The Universe had (probably) an origin: on the theorem of Borde-Guth-Vilenkin

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Emilio Elizalde

ICE/CSIC & IEEC

Campus UAB, Barcelona

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Trento, July 26, 2016*

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*XXV Years of fruitful and wonderful collaboration Trento-Barcellona

Some facts (a few rather surprising...)

• Adam Riess, NP 2011, at Starmus (Tenerife), about Hubble:
• “Hubble obtained the distances and redshifts of distant nebulae…”
• “Hubble discovered that the Universe was expanding…”
• No mention to Vesto Slipher, an extraordinary astronomer
• Brian Schmidt, NP 2011, at Starmus (Tenerife) & Lisa Randall, Harvard U, in Barcelona, about Einstein:
• “Einstein was the first to think about the possibility of a ‘dark energy’…”
• No mention to Fritz Zwicky, another extraordinary astronomer
• Fritz Zwicky discovered dark matter in the early 1930s while studying how galaxies move within the Coma Cluster
• He was also the first to postulate and use nebulae as gravitational lenses (1937)
• How easily* brilliant astronomers get dismissed
• How easily* scientific myths arise *in few decades

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SHOES-Supernovae

How did the “Big Bang” get its name ?

• Sir Fred Hoyle (1915–2001) English astronomer noted primarily for the theory of stellar nucleosynthesis (1946,54 groundbreaking papers)
• Work on Britain's radar project with Hermann Bondi and Thomas Gold
• William Fowler NP’83: “The concept of nucleosynthesis in stars was first established by Hoyle in 1946”
• He found the idea universe had a beginning to be pseudoscience, also arguments for a creator, “…for it's an irrational process, and can't be described in scientific terms“; “…belief in the first page of Genesis"
• Hoyle-Gold-Bondi 1948 steady state theory, "creation or C-field"
• BBC radio's Third Programme broadcast on 28 Mar 1949: “… for this to happen you would need such a Big Bang!”

http://www.bbc.co.uk/science/space/universe/scientists/fred_hoyle

Thus:

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Big Bang = Impossible !!

But now:

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Big Bang ≈ Inflation !

• Same underlying physics as in steady state theory, “creation or C-field”
• Richard C. Tolman, 1934: Relativity, Thermodynamics, and Cosmology. Explained how a closed universe could equal zero energy: how all mass/energy is positive and all gravitational energy is negative and how they may cancel each other out, leading to a universe of zero energy
• Tolman–Oppenheimer–Volkoff (TOV) equation: constrains in GR the structure of a spherically symmetric body of isotropic material in static equilibrium

Gravitational Waves, as Einstein predicted

Febr 11, 2016; detected on Sept 14, 2015

…and Dec 2015 !

These plots show the signals of gravitational waves detected by the twin LIGO observatories at Livingston, Louisiana, and Hanford, Washington.

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The signals came from two merging black holes, of about 36 and 29 times the mass of our Sun, lying 1.3 billion light-years away.

The top two plots show data received at Livingston and Hanford, along with the predicted shapes for the waveform.

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As the plots reveal, the LIGO data very closely match Einstein's predictions.

But there is another meaning for BB:

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The Big Bang Singularity !!

Singularity Theorems: Roger Penrose, Stephen Hawking, …

Theorem 1 (Big Bang). Let (M,g) a global hyperbolic spacetime satisfying R_ab χ^aχ^b ≥ 0 for all temporal vectors χ^a (Einstein’s Eqs. with the strong energy condit.) If there exists a spatial Cauchy C² hypersurface, Σ, for which the trace of the intrinsic curvature satisfies K<C<0, C const., then no temporal curve starting from Σ and going towards the past can have a length that is larger than 3/|C|. All temporal geodesics to the past are incomplete.

[This is to say, under the conditions observed in our Universe (Hubble’s law) and admitting the validity of General Relativity, our Universe had a begining]

Theorem 2 (Black Holes). Let (M,g) a global hyperbolic space-time satisfying R_ab k^ak^b ≥ 0 for all lightlike vectors k^a (Einstein’s Eqs. with the strong or the weak energy condit’s.). Let us assume that there exists a spatial Cauchy C² hypersurface, Σ, and a trapped surface, and let θ₀ be the maximum value of the expansion over it. If θ₀ < 0, then there exists at least a lightlike geodesic, which cannot be extended to the future, and which is orthogonal to the trapped surface. Moreover, the value of the affine parameter, up to the point where the geodesic is no further extensible, is less than 2/|θ₀|. [The existence of a non-extensible lightlike geodesic implies that there will be a photon which, starting from that surface and after a time of travel proportional to 2/c|θ₀|, will fall into a future time singularity. In absence of a theory of quantum gravity we cannot know the physical nature of the singularity.]

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R Penrose,”Gravitational collapse and space-time singularities”, Phys Rev Lett 14 (1965) 57

S Hawking, GFR Ellis, “The Large Scale Structure of Space-Time” (Cambridge U P, 1973) 

RM Wald, “General Relativity” (U Chicago P, 1984); R Geroch, Ann Phys 48 (1968) 526

http://www.hawking.org.uk/the-beginning-of-time.html

Creation of the Universe

… out of nothing !

What’s ‘nothing’ ??

• In fundamental classical physics, GR:

de Sitter sol. is the zero-energy sol. of Einstein’s Eqs.

• In quantum physics:

The vacuum state of a quantum system

• Of quantum spacetime Krauss-Wilczek ‘15, nope!
• Of a scalar field Hamilt (Higgs, inflaton, …)

On the BGV Theorem: Introduction

• Inflationary cosmological models are generically eternal in their future evolution [1981-83: Guth, Linde, Albrecht, Steinhardt, Vilenkin]
• In inflation one cannot in principle conclude for any future time that the whole Universe will reach thermal equilibrium
• We could ask ourselves if it would be possible to construct a model with the Universe being eternal towards the past, without ever ocurring an initial singularity
• In this case the Universe would not be created at a certain moment, it would have always existed
• This possibility was already discussed in the 80’s, but a specific model starting from an exact dS solution could not be formulated

• What could be proven is a theorem which states that inflationary spacetimes are past geodesic-ally incomplete [1994: Borde, Vilenkin] 🡪 an initial singularity
• Key assumption: the energy-momentum tensor obeys the weak energy condition (WEC)
• But: quantum corrections to inflationary models seem to violate such condition

• This happens when quantum fluctuations result in an increase of the Hubble parameter: dH/dt > 0
• And this point is essential for chaotic inflation to be eternal !
• Therefore, the WEC must be generically violated in those models !
• What opens the door to a scape from the BV theorem, and from the initial singularity

• This was the motivation for Borde, Guth & Vilenkin: “Inflationary Spacetimes Are Incomplete in Past Directions,” PRL 90 (2003) 151301
• As the title clearly indicates, they eventually recovered the same result of BV’94
• In few words, they proved that the situation is very similar to the case of an initial (classical) de Sitter space

• The intuitive reason why de Sitter inflation cannot be past eternal is that in dS space exponential inflation is always preceded by an exponential contraction
• But an exponential contraction is not consistent with the physics of inflation: it thermalizes the Universe in the past infinity and prevents the appearance of inflation itself
• In the work, initial inflation is saved by imposing an “almost dS” space together with a minimal condition of “averaged expansion”: H_av > 0
• The authors proved that the condition of averaged expansion implies, again, past incompleteness
• The theo can be extended to cosmologies with extra dimensions

• To finish, a comment about ciclic models [2002: Steinhardt-Turok]
• It is not true that ciclic models do no require initial conditions
• Most usual is that H_av > 0 is fulfilled and, then the BGV theorem applies
• As a consequence, also in these models one gets initial geodesic incompleteness ! 🡪 an origin

Recent work:

• INFLATIONARY UNIV. WITH A VISCOUS FLUID AVOIDING SELF-REPRODUCTION, I Brevik, EE et al, 2016
• Self-reproduction of the universe means that the inflationary process has no way to finish; it can be also seen as a major problem: inflation would never end
• Recently, an inflationary scenario which avoids the self-reproduction problem has been proposed in:

V. Mukhanov, Fortschritte der Physik 63, 1 (2014), arXiv:1409.2335

• We have considered what we believe is the simplest possibility for inflation without self-reproduction
• We consider in the paper a universe with a bulk viscous cosmic fluid in flat FLRW
• We derive the conditions for the existence of inflation, and those which at the same time prevent the occurrence of self-reproduction
• Our theoretical model gives results which are in agreement with data from BICEP2 and PLANCK

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Thank You ���Mille Grazie��

Age of the universe

• In the context of the ΛCDM model
• Three density parameters Ω_m, Ω_r, Ω_Λ + the Hubble parameter H₀

Cosmological parameters from 2015 Planck results

Null energy condition

The null energy condition stipulates that for every future-pointing null vector field k,

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Each of these has an averaged version, in which the properties noted above are to hold only on average along the flowlines of the appropriate vector fields. Otherwise, the Casimir effect leads to exceptions. For example, the averaged null energy condition states that for every flowline (integral curve) C of the null vector field k, we must have

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Weak energy condition

The weak energy condition stipulates that for every timelike vector field X, the matter density observed by the corresponding observers is always non-negative:

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Dominant energy condition

The dominant energy condition stipulates that, in addition to the weak energy condition holding true, for every future-pointing causal vector field (either timelike or null) Y, the vector field –T^a_b Y^b must be a future-pointing causal vector. That is, mass-energy can never be observed to be flowing faster than light.

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Strong energy condition

The strong energy condition stipulates that for every future-pointing timelike vector field X, the trace of the tidal tensor measured by the corresponding observers is always non-negative:

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There are many matter configurations which violate the strong energy condition, at least from a mathematical perspective. It is not clear whether these violations are physically possible in a classical regime. For instance, a scalar field with a positive potential can violate this condition. Moreover, it is violated in any cosmological inflationary process. However, it is clear that such a violation would violate the classical regime of general relativity, and one would be required to use an alternative theory of gravity.

ρ is the energy density and p is the pressure

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The energy conditions can then be reformulated in terms of these eigenvalues:

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The weak energy condition stipulates that

ρ ≥ 0, ρ + p ≥ 0

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The null energy condition stipulates that

ρ + p ≥ 0

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The strong energy condition stipulates that

ρ + p ≥ 0, ρ + 3p ≥ 0

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The dominant energy condition stipulates that ρ ≥ |p|

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Despite the names the strong energy condition does not imply the weak energy condition even in the context of perfect fluids.