Inflationary cosmology, fundamental aspects ofquantum theory, and the emergence of the seeds of

structure i.e. of nontrivial shapes.

Daniel Sudarsky, Inst. for Nuclear Sciences, UNAM, MexicoCollaborations with: M. Castagnino (U. Buenos Aires, Arg.), P.

Pearle (Hamilton College, USA.), R. Laura (U. Rosario, Arg.), H.Sahlman ( U. Utrecht, Ndr.), A. Perez (U. of Marseille Fr.), A. de

Unanue (ICN- UNAM), G. Garcia (ICN-UNAM), P Canate(ICN-UNAM) A. Diez-Tejedor (U. Guanajuato, Mx.), S. Landau(U. Buenos Aires, Arg.) & C. Scoccola (Inst. Astr. Canarias, Sp.)

Shape Dynamics Workshop, May 8th, 2014

CONCEPTUAL DIFFICULTIES IN QUANTUM THEORYQM is extremely successful theory, however it still presents someserious shortcomings.Instead of a long discussion let us illustrate the situation with a fewquotes:“Either the wave function as given by Schrodinger equation is noteverything, or it is not right” Bell, J. S. , “Are there quantumjumps?”, in Speakable and unspeakable in quantum mechanics.Cambridge: Cambridge University Press, 201–212 (1987)

I , personally, have been strongly influenced by R. Penrose writings inthis regard.

Also, in a very recent paper entitled Collapse of the State Vector by S.Weinberg (arXiv:1109.6462v3 [quant-ph] ).

“There is now in my opinion no entirely satisfactory interpretation ofquantum mechanics”.

One of the most promising class of approaches to the issue involvesmodifications of Q.T. known as dynamical reduction or collapsetheories

Collapse Theories: There is a large amount of work in this direction:GRW, Pearle, Diosi, Penrose, Bassi (recent advances to make itcompatible with S.R. : Tumulka ( Th. of flashes), Bedningham (Q Fwith stoch. dynamics)).

Example, Continuous Spontaneous Localization , P. Pearle. ( CSL):

d|ψ〉 = −[iH − λ2

2 (A− 〈ψ|A|ψ〉)2]dt + λ(A− 〈ψ|A|ψ〉)dWt|ψ〉.where Wt is a Wiener process. ( W2

t = t ).

It includes the U and R evolution processes (for measuring A) in a

unified fashion. The proposal for particles assumes A = ~X.λ is taken to be small enough that particle physics is not stronglyaffected but leads to localization of macroscopic objects.New versions involving application to quantum fields.

The problems on Q.T. become exacerbated in cosmology.

In fact it its worthwhile noting that one of the most importantmotivations for J. Hartle et. al. in pursuing the decoherent historiesapproach, actually came from cosmology.In the usual, “Copenhagen” formulations of quantum mechanicspresented in textbooks, probabilities can be assigned to the historiesof alternatives of a subsystem that were “measured” by an“observer”. But such formulations are not general enough forquantum cosmology which seeks to describe the early universe wherethere were neither measurements nor observers present.( Quantum cosmology: Problems for the 21st century James B. Hartlein “Nishinomiya 1996, Physics in the 21st century” 179-199).

On the other hand modern cosmology has transformed itself into ascience dealing with precise observational data. This indicates thatone might use it to further our understanding of Quantum Theoryitself.

1) Cosmic Inflation:Contemporary cosmology includes inflation as one of its mostattractive components. The inclusion of an early inflationary stage incosmology leads to a natural account for the seeds of cosmic structurein terms of quantum fluctuations and a correct estimate of thecorresponding spectrum.

The starting point of the analysis is a FRW space-time background

dS2 = a(η)2−dη2 + d~x2inflating under the influence of an inflaton background fieldφ = φ0(η).

On top of this, one considers quantum fluctuations: δφ, δψ, ...., δhij

assumed to be characterized by the “ vacuum state” (essentially theBunch-Davies (BD) vacuum) |0〉.

From these, one argues, the primordial inhomogeneities andanisotropies ‘arise”.

THE DATA: δTT0

(θ, ϕ) = 13ψ(ηD,~xD), (ignoring some modifications)

gives us a picture of the Newtonian Potential on the LSS.

We characterize this map in terms of the spherical harmonicfunctions, and write: δT

T0(θ, ϕ) =

∑lm αlmYlm(θ, ϕ), so that

αlm =13

∫dΩ2ψ(ηD,~xD)Y∗lm(θ, ϕ) (1)

The quantity that is often the focus of the analysis is:

Cl ≡1

2l + 1

∑m

|αlm|2. (2)

As we now the analysis leads to a remarkable agreement withobservations. However, there is a difficulty that is not often addressed.

These are supposed to represent the primordial inhomogeneitieswhich evolved into all the structure in our Universe: galaxies, starsplanets, etc...

As THEORY FITS VERY WELL WITH THE OBSERVATIONS, oneis then very tempted to say “well that is it. What else do we want?”.

However, let us consider the following: The analysis starts with anHomogeneous and Isotropic (H&I) region, (both in the part that couldbe described at the “classical level”, and the quantum level) thatgrows into our causal Universe. But we end up with a situation whichis not H&I : It contains the primordial inhomogeneities which willresult in our Universe’s structure, and the conditions that permit ourown existence.

How does this happen if the dynamics of the closed system does notbreak those symmetries?

There are many proposed answers but we find they are allproblematic. ( We can discuss this at the end!)

3) OUR APPROACH: The situation we face here is unique(Quantum + Gravity (GR) + Observations).

We want to be able to point to a physical process that occurs in timeas explaining the emergence of the seeds of structure. After all,emergence (in this context) means : Something that was not there at acertain time is there at a later time. We need to explain the breakdownof the symmetry of the initial state: Collapse can do this.

Collapse Theories: As I said there is important existing work in thisdirection but in this talk we will NOT start from any of those, as wefirst want to learn what are the relevant issues and how can we dealwith them?.

We consider here adding, to the standard inflationary paradigm, aquantum collapse of the wave function as a self-induced instantaneousprocess.

How could this possibly fit with our current theoretical views? Thebig question. Let’s recall some issues and conceptual difficulties stilloutstanding (which I think might offer some hope):

I) The Problem of Time. In following the usual quantization rules inGR one often ends up with a timeless theory ( canonical approaches).

II) More generally: how do we recover space-time from differentapproaches to QG ?

Solutions to I) usually consider using some dynamical variable as aphysical clock and then considering relative probabilities (and wavefunctions). However it seems that one recovers only an approx.Schrodinger eq. with corrections that violate unitarity (Pullin &Gambini). Could something like this underlie collapse theories?Regarding II), there are many suggestions indicating space-timemight be an emergent phenomena... ( T. Jacobson, R. Sorkin, N.Seiberg and many others...). In that case, the level at which one cantalk about space-time concepts, might well be the classicaldescription. However some QG aspects could remain as traces, and“look” like collapse. Hydrodynamic analogy.

4) The Proposal:The idea is that at the quantum level gravity is VERY different, and atmacro scales leaves something that looks like a collapse of thequantum wave function of the matter fields. ( Inspired by Penrose andDiosi’s ideas).

Thus, the inflationary regime is one where gravity already has a goodclassical description, but matter fields still require a quantumtreatment.

The setting will thus naturally be semiclassical Einstein gravity (withthe extra element: THE COLLAPSE): i.e., besides U we have,sometimes, spontaneous jumps:....|0〉k1 ⊗ |0〉k2 ⊗ |0〉k3 ⊗ ....→ ....|Ξ〉k1 ⊗ |0〉k2 ⊗ |0〉k3 ⊗ .....

There is an underlying Quantum Theory of Gravity (probably with nonotion of time).

But, by the “time” one recovers space-time concepts, thesemiclassical treatment is a very good one, its regime of validityincludes the inflationary regime as long as R << 1/l2Planck.

More precisely, we will rely on the notion of SemiclassicalSelf-Consistent Configuration (SSC).

DEFINITION: The set gµν(x), ϕ(x), π(x),H, |ξ〉 ∈ H represents aSSC if and only if ϕ(x), π(x) andH correspond a to quantum fieldtheory constructed over a classical space-time with metric gµν(x) andthe state |ξ〉 inH is such that:

# Gµν [g(x)] = 8πG〈ξ|Tµν [g(x), ϕ(x), π(x)]|ξ〉.It is, in a sense, the GR version of Schrodinger-Newton equation.

This, however, can not describe the transition from a H&I SSC to onethat is not. For that, we need to add a collapse: However, a collapsewill be a transition from one SSC to another, not simply from onestate to another. So we need: ....SSC1....→ ....SSC2.... During thecollapse eq. # must be modified (as in W. Israel’s matching)!.

Say for instance we want to describe a transition from an H&I SSC toone that is not. That involves changing the state, and thus thespace-time, and thus the Hilbert space where the state “lives” and isnon trivial (arXiv:1108.4928, paper in JCAP), but seems to teach ussomething about the required modifications in #.

5) PRACTICAL TREATMENT:

As we said, space-time is thus treated in classical language, and in ourcase (working in a specific gauge and ignoring the tensorperturbations) the metric is:

ds2 = a2(η)[−(1 + 2Ψ)dη2 + (1− 2Ψ)δijdxidxj

],Ψ(η,~x) 1

But now, in the practical approach, the field is split φ = φ0 + δφ: Thehomogeneous scalar background φ0(η) and a perturbation δφ to betreated with QFT. In the previous, more precise treatment, the formercorresponds to a specific mode (or combination) of the quantum field.

During Inflation (slow roll regime), we will have a(η) ≈ −1HIη

.

We will set atoday = 1, and assume that the inflationary regime ends atη = η0 which is negative and very small in absolute terms.

The semiclassical Einstein equations, at lowest order, lead to∇2Ψ = 4πGφ0〈 ˆ

δφ〉 = s〈 ˆδφ〉, s ≡ 4πGφ0

Consider the quantum theory of the field δφ, and work with therescaled field variable y = aδφ and its conjugate momentumπ =

ˆδφ/a. (Set the problem in a box of side L, and L→∞ at the

end).

We decompose the field and momentum operators as:y(η,~x) = 1

L3

∑~k ei~k·~xyk(η), πy(η,~x) = 1

L3

∑~k ei~k·~xπk(η),

whereyk(η) ≡ fk(η)ak + fk(η)a+

−k, πk(η) ≡ gk(η)ak + gk(η)a†−k

with the usual choice of modes: fk(η) = 1√2k

(1− i

ηk

)exp(−ikη)

gk(η) = −i√

k2 exp(−ikη), which leads to what is known as the

Bunch-Davies vacuum: the state defined by ak|0〉 = 0 .

Note that 〈0|yk(η)|0〉 = 〈0|πk(η)|0〉 = 0 , so Ψ = 0.

The collapse will modify the state and, thus, the expectation values ofthe operators yk(η) and πk(η). We need to specify the “rules”according to which “collapse happens”. We must make some guess.HERE I’LL SHOW THE SIMPLEST, MOST NAIVE SCHEME:Single Instantaneous collapse (per mode) to a new state |Θ〉. We willassume that, after the collapse, the expectation value of themomentum operator (or/and the field operator) in each mode is relatedto the uncertainties of the pre-collapse state.

Specifically, in the vacuum state yk and πk are characterized byGaussian wave functions centered at 0 with spread ∆yk and ∆πyk, so

〈yR,I~k

(ηck)〉Θ = 0, 〈πR,I

~k(ηc

k)〉Θ = xR,I~k

√(∆πR,I

~k)2

0,

where ηc~k

is the time of collapse for each mode.

The xR,I~k

are numbers selected randomly from a Gaussian distributioncentered at 0 and with unit dispersion. Our Universe corresponds to asingle realization of these random variables, so each of thesequantities has a single specific value.

Then, as a result of Einstein’s semiclassical equation,

Ψ~k(η) =

√ε

2H

MPk2 〈π~k(η)〉.

In terms of the random variables,

Ψ~k(η) =

√εH

MP

(L2k

)3/2(cos ∆k +

sin ∆k

zk

)X~k,

where X~k ≡ xR~k

+ ixI~k, zk ≡ kηc

~k, ∆k ≡ kη − zk, and L is the size of the

fiducial quantization box.

The value of all theX~k corresponding to our Universe completely fixthe value of the temperature fluctuations ( ignoring here some otherlate physics effects) δT

T0(θ, ϕ) ' 1

3Ψmatt(ηD,~xD), where~xD is the pointat angular position (θ, ϕ) on the last scattering two sphere.

Thus using δTT0

(θ, ϕ) =∑

l,m almYlm(θ, ϕ), the coefficients alm can beexpressed as

alm =

∫δTT0

(θ, ϕ)Y?lm(θ, ϕ)dΩ.

The relation between the metric perturbation during the matterdominated epoch Ψmatt and that during inflation Ψ is simplyΨmatt = (3/5ε)Ψ (aside from the effects codified in the transferfunctions, which encode for instance, the acoustic peaks).

Then we can write the coefficients as

alm =il

L3/2

∑~k

g(zk)jl(kRD)Y?lm(k)

k3/2 T(k)X~k

where g(zk) ≡ πH5MP

√2ε (cos zk − sin zk/zk) (recall zk = kηc

k ) and T(k)

are the transfer functions.

The X~k are random, and the coefficients alm, are weighted sums ofthem, i.e. a kind of two-dimensional random walk. We estimate themost likely value of the total displacement. To evaluate it we use a‘fiducial ensemble of realizations of the random walk” and identifythe most likely value with the ensemble average: |alm|M.L. = |alm|.Taking a normalized Gaussian assumption for the ensemble(X~kX?~k′ = 2δ~k,~k′), and going to the continuum we find :

|alm|2M.L. =

∫d3k

4π3k3 jl(kRD)2|Ylm(k)|2T(k)2g(zk)2.

The observational quantity usually considered is the angular spectrumCl defined by Cl ≡ 1

2l+1∑

m |alm|2. Within our proposal, if we assumethe time of collapse satisfies ηc

~k∝ k−1, i.e., zk = z independent of k,

we recover an exact scale-invariant spectrum (aside from the effectsencoded in T(k)), this is,

l(l + 1)CM.L.l ≡ A =

g(z)2

(2π)3 =H2

102πM2Pε

(cos z− sin z

z

)2

.

The quantity A is fixed by the observational data to be A ' 10−10.

So far we have only arranged to fit observations. Now let us look at anovel prediction.The standard way to search for non-trivial statistical aspects, such asnon-Gaussianities, in the CMB is to analyze the bispectrum, which isusually considered as intimately connected with the three-pointfunction of the inflaton field. In particular, one usually focuses on the“angle-averaged bispectrum,” defined as

Bl1l2l3 ≡∑mi

(l1 l2 l3m1 m2 m3

)a11m1a12m2a13m3 ,

where the object(

l1 l2 l3m1 m2 m3

)is the Wigner 3-j symbol. Note

that the overline denotes the average over an ensemble of universes.However, within our approach, we focus on the value of this quantityin our own Universe; therefore, the quantity of interest is obtained bysubstituting here the explicit expression for the alm, leading to:

Bobsl1l2l3 =

∑mi

(l1 l2 l3m1 m2 m3

)1

L9/2

∑~k1,~k2,~k3

×g(zk1)g(zk2)g(zk3)jl1(k1RD)jl2(k2RD)jl3(k3RD)

(k1k2k3)3/2

×Y?l1m1(k1)Y?l2m2

(k2)Y?l3m3(k3)T(k1)T(k2)T(k3)X~k1

X~k2X~k3.

Just as with the quantity alm, the bispectrum Bobsl1l2l3 is characterized by

a sum of random numbers, and thus we cannot give a specificprediction for its value. However, the same reasoning we usedregarding alm enables us to focus on the most likely value of|Bobs

l1l2l3 |2M.L., and we estimate it again using the fiducial ensemble.

Again we use the estimate |Bobsl1l2l3 |

2M. L. = |Bobs

l1l2l3|2. The expression for

|Bobsl1l2l3|2 contains term of the form X~k1

X~k2X~k3

X?~k4X?~k5

X?~k6. Taking the

random variables X~k as Gaussian-distributed, the six-product averagebecomes a sum of terms of the form X~k1

X~k2· X~k3

X?~k4· X?~k5

X?~k6. Thus, by

assuming ηc~k∝ k−1 and setting for simplicity T(k) = 1, our model

predicts:

|Bobsl1l2l3|2 =

g(z)6(1 + ∆l1l2l3)

(2π)9l1(l1 + 1)l2(l2 + 1)l3(l3 + 1),

where

∆l1l2l3 = δl1,l2

(2l3 + 1)

(∑m1

(l1 l1 l3m1 −m1 0

)(−1)m1

)2

+ (−1)l3

+cyclic perms.

+δl1,l2δl2,l3

2 + 3(1 + (−1)l1)(2l1 + 1)

(∑m1

(l1 l1 l1m1 −m1 0

)(−1)m1

)2 ,

Consequently, the most likely value for the magnitude of the collapse

bispectrum |Bobsl1l2l3 |M.L. =

(|Bobs

l1l2l3|2)1/2

is

|Bobsl1l2l3 |M.L. = A2 f

(1 + ∆l1l2l3

l1(l1 + 1)l2(l2 + 1)l3(l3 + 1)

)1/2

,

where f ≡ A−1/2.

In the standard treatment, the prediction for the bispectrum has theform Bl1l2l3 = A2fNLF(l1, l2, l3) with F(l1, l2, l3) and fNL, respectively,characterizing its shape and amplitude. The prediction depends on theassumptions made about the three-point function for the inflaton field.

We predict a non-vanishing bispectrum with a specific functionalform and fixed amplitude (without requiring a non vanishing thethree-point function).

The observed angular spectrum fixes A ' 10−10, so the prediction ofthe collapse model is f ' 105, for this functional form, but there areno adjustable parameters in the model!.

7) SOME OF OUR OTHER STUDIES ALONG THESE LINESi) If tensor modes are confirmed, we would need to go to beyond thesemiclassical approach.ii) This approach might offer a solution to the Fine Tuning problemfor the inflaton Potential. [ CQG, 27, 225017 (2010)].i ii) Studies on dependence of time of collapse [PRD 78, 043510,(2008) arXiv:0801.4702 [gr-qc]. PRD 85, 123001, (2012)arXiv:1112.1830 [astro-ph.CO].iv) Multiple collapses. More information about the post-collapsestates [ CQG, 28, 155010 (2011)]v) Framing Inflation within explicit collapse theory: CSL naivelyadapted to QFT,

|ψ, η〉 = T e−i∫ η−T dη′H− 1

4λ

∫ η−T dη′

∫d3x[w(~x,η′)−2λπ(~x)]2 |ψ,−T 〉.

where π(~x) ≡ (−∇2)−1/4π(~x) (another option is (−∇2)1/4y(~x) )Standard inflationary values lead to the estimateλ ∼ 10−5MpC−1 ≈ 10−19sec−1. (Not too different from the GRWpreferred value).Other studies of this by T.P. Singh et. al. and J. Martin et. al. .

Directions for further analysis and some ongoing research :

i) Black Hole Information Paradox: “Dynamical reduction” or“Collapse theories” seem to have the potential to remove the issue.The fundamental evolution involves non-unitarity and loss ofinformation. Ongoing detail analysis by our group.ii) Very Speculative Ideas connecting with QG and the problem oftime: Wheeler-DeWitt or LQG are timeless theories.Can the recovery of time be connected with collapse theories? In whatdirection would the connection be?

OPEN ISSUES:

a) The precise analysis led to jump in extrinsic curvature at the SSCjunction. Can this be resolved in a QG theory? Perhaps in analogywith the BH singularity resolution in LQG?

b) Framing within a fully covariant approach (which might involvenon-locality) but which ensures no conflicts with causality.

FINAL WORDSQuantum Theory is still very mysterious and relativity (both Specialand General) makes it more so. However people are often advisedagainst worrying about these issues.

Perhaps this is because we have all grown used to employingQuantum Theory by following, in practice, the CopenhagenInterpretation or some other FAPP kind of approach (as described byJohn Bell). We can, in fact, do this quite generally when dealing withlaboratory experiments where there are no doubts about: what isbeing measured? when? and by whom?The fact that we are nowadays applying QT to cosmology forces us,in my view, to reconsider such approach.

We could see this as a problem, but it might be an opportunityinstead!

It might lead to fundamental new insights about the Quantum/Gravityinterface!. And it is very interesting. END.