SUTTON, J., delivered the opinion of the court, in which SILER, J., joined. MARTIN, J. (pp. 642-46), delivered a separate concurring opinion.
OPINION
SUTTON, Circuit Judge.Mark Wiles murdered a fifteen-year-old boy with a kitchen knife during a botched burglary in 1985. After he waived his right to a jury trial, a panel of three Ohio judges convicted him of aggravated murder and aggravated burglary, then sentenced him to death. After exhausting his state-court appeals and post-conviction remedies, Wiles sought a writ of habeas corpus under 28 U.S.C. § 2254, arguing (among other things) that he was denied the effective assistance of counsel under the Sixth and Fourteenth Amendments. Because Wiles has not shown that he was prejudiced by his counsel’s alleged shortcomings, we affirm.
I.
In 1982, Wiles went to work as a part-time laborer for Charles and Carol Klima on their horse farm, where they lived with their son Mark. State v. Wiles, 59 Ohio St.3d 71, 571 N.E.2d 97, 103 (1991). One day in early 1983, the family learned that $200 in cash was missing. That same day, Wiles had reported for work, but he could not be found after the Klimas learned of the missing cash, and he did not return to collect his paycheck or for that matter return to work any longer on the farm. Id. at 103-04. In the spring of that year, Wiles began serving a 4-25 year sentence in an Ohio prison for an unrelated burglary he had committed the previous year.
On August 7, 1985, after serving eighteen months of this sentence, Wiles returned to the Klima farm, entered the unlocked house while the family was gone and began to search the house for valuables. Wiles, 571 N.E.2d at 104. While he was still in the house, Mark Klima returned and confronted him. Id. Wiles stabbed the boy 24 times with a kitchen knife, stole approximately $260 and fled. Id. Carol Klima returned home to find her unconscious son lying on the floor with a knife buried in his back. Id. Later that day, Mark Klima died in a hospital emergency room. Id.
Wiles initially fled from the authorities. Five days after the murder, however, he turned himself in to the police in Savannah, Georgia, telling them that he was wanted for murder in Ohio. Id. at 105. After being informed of his rights, he told the police what he had done and signed a confession admitting that he had killed Klima. Id.
*638A state grand jury indicted Wiles for aggravated murder and two counts of aggravated burglary — one for the 1985 home invasion, one for the 1983 $200 theft. Id. at 105-06. He waived his right to a jury, and a three-judge panel heard his case. Id. at 106-07. After the guilt phase of the proceedings, the court determined that there was insufficient evidence that he had committed the 1983 burglary but convicted him on the aggravated-murder and the other aggravated-burglary count. Id. at 107. After a mitigation hearing, the court determined that neither Wiles’ youth (he was 22-years old at the time of the murder) nor his confession outweighed the aggravating circumstances of his crime. Id. at 107-08. The court imposed a death sentence, and the Ohio Court of Appeals and the Ohio Supreme Court affirmed his conviction and sentence. Id. at 108, 125; State v. Wiles, No. 1675, 1988 WL 59838, at *10 (Ohio Ct.App. June 3, 1988).
Wiles filed a state post-conviction petition, which included a claim that his trial counsel had provided constitutionally inadequate assistance at the mitigation phase of the trial. The state trial and appellate courts rejected the petition. In rejecting his ineffective-assistance claim, the court of appeals concluded that Wiles “ha[d] failed to demonstrate ineffective assistance of his trial counsel at the ... penalty phase” and that “he [was] unable to demonstrate with a reasonable probability that the result at trial would have been different” if his counsel had not made the alleged errors. JA 918. The Ohio Supreme Court declined review. State v. Wiles, 93 Ohio St.3d 1412, 754 N.E.2d 260, 260 (2001).
In 2002, Wiles filed a petition for a writ of habeas corpus in federal court, raising 36 claims. The district court denied the petition in 2005 and declined to issue a certificate of appealability on any of the claims. Wiles sought a COA from us, which we granted with respect to the claim that his attorneys failed him at the penalty phase of his trial.
II.
To establish ineffective assistance of counsel, a claimant must show two things. He must establish that his attorneys’ performance was “deficient,” which “requires showing that [they] made errors so serious that [they were] not functioning as the ‘counsel’ guaranteed ... by the Sixth Amendment.” Strickland v. Washington, 466 U.S. 668, 687, 104 S.Ct. 2052, 80 L.Ed.2d 674 (1984). And he must show that “there is a reasonable probability that, but for counsel’s [failure to investigate], the result of the [mitigation hearing] would have been different.” Id. at 694, 104 S.Ct. 2052.
Like all claimants seeking federal habe-as relief after 1996, Wiles faces another hurdle: the Antiterrorism and Effective Death Penalty Act of 1996, Pub.L. No. 104-132, 110 Stat. 1214. Under that legislation, we may grant the writ only if the state court of appeals’ decision “was contrary to, or involved an unreasonable application of, clearly established federal law.” 28 U.S.C. § 2254(d)(1).
A.
In maintaining that his trial counsel did not adequately prepare for the mitigation hearing, Wiles claims that his attorneys failed (1) to uncover abuse in his childhood, (2) to uncover that he had taken barbiturates before entering the Klimas’ house on the day of the murder and (3) to investigate a head injury he received twelve days before the murder. Even if we grant for the sake of argument that these claimed lapses meet the first prong of Strickland (ineffective assistance), they do not meet its second prong (prejudice). *639See Poindexter v. Mitchell, 454 F.3d 564, 572 (6th Cir.2006).
Wiles has not shown that “there is a reasonable probability” that, but for this alleged absence of investigation, “the result” of the mitigation hearing “would have been different.” Strickland, 466 U.S. at 694, 104 S.Ct. 2052. Above all, the new evidence does not “differ[ ] markedly from the testimony and evidence the [three-judge panel] in fact considered.” Hill v. Mitchell, 400 F.3d 308, 332 (6th Cir.2005).
First, the new family-history evidence about Wiles’ father and mother adds little to what was introduced at the mitigation hearing. As to his father: Counsel introduced evidence at the trial showing that his father was “stern,” JA 1037, that he was “less tolerant with [Wiles] than he was [with] the other[ ] [children]” and that he viewed Wiles as an “interference ... in [the family’s] life” because “[Wiles] was not a planned child,” JA 1302-03. A counselor’s report also described the following incident between Wiles and his father when Wiles was 17:
Mark has been high everyday during the past week. Sat. 1-10 his father and him got into a fight. His father got rough w/ him. Mark got pushed down some steps. Mark[’]s father called the police and had him charged w/ intoxication. Mark was placed in detention....
JA 981.
Most of the ostensibly new evidence represents variations on this same theme. For example: Wiles claims that the court should have heard that his father was emotionally distant — that he was “not one to touch much,” JA 783; that, after Wiles began serving time in prison for his first burglary in 1983, his father “washed [his] hands of him and did not visit him,” JA 744; that his father stated that he “never wanted that bastard anyway,” JA 746; that he told Wiles that he “would never amount to anything and that he did not want him,” id.; and that Wiles suffered “emotional[] and physical ] abuse” from his father, JA 747, though the affidavit to this effect (from his sister Jona) offers no details about any incidents of physical violence. All of this evidence adds little to the picture that counsel already had painted of Wiles’ father: a man who did not like Wiles, who resented him as an unwanted addition to the family and who, on occasion, “got rough” with him, JA 981. This is precisely the “kind of cumulative evidence that does not show prejudice,” Brooks v. Bagley, 513 F.3d 618, 626 (6th Cir.2008), because it does not “differ in a substantial way — in strength [or in] subject matter — from the evidence actually presented at sentencing,” Hill, 400 F.3d at 319.
Wiles, however, does identify one new piece of evidence that was not covered at the trial- — the allegation that his father was “sexually inappropriate with his sisters.” JA 754. According to Wiles’ post-conviction psychological expert, Robert Smith, “Wiles reluctantly disclosed that his father was ‘sexually inappropriate’ with his sisters.” JA 754. New though this allegation may be, it is not corroborated: Neither Wiles’ father, his sister, nor his mother mentions any such abuse in their affidavits. Even if we accept the allegation as credible, moreover, it still amounts to exceedingly weak mitigation evidence, because nothing shows that Wiles was aware of this abuse at the time of the murder, and there is no evidence that it caused him any psychological harm beyond what he had already experienced at the hands of a distant and sometimes abusive father. Confirming the point, Smith does not refer to these allegations in discussing Wiles’ troubled relationships, his overall diagnostic impressions of Wiles or his impressions of Wiles’ mental state at *640the time of the crime. Absent evidence that Wiles knew of the abuse or even evidence about how it might have affected his psychological profile, we see no tenable basis on which it could have altered the three-judge panel’s sentence.
As to Wiles’ mother: Counsel offered evidence at the mitigation hearing from a school psychologist who said that his “mother [was] ignoring” and that the family was “not close.” JA 1037. Wiles claims that his attorneys also should have introduced evidence that his mother “was quite depressed and cleaned the house all the time for [their] father.” JA 746. The expert witness, Smith, likewise spoke of the mother’s “long-standing history of depression,” which prompted her to “spen[d] a considerable amount of time sleeping and withdrawing from the family and others.” JA 755. But this evidence adds little to what the three-judge panel heard. There is no indication that the depression affected Wiles in any way beyond causing his mother to “withdraw[ ]” from the family, id., and a similar theory was put before the three-judge court when it heard evidence that she “ignor[ed]” him, JA 1037.
Second, the ostensibly new evidence of drug use suffers from a similar flaw. Wiles argues that his counsel failed to introduce evidence about his “drug problems, which escalated after an industrial accident which caused the tragic death of his brother.” Br. at 20. Yet this evidence, once again, largely duplicates what the judges already heard. Wiles’ attorneys presented ample evidence at the mitigation hearing of Wiles’ history of drug and alcohol abuse, the death of his brother Randy and the impact these events had on him.
Here, too, there is an exception. Wiles argues that his prior attorneys should have discovered — and should have presented to the court — evidence that he ingested “3 or 4 barbiturates shortly before entering the Klima home,” which prevented him from thinking “clearly” at the time of the offense. Br. at 11-12. Even if we accept this new evidence as true, even if we accept that his counsel did not know about this fact and even if we overlook the conspicuous oddity that Wiles himself did not tell his counsel about this background information, there is no prejudice and indeed the omission of this evidence likely benefited Wiles. For one thing, this evidence directly contradicts his confession to the crime, in which he denied consuming drugs on the day of the murder or within the previous twelve days. The claim therefore was readily impeachable, making it unlikely to change the outcome of the hearing for that reason alone. See Owens v. Guida, 549 F.3d 399, 411 (6th Cir.2008). For another, Wiles’ mitigation strategy emphasized, quite understandably, that he had confessed truthfully in all respects to the crime, see Wiles, 1988 WL 59838, at *8, but this evidence would have undercut that mitigation theme. For still another, it is hardly self-evident that getting high on barbiturates before stabbing someone to death is the kind of evidence that makes a capital defendant look better in the eyes of a court as opposed to “mak[ing] him look even worse.” Carter v. Mitchell, 443 F.3d 517, 532 (6th Cir.2006). Wiles had little to gain from this new evidence and much to lose by introducing it. He thus cannot show prejudice by its omission.
Third, Wiles contends that his counsel failed him by not investigating a head injury he sustained twelve days before the murder. As Wiles tells it, a man named Joe Kelly “jumped [him] and hit [him] on the head with a tire iron” when he was leaving a bar, knocking him unconscious. Br. at 12-13. He arrived at the emergency room with cuts on his face and with his right eye “swollen shut,” JA 568, and an *641examination showed that he had “multiple facial bone [fractures],” id. After doctors cleaned and closed his wounds, Wiles left the hospital against medical advice, only to return five days later complaining of “dizziness, somnolence and difficulty walking.” Id.
Notably absent from the record is any evidence that Wiles was still experiencing symptoms from his head injury on the day he murdered Mark Klima. Wiles argues only that investigation by his counsel prior to the mitigation hearing, including “retention of a neurologist and a follow up CAT SCAN, may have assisted counsel ... in explaining to the fact finders a causal ... connection between the head injury and [his] uncharacteristic violent behavior.” Br. at 13-14. But such “speculation” as to the effects still-further investigation might have on the outcome of this theory at the mitigation hearing does not by itself establish prejudice. See Slaughter v. Parker, 450 F.3d 224, 234 (6th Cir.2006). This claim, too, necessarily fails. All of these arguments considered, the state court of appeals reasonably applied Strickland in concluding that Wiles failed to show prejudice from the failure to present this additional evidence.
B.
Wiles separately argues that his attorneys did not adequately prepare Dr. Willis Carpenter Driscoll, a psychologist, for the mitigation hearing: They gave Driscoll too little time to prepare for the hearing because they hired him a week before the hearing; Driscoll interviewed Wiles for just two hours and never spoke to any of his family, friends or coworkers; and Driscoll never reviewed Wiles’ educational records. Driscoll’s inadequate preparation, according to Wiles, led him to testify in a way that failed to convey a useful mitigation theory to the three-judge panel. Even if we accept this theory of ineffective assistance for the sake of argument, Wiles again has failed to show prejudice.
Driscoll’s inadequate preparation, Wiles urges, caused Driscoll to admit a damning fact on cross-examination, namely that “one of the many factors” that motivated Wiles to kill Klima was that the boy was “the only witness” to the burglary. JA 1421. But it is hard to see how this admission could have affected the death sentence. By this stage in the ease, the three-judge panel already had determined that Wiles killed Klima “for the purpose of escaping detection, apprehension, trial or punishment for another crime.” Wiles, 571 N.E.2d at 106. Once the panel found that fact to be true beyond a reasonable doubt, defense counsel had every reason to accept, rather than challenge, that premise of the guilt determination and to work with, rather than reargue, the point during the mitigation hearing. A psychologist’s “admission” of a fact at a mitigation hearing, one already established beyond a reasonable doubt at the liability hearing, does not establish prejudice.
Nor has Wiles succeeded in showing that a better-prepared expert would have given more useful testimony. As an example of what a well-prepared expert would have said, he offers the affidavit of Robert Smith, who has a Ph.D., a license in clinical psychology and extensive clinical experience in that field, and who has interviewed Wiles and reviewed the relevant records in this case. But Smith’s submission, too, adds little to what the court heard at the hearing.
Smith opines that “the antisocial behaviors reported by Mr. Wiles are directly related to his substance dependence” and says that the murder was a “direct result” of Wiles ingestion of barbiturates. JA 765, 767. As explained, however, linking Wiles’ *642conduct to drug abuse was already in front of the panel and linking the crime to the recent consumption of barbiturates was inconsistent with his confession and with the prevailing theme of his case for leniency.
The other explanation Smith offers for the murder was that Wiles had “interpersonal difficulties,” in particular a poor relationship with his parents and a brother who died shortly before Wiles began high school. JA 767. But, as mentioned, the three-judge panel heard considerable testimony regarding Wiles’ difficult family circumstances, including Driscoll’s opinion that Wiles’ “emotional development ha[d] been arrested at the age of 10[to] 12” due, in part, to the “dearth of warmth from his father,” JA 1399, and that Wiles’ desire “to punish himself’ due to an overabundance of guilt led him to kill Klima, JA 1424. In the end, Wiles has not shown that he was prejudiced by his counsel’s preparation of and reliance on Driscoll or that the state court of appeals unreasonably concluded otherwise.
III.
For these reasons, we affirm.