Inflation with a stringy minimal length (reworked)

Transcription

1 Humboldt Universität zu Berlin Nordic String Meeting, Bremen, March 27 th 2009

2 Acknowledgements Part of an ongoing collaboration with Gonzalo A. Palma. This work reported on in arxiv : , to appear in JHEP.

3 Alternative title: Trans-Planckian problem revisited Inflation provides us with a complimentary microscope to particle accelerators. We know that the universe expanded at least e fold during inflation. l pl = m 10 9 m i.e. during inflation, the comoving Planck scale gets stretched to molecular sizes.

4 Alternative title: Trans-Planckian problem revisited Inflation provides us with a complimentary microscope to particle accelerators. We know that the universe expanded at least e fold during inflation. l pl = m 10 9 m i.e. during inflation, the comoving Planck scale gets stretched to molecular sizes. After inflation, universe expanded T reheat /T CMB fold. l pl m, i.e. all scales 10 5 l/y were sub Planckian.

5 Alternative title: Trans-Planckian problem revisited Inflation provides us with a complimentary microscope to particle accelerators. We know that the universe expanded at least e fold during inflation. l pl = m 10 9 m i.e. during inflation, the comoving Planck scale gets stretched to molecular sizes. After inflation, universe expanded T reheat /T CMB fold. l pl m, i.e. all scales 10 5 l/y were sub Planckian. Could there be signatures of Trans-Planckian physics in the CMB?

6 Alternative title: Trans-Planckian problem revisited Inflation provides us with a complimentary microscope to particle accelerators. We know that the universe expanded at least e fold during inflation. l pl = m 10 9 m i.e. during inflation, the comoving Planck scale gets stretched to molecular sizes. After inflation, universe expanded T reheat /T CMB fold. l pl m, i.e. all scales 10 5 l/y were sub Planckian. Could there be signatures of Trans-Planckian physics in the CMB? Only an intuitive argument. Other ways of phrasing the question: why does our EFT treatment seem to work so well? Could there be a breakdown of EFT during inflation? Could observation be sensitive to higher dimensional operators? Can we infer any UV physics from the CMB?

7 Alternative title: Trans-Planckian problem revisited Inflation provides us with a complimentary microscope to particle accelerators. We know that the universe expanded at least e fold during inflation. l pl = m 10 9 m i.e. during inflation, the comoving Planck scale gets stretched to molecular sizes. After inflation, universe expanded T reheat /T CMB fold. l pl m, i.e. all scales 10 5 l/y were sub Planckian. Could there be signatures of Trans-Planckian physics in the CMB? Only an intuitive argument. Other ways of phrasing the question: why does our EFT treatment seem to work so well? Could there be a breakdown of EFT during inflation? Could observation be sensitive to higher dimensional operators? Can we infer any UV physics from the CMB? Scale of inflation could be as high as GeV.

8 Precedent in Black Hole Physics It seems that a similar Trans-Planckian issue crops up when we compute the spectrum of a radiating Black Hole photons which began (tunnelled out) arbitrarily close to the horizon are infinitely red shifted on their way to the asymptotic observer. Photon propagation is extrapolated to arbitrarily high frequencies. Unruh, Corley and Jacobson: unknown physics ω 2 = k 2 ω 2 (k 2 ), where there is a deviation from the linear dispersion relations above some critical wavelength k c.

9 Precedent in Black Hole Physics It seems that a similar Trans-Planckian issue crops up when we compute the spectrum of a radiating Black Hole photons which began (tunnelled out) arbitrarily close to the horizon are infinitely red shifted on their way to the asymptotic observer. Photon propagation is extrapolated to arbitrarily high frequencies. Unruh, Corley and Jacobson: unknown physics ω 2 = k 2 ω 2 (k 2 ), where there is a deviation from the linear dispersion relations above some critical wavelength k c. c.f. condensed mater systems a 1d lattice has ω 2 = sin 2 [πk a], where a is the lattice spacing.

10 Precedent in Black Hole Physics It seems that a similar Trans-Planckian issue crops up when we compute the spectrum of a radiating Black Hole photons which began (tunnelled out) arbitrarily close to the horizon are infinitely red shifted on their way to the asymptotic observer. Photon propagation is extrapolated to arbitrarily high frequencies. Unruh, Corley and Jacobson: unknown physics ω 2 = k 2 ω 2 (k 2 ), where there is a deviation from the linear dispersion relations above some critical wavelength k c. c.f. condensed mater systems a 1d lattice has ω 2 = sin 2 [πk a], where a is the lattice spacing. No effect found: thermalization erases any initial state effects.

11 Precedent in Black Hole Physics It seems that a similar Trans-Planckian issue crops up when we compute the spectrum of a radiating Black Hole photons which began (tunnelled out) arbitrarily close to the horizon are infinitely red shifted on their way to the asymptotic observer. Photon propagation is extrapolated to arbitrarily high frequencies. Unruh, Corley and Jacobson: unknown physics ω 2 = k 2 ω 2 (k 2 ), where there is a deviation from the linear dispersion relations above some critical wavelength k c. c.f. condensed mater systems a 1d lattice has ω 2 = sin 2 [πk a], where a is the lattice spacing. No effect found: thermalization erases any initial state effects. For inflation? Parentani & Niemeyer: if H < k c and if ω /ω 2 << 1, no measurable effect.

12 Precedent in Black Hole Physics It seems that a similar Trans-Planckian issue crops up when we compute the spectrum of a radiating Black Hole photons which began (tunnelled out) arbitrarily close to the horizon are infinitely red shifted on their way to the asymptotic observer. Photon propagation is extrapolated to arbitrarily high frequencies. Unruh, Corley and Jacobson: unknown physics ω 2 = k 2 ω 2 (k 2 ), where there is a deviation from the linear dispersion relations above some critical wavelength k c. c.f. condensed mater systems a 1d lattice has ω 2 = sin 2 [πk a], where a is the lattice spacing. No effect found: thermalization erases any initial state effects. For inflation? Parentani & Niemeyer: if H < k c and if ω /ω 2 << 1, no measurable effect. Brandenberger and Martin: If we allow for non-standard dispersion relations (c.f. rotons in liquid helium), then there is a possibility of a measurable effect (although unlikely to be realized in UV complete theory).

13 Measurable initial state effects? Were we to take k c to be M pl in the above, we will generically find O(H 2 /Mpl 2 ) corrections to the correlation functions we compute and measure in cosmology. Given that H GeV, this is far too small an effect.

14 Measurable initial state effects? Were we to take k c to be M pl in the above, we will generically find O(H 2 /Mpl 2 ) corrections to the correlation functions we compute and measure in cosmology. Given that H GeV, this is far too small an effect. Once our modes have been redshifted to below the Planckian regimes, the usual EFT analysis applies. Could the net effect of super-planckian physics be to modify the usual initial state of the inflaton modes away from the usual Bunch-Davies vacuum?

15 Measurable initial state effects? Were we to take k c to be M pl in the above, we will generically find O(H 2 /Mpl 2 ) corrections to the correlation functions we compute and measure in cosmology. Given that H GeV, this is far too small an effect. Once our modes have been redshifted to below the Planckian regimes, the usual EFT analysis applies. Could the net effect of super-planckian physics be to modify the usual initial state of the inflaton modes away from the usual Bunch-Davies vacuum? O(H/M pl ) effects are then possible.

16 Measurable initial state effects? Were we to take k c to be M pl in the above, we will generically find O(H 2 /Mpl 2 ) corrections to the correlation functions we compute and measure in cosmology. Given that H GeV, this is far too small an effect. Once our modes have been redshifted to below the Planckian regimes, the usual EFT analysis applies. Could the net effect of super-planckian physics be to modify the usual initial state of the inflaton modes away from the usual Bunch-Davies vacuum? O(H/M pl ) effects are then possible. Banks and Mannelli: Not possible consistent with locality.

17 Measurable initial state effects? Were we to take k c to be M pl in the above, we will generically find O(H 2 /Mpl 2 ) corrections to the correlation functions we compute and measure in cosmology. Given that H GeV, this is far too small an effect. Once our modes have been redshifted to below the Planckian regimes, the usual EFT analysis applies. Could the net effect of super-planckian physics be to modify the usual initial state of the inflaton modes away from the usual Bunch-Davies vacuum? O(H/M pl ) effects are then possible. Banks and Mannelli: Not possible consistent with locality. Kaloper, Kleban, Lawrence, Shenker: all signatures of short distance physics consistent with low energy locality are to be corrections of the form H 2 /M 2, where M is the cutoff of the theory.

18 Measurable initial state effects? Were we to take k c to be M pl in the above, we will generically find O(H 2 /Mpl 2 ) corrections to the correlation functions we compute and measure in cosmology. Given that H GeV, this is far too small an effect. Once our modes have been redshifted to below the Planckian regimes, the usual EFT analysis applies. Could the net effect of super-planckian physics be to modify the usual initial state of the inflaton modes away from the usual Bunch-Davies vacuum? O(H/M pl ) effects are then possible. Banks and Mannelli: Not possible consistent with locality. Kaloper, Kleban, Lawrence, Shenker: all signatures of short distance physics consistent with low energy locality are to be corrections of the form H 2 /M 2, where M is the cutoff of the theory. Instead of considering ad-hoc modifications in the UV, what about string theory? M m s.

19 A minimal length in string theory In string theory, we are presented with a UV complete theory of quantum gravity with the string length as its only scale. At the level of the low energy degrees of freedom (particles), this implies the Generalized Uncertainty Principle (GUP): x p 1 2 [1 + β( p)2 +...].

20 A minimal length in string theory In string theory, we are presented with a UV complete theory of quantum gravity with the string length as its only scale. At the level of the low energy degrees of freedom (particles), this implies the Generalized Uncertainty Principle (GUP): x p 1 2 [1 + β( p)2 +...]. [β] = L 2 (comensurate to α ).

21 A minimal length in string theory In string theory, we are presented with a UV complete theory of quantum gravity with the string length as its only scale. At the level of the low energy degrees of freedom (particles), this implies the Generalized Uncertainty Principle (GUP): x p 1 2 [1 + β( p)2 +...]. [β] = L 2 (comensurate to α ). Arises from considering string scattering amplitudes at high energies (Gross and Mende; Amati, Ciafaloni and Veneziano).

22 A minimal length in string theory In string theory, we are presented with a UV complete theory of quantum gravity with the string length as its only scale. At the level of the low energy degrees of freedom (particles), this implies the Generalized Uncertainty Principle (GUP): x p 1 2 [1 + β( p)2 +...]. [β] = L 2 (comensurate to α ). Arises from considering string scattering amplitudes at high energies (Gross and Mende; Amati, Ciafaloni and Veneziano). Arises from worldsheet renormalization considerations (Konishi, Prafutti and Provero).

23 A minimal length in string theory In string theory, we are presented with a UV complete theory of quantum gravity with the string length as its only scale. At the level of the low energy degrees of freedom (particles), this implies the Generalized Uncertainty Principle (GUP): x p 1 2 [1 + β( p)2 +...]. [β] = L 2 (comensurate to α ). Arises from considering string scattering amplitudes at high energies (Gross and Mende; Amati, Ciafaloni and Veneziano). Arises from worldsheet renormalization considerations (Konishi, Prafutti and Provero). Implies the minimum spatial uncertainty x β.

24 A minimal length in string theory In string theory, we are presented with a UV complete theory of quantum gravity with the string length as its only scale. At the level of the low energy degrees of freedom (particles), this implies the Generalized Uncertainty Principle (GUP): x p 1 2 [1 + β( p)2 +...]. [β] = L 2 (comensurate to α ). Arises from considering string scattering amplitudes at high energies (Gross and Mende; Amati, Ciafaloni and Veneziano). Arises from worldsheet renormalization considerations (Konishi, Prafutti and Provero). Implies the minimum spatial uncertainty x β. Heuristically, since P 2 c.o.m. N, we lose resolution the higher energies we probe, implying a minimal length scale we can resolve if we only have strings as our probes of geometry.

25 Representing a stringy minimal length We will attempt to write down a field theory that represents the GUP and study its quantization. In doing so, we will analytically compute H 2 /ms 2 corrections to observables and comment on the likelihood of measuring these corrections. In what follows, we begin in the footsteps of Kempf and collaborators. ) Try [X i, P j ] = i (f (P 2 )δj i + g(p2 )P i P j.

26 Representing a stringy minimal length We will attempt to write down a field theory that represents the GUP and study its quantization. In doing so, we will analytically compute H 2 /ms 2 corrections to observables and comment on the likelihood of measuring these corrections. In what follows, we begin in the footsteps of Kempf and collaborators. ) Try [X i, P j ] = i (f (P 2 )δj i + g(p2 )P i P j. Jacobi identities impose g(s) = 2ff f 2sf, s = P 2

27 Representing a stringy minimal length We will attempt to write down a field theory that represents the GUP and study its quantization. In doing so, we will analytically compute H 2 /ms 2 corrections to observables and comment on the likelihood of measuring these corrections. In what follows, we begin in the footsteps of Kempf and collaborators. ) Try [X i, P j ] = i (f (P 2 )δj i + g(p2 )P i P j. Jacobi identities impose g(s) = 2ff f 2sf, s = P 2 Solved by g = 2β, f (s) = 2βs/( 1 + βs 1).

28 Representing a stringy minimal length We will attempt to write down a field theory that represents the GUP and study its quantization. In doing so, we will analytically compute H 2 /ms 2 corrections to observables and comment on the likelihood of measuring these corrections. In what follows, we begin in the footsteps of Kempf and collaborators. ) Try [X i, P j ] = i (f (P 2 )δj i + g(p2 )P i P j. Jacobi identities impose g(s) = 2ff f 2sf, s = P 2 Solved by g = 2β, f (s) = 2βs/( 1 + βs 1). Introduce auxiliary variables to represent the above as: X i φ( ρ) = i ρ i φ( ρ), P i φ( ρ) = ρ i φ( ρ) 1 βρ 2

29 Representing a stringy minimal length We will attempt to write down a field theory that represents the GUP and study its quantization. In doing so, we will analytically compute H 2 /ms 2 corrections to observables and comment on the likelihood of measuring these corrections. In what follows, we begin in the footsteps of Kempf and collaborators. ) Try [X i, P j ] = i (f (P 2 )δj i + g(p2 )P i P j. Jacobi identities impose g(s) = 2ff f 2sf, s = P 2 Solved by g = 2β, f (s) = 2βs/( 1 + βs 1). Introduce auxiliary variables to represent the above as: X i φ( ρ) = i ρ i φ( ρ), P i φ( ρ) = ρ i φ( ρ) 1 βρ 2 N.B. ρ < 1/ β in order for this representation to be well defined.

30 Representing a stringy minimal length We will attempt to write down a field theory that represents the GUP and study its quantization. In doing so, we will analytically compute H 2 /ms 2 corrections to observables and comment on the likelihood of measuring these corrections. In what follows, we begin in the footsteps of Kempf and collaborators. ) Try [X i, P j ] = i (f (P 2 )δj i + g(p2 )P i P j. Jacobi identities impose g(s) = 2ff f 2sf, s = P 2 Solved by g = 2β, f (s) = 2βs/( 1 + βs 1). Introduce auxiliary variables to represent the above as: X i φ( ρ) = i ρ i φ( ρ), P i φ( ρ) = ρ i φ( ρ) 1 βρ 2 N.B. ρ < 1/ β in order for this representation to be well defined. Symmetric w.r.t. the scalar product (φ 1, φ 2 ) = βρ 2 <1 d 3 ρ φ 1 ( ρ)φ 2( ρ)

31 Representing a stringy minimal length We will attempt to write down a field theory that represents the GUP and study its quantization. In doing so, we will analytically compute H 2 /ms 2 corrections to observables and comment on the likelihood of measuring these corrections. In what follows, we begin in the footsteps of Kempf and collaborators. ) Try [X i, P j ] = i (f (P 2 )δj i + g(p2 )P i P j. Jacobi identities impose g(s) = 2ff f 2sf, s = P 2 Solved by g = 2β, f (s) = 2βs/( 1 + βs 1). Introduce auxiliary variables to represent the above as: X i φ( ρ) = i ρ i φ( ρ), P i φ( ρ) = ρ i φ( ρ) 1 βρ 2 N.B. ρ < 1/ β in order for this representation to be well defined. Symmetric w.r.t. the scalar product (φ 1, φ 2 ) = βρ 2 <1 d 3 ρ φ 1 ( ρ)φ 2( ρ) B.C. s for this representation: φ(ρ = β 1/2 ) = 0.

32 Representing a stringy minimal length We begin by considering Minkowski space, where the action for a scalar field can be written in the form: S = 1 [ 2 dt (φ, 2 t φ) + (φ, P 2 φ) ], where (f, g) is the scalar product d 3 x f (x)g(x).

33 Representing a stringy minimal length We begin by considering Minkowski space, where the action for a scalar field can be written in the form: S = 1 [ 2 dt (φ, 2 t φ) + (φ, P 2 φ) ], where (f, g) is the scalar product d 3 x f (x)g(x). Suggests a natural way to incorporate the GUP replace the corresponding representation where P appears: S = 1 [ 2 ρ 2 β<1 dt d 3 ρ φ ] 2 ρ2 φ 2. (1 βρ 2 ) 2

34 Representing a stringy minimal length We begin by considering Minkowski space, where the action for a scalar field can be written in the form: S = 1 [ 2 dt (φ, 2 t φ) + (φ, P 2 φ) ], where (f, g) is the scalar product d 3 x f (x)g(x). Suggests a natural way to incorporate the GUP replace the corresponding representation where P appears: S = 1 [ 2 ρ 2 β<1 dt d 3 ρ φ ] 2 ρ2 φ 2. (1 βρ 2 ) 2 Generalizing to a curved background: ds 2 = a 2 (τ) ( dτ 2 + δ ij dx i dx j).

35 Representing a stringy minimal length We begin by considering Minkowski space, where the action for a scalar field can be written in the form: S = 1 [ 2 dt (φ, 2 t φ) + (φ, P 2 φ) ], where (f, g) is the scalar product d 3 x f (x)g(x). Suggests a natural way to incorporate the GUP replace the corresponding representation where P appears: S = 1 [ 2 ρ 2 β<1 dt d 3 ρ φ ] 2 ρ2 φ 2. (1 βρ 2 ) 2 Generalizing to a curved background: ds 2 = a 2 (τ) ( dτ 2 + δ ij dx i dx j). S = 1 2 dτd 3 x a 2 (τ) [( τ φ) 2 ] 3 i=1 ( x i φ)2.

36 Representing a stringy minimal length We begin by considering Minkowski space, where the action for a scalar field can be written in the form: S = 1 [ 2 dt (φ, 2 t φ) + (φ, P 2 φ) ], where (f, g) is the scalar product d 3 x f (x)g(x). Suggests a natural way to incorporate the GUP replace the corresponding representation where P appears: S = 1 [ 2 ρ 2 β<1 dt d 3 ρ φ ] 2 ρ2 φ 2. (1 βρ 2 ) 2 Generalizing to a curved background: ds 2 = a 2 (τ) ( dτ 2 + δ ij dx i dx j). S = 1 2 dτd 3 x a 2 (τ) [( τ φ) 2 ] 3 i=1 ( x i φ)2. Expressing fields in terms of physical co-ordinates y i = ax i, i.e. φ(τ, x) φ(τ, a(τ)x i ), we obtain S = 1 2 dτd 3 y a 2 (τ) [(Aφ) 2 a 2 (τ) ] 3 i=1 ( y i φ)2, where A = τ + i a a P iy i 3 a a, P i = i y i represents the convective derivative.

37 Representing a stringy minimal length In an FRW background, the replacement of the GUP representation of P i in A = τ + i a a P iy i 3 a a results in the action: [ S = 1 ( ) ] 2 βρ 2 <1 dτ d 3 ρ 1 a τ a ρ i a 1 βρ 2 ρi 3 a a φ 2 a2 ρ 2 φ 2 (1 βρ 2 )

38 Representing a stringy minimal length In an FRW background, the replacement of the GUP representation of P i in A = τ + i a a P iy i 3 a a results in the action: [ S = 1 ( ) ] 2 βρ 2 <1 dτ d 3 ρ 1 a τ a ρ i a 1 βρ 2 ρi 3 a a φ 2 a2 ρ 2 φ 2 (1 βρ 2 ) This action mixes different ρ modes. The change of variable k i = aρ i e βρ2 /2 decouples these modes, from which we derive the equation of motion for our mode functions: φ k + ν ν φ k + [µ 3 ( a a ) 9 ( a a ) 2 3a ν aν µ(τ, k) := a2 W ( βk 2 /a 2 ) β(1+w ( βk 2 /a 2 )) 2 = a2 ρ 2 (1 βρ 2 ) 2 a 4 (1+W ( βk 2 /a 2 )) = e 2 3 βρ 2 a 4 (1 βρ 2 ). ν(τ, k) := e 3 2 W ( βk2 /a 2 ) ] φ k = 0, with and

39 Representing a stringy minimal length In an FRW background, the replacement of the GUP representation of P i in A = τ + i a a P iy i 3 a a results in the action: [ S = 1 ( ) ] 2 βρ 2 <1 dτ d 3 ρ 1 a τ a ρ i a 1 βρ 2 ρi 3 a a φ 2 a2 ρ 2 φ 2 (1 βρ 2 ) This action mixes different ρ modes. The change of variable k i = aρ i e βρ2 /2 decouples these modes, from which we derive the equation of motion for our mode functions: φ k + ν ν φ k + [µ 3 ( a a ) 9 ( a a ) 2 3a ν aν µ(τ, k) := a2 W ( βk 2 /a 2 ) β(1+w ( βk 2 /a 2 )) 2 = a2 ρ 2 (1 βρ 2 ) 2 a 4 (1+W ( βk 2 /a 2 )) = e 2 3 βρ 2 a 4 (1 βρ 2 ). ν(τ, k) := e 3 2 W ( βk2 /a 2 ) ] φ k = 0, with and Extremely complicated to solve, numerical solutions at best. Not entirely unique prescription allows us to wonder if this is the only way to proceed.

40 Representing a stringy minimal length (reworked) ) Recalling that [X i, P j ] = i (f (P 2 )δj i + g(p2 )P i P j represents the GUP with g = 2β, f (s) = 2βs/( 1 + βs 1).

41 Representing a stringy minimal length (reworked) ) Recalling that [X i, P j ] = i (f (P 2 )δj i + g(p2 )P i P j represents the GUP with g = 2β, f (s) = 2βs/( 1 + βs 1). We find a new position space representation: ˆX i = x i, ˆP j = i j 1+β 2.

42 Representing a stringy minimal length (reworked) ) Recalling that [X i, P j ] = i (f (P 2 )δj i + g(p2 )P i P j represents the GUP with g = 2β, f (s) = 2βs/( 1 + βs 1). We find a new position space representation: ˆX i = x i, ˆP j = i j 1+β 2. This representation applies just as well to Minkowski as well as FRW backgrounds as [ i, j ] = 0 in both cases.

43 Representing a stringy minimal length (reworked) ) Recalling that [X i, P j ] = i (f (P 2 )δj i + g(p2 )P i P j represents the GUP with g = 2β, f (s) = 2βs/( 1 + βs 1). We find a new position space representation: ˆX i = x i, ˆP j = i j 1+β 2. This representation applies just as well to Minkowski as well as FRW backgrounds as [ i, j ] = 0 in both cases. In order for the representation to be well defined, βk 2 /a 2 < 1.

44 Representing a stringy minimal length (reworked) ) Recalling that [X i, P j ] = i (f (P 2 )δj i + g(p2 )P i P j represents the GUP with g = 2β, f (s) = 2βs/( 1 + βs 1). We find a new position space representation: ˆX i = x i, ˆP j = i j 1+β 2. This representation applies just as well to Minkowski as well as FRW backgrounds as [ i, j ] = 0 in both cases. In order for the representation to be well defined, βk 2 /a 2 < 1. Non-locality of the operator P j introduces spatial fuzziness β.

45 Representing a stringy minimal length (reworked) ) Recalling that [X i, P j ] = i (f (P 2 )δj i + g(p2 )P i P j represents the GUP with g = 2β, f (s) = 2βs/( 1 + βs 1). We find a new position space representation: ˆX i = x i, ˆP j = i j 1+β 2. This representation applies just as well to Minkowski as well as FRW backgrounds as [ i, j ] = 0 in both cases. In order for the representation to be well defined, βk 2 /a 2 < 1. Non-locality of the operator P j introduces spatial fuzziness β. The action for a scalar field is then S = 1 2 dτd 3 x a 2 (τ) [ ( τ φ) 2 3 i=1 ( ) ] 2 i φ. 1+β 2

46 Representing a stringy minimal length (reworked) ) Recalling that [X i, P j ] = i (f (P 2 )δj i + g(p2 )P i P j represents the GUP with g = 2β, f (s) = 2βs/( 1 + βs 1). We find a new position space representation: ˆX i = x i, ˆP j = i j 1+β 2. This representation applies just as well to Minkowski as well as FRW backgrounds as [ i, j ] = 0 in both cases. In order for the representation to be well defined, βk 2 /a 2 < 1. Non-locality of the operator P j introduces spatial fuzziness β. The action for a scalar field is then S = 1 2 dτd 3 x a 2 (τ) [ ( τ φ) 2 3 i=1 ( ) ] 2 i φ. 1+β 2 Which yields the equations of motion for the mode functions φ 2 τ φ p + 2 φ = 0. (1 a 2 βp 2 ) 2

47 Exact solutions We find the exact solutions for the mode functions to be φ ± (τ, p) = H2 1 βh 2 p 2 τ 2 [1 + pτ ( βh 2 pτ iγ ) ]e ±i γ(1+3βh 2 ) 2p 3 with γ = 1 βh 2. γ βh tanh 1 ( βhpτ)

48 Exact solutions We find the exact solutions for the mode functions to be φ ± (τ, p) = H2 1 βh 2 p 2 τ 2 [1 + pτ ( βh 2 pτ iγ ) ]e ±i γ(1+3βh 2 ) 2p 3 with γ = 1 βh 2. γ βh tanh 1 ( βhpτ) Can immediately read off the usual Bunch-Davies vacuum solutions in the limit β 0, also when ap 1 >> β.

49 Exact solutions We find the exact solutions for the mode functions to be φ ± (τ, p) = H2 1 βh 2 p 2 τ 2 [1 + pτ ( βh 2 pτ iγ ) ]e ±i γ(1+3βh 2 ) 2p 3 with γ = 1 βh 2. γ βh tanh 1 ( βhpτ) Can immediately read off the usual Bunch-Davies vacuum solutions in the limit β 0, also when ap 1 >> β. N.B. When H > β 1/2, no normalizable mode functions!

50 Exact solutions We find the exact solutions for the mode functions to be φ ± (τ, p) = H2 1 βh 2 p 2 τ 2 [1 + pτ ( βh 2 pτ iγ ) ]e ±i γ(1+3βh 2 ) 2p 3 with γ = 1 βh 2. γ βh tanh 1 ( βhpτ) Can immediately read off the usual Bunch-Davies vacuum solutions in the limit β 0, also when ap 1 >> β. N.B. When H > β 1/2, no normalizable mode functions! This is sensibly signalling the breakdown of EFT towards the string scale.

51 Exact solutions We find the exact solutions for the mode functions to be φ ± (τ, p) = H2 1 βh 2 p 2 τ 2 [1 + pτ ( βh 2 pτ iγ ) ]e ±i γ(1+3βh 2 ) 2p 3 with γ = 1 βh 2. γ βh tanh 1 ( βhpτ) Can immediately read off the usual Bunch-Davies vacuum solutions in the limit β 0, also when ap 1 >> β. N.B. When H > β 1/2, no normalizable mode functions! This is sensibly signalling the breakdown of EFT towards the string scale. Since inflation is described by the spatial zero mode, our background equation of motions are the same.

52 Exact solutions We find the exact solutions for the mode functions to be φ ± (τ, p) = H2 1 βh 2 p 2 τ 2 [1 + pτ ( βh 2 pτ iγ ) ]e ±i γ(1+3βh 2 ) 2p 3 with γ = 1 βh 2. γ βh tanh 1 ( βhpτ) Can immediately read off the usual Bunch-Davies vacuum solutions in the limit β 0, also when ap 1 >> β. N.B. When H > β 1/2, no normalizable mode functions! This is sensibly signalling the breakdown of EFT towards the string scale. Since inflation is described by the spatial zero mode, our background equation of motions are the same. Can compute the curvature perturbations as ζ(p)ζ(p ) = (2π) 3 δ 3 (p + p ) H4 2p 3 φ 1 2 γ (1+3βH ), N.B. 2 H 2 β = H 2 /ms 2 corrections.

53 Comparison with observation ( ) ( ) P R (p) = 1 H 2 ns 1 1 p 2ɛ 2π γ (1+3βH 2) ah

54 Comparison with observation ( ) P R (p) = 1 H 2 2ɛ 2π [ n s 1 = 2(η 3ɛ) H 2 β ( ) ns 1 1 p γ (1+3βH 2) ah 1 1 βh βH 2 ]

55 Comparison with observation ( ) P R (p) = 1 H 2 2ɛ 2π [ n s 1 = 2(η 3ɛ) H 2 β [ n T 1 = 2ɛ H 2 β ( ) ns 1 1 p γ (1+3βH 2) ah 1 1 βh βh βH βH 2 ] ɛ ]

56 Comparison with observation ( ) P R (p) = 1 H 2 2ɛ 2π [ n s 1 = 2(η 3ɛ) H 2 β [ n T 1 = 2ɛ H 2 β ( ) ns 1 1 p γ (1+3βH 2) ah 1 1 βh βh βH βH 2 ] ɛ Can we measure any of these corrections? ]

57 Comparison with observation ( ) P R (p) = 1 H 2 2ɛ 2π [ n s 1 = 2(η 3ɛ) H 2 β [ n T 1 = 2ɛ H 2 β ( ) ns 1 1 p γ (1+3βH 2) ah 1 1 βh βh βH βH 2 ] ɛ Can we measure any of these corrections? Experimentally P A where 0.6 < A < 1 is a model dependent number. This implies H ɛ10 15 GeV. ]

58 Comparison with observation ( ) P R (p) = 1 H 2 2ɛ 2π [ n s 1 = 2(η 3ɛ) H 2 β [ n T 1 = 2ɛ H 2 β ( ) ns 1 1 p γ (1+3βH 2) ah 1 1 βh βh βH βH 2 ] ɛ Can we measure any of these corrections? Experimentally P A where 0.6 < A < 1 is a model dependent number. This implies H ɛ10 15 GeV. If we take ɛ 10 2, then H 2 /m 2 s = 10 4 (M 2 GUT /m2 s ). ]

59 Comparison with observation ( ) P R (p) = 1 H 2 2ɛ 2π [ n s 1 = 2(η 3ɛ) H 2 β [ n T 1 = 2ɛ H 2 β ( ) ns 1 1 p γ (1+3βH 2) ah 1 1 βh βh βH βH 2 ] ɛ Can we measure any of these corrections? Experimentally P A where 0.6 < A < 1 is a model dependent number. This implies H ɛ10 15 GeV. If we take ɛ 10 2, then H 2 /m 2 s = 10 4 (M 2 GUT /m2 s ). If m s GeV then these corrections will be observable at present. ]

60 Comparison with observation ( ) P R (p) = 1 H 2 2ɛ 2π [ n s 1 = 2(η 3ɛ) H 2 β [ n T 1 = 2ɛ H 2 β ( ) ns 1 1 p γ (1+3βH 2) ah 1 1 βh βh βH βH 2 ] ɛ Can we measure any of these corrections? Experimentally P A where 0.6 < A < 1 is a model dependent number. This implies H ɛ10 15 GeV. If we take ɛ 10 2, then H 2 /m 2 s = 10 4 (M 2 GUT /m2 s ). If m s GeV then these corrections will be observable at present. If m s GeV = M GUT, then we will forever be one order of magnitude away from experimental sensitivity. ]

61 Conclusions and future prospects Unless we are exceedingly lucky, corrections from stringy physics to the CMB are likely to be forever out of our reach at the level of this analysis.

62 Conclusions and future prospects Unless we are exceedingly lucky, corrections from stringy physics to the CMB are likely to be forever out of our reach at the level of this analysis. Future experiments (e.g. Planck) will likely be the state of the art for some time to come in terms of accuracy of measuring the spectrum of primordial fluctuations.

63 Conclusions and future prospects Unless we are exceedingly lucky, corrections from stringy physics to the CMB are likely to be forever out of our reach at the level of this analysis. Future experiments (e.g. Planck) will likely be the state of the art for some time to come in terms of accuracy of measuring the spectrum of primordial fluctuations. Significant progress will be made in the statistical significance of our observations.

64 Conclusions and future prospects Unless we are exceedingly lucky, corrections from stringy physics to the CMB are likely to be forever out of our reach at the level of this analysis. Future experiments (e.g. Planck) will likely be the state of the art for some time to come in terms of accuracy of measuring the spectrum of primordial fluctuations. Significant progress will be made in the statistical significance of our observations. If we find any significant outlier points for the primordial spectrum, then we could perhaps deduce non-trivial effects on inflaton dynamics from UV physics.

65 Conclusions and future prospects Unless we are exceedingly lucky, corrections from stringy physics to the CMB are likely to be forever out of our reach at the level of this analysis. Future experiments (e.g. Planck) will likely be the state of the art for some time to come in terms of accuracy of measuring the spectrum of primordial fluctuations. Significant progress will be made in the statistical significance of our observations. If we find any significant outlier points for the primordial spectrum, then we could perhaps deduce non-trivial effects on inflaton dynamics from UV physics. Effective field theory analysis of inflaton dynamics is a current and active field of research. Stay tuned for more progress and interplay with results of observations.