EFTA00615287.pdf

Killing the Straw Man: Does BICEP Prove Inflation? James 13. Dent Department of Physics, University of Louisiana at Lafayette, Lafayette, LA 70504, USA, Lawrence M. Krauss Department of Physics and School of Earth and Space Exploration, Arizona State University, Tempe, AZ 85287, USA, and Mount Stromlo Observatory, Research School of Astronomy and Astrophysics, Australian National University, Weston, ACT, Australia, 2611 Harsh Mathur Department of Physics, Case Western Reserve University, Cleveland, Ohio 44106-7079 The surprisingly large value of r, the ratio of power in tensor to scalar density perturbations in the CMB reported by the BICEP2 Collaboration provides strong evidence for Inflation at the GUT scale. In order to provide compelling evidence, other possible sources of the signal need to be ruled out. While the Inflationary signal remains the best motivated source, the current measurement unfortunately still allows for the possibility that a comparable gravitational wave background might result from a self ordering scalar field transition that takes place later at somewhat lower energy. However even marginally improved limits on the possible isocurvature contribution to CMB anistropies could rule out this possibility, and essentially all other sources of the observed signal other than Inflation. The recent claimed observation of primordial gravita- tional waves [1] provides a dramatic new empirical win- dow on the early universe. In particular, it provides the opportunity, in principle, to definitively test the inflation- ary paradigm[2, 3], and to explore the specific physics of inflationary models. However, while there is little doubt that inflation at the Grand Unified Scale is the best mo- tivated source of such primordial waves (e.g. [4-7], it is important to demonstrate that other possible sources cannot account for the current BICEP2 data before def- initely claiming Inflation has been proved. A surprisingly large value of r, the ratio of power in tensor modes to scalar density perturbations provides a challenge for other possible primordial sources, as such sources would have to generate gravitational waves effi- ciently without altering the observed adiabatic density fluctuations that are so consistent with inflationary pre- dictions. Here we explore to what extent that challenge might rule out other possibilities. We have previously explored a relatively generic pos- sible competing source of a scale invariant spectrum of tensor modes [8-10], a simple self ordering scalar field (SOSF) in the early universe, and frankly had hoped that the BICEP2 observation would rule out this possibility, thus allowing a cleaner interpretation of the the existing data in terms of inflation. As we describe here unfortu- nately the measured value of 7' falls just short of ruling out this other source as the dominant contribution of the observed effect. Nevertheless, as we also show, reducing the bound on any possible isocurvature component of the scalar power spectrum can rule out this possibility, and therefore any likely candidate source after inflation that produces gravitational waves. This would then imply the BICEP2 result definitely reflects gravitational waves from inflation, with all of the exciting concomitant im- plications (i.e. quantization of gravity [11]). In the following we assume inflation occurs, and pro- vides the measured adiabatic scalar density fluctua- tions inferred from CMB measurements (because that is strongly suggested by the data), but that a SOSF phase transition occurs after inflation, producing a grav- itational wave signature that might overwhelm the infla- tionary signal. Let Si and Ti denote the scalar and tensor power gen- erated by inflation and S,, and To the same quanti- ties for the self-ordered scalar field. Out of these four quantities one can form several ratios of interest: (i) r uff = (Ti + T41($1 -F Sc,,) is the tensor to scalar ratio in- corporating both sources that has just been observed to have a central value of 0.2. (ii) The self-ordering scalar field produces isocurvature scalar fluctuations whereas inflation produces adiabatic ones. Measurements of the temperature anisotropies constrain the isocurvature frac- tion x = + $O to lie in the range 0 < x < 0.09 [12]. (iii) = To/$w, the tensor-to-scalar ratio for the SOSF case, can be calculated within the self-ordering scalar field model using the scalar power spectrum de- scribed in [13] along with the tensor power given in [9, 10], and is found to be 2.34 (iv) f = TaTi, the ratio of the tensor contributions from the SOSF mechanism to that produced by inflation, is given by (140/N)(Vd1/i ) [9, 10] where N denotes the number of components of the self- ordering scalar field (presumed to be large and definitely greater than three), V is the symmetry breaking scale for the self-ordering field and Vi is the scale of inflation. We need Vp < 14 to ensure that symmetry breaking occurs after inflation (otherwise evidence of it would be oblit- erated by inflation). This inequality constrains the ratio f. (v) The tensor to scalar ratio for inflation r i = Ti /S1 is the quantity of interest for inflationary models. In the EFTA00615287 2 absence of the self-ordering scalar fields, ri is equal to the measured quantity rah but the present measurement currently allows ri to have a considerably lower value if self-ordering scalar fields dominate the observed signal. A priori, this need not have been the case. Since only three of these ratios are independent, but there are now constraints on four of them, in principle, the data is capa- ble of ruling out the existence of self-ordering scalar fields as a source. To explicitly determine the constraints we express f in terms of reff, x and rv, xrs, f — raw — xr, (1) Fig.1 shows a plot of f as a function of x reveals that f grows monotonically with x, diverging at x©© = r effir tio 0.085. This corresponds to a situation where the SOSF contribution essentially accounts for all of the observed BICEP2 polarization, and therefore contributes a frac- tion 0.2/2.34 of the (isocurvature) power in scalar density perturbations. Since x©© is less than the maximum iso-curvature ra- tio compatible with the temperature anisotropy data we arrive at the disappointing conclusion that the new mea- surement of rat does not additionally constraint self- ordering scalar fields. Had rat been larger, the isocurva ture contribution of SOSF to scalar density perturbations would have to have been larger to account for the entire tensor signal, and existing constraints on this contribu- tion would have therefore constrained f, and thereby the symmetry breaking scale, While this is disappointing, it is cause for hope. A small improvement on the iso-curvature fraction in CMB temperature fluctuations would imply that $0$F cannot give the full measured contribution to rat and therefore the signal front inflation is observable in the data. Alter- natively a non-zero measured isocurvature fraction might be suggestive that an SOSF has occurred and contributes to the BICEP2 signal. 30 20 %•-• 10 0.02 0.04 X 0.06 0.08 FIG. 1: Plot of f, the ratio of tensor contributions from SOSF to those of inflation as a function of the isocurvature fraction, x. Note f must lie below a maximum value, f ma.„, so x is actually constrained to lie in the range 0 < x < 0.2 C 0.1 00. 0.02 0.04 x 0.06 0.08 FIG. 2: The inflationary tensor-to-scalar ratio. r, as a func- tion of the isocurvature fraction, x. rather than 0 < x < xce. Here i max x Xmax fm.) ca' (2) obtained by inverting eq (1) and setting f We estimate f„,.. At-- 35 by taking N = 4, and setting Ka = V1 leading to x„,. %.•:: 0.083. Finally for completeness we display the inflationary tensor to scalar ratio ri that may be inferred from the data as a function of the isocurvature fraction of scalar density perturbations induced by SOSF. This allows a quantitative estimate of how future constraints on this fraction can then allow one to infer the fraction of the BICEP2 signal that must result from Inflation. Fig.2 shows a plot of ri as a function of x over its al- lowed range. As can be seen, if the current upper limit of 0.09 is reduced by a factor about 2, then the inflationary contribution must dominate. However, even a reduction by only 20% or so would imply a clear non-zero inflation- ary component to the observed BICEP2 signal While it is perhaps frustrating that the current ob- servation cannot unambiguously rule out this toy model straw man as a source of gravitational waves that could polarize the CNIB signal as observed by BICEP2. How- ever, as we have described, we are at the threshold of being able to argue that Inflation unambiguously pro- vides at the very least a non-zero component of the sig- nal. Note that because the scale of inflation varies as the fourth root of r, the scale of inflation will remain essen- tially identical to the Grand Unified Scale independent of whether it contributes all, or only a fraction of the observed polarization signal. We also note that the current analysis has not in- cluded the passible contribution from vector modes due to SOSF. However since such modes are known to con- tribute roughly equally to scalar and tensor modes in the CMB it should not significantly affect ratios, although it would need to be calculated and included in a more complete future quantitative analysis. Filially we note that while current data cannot defini- tively rule out a SOW transition as the source of gravita- EFTA00615288 3 tional waves, it nevertheless does imply that the source for such waves is at, or near the Grand Unified Scale. Thus, it allows an exploration of physics at a scale far larger than we can currently constrain at terrestrial ex- periments. This will be very important for constraining physics beyond the standard model, whether or not in- flation is responsible for the entire BICEP2 signal, even though existing data from cosmology is strongly sugges- tive that it does. We acknowledge discussions with Kate Jones-Smith and Paul J. Steinhardt at an early stage of this work. (1] BICEP2 Collaboration (PAIL Ade et at), arXiv:1403.3985 [astro-ph.CO]. (2] A.H. Guth, Phys.Rev. D23 (1981) 347-356. (3] A. Linde, Phys.Lett. 8108 (1982) 389-393. (4] V.A. Rubakov, M.V. Sazhin, and A.V. Veryaskin, Phys.Lett. 13115 (1982) 189-192. [5] R. Fabbri and M.d. Pollock, Phys.Lett. B125 (1983) 445- 448. [6] L.F. Abbott and M.B. Wise, Nucl.Phys. B244 (1984) 541-548. [7] L.M. Krauss and M.J. White, Phys.Rev.Lett. 69 (1992) 869-872, hep-ph/9205212. [8] L.M. Krauss, Phys.Lett. B284 (1992) 229-233. [9] K. Jones-Smith, L.M. Krauss, and H. Mathur, Phys.Rev.Lett. 100 (2008) 131302. arXiv:0712.0778 [astro-ph]. [10] L.M. Krauss, K. Jones-Smith, H. Mathur, and J. B. Dent, Phys.Rev. D82 (2010) 044001, arXiv:1003.1735. [11] L.M. Krauss and F. Wilczek, Phys.Rev. D89 (2014) 047501, arXiv:1309.5343 [hep-th]. [12] Planck Collaboration (P.A.R_ Ade et at.), arXiv:1303.5082 [astro-ph.CO]. [13] D.G. Figueroa, R.R. Caldwell, and M. Kamionkowski, Phys.Rev. D81 (2010) 123504, arXiv:1003.0672 [astro- ph.00]. EFTA00615289