In this paper we investigate tensor modes of perturbations in the universe governed by Loop Quantum Cosmology. We derive the equation for tensor modes and investigate numerically effects of quantum corrections. This investigation reveals that the region of super-adiabatic amplification of tensor modes is smaller in comparison with the classical case. Neglecting quantum corrections to the equation for tensor modes and holding underlying loop dynamics we study analytically the creation of gravitons. We calculate the power spectrum of tensor perturbations during the super-inflationary phase induced by Loop Quantum Gravity. The main result obtained is the spectrum of gravitons, produced in the transition from the quantum to classical regime of the Universe. Obtained spectrum is characterized by a hard branch. The numerical investigation shows the strong dependence of $ν_{\text{max}}$ frequency with respect to quantum numbers. The parameter $Ω_{\text{gw}}(ν_{\text{max}})$ approaches even to $\sim 10^{-1}$ for highest frequencies. We compare our results with recent constraints from the Laser Interferometer Gravitational-wave Observatory (LIGO) and find that it is possible to test the quantum effects in the early Universe.
The semiclassical approach to quantum gravity would yield the Schroedinger formalism for the wave function of metric perturbations or gravitons plus quantum gravity correcting terms in pure gravity; thus, in the inflationary scenario, we should expect correcting effects to the relic graviton (Zel'dovich) spectrum of the order (H/mPl)^2.
The spectra of relic gravitational waves produced as a result of cosmological expansion of the inflationary models are derived in Brans-Dicke theory of gravity.The time dependence of the very early Hubble parameter and matter energy density are derived from frequency dependent spectrum of relic gravitational waves.Also it is found that Brans-Dicke scalar field contributes to the energy density of relic gravitons.
The super-inflationary phase is predicted by the Loop Quantum Cosmology. In this paper we study the creation of gravitational waves during this phase. We consider the inverse volume corrections to the equation for the tensor modes and calculate the spectrum of the produced gravitons. The amplitude of the obtained spectrum as well as maximal energy of gravitons strongly depend on the evolution of the Universe after the super-inflation. We show that a further standard inflationary phase is necessary to lower the amount of gravitons below the present bound. In case of the lack of the standard inf…
We discuss the production of massive relic coherent gravitons in a particular class of f(R) gravity which arises from string theory and their possible imprint in Cosmic Microwave Background. In fact, in the very early universe these relic gravitons could have acted as slow gravity waves. They may have then acted to focus the geodesics of radiation and matter. Therefore, their imprint on the later evolution of the universe could appear as filaments and domain wall in the Universe today. In that case, the effect on Cosmic Microwave Background should be analogous to the effect of water waves, whi…
We present a simple model in which a brane with anisotropic metric is embedded in an AdS bulk. We discuss the localization of the massless mode and the amplification of both the massless and the massive modes on the branes, paying particular attention to the normalization of the perturbed action and to the evaluation of the effective coupling constant that controls the amplitude of the spectrum. In the model under investigation there is no mass gap between massless and massive modes, and the massive modes can be amplified, with mass-dependent amplitudes.
I discuss the status of present knowledge about a possible background of relic gravitons left by an early, pre-big bang cosmological epoch, whose existence in the past of our Universe is suggested by the duality symmetries of string theory.
The spectrum and amplitude of the stochastic background of relic gravitons produced in a bouncing universe is calculated. The matter content of the model consists of dust and radiation fluids, and the bounce occurs due to quantum cosmological effects when the universe approaches the classical singularity in the contracting phase. The resulting amplitude is very small and it cannot be observed by any present and near future gravitational wave detector. Hence, as in the ekpyrotic model, any observation of these relic gravitons will rule out this type of quantum cosmological bouncing model.
We discuss some peculiar properties of the stochastic graviton background predicted by string cosmology. At Planckian times, for the values of the parameters of the model which are more interesting for the detection in gravitational wave experiments, the number density of gravitons is parametrically large compared to Planck density, and is peaked at small energies. The large parameter is related to the duration of the string phase and is a characteristic of string cosmology. The typical interaction time is parametrically larger than the Planck or string time. Therefore the shape of the gravito…
The polarizations of the tensor modes of the geometry evolving in cosmological backgrounds are treated as the components of a bispinor whose dynamics follows from an appropriate gauge-invariant action. This novel framework bears a close analogy with the (optical) Jones calculus and leads to a compact classification of the various interactions able to polarize the relic gravitons.
The energy spectra of gravitational waves (GW) produced in quintessential inflationary models increase in frequency and exhibit a sharp spike around 170 GHz where the associated fraction of critical energy density today stored in relic gravitons is of the order of $10^{-6}$. We contrast our findings with the spectra of ordinary inflationary models and we comment about possible detetction strategies of the spike.
The tensor contribution to the $V$-mode polarization induced by a magnetized plasma at last scattering vanishes exactly. Conversely a polarized background of relic gravitons cannot generate a $V$-mode polarization. The reported results suggest that, in the magnetized $Λ$CDM paradigm, the dominant source of circular dichroism stems from the large-scale fluctuations of the spatial curvature.
The diffuse backgrounds of relic gravitons with frequencies ranging between the aHz band and the GHz region encode the ultimate information on the primeval evolution of the plasma and on the underlying theory of gravity well before the electroweak epoch. While the temperature and polarization anisotropies of the microwave background radiation probe the low-frequency tail of the graviton spectra, during the next score year the pulsar timing arrays and the wide-band interferometers (both terrestrial and hopefully space-borne) will explore a much larger frequency window encompassing the nHz domai…
Stochastic backgrounds or relic gravitons, if ever detected, will constitute a prima facie evidence of physical processes taking place during the earliest stages of the evolution of the plasma. The essentials of the stochastic backgrounds of relic gravitons are hereby introduced and reviewed. The pivotal observables customarily employed to infer the properties of the relic gravitons are discussed both in the framework of the $Λ$CDM paradigm as well as in neighboring contexts. The complementarity between experiments measuring the polarization of the Cosmic Microwave Background (such as, for ins…
The dynamical effects of the tensor modes of the geometry are investigated in the context of curvature bounces. Since the bouncing behaviour implies sharp deviations from a radiation-dominated evolution, significant back-reaction effects of relic gravitons may be expected at short wavelengths. After developing a general iterative framework for the calculation of dynamical back-reaction effects, explicit analytical and numerical examples are investigated for different parametrizations of the energy-momentum pseudo-tensor(s) of the produced gravitons. The reported results suggest that dynamical …
The energy density, the pressure and the anisotropic stress of the relic gravitons do not have a unique gauge-invariant and frame-invariant expression since the equivalence principle ultimately forbids the localization of the energy-momentum of the gravitational field. All the strategies proposed so far suggest compatible descriptions inside the effective horizon but they lead to sharply different answers when the wavelengths of the gravitons exceed the Hubble radius by remaining shorter than the typical extension of causally connected patches. We present a coherent discussion of the energy-mo…
Stochastic backgrounds of relic gravitons of cosmological origin extend from frequencies of the order of the aHz up to the GHz range. Since the temperature and polarization anisotropies constrain the low frequency normalization of the spectra, in the concordance paradigm the strain amplitude corresponding to the frequency window of wide-band interferometers turns out to be, approximately, nine orders of magnitude smaller than the astounding signal recently reported and attributed to a binary black hole merger. The backgrounds of relic gravitons expected from the early Universe are compared wit…
If the energy momentum tensor contains bulk viscous stresses violating the dominant energy condition (DOC) the energy spectra of the relic gravitons (produced at the time of the DOC's violation) increase in frequency in a calculable way. In a general relativistic context we give examples where the DOC is only violated for a limited amount of time after which the ordinary (radiation dominated) evolution takes place. We connect our discussion to some recent remarks of Grishchuk concerning the detectability of the stochastic gravitational wave background by the forthcoming interferometric detecto…
In this paper we discuss some consequences of cosmological models in which the primordial cosmic matter is described by a relativistic imperfect fluid. The latter takes into account the dissipative effects (bulk viscosity) arising from different cooling rates of the fluid components in the expanding Universe. We discuss, in particular, the effects of the bulk viscosity on Big Bang Nucleosynthesis and on the thermal relic abundance of particles, looking at recent results of PAMELA experiment. The latter has determined an anomalous excess of positron events, that cannot be explained by the conve…
In conformally flat background geometries the long wavelength gravitons can be described in the fluid approximation and they induce scalar fluctuations both during inflation and in the subsequent radiation-dominated epoch. While this effect is minute and suppressed for a de Sitter stage of expansion, the fluctuations of the energy-momentum pseudo-tensor of the graviton fluid lead to curvature perturbations that increase with time all along the post-inflationary evolution. An explicit calculation of these effects is presented for a standard thermal history and it is shown that the growth of the…
The coherence of the relic gravitons is investigated within a quantum mechanical perspective. After introducing the notion and the properties of the generalized Glauber correlators valid in the tensor case, the degrees of first- and second-order coherence are evaluated both inside and beyond the effective horizon. The inclusive approach (encompassing the polarizations of the gravitons) is contrasted with the exclusive approximation where the total intensity is calculated either from a single polarization or even from a single mode of the field. While the relic gravitons reentering the effectiv…
We propose to uncover the signature of a stringy era in the primordial Universe by searching for a prominent peak in the relic graviton spectrum. This feature, which in our specific model terminates an $ω^3$ increase and initiates an $ω^{-7}$ decrease, is induced during the so far overlooked bounce of the scale factor between the collapsing deflationary era (or pre-Big Bang) and the expanding inflationary era (or post-Big Bang). We evaluate both analytically and numerically the frequency and the intensity of the peak and we show that they may likely fall in the realm of the new generation of i…
The early expansion history of the Universe is constrained by combining the most recent limits on the cosmic gravitons in the audio band and the claimed evidences of the nHz domain. The simplest scenario stipulates that between the end of inflation and the formation of light nuclei the evolution consists of a single phase expanding at a rate that is either faster or slower than the one of radiation. If there are instead multiple post-inflationary stages evolving at different rates, the spectral energy density always undershoots the signals potentially attributed to relic gravitons by the pulsa…
We investigate the impact of the post-inflationary thermal histories on the cosmic graviton spectrum caused by the inflationary variation of their refractive index. Depending on the frequency band, the spectral energy distribution can be mildly red, blue or even violet. Wide portions of the parameter space lead to potentially relevant signals both in the audio range (probed by the advanced generation of terrestrial interferometers) and in the mHz band (where space-borne detectors could be operational within the incoming score year). The description of the refractive index in conformally relate…
The tensor modes reentering the Hubble radius when the plasma is dominated by a stiff fluid lead to a spectral energy density whose blue slope depends on the total post-inflationary sound speed. This result gets however corrected by a secondary (gauge-dependent) term coming from the curvature inhomogeneities that reenter all along the same stage of expansion. In comparison with the first-order result, the secondary contribution is shown to be always suppressed inside the sound horizon and its effect on the total spectral energy density of the relic gravitons is therefore negligible for all phe…
In the same way as the energy density associated with the tensor modes of the geometry modifies the evolution of the curvature perturbations, the scalar modes may also indirectly affect the cosmic backgrounds of relic gravitons by inducing higher-order corrections that are only superficially gauge-invariant. This spurious gauge-invariance gets manifest when the effective anisotropic stresses, computed in different coordinate systems, are preliminarily expressed in a form that only depends on the curvature inhomogeneities defined on comoving orthogonal hypersurfaces and on their corresponding t…
The degree of second-order coherence of the relic gravitons produced from the vacuum is super-Poissonian and larger than in the case of a chaotic source characterized by a Bose-Einstein distribution. If the initial state does not minimize the tensor Hamiltonian and has a dispersion smaller than its averaged multiplicity, the overall statistics is by definition sub-Poissonian. Depending on the nature of the sub-Poissonian initial state, the final degree of second-order coherence of the quanta produced by stimulated emission may diminish (possibly even below the characteristic value of a chaotic…
This paper summarizes the contribution presented at the {\sl IX Marcel Grossmann Meeting} (Rome, July 2000). It is stressed, in particular, that a non-relativistic background of ultra-light dilatons, produced in the context of string cosmology, could represent today a significant fraction of cold dark matter. If the dilaton mass lies within the resonant band of present gravitational antennas, a stochastic dilaton background with a nearly critical density could be visible, in principle, already at the level of sensitivity of the detectors in operation and presently under construction.
Absent any indirect tests on the thermal history of the Universe prior to the formation of light nuclear elements, it is legitimate to investigate situations where, before nucleosyntheis, the sound speed of the plasma was larger than $c/\sqrt{3}$, at most equalling the speed of light $c$. In this plausible extension of the current cosmological paradigm, hereby dubbed Tensor-$Λ$CDM (i.e. T$Λ$CDM) scenario, high-frequency gravitons are copiously produced. Without conflicting with the bounds on the tensor to scalar ratio stemming from the combined analysis of the three standard cosmological data …
Using general arguments we determine the allowed region for the end point frequency and the peak energy density of the stochastic background of gravity waves expected in string cosmology. We provide an accurate estimate of the minimal experimental sensitivity required to detect a signal in the Hz to GHz range.
The discrepancy between the results of covariant and non-covariant one-loop calculations for higher-spin fields in quantum cosmology is analyzed. A detailed mode-by-mode study of perturbative quantum gravity about a flat Euclidean background bounded by two concentric 3-spheres, including non-physical degrees of freedom and ghost modes, leads to one-loop amplitudes in agreement with the covariant Schwinger-DeWitt method. This calculation provides the generalization of a previous analysis of fermionic fields and electromagnetic fields at one-loop about flat Euclidean backgrounds admitting a well…
We discuss the cosmological amplification of tensor perturbations in a simple example of brane-world scenario, in which massless gravitons are localized on a higher-dimensional Kasner-like brane embedded in a bulk AdS background. Particular attention is paid to the canonical normalization of the quadratic action describing the massless and massive vacuum quantum fluctuations, and to the exact mass-dependence of the amplitude of massive fluctuations on the brane. The perturbation equations can be separated. In contrast to de Sitter models of brane inflation, we find no mass gap in the spectrum …
The dynamical evolution of the refractive index of the tensor modes of the geometry produces a specific class of power spectra characterized by a blue (i.e. slightly increasing) slope which is directly determined by the competition of the slow-roll parameter and of the rate of variation of the refractive index. Throughout the conventional stages of the inflationary and post-inflationary evolution, the microwave background anisotropies measurements, the pulsar timing limits and the big-bang nucleosythesis constraints set stringent bounds on the refractive index and on its rate of variation. Wit…
If the refractive index of the relic gravitons increases during a conventional stage of inflationary evolution the spectral energy density is blue at intermediate frequencies above the fHz and then flattens out after a knee that is typically smaller than the mHz. We investigate here the conditions leading to a sufficiently large spectral energy density in the nHz range where some peculiar signatures observed with the pulsar timing arrays have been recently attributed to cosmic gravitons. If these potential evidences are combined with the most recent bounds provided by wide-band interferometers…
This is a contracted version of a series of lectures for graduate and undergraduate students given at the {\sl "VI Seminario Nazionale di Fisica Teorica"} (Parma, September 1997), at the Second Int. Conference {\sl "Around VIRGO"} (Pisa, September 1998), and at the Second {\sl SIGRAV} School on {\sl "Gravitational Waves in Astrophysics, Cosmology and String Theory"} (Center "A. Volta", Como, April 1999). The aim is to provide an elementary, self-contained introduction to string cosmology and, in particular, to the background of relic cosmic gravitons predicted in the context of the so-called "…
The spectrum of relic gravitational wave (RGW) contains high-frequency divergences, which should be removed. We present a systematic study of the issue, based on the exact RGW solution that covers the five stages, from inflation to the acceleration, each being a power law expansion. We show that the present RGW consists of vacuum dominating at $f>10^{11}$Hz and graviton dominating at $f<10^{11}$Hz, respectively. The gravitons are produced by the four cosmic transitions, mostly by the inflation-reheating one. We perform adiabatic regularization to remove vacuum divergences in three schemes: at …
Conversion of gravitational waves into electromagnetic radiation is discussed. The probability of transformations of gravitons into photons in presence of cosmological background magnetic field is calculated at the recombination epoch and during subsequent cosmological stages. The produced electromagnetic radiation is concentrated in the X-ray part of the spectrum. It is shown that if the early Universe was dominated by primordial black holes (PBHs) prior to Big Bang Nucleosynthesis (BBN), the relic gravitons emitted by PBHs would transform to an almost isotropic background of electromagnetic …
Contents: Introduction. The Present State of the Universe. What Can We Expect From a Complete Cosmological Theory? An Overview of Quantum Effects in Cosmology. Parametric (Superadiabatic) Amplification of Classical Waves. Graviton Creation in the Inflationary Universe. Quantum States of a Harmonic Oscillator. Squeezed Quantum States of Relic Gravitons and Primordial Density Perturbations. Quantum Cosmology, Minisuperspace Models and Inflation. From the Space of Classical Solutions to the Space of Wave Functions. On the Probability of Quantum Tunneling From "Nothing". Duration of Inflation
We discuss the creation of massless particles in a Universe, which transits from a radiation-dominated era to any other expansion law. We calculate in detail the generation of gravitons during the transition to a matter dominated era. We show that the resulting gravitons generated in the standard radiation/matter transition are negligible. We use our result to constrain one or more previous matter dominated era, or any other expansion law, which may have taken place in the early Universe. We also derive a general formula for the modification of a generic initial graviton spectrum by an early m…
We consider the production of gravitons in an inflationary cosmology by approximating each epoch of change in the equation of state as sudden, from which a simple analytic graviton mode function has been derived. We use this mode function to compute the graviton spectral energy density and the tensor-induced cosmic microwave background anisotropy. The results are then compared to the numerical calculations which incorporate a smooth radiation-matter phase transition. We find that the sudden approximation is a fairly good method. Besides, in determining the frequency range and amplitude of the …
A large class of quintessential inflationary models, recently proposed by Peebles and Vilenkin, leads to post-inflationary phases whose effective equation of state is stiffer than radiation. The expected gravitational waves logarithmic energy spectra are tilted towards high frequencies and characterized by two parameters: the inflationary curvature scale at which the transition to the stiff phase occurs and the number of (non conformally coupled) scalar degrees of freedom whose decay into fermions triggers the onset of a gravitational reheating of the Universe. Depending upon the parameters of…
A characteristic spectrum of relic gravitational radiation is produced by a period of ``stringy inflation" in the early universe. This spectrum is unusual, because the energy-density rises rapidly with frequency. We show that correlation experiments with the two gravitational wave detectors being built for the Laser Interferometric Gravitational Observatory (LIGO) could detect this relic radiation, for certain ranges of the parameters that characterize the underlying string cosmology model.
Although matter contributions to the graviton self-energy $-i[\mbox{}^{μν} Σ^{ρσ}](x;x')$ must be separately conserved on $x^μ$ and ${x'}^μ$, graviton contributions obey the weaker constraint of the Ward identity, which involves a divergence on both coordinates. On a general homogeneous and isotropic background this leads to just four structure functions for matter contributions but nine structure functions for graviton contributions. We propose a convenient parameterization for these nine structure functions. We also apply the formalism to explicit one loop computations of $-i[\mbox{}^{μν} Σ^…
The spectra of the relic gravitons are customarily normalized in the low-frequency domain where the signal of the concordance paradigm is expected to peak and this is why their contribution to the temperature and polarization anisotropies of the microwave background is only described by the tensor to scalar ratio. If the consistency relations are broken, the same strategy is accomplished by introducing the tensor spectral index as a further independent parameter. When the dominant component of the spectral energy density is distributed for frequencies much larger than the aHz, the logic behind…
A large class of string-cosmology backgrounds leads to a spectrum of relic stochastic gravitational waves, strongly tilted towards high frequencies, and characterized by two basic parameters of the cosmological model. We estimate the required sensitivity for detection of the predicted gravitational radiation and show that a region of our parameter space is within reach for some of the planned gravitational-wave detectors.
We discuss the effect of Kaluza-Klein (KK) modes of bulk metric perturbations on the second Randall-Sundrum (RS II) type brane cosmology, taking the possible backreaction in the bulk and on the brane into account. KK gravitons may be produced via quantum fluctuations during a de Sitter (dS) inflating phase of our brane universe. In an effective 4-dimensional theory in which one integrates out the extra-dimensional dependence in the action, KK gravitons are equivalent to massive gravitons on the brane with masses $m>3H/2$, where $H$ represents the expansion rate of a dS brane. Thus production o…
Provided the consistency relations are not violated, the recent Bicep2 observations pin down the absolute normalization, the spectral slope and the maximal frequency of the cosmic graviton background produced during inflation. The properly normalized spectra are hereby computed from the lowest frequencies (of the order of the present Hubble rate) up to the highest frequency range in the GHz region. Deviations from the conventional paradigm cannot be excluded and are examined by allowing for different physical possibilities including, in particular, a running of the tensor spectral index, an ex…
We compute the spectrum of relic gravitons in a model of string cosmology. In the low- and in the high-frequency limits we reproduce known results. The full spectrum, however, also displays a series of oscillations which could give a characteristic signature at the planned LIGO/VIRGO detectors. For special values of the parameters of the model the signal reaches its maximum already at frequencies accessible to LIGO and VIRGO and it is close to the sensitivity of first generation experiments.
We study braneworlds in a five dimensional bulk, where cosmological expansion is mimicked by motion through AdS$_5$. We show that the five dimensional graviton reduces to the four dimensional one in the late time approximation of such braneworlds. Inserting a fixed regulator brane far from the physical brane, we investigate quantum graviton production due to the motion of the brane. We show that the massive Kaluza-Klein modes decouple completely from the massless mode and they are not generated at all in the limit where the regulator brane position goes to infinity. In the low energy limit, th…
The leading quantum correction to Einstein-Hilbert Hamiltonian coming from large volume vacuum isotropic loop quantum cosmology, is independent of quantization ambiguity parameters. It is shown here that this correction can be viewed as finite volume gravitational Casimir energy due to one-loop `graviton' contributions. In vacuum case sub-leading quantum corrections and in non-vacuum case even leading quantum correction depend on ambiguity parameters. It may be recalled that these are in fact analogous features of perturbative quantum gravity where it is well-known that pure gravity (on-shell)…
The effective anisotropic stresses induced by the scalar modes of the geometry depend on the coordinate system so that the comparison of the competing results is ultimately determined by the evolution of the pivotal variables in each particular gauge. After arguing that the only reasonable physical coordinate systems for this problem are the ones where the gauge freedom is completely fixed (like the longitudinal and the uniform curvature gauges), we propose a novel gauge-invariant strategy for the comparison of gauge-dependent results. Instead of employing the pivotal variables of a given coor…
