κατὰ πρόγνωσιν θεοῦ πατρὸς ἐν ἁγιασμῷ πνεύματος εἰς ὑπακοὴν καὶ ῥαντισμὸν αἵματος Ἰησοῦ Χριστοῦ, χάρις ὑμῖν καὶ εἰρήνη πληθυνθείη.I connect κατὰ πρόγνωσιν θεοῦ πατρὸς with ἐκλεκτοῖς in v. 1, and understand foreknowledge as God's active, distinguishing love set on persons, not his passive awareness of events. So sovereign election is in line with (κατὰ) the Father's distinguishing love.
Note then that this sovereign-grace election is in connection with the sanctifying work of the Holy Spirit (ἐν ἁγιασμῷ πνεύματος), setting the elect apart from the mass of humanity to God's ownership and service.
This sanctification has a twofold effect or result (εἰς 2X), the first of which is ὑπακοὴν. This ὑπακοή refers to listening-from-under, listening submissively and responsively. I don't think the object of the submission is expressed, but can be inferred from the other two uses in 1 Peter, both of which are in this chapter:
1:14 ὡς τέκνα ὑπακοῆς μὴ συσχηματιζόμενοι ταῖς πρότερον ἐν τῇ ἀγνοίᾳ ὑμῶν ἐπιθυμίαις
1:22 Τὰς ψυχὰς ὑμῶν ἡγνικότες ἐν τῇ ὑπακοῇ τῆς ἀληθείας εἰς φιλαδελφίαν ἀνυπόκριτον, ἐκ [καθαρᾶς] καρδίας ἀλλήλους ἀγαπήσατε ἐκτενῶς
It is submission to the Gospel, which is to say saving faith. See similar uses of the noun also in Peter's man Paul, in Romans 1:5; 16:26. See also the similar uses of the verb ὑπακούω in Acts 6:7; Romans 6:17; 10:16; 2 Thessalonians 1:8; Hebrews 5:9. It isn't the obedience that should characterize the Christian life that is in view, but the submission to the Gospel that begins and characterizes the Christian life.
The second effect of the sanctifying work of the Holy Spirit is ῥαντισμὸν αἵματος Ἰησοῦ Χριστοῦ. This is an obvious allusion to OT sacrifical imagery. I take it that this means the personal application of the benefits of Christ's redemption, purchased on the Cross by the shedding of His blood.
In eternity past the Father sets His distinguishing love on us in sovereign electionGlorious passage.
This involves the Spirit setting us apart to God, with two results:
—The first is our submission to the Gospel
—The second is God applying Christ's blood to us personally