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We apply the gradient expansion approximation to the light-cone gauge, obtaining a separate universe picture at non-linear order in perturbation theory within this framework. Thereafter, we use it to generalize the δ N formalism in terms of light-cone perturbations. As a consistency check, we demonstrate the conservation of the gauge invariant curvature perturbation on uniform density hypersurface ζ at the completely non-linear level. The approach studied provides a self-consistent framework to connect at non-linear level quantities from the primordial universe, such as ζ , written in terms of the light-cone parameters, to late time observables. [ABSTRACT FROM AUTHOR]

The optical depth τ is the least well determined parameter in the standard model of cosmology, and one whose precise value is important for both understanding reionization and for inferring fundamental physics from cosmological measurements. We forecast how well future epoch of reionization experiments could constraint τ using a symmetries-based bias expansion that highlights the special role played by anisotropies in the power spectrum on large scales. Given a parametric model for the ionization evolution inspired by the physical behavior of more detailed reionization simulations, we find that future 21 cm experiments could place tight constraints on the timing and duration of reionization and hence constraints on τ that are competitive with proposed, space-based CMB missions provided they can measure k ≈ 0.1 h Mpc-1 with a clean foreground wedge across redshifts spanning the most active periods of reionization, corresponding to ionization fractions 0.2 ≲ x ≲ 0.8. Significantly improving upon existing CMB-based measurements with next-generation 21 cm surveys would require substantially longer observations (∼ 5 years) than standard O(1000 hour) integration times. Precise measurements of smaller scales will not improve constraints on τ until a better understanding of the astrophysics of reionization is achieved. In the presence of noise and foregrounds even future 21 cm experiments will struggle to constrain τ if the ionization evolution deviates significantly from simple parametric forms.

We study the second-order scalar and density perturbations generated by Gaussian curvature perturbations and primordial gravitational waves in the radiation-dominated era. After presenting all the possible second-order source terms, we obtain the explicit expressions of the kernel functions and the power spectra of the second-order scalar perturbations. We show that the primordial gravitational waves might affect second-order energy density perturbation δ (2) = δ ρ (2) / ρ (0) significantly. The effects of primordial gravitational waves are studied in terms of different kinds of primordial power spectra. [ABSTRACT FROM AUTHOR]

An observational program focused on the high redshift (2

We present the first measurement of the Ly α forest baryon acoustic oscillations (BAO) shift parameter from cosmological simulations. In particular, we generate a suite of 1000 accurate effective field-level bias-based Ly α forest simulations of volume $V={(1\,{h}^{-1}\,\mathrm{Gpc})}^{3}$ at z = 2, both in real and redshift space, calibrated upon two fixed-and-paired cosmological hydrodynamic simulations. To measure the BAO, we stack the 3D power spectra of the 1000 different realizations, compute the average, and use a model accounting for a proper smooth-peak component decomposition of the power spectrum, to fit it via an efficient Markov Chain Monte Carlo scheme estimating the covariance matrices directly from the simulations. We report the BAO shift parameters to be $\alpha ={0.9969}_{-0.0014}^{+0.0014}$ and $\alpha ={0.9905}_{-0.0027}^{+0.0027}$ in real and redshift space, respectively. We also measure the bias b _lya and the BAO broadening parameter Σ _nl , finding ${b}_{\mathrm{lya}}=-{0.1786}_{-0.0001}^{+0.0001}$ and ${{\rm{\Sigma }}}_{\mathrm{nl}}={3.87}_{-0.20}^{+0.20}$ in real space, and ${b}_{\mathrm{lya}}=-{0.073}_{-0.004}^{+0.005}$ and ${{\rm{\Sigma }}}_{\mathrm{nl}}={6.55}_{-0.22}^{+0.23}$ in redshift space. Moreover, we measure the linear Kaiser factor ${\beta }_{\mathrm{lya}}={1.39}_{-0.18}^{+0.24}$ from the isotropic redshift space fit. Overall, we find evidence for a negative shift of the BAO peak at the ∼2.2 σ and ∼3.5 σ levels in real and redshift space, respectively. This work sets new important theoretical constraints on the Ly α forest BAO scale and offers a potential solution to the tension emerging from previous observational analysis, in light of ongoing and upcoming Ly α forest spectroscopic surveys, such as DESI, the Prime Focus Spectrograph Survey, and WEAVE-QSO.

The standard model of cosmology,, has been successful in describing many observations. With the improvement of the number and the accuracy of observations, some inconsistencies among key cosmological parameters of the model have emerged. Many alternative models are proposed to alleviate these tensions. On the other hand, some observations of peculiar velocity show higher values than expected in a universe which may contradict the cosmological principle. In this work, we used linear perturbation theory to measure bulk flow and parameter in two alternative cosmological models and XCDM. We compared measured bulk flows with the predictions and some observations. We did a analysis to see which model is preferred by data. We find that model predicts higher bulk flows and is more consistent with observational data but does not reduce tension. Bulk flows measured in the XCDM model are lower compared to . However, this model can reconcile tension.

Unimodular gravity is one of the oldest geometric gravity theories and alternatives to general relativity. Essentially, it is based on the Einstein–Hilbert Lagrangian with an additional constraint on the determinant of the metric. It can be explicitly shown that unimodular gravity can be recast as general relativity in the presence of a cosmological constant. This fact has led to many discussions on the equivalence of both theories at the classical and quantum levels. Here, we present an analysis focused on the classical scalar perturbations around a cosmological background. We focus on the unusual situation in which the typical conservation laws are not adopted. The discussion is extended to the case where a non-minimal coupled scalar field is introduced. We also present a gauge-invariant analysis showing that perturbations in unimodular gravity display instabilities. Our results reinforce that the equivalence is not verified completely at a cosmological perturbative level. [ABSTRACT FROM AUTHOR]

The peculiar velocities of biased tracers of the cosmic density field contain important information about the growth of large scale structure and generate anisotropy in the observed clustering of galaxies. Using N-body data, we show that velocity expansions for halo redshift-space power spectra are converged at the percent-level at perturbative scales for most line-of-sight angles μ when the first three pairwise velocity moments are included, and that the third moment is well-approximated by a counterterm-like contribution. We compute these pairwise-velocity statistics in Fourier space using both Eulerian and Lagrangian one-loop perturbation theory using a cubic bias scheme and a complete set of counterterms and stochastic contributions. We compare the models and show that our models fit both real-space velocity statistics and redshift-space power spectra for both halos and a mock sample of galaxies at sub-percent level on perturbative scales using consistent sets of parameters, making them appealing choices for the upcoming era of spectroscopic, peculiar-velocity and kSZ surveys.

We examine the effects of massive primordial black holes (PBHs) on cosmic structure formation, employing both a semianalytical approach and cosmological simulations. Our simulations incorporate PBHs with a monochromatic mass distribution centered around 10 ^6 M _⊙ , constituting a fraction of 10 ^−2 to 10 ^−4 of the dark matter (DM) in the Universe, with the remainder being collisionless particle DM. Additionally, we conduct a ΛCDM simulation for comparative analysis with runs that include PBHs. At smaller scales, halos containing PBHs exhibit similar density and velocity dispersion profiles to those without PBHs. Conversely, at larger scales, PBHs can expedite the formation of massive halos and reside at their centers owing to the “seed effect.” To analyze the relative distribution of PBH host halos compared to non-PBH halos, we apply nearest neighbor statistics. Our results suggest that PBH host halos, through gravitational influence, significantly impact the structure formation process, compared to the ΛCDM case, by attracting and engulfing nearby newly formed minihalos. Should PBHs constitute a fraction of DM significantly larger than ∼10 ^−3 , almost all newly formed halos will be absorbed by PBH-seeded halos. Consequently, our simulations predict a bimodal feature in the halo mass function, with most of the massive halos containing at least one PBH at their core and the rest being less massive non-PBH halos.

We study the second-order scalar and density perturbations generated by Gaussian curvature perturbations and primordial gravitational waves in the radiation-dominated era. After presenting all the possible second-order source terms, we obtain the explicit expressions of the kernel functions and the power spectra of the second-order scalar perturbations. We show that the primordial gravitational waves might affect second-order energy density perturbation δ(2)=δρ(2)/ρ(0) significantly. The effects of primordial gravitational waves are studied in terms of different kinds of primordial power spectra.

Density-reconstruction sharpens the baryon acoustic oscillations signal by undoing some of the smoothing incurred by nonlinear structure formation. In this paper we present an analytical model for reconstruction based on the Zeldovich approximation, which for the first time includes a complete set of counterterms and bias terms up to quadratic order and can fit real and redshift-space data pre- and post-reconstruction data in both Fourier and configuration space over a wide range of scales. We compare our model to n-body data at z = 0 from the DarkSky simulation [1], finding sub-percent agreement in both real space and in the redshift-space power spectrum monopole out to k = 0.4 h Mpc-1, and out to k = 0.2 h Mpc-1 in the quadrupole, with comparable agreement in configuration space. We compare our model with several popular existing alternatives, updating existing theoretical results for exponential damping in wiggle/no-wiggle splits of the BAO signal and discuss the usually-ignored effect of higher bias contributions on the reconstructed signal. In the appendices, we re-derive the former within our formalism, present exploratory results on higher-order corrections due to nonlinearities inherent to reconstruction, and present numerical techniques with which to calculate the redshift-space power spectrum of biased tracers within the Zeldovich approximation.

We explore Lagrangian perturbation theory (LPT) for biased tracers in the presence of two fluids, focusing on the case of cold dark matter (CDM) and baryons. The presence of two fluids induces corrections to the Lagrangian bias expansion and tracer advection, both of which we formulate as expansions in the three linear modes of the Lagrangian equations of motion. We compute the linear-order two-fluid corrections in the Zeldovich approximation, finding that modifications to the bias expansion and tracer advection both enter as percent-level corrections over a large range of wavenumbers at low redshift and draw parallels with the Eulerian formalism. We then discuss nonlinear corrections in the two-fluid picture, and calculate contributions from the relative velocity effect (vr2) at one loop order. Finally, we conduct an exploratory Fisher analysis to assess the impact of two-fluid corrections on baryon acoustic oscillations (BAO) measurements, finding that while modest values of the relative bias parameters can introduce systematic biases in the measured BAO scale of up to 0.5 σ, fitting for these effects as additional parameters increases the error bar by less than 30% across a wide range of bias values.

We explore and compare different ways large-scale structure observables in redshift-space and real space can be connected. These include direct computation in Lagrangian space, moment expansions and two formulations of the streaming model. We derive for the first time a Fourier space version of the streaming model, which yields an algebraic relation between the real- and redshift-space power spectra which can be compared to earlier, phenomenological models. By considering the redshift-space 2-point function in both configuration and Fourier space, we show how to generalize the Gaussian streaming model to higher orders in a systematic and computationally tractable way. We present a closed-form solution to the Zeldovich power spectrum in redshift space and use this as a framework for exploring convergence properties of different expansion approaches. While we use the Zeldovich approximation to illustrate these results, much of the formalism and many of the relations we derive hold beyond perturbation theory, and could be used with ingredients measured from N-body simulations or in other areas requiring decomposition of Cartesian tensors times plane waves. We finish with a discussion of the redshift-space bispectrum, bias and stochasticity and terms in Lagrangian perturbation theory up to 1-loop order.

We develop a complete Hamiltonian approach to the theory of perturbations around any spatially homogeneous spacetime. We employ the Dirac method for constrained systems which is well-suited to cosmological perturbations. We refine the method via the so-called Kuchař parametrization of the kinematical phase space. We separate the gauge-invariant dynamics of the three-surfaces from the three-surface deformations induced by linear coordinate transformations. The canonical group of the three-surface deformations and the complete space of gauge-fixing conditions are explicit in our approach. We introduce a frame in the space of gauge-fixing conditions and use it to considerably simplify the prescription for gauge-fixing, partial gauge-fixing and spacetime reconstruction. Finally, we illustrate our approach by considering the perturbed Kasner Universe, for which we discuss two kinds of gauges that correspond respectively to the Coulomb-like and the Lorenz-like gauge in electrodynamics. [ABSTRACT FROM AUTHOR]

We study the dynamics of gauge-invariant scalar perturbations in cosmological scenarios with a modified Friedmann equation, such as quantum gravity bouncing cosmologies. We work within a separate universe approximation which captures wavelengths larger than the cosmological horizon; this approximation has been successfully applied to loop quantum cosmology and group field theory. We consider two variables commonly used to characterise scalar perturbations: the curvature perturbation on uniform-density hypersurfaces ζ and the comoving curvature perturbation R . For standard cosmological models in general relativity as well as in loop quantum cosmology, these quantities are conserved and equal on super-horizon scales for adiabatic perturbations. Here we show that while these statements can be extended to a more general form of modified Friedmann equations similar to that of loop quantum cosmology, in other cases, such as the simplest group field theory bounce scenario, ζ is conserved across the bounce whereas R is not. We relate our results to approaches based on a second-order equation for a single perturbation variable, such as the Mukhanov–Sasaki equation. [ABSTRACT FROM AUTHOR]

Unimodular gravity is one of the oldest geometric gravity theories and alternatives to general relativity. Essentially, it is based on the Einstein–Hilbert Lagrangian with an additional constraint on the determinant of the metric. It can be explicitly shown that unimodular gravity can be recast as general relativity in the presence of a cosmological constant. This fact has led to many discussions on the equivalence of both theories at the classical and quantum levels. Here, we present an analysis focused on the classical scalar perturbations around a cosmological background. We focus on the unusual situation in which the typical conservation laws are not adopted. The discussion is extended to the case where a non-minimal coupled scalar field is introduced. We also present a gauge-invariant analysis showing that perturbations in unimodular gravity display instabilities. Our results reinforce that the equivalence is not verified completely at a cosmological perturbative level.

Two recent articles (Moncrief V. 2015 In General relativity and gravitation-A centennial perspective (eds A Asthekar, B Berger, J Isenberg, M MacCallum), pp. 480–498. Cambridge, UK: Cambridge University Press; Moncrief V, Mondal P. 2019 Pure Appl. Math. Q.15, 921–965. (doi:10.4310/PAMQ.2019.v15.n3.a7)) suggested an interesting dynamical mechanism within the framework of the vacuum Einstein flow (or Einstein-Λ flow if a positive cosmological constant Λ is included) which suggests that many closed (compact without boundary) manifolds that do not support homogeneous and isotropic metrics at all will nevertheless evolve to be asymptotically compatible with the observed approximate homogeneity and isotropy of the physical universe. These studies however did not include matter sources. Therefore, the aim of the present study is to include suitable matter sources and investigate whether one is able to draw a similar conclusion. This article is part of the theme issue 'The future of mathematical cosmology, Volume 1'. [ABSTRACT FROM AUTHOR]

We apply an orthogonalization procedure on the effective field theory of large scale structure (EFT of LSS) shapes, relevant for the angle-averaged bispectrum and non-Gaussian covariance of the matter power spectrum at one loop. Assuming natural-sized EFT parameters, this identifies a linear combination of EFT shapes - referred to as the principal shape - that gives the dominant contribution for the whole kinematic plane, with subdominant combinations suppressed by a few orders of magnitude. For the covariance, our orthogonal transformation is in excellent agreement with a principal component analysis applied to available data. Additionally we find that, for both observables, the coefficients of the principal shapes are well approximated by the EFT coefficients appearing in the squeezed limit, and are thus measurable from power spectrum response functions. Employing data from N-body simulations for the growth-only response, we measure the single EFT coefficient describing the angle-averaged bispectrum with O(10%) precision. These methods of shape orthogonalization and measurement of coefficients from response functions are valuable tools for developing the EFT of LSS framework, and can be applied to more general observables.

Sufficiently large scalar perturbations in the early universe can create overdense regions that collapse into primordial black holes (PBHs). This process is accompanied by the emission of scalar-induced gravitational waves that behave like an extra radiation component, thus contributing to the relativistic degrees of freedom ( N _eff ). We show that the cosmological constraints on N _eff can be used to pose stringent limits on PBHs created from this particular scenario as well as the relevant small-scale curvature perturbation ( ${{ \mathcal P }}_{{ \mathcal R }}(k)$ ). We show that the combination of cosmic microwave background (CMB), baryon acoustic oscillation, and Big Bang nucleosynthesis data sets can exclude supermassive PBHs with peak mass M _• ∈ [5 × 10 ^5 , 5 × 10 ^10 ] M _⊙ as the major component of dark matter, while the detailed constraints depend on the shape of the PBHs’ mass distribution. A future CMB mission such as CMB-S4 can greatly broaden this constraint window to M _• ∈ [8 × 10 ^−5 , 5 × 10 ^10 ] M _⊙ , covering substellar masses. These limits on PBHs correspond to a tightened constraint on ${{ \mathcal P }}_{{ \mathcal R }}$ on scales of k ∈ [10, 10 ^22 ] Mpc ^−1 , much smaller than those probed by direct CMB and large-scale structure power spectra.

We study a model of inflation with multiple pseudo-scalar fields coupled to a U (1) gauge field through Chern–Simons interactions. Because of parity-violating interactions, one polarization of the gauge field is amplified, yielding to an enhanced curvature perturbation power spectrum. Inflation proceeds in multiple stages, as each pseudo-scalar field rolls toward its minimum, yielding to distinct multiple peaks in the curvature perturbation power spectra at various scales during inflation. The localized peaks in the power spectra generate primordial black holes that can furnish a large fraction of dark matter abundance. In addition, gravitational waves with nontrivial spectra are generated that are in the sensitivity ranges of various forthcoming GW observatories.

Fermi Normal Coordinates (FNC) are a useful frame for isolating the locally observable, physical effects of a long-wavelength spacetime perturbation. Their cosmological application, however, is hampered by the fact that they are only valid on scales much smaller than the horizon. We introduce a generalization that we call Conformal Fermi Coordinates (CFC). CFC preserve all the advantages of FNC, but in addition are valid outside the horizon. They allow us to calculate the coupling of long- and short-wavelength modes on all scales larger than the sound horizon of the cosmological fluid, starting from the epoch of inflation until today, by removing the complications of the second order Einstein equations to a large extent, and eliminating all gauge ambiguities. As an application, we present a calculation of the effect of long-wavelength tensor modes on small scale density fluctuations. We recover previous results, but clarify the physical content of the individual contributions in terms of locally measurable effects and "projection" terms.

We study the dynamics of gauge-invariant scalar perturbations in cosmological scenarios with a modified Friedmann equation, such as quantum gravity bouncing cosmologies. We work within a separate universe approximation which captures wavelengths larger than the cosmological horizon; this approximation has been successfully applied to loop quantum cosmology and group field theory. We consider two variables commonly used to characterise scalar perturbations: the curvature perturbation on uniform-density hypersurfaces ζ and the comoving curvature perturbation R. For standard cosmological models in general relativity as well as in loop quantum cosmology, these quantities are conserved and equal on super-horizon scales for adiabatic perturbations. Here we show that while these statements can be extended to a more general form of modified Friedmann equations similar to that of loop quantum cosmology, in other cases, such as the simplest group field theory bounce scenario, ζ is conserved across the bounce whereas R is not. We relate our results to approaches based on a second-order equation for a single perturbation variable, such as the Mukhanov–Sasaki equation.

In cosmological perturbation theory it is convenient to use the scalar, vector, tensor basis as defined according to how these components transform under three-dimensional rotations. In attempting to solve the fluctuation equations that are automatically written in terms of gauge-invariant combinations of these components, the equations are taken to break up into separate scalar, vector and tensor sectors, the decomposition theorem. Here, without needing to specify a gauge, we solve the fluctuation equations exactly for some standard cosmologies, to show that in general the various gauge-invariant combinations only separate at a higher-derivative level. To achieve separation at the level of the fluctuation equations themselves one has to assume boundary conditions for the higher-derivative equations. While spatially asymptotic boundary conditions suffice for fluctuations around a de Sitter background or a spatially flat Robertson–Walker background, for fluctuations around a spatially non-flat Robertson–Walker background one additionally has to require that the fluctuations be well-behaved at the origin. We show that in certain cases the gauge-invariant combinations themselves involve both scalars and vectors. For such cases there is no decomposition theorem for the individual scalar, vector and tensor components themselves as that would violate gauge invariance, but for the gauge-invariant combinations there still can be. Given the lack of manifest covariance (though not of covariance itself) in defining a basis with respect to three-dimensional rotations, we introduce an alternate scalar, vector, tensor basis whose components are defined according to how they transform under four-dimensional general coordinate transformations. With this basis the fluctuation equations greatly simplify, and while one can again break them up into separate gauge-invariant sectors at the higher-derivative level, in general we find that even with boundary conditions we do not obtain a decomposition theorem in which the fluctuations separate at the level of the fluctuation equations themselves. [ABSTRACT FROM AUTHOR]

In this paper, the criterion used to determine whether a density perturbation will collapse to form a primordial black hole (PBH) is re-examined in respect of its use to determine the abundance of PBHs. There is particular focus on which parameter to use, the time at which the abundance should be calculated, and the use of different smoothing functions. It is concluded that, with the tools currently available, the smoothed density contrast should be used rather than the peak value, and should be calculated from the time-independent component of the density contrast in the super-horizon limit (long before perturbations enter the horizon) rather than at horizon crossing. For the first time, the effect of the choice of smoothing function upon the formation criterion is calculated, and, for a given abundance of PBHs, it is found that the uncertainty in the amplitude of the power spectrum due to this is 𝒪 (1 0 %) , an order of magnitude smaller than that suggested by previous calculations. The relation between the formation criterion stated in terms of the density contrast and the curvature perturbation ℛ is also discussed. [ABSTRACT FROM AUTHOR]

Can dark matter be stabilized by charge conservation, just as the electron is in the standard model? We examine the possibility that dark matter is hidden, that is, neutral under all standard model gauge interactions, but charged under an exact U(1) gauge symmetry of the hidden sector. Such candidates are predicted in WIMPless models, supersymmetric models in which hidden dark matter has the desired thermal relic density for a wide range of masses. Hidden charged dark matter has many novel properties not shared by neutral dark matter: (1) bound state formation and Sommerfeld-enhanced annihilation after chemical freeze out may reduce its relic density, (2) similar effects greatly enhance dark matter annihilation in protohalos at redshifts of z ∼ 30, (3) Compton scattering off hidden photons delays kinetic decoupling, suppressing small scale structure, and (4) Rutherford scattering makes such dark matter self-interacting and collisional, potentially impacting properties of the Bullet Cluster and the observed morphology of galactic halos. We analyze all of these effects in a WIMPless model in which the hidden sector is a simplified version of the minimal supersymmetric standard model and the dark matter is a hidden sector stau. We find that charged hidden dark matter is viable and consistent with the correct relic density for reasonable model parameters and dark matter masses in the range 1GeVm ≲ m X ≲ 10 TeV. At the same time, in the preferred range of parameters, this model predicts cores in the dark matter halos of small galaxies and other halo properties that may be within the reach of future observations. These models therefore provide a viable and well-motivated framework for collisional dark matter with Sommerfeld enhancement, with novel implications for astrophysics and dark matter searches. © 2009 IOP Publishing Ltd and SISSA.

Generally the Brans–Dicke (BD) theory reduces to general relativity (GR) in the limit ω → ∞ if the scalar field goes as ϕ ∝ 1 / ω. However, it is also known that there are examples with ϕ ∝ 1 / ω that does not tend to GR. We discuss another case: a homogeneous and isotropic universe filled with stiff matter. The power of time dependence of these solutions does not depend on ω , and there is no GR limit even though we have ϕ ∝ 1 / ω. A perturbative and a dynamical system analysis of this exotic case are carried out. [ABSTRACT FROM AUTHOR]

We provide two derivations of the baryonic equations that can be straightforwardly implemented in existing Einstein−Boltzmann solvers. One of the derivations begins with an action principle, while the other exploits the conservation of the stress-energy tensor. While our result is manifestly covariant and satisfies the Bianchi identities, we point out that this is not the case for the implementation of the seminal work by Ma and Bertschinger and in the existing Boltzmann codes. We also study the tight coupling approximation up to the second order without choosing any gauge using the covariant full baryon equations. We implement the improved baryon equations in a Boltzmann code and investigate the change in the estimate of cosmological parameters by performing an MCMC analysis. With the covariantly correct baryon equations of motion, we find 1 % deviation for the best fit values of the cosmological parameters that should be taken into account. While in this paper, we study the Λ CDM model only, our baryon equations can be easily implemented in other models and various modified gravity theories.

We use the method of the Lewis-Riesenfeld invariant to analyze the dynamical properties of the Mukhanov-Sasaki Hamiltonian and, following this approach, investigate whether we can obtain possible candidates for initial states in the context of inflation considering a quasi-de Sitter spacetime. Our main interest lies in the question of to which extent these already well-established methods at the classical and quantum level for finitely many degrees of freedom can be generalized to field theory. As our results show, a straightforward generalization does in general not lead to a unitary operator on Fock space that implements the corresponding time-dependent canonical transformation associated with the Lewis-Riesenfeld invariant. The action of this operator can be rewritten as a time-dependent Bogoliubov transformation, where we also compare our results to already existing ones in the literature. We show that its generalization to Fock space has to be chosen appropriately in order to not violate the Shale-Stinespring condition. Furthermore, our analysis relates the Ermakov differential equation that plays the role of an auxiliary equation, whose solution is necessary to construct the Lewis-Riesenfeld invariant, as well as the corresponding time-dependent canonical transformation, to the defining differential equation for adiabatic vacua. Therefore, a given solution of the Ermakov equation directly yields a full solution of the differential equation for adiabatic vacua involving no truncation at some adiabatic order. As a consequence, we can interpret our result obtained here as a kind of non-squeezed Bunch-Davies mode, where the term non-squeezed refers to a possible residual squeezing that can be involved in the unitary operator for certain choices of the Bogoliubov map.