Reconstructing Inflation

By Sean Carroll | October 25, 2006 9:56 am

All sorts of responsibilities have been sadly neglected, as I’ve been zooming around the continent — stops in Illinois, Arizona, New York, Ontario, New York again, and next Tennessee, all within a matter of two weeks. How is one to blog under such trying conditions? (Airplanes and laptops are involved, if you must know.)

But the good news is that I’ve been listening to some very interesting physics talks, the kind that actually put ideas into your head and set off long and convoluted bouts of thinking. Possibly conducive to blogging, but only if one pauses for a moment to stop thinking and actually write something. Which is probably a good idea in its own right.

One of the talks was a tag-team performance by Dick Bond and Lev Kofman, both cosmologists at the Canadian Institute for Theoretical Astrophysics at the University of Toronto. The talk was part of a brief workshop at the Perimeter Institute on “Strings, Inflation, and Cosmology.” It was just the right kind of meeting, with only about twenty people, fairly narrowly focused on an area of common interest (although the talks themselves spanned quite a range, from a typically imaginative propsoal by Gia Dvali about quantum hair on black holes to a detailed discussion of density fluctuations in inflation by Alan Guth).

Dick and Lev were interested in what we should expect inflationary models to predict, and what data might ultimately teach us about the inflationary era. The primary observables connected with inflation are primordial perturbations — the tiny deviations from a perfectly smooth universe that were imprinted at early times. These deviations come in two forms: “scalar” perturbations, which are fluctuations in the energy density from place to place, and which eventually grow via gravitational instability into galaxies and clusters; and the “tensor” perturbations in the curvature of spacetime itself, which are just long-wavelength gravitational waves. Both arise from the zero-point vacuum fluctuations of quantum fields in the very early universe — for scalar fluctuations, the relevant field is the “inflaton” φ that actually drives inflation, while for tensor fluctuations it’s the spacetime metric itself.

The same basic mechanism works in both cases — quantum fluctuations (due ultimately to Heisenberg’s uncertainty principle) at very small wavelengths are amplified by the process of inflation to macroscopic scales, where they are temporarily frozen-in until the expansion of the universe relaxes sufficiently to allow them to dynamically evolve. But there is a crucial distinction when it comes to the amount of such fluctuations that we would ultimately see. In the case of gravity waves, the field we hope to observe is precisely the one that was doing the fluctuating early on; the amplitude of such fluctuation is related directly to the rate of inflation when they were created, which is in turn related to the energy density, which is given simply by the potential energy V(φ) of the scalar field. But scalar perturbations arise from quantum fluctuations in φ, and we aren’t going to be observing φ directly; instead, we observe perturbations in the energy density ρ. A fluctuation in φ leads to a different value of the potential V(φ), and consequently the energy density; the perturbation in ρ therefore depends on the slope of the potential, V’ = dV/dφ, as well as the potential itself. Once one cranks through the calculation, we find (somewhat counterintuitively) that a smaller slope yields a larger density perturbation. Long story short, the amplitude of tensor perturbations looks like

T 2 ~ V ,

while that of the scalar perturbations looks like

S 2 ~ V 3/(V’ )2 .

Of course, such fluctuations are generated at every scale; for any particular wavelength, you are supposed to evaluate these quantities at the moment when the mode is stretched to be larger than the Hubble radius during inflation.

To date, we are quite sure that we have detected the influence of scalar perturbations; they are responsible for most, if not all, of the temperature fluctuations we observe in the Cosmic Microwave Background. We’re still looking for the gravity-wave/tensor perturbations. It may someday be possible to detect them directly as gravitational waves, with an ultra-sensitive dedicated satellite; at the moment, though, that’s still pie-in-the-sky (as it were). More optimistically, the stretching caused by the gravity waves can leave a distinctive imprint on the polarization of the CMB — in particular, in the type of polarization known as the B-modes. These haven’t been detected yet, but we’re trying.

Problem is, even if the tensor modes are there, they are probably quite tiny. Whether or not they are substantial enough to produce observable B-mode polarization in the CMB is a huge question, and one that theorists are presently unable to answer with any confidence. (See papers by Lyth and Knox and Song on some of the difficulties.) It’s important to get our theoretical expectations straight, if we’re going to encourage observers to spend millions of dollars and years of their time building satellites to go look for the tensor modes. (Which we are.)

So Dick and Lev have been trying to figure out what we should expect in a fairly model-independent way, given our meager knowledge of what was going on during inflation. They’ve come up with a class of models and possible behaviors for the scalar and tensor modes as a function of wavelength, and asked which of them could fit the data as we presently understand it, and then what they would predict for future experiments. And they hit upon something interesting. There is a well-known puzzle in the anisotropies of the CMB: on very large angular scales (small l, in the graph below), the observed anisotropy is much smaller than we expect. The red line is the prediction of the standard cosmology, and the data come from the WMAP satellite. (The gray error bars arise from the fact that there are only a finite number of observations of each mode at large scales, while the predictions are purely statistical — a phenomenon known as “cosmic variance.”)

WMAP CMB power spectrum

It’s hard to tell how seriously we should take that little glitch, especially since it is at one end of what we can observe. But the computers don’t care, so when Dick and Leve fit models to the data, the models like to do their best to fit that point. If you have a perfectly flat primordial spectrum, or even one that is tilted but still a straight line, there’s not much you can do to fit it. But if you allow some more interesting behavior for the inflaton field, you have a chance.

Let’s ask ourselves, what would it take for the inflaton to be generating smaller perturbations at earlier times? (Larger wavelengths are produced earlier, as they are the first to get stretched outside the Hubble radius during inflation.) We expect the value of the inflaton potential V to monotonically decrease during inflation, as the scalar field rolls down. So, from the second equation above, the only way to get a smaller scalar amplitude S at early times is to have a substantially larger value of the slope V’. So the inflaton potential might look something like this.


Maybe it’s a little contrived, but it seems to fit the data, and that’s always nice. And the good news is that a large slope at early times implies that the actual value of the potential V was also large at early times (because the field was higher up a steep slope). Which means, from the equation for T above, that we expect (relatively) large tensor modes at large scales! Which in turn is exactly where we have some hope to look for them.

This is all a hand-waving reconstruction of the talk that Dick and Lev gave, which involved a lot more equations and Monte Carlo simulations. The real lesson, to me, is that we are still a long way from having a solid handle on what to expect in terms of the inflationary perturbations, and shouldn’t fool ourselves into thinking that our momentary theoretical prejudices are accurate reflections of the true space of possibilities. If it’s true that we have a decent shot at detecting the tensor modes at large scales, it would represent an incredible triumph of our ability to extend our thinking about the universe back to its earliest moments.

  • onymous

    You wrote:

    And the good news is that a large slope at early times implies that the actual value of the potential V was also large at early times

    How large is “large”? Was it trans-Planckian? If so, do you trust the theory?

  • Richard E.

    Re #1: Generically, if you have single field slow roll inflation, it is hard to avoid having trans-Planckian vevs at some point during the inflationary epoch. This is a real issue for model building inside string theory. (In general you can push down delta phi by lowering the inflationary energy scale. However, at the end of inflation epsilon approaches 1, and delta phi can easily be large over the last e-fold or two, unless you have some sort of sudden waterfall-style transition — see astro-ph/0601276)

    However, if the inflaton is not a single fundamental scalar, the picture changes somewhat. For instance, “assisted inflation” has inflation arise as a co-operative effect sourced by many independent fields. This can have an effective description as a single scalar field with a large vev, even though the component fields remain safely sub-Planckian. Alternatively, you could imagine a multi-field model where the inflationary trajectory was “folded” into a more complicated potential.

    In this case, what one recovers from the sort of analysis Sean describes is the effective single field description, not the underlying “microphysics” that gave rise to it.

  • Plato

    Sometimes it is useful to find analogies to this process so I suggest sometimes to look at the “Chaldni plate” and then reassess your thinking to what Wayne Hu imparts as you travel to the B modes.

    This is part of the process to encourage model thinking in relation to WMAP. Tying Wayne Hu and WMAP “mapping” is very important in my view.

    “Omega” of course.

    Hopefully you were able to join chanda, Stefan and Bee of Backreaction together, as these would make a good team?:)

  • Louise

    Inflation’s predicted spectrum falls outside even generous error bars. After 25 years of work there is no agreement as to causes of inflation, just a divergence of theories. Inflation does have a lot in common with strings.

  • anonymous

    Much progress in physics has been made when theorists make utterly rosy predictions of what proposed experiments should see. This gets the experiments built, at which point the theorists immediately reduce their predictions before the data can rule them out! I suppose this is going to be more of the same.

  • Richard E.

    Also, I should add that I have seen this work described several times by Dick and Lev and their student(s). So far as I know, there is no paper on the archive that describes it in detail. I would very much like to understand what they have done — however, it is worth pointing out that related ideas have been floating around for several years.

    In particular, the idea of recovering the inflaton potential from data received a lot of coverage in the mid-90s — see: (This review gives a very nice account of perturbation generation in inflation, and is worth reading for that alone.) However, it is possible to contruct potentials for which this approach fails, and after a flurry of activity work in this area dropped away. Morever, it has taken another decade for the data to become good enough to put really nontrivial constraints on the inflationary model space, so in this sense the whole discussion was somewhat before its time.

    More recently, Will Kinney and I described an approach to generating “random” inflationary models by specifying the slow roll parameters at an initial time, and then using the inflationary slow roll hierarchy to reconstruct the potential: Having done this, one can then “filter” models which meet some specified set of observational constraints. Recently, Will (along with Kolb, Melchiorri and Riotto) used this approach to examine the impact of WMAPII on the inflationary parameter space.

    Hiranya Peiris and I have taken this work in a different direction, adding the inflationary slow roll parameters (from which the potential can then be computed) to Monte Carlo Markov Chain fits ( and By doing this we avoid using any explicit parametrization of spectrum (in terms of the spectral index or the running) and we again make use of the flow equations to reconstruct the entire potential. The nice thing about this approach is that we can compute (subject to some priors) the number of remaining e-folds N, and then feed this information back into our parameter constraints.

    The impact of the “glitch” at l~30 on all these schemes is a somewhat complicated question. If I understand Sean’s precis, it sounds as though Dick and Lev are finding support for some for a version the “fast roll” proposal made by Contaldi et al ( However, this is typically invoked to fix the low quadrupole (a problem that seems more benign now than it did when it first appeared, thanks to a better understanding of the extraction of the lowest C_l from the CMB maps) and not the glitch.

    Typically, one might do a better job of fitting the glitch with a localized feature in the potential, and this is hard to do without adding some sort of fairly sharp change in V(phi) – and this is typically implicitly ruled out when one expands in the first few slow roll parameters. The spectrum for this sort of model often has to be computed numerically (, and the original WMAP inflation paper by Peiris et al actually checked to see whether a step improved the fit to the data ( More recently, Covi et al ( have repeated this treatment for the 3-year data, and find a similar level of support for a “step” around l~30 (which is not surprising, as the data has not changed much here). Obviously, if the inflationary potential really looks like this, it will be non-trivial to produce it some fundamental theory.

    OK, sorry for the long discussion but this is an interesting topic for me….

  • Sean

    Richard, thanks for the extra links, especially as I am not really an expert on this stuff. I was calling the low quadrupole a “glitch,” which I think is what is driving the results that Lev and Dick are getting. Also, they were pretty cautious about what might be the take-home message of their work, so nobody should blame them for my interpretation of it.

  • Rob Knop

    It’s hard to tell how seriously we should take that little glitch, especially since it is at one end of what we can observe.

    What’s always bothered me more is that the glitch is only about 1.3sigma, where sigma is dominated by the model uncertainty from cosmic variance.

    Given that — is it even a real glitch? I’m not saying I don’t think it’s worth thinking about what might suppress the largest-scale anisotropies– because they could be unambiguously detected in the future via potential tensor modes. However, from my perhaps-not-informed-enough outsider’s perspective, I don’t see how anything that better fits that point would be enough statistically better than the fit shown in the plot above that we should accept it as evidence of a better model, given that it’s just barely outside the error bars as it is.


  • TomC

    Rob –

    Rest assured that there has been much soul-searching in the field about exactly this question (,,, and are just a few examples). Most of the groups that have proposed alternative theories have tried to be rigorous about quantifying the significance of any improvement in fitting the data, and generally the answers have been about what you’d expect: delta-chi^2 of a few for one or two new parameters.

    Of course, this was all for first-year WMAP data; the 3-year WMAP results (which I believe are what’s plotted above) have put a damper on most of this line of research, as the re-analyzed low-l points have moved closer to the best-fit Lambda-CDM model, leaving even less room for improvement by alternative theories.

  • Hiranya

    Sean, I would add to the previous comments about the low statistical significance of the “lowness” of the quadrupole, that its hard to imagine any actual result being *driven* by its lowness. Cosmic variance is so large there that the contribution of the quadrupole to the overall WMAP likelihood is tiny. Perhaps this is not what Lev/Dick meant to say.

    Louise, I have no idea what your comment is based on.

  • Stu Savory

    I assume you (and Gia Dvali) know that “quantum hair on black holes” sounds very rude when translated into Russian as Dvali’s paper may be 😉

  • Louise

    Hiranya, the TT/TE graph given to me by Ned Wright is reproduced here. Inflation’s prediction is ruled out by both COBE and WMAP.

  • Hiranya

    Louise, that’s not a TT/TE graph but the angular correlation function. I have no idea what “unified spacetime” could be, and I am not sure its something Ned is working on. The whole point is that to “rule something out” you need to know what the error on your observation is. Errorbars are lacking in your graph, and if they were present, it should show that at large angular separations (corresponding the quadrupole) the errorbar is very big. This is not experimental noise but a fundamental limitation of our inability to observe more than one CMB sky (i.e. we only see one statistical realization of the primordial power spectrum). There is a lot of work being done on this but no study has concluded that “inflation is ruled out” as you claim. It may not be the best or only theory but the reason it has survived so long is that there is a lot of observational evidence to support it. Of course, it is unlikely to be the full picture, but the evidence is strong that something *like* inflation took place in the early universe.

  • Haelfix

    Any word on some of the more contrived inflationary models where we can relax the slow roll approximation. (eg two or more bouts of inflation), last I heard it was possible to fit just about anything with that

  • Hiranya

    Haelfix, yes you can fit a lot with a contrived model (even an open universe) but the data does not require it. The data can be fit adequately with with the simplest slow roll models. You can use a mathematical formulation of the concept of Occam’s razor to decide when the data requires something more contrived. At the moment it doesn’t.

  • Plato

    With the discovery of sound waves in the CMB, we have entered a new era of precision cosmology in which we can begin to talk with certainty about the origin of structure and the content of matter and energy in the universe.
    Wayne Hu

  • Louise

    Glenn Starkman and Dominik Schwarz give us a cleaner graph of the angular correlation function with error bars, which is reproduced here. Inflation’s prediction is ruled out by both WMAP and COBE. We all agree that something like inflation is needed, but a paradigm is just one step toward the solution.

  • Paul Valletta

    On the basis of Occam’s Razor, does the WMAP team have a


    the long awaited publication of the data, invoked a hornets nest of suppositions:

    Sean posted a number of threads dealing with WMAP:

    I admire the WMAP team ability to make something out of the immense amount of data they had, and the recent nobel prize award further justifies the need for experimental status for cosmology in the whole.

    But, It would have been more useful if the data was independantly analyzed by a number of groups, similaineously?

  • Paul Valletta

    Louise, the paper is on Xarchive, I have read the Glenn Starkman and Dominik Schwarz paper a while ago, and recall the paper made a very good case, with a number of further response’s appear after the publication, drawing similar conclusions?

  • Hiranya

    Louise, I know that analysis, its a nice paper, but the problem many people have with the statistics they quote is that they are a posteriori. In other words, given such a huge dataset, in just how many places did you look for weirdness till you found something that was weird, and is it right to then focus on that one thing for the statistical significance you quote without accounting for the fact that you biased yourself by focussing on it? Other analyses don’t come to their extreme conclusion (look on the arxiv for citations to their re-analysis of WMAP3). Also look in the Spergel et al WMAP3 paper about how you can arrive at a wrong conclusion by using a posteriori statistics.

    The so called “axis of evil” is indeed very intriguing but there is definitely no consensus yet that we have throw away the concept of statistical isotropy of the universe because of these results. The beauty of science is that there *can* be a debate about these things and people are coming up with novel and interesting ways to test this very important concept.

    Paul, I thought this was widely known, but the entirety of the WMAP data *is* public, from the raw timestream from the satellite down to the parameter analysis statistics. *Anyone* can download and reanalyze our data, and indeed have done so. Mostly the reanalyses come to very similar conclusions to the WMAP team, and some make very nice improvements to our analysis which the team have then adopted in their own analysis. There is a degree of openness about the whole procedure which I am not sure is found even in the particle physics experimental community.

  • Plato

    a posteriori

    At least someone understands the “Aristotlean” view?:)

    I still think we have to be “introspective” about these calculations or you wouldn’t have any evidence in whch to compare?

    You had to look for asymmetrical breaking values from “reductionistic attitudes” in order to discern the value of any universe created?

  • Amara

    Sean: “It may someday be possible to detect them directly as gravitational waves, with an ultra-sensitive dedicated satellite; at the moment, though, that’s still pie-in-the-sky (as it were). More optimistically, the stretching caused by the gravity waves can leave a distinctive imprint on the polarization of the CMB — in particular, in the type of polarization known as the B-modes. These haven’t been detected yet, but we’re trying.”

    BICEP‘s main goal? That group’s Antarctica wintover finishes next week, so maybe (after a well-deserved holiday and after they’ve had time to reduce their data), you can update us on their results …

  • Plato

    By simple analogies the complicated views of science has somehow been answered for the lay person? For the scientist as well?

    How far did you want to take this in terms of empirical views? Or, be left with insulting comments about the way in which we look at the cosmos from the very beginning?

    In cosmology, the early Universe was crossed by real acoustic waves generated soon after Big Bang. Such vibrations left their imprints 300 000 years later as tiny density fluctuations in the primordial plasma. Hot and cold spots in the present-day 2.7 K CMB radiation reveal those density fluctuations. Thus the CMB temperature fluctuations look like Chaldni patterns resulting from a complicated three-dimensional drumhead that

    Not a soccer ball? Yet some would paint a Platonic image and give monte carlo demonstrations. :)

  • Paul Valletta

    Hiranya, I was ill-informed, and that is “my fault”, my apologies if I made a slightly negative hand_waving in my post. Having gone over a vast number of sites detailing the WMAP data, the number of years these sites have been active, allows slight variations to be mis-interpreted.

    The length of time (in years) just shows how difficult the task at hand must have been for all those involved in WMAP, and initally the COBE team. WMAP is rightly so, a fantastic success and will remain, one of the amazing achievements of Astronomical experiments, a minor miracle if ever there was one !

  • Hiranya

    Paul, no problem :) You can find all the data here at the LAMBDA site, this is and will continue to be the central repository for the data. Enjoy!

  • Tyler

    Articles and comments like these are why I read Cosmic Variance. It’s all one full step above my head – I am a fascinated physics voyeur with mathematical limitations, so I can follow the arguements but get lost in the equations and am therefore unable to judge when there is disagreement – but that’s how learning happens, for me anyway.

    The articles on society et. al. are interesting enough, but this is very valuable stuff for me. It strikes a middle ground between struggling through technical papers on arxiv and sighing at the handwaving meathead analysis of the pop sci press.

    So my comment is: thanks for the article, and thanks very much to everyone who has posted in comments. I feel my brain expanding as I read: my favorite sensation.

  • Ambitwistor

    Sean: somewhat off-topic, but have you noticed that your article on the cosmological constant has been cited by a U.S. patent on an inflation-powered warp drive? (Saw this on Georg von Hippel’s blog.) I guess you missed out on the trillions of dollars this gadget will earn for the patent-holder.

  • Shantanu

    Sean or anyone else, could you provide a web link to this workshop and also the slides/videos
    of the talks if they are available?

  • Sean

    Shantanu, there is no web link for the workshop or the talks; it was small and informal.

    Ambitwistor, I hadn’t seen the patent. I’ll just have to make my millions some other way.

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Cosmic Variance

Random samplings from a universe of ideas.

About Sean Carroll

Sean Carroll is a Senior Research Associate in the Department of Physics at the California Institute of Technology. His research interests include theoretical aspects of cosmology, field theory, and gravitation. His most recent book is The Particle at the End of the Universe, about the Large Hadron Collider and the search for the Higgs boson. Here are some of his favorite blog posts, home page, and email: carroll [at] .


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