Cosmology with the Compact Phase Space of Matter

Homogeneous cosmology Perturbative inhomogeneities

Cosmology with the compact phase space of matter

Tomasz Trzesniewski´ ∗

Institute for Theoretical Physics, Wrocław University, Poland / Institute of Physics, Jagiellonian University, Poland

September 26, 2017

∗J. Mielczarek & T.T., Phys. Rev. D 96, 043522 (2017) T. Trzesniewski´ Compact phase space and cosmology Homogeneous cosmology Perturbative inhomogeneities

Outline:

1 Homogeneous cosmological model Spherical phase space for the ﬁeld Classical dynamics of the model

2 Perturbative quantum inhomogeneities Quantization of the linearized model First order corrections to the standard case

T. Trzesniewski´ Compact phase space and cosmology 1 / 14 Homogeneous cosmology Perturbative inhomogeneities

Outline:

1 Homogeneous cosmological model Spherical phase space for the ﬁeld Classical dynamics of the model

2 Perturbative quantum inhomogeneities Quantization of the linearized model First order corrections to the standard case

T. Trzesniewski´ Compact phase space and cosmology 1 / 14 Homogeneous cosmology Perturbative inhomogeneities Context and the existing work

Momentum spaces, or phase spaces, with nontrivial geometry that appear in quantum gravity Born reciprocity and three-dimensional gravity Quantum gravity phenomenology, relative locality framework Group field theory, loop quantum cosmology etc. Target manifolds for field values of non-linear sigma models and the Tseytlin string action The principle of finiteness of physical quantities, at the base of the Born-Infeld theory Potential connections between quantum gravity, cosmology and condensed matter physics Nonlinear Field Space Theory: J. Mielczarek and T.T., Phys. Lett. B 759, 424 (2016) J. Mielczarek, Universe 3, 29 (2017) T.T., Acta Phys. Pol. B Proc. Suppl. 10, 329 (2017) J. Bilski, S. Brahma, A. Marcianò and J. Mielczarek, arXiv:1708.03207 [hep-th]

T. Trzesniewski´ Compact phase space and cosmology 2 / 14 Homogeneous cosmology Spherical phase space Perturbative inhomogeneities Dynamics of the model Phase space variables

Phase space Γ = R2 is formed by values of a scalar ﬁeld ϕ and its conjugate momentum πϕ at every point of space Σ. We assume that Γ is actually a sphere, parametrized in terms of usual angles φ and θ or the spin-like vector S = (Sx , Sy , Sz ), so that πϕ ϕ Sx := S sin θ cos φ = S cos cos , (1) R2 R1 πϕ ϕ Sy := S sin θ sin φ = S cos sin , (2) R2 R1 πϕ Sz := S cos θ = S sin , (3) R2

where R1, R2 are certain dimensionful constants and ϕ/R1 ∈ [−π, π), π π πϕ/R2 ∈ [− 2 , 2 ]. For a ﬁeld deﬁned on the Minkowski background we have the limiting condition R1R2 = S.

T. Trzesniewski´ Compact phase space and cosmology 3 / 14 Homogeneous cosmology Spherical phase space Perturbative inhomogeneities Dynamics of the model Phase space algebra

In the Minkowski case the symplectic form is given by the area 2-form

πϕ ωM = S sin θ dφ ∧ dθ = cos dπϕ ∧ dϕ , (4) R2M R satisfying S2 ωM = 4πS. For the FRW background we introduce the 3 gravitational ﬁeld variable q ≡ V0a (here a is the scale factor and V0 a ﬁducial spatial volume) and its conjugate momentum p. Then we assume that the generalized symplectic total form is

πϕ ω := dp ∧ dq + cos dπϕ ∧ dϕ . (5) R2(q)

However, for ω to be a closed form we need R2(q) = R2. The corre- sponding Poisson bracket has the form

∂· ∂· ∂· ∂· 1 ∂· ∂· ∂· ∂· {·, ·} = − + πϕ − . (6) ∂q ∂p ∂p ∂q cos ∂ϕ ∂πϕ ∂πϕ ∂ϕ R2

T. Trzesniewski´ Compact phase space and cosmology 4 / 14 Homogeneous cosmology Spherical phase space Perturbative inhomogeneities Dynamics of the model Hamiltonian from the Heisenberg model

The Hamiltonian of the continuous XXZ Heisenberg model coupled to a magnetic ﬁeld B (for convenience, B := (Bx , 0, 0)) has the form Z 3 ˜ 2 2 2 HXXZ = − d x J (∇Sx ) + (∇Sy ) + ∆(∇Sz ) +µ ˜ B · S , (7)

where J˜, µ˜ are coupling constants and ∆ the anisotropy parameter. The homogeneous ﬁeld corresponds to the term ∝ B, which we adapt to the FRW background multiplying the measure d 3x by Na3, obtaining

HSmatmat = −Nq µ˜Bx Sx (8) µ˜Bx S 2 µ˜Bx S 2 4−n n = Nq −µ˜Bx S + 2 πϕ + 2 ϕ + O(ϕ πϕ) . 2R2 2R1 The ordinary scalar ﬁeld is recovered (up to a shift ∝ S) in the limit S → ∞ for the following identiﬁcation of the model’s parameters: r q0m 1 Sq0 p µ˜B ≡ , R ≡ , R ≡ Sq m . (9) x q2 1 q m 2 0

T. Trzesniewski´ Compact phase space and cosmology 5 / 14 Homogeneous cosmology Spherical phase space Perturbative inhomogeneities Dynamics of the model Total Hamiltonian

As the result, the matter Hamiltonian acquires the form ! q q π2 1 H = −Nm 0 S = Nq −Sm 0 + ϕ + m2ϕ2 + O(4) . (10) Smat q x q2 2q2 2 The ﬁrst term in the expansion will lead to a cosmic bounce, while the negative energy density that occurs for Sx > 0 should be balanced by some additional matter content. Nevertheless, in what follows we will use the positive-deﬁnite Hamiltonian q H := Nm 0 (S − S ) (11) mat q x and then the total Hamiltonian is 3κ H = H + H , H = − Nqp2 , (12) tot FRW mat FRW 4 ∂ where κ ≡ 8πG. It generates the constraint ∂N Htot = 0, equivalent to q 3κ m 0 (S − S ) = p2 . (13) q2 x 4

T. Trzesniewski´ Compact phase space and cosmology 6 / 14 Homogeneous cosmology Spherical phase space Perturbative inhomogeneities Dynamics of the model Friedmann equation

Introducing the Hubble factor h ≡ q˙ /(3q), we now ﬁnd that the Fried- mann equation is given by (for the gauge N = 1) q κ h2 = m 0 (S − S ) ≡ ρ , (14) q2 x 3

where ρ denotes the matter energy density. If we express it in terms of the energy density and pressure of an ordinary scalar ﬁeld

π2 1 π2 1 ρ := ϕ + m2ϕ2 , P := ϕ − m2ϕ2 , (15) ϕ 2q2 2 ϕ 2q2 2

we may obtain corrections to the usual Friedmann equation

2 2 κ κ q 2 1 2 2 h = ρϕ − ρϕ − Pϕ + O(1/S ) . (16) 3 9 Sq0m 2

They become relevant for large q and then can trigger a recollapse.

T. Trzesniewski´ Compact phase space and cosmology 7 / 14 Homogeneous cosmology Spherical phase space Perturbative inhomogeneities Dynamics of the model Evolution equations

Our Hamiltonian Htot also leads to the following equations for gravity 3κ q˙ = − Nqp , 2 3κ q S ˙ = 2 + 0 − − y p Np Nm 2 S Sx Sy arctan (17) 4 q Sx and for the ﬁeld

˙ 3κ Sy Sx = Np Sy arctan , (18) 2 Sx ˙ 3κ Sy ˙ Sy = Nm Sz − Np Sx arctan , Sz = −Nm Sy . (19) 2 Sx π π π π Alternatively, on the hemisphere ϕ/R1 ∈ (− 2 , 2 ), πϕ/R2 ∈ (− 2 , 2 ) one may use the angular-like variables, which are governed by

NR2 πϕ ϕ 2 ϕ ϕ˙ = tan cos , π˙ ϕ = −NR1qm sin . (20) q R2 R1 R1

T. Trzesniewski´ Compact phase space and cosmology 8 / 14 Homogeneous cosmology Quantization of the model Perturbative inhomogeneities First order corrections Hamiltonian for the inhomogeneous ﬁeld

In this case let us restrict to the regime of S → ∞ but with ∆ 6= 0. Then our matter ﬁeld Hamiltonian becomes (we also choose N = a) Z 3 Hϕ = d x Hϕ " # Z π2 (∇ϕ)2 1 ∆ = d 3x a4 ϕ + + m2ϕ2 + (∇π )2 . (21) 2a6 2a2 2 2m2a8 ϕ

0 Changing the variables to v := a ϕ and πv := ∂Lv /∂v we can derive

π2 (∇v)2 1 ∆ H = v + + m2 v 2 + (∇π − h ∇v)2 v 2 2 2 eff 2m2a2 v + O(∆2) , (22)

2 2 2 00 0 where meff ≡ m a −a /a is the effective mass, h ≡ a /a the conformal Hubble factor and we make an expansion around ∆ = 0.

T. Trzesniewski´ Compact phase space and cosmology 9 / 14 Homogeneous cosmology Quantization of the model Perturbative inhomogeneities First order corrections Quantum ﬁeld operators

Since in the considered limit S → ∞ we simply have

(3) δ (x − y) (3) {ϕ(x), πϕ(y)} = −→ δ (x − y) , (23) cos(πϕ(x)/R2) the standard quantization can be applied, leading to (3) ˆ [vˆ(x), πˆv (y)] = iδ (x − y) I . (24) Furthermore, we Fourier expand the ﬁeld operators Z d 3k Z d 3k vˆ(x) = eik·xvˆ , πˆ (x) = eik·xπˆ (25) (2π)3/2 k v (2π)3/2 vk and decompose their modes in the basis of creation and annihilation † (3) operators, satisfying [aˆk, aˆq] = δ (k − q),(τ is the conformal time) ˆ ˆ ∗ ˆ† vk(τ) = fk (τ) ak + fk (τ) a−k , (26) ˆ ∗ ˆ† πˆvk(τ) = gk (τ) ak + gk (τ) a−k . (27)

T. Trzesniewski´ Compact phase space and cosmology 10 / 14 Homogeneous cosmology Quantization of the model Perturbative inhomogeneities First order corrections Dynamics of mode functions

Next we may write down the (symmetrized) quantum Hamiltonian 1 Z ∆k 2 Hˆ = O(∆2) + d 3k 1 + πˆ πˆ† +π ˆ† πˆ v 4 m2a2 vk vk vk vk 1 Z ∆k 2 + d 3k ω2 + h2 vˆ vˆ† + vˆ†vˆ 4 k m2a2 k k k k 1 Z ∆k 2 − d 3k h vˆ πˆ† + vˆ†πˆ +π ˆ vˆ† +π ˆ† vˆ , (28) 4 m2a2 k vk k vk vk k vk k

2 2 2 with ωk ≡ k + meff. It determines the evolution equations of vˆk and πˆvk, which together give us equations for mode functions: 2 2 00 ∆k 0 ∆k f + 2h f + ω2 + k 2 + m2a2 − 2h2 f = 0 , (29) k m2a2 k k m2a2 k as well as 2 0 ∆k 0 g = f + (h f − f ) + O(∆2) . (30) k k m2a2 k k

T. Trzesniewski´ Compact phase space and cosmology 11 / 14 Homogeneous cosmology Quantization of the model Perturbative inhomogeneities First order corrections Vacuum state normalization

Therefore, the Wronskian condition also becomes modiﬁed

2 0 0 ∆k f (f ∗) − f ∗f = i 1 + + O(∆2) . (31) k k k k m2a2 We now calculate that energy of the initial ground state is given by 1 Z h0|Hˆ |0i = δ(3)(0) d 3k E , v 2 k ∆k 2 ∆k 2 2∆k 2 E ≡ 1 + |g |2 + ω2 + h2 |f |2 + h f g∗ . (32) k m2a2 k k m2a2 k m2a2 k k

iα The form of fk can be found by applying the decomposition fk = rk e k 2 2 and looking for a minimum of Ek . In particular, for such k that k meff ∆k 2 and m2a2 1 we obtain the corrected Bunch-Davies vacuum −ikτ 2 Z e ∆k 1 2 dτ 2 fk = √ 1 + − ika + O(∆ ) . (33) 2k m2a2 4 a2

T. Trzesniewski´ Compact phase space and cosmology 12 / 14 Homogeneous cosmology Quantization of the model Perturbative inhomogeneities First order corrections Spectrum of perturbations

Simple quantum correlations are captured by a two-point function

Z ∞ sin(k|x − y|) h |ϕˆ( , τ)ϕ ˆ( , τ)| i = P ( , η) , 0 x y 0 dk 2 ϕ k (34) 0 k |x − y|

1 3 2 where Pϕ(k, τ) ≡ 2π2 k |fk (τ)/a(τ)| is the power spectrum. In particular, in the de Sitter regime (i.e. for a = −(h τ)−1) it simpliﬁes to

h 2 ∆ P (k, τ) = x 2 1 + x 2 + O(∆2), (35) ϕ 2π 6η

with x ≡ −kτ. Consequently, the spectral index is found to be

d ln P (x = 1) n := ϕ = 0 . (36) S d ln k

T. Trzesniewski´ Compact phase space and cosmology 13 / 14 Homogeneous cosmology Perturbative inhomogeneities Work in progress and outlook

Joint treatment for the ﬁeld’s background and perturbations Calculation of the tensor to scalar ratio Analysis of the phase space trajectories Quantum theory in the case of the ﬁnite size of phase space Investigation of relations with condensed matter physics ...

T. Trzesniewski´ Compact phase space and cosmology 14 / 14