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Inflation on the back of an envelope

Guest blog by John Preskill of Caltech, a trained particle physicist, cosmologist, a top quantum computing expert, and a teacher of nations

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 \(r\sim 0.2\).

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. Click to zoom in.

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 \(2.726\,{\rm K}\) 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\[

\zav{ \frac{\dot a}{a} }^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 \(\sim\) 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 \(50/H\).

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 \(\exp(Ht)\), and once the wavelength gets large compared to \(1/H\), 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 \(1/H\) 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 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 \zav{ \frac{\delta \rho}{\rho} }_{\rm hor} \sim \frac{\delta a}{a} \sim \frac{\dot a}{a} \delta t \sim \frac{H^2}{\dot \phi}.

\] The subscript \({\rm hor}\) 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^{\prime 2}} \sim \frac{1}{m_P^6} \frac{V^3}{V^{\prime 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 \zav{ \frac{r}{0.2} }^{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\\ \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 the CMB anisotropy data already supported a scalar spectral tilt of about \(0.04\), which suggested something like \(r \sim 0.16\). 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.

John Preskill also blogs at QuantumFrontiers.com.

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snail feedback (28) :

reader Umesh said...

Incredible! That's really nice Prof. Preskill, thanks so much. Handsome or what!

reader PlatoHagel said...

Yes thank you Prof. Preskill

reader Rich B. said...

And how exactly does this happen (in 100 words or less!)? :"The universe is different — it can expand without diluting its mass density, ..." Because energy is being converted to mass during the expansion?

reader lukelea said...

I am also intrigued by Brandenberger's idea, not that I understand it, but by the the fact that it begins with strings. If string theory is true, shouldn't stringiness be part of the initial moment? I am looking forward to Lubos's further reflections on this general topic.

reader lukelea said...

Hey! I liked that too. And I was a lit major! One nagging question though: in several places you made "simplifying" assumptions -- that something changing slowly could be treated as constant, and similar things like that. I am wondering if you make too many such assumptions that, though no one of them makes much difference, they all together lead to more significant error. I think the same thing happens in perturbative (sp?) theory where all powers of a variable higher than one are ignored as being too small to matter much. But if nature really does care, how do we know they don't matter? I'm sure you real physicists have more than good enough answers to my naive question, but I thought I would ask it anyway.

reader Gordon said...

Really nicely written and clear post.

reader david55 said...

Plugging the hole in Hawking’s black hole theory

reader Dilaton said...

Thanks so very much for this immensely nice summery :-)

Having watched basically only Lenny Susskinds cosmology lectures some time ago, it was exactly what I needed to understand the fun exciting results of BICEP2 at a slightly technical level!

Having people like you, at best in the row before me such that I can peek over their shoulder, doing such nice stuff would make long flights definitively less boring ... ;-)


reader kashyap vasavada said...

@MJ. OK! This was a joke but it is quite true. Here,people do shoot others for trivial reasons. I am sure, Lubos also meant as a joke.You are probably right. But I am not sure if people notice which finger you are showing.Nobody thinks that Italian physicist meant any insult to Guth. He just showed his finger to make a point. I am curious if there is any difference between Europe and U.S. about this point. For example, in India showing finger does not mean anything. Since I came to U.S. I use all the fingers just to be on safe side!!

reader Giotis said...

Thanks for this analysis but I'm missing SUSY from this picture. What are the implication of breaking SUSY at that scale?

reader PlatoHagel said...

Strange things come to me while I sleep.... and I was thinking about your, "Back of the envelope" writings on good paper.....you prepared yourself this time.:) So I am constructing too, but not in the same way. Your method is encouraging.

See: Parameters

While one might think I have something profound to say, it was more the method by which such examination was thought about and entertained, that you selected the 5hrs to do something?

Okay, from a layman perspective the intuitive journey through the information, your work and writing, qualitatively exemplifies this approach through such a journey? Then you "break it down" in the post. While this look into "your method," as a layman, what I write said nothing about your writings on the page, for myself it gives a good glimpse on the way some introspective scientists use their time.


reader pendulum said...

Great post. Hey Motl, don't you think other physicists could post here instead of your choppy writings?

reader kashyap vasavada said...

Great notes. Thanks. I hope he gets an assistant to type these and makes a review article with the notes and the bottom part. In red ink it takes a long time to read on a computer screen.

reader Honza said...

The museum is a great idea. I just think that instead of wax figures, there should be cages, or better yet, pillory, with real Al Gore, James Hansen and Michael Mann in it.

reader cynholt said...

Wow, and to think, it was just this passed January that Hawking went on the record as saying that Hawking radiation, at least at the event horizon of black holes, is nothing but a product of pure fiction. Hopefully he'll have the good sense to retract this statement. Otherwise, he's gonna have a mighty tough time explaining how gravitons behave quantumly just moments after the Big Bang but don't behave that way teetering on the edge of a black hole. The laws of nature simply require that they behave the same regardless of time or place.

reader cynholt said...

Handsome, indeed. That's the first thing I noticed about him too. :~)

reader Ann said...

Love it! Or maybe taxidermy? If it was good enough for Trigger, Roy Rogers horse, these guys should be honored to be sort of present in a museum.

reader aaron said...

Wax statues of Hansen and Gore would melt from global warming.

reader papertiger0 said...

Nerf balls. We're not savages.

reader Mervyn said...

Over the many decades, some high profile scientists have made alarming statements about future climate, which have been spectacularly wrong. The cacophony of climate alarmism by the IPCC is proving no different.

reader papertiger0 said...

I have no problem with redwoods growing in Seattle.
If that's the worst case senario, we have better things to worry about.

reader anna v said...

interesting that one could do back of the envelope on GR problems :). As an experimentalist reading theoretical notes I am often reminded of Lady Catherin, an authoritarian character in "Pride and Prejudice" by Jane Austen.

Hearing a discussion on music ( chapter 31)

"What is that you are saying, Fitzwilliam? What is it you are
talking of? What are you telling Miss Bennet? Let me hear
what it is."

"We are speaking of music, madam," said he, when no longer able to avoid a reply.

"Of music! Then pray speak aloud. It is of all subjects my
delight. I must have my share in the conversation if you are
speaking of music. There are few people in England, I suppose,who have more true enjoyment of music than myself, or a better natural taste. If I had ever learnt, I should have been a great proficient. And so would Anne, if her health had allowed her toapply. I am confident that she would have performed delightfully. "

In this case :

"of General Relativity? "

"If I had ever learnt, I should have been a great proficient."

reader M J said...

Americans generally do not consider it impolite to point with or raise the index finger, though the older generation considered it impolite to point at a person that way. It is the middle finger that causes trouble! But your procedure of pointing with the whole hand is a good idea, since no ethnicity I know of considers that impolite.

reader John Archer said...

Me too.

I could have been a contender!


reader M J said...

OTOH, yes, I have to assume the good doctor knows what he is talking about, and I see him getting a lot of praise for this explanation. But I still cannot imagine what he thought he was saying when he wrote, "The inflation rolled slowly for a while, while the dark energy stored in V(phi) caused the universe to expand exponentially. This -rapid- (emphasis mine) cosmic inflation lasted..."

How can it "roll slowly" and yet be 'rapid'? Or what distinction could he have had in mind between "inflation rolling" and "universe expanding"? He just admitted that the inflation field phi filled the WHOLE universe. At what point did it no longer fill it? I thought that only came much later.

reader Alexander Ač said...

Of course global warming IS benefical!

Killing poor people is benefical for the rich. Why even discuss about it?

reader Luboš Motl said...

Are you still around, AA? I thought that global warming has killed you something like 5 years ago already, like most prophets of doom.

The word is "beneficial", not "benefical".

reader dave1964b said...

"The other parts of the IPCC may very well see the light before WG1 and they may conclude that there is no problem before others do." After reading Tim Ball's new book on corrupted climate science, it is obvious this would not come to pass. The other parts of the IPCC pay no attention to WG1 unless WG1 serves up alarmism. The IPCC does not do or present objective science, they exist solely to promote alarmism and funnel money to the right places.