Sachs-Wolfe Research Articles

On Low Hubble Expansion Rate from Planck Data Anomalies

From the linear perturbations of Nash–Greene fluctuations of a background metric, we obtain profiles of Hubble function evolution H(z) and fσ8(z) measurements as compared with the ΛCDM results at intermediate redshifts 0.1<z<1. For parameter estimation, we use joint data from Planck Cosmic Microwave Background (CMB) likelihoods of CMB temperature and polarization angular power spectra, Barionic Acoustic Oscillations (BAO) and local measurements of Hubble constant H0 from the Hubble Space Telescope (HST). We analyze the stability of the effective Newtonian constant Geff and its agreement with Big Bang Nucleosynthesis (BBN) constraints. We show that our results are highly compatible with the ΛCDM paradigm, rather extending the perspective for further studies on redshift-space galaxy clustering data. Moreover, we obtain the CMB TT angular spectra with the Integrated Sachs–Wolfe (ISW) effect, which is weakened on low-l scales. The resulting linear matter power spectrum P(k) profile is also compatible with ΛCDM results but somewhat degenerate with an early dark energy (DE) contribution. Finally, posing a dilemma to the solution of Hubble tension, our results indicate a low Hubble expansion rate suggesting possible anomalies in Planck data in consonance with the recent South Pole Telescope (SPT-3G) data.

Planck integrated Sachs-Wolfe-lensing likelihood and the CMB temperature

We present a new Planck CMB lensing-CMB temperature cross-correlation likelihood that can be used to constrain cosmology via the Integrated Sachs-Wolfe (ISW) effect. CMB lensing is an excellent tracer of ISW, and we use the latest PR4 Planck data maps and lensing reconstruction to produce the first public Planck likelihood to constrain this signal. We demonstrate the likelihood by constraining the CMB background temperature from Planck data alone, where the ISW-lensing cross-correlation is a powerful way to break the geometric degeneracy, substantially improving constraints from the CMB and lensing power spectra alone.

Constraining CMB physical processes using Planck 2018 data

This paper aims to perform a phenomenological parametrisation of the standard cosmological model, ΛCDM, to weigh the different physical processes that define the pattern of the angular power spectra of the Cosmic Microwave Background (CMB) anisotropies. We use six phenomenological amplitudes to account for the Sachs-Wolfe, early and late Integrated Sachs-Wolfe, polarization contribution, Doppler and lensing effects. To do so, we have adapted the CLASS Boltzmann code and used the Markov Chain Monte Carlo (MCMC) sampler of Cobaya to explore the Planck 2018 likelihood to constrain different combinations of cosmological and phenomenological parameters. Observing deviations of the mean values of the phenomenological amplitudes from the predictions of the ΛCDM model could be useful to resolve existing cosmological tensions. For the first time, a comprehensive analysis of the physical processes of the CMB has been performed using the Planck 2018 temperature, polarization and lensing power spectra. In a previous work, the phenomenological amplitudes were constrained using only the TT data, however, by including the polarization and lensing data we find that the constraints on these physical contributions are tighter. In addition, some degeneracies that appear only when considering TT data are completely broken by taking into account all Planck 2018 data. Consequently, models with more than three phenomenological amplitudes can be studied, which is prohibitive when only the temperature power spectrum is used. The results presented in this paper show that the Planck experiment can constrain all phenomenological amplitudes except the late Integrated Sachs-Wolfe effect. No inconsistencies were found with the ΛCDM model, and the largest improvements were obtained for the models that include the lensing parameter, AL .

Noether currents in theories with higher derivatives and theories with differential field transformation in action (DFTA)

The article is devoted to the investigation of the Noether currents and integrals of motion in the special subclass of theories with higher field derivatives -- theories under differential field transformations in action (DFTA). Under fairly general assertions, we derive a simple representation for the integrals as sums of "old" integrals of motion, terms, that vanishes on the "old" equations of motion and surface integrals. We show that for some cosmological theories of that kind with high order derivatives the last contribution can violate gauge invariance, specifically, diffeomorphisms. Then we investigate ambiguity of the Noether procedure for higher derivative theories and fix it in the way the problem resolves in quite general setting. The obtained results are discussed for the several extensions of the General Relativity, which are obtained by DFTA, in particular, for mimetic, disformal and Regge-Teitelboim theories of gravity.

Novel cosmological tests from combining galaxy lensing and the polarized Sunyaev-Zel’dovich effect

The polarized Sunyaev-Zel'dovich (pSZ) effect is sourced by the Thomson scattering of CMB photons from distant free electrons and yields a novel view of the CMB quadrupole throughout the observable Universe. Galaxy shear measures the shape distortions of galaxies, probing both their local environment and the intervening matter distribution. Both observables give interesting constraints on the cosmological model; here we ask: what can be learnt from their combination? The pSZ-shear cross-spectrum measures the shear-galaxy-polarization bispectrum and contains contributions from (1) the Sachs-Wolfe (SW) effect, (2) the integrated Sachs-Wolfe (ISW) effect, and (3) inflationary gravitational waves. Since the modes contributing to the pSZ signal are not restricted to the Earth's past lightcone, the low-redshift cross-spectra could provide a novel constraint on dark energy via the ISW effect, whilst the SW signal couples scalar modes at very different times but at similar positions; this provides a unique probe of the Universe's homogeneous time evolution. We give expressions for all major contributions to the shear, galaxy, and pSZ auto- and cross-spectra, and evaluate their detectability via Fisher forecasts. Despite significant theoretical utility, the cross-spectra will be challenging to detect: combining CMB-S4 with Rubin yields a $1.6\sigma$ detection of the ISW contribution, which increases to $5.2\sigma$ for a futuristic experiment involving CMB-HD and a higher galaxy sample density. For parity-even (parity-odd) tensors, we predict a $1\sigma$ limit of $\sigma(r) = 0.9$ ($0.2$) for CMB-S4 and Rubin, or $0.3$ ($0.06$) for the more futuristic setup. Whilst this is significantly better than the constraints from galaxy shear alone (and less sensitive to systematics), it is unlikely to be competitive, but may serve as a useful cross-check.

The First Direct Measurement of Gravitational Potential Decay Rate at Cosmological Scales and Improved Dark Energy Constraint

The integrated Sachs–Wolfe (ISW) effect probes the decay rate (DR) of large-scale gravitational potential and therefore provides a unique constraint on dark energy (DE). However, its constraining power is degraded by the ISW measurement, which relies on cross-correlating with the large-scale structure (LSS) and suffers from uncertainties in galaxy bias and matter clustering. In combination with lensing-LSS cross correlation, DR can be isolated in a way free of uncertainties in galaxy bias and matter clustering. We applied this proposal to the combination of the Data Release 8 galaxy catalog of DESI imaging surveys and Planck cosmic microwave background maps. We achieved the first DR measurement, with a total significance of 3.2σ. We verified the measurements at three redshift bins ([0.2, 0.4), [0.4, 0.6), [0.6, 0.8]), with two LSS tracers (the “low-density points” and the conventional galaxy positions). Despite its relatively low signal-to-noise ratio, the addition of DR significantly improves dark energy constraints, over Sloan Digital Sky Survey baryon acoustic oscillation (BAO) data alone or Pantheon supernovae (SNe) compilation alone. For flat wCDM cosmology, the improvement in the precision of Ω m is a factor of 1.8 over BAO and 1.5 over SNe. For the DE equation of state w, the improvement factor is 1.3 over BAO and 1.4 over SNe. These improvements demonstrate DR as a useful cosmological probe, and therefore we advocate its usage in future cosmological analysis.

Relativistic angular redshift fluctuations embedded in large scale varying gravitational potentials

We compute the linear order, general relativistic corrections to angular redshift fluctuations (ARF), a new cosmological observable built upon density-weighted two-dimensional (2D) maps of galaxy redshifts. We start with an existing approach for galaxy/source counts developed in the Newtonian gauge, and generalize it to ARF, modifying for this purpose a standard Boltzmann code. Our calculations allow us identifying the velocity terms as the leading corrections on large scales, emphasizing the sensitivity of ARF to peculiar, cosmological velocity fields. Just like for standard 2D clustering, the impact of gravitational lensing on ARF is dominant on small angular scales and for wide redshift shells, while the signatures associated to gravitational potentials are extremely small and hardly detectable. The ARF also present interesting correlation properties to anisotropies of the Cosmic Microwave Background (CMB): they are highly correlated to CMB lensing potential fluctuations, while also exhibiting a significant (S/N∼ 4–5) anti-correlation with the Integrated Sachs-Wolfe effect (ISW). This negative ARF×ISW signal is quite complementary to the standard 2D clustering×ISW correlation, since the former appears mostly at higher redshift (z ∼ 2) than the latter (z ≲ 1), and the combination of the two observables significantly increases the χ 2 statistics testing the null (no ISW) hypothesis. We conclude that ARF constitute a novel, alternative, and potentially powerful tool to constrain the nature of Dark Energy component that gives rise to the ISW.

Probing modified gravity with integrated Sachs-Wolfe CMB and galaxy cross-correlations

We use the cross-correlation power spectrum of the integrated Sachs-Wolfe (ISW) effect in the cosmic microwave background (CMB) temperature anisotropy and galaxy fluctuations to probe the physics of late-time cosmic acceleration. For this purpose, we focus on three models of dark energy that belong to a sub-class of Horndeski theories with the speed of gravity equivalent to that of light: Galileon Ghost Condensate (GGC), Generalized Cubic Covariant Galileon (GCCG), and K-mouflage. In the GGC and GCCG models, the existence of cubic-order scalar self-interactions allows a possibility for realizing negative ISW-galaxy cross-correlations, while the K-mouflage model predicts a positive correlation similar to the Λ-cold-dark-matter (ΛCDM) model. In our analysis, we fix the parameters of each model to their best-fit values derived from a baseline likelihood analysis with observational data from CMB, baryon acoustic oscillations, and supernovae type Ia. Then we fit those best-fit models to the ISW-galaxy cross-correlation power spectrum extracted from a collection of photometric redshift surveys. We find that both GGC and GCCG best-fit models degrade the fit to the ISW-galaxy cross-correlation data compared to ΛCDM best-fit model. This is attributed to the fact that, for their best-fit values constrained from the baseline likelihood, the cubic-order scalar self-interaction gives rise to suppressed ISW tails relative to ΛCDM. The K-mouflage best-fit model is largely degenerate with the ΛCDM best-fit model and has a positively correlated ISW-galaxy power close to that of ΛCDM.

Redshift-space fluctuations in stochastic gravitational wave background

We study the redshift-space fluctuations induced by a stochastic gravitational wave background (SGWB) via the Sachs-Wolfe effect. The redshift-space fluctuations can be encapsulated in a line-of-sight integral that is useful for studying the imprint of short-wavelength gravitational waves on the cosmic microwave background (CMB) anisotropy. We thus derive constraints on the SGWB from small-scale CMB anisotropy measurements. Our results reproduce the constraint on the short-wavelength SGWB, previously derived from the Planck and BICEP/Keck array CMB data with a CMB Boltzmann numerical code. Furthermore, we improve the constraint and extend it to shorter wavelengths by using the CMB measurements made by the Atacama Cosmology Telescope and the South Pole Telescope. Also, the integral provides us with a precise redshift fluctuation correlation between a pair of pulsars in pulsar timing measurements, which conveniently incorporates the effect of the pulsar term into a small-angle correlation. We further discuss the observation of pulsar pairs in globular clusters to look for this small-angle correlation.

A measurement of the integrated Sachs–Wolfe effect with the Rapid ASKAP Continuum Survey

ABSTRACT The evolution of the gravitational potentials on large scales due to the accelerated expansion of the Universe is an important and independent probe of dark energy, known as the integrated Sachs–Wolfe (ISW) effect. We measure this ISW effect through cross-correlating the cosmic microwave background maps from the Planck satellite with a radio continuum galaxy distribution map from the recent Rapid ASKAP Continuum Survey (RACS). We detect a positive cross-correlation at $\sim 2.8\, \sigma$ relative to the null hypothesis of no correlation. We parametrize the strength of the ISW effect through an amplitude parameter and find the constraints to be $A_{\mathrm{ISW}} = 0.94^{+0.42}_{-0.41}$, which is consistent with the prediction of an accelerating universe within the current concordance cosmological model, ΛCDM. The credible interval on this parameter is independent of the different bias models and redshift distributions that were considered when marginalizing over the nuisance parameters. We also detect a power excess in the galaxy autocorrelation angular power spectrum on large scales (ℓ ≤ 40), and investigate possible systematic causes.

CoLoRe: fast cosmological realisations over large volumes with multiple tracers

We present CoLoRe, a public software package to efficiently generate synthetic realisations of multiple cosmological surveys.CoLoRe can simulate the growth of structure with different degrees of accuracy, with the current implementation supporting lognormal fields, first, and second order Lagrangian perturbation theory.CoLoRe simulates the density field on an all-sky light-cone up to a desired maximum redshift, and uses it to generate multiple 2D and 3D maps: galaxy positions and velocities, lensing (shear, magnification, convergence), integrated Sachs-Wolfe effect, line intensity mapping, and line of sight skewers for simulations of the Lyman-α forest. We test the accuracy of the simulated maps against analytical theoretical predictions, and showcase its performance with a multi-survey simulation including DESI galaxies and quasars, LSST galaxies and lensing, and SKA intensity mapping and radio galaxies. We expect CoLoRe to be particularly useful in studies aiming to characterise the impact of systematics in multi-experiment analyses, quantify the covariance between different datasets, and test cross-correlation pipelines for near-future surveys.

Anisotropy of phase transition gravitational wave and its implication for primordial seeds of the Universe

We quantitatively study how the primordial density fluctuations are imprinted on the anisotropy of the phase transition gravitational wave (PTGW). Generated long before recombination and free from Silk damping, the anisotropic PTGW might reveal the density perturbation seeded from inflation or alternatives. We find new behaviors of the PTGW anisotropy power spectrum. The PTGW anisotropy is stronger than the anisotropy of the cosmic microwave background temperature at all scales, and the high-$\ell$ multiples are enhanced about 1 order due to the early integrated Sachs-Wolfe effect. Furthermore, differences in primordial power spectra at small scales manifest themselves more significantly on the angular power spectrum of PTGW anisotropy compared to that of the cosmic microwave background. These properties might provide a novel clue to understanding the primordial density perturbation of our early Universe and thereby complete our understanding of inflation theory. Taking nanohertz PTGW from dark matter models as a typical example, we obtain amplitudes of PTGW anisotropy which are about 4 or 3 orders weaker than the isotropic PTGW energy spectra.

Evidence for a high-zISW signal from supervoids in the distribution of eBOSS quasars

ABSTRACTThe late-time integrated Sachs-Wolfe (ISW) imprint of $R\gtrsim 100~h^{-1}\, \mathrm{Mpc}$ superstructures is sourced by evolving large-scale potentials due to a dominant dark energy component in the ΛCDM model. The aspect that makes the ISW effect distinctly interesting is the repeated observation of stronger-than-expected imprints from supervoids at z ≲ 0.9. Here we analyse the un-probed key redshift range 0.8 < z < 2.2 where the ISW signal is expected to fade in ΛCDM, due to a weakening dark energy component, and eventually become consistent with zero in the matter dominated epoch. On the contrary, alternative cosmological models, proposed to explain the excess low-z ISW signals, predicted a sign-change in the ISW effect at z ≈ 1.5 due to the possible growth of large-scale potentials that is absent in the standard model. To discriminate, we estimated the high-z ΛCDM ISW signal using the Millennium XXL mock catalogue, and compared it to our measurements from about 800 supervoids identified in the eBOSS DR16 quasar catalogue. At 0.8 < z < 1.2, we found an excess ISW signal with AISW ≈ 3.6 ± 2.1 amplitude. The signal is then consistent with the ΛCDM expectation (AISW = 1) at 1.2 < z < 1.5 where the standard and alternative models predict similar amplitudes. Most interestingly, we also observed an opposite-sign ISW signal at 1.5 < z < 2.2 that is in 2.7σ tension with the ΛCDM prediction. Taken at face value, these recurring hints for ISW anomalies suggest an alternative growth rate of structure in low-density environments at $\sim 100~h^{-1}\, \mathrm{Mpc}$ scales.

The Integrated Sachs Wolfe effect: unWISE and Planck constraints on dynamical dark energy

CMB photons redshift and blueshiftas they move through gravitational potentials Φwhile propagating across the Universe. If the potential is not constant in time, the photons will pick up a net redshift or blueshift, known as the Integrated Sachs-Wolfe (ISW) effect.In the z ≪ 1000 universe, Φ̇ is nonzero on large scales when the Universe transitions from matter to dark energy domination. This effect is only detectable in cross-correlation with large-scale structure at z ∼ 1. In this paper we present a 3.2σ detection of the ISW effect using cross-correlations between unWISE infrared galaxies and Planck CMB temperature maps. We use 3 tomographic galaxy samples spanning 0 < z < 2, allowing us to fully probe the dark energy domination era and the transition into matter domination. This measurement is consistent with ΛCDM (AISW = 0.96 ± 0.30).We study constraints on a particular class of dynamical dark energy models(where the dark energy equation of state is different in matter and dark energy domination),finding that unWISE-ISW improves constraintsfrom type Ia supernovae due to improved constraints on the time evolution of dark energy. When combining with BAO measurements, we obtain the tightest constraints on specific dynamical dark energy models. In the context of a phenomenological model for freezing quintessence, the Mocker model, we constrain the dark energy density within 10% at z < 2 using ISW, BAO and supernovae. Moreover, the ISW measurement itself provides an important independent check when relaxing assumptions about the theory of gravity, as it is sensitive to the gravitational potential rather than the expansion history.

The integrated Sachs–Wolfe effect in 4D Einstein–Gauss–Bonnet gravity

A novel 4-dimensional Einstein-Gauss-Bonnet (4D EGB) gravity has been proposed that asserts to bypass the Lovelock's theorem and to result in a non-trivial contribution to the gravitational dynamics in four-dimensional spacetime. In this work, we study the integrated Sachs-Wolfe (ISW) effect in the 4D EGB model. For this purpose, we calculate the evolution of the gravitational potential and the linear growth factor as a function of redshift for the 4D EGB model and compare it with the corresponding result obtained from the $\Lambda$-cold dark matter ($\Lambda$CDM) model. We also calculate the ISW-auto power spectrum and the ISW-cross power spectrum as functions of cosmic microwave background (CMB) multipoles for the 4D EGB model and compare those with the one obtained from the $\Lambda$CDM model. To do this, we use the strongest constraint on the coupling parameter proposed for the 4D EGB model. Additionally, to calculate the ISW effect for the 4D EGB model, we employ three large-scale structure surveys from different wavelengths. The results exhibit that the ISW effect in the 4D EGB model is higher than the one obtained from the $\Lambda$CDM model. Hence, we show that the 4D EGB model can amplify the amplitude of the ISW power spectrum, which can be considered as a relative advantage of the 4D EGB model comparing the $\Lambda$CDM one. Also, we indicate that the deviation from the $\Lambda$CDM model is directly proportional to the value of the dimensionless coupling parameter $\beta$.

Observational constraints and predictions of the interacting dark sector with field-fluid mapping

We consider an interacting field theory model that describes the interaction between dark energy-dark matter interaction. Only for a specific interaction term, this interacting field theory description has an equivalent interacting fluid description. For inverse power law potentials and linear interaction function, we show that the interacting dark sector model with field-fluid mapping is consistent with four cosmological data sets — Hubble parameter measurements (Hz), Baryonic Acoustic Oscillation data (BAO), Supernova Type Ia data (SN), and High redshift HII galaxy measurements (HIIG). More specifically, these data sets prefer a negative value of interaction strength in the dark sector and lead to consistent best-fit values of Hubble constant and other cosmological parameters. Having established that this interacting field theory model is consistent with cosmological observations, we obtain quantifying tools to distinguish between the interacting and non-interacting dark sector scenarios. We focus on the variation of the scalar metric perturbed quantities as a function of redshift related to structure formation, weak gravitational lensing, and the integrated Sachs-Wolfe effect. We show that the difference in the evolution becomes significant for z < 20, for all length scales, and the difference peaks at smaller redshift values z < 5. We then discuss the implications of our results for the upcoming missions.

The Integrated Sachs-Wolfe Effect in Interacting Dark Matter-Dark Energy Models

The Integrated Sachs-Wolfe Effect in Interacting Dark Matter-Dark Energy Models

Hint of a truncated primordial spectrum from the CMB large-scale anomalies

Context. Several satellite missions have uncovered a series of potential anomalies in the fluctuation spectrum of the cosmic microwave background temperature, including: (1) an unexpectedly low level of correlation at large angles, manifested via the angular correlation function, C(θ); and (2) missing power in the low multipole moments of the angular power spectrum, Cℓ. Aims. Their origin is still debated, however, due to a persistent lack of clarity concerning the seeding of quantum fluctuations in the early Universe. A likely explanation for the first of these appears to be a cutoff, kmin = (3.14 ± 0.36)×10−4 Mpc−1, in the primordial power spectrum, 𝒫(k). Our goal in this paper is twofold: (1) we examine whether the same kmin can also self-consistently explain the missing power at large angles, and (2) we confirm that the introduction of this cutoff in 𝒫(k) does not adversely affect the remarkable consistency between the prediction of Planck-ΛCDM and the Planck measurements at ℓ &gt; 30. Methods. We have used the publicly available code CAMB to calculate the angular power spectrum, based on a line-of-sight approach. The code was modified slightly to include the additional parameter (i.e., kmin) characterizing the primordial power spectrum. In addition to this cutoff, the code optimized all of the usual standard-model parameters. Results. In fitting the angular power spectrum, we found an optimized cutoff, kmin = (2.04−0.79+1.4) × 10−4 Mpc−1, when using the whole range of ℓ’s, and kmin = (3.3−1.3+1.7) × 10−4 Mpc−1, when fitting only the range ℓ ≤ 30, where the Sachs-Wolfe effect is dominant. Conclusions. These are fully consistent with the value inferred from C(θ), suggesting that both of these large-angle anomalies may be due to the same truncation in 𝒫(k).

Consistency tests of ΛCDM from the early integrated Sachs-Wolfe effect: Implications for early-time new physics and the Hubble tension

New physics increasing the expansion rate just prior to recombination is among the least unlikely solutions to the Hubble tension and would be expected to leave an important signature in the early integrated Sachs-Wolfe (eISW) effect, a source of cosmic microwave background (CMB) anisotropies arising from the time variation of gravitational potentials when the Universe was not completely matter dominated. Why, then, is there no clear evidence for new physics from the CMB alone, and why does the $\mathrm{\ensuremath{\Lambda}}$ cold dark matter ($\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$) model fit CMB data so well? These questions and the vastness of the Hubble tension theory model space provide the motivation for general consistency tests of $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$. I perform an eISW-based consistency test of $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ introducing the parameter ${A}_{\mathrm{eISW}}$, which rescales the eISW contribution to the CMB power spectra. A fit to Planck CMB data yields ${A}_{\mathrm{eISW}}=0.988\ifmmode\pm\else\textpm\fi{}0.027$, in perfect agreement with the $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ expectation ${A}_{\mathrm{eISW}}=1$ and posing an important challenge for early-time new physics, which I illustrate in a case study focused on early dark energy (EDE). I explicitly show that the increase in ${\ensuremath{\omega}}_{c}$ needed for EDE to preserve the fit to the CMB, which has been argued to worsen the fit to weak lensing and galaxy clustering measurements, is specifically required to lower the amplitude of the eISW effect, which would otherwise exceed $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$'s prediction by $\ensuremath{\approx}20%$: this is a generic problem beyond EDE that likely applies to most models enhancing the expansion rate around recombination. Early-time new physics models invoked to address the Hubble tension are therefore faced with the significant challenge of making a similar prediction to $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ for the eISW effect while not degrading the fit to other measurements in doing so.

Fluid approach of linear cosmological Nash–Greene perturbations

In this paper, we obtain numerical solutions of the linear cosmological Nash-Greene perturbation equations in an embedded four dimensional space-time. We propose an effective fluid approach with the full anisotropic stresses to interpret the consequences of the extrinsic curvature on the cosmic dynamics. The effective Newtonian function G e f f is used to estimate the model parameters once it must obey Big Bang Nucleosynthesis and solar constraints. To obtain sufficiently accurate precision on the cosmic fluid parameter w , we use the Dynamic Nested Sampling algorithm with DYNESTY code to the obtainment of viable solutions of numerical evolutions of the matter growth rate, scalar fluid velocity, anisotropic quantities and on the analysis of effective sound speed. By means of a modified version of CLASS Einstein-Boltzmann solver, called EFCLASS , we obtain prospects on the resulting CMB TT angular spectra with the integrated Sachs-Wolfe effect, which is weakened on low- l scales as compared with the Λ CDM profile. Moreover, the obtained linear matter power spectrum P ( k ) profiles are compatible with Λ CDM standard results. • Dynamical approach for extrinsic curvature by means of Nash-Greene theorem. • Subhorizon linear perturbations in the embedding context. • A fluid approach for the extrinsic curvature with the full presence of anisotropic stresses. • The related ISW effect is suppressed as compared with the Λ CDM profiles.