeshuos Yaakov; Piskeiy Teshuvos 240:10; See Sheyikadesh Atzmo 8:4 footnote 5
 See K’neses Hagedola 581; Setimas Haposkim
 Seder Hayom, brought in Sh’lah p. 213; Chayeh Adam 139:7; M”B 240:7 in name of Chochmas Adam; M”B 581:26; Kaf Hachaim 581:86; Sheyikadesh Atzmo 8:8
 Kerem Shlomo, brought in Kaf Hachaim 581:89; Chayeh Adam 139:7
 Mateh Efraim 581:48; Alef Lamagen 581:99