CMB Spectrum Research Articles

Observability of CMB spectrum distortions from dark matter annihilation

Even after dark matter chemically freezes out in the early universe, electromagnetic cascades from dark matter annihilation can still perturb the background photon spectrum when the universe temperature cools down to 0.5 keV. We revisit the CMB spectrum distortions caused by s-wave dark matter annihilation under the updated Planck data and the future CMB sensitivity, concluding that s-wave annihilation cannot create observable distortions under forecast sensitivities of the (Super-)PIXIE missions. We further detail the case of p-wave dark matter annihilation, demonstrating the observability of the primordial μ-distortion. Taking current constraints from primordial light elements, structure formations, cosmic electron-positron rays, and gamma rays, we find that the μ-distortion reaching the observational limit as large as μ ≃ 3 × 10-8 can only be realized with a dark matter mass at 10–50 MeV and a kinetic decoupling temperature around 1 keV. The upper bound of the p-wave annihilation cross section can be strengthened by an order of magnitude if the μ-distortion is not detected.

CMB spectral distortions from an axion-dark photon-photon interaction

The presence of a plethora of light spin 0 and spin 1 fields is motivated in a number of BSM scenarios, such as the axiverse. The study of the interactions of such light bosonic fields with the Standard Model has focused mostly on interactions involving only one such field, such as the axion (ϕ) coupling to photons, \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi F\\widetilde{F}$$\\end{document}, or the kinetic mixing between photon and the dark photon, FFD. In this work, we continue the exploration of interactions involving two light BSM fields and the standard model, focusing on the mixed axion-photon-dark-photon interaction \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi F{\\widetilde{F}}_{D}$$\\end{document}. If either the axion or dark photon are dark matter, we show that this interaction leads to conversion of the CMB photons into a dark sector particle, leading to a distortion in the CMB spectrum. We present the details of these unique distortion signatures and the resulting constraints on the \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi F{\\widetilde{F}}_{D}$$\\end{document} coupling. In particular, we find that for a wide range of masses, the constraints from these effect are stronger than on the more widely studied axion-photon coupling.

Axionic defects in the CMB: birefringence and gravitational waves

The evidence for a non-vanishing isotropic cosmic birefringence in recent analyses of the CMB data provides a tantalizing hint for new physics. Domain wall (DW) networks have recently been shown to generate an isotropic birefringence signal in the ballpark of the measured value when coupled to photons. In this work, we explore the axionic defects hypothesis in more detail and extending previous results to annihilating and late-forming networks, and by pointing out other smoking-gun signatures of the network in the CMB spectrum such as the anisotropic birefringent spectrum and B-modes. We also argue that the presence of cosmic strings in the network does not hinder a large isotropic birefringence signal because of an intrinsic environmental contribution coming from low redshifts thus leaving open the possibility that axionic defects can explain the signal. Regarding the remaining CMB signatures, with the help of dedicated 3D numerical simulations of DW networks, that we took as a proxy for the axionic defects, we show how the anisotropic birefringence spectrum combined with a tomographic approach can be used to infer the formation and annihilation time of the network. From the numerical simulations, we also computed the spectrum of gravitational waves (GWs) generated by the network in the post-recombination epoch and use previous searches for stochastic GW backgrounds in the CMB to derive for the first time a bound on the tension and abundance of networks with DWs that annihilate after recombination. Our bounds extend to the case where the network survives until the present time and improve over previous bounds by roughly one order of magnitude. Finally, we show the interesting prospects for detecting B-modes of DW origin with future CMB experiments.

Primordial black holes and inflation from double-well potentials

We investigate the formation of large peaks in the inflationary curvature power spectrum from double-well potentials. In such scenarios, the initial CMB spectrum is created at large field values. Subsequently, the inflaton will cross one of the minima and will decelerate rapidly as it reaches the local maximum at the origin, either falling back or crossing it. During this final phase, a significant peak in the curvature power spectrum can be generated. Our analysis reveals that this class of models produces more pronounced peaks than most quasi-inflection point scenarios with less tuning for the model parameters. Finally, we construct an explicit theoretically motivated inflationary scenario that is consistent with the latest CMB observations and capable of generating sufficiently large curvature perturbations for primordial black holes.

Untying the Growth Index to Relieve the σ8 Discomfort

The matter fluctuation parameter σ8 is, by model construction, degenerate with the growth index γ. Here, we study the effect on the cosmological parameter constraints by treating each independently from one another, considering σ8 as a free and non-derived parameter along with a free γ. We then try to constrain all parameters using three probes that span from deep to local redshifts, namely the CMB spectrum, the growth measurements from redshift space distortions fσ8, and the galaxy cluster counts. We also aim to assess the impact of this relaxation on the σ8 tension between its inferred CMB value in comparison to that obtained from local cluster counts. We also propose a more sophisticated correction, along with the classical one, that takes into account the impact of cosmology on the growth measurements when the parameters are varied in the Monte Carlo process, which consist in adjusting the growth to keep the observed power spectrum, integrated over all angles and scales, as invariant with the background evolution. We found by using the classical correction that untying the two parameters does not shift the maximum likelihood of either σ8 or γ, but it rather enables larger bounds with respect to when σ8 is a derived parameter, and that when considering CMB + fσ8, or when further combining with cluster counts albeit with tighter bounds. Precisely, we obtain σ8=0.809±0.043 and γ=0.613±0.046 in agreement with Planck’s constraint for the former and compatible with ΛCDM for the latter but with bounds wide enough to accommodate both values subject to the tensions. Allowing for massive neutrinos does not change the situation much. On the other hand, considering a tiered correction yields σ8=0.734±0.013 close to ∼1 σ for the inferred local values albeit with a growth index of γ=0.636±0.022 at ∼2 σ from its ΛCDM value. Allowing for massive neutrinos in this case yielded σ8=0.756±0.024, still preferring low values but with much looser constraints on γ=0.549±0.048 and a slight preference for Σmν∼0.19. We conclude that untying σ8 and γ helps in relieving the discomfort on the former between the CMB and local probes, and that careful analysis should be followed when using data products treated in a model-dependent way.

Cosmological constraints on the multiscalar field dark matter model

The main aim of this paper is to provide cosmological constraints on the Multi Scalar Field Dark Matter model (MSFDM), in which we assume the dark matter is made up of different ultra-light scalar fields. As a first approximation, we consider they are real and do not interact with each other. We study the equations for both the background and perturbations for $N$-fields and present the evolution of the density parameters, the mass power spectrum and the CMB spectrum. In particular, we focus on two scalar fields with several combinations for the potentials $V(\phi) = 1/2 m_{\phi}^2 \phi^2$, $V(\phi) = m_{\phi}^2f^2\left[1+\cos(\phi/f)\right]$ and $V(\phi) = m_{\phi}^2f^2\left[\cosh(\phi/f)-1\right]$, however the work, along with the code, could be easily extended to more fields. We use the data from BAO, Big Bang Nucleosynthesis, Lyman-$\alpha$ forest and Supernovae to find constraints on the sampling parameters for the cases of a single field and double field, along with the Bayesian evidence. We found that some combinations of the potentials get penalized through the evidence, however for others there is a preference as good as for the cold dark matter.

Modifications of CMB spectrum by nonextensive statistical mechanics

Cosmic microwave background radiation can supply us some most significant parts of the information on the universe. Some researchers believe that the gravitational system cannot be decribed by the standard statistical mechanics. In this article we apply Tsallis nonextensive statistical mechanics to investigate CMB spectrum and related cosmological processes. Based on recent observational data we find that the nonextensive statistical mechanics can modify the values of related physical quantities. Since the value of physical quantites have changed, some processes, such as recombination, can be affected. We have investigated the anisotropy of the CMB for two effects: the dipole anisotropy of CMB and the Sunyaev-Zel’dovich effect. We find that the dipole anisotropy of CMB cannot be modified by the nonextensive statistical mechanics. However, the standard result of the Sunyaev-Zel’sdovich effect should be modified by nonextensive statistical mechanics. In principle, future work can distinguish these effects.

Lensing power spectrum of the cosmic microwave background with deep polarization experiments

Precise reconstruction of the cosmic microwave background lensing potential can be achieved with deep polarization surveys by iteratively removing lensing-induced $B$ modes. We introduce a lensing spectrum estimator and its likelihood for such optimal iterative reconstruction. Our modeling shares similarities to the state-of-the-art likelihoods for quadratic estimator (QE)-based lensing reconstruction. In particular, we generalize the ${N}^{(0)}$ and ${N}^{(1)}$ lensing biases, and design a realization-dependent spectrum debiaser, making this estimator robust to uncertainties in the data modeling. We demonstrate unbiased recovery of the cosmology using map-based reconstructions, focussing on lensing-only cosmological constraints and neutrino mass measurement in combination with CMB spectra and acoustic oscillation data. We find this spectrum estimator is essentially optimal and with a diagonal covariance matrix. For a CMB-S4 survey, this likelihood can double the constraints on the lensing amplitude compared to the QE on a wide range of scales, while at the same time keeping numerical cost under control and being robust to errors.

μ-distortion around stupendously large primordial black holes

In a variety of mechanisms generating primordial black holes, eachblack hole is expected to form along with a surrounding underdense region thatroughly compensates the black hole mass. This region will propagateoutwards and expand as a shell at the speed of sound in the homogeneous background. Dissipation of the shell due to Silk damping could lead todetectable μ-distortion in the CMB spectrum: if black holes are rare on the last scattering surface, the signal(s) would be pointlike; whereas if there are a sufficient number of them, we could have a uniform distortion in the CMB sky. While the currentbound on the average μ-distortion is |μ̅| ≲ 10-4,the standard ΛCDM model predicts |μ̅| ∼ 10-8,which could possibly be detected in future missions. It is shown inthis work that the non-observation of μ̅ beyond ΛCDMcan place a new upper bound on the density of supermassive primordialblack holes within the mass range 106 M ☉≲ M ≲ 1015 M ☉.Furthermore, black holes with initial mass M ≳ 1012 M ☉could leave a pointlike distortion with μ ≳10-8 at anangular scale ∼ 1° in CMB, and its non-observation wouldimpose an even more stringent bound on the population of these stupendously large primordialblack holes.

Toward a concordance teleparallel cosmology. Part II. Linear perturbation

Late time cosmic acceleration may be achieved by modifying gravity on large scales.This should also have consequences on the evolution of perturbations.We thus extend our study of exponential infrared f(T) teleparallel gravity to examine the viability of the theory at the linear perturbation level, evaluating the full CMB and matter power spectra. As the theory does not introduce extra free parameters, it fits within the minimal six parameter space of standard . Using Planck 2018 CMB (TT+TE+EE+lensing) alone,best fits predict those parameters to be almost identical to ΛCDM, with slightly smaller χ2 min. The resulting H 0=72.24 ± 0.64 km/s/Mpc, which virtually alleviates the tension with local measurements, due to late time phantom behaviour. Inclusion of BAO data however reduces H 0, reflecting furthermore systematic deviations from data that are also present in supernova distances and the growth rate of structure (increasing the apparent tension in the latter case). As the theory, unlike other viable f(T) models, does not reduce to ΛCDM through extra free parameters, those conclusions are generic; applying to any modified gravity or dynamical dark energy with phantom behaviour. With best fit parameters, the present scenario produces a CMB spectrum almost identical to ΛCDM, with slight deviation at low-multipole ℓ < 30, where cosmic variance is large. The matter power spectrum is also quite close to ΛCDM; with percent level scale free modifications affecting modes significantly smaller than the horizon, arising primarily from modified background evolution. More significant deviations appear on larger scales, and may in principle distinguish modified gravity scenarios of the type studied here from dynamical dark energy.

Energy injection in pre-recombination era and EDGES detection

We study the possibility of explaining the recent EDGES detection by an energy injection in the pre-recombination era. Our aim is to show that the residue of this energy injection could give the resultant increase in the energy density at frequencies xe ≡ hν/(kT) ≃ 10−3, which is needed to explain the EDGES result. We consider two models of energy injection: Gaussian profile with a fixed redshift of injection and radiative decay of a non-relativistic particle. We show that the energy injection should occur in the redshift range z≲ 4 × 103 to prevent free-free processes from thermalizing the injected energy. The injected energy should be nearly 200–1000 times the CMB intensity at xe ≃ 10−3 to obtain the requisite residue. A large fraction of the injected energy gets thermalized and therefore distorts the CMB spectrum. We compute CMB spectral distortion for both the models and show that the fractional change in CMB energy density, Δ ρ CMB/ρ CMB ≃ 10−6, which might be detectable with the proposed experiment PIXIE . We also outline the implication of our proposed scenario for CMB anisotropies.

Investigating the relationship between cosmic curvature and dark energy models with the latest supernova sample

We investigate the relationship between cosmic curvature and model of dark energy (hereafter DE) with recent Type Ia supernovae (hereafter SNe Ia) data, i.e., the Pantheon sample including 1048 SNe Ia with 0.01 < z < 2.3. We obtain measurements of the dimensionless spatial curvature density today, i.e., and at 68% confidence level (CL), respectively, in the scenarios of ΛCDM, ϕCDM (i.e., scalar field DE), ωCDM and ω0ωaCDM models. In the scenario of ΛCDM model, a closed universe is preferred by the Pantheon sample, which is consistent with that from the Planck CMB spectra. However, the uncertainty of Ωk0 from the Pantheon SNe sample is about 8 times larger than that from the Planck data, so the former one supports a closed universe at a much lower CL than that from the latter one. An open unverse is supported by the Pantheon sample at ∼32% and ∼78% CLs, respectively, in the ωCDM and ω0ωaCDM models. Among these models, the ϕCDM model is the one which supports the flat universe most strongly. It shows that Ωk0 is significantly dependent on the adopted model of DE, and there is a negative correlation between Ωk0 and the equation of state of DE.

H0 tension, phantom dark energy, and cosmological parameter degeneracies

Phantom dark energy can produce amplified cosmic acceleration at late times, thus increasing the value of $H_0$ favored by CMB data and releasing the tension with local measurements of $H_0$. We show that the best fit value of $H_0$ in the context of the CMB power spectrum is degenerate with a constant equation of state parameter $w$, in accordance with the approximate effective linear equation $H_0 + 30.93\; w - 36.47 = 0$ ($H_0$ in $km \; sec^{-1} \; Mpc^{-1}$). This equation is derived by assuming that both $\Omega_{0 \rm m}h^2$ and $d_A=\int_0^{z_{rec}}\frac{dz}{H(z)}$ remain constant (for invariant CMB spectrum) and equal to their best fit Planck/$\Lambda$CDM values as $H_0$, $\Omega_{0 \rm m}$ and $w$ vary. For $w=-1$, this linear degeneracy equation leads to the best fit $H_0=67.4 \; km \; sec^{-1} \; Mpc^{-1}$ as expected. For $w=-1.22$ the corresponding predicted CMB best fit Hubble constant is $H_0=74 \; km \; sec^{-1} \; Mpc^{-1}$ which is identical with the value obtained by local distance ladder measurements while the best fit matter density parameter is predicted to decrease since $\Omega_{0 \rm m}h^2$ is fixed. We verify the above $H_0-w$ degeneracy equation by fitting a $w$CDM model with fixed values of $w$ to the Planck TT spectrum showing also that the quality of fit ($\chi^2$) is similar to that of $\Lambda$CDM. However, when including SnIa, BAO or growth data the quality of fit becomes worse than $\Lambda$CDM when $w< -1$. Finally, we generalize the $H_0-w(z)$ degeneracy equation for $w(z)=w_0+w_1\; z/(1+z)$ and identify analytically the full $w_0-w_1$ parameter region that leads to a best fit $H_0=74\; km \; sec^{-1} \; Mpc^{-1}$ in the context of the Planck CMB spectrum. This exploitation of $H_0-w(z)$ degeneracy can lead to immediate identification of all parameter values of a given $w(z)$ parametrization that can potentially resolve the $H_0$ tension.

Scrutinizing various phenomenological interactions in the context of holographic Ricci dark energy models

In this paper, we examine two types of interacting holographic dark energy model using Pantheon supernova data, BAO BOSS DR12, CMB Planck 2015, fgas (gas mass fraction) and SZ/Xray (Sunyaev–Zeldovich effect and X-ray emission) data from galaxy clusters (GC). In particular, we considered the Holographic Ricci dark energy and Extended holographic Ricci dark energy models. During this analysis, we considered seven types of phenomenological interaction terms (three linear and four non-linear) Q_1=3Hbleft( rho _{D}+rho _mright) , Q_2=3Hbrho _{D}, Q_3=3Hbrho _{m}, Q_4=3Hbleft( rho _{D}+frac{rho _{D}^2}{rho _{D}+rho _m}right) , Q_5=3Hbleft( rho _{m}+frac{rho _{m}^2}{rho _{D}+rho _m}right) , Q_6=3Hbleft( rho _{D}+rho _{m}+frac{rho _{D}^2}{rho _{D}+rho _m}right) , Q_7=3HbBig (rho _{D}+rho _{m}+frac{rho _{m}^2}{rho _{D}+rho _m}Big ) respectively. To find the best model we apply Bayesian Inference (BI) and use the Lambda CDM as the referring model for comparison. Using the Bayesian Evidence model selection method we note that the Q_3 and Q_5 interaction terms are favored by observational data among the other ones. The obtained results also demonstrated that the evidence from the Bayesian inference method against the considered types of holographic Ricci dark energy model is strong since the Lambda CDM is considered as the reference model and also the Lambda CDM is preferred over the models. We also observed that the values of the deceleration parameter and the transition redshift for all models are compatible with the latest observational data and Planck 2015. In addition, we studied the jerk parameter for all models. Using our modified CAMB code, we observed that the interacting models suppress the CMB spectrum at low multipoles and enhances the acoustic peaks.

Analytic Calculation of Covariance between Cosmological Parameters from Correlated Data Sets, with an Application to SPTpol.

Consistency checks of cosmological data sets are an important tool because they may suggest systematic errors or the type of modifications to ΛCDM necessary to resolve current tensions. In this work, we derive an analytic method for calculating the level of correlations between model parameters from two correlated cosmological data sets, which complements more computationally expensive simulations. This method is an extension of the Fisher analysis that assumes a Gaussian likelihood and a known data covariance matrix. We apply this method to the South Pole Telescope Polarimeter (SPTpol) temperature and polarization cosmic microwave background (CMB) spectra (TE and EE). We find weak correlations between ΛCDM parameters with a 9% correlation between the TE-only and EE-only constraints on H 0 and a 25% and 32% correlation for log(A s ) and n s respectively. The TE-EE parameter differences are consistent with zero, with a probability to exceed of 0.53. Using simulations we show that this test is independent of the consistency of the SPTpol TE and EE band powers with the best-fit ΛCDM model spectra. Despite the negative correlations between the TE and EE power spectra, the correlations between TE-only and EE-only ΛCDM parameters are positive. Ignoring correlations in the TT-TE and TE-EE comparisons biases the χ 2 low, artificially making parameters look more consistent. Therefore, we conclude that these correlations need to be accounted for when performing internal consistency checks of the TT versus TE versus EE power spectra for future CMB analyses.

Planck evidence for a closed Universe and a possible crisis for cosmology

The recent Planck Legacy 2018 release has confirmed the presence of an enhanced lensing amplitude in CMB power spectra compared to that predicted in the standard $\Lambda$CDM model. A closed universe can provide a physical explanation for this effect, with the Planck CMB spectra now preferring a positive curvature at more than $99 \%$ C.L. Here we further investigate the evidence for a closed universe from Planck, showing that positive curvature naturally explains the anomalous lensing amplitude and demonstrating that it also removes a well-known tension within the Planck data set concerning the values of cosmological parameters derived at different angular scales. We show that since the Planck power spectra prefer a closed universe, discordances higher than generally estimated arise for most of the local cosmological observables, including BAO. The assumption of a flat universe could, therefore, mask a cosmological crisis where disparate observed properties of the Universe appear to be mutually inconsistent. Future measurements are needed to clarify whether the observed discordances are due to undetected systematics, or to new physics, or simply are a statistical fluctuation.

Observational constraints on interacting Tsallis holographic dark energy model

In this paper, we investigate a recent proposed model – so called the Tsallis holographic dark energy (THDE) model with consideration of the Hubble and the event future horizon as IR cutoffs. In this case, we consider the non-gravitational and phenomenological interaction between dark sectors. We fit the free parameters of the model using Pantheon Supernovae Type Ia data, Baryon Acoustic Oscillations, Cosmic Microwave Background, Gamma-Ray burst and the the local value of the Hubble constant. We examine the THDE model to check its compatibility with observational data using objective Information Criterion (IC). We find that the THDE models cannot be supported by observational data once the Lambda CDM is considered as the referring model. Therefore we re-examine the analysis with the standard holographic dark energy model (HDE) as another reference. Changing the Lambda CDM to main standard dark energy model (HDE), we observe the compatibility of the THDE models. Using the Alcock–Paczynski (AP) test we check the deviation of the model compared to Lambda CDM and HDE. Surveying the evolution of squared of sound speed v^2_s as an another test we check the stability of the interacting and non-interacting THDE models and we find that while the THDE model with the Hubble horizon as IR cutoff is unstable against the background perturbation, the future event horizon as IR cutoff show stability at the late time. In addition, using the modified version of the CAMB package, we observe the suppressing the CMB spectrum at small K-modes and large scale.

Polarisation as a tracer of CMB anomalies: Planck results and future forecasts

The lack of power anomaly is an intriguing feature at the largest angular scales of the CMB anisotropy temperature pattern, whose statistical significance is not strong enough to claim any new physics beyond the standard cosmological model. We revisit the former statement by also considering polarisation data. We propose a new one-dimensional estimator which takes jointly into account the information contained in the TT, TE and EE CMB spectra. By employing this estimator on Planck 2015 low-$\ell$ data, we find that a random $\Lambda$CDM realisation is statistically accepted at the level of $3.68 \%$. Even though Planck polarisation contributes a mere $4 \%$ to the total information budget, its use pushes the lower-tail-probability down from the $7.22 \%$ obtained with only temperature data. Forecasts of future CMB polarised measurements, as e.g. the LiteBIRD satellite, can increase the polarisation contribution up to $6$ times with respect to Planck at low-$\ell$. We argue that the large-scale E-mode polarisation may play an important role in analysing CMB temperature anomalies with future mission.

Observational status of the Galileon model general solution from cosmological data and gravitational waves

The Galileon model is a tensor-scalar theory of gravity which explains the late acceleration of the Universe expansion with no instabilities and recovers General Relativity in the strong field limit. Most constraints obtained so far on Galileon model parameters from cosmological data were derived for the limited subset of tracker solutions and reported tensions between the model and data. This paper presents the first exploration of the general solution of the Galileon model, which is confronted against recent cosmological data for both background observables and linear perturbations, using Monte-Carlo Markov chains. As representative scenarios of the Galileon models, we study the Full galileon model with disformal coupling to matter and the uncoupled Cubic galileon model. We find that the general solution of the Full galileon model provides a good fit to CMB spectra, while the Cubic galileon model does not. When extending the comparison to BAO and SNIa data, even the general solution of the Full galileon model fails at providing a good fit to all datasets simultaneously. Tensions remain if the models are extended with an additional free parameter, such as the sum of active neutrino masses or the normalization of the CMB lensing spectrum. Finally, the multi-messenger observation of GW170817 is also discussed in the framework of the scenarios considered. The time delay between the gravitational wave signal and its electromagnetic counterpart was computed a posteriori in every scenario of the Full galileon model cosmological fit chains and found to be ruled out by this observation.

Constraints on f(R) theories of gravity from GW170817

A novel constraint on $f(R)$ theories of gravity is obtained from the gravitational wave signal emitted from the binary neutron star merger event GW170817. The $f(R)$ theories possess an additional massive scalar degree of freedom apart from the massless spin-2 modes. The corresponding scalar field contributes an additional attractive, short-ranged "fifth" force affecting the gravitational wave radiation process. We realize that chameleon screening is necessary to conform with the observation. A model independent bound $\vert f'(R_0) -1\vert < 3\times 10^{-3}$ has been obtained, where the prime denotes the derivative with respect $R$ and $R_0$ is the curvature of our Universe at present. Though we use the nonrelativistic approximations and obtain an order of magnitude estimate of the bound, it comes from direct observations of gravitational waves and thus it is worth noting. This bound is stronger/equivalent compared to some earlier other bounds such as from the Cassini mission in the Solar-System, Supernova monopole radiation, the observed CMB spectrum, galaxy cluster density profile, etc., although it is weaker than best current constraints ($\vert f'(R_0) -1\vert \lesssim 10^{-6}$) from cosmology. Using the bound obtained, we also constrain the parameter space in the $f(R)$ theories of dark energy like Hu-Sawicki, Starobinsky, and Tsujikawa models.