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Gravitational-Wave Cosmology Across 29 Decades In Frequency
P. D. Lasky
C. M.F. Mingarelli
Tristan L. Smith Swarthmore College, [email protected]
J. T. Giblin Jr.
E. Thrane
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Recommended Citation P. D. Lasky, C. M.F. Mingarelli, Tristan L. Smith, J. T. Giblin Jr., E. Thrane, D. J. Reardon, R. Caldwell, M. Bailes, N.D. R. Bhat, S. Burke-Spolaor, S. Dai, J. Dempsey, G. Hobbs, M. Kerr, Y. Levin, R. N. Manchester, S. Osłowski, V. Ravi, P. A. Rosado, R. M. Shannon, R. Spiewak, W. Van Straten, L. Toomey, J. Wang, L. Wen, X. You, and X. Zhu. (2016). "Gravitational-Wave Cosmology Across 29 Decades In Frequency". Physical Review X. Volume 6, Issue 1. DOI: 10.1103/PhysRevX.6.011035 https://works.swarthmore.edu/fac-physics/261
This work is brought to you for free by Swarthmore College Libraries' Works. It has been accepted for inclusion in Physics & Astronomy Faculty Works by an authorized administrator of Works. For more information, please contact [email protected]. Authors P. D. Lasky, C. M.F. Mingarelli, Tristan L. Smith, J. T. Giblin Jr., E. Thrane, D. J. Reardon, R. Caldwell, M. Bailes, N.D. R. Bhat, S. Burke-Spolaor, S. Dai, J. Dempsey, G. Hobbs, M. Kerr, Y. Levin, R. N. Manchester, S. Osłowski, V. Ravi, P. A. Rosado, R. M. Shannon, R. Spiewak, W. Van Straten, L. Toomey, J. Wang, L. Wen, X. You, and X. Zhu
This article is available at Works: https://works.swarthmore.edu/fac-physics/261 PHYSICAL REVIEW X 6, 011035 (2016)
Gravitational-Wave Cosmology across 29 Decades in Frequency
Paul D. Lasky,1* Chiara M. F. Mingarelli,2,3 Tristan L. Smith,4 John T. Giblin, Jr.,5,6 Eric Thrane,1 Daniel J. Reardon,1 Robert Caldwell,7 Matthew Bailes,8 N. D. Ramesh Bhat,9 Sarah Burke-Spolaor,10 Shi Dai,11,12 James Dempsey,13 George Hobbs,11 Matthew Kerr,11 Yuri Levin,1 Richard N. Manchester,11 Stefan Osłowski,14,3 Vikram Ravi,15 Pablo A. Rosado,8 Ryan M. Shannon,11,9 Renée Spiewak,16 Willem van Straten,8 Lawrence Toomey,11 Jingbo Wang,17 Linqing Wen,18 Xiaopeng You,19 and Xingjiang Zhu18 1Monash Centre for Astrophysics, School of Physics and Astronomy, Monash University, Victoria 3800, Australia 2TAPIR Group, MC 350-17, California Institute of Technology, Pasadena, California 91125, USA 3Max Planck Institute for Radio Astronomy, Auf dem Hügel 69, D-53121 Bonn, Germany 4Department of Physics and Astronomy, Swarthmore College, Swarthmore, Pennsylvania 19081, USA 5Department of Physics, Kenyon College, Gambier, Ohio 43022, USA 6Department of Physics and CERCA, Case Western Reserve University, Cleveland, Ohio 44106, USA 7Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire 03755 USA 8Centre for Astrophysics and Supercomputing, Swinburne University of Technology, P.O. Box 218, Hawthorn, Victoria 3122, Australia 9International Centre for Radio Astronomy Research, Curtin University, Bentley, Western Australia 6102, Australia 10National Radio Astronomy Observatory Array Operations Center, P.O. Box O, Soccoro, New Mexico 87801, USA 11Australia Telescope National Facility, CSIRO Astronomy and Space Science, P.O. Box 76, Epping, New South Wales 1710, Australia 12School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China 13CSIRO Information Management and Technology, P.O. Box 225, Dickson, Australian Capital Territory 2602, Australia 14Fakultät für Physik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany 15Cahill Center for Astronomy and Astrophysics, MC 249-17, California Institute of Technology, Pasadena, California 91125, USA 16Department of Physics, University of Wisconsin-Milwaukee, P.O. Box 413, Milwaukee, Wisconsin 53201, USA 17Xinjiang Astronomical Observatory, Chinese Academy of Sciences, 150 Science 1-Street, Urumqi, Xinjiang 830011, China 18School of Physics, University of Western Australia, Crawley, Western Australia 6009, Australia 19School of Physical Science and Technology, Southwest University, Chongqing, 400715, China (Received 13 October 2015; revised manuscript received 8 January 2016; published 31 March 2016) Quantum fluctuations of the gravitational field in the early Universe, amplified by inflation, produce a primordial gravitational-wave background across a broad frequency band. We derive constraints on the spectrum of this gravitational radiation, and hence on theories of the early Universe, by combining experiments that cover 29 orders of magnitude in frequency. These include Planck observations of cosmic microwave background temperature and polarization power spectra and lensing, together with baryon acoustic oscillations and big bang nucleosynthesis measurements, as well as new pulsar timing array and ground-based interferometer limits. While individual experiments constrain the gravitational-wave energy density in specific frequency bands, the combination of experiments allows us to constrain cosmological parameters, including the inflationary spectral index nt and the tensor-to-scalar ratio r. Results from individual experiments include the most stringent nanohertz limit of the primordial background to date −10 from the Parkes Pulsar Timing Array, ΩGWðfÞ < 2.3 × 10 . Observations of the cosmic microwave background alone limit the gravitational-wave spectral index at 95% confidence to nt ≲ 5 for a tensor-to- scalar ratio of r ¼ 0.11. However, the combination of all the above experiments limits nt < 0.36. Future
Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
2160-3308=16=6(1)=011035(11) 011035-1 Published by the American Physical Society PAUL D. LASKY et al. PHYS. REV. X 6, 011035 (2016)
Advanced LIGO observations are expected to further constrain nt < 0.34 by 2020. When cosmic microwave background experiments detect a nonzero r, our results will imply even more stringent constraints on nt and, hence, theories of the early Universe.
DOI: 10.1103/PhysRevX.6.011035 Subject Areas: Cosmology, Gravitation
I. INTRODUCTION A red spectrum, combined with observational constraints Gravitational-wave astronomy is now a reality. The on the amplitude of GWs from the CMB, imply that GW LIGO Scientific Collaboration has recently announced detectors such as pulsar timing arrays (PTAs) and ground- the first direct detection of gravitational waves (GWs) based interferometers such as the Laser Interferometer coming from the merger of a binary black hole [1]. Other Gravitational-wave Observatory (LIGO) [7] and Virgo experiments worldwide are ready to measure gravitational [8] are not sufficiently sensitive to detect primordial radiation across a wide range of frequencies. From the GWs predicted by the simplest model of inflation (see, cosmic microwave background (CMB) to ground-based e.g., Ref. [9]). Detection at frequencies at or above PTAs GW interferometers, these experiments cover more than 21 may require extremely ambitious detectors such as the Big orders of magnitude in frequency—29 with complementary Bang Observer [10] or DECIGO [11]. However, some but indirect bounds from big bang nucleosynthesis (BBN), nonstandard models for the early Universe predict blue GW CMB temperature and polarization power spectra and spectra, which could be detected by PTAs and/or LIGO (see below). lensing, and baryon acoustic oscillation (BAO) measure- A blue spectrum can be generated from inflation depend- ments. Each of these experiments is sensitive to a primor- ing on what happens when GW modes exit the horizon, dial stochastic GW background, originating from quantum either by nonstandard evolution of the Universe during fluctuations in the early Universe, and amplified by an inflation or if there is nonstandard power in these modes inflationary phase [2–5]. Standard inflationary models when they exit. This idea gained recent popularity in the predict a primordial GW background whose amplitude is wake of some early interpretations of the BICEP2 obser- proportional to the energy scale of inflation [6]. vations [12], where a flat GW spectrum was unable to Observations of primordial GWs therefore provide unique simultaneously explain both the lower-frequency Planck insights into poorly understood processes in the very early observations [13] and the higher-frequency BICEP2 results Universe and its evolution from 10−32 s after the big bang (see, e.g., Refs. [14–16]). through to today. Standard models of inflation suggest that the slope of In standard inflationary theories, the GW energy spec- the GW spectrum should be approximately equal to the trum is expected to be nearly scale invariant—above a slope of the power spectrum of density perturbations. This certain frequency, the GW energy density decreases mono- prediction can be modified by having more than just a tonically with increasing frequency [6]. The gravitational simple scalar field driving inflation. These nonminimal field has quantum mechanical fluctuations, which are models can predict either red GW spectra whose spectral dynamic at wavelengthspffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi smaller than the cosmological index varies from that of standard inflation [17] or blue −1 2 horizon, H ¼ 3c =ð8πGρÞ, and static due to causality spectra (see, e.g., Refs. [18,19]). The latter modification is at wavelengths larger than the horizon. During inflation, so dramatic that the system violates the null-energy con- modes are redshifted and pulled outside the horizon, where dition, a desirable, but by no means compulsory, property their power is frozen in with an amplitude that corresponds of the stress-energy tensor. Alternatively, blue spectra can to the size of the cosmological horizon, and, hence, to the be generated if the propagation speed of primordial GWs energy density of the Universe at that time. As inflation varies during inflation [20], or by introducing new inter- progresses, the energy scale of the Universe decreases, and actions between the scalar field and gravity, where these the cosmological horizon grows. This is a consequence of interactions are low-energy remnants of some (unknown) the null energy condition, which posits that the energy modification of general relativity at much higher energy density of the Universe cannot increase as a function of scales, such as the Planck scale. Couplings of this form do time. Modes that freeze out at larger physical wavelength not change any of the standard predictions of general have less power in them. Therefore, the slowly and relativity, but the theories that predict them allow us to treat monotonically decreasing energy density of the Universe the (unknown) high-energy theory of gravity in an effective during inflation is responsible for the monotonically low-energy limit for some energy scales. The simplest of decreasing shape of the primordial power spectrum of all such effective field theories produce a blue spectrum [21]. fields. Spectra that decrease with increasing frequency It is also possible to abandon inflation altogether and are referred to as “red” spectra, and those that grow with replace it with a scenario that preserves the observed increasing frequency are “blue.” spectrum of density perturbations. Two classic examples
011035-2 GRAVITATIONAL-WAVE COSMOLOGY ACROSS 29 … PHYS. REV. X 6, 011035 (2016) are string-gas [22] and ekpyrotic cosmologies [23]. In the Equation (1) is the simplest approximation one can make former, an ensemble of fundamental strings has thermo- about the primordial GW spectrum. Most early-Universe dynamic properties that produce a high-temperature, qua- theories predict only a small deviation from pure power-law sistatic state, which produces a blue GW spectrum, whose behavior. The next level of complexity is to replace nt with n þ α ðf=f Þ=2 α size is comparable in magnitude to the standard red t t ln CMB in Eq. (1), where t is known as the spectrum [24,25]. Ekpyrosis posits that the primordial running of the spectral index. For example, single-field, 2 spectrum of perturbations is a result of a pre-big-bang slow-roll inflationary models predict αt ≃ ð1 − nsÞ (see, contracting phase. Such a phase has an increasing e.g., Ref. [6]), where ns ¼ 0.9645 0.0049 is the mea- energy density and would create a blue power GW sured value of the scalar spectral index [13]. Therefore, spectrum [23,26]. within this class of theories, we expect a correction to the Importantly, a blue primordial GW spectrum may yield a total GW power-law index of ≃10−2. Over 29 decades in primordial background immediately below present-day frequency, this can have a marginal effect on the results limits, which may be detectable in the near future. presented here, a point we discuss in more detail below. While CMB experiments are likely to make direct mea- Because of the expansion of the Universe, the primordial r surements of the tensor-to-scalar ratio , they will poorly GW spectrum that we observe today has evolved since it n constrain the tensor index t. In this paper, we show how was created. This evolution is expressed in terms of a the combination of constraints on the primordial GW transfer function, T ðfÞ, which encodes information about background from CMB, PTA, BBN, BAO, and ground- how GWs change as a function of frequency [33]. The based interferometer GW experiments can place stringent energy density of GWs today is given by constraints on nt, yielding insights into the physics of the Z early Universe not accessible by any other means. 4 3 dff ð2πÞ 2 ρ ¼ PtðfÞT ðfÞ : ð2Þ GW c5 II. GRAVITATIONAL WAVE EXPERIMENTS Current results from experiments trying to measure the The PTA and LIGO communities commonly present the primordial GW background do little to constrain the GW spectrum in terms of the energy density in GWs as a possible tilt of the spectrum. However, combining GW fraction of the closure energy density per logarithmic experiments over all frequencies allows us to constrain frequency interval [31,34], cosmological parameters from nonstandard inflationary 1 dρ cosmologies [27–30]. Combined CMB observations Ω ðfÞ ≡ GW ; ð Þ GW 3 from the Planck satellite and the BICEP2 experiment ρc d ln f constrain the stochastic GW background at frequencies −20 −16 ρ ≡ 3c2H2=ð8πGÞ H ¼ 100h −1 −1 of ∼10 –10 Hz, while PTAs are sensitive to GW where c 0 , 0 km s Mpc is −9 −7 h ¼ 0 67 frequencies of ∼10 –10 Hz and ground-based interfer- the Hubble expansion rate, and . is the dimension- ometers are sensitive at ∼10–103 Hz [31]. As we show less Hubble parameter [13]. Indirect constraints on the “ ” Ω ≡ below, constraints on the total energy density of GWs from GWR background are typically integral bounds on GW d fΩ ðfÞ BBN, gravitational lensing, CMB power spectra, and BAO ln GW . are sensitive to GWs as high as 109 Hz. Therefore, even a Assuming a standard expansion history that includes small blue tilt in the GW spectrum may be detectable in the nonrelativistic matter and radiation, the GW spectrum – GW frequency band covered from the CMB to LIGO and today is given by [33,35 37] Virgo and provide more stringent constraints on the overall f nt 1 f 2 16 shape of the GW background [30]. Ω ðfÞ¼ΩCMB eq þ ; ð Þ GW GW f 2 f 9 4 A first step in our effort to apply experimental constraints CMB to the GW energy-density spectrum is to assume that it can f be well approximated by a power law: where eq is the frequency of the mode whose correspond- ing wavelength is equal to the size of the Universe at the f nt time of matter-radiation equality with frequency f ¼ P ðfÞ¼A ; ð Þ pffiffiffi pffiffiffiffiffiffi eq t t f 1 2cH Ω =2π Ω Ω Ω CMB 0 m r. Here, m and r are, respectively, the total matter and radiation energy density evaluated today, f where the pivot frequency CMB is taken to be the standard and f ¼ðc=2πÞ0 05 −1 value CMB . Mpc (see, e.g., Ref. [32]). It is ΩCMB ≡ 3rA Ω =128: ð Þ conventional to reexpress the amplitude of the primordial GW s r 5 GW spectrum in terms of the tensor-to-scalar ratio, r ≡ At=As, where As is the amplitude of the primordial Equation (4) is key in our analysis since it allows us to power spectrum of density perturbations, and both are combine constraints on r and nt from the CMB with Ω ðfÞ n evaluated at the pivot scale. constraints to GW and t from PTAs and LIGO. We use
011035-3 PAUL D. LASKY et al. PHYS. REV. X 6, 011035 (2016) cosmological parameters obtained from the latest Planck data preferred a slightly positive blue tilt for the tensor satellite data release [13]. power spectrum, which resulted from the inconsistency In the following sections, we combine observational between the original BICEP2 data and Planck observations. constraints spanning 29 orders of magnitude in frequency On the other hand, Liu et al. [39] presented constraints to derive stringent constraints on backgrounds with a from only CMB and PTA data, but focused on what a nonzero spectral index nt. Figure 1 highlights the key positive detection could do for our understanding of the Ω ðfÞ idea: we show the current best upper limits on GW , and early-Universe equation of state, cosmic phase transitions, a series of curves given by Eq. (4) that are constrained by and relativistic free streaming. these limits. Starting from the lowest frequency limits, we Our analysis improves on those of Refs. [30,39,40] in a summarize current upper limits before combining them to number of significant ways. First, we include the indirect derive joint constraints. GW constraints in a self-consistent way, which allows us to Note that, in evaluating the observed GW spectrum, we compare integral and nonintegral constraints with varying neglect the effects of neutrino free streaming [38] and phase spectral indices (see Sec. II D). Second, we present a new transitions occurring in the early Universe [35]. As shown analysis of Parkes Pulsar Timing Array (PPTA) data [43] Ω ðfÞ in Refs. [35,38,39], the free streaming of neutrinos and that give the best limit on GW in the PTA band by a phase transitions during the very early Universe lead to a factor of 4 over previous published results. Finally, we Ω ∼1=2–1=3 suppression of GW by a factor of for PTA and provide our own analysis of the raw PPTA time-of-arrival LIGO frequencies. However, because of the large lever arm data to allow for varying spectral indices, instead of between the frequencies probed by these detection methods assuming a constant nt for PTA observations taken from and the CMB, including this suppression in the analysis older PPTA analyses as is done in Refs. [30,39]. changes our constraints on nt by only a few percent. We therefore neglect these effects in our analysis. A. CMB intensity and polarization Three recent papers have presented combined constraints on nt and r using some combination of CMB, LIGO, and Primordial GWs imprint a characteristic signal onto the PTA data [30,39,40]. Huang and Wang [40] presented intensity and polarization of the CMB that can be measured their analysis soon after the original BICEP2 results were by ground-based and space-borne observatories. A joint reported [41,42] and, as such, focused on the fact that those analysis [13,42] of Planck satellite and BICEP2 and Keck
FIG. 1. Experimental constraints on ΩGWðfÞ. The black star is the current Parkes Pulsar Timing Array (PPTA) upper limit and all black curves and data points are current 95% confidence upper limits. The gray curve and triangle are, respectively, the predicted aLIGO sensitivity and PPTA sensitivity with 5 more years of data. The indirect GW limits are from CMB temperature and polarization power spectra, lensing, BAOs, and BBN. Models predicting a power-law spectrum that intersect with an observational constraint are ruled out at > 95% confidence. We show five predictions for the GW background, each with r ¼ 0.11, and with nt ¼ 0.68 (orange curve), nt ¼ 0.54 (blue curve), nt ¼ 0.36 (red curve), nt ¼ 0.34 (magenta curve), and the consistency relation, nt ¼ −r=8 (green curve), corresponding to minimal inflation.
011035-4 GRAVITATIONAL-WAVE COSMOLOGY ACROSS 29 … PHYS. REV. X 6, 011035 (2016) array data found that r<0.12 at 95% confidence level data sets were determined using the Boltzmann solver (C.L.) under the assumption that nt ¼ 0. The solid black CAMB and a modified version of the Monte Carlo stepper curve in Fig. 1 labeled “CMB” shows the estimated cosmoMC [53–55]. sensitivity of the Planck satellite. The CMB sensitivity curve is calculated by determining the value of the spectral B. Pulsar timing arrays Ω ðfÞ density GW that yields a marginally detectable signal given a model of the Planck satellite noise properties [44]. The incoherent superposition of primordial GWs is Observations of the CMB intensity and polarization are expected to imprint on the arrival time of pulses from analyzed by reexpressing the real-space data in a spherical the most stable millisecond pulsars. A number of PTAs harmonic expansion. The intensity measurements can be around the world are engaged in the hunt for GWs, expanded in the standard (scalar) spherical harmonics, including the Parkes Pulsar Timing Array [56], the whereas the polarization data must be expanded in spin- North American Nanohertz Observatory for Gravitational weighted spherical harmonics [45]. We can further separate Waves (NANOGrav) [57], and the European Pulsar various physical processes by dividing the polarization data Timing Array (EPTA) [58]. Here, we use recent data from into a curl-free (E-mode) and curl (B-mode) basis. In order the PPTA [43] to provide the strongest constraints Ω ðfÞ to compare these data to a theoretical model, the measured to date on GW from a primordial background in the spherical harmonic coefficients are further analyzed to PTA band. estimate their statistical correlations. The presence of a The PPTA monitors 24 pulsars with the 64-m Parkes primordial GW spectrum fundamentally alters the expected radio telescope in a bid to directly detect GWs, and correlations leading to an enhanced correlation for the currently has the most stringent upper limits on the GW intensity of the CMB on the largest angular scales as well as background from supermassive black hole binaries [43]. a nonzero correlation for the B-mode polarization [46,47]. We derive our limit on the primordial GW background by The expected effect of a nonzero primordial GW performing a similar Bayesian analysis to that in Ref. [43], spectrum on the CMB is calculated by solving the with the exception that we utilize the Bayesian pulsar Boltzmann equation for the various components of matter timing data analysis suite PAL2 [59], and allow for an that fill the Universe. The Boltzmann equation for the arbitrary strain spectral index. photons encodes all of the information about correlations in The GW spectrum in the PTA band can be approximated the intensity and polarization of the CMB. In particular, for as a power law, with each spherical harmonic multipole, the expected correla- 2 2π f nt tions can be expressed as an integral over cosmic time and Ω ðfÞ¼ A2 f2 ; ð Þ GW 3H2 GW yr f 6 frequency [48]. Therefore, the total expected CMB signal 0 yr due to primordial GWs can be expressed as an integral over A its spectrum, Ω ðfÞ. where GW is the amplitude of the characteristic strain at a GW f ≡ −1 The CMB sensitivity curve shown in Fig. 1 is calculated reference frequency of yr yr . The star in Fig. 1, by setting the total CMB signal-to-noise ratio equal to 2 labeled “PTA” is the 95% C.L. upper limit assuming a (corresponding to a 95% C.L. bound). The squared signal- spectral index of nt ¼ 0.5 (approximately the middle of to-noise ratio is calculated from a sum in quadrature of the range we are trying to constrain—see below), with Ω95%ðfÞ < 2 3 10−10 CMB B-mode multipoles divided by the estimated polari- GW . × . The black dots above the PPTA zation noise for each multipole from the Planck satellite’s limit are the upper limits from the EPTA [60] and 143-GHz detector [44,49]. Since, as discussed above, each NANOGrav [61]. Both the EPTA and NANOGrav present B-mode multipole is an integral over the GW spectrum, we limits on the GW energy density from inflationary relics express the integral over frequency as a sum so that we can assuming nt ¼ 0; our new limit for nt ¼ 0 [cf. our limit on Ω ðfÞ n ¼ 0 5 evaluate the contribution at each frequency interval. We GW for t . , which differs only in the second relate the primordial amplitude to the present-day spectral decimal place] is a factor of 4.1 better than the previous best density using Eq. (21) of Ref. [35]. The noise is calculated limit from Ref. [61]. under the hypothesis of no primordial GWs, although weak The gray triangle below the star in Fig. 1 is a predicted gravitational lensing also induces a nonzero B-mode GW upper limit derived by simulating an additional 5 years correlation, which we treat as an additional source of of PPTA data. We take the maximum likelihood red noise noise. Finally, the limit is converted into a “power-law- parameters in the existing data sets, estimate the white integrated” curve using the formalism from Thrane and noise level using the most recent data that represents Romano [50]. Any model intersecting this curve is ruled current observation quality, and assume a two-week out at 95% C.L. observing cadence to derive the 95% C.L. upper limit of r n Ω95%ðfÞ ≲ 5 10−11 Current CMB constraints to and t come from GW × . However, the PPTA limit will be measurements made by Planck [51], the BICEP2 and superseded before 2020 with limits placed from collating Keck array [42], and SPTpol [52]. Constraints from these data sets from the three existing PTAs as part of the
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International Pulsar Timing Array (IPTA) [62]. From We combine published data from Refs. [68,69] to generate Fig. 1, it becomes clear that PTAs may not play a significant a power-law-integrated curve [50], shown in Fig. 1.Any role in constraining inflationary models where the GW power-law model intersecting a power-law-integrated curve spectrum is described by Eq. (3) when aLIGO reaches is ruled out at 95% C.L. Then, utilizing the formalism from Ω ðfÞ design sensitivity, given the significant improvements in Mandic et al. [70], we obtain constraints on GW for Ω ðfÞ the latter experiment. However, PTAs can still play an arbitrary spectral indices. The limits on GW are important role for cosmological models with a varying converted into constraints on nt and r. spectral index, that is, with a non-negligible running of the At the time of writing, the LIGO experiment has begun spectral index αt. taking data for the first observing run of the advanced Giblin and Thrane [63] recently proposed a “rule of detector era, with the Virgo experiment to follow in 2016. thumb” for the maximum GW energy density for cosmo- At design sensitivity, advanced detectors are forecast to logical backgrounds based only on arguments of the energy achieve nearly 4 orders of magnitude of improvement in Ω ðfÞ “ ” budget of the Universe at early times. They presented GW ; see the curve marked aLIGO in Fig. 1, which is optimistic, realistic, and pessimistic upper limits for the projected sensitivity given two LIGO detectors operat- Ω ðfÞ GW , with the optimistic limit representing the largest ing for 1 year at design sensitivity. Ω ðfÞ value of GW possible given a reasonable set of conditions. The new PPTA limit reported here is the first time a GW limit in either the PTA or LIGO band has gone D. Indirect constraints under this optimistic threshold, thus marking the first time Indirect constraints on GW backgrounds have been the detection of cosmological GWs could actually have obtained using a variety of data including CMB temper- been possible according to arguments in Ref. [63]. ature and polarization power spectra, lensing, BAOs, and Conventional models of early-Universe particle physics BBN (see, e.g., Refs. [71–73]). Indirect bounds are “ ” Ω do not predict such a large GW background in the PTA integral bounds, which apply to GW and not to Ω ðfÞ frequency band. The temperature of the Universe at the GW ; see Eq. (3). Recently, Pagano et al. [74] combined time when such GWs are produced is ∼1 GeV (see top axis the latest Planck observations of CMB temperature and of Fig. 1), a temperature at which physics of the early polarization power spectra and lensing with BAO and BBN Universe is relatively well known. We note that the measurements (specifically, observations of the primordial possibility of first-order phase transitions that generate a deuterium abundance) to put an integral constraint on the strong GW background in the PTA frequency band is not Ω < 3 8 10−6 primordial GW background of GW . × . completely ruled out (see, e.g., Refs. [64–67]). Of course, it While there is a long history in the literature of plotting is possible that there is unknown physics that influences Ω Ω ðfÞ GW integral bounds alongside GW , they are not gravity without coupling strongly to the standard model of directly comparable. However, the two quantities can be particle physics that could produce a strong GW back- Ω ðfÞ related if we assume that GW is described by a power- ground in the PTA frequency band. law spectrum with a known cutoff frequency, which we f ¼ 1 choose to be max GHz, corresponding to an energy C. Ground-based interferometers scale typical of inflation, T ¼ 1017 GeV. Given this plau- LIGO [7] and Virgo [8] are long-baseline, ground-based sible assumption, we plot the indirect constraints in Fig. 1 GW interferometers with best sensitivity at frequencies of as power-law-integrated curves using the formalism from 102–103 Hz. Data collected during the initial phases of Ref. [50]. Any power-law model intersecting a power-law- these instruments have been used to place upper limits on a integrated curve is ruled out at 95% C.L. stochastic background of GWs from astrophysical and Inspecting Fig. 1, it is apparent that the current best cosmological sources [9,68,69]. We utilize data from the constraints on nt come from observations of the CMB initial LIGO and Virgo observatories. These data were combined with indirect bounds. collected in 2009–2010 as part of the fifth LIGO science The strength of the indirect bounds depends in part on f ¼ 1 run. Two limits were originally obtained using these our choice of max GHz; however, changing the cutoff combined observations: a lower-frequency limit from frequency by several orders of magnitude would not the combined LIGO-Virgo observations that assumed a flat, change qualitative picture. For example, in alternative i.e., nt ¼ 0, spectrum [68], and a higher-frequency limit theories of inflation, it is possible to posit an energy scale from an analysis of the two colocated LIGO detectors at as low as 106 GeV, corresponding to a cutoff frequency of f ¼ 10 Hanford, which assumed nt ¼ 3 [69]. max mHz. This choice of cutoff frequency shifts the We implement a new way to analyze LIGO and Virgo minimum of the indirect bound curve from ∼100 mHz to ∼1 μ Ω ðfÞ limits on the primordial background that allows for a Hz, while the minimum value of GW increases by varying spectral index. The analysis goes beyond a factor of ∼2. When aLIGO reaches design sensitivity, it Meerburg et al. [30] and Huang and Wang [40], which will surpass indirect constraints on primordial backgrounds both assume nt ¼ 0 for their LIGO and Virgo constraints. with nonrunning spectral indices.
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PPTA Planck LIGO and Virgo ... + BICEP2 ... + SPTpol Planck + BICEP2 + SPTpol
... + PPTA ... + LIGO and Virgo ... + PPTA ... + indirect ... + aLIGO + PPTA(2020)
FIG. 2. Combined, two-dimensional posterior distribution for the tensor-to-scalar ratio r, and the blue tilt of the GW spectrum nt, using CMB, PPTA, indirect, and LIGO observations. The contours are the 95% and 99% limits. The green, dashed curve shows the consistency relation, nt ¼ −r=8, while the red and blue triangles correspond, respectively, to the red and blue curves in Fig 1. For clarity, the right-hand panel is a zoomed-in version of the left-hand panel, with additional posterior distributions shown. See Sec. III A for a description of each posterior distribution.
III. COMBINED CONSTRAINTS ON THE converted to nt and r using Eqs. (4) and (5). Finally, in PRIMORDIAL TILT the left-hand panel of Fig. 2, we plot the combined CMB þ PPTA posterior (black). This distribution repre- A. Combined experimental constraints sents the state-of-the-art constraints one can derive from Ω ðfÞ Here, we combine the current limits on GW from the CMB and PTA experiments alone. At a reference value of individual experiments mentioned above to constrain the r ¼ 0.11, this limits nt < 0.68 with 95% confidence. In r n tensor-to-scalar ratio and the tensor index t. In Fig. 2, general, the 95% C.L. upper limit on nt as a function of r we plot these two-dimensional posterior distributions for r derived from these constraints is well approximated by the and nt. In both panels, we plot two theory points and a simple relation theory curve. The green, dashed curve corresponds to the r consistency relation from standard inflationary models nt ¼ Alog10 þ B; ð7Þ 0.11 (nt ¼ −r=8), and the red and blue triangles have the same A ¼ −0 13 B ¼ 0 68 values of nt and r as the red and blue curves in Fig. 1. where . and . . Equation (7) allows one to Figure 2 combines the constraints from each experiment extrapolate the constraints on nt to arbitrary small values of in a heuristic manner. The left-hand panel shows all of the r. With the addition of each experimental constraint, we get CMB constraints from direct detection experiments starting tighter limits on A and B; the 95% C.L. upper limits for with Planck (gray shaded region), adding BICEP2 (green), each experiment are collated in Table I. and finally adding SPTpol (red) to get the overall con- In the right-hand panel of Fig. 2, we retain the CMB (red, straints on the CMB from direct GW observations. Also labeled “Planck þ BICEP2 þ SPTpol”) and CMB þ PPTA plotted in the left-hand panel is the PPTA posterior (blue). (black) posterior distributions from the left-hand panel. As The PPTA search algorithm described in Sec. II B derives a with the PPTA analysis, the LIGO and Virgo data analysis n Ω ðfÞ algorithm described in Sec. II C constrains nt and Ω , posterior distribution in terms of t and GW , which is GW which we convert to nt and r using Eqs. (4) and (5). This distribution is shown in yellow, and the combined TABLE I. 95% C.L. upper limits on A and B as in Eq. (7). The CMB þ PPTA þ LIGO and Virgo constraints are shown value of B is therefore the 95% upper limit of nt at a reference in pink. This pink contour represents the current best value of r ¼ 0.11. constraint from the direct GW experiments covering 21 decades in frequency. At a reference value of r ¼ 0.11, the Experiment AB CMB þ PPTA þ LIGO and Virgo constraints yield an þ −0 13 CMB PPTA . 0.68 upper limit of nt < 0.54. For smaller values of r, the CMB þ PPTA þ LIGO −0.06 0.54 þ þ þ −0 04 theoretical curves at the CMB frequency take on lower CMB PPTA LIGO indirect . 0.36 values of Ω , which implies that higher-frequency experi- CMB þ PPTAð2020ÞþaLIGO −0.06 0.34 GW ments play even more of a role in constraining nt than
011035-7 PAUL D. LASKY et al. PHYS. REV. X 6, 011035 (2016) comparatively lower-frequency experiments. For example, CMB experiment (see Ref. [76] and references therein). at r ¼ 0.01, CMB þ PPTA constraints imply nt < 0.82, (One should be cautious of the projected constraints from while CMB þ LIGO and Virgo constraints imply Huang et al. [76]. Their Fisher matrix analysis necessarily nt < 0.60. The 95% C.L. from the CMB þ LIGO con- assumes the posterior distribution is Gaussian and, hence, straint is well approximated by Eq. (7), where values for A symmetric about some fiducial model. This is not the case and B can be found in Table I. for nt, which allows for significantly larger positive values Also in the right-hand panel of Fig. 2, we add the indirect than it does negative.) Such a detection, together with the constraints described in Sec. II D to the direct constraints data from PTAs and ground-based interferometers, will put (turquoise). This contour represents our total knowledge of very tight limits on nt, with larger values of r being the nt and r using all experimental constraints. In this case, at most constraining. For example, a confirmed detection of r ¼ 0.11 we find nt < 0.36 at 95% C.L., and the upper r ≈ 0.1 would put very tight bounds on nt with a strong limit as a function of r is well approximated by Eq. (7), preference for small and positive values. Such tight con- where the values for A and B can be found in Table I. straints are truly a result of CMB bounds on the low- From Fig. 2, it is clear that only direct observations of the frequency end and PTA, LIGO, and indirect bounds on the CMB constrain nt < 0. This makes sense in the context of upper end. Fig. 1: given that the lever arm for the GW theory curves, When a detection is made in any of the frequency bands f i.e., Eq. (4), are hinged at CMB, a negative spectral index is studied herein, it becomes even more pertinent to analyze not constrained by experiments that are only sensitive to all experimental data, consistently taking into account the Ω ðfÞ values of GW higher than at the CMB. spectral running αt. Indeed, in the case of a detection, upper Finally, in Fig. 2 we show the projected constraints that limits in each frequency band can be used to simulta- one can expect by the year 2020 [dark blue, labeled neously constrain nt and αt; three or more experiments are “ þaLIGO þ PPTAð2020Þ”] assuming 5 more years required to constrain both parameters. In a future work, we of PPTA observations and aLIGO at design sensitivity will present three-dimensional posterior constraints that (see Secs. II B and II C). These contours show that the include r, nt, and αt, and also incorporate predictions for constraint for the spectral index improves to nt ≲ 0.34 at future CMB experiments. Distinguishing primordial back- r ¼ 0.11, and is well approximated by Eq. (7) with A and B grounds from astrophysical foregrounds may be a daunting given in Table I. As is evident from Fig. 1, the constraint at task, though multiwavelength measurements could prove high nt will be dominated by aLIGO. Similar constraints useful toward this end. in the PTA band are not expected until the era of the Square Kilometre Array and Five hundred meter Aperture Spherical Telescope (FAST) (see, e.g., Refs. [39,75]). IV. CONCLUSION By combining limits from many different GW experi- B. Comparison with theory ments probing 29 decades in frequency, we present new n r In the previous section, we present stringent constraints constraints on cosmological parameters t and , which are on the blue tilt of the primordial GW background from intimately related to the evolution of the early Universe. This experiments spanning 29 decades in frequency. These interdisciplinary research also makes significant advances in results can be used to comment on early-Universe models. PTA and LIGO and VIRGO and indirect GW limit analysis Those models, whose spectral indices are near zero—or of techniques. Specifically, we present new PPTA data that comparable magnitude to standard inflationary models— provide the most stringent limit on the primordial gravita- Ω ðfÞ < 2 3 10−10 are consistent with the data. String-gas cosmologies and tional-wave background, GW . × , more modified inflationary scenarios with nonminimal couplings than a factor of 4 tighter than the previous best limit from to gravity seem to be the least constrained, since these Ref. [61]. Moreover, we develop and implement a method to models predict relatively small values of nt and are give the best limits on the primordial background from unconstrained for even relatively large tensor-to-scalar ground-based interferometers—a method we anticipate ratios. will become standard in future LIGO and Virgo primordial Ekpyrosis has a tendency to predict large values of the background analyses. Furthermore, we provide a new spectral tilt including nt ≈ 2 [23] that come from modes interpretation of indirect GW constraints from CMB tem- freezing out of the horizon during the contracting phase of the perature and polarization measurements, lensing, BBN, and Universe. More modern incarnations of ekpyrosis produce BAO observations that allow for a varying primordial blue tilts, but with relatively low values of tensor-to-scalar spectral index, allowing us to directly compare these “ ” Ω ratio r [26]. Here, our derivation of fitting formulas for nt as a integral constraints on GW with the usual frequency- Ω ðfÞ function of r (see Table I) allow specific ekpyrotic predictions dependent GW constraints. Our technique for compar- to be tested to arbitrary small values of r. ing direct and indirect limits can be widely adopted within Our results will have important implications following the GW community to avoid the confusion created from the detection of nonzero tensor-to-scalar ratio by a future “apples-to-oranges” comparisons.
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While Refs. [30,39,40] present constraints on nt and r supported by the National Science Foundation, PHY- using combinations of CMB, LIGO, and PTA data, the 1414479. C. M. F. M. was supported by a Marie Curie focus of their work was significantly different. Indeed, the International Outgoing Fellowship within the European work of Meerburg et al. [30] and Huang and Wang [40] Union Seventh Framework Programme. R. C. is supported were originally in response to the now defunct BICEP2 in part by DOE Grant No. DE-SC0010386. The Parkes radio results [41,42], while Liu et al. [39] presented constraints telescope is part of the Australia Telescope National Facility, from only CMB and PTA data, but focused on what a which is funded by the Commonwealth of Australia for positive detection could do for our understanding of the operation as a National Facility managed by CSIRO. Y. L. early-Universe equation of state, cosmic phase transitions, and G. H. are recipients of ARC Future Fellowships and relativistic free streaming. (respectively, FT110100384 and FT120100595). Y. L., A direct comparison between our results and that of M. B., W. v. S., P. A. R., and P. D. L. are supported by Meerburg et al. [30] is not possible for a number of reasons. ARC Discovery Project No. DP140102578. S. O. is sup- Notably, they use a linear prior on r, which, together with ported by the Alexander von Humboldt Foundation. the use of the original BICEP2 results, ends in constraints L. W. and X. Z. acknowledge support from the ARC. that are not bounded below. Figure 1, together with Eqs. (4) J. W. is supported by NSFC Project No. 11403086 and and (5), shows that r should be unbounded below given West Light Foundation of CAS XBBS201322. X. Y. is Ω ðfÞ that there are no lower limits on the amplitude of GW supported by NSFC Project No. U1231120, FRFCU from any experiments. Moreover, Meerburg et al. [30] use Project No. XDJK2015B012, and China Scholarship an unconventional pivot scale for the theoretical GW Council (CSC). E. T. is supported by ARC Future spectrum (see, e.g., Ref. [32] for a discussion of the Fellowship FT150100281. optimal pivot scale). Our results are significantly more constraining than those Note added.—Recently, Cabass et al. [77] also presented of Liu et al. [39] (see their Fig. 7), most notably due to combined constraints on nt and r, as well as forecasts on the inclusion of indirect GW constraints. Our analysis the capability of future CMB satellite experiments to quantifies how large the spectral index of the primordial constrain nt. spectrum nt can grow as a function of the tensor-to-scalar ratio r; see Fig. 2 and Table I for a summary of the results. Various theories of the early Universe predict a blue primordial gravitational-wave spectrum [19,21,23,26], and, [1] B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, indeed, some versions of ekpyrosis predict large values F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, of nt, which we can now rule out by our analysis—see R. X. Adhikari et al., Observation of Gravitational Waves Sec. III B. Observations of the CMB alone only limit the from a Binary Black Hole Merger, Phys. Rev. Lett. 116, inflationary GW spectrum to nt ≲ 5 at a reference value of 061102 (2016). the tensor-to-scalar ratio of r ¼ 0.11. Current observations [2] L. P. Grishchuk, Primordial Gravitons and Possibility of by the PPTA and initial LIGO and Virgo reduce this limit to Their Observation, JETP Lett. 23, 293 (1976). [3] L. P. Grishchuk, Reviews of Topical Problems: Gravita- nt < 0.54 with 95% confidence, and including limits from n < 0 36 tional Waves in the Cosmos and the Laboratory, Sov. Phys. indirect GW observations reduces this to t . .We Usp. 20, 319 (1977). predict that observations by aLIGO at design sensitivity [4] A. A. 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Addesso et al., Advanced LIGO, Classical Quantum Gravity 32, 074001 (2015). ACKNOWLEDGMENTS [8] F. Acernese, M. Agathos, K. Agatsuma, D. Aisa, N. Allemandou, A. Allocca, J. Amarni, P. Astone, G. Balestri, P. D. L. is grateful to Justin Ellis for valuable support with G. Ballardin et al., Advanced Virgo: A Second-Generation the pulsar timing package PAL2. We are extremely grateful to Interferometric Gravitational Wave Detector, Classical the three referees who all provided excellent feedback that Quantum Gravity 32, 024001 (2015). improved the manuscript. This work was initiated at the [9] B. P. Abbott et al., An Upper Limit on the Stochastic Aspen Center for Physics, which is supported by National Gravitational-Wave Background of Cosmological Origin, Science Foundation Grant No. PHY-1066293. J. T. G. is Nature (London) 460, 990 (2009).
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