Introduction Quantum field theory in curved spacetime ...

Introduction Quantum field theory in curved spacetime Oscillating scale factor in a spatially flat FRW background Number density and energy density for gravitons Conclusion

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Table of contents 1

Introduction

2

Quantum field theory in curved spacetime Basic formalism Particle creation by gravitational fields Perturbation calculation of graviton creation in an expanding universe

3

Oscillating scale factor in a spatially flat FRW background General features Standard general relativity plus a minimally coupled scalar field Modified Einstein’s gravity: quadratic terms in the curvature

4

Number density and energy density for gravitons Explicit calculation of graviton creation Cosmological implications

5

Conclusion Scalar field quantization in curved spacetime

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Introduction

A time-dependent spacetime metric can result in quantum particle creation, as was first discussed by Parker (1969, 1971) in the context of the expansion of the universe. The cosmological creation of gravitational waves was postulated by Grishchuk (1974) and, after that, this process has been studied specially in the context of inflation. Quantum creation of particles, including gravitons, is expected at the end of the inflation, contributing to the mater and radiation of the universe after the inflationary era (Ford, 1987).

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We focus on a graviton creation due to rapid oscillations around the average of an expanding universe. (Image from: plato.stanford.edu.) The spacetime metric in a spatially flat Friedmann-Robertson-Walker (FRW) background is given by: ds 2 = −dt 2 + a2 (t) dx 2 + dy 2 + dz 2

These kind of oscillations can arise in some specific scenarios: Standard general relativity plus a minimally coupled scalar field (GRSF). Modified gravity such as f (R) gravity or semiclassical gravity.

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Basic formalism Particle creation by gravitational fields Perturbation calculation of graviton creation in an expanding universe

Basic formalism There are four basic ingredients in the construction of a quantum field theory (Ford, 1997): The Lagrangian density or, equivalently, the equation of motion of the classical theory. A quantization procedure (canonical quantization or path integral approach). The characterization of quantum states. The physical interpretation of the quantum states and of the observables.

The real differences between flat and curved spacetime arise in the latter two steps. In general, there does not exist a unique vacuum state in a curved spacetime and the concept of particles becomes much ambiguous.

here

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Particle creation by gravitational fields Let us take a spacetime which is asymptotically flat in the past and in the future, but which is non-flat in the intermediate region: {fj } → in-region {Fj } → out-region

Set of solutions to be orthonormal and expand the field operator in terms of {fj } or {Fj }: φ=

X

aˆj fj + aˆj† fj∗ =

j

X

bˆj Fj + bˆj† Fj∗ .

j

Annihilation and creation operators: in-region : ˆ aj , ˆ aj† → ˆ aj |0iin = 0, ∀j. ˆj , b ˆ† → b ˆj |0iout = 0, ∀j. out-region : b j

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We may expand the in-modes in terms of the out-modes: fj =

X

(αjk Fk + βjk Fk∗ ) .

k

Time-dependent gravitational field: No particle were present before the gravitational field is turn on → |0iin (Heisenberg picture). ˆk = b ˆ† b ˆ . The physical number operator in the out-region → N k k The mean number of particles into mode K is ˆk i = hN

X

|βjk |2 .

j

If any of the βjk coefficients are nonzero, then particles are created by the gravitational field. here

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Perturbation calculation of graviton creation in an expanding universe The most straightforward application of this phenomenon is to particle creation by an expanding universe. In a conformally flat spacetime, gravitons, in the transverse tracefree gauge, are equivalent to a pair of massless minimally coupled scalar fields (Lifshitz, 1946). Consider a minimally coupled massless scalar field in a Rspatially flat FRW t −1 0 universe and set a complete set of solutions (here η = a (t )dt 0 ):

fj , fj

∗

⇐=

ds 2 = dt 2 − a2 (t)d 2 x = a2 (η) dη 2 − d 2 x 1 L = 21 (−g) 2 [g µν ∂µ φ∂ν φ] → g µν ∂µ ∂ν φ(x ) = 0

The positive norm solutions of the wave equation may be taken to be (here k = |k|) fk (x ) =

e ik·x

p

a(η)

(2π)3

χk (η) where

d 2 χk (η) 1 d 2 a(η) χk (η) = 0 . + k2 − 2 dη a(η) dη 2 8 / 33



Suppose that the universe is static both in the past and in the future. Choose modes which are pure positive frequency in the past. The in and out-modes will be e −ikη χk (η) ∼ χin , η → −∞ , k (η) = √ 2k 1 χk (η) ∼ χout αk e −ikη + βk e ikη , η → +∞ k (η) = √ 2k

(1)

where the coefficients αk and βk are determined by solving the equation for χk (η) for a given a(η). The number density and energy density of created particles per unit proper volume is given at late times by

n=

1 (2πa)3

Z

1 ρ= (2πa)3 a

Z

   

d 3 k |βk |2 , 3

These formulas hold in the asymptotic region.

2

  d k k |βk | . 9 / 33



We use a perturbative method described by Birrell and Davies (1982) to solve the equation for χk (η) as follows −1 χk (η) = χin k (η) + ω

Z

η

a(η 0 ),η0 η0 sin ω(η − η 0 ) χk (η 0 )dη 0 . 0 a(η )

−∞

0 Iterate this integral equation to lowest order by replacing χk (η 0 ) by χin k (η ) −ikη ikη out 1 √ + βk e : in its integrand and compare that with χk (η) = 2k αk e

βk ≈ −

i 2k

Z

∞

e −2ikη

−∞

a(η),ηη dη . a(η)

Then, the number density and energy density per unit of proper volume of created particles are (here V (η) = [a(η),ηη )/a(η)] ): n =2×[16πa3 (η)]−1

Z

∞

V 2 (η1 )dη1 ,

−∞

ρ = −2×[32π 2 a4 (η)]−1

Z

∞

−∞

Back to

Z

∞

dη1

dη2 −∞

ln [(η2 − η1 )µ]2 × V˙ (η1 )V˙ (η2 ) . 2

main 10 / 33


General features Standard general relativity plus a minimally coupled scalar field Modified Einstein’s gravity: quadratic terms in the curvature

General features We consider small oscillations around a FRW background with a scale factor of the form a(t) = a¯(t) [1 + Aeff (t)cos(ω0 t)] . Here a¯(t) is the background scale factor time-averaged over oscillations. There are some scenarios in which a scale factor like this can arise: Standard matter fields in general relativity + minimally coupled scalar field in a harmonic potential (GRSF). f (R) gravity with a term proportional to the square of the Ricci scalar in the gravitational action.

Both models produce an oscillating scale factor and graviton production, but the physics behind those models and their respective graviton creation rates are different.

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Standard general relativity plus a minimally coupled scalar field

Coherent oscillating scalars field have been widely considered in the literature in the context of inflation (Emma et al. 2015) or as dark matter candidate (Peebles and Vilenkin, 1999). We focus on the oscillating feature of the scale factor when a minimally coupled scalar field is part of the total energy density in a spatially flat FRW universe. The action for this model is given by Mpl2 S= 2

Z

√ d x −gR + 4

Z

d 4 x [LM (gµν , ΨM ) + Lscalar (gµν , ϕ)] ,

√ where Lscalar (gµν , ϕ) = ( −g/2)(−g µν ∂µ ϕ∂ν ϕ − ω 2 ϕ2 ), LM is the matter Lagrangian, and ΨM are matter fields.

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Varying the action with respect to the scalar field and the metric, we have the following cosmological equations: 3H 2 Mpl2 = ρM + ρϕ , ∂t2 ϕ + 3H∂t ϕ + ω 2 ϕ = 0 Here ρϕ = (∂t ϕ)2 /2 + (ωϕ)2 /2 refers to the energy density for the scalar field. In the regime ω H, the field rapidly oscillates around the minimum of the harmonic potential. However, these oscillations decrease as times goes on according to

ϕ(t) = ϕi

a(ti ) a(t)

3/2 cos(ωt) .

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The oscillating behavior of the scalar field causes that the scale factor in this model also has an oscillating behavior (Schiappacasse & Ford, 2016):

" a(t) = a¯(t) 1 − Di

a¯(ti ) a¯(t)

#

3

cos(2ωt) .

Here Di ≡ (ϕ2i )/(16Mpl2 ) is the initial amplitude of the metric oscillations, a¯(t) is the background scale factor time-averaged over oscillations. If we consider this a¯(t) to be the flat spacetime and Di 1, the scale factor takes the form (to leading order): a(η) = 1 + A0 (cos ω0 η) , with Di = A0 and ω = ω0 /2.

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The generation of gravitational waves in this model is ruled by the usual equation because we are working in standard general relativity. But in a FRW universe, as we already discussed, the gravitational wave equation is equivalent, in a particular gauge, to the equation of a pair of minimally coupled scalar fields (Lifshitz, 1946). WE SOLVED THIS PROBLEM BEFORE! :

d 2 χk (η) a(η),ηη + k2 − χk (η) = 0 . dη 2 a(η) here

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Modified Einstein’s gravity: quadratic terms in the curvature

We consider f (R) gravity where the Einstein-Hilbert action is taken to be 2 R Mpl √ d 4 x −gf (R) with f (R) = R + (a2 /2)R 2 . 2

SH =

The action for this model is given by Mpl2 S= 2

Z

√ d x −gf (R) + 4

Z

d 4 x LM (gµν , ΨM ) .

Apply a conformal transformation g˜µν = [df(R)/dR]g µν and introduce an √ auxiliary scalar field φ such that F (R(φ)) = e

2/3(φ/Mpl )

.

THE ACTION LOOKS LIKE STANDARD GENERAL RELATIVITY PLUS A SCALAR FIELD MINIMALLY COUPLED TO GRAVITY (but non-minimally coupled to the matter fields).

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In fact, the action of this model is given by Mpl2 S= 2

Z

√

˜ d x −g R+ 4

Z

d x LM (e 4

"

√ 2/3φ − M pl

√ Lscalar (gµν , ϕ) = ( −g/2) −g µν ∂µ ϕ∂ν ϕ −

g˜µν , ΨM ) + Lscalar (gµν , φ) ,

2 3ω 2 Mpl

2

1−e

√ 2/3φ − M pl

2 # .

In the regime |φ/Mpl | 1, the potential behaves like a harmonic potential. To leading order the equation for the ˜ ∂˜t φ + ω 2 φ = 0 . scalar field is ∂˜t2 φ + 3H ˜ the scalar field In the regime ω H, rapidly oscillates according to φ(˜t ) = φi

h ˜ i3/2 ˜ a(ti ) ˜ a(˜t )

cos(ω˜t ) .

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The oscillating behavior of the scalar field causes that the scale factor in this model also has an oscillating behavior in the original frame (Schiappacasse & Ford, 2016):

" a(t) = a¯(t) 1 − Ei

a¯(ti ) a¯(t)

#

3/2

cos(ωt) .

√ Here Ei ≡ (φi )/( 6Mpl ) is the initial amplitude of the metric oscillations, a¯(t) is the background scale factor time-averaged over oscillations. If we consider this a¯(t) to be the flat spacetime and Ei 1, the scale factor takes the form a(η) = 1 + A0 (cos ω0 η) , with Ei = A0 and ω = ω0 .

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The generation of gravitational waves in this model is ruled by a modified gravitational wave equation according to (Hwang, 1991) √ d 2 vk (η) 1 d 2 (a F ) 2 + k − √ vk (η) = 0 , (2) dη 2 dη 2 a F where F (R) = df (R)/dR = 1 + a2 R. Note that when a2 → 0 we recover the usual equation from general relativity. Laboratory tests of the inverse law of gravity place an upper bound on a2 −9 2 of a2 . 2 × √ 10 m (Berry & Gair, 2011) which is related to ω according to ω = 1/ 3a2 (Horowitz & Wald, 1978) ⇒ ω & ωB = 2 × 10−3 ev. We are more interested in the lower part of this range for ω.

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Explicit calculation of graviton creation Cosmological implications

Explicit calculation of graviton creation Both scenarios show an oscillating scale factor and quantum production of gravitons. On short time scales, both scale factors can be expressed in flat spacetime as a(η) = 1 + A0 cos (ω0 η) Here A0 and ω0 take different values for each case: 2 ) and ω = 2ω. For the GRSF: A0 = Di = (ϕ2i )/(16Mpl 0

√ For f (R)-gravity: A0 = Ei = (φi )/( 6Mpl ) and ω0 = ω.

We use the mentioned perturbative method (Birrel & Davies, 1980) to solve both gravitational wave equations using this scale factor and to calculate the number density (ng ) and energy density (ρg ) of gravitons. We take the asymptotic behavior of ng and ρg for times scales long compared to the period of oscillation. 20 / 33



For the number density and energy density of gravitons in both cases we have dng Di 2 ω04 dng 9Ei 4 ω04 ∼ and ∼ dt GRSF 16π dt f (R) 16π dρg Di 2 ω05 ∼ dt GRSF 32π

and

dρg 9Ei 4 ω05 ∼ dt f (R) 16π

Basically, the graviton energy density for each case can be extended to an expanding universe in a similar way. We have to consider two main phenomena in an expanding universe: (1) Damping of the metric oscillations. (2) Redshifting and dilution of created gravitons.

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Damping of the metric oscillations:



h

i3

Di → Deff = Di ¯a(ti ) , ¯ a(t) 1 d¯ a h i3/2 ω a¯ dt ¯ a (t Ei → E = Ei i) . eff ¯ a(t) Redshifting-dilution of gravitons: ρg (t) in both cases scales as 1/¯ a4 (t). Putting all together, we have the same main result from both cases:

6

dρg (t 0 ) a¯(ti ) ∝ ω05 dt a¯(t)

a¯(t) a¯(t 0 )

4

Assuming that oscillations start at time ti , the present graviton energy density in both cases will be (here we take t0 to be the present time): ρg (t0 ) ∝

ω05

6

Z

a¯(ti )

t0

a¯−2 (t)dt

ti

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Cosmological implications

Consider a simple model of universe which is spatially flat and contains radiation, non-relativistic matter, and a cosmological constant.

On time scales much longer than the period of oscillations, the Friedmann in both models can be expressed in terms of the background scale factor as 2 2 ¯ 2 Mpl2 ≈ ρr ,0 + ρm,0 + ρΛ,0 + ω χi 3H(t) a¯4 (t) a¯3 (t) 2

a¯(ti ) a¯(t)

3 ,

(3)

¯ where H(t) ≡ [a¯˙ (t)]/[¯ a(t)] is the Hubble parameter. Note that ρχ ≈ (ω 2 χ2i /2)(¯ ai /¯ a)3 (here χ refers to either the ϕ scalar field in the GRSF model or the φ scalar field in the f (R) gravity model). 23 / 33



We consider that oscillations of scale factor continue through the present epoch. But the scalar energy density in both cases scales like non-relativistic matter ⇒ It can grow to dominate the radiation energy density before the outset of the matter dominated epoch. Take a conservative constraint on the scalar energy density as ρχ (t) . ρm (t). This condition is equivalent to set (Ti is the energy scale at time ti ): χi . 10−11 Mpl

ωB ω

h

Ti 1 GeV

i3/2

.

This bound also constraints the graviton production since Di ∝ (ϕi /Mpl )2 and Ei ∝ (φi /Mpl ).

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Scalar field quantization in curved spacetime

Conclusion

Small oscillations of a scale factor in a FRW-background lead to quantum creation of gravitons. Such oscillations can rise either in general relativity plus a coherent oscillating scalar field or in f (R)-gravity. Both models show a similar expression for the graviton energy density in a FRW-background with some differences with respect to: In the GRSF model, the scalar field oscillates with a frequency 2ω and the initial amplitude of oscillations is proportional to the square of the initial value of the scalar field. In f (R) gravity, the scalar field oscillates with a frequency ω and the initial amplitude of oscillations is proportional to the initial value of the scalar field.

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The best way to discuss this issue is in the context of a particular model theory. Formally, field quantization in curved spacetime proceeds in close analogy to the Minkowski spacetime case. Let us consider a real, minimally coupled massless scalar field for which the Lagrangian density is L =

1 1 (−g) 2 [g µν ∂µ φ∂ν φ] . 2

Setting the variation of the action S = L (x )d n x with respect to φ equal to zero yields the scalar field equation

R

g µν ∂µ ∂ν φ(x ) = 0 .

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For a spacelike hypersurface Σ with unit normal vector nµ , the inner product on solutions to this equation, (φ1 , φ2 ), is

Z −i

[φ1 (x )∂µ φ∗2 (x ) − φ∗2 (x )∂µ φ1 ] nµ

p

|γ|d n−1 x .

Σ

This inner product is independent of the hypersurface (Hawking & Ellis, 1973): (φ1 , φ2 )Σ1 = (φ1 , φ2 )Σ2 The quantization of φ follows canonical methods: Choose a particular Σ. ∂L Define π = ∂(∂ . φ) 0

Impose [φ(t, x), π(t, x0 )] =

√ i δ (n−1) (x −g

− x0 ).

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As in flat spacetime, we should do now: Set {fj } and {fj∗ } → complete basis for solutions of equation of motion of φ(x ). Expand the field operator φ in terms of {fj , fj∗ }. Interpret the operators ˆ aj , ˆ aj† .

In Minkowski spacetime: We were able to pick up out a natural set of modes by demanding that they be positive-frequency with respect to the time coordinate. ∂t fj = −iωfj with ω > 0. The time coordinate is not unique (Lorentz transformation). But the vacuum state and total number operators are invariant under such transformations.

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In curved spacetime: In general we will not be able to find solutions that separate into time-dependent and space-dependent factors. We cannot classify modes as positive or negative frequency. We can find many set of basis modes. There are no way to prefer one over any others. Notion of vacuum or number operator depend on which set we choose.

One solution is to take an asymptotically flat spacetime. This procedure carries interesting physical consequences → particle creation. Back to

main

.

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Particle creation by gravitational fields Let us take a spacetime which is asymptotically flat in the past and in the future, but which is non-flat in the intermediate region: {fj } → in-region {Fj } → out-region

Set of solutions to be orthonormal:

fj , fj 0 = Fj , Fj 0 = δjj 0

fj∗ , fj∗0 = Fj∗ , Fj∗0 = −δjj 0

fj , fj∗0 = Fj , Fj∗0 = 0

The field operator can be expanded in terms of {fj } or {Fj }: φ=

X j

aˆj fj + aˆj† fj∗ =

X

bˆj Fj + bˆj† Fj∗ .

j

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Annihilation and creation operators: in-region : ˆ aj , ˆ aj† → ˆ aj |0iin = 0, ∀j. ˆj , b ˆ† → b ˆj |0iout = 0, ∀j. out-region : b j

We may expand the in-modes in terms of the out-modes: fj =

X

(αjk Fk + βjk Fk∗ ) .

k

Inserting this expansion into the orthogonality relations leads to the conditions:

X

αjk αj∗0 k − βjk βj∗0 k = δjj 0 ,

(4)

k

X

(αjk αj 0 k − βjk βj 0 k ) = 0 .

k

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Noting that aˆj = (φ, fj ) and bˆj = (φ, Fj ), we may expand the two sets of creation and annihilation operator in terms of one another as

( Bogolubov transformation

∗ˆ aˆj = k αjk bk − βjk∗ bˆk† , P bˆk = j αjk aˆj + βjk∗ aˆj† .

P

Time-dependent gravitational field: No particle were present before the gravitational field is turn on → |0iin (Heisenberg picture). ˆk = b ˆ† b ˆ . The physical number operator in the out-region → N k k

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Thus the mean number of particles into mode k is ˆk i = hN =

ˆ† in h0|bk bk |0iin in h0|

X

∗ † αjk aˆj

(5) + βjk aˆj

αik aî +

βik∗ aî†

|0iin

ji

=

X

=

X

=

X

(βjk βik∗ ) in h0|ˆ aj aî† |0iin

ji

(βjk βik∗ ) in h0|ˆ ai† aˆj + δji |0iin

ji

ji

(βjk βik∗ ) δji in h0|0iin =

X

|βjk |2 .

j

If any of the βjk coefficients are nonzero, then particles are created by the gravitational field. Back to

main

.

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