String Theory • String theory: A good theory of quantum gravity! • Unifies all forces and fundamental particles! • This only works if there are 9 dimensions of space! The solution: make the extra dimensions small! Thursday, 4 July, 13
String Theory • To keep the extra dimensions small, need to add energy. Thursday, 4 July, 13
String Theory • To keep the extra dimensions small, need to add energy. • The inflaton: some property of the compact extra dimesions. e.g. vary the size as a function of position. Thursday, 4 July, 13
String Theory • To keep the extra dimensions small, need to add energy. • The inflaton: some property of the compact extra dimesions. e.g. vary the size as a function of position. • Changing size changes potential energy: inflation can be driven by the energy stored in the extra dimensions. Thursday, 4 July, 13
String Inflaton • A ``proof of principle’’ exists, but how predictive is this? Thursday, 4 July, 13
String Inflaton • A ``proof of principle’’ exists, but how predictive is this? • The extra dimensions can assume many configurations: Many possible inflaton potentials! (Many possible values of the Cosmological Constant) Thursday, 4 July, 13
String Inflaton • A ``proof of principle’’ exists, but how predictive is this? • The extra dimensions can assume many configurations: Many possible inflaton potentials! (Many possible values of the Cosmological Constant) • To do list: how do we make predictions then? Thursday, 4 July, 13
Reheating • Inflation has to come to an end. ' V ( φ ) d 2 φ dt 2 + 3 H d φ dt = � dV d φ friction gradient φ Thursday, 4 July, 13
Reheating • Inflation has to come to an end. ' V ( φ ) d 2 φ dt 2 + 3 H d φ dt = � dV d φ friction gradient φ • The inflaton oscillates around the minimum, fragments and decays into standard model particles, dark matter, and perhaps other stuff. Thursday, 4 July, 13
Reheating • Inflation has to come to an end. ' V ( φ ) d 2 φ dt 2 + 3 H d φ dt = � dV d φ friction gradient φ • The inflaton oscillates around the minimum, fragments and decays into standard model particles, dark matter, and perhaps other stuff. • The standard story of the hot big-bang follows. Thursday, 4 July, 13
Number of e-folds � � � � • Total expansion of the Universe during inflation: Z φ begin N e = log a end M p d � p p ✏ ' a begin 4 3 ⇡ φ end Thursday, 4 July, 13
Number of e-folds � � � � • Total expansion of the Universe during inflation: Z φ begin N e = log a end M p d � p p ✏ ' a begin 4 3 ⇡ φ end • To solve the horizon problem, need our observable universe to come from single primordial Hubble patch: a 0 = a 0 a eq a reh = H begin ' 10 55 a begin a eq a reh a begin H 0 (GUT scale inflation) Thursday, 4 July, 13
Number of e-folds � � � � • Total expansion of the Universe during inflation: Z φ begin N e = log a end M p d � p p ✏ ' a begin 4 3 ⇡ φ end • To solve the horizon problem, need our observable universe to come from single primordial Hubble patch: a 0 = a 0 a eq a reh = H begin ' 10 55 a begin a eq a reh a begin H 0 (GUT scale inflation) 3000 T reh T eq Thursday, 4 July, 13
Number of e-folds � � � � • Total expansion of the Universe during inflation: Z φ begin N e = log a end M p d � p p ✏ ' a begin 4 3 ⇡ φ end • To solve the horizon problem, need our observable universe to come from single primordial Hubble patch: a 0 = a 0 a eq a reh = H begin ' 10 55 a begin a eq a reh a begin H 0 (GUT scale inflation) 3000 T reh e N e , N e ⇠ 60 T eq Thursday, 4 July, 13
Classical Fields • Scalar field in Minkowski space: 1 � Z 2( ∂ t φ ) 2 � 1 2( ∂ i φ ) 2 � V ( φ ) d 3 xdt S = d 2 φ dt 2 � r 2 φ + m 2 φ = 0 Thursday, 4 July, 13
Classical Fields • Scalar field in Minkowski space: 1 � Z 2( ∂ t φ ) 2 � 1 2( ∂ i φ ) 2 � V ( φ ) d 3 xdt S = d 2 φ dt 2 � r 2 φ + m 2 φ = 0 • Go to fourier space: free field theory is an infinite number of independent oscillators. d 2 φ ~ dt 2 + ω 2 k k = 0 k φ ~ ~ d 3 k Z k ( t ) e i ~ k · ~ x φ ( t, x ) = (2 π ) 3 φ ~ k = k 2 + m 2 ! 2 ~ Thursday, 4 July, 13
Quantum Fields • Promote fields to operators: φ ! ˆ φ π ! ˆ π [ˆ � ( t, ~ x ) , ˆ ⇡ ( t, ~ y )] = i � ( ~ y ) x − ~ Thursday, 4 July, 13
Quantum Fields • Promote fields to operators: φ ! ˆ φ π ! ˆ π [ˆ � ( t, ~ x ) , ˆ ⇡ ( t, ~ y )] = i � ( ~ y ) x − ~ • In fourier space: quantize the infinite number of independent oscillators: d 3 k 1 Z h x + a + x i k e i ~ k v k e − i ~ ˆ k · ~ k · ~ φ = a − k v ∗ p ~ ~ (2 π ) 3 2 Thursday, 4 July, 13
Quantum Fields • Promote fields to operators: φ ! ˆ φ π ! ˆ π [ˆ � ( t, ~ x ) , ˆ ⇡ ( t, ~ y )] = i � ( ~ y ) x − ~ • In fourier space: quantize the infinite number of independent oscillators: d 3 k 1 Z h x + a + x i k e i ~ k v k e − i ~ ˆ k · ~ k · ~ φ = a − k v ∗ p ~ ~ (2 π ) 3 2 1 e i ! k t k 0 ] = � 3 ( ~ k � ~ k , a + v k = [ a � k 0 ) p ω k ~ ~ mode function creation/annihilation operators Thursday, 4 July, 13
Quantum Fields a � k | 0 i = 0 ~ vacuum Thursday, 4 July, 13
Quantum Fields a + k | 0 i = | 1 i ~ ~ k particle = positive frequency excitation Thursday, 4 July, 13
Quantum Fields a + k a + k 0 | 0 i = | 1 i ~ k | 1 i ~ ~ ~ k 0 multi-particle states Thursday, 4 July, 13
QFT in Curved Spacetime • In curved space: S = � 1 Z d 4 x p� g g ↵� @ ↵ �@ � � + m 2 � ⇥ ⇤ 2 Thursday, 4 July, 13
QFT in Curved Spacetime • In curved space: S = � 1 Z d 4 x p� g g ↵� @ ↵ �@ � � + m 2 � ⇥ ⇤ 2 • Re-cast as canonical free scalar (for FRW): d 2 � ~ comoving k! d ⌘ 2 + ! 2 k � = a � k ( ⌘ ) � ~ k = 0 ~ Thursday, 4 July, 13
QFT in Curved Spacetime • In curved space: S = � 1 Z d 4 x p� g g ↵� @ ↵ �@ � � + m 2 � ⇥ ⇤ 2 • Re-cast as canonical free scalar (for FRW): d 2 � ~ comoving k! d ⌘ 2 + ! 2 k � = a � k ( ⌘ ) � ~ k = 0 ~ • In de Sitter, the scale factor is: a = � 1 �1 < η 0 H η ✓ m 2 ◆ com + 1 ω 2 k ( η ) = k 2 H 2 − 2 ~ η 2 ω 2 k ( η ) < 0 , k η ⌧ 1 , m ⌧ H ~ Thursday, 4 July, 13
QFT in Curved Spacetime • Classical solutions to the equation of motion: Im Re χ ~ k < η r ⇡ | ⌘ | v k = [ J n ( k ⌘ ) � iY n ( k ⌘ )] 2 r 9 4 � m 2 n = H 2 Thursday, 4 July, 13
QFT in Curved Spacetime • Classical solutions to the equation of motion: Im Re χ ~ k < η r ⇡ | ⌘ | v k = [ J n ( k ⌘ ) � iY n ( k ⌘ )] 2 r 9 4 � m 2 Initial conditions n = H 2 Thursday, 4 July, 13
QFT in Curved Spacetime • Classical solutions to the equation of motion: Im Re χ ~ k < η r ⇡ | ⌘ | v k = [ J n ( k ⌘ ) � iY n ( k ⌘ )] 2 r 9 4 � m 2 Initial conditions n = H 2 φ ~ k < η Thursday, 4 July, 13
QFT in Curved Spacetime • Classical solutions to the equation of motion: Im Re χ ~ k < η r ⇡ | ⌘ | v k = [ J n ( k ⌘ ) � iY n ( k ⌘ )] 2 r 9 4 � m 2 Initial conditions n = k η ⇠ 1 H 2 φ ~ k < η The field ``freezes in” Thursday, 4 July, 13
QFT in Curved Spacetime • Quantize just as before, but..... ✓ m 2 ◆ com + 1 ω 2 k ( η ) = k 2 H 2 − 2 ~ η 2 Thursday, 4 July, 13
QFT in Curved Spacetime • Quantize just as before, but..... ✓ m 2 ◆ com + 1 ω 2 k ( η ) = k 2 H 2 − 2 ~ η 2 • The frequency is time-dependent, so no unambiguous definition of positive frequency -- no unambiguous definition of the vacuum! Thursday, 4 July, 13
QFT in Curved Spacetime • Quantize just as before, but..... ✓ m 2 ◆ com + 1 ω 2 k ( η ) = k 2 H 2 − 2 ~ η 2 • The frequency is time-dependent, so no unambiguous definition of positive frequency -- no unambiguous definition of the vacuum! • A prescription for the vacuum: Minkowski at small scales. 1 e i ! k ⌘ , v k = k ⌘ ! �1 p ! k Bunch-Davies Thursday, 4 July, 13
QFT in Curved Spacetime • Classical solutions to the equation of motion: χ ~ k < η Initial conditions Thursday, 4 July, 13
QFT in Curved Spacetime r | v k | 2 = 1 1 ⇡ | ⌘ | (massless) k + v k = [ J n ( k ⌘ ) � iY n ( k ⌘ )] ⌘ 2 k 3 2 • Find the correlation functions: Thursday, 4 July, 13
QFT in Curved Spacetime r | v k | 2 = 1 1 ⇡ | ⌘ | (massless) k + v k = [ J n ( k ⌘ ) � iY n ( k ⌘ )] ⌘ 2 k 3 2 • Find the correlation functions: k 0 ) | v k | 2 Fourier space: k 0 | 0 i = � ( ~ k � ~ � ⇤ h 0 | ˆ k ˆ � ~ ~ 2 Z 1 (2 ⇡ ) 2 k 2 | v k | 2 sin( kL ) dk Real space: � ⇤ ( y, t ) | 0 i = h 0 | ˆ � ( x, t )ˆ kL 0 Thursday, 4 July, 13
QFT in Curved Spacetime r | v k | 2 = 1 1 ⇡ | ⌘ | (massless) k + v k = [ J n ( k ⌘ ) � iY n ( k ⌘ )] ⌘ 2 k 3 2 • Find the correlation functions: k 0 ) | v k | 2 Fourier space: k 0 | 0 i = � ( ~ k � ~ � ⇤ h 0 | ˆ k ˆ � ~ ~ 2 Z 1 (2 ⇡ ) 2 k 2 | v k | 2 sin( kL ) dk Real space: � ⇤ ( y, t ) | 0 i = h 0 | ˆ � ( x, t )ˆ kL 0 Coincident limit: Z ˜ Z 1 Z 1 k 1 dk dk dk (2 ⇡ ) 2 k 2 | v k | 2 ' � ⇤ ( x, t ) | 0 i = h 0 | ˆ � ( x, t )ˆ ⌘ 2 k + (2 ⇡ ) 2 k (2 ⇡ ) 2 ˜ 0 0 k IR UV Thursday, 4 July, 13
QFT in Curved Spacetime Z ˜ Z 1 Z 1 k 1 dk dk dk (2 ⇡ ) 2 k 2 | v k | 2 ' � ⇤ ( x, ⌘ ) | 0 i = h 0 | ˆ � ( x, ⌘ )ˆ ⌘ 2 k + (2 ⇡ ) 2 k (2 ⇡ ) 2 ˜ 0 0 k Thursday, 4 July, 13
QFT in Curved Spacetime Z ˜ Z 1 Z 1 k 1 dk dk dk (2 ⇡ ) 2 k 2 | v k | 2 ' � ⇤ ( x, ⌘ ) | 0 i = h 0 | ˆ � ( x, ⌘ )ˆ ⌘ 2 k + (2 ⇡ ) 2 k (2 ⇡ ) 2 ˜ 0 0 k Thursday, 4 July, 13
QFT in Curved Spacetime Z ˜ Z 1 Z 1 k 1 dk dk dk (2 ⇡ ) 2 k 2 | v k | 2 ' � ⇤ ( x, ⌘ ) | 0 i = h 0 | ˆ � ( x, ⌘ )ˆ ⌘ 2 k + (2 ⇡ ) 2 k (2 ⇡ ) 2 ˜ 0 0 k • Go back to original field: ✓ H ◆ 2 Z ˜ k � ⇤ ( x, ⌘ ) | 0 i = 1 dk h 0 | ˆ � ( x, ⌘ )ˆ � ⇤ ( x, ⌘ ) | 0 i = a 2 h 0 | ˆ � ( x, ⌘ )ˆ 2 ⇡ k 0 Thursday, 4 July, 13
QFT in Curved Spacetime Z ˜ Z 1 Z 1 k 1 dk dk dk (2 ⇡ ) 2 k 2 | v k | 2 ' � ⇤ ( x, ⌘ ) | 0 i = h 0 | ˆ � ( x, ⌘ )ˆ ⌘ 2 k + (2 ⇡ ) 2 k (2 ⇡ ) 2 ˜ 0 0 k • Go back to original field: ✓ H ◆ 2 Z ˜ k � ⇤ ( x, ⌘ ) | 0 i = 1 dk h 0 | ˆ � ( x, ⌘ )ˆ � ⇤ ( x, ⌘ ) | 0 i = a 2 h 0 | ˆ � ( x, ⌘ )ˆ 2 ⇡ k 0 • Assume inflation has a finite duration, count modes larger than the comoving horizon, go back to proper time: � ⇤ ( x, t ) | 0 i = H 3 h 0 | ˆ � ( x, t )ˆ 4 ⇡ 2 ( t � t 0 ) • Diverges with increasing time -- pile-up of superhorizon modes. Regulated for non-zero mass. Thursday, 4 July, 13
QFT in Curved Spacetime k 0 )(2 ⇡ ) 2 Fourier space: k 0 | 0 i = � ( ~ k � ~ � ⇤ h 0 | ˆ k ˆ P � ( k ) � ~ ~ k 3 • Transform back to the original field: H 2 P � ( k ) = P � ( m ⌧ H ) a 2 = (2 π ) 2 Thursday, 4 July, 13
QFT in Curved Spacetime k 0 )(2 ⇡ ) 2 Fourier space: k 0 | 0 i = � ( ~ k � ~ � ⇤ h 0 | ˆ k ˆ P � ( k ) � ~ ~ k 3 • Transform back to the original field: H 2 P � ( k ) = P � ( m ⌧ H ) a 2 = (2 π ) 2 • The power spectrum of a free field in dS is: • Nearly scale invariant • Gaussian uncoupled harmonic oscillators! • Small amplitude (compared to...) Thursday, 4 July, 13
QFT in Curved Spacetime k 0 )(2 ⇡ ) 2 Fourier space: k 0 | 0 i = � ( ~ k � ~ � ⇤ h 0 | ˆ k ˆ P � ( k ) � ~ ~ k 3 • Transform back to the original field: H 2 P � ( k ) = P � ( m ⌧ H ) a 2 = (2 π ) 2 • The power spectrum of a free field in dS is: • Nearly scale invariant • Gaussian uncoupled harmonic oscillators! • Small amplitude (compared to...) !!Gravitational waves!! Thursday, 4 July, 13
Quantum to Classical h 0 | ˆ � ( x, ⌘ )ˆ � ⇤ ( y, ⌘ ) | 0 i ! h ˆ � ( x, ⌘ )ˆ � ⇤ ( y, ⌘ ) i Quantum expectation Ensemble average value Spatial average ??? (pure dS might not be best example...) Thursday, 4 July, 13
Inflationary Fluctuations • The field couples to the metric - can choose a convenient coordinate system (gauge). Thursday, 4 July, 13
Inflationary Fluctuations • The field couples to the metric - can choose a convenient coordinate system (gauge). φ = const . time space Spatially varying field and metric. Thursday, 4 July, 13
Inflationary Fluctuations • The field couples to the metric - can choose a convenient coordinate system (gauge). φ = const . t = const . time Spatially varying metric, spatially uniform field. space Comoving curvature perturbation: R Thursday, 4 July, 13
Inflationary Fluctuations • The field couples to the metric - can choose a convenient coordinate system (gauge). φ = const . t = const . !!!Conserved on superhorizon scales!!! time Spatially varying metric, spatially uniform field. space Comoving curvature perturbation: R Thursday, 4 July, 13
QFT in Curved Spacetime 1 • An important scale: comoving horizon aH • Horizon crossing: k = aH a = � 1 comoving H η scale k | η | = 1 frozen oscillating conformal time η Thursday, 4 July, 13
Inflationary Fluctuations • The action: � 16 π G � 1 Z R 2 g µ ⌫ ∂ µ φ∂ ⌫ φ � V ( φ ) d 4 x S = gravity inflaton field Thursday, 4 July, 13
Inflationary Fluctuations • The action: � 16 π G � 1 Z R 2 g µ ⌫ ∂ µ φ∂ ⌫ φ � V ( φ ) d 4 x S = gravity inflaton field • Expand into background and fluctuations: S ' S 0 + S 2 Thursday, 4 July, 13
Inflationary Fluctuations • The action: � 16 π G � 1 Z R 2 g µ ⌫ ∂ µ φ∂ ⌫ φ � V ( φ ) d 4 x S = gravity inflaton field • Expand into background and fluctuations: S ' S 0 + S 2 • With a few re-definitions, looks like a free field with time- dependent mass: "✓ dv # ◆ 2 � ( r v ) 2 + d 2 z v 2 S 2 = 1 Z d 3 xd η 2 d η 2 d η z ◆ 2 z ⌘ a 2 ✓ d φ v ⌘ zM p R H 2 dt Thursday, 4 July, 13
Inflationary Fluctuations • Quantize v, choose the Bunch-Davies vacuum, and find the correlation functions: P ( k ) = Ak n s − 1 V 3 A = n s � 1 = 2 ⌘ � 6 ✏ 12 ⇡ 2 ( @ � V ) 2 M 6 P n s < 1 n s > 1 Red Blue Thursday, 4 July, 13
Inflationary Fluctuations • Quantize v, choose the Bunch-Davies vacuum, and find the correlation functions: P ( k ) = Ak n s − 1 V 3 A = n s � 1 = 2 ⌘ � 6 ✏ 12 ⇡ 2 ( @ � V ) 2 M 6 P n s < 1 n s > 1 Red Blue • Single-field slow-roll inflation: small non-gaussianity (to have slow-roll, interactions must be small). Thursday, 4 July, 13
Inflationary Fluctuations ' V ( φ ) P ( k ) = φ Thursday, 4 July, 13
Inflationary Fluctuations • Tensor modes: ✓ H ◆ 2 r ⌘ A T A T = A = 16 ✏ 2 ⇡ 0.25 Planck +WP Planck +WP+highL 0.20 Tensor-to-Scalar Ratio ( r 0.002 ) Convex Planck +WP+BAO Natural Inflation Concave 0.15 Power law inflation Low Scale SSB SUSY R 2 Inflation 0.10 V ∝ φ 2 / 3 V ∝ φ 0.05 V ∝ φ 2 V ∝ φ 3 N ∗ =50 0.00 N ∗ =60 0.94 0.96 0.98 1.00 Primordial Tilt ( n s ) Thursday, 4 July, 13
Initial Conditions for Inflation • Inflation, once it gets off the ground, can predict everything we observe about the linear universe. • Under what conditions can inflation begin? H − 1 V � ( ∂φ ) 2 To do.... (singularity? vacuum? nothing?) Thursday, 4 July, 13
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