COOK, J., delivered the opinion of the court, in which BOGGS, C.J., joined.
MARTIN, J. (pp. 257-70), delivered a separate dissenting opinion.
OPINION
COOK, Circuit Judge.
Brian Keith Moore, a Kentucky prisoner under sentence of death, appeals the district court’s denial of his petition for a writ of habeas corpus. For the following reasons, we affirm the district court and deny the writ.
I. Background and Procedural History
In 1984, a Kentucky jury convicted Moore of the 1979 kidnaping, murder, and first-degree robbery of seventy-seven-year-old Virgil Harris. In the trial’s penalty phase, the jury determined Moore committed the murder during the commission of first-degree robbery, and sentenced him to death.1 The Kentucky Supreme Court affirmed the conviction and sentence, and the United States Supreme Court denied certiorari. Moore v. Commonwealth, 771 S.W.2d 34 (Ky.1988) (“Moore I”), cert. denied, 494 U.S. 1060, 110 S.Ct. 1536, 108 L.Ed.2d 774 (1990).
In 1990, Moore filed a motion to vacate his sentence under Ky. R.Crim. P. 11.42 in the state trial court, alleging ineffective assistance of trial counsel. While that motion was pending, he filed a motion under Ky. R. Civ. P. 60.02 for a new trial based upon newly discovered evidence. The trial court allowed Moore to present evidence on this second motion during the Rule 11.42 hearing. It denied both motions in January 1997. The Kentucky Supreme Court affirmed, and the United States Supreme Court denied certiorari. Moore v. Commonwealth, 983 S.W.2d 479 (Ky.1998) (“Moore II”), cert. denied, 528 U.S. 842, 120 S.Ct. 110, 145 L.Ed.2d 93 (1999).
Moore filed his petition for habeas corpus in the district court in November 1999, raising claims of ineffective assistance of counsel, trial errors, prosecutorial misconduct, and a due-process violation. He also claimed police obtained incriminating statements used against him at trial in violation of Miranda v. Arizona, 384 U.S. 436, 86 S.Ct. 1602, 16 L.Ed.2d 694 (1966).
The magistrate judge recommended denying Moore’s petition. He concluded that each of Moore’s ineffective-assistance claims was either procedurally defaulted or meritless; that each alleged trial error was meritless; that the prosecutorial-miscon-duct claims were procedurally defaulted; that the due-process claim was procedurally defaulted and meritless; and that the Miranda claims were procedurally defaulted. The district court reviewed the portions of the magistrate judge’s report and recommendation to which Moore had objected, adopted them, and denied Moore’s petition.
Moore then obtained a certificate of ap-pealability as to the following claims: (1) ineffective assistance of counsel based on [253]*253counsel’s failure to impeach a commonwealth witness, Doris Riddle; (2) ineffective assistance of counsel based on inadequate penalty-phase preparation; (3) error in restricting Moore’s contact with his attorneys during a lunch break; (4) error in using the same jury for both the guilt and sentencing phases of the trial; and (5) cumulative error.
II. Standard of Review
Under the Antiterrorism and Effective Death Penalty Act (“AEDPA”), we review a district court’s legal conclusions de novo and its factual findings for clear error. Hill v. Hofbauer, 337 F.3d 706, 710 (6th Cir.2003). We may not grant a writ of habeas corpus unless we conclude that the state court’s decision: (1) was contrary to, or involved an unreasonable application of, clearly established federal law; or (2) was based on an unreasonable determination of the facts. Armstrong v. Morgan, 372 F.3d 778, 781 (6th Cir.2004).
III. Ineffective Assistance of Counsel
A. Failure to Impeach
Moore argues that his trial counsel performed ineffectively by falling to impeach a commonwealth witness, Doris Riddle. Riddle, an employee at the driver’s license bureau, testified concerning the alibi of a second suspect, Kenneth Blair, whom Moore tried to target at trial as the real killer. Riddle’s testimony placed Blair at the license bureau in the same general time frame as the crime (as described by witnesses) and thus hurt Moore’s attempt to blame Blair. But Riddle had told police shortly after the murder that she did not know what time Blair came in to the license bureau — and Moore’s counsel failed to impeach Riddle with this earlier inconsistent statement.
To support a Sixth Amendment ineffective-assistance claim, a defendant (or petitioner) must show (1) that counsel’s performance was deficient, and (2) that the deficient performance prejudiced the defense. Strickland v. Washington, 466 U.S. 668, 687, 104 S.Ct. 2052, 80 L.Ed.2d 674 (1984). To show prejudice, the defendant must show a reasonable probability that, but for counsel’s deficient performance, the proceeding’s result would have been different. Id. at 694, 104 S.Ct. 2052.
The Kentucky Supreme Court rejected Moore’s claim in state post-conviction proceedings, finding that, although Moore’s counsel was deficient for failing to impeach Riddle, that deficiency did not prejudice Moore, and therefore was not ineffective assistance under Strickland. See Moore II, 983 S.W.2d at 482-84. The Kentucky court gave three reasons for its lack-of-prejudice finding: (1) even considering Riddle’s testimony, Blair’s alibi was not air-tight — he could have committed the murder and still been at the license bureau during Riddle’s time-frame; (2) other witnesses confirmed Blair’s alibi; and (3) abundant evidence, including physical evidence and Moore’s confession, demonstrated that Moore — not Blair — killed the victim. Id. at 483-84.
Moore argues the Kentucky court unreasonably applied Strickland by creating a new requirement that to show prejudice, a defendant must demonstrate actual innocence or show that the jury had insufficient evidence to support its guilty verdict. But the Kentucky court did not create any such requirement. That court identified the correct standard for prejudice (“reasonable probability”), looked at the totality of the evidence (including the flaws in Blair’s alibi and the overwhelming evidence against Moore), and found no reasonable probability of a different outcome in the absence of the error. While Moore claims the state court considered the remaining evidence against him in an effort [254]*254to create a sufficieney-of-the-evidence standard, the court properly looked at that evidence only to determine whether a reasonable probability of a different outcome existed. See, e.g., Hicks v. Collins, 384 F.3d 204, 215 (6th Cir.2004) (“overwhelming evidence” of petitioner’s guilt precluded reasonable-probability determination).
And even if the Kentucky court might have undervalued Riddle’s testimony2 (so we could possibly disagree with its ultimate decision), that decision did not constitute an unreasonable application of established Supreme Court precedent, sufficient to grant the writ. See Williams v. Taylor, 529 U.S. 362, 411, 120 S.Ct.
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COOK, J., delivered the opinion of the court, in which BOGGS, C.J., joined.
MARTIN, J. (pp. 257-70), delivered a separate dissenting opinion.
OPINION
COOK, Circuit Judge.
Brian Keith Moore, a Kentucky prisoner under sentence of death, appeals the district court’s denial of his petition for a writ of habeas corpus. For the following reasons, we affirm the district court and deny the writ.
I. Background and Procedural History
In 1984, a Kentucky jury convicted Moore of the 1979 kidnaping, murder, and first-degree robbery of seventy-seven-year-old Virgil Harris. In the trial’s penalty phase, the jury determined Moore committed the murder during the commission of first-degree robbery, and sentenced him to death.1 The Kentucky Supreme Court affirmed the conviction and sentence, and the United States Supreme Court denied certiorari. Moore v. Commonwealth, 771 S.W.2d 34 (Ky.1988) (“Moore I”), cert. denied, 494 U.S. 1060, 110 S.Ct. 1536, 108 L.Ed.2d 774 (1990).
In 1990, Moore filed a motion to vacate his sentence under Ky. R.Crim. P. 11.42 in the state trial court, alleging ineffective assistance of trial counsel. While that motion was pending, he filed a motion under Ky. R. Civ. P. 60.02 for a new trial based upon newly discovered evidence. The trial court allowed Moore to present evidence on this second motion during the Rule 11.42 hearing. It denied both motions in January 1997. The Kentucky Supreme Court affirmed, and the United States Supreme Court denied certiorari. Moore v. Commonwealth, 983 S.W.2d 479 (Ky.1998) (“Moore II”), cert. denied, 528 U.S. 842, 120 S.Ct. 110, 145 L.Ed.2d 93 (1999).
Moore filed his petition for habeas corpus in the district court in November 1999, raising claims of ineffective assistance of counsel, trial errors, prosecutorial misconduct, and a due-process violation. He also claimed police obtained incriminating statements used against him at trial in violation of Miranda v. Arizona, 384 U.S. 436, 86 S.Ct. 1602, 16 L.Ed.2d 694 (1966).
The magistrate judge recommended denying Moore’s petition. He concluded that each of Moore’s ineffective-assistance claims was either procedurally defaulted or meritless; that each alleged trial error was meritless; that the prosecutorial-miscon-duct claims were procedurally defaulted; that the due-process claim was procedurally defaulted and meritless; and that the Miranda claims were procedurally defaulted. The district court reviewed the portions of the magistrate judge’s report and recommendation to which Moore had objected, adopted them, and denied Moore’s petition.
Moore then obtained a certificate of ap-pealability as to the following claims: (1) ineffective assistance of counsel based on [253]*253counsel’s failure to impeach a commonwealth witness, Doris Riddle; (2) ineffective assistance of counsel based on inadequate penalty-phase preparation; (3) error in restricting Moore’s contact with his attorneys during a lunch break; (4) error in using the same jury for both the guilt and sentencing phases of the trial; and (5) cumulative error.
II. Standard of Review
Under the Antiterrorism and Effective Death Penalty Act (“AEDPA”), we review a district court’s legal conclusions de novo and its factual findings for clear error. Hill v. Hofbauer, 337 F.3d 706, 710 (6th Cir.2003). We may not grant a writ of habeas corpus unless we conclude that the state court’s decision: (1) was contrary to, or involved an unreasonable application of, clearly established federal law; or (2) was based on an unreasonable determination of the facts. Armstrong v. Morgan, 372 F.3d 778, 781 (6th Cir.2004).
III. Ineffective Assistance of Counsel
A. Failure to Impeach
Moore argues that his trial counsel performed ineffectively by falling to impeach a commonwealth witness, Doris Riddle. Riddle, an employee at the driver’s license bureau, testified concerning the alibi of a second suspect, Kenneth Blair, whom Moore tried to target at trial as the real killer. Riddle’s testimony placed Blair at the license bureau in the same general time frame as the crime (as described by witnesses) and thus hurt Moore’s attempt to blame Blair. But Riddle had told police shortly after the murder that she did not know what time Blair came in to the license bureau — and Moore’s counsel failed to impeach Riddle with this earlier inconsistent statement.
To support a Sixth Amendment ineffective-assistance claim, a defendant (or petitioner) must show (1) that counsel’s performance was deficient, and (2) that the deficient performance prejudiced the defense. Strickland v. Washington, 466 U.S. 668, 687, 104 S.Ct. 2052, 80 L.Ed.2d 674 (1984). To show prejudice, the defendant must show a reasonable probability that, but for counsel’s deficient performance, the proceeding’s result would have been different. Id. at 694, 104 S.Ct. 2052.
The Kentucky Supreme Court rejected Moore’s claim in state post-conviction proceedings, finding that, although Moore’s counsel was deficient for failing to impeach Riddle, that deficiency did not prejudice Moore, and therefore was not ineffective assistance under Strickland. See Moore II, 983 S.W.2d at 482-84. The Kentucky court gave three reasons for its lack-of-prejudice finding: (1) even considering Riddle’s testimony, Blair’s alibi was not air-tight — he could have committed the murder and still been at the license bureau during Riddle’s time-frame; (2) other witnesses confirmed Blair’s alibi; and (3) abundant evidence, including physical evidence and Moore’s confession, demonstrated that Moore — not Blair — killed the victim. Id. at 483-84.
Moore argues the Kentucky court unreasonably applied Strickland by creating a new requirement that to show prejudice, a defendant must demonstrate actual innocence or show that the jury had insufficient evidence to support its guilty verdict. But the Kentucky court did not create any such requirement. That court identified the correct standard for prejudice (“reasonable probability”), looked at the totality of the evidence (including the flaws in Blair’s alibi and the overwhelming evidence against Moore), and found no reasonable probability of a different outcome in the absence of the error. While Moore claims the state court considered the remaining evidence against him in an effort [254]*254to create a sufficieney-of-the-evidence standard, the court properly looked at that evidence only to determine whether a reasonable probability of a different outcome existed. See, e.g., Hicks v. Collins, 384 F.3d 204, 215 (6th Cir.2004) (“overwhelming evidence” of petitioner’s guilt precluded reasonable-probability determination).
And even if the Kentucky court might have undervalued Riddle’s testimony2 (so we could possibly disagree with its ultimate decision), that decision did not constitute an unreasonable application of established Supreme Court precedent, sufficient to grant the writ. See Williams v. Taylor, 529 U.S. 362, 411, 120 S.Ct. 1495, 146 L.Ed.2d 389 (2000) (“[A] federal habeas court may not issue the writ simply because that court concludes in its independent judgment that the relevant state-court decision applied clearly established federal law erroneously or incorrectly. Rather, that application must also be unreasonable.”). Thus Moore’s first ineffective-assistance claim fails.
B. Penalty Phase
Moore also claims his counsel insufficiently prepared for the penalty phase. Again he cites Strickland as the Supreme Court precedent allegedly violated, and again he fails because he cannot show prejudice.
Moore argues that his attorneys erred by (1) allegedly spending only about three percent of their preparation time on the penalty phase; (2) remaining unaware of ninety-five letters sent to the first trial court supporting him, which could have led them to more mitigating evidence; and (3) not having another psychologist examine him after the first one they selected proved to be a fraud.
Moore cannot show prejudice here. He does not dispute that his attorneys researched his background and presented four witnesses who testified regarding his childhood at his sentencing hearing. Another psychologist did interview Moore for three to four hours, and testified that Moore had various problems that began to emerge during childhood, including impulsiveness, poor judgment, behavior control, anger, and harmful emotional attachment to others. The district court noted that this testimony “cast Moore as an easily angered, impulsive, out-of-control emotional leech with poor judgment.” So introducing more evidence of this background, as Moore desired, would likely have made him look even worse to the jury. Thus counsel’s failure to seek or present more background evidence was not even deficient performance, let alone prejudicial. See Strickland, 466 U.S. at 691, 104 S.Ct. 2052 (no deficiency where defendant gives counsel reason to believe that “pursuing certain investigations would be fruitless or even harmful”).3
[255]*255The cases Moore cites as comparable to Ms own are not. All three involve situations in which counsel failed entirely to seek or present mitigating family-background evidence. See Wiggins v. Smith, 539 U.S. 510, 123 S.Ct. 2527, 156 L.Ed.2d 471 (2003) (due to minimal investigation, counsel presented no evidence of defendant’s family history, which included severe childhood abuse); Hamblin v. Mitchell, 354 F.3d 482 (6th Cir.2003) (counsel failed to seek mitigating evidence and thus did not learn of defendant’s unpleasant childhood); Frazier v. Huffman, 343 F.3d 780 (6th Cir.2003) (counsel presented no mitigating evidence except defendant’s one-sentence statement). Here, in contrast, counsel sought and presented the above-described mitigating evidence. Because counsel did so, Moore cannot show prejudice.
IV. Denial of Counsel
Moore claims the trial court violated his Sixth Amendment right to counsel when it recessed for lunch in the middle of his testimony. The court ordered Moore not to discuss his testimony with anyone, including his attorneys, during the 45-min-ute break, but allowed him to consult counsel regarding anything else. Moore claims this order violated his right to counsel.
The Kentucky Supreme Court, considering Moore’s direct appeal, rejected this argument. See Moore I, 771 S.W.2d at 39-41. At the time of the Kentucky court’s 1988 decision, Geders v. United States was the clearly established federal law on this issue. 425 U.S. 80, 96 S.Ct. 1330, 47 L.Ed.2d 592 (1976). But while Moore’s petition for rehearing of that decision pended, the Supreme Court decided Perry v. Leeke, which further developed the law on this issue. 488 U.S. 272, 109 S.Ct. 594, 102 L.Ed.2d 624 (1989). We therefore look to Perry to determine whether the Kentucky courts denied Moore a clearly established right. See Myers v. United States, 198 F.3d 615, 618 (6th Cir.1999) (habeas petitioner could rely on a new constitutional rule announced before his conviction became final); Hardy v. Wigginton, 922 F.2d 294, 296 (6th Cir.1990) (conviction became final for habeas purposes when Kentucky Supreme Court denied petition for rehearing).
The Kentucky Supreme Court rejected Moore’s claim because it concluded that he failed to show a constitutional violation, and, even assuming á constitutional violation, he failed to show any actual prejudice. Moore I, 771 S.W.2d at 39-41.
We agree with the Kentucky court that Moore has failed to show a Sixth Amendment violation. Moore correctly points out that the Kentucky' court erroneously required him to show prejudice — a requirement Perry explicitly rejected. 488 U.S. at 278, 109 S.Ct. 594. But the Supreme Court also held in Perry that a defendant has no right to discuss his testimony with counsel during a short recess in the middle of that testimony. 488 U.S. at 280-84, 109 S.Ct. 594. Thus the Kentucky court’s conclusion that the recess here did not violate Moore’s Sixth Amendment rights was not unreasonable, and Moore’s claim fails.
V. Impartial Jury
Moore next contends that the trial court denied his right to an impartial jury at sentencing by using the same jury that convicted him. Moore argues that the jury heard prejudicial evidence during the guilt phase — in particular, testimony of a defense witness suggesting that a previous jury had convicted Moore and sentenced him to death — and therefore he deserved a new jury for the penalty phase.
On direct appeal, the Kentucky Supreme Court rejected this argument, concluding that Moore himself presented much of the [256]*256harmful evidence during the guilt phase, and that the harmful evidence “did not rise to [a] degree of pervasiveness” so as to establish prejudice and thus require a new jury for sentencing. Moore I, 771 S.W.2d at 37. Moore argues that the Kentucky court’s decision constituted an unreasonable application of the clearly established law of two Supreme Court cases: Patterson v. Colorado, 205 U.S. 454, 27 S.Ct. 556, 51 L.Ed. 879 (1907), and Leonard v. United States, 378 U.S. 544, 84 S.Ct. 1696, 12 L.Ed.2d 1028 (1964). But because neither Patterson nor Leonard clearly establishes any law relevant to Moore’s claim, this argument fails.
Patterson, which concerned a court’s ability to issue a contempt order against an individual for publishing articles and cartoons criticizing the state supreme court, presents no law bearing on Moore’s claim.
Leonard is likewise inapposite. It involved a situation in which prospective jury members sat in a courtroom and saw a jury announce a guilty verdict against the defendant. The government then tried the defendant for a second crime, impaneling some of those who were present during the first verdict. The Court held — and the government conceded — that the second jury should have been disqualified. Leonard, 378 U.S. at 544-45, 84 S.Ct. 1696. Moore argues that Leonard is clearly established law that a jury possessing knowledge of a defendant’s guilt must be disqualified from sentencing. But the Leonard Court limited that decision to its facts. Id. at 545, 84 S.Ct. 1696 (disqualifying the jurors only “under the circumstances of this case”). Because Moore’s situation is factually distinguishable, Leonard cannot serve as relevant clearly established law. The Supreme Court has certainly not clearly established that bifurcated trials are generally required in the United States, even in capital cases. And in the absence of any clearly established law, courts deny habe-as relief. See, e.g., Bugh v. Mitchell, 329 F.3d 496, 513 (6th Cir.2003) (“[T]here is no Supreme Court precedent that the trial court’s decision could be deemed ‘contrary to,’ under AEDPA.”).
VI. Cumulative Errors
Moore claims he is entitled to relief because of cumulative trial errors. But we have held that, post-AEDPA, not even constitutional errors that would not individual-' ly support habeas relief can be cumulated to support habeas relief. See Scott v. Elo, 302 F.3d 598, 607 (6th Cir.2002); Lorraine v. Coyle, 291 F.3d 416, 447 (6th Cir.2002) (death-penalty decision stating, “The Supreme Court has not held that distinct constitutional claims can be cumulated to grant habeas relief.”).
Moore cites Walker v. Engle for the proposition that cumulative evidentiary errors can warrant habeas relief. 703 F.2d 959, 963 (6th Cir.1983). But Walker is pre-AEDPA, and is not Supreme Court precedent, as AEDPA requires. See Lorraine, 291 F.3d at 447 (noting Walker was pre-AEDPA and refusing to consider errors cumulatively). Because Moore can cite no Supreme Court precedent obligating the state court to consider the alleged trial errors cumulatively, we cannot grant relief on this ground.4
[257]*257And even if we could consider the alleged errors cumulatively, Moore’s claim would fail. We already denied a certificate of appealability for two of the alleged errors: restriction on cross-examination of prosecution witnesses, and failure to instruct the jury on second-degree manslaughter. And we have above addressed and rejected Moore’s arguments regarding the other two alleged trial errors: restricted contact between Moore and counsel during a lunch recess and use of the same jury for the trial’s guilt and penalty phases. Thus the alleged errors, considered individually or cumulatively, cannot support relief.
VII. Conclusion
For the foregoing reasons, we affirm the district court’s denial of the writ of habeas corpus.