PASCOS 01 Jul 2019 Positivity in the Sky Scott Melville

PASCOS 01 Jul 2019 Positivity in the Sky Scott Melville

PASCOS 01 Jul 2019 Positivity in the Sky Energy Scott Melville

PASCOS 01 Jul 2019 Positivity in the Sky Energy Scott Melville

PASCOS 01 Jul 2019 Positivity in the Sky Energy 𝐴𝐸𝐹𝑇 Scott Melville

Positivity in the Sky Scott Melville PASCOS 01 Jul 2019 Energy 𝐴𝑈𝑉 ? 𝐴𝐸𝐹𝑇 Scott Melville

Positivity in the Sky Scott Melville PASCOS 01 Jul 2019 Energy UV Properties 𝐴𝑈𝑉 ? IR Parameters 𝐴𝐸𝐹𝑇 Scott Melville

Outline hep-ph/9607351 Donoghue Let me start by describing the literature. There have been an explosion of papers on this subject recently, and they fall neatly into two categories. Outline hep-ph/9607351 Donoghue

Positivity Constraints Dark Energy SMEFT Big Picture Positivity Constraints Dark Energy SMEFT Only some EFTs have “good” UV completions … and this gives us constraining power! Horndeski parameters improved by factor 110 VBS parameters improved by factor ~100 These constraints have really come into their own in the last few years. The central ideas were laid out in 2006, and then we got to work developing all of the formalism and technical details, and now we’re finally in a position to apply the resulting constraints to all kinds of theories. The nitty-gritty details can be found in the papers, but for today I want to paint a broad picture of what’s going on and why. Outline hep-ph/9607351 Donoghue

Positivity Constraints Dark Energy SMEFT Big Picture Positivity Constraints Dark Energy SMEFT Only some EFTs have “good” UV completions … and this gives us constraining power! Horndeski parameter space reduced by factor 110 VBS parameters improved by factor ~100 These constraints have really come into their own in the last few years. The central ideas were laid out in 2006, and then we got to work developing all of the formalism and technical details, and now we’re finally in a position to apply the resulting constraints to all kinds of theories. The nitty-gritty details can be found in the papers, but for today I want to paint a broad picture of what’s going on and why. Outline hep-ph/9607351 Donoghue

Positivity Constraints Dark Energy SMEFT Where do they come from? hep-ph/9607351 Donoghue 1605.06111 Bellazzini hep-th/0602178 Adams et al 1702.06134 SM et al hep-th/0609159 Jenkins et al 1706.02712 SM et al Big Picture Positivity Constraints Dark Energy SMEFT Only some EFTs have “good” UV completions What good are they? … and this gives us constraining power! 1204.63880 Dvali 1502.07304 Baumann et al 1702.08577 SM et al 0802.4081 Adams et al 1405.2960 Bellazzini et al 1808.00010 Zhang, Zhou 1509.00851 Bellazzini et al 1601.04068  Cheung et al 1607.06084 Bonifacio et al 1608.02942  Cheung et al 1710.02539 Bellazzini 1710.09611 SM et al 1804.10624 SM et al Horndeski parameter space reduced by factor 110 VBS parameters improved by factor ~100 References: Donoghue hep-ph/9607351 EFTs and Dispersion relations Adams et al., hep-th/0602178 Causality, Analyticity and an IR Obstruction to UV Completion SM, de Rham, Tolley, Zhou 1702.06134, 1706.02712 Positivity Bounds for Effective Field Theories hep-th/0602178 Adams, Arkani-Hamed, Dubovsky, Nicolis, Rattazzi 1702.06134 de Rham, SM, Tolley, Zhou 1811.12928 Noller, Nicola 1904.05874 SM, Noller These constraints have really come into their own in the last few years. The central ideas were laid out in 2006, and then we got to work developing all of the formalism and technical details, and now we’re finally in a position to apply the resulting constraints to all kinds of theories. The nitty-gritty details can be found in the papers, but for today I want to paint a broad picture of what’s going on and why. [1904.05874] Outline hep-ph/9607351 Donoghue

[1904.05874] Positivity

UV IR Can have classicalization in other places [1904.05874] Positivity

UV IR General Relativity Positivity [1904.05874] Can have classicalization in other places [1904.05874] Positivity

UV IR New physics General Relativity Positivity [1904.05874] Can have classicalization in other places [1904.05874] Positivity

𝑀 UV ??? IR New physics General Relativity Positivity [1904.05874] Can have classicalization in other places [1904.05874] Positivity

𝑀 UV ??? 𝑐 𝑛 𝑀 𝑛 𝓞 𝑛 + IR New physics General Relativity Positivity 𝑐 𝑛 𝑀 𝑛 𝓞 𝑛 + IR General Relativity Can have classicalization in other places [1904.05874] Positivity

𝑀 UV ??? 𝑐 𝑛 𝑀 𝑛 𝓞 𝑛 + IR New physics General Relativity 𝑐 𝑛 𝑀 𝑛 𝓞 𝑛 + Known operator basis IR General Relativity This lets us calculate things without needing to worry about the complicated underlying UV physics. e.g. 𝑐 2 𝜕 2 𝜙 4 𝑀 2 + 𝑐 4 𝜕 4 𝜙 4 𝑀 4 +… [1904.05874] Positivity

𝑀 UV ??? 𝑐 𝑛 𝑀 𝑛 𝓞 𝑛 + IR New physics General Relativity 𝑐 𝑛 𝑀 𝑛 𝓞 𝑛 + Known operator basis IR General Relativity with undetermined coefficients BUT, the price to pay is that each local operator gets its own undetermined coefficient. I’ll refer to these as EFT parameters, Wilson coefficients, or couplings. e.g. 𝑐 2 𝜕 2 𝜙 4 𝑀 2 + 𝑐 4 𝜕 4 𝜙 4 𝑀 4 +… [1904.05874] Positivity

𝑀 UV ??? 𝑐 𝑛 𝑀 𝑛 𝓞 𝑛 + IR New physics General Relativity 𝑐 𝑛 𝑀 𝑛 𝓞 𝑛 + Known operator basis IR General Relativity with undetermined coefficients Problem number 1: often there are many operators to consider, even at relatively small n. We need to make AT LEAST this many independent measurements in order to fix c_n and make the theory predictive. In the SM, working with the leading order dimension 4 operators gives 19 undetermined parameters. In GR there’s all contractions of Ricci Scalar, Ricci Tensor and Riemann and their derivatives, and so we’d need to make a lot of very precise quantum gravity measurements in order to make this GR EFT predictive beyond leading orders. e.g. 𝑐 2 𝜕 2 𝜙 4 𝑀 2 + 𝑐 4 𝜕 4 𝜙 4 𝑀 4 +… Need many measurements [1904.05874] Positivity

𝑀 UV ??? 𝑐 𝑛 𝑀 𝑛 𝓞 𝑛 + IR New physics General Relativity 𝑐 𝑛 𝑀 𝑛 𝓞 𝑛 + Known operator basis IR General Relativity with undetermined coefficients Because we’ve given up on understanding the UV, no amount of testing or experimentation can reveal fundamental truths about the underlying physics on very small length scales. This isn’t a very satisfying state of affairs. e.g. 𝑐 2 𝜕 2 𝜙 4 𝑀 2 + 𝑐 4 𝜕 4 𝜙 4 𝑀 4 +… Need many measurements (2) No deeper understanding of UV physics [1904.05874] Positivity

𝑀 UV ??? 𝑐 𝑛 𝑀 𝑛 𝓞 𝑛 + IR New physics General Relativity Unitary, Causal, Local, … ??? 𝑀 𝑐 𝑛 𝑀 𝑛 𝓞 𝑛 + Known operator basis IR General Relativity with undetermined coefficients Because we’ve given up on understanding the UV, no amount of testing or experimentation can reveal fundamental truths about the underlying physics on very small length scales. This isn’t a very satisfying state of affairs. e.g. 𝑐 2 𝜕 2 𝜙 4 𝑀 2 + 𝑐 4 𝜕 4 𝜙 4 𝑀 4 +… Need many measurements (2) No deeper understanding of UV physics [1904.05874] Positivity

𝑀 UV ??? 𝑐 𝑛 𝑀 𝑛 𝓞 𝑛 + IR New physics General Relativity Unitary, Causal, Local, … ??? 𝑀 𝑐 𝑛 𝑀 𝑛 𝓞 𝑛 𝑐 4 𝑐 2 + IR General Relativity Any questions at this stage, about what we’re going to do or why it’s important 𝑐 2 𝜕 2 𝜙 4 𝑀 2 + 𝑐 4 𝜕 4 𝜙 4 𝑀 4 +… Need many measurements (2) No deeper understanding of UV physics [1904.05874] Positivity

𝑀 UV ??? 𝑐 𝑛 𝑀 𝑛 𝓞 𝑛 + IR New physics General Relativity Unitary, Causal, Local, … ??? 𝑀 𝑐 𝑛 𝑀 𝑛 𝓞 𝑛 𝑐 4 + IR General Relativity Any questions at this stage, about what we’re going to do or why it’s important 𝑐 2 𝑐 2 𝜕 2 𝜙 4 𝑀 2 + 𝑐 4 𝜕 4 𝜙 4 𝑀 4 +… Need many measurements (2) No deeper understanding of UV physics [1904.05874] Positivity

𝑀 UV ??? 𝑐 𝑛 𝑀 𝑛 𝓞 𝑛 + IR New physics General Relativity Unitary, Causal, Local, … ??? 𝑀 𝑐 𝑛 𝑀 𝑛 𝓞 𝑛 𝑐 4 + IR General Relativity Any questions at this stage, about what we’re going to do or why it’s important 𝑐 2 𝑐 2 𝜕 2 𝜙 4 𝑀 2 + 𝑐 4 𝜕 4 𝜙 4 𝑀 4 +… Need many measurements (2) No deeper understanding of UV physics [1904.05874] Positivity

𝑀 UV ??? 𝑐 𝑛 𝑀 𝑛 𝓞 𝑛 + IR New physics General Relativity Unitary, Causal, Local, … ??? 𝑀 𝑐 𝑛 𝑀 𝑛 𝓞 𝑛 𝑐 4 + IR General Relativity Any questions at this stage, about what we’re going to do or why it’s important 𝑐 2 𝑐 2 𝜕 2 𝜙 4 𝑀 2 + 𝑐 4 𝜕 4 𝜙 4 𝑀 4 +… Need many measurements (2) No deeper understanding of UV physics [1904.05874] Positivity

𝑀 UV ??? 𝑐 𝑛 𝑀 𝑛 𝓞 𝑛 + IR New physics General Relativity Unitary, Causal, Local, … ??? 𝑀 𝑐 𝑛 𝑀 𝑛 𝓞 𝑛 𝑐 4 + IR General Relativity Any questions at this stage, about what we’re going to do or why it’s important 𝑐 2 𝑐 2 𝜕 2 𝜙 4 𝑀 2 + 𝑐 4 𝜕 4 𝜙 4 𝑀 4 +… Data more constraining (2) No deeper understanding of UV physics [1904.05874] Positivity

𝑀 UV ??? 𝑐 𝑛 𝑀 𝑛 𝓞 𝑛 + IR New physics General Relativity Unitary, Causal, Local, … ??? 𝑀 𝑐 𝑛 𝑀 𝑛 𝓞 𝑛 𝑐 4 + IR General Relativity Any questions at this stage, about what we’re going to do or why it’s important 𝑐 2 𝑐 2 𝜕 2 𝜙 4 𝑀 2 + 𝑐 4 𝜕 4 𝜙 4 𝑀 4 +… Data more constraining (2) Can infer UV properties from IR measurements [1904.05874] Positivity

𝑀 UV ??? 𝑐 𝑛 𝑀 𝑛 𝓞 𝑛 + IR New physics General Relativity Unitary, Causal, Local, … ??? 𝑀 𝑐 𝑛 𝑀 𝑛 𝓞 𝑛 𝑐 4 + IR General Relativity Any questions at this stage, about what we’re going to do or why it’s important 𝑐 2 Data more constraining Positivity Bounds (2) Can infer UV properties from IR measurements [1904.05874] Positivity

𝑎 𝑎 ? 𝑏 𝑏 Being explicit, the positivity bounds which we’ll derive and use today are: [1904.05874] Positivity

𝑎 𝑎 𝑝 1 𝑝 3 ? 𝑝 2 𝑝 4 𝑏 𝑏 Positivity [1904.05874] Being explicit, the positivity bounds which we’ll derive and use today are: [1904.05874] Positivity

𝑎 𝑎 𝑝 1 𝑝 3 ? 𝑝 2 𝑝 4 𝑏 𝑏 Positivity 𝑡=− 𝑝 1 + 𝑝 3 2 𝑠=− 𝑝 1 + 𝑝 2 2 𝑡=− 𝑝 1 + 𝑝 3 2 𝑎 𝑎 𝑝 1 𝑝 3 𝑠=− 𝑝 1 + 𝑝 2 2 ? 𝑝 2 𝑝 4 𝑏 𝑏 Being explicit, the positivity bounds which we’ll derive and use today are: [1904.05874] Positivity

𝐴 𝐸𝐹𝑇 𝑠,𝑡 = 𝑐 0 + 𝑐 𝑠𝑠 𝑠 2 𝑀 4 + 𝑐 𝑠𝑠𝑡 𝑠 2 𝑡 𝑀 6 +… 𝑡=− 𝑝 1 + 𝑝 3 2 𝑎 𝑎 𝑝 1 𝑝 3 𝑠=− 𝑝 1 + 𝑝 2 2 ? 𝑝 2 𝑝 4 𝑏 𝑏 𝐴 𝐸𝐹𝑇 𝑠,𝑡 = 𝑐 0 + 𝑐 𝑠𝑠 𝑠 2 𝑀 4 + 𝑐 𝑠𝑠𝑡 𝑠 2 𝑡 𝑀 6 +… Being explicit, the positivity bounds which we’ll derive and use today are: [1904.05874] Positivity

Unitarity, Causality, Locality of 𝑡=− 𝑝 1 + 𝑝 3 2 𝑎 𝑎 𝑝 1 𝑝 3 𝑠=− 𝑝 1 + 𝑝 2 2 ? 𝑝 2 𝑝 4 𝑏 𝑏 𝐴 𝐸𝐹𝑇 𝑠,𝑡 = 𝑐 0 + 𝑐 𝑠𝑠 𝑠 2 𝑀 4 + 𝑐 𝑠𝑠𝑡 𝑠 2 𝑡 𝑀 6 +… Being explicit, the positivity bounds which we’ll derive and use today are: Unitarity, Causality, Locality of ? ⇒ 𝑐 𝑠𝑠 >0 , 𝑐 𝑠𝑠𝑡 ≳0 [1904.05874] Positivity

Simple UV Example UV IR [1904.05874] Positivity

𝜑 𝐻 UV IR Simple UV Example 1 𝑚 2 + 𝑝 2 𝑍 𝑀 2 + 𝑝 2 Positivity 1 𝑚 2 + 𝑝 2 𝑍 𝑀 2 + 𝑝 2 IR [1904.05874] Positivity

𝜑 𝐻 𝑔 𝐻 𝜑 2 UV IR Simple UV Example 1 𝑚 2 + 𝑝 2 𝑍 𝑀 2 + 𝑝 2 Positivity 𝑔 𝐻 𝜑 2 𝑔 1 𝑚 2 + 𝑝 2 𝑍 𝑀 2 + 𝑝 2 IR [1904.05874] Positivity

𝜑 𝐻 𝑔 𝐻 𝜑 2 UV IR Simple UV Example 1 𝑚 2 + 𝑝 2 𝑍 𝑀 2 + 𝑝 2 Positivity 𝑔 𝐻 𝜑 2 𝑝 1 𝑝 3 𝑔 1 𝑚 2 + 𝑝 2 𝑍 𝑀 2 + 𝑝 2 𝑝 2 𝑔 2 𝑍 𝑀 2 −𝑠 𝑝 4 IR [1904.05874] Positivity

𝜑 𝐻 𝑔 𝐻 𝜑 2 𝜑 UV IR Simple UV Example 1 𝑚 2 + 𝑝 2 𝑍 𝑀 2 + 𝑝 2 𝑀 2 𝑔 𝐻 𝜑 2 𝑝 1 𝑝 3 𝑔 1 𝑚 2 + 𝑝 2 𝑍 𝑀 2 + 𝑝 2 𝑝 2 𝑔 2 𝑍 𝑀 2 −𝑠 𝑝 4 𝑀 2 𝜑 IR 1 𝑚 2 + 𝑝 2 [1904.05874] Positivity

𝜑 𝐻 𝑔 𝐻 𝜑 2 𝜑 UV IR Simple UV Example 1 𝑚 2 + 𝑝 2 𝑍 𝑀 2 + 𝑝 2 𝑀 2 𝑔 𝐻 𝜑 2 𝑝 1 𝑝 3 𝑔 1 𝑚 2 + 𝑝 2 𝑍 𝑀 2 + 𝑝 2 𝑝 2 𝑔 2 𝑍 𝑀 2 −𝑠 𝑝 4 𝑀 2 𝜑 IR 1 𝑚 2 + 𝑝 2 𝑔 2 𝑍 𝑀 2 1+ 𝑠 𝑀 2 + 𝑠 2 𝑀 4 +… [1904.05874] Positivity

𝜑 𝐻 𝑔 𝐻 𝜑 2 𝜑 UV IR 𝑐 0 𝜑 4 𝑐 1 𝜕 2 𝜑 4 𝑐 2 𝜕 4 𝜑 4 Simple UV Example 𝑔 𝐻 𝜑 2 𝑝 1 𝑝 3 𝑔 1 𝑚 2 + 𝑝 2 𝑍 𝑀 2 + 𝑝 2 𝑝 2 𝑔 2 𝑍 𝑀 2 −𝑠 𝑝 4 𝑀 2 𝜑 𝑐 0 𝜑 4 𝑐 1 𝜕 2 𝜑 4 𝑐 2 𝜕 4 𝜑 4 … +… = + + IR 1 𝑚 2 + 𝑝 2 𝑔 2 𝑍 𝑀 2 1+ 𝑠 𝑀 2 + 𝑠 2 𝑀 4 +… [1904.05874] Positivity

𝜑 𝐻 𝑔 𝐻 𝜑 2 𝜑 UV IR 𝑐 0 𝜑 4 𝑐 1 𝜕 2 𝜑 4 𝑐 2 𝜕 4 𝜑 4 Simple UV Example 𝑔 𝐻 𝜑 2 𝑝 1 𝑝 3 𝑔 1 𝑚 2 + 𝑝 2 𝑍 𝑀 2 + 𝑝 2 𝑝 2 𝑔 2 𝑍 𝑀 2 −𝑠 𝑝 4 𝑀 2 𝜑 𝑐 0 𝜑 4 𝑐 1 𝜕 2 𝜑 4 𝑐 2 𝜕 4 𝜑 4 … +… = + + IR 1 𝑚 2 + 𝑝 2 𝑔 2 𝑍 𝑀 2 𝑔 2 𝑍 𝑀 4 𝑠 𝑔 2 𝑍 𝑀 6 𝑠 2 𝑔 2 𝑍 𝑀 2 1+ 𝑠 𝑀 2 + 𝑠 2 𝑀 4 +… + + +… = [1904.05874] Positivity

𝜑 𝐻 𝑔 𝐻 𝜑 2 𝜑 UV IR 𝑐 0 𝜑 4 𝑐 1 𝜕 2 𝜑 4 𝑐 2 𝜕 4 𝜑 4 Simple UV Example 𝑔 𝐻 𝜑 2 𝑝 1 𝑝 3 𝑔 1 𝑚 2 + 𝑝 2 𝑍 𝑀 2 + 𝑝 2 𝑝 2 𝑔 2 𝑍 𝑀 2 −𝑠 𝑝 4 𝑀 2 𝜑 𝑐 0 𝜑 4 𝑐 1 𝜕 2 𝜑 4 𝑐 2 𝜕 4 𝜑 4 … +… = + + IR 1 𝑚 2 + 𝑝 2 𝑔 2 𝑍 𝑀 2 𝑔 2 𝑍 𝑀 4 𝑠 𝑔 2 𝑍 𝑀 6 𝑠 2 𝑔 2 𝑍 𝑀 2 1+ 𝑠 𝑀 2 + 𝑠 2 𝑀 4 +… + + +… = [1904.05874] Positivity

𝜑 𝐻 𝑔 𝐻 𝜑 2 Unitarity ⇒ 𝑍>0 𝜑 Positive 𝑐 𝑛 UV IR 𝑐 0 𝜑 4 Simple UV Example UV 𝜑 𝐻 𝑔 𝐻 𝜑 2 𝑝 1 𝑝 3 Unitarity ⇒ 𝑍>0 𝑔 1 𝑚 2 + 𝑝 2 𝑍 𝑀 2 + 𝑝 2 𝑝 2 𝑔 2 𝑍 𝑀 2 −𝑠 𝑝 4 𝑀 2 𝜑 𝑐 0 𝜑 4 𝑐 1 𝜕 2 𝜑 4 𝑐 2 𝜕 4 𝜑 4 … Positive 𝑐 𝑛 +… = + + IR 1 𝑚 2 + 𝑝 2 𝑔 2 𝑍 𝑀 2 𝑔 2 𝑍 𝑀 4 𝑠 𝑔 2 𝑍 𝑀 6 𝑠 2 𝑔 2 𝑍 𝑀 2 1+ 𝑠 𝑀 2 + 𝑠 2 𝑀 4 +… + + +… = [1904.05874] Positivity

𝜑 𝐻 𝑔 𝐻 𝜑 2 Causal + Local ⇒ Unitarity ⇒ 𝑍>0 Various Poles 𝜑 Simple UV Example UV 𝜑 𝐻 𝑔 𝐻 𝜑 2 𝑝 1 𝑝 3 Causal + Local ⇒ Unitarity ⇒ 𝑍>0 Various Poles 𝑔 1 𝑚 2 + 𝑝 2 𝑍 𝑀 2 + 𝑝 2 𝑝 2 𝑔 2 𝑍 𝑀 2 −𝑠 𝑝 4 𝑀 2 𝜑 𝑐 0 𝜑 4 𝑐 1 𝜕 2 𝜑 4 𝑐 2 𝜕 4 𝜑 4 … Positive 𝑐 𝑛 +… = + + IR 1 𝑚 2 + 𝑝 2 𝑔 2 𝑍 𝑀 2 𝑔 2 𝑍 𝑀 4 𝑠 𝑔 2 𝑍 𝑀 6 𝑠 2 𝑔 2 𝑍 𝑀 2 1+ 𝑠 𝑀 2 + 𝑠 2 𝑀 4 +… + + +… = [1904.05874] Positivity

Unitarity, Causality, Locality of 𝑡=− 𝑝 1 + 𝑝 3 2 𝑎 𝑎 𝑝 1 𝑝 3 𝑠=− 𝑝 1 + 𝑝 2 2 ? 𝑝 2 𝑝 4 𝑏 𝑏 𝐴 𝐸𝐹𝑇 𝑠,𝑡 = 𝑐 0 + 𝑐 𝑠𝑠 𝑠 2 𝑀 4 + 𝑐 𝑠𝑠𝑡 𝑠 2 𝑡 𝑀 6 +… Unitarity, Causality, Locality of ? ⇒ 𝑐 𝑠𝑠 >0 , 𝑐 𝑠𝑠𝑡 ≳0 [1904.05874] Positivity

[1904.05874] Dark Energy

ℒ= ℒ 𝐺𝑅 + ℒ Scalar−Tensor Scalar-Tensor Theories ℒ= ℒ 𝐺𝑅 + ℒ Scalar−Tensor [1904.05874] Dark Energy

ℒ= ℒ 𝐺𝑅 + ℒ Scalar−Tensor Scalar-Tensor Theories ℒ= ℒ 𝐺𝑅 + ℒ Scalar−Tensor Single Scalar Field [1904.05874] Dark Energy

ℒ= ℒ 𝐺𝑅 + ℒ Scalar−Tensor Scalar-Tensor Theories ℒ= ℒ 𝐺𝑅 + ℒ Scalar−Tensor Single Scalar Field 𝛻𝜙 2 𝛻𝜙 6 𝛻𝜙 4 … [1904.05874] Dark Energy

ℒ= ℒ 𝐺𝑅 + ℒ Scalar−Tensor Scalar-Tensor Theories ℒ= ℒ 𝐺𝑅 + ℒ Scalar−Tensor Single Scalar Field 𝛻𝜙 2 𝛻𝜙 6 𝛻𝜙 4 … 𝛻𝛻𝜙 … 𝛻𝛻𝜙 𝛻𝜙 2 [1904.05874] Dark Energy

ℒ= ℒ 𝐺𝑅 + ℒ Scalar−Tensor Scalar-Tensor Theories ℒ= ℒ 𝐺𝑅 + ℒ Scalar−Tensor Single Scalar Field 𝛻𝜙 2 𝛻𝜙 6 𝛻𝜙 4 … 𝛻𝛻𝜙 … 𝛻𝛻𝜙 𝛻𝜙 2 𝛻𝛻𝜙 2 … [1904.05874] Dark Energy

ℒ= ℒ 𝐺𝑅 + ℒ Scalar−Tensor Scalar-Tensor Theories ℒ= ℒ 𝐺𝑅 + ℒ Scalar−Tensor Single Scalar Field 𝛻𝜙 2 𝛻𝜙 6 (𝑋=− 1 2 𝛻𝜙 2 ) 𝐺 2 𝑋 𝛻𝜙 4 … 𝛻𝛻𝜙 … 𝛻𝛻𝜙 𝛻𝜙 2 𝛻𝛻𝜙 2 … [1904.05874] Dark Energy

ℒ= ℒ 𝐺𝑅 + ℒ Scalar−Tensor Scalar-Tensor Theories ℒ= ℒ 𝐺𝑅 + ℒ Scalar−Tensor Single Scalar Field 𝛻𝜙 2 𝛻𝜙 6 (𝑋=− 1 2 𝛻𝜙 2 ) 𝐺 2 𝑋 𝛻𝜙 4 … 𝛻𝛻𝜙 … 𝐺 3 𝑋 𝛻 2 𝜙 𝛻𝛻𝜙 𝛻𝜙 2 𝛻𝛻𝜙 2 … [1904.05874] Dark Energy

ℒ= ℒ 𝐺𝑅 + ℒ Scalar−Tensor Scalar-Tensor Theories ℒ= ℒ 𝐺𝑅 + ℒ Scalar−Tensor Single Scalar Field 𝛻𝜙 2 𝛻𝜙 6 (𝑋=− 1 2 𝛻𝜙 2 ) 𝐺 2 𝑋 𝛻𝜙 4 … 𝛻𝛻𝜙 … 𝐺 3 𝑋 𝛻 2 𝜙 𝛻𝛻𝜙 𝛻𝜙 2 𝛻𝛻𝜙 2 𝐺 4,𝑋 𝑋 𝛻𝛻𝜙 2 … [1904.05874] Dark Energy

Positivity in Horndeski ℒ= ℒ 𝐺𝑅 + ℒ Scalar−Tensor [1904.05874] Dark Energy

Positivity in Horndeski (𝑋=− 1 2 𝛻𝜙 2 ) ℒ= ℒ 𝐺𝑅 + 𝐺 4 𝑋 𝑅+ 𝐺 2 𝑋 + 𝐺 4,𝑋 (𝑋) (𝛻𝛻𝜙) 2 [1904.05874] Dark Energy

Positivity in Horndeski (𝑋=− 1 2 𝛻𝜙 2 ) ℒ= ℒ 𝐺𝑅 + 𝐺 4 𝑋 𝑅+ 𝐺 2 𝑋 + 𝐺 4,𝑋 (𝑋) (𝛻𝛻𝜙) 2 Part of ``Horndeski’’ class [1904.05874] Dark Energy

𝐴 ~ 𝐺 2,𝑋𝑋 𝑠 2 − 𝐺 4,𝑋𝑋 𝑠 2 𝑡+… Positivity in Horndeski (𝑋=− 1 2 𝛻𝜙 2 ) ℒ= ℒ 𝐺𝑅 + 𝐺 4 𝑋 𝑅+ 𝐺 2 𝑋 + 𝐺 4,𝑋 (𝑋) (𝛻𝛻𝜙) 2 Part of ``Horndeski’’ class 𝜙 𝜙 𝐴 ~ 𝐺 2,𝑋𝑋 𝑠 2 − 𝐺 4,𝑋𝑋 𝑠 2 𝑡+… 𝜙 𝜙 [1904.05874] Dark Energy

𝐴 ~ 𝐺 2,𝑋𝑋 𝑠 2 − 𝐺 4,𝑋𝑋 𝑠 2 𝑡+… 𝐺 2,𝑋𝑋 >0, − 𝐺 4,𝑋𝑋 ≳0 Positivity in Horndeski (𝑋=− 1 2 𝛻𝜙 2 ) ℒ= ℒ 𝐺𝑅 + 𝐺 4 𝑋 𝑅+ 𝐺 2 𝑋 + 𝐺 4,𝑋 (𝑋) (𝛻𝛻𝜙) 2 Part of ``Horndeski’’ class 𝜙 𝜙 𝐴 ~ 𝐺 2,𝑋𝑋 𝑠 2 − 𝐺 4,𝑋𝑋 𝑠 2 𝑡+… 𝐺 2,𝑋𝑋 >0, − 𝐺 4,𝑋𝑋 ≳0 𝜙 𝜙 [1904.05874] Dark Energy

𝐴 ~ 𝐺 2,𝑋𝑋 𝑠 2 − 𝐺 4,𝑋𝑋 𝑠 2 𝑡+… 𝐺 2,𝑋𝑋 >0, − 𝐺 4,𝑋𝑋 ≳0 Positivity in Horndeski (𝑋=− 1 2 𝛻𝜙 2 ) ℒ= ℒ 𝐺𝑅 + 𝐺 4 𝑋 𝑅+ 𝐺 2 𝑋 + 𝐺 4,𝑋 (𝑋) (𝛻𝛻𝜙) 2 Part of ``Horndeski’’ class 𝜙 𝜙 𝐴 ~ 𝐺 2,𝑋𝑋 𝑠 2 − 𝐺 4,𝑋𝑋 𝑠 2 𝑡+… 𝐺 2,𝑋𝑋 >0, − 𝐺 4,𝑋𝑋 ≳0 𝜙 𝜙 ℎ 𝜇ν ℎ 𝜇ν 𝐴 ~ 𝐺 4,𝑋 𝑠 2 +… 𝜙 𝜙 [1904.05874] Dark Energy

𝐴 ~ 𝐺 2,𝑋𝑋 𝑠 2 − 𝐺 4,𝑋𝑋 𝑠 2 𝑡+… 𝐺 2,𝑋𝑋 >0, − 𝐺 4,𝑋𝑋 ≳0 Positivity in Horndeski (𝑋=− 1 2 𝛻𝜙 2 ) ℒ= ℒ 𝐺𝑅 + 𝐺 4 𝑋 𝑅+ 𝐺 2 𝑋 + 𝐺 4,𝑋 (𝑋) (𝛻𝛻𝜙) 2 Part of ``Horndeski’’ class 𝜙 𝜙 𝐴 ~ 𝐺 2,𝑋𝑋 𝑠 2 − 𝐺 4,𝑋𝑋 𝑠 2 𝑡+… 𝐺 2,𝑋𝑋 >0, − 𝐺 4,𝑋𝑋 ≳0 𝜙 𝜙 ℎ 𝜇ν ℎ 𝜇ν 𝐴 ~ 𝐺 4,𝑋 𝑠 2 +… 𝐺 4,𝑋 >0 𝜙 𝜙 [1904.05874] Dark Energy

Our Assumptions [1904.05874] Dark Energy

Our Assumptions Flat space positivity continues to hold on Cosmological background [1904.05874] Dark Energy

Only 𝐺 2 (𝑋), 𝐺 4 (𝑋) Our Assumptions Flat space positivity continues to hold on Cosmological background Particular subset of Horndeski 3 2 Only 𝐺 2 (𝑋), 𝐺 4 (𝑋) 5 4 [1904.05874] Dark Energy

Only 𝐺 2 (𝑋), 𝐺 4 (𝑋) 𝐺 𝑛 𝑡 → 𝑐 𝑛 Ω DE (𝑡) Our Assumptions Flat space positivity continues to hold on Cosmological background Particular subset of Horndeski 3 2 Only 𝐺 2 (𝑋), 𝐺 4 (𝑋) 5 4 https://arxiv.org/pdf/1509.07816.pdf Particular parametrization 𝐺 𝑛 𝑡 → 𝑐 𝑛 Ω DE (𝑡) 𝑡 [1904.05874] Dark Energy

𝑐 𝐵 𝐺 2 𝑋 𝐺 4 (𝑋) 𝑐 𝑀 𝑐 𝑇 Dark Energy Parameter estimation with positivity Scott Melville 𝑐 𝐵 ‘Braiding’ (mixing of scalar and tensor) 𝐺 2 𝑋 𝐺 4 (𝑋) 𝑐 𝑀 Time-dependence of 𝑀 𝑃 (i.e. 𝐺 𝑁 ) 𝑐 𝑇 Gravitational wave speed [SM, Noller, 1904.xxxxx] Contours mark 1 and 2 condence intervals, computed using CMB, RSD, BAO and matter power spectrum measurements. [1904.05874] Dark Energy

𝑐 𝐵 𝑐 𝑀 𝑐 𝑇 Dark Energy Parameter estimation with positivity Scott Melville 𝑐 𝐵 ‘Braiding’ (mixing of scalar and tensor) 𝑐 𝑀 Time-dependence of 𝑀 𝑃 (i.e. 𝐺 𝑁 ) 𝑐 𝑇 Gravitational wave speed [SM, Noller, 1904.xxxxx] Contours mark 1 and 2 condence intervals, computed using CMB, RSD, BAO and matter power spectrum measurements. [1904.05874] Dark Energy

𝑐 𝐵 𝑐 𝑀 𝑐 𝑇 Dark Energy Parameter estimation with positivity Scott Melville 𝑐 𝐵 ‘Braiding’ (mixing of scalar and tensor) 𝑐 𝑀 Time-dependence of 𝑀 𝑃 (i.e. 𝐺 𝑁 ) 𝑐 𝑇 Gravitational wave speed [SM, Noller, 1904.xxxxx] Contours mark 1 and 2 condence intervals, computed using CMB, RSD, BAO and matter power spectrum measurements. [1904.05874] Dark Energy

Dark Energy Parameter estimation with positivity Scott Melville [SM, Noller, 1904.xxxxx] Contours mark 1 and 2 condence intervals, computed using CMB, RSD, BAO and matter power spectrum measurements. [1904.05874] Dark Energy

CMB (Planck 2015) BAO (SDSS/BOSS) RSD (BOSS/6dF) Matter (SDSS DR4 LRG) Parameter estimation with positivity Scott Melville CMB (Planck 2015) BAO (SDSS/BOSS) RSD (BOSS/6dF) Matter (SDSS DR4 LRG) [SM, Noller, 1904.xxxxx] Contours mark 1 and 2 condence intervals, computed using CMB, RSD, BAO and matter power spectrum measurements. [1904.05874] Dark Energy

CMB (Planck 2015) BAO (SDSS/BOSS) RSD (BOSS/6dF) Matter (SDSS DR4 LRG) Parameter estimation with positivity Scott Melville CMB (Planck 2015) BAO (SDSS/BOSS) RSD (BOSS/6dF) Matter (SDSS DR4 LRG) [SM, Noller, 1904.xxxxx] Contours mark 1 and 2 condence intervals, computed using CMB, RSD, BAO and matter power spectrum measurements. [1904.05874] Dark Energy

𝑐 𝑇 <0 CMB BAO RSD Matter Dark Energy Parameter estimation with positivity Scott Melville CMB BAO RSD Matter Prior I: 𝜙 ℎ 𝜇ν [SM, Noller, 1904.xxxxx] 𝑐 𝑇 <0 [1904.05874] Dark Energy

𝑐 𝑇 <0 𝑐 𝐵 <2 𝑐 𝑇 CMB BAO RSD Matter Dark Energy Parameter estimation with positivity Scott Melville CMB BAO RSD Matter Prior I: 𝜙 ℎ 𝜇ν 𝑐 𝑇 <0 Prior II: 𝜙 𝑐 𝐵 <2 𝑐 𝑇 [1904.05874] Dark Energy

𝑐 𝑇 <0 𝑐 𝐵 <2 𝑐 𝑇 CMB BAO RSD Matter Dark Energy Parameter estimation with positivity Scott Melville CMB BAO RSD Matter Prior I: 𝜙 ℎ 𝜇ν [SM, Noller, 1904.xxxxx] 𝑐 𝑇 <0 Prior II: Data is more constraining 𝜙 𝑐 𝐵 <2 𝑐 𝑇 [1904.05874] Dark Energy

Data more constraining Parameter estimation with positivity Scott Melville CMB BAO RSD Matter Prior I: 𝜙 ℎ 𝜇ν [SM, Noller, 1904.xxxxx] 𝑐 𝑇 <0 Positivity Data more constraining Prior II: ⇒ In future, more data will shrink these contours even further. 𝜙 𝑐 𝐵 <2 𝑐 𝑇 [1904.05874] Dark Energy

[1904.05874] Summary

𝐴 𝐸𝐹𝑇 𝑠 = 𝑐 0 + 𝑐 𝑠𝑠 𝑠 2 𝑀 4 + 𝑐 𝑠𝑠𝑡 𝑠 2 𝑡 𝑀 6 +… 𝑎 𝑎 𝐴 𝐸𝐹𝑇 𝑠 = 𝑐 0 + 𝑐 𝑠𝑠 𝑠 2 𝑀 4 + 𝑐 𝑠𝑠𝑡 𝑠 2 𝑡 𝑀 6 +… Unitary, causal, local 𝑐 𝑠𝑠 >0 , 𝑐 𝑠𝑠𝑡 ≳0 𝑏 𝑏 [1904.05874] Summary

𝐴 𝐸𝐹𝑇 𝑠 = 𝑐 0 + 𝑐 𝑠𝑠 𝑠 2 𝑀 4 + 𝑐 𝑠𝑠𝑡 𝑠 2 𝑡 𝑀 6 +… 𝑎 𝑎 𝐴 𝐸𝐹𝑇 𝑠 = 𝑐 0 + 𝑐 𝑠𝑠 𝑠 2 𝑀 4 + 𝑐 𝑠𝑠𝑡 𝑠 2 𝑡 𝑀 6 +… Unitary, causal, local 𝑐 𝑠𝑠 >0 , 𝑐 𝑠𝑠𝑡 ≳0 𝑏 𝑏 Dark Energy [1904.05874] [1904.05874] Summary

𝐴 𝐸𝐹𝑇 𝑠 = 𝑐 0 + 𝑐 𝑠𝑠 𝑠 2 𝑀 4 + 𝑐 𝑠𝑠𝑡 𝑠 2 𝑡 𝑀 6 +… 𝑎 𝑎 𝐴 𝐸𝐹𝑇 𝑠 = 𝑐 0 + 𝑐 𝑠𝑠 𝑠 2 𝑀 4 + 𝑐 𝑠𝑠𝑡 𝑠 2 𝑡 𝑀 6 +… Unitary, causal, local 𝑐 𝑠𝑠 >0 , 𝑐 𝑠𝑠𝑡 ≳0 𝑏 𝑏 Dark Energy Up next: [1904.05874] Improved positivity bounds Inflation Beyond Standard Model [1904.05874] Summary

Backup Slides Backup Slides

General UV IR UV Positivity Constraints

General UV 𝐴 EFT (𝑠) IR UV Positivity Constraints

General UV ??? 𝐴 EFT (𝑠) 𝐴 UV (𝑠) 𝑀 IR UV Positivity Constraints

General UV 𝑠 𝑀 𝐴 EFT (𝑠) 𝐴 UV (𝑠) IR UV Positivity Constraints

Causality 𝐴 EFT (𝑠) 𝐴 UV (𝑠) IR UV General UV 𝑀 𝐴 EFT (𝑠) 𝐴 UV (𝑠) IR UV Causality ⇒ 𝐴 𝑠 is analytic (up to known poles & branch cuts) Positivity Constraints

Causality 𝐴 EFT (𝑠) 𝐴 UV (𝑠) - - IR UV General UV 𝑀 𝐴 EFT (𝑠) 𝐴 UV (𝑠) - - −𝑡 𝑚 2 IR 3 𝑚 2 −𝑡 4 𝑚 2 UV Causality ⇒ 𝐴 𝑠 is analytic (up to known poles & branch cuts) Positivity Constraints

Causality - - General UV 𝑠 𝑀 𝐶 - - x −𝑡 𝑚 2 3 𝑚 2 −𝑡 4 𝑚 2 Causality ⇒ 𝐴 𝑠 is analytic (up to known poles & branch cuts) Positivity Constraints

= - - - - General UV (Causality) 𝑀 𝐶 𝑀 𝐶 Positivity Constraints 𝑠 −𝑡 x −𝑡 𝑚 2 3 𝑚 2 −𝑡 4 𝑚 2 = (Causality) 𝑠 𝑀 𝐶 - - x −𝑡 𝑚 2 3 𝑚 2 −𝑡 4 𝑚 2 Positivity Constraints

= - - - - General UV (Causality) 𝑀 𝐶 𝑀 Positivity Constraints 𝑠 −𝑡 𝑚 2 x −𝑡 𝑚 2 3 𝑚 2 −𝑡 4 𝑚 2 = (Causality) 𝑠 ∞ 𝑀 - - x −𝑡 𝑚 2 3 𝑚 2 −𝑡 4 𝑚 2 ∞ Positivity Constraints

𝐴 𝐸𝐹𝑇 (𝑠) = - - - General UV (Causality) 𝑀 𝐶 𝑀 Positivity Constraints 𝑚 2 4 𝑚 2 3 𝑚 2 −𝑡 −𝑡 𝐶 x General UV 𝐴 𝐸𝐹𝑇 (𝑠) = (Causality) 𝑠 ∞ 𝑀 - - x −𝑡 𝑚 2 3 𝑚 2 −𝑡 4 𝑚 2 ∞ Positivity Constraints

Poles+Im 𝐴 𝑈𝑉 (𝑠)+ 𝐴 𝑈𝑉 (∞) - 𝑠 𝑀 𝑚 2 4 𝑚 2 3 𝑚 2 −𝑡 −𝑡 𝐶 x General UV 𝐴 𝐸𝐹𝑇 (𝑠) = (Causality) 𝑠 ∞ 𝑀 Poles+Im 𝐴 𝑈𝑉 (𝑠)+ 𝐴 𝑈𝑉 (∞) - - x −𝑡 𝑚 2 3 𝑚 2 −𝑡 4 𝑚 2 ∞ Positivity Constraints

Poles+Im 𝐴 𝑈𝑉 (𝑠)+ 𝐴 𝑈𝑉 (∞) - 𝑠 𝑀 𝑚 2 4 𝑚 2 3 𝑚 2 −𝑡 −𝑡 𝐶 x General UV 𝐴 𝐸𝐹𝑇 (𝑠) = (Causality) 𝑠 ∞ 𝑀 Poles+Im 𝐴 𝑈𝑉 (𝑠)+ 𝐴 𝑈𝑉 (∞) - - x −𝑡 𝑚 2 3 𝑚 2 −𝑡 4 𝑚 2 Positive (Unitarity) ∞ Positivity Constraints

Poles+Im 𝐴 𝑈𝑉 (𝑠)+ 𝐴 𝑈𝑉 (∞) - 𝑠 𝑀 𝑚 2 4 𝑚 2 3 𝑚 2 −𝑡 −𝑡 𝐶 x General UV 𝐴 𝐸𝐹𝑇 (𝑠) = (Causality) 𝑠 ∞ 𝑀 Poles+Im 𝐴 𝑈𝑉 (𝑠)+ 𝐴 𝑈𝑉 (∞) - - x −𝑡 𝑚 2 3 𝑚 2 −𝑡 4 𝑚 2 Positive < 𝑠 2 (Unitarity) (Locality) ∞ Positivity Constraints

Poles+Im 𝐴 𝑈𝑉 (𝑠)+ 𝐴 𝑈𝑉 (∞) - 𝑠 𝑀 𝑚 2 4 𝑚 2 3 𝑚 2 −𝑡 −𝑡 𝐶 x General UV 𝑐 0 + 𝑐 𝑠𝑠 𝑠 2 𝑀 4 +… 𝑐 𝑠𝑠 >0 = 𝐴 𝐸𝐹𝑇 (𝑠) = (Causality) 𝑠 ∞ 𝑀 Poles+Im 𝐴 𝑈𝑉 (𝑠)+ 𝐴 𝑈𝑉 (∞) - - x −𝑡 𝑚 2 3 𝑚 2 −𝑡 4 𝑚 2 Positive < 𝑠 2 (Unitarity) (Locality) ∞ Positivity Constraints

ℒ= ℒ 𝐺𝑅 + ℒ Scalar−Tensor Scalar-Tensor Theories ℒ= ℒ 𝐺𝑅 + ℒ Scalar−Tensor - Uniqueness? [1904.05874] Dark Energy

ℒ= ℒ 𝐺𝑅 + ℒ Scalar−Tensor Scalar-Tensor Theories ℒ= ℒ 𝐺𝑅 + ℒ Scalar−Tensor - Uniqueness? - New Experimental Tests [1904.05874] Dark Energy

ℒ= ℒ 𝐺𝑅 + ℒ Scalar−Tensor Scalar-Tensor Theories ℒ= ℒ 𝐺𝑅 + ℒ Scalar−Tensor - Uniqueness? - New Experimental Tests - Naturalness Arguments [1904.05874] Dark Energy

ℒ= ℒ 𝐺𝑅 + ℒ Scalar−Tensor Scalar-Tensor Theories ℒ= ℒ 𝐺𝑅 + ℒ Scalar−Tensor - Uniqueness? - New Experimental Tests - Naturalness Arguments - Possible UV Completions [1904.05874] Dark Energy