Inflation on the back of an envelope

Last Monday was an exciting day!

After following the BICEP2 announcement via Twitter, I had to board a transcontinental flight, so I had 5 uninterrupted hours to think about what it all meant. Without Internet access or references, and having not thought seriously about inflation for decades, I wanted to reconstruct a few scraps of knowledge needed to interpret the implications of r ~ 0.2.

I did what any physicist would have done … I derived the basic equations without worrying about niceties such as factors of 3 or 2 \pi. None of what I derived was at all original —  the theory has been known for 30 years — but I’ve decided to turn my in-flight notes into a blog post. Experts may cringe at the crude approximations and overlooked conceptual nuances, not to mention the missing references. But some mathematically literate readers who are curious about the implications of the BICEP2 findings may find these notes helpful. I should emphasize that I am not an expert on this stuff (anymore), and if there are serious errors I hope better informed readers will point them out.

By tradition, careless estimates like these are called “back-of-the-envelope” calculations. There have been times when I have made notes on the back of an envelope, or a napkin or place mat. But in this case I had the presence of mind to bring a notepad with me.

Notes from a plane ride

Notes from a plane ride

According to inflation theory, a nearly homogeneous scalar field called the inflaton (denoted by \phi)  filled the very early universe. The value of \phi varied with time, as determined by a potential function V(\phi). The inflaton rolled slowly for a while, while the dark energy stored in V(\phi) caused the universe to expand exponentially. This rapid cosmic inflation lasted long enough that previously existing inhomogeneities in our currently visible universe were nearly smoothed out. What inhomogeneities remained arose from quantum fluctuations in the inflaton and the spacetime geometry occurring during the inflationary period.

Gradually, the rolling inflaton picked up speed. When its kinetic energy became comparable to its potential energy, inflation ended, and the universe “reheated” — the energy previously stored in the potential V(\phi) was converted to hot radiation, instigating a “hot big bang”. As the universe continued to expand, the radiation cooled. Eventually, the energy density in the universe came to be dominated by cold matter, and the relic fluctuations of the inflaton became perturbations in the matter density. Regions that were more dense than average grew even more dense due to their gravitational pull, eventually collapsing into the galaxies and clusters of galaxies that fill the universe today. Relic fluctuations in the geometry became gravitational waves, which BICEP2 seems to have detected.

Both the density perturbations and the gravitational waves have been detected via their influence on the inhomogeneities in the cosmic microwave background. The 2.726 K photons left over from the big bang have a nearly uniform temperature as we scan across the sky, but there are small deviations from perfect uniformity that have been precisely measured. We won’t worry about the details of how the size of the perturbations is inferred from the data. Our goal is to achieve a crude understanding of how the density perturbations and gravitational waves are related, which is what the BICEP2 results are telling us about. We also won’t worry about the details of the shape of the potential function V(\phi), though it’s very interesting that we might learn a lot about that from the data.

Exponential expansion

Einstein’s field equations tell us how the rate at which the universe expands during inflation is related to energy density stored in the scalar field potential. If a(t) is the “scale factor” which describes how lengths grow with time, then roughly

\left(\frac{\dot a}{a}\right)^2 \sim \frac{V}{m_P^2}.

Here \dot a means the time derivative of the scale factor, and m_P = 1/\sqrt{8 \pi G} \approx 2.4 \times 10^{18} GeV is the Planck scale associated with quantum gravity. (G is Newton’s gravitational constant.) I’ve left our a factor of 3 on purpose, and I used the symbol ~ rather than = to emphasize that we are just trying to get a feel for the order of magnitude of things. I’m using units in which Planck’s constant \hbar and the speed of light c are set to one, so mass, energy, and inverse length (or inverse time) all have the same dimensions. 1 GeV means one billion electron volts, about the mass of a proton.

(To persuade yourself that this is at least roughly the right equation, you should note that a similar equation applies to an expanding spherical ball of radius a(t) with uniform mass density V. But in the case of the ball, the mass density would decrease as the ball expands. The universe is different — it can expand without diluting its mass density, so the rate of expansion \dot a / a does not slow down as the expansion proceeds.)

During inflation, the scalar field \phi and therefore the potential energy V(\phi) were changing slowly; it’s a good approximation to assume V is constant. Then the solution is

a(t) \sim a(0) e^{Ht},

where H, the Hubble constant during inflation, is

H \sim \frac{\sqrt{V}}{m_P}.

To explain the smoothness of the observed universe, we require at least 50 “e-foldings” of inflation before the universe reheated — that is, inflation should have lasted for a time at least 50 H^{-1}.

Slow rolling

During inflation the inflaton \phi rolls slowly, so slowly that friction dominates inertia — this friction results from the cosmic expansion. The speed of rolling \dot \phi is determined by

H \dot \phi \sim -V'(\phi).

Here V'(\phi) is the slope of the potential, so the right-hand side is the force exerted by the potential, which matches the frictional force on the left-hand side. The coefficient of \dot \phi has to be H on dimensional grounds. (Here I have blown another factor of 3, but let’s not worry about that.)

Density perturbations

The trickiest thing we need to understand is how inflation produced the density perturbations which later seeded the formation of galaxies. There are several steps to the argument.

Quantum fluctuations of the inflaton

As the universe inflates, the inflaton field is subject to quantum fluctuations, where the size of the fluctuation depends on its wavelength. Due to inflation, the wavelength increases rapidly, like e^{Ht}, and once the wavelength gets large compared to H^{-1}, there isn’t enough time for the fluctuation to wiggle — it gets “frozen in.” Much later, long after the reheating of the universe, the oscillation period of the wave becomes comparable to the age of the universe, and then it can wiggle again. (We say that the fluctuations “cross the horizon” at that stage.) Observations of the anisotropy of the microwave background have determined how big the fluctuations are at the time of horizon crossing. What does inflation theory say about that?

Well, first of all, how big are the fluctuations when they leave the horizon during inflation? Then the wavelength is H^{-1} and the universe is expanding at the rate H, so H is the only thing the magnitude of the fluctuations could depend on. Since the field \phi has the same dimensions as H, we conclude that fluctuations have magnitude

\delta \phi \sim H.

From inflaton fluctuations to density perturbations

Reheating occurs abruptly when the inflaton field reaches a particular value. Because of the quantum fluctuations, some horizon volumes have larger than average values of \phi and some have smaller than average values; hence different regions reheat at slightly different times. The energy density in regions that reheat earlier starts to be reduced by expansion (“red shifted”) earlier, so these regions have a smaller than average energy density. Likewise, regions that reheat later start to red shift later, and wind up having larger than average density.

When we compare different regions of comparable size, we can find the typical (root-mean-square) fluctuations \delta t in the reheating time, knowing the fluctuations in \phi and the rolling speed \dot \phi:

\delta t \sim \frac{\delta \phi}{\dot \phi} \sim \frac{H}{\dot\phi}.

Small fractional fluctuations in the scale factor a right after reheating produce comparable small fractional fluctuations in the energy density \rho. The expansion rate right after reheating roughly matches the expansion rate H right before reheating, and so we find that the characteristic size of the density perturbations is

\delta_S\equiv\left(\frac{\delta \rho}{\rho}\right)_{hor} \sim \frac{\delta a}{a} \sim \frac{\dot a}{a} \delta t\sim \frac{H^2}{\dot \phi}.

The subscript hor serves to remind us that this is the size of density perturbations as they cross the horizon, before they get a chance to grow due to gravitational instabilities. We have found our first important conclusion: The density perturbations have a size determined by the Hubble constant H and the rolling speed \dot \phi of the inflaton, up to a factor of order one which we have not tried to keep track of. Insofar as the Hubble constant and rolling speed change slowly during inflation, these density perturbations have a strength which is nearly independent of the length scale of the perturbation. From here on we will denote this dimensionless scale of the fluctuations by \delta_S, where the subscript S stands for “scalar”.

Perturbations in terms of the potential

Putting together \dot \phi \sim -V' / H and H^2 \sim V/{m_P}^2 with our expression for \delta_S, we find

\delta_S^2 \sim \frac{H^4}{\dot\phi^2}\sim \frac{H^6}{V'^2} \sim \frac{1}{{m_P}^6}\frac{V^3}{V'^2}.

The observed density perturbations are telling us something interesting about the scalar field potential during inflation.

Gravitational waves and the meaning of r

The gravitational field as well as the inflaton field is subject to quantum fluctuations during inflation. We call these tensor fluctuations to distinguish them from the scalar fluctuations in the energy density. The tensor fluctuations have an effect on the microwave anisotropy which can be distinguished in principle from the scalar fluctuations. We’ll just take that for granted here, without worrying about the details of how it’s done.

While a scalar field fluctuation with wavelength \lambda and strength \delta \phi carries energy density \sim \delta\phi^2 / \lambda^2, a fluctuation of the dimensionless gravitation field h with wavelength \lambda and strength \delta h carries energy density \sim m_P^2 \delta h^2 / \lambda^2. Applying the same dimensional analysis we used to estimate \delta \phi at horizon crossing to the rescaled field h/m_P, we estimate the strength \delta_T of the tensor fluctuations as

\delta_T^2 \sim \frac{H^2}{m_P^2}\sim \frac{V}{m_P^4}.

From observations of the CMB anisotropy we know that \delta_S\sim 10^{-5}, and now BICEP2 claims that the ratio

r = \frac{\delta_T^2}{\delta_S^2}

is about r\sim 0.2 at an angular scale on the sky of about one degree. The conclusion (being a little more careful about the O(1) factors this time) is

V^{1/4} \sim 2 \times 10^{16}~GeV \left(\frac{r}{0.2}\right)^{1/4}.

This is our second important conclusion: The energy density during inflation defines a mass scale, which turns our to be 2 \times 10^{16}~GeV for the observed value of r. This is a very interesting finding because this mass scale is not so far below the Planck scale, where quantum gravity kicks in, and is in fact pretty close to theoretical estimates of the unification scale in supersymmetric grand unified theories. If this mass scale were a factor of 2 smaller, then r would be smaller by a factor of 16, and hence much harder to detect.

Rolling, rolling, rolling, …

Using \delta_S^2 \sim H^4/\dot\phi^2, we can express r as

r = \frac{\delta_T^2}{\delta_S^2}\sim \frac{\dot\phi^2}{m_P^2 H^2}.

It is convenient to measure time in units of the number N = H t of e-foldings of inflation, in terms of which we find

\frac{1}{m_P^2} \left(\frac{d\phi}{dN}\right)^2\sim r;

Now, we know that for inflation to explain the smoothness of the universe we need N larger than 50, and if we assume that the inflaton rolls at a roughly constant rate during N e-foldings, we conclude that, while rolling, the change in the inflaton field is

\frac{\Delta \phi}{m_P} \sim N \sqrt{r}.

This is our third important conclusion — the inflaton field had to roll a long, long, way during inflation — it changed by much more than the Planck scale! Putting in the O(1) factors we have left out reduces the required amount of rolling by about a factor of 3, but we still conclude that the rolling was super-Planckian if r\sim 0.2. That’s curious, because when the scalar field strength is super-Planckian, we expect the kind of effective field theory we have been implicitly using to be a poor approximation because quantum gravity corrections are large. One possible way out is that the inflaton might have rolled round and round in a circle instead of in a straight line, so the field strength stayed sub-Planckian even though the distance traveled was super-Planckian.

Spectral tilt

As the inflaton rolls, the potential energy, and hence also the Hubble constant H, change during inflation. That means that both the scalar and tensor fluctuations have a strength which is not quite independent of length scale. We can parametrize the scale dependence in terms of how the fluctuations change per e-folding of inflation, which is equivalent to the change per logarithmic length scale and is called the “spectral tilt.”

To keep things simple, let’s suppose that the rate of rolling is constant during inflation, at least over the length scales for which we have data. Using \delta_S^2 \sim H^4/\dot\phi^2, and assuming \dot\phi is constant, we estimate the scalar spectral tilt as

-\frac{1}{\delta_S^2}\frac{d\delta_S^2}{d N} \sim - \frac{4 \dot H}{H^2}.

Using \delta_T^2 \sim H^2/m_P^2, we conclude that the tensor spectral tilt is half as big.

From H^2 \sim V/m_P^2, we find

\dot H \sim \frac{1}{2} \dot \phi \frac{V'}{V} H,

and using \dot \phi \sim -V'/H we find

-\frac{1}{\delta_S^2}\frac{d\delta_S^2}{d N} \sim \frac{V'^2}{H^2V}\sim m_P^2\left(\frac{V'}{V}\right)^2\sim \left(\frac{V}{m_P^4}\right)\left(\frac{m_P^6 V'^2}{V^3}\right)\sim \delta_T^2 \delta_S^{-2}\sim r.

Putting in the numbers more carefully we find a scalar spectral tilt of r/4 and a tensor spectral tilt of r/8.

This is our last important conclusion: A relatively large value of r means a significant spectral tilt. In fact, even before the BICEP2 results, the CMB anisotropy data already supported a scalar spectral tilt of about .04, which suggested something like r \sim .16. The BICEP2 detection of the tensor fluctuations (if correct) has confirmed that suspicion.

Summing up

If you have stuck with me this far, and you haven’t seen this stuff before, I hope you’re impressed. Of course, everything I’ve described can be done much more carefully. I’ve tried to convey, though, that the emerging story seems to hold together pretty well. Compared to last week, we have stronger evidence now that inflation occurred, that the mass scale of inflation is high, and that the scalar and tensor fluctuations produced during inflation have been detected. One prediction is that the tensor fluctuations, like the scalar ones, should have a notable spectral tilt, though a lot more data will be needed to pin that down.

I apologize to the experts again, for the sloppiness of these arguments. I hope that I have at least faithfully conveyed some of the spirit of inflation theory in a way that seems somewhat accessible to the uninitiated. And I’m sorry there are no references, but I wasn’t sure which ones to include (and I was too lazy to track them down).

It should also be clear that much can be done to sharpen the confrontation between theory and experiment. A whole lot of fun lies ahead.

Added notes (3/25/2014):

Okay, here’s a good reference, a useful review article by Baumann. (I found out about it on Twitter!)

From Baumann’s lectures I learned a convenient notation. The rolling of the inflaton can be characterized by two “potential slow-roll parameters” defined by

\epsilon = \frac{m_p^2}{2}\left(\frac{V'}{V}\right)^2,\quad \eta = m_p^2\left(\frac{V''}{V}\right).

Both parameters are small during slow rolling, but the relationship between them depends on the shape of the potential. My crude approximation (\epsilon = \eta) would hold for a quadratic potential.

We can express the spectral tilt (as I defined it) in terms of these parameters, finding 2\epsilon for the tensor tilt, and 6 \epsilon - 2\eta for the scalar tilt. To derive these formulas it suffices to know that \delta_S^2 is proportional to V^3/V'^2, and that \delta_T^2 is proportional to H^2; we also use

3H\dot \phi = -V', \quad 3H^2 = V/m_P^2,

keeping factors of 3 that I left out before. (As a homework exercise, check these formulas for the tensor and scalar tilt.)

It is also easy to see that r is proportional to \epsilon; it turns out that r = 16 \epsilon. To get that factor of 16 we need more detailed information about the relative size of the tensor and scalar fluctuations than I explained in the post; I can’t think of a handwaving way to derive it.

We see, though, that the conclusion that the tensor tilt is r/8 does not depend on the details of the potential, while the relation between the scalar tilt and r does depend on the details. Nevertheless, it seems fair to claim (as I did) that, already before we knew the BICEP2 results, the measured nonzero scalar spectral tilt indicated a reasonably large value of r.

Once again, we’re lucky. On the one hand, it’s good to have a robust prediction (for the tensor tilt). On the other hand, it’s good to have a handle (the scalar tilt) for distinguishing among different inflationary models.

One last point is worth mentioning. We have set Planck’s constant \hbar equal to one so far, but it is easy to put the powers of \hbar back in using dimensional analysis (we’ll continue to assume the speed of light c is one). Since Newton’s constant G has the dimensions of length/energy, and the potential V has the dimensions of energy/volume, while \hbar has the dimensions of energy times length, we see that

\delta_T^2 \sim \hbar G^2V.

Thus the production of gravitational waves during inflation is a quantum effect, which would disappear in the limit \hbar \to 0. Likewise, the scalar fluctuation strength \delta_S^2 is also O(\hbar), and hence also a quantum effect.

Therefore the detection of primordial gravitational waves by BICEP2, if correct, confirms that gravity is quantized just like the other fundamental forces. That shouldn’t be a surprise, but it’s nice to know.

20 thoughts on “Inflation on the back of an envelope

  1. thanks for this succinct account..i have been looking for something like this for sometime and all i got were tediums of jargon and calculations..

    • The story is not overwhelmingly impressive from that point of view, although it’s encouraging that the observed value of r seems to fit together reasonably well with the observed scalar spectral tilt. The more relevant question, I think, is whether the signal detected by BICEP2 is really of cosmological origin, and if so whether there is any reasonable explanation other than gravitational waves produced during inflation. When other experiments see the signal at other frequencies we’ll be more convinced about the first point, and meanwhile theorists will be working hard to address the second point.

  2. “Compared to last week, we have strong evidence now that inflation occurred …” My impression is that even before the BICEP2 results, Linde claimed that inflation (in some form, at least) had been proven beyond a reasonable doubt. Also, Witten has been convinced for many years that inflation occurred. The inflation theories of Guth, Linde, et al. imply that Newton-Einstein gravitational theory is 100% correct, apparently contradicting Milgrom’s acceleration law.
    On 14 March 2014 Kroupa wrote, “The falsification of the standard model of cosmology (to be more precise, of the existence of cold or warm dark matter particles) is very robust indeed. It is completely consistent with all existing data.”

    http://www.scilogs.com/the-dark-matter-crisis/2013/11/22/pavel-kroupa-on-the-vast-polar-structures-around-the-milky-way-and-andromeda/

    I conjecture that the inflation theories of Guth, Linde, et al. will find it very difficult to explain the space roar. In my theory, inflation is due to a statistically significant escape of gravitons from the boundary of the multiverse into the interior of the multiverse, and the space roar is entirely due to electromagnetic radiation emitted by the inflaton field. My guess is that Linde’s theory of chaotic inflation is correct if and only if dark matter particles exist.

  3. Pingback: Inflation on the back of an envelope | Φ&up...

  4. I’m guessing that when all of the matter in the universe is in one place the gravitational force has defeated all the other forces of nature; locked them up and thrown away the key. Maybe that kind of reality is a big casino with an infinite number of slot machines. If the right numbers come up on any machine the other forces get a get out of jail card. But, in lieu of that, what force could possible push back against the force of gravity in that state?

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  6. “What inhomogeneities remained arose from quantum fluctuations in the inflaton and the spacetime geometry occurring during the inflationary period.” The preceding statement should be true in both Newton-Einstein inflation and Milgrom inflation.
    “Einstein’s field equations tell us how the rate at which the universe expands during inflation is related to energy density stored in the scalar field potential.” The preceding statement is true in Newton-Einstein inflation but might not be true in Milgrom inflation.
    What is the simplest way that Einstein’s field equations might fail? Replace the -1/2 in the standard form of Einstein’s field equations by -1/2 + dark-matter-compensation-constant. This idea might not work. However, in any case, my guess is that Milgrom is the Kepler of contemporary cosmology. I also guess that the main problem with string theory is that string theorists fail to realize that Milgrom is the Kepler of contemporary cosmology. Are string vibrations confined to 3 copies of the Leech lattice? Perhaps not. However, I predict that the inflation theories of Guth, Linde, et al. are not quite correct, because these theories are incompatible with Milgromian dynamics.

  7. Thank you for showing your actual “back of the envelope.” It brought back memories of Ed Purcell teaching E&M (“Physics 12B”) in the spring of 1976. One day the class laughed when Ed set c=1 and e=1 and pi=1 to do an order-of-magnitude calculation. Ed turned from the board surprised and a bit crestfallen, sharing with the class that “until you are able to back-of-the-envelope calculations, you are not ready to do physics.” The next class Ed actually brought in an envelope and an overhead projector, and he taught that day’s class by writing on the envelope rather than on the blackboard. I can’t remember who the TA was in 1976 (he was also an outstanding teacher), but Ed’s passion for order-of-magnitude calculations remains indelible. The notes from your airplane trip are very much in this spirit.

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  9. Dear John,

    Thank you for the nice summary and the calculations. It does give a good overview to non-specialists.

    However, I wonder if one can really deduce anything about the quantization of the gravitational waves. The model you presented does assume that the fluctuations are quantum, but could there be a model with just classical fluctuations which may have produced the same B-mode pattern? While the current predictions are consistent with a quantum model, they may also be consistent with some classical model as well. The classical model might need some strange initial stochastic fluctuations to reproduce the observed pattern, but maybe these can be reasonably assumed for the very early stages of the universe. Or can it be said without reasonable doubt that all meaningful classical models are excluded? And if so, why?

      • Is ‘t Hooft determinism the best hope for a model of deterministic origin for gravitational waves? Is Newton-Einstein inflation versus Milgrom inflation the key to understanding ‘t Hooft determinism?
        http://arxiv.org/abs/1207.3612 “Discreteness and Determinism in Superstrings” by Gerard ‘t Hooft, 2012
        VIEWPOINT 1: Classical reality is an approximation to quantum reality. Measurement modifies quantum reality.
        VIEWPOINT 2: Classical reality is an approximation to quantum reality, and quantum reality is an approximation generated by Wolfram’s automaton. Measurement is a natural process that creates quantum reality using 3 copies of the Leech lattice and Fredkin-Wolfram information below the Planck scale. String vibrations are confined to 3 copies of the Leech lattice, and this confinement enables string theory to make testable predictions by means of ‘t Hooft determinism.

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  13. What do you think of physics-by-press-conference? I recall Caltech being involved in the reaction to Pons-Fleischmann, so many years ago. Now it looks like my dear alma mater is on the other side of the bar re BICEP2. V painful to observe…

  14. Pingback: BICEP2 Redux: How the Sausage is Made | Whiskey…Tango…Foxtrot?

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