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11-15-2016

Probing A Panoply Of Curvaton-Decay Scenarios Using CMB Data

Tristan L. Smith Swarthmore College, [email protected]

D. Grin

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Recommended Citation Tristan L. Smith and D. Grin. (2016). "Probing A Panoply Of Curvaton-Decay Scenarios Using CMB Data". Physical Review D. Volume 94, Issue 10. DOI: 10.1103/PhysRevD.94.103517 https://works.swarthmore.edu/fac-physics/287

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]. PHYSICAL REVIEW D 94, 103517 (2016) Probing a panoply of curvaton-decay scenarios using CMB data

† Tristan L. Smith1,* and Daniel Grin2, 1Swarthmore College, Department of Physics and Astronomy, Swarthmore, Pennsylvania 19081, USA 2Department of Astronomy & Astrophysics, Kavli Institute for Cosmological Physics, University of Chicago, Chicago, Illinois 60637, USA (Received 3 December 2015; published 17 November 2016) In the curvaton scenario, primordial curvature perturbations are produced by a second field that is subdominant during inflation. Depending on how the curvaton decays [possibly producing baryon number, lepton number, or cold dark matter (CDM)], mixtures of correlated isocurvature perturbations are produced, allowing the curvaton scenario to be tested using cosmic microwave background (CMB) data. Here, a full range of 27 curvaton-decay scenarios is compared with CMB data, placing limits on the ξ curvaton fraction at decay, rD, and the lepton asymmetry, lep. If baryon number is generated by curvaton decay and CDM before (or vice versa), these limits imply specific predictions for non-Gaussian signatures testable by future CMB experiments and upcoming large-scale-structure surveys.

DOI: 10.1103/PhysRevD.94.103517

I. INTRODUCTION number, lepton number, and cold dark matter (CDM), primordial fluctuations in different species could carry The observed cosmic microwave background (CMB) an isocurvature component, in which the relative number anisotropies and large-scale structure of the Universe are thought to result from primordial curvature perturbations. densities of different species fluctuate in space. In the The prevailing model is that these perturbations are simplest models, these isocurvature fluctuations are produced during inflation, an epoch of accelerated cosmo- totally correlated (or anticorrelated) with the dominant logical expansion preceding the radiation-dominated era. In adiabatic component. Curvaton density fluctuations are the simplest scenarios, both the accelerated expansion and non-Gaussian, and so the curvature perturbation is non- the curvature perturbations result from the dynamics of a Gaussian [20,23]. Less phenomenological realizations of single field (the inflaton) [1–3]. At the end of inflation, the the curvaton model in higher-dimensional theories may inflaton field is thought to decay and initiate the radiation- break the instantaneous decay approximation, modifying dominated era, a process known as reheating [4,5]. non-Gaussian and possibly isocurvature signatures [26]. Standard single-field models of inflation produce The level of non-Gaussianity is set by rD, a parameter nearly scale-invariant, Gaussian, and adiabatic primordial describing the curvaton energy density. The level of ξ fluctuations [2,3]. It may be challenging for the dynamics isocurvature is set by rD and lep, the chemical potential of a single field to satisfy observational constraints to the describing cosmological lepton number [7]. Both param- amplitude and scale dependence of the curvature pertur- eters are constrained by observations. bations as well as constraints to the amplitude of a Isocurvature perturbations alter the phase structure and background of primordial gravitational waves [6,7].In large-scale amplitude of CMB power spectra [27–32]. order to ease these requirements, a second field (the Planck satellite observations therefore indicate that CMB curvaton) could source curvature perturbations and anisotropy power spectra are consistent with adiabatic later decay [3,6–9]. There are a variety of candidates fluctuations, requiring that isocurvature fluctuations con- for the curvaton motivated by high-energy particle theory tribute a fraction ≲10−3–0.1 of the total observed power, [10–17]. In the curvaton scenario, constraints are more depending on various assumptions [33–35]. Big-bang ξ2 0 permissive because the inflaton need only produce a nucleosynthesis abundances are altered if lep > , and sufficiently long epoch of acceleration to dilute topologi- so the primordial 4He and deuterium abundances impose cal defects and does not have to be the main source of the limit ξ ≤ 0.03 [33,36–39]. – j lepj perturbations [3,6 8,18]. In past work comparing curvaton model predictions with This scenario is distinct from single-field models in CMB data, isocurvature constraints were obtained consid- predicting a nonadiabatic and non-Gaussian component – ering a single mode (neutrino, CDM, or baryon) at a time, to primordial fluctuations [7,18 25]. Depending on when with consideration limited to several curvaton-decay sce- the curvaton decays relative to the production of baryon narios [33–35,40–42]. Priors and parameter-space explora- tion were implemented on the cross-/auto-power spectrum *[email protected] amplitudes and correlation coefficients of single isocurva- † ξ [email protected] ture modes, rather than rD and lep, and then mapped to the

2470-0010=2016=94(10)=103517(18) 103517-1 © 2016 American Physical Society TRISTAN L. SMITH and DANIEL GRIN PHYSICAL REVIEW D 94, 103517 (2016) curvaton parameter space. Neutrino isocurvature perturba- tions were not included.1 In fact, each curvaton-decay scenario makes specific predictions for the amplitudes and cross-correlations (with ζ) of each isocurvature mode: the baryon isocurvature mode, the CDM isocurvature mode, and the neutrino isocurvature density mode [6,22,43]. We take a different approach and separately consider all 27 curvaton-decay scenarios. We use 2015 Planck CMB temperature and polarization data to determine the allowed parameter space FIG. 1. Prediction for the amplitude fnl of primordial non- ξ of rD and lep (breaking degeneracies with other data), Gaussianity in curvaton-decay scenarios allowed by isocurvature ∼ 1 computing the full set of isocurvature mode amplitudes and constraints. The left panel shows models with fnl , which are cross-correlation spectra (with ζ) for each set of parameter potentially testable by future high-redshift 21-cm surveys – values. We use a Monte Carlo Markov Chain (MCMC) [48 50]. The solid green curve shows the case in which baryon number/CDM are generated after/by curvaton decay. The dotted analysis to obtain constraints to all these scenarios. We also blue curve shows the case in which baryon number/CDM are perform a Fisher-matrix analysis to determine the sensi- generated by/after curvaton decay. The solid black curve shows tivity of a future cosmic-variance limited experiment to the case in which baryon number/CDM are generated before/by these curvaton-decay scenarios. curvaton decay. The right panel shows the predicted fnl values if The models fall into several categories. Some decay baryon number/CDM are generated by/before curvaton decay, scenarios generate purely adiabatic perturbations, and these which could be tested using scale-dependent bias measurements Δ ≃ 1 are always allowed, and these are unconstrained by limits from future galaxy surveys with sensitivity fnl . to isocurvature perturbations. Some generate order-unity isocurvature fluctuations between nonrelativistic matter We begin in Sec. II by reviewing basic aspects of the and radiation, independent of r and ξ values, and these D lep curvaton model, including the production of curvature and are not allowed by the CMB data. Others generate isocurvature perturbations. In Sec. III we continue with a 1 isocurvature perturbations that vanish when rD ¼ . Here, detailed discussion of curvaton-decay scenarios and the the data impose lower limits to rD, with 95% confidence resulting mixtures of curvature and isocurvature fluctua- 0 93–0 99 regions given by rD > . . , depending on precise tions. The data sets, methodology, and resulting constraints model assumptions. on these scenarios are presented in Sec. IV. We present our Finally, two cases lead to nonzero isocurvature pertur- conclusions in Sec. V. bations in both the baryon and CDM. The only way for these scenarios to agree with the CMB data is for the II. THE CURVATON MODEL baryon and CDM isocurvature modes to have opposite signs and nearly equal amplitudes, producing what is The family of inflationary models is extremely rich. known as a compensated isocurvature perturbation [44]. Nonetheless, a successful inflationary model must meet This naturally leads to a measured value of rD which is some fairly stringent requirements, producing a sufficient significantly different from unity. For the curvaton-decay number (∼60) of e-foldings to dilute dangerous early relics, −9 scenario in which baryon number/CDM are generated generating the observed value of As ¼ 2.2 × 10 , and 0 1602þ:0051 by/before curvaton decay, we find that rD ¼ . −0.0047, agreeing with ever more precise measurements of the scalar while for the scenario in which baryon number/CDM are spectral index ns ≃ 0.96 [34]. Limits to the tensor-to-scalar 0 8492þ0.0099 ratio (r<0.11 [34]) must also be met. If these limits turn generated before/by curvaton decay, rD ¼ . −0.0096 . All of these decay scenarios (except the one where both into detections, single-field slow-roll models must further CDM and baryon number are produced after curvaton obey a consistency relation, r ¼ 16ϵ (see Ref. [51] and decay) make specific predictions for the amplitude f of references therein), which relates r to the slow-roll nl ϵ local-type primordial non-Gaussianity, shown by the dis- parameter . In fact, current data already rule out the tributions in Fig. 1. These are all consistent with Planck simplest of inflationary models [34]. One alternative to simple inflationary models is the limits to fnl [45]. Future measurements of scale-dependent Δ ≃ 1 curvaton scenario, in which the inflaton (ϕ) drives bias in galaxy surveys (with sensitivity fnl ) [46,47] and high-redshift 21-cm surveys (with sensitivity exponential cosmic expansion but is not the primary source Δf ≃ 0.03) [48–50] could rule out these decay of the observed cosmological fluctuations. Instead, a nl σ scenarios. subdominant spectator field, the curvaton ( ), acquires quantum fluctuations that are frozen after σ perturbation modes cross the horizon during inflation. The curvaton 1Exceptions are Refs. [33,37,42], which included isocurvature field then has a dimensionless fluctuation power-spectrum in neutrinos but not other species. of [7,18,40,52]

103517-2 PROBING A PANOPLY OF CURVATON-DECAY SCENARIOS … PHYSICAL REVIEW D 94, 103517 (2016) 2 ϕ σ Δ2 HI potential energies Vð Þ and Vð Þ [55]. A survey of the σσðkÞ¼ 2π ; ð1Þ literature shows that for a wide range of curvaton potentials k¼aH that yield ns ≃ 0.96, the resulting αs value is not detectable where HI is the inflationary Hubble parameter when the at Planck sensitivity levels [with a 68% confidence level mode with wave number k freezes out. Initially, these (C.L.) of Δα ≃ 0.01] [34,56,57]. To isolate the effect of fluctuations are isocurvature perturbations, as the curvaton isocurvature perturbations, to simplify our analysis, and to is energetically subdominant to the thermal bath (with leave our analysis unpinned to specific curvaton potentials, ρ energy density R) produced at the end of inflation we thus impose the restriction αs ¼ 0, which is consistent [3,7–9,20,22]. The curvaton has mass mσ, and once the with Planck data. In future work, it would be interesting to condition mσ ≫ 3H is met (where H is the Hubble simultaneously explore the diversity of curvaton potentials parameter), σ begins to coherently oscillate. The curvaton and decay scenarios (including α ≠ 0) in the analysis, with −3 s energy-density then redshifts as ρσ ≃ a , where a is the an eye towards future cosmological data sets. −3 cosmological scale factor. As the scaling ρσ ∼ a is slower The curvaton is a massive scalar field, and so for the −4 than ρR ∼ a , the curvaton becomes increasingly ener- simplest quadratic curvaton potentials, the curvaton energy 2 2 getically important, converting the initial isocurvature density is ρσ ∼ σ¯ þ 2¯σδσ þðδσÞ , where σ¯ is the homo- fluctuation into a gauge-invariant curvature perturbation geneous value of σ and δσ is a spatial perturbation. As σ ζ [7,18,40,52]. Eventually the curvaton decays, initiating itself is a Gaussian random field, ρσ is non-Gaussian. the usual epoch of radiation domination. The resulting non-Gaussianity is of local type—that is, ζ ζ 3 ζ2 − ζ2 During radiation domination, the cosmic equation of ¼ gðx~Þþ5 fnl½ gðx~Þ h gðx~Þi, with state is constant, and it can be shown that this implies the ζ 5 5r 5 conservation of superhorizon modes of , with value f ¼ − D − ; ð5Þ nl 4r 6 3 ζ 1 − ζ ζ D ¼ð rDÞ ϕjD þ rD σjD; ð2Þ where ζgðx~Þ is a Gaussian random field [6,7,23,24,34,40]. where The stringent limits to local-type non-Gaussianity from 2 7 5 8 ρ Planck temperature data, fnl ¼ . . , impose the con- σ straint r > 0.12 [34,45]. These constraints do not depend rD ¼ ρ 4ρ 3 ð3Þ D ð σ þ R= Þ D on the curvaton-decay scenario, and are thus relatively is the fractional contribution of the curvaton to the trace of model independent. In some curvaton-decay scenarios, residual isocurvature perturbations would be excited, mak- the stress-energy tensor just before curvaton decay. Here ζx denotes the spatial-curvature perturbation on hypersurfaces ing more stringent limits to rD possible. Additionally, limits of constant x energy density (or equivalently, the energy- to or a detection of curvaton-type isocurvature would make density perturbation on surfaces of constant total ζ). The it possible to test the decay physics of the curvaton. ζ ζ If the densities of all species are determined after notation xjD indicates that x is evaluated at the moment of curvaton decay. For the duration of this paper, we neglect curvaton decay, then the density perturbations in all species the time dependence of the curvaton-decay rate [53] and are set by ζ alone, leading to purely adiabatic fluctuations. assume the usual instantaneous-decay approximation. On the other hand, if some conserved quantum numbers are In principle, as we can see from Eq. (2), ζ has inflationary generated by or before curvaton decay while others are not, and curvaton contributions. We follow the usual practice of there is a mismatch in density fluctuations, leading to a considering the scenario where the curvaton dominates the gauge-invariant entropy (or isocurvature) perturbation. In curvature perturbation—that is, r ζσ ≫ ð1 − r Þζϕ—and particular [6,7,40], D D 8 so we may use the approximation [21,22] > <> 0; if x is produced before σ decay; ζ ≃ r ζσ: ð4Þ ζ~ ζ~ σ D x ¼ > σ; if x is produced by decay; ð6Þ :> Using expressions found in Ref. [54], and assuming a low ζ; if x is produced after σ decay: tensor-to-scalar ratio but detectable isocurvature and primor- dial non-Gaussianity, it is straightforward to show that this Here the index x denotes b (baryon number), L (lepton ζ~ limit implies that the homogeneous curvaton field values are number), or c (CDM). The x indicates initial curvature sub-Planckian at horizon crossing. fluctuations on hypersurfaces of constant particle number Note that in this limit the spectral index of all perturba- (for CDM) or conserved quantum number (in the case of tion spectra (adiabatic and isocurvature) is given by one baryons or leptons). The curvaton is assumed to behave as ~ value, ns. In general, there could also be a nonzero running matter at the relevant epochs, and so ζσ ¼ ζσ. of the spectral index, αs ≡ dns=d ln k ≠ 0. In the context of We distinguish between quantum numbers (like baryon curvaton scenarios, αs is not a free parameter, but rather and lepton number) and densities, as baryon and lepton depends on the functional form of the inflaton and curvaton number could be generated at very early times, long before

103517-3 TRISTAN L. SMITH and DANIEL GRIN PHYSICAL REVIEW D 94, 103517 (2016) quarks bind to produce actual baryons. Indeed, baryo- The constant superhorizon values of ζ and Sxγ are “initial genesis (which refers to the creation of baryon number) conditions” which precede horizon entry and determine the could be related to curvaton physics, even if the production spectra of CMB anisotropies, as computed by CAMB [60] or of actual baryons happens much later. any other CMB Boltzmann code. We take the initial values The gauge-invariant entropy fluctuation between x and Sxγ to be defined at some time after the relevant species photons is given by thermally decouple and reach their final equation of state (for example, if x ¼ c, we consider S γ at some time after ≡ 3 ζ − ζ c Sxγ ð x γÞð7Þ CDM has become nonrelativistic). After the quantum ~ number associated with x thermally freezes out, ζx is and is conserved on superhorizon scales [3,58,59], as long conserved on superhorizon scales because the relevant as the equation of state of the species i (or the carriers of the quantum numbers are conserved. If x ∈ fc; bg, S γ is set relevant quantum number) is constant and the quantum x long after actual baryons and CDM become nonrelativistic, numbers are conserved. Photon perturbations are described and so ζ ζ~ , because surfaces of constant energy and by ζγ, the spatial curvature perturbation on hypersurfaces of x ¼ x number density coincide. We discuss the subtler case of constant photon energy-density. For baryons or leptons, ζx is the curvature perturbation on surfaces of constant energy lepton-number fluctuations and neutrino isocurvature in density of whichever species carries the quantum number Sec. III. (at late times, these would be actual surfaces of constant For any quantum number/species, there are then three baryon energy density). scenarios [22]:

8 > −3ζ − 3ðζγ − ζÞ; if x is produced before σ decay; <> 1 Sxγ ¼ 3 − 1 ζ − 3ðζγ − ζÞ; if x is produced by σ decay; ð8Þ > rD :> −3ðζγ − ζÞ; if x is produced after σ decay:

When fluctuations are set by the curvaton, as we can see distinct scenarios. As discussed in the previous section, from Eq. (8), entropy fluctuations are set completely by the curvaton decay can occur at any time after inflation ends. adiabatic fluctuation (as we would expect when only Curvaton decay must certainly also occur before big-bang fluctuations in a single field are important), and are thus nucleosynthesis (BBN). This means that within the single- totally correlated or anticorrelated to ζ. field slow-roll inflationary models, the curvaton may decay Anticorrelated isocurvature perturbations can lower the at temperatures ranging from 1016 GeV [34] down to observed CMB temperature anisotropy at low multipole l, ∼4 MeV [63], at which point the primordial light elements improving the mild observed tension between the best-fit must be produced. In order for all 27 scenarios to be ΛCDM model and large-scale CMB observations [61,62]. realized, there must be mechanisms that generate baryon To see what this fact implies for curvaton physics, and to number, lepton number, and CDM over this wide range of more broadly test the curvaton model using CMB obser- energy scales, as we now discuss. vations, we now derive the isocurvature amplitudes in A persistent mystery is the origin of baryon number— different curvaton-decay scenarios. To simplify the dis- i.e., the observed net asymmetry of baryons over anti- cussion, we will describe curvaton-decay scenarios with the baryons in the Universe. Plausible models bracket a range ∈ notation ðbyb ;cyc ;LyL Þ, where yL fbefore; by; afterg.For of energy scales, from baryogenesis at the electroweak example, ðbby;cafter;LbeforeÞ indicates a curvaton-decay scale [64] to direct production of baryon number through a scenario in which baryon number is generated by curvaton coupling to the inflaton or curvaton (see Ref. [65] and decay, cold dark-matter after curvaton decay, and lepton references therein). The energy scale of baryogenesis could number before curvaton decay. thus be anywhere in the range 1 TeV–1016 GeV. Since both the inflationary energy scale and the energy scale of curvaton relevance/decay are poorly constrained, it is III. CURVATON-DECAY SCENARIOS possible for baryon number to be produced before, by, The various curvaton-decay scenarios can be divided or after curvaton decay. into cases where the production of either the baryon The observed baryon asymmetry could be produced number, lepton number, or CDM occurs before the curva- through partial conversion of a much larger primordial ton decays, by the curvaton decay, or after the curvaton lepton asymmetry. One of the ways (reviewed at length in decays. This naturally leads to a total of 3 × 3 × 3 ¼ 27 Ref. [66]) to account for the observed nonzero neutrino

103517-4 PROBING A PANOPLY OF CURVATON-DECAY SCENARIOS … PHYSICAL REVIEW D 94, 103517 (2016) TABLE I. Baryon and CDM isocurvature amplitudes (in terms of the curvature perturbation ζ) for the various curvaton-decay scenarios. If the lepton chemical potential ξlep ¼ 0, then Sνγ ¼ 0. Otherwise, if ξlep ≠ 0 and there is a net lepton number L ≠ 0, then Sνγ is given by Eq. (21), taking nonzero values only if L is generated by curvaton — decay that is, yL ¼ by. This is discussed in detail in Secs. III A and III B. The notation ðbyb ;cyc ;LyL Þ for various curvaton-decay scenarios is introduced in Sec. II.

Scenario Sbγ Scγ Smγ ζ ζ ζ

Sνγ Sνγ Ω Sνγ b ;c ;L 1 b ð by before yL Þ 3ð − 1ÞþRν −3 þ Rν 3ð − 1ÞþRν rD ζ ζ ΩmrD ζ Sνγ Sνγ Ω Sνγ b ;c ;L 1 c ð before by yL Þ −3 þ Rν 3ð − 1ÞþRν 3ð − 1ÞþRν ζ rD ζ ΩmrD ζ Sνγ Sνγ Ω Sνγ b ;c ;L 1 b 1 ð by after yL Þ 3ð − 1ÞþRν Rν 3 ð − 1ÞþRν rD ζ ζ Ωm rD ζ Sνγ S γ Ω Sνγ b ;c ;L 1 b c 1 ð after by yL Þ Rν 3ð − 1Þþ 3 ð − 1ÞþRν ζ rD ζ Ωm rD ζ Sνγ Sνγ Ω Sνγ b ;c ;L b ð before after yL Þ −3 þ Rν Rν −3 þ Rν ζ ζ Ωm ζ Sνγ S γ Ω Sνγ b ;c ;L b c ð after before yL Þ Rν −3 þ −3 þ Rν ζ ζ Ωm ζ ðb ;c ;L Þ −3 Sνγ Sbγ Sbγ before before yL þ Rν ζ ζ ζ b ;c ;L 1 Sνγ Sbγ Sbγ ð by by yL Þ 3ð − 1ÞþRν rD ζ ζ ζ ðb ;c ;L Þ Sνγ Sνγ Sνγ after after yL Rν ζ Rν ζ Rν ζ mass is to invoke the seesaw mechanism [67,68]. The standard lighter supersymmetric WIMP could be produced seesaw mechanism generically introduces a hierarchy of by curvaton decay if WIMPs couple to the curvaton field neutrinos with masses above the electroweak scale, leading [7]. Just as with baryon and lepton numbers, CDM could to the generation of lepton number at temperatures greater thus be produced before, by, or after curvaton decay [40]. than ∼100 GeV. Altogether, there is a variety of logically possible Alternatively, lepton number could be produced near scenarios for producing the correlated isocurvature fluctu- the end of inflation (at energies as high as ∼1016 GeV), ations discussed in Sec. II. Our goal in this work is to test perhaps by Chern-Simons (parity-violating) terms in the these scenarios using CMB data. We assemble for the first gravitational sector [69] or by a novel coupling of chiral time in one work expressions for the amplitude of corre- fermions to an axion-like field [70]. On the other hand, lated baryon, CDM, and neutrino isocurvature-density the νMSM model [71] allows for lepton number to be (NID) perturbations in all 27 possible curvaton-decay generated at lower energies. Finally, as discussed in the scenarios, as shown in Table I and Eq. (22). This allows previous section, it is possible that the decay of the curvaton us to build on past work, which explored only one field produces lepton number, leading to isocurvature isocurvature mode at a time [42] or neglected NID perturbations in the neutrino density perturbations. perturbations [40,79], and self-consistently test for the first ξ The identity and production mechanism of the CDM is time the full parameter space of rD and lep in all 27 also a mystery [72–75]. One possibility is that the CDM curvaton-decay scenarios. consists of weakly interacting massive particles (WIMPs) We recognize that in the context of specific particle thermally produced by physics at the ∼TeV scale [76]. physics models for baryogenesis, leptogenesis, or dark If this is so, the CDM would be produced around the matter production, some of these scenarios are more viable electroweak energy scale. Direct-detection experiments, than others. For example, in the νMSM [71], the lepton however, have placed increasingly stringent limits on asymmetry and dark matter are produced nearly concur- WIMP couplings. The most natural WIMP candidate rently, and so scenarios of the form ðbbefore;cbefore;LafterÞ do (a stable superpartner in supersymmetric models) is also not apply. With this caveat in mind, we have considered under increasing pressure from experiment, due to the all possible curvaton-decay scenarios without theoretical lack of evidence for low-energy supersymmetry from the restrictions, in order to determine the most general Large Hadron Collider (LHC) (see Ref. [77] and references constraints. therein). This motivates the consideration of other CDM At the level of observable power spectra in linear candidates. perturbation theory, the CDM and baryon isocurvature One possible CDM candidate is a stable extremely modes are indistinguishable [40,44,60,79,80], but the NID massive particle (or wimpzilla) with mass in the range mode has a distinct physical imprint [42,81] from the others 1012 GeV

103517-5 TRISTAN L. SMITH and DANIEL GRIN PHYSICAL REVIEW D 94, 103517 (2016) 0 where nL and nL¯ denote the number densities of the lepton respectively) by the values shown in Table I with Sνγ ¼ . number and antilepton number, respectively. On the other hand, if the lepton asymmetry is generated by σ decay, there is a residual neutrino isocurvature perturba- A. No lepton asymmetry tion Sνγ [6,22,43]. During radiation domination, the total curvature pertur- bation is given by B. Lepton asymmetry Each neutrino species carries the lepton number of the ζ ¼ð1 − RνÞζγ þ Rνζν; ð9Þ corresponding lepton flavor, and so in the presence of a lepton asymmetry, fluctuations in Δn result in neutrino where Rν ≡ ρν=ðργ þ ρνÞ is the energy fraction in massless L isocurvature perturbations. For massless neutrinos, the neutrinos, a constant after electron-positron annihilation. occupation number is Neutrinos carry lepton number and thermally decouple near temperatures T ∼ 2 MeV. If there is no lepton asymmetry, ∓ξ −1 f E eE=Tν j 1 ; 11 spatial fluctuations in lepton-number density track the total jð Þ¼½ þ ð Þ energy density, and so ζν ¼ ζ. From Eq. (9) we then see where the flavor label takes values j ¼ e, μ,orτ; that ζ ¼ ζγ ¼ ζν, and thus Sνγ ¼ 0. It is then straightfor- the corresponding chemical potential ξ parameterizes the ward to obtain the relationships between baryon/CDM j entropy fluctuations and curvature fluctuations for a variety primordial lepton asymmetry; the minus sign applies for of curvaton-decay scenarios by applying Eq. (8). The neutrinos; and the plus sign applies for antineutrinos. Unlike η ≃ 6 10−10 ξ resulting amplitudes are shown in Table I. Later, to interpret the cosmological baryon asymmetry × , j is constraints, it is useful to define the total isocurvature in rather poorly constrained. Some models of baryogenesis nonrelativistic matter: require comparable levels of lepton and baryon asymmetry, but others convert a much larger lepton asymmetry into the Ωb Ωc experimentally known baryon asymmetry. Electron neutri- S γ ≡ 3 ζ þ ζ − ζ m b c nos (whose number density depends on ξe− ) set the rates of Ωm Ωm β-decay processes active during BBN, and so the value of Ω Ω b c ξ − ∝ ¼ S γ þ S γ : ð10Þ e affects the primordial neutron-to-proton ratio n=p Ω b Ω c 4 m m expð−ξe− Þ and the resulting abundance of He [36,37]. A lepton asymmetry also alters N , the number of Ω Ω eff Here b and c are the usual relic densities of baryons and relativistic degrees of freedom during BBN, although this CDM relative to the cosmological critical density. effect is less important for setting abundances than the We note that the scenarios ðb ;c ;L Þ and by before yL altered n=p ratio. Neutrinos are now known to have mass ðb ;c ;L Þ lead to correlated (or anticorrelated) before by yL and as a result exhibit flavor oscillations. Independent of isocurvature perturbations. These scenarios mitigate some initial conditions, solar neutrino observations and results ≲ 50 of the tension between CMB data (for l ) and the from the KamLAND experiment [36,38] indicate ν mass Λ best-fit CDM model [35,61,62]. We discuss this further splittings and mixing angles that would lead to near flavor in Sec. IV. ξ − ξ ξ ≡ ξ equilibrium early on, and so e ¼ μ ¼ τ lep. BBN The near cancellation of baryon and CDM isocurvature 4 abundances (including the He abundance YHe) depend not contributions to Smγ in these scenarios requires fine-tuned ξ only on the primordial values lep, but also on the mixing values of rD ∼ Ωb=Ωm and rD ∼ Ωc=Ωm. This yields a 3 ζ − ζ angles between neutrinos [36], and in particular on the relatively large CIP amplitude of Sbc ¼ ð b cÞ¼ θ 3ζ −3ζ value of 13. Current reactor and long-baseline neutrino =rD and =rD in the ðbby;cbefore;L=yLÞ and 2 experiments indicate that sin ðθ13Þ ≃ 0.03, giving a ðb ;c ;L Þ scenarios, respectively, or more explicitly, before by yL 95% confidence BBN limit of ξ ≤ 0.03 [39]. S ≃ 20ζ and S ≃ 3.5ζ. These CIP amplitudes could lep bc bc The resulting ν energy and lepton-number densities are leave observable imprints on off-diagonal correlations [7,42,43] (or equivalently, the CMB bispectrum and trispectrum), a possibility discussed further in Refs. [44,80]. 4 ρi 7 Tν The cases ðbbefore;cafter;LyL Þ, ðbafter;cbefore;LyL Þ, and ¼ Ai; ð12Þ ργ 8 Tγ ðbbefore;cbefore;Ly Þ are completely ruled out by the data, L as already shown in Ref. [40]. We do not consider them 3 Δni Tν further. ¼ 2.15 Bi; ð13Þ The situation is considerably richer if there is a net lepton nγ Tγ asymmetry. As we see in Sec. III B, if the lepton symmetry 30 ξ 2 15 ξ 4 is generated before or after σ decay, the ratios S γ=ζ, S γ=ζ, 1 lep lep b c Ai ¼ þ 7 π þ 7 π ; ð14Þ and Smγ=ζ are given (to very good or perfect accuracy,

103517-6 PROBING A PANOPLY OF CURVATON-DECAY SCENARIOS … PHYSICAL REVIEW D 94, 103517 (2016)

1 0 δξ ~ Bi lep ζi − ζγ ¼ : ð18Þ 3 Bi π

Neutrinos inherit the lepton asymmetry and its fluctuations, δξ π ~ 0 ~ 3 ζ~ − ζ and so lep ¼ BiSL=Bi, where SL ¼ ð L γÞ. We then see that 15 X ξ 2 ≃ lep ~ Sνγ 7 π SL; ð19Þ i¼μ;e;τ

ξ π ≪ 1 where we have assumed that lep= and assumed that flavor mixing of the cosmic neutrino background is negligible after neutrino decoupling. We thus have that ξ2 FIG. 2. The relationship between Neff and lep [Eq. (16)]. The [42,43] black dashed line indicates the 95% C.L. upper limit to ΔN eff from the 2015 Planck analysis using TT þ LowP þ BAO and 45 ξ2 135 ξ2 2 lep ~ lep ~ corresponds to a 95% C.L. upper limit ξ ≤ 0.5; the red dashed Sνγ ≃ S ζ − ζγ : lep 7 π2 L ¼ 7 π2 ð L Þ ð20Þ line indicates the 95% C.L. upper limit using TT þ AllP þ BAO.

To proceed further, we must specify when lepton number ξ 2 ξ 4 (L) is generated. Applying Eq. (6), we obtain [42,43] B ¼ lep þ lep ; ð15Þ i π π 8 ξ 2 >−135 lep ζ σ > 7 π γ if Lis generated before decay; which can also be parameterized as [7] <> 2 135 ξlep ζ 30 ξ 2 15 ξ 4 Sνγ ¼ 7 π −ζγ if Lis generated byσdecay; 3 046 3 lep lep > rD Neff ¼ . þ þ : ð16Þ > 7 π 7 π > ξ 2 :>135 lep 7 π ðζ −ζγÞ if Lis generated afterσdecay: Past forecasts and recent analyses of Planck data ξ ð21Þ show that if the only effects of lep are to alter Neff and the free-electron fraction (by altering YHe) at decoupling, CMB constraints to ξ (shown in Fig. 2) will remain less Substituting into Eq. (9) and solving for Sνγ, we then lep ξ π 2 sensitive than constraints from astronomical measurements obtain [to lowest order in ð lep= Þ ] [43]: of primordial element abundances [82–88]. In the curvaton 8 scenario, however, if the lepton asymmetry is generated by >0 if Lis generated beforeσdecay; <> curvaton decay, the amplitude of neutrino-isocurvature- Sνγ ξ 2 135 lep 1 −1 σ density fluctuations depends on the values of ξ , offering ¼ 7 π if Lis generated by decay; lep ζ > rD :> an additional possible channel for constraining this param- 0 if Lis generated afterσdecay: eter. Neutrino experiments may still yield surprises as to the precise values of quantities like θ13. We thus explore ð22Þ what constraints to the neutrino sector are possible from CMB observations alone. In the future, measurements of The expression for the case of L generated before σ the 21 cm emission/absorption power spectrum from decay is approximate, and has corrections of order −2 neutral hydrogen (during the epoch of reionization or Sνγ ∼ 10 ζ which are negligible at the level of accuracy during the cosmic dark ages) could be useful probes of needed for the MCMC analysis of Sec. IV. The expression ξ the value of lep [89]. for the case of L generated after σ decay results from the We assume that the cosmic thermal history is conven- requirement that the penultimate equation hold independent tional between neutrino decoupling and electron-positron ξ of the true values of Rν and lep. annihilation, and thus neglect fluctuations in the neutrino- In scenarios where L is generated by σ decay, there is a 1=3 photon temperature ratio Tν=Tγ ≃ ð4=11Þ . It is then mismatch between the total ζ (which has contributions straightforward to show that for neutrinos [43] from neutrinos and photons) and ζγ. This must be self- consistently included in Eq. (9) to obtain the correct 0 δξ 1 A lep ζ − ζ i expressions for the relationships between Sbγ (or Scγ) i γ ¼ 4 π ; ð17Þ Ai and ζ, shown in Table I. If lepton number is generated

103517-7 TRISTAN L. SMITH and DANIEL GRIN PHYSICAL REVIEW D 94, 103517 (2016) before or after σ decay, the amplitudes are given as before CMB data and thus improve the sensitivity of the Planck in Table I with Sνγ ¼ 0. data to isocurvature perturbations. The introduction of matter isocurvature modes, shown by the blue curves in Fig. 3, has its most significant effect IV. DATA on the large-scale TT and TE power spectra, where it The main effect of the curvaton model is to introduce changes the height of the Sachs-Wolfe plateau and alters totally correlated (or anticorrelated) isocurvature modes the shape/amplitude of the Integrated Sachs-Wolfe (ISW) into the initial conditions of the cosmological perturbations. effect. On the other hand, neutrino-density isocurvature In order to test the various curvaton decay channels, we use with a comparable amplitude, shown by the orange curves the CMB temperature and E-mode polarization power- in Fig. 3, affects CMB anisotropies more dramatically at all spectra measured by the Planck satellite [35,90,91]. The scales. large-scale E-mode measurements mainly constrain the The Planck data have been divided up into a large optical depth to the surface of last scattering, τ, while angular-scale data set (low multipole number) and a small the small-scale E-mode measurements provide additional angular-scale data set (high multipole number) [91]. For all constraints on the allowed level of isocurvature [92].We constraints, we use the entire range of measurements for the also use measurements of baryon acoustic oscillations TT power spectrum as well as the low multipole polari- (BAOs) [93,94] to break geometric degeneracies in the zation (TE and EE) data, which we denote as LowP. We

FIG. 3. A comparison of the differences between a purely adiabatic mode and a totally correlated (solid) or anticorrelated XY (dashed) matter- (blue) or neutrino-density (orange) isocurvature mode. Each panel shows the binned residuals ΔDl ≡ XY lðl þ 1ÞΔCl =ð2πÞ (see Ref. [91] for details on the binning procedure). The matter isocurvature has an amplitude Smγ ¼ 0.2, and the neutrino-density isocurvature has an amplitude Sνγ ¼ 0.1. We also show the residuals for the power spectrum measured by the Planck satellite [91]. Note that the horizontal scale is logarithmic up to l ¼ 29 and then is linear; the vertical scales on the left- and right- hand sides are different.

103517-8 PROBING A PANOPLY OF CURVATON-DECAY SCENARIOS … PHYSICAL REVIEW D 94, 103517 (2016) also compute constraints using the entire multipole range of impose the BBN constraint by placing a Gaussian prior on ξ polarization measurements, denoted by AllP. The division with a mean of zero and a standard deviation of 0.03; third, ξ 0 between these two data sets is the multipole number we consider the case where lep ¼ , removing the neutrino l ¼ 29, which approximately corresponds to an angular isocurvature mode. We find that both current CMB mea- scale of ≃5°. surements by Planck and a future cosmic-variance limited As demonstrated in Fig. 3, polarization data can break experiment (with maximum l ¼ 2200) are less sensitive to degeneracies present in a temperature-only analysis. This ξ2 lep than measurements of the light-element abundances. statement is especially true for tests of the adiabaticity of The observed CMB power spectra can be written in terms the initial conditions [92]. The analysis in Sec. 11 of of the primordial curvature perturbation power spectrum, Ref. [34] and Sec. VI. 2. 3 of Ref. [35] includes constraints 2 i;X Δ ðkÞ, and the photon transfer function Θ ðkÞ for each to isocurvature modes using the Planck 2015 data. As they ζ l initial condition i,as point out, the addition of AllP greatly improves the Z constraint to isocurvature modes which are correlated to ∞ X XY dk 2 i;X the adiabatic mode. Cl ¼ 4π ΔζðkÞ AiγΘl ðkÞ 0 k For example, the fractional contribution to the temper- i α −0 0025þ0.0035 X ature power spectrum is constrained to ¼ . −0.0047 j;Y × A γΘ ðkÞ ; ð23Þ at 95% C.L. using Planck TT þ LowP, where the sign j l j of α indicates whether the isocurvature contribution is α 0 α 0 totally correlated ( > ) or anticorrelated ( < ) with the where X ∈ fT;Eg denotes the relevant observable (CMB adiabatic mode. The preference for an anticorrelated mode temperature or E-mode polarization anisotropy). comes from the well-known deficit of power on large The primordial curvature perturbation is given in terms angular scales [90,95]. When all polarization data are of the amplitude parameter As: included in the analysis, the centroid shifts upward and α 3 −1 the overall uncertainty on is reduced by more than 50%: k k ns 0 0016 Δ2 ≡ α 0 0003þ . ζðkÞ 2 PζðkÞ¼As ; ð24Þ ¼ . −0.0012 at 95% C.L. As noted by the Planck 2π k0 team [35], these effects may both be driven by a signifi- cantly low point in the TE cross power spectrum, which where PζðkÞ is the dimensional power spectrum of ζ, As may be due to unidentified systematic effects (see, is the primordial scalar amplitude and ns is the primordial e.g., Ref. [96]). scalar spectral index, and the pivot wave number is taken to −1 In order to highlight the effects of including all of the be k0 ¼ 0.05 Mpc . As discussed in Sec. II, we set the publicly available Planck data, we divide our analysis into running αs ¼ 0. The amplitude parameters two sets of data: Planck TT þ BAO þ LowP and Planck TT þ BAO þ AllP. Given the uncertainty around system- Aiγ ≡ fAad;Acγ;Abγ;Aνγgð25Þ atic effects in the high-l polarization power spectrum, we take the Planck TT þ BAO þ LowP constraints to be more are used to set the mixture of adiabatic and isocurvature robust. modes in the CMB Boltzman code CAMB. It is important to In order to compare the data to our model, we use a set all these amplitudes correctly in the presence of neutrino modified version of the publicly available Boltzmann isocurvature, as neutrinos contribute to the relativistic code COSMOMC [97] along with the publicly available energy density at early times, and the neutrino isocurvature Planck likelihood code [91] included with the 2015 Planck density mode is excited in the curvaton model, as we saw in data release. We made modifications to these codes in order Sec. III. As discussed in the Appendix, using the initial to include the two curvaton parameters r and ξ .As D lep perturbation values δc, δb, δγ, and δν for each perturbation discussed previously, the parameter r only affects the D mode used in CAMB, we have that Abγ ¼ Sbγ=ζ − RνSνγ=ζ, initial conditions, whereas the lepton asymmetry, ξ , lep Acγ ¼ Scγ=ζ − RνSνγ=ζ, and Aνγ ¼ 3SνγRγ=4ζ, where affects both the initial conditions and the effective number Rγ ≡ 1 − Rν is the fraction of relativistic energy in photons. β of neutrino species, as well as -decay processes occurring We apply these relations when running our MCMCs for during BBN. This latter effect alters the primordial each of the curvaton-decay scenarios enumerated in light-element abundances, so that from measurements of 4 Sec. III, along with Table I and Eq. (16). primordial He and deuterium abundances, we have an Before presenting constraints to r and ξ ,itis ξ ≲ 0 03 D lep independent constraint j lepj . at 95% C.L. [36,39],as instructive to consider a “model independent” parameter- discussed in Sec. III. ization of the totally correlated (or anticorrelated) isocur- ξ In our analysis, we try three different priors on lep: first, vature modes. Figure 3 gives us a sense of what to expect ξ we consider the constraints to lep from the CMB only from this exercise. First, note that a 20% contribution ξ2 0 ≤ ξ2 ≤ 4 imposing a flat prior on lep of lep ; second, we from totally correlated CDM isocurvature (blue curves) can

103517-9 TRISTAN L. SMITH and DANIEL GRIN PHYSICAL REVIEW D 94, 103517 (2016) produce a deficit of power on large scales while also As was noted in Ref. [35], the difference between these causing a significant change at around the first peak in the constraints may be driven by a handful of data points TT power spectrum. Given the mild tension between around l ≃ 160. This can be seen by eye in Fig. 3: in the the best-fit theoretical power spectrum and the relatively top panel, which shows the TT spectrum, the large-scale low-temperature quadrupole (at the level of a little more residuals are significantly below zero, preferring totally than 1 standard deviation), we expect the data to prefer a anticorrelated matter and neutrino-density isocurvature slightly negative value for Smγ. The matter isocurvature (dashed curves); in the TE spectrum, there are a few data also has a significant effect on the TE power spectrum points around l ≃ 160 which have residuals significantly between 100 ≲ l ≲ 500. above and below zero. As the isocurvature curves show, The introduction of a 10% contribution from neutrino these data introduce a tension between totally correlated isocurvature (orange curves) significantly changes the TT and anticorrelated isocurvature modes. We note that this power spectrum at nearly all scales, as well as the TE and tension may be a significant driver in the difference EE power spectrum on scales with l ≳ 100. We therefore between the LowP and AllP constraints on isocurvature expect that the CMB data will be more sensitive to Sνγ than perturbations, although we do not explore this issue further. to Smγ. As the data are well fit by a Universe with purely The results presented in Fig. 4 confirm our expectations: adiabatic perturbations, curvaton scenarios that fit have rD ξ the Planck TT þ BAO þ LowP (red curves) prefers a and lep values that produce adiabatic perturbations. This slightly anticorrelated matter isocurvature amplitude, and immediately eliminates the scenarios ðbafter;cafter;LyL Þ and when all of the polarization data are included (blue curves), ðbbefore;cbefore;LyL Þ. We also note that the case where the constraints shift towards a purely adiabatic spectrum. ðbbefore;cafter;LyL Þ will produce a huge isocurvature per- We find that the Planck TT þ BAO þ LowP places a turbation, unless rD exceeds the bounds from non- −0 19 0 18 0 04þ0.18 ξ constraint Smγ ¼ . . and Sνγ ¼ . −0.16 , whereas Gaussianity and lep exceeds the BBN bounds. This Planck TT þ BAO þ AllP gives Smγ ¼ −0.06 0.16 and scenario is thus ruled out to high significance as well. Sνγ ¼ 0.05 0.11 at 68% C.L. We are then left with 18 scenarios which may be consistent with the data. Each of the allowed 18 scenarios yields zero-isocurvature contributions to CMB power spectra if Amγ ¼ 0—i.e., as long as they correspond to a compensated isocurvature mode. We show the value of rD in each of these scenarios for which Amγ ¼ 0 in Table II, along with the constraints to ξ 0 rD when lep ¼ . In addition to running MCMCs to obtain constraints, we perform a Fisher-matrix analysis to forecast the sensitivity ξ2 of CMB data to lnðrDÞ and lep. We include these parameters, as well as the standard six ΛCDM parameters. We apply the Fisher-matrix formalism as described in Ref. [98]. In this analysis, we also include a BBN prior 4 σ 0 005 on the primordial He abundance, with error YHe ¼ . .

ξ2 0 TABLE II. Constraints to rD (with lep ¼ ) using Planck TT þ BAO þ LowP to those models which can yield vanishing isocurvature perturbations as seen in any two-point correlation

function. Note that in the scenario ðbafter;cafter;LyL Þ, we quote a constraint to χD which is related to rD, as discussed in more detail in Sec. IV B.

Scenario rD (Amγ ¼ 0) rD (95% CL) FIG. 4. The posteriors for the correlated isocurvature ampli- ðb ;c ;L Þ 0 1580þ0.0042 0 1602þ0.0051 by before yL . −0.0040 . −0.0047 tudes Smγ and Sνγ; the red curves show constraints using Planck ðb ;c ;L Þ 0 8373þ0.0042 0 8492þ0.0099 TT þ BAO þ LowP, and the blue curves show constraints using before by yL . −0.0043 . −0.0096 Planck TT þ BAO þ AllP. Note that at the level of about 1 0 9578 ðbby;cafter;LyL Þ 1 > . standard deviation, the LowP case is better fit by a totally 0 9919 ðbafter;cby;LyL Þ 1 > . anticorrelated matter isocurvature component, which leads to a 0 9931 ðbby;cby;LyL Þ 1 > . suppression of power on large angular scales. When all of the ðb ;c ;L Þ 1 >0.9973 polarization data are included, this preference is less dramatic. after after yL

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Numerical derivatives are evaluated using a standard Constraints to rD in these two scenarios are significantly two-sided two-point numerical derivative, except for the different when all of the polarization data are included. ξ2 ξ2 parameters ln rD and lep, for which a one-sided seven-point In this case, marginalizing over lep for ðbby;cbefore;LyL Þ, 0 1595þ0.0044 rule was applied to obtain sufficiently convergent numerical we find that at 95% C.L., rD ¼ . −0.0041 and 2 0.0035 derivatives. Additionally, for Ωbh , a two-sided seven-point Ω Ω 0 1570þ b= m ¼ . −0.0033 ; for ðbbefore;cby;LyL Þ we find that rule was used to guarantee numerical convergence. For r ¼ 0.853þ0.015 and Ω =Ω ¼ 0.8455þ0.0052. We can see 1010 XY D −0.014 c m −0.0045 ln ð AsÞ, the derivative dCl =dAs was evaluated ana- that in both scenarios, r is constrained to be significantly XY ∝ D lytically, as Cl As, obviating the need to compute a closer to its compensated values when all of the polariza- numerical derivative for this parameter. These results are tion data are used. used both to verify that our MCMC results for Planck data The constraint to ξ2 in these two scenarios is particu- are reasonable and to forecast the ideal sensitivity of a lep larly interesting, since the compensated isocurvature leads cosmic-variance limited CMB polarization experiment to to a stricter Planck/BAO constraint. Looking at Eq. (22), curvaton-generated isocurvature perturbations. 2 2 we can see that the smaller rD is, the larger the neutrino Fiducial values for Ω h , Ω h , ΩΛ, A , n , and τ were b c s s isocurvature contribution. This means that the Planck/BAO set to the marginalized means for these parameters in a constraints to ξ2 for the scenario ðb ;c ;L Þ are the Λ lep by before yL CDM-only MCMC run. For the lepton asymmetry, 2 2 ξ ≤ 0 0164 ξ 0 most constraining with lep . at 95% C.L., as seen we used lep ¼ as the fiducial value. For all curvaton in Fig. 5. Although this is not competitive with constraints scenarios except (bby, cbefore, LyL ) and (bbefore, cby, LyL ), we used the fiducial value r ¼ 1, guaranteeing that the inferred from measurements of the primordial light-element D – ξ2 ≤ 0 001 fiducial model has adiabatic perturbations. For the scenar- abundances [33,36 39], lep . at 95% C.L., it is the tightest constraint to ξ2 using only Planck/BAO data. ios (bby, cbefore, LyL ) and (bbefore, cby, LyL ), we used fiducial lep values corresponding to zero isocurvature between radia- Since the value of rD is larger in the scenario 0 ξ2 tion and nonrelativistic matter (i.e., Amγ ¼ ), correspond- ðbbefore;cby;LyL Þ, the constraint to lep in this case is not ing to the r values given in the middle column of Table II. ξ2 ≤ 0 368 D as restrictive, giving lep . at 95% C.L. As shown in We now present our constraints to curvaton-decay Fig. 2, however, this is more restrictive than the upper limit scenarios, grouped by the character of their effects on placed on ξ2 ≤ 0.5 from its contribution to the total CMB power spectra. We begin by discussing scenarios for lep radiative energy density of the Universe, showing that this which there is nonzero isocurvature unless ξ ¼ 0 and constraint is driven by the effect the lepton asymmetry has r ¼ Ω =Ω , where i denotes baryons or CDM. We then D i m on neutrino isocurvature perturbations. move on to a scenario showing a total degeneracy between ξ2 2 The constraint to lep does not change significantly when ξ and rD at the level of isocurvature amplitudes. We lep including all of the polarization data: for the scenario finish by discussing scenarios for which all isocurvature ðb ;c ;L Þ, the constraint becomes ξ2 ≤ 0.0165; modes vanish when r ¼ 1. before by yL lep D ξ2 ≤ 0 348 and for ðbbefore;cby;LyL Þ, it becomes lep . . A. Constraints to baryon number or CDM As shown in the bottom panels of Figs. 5 and 6, the production before curvaton decay marginalized 1D constraint on rD is fairly insensitive to ξ2 The two decay scenarios which produce compen- how we treat lep. In those panels, the red curve shows the constraint arising from flat priors on ξ2 . The blue curve sated isocurvature modes are ðbby;cbefore;LyL Þ and lep shows the constraint that arises when ξ2 has the BBN prior ðbbefore;cby;LyL Þ. As shown in Table I, the isocurvature lep ξ2 ≤ 0 001 contribution vanishes (i.e., is purely compensated) when lep . at 95% C.L. The orange curve shows the ξ ¼ 0 and r ¼ Ω =Ω for ðb ;c ;L Þ or r ¼ 2 lep D b m by before yL D constraint obtained when we assume ξ ¼ 0. Ω Ω lep c= m for ðbbefore;cby;Ly Þ. In addition to this, if rD is L The values of rD allowed by Planck/BAO data in these greater than the previous values, the matter isocurvature is scenarios also imply a non-Gaussian signature in the anticorrelated with the adiabatic mode, leading to a CMB. The predicted level of this signature can be suppression of the large-scale temperature power spectrum. determined through Eq. (5). We show the predicted As expected, constraints from Planck TT þ BAO þ LowP ranges for the amplitude of this signal, fnl, in Fig. 7. lead to values of rD which are slightly larger than 5 92 0 26 The scenario ðbbefore;cby;LyL Þ predicts fnl ¼ . . , the purely compensated case, since that leads to a þ0.034 and ðb ;c ;L Þ predicts that f ¼ −0.919−0 040 suppression of the large-scale temperature power spectrum. before by yL nl . 2 at 95% C.L. Marginalizing over ξ for ðb ;c ;L Þ, we find that at lep by before yL Current data impose the constraint f ¼ 2.5 5.7 [45]. 0 1619þ0.0055 Ω Ω 0 1580þ0.0043 nl 95% C.L., rD ¼ . −0.0053 and b= m ¼ . −0.0041 ; The scenario ðbby;cbefore;LyL Þ implies a particularly large for ðb ;c ;L Þ we find that r ¼ 0.856þ0.015 and before by yL D −0.014 fnl value, which could be sensitively tested using mea- Ω Ω 0 8401þ0.0063 c= m ¼ . −0.0059 . surements of scale-dependent bias in future galaxy surveys

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ξ2 FIG. 6. Constraints to rD and lep for the scenario ξ2 b ;c ;L . Top: The marginalized 2D constraints to both FIG. 5. Constraints to rD and lep for scenario ðbby;cbefore;Ly Þ. ð before by yL Þ L ξ Top: The marginalized 2D constraints to both rD and ξlep. The red rD and lep. The red regions show the current constraints using regions show the current constraints using Planck TT þ BAO þ Planck TT þ BAO þ LowP data, the blue regions show con- LowP data, the blue regions show constraints using Planck straints using Planck TT þ BAO þ AllP, and the black regions TT þ BAO þ AllP, and the black regions show the projected show the projected constraints for a cosmic-variance limited l 2200 constraints for a cosmic-variance limited CMB experiment which CMB experiment which measures out to max ¼ , obtained l 2200 measures out to max ¼ , obtained from a Fisher-matrix from a Fisher-matrix analysis. In this panel, a flat prior is imposed ξ2 ξ2 analysis. In this panel, a flat prior is imposed on lep, as discussed on lep, as discussed in the text. Bottom: Marginalized 1D in the text. The dashed vertical green line gives the value of rD for constraints to rD using Planck TT þ BAO þ LowP under a 0 ξ ξ2 which the isocurvature is totally compensated (i.e. Amγ ¼ ); the variety of assumptions for lep: flat prior on lep (red), BBN dashed horizontal purple line gives the 95% C.L. upper limit on prior on ξ2 (blue), and ξ2 ¼ 0 (orange). ξ2 lep lep lep from measurements of the primordial light element abun- dances. Bottom: Marginalized 1D constraints to rD using Planck ξ TT þ BAO þ LowP under a variety of assumptions for lep: flat the increased sensitivity, CMB/BAO measurements of ξ2 prior on ξ2 (red), BBN prior on ξ2 (blue), and ξ2 0 (orange). lep lep lep lep ¼ are still not as sensitive as measurements of the primordial light-element abundances. [46,47] or measurements of the matter bispectrum from high-redshift 21 cm experiments [48–50]. The scenario B. Constraints to baryon and CDM production ðbbefore;cby;LyL Þ, which makes more modest predictions, could be tested with high-redshift 21 cm experiments after curvaton decay [48–50]. In the scenario where both the baryon number and CDM Future CMB measurements will greatly improve upon are produced after curvaton decay, while lepton number is these constraints. As shown by the black ellipses in Figs. 5 produced by its decay, the initial conditions are completely and 6, a cosmic-variance limited CMB experiment which determined by the level of neutrino isocurvature alone, as measures both the temperature and polarization power shown in Table I. Looking at Eq. (22), we can see this spectra out to l 2200 will give a factor of 4.3 increase ξ2 max ¼ produces a perfect degeneracy between rD and lep: the ξ2 in sensitivity to lep, a factor of 3.5 increase in sensitivity to level of isocurvature can be made to be arbitrarily small for r for the scenario ðb ;c ;L Þ, a factor of 11 increase ≤ 1 ξ2 D by before yL any value of rD with a small enough value for lep.In ξ2 in sensitivity to lep, and a factor of 4 increase in sensitivity order to determine the allowed region of parameter space, it χ to rD for the scenario ðbbefore;cby;LyL Þ. Note that even with is convenient to define a new parameter, D:

103517-12 PROBING A PANOPLY OF CURVATON-DECAY SCENARIOS … PHYSICAL REVIEW D 94, 103517 (2016) ξ2 ≤ 0 49 constraint lep . (at 95% C.L.), due to the effect of ξ2 lep on Neff. We have also seen that measurements of the primordial light-element abundances further constrain ξ2 ≤ 0 001 lep . at 95% C.L. The unshaded region of the rightmost panel of Fig. 8 shows the currently allowed − ξ2 region of the rD lep parameter space in this scenario. The Planck TT þ BAO þ LowP data place the con- 1 −χ ≤ 0 0027 ξ2 ≤ 0 5 straints D . and lep . at 95% C.L. These ξ2 data have a slight preference for nonzero lep due to its additional contribution to the radiative energy density. This preference has been seen in previous analyses [35,99,100]. When all the polarization data are used, the preference for a ξ2 nonzero lep disappears but is replaced by a slight prefer- ence for χD < 1, as can be seen in the blue curves in the FIG. 7. Predicted value of the non-Gaussianity parameter f nl left panel of Fig. 8. In this case, we have 1 − χD ≤ 0.0025 for the scenarios ðb ;c ;L Þ and ðb ;c ;L Þ for 2 by before yL before by yL and ξ ≤ 0.33 at 95% C.L. parameter values which are consistent with our limits (on lep isocurvature and the radiative energy density at decoupling) Given that any value of rD is consistent with the Planck/ from Planck/BAO data (red). The vertical dashed lines indicate BAO data, this scenario does not make a specific prediction the 95% C.L. range of these predictions. for a level of non-Gaussianity. Instead, current data 2 5 5 7 ≥ 0 12 (fnl ¼ . . [45]) allow us to conclude that rD . 1 ξ2 1 at 95% C.L. This constraint is shown in the left-hand panel − 1 ≡ lep − 1 : of Fig. 8 as the vertical dashed line. χ π2 ð26Þ D rD Future measurements of the CMB will more sensitive to this curvaton-decay scenario, as shown by the black curves χ ξ2 The constraints to D and lep are shown in Fig. 8. in the left and center panels of Fig. 8. Using a Fisher-matrix As discussed in Sec. III B, even in the absence of analysis, we find that a cosmic-variance limited experiment neutrino isocurvature, Planck/BAO data impose the which measures both the temperature and polarization

FIG. 8. Marginalized 1D constraints to a scenario in which lepton number is produced by curvaton decay, while baryon number and CDM are produced after curvaton decay. For the left and middle panels, the red curve shows constraints using Planck TT þ BAO þ LowP, the blue curve shows constraints using Planck TT þ BAO þ AllP, and the black curve shows projected l 2200 constraints for a cosmic-variance limited CMB experiment which measures out to max ¼ , obtained from a Fisher-matrix analysis. The vertical dashed lines indicate the 95% C.L. upper limit to each parameter using the Planck TT þ BAO þ LowP data. Left-hand 1 − χ χ ξ2 panel: The 1D marginalized posterior for D, where D is defined in Eq. (26). Middle panel: The 1D marginalized posterior on lep χ ξ2 from CMB/BAO observations only. Right-hand panel: A contour plot showing the relationship between D, rD, and lep. The dotted red contour shows the 68% C.L. upper limit on 1 − χD from Planck TT þ BAO þ LowP; the dashed red contour shows the 95% C.L. upper limit on 1 − χD from Planck TT þ BAO þ LowP. The vertical dashed line shows the 95% C.L. lower limit on rD from constraints to the ξ2 level of non-Gaussianity in the CMB; the horizontal dashed lines show the 95% C.L. upper limits on lep from the Planck TT þ BAO þ LowP data (black) and measurements of the primordial light-element abundances (red). The shaded region is currently ruled out at 95% C.L.

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FIG. 9. Constraints to the three cases where the baryon number and/or CDM is produced after the curvaton decays. The red regions show the current constraints using Planck TT þ BAO þ LowP data, the blue regions show constraints using Planck TT þ BAO þ AllP, and the black regions show the projected constraints for a cosmic-variance limited CMB experiment which measures out to lmax ¼ 2200, obtained from a Fisher-matrix analysis. In all cases, the inner contour corresponds to 68% C.L., and the outer contour ξ2 corresponds to 95% C.L. In this panel, a flat prior is imposed on lep, as discussed in the text. In these three cases, the initial conditions are purely adiabatic when rD ¼ 1.

l 2200 power spectra out to ¼ will be 4 times more in Fig. 9. The 95% C.L. lower limit for ðbafter;cby;LyL Þ is sensitive to χ and 3 times more sensitive to ξ2 . ≥ 0 992 ≥ 0 963 D lep rD . .Forðbby;cafter;LyL Þ, the limit is rD . . ≥ 0 993 Finally, for ðbby;cby;LyL Þ, the limit is rD . . When C. Constraints to remaining scenarios using all of the polarization data (blue regions in Fig. 9), the As shown in Table I, unlike the other cases considered, sensitivity to rD is nearly unchanged. these three scenarios yield purely adiabatic initial condi- The values of rD in these scenarios which are consistent with the Planck/BAO data lead to a non-Gaussian signal. tions when rD ¼ 1. This has important implications for Planck/BAO constraints to ξ2 in these scenarios. From The predicted level of this signal can be determined through lep Eq. (5). Note that the predicted values of f are bounded Eq. (22), it is clear that the level of neutrino isocurvature in nl from below, since when r ¼ 1 we have f ¼ −1.25.We these models is negligible. As a result, the sensitivity of D nl ξ2 show the predicted ranges for the amplitude of this signal, Planck/BAO data to lep comes solely from its contribution to the total radiative energy density of the Universe. This expectation is borne out in Fig. 9, since all three scenarios give nearly the same 95% C.L. upper limit ξ2 from the Planck/BAO data for lep: for ðbafter;cby;LyL Þ, ξ2 ≤ 0 42 ξ2 ≤ 0 40 lep . ; for ðbby;cafter;LyL Þ, lep . ; for ξ2 ≤ 0 44 ðbby;cby;LyL Þ, lep . . When using all of the polari- ξ2 zation data (blue regions in Fig. 9), the sensitivity to lep is significantly improved. In all three cases, the 95% con- ξ2 ≃0 75 fidence upper limit to lep is a factor of . of its value for less complete polarization data. The constraint to rD in each scenario varies because of the specific prefactor generated in each case. Looking at Table I, we can see that the overall matter isocurvature in scenario ðbby;cafter;LyL Þ is suppressed by the small factor Ωb=Ωm ≃ 0.15. Because of this, we expect the constraint to r in that case to be the least restrictive. The factor D FIG. 10. Predicted value of the non-Gaussianity parameter f Ω =Ω ≃ 0.8 appears in the expression for the isocurvature nl c m for the scenarios ðb ;c ;L Þ and ðb ;c ;L Þ for param- amplitude in the scenario ðb ;c ;L Þ, leading to a after by yL by after yL after by yL eter values which are consistent with our limits (on isocurvature moderate suppression of the matter isocurvature. Finally, and the radiative energy density at decoupling) from Planck/BAO since the scenario ðbby;cby;LyL Þ contains no suppression, data (red). The vertical dashed lines indicate the 95% C.L. range we expect the most restrictive constraint on rD to occur in of these predictions. The results for the ðbby;cby;LyL Þ scenario this case. All of these expectations are borne out, as shown are indistinguishable from those for the ðbafter;cby;LyL Þ scenario.

103517-14 PROBING A PANOPLY OF CURVATON-DECAY SCENARIOS … PHYSICAL REVIEW D 94, 103517 (2016) fnl, in Fig. 10. Since the upper limit on rD for the scenarios scenario in which both CDM and baryons are produced ∼−1 25 ðbafter;cby;LyL Þ and ðbby;cby;LyL Þ is more restrictive, the after curvaton decay) predict fnl . , a value which 95% C.L. upper limit on the predicted level of non- could be tested by future high-redshift 21-cm surveys Gaussianity in these scenarios is more restrictive with [48–50]. When both CDM and baryons are produced after −1 25 ≤ ≤−1 23 curvaton decay, r and ξ2 are completely degenerate and . fnl . , whereas for ðbby;cafter;LyL Þ,we D lep −1 25 ≤ ≤−1 17 have . fnl . . Current Planck data indicate no specific prediction for fnl can be made. 2 5 5 7 that fnl ¼ . . [45], and so both of these scenarios The most interesting cases from an observational point of are consistent with current constraints to primordial view are those in which baryon number is produced by non-Gaussianity. These fnl values could, however, be tested curvaton decay, and CDM before, or vice versa. The data 0 160 0 004 0 850 using future measurements of the matter bispectrum from then require that rD ¼ . . or rD ¼ . high-redshift 21 cm experiments [48–50]. 0.009 at 95% confidence for these two cases, respectively. Future CMB measurements will greatly improve these This window results from the requirement that the baryon constraints. As shown by the black ellipses in Fig. 9,a and CDM isocurvature fluctuations nearly cancel, making cosmic-variance limited CMB experiment which mea- testable predictions for future experiments. sures both the temperature and polarization power spectra First of all, there must be a compensated isocurvature l 2200 out to max ¼ will give a factor of 3.5 increase in perturbation between baryons and CDM to obtain a small ξ2 sensitivity to lep and a factor of 2 increase in sensitivity overall isocurvature amplitude [41]. In the curvaton model, ζ to rD for each of the three scenarios considered in this this CIP must be totally correlated with , and a future subsection. CMB experiment (such as CMB Stage IV [101]) could test the scenario in which baryon number is generated by V. CONCLUSIONS curvaton decay and CDM before [79]. The scenario in which CDM is generated by curvaton decay and baryon The curvaton scenario presents a rich and interesting number before is inaccessible to CMB searches for com- alternative to standard single-field slow-roll inflationary pensated isocurvature perturbations [79]. models of early-Universe physics. There are 27 curvaton- Second of all, in these decay scenarios the perturbations decay scenarios, distinguished by whether baryon number, are non-Gaussian. The non-Gaussian signal is larger than lepton number, and CDM are produced before, by, or after in the cases where rD ¼ 1, since the limit of adiabatic curvaton decay. Although some are better motivated perturbations corresponds to rD < 1 in these scenarios. We theoretically than others, we have presented constraints find that when baryon number is produced by curvaton to all logical possibilities to fully explore the curvaton decay and CDM before, the parameter values allowed parameter space. Of these, 18 are currently allowed by by the CMB power spectra/BAO data predict that CMB and large-scale structure measurements. 5 92 0 26 fnl ¼ . . . This is still within the current limits Sensitivity to r , a parameter describing the curvaton D to fnl from the CMB bispectrum and may be detected by energy density, comes from the effects of nonadiabatic future galaxy surveys [46] (through scale-dependent bias) initial conditions on the CMB, as well as the introduction and high-redshift 21 cm experiments [48–50]. If CDM ξ2 of non-Gaussian statistics. Constraints on lep, the lepton- is produced by and baryon number is produced before number chemical potential, come from the effects of −0 919þ0.034 curvaton decay, the model predicts fnl ¼ . −0.04 ; nonadiabatic initial conditions on the CMB, as well as detection is more challenging, but perhaps possible with its contribution to the total radiative energy density. high-redshift 21 cm experiments [48–50]. We compared predictions for CMB anisotropy power If lepton number is produced by curvaton decay, spectra in these 18 scenarios with Planck CMB measure- the requirement that neutrino isocurvature perturbations ξ ments and the location of the BAO peak. The CMB data is satisfy constraints imposes a limit on lep. If baryon number divided between large-scale and small-scale measurements. is produced by curvaton decay, CDM after, and lepton As noted in Refs. [34,35], the inclusion of the small-scale ξ ≤ 0 13 number by the decay, the Planck data require lep . , polarization data significantly improves sensitivity to iso- ξ much tighter than the constraint to lep obtained from the curvature perturbation. We find that, when the small-scale overall radiation energy density at the surface of last polarization data is also used to measure the curvaton scattering. scenario parameters, the improved sensitivity is less sig- Conservatively speaking, future CMB experiments may nificant, due, in part, to degeneracies between parameters. bring an additional factor of ∼3 improvement in sensitivity For cases where rD ¼ 1 restores totally adiabatic 0 96 − 0 997 to deviations of rD from values consistent with purely perturbations, we find limits of rD > . . at adiabatic fluctuations. Depending on the precise character 95% C.L., depending on the precise decay scenario. In of small-scale polarized foregrounds [102], primary CMB ξ2 these cases, constraints to lep are primarily driven by its polarization anisotropies could be measured at multipole ξ2 ≤ 0 5 effect on the relativistic energy density with lep . at scales as high as l ∼ 4000, further improving sensitivity 95% C.L. These scenarios (with the exception of the decay to curvaton-generated isocurvature. As such, it would be

103517-15 TRISTAN L. SMITH and DANIEL GRIN PHYSICAL REVIEW D 94, 103517 (2016) interesting to repeat the Fisher analysis of this paper for a 3 3 Δ ¼ Δ ¼ Δγ ¼ Δν: ðA4Þ variety of specific curvaton potentials, self-consistently c b 4 4 including isocurvature as well as variations in the Now we can also see how to translate the conditions n α spectral index s and running s of primordial density given here to the initial conditions specified in a Boltzmann perturbations. solver such as CAMB. For example, with CDM isocurvature Furthermore, primordial initial conditions should have we have an imprint on the shape of the BAO peak, going beyond 3 the simple location of the peak in real space. This effect Δ − Δ Scγ ¼ c 4 γ could yield an additional test of the curvaton model, 3 8 if it can be disentangled from redshift-space distortions 1 − 2Ω τ 3Ω τ2 − − Ω τ 4Ω τ2 and nonlinearities. ¼ Acγ c;0 þ c;0 4 3 c;0 þ c;0

¼ Acγζ; ðA5Þ ACKNOWLEDGMENTS We acknowledge useful conversations with W. Hu, C. where we have applied the superhorizon power-series He, and R. Hložek. We thank Y. Ali-Haïmoud for useful solution for the CDM isocurvature mode from Ref. [103] τ 0 conversations and a thorough reading of the manuscript. and then evaluated it at initial conformal time ¼ .This D. G. is funded at the University of Chicago by a National means that if this mode is excited with an amplitude Acγ ζ Science Foundation Astronomy and Astrophysics (relative to the adiabatic mode), then Scγ ¼ Acγ . Postdoctoral Fellowship under Grant No. AST-1302856. When we excite multiple isocurvature modes, then the We acknowledge support from the Kavli Foundation and its overall isocurvature is the linear combination of each mode. founder Fred Kavli, and by U.S. Department of Energy Exciting both the CDM (with amplitude Acγ) and baryon Contract No. DE-FG02-13ER41958, which made possible isocurvature (with amplitude Abγ) modes leads to visits by T. L. S. to the Kavli Institute for Cosmological 3 Physics at the University of Chicago. Δ − Δ ζ Scγ ¼ c 4 γ ¼ Acγ ; ðA6Þ

3 APPENDIX: DERIVATION OF RELATION Δ − Δ ζ BETWEEN ISOCURVATURE AMPLITUDE AND Sbγ ¼ b 4 γ ¼ Abγ : ðA7Þ INITIAL MODE AMPLITUDE Things get more interesting when we consider the excita- We now derive relationships between the mode ampli- tion of both matter and neutrino-density isocurvature. The tudes A γ used in CAMB and the physical isocurvature i linear combination of CDM, baryon, and neutrino-density amplitudes Siγ predicted by the curvaton-decay scenarios in isocurvature gives initial density contrasts (applying the Table I. In terms of the curvature perturbation on hyper- power-series solutions from Ref. [103] again): surfaces of constant single-species energy density (ζi), we have Δ Δ Rb Δ − Rν γ ¼ Acγ γ;0 þ Abγ γ;0 Aνγ ; ðA8Þ δρ Rc Rγ δρi γ S γ ≡ 3 ζ − ζγ −3H − ; A1 i ð i Þ¼ ρ0 ρ0 ð Þ i γ 3 3 Rb Δc ¼ Acγ 1 þ Δγ;0 þ Δγ;0Abγ; ðA9Þ 4 4 Rc where δρi ¼ ρiΔi, the prime indicates a derivative with respect to conformal time, and H is the conformal Hubble 3 3 rate (and H ¼ aH). Rb Δb ¼ Δγ;0Acγ þ 1 þ Δγ;0 Abγ; ðA10Þ The continuity equation dictates that 4 4 Rc

ρ_ −3 ρ 1 → ρ0 −3Hρ 1 R i ¼ H ið þ wiÞ i ¼ ið þ wiÞ: ðA2Þ b Δν ¼ AcγΔγ;0 þ Δγ;0Abγ þ Aνγ; ðA11Þ Rc Therefore, we can write the isocurvature perturbation in Δ ρ ρ ρ terms of the relative energy-density perturbation Δi: where γ;0 is a constant, Rc ¼ c=ð c þ bÞ, Rb ¼ ρ =ðρ þ ρ Þ, and Rν ¼ ρν=ðργ þ ρνÞ. 1 3 b c b S γ ¼ Δ − Δγ: ðA3Þ Applying Eq. (A3), we then find that i 1 þ w i 4 i 3 Rν We can now see that adiabatic initial conditions take the ζ Scγ= ¼ Acγ þ 4 Aνγ ; ðA12Þ usual form Rγ

103517-16 PROBING A PANOPLY OF CURVATON-DECAY SCENARIOS … PHYSICAL REVIEW D 94, 103517 (2016) 3 Rν Acγ ¼ Scγ=ζ þðRγ − 1ÞSνγ=ζ; ðA15Þ Sbγ=ζ ¼ Abγ þ Aνγ ; ðA13Þ 4 Rγ

3 Rν Abγ ¼ Sbγ=ζ þðRγ − 1ÞSνγ=ζ; ðA16Þ Sνγ=ζ ¼ AbγRb þ AcγRc þ : ðA14Þ 4 Rγ 3 Solving this set of equations for the initial condition ζ Aνγ ¼ 4 RγSνγ= : ðA17Þ amplitudes in terms of the isocurvature amplitudes, we obtain

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