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In footnotes or endnotes please cite AIP interviews like this:
Interview of Marc Davis by Alan Lightman on 1988 October 14,
Niels Bohr Library & Archives, American Institute of Physics,
College Park, MD USA,
For multiple citations, "AIP" is the preferred abbreviation for the location.
Marc Davis discusses his childhood in Canton Ohio and family background; early reading; education at Massachusetts Institute of Technology (MIT) and at Princeton University; thesis work with Jim Peebles and discussion of Peebles; early work on the correlation function of galaxies; creation of the Center for Astrophysics (CFA) redshift survey in 1978; attitude toward the horizon problem; attitude toward the inflationary universe model; biasing, cold dark matter, and models of the formation of large-scale structure; attitude toward the flatness problem; attitude toward the CFA redshift surveys by de Lapparent, Geller, and Huchra; the question of whether the universe is homogeneous; relationship of theory and observation; important outstanding problems in cosmology: the Great Attractor, biasing, dark matter, galaxy formation; the ideal design of the universe; the question of whether the universe has a point.
I wanted to start with your childhood and ask whether you remember any particularly impressionable experiences that you had?
In terms of my interest in science, which I guess is what you’re most interested in; I recall vividly attending a NASA exhibit in Cleveland. I was raised in Canton, Ohio, about 60 miles to the south. I’d been interested in science as a young child. My parents took me to a big NASA exhibit for the public. I must have been 10 or 12. They had lots of material to give away, and they had a big chalked sign on a blackboard saying “More material to anyone who knows what an ion is.” I answered the question of what an ion was, and they gave me a pile of material.
How old were you?
I’m not certain. At least 12, I imagine. I have a very vivid memory of a chemistry set, as a young child that interested me. I kept trying to make explosives, but it was a very safe set and I couldn’t make anything explode.
Tell me a little bit about your parents and what they did when you were growing up.
My parents encouraged me to read. I used to go to the library a lot. Neither of my parents had gone to college. My father and mother were raised in a poor family. They’re second generation immigrants. My mother had been a very good student in school. We used to read a lot of books together and talk about books all the time. When my father was a student, he had to work part-time and help support his family and had to go to school only part-time. He never had the chance to go to college, but [he] was very enterprising and was a business man. He started his own business. He was a wholesaler. He ran a business that grew over the years, eventually employing 40 or 50 people, I believe. It was pretty successful. He retired after they sold the business, about 15 years ago, maybe 10 years ago.
What about your mother?
My mother worked for him as a bookkeeper. She was always working, but usually at home. For a long time, the only office for the business was in the living room of our house. When I was a very young child, my parents owned a truck and no car. My father bought things and sold them using the truck. I was told that he would be gone most of the week and only come home on weekends. Then the business grew and they eventually got a car. The first car I remember had a push button starter. That must have been an early 1950s car. I don’t remember what it was.
But you remember that they took you to this exhibit in Cleveland?
You must have already been interested in science at that point?
Oh yes, I was quite interested in science by then. I’m trying to remember the first inkling...
Do you remember any books that you read at that age or earlier that were memorable?
Well, I must have been older. I read a lot of science books at the 8th and 9th grade level — books on relativity and on physics and chemistry.
In school or out of school?
No, it was out of school. I remember in the fifth grade I had an influential teacher, Mr. Friedman. We had just moved to a different school district. He saw my interest in science and he gave me a chemistry book to read, a high school chemistry text, [when I was] in the fifth grade. I read that over the summer and thought it was great. That was my first serious science book, I suppose. I must have had a strong interest before that. I do recall very vividly that later in high school, George Gamow’s books, One Two Three... Infinity in particular, were really influential. And I really was quite intrigued by The Creation of the Universe. But particularly One Two Three... Infinity was just spectacularly great for me. It really made me decide that I wanted to study physics.
Did it make you interested in cosmology in particular, or just physics?
That one really turned me on to physics. I think all the relativity books at the lay level that I looked through made me particularly interested in gravitation and cosmology. Even in high school I thought I was interested in relativistic questions. Even though I’m in the astronomy department, I wasn’t really an amateur astronomer. I never really knew anybody who knew anything about telescopes. Maybe I would have been interested in telescopes, if I knew they existed. That part I’ve never experienced — building my own telescope, for example, which a lot of my peers have done. It’s something I had never even heard of. I still don’t know the constellations.
Neither do I.
Neither do most of my colleagues.
I learned one of the constellations once. The sole purpose was to impress a new girl that I was going out with.
I knew them at one time, but I just can’t remember them. So they must not be too useful. I remember at one point studying chemistry and deciding I wanted to be a chemist. Then the next year in high school I studied physics in my senior year and decided I wanted to be a physicist. I didn’t really know anything about physics before that.
Had you been exposed to theories of cosmology at this age in any of your popular readings?
Well, certainly The Creation of the Universe was about a theory of cosmology, but I don’t recall understanding, or even reading in any detail, more than a few buzz words — big bang versus steady state — and not really understanding anything at that point.
You didn’t have any ideas about that?
I can’t recall any books that would have discussed it in detail. I think Gamow’s book was written in the very early 1950s or late 1940s, and I think it was before the steady state theory had been invented. I don’t recall other books that discussed it, but I could have mixed them up.
Do you recall ever thinking of the universe as a whole at this age?
Yes, I did. I definitely did, particularly after reading relativity books. I get confused as to when I read what. Didn’t Bertrand Russell write a book?
Yes, The ABC of Relativity, but I think that was special relativity.
Yes, it was. Even so, that was pretty amazing. I can’t remember when I read that, so I’m completely mixed up. I still have a few of them around. Let me see if I can see a date on it. Here it is. [Davis is referring to The ABC of Relativity by Bertrand Russell.] It’s got my home address in Canton, Ohio, so I must have read this in high school. That’s amazing. I think I have One, Two, Three ... Infinity here also. Here it is. It also has my home address in it.
And you have these books here with you in your office right now?
Yes. On occasion, I’ve gone through my various collections of books and sorted them. These went into my office, but I don’t refer to them terribly often. I’ve had them here for a reference to undergraduates who come by and want to learn more about science at an introductory level. I’ll recommend books like this to them. These are still really good.
Yes. Let me move on and ask you a little about your undergraduate education at MIT. Do you remember any particularly influential experiences there?
The whole experience at MIT was staggeringly influential to me because I had come from a modestly-sized town in the Midwest and was somewhat unusual in my interest and talents for science. My school didn’t really have that many peers to talk to — no one at all, really, with my interest in science. Then I go to MIT and discover that the entire class seems to be made out of class valedictorians like I was. I remember the introductory week at MIT, when the freshman dean said, “Look around the room. Half of you scored 800 on the advanced math aptitude test.” I came from a school where I didn’t even have preparation to take that test. I had to take the intermediate math test because I didn’t have an advanced course in high school to prepare me for it. So I felt very bad — [thinking] here I was, perhaps in over my head. And MIT was definitely a lot of work. It was not like high school, where you could just breeze through without worrying about anything. MIT required a serious effort. But it was a fantastic environment to be with that class of peers, people who really were interested in science, people who were really cultured, who knew about music. I didn’t know a damn thing about classical music when I got to college. I didn’t know anything about anything, it seems, compared to what all these kids knew. I felt I had a lot of learning to do. It was an incredible experience. It required a lot of effort, but I managed to make my way through.
Did you take any courses in physics or astronomy that got you particularly fired-up?
Well, practically the whole curriculum. I started off as a physics major and, course after course; I was really fired-up — even the introductory freshman physics with 600 students in it. The whole freshman class had to take an advanced physics course. At MIT, it used to be you took 8.01, 8.02, 8.03, and 8.04 through your sophomore year. Everybody took physics. There wasn’t any distinction between the future economists and the future physicists. They all took the same thing. They’ve since changed that, but it was impressive. I remember Professor [Anthony] French, who was very, very good when we were just learning calculus and physics and mechanics. I had a very inspirational course with Ray Weiss, who taught 8.03, the first introduction to electricity and magnetism. He was a great lecturer. There was any number of really good courses in physics.
Did you know at this time that you wanted to become a scientist?
Oh, that was clear, yes.
Did you know before you went to college?
Yes, I knew before I went to college that I wanted to study science and that’s why I chose MIT.
Were you still interested in gravitational physics?
I wanted to remain open. I thought I was interested in physics. I didn’t know enough to specify my interest beyond that, but it always had been clear that I was more interested in, say, gravitational physics than solid state physics, for example. So I felt that was what I would probably end up doing, and it is what I ended up directing myself to. As a senior, I worked with Bill Rose, who was an assistant professor at MIT at the time. We did a senior thesis project together that went pretty well. At the same time, though, given my background and my parents’ background, they were clearly struggling to pay for me in college, and they wanted me to work while I was there, which is only fair. The university paid rather poorly, and I was able to work outside for a software firm, actually, a computer firm called Adage, Incorporated. I became a systems programmer and made a lot more money than I would have made working in the dining hall. That took a fair amount of time. I was working as much as 20 hours a week in my junior and senior years.
It’s amazing that you had time to do your studies.
I was really busy. I also had a girlfriend. I had plenty to do.
Do you have any opinion why gravitational physics would have appealed to you more than solid state physics, for example, at this age?
I felt that it addressed large questions that I had read about as a child, questions [related to] great philosophical issues. In my humanities, I minored in philosophy. I had read philosophical discussions — Kant, for example — on the early notions of the universe. That [topic] used to be the domain of philosophy. These were large questions that always appealed to me, and gravitational physics seemed to address them. That was it. How can a high school kid who doesn’t know any science at all say he wants to study solid state physics? He doesn’t even know what solid state physics is. It’s a much more technical subject that requires knowing the minutiae.
Let me ask you a little bit now about your graduate work at Princeton. Can you tell me about the influences that you, got there?
By the time I had gotten through my senior year at MIT, I was doing pretty well. My grades were good, and I applied to a bunch of graduate schools. When I went to Princeton, I encountered, once again, a culture shock. I thought I knew some physics, but here the class was full of people that were just unbelievable. There was Claudio Teitelbaum, there was Jacob Bekenstein, and there was Bob Wald. There was Claude Swanson, who has since left physics, but he was a friend of mine from MIT. He was talking about starting to work with Wightman as soon as he got there. He was reading about Banuch algebras and all this stuff. I didn’t know what the hell he was talking about, frankly, and these guys just left me in the dust. Now, partly, I felt that here I was paying the price for the fact that I had worked while at MIT and hadn’t taken quite so many graduate physics courses as some of my peers had taken. That put me at some disadvantage, particularly in theory, where people seemed to really have a head start.
Is that what you wanted to do — theory?
I wasn’t sure. I never could quite decide what I wanted to do. I thought I might want to do theory, but I like tinkering with my hands as well. Because I was interested in gravitational physics, I started to associate myself with the gravity group. That was Bob Dicke’s group at the time, and, of course, Dave Wilkinson and Jim Peebles were major players. Even at this point, Bob Dicke was mired in all the controversy of the solar oblateness experiment. I started to work with Dave and also Paul Boynton, who was there. Bruce Partridge was also there. This was just before they had moved to the new Jadwin lab.
Let’s see, I graduated in 1970 and they had just moved about that time. Partridge was an assistant professor when I was there.
Yes, I came in 1969, so we overlapped one year. It was amazing, even more amazing than MIT because here the graduate class was really, truly impressive. We used to do our problem sets together. I remember Bob Tribble, who’s a nuclear physicist. I don’t have his last name correct. He and I used to work together. We were both married at the time, living in Lawrence apartments and we used to work like the dickens to do our problem sets. I was taking an intermediate quantum mechanics course with John Schwarz. That was when he was there.
Before he became famous.
That’s right, before any of these people became famous. He certainly knew what he was talking about, more than I did. I remember having trouble calculating Coulomb scattering integrals and other things. Unfortunately, just the way astrophysics is, most of us don’t use quantum mechanics at all, and I’ve forgotten everything I ever knew. Sam Treiman was a particularly effective teacher. He was amazingly brilliant as a lecturer. Even though I thought I knew statistical physics, taking a course from him was just wonderful. Then he taught a course in field theory that I took that was just great. He just makes it all seem so simple. Of course, then you get out of the class and you discover that you can’t calculate anything, but you sure could understand what he was talking about.
Did you take any astronomy courses?
I took relatively few. I sat in on an essentially shotgun seminar with Lyman Spitzer, which was a bit intimidating because I really wasn’t up to speed with the subject. Jerry Ostriker was teaching a stellar interiors course. No, that must have been Martin Schwarzschild. Yes, Martin was teaching a stellar interiors course, and I sat in on a seminar with that. During my first year, Jim Peebles was offering a course in physical cosmology, which led to his first book. At the time, I sat in on it a bit, but I was so busy with other courses that I didn’t take it regularly. It was really geared for more advanced students. I had trouble following some of it.
Didn’t you ultimately do your thesis with him?
No, actually, I did it with Wilkinson. People think that I worked with Peebles, but the project to do with Pebbles was later. So there were some really great people around. You start to associate with these really high powered physicists, and it’s hard to maintain an air of superiority like you could in high school. So it was good. It was humbling and I worked hard.
Was cosmology at this time particularly attractive to you, or was it just another possible discipline?
It was another possible discipline. It hadn’t branched out to the point that [it has now]. Jim Peebles certainly made it interesting. His course was great; his book was terrific. He was truly inspirational. He basically invented the subject, for the Westerners at least — he and Zeldovich. My thesis was inspired by some crazy paper he had written with Bruce Partridge about how primeval galaxies might appear in the sky. Looking back on my thesis, it’s almost embarrassing. I did a project with Dave Wilkinson. I built some hardware to search for faint, fuzzy extended objects on the sky. We took it to Hawaii. It was nice to go to Hawaii and use the telescopes. We searched in blank areas of the sky, looking for faint, fuzzy objects. We didn’t find any. Of course, in those days we didn’t have any CCD detectors or anything. We had a photo-multiplier; we had no multiplexing advantage. It’s ludicrous, the limits that we could set then compared to what people can do now. The Partridge and Pebbles models are no longer considered relevant classes of models for forming galaxies, but I got a thesis out of it and went on to do other things.
Let me ask you one thing that would probably be hard to remember, but I’ll ask anyway before we go on. You certainly had been exposed to various cosmological models at this time, I would expect. As a graduate student, do you remember having a preference for any particular kind of model that is open versus closed or homogeneous versus inhomogeneous?
Well, the question which had just been settled, of course, five years earlier, was the hot big bang versus the steady state. Of course, being at Princeton there was no question that the hot big bang was right. So that was over. It was interesting to see that cosmology, because of the microwave background radiation, had now advanced to the point of having data. It was a real science. It wasn’t just philosophical speculation. It rapidly became apparent, as Wilkinson, Partridge, Boynton and their students did one experiment after another, that the radiation was quite isotropic. I remember the first several attempts with Dave’s students. They couldn’t detect any anisotropy at all. Eventually, they succeeded in detecting dipole anisotropy after a lot of hard work. So the evidence was clearly pointing toward an isotropic universe, and by the Copernican Principle, unless we’re at a special place in that universe, you would infer that it’s homogenous as well. Of course, that’s the simplest universe, the simplest mathematical model. [On] the question of open versus closed, much of the data we use now was somewhat in existence at that point. The evidence was that [the universe] was open, based on masses of clusters and groups of galaxies. Even though the data is greatly improved in the last decade or two, the conclusions have basically remained the same.
Did you accept that or did you think about it?
I thought about it. We thought about it a lot. I had some preference for a fiat universe or closed universe. I’ve always had trouble visualizing an infinite, open universe and still do. I understand what it’s saying, but somehow an infinite number of galaxies seem a little bit strange. But that was just a problem that I had.
So you thought that maybe a higher density might be discovered in the future or something like that?
We always knew the standard arguments that you can recite today. The omega estimates of masses of clusters were sensitive only to the cluster component and were insensitive to a uniform component that could be there for reasons unknown. It still could. The other test, which at that time was considered more promising — the curvature tests at large scales — were variously debated one way or the other. [Allan] Sandage wrote these elegant papers in the early 1960’s that I read closely. I understand he has another article in Annual Reviews, which I haven’t seen yet, which I guess is a restatement of these papers. The issues were laid before us. We all followed the data coming from Palomar with intense interest — the question of magnitude-redshift tests and all that stuff. Then Jerry Ostriker and his students, particularly Scott Tremaine, and also Beatrice Tinsley at Yale, started to calculate evolutionary corrections of one sort or another. By the time I left graduate school, it was clear that it was hopeless. The evolutionary corrections were going in both directions. They didn’t know which one was greater amplitude. It was obviously going to be subject to the accuracy with which you could do this. [There were] too many astrophysical parameters. That cast a pall, over global tests. That must have been 1973 or so when those tests were really in question.
Scott and I overlapped entirely except for maybe one year at Princeton. Our offices were next to each other. I remember we were both in the gravity group. Scott worked with Jerry, and I was working with Wilkinson and Peebles. We talked all the time. It was great. We used to have these high level discussions on everything. It was a fantastic environment. It was a good place to go to school.
So you were already questioning those global tests by the end of your graduate school?
It was becoming clear that the corrections could be substantial and that they had to be calculated carefully. So we were questioning them, yes.
If you thought of ten effects, you don’t know that there isn’t an eleventh or twelfth that you haven’t thought of yet.
That’s right. Jerry was inventing one effect after another, as he is so good at doing.
So that would leave you with just the clustering tests.
Or something completely different that we hadn’t thought of.
And your own prejudice, whatever that might be.
In fact, Peebles and I then started working on these questions. I had come to Peebles when I was searching around for a thesis topic, and he had tried to turn me off saying that what the world needs is good data. It doesn’t need another crummy theory. That has always been his attitude — good observations are worth more than another mediocre theory. I’ve been told that by quite a few people. I remember John Bahcall had [a similar] conversation with me once.
It’s remarkable that Peebles would say that, given that he has been one of the leading theorists in cosmology.
Well, there are two reasons for that. One, the subject is data—starved and good data is worth its weight in gold. Secondly, in all honesty, I think that Jim would prefer not to have too much competition from other theorists He liked it when he had the field all to himself, and he did such a good job of selling it that he had all these young Turks come and be with him. He never considered me as competition because he knew I was primarily an experimentalist. But I think he felt a lot of other people were competition. Maybe I’m misreading him. In any event, regarding other tests, Jim and I started to follow-up some kinetic theory calculations that he had done while he was on sabbatical at Berkeley. He had showed me a notebook of calculations he had done related to what is known as the BBGKY hierarchy. He had thought about applying it to cosmology, and he and I started working on that. I decided that the nonlinear integral differential equations that represented the hierarchy, an infinite series of equations nonetheless, was not completely out of the question for a numerical solution. We could attempt to solve those equations in certain limits. So we started a project to understand if we could cast the equations into a form suitable for really solving and not just writing them out formally. It was clear that we needed to somehow truncate the series in higher order correlations, and we just made mathematically expedient steps to do so. That was fair, mathematically. It’s never been justifiable physically because we couldn’t argue convincingly, and to this day you still can’t, why you can truncate in a particular way. We did what we could. We attempted to solve these equations under various approximations. We reached conclusions which, in fact, have been at variance with all the simulations. If you look at them in detail, the important part of what we did was scaling solutions. We were able to make self-similarity transformations and show, under scaling laws, what is expected when the behavior is self-similar. Those scaling laws are correct, but the details of the solution have not been confirmed by direct numerical experiments. So, therefore, our truncation and mathematical approximations aren’t correct in detail. I don’t want to go into technical issues. However, one of the things that were interesting out of that was that we realized that the shape of the correlation function — the 2-point galaxy correlation function, which was one of the statistics in this hierarchy — was going to be sensitive to cosmological density. If we knew the initial spectrum of perturbations, we could more or less, as Ed Turner said, read omega off in the sky just by looking at the shape of the correlation. We had a lot of flak for what we said in that paper, and I think fairly so, but that got me a little inspired that maybe you can infer omega in rather different schemes than the way we had always thought of — viral analyses in isolated groups. So, coming out of this statistical approach were a couple of useful theorems. The cosmic energy theorem is a trivial limit to the first equation, that’s otherwise known as the Layzer-Irvine equation. You can use that to solve for omega if you know velocities and correlation functions. There’s also a cosmic viral theorem which comes out of the next order equation. That can also be used to solve for omega if you know two and three point correlations and if you know relative pair velocities as a function of projective separation. These were two tests. When you applied those tests together to the available data, again you concluded omega was 0.1 to 0.2. I think you were at Harvard at the time I was finishing this work with Jim. We wrote a couple of long papers, one infamous one in Astrophysical Journal Supplement. At that point, we realized that we had some interesting statistical tests related to velocity fields and relative velocities of galaxies in space and it sure would be nice to have some data to test this with.
Which leads to the redshift survey, which I was going to ask you about?
Yes, actually, I had this theoretical interest that leads me into redshift surveys. It was becoming clearer that these statistical procedures could be very powerful, but they just needed better data.
This was real data, and not N-body simulations?
No, real data. N-body simulations didn’t exist except...
Well, some people were starting to do them.
I think the Aarseth, Gott and Turner stuff was later than this.
I believe Peebles and Groth were starting to do N-body simulations.
Yes, they were.
They started when Ed Turner was a graduate student at Caltech, so it must have been around 1973 approximately.
Yes, but in fact they never published too much.
No, they didn’t, but they were going around talking about it.
That’s right. Aarseth’s, Gott’s and Turner’s work must have been about 1976 or something like that.
Yes, they were inspired by a talk that Peebles gave.
I see. All right, 1976 or 1977, something in that range.
Did it occur to you to get N-body data as opposed to real data, or did you just immediately decide that the real data was by far preferable?
The N-body data was cooked. You knew what omega was. In fact, I don’t know when it first occurred to me that the N-body data would be a good way to calibrate the sky. You really wanted to have both. In any event, you clearly needed to look at the sky in some detail. Margaret Geller, John Huchra and I started talking about this at length. We did some analysis with the Shapley-Ames catalog, which is a whole sky survey of about 1100 galaxies to the 13th magnitude.
Would this have been around 1977 or so?
Yes, I think it was finished at about that time, maybe 1978. We started looking at that and looked at the three-dimensional distribution of galaxies there. We didn’t do very much with it. It was clearly not a very fair sample because there were three times as many galaxies in the north as in the south.
So you were using other people’s data?
That data had just been made available by Sandage and Tammann. So we were using that data. We studied correlation functions of it and wrote a paper about it, but didn’t do too much. Then, Mike Fall was a postdoc at Harvard at the time. We had some chats with him about using some of this. I remember a workshop we once had at the Radcliffe Quad, where Bill Press had this very interesting way of [describing] the available surveys of redshift space. He said it was like a mace — a mace being an iron ball with spikes — and the height of the location at any angle is the depth of the survey information at that region. A few regions had been surveyed deeply, [but] most had been surveyed very shallowly. I think at the time he gave that talk — I remember Jim Gunn being around then — we were already getting started to get on with the CfA survey.
So you had decided then that you were going to collect your own data?
Yes. Let’s see, I had been at Harvard for about a year, and I knew that I could count on a six year term hopefully without too much problem. I had the time to do a big project and decided that it wasn’t worth fiddling around with just a few observing runs. We would do something serious. John Huchra and I were chatting with everybody to try to get some money. I had gone out to Mount Hopkins to use their spectrograph and see what it was like, and it was a disaster, totally useless. It had a one-stage image tube that was good for taking spectra of stars, but we couldn’t even get a spectrum of M81, which is a very nearby bright spiral. It was hopeless for doing survey work. And that was image tube onto plates. It would have required terrible analysis. It was clear that we were going to have to do a lot of work. Here is where the Center came into its own. It was a project that we were able to do at Harvard—Smithsonian because of its size and resources. It would have been impossible at any normal university setting. I remember writing a proposal to the National Science Foundation asking for an enormous amount of money, I think it was $300,000. I had no idea what was a reasonable budget to ask for. At the time, I had been chatting with Gethyn Timothy, who was building what are called MAMA detectors — micromultianode micro channel arrays — that he was [saying] would be great for a detector on these things.
Had the Z-machine been designed at that time?
No. We were focusing on designing at that time. We wrote this proposal to the NSF for $300,000. I remember it was only when [George] Field and Alexander] Dalgarno got on the phone and badgered Jim Wright, or whoever it was, that I got $20,000 out of them. If it were up to me alone, it would have been zip, absolutely nothing. Herb Gursky had just [become] associate director of the Optical and Infrared group. He was used to spending money because he came from the X-ray division. So he didn’t think anything was amiss when we said we needed lots of bucks to build a real computer-driven system and we needed the resources and manpower to make it work. Fortunately, Tom Stevenson, who had written the drive and control system for the Multiple Mirror Telescope, was available to help with our system software, I latched onto John Tonry, who I’d known at Princeton. He overlapped with you too, didn’t he?
I guess so.
He was a little younger than you. I had him as a student at Princeton. He was a math student at Princeton and then went to Harvard physics. He was interested in the project from the beginning. He came to talk to me when we were just getting started on building it. We had the design in hand at that point.
John Geary got in at some point, didn’t he?
No, that was much later. I’ll mention him later. I had met Dave Latham when I first went to Harvard. We had done some silly project on grid photography. He was a photographic expert, but no one was utilizing his talents at the time. He got interested in the project. So we had Smithsonian people as well as myself and a few others, and we started thinking about doing this for real with real dollars. The Smithsonian put an enormous amount of money into rebuilding the spectrograph which was there, which Rudy Schild had had built by an instrument maker in Wisconsin. We brought it back and removed the vignetting. We had all kinds of problems; we had to fix up a lot of it. Dave Latham took responsibility for building the image tube packages which we needed for this. By this point, when the dollars had been cut so badly, it was clear that we weren’t going to be able to afford Gethyn Timothy’s device, which really wasn’t ready. It was in the engineering development stage. I saw that what I thought was the most suitable device was, the photon counting Reticon systems that Steve Shectman had built. One summer I went out to his lab and spent the summer copying his electronics. We had some circuit boards, and I wired them up and got the electronics working. I learned how to jimmie the boards. It was really a cut and paste operation.
This is around 1978?
Yes, it must have been. He had taken these boards supplied by Reticon and cut them up a bit to modify them for his purposes. There were wires running all over underneath the circuit boards and various components removed and replaced. It was amazing. It was amazingly chopped up. I wire-wrapped a bunch of other boards that we needed. I’m no expert at any of this so I didn’t do the neatest job in the world, but it did work.
I apologize for interrupting, but there is just so much more that I want to get into.
That’s all right, let’s go on.
I wanted to ask you one more question about the CfA redshift survey. You thought it up, designed it, and got it running. I wanted to know why you then moved on to other things, why you didn’t keep pushing it beyond the initial survey. Maybe you decided that it could be done well by other people at that point?
Okay. We had a great time doing the survey, and we learned an enormous amount about the large scale clustering of galaxies. When I wrote our series of papers, I started to compare [the data] with the best available N-body simulations, since it was clear that they were a useful tool for comparing to these data. The models were so bad in comparison to the observations that it was clear that the data was way ahead of the theory. As I had been interested in theory anyhow, I decided it was time to focus a little more on theoretical questions. If we repeated the survey, we could work three more years and double the database, but I didn’t think qualitatively it was going to change. I think I’ve been proven wrong by some of the Harvard “stick-man” [results] that Margaret Geller, Valerie de Lapparent and John Huchra have shown. They’ve shown that the clustering is even a little more dramatic than we thought it was. They basically have better signal-to-noise on the structures, and you’re really quite struck with how empty and round some of these things are. But, in any event, I felt qualitatively — and I still more or less believe it — that we had a pretty good feel for what the clustering was, but we had a very poor intuition about what it meant. I was therefore inspired to try to focus on that.
That answers my question pretty well. I want to move into your reactions to some recent discoveries. First, I want to go back a little bit for background. Do you remember when you first heard about the horizon problem? Do you know approximately when that was?
Oh, I probably didn’t hear about it in any serious way until I took a relativity course. Probably in graduate school.
In graduate school, so that would have been in the 1970’s.
I think so. 1971 or so.
Do you remember how you reacted to it when you heard about it? Did you regard it as a serious problem?
You mean the question of why the universe is homogeneous at all, given the [existence of] horizons?
I never paid great attention to it because it was clear that in some sense it was an initial value problem, at least from all the theory that we knew of. It was an initial value problem, and I didn’t have any basis for understanding how I could argue one way or another about initial values. Well, that’s it. I don’t know how to address the question scientifically. It seems a little bit bizarre, but so be it. That’s about as deep as I went on it.
Did your view of the horizon problem change any as a result of the inflationary universe model?
I have to say that I was so impressed with the inflationary model because it had promoted the horizon problem to a tractable problem. I was so amazed. [I thought], “My God, this actually can explain the horizon problem.” That sold me immediately. Here we had a real explanation for these things that we didn’t think could be addressed at all. In a sense, when people ask about the role of God in science, scientists frequently answer that God is just over the horizon, or just beyond. He was pushed out of the solar system when Newton made celestial mechanics tractable with his inverse square law. It was pushed back to initial conditions when the big bang model came around, because we didn’t need God once the initial conditions were set, but we sorted of needed Him to set them up. But now inflation has pushed Him back to the Planck time. Maybe if Linde’s ideas of eternal inflation are correct, and [ideas about] higher dimensional space time [are correct], then God’s in 10 dimensional space time or something. But the progress of science pushes back these questions that you think aren’t even addressable. That’s what is so impressive — when you can actually push back your ignorance to a point where you can address a question that you didn’t think was in the bounds of science at all. So that was a real step forward. Although it’s not proven that [the inflationary universe model] is right, it certainly has a strong appeal to all of us.
Why do you think the inflationary universe model has caught on so widely given that there is so little very direct observational support for it?
I guess it’s because it solves the horizon problem and flatness problem. Perhaps it solves the flatness problem too well, but it seems to solve it in what appears to be a dynamical and physically intuitive method. I think what people like are the results of inflation. They like what it does, that we have a dynamical explanation of these amazing observations. I think a lot of us would be happy to use whatever mechanism is shown to work to give these results. It doesn’t have to be inflation, if you can think of something else that produces these good results So far no one has, but maybe they will someday.
Do you think that the inflationary universe model or some version of it has a good chance of being correct?
Yes, although to say that authoritatively probably requires field theory credentials that I don’t have. I know that while the particle theorists are impressed with the results of inflation, as is everyone, they are less impressed with the means by which [the results are obtained.] There is no such thing as a fundamental scalar field that anyone has ever seen, and that’s what you need. It has to be weakly coupled in an extremely strange way. You need rather refined parameters [in order to] not overproduce fluctuations when you end inflation, and simultaneously get to a sufficient temperature to do baryogenesis, which must occur after inflation. So the fine tuning problem in the current inflation models is pretty extreme and not considered very compelling by any particle theorists. That’s just the state of affairs. It’s not satisfactory. I don’t know how we’re going to make progress. I’m not going to make progress on those questions.
If you accept the inflationary universe model, how do you reconcile the required mass density with what we observe?
Ah, yes. Right. The reconciliation of that was — and is — mysterious, and in fact, is quite controversial to date. This is a subject that I hope I can make contributions on. I’ve made some, so far. Nick Kaiser, when he was a postdoc here at Berkeley, started reading about the statistics of Gaussian fields and read some of the original work by the electrical engineer in the forties. What was his name? [S.O. Rice] Anyhow, this was one-dimensional signal processing. [Nick Kaiser] started studying the statistics of rare events and realized that the statistics of peaks was not the same as the statistics of the mean field, and that conceivably there could be bias in the galaxy distribution. He then applied it to Abell clusters, which are high-density rare events in the galaxy distribution. Abell clusters are sufficiently rare events that they would have correlation properties not representative of the mean field. Mathematically, it has to work. If the noise field, [that is] the random field, is Gaussian, it clearly works. It’s mathematically easy to demonstrate. Comparing to the available data on the Abel clusters, you can set constraints of what sort of power spectrum of perturbations is most suitable to explain the rather substantial boost of the. Abel clusters relative to the other galaxies. It’s not easy to explain in most models, but there is some effect. Not as big as reported in the literature, but there’s something. The notion of bias was then used by us, that is, my colleagues and I, Simon White, Carlos Frenk and George Efstathiou. We applied it to solve the problem we were seeing in the numerical simulations, which by that point had made enormous progress. When I came to Berkeley, we started working on numerical simulations, and within a year or two demonstrated hot dark matter models with massive neutrinos had irredeemable problems that were just fatal — and most people since agreed — for analytical reasons as well as the [results] of simulations that [show that] the model’s hopeless unless you have seeds that preserve galaxies. Cold dark matter was the next simplest model. It is, in a sense, a minimal model consistent with inflation and more or less any dark matter. It does require some dark matter candidate, which we haven’t detected in a laboratory, but it could be almost anything. We calculated the models with omega equals unity. That’s what inflation expects. That’s the only thing that makes any sense.
You put in biasing?
Without biasing, the model made no sense whatsoever. It was a terrible match to the observations. We knew that would be the case. It’s clear that the dynamic cluster masses [would not give a large enough omega]. So that was clearly going to fail and it did, rather spectacularly.
So you need the biasing plus [some form of dark matter]?
You need a biasing plus cold dark matter. At the time we wrote our first paper on this, we demonstrated that if you just did the biasing in a statistical sense by saying that you only look at the distribution of points that were 2.5 sigma fluctuations in the initial noise field — not in the final noise field, but in the initial noise field, when it’s Gaussian, and this should be thought of as an epoch when galaxies were forming — that they were sufficiently biased to reconcile the apparent observed omega of 0.1 or 0.2 with a real omega of unity. That begged the question of how the bias worked and why the bias would be there in the first place.
And why you should choose 2.5 sigma.
Oh yes, and why we filtered at a given scale. We chose 2.5 sigma. The argument [for that] was that [we] filtered on a scale appropriate to the mass of a galaxy, [we] chose a threshold so that you get the right number density of galaxies compared to what’s observed, and [we] measured it, and it worked. We didn’t actually do much of a parameter search. That was basically the first parameter that we tried. It worked and we published it. Since then, a cottage industry has been born about bias. There have been lots of theoretical mechanisms suggested for how bias may occur or whether it may occur sufficiently. None of them are very compelling. We later looked at simulations in detail, very high resolution simulations where we thought we could see individual galaxies forming as multiple components. We were able to show that there is a natural biasing that should occur in gravitational models. Deep potential wells form relatively late in cold dark matter universes, and the efficiency of matter going into a deep well, like a galaxy’s potential well, is dependent on the background density. In a proto-void, the efficiency of agglomerating lumps into these deep wells is less than in a proto-cluster. That introduces a natural bias. We ran some models and demonstrated that if galaxies are associated with the wells, a potential well depth of 200 kilometers a second, that is, a circular velocity of 200 kilometers a second — which they typically are because the knee in the luminosity function has that circular velocity — that they are biased relative to the matter distribution. In fact, this bias is expected in any hierarchical model of structure formation. It’s purely gravitational. There’s no additional physics and it’s unavoidable. So that, in a sense, provides some degree of bias. We are still not absolutely certain that that’s sufficient to reconcile the apparent low omega.
That allows you to think that is a possibility for explaining omega equal one?
Let me move on to the flatness problem, which you mentioned a few minutes ago. You said that you first heard of the horizon problem in the early 1970’s. Do you remember when you first heard of the flatness problem?
Oh, it must have been about the same time. It might have been earlier. Surely that was earlier, because in popular books on general relativity, they talk about the universe expanding or contracting eventually, so the flatness problem comes in there. I must have read about that.
Was it stated as a problem?
No, it was stated as options.
But I mean the flatness problem.
Oh, as a dynamical instability, it was not appreciated.
I mean the way that Guth talks about it in his paper.
No, that wasn’t impressed upon me until later.
Was it after you heard about the horizon problem?
Probably. Well, maybe at the same time. I’m not sure.
You don’t remember when you heard about it?
My first recollection of it was the analogy that Jim Peebles used to make with Bob Dicke, namely, that the flatness problem is like the problem of a pencil standing on its point. It’s okay; it’s great until it’s time to go, and then over it goes. I think Dicke has a little example of how to calculate the time you expect based on uncertainty principle consideration. You can actually calculate it precisely.
The question is, with the uncertainty principle, how finely you can balance it?
That’s right. I don’t remember how long it is.
We can figure that out. You think you heard about the flatness problem when you were a graduate student at Princeton?
Yes, not before.
Did you consider it to be a serious problem?
I did. I considered it to be a very serious problem and my heart was not sold on the famous paper by Gunn, Gott, Schramm, and Tinsley, in which they presented four lines of argument for an open universe. I knew the arguments, but philosophically, I wasn’t convinced.
That was around 1974.
Was it that early?
Yes, something like that. The arguments were well-known. Peebles and I at the time were arguing that the shape of the correlation function dictated against an open universe. –
When you say that you took the flatness problem so seriously that you were not persuaded by the Gunn, Gott, Schram, and Tinsley article, does that mean that the flatness problem made you think that omega was equal to one or that k was equal to zero?
Yes. That’s right. I had a reputation, I think, of being one of the people trying to defend an omega of one.
Did you ever wonder how it was that the universe began as an omega equal one universe or k equal zero universe? Did that concern you? Or was that just another question of initial conditions?
In the same class. The reason that the flatness problem wasn’t wholly compelling was that we couldn’t justify why omega started off at one in the first place, so we couldn’t really justify why it had decided to deviate from one a few Hubble times ago. It’s just one of those things. Unless you have a dynamical argument, you’re arguing about non-physical questions. You can’t address the question.
Let me restate what you said and see if I understand it. In your own mind, you approached the horizon problem in the same way that you approached the flatness problem, that there were some initial conditions which we couldn’t calculate that determined both the homogeneity and the fact that the universe was set up with omega equals one or very, very close to one?
If you took omega equals one as an implication of the flatness problem, then did you worry about where all of that extra mass was?
Yes, I definitely worried about it. It was clear that we had a lot of loose ends, that we simply didn’t understand how we could make these mistakes unless there were things like Peebles was suggesting — dead galaxies out in space. In his first book, I think he mentions dead galaxies. I don’t think he’ll talk about that anymore, but you have to hide the stuff somewhere. He talks about all the ways to hide it. It had to be something we didn’t know about. Relativistic background sounded really favorable for a long time.
So you were willing to entertain the possibility that there was all of this stuff out there that was unmeasured?
Yes, I was willing to entertain that.
Based upon this theoretical argument.
Yes, the theory was a strong [influence]. I had no data to support a high omega so it had to be a theoretical argument or a philosophical argument that drove it.
What would the philosophical argument have been?
I guess the uniqueness of an omega of one. If it’s not one, then why isn’t it 10-6 or something completely different? Why is it so close? That sort of thing. Why are we so special?
Let me go on to ask you about de Lapparent, Geller and Huchra’s results — what you called it, the Harvard “stick-man” — that you mentioned a little while ago. When you first got whiff of those results, which came out of a program that you had created yourself, were you surprised by the results that they found?
Yes, I have to say that I was a little surprised that they seemed to find a case of such clean delineation of on or off voids, on or off filaments. That didn’t actually square with my perception of what I had seen in all the different slices of the CfA survey. I think, in fact, the reality is that that’s a fairly exceptional slice of the universe, that a more typical slice of the universe is muddier.
We will know more in the next few years.
Well, there is a lot of data that is available from them and from people such as Giovanelli and Haynes. We finished our southern sky survey, which just was published this year. Generally, we don’t see quite such compelling on and off slice distributions. Rather, we see what appears to be heterogeneity of structure. Different zones are cleaner than others, and you see all types of things out there. In some cases, you just see a loose agglomeration of stuff with no well-defined structure to it at all. I think the reality is that once we have put all these data sets together, we will perhaps be able to characterize the frequency with which you see such clean structures as that, and it isn’t going to be all that high. It’s going to be a mixed bag. So I was surprised that they found such a good one. I think, I have to say, that I felt that the data got to be over interpreted, because they started to argue for bubbles. We had actually looked at lots of regions in space. We had looked at bigger volumes than they have, and we haven’t seen things that I could convince myself were bubbles in the sense of surrounding, 4π enclosures. I immediately liked the term “sponges,” when Rich Gott made it up. I think that’s a fair description. We have interconnected voids and interconnected filaments. That means that nothing is completely surrounded, and that I think is a better representation.
Given this large-scale structure, this sponge like structure, based upon many different surveys, do you think that these structures suggest that the universe may not be really homogeneous on large scale. When I say not really homogeneous, I mean not homogeneous enough to [satisfy] the assumptions that go into the models.
The Friedmann universe?
Yes, the Friedmann universe. Are you worried at all?
No, I’m not.
You still think that the rough homogeneity is a good approximation?
Yes, and the reasons for that are more than just philosophical desire. There are constraints. If inhomogeneity persists on too large a scale, you start getting fluctuations in the microwave sky, which we don’t see. If the inhomogeneity is gravitationally attracting, you would expect large-scale coherence in the velocity field — more than is observed. We are working on a paper now on this material, in which we computed velocity correlation functions, and the coherence length is not all that excessive. In fact, it matches the cold dark matter models, believe it or not. The Seven Samurai amplitude is very large. That’s hard to match with cold dark matter. But other catalogs of velocity fields, such as the Aaronson, Huchra, Mould study of the Local super cluster flow field, have the same coherence length as the Seven Samurai, but not the same amplitude. There are other data sets, such as the IRAS velocity field, or IRAS gravity field, that I’ve been working on with Michael Strauss. It matches the Aaronson, Huchra, Mould velocity field very well. None of these things have a coherence length that’s excessive. Also, just recently, I’ve run models from Peebles’s are curvature baryon universe. It’s not a theoretically well-motivated model, but at least it’s specific. It has very large scale structures. That one is almost challenging the notion of a Friedmann limit, and in a model such as that, you expect to see more coherence in the velocity field more than is observed. So I think there are real tests now. Jerry [Ostriker] and I apparently disagree completely from the same data. I think the data supports the notion that we are approaching the Friedmann limit.
That is an approximately homogeneous universe.
So your faith in the big bang has not been shaken by any of this?
No, I’m not willing to claim, as he apparently is, that the dipole anisotropy may not, in fact, be a result of motion. It may be intrinsic. I don’t think we need to take such a Draconian step just yet.
You are both an observer and a theorist yourself, so you are in an ideal position to answer this next question. Do you think that the theory and observations have worked well together in cosmology in the last 10 or 15 years?
Oh, absolutely. I have enjoyed it immensely, because I’ve been fortunate to be in the field before it became so popular. We’ve seen the situation shift from one side to the other. In a sense, cosmology was originally thought to be a science in search of two numbers. [Indeed] it was finally promoted to being science at all when the microwave radiation was discovered. Now all this industry of people searching for fluctuations has kept an army of theorists busy trying to calculate the expected small-scale and large-scale fluctuations in various models. It has already ruled out several classes of models. The large-scale structure dovetails in nicely with this because it provides normalization to the amplitude of fluctuations expected now and then. That’s observational input that the theorists absolutely need to make their calculations for a whole different class of observers. So the interaction is quite good. Now this interaction with particle theory is just amazing. In fact, that there is any interconnection at all took us all by surprise. Before the invention of inflation and grand unification, none of us thought too much about the intimate connection that now exists. [However], I think the particle physics connection is dropping by the wayside. The particle theorists have their problems with grand unification and supersymmetry and superstrings. The astrophysicists are worried about the details of galaxy formation and whether cold dark matter works or not. They’re not communicating as much as they were a few years ago.
What do you think are some of the major problems in cosmology right now?
In observational cosmology?
You can spend a minute in observations and a minute in theory.
All right. On the observational side, we still need to understand the Seven Samurai result. That’s very ill-explained. The data is not secure because the different groups do not agree on the flow field in detail. That has to be developed. In the next few years, I think we are going to see an enormous growth of knowledge in the flow field given by peculiar velocity studies using Fisher-Tully or Faber-Jackson relations, or improved relations. That will finally allow us to measure a gravity field. That will be great, because then we’ll have real information about mass distribution and not just the galaxy distribution. Another serious question is whether there is or is not any evidence that this bias mechanism works. There are observational consequences of our explanation for natural bias. There should be tracers of the mass distribution. For example, we would expect weak, small galaxies to be less biased than rich galaxies. Some of us have written papers saying that we can see it in various data sets. Other people have criticized those data sets for various incompleteness problems and have said their own data shows no such effect. This is all very new. It’s just this year that this controversy has really been coming to light. That has to be resolved because the bias predictions are very clear. There has to be something out there that’s clustered, but not as clustered as the galaxies. If that’s wrong, then the model’s basically irretrievable.
You can’t explain omega equal to one.
No, it isn’t going to work. So that’s fundamental.
What do you think about in theory?
There is another experimental question — find the dark matter. No physicist worth his salt is going to like this until you can actually discover this dark matter in a laboratory. It’s got to be there in some form or another, but unfortunately, there are too many candidates and none of them are well enough motivated. Theoretically, I guess the particle theory question is very serious. I hear lots of problems with all the supersymmetric particles. They’re all in bad shape. None of the inflationary models make sense. None of the dark matter candidates are particularly compelling. Maybe axioms are the best. It’s not a good [situation]. Theoretically, for the astrophysics, we think — prejudicially — that the cold dark matter model has provided a good motivation for looking at galaxy formation with real initial conditions that you think make sense. The technical challenge is to take these initial conditions and try to form galaxies with real physics. This will probably have to be done numerically, but it’s going to be extremely difficult. There is too much physics that happens all at once to likely lead to believable results. But work will pick up on that in the next several years as people really get geared up for it. We’re getting geared up to do it. Other groups are really competing now. We’ll see a lot of output on that in the next few years. I don’t know if it will lead to anything.
Let me ask you to take a big step backwards. If you could have designed the universe any way that you wanted to, how would you have done it?
How would I have done it? How would I have done it differently? Do you want to do it the same or different?
That’s up to you.
I know what I would definitely have done. I would have provided a non-compact dimension that would permit you to do time travel — or some mechanism to do time travel — because here in astrophysics we see all these staggering things. I liked Carl Sagan’s recent novel, Contact, where they managed to travel to the center of the galaxy by going in some sort of space warp. [I would like] something of that sort. The fact that we can’t travel faster than the speed of light is obviously essential to make physics work. It wouldn’t be a very sensible universe if it were otherwise. But our inability to get to astrophysical interesting sights and see an accretion disk around the black hole or any of that stuff is really disappointing. We can speculate on all this, but we can’t get there. I fear that we’re never going to be able to travel the galaxies. I’d like to design a universe that would permit that — somehow allow you to do it in some sort of higher dimensional space time, if necessary. Maybe it’s possible in seven or eight dimensions in our universe, if it folds back on itself in some complex way. I imagine the energies required to get to that are probably beyond our means. So life is tough.
Let me ask you one final question. There is a place in Steve Weinberg’s book The First Three Minutes where he says the more the universe is comprehensible, the more it also seems pointless. Have you ever thought, yourself, about whether the universe has a point or not?
The question is raised most often in terms of the human endeavor, [in terms] of the pointlessness or point of life on earth. Does the universe have a point in being? I try not to think about the question too much because all too often I agree with Steven Weinberg, and it’s rather depressing. Philosophically, I have to agree that I see no arguments against his attitude, that we certainly don’t see a point. And to even answer in the alternative sense really requires you to invoke the principle of God, I think. At least, that’s the way I would view it and there’s no evidence that He’s around, or it’s around. On the other hand, that doesn’t mean that you can’t enjoy your life. I feel very privileged as a physicist. I think about the question a lot — what is the point of most people’s lives? They make a living and does some work that’s moderately interesting, but physicists get to do these amazing things. We even get paid for learning how the universe ticks. I’m very exhilarated when I wake up in the morning because we can go back and knock on these questions a little more and eventually make a little progress. I mean, that’s got a real point. I’m very enthusiastic and excited by it. I’m sure that many other people in different lines of work have similar good feelings about what they do, but being a physicist, I think, is very special.
 G. Gamow, One Two Three…Infinity (New York: Viking, 1947)
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 Bertrand Russell, The ABC of Relativity, rev. Felix Pirani, (1925; rpt. London: G. Allen & Unwin, 1958)
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 M. Davis and D.T. Wilkinson, “Search for Primeval Galaxies,” The Astrophysical Journal, vol. 192, pg. 251 (1974)
 R.B. Partridge and P.J.E. Peebles, “Are Young Galaxies Visible?” The Astrophysical Journal, vol. 147, pg. 868, (1967); “Are Young Galaxies Visible II?” The Astrophysical Journal, vol. 148, pg. 377, (1968)
 See, for example, R.B. Partridge and D.T. Wilkinson, Physical Review Letters, vol. 18, pg. 37 (1969)
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 There were several papers in this series. The first was S.J. Aarseth, J.R. Gott III, and E.L. Turner, “N-Body Simulations of Galaxy Clustering. I. Initial Conditions and Galaxy Collapse Times,” Astrophysical Journal, vol. 228, pg. 664 (1979)
 M. Davis, M. Geller, and J. Huchra, “The Local Mean Mass Density of the University: New Methods for Studying Galaxy Clustering,” The Astrophysical Journal, vol. 221, pg. 1 (1978)
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 A. Guth, “Inflationary Universe: A possible solution to the horizon and flatness problems,” Physical Review D, vol. 23, pg. 347 (1981)
 See A.D. Linde “Particle Physics and Inflationary Cosmology,” Physics Today, September (1987) and references therein.
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 N. Kaiser, “On the Spatial Correlations of Abell Clusters,” The Astrophysical Journal Letters, vol. 284, pg. L9 (1984)
 S. White, C. Frenk, M. Davis, “Is the Universe Made of Massive Neutrinos,” Third Morian Astrophysics Meeting, pg. 117, March, 1983
 M. Davis, G. Efstathious, C. Frenk, S. White, “The Evolution of Structure in a Universe Dominated by Cold Dark Matter,” The Astrophysical Journal, vol, 292, pg. 371 (1985)
 J.R. Gott, III, J.E. Gunn, D.N. Schramm, and B.M. Tinsley, “An Unbound Universe?,” Astrophysical Journal, vol. 194, pg. 543 (1974)
 See reference 3.
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