After approximately seven weeks of jury trial, the defendants were each convicted on indictments charging two counts of murder in the first degree, two counts *721of armed robbery, and two counts of armed assault in a dwelling. Although the defendants Daye and Costa were tried by one jury and the defendant DeNictolis was tried by another, the same judge presided over both trials in the same courtroom. All the witnesses, with the following two exceptions, testified to the two juries simultaneously. Karen Moodie, who was a critical Commonwealth witness, testified to the Daye-Costa jury in the absence of the DeNictolis jury, and to the DeNictolis jury in the absence of the Daye-Costa jury. Lieutenant Spartichino testified on direct examination by the prosecutor to the two juries simultaneously, but he testified on cross-examination to the two juries separately. Also, counsels’ opening statements and summations were made, and the judge’s final jury instructions were given, to the two juries separately. The defendants appealed from all the convictions. While the appeals were pending in this court, the defendants moved for a new trial and requested an evidentiary hearing in connection with their motion. A single justice of this court ordered that the motion and request for an evidentiary hearing be transferred to the Superior Court. A judge of that court, who had been the trial judge, denied both the motion and the request, and the defendants appealed. Thus, we have before us the defendants’ appeals from the convictions and from the denial of the motion for a new trial.
On appeal, the defendants argue that they were denied their Federal and State constitutional rights to a fair trial (1) by the failure of the prosecution to make timely disclosure of exculpatory evidence relating to “other suspects,” and (2) by the trial judge’s undue limitation of their cross-examination primarily with regard to other suspects. They also argue that (3) the integrity of the grand jury was impaired by false or deceptive police testimony, (4) the judge’s refusal to hold an evidentiary hearing concerning the voluntariness of out-of-court statements made by Karen Moodie and concerning her proposed testimony violated their rights to due process, (5) the judge’s failure to give promised jury instructions concerning Moodie’s credibility violated their due process rights, (6) *722under Commonwealth v. Bowden, 379 Mass. 472, 485-486 (1980), they were entitled to but did not receive instructions to the juries that they could consider the failure of the police to carry out certain investigative procedures, and (7) the judge erred in admitting certain opinion evidence concerning analysis of bullet lead. We address these several issues below. We affirm the judgments.
The jury could have found that on January 7, 1985, the bodies of Robert Paglia and his wife, Patricia Paglia, were discovered in their home in Lynnfield. Robert had been killed by a single shotgun blast to the head and Patricia had been killed by multiple gunshot wounds to the head. Robert was a major cocaine dealer, and the police found drugs, drug paraphernalia, and $90,920 in cash in the Paglia home. The jury could have found that the Paglias were murdered in the course of an armed robbery.
As we have said, Karen Moodie was a very important witness. She gave testimony over a period of several days, the relevant portions of which are set forth in this and the next eleven paragraphs. We make no distinction between the testimony she gave to the Daye-Costa jury and the testimony she gave to the DeNictolis jury because whatever differences exist are of no consequence to these appeals. Also, we omit any reference to Moodie’s testimony on cross-examination. On direct examination, Moodie testified that, in exchange for her cooperation and truthful testimony, she would be kept in protective custody, would receive treatment and counseling for her drug addiction, and would not be prosecuted for any involvement in the Paglia murders or for any crimes that she revealed during the course of the investigation. At the time of her testimony, Moodie was participating in a methadone maintenance program.
Moodie met the defendant Costa in 1980 and became romantically involved with him. They lived in various places and had a child together in 1982. In November, 1984, Moodie began living with Costa and their son in Rye, New Hampshire, at 2320 Ocean Boulevard (the Rye house). Some time later, the defendant DeNictolis moved into the house. *723In late November or early December, 1984, Moodie had a conversation with Costa about how they were going to make ends meet. When Moodie asked Costa what they were going to do for money, Costa replied that he was going to “make a move” on a cocaine dealer. In the next few weeks, Costa and DeNictolis held additional conversations about moving on a cocaine dealer. They discussed gaining entry to the house by disguising themselves as police officers. Costa first referred to the cocaine dealer as “Pag, that maggot,” but later referred to him as “Paglia.”
In December, 1984, Costa began to speak to Moodie about a third participant in the plan. He mentioned the name of Dennis Daye. Daye arrived at the Rye house during the early morning hours of a day late in December. A conversation ensued among Daye, DeNictolis, and Costa, and Costa mentioned a move on a cocaine dealer. Moodie also heard additional talk about the plan. She heard Costa on the telephone asking what the patrolmen were wearing for outer coats in Boston. Within a few days, two men showed up at the house with police uniforms.
At some point in December, Costa asked Moodie if her brother, who had a New Hampshire identification card, would go to a gun store and buy some ammunition. Moodie picked up her brother and took him to a gun store in New Hampshire, where he purchased some ammunition. At least one box of bullets was of the “hollow type.” In the few remaining days before the murders, the three defendants spoke of the Paglias. Moodie hemmed the pants on one of the uniforms because they were too long. Costa made another call about what Boston police were wearing for winter coats.
On January 5, 1985, the defendants began to talk about the specifics of the plan. They had conversations about who would enter the house first. They also talked about what they would do about the Paglias’ dogs. Costa asked DeNictolis, who was a friend of the Paglias, what time the babysitter came to the Paglias’ house. In ,the late afternoon, the men practiced shooting guns in the cellar. They had a carbine, a .38 revolver, and another small handgun. A few weeks prior *724to this, Costa and Moodie had fired guns in 'the cellar. When they finished, Costa picked up the spent bullets, put them in an empty felt bag and buried it near a rock in the back of the house.
On January 5, Costa told Moodie that he “wanted to make the move the next day” and that they would “take off the coke dealer.” Costa told Moodie that he planned to rob the dealer of his money. When Moodie asked Costa what would happen if he were recognized, Costa replied, “I’ll have to blow him away.”
On January 6, 1985, Costa told Moodie to wrap some empty gift boxes. When the boxes were wrapped, Costa, DeNictolis, and Daye left the house. Costa was wearing a suit and DeNictolis was wearing casual clothes. Daye was wearing a police uniform. Daye had a .38 caliber pistol and Costa had a smaller handgun. Costa’s carbine was in- one of the gift-wrapped boxes. The three left the house at about 2:00 p.m. and returned about two and one-half hours later.
Upon returning, the three defendants entered the Rye house hurriedly, and Costa called Moodie into the bedroom. He began to take money in the form of packages of bills with elastic bands around them out of his pockets. Costa separated out a few packages and told Moodie, “This is for us. Just put that in my top drawer. They don’t know I have this.” Moodie asked him about the amount of money and he replied, “We didn’t get what we expected.” Later in the evening, at approximately 8:30 p.m., Costa took the police uniform that Daye had worn, rolled it up in a ball, and put it in a paper bag. He then placed in the bag a rock from the fireplace. He also put one of the holsters, a gun, a hat, and some loose bullets from his right hand pocket into the bag. He instructed Moodie to throw the bag off the “old bridge down the street,” and Moodie did so.
When Moodie returned to the house, the three defendants were engaged in conversation, and Moodie heard “bits and pieces” of it. The conversation was about “who would find the bodies.” Costa asked DeNictolis when the babysitter *725came to the Paglias’. The three had some drinks and watched television, including the news.
While Costa and Moodie were in bed that night, Costa revealed to Moodie that Robert Paglia had recognized him. Costa said that Daye was with Patricia Paglia and he, Costa, was with Robert. Upon hearing a gunshot, Robert asked if that was his wife. Costa replied that it was and then told Robert to “[t]ake a good look at [Costa], because it’s the last face you’ll see.” Costa then shot Robert.
Costa also told Moodie that he had entered the house first. Daye was behind him. They knocked on the door and Patricia Paglia opened the door. Costa told Patricia that he wanted to speak to her husband regarding a DEA investigation. Robert was taking a shower but came downstairs. Daye then took Patricia into the bathroom. Robert was made to lie on the floor with his hands behind him. Moodie asked Costa if it was messy, to which he replied, “Not really.” Costa said that Robert was killed in a family room or a cellar.
Following an argument with Costa, Moodie left the Rye house in February, 1985. She was arrested in Seabrook, New Hampshire, on July 5, 1986, on an unrelated offense, and on that date she was interviewed by Lieutenant Spartichino of the Massachusetts State Police. Moodie took Spartichino to the Rye house and the Rye bridge from which she said she had thrown some item on January 6, 1985.
We need not summarize the remainder of the evidence. We shall refer to as much of it as is necessary to our discussion of the issues. We observe, however, that there was considerable evidence that, at the probable time of the murders, the defendant Daye was at a wake. That alibi, had it been believed by the juries, would have contradicted Moodie’s testimony and thus would have benefited the defense of Costa and DeNictolis as well as Daye.
1. Nondisclosure of Exculpatory Evidence.
We turn now to a discussion of the issues raised by these appeals. The first issue is whether the defendants’ Federal or State constitutional rights to due process, to confront witnesses, and to present all matters of proof favorable to their *726defense were violated by the prosecutor’s withholding of exculpatory evidence concerning other suspects. The defendants point to several reports authored by police personnel as containing the exculpatory evidence that they claim was wrongfully withheld. One of those reports is the so-called Zuk report, which the prosecution furnished to the defendants on May 4, 1989, nearly two years after the trial. The trial had taken place in the summer of 1987. Other reports asserted to have been wrongfully withheld are the Macchia report, which the prosecution provided to the defendants on March 31, 1989, and the Murphy report, which one of the defense counsel has informed us was sent to him by a lawyer in another case in September, 1988. In addition, the defendants assert that their rights were violated by the Commonwealth’s failure to furnish until after trial a second Murphy report, the so-called Cox report, and a transcribed interview between Boston police detective Murphy and Elizabeth DeNunzio.
The Zuk report was authored by State police Trooper Norman C. Zuk. Dated July 25, 1986, approximately one year before the trial, it related to a telephone call Zuk had received on July 20, 1986, from Peter J. Early and a followup interview between Zuk and Early on July 21. Early stated in the telephone call that he had “first hand knowledge that [John Doe] who is an attorney in Boston was responsible for the murder of the Paglias.” According to the report, Zuk met with Early the next day and Early told Zuk that Doe was in the cocaine supply business, that “Paglia had done business with [Doe] and . . . Paglia owed money to [Doe] for cocaine transactions.” The report says that Early told Zuk that Doe had stated to Early in a telephone conversation that “Paglia got it in the face” (actually, Paglia was shot in the head), and that Doe “had earlier advised [Early] that Paglia was in trouble with some people.” Zuk orally communicated this information to Lieutenant Thomas Spartichino of the Essex County crime prevention and control unit (CPAC), the officer in charge of the Paglia murder investigation, which was being conducted in conjunction with the Essex County district attorney’s office. Zuk forwarded the written report to *727his supervisor at that time but not to Spartichino. Trooper Zuk did not become a member of the Essex County CP AC until approximately three months after July 25, 1986. Thus, Zuk was not a member of that unit when he talked with Early, wrote his report, or talked with Spartichino about it. The report was not in the Paglia murder investigation file and the prosecution did not learn of it until May, 1989.
The Commonwealth does not contend that, because Zuk was not a member of the Essex County CP AC when he talked to Early and then provided a written report of that conversation to his supervisor and told Spartachino about it, the prosecutor owed no duty to provide Zuk’s report to the defendants before trial. Instead, the Commonwealth’s position is that, in any event, the written report and oral information were not exculpatory and material and therefore there was no duty of disclosure. We shall address that issue.
The defendants’ argument with respect to the alleged suppression of the Zuk and other reports focuses on two trial court orders that were issued before trial in response to the defendants’ motions for specific discovery regarding other suspects and for exculpatory evidence. The defendants claim that the prosecution did not comply with those orders. The relevant part of the order with respect to the first motion was as follows: “[T]he prosecution shall furnish any evidence that either may be within its possession or which may be available to it, that 1) any other person may have committed the offenses that are the subject of these indictments; and 2) any other person had a motive, opportunity, and ability to commit the offenses that are the subject of these indictments.” With reference to the defendants’ motion for exculpatory evidence, the judge ordered the Commonwealth “to disclose, including matters not presently known but hereinafter acquired all evidence which the prosecution knows, or in the exercise of due diligence ought to know 1) which is favorable to the defendants and which is material to either guilt or innocence, punishment or calls into question a material element of the prosecution’s version; 2) challenges the credibility of a Com*728monwealth witness; or 3) [corroborates] the defendant’s denial of guilt.”
In Brady v. Maryland, 373 U.S. 83, 87 (1963), the United States Supreme Court held that “suppression by the prosecution of evidence favorable to an accused upon request violates due process where the evidence is material either to guilt or to punishment, irrespective of the good faith or bad faith of the prosecution.” The goal of that rule is avoidance of unfair trials. As the Court explained, “[s]ociety wins not only when the guilty are convicted but when criminal trials are fair; our system of the administration of justice suffers when any accused is treated unfairly. An inscription on the walls of the Department of Justice states the proposition candidly for the federal domain: ‘The United States wins its point whenever justice is done its citizens in the courts.’ A prosecution that withholds evidence on demand of an accused which, if made available, would tend to exculpate him or reduce the penalty helps shape a trial that bears heavily on the defendant. That casts the prosecutor in the role of an architect of a proceeding that does not comport with standards of justice . . . .” (Footnote omitted.) Id. at 87-88.
Thirteen years after Brady, the Supreme Court held in United States v. Agurs, 427 U.S. 97, 107-108 (1976), that, just as the prosecution must disclose to a defendant obviously exculpatory evidence in its possession when the defendant makes a general request for such material, the prosecution must make a similar disclosure even in the absence of a request. The Court observed that “the prudent prosecutor will resolve doubtful questions in favor of disclosure.” Id. at 108. However, the Court also made the point, which is especially relevant to our discussion of the Zuk report, that nondisclosure does not require a posttrial remedy unless, based on the entire record, the nondisclosure was constitutionally material. The Court stated: “But to reiterate a critical point, the prosecutor will not have violated his constitutional duty of disclosure unless his omission is of sufficient significance to result in the denial of the defendant’s right to a fair trial.” Id.
*729The defendants say that, for the purpose of establishing the appropriate standard of constitutional materiality, the orders allowing their pretrial discovery motions ought to be treated as specific, not general, requests for exculpatory evidence. We concur. We also agree with the defendants that the standard of materiality under Massachusetts law is more favorable to defendants when the omitted material was specifically requested than when the request was a general request. In Commonwealth v. Gallarelli, 399 Mass. 17, 21 n.5 (1987), we declined to follow the unitary approach adopted for use in the Federal courts by the United States Supreme Court in United States v. Bagley, 473 U.S. 667 (1985). In Bagley, the Court held that, regardless of whether there was no request, a general request, or a specific request, “[t]he evidence is material only if there is a reasonable probability that, had the evidence been disclosed to the defense, the result of the proceeding would have been different. A ‘reasonable probability’ is a probability sufficient to undermine confidence in the outcome.” Id. at 682. In Commonwealth v. Gallarelli, supra at 21 n.5, we expressed our decision to adhere to the approach to materiality that we had taken in Commonwealth v. Ellison, 376 Mass. 1, 23-25 (1978) — an approach that we characterized in Gallarelli as including “more prudent safeguards of defendants’ rights.” Therefore, consistent with Ellison and Gallarelli, our inquiry with respect to the materiality of the Zuk report is whether we can be confident that, even if the prosecution had supplied the report to the defendants in timely fashion, the report or available evidence disclosed by it would not have influenced the jury. Commonwealth v. Gallarelli, supra at 23. Commonwealth v. Ellison, supra at 24-25.
Applying that standard, we conclude that the Zuk report was not material. We are confident that, even if the defendants had had the report before trial, the outcome of the trial would not have been different. The significance of Early’s telephone report to Zuk that Early had first hand knowledge that Doe was responsible for the Paglia murders was considerably diminished by the following day’s interview in which *730it became clear that Early’s so-called first hand knowledge was no more than his conclusion based on Doe’s statements that “Paglia owed money to [Doe] for cocaine transactions,” “Paglia got it in the face,” which was not entirely accurate, and “Paglia was in trouble with some people.” Nothing in the record suggests that Early would have been a source of admissible evidence favorable to the defendants. See Commonwealth v. Cinelli, 389 Mass. 197, 214-215, cert. denied, 464 U.S. 860 (1983). Surely it is unlikely that Doe would have testified that he had committed the murders. The defendants do not demonstrate how they would have been aided by the Zuk report. We are not persuaded by the defendants’ speculation that, had the report surfaced before trial or at trial, the trial judge would have allowed the defendants more latitude in cross-examining police witnesses concerning “other suspects.”
The Zuk report is only one of several reports that the defendants argue were unlawfully suppressed. We next discuss the Macchia report. Vincent Macchia was the chief investigating officer for the Lynnfield police department in connection with the Paglia murders. He authored a report of a telephone conversation that he had engaged in with Elizabeth DiNunzio on September 1, 1986. According to the Macchia report, DiNunzio identified herself to Macchia on the condition that her identity remain confidential. DiNunzio told Macchia that someone “very high up in the underworld” had told her that Costa, one of the defendants here, had killed her brother, Vincent Limoli, and had also killed the Paglias. DiNunzio stated that, on hearing the news of the Paglia homicides on television, Limoli said, “Oh my God, Bobby was a good friend of mine. It must have been over Cocaine. Bobby was a big Cocaine dealer.” DiNunzio also told Macchia, according to the report, that “Limoli and one Frank [Salemme] Jr. were doing drug rip-offs,” and that Limoli’s murder was in conjunction with his having been set up as part of a rip-off. Lastly, the Macchia report stated that DiNunzio “has also been in contact with a Detective Murphy at the Boston [police department] Homicide Unit . . . .” Al*731though the record is unclear, it appears that the prosecution was in possession of the Macchia report long before the trial, but first provided it to the defendants on March 31, 1989, long after the trial.
As we have said in connection with our discussion of the Zuk report, we treat the judge’s orders regarding the defense motions for specific discovery regarding other suspects and for exculpatory evidence as the equivalent of specific requests for exculpatory evidence. Again, applying the Gallarelli materiality test to determine whether the Commonwealth’s withholding of the Macchia report deprived the defendants of a fair trial, we inquire whether we can be confident that, if the prosecution had supplied the report to the defendants in timely fashion, the report or available evidence disclosed by it would not have influenced the jury. Commonwealth v. Gallarelli, supra at 23. We have that confidence. The main thrust of the report is that DiNunzio had blamed the Paglia killings, as well as the murder of Limoli, on the defendant Costa. Such a statement would have provided no “significant aid” to the defendants’ case. Commonwealth v. Gregory, 401 Mass. 437, 442 (1988). Therefore, the Macchia report was not constitutionally material, and its suppression did not result in an unfair trial.
In addition to the orders we have discussed above in connection with the Zuk report, the judge, who ultimately was the trial judge, allowed two other pretrial defense motions that are also relevant to the Macchia report. The first was Costa’s motion to compel production of police department reports. The judge allowed that motion, but “as to the reports having to do with, or related to these defendants only.” The judge denied the motion “as to the reports with respect to time, date and manner in which any undisclosed witness may have given information which resulted in the arrest or indictment returned, except insofar as the same may be called as a witness for the Commonwealth.” The judge also allowed another pretrial defense motion for police reports in part, ordering the Commonwealth to make available “any and all reports or written memorandum of Vincent Macchia . . . *732concerning [his] investigation of the Paglia homicides made on or about January 7, 1985, or if concerning these defendants with respect to the Paglia homicide at any time thereafter.”
The defendants are not aided with respect to the Macchia report by the judge’s having allowed their motions for police reports and for reports and memoranda of Macchia concerning his investigation of the Paglia murders. Not only was the Macchia report protected by the judge’s denial of Costa’s motion to compel production of police department reports to the extent that the reports contained information obtained from an “undisclosed witness,” but also the defendants have shown no prejudice as a result of their not having the Macchia report at the time of trial.
Because the Macchia report stated that Elizabeth DiNunzio had spoken with Detective Murphy at the Boston police department homicide unit, the defendants argue that timely disclosure of the Macchia report would have led them to the Murphy report and that they would have benefited from having the Murphy report. The Commonwealth did not furnish the Murphy report to the defendants at any time. We were informed at oral argument that defense counsel acquired the Murphy report from another attorney who sent it to him when it surfaced in another case. The Murphy report was authored by Detective Paul J. Murphy of the Boston police department on November 27, 1985, in connection with the Boston police department’s investigation of the murder of Vincent J. Limoli, Jr. On page 4 of the report, Murphy states that “during our investigation information has been received concerning the Murder of Robert Paglia and his wife which occurred in Lynnfield, Mass, during the month of December of 1984. Our information has revealed that Pasquale Barone, Vincent Limoli and one Francis P. Salemme ... of 113 Cimmeron Dr. Seabrook, N.H. . . . were involved in the murder of Paglia over drugs. . . .”
The defendants raise numerous claims, both factual and legal, regarding the Murphy report. First, the report, which was retained by the Boston police, gave “113 Cimmeron *733Drive, Seabrook, New Hampshire,” as the address of Francis P. Salemme. Karen Moodie, the main Commonwealth witness, also lived at this address, allegedly in an apartment in the same complex as Francis Salemme. Moodie states in a posttrial affidavit that she never knew Salemme and was never aware that they resided in the same apartment complex. In addition, an affidavit of Jean Paquette, the keeper-of-the-records for the Cimarron Realty Trust, which owns and manages the Cimarron Apartment Complex in Sea-brook, states that the records indicate that no one by the name of Francis P. Salemme has ever been listed as an occupant of any apartment within the Cimarron Apartment Complex. The defendants allege a connection between Moodie, Limoli and Salemme.
The defendants also point out some additional reports generated by the Boston police department in connection with the murder of Limoli. A report from Detective Lieutenant James Cox states that Limoli “was murdered as a result of a dispute over a drug deal.” Another report written by Detective Murphy concerns a conversation that he had with State Trooper Lynch, who assisted in the investigation of the Paglia murders, in which Lynch informed Murphy that the Essex County district attorney’s office was planning to indict Richard Costa and Dennis Daye for the murder of Robert Paglia. The defendants allege that this report by Detective Murphy “establishes a significant liaison or coordinating relationship between Murphy and [Lynch] in the nature of a joint, or cooperative, investigation with respect to . . . Costa.” In addition, the defendants point to other evidence, which we need not repeat, to support their contention that the Boston police and the Essex County district attorney’s office were conducting a joint investigation. We have examined the record and we are satisfied that it does not warrant the conclusion that a joint investigation was conducted. The Boston police were investigating the Limoli murder. The Essex County district attorney’s office was investigating the Paglia murders. Nothing in the record suggests that the prosecutor in this case has access to the Boston police department files. *734The result is that the Commonwealth did not suppress the Boston police department reports, and thus we need not inquire whether those reports were exculpatory and material. It is clear that a “prosecutor cannot be said to suppress that which is not in his possession or subject to his control,” Commonwealth v. Donahue, 396 Mass. 590, 596 (1986), and thus “[ojrdinarily the prosecutor’s obligation to disclose information is limited to that in the possession of the prosecutor or police.” Commonwealth v. Liebman, 379 Mass. 671, 675 (1980). The prosecutor is not expected “to produce exculpatory evidence held by government agencies other than the prosecutor and police.” Commonwealth v. Donahue, supra at 597. The “police” to which that rule applies are those police who are participants in the investigation and presentation of the case. “The prosecuting attorney’s obligations . . . extend to material and information in the possession or control of members of his staff and of any others who have participated in the investigation or evaluation of the case and who either regularly report or with reference to the particular case have reported to his office.” Commonwealth v. St. Germain, 381 Mass. 256, 261-262 n.8 (1980), quoting A.B.A. Standards for Criminal Justice, Standards Relating to Discovery and Procedure Before Trial 2.1 (d) (Approved Draft 1970). See Commonwealth v. Waters, 410 Mass. 224, 229 (1991); Commonwealth v. Manning, 373 Mass. 438, 442 n.5 (1977). The Boston police reports were not within the control of the Essex County prosecution team. Therefore, the defendants’ contentions with respect to those reports lack merit.
Because of our conclusion that the defendants had no right to the Boston police reports prior to trial, we review the reports as newly discovered evidence. “A defendant seeking a new trial on the ground of newly discovered evidence must establish both that the evidence is newly discovered and that it casts real doubt on the justice of the conviction.” Commonwealth v. Grace, 397 Mass. 303, 305 (1986). The evidence “not only must be material and credible . . . but also must carry a measure of strength in support of the defendant’s position” (citation omitted). Id. We note also that the *735motion judge was the trial judge and thus is entitled to special deference. Id. at 307. We accept the motion judge’s conclusion that “there is no[] realistic possibility that [the Boston police department reports] would or could have been utilized by any defendant in a way in which either of the two juries would have been influenced in any significant way.”
2. Limitations on Defendants’ Cross-Examination.
We turn now to the second issue, whether the defendants were denied a fair trial by the judge’s undue limitation of their cross-examination especially, but not exclusively, with regard to police investigation of other individuals characterized by the defense as “other suspects.” The defendants contend that they should have been, but were not, permitted to elicit from Trooper Lynch on cross-examination the identities of certain persons whom the police had information about or had investigated. The defendants also assert that they were improperly denied cross-examination concerning the locations in which investigative activity occurred. “A defendant ... is entitled to a reasonable cross-examination of witnesses against him, but, consistent with that principle, the scope of cross-examination rests in the sound discretion of the trial judge. The burden of showing an abuse of that discretion, an abuse that must be shown on the trial record, rests on the party claiming it, in this case the defendant[s].” Commonwealth v. Weichel, 403 Mass. 103, 105 (1988). The judge acted well within his discretion. He permitted the defendants to elicit testimony from Trooper Lynch that, as a result of an anonymous telephone call, he did “numerous checks of various names supplied to [him],” and that he visited various towns during his investigation. Lynch also testified that the approximately twenty-five people he investigated as a result of telephone information were persons other than the defendants. The defendants were not entitled, in addition, to the names of people and places in the absence of some demonstration by the defendants that such evidence would tend to show that persons other than the defendants had murdered the Paglias or that the police bungled the investigation or otherwise avoided pursuit of the truth. No such showing was *736made or promised. For all that appears in the record, defense counsel had no legitimate purpose in seeking the disclosure of names of persons concerning whom there was no reliable evidence beneficial to the defendants.
Our analysis of the cross-examination of Trooper Lynch applies equally to the defendants’ claims of improper restrictions on the cross-examination of Lieutenant Spartichino. The defendants elicited testimony from Lieutenant Spartichino that in the course of his supervision of the Paglia murder investigation, he had investigated other suspects. The judge sustained objections to questions relative to methods of police investigation, the names and number of people investigated, and locations of sites investigated. Here, too, the judge acted well within his discretion. The record shows that the defendants vigorously cross-examined Spartichino and were allowed to explore in detail the investigation of the Paglia homicides. We cannot say on this record that “the substantial rights of [the defendants] are clearly shown to have been prejudiced.” Commonwealth v. Fuller, 399 Mass. 678, 684 (1987).
Finally, with respect to the defendants’ assertion that their cross-examination was unfairly restrained, we mention briefly the claim that the judge improperly disallowed defense questions put to Trooper Lynch regarding his opinion as to whether drug paraphernalia seized from the Paglia residence were consistent with mere personal use or, instead, with distribution of cocaine. No one at the trial disputed that the Paglias were cocaine dealers. The trooper’s opinion would not have aided the defense. Its exclusion was not error.
3. Impairment of the Grand Jury.
We next address the defendants’ argument that the misconduct of the police and prosecutor substantially impaired the integrity of the grand jury in violation of the defendants’ rights under art. 12 of the Declaration of Rights of the Massachusetts Constitution and the Fifth and Fourteenth Amendments to the United States Constitution. Our rejection above of the defendants’ claims that the Commonwealth withheld exculpatory and material evidence substantially dis*737poses of their further claim that the “deliberate omission of exculpatory evidence” impaired the integrity of the grand jury. “[W]e have indicated that a grand jury should be told of known exculpatory evidence that would greatly undermine the credibility of an important witness. See Commonwealth v. Connor, 392 Mass. 838, 854 (1984).” Commonwealth v. Mayfield, 398 Mass. 615, 620-621 (1986). However, after reviewing the several contentions made by the defendants concerning matters not brought to the grand jury’s attention, we are satisfied that there is no danger that this principle has been violated in this case.
In addition to their contention that the prosecution withheld exculpatory evidence from the grand jury, the defendants argue that the integrity of the grand jury was impaired by false testimony given by Lieutenant Spartichino. Specifically, the defendants assert that Spartichino falsely testified to the grand jury when he said that Karen Moodie had come forward because she was “conscience stricken.” We cannot say that Moodie was not conscience stricken or that Spartichino did not think she was. Most importantly, however, in light of the fact that the grand jury was informed that Moodie was under arrest for larceny at the time she agreed to speak to the police, we are content that there was no distortion of the truth that was significant enough to reach the level of impairment of the integrity of the grand jury proceedings within the rule of Commonwealth v. Mayfield, supra at 621, and Commonwealth v. O’Dell, 392 Mass. 445, 446-447 (1984).
4. Denial of Pretrial Evidentiary Hearing.
We turn now to the defendants’ argument that they were entitled to a pretrial evidentiary hearing at which the Commonwealth would have had the burden of proving that Karen Moodie’s out-of-court statements and testimony were voluntary. No such hearing was granted — nor was it required. In their brief, the defendants have cited several Supreme Judicial Court and Federal cases to support their position. Those cases, however, are not on point. In each of the cases decided by this court, the police discovered a witness through an ille*738gality perpetrated on the defendant. See Commonwealth v. Sperrazza, 399 Mass. 1001, 1002 (1987); Commonwealth v. Lahti, 398 Mass. 829, 837 (1986), cert. denied, 481 U.S. 1017 (1987); Commonwealth v. Caso, 377 Mass. 236, 237-240 (1979). That did not occur in this case. The Federal cases cited by the defendants involve out-of-court statements by witnesses used by the government at trial either to impeach or rehabilitate witnesses. See LaFrance v. Bohlinger, 499 F.2d 29, 31 (1st Cir.), cert. denied sub nom. LaFrance v. Meachum, 419 U.S. 1080 (1974); Vargas v. Brown, 512 F. Supp. 271, 275 (D.R.I. 1981); United States ex rel. Blackwell v. Franzen, 546 F. Supp. 151, 153 (N.D. Ill. 1981). Those cases provide no support for the defendants’ position. Here, the Commonwealth never tried to introduce a prior out-of-court statement of Moodie.
5. Jury Instructions Concerning Witness Credibility.
The next issue for discussion is whether the judge’s failure to give promised jury instructions concerning Moodie’s credibility violated the defendants’ due process rights. Evidence was presented at trial that the prosecutor promised not to prosecute Moodie for her involvement in any crimes revealed during the course of interviews related to the Paglia murder investigation, and that she would receive drug rehabilitation treatment in exchange for her truthful testimony at trial in this case. Over defense counsel’s objection, the juries also heard evidence that, as part of this agreement between the prosecutor and Moodie, she would be placed in protective custody by the State police if she agreed to testify against the defendants. The defendants jointly requested that the juries be instructed that in determining the credibility of a witness and the weight to be given to his or her testimony, the jury may consider, among other things, “the witness’s motive for testifying; and, of course the interest or lack of interest the witness may have in the outcome of the case.” In addition, Daye and Costa requested an instruction “[tjhat the jury should take great care in evaluating the testimony of a witness who has been promised the inducement of non-prosecution.” At the jury charge conference, in response to the *739defendants’ requests for the above-mentioned instructions, the judge said, “I’m going to give, ... in essence, all of those [instructions] that you asked.”
The judge instructed the Daye-Costa jury as follows: “You consider from — where is the witness coming from? Has he got any interest in the outcome? Does he have any favor to one side or the other?” The judge instructed the DeNictolis jury separately: “You can also consider . . . whether [the witness] has got bias or favor for one side or the other.” The judge also instructed each jury to consider such factors as the witness’s demeanor, prior inconsistent statements, and criminal records in determining the credibility of the witness. Following the charge to the Daye-Costa jury, the defendants objected to the judge’s instruction relating to the credibility of a witness and requested that the judge instruct the jury that they could not reach their verdict on the uncorroborated testimony of an immunized witness. No such instruction was given.
We have ruled that “[t]he judge is not required to grant a particular instruction so long as the charge, as a whole, adequately covers the issue,” Commonwealth v. Anderson, 396 Mass. 306, 316 (1985). Furthermore, “[i]n this Commonwealth a ‘judge is not required to give a cautionary instruction to the jury’ on accomplice credibility. Commonwealth v. Watkins, 377 Mass. 385, 389 [cert. denied, 442 U.S. 932] (1979) . Commonwealth v. Liebman, 379 Mass. 671, 679 (1980) . Commonwealth v. Allen, 379 Mass. 564, 584-585 (1980).” Commonwealth v. Griffith, 404 Mass. 256, 265 (1989).
The defendants argue that our decision in Commonwealth v. Ciampa, 406 Mass. 257 (1989), more than two years after the defendants’ trial concluded, requires that the convictions be reversed because, in light of the nonprosecution agreement between the Commonwealth and Moodie, the judge was required, but failed, to caution the juries to study Moodie’s testimony with great care. The Ciampa case, however, does not help the defendants. In Ciampa, we held that the prejudicial admission of certain portions of a written plea agreement be*740tween the prosecution and a witness, including repeated references to the witness’s obligation to tell the truth, was not cured by the judge’s charge. Id. at 262-263. The danger the court feared in Ciampa was that the jury would believe the witness was telling the truth because his benefiting from the agreement depended on his doing so. Id. at 260. Here, however, with the exception that the judge should have redacted from the agreement the Commonwealth’s promise to provide protection to Moodie, there was no error in the admission of the agreement between the prosecution and Moodie and none is claimed by the defendants. This case does not present the question presented in Ciampa, that is, whether the jury instructions cured the errors in failing to redact the repeated references to the witness’s obligation to tell the truth. Here, it does not appear that the evidence heard by either jury presented a realistic possibility that they would believe Moodie’s testimony based on her agreement to tell the truth. See id. at 260-261, 265-266. We conclude, therefore, that the defendants were not entitled to a specific instruction that the jury should exercise special caution in evaluating Moodie’s testimony. We note in this connection that defense counsel vigorously cross-examined Moodie and vigorously argued to each jury her lack of credibility.
6. Jury Instructions About Police Investigation.
Next, relying on Commonwealth v. Bowden, 379 Mass. 472, 485-486 (1980), the defendants argue that they were entitled to, but did not receive, instructions to the juries that they could consider the failure of the police to carry out certain investigative procedures. Bowden does not go nearly so far as the defendants claim. There, the judge instructed the jury that the nonexistence of certain scientific tests and other evidence was not to be considered by them. Id. at 485. We held that, in the circumstances of that case, the “failure of the authorities to conduct certain tests or produce certain evidence was a permissible ground on which to build a defense” because such a failure “could raise a reasonable doubt as to the defendant’s guilt in the minds of the jurors.” Id. at 485-486. We concluded that “[t]he judge should not have re*741moved this evidence from the jury’s consideration, and in so doing invade the province of the jury to decide what inferences to draw from certain evidence.” Id. at 486. No claim is made in this case that the judge invaded the province of the jury by instructing them to ignore evidence of police investigative failures. The claim is only that the defendants were entitled to an instruction that the jury could consider such failures. We have never held, and decline to do so now, that a judge must give instructions on the claimed inadequacies of a police investigation.
7. Admission of Opinion Evidence.
Finally, the defendants argue that the judge erred in admitting certain opinion evidence concerning analysis of bullet lead. Special agent John Riley of the Federal Bureau of Investigation testified that two bullet fragments found in Patricia Paglia’s body came from the same box of ammunition or from different boxes that were manufactured at the same place on or about the same date as a bullet retrieved from the basement of the Rye house. Riley further testified that three other bullets found in Patricia Paglia’s body “could have come from the same box of ammunition” as the two bullet fragments mentioned above. The defendants contend that the admission of that evidence, and of tests which formed the basis of Riley’s opinion, was erroneous because the tests are not generally accepted by the appropriate scientific community. See Commonwealth v. Mendes, 406 Mass. 201, 204-206 (1989). At trial, the defendants did not object to Riley’s testimony on this ground, nor did the defendants request that a voir dire hearing be conducted to determine the existence or nonexistence of such acceptance. Thus, our review is confined to the question whether the record shows a substantial risk of a miscarriage of justice as a result of the evidence having been admitted. G. L. c. 278, § 33E (1990 ed.). The record shows that Riley testified that the methods used in this case in conducting elemental analysis of bullet lead are generally accepted by the relevant scientific community. The record also shows contrary evidence introduced through an expert defense witness. We cannot fairly say, *742based on the evidence offered by the experts, that admission of the tests conducted in this case, and Riley’s opinion based thereon, creates a substantial risk of a miscarriage of justice.
In addition to their contention that the testing methods used in this case are not generally accepted by the appropriate scientific community, the defendants argue that, in giving his testimony, Riley was relying on facts and data gathered by others who were not before the court, thus depriving the defendants of the confrontation and cross-examination to which they were entitled. Riley testified on cross-examination that he prepares bullet lead samples in his own laboratory, as he did in the present case. After that, they are “packaged in plastic and sealed, heat sealed and initialed by me over the seal, then transported to the reactor by one of our technicians in the sealed condition and sent through a pneumatic tube into the core of the reactor by the technician. They are radiated, and after the samples are radioactive, returned to the laboratory where they are analyzed on our equipment and processed by the computer. When they come back to me, they are in the same sealed condition as I sent them out.” Riley further testified that the entire “round trip probably takes about three hours.”
In Department of Youth Servs. v. A Juvenile, 398 Mass. 516, 527-532 (1986), we took “a modest step by permitting an expert to base an opinion on facts or data not in evidence if the facts or data are independently admissible and are a permissible basis for an expert to consider in formulating an opinion.” Id. at 531. We said that such a change will eliminate the necessity of producing exhibits and witnesses whose sole function is to construct a proper foundation for the expert’s opinion.
“If a party believes that an expert is basing an opinion on inadmissible facts or data, the party may request a voir dire to determine the basis of the expert opinion. If the facts or data are admissible and of the sort that experts in that specialty reasonably rely on in forming their opinions, then the expert may state that opinion without the facts or data being admitted in evidence.” Id. at 532. Then we said: “Because *743the facts or data on which an expert may rely in formulating an opinion may not be in evidence, we next consider Proposed Rule 705. Proposed Mass. R. Evid. 705 provides: ‘The expert may testify in terms of opinion or inference and give his reasons therefor without prior disclosure of the underlying facts or data, unless the court requires otherwise. The expert may in any event be required to disclose the underlying facts or data on cross-examination.’ ” Id. at 532. Nothing in the record shows that Riley’s reliance on facts and data gathered by others not before the court fell outside the limits of the new rule laid down in Department of Youth Servs. v. A Juvenile, supra. The defendants were not deprived of their rights of confrontation and cross-examination. Indeed, as the Advisory Committee’s Note on Proposed Mass. R. Evid. 705 states, “[t]he thrust of the rule is to leave inquiry regarding the, basis of expert testimony to cross examination, which is considered an adequate safeguard.”
We have reviewed the entire case on the law and the evidence pursuant to G. L. c. 278, § 33E. Finding no cause to reduce the verdicts or to grant a new trial, we affirm the convictions.
Judgments affirmed.