Cosmology of the string tachyon - University of · PDF fileMotivations The DBI tachyon The...

Motivations The DBI tachyon The CSFT tachyon Cosmology Nonlocal solutions Conclusions

Cosmology of the string tachyonProgress in cubic string field theory

Gianluca Calcagni

March 28th, 2007

Outline

1 Motivations

2 The DBI tachyon

3 The CSFT tachyon

4 Cosmology

5 Nonlocal solutions

Outline

1 Motivations

2 The DBI tachyon

3 The CSFT tachyon

4 Cosmology

Outline

1 Motivations

2 The DBI tachyon

3 The CSFT tachyon

4 Cosmology

Outline

1 Motivations

2 The DBI tachyon

3 The CSFT tachyon

4 Cosmology

Outline

1 Motivations

2 The DBI tachyon

3 The CSFT tachyon

4 Cosmology

The many faces of the string tachyon

Conformal field theory [Sen 2002]

Dirac–Born–Infeld (DBI) effective action [Garousi 2000;

Bergshoeff et al. 2000; Kluson 2000; Gibbons, Hori, and Yi 2001]

Boundary string field theory (BSFT) [Witten 1992,1993;

Shatashvili 1993a,b]

Cubic string field theory (CSFT) [Witten 1986a,b; Preitschopf et al.

1990; Aref’eva et al. 2002; Aref’eva et al. 1990a,b; Berkovits 1996,1999]

For a review see Sen (hep-th/0410103).

Conformal field theory

[Sen 2002]

Dirac–Born–Infeld (DBI) effective action

[Garousi 2000;

Boundary string field theory (BSFT)

[Witten 1992,1993;

Cubic string field theory (CSFT)

[Witten 1986a,b; Preitschopf et al.

Tachyon condensation (in Minkowski)

All methods agree: as the tachyon rolls down towards theasymptotic minimum of the potential, the brane it lives ondecays into a lower-dimensional brane or the closed stringvacuum

( Sen’s conjecture... proven by Schnabl 2005)

All methods agree: as the tachyon rolls down towards theasymptotic minimum of the potential, the brane it lives ondecays into a lower-dimensional brane or the closed stringvacuum ( Sen’s conjecture...

proven by Schnabl 2005)

All methods agree: as the tachyon rolls down towards theasymptotic minimum of the potential, the brane it lives ondecays into a lower-dimensional brane or the closed stringvacuum ( Sen’s conjecture... proven by Schnabl 2005)

Two actions for one tachyon (in Minkowski)

DBI tachyon (negligible higher-than-first-order derivatives):

ST = −∫

dDxV(T)√

1 + ∂µT∂µT

Cubic string field theory:

S = − 1g2

∫ (1

2α′Φ ∗ QBΦ +

13Φ ∗ Φ ∗ Φ

)At level (0, 0): Φ ∼= |Φ〉 = φ(x)|↓〉 and (metric −+ + + . . . )

Sφ =1g2

∫dDx

2α′φ(α′∂µ∂

µ + 1)φ− λ

(λα′∂µ∂µ/3φ

)3− Λ

]where λ = 39/2/26 ≈ 2.19. See Ohmori (hep-th/0102085).

ST = −∫

dDxV(T)√

1 + ∂µT∂µT

S = − 1g2

∫ (1

2α′Φ ∗ QBΦ +

13Φ ∗ Φ ∗ Φ

Sφ =1g2

∫dDx

µ + 1)φ− λ

)3− Λ

ST = −∫

dDxV(T)√

1 + ∂µT∂µT

S = − 1g2

∫ (1

2α′Φ ∗ QBΦ +

13Φ ∗ Φ ∗ Φ

At level (0, 0): Φ ∼= |Φ〉 = φ(x)|↓〉 and (metric −+ + + . . . )

Sφ =1g2

∫dDx

µ + 1)φ− λ

)3− Λ

ST = −∫

dDxV(T)√

1 + ∂µT∂µT

S = − 1g2

∫ (1

2α′Φ ∗ QBΦ +

13Φ ∗ Φ ∗ Φ

Sφ =1g2

∫dDx

µ + 1)φ− λ

)3− Λ

ST = −∫

dDxV(T)√

1 + ∂µT∂µT

S = − 1g2

∫ (1

2α′Φ ∗ QBΦ +

13Φ ∗ Φ ∗ Φ

Sφ =1g2

∫dDx

µ + 1)φ− λ

)3− Λ

]where λ = 39/2/26 ≈ 2.19.

See Ohmori (hep-th/0102085).

ST = −∫

dDxV(T)√

1 + ∂µT∂µT

S = − 1g2

∫ (1

2α′Φ ∗ QBΦ +

13Φ ∗ Φ ∗ Φ

Sφ =1g2

∫dDx

µ + 1)φ− λ

)3− Λ

Do they give the same predictions?

DBI tachyon: extensively studied both as inflaton and darkenergy field (problematic or ineffective in both cases)SFT tachyon: poor control both on Minkowski and FRWbackgrounds... ?

⇒ A comparison would open up interesting possibilities!

DBI tachyon: extensively studied both as inflaton and darkenergy field

(problematic or ineffective in both cases)SFT tachyon: poor control both on Minkowski and FRWbackgrounds... ?

DBI tachyon: extensively studied both as inflaton and darkenergy field (problematic or ineffective in both cases)

SFT tachyon: poor control both on Minkowski and FRWbackgrounds... ?

Friedmann–Robertson–Walker background

Flat FRW metric:

ds2 = −dt2 + a2(t) dxidxi.

The Hubble parameter is defined as H ≡ a/a = dta/a.

Dynamics

DBI action:

ST = −∫

dDx√−gV(T)

√1 + ∂µT∂µT

Perfect fluid (FRW):

pT = −V(T)√

1− T2

ρT = V(T)/√

1− T2

w ≡ pT/ρT = T2 − 1 < 0

Equations of motion (EH action):

H2 = ρT + ρm + ρr

T1− T2

+ 3HT +V,T

Dynamics

DBI action:

ST = −∫

dDx√−gV(T)

√1 + ∂µT∂µT

pT = −V(T)√

1− T2

ρT = V(T)/√

1− T2

w ≡ pT/ρT = T2 − 1 < 0

T1− T2

+ 3HT +V,T

Dynamics

DBI action:

ST = −∫

dDx√−gV(T)

√1 + ∂µT∂µT

pT = −V(T)√

1− T2

ρT = V(T)/√

1− T2

w ≡ pT/ρT = T2 − 1

T1− T2

+ 3HT +V,T

Dynamics

DBI action:

ST = −∫

dDx√−gV(T)

√1 + ∂µT∂µT

pT = −V(T)√

1− T2

ρT = V(T)/√

1− T2

w ≡ pT/ρT = T2 − 1 < 0

T1− T2

+ 3HT +V,T

Dynamics

DBI action:

ST = −∫

dDx√−gV(T)

√1 + ∂µT∂µT

pT = −V(T)√

1− T2

ρT = V(T)/√

1− T2

w ≡ pT/ρT = T2 − 1 < 0

T1− T2

+ 3HT +V,T

Tachyon condensation

Condensation into the closed string vacuum (Carrollianlimit).

Near the minimum gs = O(1) and the perturbativedescription may fail down (for simple compactification).This is a toy model valid on cosmological scales.More problematic to justify at late times.

Condensation into the closed string vacuum (Carrollianlimit).Near the minimum gs = O(1) and the perturbativedescription may fail down (for simple compactification).

This is a toy model valid on cosmological scales.More problematic to justify at late times.

Condensation into the closed string vacuum (Carrollianlimit).Near the minimum gs = O(1) and the perturbativedescription may fail down (for simple compactification).This is a toy model valid on cosmological scales.

More problematic to justify at late times.

Condensation into the closed string vacuum (Carrollianlimit).Near the minimum gs = O(1) and the perturbativedescription may fail down (for simple compactification).This is a toy model valid on cosmological scales.More problematic to justify at late times.

DBI tachyon inflation

No reheating with runaway string effective potentials.Large density perturbations with such potentials.Reheating can be achieved with a negative KKLT-like Λ.Anisotropies adjusted with small warp factor [Garousi, Sami,

Tsujikawa 2004]

With phenomenological potentials, good inflation andnon-Gaussianity but non-characteristic predictions.

No reheating with runaway string effective potentials.

Large density perturbations with such potentials.Reheating can be achieved with a negative KKLT-like Λ.Anisotropies adjusted with small warp factor [Garousi, Sami,

Tsujikawa 2004]

No reheating with runaway string effective potentials.Large density perturbations with such potentials.

Reheating can be achieved with a negative KKLT-like Λ.Anisotropies adjusted with small warp factor [Garousi, Sami,

Tsujikawa 2004]

DBI tachyon as dark energywith Andrew R. Liddle – PRD 74, 043528 (2006), astro-ph/0606003

Different predictions between the DBI tachyon andcanonical quintessence?For a wide choice of potentials (ad-hoc or motivated), finetuning on either the i.c. or the parameters of the potential.High-precision observational cosmology allows toconstrain the theory in a remarkable way.The tachyon is poorly effective as dark energy (thecosmological constant problem is NOT solved).The tachyon cannot decay faster than dust matter,ρT ∼ a−3(1+wT) > ρm ∼ a−3 ⇒ cannot be used asquintessential inflaton.

Different predictions between the DBI tachyon andcanonical quintessence?

For a wide choice of potentials (ad-hoc or motivated), finetuning on either the i.c. or the parameters of the potential.High-precision observational cosmology allows toconstrain the theory in a remarkable way.The tachyon is poorly effective as dark energy (thecosmological constant problem is NOT solved).The tachyon cannot decay faster than dust matter,ρT ∼ a−3(1+wT) > ρm ∼ a−3 ⇒ cannot be used asquintessential inflaton.

Different predictions between the DBI tachyon andcanonical quintessence?For a wide choice of potentials (ad-hoc or motivated), finetuning on either the i.c. or the parameters of the potential.

High-precision observational cosmology allows toconstrain the theory in a remarkable way.The tachyon is poorly effective as dark energy (thecosmological constant problem is NOT solved).The tachyon cannot decay faster than dust matter,ρT ∼ a−3(1+wT) > ρm ∼ a−3 ⇒ cannot be used asquintessential inflaton.

Different predictions between the DBI tachyon andcanonical quintessence?For a wide choice of potentials (ad-hoc or motivated), finetuning on either the i.c. or the parameters of the potential.High-precision observational cosmology allows toconstrain the theory in a remarkable way.

The tachyon is poorly effective as dark energy (thecosmological constant problem is NOT solved).The tachyon cannot decay faster than dust matter,ρT ∼ a−3(1+wT) > ρm ∼ a−3 ⇒ cannot be used asquintessential inflaton.

Different predictions between the DBI tachyon andcanonical quintessence?For a wide choice of potentials (ad-hoc or motivated), finetuning on either the i.c. or the parameters of the potential.High-precision observational cosmology allows toconstrain the theory in a remarkable way.The tachyon is poorly effective as dark energy (thecosmological constant problem is NOT solved).

The tachyon cannot decay faster than dust matter,ρT ∼ a−3(1+wT) > ρm ∼ a−3 ⇒ cannot be used asquintessential inflaton.

Different predictions between the DBI tachyon andcanonical quintessence?For a wide choice of potentials (ad-hoc or motivated), finetuning on either the i.c. or the parameters of the potential.High-precision observational cosmology allows toconstrain the theory in a remarkable way.The tachyon is poorly effective as dark energy (thecosmological constant problem is NOT solved).The tachyon cannot decay faster than dust matter,ρT ∼ a−3(1+wT) > ρm ∼ a−3 ⇒ cannot be used asquintessential inflaton.

Cubic SFT action

S = Sg + Sφ,

∫dDx

√−g R

Sφ =∫

dDx√−g

[12φ(�− m2)φ− U(φ)− Λ

]� ≡ 1√

−g∂µ(

√−g ∂µ)

φ ≡ λ�/3φ ≡ er∗�φ , r∗ ≡lnλ

3≈ 0.2616

er∗� =+∞∑`=0

r`∗`!

�` ≡+∞∑`=0

c`�`

Cubic SFT action

S = Sg + Sφ, Sg =1

∫dDx

√−g R

Sφ =∫

dDx√−g

[12φ(�− m2)φ− U(φ)− Λ

]� ≡ 1√

−g∂µ(

√−g ∂µ)

φ ≡ λ�/3φ ≡ er∗�φ , r∗ ≡lnλ

3≈ 0.2616

er∗� =+∞∑`=0

r`∗`!

�` ≡+∞∑`=0

c`�`

Cubic SFT action

S = Sg + Sφ, Sg =1

∫dDx

√−g R

Sφ =∫

dDx√−g

[12φ(�− m2)φ− U(φ)− Λ

� ≡ 1√−g

∂µ(√−g ∂µ)

φ ≡ λ�/3φ ≡ er∗�φ , r∗ ≡lnλ

3≈ 0.2616

er∗� =+∞∑`=0

r`∗`!

�` ≡+∞∑`=0

c`�`

Cubic SFT action

S = Sg + Sφ, Sg =1

∫dDx

√−g R

Sφ =∫

dDx√−g

[12φ(�− m2)φ− U(φ)− Λ

]� ≡ 1√

−g∂µ(

√−g ∂µ)

φ ≡ λ�/3φ ≡ er∗�φ , r∗ ≡lnλ

3≈ 0.2616

er∗� =+∞∑`=0

r`∗`!

�` ≡+∞∑`=0

c`�`

Cubic SFT action

S = Sg + Sφ, Sg =1

∫dDx

√−g R

Sφ =∫

dDx√−g

[12φ(�− m2)φ− U(φ)− Λ

]� ≡ 1√

−g∂µ(

√−g ∂µ)

φ ≡ λ�/3φ ≡ er∗�φ , r∗ ≡lnλ

3≈ 0.2616

er∗� =+∞∑`=0

r`∗`!

�` ≡+∞∑`=0

c`�`

Cubic SFT action

S = Sg + Sφ, Sg =1

∫dDx

√−g R

Sφ =∫

dDx√−g

[12φ(�− m2)φ− U(φ)− Λ

]� ≡ 1√

−g∂µ(

√−g ∂µ)

φ ≡ λ�/3φ ≡ er∗�φ , r∗ ≡lnλ

3≈ 0.2616

er∗� =+∞∑`=0

r`∗`!

�` ≡+∞∑`=0

c`�`

Some other definitions

V(φ) ≡ 12 m2φ2 + U(φ) + Λ

U′ ≡ δUδφ

= er∗�U′

U′ ≡ ∂U

V(φ) ≡ 12 m2φ2 + U(φ) + Λ

U′ ≡ δUδφ

= er∗�U′

U′ ≡ ∂U

V(φ) ≡ 12 m2φ2 + U(φ) + Λ

U′ ≡ δUδφ

= er∗�U′

U′ ≡ ∂U

Choice for the potential

Pure monomial potential:

U(φ) =σ

U′ = σer∗�φn−1

Bosonic CSFT when m2 = −1, σ = λ, n = 3.

Susy CSFT when m2 = −1/2, σ = σ(λ), n = 4.

U(φ) =σ

Equations of motion: scalar field

−(�− m2)φ+ U′ = 0

In terms of φ:−(�− m2)e−2r∗�φ+ U′ = 0

Equations of motion: scalar field

−(�− m2)φ+ U′ = 0

In terms of φ:−(�− m2)e−2r∗�φ+ U′ = 0

Energy density and pressure

ρ = −T00 =

2(1−O2) + V −O1

p = Tii =

2(1−O2)− V +O1

O1 =∫ r∗

0ds (es�U′)(�e−s�φ),

O2 =2φ2

∫ r∗

0ds (es�U′).(e−s�φ)..

ρ = −T00 =

2(1−O2) + V −O1

p = Tii =

2(1−O2)− V +O1

O1 =∫ r∗

O2 =2φ2

∫ r∗

0ds (es�U′).(e−s�φ)..

ρ = −T00 =

2(1−O2) + V −O1

p = Tii =

2(1−O2)− V +O1

O1 =∫ r∗

O2 =2φ2

∫ r∗

0ds (es�U′).(e−s�φ)..

Equation of state

Pseudo slow-roll condition:

φ2 � V and φ2|O2| � |O1| ⇒ wφ ∼ −1

A new condition for acceleration:

|φ2(1−O2)/2 + V| � |O1| ⇒ wφ ∼ −1

Pseudo kinetic regime (beyond the DBI barrier):

φ2 � V and φ2|O2| � |O1| ⇒ wφ ∼ 1

Pseudo phantom regime:

φ2 � |V −O1| and O2 > 1 ⇒ wφ . −1

Equation of state

φ2 � V and φ2|O2| � |O1|

⇒ wφ ∼ −1

|φ2(1−O2)/2 + V| � |O1| ⇒ wφ ∼ −1

φ2 � V and φ2|O2| � |O1| ⇒ wφ ∼ 1

φ2 � |V −O1| and O2 > 1 ⇒ wφ . −1

Equation of state

φ2 � V and φ2|O2| � |O1| ⇒ wφ ∼ −1

|φ2(1−O2)/2 + V| � |O1| ⇒ wφ ∼ −1

φ2 � V and φ2|O2| � |O1| ⇒ wφ ∼ 1

φ2 � |V −O1| and O2 > 1 ⇒ wφ . −1

Equation of state

φ2 � V and φ2|O2| � |O1| ⇒ wφ ∼ −1

|φ2(1−O2)/2 + V| � |O1|

⇒ wφ ∼ −1

φ2 � V and φ2|O2| � |O1| ⇒ wφ ∼ 1

φ2 � |V −O1| and O2 > 1 ⇒ wφ . −1

Equation of state

φ2 � V and φ2|O2| � |O1| ⇒ wφ ∼ −1

|φ2(1−O2)/2 + V| � |O1| ⇒ wφ ∼ −1

φ2 � V and φ2|O2| � |O1| ⇒ wφ ∼ 1

φ2 � |V −O1| and O2 > 1 ⇒ wφ . −1

Equation of state

φ2 � V and φ2|O2| � |O1| ⇒ wφ ∼ −1

|φ2(1−O2)/2 + V| � |O1| ⇒ wφ ∼ −1

φ2 � V and φ2|O2| � |O1|

⇒ wφ ∼ 1

φ2 � |V −O1| and O2 > 1 ⇒ wφ . −1

Equation of state

φ2 � V and φ2|O2| � |O1| ⇒ wφ ∼ −1

|φ2(1−O2)/2 + V| � |O1| ⇒ wφ ∼ −1

φ2 � V and φ2|O2| � |O1| ⇒ wφ ∼ 1

φ2 � |V −O1| and O2 > 1 ⇒ wφ . −1

Equation of state

φ2 � V and φ2|O2| � |O1| ⇒ wφ ∼ −1

|φ2(1−O2)/2 + V| � |O1| ⇒ wφ ∼ −1

φ2 � V and φ2|O2| � |O1| ⇒ wφ ∼ 1

φ2 � |V −O1| and O2 > 1

⇒ wφ . −1

Equation of state

φ2 � V and φ2|O2| � |O1| ⇒ wφ ∼ −1

|φ2(1−O2)/2 + V| � |O1| ⇒ wφ ∼ −1

φ2 � V and φ2|O2| � |O1| ⇒ wφ ∼ 1

φ2 � |V −O1| and O2 > 1 ⇒ wφ . −1

Non-locality

The cosmological equations of motion are nonlinear andinvolve all the derivatives φ(n) and H(n), that is, all theinfinite SR tower.

Non-standard Cauchy problem: infinite initial conditions, toknow them means to find the solution! [Moeller and Zwiebach

Nonlocal theories are at odds with the inflationaryparadigm: while the latter tends to erase any memory ofthe initial conditions, the formers do preserve this memory.

Non-locality

The cosmological equations of motion are nonlinear andinvolve all the derivatives φ(n) and H(n), that is, all theinfinite SR tower.Non-standard Cauchy problem: infinite initial conditions, toknow them means to find the solution! [Moeller and Zwiebach

Non-locality

The cosmological equations of motion are nonlinear andinvolve all the derivatives φ(n) and H(n), that is, all theinfinite SR tower.Non-standard Cauchy problem: infinite initial conditions, toknow them means to find the solution! [Moeller and Zwiebach

Unviable cosmologies?

Very important class of dynamics:

φ = tp, H = H0tq

Constant SR parameters when q = −1.

�`φ = (−1)`tp−2``−1∏n=0

(p− 2n)(p− 2n− 1 + 3H0).

If p ∈ R \ N+, φ is ill-defined

lim`→∞

∣∣∣∣c`+1�`+1φ

c`�`φ

∣∣∣∣ = lim`→∞

|c1(p−2`)(p−2`−1+3H0)|t−2

`+ 1= +∞ ,

Possibility: p is a positive even number, so that the series endsat `∗ ≡ p/2 and φ ∼ tp. Another case is p− 1 + 3H0 = 2n.

φ = tp, H = H0tq

�`φ = (−1)`tp−2``−1∏n=0

(p− 2n)(p− 2n− 1 + 3H0).

lim`→∞

∣∣∣∣c`+1�`+1φ

c`�`φ

∣∣∣∣ = lim`→∞

|c1(p−2`)(p−2`−1+3H0)|t−2

`+ 1= +∞ ,

φ = tp, H = H0tq

�`φ = (−1)`tp−2``−1∏n=0

(p− 2n)(p− 2n− 1 + 3H0).

lim`→∞

∣∣∣∣c`+1�`+1φ

c`�`φ

∣∣∣∣ = lim`→∞

|c1(p−2`)(p−2`−1+3H0)|t−2

`+ 1= +∞ ,

φ = tp, H = H0tq

�`φ = (−1)`tp−2``−1∏n=0

(p− 2n)(p− 2n− 1 + 3H0).

lim`→∞

∣∣∣∣c`+1�`+1φ

c`�`φ

∣∣∣∣ = lim`→∞

|c1(p−2`)(p−2`−1+3H0)|t−2

`+ 1= +∞ ,

φ = tp, H = H0tq

�`φ = (−1)`tp−2``−1∏n=0

(p− 2n)(p− 2n− 1 + 3H0).

lim`→∞

∣∣∣∣c`+1�`+1φ

c`�`φ

∣∣∣∣ = lim`→∞

|c1(p−2`)(p−2`−1+3H0)|t−2

`+ 1= +∞ ,

φ = tp, H = H0tq

�`φ = (−1)`tp−2``−1∏n=0

(p− 2n)(p− 2n− 1 + 3H0).

lim`→∞

∣∣∣∣c`+1�`+1φ

c`�`φ

∣∣∣∣ = lim`→∞

|c1(p−2`)(p−2`−1+3H0)|t−2

`+ 1= +∞ ,

φ = tp, H = H0tq

�`φ = (−1)`tp−2``−1∏n=0

(p− 2n)(p− 2n− 1 + 3H0).

lim`→∞

∣∣∣∣c`+1�`+1φ

c`�`φ

∣∣∣∣ = lim`→∞

|c1(p−2`)(p−2`−1+3H0)|t−2

`+ 1= +∞ ,

Possibility: p is a positive even number, so that the series endsat `∗ ≡ p/2 and φ ∼ tp.

Another case is p− 1 + 3H0 = 2n.

φ = tp, H = H0tq

�`φ = (−1)`tp−2``−1∏n=0

(p− 2n)(p− 2n− 1 + 3H0).

lim`→∞

∣∣∣∣c`+1�`+1φ

c`�`φ

∣∣∣∣ = lim`→∞

|c1(p−2`)(p−2`−1+3H0)|t−2

`+ 1= +∞ ,

Can we conclude that power-law cosmology cannot be used asa base for nonlocal solutions?

NO!What is not defined is the nonlocal solution expressed as aninfinite series of powers of the d’Alembertian.

Can we conclude that power-law cosmology cannot be used asa base for nonlocal solutions?

NO!What is not defined is the nonlocal solution expressed as aninfinite series of powers of the d’Alembertian.

LocalizationGC, G. Nardelli, and M. Montobbio, to appear; GC et al., to appear

Interpret r∗ as a fixed value of an auxiliary evolutionvariable r. The scalar field φ(r, t) is thought to live in 1 + 1dimensions.The solution φloc(t) = φ(r∗ = 0, t) of the local system(r∗ = 0) is the “initial condition”.Define

φ(r, t) ≡ er(β+�/α)φloc(t)

Interpret r∗ as a fixed value of an auxiliary evolutionvariable r.

The scalar field φ(r, t) is thought to live in 1 + 1dimensions.The solution φloc(t) = φ(r∗ = 0, t) of the local system(r∗ = 0) is the “initial condition”.Define

Interpret r∗ as a fixed value of an auxiliary evolutionvariable r. The scalar field φ(r, t) is thought to live in 1 + 1dimensions.

The solution φloc(t) = φ(r∗ = 0, t) of the local system(r∗ = 0) is the “initial condition”.Define

Interpret r∗ as a fixed value of an auxiliary evolutionvariable r. The scalar field φ(r, t) is thought to live in 1 + 1dimensions.The solution φloc(t) = φ(r∗ = 0, t) of the local system(r∗ = 0) is the “initial condition”.

Defineφ(r, t) ≡ er(β+�/α)φloc(t)

Interpret r∗ as a fixed value of an auxiliary evolutionvariable r. The scalar field φ(r, t) is thought to live in 1 + 1dimensions.The solution φloc(t) = φ(r∗ = 0, t) of the local system(r∗ = 0) is the “initial condition”.Define

Localization

Property 1: it satisfies the diffusion equation:

α∂rφ(r, t) = αβ φ(r, t) + �φ(r, t)

Property 2: eq� is simply a shift of the auxiliary variable r.

eq�φ(r, t) = eαq ∂rφ(r, t) = φ(r + αq, t)

⇒ The system becomes local in t!

(�− m2)φ(r, t) = e−rβU′[erβφ((1 + 2α)r, t)]

Localization

α∂rφ(r, t) = αβ φ(r, t) + �φ(r, t)

Localization

α∂rφ(r, t) = αβ φ(r, t) + �φ(r, t)

Localization

α∂rφ(r, t) = αβ φ(r, t) + �φ(r, t)

Steps towards localized solutions

1 Find the eigenstates of the d’Alembertian operator:

�G(µ, t) = −G(µ, t)− 3HG(µ, t) = µ2G(µ, t) .

2 Write the local solution as an expansion in the basis ofeigenstates of the � (Mellin–Barnes transform):

φ(0, t) =∫

dµ [C1G1(µ, t) + C2 G2(µ, t)] f (µ) .

3 Write the nonlocal “solution” (Gabor transform):

φ(r, t) =∫

dµ er(β+µ2/α)[C1G1(µ, t) + C2 G2(µ, t)] f (µ) .

It satisfies the heat equation by definition!4 Check that φ(r, t) is an approximated solution of the

nonlocal e.o.m.s for some α, β, Ci.

�G(µ, t) = −G(µ, t)− 3HG(µ, t) = µ2G(µ, t) .

φ(0, t) =∫

dµ [C1G1(µ, t) + C2 G2(µ, t)] f (µ) .

φ(r, t) =∫

�G(µ, t) = −G(µ, t)− 3HG(µ, t) = µ2G(µ, t) .

φ(0, t) =∫

dµ [C1G1(µ, t) + C2 G2(µ, t)] f (µ) .

φ(r, t) =∫

�G(µ, t) = −G(µ, t)− 3HG(µ, t) = µ2G(µ, t) .

φ(0, t) =∫

dµ [C1G1(µ, t) + C2 G2(µ, t)] f (µ) .

φ(r, t) =∫

�G(µ, t) = −G(µ, t)− 3HG(µ, t) = µ2G(µ, t) .

φ(0, t) =∫

dµ [C1G1(µ, t) + C2 G2(µ, t)] f (µ) .

φ(r, t) =∫

�G(µ, t) = −G(µ, t)− 3HG(µ, t) = µ2G(µ, t) .

φ(0, t) =∫

dµ [C1G1(µ, t) + C2 G2(µ, t)] f (µ) .

φ(r, t) =∫

�G(µ, t) = −G(µ, t)− 3HG(µ, t) = µ2G(µ, t) .

φ(0, t) =∫

dµ [C1G1(µ, t) + C2 G2(µ, t)] f (µ) .

φ(r, t) =∫

�G(µ, t) = −G(µ, t)− 3HG(µ, t) = µ2G(µ, t) .

φ(0, t) =∫

dµ [C1G1(µ, t) + C2 G2(µ, t)] f (µ) .

φ(r, t) =∫

It satisfies the heat equation by definition!

4 Check that φ(r, t) is an approximated solution of thenonlocal e.o.m.s for some α, β, Ci.

�G(µ, t) = −G(µ, t)− 3HG(µ, t) = µ2G(µ, t) .

φ(0, t) =∫

dµ [C1G1(µ, t) + C2 G2(µ, t)] f (µ) .

φ(r, t) =∫

Present status and advantages

1 SUSY Minkowski solution successfully found (to appear)2 Only known method allowing to keep the whole series e�

3 Quantization is well-defined, as in the closely related 1 + 1Hamiltonian formalism [Llosa and Vives 1994; Gomis et al. 2001,2004;

Cheng et al. 2002]

1 SUSY Minkowski solution successfully found (to appear)

2 Only known method allowing to keep the whole series e�

Cheng et al. 2002]

“Unviable” cosmologies revisiteda = tp, H = H0t−1

ψ(r, t) ∝(

Ψ(−p

2; 1− ν;

ν = (1− 3H0)/2

Ψ(α; β; z) =π

sinπβ

[Φ(α; β; z)

Γ(1 + α− β)Γ(β)

−z1−β Φ(1 + α− β; 2− β; z)Γ(α)Γ(2− β)

Φ(α; β; x) =Γ(β)Γ(α)

+∞∑k=0

Γ(α+ k)Γ(β + k)

ψ(r, t) ∝(

Ψ(−p

2; 1− ν;

ν = (1− 3H0)/2

Ψ(α; β; z) =π

sinπβ

[Φ(α; β; z)

Γ(1 + α− β)Γ(β)

−z1−β Φ(1 + α− β; 2− β; z)Γ(α)Γ(2− β)

+∞∑k=0

Γ(α+ k)Γ(β + k)

ψ(r, t) ∝(

Ψ(−p

2; 1− ν;

ν = (1− 3H0)/2

Ψ(α; β; z) =π

sinπβ

[Φ(α; β; z)

Γ(1 + α− β)Γ(β)

−z1−β Φ(1 + α− β; 2− β; z)Γ(α)Γ(2− β)

+∞∑k=0

Γ(α+ k)Γ(β + k)

ψ(r, t) ∝(

Ψ(−p

2; 1− ν;

ν = (1− 3H0)/2

Ψ(α; β; z) =π

sinπβ

[Φ(α; β; z)

Γ(1 + α− β)Γ(β)

−z1−β Φ(1 + α− β; 2− β; z)Γ(α)Γ(2− β)

+∞∑k=0

Γ(α+ k)Γ(β + k)

ψ(r, t) ∝(

Ψ(−p

2; 1− ν;

ν = (1− 3H0)/2

Ψ(α; β; z) =π

sinπβ

[Φ(α; β; z)

Γ(1 + α− β)Γ(β)

−z1−β Φ(1 + α− β; 2− β; z)Γ(α)Γ(2− β)

+∞∑k=0

Γ(α+ k)Γ(β + k)

ψ(r, t) ∝(

Ψ(−p

2; 1− ν;

ν = (1− 3H0)/2

Ψ(α; β; z) =π

sinπβ

[Φ(α; β; z)

Γ(1 + α− β)Γ(β)

−z1−β Φ(1 + α− β; 2− β; z)Γ(α)Γ(2− β)

+∞∑k=0

Γ(α+ k)Γ(β + k)

“Unviable” cosmologies revisited

Figure: p = 1/2, ν = −3/2, r = 1

“Unviable” cosmologies revisited

Figure: p = 1/2, ν = −3/2, r = −1

Conclusions

Non-locality generates new dynamics (at classical andquantum level)

The CSFT tachyon is cosmologically inequivalent to (andmaybe more viable than) the DBI one

Open issuesSearch for analytic cosmological solutions in progress (toappear)Evidence for a nontrivial relation between SBFT andCSFT! (to appear)

Conclusions

Non-locality generates new dynamics (at classical andquantum level)The CSFT tachyon is cosmologically inequivalent to (andmaybe more viable than) the DBI one

Conclusions

Open issues

Search for analytic cosmological solutions in progress (toappear)Evidence for a nontrivial relation between SBFT andCSFT! (to appear)

Conclusions

Open issuesSearch for analytic cosmological solutions in progress (toappear)

Evidence for a nontrivial relation between SBFT andCSFT! (to appear)

Conclusions

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