We present the details for the covariant renormalization of the stress tensor for vacuum tensor perturbations at the level of the effective action, adopting Hadamard regularization techniques to isolate short distance divergences and gauge fixing via the Faddeev-Popov procedure. The subsequently derived renormalized stress tensor can be related to more familiar forms reliant upon an averaging prescription, such as the Isaacson or Misner-Thorne-Wheeler forms. The latter, however, are premised on a prior scale separation (beyond which the averaging is invoked) and therefore unsuited for the purposes of renormalization. This can lead to potentially unphysical conclusions when taken as a starting point for the computation of any observable that needs regularization, such as the energy density associated to a stochastic background. Any averaging prescription, if needed, should only be invoked at the end of the renormalization procedure. The latter necessarily involves the imposition of renormalization conditions via a physical measurement at some fixed scale, which we retrace for primordial gravitational waves sourced from vacuum fluctuations through direct or indirect observation.
Many cosmological observables of interest derive from primordial vacuum fluctuations evolved to late times. These observables represent statistical draws from some underlying quantum or statistical field theoretic framework where infinities arise and require regularization. After subtracting divergences, renormalization conditions must be imposed by measurements or observations at some scale, mindful of scheme and background dependence. We review this process on backgrounds that transition from finite duration inflation to radiation domination, and show how in spite of the ubiquity of scaleless integrals, UV divergences can still be meaningfully extracted from quantities that nominally vanish when dimensionally regularized. In this way, one can contextualize calculations with hard cutoffs, distinguishing between UV and IR scales corresponding to the beginning and end of inflation from UV and IR scales corresponding the unknown completion of the theory and its observables. This distinction has significance as observable quantities cannot depend on the latter although they will certainly depend on the former. One can also explicitly show the scheme independence of the coefficients of UV divergent logarithms. Furthermore, certain IR divergences can be shown to be an artifact of the de Sitter limit and are cured for finite duration inflation. For gravitational wave observables, we stress the need to regularize stress tensors that do not presume a prior scale separation in their construction (as with the standard Isaacson form), deriving an improved stress tensor fit to purpose. We conclude by highlighting the inextricable connection between inferring $N_{\rm eff}$ bounds from vacuum tensor perturbations and the process of background renormalization.
Santosh K. Das, Prabhakar Palni, Jhuma Sannigrahi, Jan-e Alam, Cho Win Aung, Yoshini Bailung, Debjani Banerjee, Gergely Gábor Barnaföldi, Subash Chandra Behera, Partha Pratim Bhaduri, Samapan Bhadury, Rajesh Biswas, Pritam Chakraborty, Vinod Chandra, Prottoy Das, Sadhana Dash, Saumen Datta, Sudipan De, Vaishnavi Desai, Suman Deb, et al (47) The discovery and characterization of hot and dense QCD matter, known as Quark Gluon Plasma (QGP), remains the most international collaborative effort and synergy between theorists and experimentalists in modern nuclear physics to date. The experimentalists around the world not only collect an unprecedented amount of data in heavy-ion collisions, at Relativistic Heavy Ion Collider (RHIC), at Brookhaven National Laboratory (BNL) in New York, USA, and the Large Hadron Collider (LHC), at CERN in Geneva, Switzerland but also analyze these data to unravel the mystery of this new phase of matter that filled a few microseconds old universe, just after the Big Bang. In the meantime, advancements in theoretical works and computing capability extend our wisdom about the hot-dense QCD matter and its dynamics through mathematical equations. The exchange of ideas between experimentalists and theoreticians is crucial for the progress of our knowledge. The motivation of this first conference named "HOT QCD Matter 2022" is to bring the community together to have a discourse on this topic. In this article, there are 36 sections discussing various topics in the field of relativistic heavy-ion collisions and related phenomena that cover a snapshot of the current experimental observations and theoretical progress. This article begins with the theoretical overview of relativistic spin-hydrodynamics in the presence of the external magnetic field, followed by the Lattice QCD results on heavy quarks in QGP, and finally, it ends with an overview of experiment results.
The primordial gravitational wave background (GWB) offers an exciting future avenue of discovery for new physics. Its information content encodes multiple eras in the early Universe's history, corresponding to many orders of magnitude in frequency and physical scale to be measured today. By numerically solving for the GW transfer functions we provide simple yet accurate formulas describing the average power of the large-scale energy spectrum of the GWB for arbitrary primordial tensor power spectra. In doing so we can pedagogically explain and clarify previous GWB literature, highlight the important cosmological parameters of various GWB features, and reveal multiple ways in which cancelling conceptual errors can give deceptively accurate results. The scales considered here are particularly important for CMB probes of the GWB, via $B$-modes and spectral distortions. In particular, we carefully study the effects of both neutrino damping, and the precise nature of the transition between the radiation-dominated (RD) and matter-dominated (MD) eras. A byproduct of numerically solving the problem is the ability to study the robustness of common approximations in the literature. Specifically, we show that a numerical treatment is especially important around the RD--MD transition, and for a brief moment of history where neutrino damping occurs during MD. In passing we also discuss the effects of late acceleration caused by dark energy -- showing that this can be neglected in most practical GWB applications -- and the effects of changing relativistic degrees of freedom on the GWB at very small-scales.
Particle-antiparticle pairs can be produced by background electric fields via the Schwinger mechanism provided they are unconfined. If, as in QED in (3+1)-$d$ these particles are massive, the particle production rate is exponentially suppressed below a threshold field strength. Above this threshold, the energy for pair creation must come from the electric field itself which ought to eventually relax to the threshold strength. Calculating this relaxation in a self-consistent manner, however, is difficult. Chu and Vachaspati addressed this problem in the context of capacitor discharge in massless QED$_2$ [1] by utilizing bosonization in two-dimensions. When the bare fermions are massless, the dual bosonized theory is free and capacitor discharge can be analyzed exactly [1], however, special care is required in its interpretation given that the theory exhibits confinement. In this paper we reinterpret the findings of [1], where the capacitors Schwinger-discharge via electrically neutral dipolar meson-production, and generalize this to the case where the fermions have bare masses. Crucially, we note that when the initial charge of the capacitor is large compared to the charge of the fermions, $Q \gg e$, the classical equation of motion for the bosonized model accurately characterizes the dynamics of discharge. For massless QED$_2$, we find that the discharge is suppressed below a critical plate separation that is commensurate with the length scale associated with the meson dipole moment. For massive QED$_2$, we find in addition, a mass threshold familiar from (3+1)-$d$, and show the electric field relaxes to a final steady state with a magnitude proportional to the initial charge. We discuss the wider implications of our findings and identify challenges in extending this treatment to higher dimensions.
Gravitational waves (GWs) have the potential to probe the entirety of cosmological history due to their nearly perfect decoupling from the thermal bath and any intervening matter after emission. In recent years, GW cosmology has evolved from merely being an exciting prospect to an actively pursued avenue for discovery, and the early results are very promising. As we highlight in this paper, spectral distortions (SDs) of the cosmic microwave background (CMB) uniquely probe GWs over six decades in frequency, bridging the gap between astrophysical high- and cosmological low-frequency measurements. This means SDs will not only complement other GW observations, but will be the sole probe of physical processes at certain scales. To illustrate this point, we explore the constraining power of various proposed SD missions on a number of phenomenological scenarios: early-universe phase transitions (PTs), GW production via the dynamics of SU(2) and ultra-light U(1) axions, and cosmic string (CS) network collapse. We highlight how some regions of parameter space were already excluded with data from COBE/FIRAS, taken over two decades ago. To facilitate the implementation of SD constraints in arbitrary models we provide GW2SD. This tool calculates the window function, which easily maps a GW spectrum to a SD amplitude, thus opening another portal for GW cosmology with SDs, with wide reaching implications for particle physics phenomenology.
Next generation cosmic microwave background spectral distortion and pulsar timing array experiments have the potential to probe primordial fluctuations at small scales with remarkable sensitivity. We demonstrate the potential of these probes to either detect signatures of primordial black holes (PBHs) sourced from primordial overdensities within the standard thermal history of the universe over a 13-decade mass range ${\cal O}(0.1-10^{12})M_\odot$, or constrain their existence to a negligible abundance. Our conclusions are based only on global cosmological signals, and are robust under changes in i) the statistical properties of the primordial density fluctuations (whether Gaussian or non-Gaussian), ii) the merger and accretion history of the PBHs and assumptions about associated astrophysical processes, and iii) clustering statistics. Any positive detection of enhanced primordial fluctuations at small scales would have far-reaching implications from the content of dark matter to origin of BHs in the centers of galaxies, and to the field content of the inflation. On the other hand, their non-detection would also have important corollaries. For example, non-detection up to forecast sensitivities would tell us that PBHs larger than a fraction of a solar mass can constitute no more than a negligible fraction of dark matter. Moreover, non-detection will also rule out the scenario that PBHs generated by primordial overdensities could be the progenitors of super-massive black holes (SMBHs), of topical interest as there are only a few widely accepted proposals for the formation of SMBHs, an even more pressing question after the detection of active galactic nuclei over a billion solar masses at redshifts $z \geq 7$. Finally, non-detection sets the strongest bounds on the amplitude of small scale inflationary fluctuations for over 6 decades.
Reconstructions of the primordial power spectrum (PPS) of curvature perturbations from cosmic microwave background anisotropies and large-scale structure data suggest that the usually assumed power-law PPS has localised features (up to $\sim 10\%$ in amplitude), although of only marginal significance in the framework of $\Lambda$CDM cosmology. On the other hand if the underlying cosmology is assumed to be Einstein-de Sitter, larger features in the PPS (up to $\sim 20\%$) are required to accurately fit the observed acoustic peaks. Within the context of single clock inflation, we show that any given reconstruction of the PPS can be mapped on to functional parameters of the underlying effective theory of the adiabatic mode within a 2nd-order formalism, provided the best fit fractional change of the PPS, $\Delta\mathcal{P}_\mathcal{R}/\mathcal{P}_\mathcal{R}$ is such that $(\Delta\mathcal{P}_\mathcal{R}/\mathcal{P}_\mathcal{R})^3$ falls within the $1\,\sigma$ confidence interval of the reconstruction for features induced by variations of either the sound speed $c_\mathrm{s}$ or the slow-roll parameter $\epsilon$. Although there is a degeneracy amongst these functional parameters (and the models that project onto them), we can identify simple representative inflationary models that yield such features in the PPS. Thus we provide a dictionary (more accurately, a thesaurus) to go from observational data, via the reconstructed PPS, to models that reproduce them to per cent level precision.
We derive analytic bounds on the shape of the primordial power spectrum in the context of single-field inflation. In particular, the steepest possible growth has a spectral index of $n_s - 1 = 4$ once transients have died down. Its primary implication is that any constraint on the power spectrum at a particular scale can be extrapolated to an upper bound over an extended range of scales. This is important for models which generate relics due to an enhanced amplitude of the primordial scalar perturbations, such as primordial black holes. In order to generate them, the power spectrum needs to grow many orders of magnitude larger than its observed value on CMB scales - typically achieved through a phase of ultra slow-roll inflation - and is thus subject to additional constraints at small scales. We plot all relevant constraints including CMB spectral distortions and gravitational waves sourced by scalar perturbations at second order. We show how this limits the allowed mass of PBHs, especially for the large masses of interest following recent detections by LIGO and prospects for constraining them further with future observations. We show that any transition from approximately constant $\epsilon$ slow-roll inflation to a phase where the power spectrum rapidly rises necessarily implies an intervening dip in power. We also show how to reconstruct a potential that can reproduce an arbitrary time-varying $\epsilon$, offering a complementary perspective on how ultra slow-roll can be achieved.
During single clock inflation, hidden fields (i.e. fields coupled to the inflaton only gravitationally) in their adiabatic vacua can ordinarily only affect observables through virtual effects. After renormalizing background quantities (fixed by observations at some pivot scale), all that remains are logarithmic runnings in correlation functions that are both Planck and slow roll suppressed. In this paper we show how a large number of hidden fields can partially compensate this suppression and generate a potentially observable running in the tensor two point function, consistently inferable courtesy of a large $N$ resummation. We detour to address certain subtleties regarding loop corrections during inflation, extending the analysis of [1]. Our main result is that one can extract bounds on the hidden field content of the universe from bounds on violations of the consistency relation between the tensor spectral index and the tensor to scalar ratio, were primordial tensors ever detected. Such bounds are more competitive than the naive bound inferred from requiring inflation to occur below the strong coupling scale of gravity if deviations from the consistency relation can be bounded to within the sub-percent level. We discuss how one can meaningfully constrain the parameter space of various phenomenological scenarios and constructions that address naturalness with a large number of species (such as `N-naturalness') with CMB observations up to cosmic variance limits, and possibly future 21cm and gravitational wave observations.
We consider the renormalization of quenched bond disorder in the Ising model in the limit that it is sparse -- highly localized and vanishing in the thermodynamic limit. We begin in 1D with arbitrary disorder assigned to a finite number of bonds and study how the system renormalizes, finding non-trivial fixed point structure for any given bond with a separatrix at zero bond strength, equivalent to inserting a break in the chain. Either side of this critical line, renormalization group (RG) trajectories flow towards one of two attractors on which the disordered bonds settle onto ferromagnetic or anti-ferromagnetic (AF) couplings of equal and opposite magnitude. Bonds that settle on an AF attractor are equivalent to inserting a twist in the chain at the location of the bond, implying a multi-kink ground state solution. Qualitatively different behavior emerges at the RG step when bonds start to coalesce, with the chain `untwisting' whenever two AF bonds coalesce. Our findings generalize to higher dimensions for codimension one defects that are sparse from the perspective of the orthogonal complement lattice. In 2D, the $\mathbb Z_2$ symmetry of the model has an IR manifestation where one can construct field strengths and Wilson loops (which characterize frustration) from fundamental plaquette variables. In the non-sparse limit where the disorder parameter is drawn from an arbitrary but homogeneously assigned probability distribution function, we recover previously found fixed distributions as special cases, but find only the trivial paramagnetic distribution to be an attractor. For non-homogeneously assigned disorder that is sufficiently dilute however, we find the Edwards-Anderson model with equal probability $\pm J$ bonds to also be an IR attractor.
We develop a unified description, via the Boltzmann equation, of damping of gravitational waves by matter, incorporating collisions. We identify two physically distinct damping mechanisms -- collisional and Landau damping. We first consider damping in flat spacetime, and then generalize the results to allow for cosmological expansion. In the first regime, maximal collisional damping of a gravitational wave, independent of the details of the collisions in the matter is, as we show, significant only when its wavelength is comparable to the size of the horizon. Thus damping by intergalactic or interstellar matter for all but primordial gravitational radiation can be neglected. Although collisions in matter lead to a shear viscosity, they also act to erase anisotropic stresses, thus suppressing the damping of gravitational waves. Damping of primordial gravitational waves remains possible. We generalize Weinberg's calculation of gravitational wave damping, now including collisions and particles of finite mass, and interpret the collisionless limit in terms of Landau damping. While Landau damping of gravitational waves cannot occur in flat spacetime, the expansion of the universe allows such damping by spreading the frequency of a gravitational wave of given wavevector.
We explore the mechanics of inflation in simplified extra-dimensional models involving an inflaton interacting with the Einstein-Maxwell system in two extra dimensions. The models are Goldilocks-like in that they are just complicated enough to include a mechanism to stabilize the extra-dimensional size, yet simple enough to solve the full 6D field equations using basic tools. The solutions are not limited to the effective 4D regime with H << m_KK (the latter referring to the mass splitting of the Kaluza-Klein excitations) because the full 6D Einstein equations are solved. This allows an exploration of inflationary physics in a controlled regime away from the usual 4D lamp-post. The inclusion of modulus stabilization is important as experience with string models teaches that this is usually what makes models fail: stabilization energies dominate the shallow potentials required by slow roll and open up directions to evolve that are steeper than those of the putative inflationary direction. We explore three representative inflationary scenarios within this simple setup. In one the radion is trapped in an inflaton-dependent local minimum whose non-zero energy drives inflation. Inflation ends as this energy relaxes to zero when the inflaton finds its minimum. The others involve power-law solutions during inflation. One is an attractor whose features are relatively insensitive to initial conditions but whose slow-roll parameters cannot be arbitrarily small; the other is not an attractor but can roll much more slowly, until eventually decaying to the attractor. These solutions can satisfy H > m_KK, but when they do standard 4D fluctuation calculations need not apply. When in a 4D regime the solutions predict eta ~ 0 hence r ~ 0.11 when n_s ~ 0.96 and so are ruled out if tensor modes remain unseen. Analysis of general parameters is difficult without a full 6D fluctuation calculation.
We find multi-scalar effective field theories (EFTs) that can achieve a slow inflationary roll despite having a scalar potential that does not satisfy the usual slow-roll condition (d V)^2 << V^2/Mp^2. They evade the usual slow-roll conditions on $V$ because their kinetic energies are dominated by single-derivative terms rather than the usual two-derivative terms. Single derivatives dominate during slow roll and so do not require a breakdown of the usual derivative expansion that underpins calculational control in much of cosmology. The presence of such terms requires some sort of UV Lorentz-symmetry breaking during inflation (besides the usual cosmological breaking). Chromo-natural inflation provides an example of a UV theory that can generate the multi-field single-derivative terms we consider, and we argue that the EFT we find indeed captures the slow-roll conditions for the background evolution for Chromo-natural inflation. We also show that our EFT can be understood as a multi-field generalization of the single-field Cuscuton models. The multi-field case introduces a new feature, however: the scalar kinetic terms define a target-space 2-form, F_ab, whose antisymmetry gives new ways for slow roll to be achieved.
Oct 26 2015
hep-th arXiv:1510.06759v1
For any given momentum transfer, gravitational interactions have a strength set by a characteristic scale $M_*$ inferred from amplitudes calculated in an effective theory with a strong coupling scale $M_{**}$. These are in general different from each other and $M_{\rm pl}$, the macroscopic strength of gravity as determined by (laboratory scale) Cavendish experiments. During single field inflation, $M_*$ can differ from $M_{\rm pl}$ due to the presence of any number of (hidden) universally coupled species between laboratory and inflationary scales. Although this has no effect on dimensionless (i.e. observable) quantities measured at a fixed scale such as the amplitude and spectral properties of the CMB anisotropies, it complicates the inference of an absolute scale of inflation given any detection of primordial tensors. In this note we review and elaborate upon these facts and address concerns raised in a recent paper.
Recently, a novel mechanism to address the hierarchy problem has been proposed \citeGraham:2015cka, where the hierarchy between weak scale physics and any putative `cutoff' $M$ is translated into a parametrically large field excursion for the so-called relaxion field, driving the Higgs mass to values much less than $M$ through cosmological dynamics. In its simplest incarnation, the relaxion mechanism requires nothing beyond the standard model other than an axion (the relaxion field) and an inflaton. In this note, we critically re-examine the requirements for successfully realizing the relaxion mechanism and point out that parametrically larger field excursions can be obtained for a given number of e-folds by simply requiring that the background break exact de Sitter invariance. We discuss several corollaries of this observation, including the interplay between the upper bound on the scale $M$ and the order parameter $\epsilon$ associated with the breaking of dS symmetry, and entertain the possibility that the relaxion could play the role of a curvaton. We find that a successful realization of the mechanism is possible with as few as $\mathcal O (10^3)$ e-foldings, albeit with a reduced cutoff $M \sim 10^6$ GeV for a dark QCD axion and outline a minimal scenario that can be made consistent with CMB observations.
All cosmological observations to date are consistent with adiabatic, Gaussian and nearly scale invariant initial conditions. These findings provide strong evidence for a particular symmetry breaking pattern in the very early universe (with a close to vanishing order parameter, $\epsilon$), widely accepted as conforming to the predictions of the simplest realizations of the inflationary paradigm. However, given that our observations are only privy to perturbations, in inferring something about the background that gave rise to them, it should be clear that many different underlying constructions project onto the same set of cosmological observables. Features in the primordial correlation functions, if present, would offer a unique and discriminating window onto the parent theory in which the mechanism that generated the initial conditions is embedded. In certain contexts, simple linear response theory allows us to infer new characteristic scales from the presence of features that can break the aforementioned degeneracies among different background models, and in some cases can even offer a limited spectroscopy of the heavier degrees of freedom that couple to the inflaton. In this review, we offer a pedagogical survey of the diverse, theoretically well grounded mechanisms which can imprint features into primordial correlation functions in addition to reviewing the techniques one can employ to probe observations. These observations include cosmic microwave background anisotropies and spectral distortions as well as the matter two and three point functions as inferred from large-scale structure and potentially, 21 cm surveys.
Observable quantities in cosmology are dimensionless, and therefore independent of the units in which they are measured. This is true of all physical quantities associated with the primordial perturbations that source cosmic microwave background anisotropies such as their amplitude and spectral properties. However, if one were to try and infer an absolute energy scale for inflation-- a priori, one of the more immediate corollaries of detecting primordial tensor modes-- one necessarily makes reference to a particular choice of units, the natural choice for which is Planck units. In this note, we discuss various aspects of how inferring the energy scale of inflation is complicated by the fact that the effective strength of gravity as seen by inflationary quanta necessarily differs from that seen by gravitational experiments at presently accessible scales. The uncertainty in the former relative to the latter has to do with the unknown spectrum of universally coupled particles between laboratory scales and the putative scale of inflation. These intermediate particles could be in hidden as well as visible sectors or could also be associated with Kaluza-Klein resonances associated with a compactification scale below the scale of inflation. We discuss various implications for cosmological observables.
The BICEP-2 team has reported the detection of primordial cosmic microwave background B-mode polarization, with hints of a suppression of power at large angular scales relative to smaller scales. Provided that the B-mode polarization is due to primordial gravitational waves, this might imply a blue tilt of the primordial gravitational wave spectrum. Such a tilt would be incompatible with standard inflationary models, although it was predicted some years ago in the context of a mechanism that thermally generates the primordial perturbations through a Hagedorn phase of string cosmology. The purpose of this note is to encourage greater scrutiny of the data with priors informed by a model that is immediately falsifiable, but which \textitpredicts features that might be favoured by the data-- namely a blue tensor tilt with an induced and complimentary red tilt to the scalar spectrum, with a naturally large tensor to scalar ratio that relates to both.
We re-examine the predictiveness of single-field inflationary models and discuss how an unknown UV completion can complicate determining inflationary model parameters from observations, even from precision measurements. Besides the usual naturalness issues associated with having a shallow inflationary potential, we describe another issue for inflation, namely, unknown UV physics modifies the running of Standard Model (SM) parameters and thereby introduces uncertainty into the potential inflationary predictions. We illustrate this point using the minimal Higgs Inflationary scenario, which is arguably the most predictive single-field model on the market, because its predictions for $A_s$, $r$ and $n_s$ are made using only one new free parameter beyond those measured in particle physics experiments, and run up to the inflationary regime. We find that this issue can already have observable effects. At the same time, this UV-parameter dependence in the Renormalization Group allows Higgs Inflation to occur (in principle) for a slightly larger range of Higgs masses. We comment on the origin of the various UV scales that arise at large field values for the SM Higgs, clarifying cut off scale arguments by further developing the formalism of a non-linear realization of $\rm SU_L(2) \times U(1)$ in curved space. We discuss the interesting fact that, outside of Higgs Inflation, the effect of a non-minimal coupling to gravity, even in the SM, results in a non-linear EFT for the Higgs sector. Finally, we briefly comment on post BICEP2 attempts to modify the Higgs Inflation scenario.
Space-filling S-branes can mediate a transition between a contracting and an expanding universe in the Einstein frame. Following up on previous work that uncovered such bouncing solutions in the context of weakly coupled thermal configurations of a certain class of type II superstrings, we set up here the formalism in which we can study the evolution of metric fluctuations across such an S-brane. Our work shows that the specific nature of the S-brane, which is sourced by non-trivial massless thermal string states and appears when the universe reaches a maximal critical temperature, allows for a scale invariant spectrum of curvature fluctuations to manifest at late times via a stringy realization of the matter bounce scenario. The finite energy density at the transition from contraction to expansion provides calculational control over the propagation of the curvature perturbations through the bounce, furnishing a working proof of concept that such a stringy universe can result in viable late time cosmology.
Heavy fields coupled to the inflaton reduce the speed of sound in the effective theory of the adiabatic mode each time the background inflationary trajectory deviates from a geodesic. This can result in features in the primordial spectra. We compute the corresponding bispectrum and show that if a varying speed of sound induces features in the power spectrum, the change in the bispectrum is given by a simple formula involving the change in the power spectrum and its derivatives. In this manner, we provide a uniquely discriminable signature of a varying sound speed for the adiabatic mode during inflation that indicates the influence of heavy fields. We find that features in the bispectrum peak in the equilateral limit and, in particular, in the squeezed limit we find considerable enhancement entirely consistent with the single field consistency relation. From the perspective of the underlying effective theory, our results generalize to a wide variety of inflationary models where features are sourced by the time variation of background quantities. A positive detection of such correlated features would be unambiguous proof of the inflaton's nature as a single light scalar degree of freedom embedded in a theory that is UV completable.
We examine the motion of light fields near the bottom of a potential valley in a multi-dimensional field space. In the case of two fields we identify three general scales, all of which must be large in order to justify an effective low-energy approximation involving only the light field, $\ell$. (Typically only one of these -- the mass of the heavy field transverse to the trough -- is used in the literature when justifying the truncation of heavy fields.) We explicitly compute the resulting effective field theory, which has the form of a $P(\ell,X)$ model, with $X = - 1/2(\partial \ell)^2$, as a function of these scales. This gives the leading ways each scale contributes to any low-energy dynamics, including (but not restricted to) those relevant for cosmology. We check our results with the special case of a homogeneous roll near the valley floor, placing into a broader context recent cosmological calculations that show how the truncation approximation can fail. By casting our results covariantly in field space, we provide a geometrical criterion for model-builders to decide whether or not the single-field and/or the truncation approximation is justified, identify its leading deviations, and to efficiently extract cosmological predictions.
We discuss and clarify the validity of effective single field theories of inflation obtained by integrating out heavy degrees of freedom in the regime where adiabatic perturbations propagate with a suppressed speed of sound. We show by construction that it is indeed possible to have inflationary backgrounds where the speed of sound remains suppressed and slow-roll persists for long enough. In this class of models, heavy fields influence the evolution of adiabatic modes in a manner that is consistent with decoupling of physical low and high energy degrees of freedom. We emphasize the distinction between the effective masses of the isocurvature modes and the eigenfrequencies of the propagating high energy modes. Crucially, we find that the mass gap that defines the high frequency modes increases with the strength of the turn, even as the naive heavy (isocurvature) and light (curvature) modes become more strongly coupled. Adiabaticity is preserved throughout, and the derived effective field theory remains in the weakly coupled regime, satisfying all current observational constraints on the resulting primordial power spectrum. In addition, these models allow for an observably large equilateral non-Gaussianity.
We discuss the implications for cosmic microwave background (CMB) observables, of a class of pre-inflationary dynamics suggested by string models where SUSY is broken due to the presence of D-branes and orientifolds preserving incompatible portions of it. In these models the would-be inflaton is forced to emerge from the initial singularity climbing up a mild exponential potential, until it bounces against a steep exponential potential of "brane SUSY breaking" scenarios, and as a result the ensuing descent gives rise to an inflationary epoch that begins when the system is still well off its eventual attractor. If a pre-inflationary climbing phase of this type had occurred within 6-7 e-folds of the horizon exit for the largest observable wavelengths, displacement off the attractor and initial-state effects would conspire to suppress power in the primordial scalar spectrum, enhancing it in the tensor spectrum and typically superposing oscillations on both. We investigate these imprints on CMB observables over a range of parameters, examine their statistical significance, and provide a semi-analytic rationale for our results. It is tempting to ascribe at least part of the large-angle anomalies in the CMB to pre-inflationary dynamics of this type.
We compute the low energy effective field theory (EFT) expansion for single-field inflationary models that descend from a parent theory containing multiple other scalar fields. By assuming that all other degrees of freedom in the parent theory are sufficiently massive relative to the inflaton, it is possible to derive an EFT valid to arbitrary order in perturbations, provided certain generalized adiabaticity conditions are respected. These conditions permit a consistent low energy EFT description even when the inflaton deviates off its adiabatic minimum along its slowly rolling trajectory. By generalizing the formalism that identifies the adiabatic mode with the Goldstone boson of this spontaneously broken time translational symmetry prior to the integration of the heavy fields, we show that this invariance of the parent theory dictates the entire non-perturbative structure of the descendent EFT. The couplings of this theory can be written entirely in terms of the reduced speed of sound of adiabatic perturbations. The resulting operator expansion is distinguishable from that of other scenarios, such as standard single inflation or DBI inflation. In particular, we re-derive how certain operators can become transiently strongly coupled along the inflaton trajectory, consistent with slow-roll and the validity of the EFT expansion, imprinting features in the primordial power spectrum, and we deduce the relevant cubic operators that imply distinct signatures in the primordial bispectrum which may soon be constrained by observations.
The computation of the primordial power spectrum in multi-field inflation models requires us to correctly account for all relevant interactions between adiabatic and non-adiabatic modes around and after horizon crossing. One specific complication arises from derivative interactions induced by the curvilinear trajectory of the inflaton in a multi-dimensional field space. In this work we compute the power spectrum in general multi-field models and show that certain inflaton trajectories may lead to observationally significant imprints of `heavy' physics in the primordial power spectrum if the inflaton trajectory turns, that is, traverses a bend, sufficiently fast (without interrupting slow roll), even in cases where the normal modes have masses approaching the cutoff of our theory. We emphasise that turning is defined with respect to the geodesics of the sigma model metric, irrespective of whether this is canonical or non-trivial. The imprints generically take the form of damped superimposed oscillations on the power spectrum. In the particular case of two-field models, if one of the fields is sufficiently massive compared to the scale of inflation, we are able to compute an effective low energy theory for the adiabatic mode encapsulating certain relevant operators of the full multi-field dynamics. As expected, a particular characteristic of this effective theory is a modified speed of sound for the adiabatic mode which is a functional of the background inflaton trajectory and the turns traversed during inflation. Hence in addition, we expect non-Gaussian signatures directly related to the features imprinted in the power spectrum.
In this work we study the effects of field space curvature on scalar field perturbations around an arbitrary background field trajectory evolving in time. Non-trivial imprints of the 'heavy' directions on the low energy dynamics arise when the vacuum manifold of the potential does not coincide with the span of geodesics defined by the sigma model metric of the full theory. When the kinetic energy is small compared to the potential energy, the field traverses a curve close to the vacuum manifold of the potential. The curvature of the path followed by the fields can still have a profound influence on the perturbations as modes parallel to the trajectory mix with those normal to it if the trajectory turns sharply enough. We analyze the dynamical mixing between these non-decoupled degrees of freedom and deduce its non-trivial contribution to the low energy effective theory for the light modes. We also discuss the consequences of this mixing for various scenarios where multiple scalar fields play a vital role, such as inflation and low-energy compactifications of string theory.
In this report, we discuss a candidate mechanism through which one might address the various cosmological constant problems. We first observe that the renormalization of gravitational couplings (induced by integrating out various matter fields) manifests non-local modifications to Einstein's equations as quantum corrected equations of motion. That is, at the loop level, matter sources curvature through a gravitational coupling that is a non-local function of the covariant d'Alembertian. If the functional form of the resulting Newton's `constant' is such that it annihilates very long wavelength sources, but reduces to $1/M^2_{pl}$ ($M_{pl}$ being the 4d Planck mass) for all sources with cosmologically observable wavelengths, we would have a complimentary realization of the degravitation paradigm-- a realization through which its non-linear completion and the corresponding modified Bianchi identities are readily understood. We proceed to consider various theories whose coupling to gravity may a priori induce non-trivial renormalizations of Newton's constant in the IR, and arrive at a class of non-local effective actions which yield a suitably degravitating filter function for Newton's constant upon subsequently being integrated out. We motivate this class of non-local theories through several considerations, discuss open issues, future directions, the inevitable question of scheme dependence in semi-classical gravitational calculations and comment on connections with other meditations in the literature on relaxing of the cosmological constant semi-classically.
It is well understood that spatial non-commutativity, if indeed realized in nature, is a phenomenon whose effects are not just felt at energy scales comparable to the non-commutativity scale. Loop effects can transmit signatures of any underlying non-commutativity to macroscopic scales (a manifestation of a phenomenon that has come to be known as UV/IR mode mixing) and offer a potential lever to constrain the amount of non-commutativity present in nature, if present at all. Field theories defined on non-commutative spaces (realized in string theory when D-branes are coupled to backgrounds of non-trivial RR background flux), can exhibit strong UV/IR mode mixing, manifesting in a non-local one loop effective action. In the context of inflation in the presence of any background non-commutativity, we demonstrate how this UV/IR mixing at the loop level can allow us to place severe constraints on the scale of non-commutativity if we presume inflation is responsible for large scale structure. We demonstrate that any amount of non-commutativity greatly suppresses the CMB power at all observable scales, \it independent of the scale of inflation, and independent of whether or not the non-commutativity tensor redshifts during inflation, therefore nullifying a very salient and successful prediction of inflation.
In this paper we revisit the formulation of scalar field theories on de Sitter backgrounds subject to the generalized uncertainty principle (GUP). The GUP arises in several contexts in string theory, but is most readily thought of as resulting from using strings as effective probes of geometry, which suggests an uncertainty relation incorporating the string scale $l_s$. After reviewing the string theoretic case for the GUP, which implies a minimum length scale $l_s$, we follow in the footsteps of Kempf and concern ourselves with how one might write down field theories which respect the GUP. We uncover a new representation of the GUP, which unlike previous studies, readily permits exact analytical solutions for the mode functions of a scalar field on de Sitter backgrounds. We find that scalar fields cannot be quantized on inflationary backgrounds with a Hubble radius $H^{-1}$ smaller than the string scale, implying a sensibly stringy (as opposed to Planckian) cutoff on the scale of inflation resulting from the GUP. We also compute $(H l_s)^2$ corrections to the two point correlation function analytically and comment on the future prospects of observing such corrections in the fortunate circumstance our universe is described by a very weakly coupled string theory.
In this report, we adopt the phenomenological approach of taking the degravitation paradigm seriously as a consistent modification of gravity in the IR, and investigate its consequences for various cosmological situations. We motivate degravitation-- where Netwon's constant is promoted to a scale dependent filter function-- as arising from either a small (resonant) mass for the graviton, or as an effect in semi-classical gravity. After addressing how the Bianchi identities are to be satisfied in such a set up, we turn our attention towards the cosmological consequences of degravitation. By considering the example filter function corresponding to a resonantly massive graviton (with a filter scale larger than the present horizon scale), we show that slow roll inflation, hybrid inflation and old inflation remain quantitatively unchanged. We also find that the degravitation mechanism inherits a memory of past energy densities in the present epoch in such a way that is likely significant for present cosmological evolution. For example, if the universe underwent inflation in the past due to it having tunneled out of some false vacuum, we find that degravitation implies a remnant `afterglow' cosmological constant, whose scale immediately afterwards is parametrically suppressed by the filter scale ($L$) in Planck units $\Lambda \sim l^2_{pl}/L^2$. We discuss circumstances through which this scenario reasonably yields the presently observed value for $\Lambda \sim O(10^{-120})$. We also find that in a universe still currently trapped in some false vacuum state, resonance graviton models of degravitation only degravitate initially Planck or GUT scale energy densities down to the presently observed value over timescales comparable to the filter scale.
String gas cosmology is rewritten in the Einstein frame. In an effective theory in which a gas of closed strings is coupled to a dilaton gravity background without any potential for the dilaton, the Hagedorn phase which is quasi-static in the string frame corresponds to an expanding, non-accelerating phase from the point of view of the Einstein frame. The Einstein frame curvature singularity which appears in this toy model is related to the blowing up of the dilaton in the string frame. However, for large values of the dilaton, the toy model clearly is inapplicable. Thus, there must be a new string phase which is likely to be static with frozen dilaton. With such a phase, the horizon problem can be successfully addressed in string gas cosmology. The generation of cosmological perturbations in the Hagedorn phase seeded by a gas of long strings in thermal equilibrium is reconsidered, both from the point of view of the string frame (in which it is easier to understand the generation of fluctuations) and the Einstein frame (in which the evolution equations are well known). It is shown that fixing the dilaton at some early stage is important in order to obtain a scale-invariant spectrum of cosmological fluctuations in string gas cosmology.
It has recently been shown that a Hagedorn phase of string gas cosmology may provide a causal mechanism for generating a nearly scale-invariant spectrum of scalar metric fluctuations, without the need for an intervening period of de Sitter expansion. A distinctive signature of this structure formation scenario would be a slight blue tilt of the spectrum of gravitational waves. In this paper we give more details of the computations leading to these results.
It has recently been shown that a Hagedorn phase of string gas cosmology can provide a causal mechanism for generating a nearly scale-invariant spectrum of scalar metric fluctuations, without the need for an intervening period of de Sitter expansion. In this paper we compute the spectrum of tensor metric fluctuations (gravitational waves) in this scenario, and show that it is also nearly scale-invariant. However, whereas the spectrum of scalar modes has a small red-tilt, the spectrum of tensor modes has a small blue tilt, unlike what occurs in slow-roll inflation. This provides a possible observational way to distinguish between our cosmological scenario and conventional slow-roll inflation.
We introduce in this paper a new framework for obtaining a period of exponential inflation that is entirely driven by the quadratic kinetic energy of a scalar field. In contrast to recent attempts to realize scalar field inflation without potentials (such as k-inflation or ghost inflation), we find that it is possible to obtain exponential inflation, without invoking any higher derivative actions, or modifying gravity, and that unlike all previous approaches, we do not require a $\rho = -p$ phase in order to realize inflation. The inflaton in our proposed framework is a scalar field with a quadratic kinetic energy term, but with the `wrong sign'. We take the perspective that this is due to some temporary instability in our system at high energies, and provide physical examples of situations where a modulus field might temprorarily exhibit such behaviour. The deflation of extra dimensions is a neccesary feature of our framework. However, unlike in previous attempts at Kaluza-Klein inflation, it is possible to obtain exponential (as opposed to power law or pole) inflation. We provide several indications of how one can gracefully exit from this type of inflation.
In this report, we study within the context of general relativity with one extra dimension compactified either on a circle or an orbifold, how radion fluctuations interact with metric fluctuations in the three non-compact directions. The background is non-singular and can either describe an extra dimension on its way to stabilization, or immediately before and after a series of non-singular bounces. We find that the metric fluctuations transfer undisturbed through the bounces or through the transients of the pre-stabilization epoch. Our background is obtained by considering the effects of a gas of massless string modes in the context of a consistent 'massless background' (or low energy effective theory) limit of string theory. We discuss applications to various approaches to early universe cosmology, including the ekpyrotic/cyclic universe scenario and string gas cosmology.
Finding a consistent way to stabilize the various moduli fields which generically appear in string theory compactifications, is essential if string theory is to make contact with the physics we see around us. We present, in this paper, a mechanism to stabilize the dilaton within a framework that has already proven itself capable of stabilizing the volume and shape moduli of extra dimensions, namely string gas cosmology. Building on previous work, which uncovered the special role played by massless F-string modes in stabilizing extra dimensions once the dilaton has stabilized, we find that the string gas cosmology of such modes also offers a consistent mechanism to stabilize the dilaton itself, given the stabilization of the extra dimensions. We then generalize the model to include D-string gases, and find that in the case of bosonic string theory, it is possible to simultaneously stabilize all the moduli we consider consistent with weak coupling. We find that our stabilization mechanism is robust, phenomenologically consistent and evades certain difficulties which might previously have seemed to generically plague moduli stabilization mechanisms, without the need for any fine tuning.
We consider the spacetime dynamics of a gas of closed strings in the context of General Relativity in a background of arbitrary spatial dimensions. Our motivation is primarily late time String Gas Cosmology, where such a spacetime picture has to emerge after the dilaton has stabilized. We find that after accounting for the thermodynamics of a gas of strings, only string modes which are massless at the self-dual radius are relevant, and that they lead to a dynamics which is qualitatively different from that induced by the modes usually considered in the literature. In the context of an ansatz with three large spatial dimensions and an arbitrary number of small extra dimensions, we obtain isotropic stabilization of these extra dimensions at the self-dual radius. This stabilization occurs for fixed dilaton, and is induced by the special string states we focus on. The three large dimensions undergo a regular Friedmann-Robertson-Walker expansion. We also show that this framework for late-time cosmology is consistent with observational bounds.
Jul 22 2004
hep-th arXiv:hep-th/0407182v3
We consider the physics of a matrix model describing D0-brane dynamics in the presence of an RR flux background. Non-commuting spaces arise as generic soltions to this matrix model, among which fuzzy spheres have been studied extensively as static solutions at finite N. The existence of topologicaly distinct static configurations suggests the possibility of D-brane topology change within this model, however a dynamical solution interpolating between topologies is still somewhat elusive. In this paper, we study this model in the limit of infinite dimensional matrices, where new solutions-- the fuzzy cylinder and the fuzzy plane among them-- appear. We argue that any dynamics which involves topology change will likely only occur in this limit, after which we study the decay of a fuzzy cylinder into an infinite collection of fuzzy spheres as both a classical and a quantum phenomenon. We conclude from this excercise that in certain limits, matrix models offer a viable framework in which to study topological dynamics, and could perhaps be a precursor to a viable theory of space-time topological dynamics.
Jun 25 2004
hep-th arXiv:hep-th/0406219v2
We wish to consider in this report the large N limit of a particular matrix model introduced by Myers describing D-brane physics in the presence of an RR flux background. At finite N, fuzzy spheres appear naturally as non-trivial solutions to this matrix model and have been extensively studied. In this report, we wish to demonstrate several new classes of solutions which appear in the large N limit, corresponding to the fuzzy cylinder,the fuzzy plane and a warped fuzzy plane. The latter two solutions arise from a possible "central extension" to our model that arises after we account for non-trivial issues involved in the large N limit. As is the case for finite N, these new solutions are to be interpreted as constituent D0-branes forming D2 bound states describing new fuzzy geometries.
We consider the effects of a gas of closed strings (treated quantum mechanically) on a background where one dimension is compactified on a circle. After we address the effects of a time dependent background on aspects of the string spectrum that concern us, we derive the energy-momentum tensor for a string gas and investigate the resulting space-time dynamics. We show that a variety of trajectories are possible for the radius of the compactified dimension, depending on the nature of the string gas, including a demonstration within the context of General Relativity (i.e. without a dilaton) of a solution where the radius of the extra dimension oscillates about the self-dual radius, without invoking matter that violates the various energy conditions. In particular, we find that in the case where the string gas is in thermal equilibrium, the radius of the compactified dimension dynamically stabilizes at the self-dual radius, after which a period of usual Friedmann-Robertson-Walker cosmology of the three uncompactified dimensions can set in. We show that our radion stabilization mechanism requires a stringy realization of inflation as scalar field driven inflation invalidates our mechanism. We also show that our stabilization mechanism is consistent with observational bounds.
The question of how perturbations evolve through a bounce in the Cyclic and Ekpyrotic models of the Universe is still a matter of ongoing debate. In this report we show that the collision between boundary branes is in most cases singular even in the full 5-D formalism, and that first order perturbation theory breaks down for at least one perturbation variable. Only in the case that the boundary branes approach each other with constant velocity shortly before the bounce, can a consistent, non singular solution be found. It is then possible to follow the perturbations explicitly until the actual collision. In this case, we find that if a scale invariant spectrum developed on the hidden brane, it will get transferred to the visible brane during the bounce.
Braneshop, and
the US National Science Foundation.
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