A generalization of the canonical and non-canonical theory of inflation is introduced in which the kinetic energy term in action is written as non-local term. The inflationary universe within the framework of considering this non-locality will be studied. To investigate the effects of non-locality on the inflationary parameters we consider two well known models of inflationary scenario includes of chaotic and exponential inflation proposals. For such scenarios some important parameters include slow roll parameters, scalar and tensor power spectra, spectral indices, the tensor-to-scalar ratio and so on for both mentioned models, chaotic and exponential inflationary scenarios, will be calculated. Also the Hamilton-Jacobi formalism, as an easiest way to study the effect of perturbation based on e-folding number $N$, to investigate inflationary attractors will be used. The free theoretical parameters of this model will be compared with observations by means of Planck $2013$, $WMAP9+eCMB+BAO+H_0$ data sets in addition to $BICEP2$ data surveying. It will be shown that our theoretical results are in acceptable range in comparison to observations. For instance the tensor-to-scalar ratio for exponential potential, by considering $BICEP2$ is in best agreement in comparison with chaotic inflation.
In the present work we show that warm chaotic inflation characterized by a simple $\fracλ{4}ϕ^{4}$ self-interaction potential for the inflaton, excluded by current data in standard cold inflation, and by an inflaton decay rate proportional to the temperature, is in agreement with the latest Planck data. The parameters of the model are constrained, and our results show that the model predicts a negligible tensor-to-scalar ratio in the strong dissipative regime, while in the weak dissipative regime the tensor-to-scalar ratio can be large enough to be observed.
The pre-big bang's kinetic driven inflationary mechanism is not an adequate form of inflation: the Planck length grows more rapidly than the scale factor. In order to explain our large universe, the resulting post-big bang universe requires the same unnatural constants (Planck problem) as those of any other non-inflationary big bang model.
The warm inflation scenario in view of the modified Chaplygin gas is studied. We consider the inflationary expansion is driven by a standard scalar field whose decay ratio $Γ$ has a generic power law dependence with the scalar field $ϕ$ and the temperature of the thermal bath $T$. By assuming an exponential power law dependence in the cosmic time for the scale factor $a(t)$, corresponding to the intermediate inflation model, we solve the background and perturbative dynamics considering that our model evolves according to (i) weak dissipative regime and (ii) strong dissipative regime. Specifica…
We study the main properties of the warm inflationary model based on Barrow's solution for the scale factor of the universe. Within this framework we calculate analytically the basic slow roll parameters for different versions of warm inflation. We test the performance of this inflationary scenario against the latest observational data and we verify that the predicted spectral index and the tensor-to-scalar fluctuation ratio are in excellent agreement with those of {\it Planck 2015}. Finally, we find that the current predictions are consistent with those of viable inflationary models.
Within the inflationary scenario, Planck scale physics should have affected the comoving modes' initial conditions and early evolution, thereby potentially affecting the inflationary predictions for the cosmic microwave background (CMB). This issue has been studied extensively on the basis of various models for how quantum field theory (QFT) is modified and finally breaks down towards the Planck scale. In one model, in particular, an ultraviolet cutoff was implemented into QFT through generalized uncertainty relations which have been motivated from general quantum gravity arguments and from st…
We examine some inflationary models based on modifications of gravity in the light of Planck 2015 data, such as the generalised Chaplygin inspired inflation, models based in $N=1$ supergravity and braneworld scenarios. We also show that, conversely, potentials with a very flat plateau yield a primordial spectrum similar to that of the Starobinsky model with no need to modify general relativity.
The accelerated expansion of space during the period of cosmological inflation leads to trans-Planckian issues which need to be addressed. Most importantly, the physical wavelength of fluctuations which are studied at the present time by means of cosmological observations may well originate with a wavelength smaller than the Planck length at the beginning of the inflationary phase. Thus, questions arise as to whether the usual predictions of inflationary cosmology are robust considering our ignorance of physics on trans-Planckian scales, and whether the imprints of Planck-scale physics are at …
The present work is a sequel of our previous work Phys.Rev.D { 93}, 084018 (2016) [arXiv:1601.08175 [gr-qc]] cite{hap} which depicted a simple version of an inflationary quintessential model whose inflationary stage was described by a Higgs type potential and the quintessential phase was responsible due to an exponential potential. Additionally, the model predicted a nonsingular universe in past which was geodesically past incomplete. Further, it was also found that the model is in agreement with the Planck 2013 data when running is allowed. But, this model was found to be unsuccessful with Pl…
We use the Hamilton--Jacobi formalism to constrain the space of possible single field, inflationary Hubble flow trajectories when compared to the WMAP and Planck satellites Cosmic Microwave Background (CMB) results. This method yields posteriors on the space of Hubble Slow Roll (HSR) parameters that uniquely determine the history of the Hubble parameter during the inflating epoch. The trajectories are used to numerically determine the observable primordial power spectrum and bispectra that can then be compared to observations. Our analysis is used to infer the most likely shape of the inflaton…
Three-dimensional neutral hydrogen mapping using the redshifted 21 cm line has recently emerged as a promising cosmological probe. Within the framework of slow-roll reconstruction, we analyze how well the inflationary potential can be reconstructed by combining data from 21 cm experiments and cosmic microwave background data from the Planck satellite. We consider inflationary models classified according to the amplitude of their tensor component, and show that 21 cm measurements can significantly improve constraints on the slow-roll parameters and determine the shape of the inflationary potent…
Making use of the primordially isocurvature fluctuations, which are generated in inflationary models with multiple scalar fields, we make a phenomenological model that predicts formation of primordial black holes which can account for the massive compact halo objects recently observed.
We describe in this paper the observables of inflationary models, in particular the spectrum index of torsion scalar perturbations, the tensor-to-scalar ratio, and the running of the spectral index, in the framework of perfect fluid models and F(T) gravity theories through the reconstruction methods. Then, our results on the perfect fluid and F(T) gravity theories of inflation are compared with recent cosmological observations such as the Planck satellite and BICEP2 experiment. Ours studies prove that the perfect fluid and F(T) gravity models can reproduce the inflationary universe consistent …
We consider an inflationary model inspired in the low energy limit of string theory. In this model, the scale factor grows exponentially with time. A perturbation study is performed, and we show that there is a mode which displays an exponential growth in the perturbation of the scalar field.
We investigate the observational signatures of quantum cosmology in the Cosmic Microwave Background data provided by Planck collaboration. We apply the warm inflationary paradigm with a tachyon scalar field to the loop quantum cosmology. In this context, we first provide the basic cosmological functions in terms of the tachyon field. We then obtain the slow-roll parameters and the power spectrum of scalar and tensor fluctuations respectively. Finally, we study the performance of various warm inflationary scenarios against the latest Planck data and we find a family of models which are in agree…
Neutron stars are perfect candidates to investigate the effects of a modified gravity theory, since the curvature effects are significant and more importantly, potentially testable. In most cases studied in the literature in the context of massive scalar-tensor theories, inflationary models were examined. The most important of scalar-tensor models is the Higgs model, which, depending on the values of the scalar field, can be approximated by different scalar potentials, one of which is the inflationary. Since it is not certain how large the values of the scalar field will be at the near vicinit…
We consider an inflationary universe model in the context of generalized cosmic Chaplygin gas by taking matter field as standard and tachyon scalar fields. We evaluate the corresponding scalar fields and scalar potentials during intermediate and logamediate inflationary regimes by modifying the first Friedmann equation. In each case, we evaluate the number of e-folds, scalar as well as tensor power spectra, scalar spectral index and important observational parameter, i.e., tensor-scalar ratio in terms of inflatons. The graphical behavior of this parameter shows that the model remains incompati…
I give a general review of inflationary cosmology and of its present status, in view of the 2013 data release by the Planck satellite. A specific emphasis is given to the new broad class of theories, the cosmological attractors, which have nearly model-independent predictions converging at the sweet spot of the Planck data in the (n_s,r) plane. I also discuss the problem of initial conditions for the theories favored by the Planck data.
In this work, we study an inflationary scenario in the presence of Generalized Chaplygin Gas (GCG). We show that in Einstein gravity, GCG is not a suitable candidate for inflation; but in a five dimensional brane world scenario, it can work as a viable inflationary model. We calculate the relevant quantities such as $n_{s}$, $r$ and $A_{s}$ related to the primordial scalar and tensor fluctuations, and using their recent bounds from Planck and BICEP2, we constrain the model parameters as well as the five-dimensional Planck mass. But as a slow-roll inflationary model with a power-law type scalar…
We study the dynamics of two concrete inflationary models, namely Spontaneous Symmetry Breaking Inflation as well as Loop Inflation. We constrain the parameters for which a viable inflationary Universe emerges using the latest PLANCK results of last year, and we give predictions for the duration of reheating as well as for the reheating temperature after inflation. Our numerical results show that baryogenesis via leptogenesis may be realized within the inflationary models considered in this work.
We derive the semiclassical evolution of massless minimally coupled scalar matter in the de Sitter space-time from the Born-Oppenheimer reduction of the Wheeler-DeWitt equation. We show that the dynamics of trans-Planckian modes can be cast in the form of an effective modified dispersion relation and that high energy corrections in the power spectrum of the cosmic microwave background radiation produced during inflation remain very small if the initial state is the Bunch-Davies vacuum.
We study the implications of the recently proposed Trans-Planckian Censorship Conjecture (TCC) for early universe cosmology and in particular inflationary cosmology. The TCC leads to the conclusion that if we want inflationary cosmology to provide a successful scenario for cosmological structure formation, the energy scale of inflation has to be lower than $10^9$ GeV. Demanding the correct amplitude of the cosmological perturbations then forces the generalized slow-roll parameter $ε$ of the model to be very small ($<10^{-31}$). This leads to the prediction of a negligible amplitude of primordi…
We investigate the insensitivity of the predictions of inflationary models with respect to modifications of Planck energy physics. The modification we consider consists in replacing the usual dispersion relation by nonlinear ones. This way of addressing the problem has recently recieved attention and contradictory results were found. Our main result is to show that the adiabaticity of the mode propagation and the separation of two scales of interest, the Planck scale and the cosmological horizon scale, are sufficient conditions for the predictions to be unchanged. We then show that almost all …
In the present paper we discuss how trans-Planckian physics affects inflationary vacuum fluctuations and primordial density perturbations. The trans-Planckian problem during inflation has been widely discussed in literature, but it is still under debate. We reconsider this problem by using the two-point correlation function of the non-minimally coupled scalar fields and constructing the effective potential with the adiabatic (WKB) regularization or approximation. First, we clearly show that the cut-off divergence of the quantum fluctuations does not drastically change during inflation under re…
Corrections to inflationary models induced by non-minimal coupling between scalar field and curvature
Bauman Moscow State Technical University（鲍曼莫斯科国立技术大学）；Ulyanovsk State Pedagogical University（乌里扬诺夫斯克州立师范学院）；Birla Institute of Technology and Science-Pilani, Hyderabad Campus（比拉理工学院皮拉尼校区海德拉巴分校）
The Planck collaboration has provided us rich information about the early universe, and a host of new observational missions will soon shed further light on the `anomalies' that appear to exist on the largest angular scales. From a quantum gravity perspective, it is natural to inquire if one can trace back the origin of such puzzling features to Planck scale physics. Loop quantum cosmology provides a promising avenue to explore this issue because of its natural resolution of the big bang singularity. Thanks to advances over the last decade, the theory has matured sufficiently to allow concrete…
Due to the quasi-exponential red-shifting which occurs during an inflationary period in the very early Universe, wavelengths which at the present time correspond to cosmological lengths are in general sub-Planckian during the early stages of inflation. This talk discusses two approaches to addressing this issue which both indicate that the standard predictions of inflationary cosmology - made using classical general relativity and weakly coupled scalar matter field theory - are not robust against changes in the physics on trans-Planckian scales. One approach makes use of modified dispersion re…
We present three non-parametric Bayesian primordial reconstructions using Planck 2018 polarization data: linear spline primordial power spectrum reconstructions, cubic spline inflationary potential reconstructions and sharp-featured primordial power spectrum reconstructions. All three methods conditionally show hints of an oscillatory feature in the primordial power spectrum in the multipole range $\ell\sim20$ to $\ell\sim50$, which is to some extent preserved upon marginalization. We find no evidence for deviations from a pure power law across a broad observable window ($50\lesssim\ell\lesssi…
We present a simple, exact and self-consistent cosmology with a phenomenological model of quantum creation of radiation due to decay of the scalar field. The decay drives a non-isentropic inflationary epoch, which exits smoothly to the radiation era, without reheating. The initial vacuum for radiation is a regular Minkowski vacuum. The created radiation obeys standard thermodynamic laws, and the total entropy produced is consistent with the accepted value. We analyze the difference between the present model and a model with decaying cosmological constant previously considered.
Starting with a study of the cosmological solution to the Einstein equations for the internal spacetime of an extreme supermassive cosmic string kink, and by evaluating the probability measure for the formation of such a kink in semiclassical approximation using a minisuperspace with the appropriate symmetry, we have found a set of arguments in favor of the claim that the kinked extreme string can actually be regarded as a unbounded chain of pairs of Planck- sized universes. Once one such universe pairs is created along a primordial phase transition at the Planck scale, it undergoes an endless…
Since the standard inflationary paradigm is based on quantum field theory on classical space-times, it excludes the Planck era. Using techniques from loop quantum gravity, the paradigm is extended to a self-consistent theory from the Planck scale to the onset of slow roll inflation, covering some 11 orders of magnitude in energy density and curvature. This pre-inflationary dynamics also opens a small window for novel effects, e.g. a source for non-Gaussianities, which could extend the reach of cosmological observations to the deep Planck regime of the early universe.
We study effect of superhorizon tensor perturbations on scalar perturbations, so called effect of clustering fossils, in cosmological model in which inflation is driven by both solid matter and scalar field. The effect deforms primordial scalar power spectrum causing asymmetry in it, which leaves imprint on CMB anisotropies and the cosmic structure. Parameter space of this combined model allows for enhancement of logarithmic scale dependence of this deformation, as opposed to simpler models in which such scale dependence is suppressed by slow-roll parameters.
The current available CMB data show an anomalously low value of the CMB temperature fluctuations at large angular scales (l < 40). This lack of power is not explained by the minimal LCDM model, and one of the possible mechanisms explored in the literature to address this problem is the presence of features in the primordial power spectrum (PPS) motivated by the early universe physics. In this paper, we analyse a set of cutoff inflationary PPS models using a Bayesian model comparison approach in light of the latest Cosmic Microwave Background (CMB) data from the Planck Collaboration. Our result…
It is investigated if predictions of the inflationary scenario regarding spectra of scalar and tensor perturbations generated from quantum vacuum fluctuations are robust with respect to a modification of the dispersion law for frequencies beyond the Planck scale. For a large class of such modifications of special and general relativity, for which the WKB condition is not violated at super-high frequencies, the predictions remain unchanged. The opposite possibility is excluded by the absence of large amount of created particles due to the present Universe expansion. Creation of particles in the…
We give a detailed presentation of a recently proposed mechanism of generating the energy scale of inflation by loop effects in quantum cosmology. We discuss the quantum origin of the early inflationary Universe from the no-boundary and tunneling quantum states and present a universal effective action algorithm for the distribution function of chaotic inflationary cosmologies in both of these states. The energy scale of inflation is calculated by finding a sharp probability peak in this distribution function for a tunneling model driven by the inflaton field with large negative constant $ξ$ of…
There are good indications that fundamental physics gives rise to a modified space-momentum uncertainty relation that implies the existence of a minimum length scale. We implement this idea in the scalar field theory that describes density perturbations in flat Robertson-Walker space-time. This leads to a non-linear time-dependent dispersion relation that encodes the effects of Planck scale physics in the inflationary epoch. Unruh type dispersion relations naturally emerge in this approach, while unbounded ones are excluded by the minimum length principle. We also find red-shift induced modifi…
The recent CMB data from Planck and BICEP2 observations have opened a new window for inflationary cosmology. In this Essay we compare three Starobinsky-like inflationary scenarios: (i) the original Starobinsky proposal; (ii) a family of dynamically broken SUGRA models; and (iii) a class of "decaying" vacuum $Λ(H)$ cosmologies. We then focus on the $Λ(H)$ variant, which spans the complete cosmic history of the universe from an early inflationary stage, followed by the "graceful exit" into the standard radiation regime, the matter epoch and, finally, the late-time accelerated expansion. Computin…
We have developed a formalism to study an inflationary scenario driven by a bulk inflaton in the two-brane system. The 4-dimensional low energy effective action is obtained using the gradient expansion method. It is also found that the dark radiation and the dark scalar source are expressed by the radion. In the single-brane limit, we find these dark components disappear. Therefore, it turns out that the inflation due to the bulk inflaton successfully takes place. Kaluza-Klein corrections are also taken into account in this case.
We compute the Bayesian evidence and complexity of 193 slow-roll single-field models of inflation using the Planck 2013 Cosmic Microwave Background data, with the aim of establishing which models are favoured from a Bayesian perspective. Our calculations employ a new numerical pipeline interfacing an inflationary effective likelihood with the slow-roll library ASPIC and the nested sampling algorithm MULTINEST. The models considered represent a complete and systematic scan of the entire landscape of inflationary scenarios proposed so far. Our analysis singles out the most probable models (from …
A warm inflationary Universe in the Randall-Sundrum II model during intermediate inflation is studied. For this purpose, we consider a general form for the dissipative coefficient $Γ(T,ϕ)=C_ϕ\,\frac{T^{m}}{ϕ^{m-1}}$, and also analyze this inflationary model in the weak and strong dissipative regimes. We study the evolution of the Universe under the slow-roll approximation and find solutions to the full effective Friedmann equation in the brane-world framework. In order to constrain the parameters in our model, we consider the recent data from the BICEP2-Planck 2015 data together with the neces…
We study inflation in the framework of $f(T)$-gravity in the presence of a canonical scalar field. After reviewing the basic equations governing the background cosmology in $f(T)$-gravity, we turn to study the cosmological perturbations and obtain the evolutionary equations for the scalar and tensor perturbations. Solving those equations, we find the power spectra for the scalar and tensor perturbations. Then, we consider a power-law form for the $f(T)$ function in the action and examine the inflationary models with the power-law and intermediate scale factors. We see that in contrast with the…
Due to the accelerated expansion of the universe, the possibility of formation of singularities has changed from the classical Big Bang and Big Crunch singularities to include a number of new scenarios. In recent papers it has been shown that such singularities may appear in inflationary cosmological models with a fractional power scalar field potential. In this paper we enlarge the analysis of singularities in scalar field cosmological models by the use of generalised power expansions of their Hubble scalar and their scalar field in order to describe all possible models leading to a singulari…
This paper explores the dynamics of warm intermediate and logamediate inflationary models during weak dissipative regime with a general form of dissipative coefficient. We analyze these models within the framework of locally rotationally symmetric Bianchi type I universe. In both cases, we evaluate solution of inflaton, effective scalar potential, dissipative coefficient, slow-roll parameters, scalar and tensor power spectra, scalar spectral index and tensor to scalar ratio under slow-roll approximation. We constrain the model parameters using recent data and conclude that anisotropic inflatio…
We discuss inflationary solutions of the coupled Einstein-Klein-Gordon equations for a complex field in a five dimensional spacetime with a compact $x^5$-dimension. As a new feature, the scalar field contains a dependence on the extra dimension of the form $\exp(i m x^5) $, corresponding to Kaluza-Klein excited modes. In a four dimensional picture, a nonzero $m$ implies the presence of a new term in the scalar field potential. An interesting feature of these solutions is the possible existence of several periods of oscillation of the scalar field around the equilibrium value at the minimum of …
Various scenarios of the initial inflation of the universe are distinguished by the choice of a scalar field {\em potential} $U(ϕ)$ which simulates a {\it temporarily} non--vanishing {\em cosmological term}. Our new method, which involves a reparametrization in terms of the Hubble expansion parameter $H$, provides a classification of allowed inflationary potentials and of the stability of the critical points. It is broad enough to embody all known {\it exact} solutions involving one scalar field as special cases. Inflation corresponds to the evolution of critical points of some catastrophe man…
We study cosmology in scalar-tensor (Bergmann Wagoner) gravity, restricting the coupling function, $ω(ϕ)$ to be constant. Rather than specify the form of the cosmological function, $λ(ϕ)$, the scalar field is modelled as a function which decays to its present value $ϕ=1$. Solutions of the field equations are found for which $λ(ϕ)$ evolves from a large value (approximately 1) near the singularity to a small, non-zero value at later times, avoiding the problem of pre-inflationary collapse in standard general relativistic cosmology. Interpreting the model within the framework of Brans-Dicke theor…
We investigate the implication of Trans-Planckian Censorship Conjecture (TCC) for the initial state of primordial perturbations. It is possible to set the state of perturbation modes in the infinite past as the Minkowski vacuum, only if the pre-inflationary era is past-complete. We calculate the evolution of the perturbation modes in such a pre-inflationary era and show that at the beginning of inflation the perturbation modes with wavelengths much shorter than the Hubble scale (but still larger than the Planck length scale) will behave as they are in the Bunch-Davis state. Therefore, a past-c…
In the present paper we study warm inflationary universe models in the context of a polytropic gas. We derive the characteristics of this model in slow-roll approximation and develop our model in two cases, 1- For a constant dissipative parameter $Γ$. 2- $Γ$ as a function of scalar field $ϕ$. In these cases we will obtain exact solution for the scalar field and Hubble parameter. We will also obtain explicit expressions for the tensor-scalar ratio $R$, scalar spectrum index $n_s$ and its running $α_s$, in slow-roll approximation.
We study the effects of the trans-Planckian dispersion relation on the spectrum of the primordial density perturbations during inflation. In contrast to the earlier analyses, we do not assume any specific form of the dispersion relation and allow the initial state of the field to be arbitrary. We obtain the spectrum of vacuum fluctuations of the quantum field by considering a scalar field satisfying the linear wave equation with higher spatial derivative terms propagating in the de Sitter space-time. We show that the power spectrum does not strongly depend on the dispersion relation and that t…
I review the "Trans-Planckian Censorship Conjecture" (TCC) and its implications for cosmology, in particular for the inflationary universe scenario. Whereas the inflationary scenario is tightly constrained by the TCC, alternative early universe scenarios are not restricted.
The 2015 Planck data release has placed tight constraints on the allowed class of inflationary models. The current data favors concave downwards inflationary potentials while offering interesting hints on possible deviations from the standard picture of CMB perturbations. We here test the predictions of the theory of the origin of the universe from the landscape multiverse, against the most recent Planck data, for the case of concave downwards inflationary potentials, such as the Starobinsky model of inflation. By considering the quantum entanglement correction of the multiverse, we can place …
In most current models of inflation based on a weakly self-coupled scalar matter field minimally coupled to gravity, the period of inflation lasts so long that, at the beginning of the inflationary period, the physical wavelengths of comoving scales which correspond to the present large-scale structure of the Universe were smaller than the Planck length. Thus, the usual computations of the spectrum of fluctuations in these models involve extrapolating low energy physics (both in the matter and gravitational sector) into regions where this physics is not applicable. In this paper we demonstrate…
In this paper my prime objective is to analyze the constraints on a sub-Planckian excursion of a single inflaton field within Effective Field Theory framework in a model independent fashion. For a generic single field inflationary potential, using the various parameterization of the primordial power spectrum I have derived the most general expression for the field excursion in terms of various inflationary observables, applying the observational constraints obtained from recent Planck 2015 and Planck 2015 +BICEP2/Keck Array data. By explicit computation I have reconstructed the structural form…
We study the primordial magnetic field generated by the simple model $f^2 FF$ in Starobinsky, $R^2$-inflationary, model. The scale invariant PMF is achieved at relatively high power index of the coupling function, $\left| α\right| \approx 7.44$. This model does not suffer from the backreaction problem as long as, the rate of inflationary expansion, $H$, is in the order of or less than the upper bound reported by Planck ($\le 3.6 \times 10^{-5} M_\rm{Pl}$) in both de Sitter and power law expansion, which show similar results. We calculate the lower limit of the reheating parameter, $R_\rm{rad} …
Depending on various assumptions on the energy scale of inflation and assuming a primordial power spectrum of a Broken Scale Invariance (BSI) type, we explore the possibility for Primordial Black Holes (PBH) and Planck relics to contribute substantially to cold dark matter in the Universe. A recently proposed possibility to produce planck relics in 4-dimensional string gravity is considered. Possible experimental detection through gravitational waves is further explored. We stress that inflation with a low energy scale, and also possibly when Planck relics are produced, leads unavoidably to re…
Models of cosmic inflation posit an early phase of accelerated expansion of the universe, driven by the dynamics of one or more scalar fields in curved spacetime. Though detailed assumptions about fields and couplings vary across models, inflation makes specific, quantitative predictions for several observable quantities, such as the flatness parameter ($Ω_k = 1 - Ω$) and the spectral tilt of primordial curvature perturbations ($n_s - 1 = d \ln {\cal P}_{\cal R} / d \ln k$), among others---predictions that match the latest observations from the {\it Planck} satellite to very good precision. In…
