Modular invariant Starobinsky inflation and the Species Scale

A bstract Potentials in cosmological inflation often involve scalars with trans-Planckian ranges. As a result, towers of states become massless and their presence pushes the fundamental scale not to coincide with M P but rather with the species scale , Λ. This scale transforms as an automorphic form of the theory’s duality symmetries. We propose that the inflaton potential should be 1) an automorphic invariant form, non-singular over all moduli space, 2) depending only on Λ and its field derivatives, and 3) approaching constant values in the region of large moduli VEVs to ensure a long period of inflation. These conditions lead to the proposal V ~ λ ( ϕ, ϕ * ), with $$ \lambda ={G}^{i\overline{j}}\left({\partial}_i\Lambda \right)\left({\partial}_{\overline{j}}\Lambda \right)/{\Lambda}^2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>G</mml:mi> <mml:mrow> <mml:mi>i</mml:mi> <mml:mover> <mml:mi>j</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> </mml:msup> <mml:mfenced> <mml:mrow> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mi>Λ</mml:mi> </mml:mrow> </mml:mfenced> <mml:mfenced> <mml:mrow> <mml:msub> <mml:mi>∂</mml:mi> <mml:mover> <mml:mi>j</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:msub> <mml:mi>Λ</mml:mi> </mml:mrow> </mml:mfenced> <mml:mo>/</mml:mo> <mml:msup> <mml:mi>Λ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> , determining the ‘species scale convex hull’. For a single elliptic complex modulus with SL(2 , Z ) symmetry, this results in an inflaton potential $$ V\simeq {\left(\operatorname{Im}\tau \right)}^2{\left|{\overset{\sim }{G}}_2\right|}^2/{N}^2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> <mml:mo>≃</mml:mo> <mml:msup> <mml:mfenced> <mml:mrow> <mml:mo>Im</mml:mo> <mml:mi>τ</mml:mi> </mml:mrow> </mml:mfenced> <mml:mn>2</mml:mn> </mml:msup> <mml:msup> <mml:mfenced> <mml:msub> <mml:mover> <mml:mi>G</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mn>2</mml:mn> </mml:msub> </mml:mfenced> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>/</mml:mo> <mml:msup> <mml:mi>N</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> , with N ≃ − log(Im τ | η ( τ )| 4 ), where η is the Dedekind function and $$ {\overset{\sim }{G}}_2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mi>G</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mn>2</mml:mn> </mml:msub> </mml:math> the Eisenstein modular form of weight 2. Surprisingly, this potential at large modulus VEV resembles that of the Starobinsky model. We compute inflationary parameters yielding results similar to Starobinsky’s, but extended to modular invariant expressions. Interestingly, the number of e-folds is proportional to the number of species in the tower, N e ≃ N , and ϵ ≃ Λ 4 at large moduli VEV, suggesting that the tower of states plays an important role in the inflation process.