Responses on Sample Covariance G+SSC+cNG Cov(k1,k2) Alexandre Barreira MPA P(k) R(k) T ( k 1 , - k 1 with Elisabeth Krause & , k 2 , - k 2 ) Fabian Schmidt arXiv:1703.09212 arXiv:1705.01092 arXiv:1711.07467 Statistical challenges for large-scale structure in the era of LSST Oxford 2018
Covariances in our life ● The Gaussian likelihood of a certain set of parameters given a hypothetical survey measurement of the 3D matter power spectrum P(k) :
Covariances in our life ● The Gaussian likelihood of a certain set of parameters given a hypothetical survey measurement of the 3D matter power spectrum P(k) : Measured data
Covariances in our life ● The Gaussian likelihood of a certain set of parameters given a hypothetical survey measurement of the 3D matter power spectrum P(k) : Measured data Theoretical prediction
Covariances in our life ● The Gaussian likelihood of a certain set of parameters given a hypothetical survey measurement of the 3D matter power spectrum P(k) : Measured data Theoretical prediction Covariance matrix ● Don't know how to compute it accurately/efficiently; ● By far, the least well understood piece of this likelihood : what is its redshift and cosmological dependence; baryonic effects?
Covariances in our life ● The Gaussian likelihood of a certain set of parameters given a hypothetical survey measurement of the 3D matter power spectrum P(k) : Measured data Theoretical prediction Covariance matrix We'll address this! ● Don't know how to compute it accurately/efficiently; ● By far, the least well understood piece of this likelihood : what is its redshift and cosmological dependence; baryonic effects?
In this talk … 1) Response Approach to Perturbation Theory Barreira, Schmidt , 1703.09212 2) An application to the lensing covariance Barreira, Schmidt , 1705.01092 Barreira, Krause, Schmidt, 1711.07467
Response Approach to PT Barreira, Schmidt , 1703.09212
What are responses? R e s p o n s e s d e s c r i b e h o w t h e p o w e r s p e c t r u m r e s p o n d s t o t h e p r e s e n c e o f l a r g e - s c a l e p e r t u r b a t i o n s . Observed patch Density or tidal field perturbation
What are responses? R e s p o n s e s d e s c r i b e h o w t h e p o w e r s p e c t r u m r e s p o n d s t o t h e p r e s e n c e What are they good for? o f l a r g e - s c a l e p e r t u r b a t i o n s . To describe squeezed N-point functions How do we evaluate them? With separate universe simulations Observed patch Density or tidal field perturbation
Responses and N-point functions Power spectrum, Bispectrum, Trispectrum …
Responses and N-point functions Power spectrum, Bispectrum, Trispectrum … Small scale (hard) modes Large scale (soft) modes
Responses and N-point functions Power spectrum, Bispectrum, Trispectrum … Small scale (hard) modes Large scale (soft) modes Mo d u l a t i o n o f t h e p o w e r s p e c t r u m P ( k ) b y l a r g e - s c a l e m o d e s R e s p o n s e s ! i . e .
Responses and N-point functions Power spectrum, Bispectrum, Trispectrum … N+2 squeezed correlations described by the N-th response hard soft Response Small scale (hard) modes Large scale (soft) modes Mo d u l a t i o n o f t h e p o w e r s p e c t r u m P ( k ) b y l a r g e - s c a l e m o d e s R e s p o n s e s ! i . e .
Squeezed bispectrum example hard soft
Squeezed bispectrum example Result is valid only if all modes are linear With Standard Perturbation Theory hard soft
Squeezed bispectrum example Result is valid only if all modes are linear With Standard Perturbation Theory hard soft With responses Result is valid for linear p, but any nonlinear k, k' !
Squeezed bispectrum example Responses as an extension of perturbation theory … Result is valid only if all modes are linear With Standard Perturbation Theory = + hard soft Responses are a Accessible with Analytical, but insufficient. With responses resummed interaction simulations Result is valid for linear p, but any nonlinear k, k' !
Response decomposition Wr i t e t h e r e s p o n s e i n t e r m s o f a l l p o s s i b l e l o c a l g r a v i t a t i o n a l o b s e r v a b l e s
Response decomposition Wr i t e t h e r e s p o n s e i n t e r m s o f a l l p o s s i b l e l o c a l g r a v i t a t i o n a l o b s e r v a b l e s All possible configurations of large-scale density/tidal fields; Given by perturbation theory.
Response decomposition Wr i t e t h e r e s p o n s e i n t e r m s o f a l l p o s s i b l e l o c a l g r a v i t a t i o n a l o b s e r v a b l e s Measure the response to each All possible configurations of specific large-scale configuration; large-scale density/tidal fields; What we will get from simulations. Given by perturbation theory.
Response decomposition Large-scale overdensity Large-scale tidal field Response to tidal field Response to overdensity
Response decomposition Response coefficients All 2nd order large-scale operators Generalizations to any order are always straightforward, just more cumbersome.
Separate universe simulations Nitty-gritty: Li et al (1401.0385) ; Wagner et al (1409.6294); Schmidt et al (1803.03274); All possible configurations of Response to specific large-scale density/tidal fields; perturbations
Separate universe simulations Nitty-gritty: Li et al (1401.0385) ; Wagner et al (1409.6294); Schmidt et al (1803.03274); All possible configurations of Response to specific large-scale density/tidal fields; perturbations 2) Compare to “mean” 1) Induce these spectrum to measure in simulations responses
Separate universe simulations Response to tidal field Response to overdensity Li et al (1401.0385) ; Wagner et al (1409.6294) Schmidt et al (1803.03274)
To keep in mind then … Responses describe the coupling of large-to-small scale modes in the nonlinear regime response hard soft Measurable with a few Separate Universe simulations.
Covariances with Responses Barreira, Schmidt, arXiv:1705.01092 Barreira, Krause, Schmidt, arXiv:1711.07467
3D covariance decomposition ● Observed, 'windowed' density field Takada&Hu (1302.6994) ● The power spectrum W(x): window function
3D covariance decomposition ● Observed, 'windowed' density field Takada&Hu (1302.6994) ● The power spectrum W(x): window function ● The power spectrum covariance + + Connected Gaussian Super-sample non-Gaussian
The Gaussian term : G ● It is the only contribution during the linear regime of structure formation Diagonal Trivially given by P(k)
The Gaussian term : G ● It is the only contribution during the linear regime of structure formation Diagonal Trivially given by P(k) Window function can be included by using the convolved P(k) . The Gaussian term is well understood !
The Gaussian term : G ● It is the only contribution during the linear regime of structure formation Corresponding lensing formulae Assuming Limber's approx., which is okay for l > 20 Diagonal Trivially given by P(k) ● Windowed lensing convergence ● Lensing power spectrum Window function can be included ● Gaussian lensing covariance by using the convolved P(k) . The Gaussian term is well understood !
Connected non-Gaussian term : cNG ● Describes the coupling of different Fourier modes due to nonlinear structure formation . Parallelogram trispectrum
Connected non-Gaussian term : cNG ● Describes the coupling of different Fourier modes due to nonlinear structure formation . Parallelogram trispectrum ● Extend to the nonlinear regime with responses if k1 >> k2: Valid for any nonlinear value of k1 ! hard soft response
cNG : response vs simulations non-Gaussian covariance matrix tree and partial 1-loop
cNG : response vs simulations non-Gaussian covariance matrix tree and partial 1-loop Black : Blot + (2015); over 12000 sims. Red : response k2 = 0.06 h/Mpc ` I f o n e m o d e i s l i n e a r : r e s p o n s e s c a p t u r e a l l t h e r e i s
cNG : response vs simulations k2 = 1 h/Mpc non-Gaussian covariance matrix tree and partial 1-loop Black : Blot + (2015); over 12000 sims. Red : response k2 = 0.06 h/Mpc ` I f o n e m o d e i s l i n e a r : r e s p o n s e s c a p t u r e a l l t h e r e i s
cNG : response vs simulations k2 = 1 h/Mpc non-Gaussian covariance matrix tree and partial 1-loop Up to 70%, but can be improved. Black : Blot + (2015); over 12000 sims. Red : response k2 = 0.06 h/Mpc ` I f o n e m o d e i s l i n e a r : r e s p o n s e s c a p t u r e a l l t h e r e i s
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