Three-form Cosmology

Transcription

1 Three-form Cosmology Nelson Nunes Centro de Astronomia e Astrofísica Universidade de Lisboa Koivisto & Nunes, PLB, arxiv: Koivisto & Nunes, PRD, arxiv:98.92 Mulryne, Noller & Nunes, JCAP, arxiv: Koivisto & Nunes, arxiv:22.254

2 What is a three-form? Totally antisymmetric tensor with three indeces A ijk = A jik For example, a three-form defines the cross product where ǫ ijk is the Levi-Civita symbol. ( a b) i = ǫ ijk a j b k

3 Why forms?. To test the robustness of scalars. Is it the only natural possibility? Scalars have not been detected yet. 2. Vectors and two-forms, Support anisotropy; Require non-minimal couplings possible instability. Three-forms, Viable isotropic inflation. 3. They exist in fundamental theories String theory.

4 Three-form action Action for the three-form A µνρ S = d 4 x g ( R 2κ ) 2 48 F 2 (A) V (A 2 ) where F µνρσ = 4 [µ A νρσ] = µ A νρσ σ A µνρ + ρ A σµν ν A ρσµ We have the equations of motion: F = 2V (A 2 )A and due to antisymmetry we have the additional constraints: V (A 2 )A =

5 Equations of motion Consider flat FRW cosmology: ds 2 = dt 2 + a 2 (t)dx 2 Most general three-form compatible with FRW: A ijk = a 3 (t)ǫ ijk χ(t) Equations of motion of the field X: χ + 3H χ + V,χ + 3Ḣχ = Equation of motion of background fluid: ρ B = 3γHρ B

6 Equations of motion Friedmann equation (ȧ ) 2 H 2 = κ2 a 3 ( ) 2 ( χ + 3Hχ)2 + V (χ) + ρ B can also write: H 2 = κ2 3 V + ρ B κ 2 (χ + 3χ) 2 /6 κ χ + 3χ < 6. = d/dlna. Evolution of the Hubble rate: Equation of state parameter of X: Ḣ = κ2 2 (V,χχ + γρ B ) w χ = + V,χχ ρ χ Universe de Sitter with V = ; Superinflation when V,χ χ <.

7 Critical points Rewriting the equations of motion in the form of system of first order differential equations: x = 3 y ( ) 2 3 y x = 3 2 λ(x) ( y 2 w 2) [ xy ] 2 3 w = 3 2 w ( γ + λ(x) ( y 2 w 2) x γw 2) γw2 y where x κχ, y κ 6 (χ + 3χ), z 2 = κ2 V 3H 2 w 2 κ2 ρ B 3H 2, λ(x) V,χ κ V The Friedmann equation y 2 + z 2 + w 2 =

8 Critical points x y w Ḣ/H 2 λ description A ± 3γ/2 any matter domination B ± ± 2/3 ± any kinetic domination C x ext 3/2 xext potential extrema

9 Effective potentials 2.5 V = X V eff X V eff,χ = V,χ + 3Ḣχ = V,χ ( 3 2 (κχ)2 ) 3 2 γκ2 ρ B χ

10 Phase space V = X 2.5 y tanh(x)

11 Example evolutions: V = χ 2 2 X y energy densities 2 3 N N w X 2 X N

12 Example evolutions: V = χ n Epoch of tracking: Point of turn around: N s = ( ) 3n ln Vi 3Hi 2y2 i ( ) 2 N t = 3( + γ/2) ln B γ A where A = 2/3 y i /( + γ/2) and B = χ i A. Epoch of inflation: N = 2 9n 2/3 (κχ init ) 2 2 Oscillations: w χ = n 2 n + 2 Thus for n = 2 the field behaves as dust, w χ = and for n = 4 it mimics radiation, w χ = /3.

13 Effective potentials.3 C =.5 C =.5.2 V = (X 2 C 2 ) 2 V = (X 2 C 2 ) V eff X V eff X V eff,χ = V,χ + 3Ḣχ = V,χ ( 3 2 (κχ)2 ) 3 2 γκ2 ρ B χ

14 Phase space V = (X 2 C 2 ) 2, C =.5 V = (X 2 C 2 ) 2, C = y y tanh(x).5.5 tanh(x)

15 Example evolutions: V = (χ 2 C 2 ) 2 X 2 C =.5 C =.5 2 y X y energy densities N N 2 4 X 6 8 w X N energy densities N N 2 4 X 6 8 w X N

16 Stability analysis V (χ) A B C exp( βχ) U B + stable for β > S B stable for β < exp( βχ 2 ) U stable for β > stable for β < χ 2 U U S χ 4 U U S ( χ 2 C 2) 2 U stable for C > 2/3 C stable, C 2 unstable C are the minima C 2 is the local maximum

17 Gauge invariance and stability L = 48 F 2 (A) V (A 2 ) F 2 is invariant under A A + C. V (A 2 ) breaks this symmetry resulting in extra degrees of freedom. To see this we can make an expansion in Stückelberg fields s.t. A = à + 4[ Σ] L = 48 F 2 (Ã) V ((à + F(Σ)) 2 ) is now invariant under à à + [ C]; Σ Σ C/4. Expanding the potential around à L = L V (à 2 )F 2 (Σ) Presence of ghost field for V (Ã2 ) < V,χ χ < i.e. when there is phantom behaviour (w χ < ).

18 Part I: Inflation

19 Cosmological perturbations General perturbations about FRW background: ds 2 = N 2 dt 2 + h ij ( dx i + N i dt )( dx j + N j dt ) h ij = a 2 e 2ζ δ ij ζ is the curvature perturbation, and we expand N i and N as: N i = ψ,i + Ñ i N = + α Perturbations of the three-form: A ij = a(t)ǫ ijk α,k A ijk = a 3 (t)ǫ ijk (χ(t) + α ) Vector perturbations are decaying and can be ignored.

20 The second order action The second order action in scalar perturbations: S 2 = dtd 3 x [a 3 ΣH ] ζ 2 2 aǫ( ζ) 2 where the speed of sound and ǫ = Ḣ H 2 c 2 s = V,χχχ V,χ ζ is conserved on the large scales. Σ = H2 ǫ c 2 s ζ = c2 s ǫ 2 a 2 ( ψ + ζ ) H

21 Power spectrum of scalar perturbations The 2-point correlation function ζ(k )ζ(k 2 ) = (2π) 5 δ 3 (k + k 2 ) P ζ(k ) 2k 3, The power spectrum P ζ 2π 2k3 ζ k 2 = 2(2π) 2 ǫc s H 2 MPl 2 indicates horizon crossing c s k = ah. The spectral index n s is n s = 2ǫ + ǫ ǫh + ċs c s H

22 Power spectrum of tensor perturbations Since the three-form does not generate tensor perturbations, evolution equation is as usual, their ḧ + 3Hḣ 2 a 2 h = Tensor power spectrum: P T = 2 H 2 π 2 The tensor spectral index is then MPl 2 n T = 2ǫ

23 Consistency relation Ratio of tensor to scalar perturbations r P T P ζ = 6 c S ǫ Thus it is in principle possible to distinguish the three-form inflation from scalar field already from the spectra of linear perturbations.

24 The third order action The third order action of scalar perturbations: { S 3 = dtd 3 x[ ǫaζ( ζ) 2 a 3 (Σ + 2λ) ζ 3 H 3 + 3a3 ǫ c 2 ζ ζ 2 s ( + 3ζ ζ ) ( i j ψ i j ψ 2 ψ 2 ψ ) 2a i ψ i ζ ψ} 2 2a H where λ is λ = 2 V 3,χV,χχχ V 3,χχ At tree level in quantum field theory, and in the interaction picture, the In-In (equal time) three-point correlation function is given by the expression ζ(t, k )ζ(t, k 2 )ζ(t, k 3 ) = i t t dt [ζ(t, k )ζ(t, k 2 )ζ(t, k 3 ), H int (t )]

25 The bispectrum and non-gaussianity The non-gaussianity of the CMB in the WMAP observations is analyzed by assuming ζ = ζ L 3 5 f NLζ 2 L where ζ L is the linear Gaussian part the perturbations, and f NL is an estimator parameterizing the size of the non-gaussianity. The three-point correlation function: ζ(k )ζ(k 2 )ζ(k 3 ) = (2π) 7 δ 3 (k + k 2 + k 3 ) k 3 k3 2 k3 3 P 2 ζ A(k,k 2,k 3 ) f NL = 3 k 3 + k3 2 + A k3 3 where K/3 = k = k 2 = k 3. f equil NL = 3 K 3A f equil NL 5 8 ( c 2 2 λ ) s Σ 35 8 ( c 2 s ) +...

26 The dual theory We define duals as: ( F) = 4! ǫ αβγδf αβγδ Φ F αβγδ = ǫ αβγδ Φ ( A) α = 3! ǫ αβγδa βγδ B α A βγδ = ǫ αβγδ B α which allows us to write the equivalent formulations of the Lagrangians: L IV (F, F) = 48 F 2 + 2A 2 ( F)V ( A 2 ( F) ) V (A 2 ( F)) L III (A, A)) = 3 [ A]2 V (A 2 ) L I (B, B) = 2 ( B)2 V ( 6B 2) L (Φ, Φ) = 2 Φ2 2B 2 ( Φ)V ( 6B 2 ( Φ) ) V ( 6B 2 ( Φ) ) We recognise L as a p(x,φ) theory or k-inflation/essence. X = µ Φ µ Φ.

27 Equivalent formulations Equivalence between Lagrangian descriptions: g f B = B( C) g 2 f 3 A = A( F) g 4 C = C( B) F = F( A) * * * * * C = C( A) Φ = Φ( B) 4 f 3 g A = A( C) 2 f g B = B( Φ) f Faraday formulation Gauge fixing formulation L f = f ( F 2 (x) ) V (x 2 ), L g = g ( ( x) 2) U(x 2 )

28 Example I: Power law potential In the original three-form theory L = 48 F 2 V A 2p and in the p(x,φ) theory ( L φ = (2p ) V ) /(2p ) ( ) p X 2p 24p 2 2 φ2 In either approach we obtain: c 2 s = 2p N e-folds before the end of inflation: χ 2 N = p + 2N The spectral index for N = 6 gives ǫ N + 2N n s 4ǫ.97

29 Power law potential Bounds from Planck in blue. Lines are for N = 5, 6, 7. f equil N L r

30 Power law potential Amplitude dominant in the equilateral shape.

31 Example II: Exponential potential In the original three-form theory L = 48 F 2 V exp(βa 2 ) and in the p(x,φ) theory: L = (W(x) ) V exp ( ) 2 W(x) 2 φ2 where W(x) is the Lambert-W function and x = X/2βV 2. N e-folds before the end of inflation χ 2 N = 2 3 8β + 6N and the spectral index gives for N = 6 n s.97

32 Exponential potential.5 f equil NL r

33 Exponential potential Amplitude dominant in the equilateral shape.

34 Part II: Dark Energy

35 Three-form action Action for the three-form A µνρ S = d 4 x g R 2κ 2 48 F 2 (A) V (A 2 ) a m a (A 2 )δ(x x(λ)) ẋ 2 g where F µνρσ = 4 [µ A νρσ] = µ A νρσ σ A µνρ + ρ A σµν ν A ρσµ We have the equations of motion: F = 2 ( V (A 2 ) + 2ρ m (A 2 ) ) A m(a 2 ) and due to antisymmetry we have the additional constraints: ( V (A 2 ) + 2ρ m (A 2 ) ) A = m(a 2 )

36 Equations of motion Consider flat FRW cosmology: ds 2 = dt 2 + a 2 (t)dx 2 Most general three-form compatible with FRW: A ijk = a 3 (t)ǫ ijk χ(t) Equations of motion of the field χ with f 2m,χ /m χ + 3H χ + V,χ + 3Ḣχ = κρ mf Equation of motion of dark matter fluid: ρ m + 3Hρ m = κρ m f

37 Equations of motion Friedmann equation (ȧ ) 2 H 2 = κ2 a 3 ( ) 2 ( χ + 3Hχ)2 + V (χ) + ρ m can also write: H 2 = κ2 3 V + ρ m κ 2 (χ + 3χ) 2 /6 κ χ + 3χ < 6. = d/dlna. Evolution of the Hubble rate: Equation of state parameter of χ: Ḣ = κ2 2 (V,χχ + ( + κfχ)ρ m ) w χ = + V,χ + κfρ m χ ρ χ

38 Effective potential χ + 3H χ + V,χ + 3Ḣχ = κρ mf V eff,χ = V,χ ( 3 2 (κχ)2 ) 3 2 κ2 ρ m χ + κfρ m ( 3 2 (κχ)2 ) We are going to study 4 cases: (i) V =, f = ; (ii) V =, f ; (iii) V, f = ; (iv) V, f.

39 Case: V =, f = ; y i χ i + 3χ i otherwise ρ χ. It is a cosmological constant! w χ X y 5 log( + z) 5 log( + z) energy densities 5 w X 5 log( + z) 5 log( + z)

40 Case: V =, f ; w χ = + κfρ m ρ χ X y 5 log( + z) 5 log( + z) energy densities 5 w X 5 log( + z) 5 log( + z)

41 Case: V, f = ; With V = V χ 2 ; w χ = + V,χ ρ χ χ X y 5 log( + z) 5 log( + z) energy densities 5 w X 5 log( + z) 5 log( + z)

42 Case: V, f ; With V = V χ 2 ; w χ = + V,χ + κfρ m χ ρ χ X y 5 log( + z) 5 log( + z) energy densities 5 w X 5 log( + z) 5 log( + z)

43 Case: V, f ; With V = V χ 2 and χ i = χ i = w χ = + V,χ + κfρ m χ ρ χ X y 5 log( + z) 5 log( + z) energy densities 5 w X 5 log( + z) 5 log( + z)

44 Newtonian limit of linear perturbations Linear evolution of matter density perturbations δ m + ( 2H + κf χ ) 2 F F ( ) κ 2 δ m = eff 2 ρ m k2 a 2c2 eff δ m κ 2 eff = κ2 F [ + 2 κ 2 ρ m ( F + (2H + κf χ) F κ2 2 V,χ κfρ m V,χχ + κf,χ ρ m )] c 2 eff = F F, F κ2 f 2 ρ m V,χχ + κf,χ ρ m F < c 2 eff < for sufficiently large modes there is extra source term for growth of perturbations!

45 Growth at a given time f =.7 2 f =.7 2 δ m (z = )/δ m (z = 3 ) k/a H

46 Growth for a given mode 5 4 f =.7 2 f =.7 2 δ m (z)/δ m (z = 3 ) log( + z).5

47 Summary Three-forms possess accelerating attractors and saddle points which can describe three-form driven inflation or dark energy; Three-forms can give rise to viable cosmological scenarios with potentially observable signatures distinct from standard single scalar field models; Equivalence between models with alternative Lagrangian descriptions. Scalar spectral index predicted to be n s.97 for N = 6. Models have typically either large non-gaussianity, e.g. potential or large ratio of tensor to scalar perturbations. power law In the presence of a coupling to dark matter, growth of structure is enhanced for small scales.

48 Further afield Reheating/preheating [De Felice et al.]; Multiple 3-forms (assisted inflation/quintessence); Quintessential inflation; Formation of structure (analytical and numerical methods); Screening mechanisms.