Last Monday was an exciting day!




Notes from a plane ride

Notes from a plane ride

According to inflation theory, a nearly homogeneous scalar field called the inflaton (denoted by $latex phi$)  filled the very early universe. The value of $latex phi$ varied with time, as determined by a potential function $latex V(phi)$. The inflaton rolled slowly for a while, while the dark energy stored in $latex V(phi)$ 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.



Exponential expansion


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

Here $latex dot a$ means the time derivative of the scale factor, and $latex m_P = 1/sqrt{8 pi G} approx 2.4 times 10^{18}$ 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 $latex hbar$ 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 $latex dot a / a$ does not slow down as the expansion proceeds.)

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

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

where $latex H$, the Hubble constant during inflation, is

$latex H sim frac{sqrt{V}}{m_P}.$

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 $latex 50 H^{-1}$.

Slow rolling

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

$latex H dot phi sim -V'(phi).$

Here $latex V'(phi)$ 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 $latex dot phi$ has to be $latex H$ on dimensional grounds. (Here I have blown another factor of 3, but let’s not worry about that.)

Density perturbations


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 $latex e^{Ht}$, and once the wavelength gets large compared to $latex H^{-1}$, 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 $latex H^{-1}$ and the universe is expanding at the rate $latex H$, so $latex H$ is the only thing the magnitude of the fluctuations could depend on. Since the field $latex phi$ has the same dimensions as $latex H$, we conclude that fluctuations have magnitude

$latex delta phi sim H.$

From inflaton fluctuations to density perturbations


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

$latex delta t sim frac{delta phi}{dot phi} sim frac{H}{dotphi}.$

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

$latex delta_Sequivleft(frac{delta rho}{rho}right)_{hor} sim frac{delta a}{a} sim frac{dot a}{a} delta tsim frac{H^2}{dot phi}.$

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 $latex H$ and the rolling speed $latex dot phi$ 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 $latex delta_S$, where the subscript $latex S$ stands for “scalar”.

Perturbations in terms of the potential

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

$latex delta_S^2 sim frac{H^4}{dotphi^2}sim frac{H^6}{V’^2} sim frac{1}{{m_P}^6}frac{V^3}{V’^2}.$


Gravitational waves and the meaning of r


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

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

From observations of the CMB anisotropy we know that $latex delta_Ssim 10^{-5}$, and now BICEP2 claims that the ratio

$latex r = frac{delta_T^2}{delta_S^2}$

is about $latex rsim 0.2$ 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

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

This is our second important conclusion: The energy density during inflation defines a mass scale, which turns our to be $latex 2 times 10^{16}~GeV$ for the observed value of $latex r$. 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 $latex r$ would be smaller by a factor of 16, and hence much harder to detect.

Rolling, rolling, rolling, …

Using $latex delta_S^2 sim H^4/dotphi^2$, we can express $latex r$ as

$latex r = frac{delta_T^2}{delta_S^2}sim frac{dotphi^2}{m_P^2 H^2}.$

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

$latex frac{1}{m_P^2} left(frac{dphi}{dN}right)^2sim r;$

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

$latex frac{Delta phi}{m_P} sim N sqrt{r}.$


Spectral tilt

As the inflaton rolls, the potential energy, and hence also the Hubble constant $latex H$, 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 $latex delta_S^2 sim H^4/dotphi^2$, and assuming $latex dotphi$ is constant, we estimate the scalar spectral tilt as

$latex -frac{1}{delta_S^2}frac{ddelta_S^2}{d N} sim – frac{4 dot H}{H^2}.$

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

From $latex H^2 sim V/m_P^2$, we find

$latex dot H sim frac{1}{2} dot phi frac{V’}{V} H,$

and using $latex dot phi sim -V’/H$ we find

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

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

This is our last important conclusion: A relatively large value of $latex r$ 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 $latex r sim .16$. The BICEP2 detection of the tensor fluctuations (if correct) has confirmed that suspicion.

Summing up




Added notes (3/25/2014):



$latex epsilon = frac{m_p^2}{2}left(frac{V’}{V}right)^2,quad eta = m_p^2left(frac{V”}{V}right).$

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

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

$latex 3Hdot phi = -V’, quad 3H^2 = V/m_P^2,$


It is also easy to see that $latex r$ is proportional to $latex epsilon$; it turns out that $latex r = 16 epsilon$. 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 $latex r/8$ does not depend on the details of the potential, while the relation between the scalar tilt and $latex r$ 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 $latex r$.


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

$latex delta_T^2 sim hbar G^2V.$

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