We study the running of power spectra in inflationary cosmology as a renormalization-group flow from the de Sitter fixed point. The beta function is provided by the equations of the background metric. The spectra of the scalar and tensor fluctuations obey RG evolution equations with vanishing anomalous dimensions in the superhorizon limit. By organizing the perturbative expansion in terms of leading and subleading logs, we calculate the spectral indices, their runnings, the runnings of the runnings, etc., to the next-to-leading log order in quantum gravity with fakeons (i.e., the theory $R+R^{2}+C^{2}$ with the fakeon prescription/projection for $C^{2}$). We show that these quantities are related to the spectra in a universal way. We also compute the first correction to the relation $r=-8n_{T}$ and provide a number of quantum gravity predictions that can be hopefully tested in the forthcoming future.
The possibility to construct an inflationary scenario for renormalization-group improved potentials corresponding to the Higgs sector of quantum field models is investigated. Taking into account quantum corrections to the renormalization-group potential which sums all leading logs of perturbation theory is essential for a successful realization of the inflationary scenario, with very reasonable parameters values. The scalar electrodynamics inflationary scenario thus obtained are seen to be in good agreement with the most recent observational data.
In single field slow-roll inflation, one expects that the spectral index $n_s -1$ is first order in slow-roll parameters. Similarly, its running $α_s = dn_s/d \log k$ and the running of the running $β_s = dα_s/d \log k$ are second and third order and therefore expected to be progressively smaller, and usually negative. Hence, such models of inflation are in considerable tension with a recent analysis hinting that $β_s$ may actually be positive, and larger than $α_s$. Motivated by this, in this work we ask the question of what kinds of inflationary models may be useful in achieving such a hiera…
Fixing the Renormalization of Inflationary Loops via Ward Identities
Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, University of Chinese Academy of Sciences（中国科学院大学杭州高等研究院基础科学与数学学院）
In this paper, we carefully calculated the tensor-to-scalar ratio, the running spectral index, and the running of running spectrum for (extra) natural inflation in order to compare with recent BICEP2 data, PLANCK satellite data and future 21 cm data. We discovered that the prediction for running spectral index and the running of running spectrum in natural inflation is different from that in the case of extra natural inflation. Near future observation for the running spectral index may achieve enough accuracy to allow us distinguishing between extra natural inflation from natural inflation. Di…
We compute the running of the scalar spectral index in general multi-field slow-roll inflation. By incorporating explicit momentum dependence at the moment of horizon crossing, we can find the running straightforwardly. At the same time, we can distinguish the contributions from the quasi de Sitter background and the super-horizon evolution of the field fluctuations.
We point out that if quantum field renormalization is taken into account the predictions of slow-roll inflation for both the scalar and tensorial power spectra change significantly for wavelengths that today are at observable scales
We extend previous classifications of inflationary models by means of their behaviour at large-N, where N is the number of e-foldings. In addition to the perturbative 1/N case, whose slow-roll parameters fall off as powers of 1/N, we introduce the constant, non-perturbative and logarithmic classes. This covers the large majority of inflationary models. Furthermore, we calculate the running of the spectral tilt for all these classes. Remarkably, we find that the tilt's runnings essentially cluster around the permil level. We comment on the implications for future experiments.
The detection of B-mode power spectrum by the BICEP2 collaboration constrains the tensor-to-scalar ratio $r=0.20^{+0.07}_{-0.05}$ for the lensed-$Λ$CDM model. The consistency of this big value with the {\em Planck} results requires a large running of the spectral index. The large values of the tensor-to-scalar ratio and the running of the spectral index put a challenge to single field inflation. For the chaotic inflation, the larger the value of the tensor-to-scalar ratio is, the smaller the value of the running of the spectral index is. For the natural inflation, the absolute value of the run…
Employing an effective field theory approach to inflationary perturbations, we analyze in detail the effect of curvature-generated Lagrangian operators on various observables, focusing on their running with scales. At quadratic order, we solve the equation of motion at next-to-leading leading order in a generalized slow-roll approximation for a very general theory of single-field inflation. We derive the resulting power spectrum, its tilt and running. We then focus on the contribution to the primordial non-Gaussianity amplitude f_{NL} sourced by a specific interaction term. We show that the ru…
The BICEP2 experiment confirms the existence of primordial gravitational wave with the tensor-to-scalar ratio $r=0$ ruled out at $7σ$ level. The consistency of this large value of $r$ with the {\em Planck} data requires a large negative running $n'_s$ of the scalar spectral index. Herein we propose two types of the single field inflation models with simple potentials to study the possibility of the consistency of the models with the BICEP2 and {\em Planck} observations. One type of model suggested herein is realized in the supergravity model building. These models fail to provide the needed $n…
If a sizeable tensor-to-scalar ratio ~0.1 turns out to be detected and a negative running of the scalar spectral index ~0.01 is significantly required by the data, the vast majority of single field models of inflation will be ruled out. We show that a flat tree-level effective potential, lifted by radiative corrections and by the imprints of a high energy scale (in the form of non-renormalizable operators) can explain those features and produce enough inflation in the slow-roll regime.
We derive one- and two-loop renormalization group equations (RGEs) of Higgs-$R^2$ inflation. This model has a non-minimal coupling between the Higgs and the Ricci scalar and a Ricci scalar squared term on top of the standard model. The RGEs derived in this paper are valid as long as the energy scale of interest (in the Einstein frame) is below the Planck scale. We also discuss implications to the inflationary predictions and the electroweak vacuum metastability.
Inflationary cosmology proposes that the early Universe undergoes accelerated expansion, driven, in simple scenarios, by a single scalar field, or inflaton. The form of the inflaton potential determines the initial spectra of density perturbations and gravitational waves. We show that constraints on the duration of inflation together with the BICEP3/Keck bounds on the gravitational wave background imply that higher derivatives of the potential are nontrivial with a confidence of 99%. Such terms contribute to the scale-dependence, or running, of the density perturbation spectrum. We clarify the…
ACT Data and Positive Running of the Spectral Index for Scalar Theory and Modified Gravity
ICREA（加泰罗尼亚高级研究院）；Institute of Space Sciences (IEEC-CSIC)（空间科学研究所）；Department of Physics, Aristotle University of Thessaloniki（萨洛尼卡亚里士多德大学物理系）；Center for Theoretical Physics, Khazar University（阿塞拜疆可汗大学理论物理中心）
We propose a class of inflation models in which the coefficient of the inflaton kinetic term rapidly changes with energy scale. This may occur especially if the inflaton moves over a long distance during inflation as in the case of large-scale inflation. The peculiar behavior of the kinetic term opens up a new way to construct an inflation model. As a concrete example we construct a linear inflation model in supergravity. It is straightforward to build a chaotic inflation model with a fractional power along the same line. Interestingly, the potential takes a different form after inflation beca…
Inflation models can be examined by the cosmological observations, WMAP, Planck, BICEP2 and so on. These observations directly constrain the spectral index, $n_s$, and the tensor-to-scalar ratio, $r$. Besides, from a theoretical point of view, it has been shown that any inflation models asymptote a universal attractor in $(n_s,r)$ plane for a larger scalar-gravity coupling. In this work we consider a simple chaotic inflation model with a scalar quartic and a scalar-curvature interactions. The quantum corrections are introduced for these interactions through the renormalization group. The infla…
We propose to slightly generalize the DeWitt-Schwinger adiabatic renormalization subtractions in curved space to include an arbitrary renormalization mass scale $μ$. The new predicted running for the gravitational couplings are fully consistent with decoupling of heavy massive fields. This is a somewhat improvement with respect to the more standard treatment of minimal (DeWitt-Schwinger) subtractions via dimensional regularization. We also show how the vacuum metamorphosis model emerges from the running couplings.
We consider 4D quantum-dilaton gravity with the most general coupling in a homogeneous and isotropic universe, especially an inflationary one, which is essentially characterized by an exponentially expanding scale factor with time. We show that on the inflationary background this theory can be miraculously renormalized, at least at the one-loop level, which must be an effective theory during the inflation of the un-constructed complete quantum theory of gravity.
We study possibilities to explain the whole dark matter abundance by primordial black holes (PBHs) or to explain the merger rate of binary black holes estimated from the gravitational wave detections by LIGO/Virgo. We assume that the PBHs are originated in a radiation- or matter-dominated era from large primordial curvature perturbation generated by inflation. We take a simple model-independent approach considering inflation with large running spectral indices which are parametrized by $n_\text{s}, α_\text{s}$, and $β_\text{s}$ consistent with the observational bounds. The merger rate is fitte…
In this paper we operate under the assumption that no tensors from inflation will be measured in the future by the dedicated experiments and argue that, while for single-field slow-roll models of inflation the running of the spectral index will be hard to be detected, in multi-field models the running can be large due to its strong correlation with non-Gaussianity. A detection of the running might therefore be related to the presence of more than one active scalar degree of freedom during inflation.
As a contribution to a viable candidate for a standard model of cosmology, we here show that pre-inflationary quantum fluctuations can provide a scenario for the long-sought initial conditions for the inflaton field. Our proposal is based on the assumption that at very high energies (higher than the energy scale of inflation) the vacuum-expectation value (VeV) of the field is trapped in a false vacuum and then, due to renormalization-group (RG) running, the potential starts to flatten out toward low energy, eventually tending to a convex one which allows the field to roll down to the true vacu…
An intriguing hypothesis is that gravity may be non-perturbatively renormalizable via the notion of asymptotic safety. We show that the Higgs sector of the SM minimally coupled to asymptotically safe gravity can generate the observed near scale-invariant spectrum of the Cosmic Microwave Background through the curvaton mechanism. The resulting primordial power spectrum places an upper bound on the Higgs mass, which for finely tuned values of the curvaton parameters, is compatible with the recently released Large Hadron Collider data.
Dynamics of eternal inflation on the landscape admits description in terms of the Martin-Siggia-Rose (MSR) effective field theory that is in one-to-one correspondence with vacuum dynamics equations. On those sectors of the landscape, where transport properties of the probability measure for eternal inflation are important, renormalization group fixed points of the MSR effective action determine late time behavior of the probability measure. I argue that these RG fixed points may be relevant for the solution of the gauge invariance problem for eternal inflation.
We describe the primeval inflationary phase of the early Universe within a quantum field theoretical (QFT) framework that can be viewed as the effective action of vacuum decay in the early times. Interestingly enough, the model accounts for the "graceful exit" of the inflationary phase into the standard radiation regime. The underlying QFT framework considered here is Supergravity (SUGRA), more specifically an existing formulation in which the Starobinsky-type inflation (de-Sitter background) emerges from the quantum corrections to the effective action after integrating out the gravitino field…
We examine renormalization of Higgs inflation in the context of the full Standard Model. In the fermionic sector of the theory there is a parametrically large top-Goldstone coupling which prevents renormalization of the theory. Using a simplified model with a global U(1) symmetry, a Higgs and a fermion, we show that the one-loop contribution to 4-Goldstone scattering cannot be absorbed in any tree level terms, and hence forbids a consistent renormalization of the theory. Our results apply for large non-minimal Higgs-gravity coupling in the large field regime, and indicate that Higgs inflation …
We study 1-loop corrections to the primordial stochastic background of gravitational waves produced during inflation. While in single-clock, at the leading order in slow-roll, quantum corrections keep the amplitude scale-free this is not the case when the pattern of symmetry breaking is different. In particular, when spatial diffeomorphisms are also broken during inflation as for solid inflation, a log-running in the external momentum is generated. We relate the appearance of a log-running to the spontaneous breaking of dilatation invariance. The running could be instrumental to distinguish si…
We study primordial cosmology with two scalar fields that participate in inflation at the same time, by coupling quantum gravity (i.e., the theory $R+R^{2}+C^{2}$ with the fakeon prescription/projection for $C^{2}$) to a scalar field with a quadratic potential. We show that there exists a perturbative regime that can be described by an asymptotically de Sitter, cosmic RG flow in two couplings. Since the two scalar degrees of freedom mix in nontrivial ways, the adiabatic and isocurvature perturbations are not RG invariant on superhorizon scales. It is possible to identify the correct perturbati…
In scale-invariant models of fundamental physics, mass scales are generated by spontaneous symmetry breaking. In this work, we study inflation in scale-invariant $R^2$ gravity, in which the Planck mass is generated by a scalar field, which is responsible for spontaneous breaking of scale--symmetry. If the self-interactions of the scalar field are non-zero, a cosmological constant is generated, which can be potentially quite large. To avoid fine-tuning at late times, we introduce another scalar field which drives the classical cosmological constant to zero during inflation. Working in the Einst…
We study inflation as a "cosmic" renormalization-group flow. The flow, which encodes the dependence on the background metric, is described by a running coupling $α$, which parametrizes the slow roll, a de Sitter free, analytic beta function and perturbation spectra that are RG invariant in the superhorizon limit. Using RG invariance as a guiding principle, we classify the main types of flows according to the properties of their spectra, without referring to their origins from specific actions or models. Novel features include spectra with essential singularities in $α$ and violations of the re…
We revisit large field inflation models with modulations in light of the recent discovery of the primordial B-mode polarization by the BICEP2 experiment, which, when combined with the Planck + WP + highL data, gives a strong hint for additional suppression of the CMB temperature fluctuations at small scales. Such a suppression can be explained by a running spectral index. In fact, it was pointed out by two of the present authors (TK and FT) that the existence of both tensor mode perturbations and a sizable running of the spectral index is a natural outcome of large inflation models with modula…
We present a new inflation model, known as noncommutative decrumpling inflation, in which space has noncommutative geometry with time variability of the number of spatial dimensions. Within the framework of noncommutative decrumpling inflation, we compute both the spectral index and its running. Our results show that the effects of both time variability of the number of spatial dimensions and noncommutative geometry on the spectral index and its running. Two classes of examples have been studied and comparisons made with the standard slow-roll formulae. We conclude that the effects of noncommu…
While dust contamination now appears as a likely explanation of the apparent tension between the recent BICEP2 data and the Planck data, we will here explore the consequences of a large running in the spectral index as suggested by the BICEP2 collaboration as an alternative explanation of the apparent tension, but which would be in conflict with prediction of the simplest model of chaotic inflation. The large field chaotic model is sensitive to UV physics, and the nontrivial running of the spectral index suggested by the BICEP2 collaboration could therefore, if true, be telling us some additio…
We argue that a large negative running spectral index, if confirmed, might suggest that there are abundant structures in the inflaton potential, which result in a fairly large (both positive and negative) running of the spectral index at all scales. It is shown that the center value of the running spectral index suggested by the recent CMB data can be easily explained by an inflaton potential with superimposed periodic oscillations. In contrast to cases with constant running, the perturbation spectrum is enhanced at small scales, due to the repeated modulations. We mention that such features a…
A model independent analysis shows that the running of the spectral index of the three year WMAP results can be nicely realized in noncommutative inflation. We also re-examine some concrete noncommutative inflation models. We find that a large tensor-scalar ratio is required, corresponding to a low number of e-folds before the end of inflation in some simple models.
In conventional model independent approaches, the power spectrum of primordial perturbations is characterized by such free parameters as spectral index, its running, the running of running, and the tensor-to-scalar ratio. In this work we, show that, at least for simple inflationary potentials, one can find the primordial scalar and tensor power spectra exactly by resumming over all the running terms. In this model-dependent method, we expand the power spectra about the pivot scale to find the series terms as functions of the e-folding number for some single field models of inflation. Interesti…
We study a class of models in which the inflaton is minimally coupled to gravity with a term $f(R)\vp^2$. We focus in particular on the case when $f\sim R^2$, the expansion of the scale factor is driven by the usual potential energy, while the rolling of the inflaton is driven by the nonminial coupling. We show that the power spectrum is in general blue, and the problem of getting a running spectral index is eased. However, the inflaton potential must have a large second derivative in order to get a large running.
We compare Higgs inflation in the metric and Palatini formulations of general relativity, with loop corrections are treated in a simple approximation. We consider Higgs inflation on the plateau, at a critical point, at a hilltop and in a false vacuum. In the last case there are only minor differences. Otherwise we find that in the Palatini formulation the tensor-to-scalar ratio is consistently suppressed, spanning the range $1\times10^{-13}<r<7\times10^{-5}$, compared to the metric case result $2\times10^{-5}<r<0.2$. Even when the values of $n_s$ and $r$ overlap, the running and running of the…
We study a recently proposed running kinetic inflation model in which the inflaton potential becomes flat due to rapid growth of the kinetic term at large inflaton field values. As concrete examples, we build a variety of chaotic inflation models in supergravity with e.g. quadratic, linear, and fractional-power potentials. The power of the potential generically increases after inflation, and the inflaton is often massless at the potential minimum in the supersymmetric limit, which leads to many interesting phenomena. First, the light inflaton mass greatly relaxes severe thermal and non-thermal…
We consider the renormalization group(RG) improved inflaton potential in unitarized Higgs inflation where the original Higgs inflation is unitarized by the addition of a real singlet scalar of sigma-model type. The sigma field coupling to the Higgs, which is introduced to reproduce a large non-minimal coupling of the Higgs below the sigma scalar threshold, also improves the Standard Model vacuum stability due to the RG running. Furthermore, the same sigma field coupling determines the reheating temperature or the number of efoldings. Considering the uncertainties in the number of efoldings in …
Within the framework of RG improved inflationary cosmology motivated by asymptotically safe gravity, we study the dynamics of a scalar field which can be interpreted as the Higgs field. The background trajectories of this model can provide sufficient inflationary e-folds and a graceful exit to a radiation dominated phase. We study the possibility of generating primordial curvature perturbations through the Standard Model Higgs boson. This can be achieved under finely tuned parameter choices by making use of the modulated reheating mechanism. The primordial non-gaussianity is expected to be siz…
The possibility to construct inflationary models for the renormalization-group improved potentials corresponding to scalar electrodynamics and to $SU(2)$ and $SU(5)$ models is investigated. In all cases, the tree-level potential, which corresponds to the cosmological constant in the Einstein frame, is seen to be non-suitable for inflation. Rather than adding the Hilbert-Einstein term to the action, quantum corrections to the potential, coming from to the RG-equation, are included. The inflationary scenario is analyzed with unstable de Sitter solutions which correspond to positive values of the…
An inflationary model in the framework of noncommutative space-time may generate a nontrivial running of the scalar spectral index, but usually induces a large tensor-to-scalar ratio simultaneously. With the latest observational data from the Planck mission, we reexamine the inflationary scenarios in a noncommutative space-time. We find that either the running of the spectral index is tiny compared with the recent observational result, or the tensor-to-scalar ratio is too large to allow a sufficient number of $e$-folds. As examples, we show that the chaotic and power-law inflation models with …
We apply the functional renormalization group to Starobinsky's stochastic equation describing the local dynamics of a light scalar field in de Sitter. After elaborating on the over-damped regime of stochastic dynamics, we introduce an effective average action for the stochastic field, resulting by progressively integrating out frequencies, and study its flow equation in the local potential approximation (LPA). This effective action determines the approach to equilibrium and allows for the computation of unequal time correlators $\left\langleϕ(t)ϕ(t+Δt)\right\rangle$ for large values of $Δt$. T…
We discuss the impact of adiabatic renormalization on the power spectrum of scalar and tensor perturbations from inflation. We show that adiabatic regularization is ambiguous as it leads to very different results, for different adiabatic subtraction schemes, both in the range $v\equiv k/(aH) \gsim 0.1$ and in the infrared regime. All these schemes agree in the far ultraviolet, $v\gg 1$. Therefore, we argue that in the far infrared regime, $v\ll 1$, the adiabatic expansion is no longer valid, and the unrenormalized spectra are the physical, measurable quantities. These findings cast some doubt …
We analyze the weak (2 sigma) evidence for a running spectral index seen in the three-year WMAP dataset and its implications for single field, slow roll inflation. We assume that the running is comparable to the central value found from the WMAP data analysis, and use the Hubble Slow Roll formalism to follow the evolution of the slow roll parameters. For all parameter choices consistent with a large, negative running, single field, slow roll inflation lasts less than 30 efolds after CMB scales leave the horizon. Thus, a definitive observation of a large negative running would imply that any in…
We explore some thermodynamical consequences of accelerated universes driven by a running cosmological constant (CC) from the renormalization group (RG). Application of the generalized second law (GSL) of gravitational thermodynamics to a framework where the running of the CC goes at the expense of energy transfer between vacuum and matter, strongly restricts the mass spectrum of a (hypothetical) theory controlling the CC running. We find that quantum effects driving the running of the CC should be dominated by a trans-planckian mass field, in marked contrast with the GUT-scale upper mass boun…
At the beginning of inflation there could be extra dynamical scalar fields that will soon disappear (become static) before the end of inflation. In the light of multi-field inflation, those extra degrees of freedom may alter the time-dependence of the original spectrum of the curvature perturbation. It is possible to remove such fields introducing extra number of e-foldings prior to $N_e\sim 60$, however such extra e-foldings may make the trans-Planckian problem worse due to the Lyth bound. We show that such extra scalar fields can change the running of the spectral index to give correction of…
I discuss folded inflation, an inflationary model embedded in a multi-dimensional scalar potential, such as the stringy landscape. During folded inflation, the field point evolves along a path that turns several corners in the potential. Folded inflation can lead to a relatively large tensor contribution to the Cosmic Microwave Background, while keeping all fields smaller than the Planck scale. I conjecture that if folded inflation generates a significant primordial tensor amplitude, this will generically be associated with non-trivial scale dependence in the spectrum of primordial scalar pert…
Running into tension: primordial black holes from ultra-slow-roll inflation, spectral running, and the Hubble tension
University of Trento（特伦托大学）；Trento Institute for Fundamental Physics and Applications (TIFPA)-INFN（特伦托基础物理与应用研究所（TIFPA）-意大利国家核物理研究所）；New York University Abu Dhabi（纽约大学阿布扎比分校）
Holographic renormalization group flows can be interpreted in terms of effective field theory. Based on such an interpretation, a formula for the running scaling dimensions of gauge-invariant operators along such flows is proposed. The formula is checked for some simple examples from the AdS/CFT correspondence, but can be applied also in non-AdS/non-CFT cases.
