Testing Multi-Field Inflation: A Geometric Approach
Courtney M. Peterson, Max Tegmark
备注：
21 pages, 4 figures + tables. Revised to clarify several points and reorganized Section III for pedagogical reasons. Error in one equation and typos were corrected, as well as additional references added
We develop an approach for linking the power spectra, bispectrum, and trispectrum to the geometric and kinematical features of multifield inflationary Lagrangians. Our geometric approach can also be useful in determining when a complicated multifield model can be well approximated by a model with one, two, or a handful of fields. To arrive at these results, we focus on the mode interactions in the kinematical basis, starting with the case of no sourcing and showing that there is a series of mode conservation laws analogous to the conservation law for the adiabatic mode in single-field inflation. We then treat the special case of a quadratic potential with canonical kinetic terms, showing that it produces a series of mode sourcing relations identical in form to that for the adiabatic mode. We build on this result to show that the mode sourcing relations for general multifield inflation are extension of this special case but contain higher-order covariant derivatives of the potential and corrections from the field metric. In parallel, we show how these interactions depend on the geometry of the inflationary Lagrangian and on the kinematics of the associated field trajectory. Finally, we consider how the mode interactions and effective number of fields active during inflation are reflected in the spectra and introduce a multifield consistency relation, as well as a multifield observable that can potentially distinguish two-field scenarios from scenarios involving three or more effective fields.
A simple inflationary model based on loop quantum cosmology is considered. Within this framework, we show that inflation does not necessarily erase the infor- mation prior to its onset, but that such information may leave its imprint in the energy-spectrum of the gravitational-waves generated at these earliest of times.
The scale of small-field inflation cannot be constrained via primordial gravitational waves through measurement of tensor-to-scalar ratio $r$. In this study, I show that if cosmic strings are produced after symmetry breaking at the end of hilltop supernatural inflation, this small-field inflation model can be tested through the production of gravitational waves from cosmic strings. Future experiments of gravitational wave detectors will determine or further constrain the parameter space in the model.
Testing the so-called consistency relations plays an important role for distinguishing the different classes of inflation models. In this paper, we investigate the possible testing based on the potential observations of the cosmic microwave background (CMB) radiation, including the planned CMBPol mission and the ideal CMB experiment where only the reduced cosmic weak lensing contamination for the B-mode polarization is considered. We find that for the canonical single-field inflation and the phantom inflation, the consistency relations are quite hard to be tested: the testing is possible only …
Even simple inflationary scenarios have many free parameters. Beyond the variables appearing in the inflationary action, these include dynamical initial conditions, the number of fields, and couplings to other sectors. These quantities are often ignored but cosmological observables can depend on the unknown parameters. We use Bayesian networks to account for a large set of inflationary parameters, deriving generative models for the primordial spectra that are conditioned on a hierarchical set of prior probabilities describing the initial conditions, reheating physics, and other free parameters…
Westudythebehaviourofisocurvatureperturbationsinnon-linearsigmamodels which naturally emerge in supergravity and string inflationary scenarios. We focus on the case of negatively curved field manifolds which can potentially lead to a geometrical destabilisation of isocurvature modes. We find however that heavy fields are stable when their effective mass is computed on the attractor background solution. On the other hand, we show that ultra-light fields can potentially suffer from a geometrical instability when the background trajectory is a geodesic in field space. In this case a full understa…
We investigate multi-field signatures of the nonminimally coupled supersymmetric Higgs inflation-type cosmological scenario, focusing on the two-field Higgs-lepton inflation model as a concrete example. This type of inflationary model is realized in a theory beyond the Standard Model embedded in supergravity with a noncanonical Kähler potential. We employ the backward $δN$ formalism to compute cosmological observables, including the scalar and tensor power spectra, the spectral indices, the tensor-to-scalar ratio and the local-type nonlinearity parameter. The trajectory of the inflaton is cont…
The inflationary models developed in presence of a background radiation can be a solution to the reheating problem faced by common cold (isentropic) inflationary scenario. A D-brane system comprising of k Neuvo-Schwarz(NS) 5-branes with a transverse circle and BPS D3-branes with world volume parallel to the NS 5-branes, placed at a point on the transverse circle diametrically opposite to the NS 5-branes has a point of unstable equilibrium and the D3 brane has a geometric tachyonic mode associated with displacement of the brane along the circle. Cold inflationary scenario has been studied in co…
Multi-field models of inflation predict an inequality between the amplitude tauNL of the collapsed limit of the four-point correlator of the primordial curvature perturbation and the amplitude fNL of the squeezed limit of its three-point correlator. While a convincing detection of non-Gaussianity through the squeezed limit of the three-point correlator would rule out all single-field models, a robust confirmation or disproval of the inequality between tauNL and fNL would provide crucial information about the validity of multi-field models of inflation. In this paper, we discuss to which extent…
We consider a model of two-field inflation, containing an ordinary scalar field and a DBI field. We work beyond the slow-roll approximation, but we assume a separable Hubble parameter. We then derive the form of potential in this framework and study the spectrum of the primordial perturbations in details. We also study the amplitude of the non-Gaussianity of the primordial perturbations both in equilateral and orthogonal configurations in this setup. We test the model with recent observational data and find some constraints on the model parameters. Our study shows that for some ranges of the D…
Split supersymmetry (SUSY) -- in which SUSY is relevant to our universe but largely inaccessible at current accelerators -- has become increasingly plausible given the absence of new physics at the LHC, the success of gauge coupling unification, and the observed Higgs mass. Indirect probes of split SUSY such as electric dipole moments (EDMs) and flavor violation offer hope for further evidence but are ultimately limited in their reach. Inflation offers an alternate window into SUSY through the direct production of superpartners during inflation. These particles are capable of leaving imprints …
Here we test the predictions of the theory of the origin of the universe from the landscape multiverse, against the 2015 Planck data, for the case of the Hilltop class of inflationary models, for $p=4$ and $p=6$. By considering the quantum entanglement correction of the multiverse, we can place just a lower limit on the local 'SUSY-breaking' scale, respectively $b>8.7\times10^6 GeV$ at $95 \%$ c.l. and $b>1.3\times10^8 GeV$ at $95 \%$ c.l. from Planck TT+lowP, so the case with multiverse correction is statistically indistinguishable from the case with an unmodified inflation. We find that the …
We study the geometrical properties of scale-invariant two-field models of inflation. In particular, we show that when the field-derivative space in the Einstein frame is maximally symmetric during inflation, the inflationary predictions can be universal and independent of the details of the theory.
We study the geometrical instability arising in multi-field models of inflation with negatively-curved field space. We analyse how the homogeneous background evolves in presence of geometrical destabilisation, and show that, in simple models, a kinematical backreaction effect takes place that shuts off the instability. We also follow the evolution of the unstable scalar fluctuations. We show that they assist the kinematical backreaction while remaining in the perturbative regime. We conclude that, in the simplest models of geometrical destabilisation, inflation does not end prematurely, but ra…
We use the most recent cosmic microwave background (CMB) data to perform a Bayesian statistical analysis and discuss the observational viability of inflationary models with a non-minimal coupling,~$ξ$, between the inflaton field and the Ricci scalar. We particularize our analysis to two examples of small and large field inflationary models, namely, the Coleman-Weinberg and the chaotic quartic potentials. We find that (i) the $ξ$ parameter is closely correlated with the primordial amplitude; (ii) although improving the agreement with the CMB data in the $r - n_s$ plane, where $r$ is the tensor-…
If observations confirm BICEP2's claim of a tensor-scalar ratio $r\approx 0.2$ on CMB scales, then the inflationary consistency relation $n_{t}=-r/8$ predicts a small negative value for the tensor spectral index $n_t$. We show that future CMB polarization experiments should be able to confirm this prediction at several sigma. We also show how to properly extend the consistency relation to solar system scales, where the primordial gravitational wave density $Ω_{gw}$ could be measured by proposed experiments such as the Big Bang Observer. This would provide a far more stringent test of the consi…
In this paper we investigate the cosmological dynamics of geometric inflation by means of the tools of the dynamical systems theory. We focus in the study of two explicit models where it is possible to sum the infinite series of higher curvature corrections that arises in the formalism. These would be very interesting possibilities since, if regard gravity as a quantum effective theory, a key feature is that higher powers of the curvature invariants are involved at higher loops. Hence, naively, consideration of the whole infinite tower of curvature invariants amounts to consideration of all of…
We study the dynamics of a generalized inflationary model in which both the scalar field and its derivatives are coupled to the gravity. We consider a general form of the nonminimal derivative coupling in order to have a complete treatment of the model. By expanding the action up to the second order in perturbation, we study the spectrum of the primordial modes of the perturbations. Also, by expanding the action up to the third order and considering the three point correlation functions, the amplitude of the non-Gaussianity of the primordial perturbations is studied both in equilateral and ort…
We discuss a general bound on the possibility to realise inflation in any minimal supergravity with F-terms. The derivation crucially depends on the sGoldstini, the scalar field directions that are singled out by spontaneous supersymmetry breaking. The resulting bound involves both slow-roll parameters and the geometry of the Kähler manifold of the chiral scalars. We analyse the inflationary implications of this bound, and in particular discuss to what extent the requirements of single field and slow-roll can both be met in F-term inflation.
We show the existence of a general mechanism by which heavy scalar fields can be destabilized during inflation, relying on the fact that the curvature of the field space manifold can dominate the stabilizing force from the potential and destabilize inflationary trajectories. We describe a simple and rather universal setup in which higher-order operators suppressed by a large energy scale trigger this instability. This phenomenon can prematurely end inflation, thereby leading to important observational consequences and sometimes excluding models that would otherwise perfectly fit the data. More…
We analyse Hybrid Natural Inflation in view of the recent results for the tensor index reported by BICEP2. We find that it predicts a large running of the scalar spectrum which is potentially detectable by large scale structure and $21\, \mathrm{cm} $ observations. The running of the running is also relatively large becoming close to $10^{-2}$. Along the way, we find general consistency relations at which observables are subject if the slow-roll approximation is imposed. Failure to satisfy these equations by the values obtained for the observables in surveys would be a failure of the slow-roll…
Effective supergravity inflationary models induced by anti-D3 brane interaction with the moduli fields in the bulk geometry have a geometric description. The Kähler function carries the complete geometric information on the theory. The non-vanishing bisectional curvature plays an important role in the construction. The new geometric formalism, with the nilpotent superfield representing the anti-D3 brane, allows a powerful generalization of the existing inflationary models based on supergravity. They can easily incorporate arbitrary values of the Hubble parameter, cosmological constant and grav…
We derive semi-analytic formulae for the power spectra of two-field inflation assuming an arbitrary potential and non-canonical kinetic terms, and we use them both to build phenomenological intuition and to constrain classes of two-field models using WMAP data. Using covariant formalism, we first develop a framework for understanding the background field kinematics and introduce a "slow-turn" approximation. Next, we find covariant expressions for the evolution of the adiabatic/curvature and entropy/isocurvature modes, and we discuss how the mode evolution can be inferred directly from the back…
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We show that a new class of helical phase inflation models can be simply realized in minimal supergravity, wherein the inflaton is the phase component of a complex field and its potential admits a deformed helicoid structure. We find a new unique complex-valued index $χ$ that characterizes almost the entire region of the $n_s-r$ plane favored by new Planck observations. Continuously varying the index $χ$, predictions interpolate from quadratic/natural inflation parameterized by a phase/axion decay constant to Starobinsky-like inflation parameterized by the $α$-parameter. We demonstrate that th…
We perform a general analysis of the cosmological viability of Geometric Inflation. We show that the evolution of the universe, from inflation to the present day, can be seen from the addition of an infinite tower of curvature invariants into the Hilbert-Einstein action. The main epochs of the Universe can be reproduced: Inflation, Big Bang Nucleosynthesis, and Late-time acceleration driven by the cosmological constant. The slow-roll condition is a robust prediction of the theory. Inflation possesses a graceful exit with enough number of $e-$folds between the limit imposed by the Planck densit…
Cosmic inflation provides a mechanism for generating the early density perturbations that seeded the large-scale structures we see today. Primordial non-Gaussianity is among the most promising of few observational tests of physics at this epoch. At present non-Gaussianity is best constrained by the cosmic microwave background, but in the near term large-scale structure data may be competitive so long as the effects of primordial non-Gaussianity can be modeled through the non-linear process of structure formation. We discuss recent work modeling effects of a few types of primordial non-Gaussian…
A new effect is described by which primordial gravity waves leave a permanent signature in the large scale structure of the Universe. The effect occurs at second order in perturbation theory and is sensitive to the order in which perturbations on different scales are generated. We derive general forecasts for the detectability of the effect with future experiments, and consider observations of the pre-reionization gas through the 21 cm line. It is found that the Square Kilometre Array will not be competitive with current cosmic microwave background constraints on primordial gravity waves from …
We present an up to cubic curvature correction to General Relativity with the following features: (i) its vacuum spectrum solely consists of a graviton and is ghost-free, (ii) it possesses well-behaved black hole solutions which coincide with those of Einsteinian cubic gravity, (iii) its cosmology is well-posed as an initial value problem and, most importantly, (iv) it entails a geometric mechanism triggering an inflationary period in the early universe (driven by radiation) with a graceful exit to a late-time cosmology arbitrarily close to $Λ$CDM.
An inflationary scenario is expected to be embedded into an ultraviolet (UV) complete theory such as string theory. The effect of UV complete theories may appear as nontrivial kinetic terms in the low energy effective field theory, which provides a nontrivial geometry in field space. In this paper, we study the effect of the geometry of multi-form-field space on an inflationary scenario. In particular, we focus on the geometric destabilization mechanism which induces the phase transition from the conventional slow-roll inflation to a novel inflationary scenario. Anisotropic inflation is a typi…
In this thesis we will primarily demonstrate how additional scalar degrees of freedom - which are motivated from many high-energy embeddings - open up new observational windows onto the physics of inflation. We construct a Bayesian framework to statistically compare models with additional fields given the current astronomical data. Putting inflation to the test, we perform our analysis on the quadratic curvaton accompanying a range of inflationary potentials, where we find that only one potential remains as a viable candidate. Furthermore, if the curvaton mechanism were to be confirmed by futu…
It is widely believed that the leading secular loop corrections from quantum gravity can be subsumed into a coordinate redefinition. Hence the apparent infrared logarithm corrections to any quantity would be just the result of taking the expectation value of the tree order quantity at the transformed coordinates in the graviton vacuum. We term this the Transformation Ansatz and we compare its predictions against explicit one loop computations in Maxwell + Einstein and Dirac + Einstein on de Sitter background. In each case the ansatz fails.
Inflation scenarios kinetically coupled with the Einstein tensor have been widely studied. They can be consistent with current observational data. Future experiments on the measurement on CMB distortions will potentially extend information about the scalar spectrum to small scales $1 \Mpc^{-1} \lesssim k \lesssim 2 \times 10^4 \Mpc^{-1}$. By taking the sensitivity of the PIXIE experiment as the criterion, we perform a model-oriented analysis of the observational prospects of spectral distortions for kinetically coupled inflation. There are five models that possibly generate a detectable level …
Testing the Starobinsky model of inflation with resonant cavities
Indian Institute of Technology Kanpur（坎普尔印度理工学院）；Indian Institute of Science Education and Research Bhopal（博帕尔印度科学教育与研究机构）；Indian Institute of Technology Bombay（孟买印度理工学院）
By means of Bayesian techniques, we study how a premature ending of inflation, motivated by geometrical destabilization, affects the observational evidences of typical inflationary models. Large field models are worsened, and inflection point potentials are drastically improved for a specific range of the field-space curvature characterizing the geometrical destabilization. For other models we observe shifts in the preferred values of the model parameters. For quartic hilltop models for instance, contrary to the standard case, we find preference for theoretically natural sub-Planckian hill wid…
The statistics of primordial curvature fluctuations are our window into the period of inflation, where these fluctuations were generated. To date, the cosmic microwave background has been the dominant source of information about these perturbations. Large scale structure is however from where drastic improvements should originate. In this paper, we explain the theoretical motivations for pursuing such measurements and the challenges that lie ahead. In particular, we discuss and identify theoretical targets regarding the measurement of primordial non-Gaussianity. We argue that when quantified i…
Perhaps the most controversial idea in modern cosmology is that our observable universe is contained within one bubble among many, all inhabiting the eternally inflating multiverse. One of the few way to test this idea is to look for evidence of the relic inhomogeneities left by the collisions between other bubbles and our own. Such relic inhomogeneities induces a coherent bulk flow over gigaparsec scales. Therefore, bubble collisions leave unique imprints in the cosmic microwave background (CMB) through the kinetic Sunyaev Zel'dovich (kSZ) effect, temperature anisotropies induced by the scatt…
We present a cosmological model arising from a gravitational theory with an infinite tower of higher-order curvature invariants that can reproduce the entire evolution of the Universe: from inflation to late-time acceleration, without invoking an inflaton nor a cosmological constant. The theory is Einsteinian-like. The field equations for a Friedmann-Lemaître-Robertson-Walker metric are of second-order and can reproduce a late-time evolution that is consistent with the acceleration provided by the cosmological constant at low redshift. Our results force us to reinterpret the nature of dark ene…
Prior to recombination, Silk damping causes the dissipation of energy from acoustic waves into the monopole of the Cosmic Microwave Background (CMB), resulting in spectral distortions. These can be used to probe the primordial scalar power spectrum on smaller scales than it is possible with CMB anisotropies. An enhancement of power on these scales is nevertheless required for the resulting distortions to be detectable by future experiments like PIXIE. In this paper, we examine all 49 single-field inflation models listed by Martin et al. in the Encyclopaedia Inflationaris [1] and find that only…
We provide a simple transfer function that determines the effect of an early matter dominated era on the gravitational wave background and show that a large class of compactifications of superstring theory might be tested by observations of the gravitational wave background from inflation. For large enough reheating temperatures $> 10^9 \GeV$ the test applies to all models containing at least one scalar with mass $< 10^{12}\GeV$ that acquires a large initial oscillation amplitude after inflation and has only gravitational interaction strength, i.e., a field with the typical properties of a mod…
Many extensions of Einstein's theory of gravity have been studied and proposed with various motivations like the quest for a quantum theory of gravity to extensions of anomalies in observations at the solar system, galactic and cosmological scales. These extensions include adding higher powers of Ricci curvature $R$, coupling the Ricci curvature with scalar fields and generalized functions of $R$. In addition when viewed from the perspective of Supergravity (SUGRA) many of these theories may originate from the same SUGRA theory interpreted in different frames. SUGRA therefore serves as a good …
School of Physics, Sun Yat-Sen University（中山大学物理学院）；Department of Physics, Babeş-Bolyai University（巴比什-博雅伊大学物理系）；Astronomical Observatory（天文台）；Key Laboratory of Atomic and Subatomic Structure and Quantum Control (Ministry of Education), Guangdong Basic Research Center of Excellence for Structure and Fundamental Interactions of Matter, School of Physics, South China Normal University（华南师范大学物理学院原子与亚原子结构及量子控制教育部重点实验室、广东省物质结构与基本相互作用卓越基础研究中心）；Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, Guangdong-Hong Kong Joint Laboratory of Quantum Matter, South China Normal University（华南师范大学广东省量子工程与量子材料重点实验室、粤港量子物质联合实验室）
According to the famous Lyth bound, one can confirm large field inflation by finding tensor modes with sufficiently large tensor-to-scalar ratio $r$. Here we will try to answer two related questions: Is it possible to rule out all large field inflationary models by not finding tensor modes with $r$ above some critical value, and what can we say about the scale of inflation by measuring $r$? However, in order to answer these questions one should distinguish between two different definitions of the large field inflation and three different definitions of the scale of inflation. We will examine t…
We present a model of $F(R)$ gravity in the presence of a string theory motivated misalignment axion like particle materialized in terms of a canonical scalar field minimally coupled with gravity, and we study the cosmological phenomenology of the model, emphasizing mainly on the late-time era. The main result of the paper is that inflation and the dark energy era may be realized in a geometric way by an $F(R)$ gravity, while the axion is the dark matter constituent of the Universe. The $F(R)$ gravity model consists of an $R^2$ term, which as we show dominates the evolution during the early ti…
The nature of the scalar field responsible for the cosmological inflation, the \qo{inflaton}, is found to be rooted in the most fundamental concept of the Weyl's differential geometry: the parallel displacement of vectors in curved space-time. The Euler-Lagrange theory based on a scalar-tensor Weyl-Dirac Lagrangian leads straightforwardly to the Einstein equation admitting as a source the characteristic energy-momentum tensor of the inflaton field. Within the dynamics of the inflation, e.g. in the slow roll transition from a \qo{false} toward a \qo{true vacuum}, the inflaton's geometry implies…
In this paper we discuss a spacetime having the topology of S^3 x R but with a different smoothness structure leading to a geometric model for inflation, called geometric inflation. In particular this spacetime is not globally hyperbolic and we obtain a time line with a spatial topology change from the 3-sphere to a homology 3-sphere and back. The topology of the spacetime remains invariant. Among the infinite possible smoothness structures of this spacetime, we choose a homology 3-sphere constructed from the knot 8_{10} with hyperbolic geometry, i.e. admitting a homogenous metric of negative …
Stochastic effects during inflation can be addressed by averaging the quantum inflaton field over Hubble-patch sized domains. The averaged field then obeys a Langevin-type equation into which short-scale fluctuations enter as a noise term. We solve the Langevin equation for a inflaton field with Dirac Born Infeld (DBI) kinetic term perturbatively in the noise and use the result to determine the field value's Probability Density Function (PDF). In this calculation, both the shape of the potential and the warp factor are arbitrary functions, and the PDF is obtained with and without volume effect…
Inspired by quantum gravitational physics, the approach of non-commutative (NC) phase space leads to a modified dispersion relation of gravitational waves. This feature, if applied to the very early universe, gives rise to a modified power spectrum of primordial tensor perturbations with a suppression of power on large scales. We confront this phenomenon with the BICEP2 and Planck experiments, and show that inflation with the modified dispersion relation can simultaneously fit the observations better than the standard inflationary paradigm. In particular, the numerical result implies that with…
We derive semi-analytic formulae for the local bispectrum and trispectrum in general two-field inflation and provide a simple geometric recipe for building observationally allowed models with observable non-Gaussianity. We use the δN formalism and the transfer function formalism to express the bispectrum almost entirely in terms of model-independent physical quantities. Similarly, we calculate the trispectrum and show that the trispectrum parameter τNL can be expressed entirely in terms of spectral observables, which provides a new consistency relation unique to two-field inflation. We show th…
We explore the Gravitational Waves (GW) phenomenology of a simple class of supergravity models that can explain and unify inflation and Primordial Black Holes (PBH) as Dark Matter (DM). Our (modified) supergravity models naturally lead to a two-field attractor-type double inflation, whose first stage is driven by Starobinsky scalaron and the second stage is driven by another scalar belonging to a supergravity multiplet. The PBHs formation in our supergravity models is efficient, compatible with all observational constraints, and predicts a stochastic GW background. We compute the PBH-induced G…
It is shown that Starobinsky-like potentials can be realized in non-geometric flux compactifications of string theory, where the inflaton involves an axion whose shift symmetry can protect UV-corrections to the scalar potential. For that purpose we evaluate the backreacted, uplifted F-term axion-monodromy potential, which interpolates between a quadratic and a Starobinsky-like form. Limitations due to the requirements of having a controlled approximation of the UV theory and of realizing single-field inflation are discussed.
We review the effects of heavy scalar fields during inflation in the framework of $\mathcal N = 1$ supergravity. Such heavy scalars can be geometric moduli from string compactifications or stabilizer fields from a different sector of the theory. Even when these fields are heavier than the Hubble scale during inflation, they generically cause backreactions which alter the dynamics of the system. Severe problems may arise when the heavy fields break supersymmetry, which is quite generic for Kähler moduli. We illustrate these effects in two examples, chaotic inflation and Starobinsky-like inflati…
If during inflation the inflaton couples to a "fossil" field, some new scalar, vector, or tensor field, it typically induces a scalar-scalar-fossil bispectrum. Even if the fossil field leaves no direct physical trace after inflation, it gives rise to correlations between different Fourier modes of the curvature or, equivalently, a nonzero curvature trispectrum, but without a curvature bispectrum. Here we quantify the effects of a fossil field on the cosmic microwave background (CMB) temperature fluctuations in terms of bipolar spherical harmonics (BiPoSHs). The effects of vector and tensor fos…
