Nishant Agarwal

McWilliams Center for Cosmology Carnegie Mellon University

Cosmology After Planck Workshop, Michigan, September 25, 2013

PROBING THE INITIAL CONDITIONS OF THE UNIVERSE USING LSS

• What is the initial state of the fluctuations generated during inflation?

• Can we observe the effects of general initial

states in the CMB and large scale structure?

N. Agarwal, R. Holman, A. J. Tolley, and J. Lin, JHEP 2013 N. Agarwal, R. Holman, and A. J. Tolley, IJMPD 2013 N. Agarwal, S. Ho, A. D. Myers, et al, arXiv:1309.2954 N. Agarwal, S. Ho, and S. Shandera, In preparation S. Ho, N. Agarwal, A. D. Myers, et al, In preparation

• Introduction • General initial states • Large scale structure • Conclusions

Introduction

-

• The primary means of learning the physics of inflation is through the statistics of the curvature perturbation, 𝜁 –

𝑔𝑖𝑖(𝑡, �⃗�) = 𝑒2𝜁(𝑡,�⃗�)𝑎2 𝑡 𝛿𝑖𝑖

• Introduction • General initial states • Large scale structure • Conclusions

Introduction

Inflaton fluctuations, 𝛿𝛿

Curvature perturbations, 𝜁

𝛿𝑇CMB 𝛿𝜌matter

Planck SDSS

Transfer functions, Growth function

• Introduction • General initial states • Large scale structure • Conclusions

Introduction

Statistics of 𝜁

• Gaussian perturbations (2-point function)

𝜁 �⃗� 𝜁 �⃗� (𝑡) – Power spectrum

• Non-Gaussian perturbations (𝑛-point functions)

𝜁 �⃗� 𝜁 �⃗� 𝜁 𝑧 (𝑡) – Bispectrum

𝑥 𝑦

𝑥 𝑦

𝑧

• Introduction • General initial states • Large scale structure • Conclusions

Introduction

• 𝑛-point functions of 𝜁 will leave observable effects in the CMB and in LSS

• Introduction • General initial states • Large scale structure • Conclusions

Introduction

• 𝑛-point functions of 𝜁 will leave observable effects in the CMB and in LSS

• How do general initial states for the perturbations affect the 𝑛-point functions of 𝜁?

• Do the modified 𝑛-point functions of 𝜁 leave distinct signatures in the CMB and LSS?

• Introduction • General initial states • Large scale structure • Conclusions

Introduction

• Time dependent expectation values of operators – In-in formalism (J. S. Schwinger 1961; L. Keldysh 1964)

• For BD initial states, 𝜌 𝑡0 = 1 • For general initial states, 𝜌 𝑡0 is quadratic at

lowest order (no initial state non-Gaussianity)

• Introduction • General initial states • Non-Gaussianity • Halo bias

General initial states

• Introduction • General initial states • Large scale structure • Conclusions

General initial states

• We can write the following generating functional for the correlation functions

• Introduction • General initial states • Large scale structure • Conclusions

General initial states

• We can write the following generating functional for the correlation functions

Effective field theory action (Cheung et al, 2008)

The initial state N. Agarwal, R. Holman, A. J. Tolley, and J. Lin, JHEP 2013 N. Agarwal, R. Holman, and A. J. Tolley, IJMPD 2013

• Introduction • General initial states • Large scale structure • Conclusions

General initial states

Effective field theory of inflation - 𝑆𝜋

• The inflaton is viewed as an effective degree of freedom, appropriate only up to some cutoff scale

• Inflation breaks time translation symmetry • An EFT for inflation is constructed by restoring

time diffeomorphism invariance using the Stuckelberg mechanism

C. Cheung, P. Creminelli, A. L. Flitzpatrick, J. Kaplan, and L. Senatore, 2008

Parameterizing the initial state - 𝒮

N is the normalization chosen such that

and

Gaussian piece

• Introduction • General initial states • Large scale structure • Conclusions

General initial states

• We had

EFT (𝑆𝜋) Initial state (𝒮2)

• Introduction • General initial states • Large scale structure • Conclusions

General initial states

N. Agarwal, R. Holman, A. J. Tolley, and J. Lin, JHEP 2013 N. Agarwal, R. Holman, and A. J. Tolley, IJMPD 2013

• We had

EFT (𝑆𝜋) Initial state (𝒮2)

• Introduction • General initial states • Large scale structure • Conclusions

General initial states

N. Agarwal, R. Holman, A. J. Tolley, and J. Lin, JHEP 2013 N. Agarwal, R. Holman, and A. J. Tolley, IJMPD 2013

Combine quadratic pieces Solve for Green’s function

• Bispectrum for general initial states (for 𝑐𝑠 < 1)

𝑥3

𝑥2

ℬnonBD

Flattened enhancement Squeezed enhancement

Consistent with constraints of R. Flauger, D. Green, and R. A. Porto, 2013

• Introduction • General initial states • Large scale structure • Conclusions

General initial states

• General initial states can – lead to enhancements in the bispectrum in the squeezed and flattened limits – lead to stronger (non-slow-roll-suppressed) enhancements in models with 𝑐𝑠 < 1

X. Chen, M.-x. Huang, S. Kachru, and G. Shiu, 2007 R. Holman and A. J. Tolley, 2008 I. Agullo and L. Parker, 2011 N. Agarwal, R. Holman, A. J. Tolley, and J. Lin, JHEP 2013

• Introduction • General initial states • Large scale structure • Conclusions

General initial states

• We can construct templates for 𝑐𝑠 < 1 and 𝑐𝑠 = 1 models that capture the enhancements in both, the flattened and squeezed limits (Work in progress: N. Agarwal, R. Bean, J. Byun, and R. Holman)

• Introduction • General initial states • Large scale structure • Conclusions

General initial states

• Introduction • General initial states • Large scale structure • Conclusions

Large scale structure

𝛿𝜌/𝜌

In the absence of local 𝑓NL

• Introduction • General initial states • Large scale structure • Conclusions

Large scale structure

𝛿𝜌/𝜌

In the presence of local 𝑓NL: 𝜁 = 𝜁𝐺 + 35𝑓NL𝜁𝐺

2

• The power spectrum of dark matter halos is given by

with

S. Matarrese, F. Lucchin, and S. A. Bonometto, 1986; N. Dalal, O. Dore, D. Huterer, and A. Shirokov, 2008; S. Matarrese and L. Verde, 2008; A. Slosar, C. Hirata, U. Seljak, S. Ho, and N. Padmanabhan, 2008

• Introduction • General initial states • Large scale structure • Conclusions

Large scale structure

• Introduction • General initial states • Large scale structure • Conclusions

Large scale structure

Power spectrum Angular power spectrum

• For non-trivial initial states, the squeezed limit bispectrum is proportional to

• This changes the scale-dependence of the non-Gaussian halo bias: ∆𝑏non−Gaussian ∝ 1/𝑘3

J. Ganc and E. Komatsu, 2012; I. Agullo and S. Shandera, 2012

• Introduction • General initial states • Large scale structure • Conclusions

Large scale structure

• Therefore, we can parameterize the bias correction as

• Exact local ansatz: 𝛼 = 2 General initial states: 𝛼 ∼ 3 Multiple fields: 0 ≤ 𝛼 ≲ 2 + 𝒪(𝜖) (S. Shandera, N. Dalal, and D. Huterer, 2011; E. Sefusatti, J. R. Fergusson, X. Chen, and E. Shellard, 2012; M. Dias, R. Ribeiro, and D. Seery, 2013)

• Introduction • General initial states • Large scale structure • Conclusions

Large scale structure

• Data: SDSS photometric luminous red galaxies (900,000) and quasars (1.6 million)

• Method: – Determine redshift distributions – Correct angular power spectrum data for systematics – Perform an MCMC analysis to constrain 𝒜NL and 𝛼

S. Ho et al, 2012; S. Ho, N. Agarwal, et al, In preparation; N. Agarwal et al, arXiv:1309.2954

• Introduction • General initial states • Large scale structure • Conclusions

Large scale structure

LRGs

• Introduction • General initial states • Large scale structure • Conclusions

Large scale structure

Quasars

• Introduction • General initial states • Large scale structure • Conclusions

Large scale structure

• Constraints on 𝒜NL and 𝛼 – LSS data

N. Agarwal, S. Ho, and S. Shandera, In preparation; T. Giannantonio et al, 2013

𝒜NL

• Introduction • General initial states • Large scale structure • Conclusions

Large scale structure

• Constraints on 𝒜NL and 𝛼 – Fisher analysis

• Introduction • General initial states • Large scale structure • Conclusions

Large scale structure

𝒜NL

N. Agarwal, S. Ho, and S. Shandera, In preparation

• Mass-dependence of 𝒜NL – analytic calculation

• Introduction • General initial states • Large scale structure • Conclusions

Large scale structure

N. Agarwal, S. Ho, and S. Shandera, In preparation

𝑏1 − 𝑝

• Using in-in and EFT techniques we can calculate the Green’s functions for general initial states

• We can then calculate the 𝑛-point correlation functions of 𝜁

• Introduction • General initial states • Large scale structure • Conclusions

Conclusions

𝑥3

𝑥2

ℬnonBD

• General initial states imprint a distinct scale-dependence in the bias of dark matter halos and a distinct mass-dependence in the amplitude of non-Gaussianity

• Introduction • General initial states • Large scale structure • Conclusions

Conclusions

𝒜NL 𝑏1 − 𝑝