Power spectrum Definition and 52 Threads

  1. ergospherical

    I Toward the spectral index in slow roll

    Getting a bit confused over a manipulation; I have a power spectrum ##P_{R}(k)## in terms of ##V## and ##\epsilon_V## (to be evaluated at ##k=aH##) for the curvature perturbation ##R## and from this need to get a spectral index, i.e. apply ##d/d(\ln{k})##. So we need to transform this operator...
  2. Skaiserollz89

    Relationship between Fourier coefficients and power spectral density

    Here, ##\Phi(f_{x_n},f_{y_m})=|\mathscr{F(\phi(x,y))}|^2 ## is the Power Spectral Density of ##\phi(x,y)## and ##\mathscr{F}## is the Fourier transform operator. Parseval's Theorem relates the phase ##\phi(x,y)## to the power spectral density ##\Phi(f_{x_n},f_{y_m})## by...
  3. SherLOCKed

    I Maximizing S/N in Angular Power Spectrum Signals

    The signal-to-noise ratio for angular power spectrum signal Cl under theoretical noise Nl, where Cl and Nl are functions of multipole l, is given as (S/N)^2= \sum (2l+1) (Cl/Nl)^2To increase the S/N we bin the power spectrum signal, if bin width \Delta l, this in principle decreases Nl by a...
  4. arcTomato

    I Is my calculation of the power spectrum correct?

    Hello PF. I am thinking about the power spectrum when observing X-rays. We are trying to obtain the power spectrum by applying a window function ##w(t)## to a light curve ##a(t)## and then Fourier transforming it. I have seen the following definition of power spectrum ##P(\omega)##. Suppose...
  5. arcTomato

    I The power spectrum of Poisson noise

    I thought that if we Fourier transformed the counts of the sum of the signal from the source and the Poisson noise, and obtained the power spectrum, we would get the following, ##P_{j}=P_{j, \text { signal }}+P_{j, \text { noise }}+\text { cross terms }## but I found the following description...
  6. F

    A Analytical expression of Cosmic Variance - Poisson distribution?

    I have an expression of Matter Angular power spectrum which can be computed numerically by a simple rectangular integration method (see below). I make appear in this expression the spectroscopic bias ##b_{s p}^{2}## and the Cosmic variance ##N^{C}##. ## \begin{aligned}...
  7. F

    A Relation between Matter Power spectrum and Angular power spectrum

    From a previous post about the Relationship between the angular and 3D power spectra , I have got a demonstration making the link between the Angular power spectrum ##C_{\ell}## and the 3D Matter power spectrum ##P(k)## : 1) For example, I have the following demonstration, ## C_{\ell}\left(z...
  8. F

    A Angular power spectrum dependence in redshift z and k

    Hi, I wanted to have a precision about a question that has been post on this relation between P(k) and C_l The author writes the ##C_\ell## like this : $$C_\ell(z,z') = \int_0^\infty dkk^2 j_\ell(kz)j_\ell(kz')P(k)$$ I don't undertstand the meaning of ##z## and ##z'## : these are not...
  9. arcTomato

    A Average of the power spectrum of Poisson noise

    I am learning about noise that follows a Poisson distribution. When I do a Fourier transform of the data with only Poisson noise to get the power spectrum, what is the average value of the power spectrum?
  10. arcTomato

    Comp Sci Power spectrum when the wave number is not an integer

    Hi all. I made a program of DFT, so I made the power spectrum of a sin wave. This is the sin wave I used. All data number ##N=100## and the frequency of sine wave is 4.5Hz. And the power spectrum is this. The wave number is not integer so the spectrum has the side lobe. But I think this is...
  11. arcTomato

    The power spectrum of Poisson noise

    Dear all. I have made the power spectrum of Poisson noise(expected value ##λ=2##), and it becomes like this I think this is not good. I don't know why the power is so high when Random variable(x axis) is 0. I tried another expected value version ,but the result didn't change. so I would like to...
  12. arcTomato

    I I would like to know the calculation process of this power spectrum

    Summary: Derive the formula of power spectrum from Discrete Fourier Transform. Hi all I don't know where should I post this, so if I am wrong, I apologize.(But this is almost math problem so ) I would like to know the calculation process when derive Eq(6.3) in this paper. Eq 2.4a is...
  13. D

    A In the power spectrum, why is the peak frequency at the first harmonic

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  14. arcTomato

    Engineering The power spectrum of a sine wave (C language)

    Hi I would like to Derive the power spectrum of sinusoid.I tried like this. But It doesn't work. <Moderator: CODE tags added> #include <stdio.h> #include <math.h> #define pi 3.1415926535 FILE *in_file, *out_file; int main() { dft(); } int dft(int argc, char *argv[]) { char...
  15. F

    I Cross-correlations: what size to select for the matrix?

    Hello, I am working on Fisher's formalism in order to get constraints on cosmological parameters. I am trying to do cross-correlation between 2 types of galaxy populations (LRG/ELG) into a total set of 3 types of population (BGS,LRG,ELG). From the following article...
  16. Rafid Mahbub

    A Numerical solution of the Mukhanov-Sasaki equation

    Hi, I am trying to figure out how to solve the Mukhanov equation numerically in Mathematica, but have some problems dealing with it. In terms of the number of efolds, the Fourier modes satisfy the following ODE in terms of the Hubble slow roll parameters: $$...
  17. Spinnor

    I ....Alternatives to Inflation in Primordial Power Spectrum

    Unique Fingerprints of Alternatives to Inflation in the Primordial Power Spectrum "Massive fields in the primordial universe function as standard clocks and imprint clock signals in the density perturbations that directly records the scale factor of the primordial universe as a function of...
  18. DoobleD

    I Vertical axis of the CMB power spectrum

    Why is the vertical axis in the CMB power spectrum usually chosen as ##l(l+1)C_l/2\pi## instead of simply ##C_l## ? The only answer I found come from this post on stackexchange, but the answer doesn't seem very complete. Anyone knows ?
  19. C

    A Angular power spectrum, bias from N weighted events

    My general question is: What is the angular power spectrum C_{l,N,ω} of N weighted (weight ω_i for event i) events from a full sky map with distribution C_l? I'm interested in: Mean of C_{l,N,ω}: <C_{l,N,ω}> Variance of C_{l,N,ω}: Var(C_{l,N,ω}) The question is important, since we observe in...
  20. S

    I Why is the Primordial Power Spectrum Defined as P(k) = (k^3)/(2π^2)|w_k|^2?

    Why is the power spectrum defined as ##P(k) = \frac{k^3}{2π^2} |w_k|^2 ## where ##w_k## is the mode function? Cosmology books and papers just states that it is defined that way but there are no details on why.
  21. F

    I How to Understand the Primordial Power Spectrum in TASI Lectures on Inflation?

    I have been reading the TASI Lectures on Inflation by William Kinney, (https://arxiv.org/pdf/0902.1529v2.pdf). I came across the mode function eq (128) (which obeys a generalization of the Klein-Gordon equation to an expanding spacetime), as I read through until eq (163), I know that it is the...
  22. W

    I Scale invariance in the power spectrum

    I understand the inflation predicts a nearly scale invariant power spectrum but some have claimed this was predicted before inflation (by Harrison and Zeldovitch?) My understanding is that perfectly scale invariance would predict ns=1 but inflation predicts ns =.96. So did the prior prediction...
  23. sunrah

    Python CAMB python convergence power spectrum code

    I'm trying to understand this python CAMB code: http://camb.readthedocs.io/en/latest/CAMBdemo.html Scroll down to In[29] and In[30] to see it. It's an integration over chi (comoving distance), yet scipy.integrate.quad is not called. It seems that the fun stuff happens in the last for-loop in...
  24. sunrah

    A Having Trouble Plotting the Cosmic Shear Power Spectrum Kernel?

    I'm having difficulty plotting the kernel I(k_{1},k) of the cosmic shear power spectrum which is defined as I(k_{1},k) = k_{1}\int^{\infty}_{0}r j_{l}(k_{1}r)dr \int^{r}_{0}\frac{r-r'}{r'}j_{l}(kr')\sqrt{P^{\Phi\Phi}(k)}dr' where the jl are spherical bessel functions. I'v tried plotting I vs...
  25. G

    Finding the One Degree Peak in a Power Spectrum

    Can someone tell me how i can determine the one degree peak from power spectrum ?
  26. J

    Cosmic acoustics -- why no intermediate waves on CMB map

    I fully understand the representation of the set of waves that are either at full compression or full rarefaction at recombination, thus, yielding a CMB map. But at this time are there no waves that are intermediate, e.g. 50% of the way to full compression or full rarefaction. Why don't these...
  27. C

    Intuition Behind Scale Invariance Power Spectrum

    In the book "Statistical physics for cosmic structures" at p. 171 a read a definition of scale invariance (leading to the so called scale invariant power spectrum) given as the requirement that ##\sigma^2_M(R=R_H(t)) = constant##, where ##R_H(t)## is the horizon, i.e. the maximal distance that...
  28. A

    The peak of the thermal radiation power spectrum (dR/dλ)

    Homework Statement The peak of the thermal radiation power spectrum (dR/dλ) is at a wacelength of about λm=hc/5kT. Why is the peak of the same power spectrum plotted as dR/df not at fm=c/λm= 5kT/h? Homework Equations dR/dλ= 2πhc2/(λ5(e(hc/λkT)-1)) f=c/λ The Attempt at a Solution
  29. ChrisVer

    How Does the CMB Power Spectrum Relate to Angular Scales?

    http://pdg.lbl.gov/2013/reviews/rpp2013-rev-cosmic-microwave-background.pdf Here one reads in sec. 26.2.4 that: However it states that a single Y_{lm} corresponds to angular variations of \theta \sim \pi /l. I am not getting these statements. Also I find it difficult to understand, since...
  30. T

    Primodial power spectrum of scalar perturbations

    For scalar modes \mathcal{R}_k originating in the Bunch-Davies vacuum at the onset of inflation, I have the following equation for their primordial power spectrum: P_{\mathcal{R}}(k)=\frac{4\pi}{\epsilon(\eta_k)}\bigg( \frac{H(\eta_k)}{2\pi} \bigg)^2, where: c = G = ħ = 1, k is the...
  31. P

    Fast Fourier Transform (FFT) power spectrum angle

    Dear Physics Buddies, How are well all, okay I hope. I was wondering if I might browse all your infinite intellects and ask you a very simple question. I am working with some medical images in MATLAB and my collaborators would like to know the orientation of the fibres that it contains...
  32. M

    Understanding the Linear Power Spectrum: How Do BAOs Create Wiggles?

    Dear Experts, I try to understand power spectra for large scale structure and CMB analyses for my exams. I constantly find the expression for linear power spectra P(k) = A\cdot k^{n_s}\cdot T^2(k). I understand that this comes from primodial primodial fluctuations and the tranfer function...
  33. M

    Why Use Wave Modes to Describe Density Fluctuations?

    Hi. I read some basic cosmology where it is always said that density fluctuations, pertubations can be described in modes of waves. In particular if you use linearised theory where δ(x,t) is Fourier transformed δ(k,t). What exactly is the reason for this? What do the wave modes describe...
  34. J

    Linear vs non linear matter power spectrum

    Hi, I have troubles understanding the difference between linear and non linear matter power spectrum. These words are commonly used in the litterature, but I have found no definitions yet. My understanding is that there is one definition of the power spectrum for matter distribution...
  35. H

    Power spectrum for real, imaginary and complex functions.

    Homework Statement What can we say about the evenness and oddness of the power spectrum (|F(s)|^{2}) if the input fuction is purely real, purely imaginary or complex? I know that a real function will give an even power spectrum. But I can't prove it! Homework Equations F(s) =...
  36. G

    How Can I Calculate the Power Spectrum for Data in Binary Cellular Automata?

    Hello! I am doing a thesis in ecology using a two dimensional binary cellular automata. I have sampled a parameter for a number of generations and I would like to calculate the power spectrum for this data. I had been planning to use a software for this, but now it seems I have to do it...
  37. A

    Why is the CMB power spectrum often plotted as l(l+1) C_l and not only C_L ?

    Hello everybody, (sorry for the eventual Engrish) I can't find any convincing answer for the following question : Why do we always (or often) plot the CMB power spectrum in this way : jb.man.ac.uk/research/cosmos/vsa/images/CMB_power_spectrum.gif I mean the y-axis is $$C_\ell \ell...
  38. P

    Power Spectrum vs. Power Spectral Density Spectrum

    Does anyone know the difference between the power spectrum of a signal and the power spectral density (PSD) spectrum of a signal? I've read on the net lots of things ranging from: i) They are identical ii) Power spectrum is units of Watts, power spectral density spectrum units of Watts/Hz...
  39. D

    What is the physical significance of the WMAP power spectrum?

    Could anybody explain what "power spectrum" in cosmology means. I am trying to understand the WMAP power spectrum graph, in particular. For example, what do the multiple moments physically represent? The graph is here: http://casa.colorado.edu/~ajsh/cosmo_04/wmapcl.html
  40. TrickyDicky

    CMB Power Spectrum and Angular Power Spectrum Plot from WMAP

    Anyone knows if the CMB map of anisotropies from WMAP is used to implement the angular power spectrum plot(acoustic peaks)? I'm not sure, but I tend to think it is not.
  41. M

    How to measure the CDM power spectrum?

    Hi all! It's my first post, but I follow this forum since a couple of years. Now I have a big question for you: how to measure the CDM (or \LambdaCDM) power spectrum (better, the variance \propto k^3 P(k)) from the observations? What parameters are necessary?
  42. C

    Why does power spectrum is calculated at the radiation/matter epochs?

    In the standard inflationary scenario, the power spectrum is evaluated at the cosmological time when one assumes an equation of state $ P= \omega \rho$ , that is, one is assuming a particular radiation or matter dominated universe. Why does it has to be in these cosmological epochs? does it...
  43. H

    Can someone explain angular power spectrum

    Hi I'm trying to get my head around this plot of the cosmic microwave background radiation. http://www.astro.ucla.edu/~wright/CMB-LCDM-w-WMAP.gif I've been searching all around to find out what exactly a angular power spectrum is. I know what an ordinary power spectrum is. I'm a second...
  44. X

    Fourier transformation: power spectrum

    Lots of works about the high-order harmonic generation in the intense laser-atom physics obtain the harmonic spectrum by Fourier transformation of the dipole moment d(t) (=\int\varphi\varphi^{*}z): p(\omega)=|\frac{1}{tf-ti}\int d(t)exp(-i\omega)dt|^{2} Here, I want to use the Monte-Carlo...
  45. Cincinnatus

    Troubleshooting Power Spectra Analysis for Time Series Data

    This is more of a data analysis question than a mathematics question so feel free to move it to wherever you think it is most likely to be answered. I have some time series data that looks (just by eye) like it should have several peaks in its power spectrum. However, when I compute the power...
  46. C

    Fourier Transform Power Spectrum

    Input: sine wave at 10Hz, amplitude 1. After the transform the plot has a spike at 10Hz with amplitude 0.5. If I vary the amplitude of the sine wave I get: sine amp. - FT spike amp. 1 - 0.5 2 - 2 4 - 8 So it seems A' = A^2/2 Is this because power is proportional to A^2 and it is...
  47. T

    Power spectrum of short signals

    I am looking for a way to calculate the power spectrum of short signals of variable duration (between 200 and 500 ms). Standard methods seem wildly inaccurate. Are there any tricks for short segments? Thanks anyone for a response.
  48. A

    Why Calculate the Power Spectrum of a 10MHz Square Wave?

    Homework Statement Calculate the power spectrum in dBm of a zero offset, 10MHz square wave with amplitude A from DC to 50MHz. Homework Equations - The Attempt at a Solution I was given this problem but am not sure how to go about solving it. Is the power spectrum the same as the...
  49. J

    Fourier transform -> power spectrum

    Fourier transform --> power spectrum Hey all! I've been learning about the discrete Fourier transform (and FFT too) recently. What I don't understand is why applying it to a signal gives its power spectrum. I am not really good in physics, so to me it just seems like a magical formulae, one...
  50. F

    Physics/Astronomy power spectrum from waveforms

    phi(t)=A_o*e^(2*pi*i*mu_0*t) (label:1) Calculate the power spectra of the following waveforms (using 1): 1) for all t 2) a pulse duration of two tau for |t|<tau, and for phi(t)=0 --- 3)an exponentially decaying sinusoid: phi(t)=A_0*(e^(-t/(2*tau)))*e^(2*pi*i*mu_0*t) for t>0 and...
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