A ΛCDM Bounce Scenario

Edward Wilson-Ewing

Albert Einstein InstituteMax Planck Institute for Gravitational Physics

Work with Yi-Fu CaiarXiv:1412.2914 [gr-qc]

Third EFI Winter Conference on Quantum Gravity, Tux

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 1 / 21

Quantum Gravity Effects in the Early Universe

One of the main difficulties of any theory of quantum gravity is toobtain predictions (that are realistically testable) and confront themto experiment or observations.

The best hope in this direction appears to lie in the very earlyuniverse, where quantum gravity effects are expected to be strongand may have left some imprints on the cosmic microwavebackground (CMB). What form could these imprints have?

In this talk, I will focus on loop quantum cosmology (LQC) and somepotential predictions concerning the CMB.

Caveat: the dynamics of LQC (just like general relativity) depend onthe matter content. Therefore the predictions of LQC will stronglydepend on what the dominant matter field (radiation, inflaton, . . . )is during the bounce.

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 2 / 21

Quantum Gravity Effects in the Early Universe

One of the main difficulties of any theory of quantum gravity is toobtain predictions (that are realistically testable) and confront themto experiment or observations.

The best hope in this direction appears to lie in the very earlyuniverse, where quantum gravity effects are expected to be strongand may have left some imprints on the cosmic microwavebackground (CMB). What form could these imprints have?

In this talk, I will focus on loop quantum cosmology (LQC) and somepotential predictions concerning the CMB.

Caveat: the dynamics of LQC (just like general relativity) depend onthe matter content. Therefore the predictions of LQC will stronglydepend on what the dominant matter field (radiation, inflaton, . . . )is during the bounce.

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 2 / 21

The CMB and the Matter Bounce

Precision measurements of the temperature anisotropies in the CMBindicate the perturbations are nearly scale-invariant, with a slight redtilt. Inflation is one model known to generate scale-invariantperturbations, but it is not the only one.

An alternative to inflation is the matter bounce scenario: Fouriermodes that are initially in the quantum vacuum state that exit theHubble radius in a contracting matter-dominated Friedmann universebecome scale-invariant. [Wands]

Then, if this contracting branch can be connected to our currentlyexpanding universe via some sort of a bounce, these scale-invariantperturbations can provide suitable initial conditions for the expandingbranch. [Finelli, Brandenberger]

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 3 / 21

The CMB and the Matter Bounce

Precision measurements of the temperature anisotropies in the CMBindicate the perturbations are nearly scale-invariant, with a slight redtilt. Inflation is one model known to generate scale-invariantperturbations, but it is not the only one.

An alternative to inflation is the matter bounce scenario: Fouriermodes that are initially in the quantum vacuum state that exit theHubble radius in a contracting matter-dominated Friedmann universebecome scale-invariant. [Wands]

Then, if this contracting branch can be connected to our currentlyexpanding universe via some sort of a bounce, these scale-invariantperturbations can provide suitable initial conditions for the expandingbranch. [Finelli, Brandenberger]

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 3 / 21

Cosmological Perturbation Theory

A commonly used gauge-invariant variable for scalar perturbations isthe comoving curvature perturbation R. In particular, the mainquantity of interest is the power spectrum

∆2R(k) =

k3

2π2|Rk |2 ∼ A ·

(k

k?

)ns−1

.

However, for calculations the Mukhanov-Sasaki variable

v = zR, z =a√ρ + P

csH,

is commonly used since the differential equation that governs itsdynamics is particularly simple:

v ′′k + c2s k

2vk −z ′′

zvk = 0.

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 4 / 21

Review of the Matter Bounce

For a contracting matter-dominated (P = 0) FLRW universe,

a(η) = η2, −∞ < η < 0.

Then, the Mukhanov-Sasaki equation becomes

v ′′k + c2s k

2vk −2

η2vk = 0,

as z ′′

z= a′′

a= 2

η2 , and the solution to this differential equation is

vk = A1

√−ηH (1)

32

(−cskη) + A2

√−ηH (2)

32

(−cskη).

At early times, |η| � 1 and the Fourier modes are inside the horizon.If one imposes quantum vacuum fluctuations as the initial conditions,then A1 ∼

√~,A2 = 0 and when the modes exit the (sound) horizon

they become scale-invariant.E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 5 / 21

The ΛCDM Bounce

In this model the matter fields are radiation and cold dark matter(CDM), and there is a positive cosmological constant Λ.

We expect the modes that exit the (sound) horizon duringmatter-domination to have a scale-invariant spectrum.

Also, those that exit when the effective equation of state is slightlynegative (due to Λ) will have a slight red tilt, in agreement withobservations of the cosmic microwave background.

We assume that loop quantum cosmology (LQC) captures therelevant high-curvature dynamics, in which case a bounce occurs atthe Planck scale [Ashtekar, Bojowald, Lewandowski, Paw lowski, Singh, . . . ] and we cancalculate the evolution of the perturbations through the bounce usingthe LQC Mukhanov-Sasaki equations.

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 6 / 21

The ΛCDM Bounce

In this model the matter fields are radiation and cold dark matter(CDM), and there is a positive cosmological constant Λ.

We expect the modes that exit the (sound) horizon duringmatter-domination to have a scale-invariant spectrum.

Also, those that exit when the effective equation of state is slightlynegative (due to Λ) will have a slight red tilt, in agreement withobservations of the cosmic microwave background.

We assume that loop quantum cosmology (LQC) captures therelevant high-curvature dynamics, in which case a bounce occurs atthe Planck scale [Ashtekar, Bojowald, Lewandowski, Paw lowski, Singh, . . . ] and we cancalculate the evolution of the perturbations through the bounce usingthe LQC Mukhanov-Sasaki equations.

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 6 / 21

The ΛCDM Bounce

In this model the matter fields are radiation and cold dark matter(CDM), and there is a positive cosmological constant Λ.

We expect the modes that exit the (sound) horizon duringmatter-domination to have a scale-invariant spectrum.

Also, those that exit when the effective equation of state is slightlynegative (due to Λ) will have a slight red tilt, in agreement withobservations of the cosmic microwave background.

We assume that loop quantum cosmology (LQC) captures therelevant high-curvature dynamics, in which case a bounce occurs atthe Planck scale [Ashtekar, Bojowald, Lewandowski, Paw lowski, Singh, . . . ] and we cancalculate the evolution of the perturbations through the bounce usingthe LQC Mukhanov-Sasaki equations.

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 6 / 21

Outline

1 Homogeneous Background

2 Perturbations

3 Predictions

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 7 / 21

Homogeneous Background I: ΛCDM Era

In the contracting branch, the dynamics of thespace-time will initially be dominated by thecosmological constant Λ and afterwards by colddark matter.

During the transition between thesetwo epochs, there will be a period oftime where the effective equation ofstate will be slightly negative.

The Fourier modes that exit the soundhorizon at this time will be nearlyscale-invariant with a slight red tilt.

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 8 / 21

Homogeneous Background II: Radiation Era

After this, the space-time will be dominated by radiation, and we willuse LQC to model the dynamics in the high curvature regime.

In LQC, the big-bang singularity is replaced by a bounce:

[Paw lowski, Pierini, WE]

Furthermore, states that areinitially sharply peaked remainsharply peaked throughouttheir entire evolution, includingat the bounce. Thus it ismeaningful to speak of an“effective geometry” anddisregard quantumfluctuations.

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 9 / 21

Homogeneous Background II: Radiation Era

After this, the space-time will be dominated by radiation, and we willuse LQC to model the dynamics in the high curvature regime.

In LQC, the big-bang singularity is replaced by a bounce:

[Paw lowski, Pierini, WE]

Furthermore, states that areinitially sharply peaked remainsharply peaked throughouttheir entire evolution, includingat the bounce. Thus it ismeaningful to speak of an“effective geometry” anddisregard quantumfluctuations.

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 9 / 21

The Key Ingredients

The key ingredients in the ΛCDM bounce scenario that thepredictions will depend upon are the following:

The effective equation of state when the Fourier modes ofinterest exit the sound horizon, ωeff = −δ with 0 < δ � 1,

The sound speed of CDM, cs = ε� 1,

The (proper) Hubble rate at the time of matter-radiationequality He ,

The energy density of the radiation field at the time of the LQCbounce, ρc ∼ ρPl.

In order to obtain an analytical solution, we assume a constantequation of state during the ΛCDM era: the scale-dependence of aparticular mode is determined by the effective equation of state whenthat mode exits the sound horizon.

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 10 / 21

The Key Ingredients

The key ingredients in the ΛCDM bounce scenario that thepredictions will depend upon are the following:

The effective equation of state when the Fourier modes ofinterest exit the sound horizon, ωeff = −δ with 0 < δ � 1,

The sound speed of CDM, cs = ε� 1,

The (proper) Hubble rate at the time of matter-radiationequality He ,

The energy density of the radiation field at the time of the LQCbounce, ρc ∼ ρPl.

In order to obtain an analytical solution, we assume a constantequation of state during the ΛCDM era: the scale-dependence of aparticular mode is determined by the effective equation of state whenthat mode exits the sound horizon.

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 10 / 21

Scalar Perturbations I: ΛCDM Background

The Mukhanov-Sasaki equation in the ΛCDM background is

v ′′k + ε2k2vk −2(1 + 3δ)

(1− 3δ)2(η − ηo)2vk = 0.

The solution (as usual for a constant equation of state) is a Hankelfunction. Assuming the quantum vacuum as the initial conditions,

vk =

√−π~(η − ηo)

4H (1)

n [−εk(η − ηo)],

where

n ≈ 3

2+ 6 δ + O(δ2).

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 11 / 21

Scalar Perturbations II: Radiation Background

The Mukhanov-Sasaki equation for a radiation-dominated space-time(in the absence of quantum gravity effects) is

v ′′k +k2

3vk = 0,

and the solutions are simply plane waves. Requiring that vk and v ′kbe continuous during the transition between the ΛCDM era and theradiation-dominated period gives

vk ∼(k−n cos

kηe√3

+ k−n−1 sinkηe√

3

)cos

kη√3

+ sinkη√

3.

Note that we have not imposed the condition that k |ηe |/√

3� 1.However, as we can see here already, this condition will be necessaryfor scale-invariance.

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 12 / 21

High-Curvature Dynamics of the Perturbations

On the previous slide, we determined the form of the perturbations inthe radiation-dominated pre-bounce era in the limit that quantumgravity effects are negligible. Of course, in order to calculate theevolution of the perturbations through the bounce, it will be necessaryto use the LQC equivalent of the Mukhanov-Sasaki equation.

There are several approaches to cosmological perturbation theory inLQC:

Effective equations from the anomaly-freedom approach, [Bojowald,

Hossain, Kagan, Shankaranarayanan; Cailleteau, Mielczarek, Barrau, Grain, Vidotto]

Hybrid quantization: LQC background, quantization of theperturbations a la Fock, [Fernandez-Mendez, Mena Marugan, Olmedo; Agullo, Ashtekar, Nelson]

Separate universe approximation. [WE]

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 13 / 21

High-Curvature Dynamics of the Perturbations

On the previous slide, we determined the form of the perturbations inthe radiation-dominated pre-bounce era in the limit that quantumgravity effects are negligible. Of course, in order to calculate theevolution of the perturbations through the bounce, it will be necessaryto use the LQC equivalent of the Mukhanov-Sasaki equation.

There are several approaches to cosmological perturbation theory inLQC:

Effective equations from the anomaly-freedom approach, [Bojowald,

Hossain, Kagan, Shankaranarayanan; Cailleteau, Mielczarek, Barrau, Grain, Vidotto]

Hybrid quantization: LQC background, quantization of theperturbations a la Fock, [Fernandez-Mendez, Mena Marugan, Olmedo; Agullo, Ashtekar, Nelson]

Separate universe approximation. [WE]

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 13 / 21

Separate Universes in Loop Quantum Cosmology

In the separate universe framework, we make the approximation thatlong-wavelength perturbations can be modeled by separate Friedmannuniverses. [Salopek, Bond; Wands, Malik, Lyth, Liddle] This can be adapted to LQC:

Take a cubic lattice and assume that each cell ishomogeneous and isotropic. (The small variationsbetween the parameters in each cell correspond to theperturbations.) Then the usual LQC quantization of aflat FLRW space-time can be done in each cell. [WE]

The resulting LQC effective equations for scalar perturbations areexpected to be valid for Fourier modes whose wavelength remainsmuch larger than `Pl. [Rovelli, WE] For these modes, the long-wavelengthMukhanov-Sasaki equation in LQC is

v ′′k −z ′′

zvk = 0, z =

a√ρ + P

cs H.

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 14 / 21

Separate Universes in Loop Quantum Cosmology

In the separate universe framework, we make the approximation thatlong-wavelength perturbations can be modeled by separate Friedmannuniverses. [Salopek, Bond; Wands, Malik, Lyth, Liddle] This can be adapted to LQC:

Take a cubic lattice and assume that each cell ishomogeneous and isotropic. (The small variationsbetween the parameters in each cell correspond to theperturbations.) Then the usual LQC quantization of aflat FLRW space-time can be done in each cell. [WE]

The resulting LQC effective equations for scalar perturbations areexpected to be valid for Fourier modes whose wavelength remainsmuch larger than `Pl. [Rovelli, WE] For these modes, the long-wavelengthMukhanov-Sasaki equation in LQC is

v ′′k −z ′′

zvk = 0, z =

a√ρ + P

cs H.

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 14 / 21

Scalar Perturbations III: Bounce

Knowing the background evolution, the effective LQCMukhanov-Sasaki equation can be solved, giving a hypergeometricfunction.

Then, by taking the limit of t � tPl, we obtain the form of the scalarperturbations after the bounce:

∆2R(k) =

k3

2π|Rk |2 ∼

√ρcρPl

· |He |`Pl

ε3·(k

k?

)−12δ

,

recall that the comoving curvature perturbation is Rk ∼ vk/a.

Here we have assumed that k |ηe |/√

3� 1 in order to ensure that theresulting spectrum is nearly scale-invariant.

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 15 / 21

Scalar Perturbations III: Bounce

Knowing the background evolution, the effective LQCMukhanov-Sasaki equation can be solved, giving a hypergeometricfunction.

Then, by taking the limit of t � tPl, we obtain the form of the scalarperturbations after the bounce:

∆2R(k) =

k3

2π|Rk |2 ∼

√ρcρPl

· |He |`Pl

ε3·(k

k?

)−12δ

,

recall that the comoving curvature perturbation is Rk ∼ vk/a.

Here we have assumed that k |ηe |/√

3� 1 in order to ensure that theresulting spectrum is nearly scale-invariant.

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 15 / 21

Tensor Perturbations

A similar calculation can be done for tensor modes, and the result isthat their resulting amplitude is predicted to be significantly smallerthan the amplitude of the scalar perturbations for two reasons:

1. The amplitude of the scalar perturbations are boosted by afactor of ε−3,

2. The amplitude of the tensor perturbations is damped by a factorof 1/4 during the bounce due to LQC effects.

The predicted tensor-to-scalar ratio is

r =∆2

h(k)

∆2R(k)

= 24ε3,

which satisfies the current observationalbound of r < 0.12 [Planck+BICEP2/Keck] for ε ∼ 0.1.

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 16 / 21

An Asymmetric Bounce

To have scale-invariance, we required that

k |ηe |√3� 1.

It is easy to check that, defining η+e to be the time of

matter-radiation equality in the expanding branch, for k observed inthe CMB today kη+

e ∼ 1.

Therefore, in order for the first condition to hold, |ηe | � η+e . This

requires an asymmetric bounce where the radiation-dominated eralasts longer in the expanding branch than in the contracting branch.

An asymmetric bounce has already been suggested to arise in LQCdue to the effect of particle production during the bounce. [Mithani, Vilenkin]

Whether this effect is strong enough to generate sufficient asymmetryis not yet clear.

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 17 / 21

An Asymmetric Bounce

To have scale-invariance, we required that

k |ηe |√3� 1.

It is easy to check that, defining η+e to be the time of

matter-radiation equality in the expanding branch, for k observed inthe CMB today kη+

e ∼ 1.

Therefore, in order for the first condition to hold, |ηe | � η+e . This

requires an asymmetric bounce where the radiation-dominated eralasts longer in the expanding branch than in the contracting branch.

An asymmetric bounce has already been suggested to arise in LQCdue to the effect of particle production during the bounce. [Mithani, Vilenkin]

Whether this effect is strong enough to generate sufficient asymmetryis not yet clear.

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 17 / 21

Are There Any LQC Effects?

In the model studied here, the bounce occurs due to LQCquantum-gravity effects. But what happens if we consider a differentΛCDM bounce, where the bounce occurs due to some other effect?

Most of the predictions are very robust. The majority of the featuresof the perturbations are determined by the physics of the contractingbranch far before the bounce and therefore the bounce will not affectthese predictions.

However, there is one exception: in LQC, we found that theamplitude of the tensor modes is suppressed by a factor of 1/4 duringthe bounce. Therefore, in other bouncing models (unless there is asimilar suppression of the tensor modes), we would expect a largervalue of r (for a given CDM sound speed ε).

This effect is a potential test for LQC versus other bouncingcosmology models.

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 18 / 21

Are There Any LQC Effects?

In the model studied here, the bounce occurs due to LQCquantum-gravity effects. But what happens if we consider a differentΛCDM bounce, where the bounce occurs due to some other effect?

Most of the predictions are very robust. The majority of the featuresof the perturbations are determined by the physics of the contractingbranch far before the bounce and therefore the bounce will not affectthese predictions.

However, there is one exception: in LQC, we found that theamplitude of the tensor modes is suppressed by a factor of 1/4 duringthe bounce. Therefore, in other bouncing models (unless there is asimilar suppression of the tensor modes), we would expect a largervalue of r (for a given CDM sound speed ε).

This effect is a potential test for LQC versus other bouncingcosmology models.

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 18 / 21

Are There Any LQC Effects?

In the model studied here, the bounce occurs due to LQCquantum-gravity effects. But what happens if we consider a differentΛCDM bounce, where the bounce occurs due to some other effect?

Most of the predictions are very robust. The majority of the featuresof the perturbations are determined by the physics of the contractingbranch far before the bounce and therefore the bounce will not affectthese predictions.

However, there is one exception: in LQC, we found that theamplitude of the tensor modes is suppressed by a factor of 1/4 duringthe bounce. Therefore, in other bouncing models (unless there is asimilar suppression of the tensor modes), we would expect a largervalue of r (for a given CDM sound speed ε).

This effect is a potential test for LQC versus other bouncingcosmology models.

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 18 / 21

Conclusions

In the ΛCDM bounce scenario, quantum vacuum fluctuations becomenearly scale-invariant with a slight red tilt for the modes that exit thesound horizon when the effective equation of state is slightly negative.

This model requires an asymmetric bounce, and makes threeimportant predictions beyond scale-invariance:

A small tensor-to-scalar ratio,

A positive running of ns , i.e., dns/dk > 0, and

nt > ns − 1.

There is an LQC-specific effect of an extra damping of thetensor-to-scalar ratio by a factor of 1/4 which is a potentialobservational test for the theory.

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 19 / 21

Open Questions

There remain four main open questions in the ΛCDM bouncescenario:

Amplitude of the running of ns: We found that dns/dk > 0,but to calculate the amplitude, more must be known about thedetails of the pre-bounce era.

Particle production during the bounce: It has been pointedout that particle production may be important during the LQCbounce, but can it provide sufficient asymmetry?

Non-Gaussianities: A small sound speed typically causes largenon-Gaussianities. How large? What about the bounce?

Anisotropies during the bounce: In the absence of anekpyrotic phase, we expect anisotropies to dominate thedynamics during the bounce. How might the presence ofanisotropies change the predictions?

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 20 / 21

Thank you for your attention!

E. Wilson-Ewing (AEI) ΛCDM Bounce February 18, 2015 21 / 21