Cosmology Beyond Homogeneity and Isotropy

Thiago PereiraUniversidade Estadual de Londrina

IV CosmoSulCosmology and Gravitation

in the Southern ConeICTP-SAIFR, 2017

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Standard Model Ingredients

● General relativity

– We extrapolate Einstein’s equations above ~ 100Mpc

● Matter content

– Standard model’s particles + Λ + CDM

● Cosmological principle

– Cosmic geometry and topology are as simple as possible

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Cosmological PrincipleSymmetries

Maximally symmetric Homogeneous Inhomogeneous

E.g. flat FLRW E.g. BianchiE.g. LTB

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Constraints from CMB

● FRW is assumed valid at all redshifts

● BUT:

– CMB is observed at z=1000

– Observations are made from a privileged point.

– Symmetric CMB does not imply symmetric geometry

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Moreover...

● Data is highly compressed

– 2ℓ+1 → 1

● Cross-correlations

contain useful information:

– Boost direction

– Topology

● LSS is a powerful probe

– Anisotropic dark energy (Koivisto&Mota MNRAS2007, JCAP2008)

– Weak-lensing in general spacetimes

(Planck,A&A2013, Notari&Quartin JCAP2012, Yoho et. al. MNRAS 2012)

(Luminet, 2016 (review paper))

(Pitrou et. al. PRD2015, Pereira et. al. A&A2016)

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Messages I planto deliver● # 1:

– Homogeneous and anisotropic are our next option to FRW

● # 2:

– Weak-lensing is a powerful probe of the cosmo’s geometry

● # 3:

– There is more in statistics than meets the eye

This talk:● Part 1: Early anisotropies

● Part 2: Late anisotropies

● Part 3: Statistics

● Conclusions

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Part 1:

Early Anisotropies:

●Models with anisotropic expansion

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Anisotropic expansion

The simplest extension to flat FRW:

Background equations:

Inflation was designed to wash away anisotropies, but one usually sets σ=0 from the beginning!

(Misner-Thorne-Wheeler – Gravitation)

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“de Sitter” anisotropic phase

In the presence of Λ one has

(Pereira&Pitrou, 2015)

What was the onset of the universe before inflation?

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Slow-roll inflation

● Chaotic inflation

● Two attractors

– Shear phase

– Slow-roll phase

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Quantization

Because the metric is time-dependent, we have

There are modes whose amplitude cannot be fixed:

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Quantization

Free field in Bianchi I:

WKB solution:

Amplitude of large perturbations cannot be fixed!

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Linear Perturbations

● “Mukhanov-Sasaki” variables

– There are three effective d.o.f generalizing the standard Mukhanov-Sasaki variables

Main characteristics:

● Vectors modes have no dynamics (if not sourced)● Dynamical modes couple at linear order!● Polarization modes of g.w. with different dynamics

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Linear Perturbations

● Dynamics:

– Coupled oscillator-like equations

– See-saw mechanism

Numerical Solutions: (Pitrou, Pereira & Uzan, JCAP2007)

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CMB signatures

● The CMB signatures of an early Bianchi I phase are

– Non-diagonal covariance matrix

– Non-zero Scalar × Tensor correlations

– Gravitational waves with different dynamics

– Anisotropic power spectrum

Even-odd correlations are forbidden!

Observational constraints from CMB give σ/H ≤ 1%

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Part 1:

Early Anisotropies:

●Models with anisotropic expansion●Models with anisotropic curvature

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Anisotropic Curvature

Bianchi III: KantowskiSachs:

Anisotropic, homogeneous and spatially curved manifold

Examples:

Since the curvature is anisotropic, the expansion canbe isotropic (Mimoso&Crawford, CCG2013), (Carneiro&Mena-Marugan, PRD2001)

Isotropic expansion if a = b. CMB will be isotropic!

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Anisotropic Curvature

Shear-free anisotropic metric:

Isotropic expansion Anisotropic

spatial curvature

● Anisotropic matter content

– Massless Scalar field (Carneiro&Mena-Marugan, PRD2001)

– Kalb-Ramon field (Koivisto et. al, PRD2010)

● Dynamics:

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Linear Perturbations

● No shear no dynamical couplings→● Polarization modes of g.w. with different dynamics

No dynamical couplings

There are three variables generalizing the Mukhanov-Sasaki variables:

Gravitationalwaves

(Franco&Pereira,to appear)

Main characteristics:

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Anisotropic CurvatureSignatures

Small curvature limit: k<<1

geodesiccorrections

eigenfunctionscorrections

Correction linear in 1/L violate parity and thus they do not appear

(Pereira et. al. JCAP2015)

Planck bounds at 95% C.L.

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Part 2: Late Time Anisotropies

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Weak Lensing in Bianchi I

Weak Lensing of the large scale structure is a powerful probe of late-time geometry

In the LCDM model, B-modes are not cosmological

In Bianchi Models, B-modes probe the geometry

screen basisat observer

screen basisat observer

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Weak Lensing in Bianchi I

General setup:

The shear evolves as:

early late

● Our motivations are

– Provide a new test for isotropy at late times

– Propose new tests for anisotropic dark energy

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Weak Lensing in Bianchi I

Weak shear approximation:

The metric perturbations are:

● We develop a perturbation scheme such that:

– Background FRW quantities are of order {0,0}

– Background FRW quantities are of order {1,0}

– Scalar FRW perturbations are of order {0,1}

– Vector and tensor perturbations are of order {1,1}

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Weak Lensing in Bianchi I

Main goal

● For this program to work we need to:

– Determine the position of the source and its geodesic direction at relevant order

– Determine the screen basis na(no,χ) at order {1,1}

– Determine the source terms at order {1,1}

– Solve metric perturbations at order {0,1}

– Perform a multipolar expansion in quantities depending on no.

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Weak Lensing in Bianchi I

We are interested in the EE and BB correlations in the cosmic shear induced by the anisotropic expansion. At small scales the dominant contribution is

Deflection angleat order {1,0}

Grav. Potentials at order {0,1}Dominant at

small scales

We employ a gradient expansion which allows us to get:

(Pereira et. al. A&A Letters, 2016)

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Weak Lensing in Bianchi IModels

We consider two phenomenological models for the anisotropic pressure:

Model (A)

Model (B)

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Weak Lensing in Bianchi IModels

Source distribution:

Euclid: SKA:

Free parameters:A, z0, β, a, b, m, n

CFHTLS:σ/H

0≤0.4

Euclidσ/H

0≤0.008

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Weak Lensing in Bianchi ICross-correlations

The convergence, E- and B-modes cross-correlations are linear in σij/H, and allow one to fully reconstruct the eigendirections of expansion.

M = -2,-1,0,1,2 = 5 components = No of components in σ

ij

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Part 3: Statistics

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Null tests of symmetry

● In the absence of symmetry it’s hard to have analytical cosmological models...

● Null tests of isotropy and/or homogeneity are a useful tool:

● ...A popular strategy is to let correlation functions be free and construct null symmetry tests with them

– C → C(n) (Pullen&Kamionkowski, PRD2007)

– C → C(n1,n2) (Hajian&Souradeep, AstrophysJ2003, AstrophysJ2004)

– C → C(n1×n2) (Pereira&Abramo, PRD2009)

– C → C(n1+n2) (Froes et. al., PRD2015)

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Correlation Functions

Spacetime symmetries uniquely fix the functional dependency of correlation functions

(Marcori&Pereira, JCAP2017)

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Correlation FunctionsExamples

● FRW: 6 Killing vectors

– Translations:

– Rotations:

● Lemaître-Tolman-Bondi: 3 Killing vectors

– Rotations:

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Correlation FunctionsGeneral solution

● Suppose e is a vector

commuting with K:

– e connects points…

– which implies that

● Bianchi models:

– Metric

– General solution

(Marcori&Pereira, JCAP2017)

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Correlation FunctionsGeneral solution

Bianchi VII0

● Examples

● Limit of small anisotropies:

Bianchi I

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Correlation FunctionsGeneral solution

Assuming SW effect:

...we find

● Features

– No perturbation theory required!

– Even-even cross-correlations in BI

– Even-even cross-correlations and damping of Cl in BVII0(Oliveira et. al. To appear)

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Non-Gaussian Correlations

We can easily extend the formalism to N-point correlation functions:

For FRW metrics:

Compare to 2pcf in LTB:

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Non-Gaussian Correlations

● Inhomogeneous universes are Non-Gaussian!

– Two points A and B will correlate with Q because

Q is privileged

● For a fixed realization, k and q are

not independent if B ≠ 0

– Non-gaussian models are spatially

inhomogeneous!

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Non-Gaussianity versusInhomogeneity

There is an infinite hierarchy between N-point functions in FRW and N-1 point functions in LTB

● FRW:● LTB:

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Non-Gaussianity versusAnisotropy

A similar degenerescence happens between NG and Statistical Anisotropy in the squeezed limit

● Three-point function:

● Squeezed limit:

(Schmidt&Hui, PRL2012)

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Final remarks

● Homogeneous and spatially anisotropic models are straightforward extension of FRW cosmologies.

● Shear-free anisotropic models are counter example to isotropy of CMB

● Perturbation theory is possible:– Can be used to model anisotropic inflation...– ...But it's not clear that it is the origin of anomalies

● E- and B- modes of weak-lensing can be measured by future surveys such as Euclid and SKA – Opens a window to new investigations of the anisotropy

of dark energy.– Eigendirections of expansions can be fully reconstructed

from weak-lensing● Correlation functions can be estimated symmetries:

– Null tests made easy!– Interplay between NG and statistical inhomogeneity…

New ways of constraining inhomogeneous models?

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Thank you!