Position-dependent Power Spectrum Eiichiro Komatsu (MPA) Information discussion, University of Milan November 11, 2016
This presentation is based on • Chiang et al. “ Position-dependent power spectrum of the large-scale structure: a novel method to measure the squeezed-limit bispectrum ”, JCAP 05, 048 (2014) • Chiang et al. “ Position-dependent correlation function from the SDSS-III BOSS DR10 CMASS Sample ”, JCAP 09, 028 (2015) • Wagner et al. “ Separate universe simulations ”, MNRAS, 448, L11 (2015) • Wagner et al. “ The angle-averaged squeezed limit of nonlinear matter N-point functions ”, JCAP 08, 042 (2015)
A Simple Question • How do the cosmic structures evolve in an over- dense region?
Simple Statistics • Divide the survey volume into many sub-volumes V L , and compare locally-measured power spectra with the corresponding local over-densities
Simple Statistics V L • Divide the survey volume into many sub-volumes V L , and compare locally-measured power spectra with the corresponding local over-densities
Simple Statistics V L ¯ δ ( r L ) • Divide the survey volume into many sub-volumes V L , and compare locally-measured power spectra with the corresponding local over-densities
Simple Statistics V L ¯ δ ( r L ) ˆ P ( k , r L ) • Divide the survey volume into many sub-volumes V L , and compare locally-measured power spectra with the corresponding local over-densities
^ Position-dependent P(k) • A clear correlation between the local over-densities and the local power spectra
Integrated Bispectrum, iB ( k ) • Correlating the local over-densities and power spectra, we obtain the “integrated bispectrum”: • This is a (particular configuration of) three-point function ! The three-point function in Fourier space is the bispectrum, and is defined as
Integrated Bispectrum, iB ( k ) • Correlating the local over-densities and power spectra, we obtain the “integrated bispectrum”: • The expectation value of this quantity is an integral of the bispectrum that picks up the contributions mostly from the squeezed limit : k q 3 ~q 1 k “taking the squeezed limit and then angular averaging”
Power Spectrum Response • The integrated bispectrum measures how the local power spectrum responds to its environment, i.e., a long-wavelength density fluctuation zero bispectrum positive squeezed-limit bispectrum
Response Function • So, let us Taylor-expand the local power spectrum in terms of the long-wavelength density fluctuation: • The integrated bispectrum is then give as response function
Response Function: N-body Results • Almost a constant, but a weak scale dependence, and clear BAO features. How do we understand this?
Non-linearity generates bispectrum • If the initial conditions were Gaussian, linear perturbations remain Gaussian • However, non-linear gravitational evolution makes density fluctuations at late times non-Gaussian, generating non- vanishing bispectrum
S tandard P erturbation T heory Illustrative Example: SPT • Second-order perturbation gives the lowest-order (“tree-level”) bispectrum as “l” stands for “linear” • Then
Illustrative Example: SPT • Standard Eulerian perturbation theory gives the lowest-order (“tree-level”) bispectrum as “l” stands for “linear” • Then
Illustrative Example: SPT • Standard Eulerian perturbation theory gives the lowest-order (“tree-level”) bispectrum as “l” stands for “linear” • Then Response, dlnP(k)/d δ
Tree-level SPT comparison
Lemaitre (1933); Peebles (1980) Separate Universe Approach • The meaning of the position-dependent power spectrum becomes more transparent within the context of the “separate universe approach” • Each sub-volume with un over-density (or under- density) behaves as if it were a separate universe with different cosmological parameters • In particular, if the global metric is a flat FLRW, then each sub-volume can be regarded as a different FLRW with non-zero curvature
Mapping between two cosmologies • The goal here is to compute the power spectrum in the presence of a long-wavelength perturbation δ . We write this as P(k,a| δ ) • We try to achieve this by computing the power spectrum in a modified cosmology with non-zero curvature. Let us put the tildes for quantities evaluated in a modified cosmology P (˜ a ) → P ( k, a | ¯ ˜ k, ˜ δ )
Separate Universe Approach: The Rules • We evaluate the power spectrum in both cosmologies at the same physical time and same physical spatial coordinates • Thus, the evolution of the scale factor is different: *tilde: separate universe cosmology
Separate Universe Approach: The Rules • We evaluate the power spectrum in both cosmologies at the same physical time and same physical spatial coordinates • Thus, comoving coordinates are different too: *tilde: separate universe cosmology
Effect 1: Dilation • Change in the comoving coordinates gives dln(k 3 P)/dlnk
Effect 2: Reference Density • Change in the denominator of the definition of δ : • Putting both together, we find a generic formula , valid to linear order in the long-wavelength δ :
Example: Linear P(k) • Let’s use the formula to compute the response of the linear power spectrum, P l (k), to the long- wavelength δ . Since P l ~ D 2 [D: linear growth], • Spherical collapse model gives
Response of P l (k) • Then we obtain: • Remember the response computed from the tree- level SPT bispectrum : • So, the tree-level SPT bispectrum gives the response of the linear P(k). Neat!!
Response of P 1-loop (k) called “1 loop” • So, let’s do the same using third-order perturbation theory! • Then we obtain:
1-loop does a decent job
This is a powerful formula • The separate universe description is powerful, as it provides physically intuitive, transparent, and straightforward way to compute the effect of a long- wavelength perturbation on the small-scale structure growth • The small-scale structure can be arbitrarily non- linear!
This is a powerful formula • How can we compute \tilde{P}(k,a) in practice? • Small N-body simulations with a modified cosmology (“Separate Universe Simulation”) • Perturbation theory • We can compute the bispectrum with n-th order PT by the power spectrum in (n–1)-th order PT!
SDSS-III/BOSS DR11 • OK, now, let’s look at the real data (BOSS DR11) to see if we can detect the expected influence of environments on the small-scale structure growth • Bottom line: we have detected the integrated bispectrum at 7.4 σ . Not bad for the first detection!
L=220 Mpc/h
L=120 Mpc/h
Results: χ 2 /DOF = 46.4/38 L=220 Mpc/h L=120 Mpc/h • Because of complex geometry of DR10 footprint, we use the local correlation function, instead of the power spectrum. Power spectrum will be presented using DR12 in the future • Integrated three-point function, i ζ (r), is just Fourier transform of iB(k):
Results: χ 2 /DOF = 46.4/38 L=220 Mpc/h L=120 Mpc/h 7.4 σ measurement of the squeezed-limit bispectrum !! • Because of complex geometry of DR11 footprint, we use the local correlation function, instead of the power spectrum. Power spectrum will be presented using DR12 in the future • Integrated three-point function, i ζ (r), is just Fourier transform of iB(k):
Nice, but what is this good for? • Primordial non-Gaussianity (“local-type f NL ”) • The constraint from BOSS is work in progress, but the Fisher matrix analysis suggests that the integrated bispectrum is a nearly optimal estimator for the local-type f NL • We no longer need to measure the full bispectrum, if we are just interested in f NLlocal !
Nice, but what is this good for? • We can also learn about galaxy bias • Local bias model: • δ g (x)=b 1 δ m (x)+( b 2 /2)[ δ m (x)] 2 +… • The bispectrum can give us b 2 at the leading (tree-level) order , unlike for the power spectrum that has b 2 at the next-to-leading order
Result on b 2 • We use the simplest, tree-level SPT bispectrum in redshift space with the local bias model to interpret our measurements • [We also use information from BOSS’s 2-point correlation function on f σ 8 and BOSS’s weak lensing data on σ 8 ] • We find: b 2 = 0.41 ± 0.41
More on b 2 • Using slightly more advanced models, we find: * * The last value is in agreement with b 2 found by the Barcelona group (Gil-Marín et al. 2014) that used the full bispectrum analysis and the same model
Separate Universe Simulation • How do we compute the response function beyond perturbation theory? • Do we have to run many big-volume simulations and divide them into sub-volumes? No. • Fully non-linear computation of the response function is possible with separate universe simulations • E.g., we run two small-volume simulations with separate- universe cosmologies of over- and under-dense regions with the same initial random number seeds, and compute the derivative dlnP/d δ by, e.g., = ln P ( k | + ¯ δ ) − ln P ( k | − ¯ d ln P ( k ) δ ) d ¯ 2¯ δ δ
Separate Universe Cosmology
R 1 =dlnP/d δ • The symbols are the data points with error bars. You cannot see the error bars!
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