A historical perspective on fine tuning: lessons from causal set cosmology Fay Dowker Oxford th December 01 Rafael Sorkin, Graham Brightwell, Djamel Dou, Fay Dowker Raquel Garcia, Joe Henson, Xavier Martin, Denjoe O Connor, David Rideout.
Take away thoughts Effective Field Theory, including some form of locality, gives us fine tuning problems of which the cosmological constant problem is the most severe. A large/small (dimensionless) number can only be explained by relating it to some other large/small number. Either a parameter, or a number that is historically large e.g. the age of the universe. Causal sets are discrete spacetimes that are Lorentz invariant and nonlocal and provide a different cosmological paradigm to add-a-scalar-field-with-such-andsuch-a-potential modelling. Causal sets were used to predict the correct order of magnitude of Lambda. Causal set cosmology point towards historical explanations of other cosmological parameters. Discussion point: is a multiverse in which universes come into being, one after another according to laws of evolution, more explanatory than one in which they just exist? (Becoming vs Being?)
Plan Causal Sets Causal Set Quantum Gravity Causal Set Lambda A Bouncing universe, evolving from epoch to epoch: self-tuning? Discussion points
Quantum Gravity: we must be radical and yet conservative To arrive at a theory of quantum gravity we must make a creative leap, from a position that is grounded in current knowledge. Identifying the grounded position is itself a creative act. The causal set approach claims that certain aspects of General Relativity and quantum theory will have direct counterparts in quantum gravity: the spacetime causal order from General Relativity and the path integral from quantum theory. It makes one main new hypothesis about the nature of the physical world: fundamental discreteness of spacetime. Taken together, these three things form the bare bones conceptual basis of causal set theory: the spacetime causal order and atomicity furnish the kinematics and the path integral provides the framework for the quantum dynamics.
Causal structure at the heart of General Relativity Spacetime in General Relativity has a causal structure. That causal structure is fundamental to understanding General Relativity. The epitome of the theory is a Black Hole and our understanding is in 4- dimensional, spacetime, causal terms. For example, Hawking s area theorem is proved using global causal analysis -- it cannot be understood in terms of three dimensional entities evolving in time. Further, this causal order unifies within itself the topology (inc. dimension), differentiable structure and 9/10 of the metric of a Lorentzian manifold (Robb, Alexandrov, Zeeman, Penrose, Kronheimer, Hawking, Malament) Spacetime in General Relativity almost is a causal order.
Causal sets: the marriage of causality and atomicity The missing tenth of the geometry, not given by causal order, is local physical scale. To account for spacetime as we know it, need to provide causal order and physical scale. Discreteness does the job because we can count the spacetime atoms: Order + Number = Geometry Now, it is widely expected that the differentiable Lorentzian manifold structure of spacetime will break down at the Planck scale. Fundamental spacetime atomicity is perhaps the simplest way to realise this expectation. It happens to be exactly what is necessary if one is drawn to conceive of spacetime as pure causal order. A causal set is a discrete causal order: Not a marriage of convenience but the uniting of soul mates
A causal set that is well approximated by d Minkowski space Future Time Hasse dimension Lorentz inv nonlocal relational austere no background Past Space On Planckian scales, this is what Minkowski space is (like) Not just Minkowski, any Lorentzian manifold
A Causal Set is a discrete order A causal set (or causet) is a set, C, with a binary relation,, referred to as precedes, which satisfies: * Transitivity: if x y and y z then x z, x, y, z C; * Acyclicity: if x y and y x then x = y, x, y C; * Local finiteness: for any ordered pair of elements x and z of C, the cardinality of the set {y x y z} is finite The first two axioms say that C is a partial order. The third axiom is what makes the set discrete. The elements of C are the atoms of spacetime The deep structure of spacetime is a causal set At macroscopic scales, the order gives rise to the spacetime causal order and the number of elements gives the spacetime Volume: Number Volume ( thooft; Myrheim; Bombelli, Lee, Meyer, Sorkin)
Everpresent Lambda (Sorkin; Ahmed, Dodelson, Green, Sorkin) In the early 1990 s Rafael Sorkin predicted the value of the Cosmological Constant using a heuristic argument based on expectations of quantum causal set theory. Other scientists were shocked by the discovery of the accelerated expansion; causal set theorists were thrilled. Since Number ~ Volume, it is natural in a path integral for causal set quantum gravity to fix N for the causal sets summed over: a justification for unimodular gravity (Weinberg) Fixing N means fluctuations in V of order V N V V and Lambda are canonically conjugate (as Time and Energy) so V Λ 1 Λ 1 V H 10 10 If Lambda fluctuates about zero, then what we see is only the fluctuation
The one successful Quantum Gravity prediction to date The nonlocality of causets is crucial. The model is homogeneous by fiat. Allowing spatial fluctuations in Lambda is the next challenge: Barrow and Zuntz showed that a local model of an inhomogeneous Lambda would be incompatible with δρ ρ 10 so if we can discover a viable model it will have to be nonlocal. Causal Set Lambda is Everpresent Lambda -- no fine tuning required to explain Why now It doesn t explain why Lambda is driven to zero but assuming this, the observed Lambda is as small as it is because the universe is as big as it is.
Causal set cosmology Path integral Quantum Theory is a species of generalised stochastic process: Quantum Mechanics is a generalisation of Brownian motion or a random walk. Therefore, as a warm up for quantum causal set theory, it makes sense to construct classical stochastic models for causal sets: random causal sets...there s a vast array of possibilities for probability distributions...how to choose something that has some chance of being physically relevant as a law of motion for a discrete spacetime? Note: others are finding discrete models of causal structure useful to gain insight into cosmological questions: Figure from Eternal Symmetree by Harlow, Shenker, Stanford, Susskind arxiv:1110.0496
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Physical conditions lead to interesting models Discrete General Covariance (label invariance) and a local causality condition (called Bell causality) severely restrict the models. A Classical Sequential Growth (CSG) model, specified by a sequence of nonnegative real numbers t 0 =1,t 1,t,t,... The CSG class contains some fascinating models, with suggestive properties
Transitive percolation In TP, each newborn chooses to be above each potential ancestor with prob p t n = t n p = t 1+t Mathematically well-studied: the Lorentzian analogue of the Erdos-Renyi random graph For p = 1, 0 this dynamics grows a bouncing universe: Picture To grow to a realistic size, p (or t ) has to be tuned to be very small
Generalised percolation In generalised percolation each newborn chooses to be above each potential subset of ancestors with relative probability t n t n = α log n n α > π α < π there are almost surely infinitely many posts there are almost surely finitely many posts There are a.s. infinitely many posts if t n grows faster than that.
Graham Brightwell at DIAS in 004!
Why the obsession with posts? The dynamics of each epoch is governed by an effective set of couplings, derived by application of a renormalisation transformation from the original ones: t n = t n + t n+1 applied N times where N is the number of elements to the past of the current epoch Under this transformation, the set of couplings tends pointwise to percolation with t = lim (t n) 1 n n For the class of models identified by Brightwell, this limit is zero. We conjecture that in each epoch, the effective dynamics is close to percolation with a decreasing value of t from epoch to epoch. This cosmic renormalisation transformation would then result in a selftuning universe (work in progress).
Summary Causal sets are models of discrete spacetime that take GR seriously, are Lorentz invariant and useful to have in your toolkit for quantum gravity phenomenology Sorkin predicted the order of magnitude of the observed cosmological constant using expectations of causal set quantum gravity: Lambda fluctuates. There are causal set cosmologies (physically motivated) which result in bouncing universes with effective laws for each epoch derived from a cosmic renormalisation group. These have the potential to be models of a self tuning universe
Some references Forks in the road on the way to quantum gravity, Sorkin, gr-qc/970600 Everpresent Lambda Ahmed, Dodelson, Green, Sorkin, astro-ph/00974 Observables in causal set cosmology Brightwell, Dowker, Garcia, Henson, Sorkin, gr-qc/010061 On the renormalisation transformations induced by cycles of expansion and contraction in causal set cosmology Martin, O Connor, Rideout, Sorkin, gr-qc/ 000906 19