
567 F.3d 246 (2009)
Kenneth W. SMITH, Petitioner-Appellant,
v.
Betty MITCHELL, Warden, Respondent-Appellee.
No. 05-4211.
United States Court of Appeals, Sixth Circuit.
Argued: March 10, 2009.
Decided and Filed: June 5, 2009.
*249 ARGUED: Kyle E. Timken, Law Office, Columbus, Ohio, for Appellant. Charles L. Wille, Office of the Ohio Attorney General, Columbus, Ohio, for Appellee. ON BRIEF: Kyle E. Timken, Law Office, Columbus, Ohio, S. Scott Haynes, Hallowes, Allen & Haynes, Reynoldsburg, Ohio, for Appellant. Charles L. Wille, Office of the Ohio Attorney General, Columbus, Ohio, for Appellee.
Before: DAUGHTREY, MOORE, and CLAY, Circuit Judges.

OPINION
KAREN NELSON MOORE, Circuit Judge.
Ohio death-row inmate Kenneth Smith ("Smith") appeals the district court's denial of his petition for habeas corpus relief from his state-court convictions and sentence. In 1996, Smith was convicted on two counts of aggravated felony murder and was sentenced to death for each aggravated-murder count. After Smith exhausted his remedies in the Ohio state courts, he filed this petition for habeas corpus pursuant to 28 U.S.C. § 2254. After the district court denied his petition, a certificate of appealability ("COA") was granted on five issues: (1) whether the prosecutor committed misconduct at trial by arguing that Smith lacked remorse and by improperly questioning Smith on cross-examination; (2) whether trial counsel provided ineffective assistance of counsel in challenging Smith's confession to police; (3) whether Smith was sentenced to death for a murder he did not commit; (4) whether the Ohio death penalty is unconstitutional as applied to Smith; and (5) whether Ohio's adoption of a one-tier system of appellate review for capital cases denied Smith due process under the United States Constitution. For the reasons *250 discussed below, we AFFIRM the district court's denial of Smith's petition for habeas relief under 28 U.S.C. § 2254.

I. BACKGROUND

A. Factual Background
The underlying facts were set forth by the Ohio Supreme Court on Smith's direct appeal:
On May 12, 1995, sometime around 11:00 p.m., defendant-appellant, Kenneth W. Smith ("defendant"), and his brother, Randy Smith ("Randy"), brutally murdered Lewis Ray and Ruth Ray in their Hamilton, Ohio home. Lewis was severely beaten, his skull was fractured, and his throat was slit, severing his windpipe and carotid arteries. Ruth died from manual strangulation. Their home was ransacked, and money and jewelry were taken. The following morning, David L. Lester, Ruth's son, discovered the bodies of his mother and stepfather and called the police.
In the Rays' home, police observed signs of a struggle, blood on the kitchen floor, and bloody footprints throughout the house. Police found a damaged white ceramic coffee pot covered with blood stains in the trash can and a green army camouflage hat on the floor. A knife had recently been removed from a butcher block set. Police found Lewis lying on the kitchen floor and Ruth lying in the doorway between the hall and bedroom. The Rays' bedroom had been ransacked, and the contents of dressers were strewn about the floor.
Earlier in the evening of May 12, 1995, defendant and Randy had gone to the Crystal Lounge, a.k.a. Crystal Bar, with a friend, Russell C. Baker. At approximately 10:20 p.m., defendant borrowed Baker's car allegedly to pick up his wife, Brenda Smith, and some friends. By midnight, defendant had not returned Russell's car. At about that time, Brenda and Lillian Canafax, Randy's live-in girlfriend, arrived at the Crystal Lounge also looking for the Smith brothers. About forty-five minutes later, Russell and the two women decided to go to Chasteens Bar. Defendant eventually showed up at Chasteens Bar at approximately 1:30 a.m. When Russell questioned defendant about the car, defendant claimed that he was late because he had been in a fight at a gas station. Defendant showed Russell a bump on his head. At the time, Russell also noticed that defendant had changed his clothes.
At approximately 2:00 a.m., defendant left Chasteens Bar in his Monte Carlo automobile with Brenda, Randy, Lillian, and Russell. Defendant drove to his house, handed his car keys to Randy, and instructed Randy to take a stuffed pillowcase from a nearby blue automobile and put it into the trunk of the Monte Carlo. Russell accused the Smith brothers of being "out thieving with my car." Defendant replied, "Russell, I wouldn't do that." The group then drove to Buckeye Street, where Russell's brother, James, was staying. Russell soon went home and to bed.
In the early hours of May 13, 1995, defendant admitted to his friend, James Baker, that he had killed Lewis Ray and that his brother, Randy, had strangled Ruth Ray. James testified that on May 12, 1995, he was staying at his mother's apartment, when defendant and Randy arrived at approximately 1:30 a.m. in Russell's automobile. The Smiths had been to the apartment earlier in the evening before going to the Crystal Lounge. Defendant told James that he had been in a fight, and James noticed that defendant had cleaned up and changed clothes. Defendant was wearing *251 a sweater and boots instead of tennis shoes. He was not wearing a hat. James further testified that defendant left the apartment again at 1:35 a.m. to go to Chasteens Bar.
When defendant returned to James's mother's apartment at approximately 2:45 a.m., he began to tell James about the murders. James testified that defendant told him that he had taken a hammer and "struck Louie Ray between his eye[s]," and that during this time, defendant had winked at his brother, Randy, who followed Ruth into a bedroom and strangled her. Defendant also told James that they took gold and jewelry in a pillowcase from the Rays' home.
James testified that when he asked defendant why he killed the Rays, defendant replied that they had killed them to prevent the Rays from identifying them. James testified that defendant "was talking how he sliced Lewis Ray's throat from ear to ear and just laughing about it." Defendant also told James that after he killed Lewis, he "kicked Ruth's brain in" to make sure she was dead. James testified that defendant brought a pillowcase stuffed with jewelry inside the apartment, but James asked him to take it back to the car.
Later that morning, James was driving around with defendant and Brenda. They stopped to buy cigarettes and marijuana. Defendant mentioned to James that he was concerned because he lost his green army camouflage hat in the struggle with Lewis. Eventually they drove to Russell's home. There, out of defendant's presence, James told Russell what defendant had admitted. Defendant then suggested to James that he hide the remaining jewelry. This prompted Russell to contact the police. Police later recovered the jewelry in the attic of a garage.
In addition to the testimony of James Baker, Lillian Canafax testified that she was outside Chasteens Bar arguing with Randy when he showed her a gun. She testified that she saw the same gun in her bedroom the following morning. Several days later, after she found the gun and money under the bed, she authorized police to search the apartment. Lillian also turned over to police three money orders she had purchased for Randy the day after these crimes occurred.
Another witness testified that around 11:15 or 11:30 p.m., he saw Randy standing outside a pizza parlor about a block from the Rays' residence. The witness testified that Randy had a hammer in his hand as he ducked behind the building. Russell testified that a hammer was missing from his car after he had loaned his car to defendant.
That afternoon, the police detained defendant for questioning. At the time, police observed cuts and scratches on defendant's face, and a long cut and bruises near his right collarbone. Police also searched Brenda's purse and discovered a cellophane bag containing rings, two $100 bills, and a quantity of nonsequentially numbered food stamps. Police knew that Lewis sold similar jewelry and suspected that he may have dealt in food stamps as currency.
At the police station, defendant waived his Miranda rights and admitted that he and Randy had killed Lewis and Ruth. Defendant said that while at the Crystal Bar, he and Randy had talked about going to rob the Rays, and decided that they would have to kill the Rays because they did not want the Rays to be able to identify them. Defendant told police that after arriving at the Rays' house, he and Lewis began to *252 argue about money that defendant supposedly owed Lewis. Defendant further admitted that he picked up an object from the kitchen counter and struck Lewis, eventually overpowering him. Defendant claimed that Lewis said, "I'm going to kill you, Kenny," so defendant grabbed a knife and cut Lewis's throat. He then rolled Lewis on his side and took his wallet. Defendant said he walked to the bedroom and saw Ruth's body on the floor. Randy had choked her to death. The two men ransacked the bedroom and left in Russell's automobile.
Police apprehended Randy Smith. They found $344 in bloodstained currency on him. Randy initially denied any knowledge of the murders. Police allowed Randy to speak with his brother, who said, "They got us brother, everybody is telling on us, tell the truth, that's what I did." Randy then explained to the police his involvement in the crimes.
Later, after again being advised of his Miranda rights, defendant gave the police a written confession. In his statement, defendant said that while playing pool at the Crystal Lounge, he talked with Randy about robbing Lewis. He borrowed Russell's car and drove to a pool hall about half a block from the Ray home. Defendant stated that he and his brother walked to the Rays' house. Lewis invited the Smiths into his home. Defendant and Lewis began to argue about $2,500 that defendant owed Lewis. The men began to fight in the kitchen and defendant grabbed something from the counter and struck Lewis's head. They continued to wrestle on the floor. Defendant knew he was going to have to kill Lewis to keep him from telling anyone what happened. Defendant then grabbed a knife and "sliced Louie across the throat."
In his written confession, defendant further admitted that he took Lewis's wallet, then walked into the bedroom. Ruth was lying on the floor in the doorway, and defendant had to step over her body. Defendant said he asked Randy what had happened, and Randy said he had choked Ruth. Defendant further admitted that he then ransacked the bedroom, taking rings, watches, and necklaces, and placed the items in a plastic bag and left.
According to his signed confession, defendant went home after the murders to shower and change clothes. He and Randy divided the money found in Lewis's wallet. Defendant's share was around $625. Defendant then put his bloody clothes, the knife, and Lewis's wallet into a green trash bag that Randy later threw into the river. The two men then drove to Chasteens Bar.
In his confession, defendant explained that after leaving Chasteens Bar, he drove to the apartment where James Baker was staying and began to go through the jewelry that the defendant and Randy had taken from the Rays' house. Defendant picked out some items he wanted to keep. The following morning he placed some rings into a plastic bag and gave them to Brenda, who put them into her purse. Defendant put the remainder of the jewelry into the trunk of his Monte Carlo. He and James then put the jewelry into the attic of James's grandmother's garage. During police questioning, defendant also admitted that the wristwatch he was wearing had belonged to Lewis.
At trial, defendant testified that he and Randy went to the Rays, intending only to steal saws and drills from the yard. They parked the car away from the house, but as they walked into the yard, Lewis opened the gate and saw them. Lewis invited them into the *253 house, and the men began to argue about money that defendant allegedly owed Lewis. Defendant testified that within ten minutes, "everything got real violent." Lewis "jumped up," told defendant he "was going to shoot" him, and hit defendant "upside the head with something." Defendant testified that he grabbed something from near the stove and struck Lewis. Defendant testified that Lewis tried to push him down the basement steps. Defendant then grabbed a knife and cut Lewis as he approached. Defendant bent down, turned Lewis on his side, and grabbed his wallet. Defendant further testified that Randy told him that he had choked Ruth. The brothers then ransacked the bedroom, taking jewelry.
Defendant denied that he intended to kill the Rays. Defendant claimed that Lewis was his best friend, and he "wouldn't cold blooded kill him for nothing." Defendant testified that he was very upset about the Rays because they were "like family" to him. He admitted that he told James about killing Lewis, but testified that he wasn't laughing or joking, but instead, he was "in tears."
Kenneth Smith was charged in two counts with the aggravated felony-murder of Lewis Ray and Ruth Ray in violation of R.C. 2903.01(B). Each murder charge contained three death specifications: the offense was committed to escape detection, apprehension, trial, or punishment for other offenses, R.C. 2929.04(A)(3); the offense was part of a course of conduct involving the purposeful killing of two or more persons, R.C. 2929.04(A)(5); and the offense was committed during the course of an aggravated robbery, R.C. 2929.04(A)(7). He was also charged with two counts of aggravated robbery that included the allegation of a prior felony conviction for attempted burglary. The jury convicted defendant as charged and recommended the death penalty on the aggravated murder counts. The trial court sentenced defendant to death.
State v. Smith, 80 Ohio St.3d 89, 684 N.E.2d 668, 675-78 (1997).

B. State Proceedings
Prior to trial, Smith's counsel filed a motion to suppress Smith's confession to the police. The motion challenged both the arrest itself and the subsequent interrogation and was described by the trial court at the hearing as a "shotgun motion." Suppression Hr'g Tr. at 69. At the hearing, the government called three detectives who participated in Smith's arrest and interrogation, and Smith's counsel did not call any witnesses. Smith's counsel did not question any of the witnesses about Smith's mental state at the time of the interrogation. A supplemental brief filed by Smith's counsel focused not on the interrogation but on the supposed lack of probable cause for the arrest. The trial court subsequently denied the motion.
Smith appealed his convictions and sentence to the Ohio Supreme Court,[1] raising several assignments of error. Among other things, Smith argued that (1) recent amendments to the Ohio constitution providing for one-tier appellate review in capital cases violate the Fourteenth and Eighth Amendments; (2) the capital specifications on his conviction for the murder of Ruth Ray required the jury inappropriately to examine the actions of the principal *254 offender, his brother Randy, rather than Smith's actions; (3) the death penalty was applied inappropriately, excessively, and disproportionately because Smith's brother Randy received only a life sentence; (4) prosecutorial misconduct at Smith's trial denied his right to due process; (5) Smith did not receive effective assistance of counsel at all stages of the proceedings; (6) Smith's statement to police was obtained involuntarily and unknowingly; and (7) Ohio's statutory provisions governing the imposition of the death penalty are unconstitutional on their face and as applied. The Ohio Supreme Court denied Smith's appeal on all grounds and affirmed his convictions and sentence.
Smith then sought postconviction relief in the Ohio state courts pursuant to Ohio Revised Code § 2953.21, raising eleven claims for relief and requesting an evidentiary hearing and the opportunity to conduct discovery. Among his grounds for relief, Smith claimed that he was denied effective assistance of counsel and that his statements to the police should have been suppressed because he was intoxicated at the time of the interrogation. The trial court denied Smith relief, and the Ohio Court of Appeals affirmed. The Ohio Court of Appeals concluded that many of Smith's claims were barred by res judicata and that Smith had not met his burden to show that an evidentiary hearing was warranted. State v. Smith, No. CA97-12-223, 1998 WL 549964 (Ohio Ct.App. Aug. 31, 1998).

C. Federal Proceedings
On October 4, 1999, Smith filed a petition for a writ of habeas corpus in the United States District Court for the Southern District of Ohio pursuant to 28 U.S.C. § 2254, asserting nineteen grounds for relief. Among these claims, Smith asserted that (1) his constitutional rights were violated by the prosecutor's misconduct on cross-examination of Smith and during closing arguments, (2) Smith's counsel was ineffective in challenging his confession, (3) Smith's constitutional rights were violated when he was sentenced to death for a murder he did not commit, (4) Smith's rights to equal protection and due process were violated by Ohio's elimination of the intermediate level of appellate review for capital defendants, and (5) Ohio's death-penalty statute is unconstitutional on its face and as applied to Smith. Smith initially sought discovery and an evidentiary hearing, but both motions were denied without prejudice. Although Smith filed a renewed motion for leave to conduct discovery, which was then denied, he never filed a renewed motion for an evidentiary hearing.
The case was referred to a magistrate judge, who recommended that the petition be denied on all grounds. Smith filed objections to the magistrate judge's report and recommendation, but the district court ultimately adopted the report and recommendation and denied relief. Smith v. Mitchell, No. C-1-99-832, 2005 WL 1969309 (S.D.Ohio Aug. 15, 2005); Smith v. Mitchell, No. C-1-99-832, 2003 WL 24136073 (S.D.Ohio Sept. 30, 2003). The district court did, however, grant a COA on one claim: whether Ohio's one-tier system of appellate review for capital cases violates the Due Process Clause of the Fourteenth Amendment. Smith, 2003 WL 24136073, at *26-28. We subsequently expanded the COA to include four additional claims: "(1) whether the prosecutor committed misconduct at trial by arguing that Smith lacked remorse and by improperly questioning Smith on cross-examination; (2) whether trial counsel ineffectively challenged Smith's confession; (3) whether Smith was sentenced to death for a murder he did not commit; (4) whether the Ohio death penalty statute is unconstitutional *255 as applied to Smith." Joint Appendix ("J.A.") at 612 (Order).

II. ANALYSIS

A. Standard of Review
"`In a habeas corpus proceeding, this Court reviews a district court's legal conclusions de novo and its factual findings for clear error.'" Smith v. Berghuis, 543 F.3d 326, 334 (6th Cir.2008) (quoting Miskel v. Karnes, 397 F.3d 446, 451 (6th Cir. 2005)). Because Smith filed his habeas petition after April 24, 1996, we review all claims that were adjudicated on the merits by the Ohio state courts under the standards set forth in the Antiterrorism and Effective Death Penalty Act of 1996 ("AEDPA"), Pub.L. No. 104-132, 110 Stat. 1214 (codified at 28 U.S.C. § 2254). Under AEDPA,
An application for a writ of habeas corpus on behalf of a person in custody pursuant to the judgment of a State court shall not be granted with respect to any claim that was adjudicated on the merits in State court proceedings unless the adjudication of the claim
(1) resulted in a decision that was contrary to, or involved an unreasonable application of, clearly established Federal law, as determined by the Supreme Court of the United States; or
(2) resulted in a decision that was based on an unreasonable determination of the facts in light of the evidence presented in the State court proceeding.
28 U.S.C. § 2254(d). The Supreme Court has explained that subsection 2254(d)(1) contains two prongs, the "contrary to" and the "unreasonable application of" clauses, each with an independent meaning. Terry Williams v. Taylor, 529 U.S. 362, 404, 120 S.Ct. 1495, 146 L.Ed.2d 389 (2000). A decision is contrary to clearly established federal law if either "the state court applies a rule that contradicts the governing law set forth in [the Supreme Court's] cases," id. at 405, 120 S.Ct. 1495, or "the state court confronts a set of facts that are materially indistinguishable from a decision of [the Supreme] Court and nevertheless arrives at a result different from [Supreme Court] precedent," id. at 406, 120 S.Ct. 1495. A decision is an unreasonable application of clearly established federal law, on the other hand, if "the state court identifies the correct governing legal rule from [the Supreme] Court's cases but unreasonably applies it to the facts of the particular state prisoner's case." Id. at 407, 120 S.Ct. 1495.

B. Prosecutorial Misconduct
Smith argues that the prosecutor committed misconduct during the guilt phase of the trial. Specifically, Smith claims that the prosecutor argued during his closing statement that Smith lacked remorse and that the prosecutor badgered Smith and made inflammatory comments about the victim on cross-examination. In the evaluation of a claim for prosecutorial misconduct, it is not enough that the prosecutor's comments were improper, but "[t]he relevant question is whether the prosecutors' comments `so infected the trial with unfairness as to make the resulting conviction a denial of due process.'" Darden v. Wainwright, 477 U.S. 168, 181, 106 S.Ct. 2464, 91 L.Ed.2d 144 (1986) (quoting Donnelly v. DeChristoforo, 416 U.S. 637, 643, 94 S.Ct. 1868, 40 L.Ed.2d 431 (1974)); accord Broom v. Mitchell, 441 F.3d 392, 412 (6th Cir.2006), cert. denied, 549 U.S. 1255, 127 S.Ct. 1376, 167 L.Ed.2d 165 (2007). "In order to satisfy the standard for prosecutorial misconduct, the conduct must be both improper and flagrant." Broom, 441 F.3d at 412; accord Boyle v. Million, 201 F.3d 711, 717 (6th Cir.2000). *256 We consider four factors in determining whether a prosecutor's improper conduct is flagrant:
(1) the likelihood that the remarks of the prosecutor tended to mislead the jury or prejudice the defendant; (2) whether the remarks were isolated or extensive; (3) whether the remarks were deliberately or accidentally made; and (4) the total strength of the evidence against the defendant.
Broom, 441 F.3d at 412 (quoting Bates v. Bell, 402 F.3d 635, 641 (6th Cir.), cert. denied, 546 U.S. 865, 126 S.Ct. 163, 163 L.Ed.2d 150 (2005)); accord Boyle, 201 F.3d at 717.
The Ohio Supreme Court rejected Smith's prosecutorial-misconduct arguments, finding that neither instance of alleged misconduct was improper. As to the prosecutor's remarks during closing arguments, the court found that "[t]he prosecutor's reference to defendant's lack of remorse may have been intended to question his credibility." Smith, 684 N.E.2d at 689. We agree that this statement was not improper in the context of this case. On direct examination, Smith testified that he cried when he told his friend James Baker about the murders; this contradicted Baker's testimony that Smith was laughing when he told Baker about slitting Lewis Ray's throat. The prosecutor's reference to Baker's testimony that Smith lacked remorse was proper to discredit Smith's contradictory testimony.
Regarding Smith's cross-examination, the state court concluded that "[b]ecause defendant claimed that parts of his written confession to police were untrue, it was not improper for the prosecutor to ask defendant to identify those parts of his confession" and that because "[d]efendant's testimony also conflicted with the crime-scene photographs ... it was not improper for the prosecutor to question defendant about the photographs in evidence." Id. at 689. Although we believe that some of the prosecutor's conduct on cross-examination may have been improper, particularly his questioning of Smith about the "kind of sounds [Ray Lewis] made when you ripped his throat from ear to ear," Trial Tr. at 889, we cannot conclude that this conduct was flagrant such that it "so infected the trial with unfairness as to make the resulting conviction a denial of due process." Darden, 477 U.S. at 181, 106 S.Ct. 2464. The comments were isolated, do not seem to have been purposeful, and any prejudice was slight, as the jury already had graphic photographs of the crime scene. Further, the evidence against Smith, particularly the crime-scene photographs, the physical evidence, Baker's testimony, and the statement Smith gave to police after he was arrested, was very strong. Overall, we do not find unreasonable the conclusion of the Ohio Supreme Court that "no misconduct occurred that would have affected the fairness of the trial." Smith, 684 N.E.2d at 689-90.

C. Ineffective Assistance of Counsel
Smith argues that his trial counsel performed ineffectively by failing to develop and introduce at the suppression hearing evidence of Smith's intoxication at the time of his arrest and interrogation and of his low mental capacity. The standard for claims of ineffective assistance of counsel was set out by the Supreme Court in Strickland v. Washington, 466 U.S. 668, 104 S.Ct. 2052, 80 L.Ed.2d 674 (1984). "Claims of ineffective assistance of counsel have `two components: A petitioner must show that counsel's performance was deficient, and that the deficiency prejudiced the defense.'" Mason v. Mitchell, 543 F.3d 766, 772 (6th Cir.2008) (quoting Wiggins v. Smith, 539 U.S. 510, *257 521, 123 S.Ct. 2527, 156 L.Ed.2d 471 (2003)). To establish that counsel was deficient, "the defendant must show that counsel's representation fell below an objective standard of reasonableness." Strickland, 466 U.S. at 688, 104 S.Ct. 2052. To establish prejudice, "[t]he defendant must show that there is a reasonable probability that, but for counsel's unprofessional errors, the result of the proceeding would have been different. A reasonable probability is a probability sufficient to undermine confidence in the outcome." Id. at 694, 104 S.Ct. 2052. After setting forth the Strickland standard, the Ohio Supreme Court denied Smith's claim because "[i]ntoxication affecting one's state of mind, absent coercive police activity, would be an insufficient reason to exclude his voluntary confession." Smith, 684 N.E.2d at 690 (citing Colorado v. Connelly, 479 U.S. 157, 107 S.Ct. 515, 93 L.Ed.2d 473 (1986)).
Although we believe that Smith has likely established the first Strickland prong, that his counsel's performance was deficient, we conclude that Smith has not shown that this deficiency prejudiced the outcome of the suppression hearing. We first note, however, that the Ohio Supreme Court's reliance solely on Colorado v. Connelly was misplaced, because Smith did not argue only that evidence of intoxication would have shown that his confession was involuntary, but also Smith contended that this evidence would have shown that he did not knowingly and intelligently waive his Miranda rights. A suspect in custody may waive his Miranda rights "provided the waiver is made voluntarily, knowingly and intelligently." Miranda v. Arizona, 384 U.S. 436, 444, 86 S.Ct. 1602, 16 L.Ed.2d 694 (1966). The Supreme Court has stated that this inquiry involves "two distinct dimensions":
First the relinquishment of the right must have been voluntary in the sense that it was the product of a free and deliberate choice rather than intimidation, coercion, or deception. Second, the waiver must have been made with a full awareness both of the nature of the right being abandoned and the consequences of the decision to abandon it. Only if the totality of the circumstances surrounding the interrogation reveal both an uncoerced choice and the requisite level of comprehension may a court properly conclude that the Miranda rights have been waived.
Colorado v. Spring, 479 U.S. 564, 573, 107 S.Ct. 851, 93 L.Ed.2d 954 (1987) (quoting Moran v. Burbine, 475 U.S. 412, 421, 106 S.Ct. 1135, 89 L.Ed.2d 410 (1986)). As this court has explained, while Connelly establishes the standard for showing that a confession was involuntary, the question of whether a Miranda waiver was knowing and intelligent is a separate question. Clark v. Mitchell, 425 F.3d 270, 283 (6th Cir.2005). In considering the merits of Smith's underlying Miranda claim, the Ohio Supreme Court improperly looked only at whether Smith would have been able to satisfy the first element of the inquiry, voluntariness.
The Ohio Supreme Court's application of Strickland, however, was not unreasonable. In evaluating whether a Miranda waiver was knowing and intelligent, a court will "apply a totality of the circumstances test to ascertain whether [the defendant] understood his right to remain silent and to await counsel." Clark, 425 F.3d at 283. Applying this standard, we conclude that Smith has not shown that he was prejudiced by his counsel's performance. In support of his claim, Smith presents his trial testimony, as well as that of James Baker, who was with Smith the day of the arrest, that Smith had ingested *258 various prescription medications, smoked marijuana, and consumed a large amount of alcohol during the hours leading up to his arrest. Smith also presents the affidavits of three experts detailing Smith's limited mental capability and long history of polysubstance abuse. Dr. Charles T. Kandiko, Ph.D., who conducted a neuropharmacological evaluation of Smith, concluded that "Mr. Smith was intoxicated at the time of his arrest and interrogation by police" and therefore "was not in a mental state to competently decide whether he wanted to talk to the police or not." J.A. at 765 (Kandiko Evaluation at 6). Even in light of this evidence however, Smith would not have been able to show that his waiver was not knowing and intelligent under the totality of the circumstances. The testimony of the officers at the suppression hearing showed that Smith was given his Miranda warnings at least three times and that he affirmatively stated that he did not need a lawyer and that he would talk about the killings. After completing a typed statement, the detective had Smith read the statement back to him to show that Smith could read, and Smith then signed the statement and initialed various edits. It is highly unlikely that this testimony would have been controverted by Smith had he testified at the suppression hearing. During his trial testimony, Smith stated that he recalled being advised of his rights and agreed that he was "willing to talk to [the detective]," "willing and ready to confess at that time to actually killing Lewis," and "had no intention of denying at all." Trial Tr. at 840-41. Smith's age and previous experience with the criminal justice system would have weighed further toward finding the waiver knowing and intelligent. See Murphy v. Ohio, 551 F.3d 485, 514 (6th Cir.2009). Moreover, Smith has submitted no evidence that his conduct during the interrogation gave the police any indication of his alleged intoxication or failure to understand his repeated waivers. See Garner v. Mitchell, 557 F.3d 257, 261-63 (6th Cir.2009) (en banc). Overall, we conclude that Smith has not shown a likelihood that the presentation of this evidence at the suppression hearing would have changed the outcome of the proceeding.[2]

D. Smith's Death Sentence for the Murder of Ruth Ray
Smith argues that his constitutional rights were violated when he was sentenced to death for the murder of Ruth Ray, who was killed by Smith's brother Randy, because the capital specifications did not require that Smith had the specific intent to murder Ruth Ray and did not genuinely narrow the class of persons eligible for the death penalty. The Ohio Supreme Court rejected these claims on direct appeal, concluding that "[t]he Eighth Amendment does not prohibit the death penalty for an accomplice whose participation in the felony murder was significant, and whose mental state is one of reckless indifference," Smith, 684 N.E.2d at 693 (citing Tison v. Arizona, 481 U.S. 137, 107 S.Ct. 1676, 95 L.Ed.2d 127 (1987)), and finding that "the jury specifically found that [Smith] purposely intended to *259 kill Ruth Ray, and that he did so with `prior calculation and design' as an aider and abettor in her aggravated murder," id.
Smith was found guilty for the murder of Ruth Ray under Ohio Revised Code § 2903.01(B) (1995), the felony-murder provision of the aggravated-murder statute in effect at the time the offenses were committed. He also was found guilty on three capital specifications, making him eligible for the death penalty: (1) "[t]he offense was committed for the purpose of escaping detection, apprehension, trial, or punishment for another offense committed by the offender," Ohio Rev.Code § 2929.04(A)(3) (1995); (2) "the offense at bar was part of a course of conduct involving the purposeful killing of or attempt to kill two or more persons by the offender," § 2929.04(A)(5); and (3) "[t]he offense was committed while the offender was committing, attempting to commit, or fleeing immediately after committing or attempting to commit ... aggravated robbery ..., and either the offender was the principal offender in the commission of the aggravated murder or, if not the principal offender, committed the aggravated murder with prior calculation and design," § 2929.04(A)(7).
In arguing that these capital specifications were deficient, Smith primarily relies on the Supreme Court's decisions in Tison v. Arizona, 481 U.S. 137, 107 S.Ct. 1676, 95 L.Ed.2d 127 (1987), and Enmund v. Florida, 458 U.S. 782, 102 S.Ct. 3368, 73 L.Ed.2d 1140 (1982). In Tison, the Court stated, "Enmund held that when `intent to kill' results in its logical though not inevitable consequencethe taking of human lifethe Eighth Amendment permits the State to exact the death penalty after a careful weighing of the aggravating and mitigating circumstances." Tison, 481 U.S. at 157, 107 S.Ct. 1676. The Court therefore held "that the reckless disregard for human life implicit in knowingly engaging in criminal activities known to carry a grave risk of death represents a highly culpable mental state, a mental state that may be taken into account in making a capital sentencing judgment when that conduct causes its natural, though also not inevitable, lethal result." Id. at 157-58, 107 S.Ct. 1676. The Supreme Court recently noted with approval that the Ohio "aggravated murder charge's intent element [does] not require any showing that [the defendant] had himself shot [the victim]. Rather, Ohio law considers aiders and abettors equally in violation of the aggravated murder statute, so long as the aiding and abetting is done with the specific intent to cause death." Bradshaw v. Stumpf, 545 U.S. 175, 184, 125 S.Ct. 2398, 162 L.Ed.2d 143 (2005).
Smith essentially argues that the state did not prove that he had the specific intent required to sentence him to death for the murder of Ruth Ray, because the capital specifications used do not require a showing of "prior calculation and design." This argument fails, however, because, the Ohio Supreme Court reasonably concluded that the jury found that Smith and his brother had planned to kill both Lewis and Ruth Ray to avoid leaving any witness, providing the requisite intent. Smith, 684 N.E.2d at 693. Smith's focus on the wording of the capital specifications ignores the fact that the specific intent to cause death is an element of the aggravated-felony-murder statute itself, which requires that "[n]o person shall purposely cause the death of another while" committing or fleeing an enumerated felony. Ohio Rev.Code § 2903.01(B) (1995) (emphasis added). The aggravated-murder statute also states that "[n]o person shall be convicted of aggravated murder unless he is specifically found to have intended to cause the death of another" and instructs that the jury *260 may not conclusively infer such intent from the fact that the person "engaged in a common design with others to commit the offense by force and violence or because the offense and the manner of its commission would be likely to produce death." Ohio Rev.Code § 2903.01(D) (1995). Therefore, regardless of the capital specifications, the jury made a finding of specific intent and purpose to cause the death of Ruth Ray when it found Smith guilty of her aggravated murder under § 2903.01. No more was required under Enmund and Tison.
Smith also argues that the jury instructions improperly added the phrase "as an aider and abettor" to the language of the third capital specification, § 2929.04(A)(7),[3] resulting in failure "genuinely [to] narrow the class of persons eligible for the death penalty or reasonably [to] justify the imposition of a more severe sentence on the defendant compared to others found guilty of murder" as required by Zant v. Stephens, 462 U.S. 862, 877, 103 S.Ct. 2733, 77 L.Ed.2d 235 (1983). Smith Br. at 35. Smith argues that this addition removed the requirement that Smith acted with prior calculation and design in the murder of Ruth Ray, which is a requirement of the capital specification for defendants, such as Smith, who were not the principal offender. This argument fails. Any confusion caused by this phrase was cleared up by the next paragraph of the jury instructions, which explained that "[p]rior calculation and design means that the purpose to cause the death was reached by a definite process of reasoning in advance of the homicide, which process of reasoning must have included a mental plan involving studied consideration of the method and the means or instrument with which to cause the death of another." Jury Instructions at 19; Trial Tr. at 978. Therefore, adding the phrase "as an aider and abettor" does not remove the requirement that Smith had "prior calculation and design" with regard to the murder of Ruth Ray. The specification required that the jury find that Smith had prior calculation and design to murder Ruth Ray, and the Ohio Supreme Court found that the jury did so find. Smith, 684 N.E.2d at 693. Neither the felony-murder provision of Ohio's aggravated-murder statute nor the capital specifications found in Smith's case violated the Supreme Court's holdings in Tison, Enmund, or Zant.[4]

E. Constitutionality of Ohio's Death-Penalty Scheme
Smith argues that Ohio's death-penalty statute is unconstitutional on its face and as applied to Smith because it imposes *261 arbitrary and unequal punishment and because it employs an inadequate system of proportionality review. The Ohio Supreme Court did not address these claims directly when raised by Smith on direct appeal, but did evaluate the appropriateness and proportionality of Smith's sentence, declining to compare it to his brother Randy's sentence. Smith, 684 N.E.2d at 694-98. To the extent that the Ohio Supreme Court did not address Smith's claims on direct appeal, we review these arguments de novo. See Howard v. Bouchard, 405 F.3d 459, 467 (6th Cir.2005), cert. denied, 546 U.S. 1100, 126 S.Ct. 1032, 163 L.Ed.2d 871 (2006).
Smith makes two arguments to support his facial challenge to Ohio's death-penalty scheme, both of which are foreclosed by prior Sixth Circuit precedent. First, Smith argues that "[p]rosecutors' virtually uncontrolled indictment discretion allows arbitrary and discriminatory imposition of the death penalty." Smith Br. at 41. We have previously rejected the same argument regarding the death-penalty system in Ohio. See Williams v. Bagley, 380 F.3d 932, 963 (6th Cir.2004), cert. denied, 544 U.S. 1003, 125 S.Ct. 1939, 161 L.Ed.2d 779 (2005). Second, Smith argues that Ohio's system of proportionality and appropriateness review, as required by sections 2929.021 and 2929.03 of the Ohio Revised Code, is inadequate. Specifically, Smith argues that the "statutes['] failure to require the jury or three-judge panel recommending life imprisonment to identify the mitigating factors undercuts adequate appellate review," Smith Br. at 43, that "[l]imiting review to only cases where the death penalty was imposed ... prevents a fair proportionality review," id., and that the Ohio Supreme Court's cursory appropriateness review violates due process. Again, we previously rejected these same challenges to Ohio's death penalty-scheme. See Buell v. Mitchell, 274 F.3d 337, 367-69 (6th Cir.2001). For these reasons, Smith's facial challenge to the Ohio death-penalty system must fail.
Smith also argues that Ohio's death-penalty scheme resulted in arbitrary and unequal punishment as applied to Smith because he was sentenced to death while his accomplice, his brother Randy, was sentenced to life in prison. This court recently rejected a similar argument in Getsy v. Mitchell, 495 F.3d 295 (6th Cir. 2007) (en banc), cert. denied, ___ U.S. ___, 128 S.Ct. 1475, 170 L.Ed.2d 299 (2008), which concluded that the Ohio Supreme Court's decision upholding the defendant's death sentence in the face of a codefendant's life sentence was neither contrary to nor an unreasonable application of clearly established federal law. 495 F.3d at 305 ("[A] defendant could not `prove a constitutional violation by demonstrating that other defendants who may be similarly situated did not receive the death penalty.'" (quoting McCleskey v. Kemp, 481 U.S. 279, 306-07, 107 S.Ct. 1756, 95 L.Ed.2d 262 (1987))). Because we are bound by the precedent established in Getsy, we must conclude that the Ohio Supreme Court's refusal to consider Randy Smith's life sentence in upholding Kenneth Smith's death sentence was neither contrary to nor an unreasonable application of clearly established federal law.

F. Constitutionality of Ohio's One-Tier System of Appellate Review of Capital Cases
Smith was the first capital inmate in Ohio subject to the amendments to the Ohio constitution that eliminated the intermediate level of appellate review for capital defendants. See Ohio Const. art. IV, § 3(B)(2). Under these amendments, which apply to all cases in which the crime was committed on or after January 1, 1995, *262 defendants sentenced to the death penalty may appeal directly to the Ohio Supreme Court as a matter of right. Ohio Const. art. IV, § 2(B)(2)(c). Smith argues that this one-tier system of appellate review for capital cases violates his equal protection and due process rights and constitutes cruel and unusual punishment under the Eighth Amendment. Because Smith was the first capital defendant in Ohio to be subject to the new system, his claims were considered in depth by the Ohio Supreme Court on direct review, which found that Ohio's system did not violate the Constitution. Smith, 684 N.E.2d at 678-85.
Regarding the equal-protection claim, Smith argues that the one-tier system unconstitutionally treats noncapital defendants more favorably than capital defendants with no rational basis for the disparate treatment. The Ohio Supreme Court correctly applied rational-basis review to Smith's claim, following the standard in Estelle v. Dorrough, 420 U.S. 534, 539, 95 S.Ct. 1173, 43 L.Ed.2d 377 (1975). As the Ohio Supreme Court explained, in certain ways capital defendants have more appeal rights than noncapital defendants under Ohio's appellate-review system, because capital defendants may appeal all issues, including noncapital convictions, to the Supreme Court as a matter of right, whereas "[o]nly two to three percent of all noncapital defendants who seek review by this court even have their cases heard." Smith, 684 N.E.2d at 682. More important, the Ohio Supreme Court articulated a rationale for treating capital defendants differently than noncapital defendants:
In addition, defendants sentenced to imprisonment are usually already serving their terms and have a great interest in expediting their appeals. Capital defendants, on the other hand, while seeking to overturn their convictions, also seek to prolong the appeal process as long as possible, using every conceivable avenue of attack, and the statistics show they have been extremely successful in Ohio.
Id. at 682. The Ohio Supreme Court's conclusion that Ohio's capital appellate system passes rational basis review is not contrary to or an unreasonable application of clearly established federal law.
Smith's due-process argument is essentially the same as his equal-protection argument: because "[d]eath is different," capital defendants should be provided more process than noncapital defendants. Smith Br. at 55. Responding to Smith's due-process argument, the Ohio Supreme Court again explained that review of right directly to the state supreme court has advantages over review of right only to the intermediate state appellate court:
Additionally, a single-tier system does not violate the capital defendant's right to due process. A direct appeal to this court affords a capital defendant significant advantages in comparison with other criminal defendants. As stated, all of a defendant's issues, both noncapital and capital, constitutional or statutory, are reviewed by the Supreme Court. A court of appeals must apply this court's existing precedents, but this court can directly reverse its own precedents. A defendant can still directly challenge existing precedent and seek change. Indeed, his or her opportunities are better, as a lower court could not reexamine precedent from the Supreme Court.
Further, this court can more readily judge both the appropriateness and proportionality of death sentences on a statewide basis, instead of the geographical limits of an appellate district. An appellate court does not have the breadth and scope of experience that this court has to review the death sentences of all eighty-eight counties and to *263 measure the appropriateness and proportionality of all cases in the state.
Smith, 684 N.E.2d at 683 (citations omitted). Further, contrary to Smith's argument, the fact that capital defendants were afforded greater rights before the amendment does not mean that the lesser rights they now have are less than the Due Process Clause requires.
Smith also argues that Ohio's one-tier system of appellate review violates the Eighth Amendment's guarantee against cruel and unusual punishment because this system does not provide for "meaningful appellate review," leading to the imposition of the death penalty in an arbitrary and capricious manner. Smith Br. at 56-57. The Supreme Court has not directly addressed the issue of whether a one-tier system of appellate review violates the Eighth Amendment, but the Court has "held that [the death penalty] could not be imposed under sentencing procedures that created a substantial risk that it would be inflicted in an arbitrary and capricious manner." Gregg v. Georgia, 428 U.S. 153, 188, 96 S.Ct. 2909, 49 L.Ed.2d 859 (1976) (citing Furman v. Georgia, 408 U.S. 238, 92 S.Ct. 2726, 33 L.Ed.2d 346 (1972)). The Supreme Court has allowed states substantial leeway in implementing procedures to assure that the death penalty is not imposed arbitrarily or capriciously, refusing "to say that there is any one right way for a State to set up its capital sentencing scheme." Spaziano v. Florida, 468 U.S. 447, 464, 104 S.Ct. 3154, 82 L.Ed.2d 340 (1984); accord Gregg, 428 U.S. at 195, 96 S.Ct. 2909.[5] One tool that states may use is an appropriateness and proportionality review by the state supreme court. See Gregg, 428 U.S. at 204-06, 96 S.Ct. 2909 (noting that the Georgia Supreme Court's proportionality review "serves as a check against the random or arbitrary imposition of the death penalty"). As discussed above, the Ohio Supreme Court reasonably rejected Smith's arguments that Ohio's scheme does not provide for meaningful adequate review for appropriateness and proportionality, noting that the Ohio Supreme Court "can more readily judge both the appropriateness and proportionality of death sentences on a statewide basis, instead of the geographical limits of an appellate district." Smith, 684 N.E.2d at 683. We therefore conclude that the Ohio Supreme Court's decision upholding Ohio's one-tier system of appellate review was neither contrary to nor an unreasonable application of clearly established federal law.

III. CONCLUSION
For these reasons, we AFFIRM the district court's denial of Smith's petition for habeas relief under 28 U.S.C. § 2254.
NOTES
[1]  Smith initially sought to appeal his convictions and sentence to the Ohio Court of Appeals, but review was denied due to then-recent changes in Ohio law limiting capital defendants to a single tier of appellate review in the Ohio Supreme Court. See Ohio Const. art. IV, §§ 2(B)(2)(c) and 3(B)(2).
[2]  We note that our analysis of Smith's ineffective-assistance-of-counsel claim would have been enhanced had an evidentiary hearing been conducted in the district court. We do not know, for example, what indication the police officers may have had that Smith was intoxicated during his interrogation. It appears from the record, however, that, after Smith's first motion for an evidentiary hearing was denied without prejudice to renew upon consideration of the Supreme Court's decision in Michael Williams v. Taylor, 529 U.S. 420, 120 S.Ct. 1479, 146 L.Ed.2d 435 (2000), his habeas counsel never filed a renewed motion. Under these circumstances, we do not find it proper to remand to the district court for an evidentiary hearing.
[3]  Section § 2929.04(A)(7) allows for imposition of the death penalty if

[t]he offense was committed while the offender was committing, attempting to commit, or fleeing immediately after committing or attempting to commit ... aggravated robbery ..., and either the offender was the principal offender in the commission of the aggravated murder or, if not the principal offender, committed the aggravated murder with prior calculation and design.
§ 2929.04(A)(7). Smith's jury instruction added the phrase "as an aider and abettor," requiring that the jury find that
[t]he aggravated murder of Ruth Ray was committed while the defendant, Kenneth Wayne Smith, acting with prior calculation and design as an aider and abettor in the commission of the aggravated murder of Ruth Ray, was committing or attempting to commit the offense of aggravated robbery.
Jury Instructions at 18; Trial Tr. at 978.
[4]  Smith also argues that § 2929.04(A)(3) violates double jeopardy because he was punished twice for the murder of Lewis Ray. This claim is not within the scope of the COA, which was concerned with whether Smith was unconstitutionally sentenced to death for the murder of Ruth Ray.
[5]  The Ohio Supreme Court noted that, at the time of Smith's direct appeal, at least thirty-three other states provided only one level of appellate review to capital defendants. Smith, 684 N.E.2d at 680-81.
