Physics 3, 103 (2010) Viewpoint Can we test inflationary expansion of the early universe? Arthur Kosowsky Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USA Published December 6, 2010 A set of proposed relations among observable quantities may allow strong tests of whether a rapid expansion of the very early universe produced the seeds of the large-scale structure we see today. Subject Areas: Particles and Fields, Astrophysics, Cosmology A Viewpoint on: Testing Inflation: A Bootstrap Approach Latham Boyle and Paul J. Steinhardt Phys. Rev. Lett. 105, 241301 (2010) – Published December 6, 2010 Inflation in the early universe is a period of exponen- scalar field the inflaton might be, or the energy scale tial expansion that stretched a small causally-connected of the field’s potential energy, or indeed if fundamental patch of the universe by a factor of at least e60 into a size classical scalar fields even exist in nature. large enough to encompass the visible universe today. Inflation is compelling for another reason as well: At the same time this expansion drives the spatial ge- during a period of exponential expansion, quantum ometry of the expanding region to flatness (for pedagog- fluctuations in both the energy density and in the grav- ical reviews, see Refs. [1, 2]). Such a period would ex- itational field will inevitably be stretched by the expan- plain why the universe today appears very nearly spa- sion into small-amplitude classical fluctuations on scales tially flat on average (even though the dynamics of the larger than the horizon (see Ref. [4] for a review). Any expanding universe generically drive it away from flat- causal influences must propagate at a velocity limited ness). It would also explain why the microwave back- by the speed of light. If the wavelength of a virtual ground today has almost the same temperature in all quantum fluctuation gets stretched to a length greater sky directions (even though in the standard cosmology, than ct, with t the age of the universe, then the fluctu- most points of origin for the microwave radiation we ation’s spatially separated parts cannot communicate, see today were never within each others’ causal hori- which prevents it from disappearing, and it becomes zon, meaning they were never in causal contact). But is frozen as a classical perturbation whose wavelength it possible to test whether such an inflationary episode continues to stretch with the expansion of the universe. actually occurred? In a paper in Physical Review Letters, These fluctuations can be precisely those we see to- Latham Boyle of the Canadian Institute for Theoretical day in the temperature fluctuations of the microwave Astrophysics in Toronto and Paul Steinhardt of Prince- background; the inflation-generated density fluctua- ton University propose connections between observable tions grow via gravitational instability into the observed quantities as a check of inflation [3]. large-scale distribution of galaxies, while the gravita- The “flatness” and “horizon” problems are intractable tional field fluctuations become a stochastic background enough in the standard cosmology that most cosmolo- of very-long-wavelength gravitational waves. The per- gists believe inflation occurred, even though we have turbations we can probe today with cosmological ob- no fundamental physical theory that provides a mecha- servations would have been generated during inflation nism to drive inflation. If general relativity is correct, ac- when the universe was around e60 times smaller than it celerating expansion occurs when p < −r/(3c2), where was at the end of inflation. If inflation occurred, it must − p is the mean pressure and r the mean energy density have started when the universe was less than 10 12 sec- of the universe. Energy density is positive by definition, onds old, corresponding to the electroweak energy scale but pressure can be negative in principle (think of a col- of 100 GeV, and may very well have begun at an age lection of masses connected by springs stretched from smaller than 10−32 seconds, corresponding to the scale their equilibrium). Cosmologists often invoke a classical at which grand unification of the strong and electroweak scalar field j, dubbed the “inflaton” field, with some po- forces is conjectured to occur (see Chapter 3 of Ref. [5] tential V(j) as a model for the contents of the universe for a classic discussion of the thermal history of the uni- during inflation. The condition for inflation would be verse). satisfied if the potential energy of the scalar field domi- The density perturbations reflected in the observed nates over its kinetic energy. But we do not know what temperature fluctuations of the cosmic microwave back- DOI: 10.1103/Physics.3.103 c 2010 American Physical Society URL: http://link.aps.org/doi/10.1103/Physics.3.103 Physics 3, 103 (2010) ground and the large-scale distribution of galaxies are consistent with a power law as a function of scale, n P(k) = Ask s , with observables of the amplitude As and power law index ns, where k is the wave number cor- responding to a given Fourier mode of the primordial density perturbations. We also consider the possibility that the power law index itself can vary with scale, a quantity defined as as = dns/dlnk [6]. The gravitational wave perturbations (or “tensor” perturbations of the FIG. 1: (Left) Design drawing of the BICEP instrument that metric in general relativity) can likewise be described measures variations in the polarization of the cosmic mi- crowave background at a level of one part in ten million. ( ) = nt+1 as a power law with power spectrum P k Atk The 25-cm aperture telescope, with an angular resolution of (with, unfortunately, a different historical convention around a half degree at a microwave frequency of 150 GHz, for the power law index!). The precise definition of am- is designed to limit spurious polarization signals from the in- plitudes depends on the scale at which they are defined. strument at this stringent level. BICEP uses 100 polarization- Assuming that both power spectra are given by these sensitive bolometers cooled to a fraction of a degree Kelvin forms, the set of observables we can hope to probe is to detect the polarized radiation. (Right) Photo of the Back- As, ns, as, At, and nt, with the first two of these be- ground Imaging of Cosmic Extragalactic Polarization (BICEP) ing well measured already. (A variation with scale of experiment, which has given the current best limits on the the tensor power law is almost surely unobservable.) It tensor-scalar ratio r from measuring microwave background has long been appreciated that in simple models of infla- polarization alone [12]. Such experiments are working to- wards measurements of r with precision of 0.01 that are nec- tion driven by a single scalar field, the two power spec- essary for strong tests of the bootstrap relations proposed by tra describing the density and gravitational-wave per- Boyle and Steinhardt. The image shows the Dark Sector Lab turbations are not independent. Expanding expressions at the South Pole, atop which sits BICEP behind the metal for measurable quantities in series of so-called “slow- ground shield. (Credit: (Left) Adapted from Takahashi et roll” parameters (formed from the expansion rate and al.[10]; (Right) Brian Keating, UCSD) its derivatives during inflation) gives a relation between r = At/As, the ratio of density to tensor perturbation amplitudes, and nt [7]. Unfortunately, obtaining precise coming experiments (Fig. 1) will likely have the sensi- measurements of nt seems unlikely. tivity and systematic error control to detect a tensor am- Boyle and Steinhardt [3] employ a simple alternate plitude as small as r = 0.01 (provided that contaminat- strategy: expand the Hubble parameter (which quanti- ing polarized emission at microwave wavelengths from fies the expansion rate of the universe) during inflation, our own galaxy can be understood well enough) [10]. H(j), as a Taylor expansion in the value of the inflaton The largest observational hurdle for decisive tests of field j around the field value generating the perturba- the full set of bootstrap relations may be measuring as tion scales of interest, j∗ [as in Eq. (2) of their paper]. to sufficient precision; note that the first bootstrap test −4 (Note that, while convenient to parameterize the evolu- predicts as = −5.3 × 10 . Future measurements of 21- tion of H in terms of the effects of some effective infla- cm radiation from neutral hydrogen prior to the reion- ton field f, this is not necessary and the results do not ization of the universe by the first stars and galaxies can, −4 depend on inflation being driven by an actual funda- in principle, probe as at the level of 10 [11], although mental scalar field.) Then, by truncating this expansion this exciting prospect depends partly on control of sys- after two, three, or four terms and requiring that the re- tematic errors that are not well understood at present. sulting functional form for H(j) be valid until the end The bootstrap tests presented here involve only quan- of inflation, Boyle and Steinhardt have generated “boot- tities that are potentially observable with current or up- strap tests”—sets of numerical relations between the ob- coming experiments. In the absence of inflation, there servables r, ns, and as. Current microwave background is no reason why r, ns, and as should have any particu- temperature power spectrum measurements by, for ex- lar relationships between them. If the relationships pre- ample, the WMAP satellite combined with the higher- sented in this paper are observed, this is strong evidence resolution ACT experiment, give ns = 0.962 ± 0.013 in favor of an inflationary origin for primordial pertur- for a model with as = 0, and ns = 1.032 ± 0.039 and bations, and constitutes a direct probe of physics at an as = −0.034 ± 0.018 for models with both ns and as as energy scale far beyond the reach of any particle accel- free parameters [8].