A Cosmic Microscope for the Preheating Era

Light fields with spatially varying backgrounds can modulate cosmic preheating, and imprint the nonlinear effects of preheating dynamics at tiny scales on large scale fluctuations. This provides us a unique probe into the preheating era which we dub the"cosmic microscope.'' We identify a distinctive effect of preheating on scalar perturbations that turns the Gaussian primordial fluctuations of a light scalar field into square waves, like a diode. The effect manifests itself as local non-Gaussianity. We present a model,"modulated partial preheating,"where this nonlinear effect is consistent with current observations and can be reached by near future cosmic probes.


Introduction
It is widely held that a period of cosmic inflation has happened in the early universe prior to the thermal big-bang phase, but it is nontrivial to connect inflation with the thermal big-bang. The transition is typically realized through reheating. Reheating is an abrupt and usually violent transition period, during which the inflaton energy is transferred to the thermal energy of other particles. 1 The thermal particles can be produced either through perturbative decays of the inflaton [1][2][3], or through various nonperturbative and outof-equilibrium dynamics [4][5][6][7][8][9][10][11]. The latter possibility is often called preheating, since it usually happens much faster and thus earlier than the perturbative decays.
The preheating and reheating era dwells in a special corner of our knowledge about the cosmic history. On the one hand, it contains rich and complex dynamics that appeals to detailed study. On the other hand, the study of this era is plagued by a lack of observable JHEP01(2021)021 (I) inflation (II) preheating (III) after preheating χ k (x) < l a t e x i t s h a 1 _ b a s e 6 4 = " p I n w 7 N O O D 3 7 2 o j 9 K Y Y h W 8 e t y 6 J w = " > A A A K 6 n i c n V Z b a 9 x G F F b S W 6 L e k v a x L 6 I b g w O b z c q 9 J C + G g E 1 J o K H u Y i c B a z E z 0 k g a d i 7 K z C j e 9 a A / 0 Z d S + t J C / 0 t / Q / 9 N z + i 2 K 9 k P p Y J d 5 n z n f G f O b U b C B a P a z O f / 3 L r 9 3 v s f f P j R n b v + x 5 9 8 + t n n 9 + 5 / 8 U r L U s X k L J Z M q j c Y a c K o I G e G G k b e F I o g j h l 5 j V d H T v / 6 H V G a S n F q N g V Z c p Q J m t I Y G Y C W U Z z T i 9 V + x H G w f n h x b z K f z e s n u L 4 I 2 8 X E a 5 + T i / t 3 / 4 4 S G Z e c C B M z p P V 5 O C / M 0 i J l a M x I 5 U e l J g W K V y g j 5

A U i B O 9 t H X U V b A H S B K k U s F P m K B G / S H F p E + X l o q i N E T E D S M t W W B k 4 H I J E q p I b N g G F i h W F D Y N 4 h w p F B v I e O D K I q d L n o Q k + X I 5 E q n I x R s n e I a q D c c V / e 2 E F h + H q Y K i c J L f n S 1 v 3 Q J B 5 6 w n w o x 9 S Q k d u U b A Q v H q H S y K F n I R V H j I D v E v 6 H f s j a I K X k 5 S g b D Y 3 J S T I E q Z Y J M m T o 3 W Y K F T m N h 9 n Y d d O v Q R x Y y h V H a q V F y T F R J D k 0 q i T T H p Y F E T
W 0 t D l 0 S i m S V n 4 Q 1 W t N T F k 0 Q w B T u 9 L T P T / Y f R x Y a w / P V Y a X d j 6 d z 7 6 B 3 / f V N B i Z u t L 9 R 9 N S s Z s t h 4 n l E H 5 G x e H s C R X T d 6 0 Q z k I q l j Y j k h O j N q N S i g T G P 6 V K G 1 C 4 c 6 j d C U w s s F x b m w p d U p N T Y c n b s j 5 4 l Y W x q B d + J M h l L D l H I r H R D 4 v q / G B p o 4 T q g q G N N h t G o h Q m 2 k Z T O w m r a F r Z y U E 1 Y q W q Y X E s 1 / b B D f Y P x g w I I B K y C Q 1 U i u w q F Q d t f T T 4 N Z 3 J S S q l E R J m 1 k Z w 1 j C N b Y 9 U c O j a I d I F O L A 2 p d l j B w + 3 h 4 E B d t 0 P 6 9 a j 6 D A r S a + v h Z E B 7 L H p D W p h Z J C z T t 2 0 O 5 z O p 7 P v 6 v g S k k Y Y + h Z h k u 1 2 p G p U J A E V A R 9 j B c 5 Q x 0 G M Z q I J u 6 G g h j L C c V Z 0 j A L K q e h 6 y y g a x g j H m e 4 Y u s R d B H p L 0 w 1 t r N x N X a o E e n o e L p s e x o j Z n 6 p 9 N w w P W z c J t V 1 / L Q T b o s d b 8 H g L d q b O T Q 0 3 + O k W P e 1 s O W 5 A n H a A b g A d q w 5 J G w Q m d N V B c e + q R W L W B x K z z n k B m R X u v d K b s c T 2 7 u 2 P f c T 5 L v y 8 h 6 8 A x U j Z q x 5 h 0 D S G R A b v q V p W I K t G b g y 0 s 9 j e n n V o U G + z j c 5 J M + c P r j R G R G b y q B S 0 X d q Q 8 5 Z m V M + J p k b B o W w V p 7 u K 0 x 3 F M T E 7 G p C 2 K i M h T p r l z W X f O Q J w M Q Y p 7 X 3 Q v n P Q M f g V O e p G r j 4 J p h P d k G c I R q m b F V A n h P V 6 4 g a 3 0 J T V V 5 d D V s B Y o a J A f V 2 5 K y z H S Y e o H C L O Z V d W i E r T r N / h C n a 4 q g N o I z x q x w j b o y 7 n R Q 8 t + k R e F v b l B V x I 8 N q z J 6 y q e v 6 C 9 G k v i L s B G / g F 7 + E X v I b b g i Q 7 l U Z J X e j x f S V x e 6 B W R I n g g J d B J O G r y l 3 2 H f j I o Z M w 2 B G r r f 3 o j F 6 a a / 4 u a U I M Z c n / d J j f 6 D B H p s U c f R J u 1 w N X v g + f f u H 4 Q + / 6 4 t X B L J z P w p + / n T x 7 2 n 4 E 3 v G + 8 r 7 2 9 r 3 Q e + I 9 8 5 5 7 J 9 6 Z F 3 t v v V + 9 P 7 w / f e b / 4 v / m / 9 6 Y 3 r 7 V c r 7 0 B o / / 1 7 9 X k / q u < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " p I n w 7 N O O D 3 7 2 o j 9 K Y Y h W 8 e t y 6 J w = " > A A A K 6 n i c n V Z b a 9 x G F F b S W 6 L e k v a x L 6 I b g w O b z c q 9 J C + G g E 1 J o K H u Y i c B a z E z 0 k g a d i 7 K z C j e 9 a A / 0 Z d S + t J C / 0 t / Q / 9 N z + i 2 K 9 k P p Y J d 5 n z n f G f O b U b C B a P a z O f / 3 L r 9 3 v s f f P j R n b v + x 5 9 8 + t n n 9 + 5 / 8 U r L U s X k L J Z M q j c Y a c K o I G e G G k b e F I o g j h l 5 j V d H T v / 6 H V G a S n F q N g V Z c p Q J m t I Y G Y C W U Z z T i 9 V + x H G w f n h x b z K f z e s n u L 4 I 2 8 X E a 5 + T i / t 3 / 4 4 S G Z e c C B M z p P V 5 O C / M 0 i J l a M x I 5 U e l J g W K V y g j 5

A U i B O 9 t H X U V b A H S B K k U s F P m K B G / S H F p E + X l o q i N E T E D S M t W W B k 4 H I J E q p I b N g G F i h W F D Y N 4 h w p F B v I e O D K I q d L n o Q k + X I 5 E q n I x R s n e I a q D c c V / e 2 E F h + H q Y K i c J L f n S 1 v 3 Q J B 5 6 w n w o x 9 S Q k d u U b A Q v H q H S y K F n I R V H j I D v E v 6 H f s j a I K X k 5 S g b D Y 3 J S T I E q Z Y J M m T o 3 W Y K F T m N h 9 n Y d d O v Q R x Y y h V H a q V F y T F R J D k 0 q i T T H p Y F E T
W 0 t D l 0 S i m S V n 4 Q 1 W t N T F k 0 Q w B T u 9 L T P T / Y f R x Y a w / P V Y a X d j 6 d z 7 6 B 3 / f V N B i Z u t L 9 R 9 N S s Z s t h 4 n l E H 5 G x e H s C R X T d 6 0 Q z k I q l j Y j k h O j N q N S i g T G P 6 V K G 1 C 4 c 6 j d C U w s s F x b m w p d U p N T Y c n b s j 5 4 l Y W x q B d + J M h l L D l H I r H R D 4 v q / G B p o 4 T q g q G N N h t G o h Q m 2 k Z T O w m r a F r Z y U E 1 Y q W q Y X E s 1 / b B D f Y P x g w I I B K y C Q 1 U i u w q F Q d t f T T 4 N Z 3 J S S q l E R J m 1 k Z w 1 j C N b Y 9 U c O j a I d I F O L A 2 p d l j B w + 3 h 4 E B d t 0 P 6 9 a j 6 D A r S a + v h Z E B 7 L H p D W p h Z J C z T t 2 0 O 5 z O p 7 P v 6 v g S k k Y Y + h Z h k u 1 2 p G p U J A E V A R 9 j B c 5 Q x 0 G M Z q I J u 6 G g h j L C c V Z 0 j A L K q e h 6 y y g a x g j H m e 4 Y u s R d B H p L 0 w 1 t r N x N X a o E e n o e L p s e x o j Z n 6 p 9 N w w P W z c J t V 1 / L Q T b o s d b 8 H g L d q b O T Q 0 3 + O k W P e 1 s O W 5 A n H a A b g A d q w 5 J G w Q m d N V B c e + q R W L W B x K z z n k B m R X u v d K b s c T 2 7 u 2 P f c T 5 L v y 8 h 6 8 A x U j Z q x 5 h 0 D S G R A b v q V p W I K t G b g y 0 s 9 j e n n V o U G + z j c 5 J M + c P r j R G R G b y q B S 0 X d q Q 8 5 Z m V M + J p k b B o W w V p 7 u K 0 x 3 F M T E 7 G p C 2 K i M h T p r l z W X f O Q J w M Q Y p 7 X 3 Q v n P Q M f g V O e p G r j 4 J p h P d k G c I R q m b F V A n h P V 6 4 g a 3 0 J T V V 5 d D V s B Y o a J A f V 2 5 K y z H S Y e o H C L O Z V d W i E r T r N / h C n a 4 q g N o I z x q x w j b o y 7 n R Q 8 t + k R e F v b l B V x I 8 N q z J 6 y q e v 6 C 9 G k v i L s B G / g F 7 + E X v I b b g i Q 7 l U Z J X e j x f S V x e 6 B W R I n g g J d B J O G r y l 3 2 H f j I o Z M w 2 B G r r f 3 o j F 6 a a / 4 u a U I M Z c n / d J j f 6 D B H p s U c f R J u 1 w N X v g + f f u H 4 Q + / 6 4 t X B L J z P w p + / n T x 7 2 n 4 E 3 v G + 8 r 7 2 9 r 3 Q e + I 9 8 5 5 7 J 9 6 Z F 3 t v v V + 9 P 7 w / f e b / 4 v / m / 9 6 Y 3 r 7 V c r 7 0 B o / / 1 7 9 X k / q u < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " p I n w 7 N O O D 3 7 2 o j 9 K Y Y h W 8 e t y 6 J w = " > A A A K 6 n i c n V Z b a 9 x G F F b S W 6 L e k v a x L 6 I b g w O b z c q 9 J C + G g E 1 J o K H u Y i c B a z E z 0 k g a d i 7 K z C j e 9 a A / 0 Z d S + t J C / 0 t / Q / 9 N z + i 2 K 9 k P p Y J d 5 n z n f G f O b U b C B a P a z O f / 3 L r 9 3 v s f f P j R n b v + x 5 9 8 + t n n 9 + 5 / 8 U r L U s X k L J Z M q j c Y a c K o I G e G G k b e F I o g j h l 5 j V d H T v / 6 H V G a S n F q N g V Z c p Q J m t I Y G Y C W U Z z T i 9 V + x H G w f n h x b z K f z e s n u L 4 I 2 8 X E a 5 + T i / t 3 / 4 4 S G Z e c C B M z p P V 5 O C / M 0 i J l a M x I 5 U e l J g W K V y g j 5

A U i B O 9 t H X U V b A H S B K k U s F P m K B G / S H F p E + X l o q i N E T E D S M t W W B k 4 H I J E q p I b N g G F i h W F D Y N 4 h w p F B v I e O D K I q d L n o Q k + X I 5 E q n I x R s n e I a q D c c V / e 2 E F h + H q Y K i c J L f n S 1 v 3 Q J B 5 6 w n w o x 9 S Q k d u U b A Q v H q H S y K F n I R V H j I D v E v 6 H f s j a I K X k 5 S g b D Y 3 J S T I E q Z Y J M m T o 3 W Y K F T m N h 9 n Y d d O v Q R x Y y h V H a q V F y T F R J D k 0 q i T T H p Y F E T
W 0 t D l 0 S i m S V n 4 Q 1 W t N T F k 0 Q w B T u 9 L T P T / Y f R x Y a w / P V Y a X d j 6 d z 7 6 B 3 / f V N B i Z u t L 9 R 9 N S s Z s t h 4 n l E H 5 G x e H s C R X T d 6 0 Q z k I q l j Y j k h O j N q N S i g T G P 6 V K G 1 C 4 c 6 j d C U w s s F x b m w p d U p N T Y c n b s j 5 4 l Y W x q B d + J M h l L D l H I r H R D 4 v q / G B p o 4 T q g q G N N h t G o h Q m 2 k Z T O w m r a F r Z y U E 1 Y q W q Y X E s 1 / b B D f Y P x g w I I B K y C Q 1 U i u w q F Q d t f T T 4 N Z 3 J S S q l E R J m 1 k Z w 1 j C N b Y 9 U c O j a I d I F O L A 2 p d l j B w + 3 h 4 E B d t 0 P 6 9 a j 6 D A r S a + v h Z E B 7 L H p D W p h Z J C z T t 2 0 O 5 z O p 7 P v 6 v g S k k Y Y + h Z h k u 1 2 p G p U J A E V A R 9 j B c 5 Q x 0 G M Z q I J u 6 G g h j L C c V Z 0 j A L K q e h 6 y y g a x g j H m e 4 Y u s R d B H p L 0 w 1 t r N x N X a o E e n o e L p s e x o j Z n 6 p 9 N w w P W z c J t V 1 / L Q T b o s d b 8 H g L d q b O T Q 0 3 + O k W P e 1 s O W 5 A n H a A b g A d q w 5 J G w Q m d N V B c e + q R W L W B x K z z n k B m R X u v d K b s c T 2 7 u 2 P f c T 5 L v y 8 h 6 8 A x U j Z q x 5 h 0 D S G R A b v q V p W I K t G b g y 0 s 9 j e n n V o U G + z j c 5 J M + c P r j R G R G b y q B S 0 X d q Q 8 5 Z m V M + J p k b B o W w V p 7 u K 0 x 3 F M T E 7 G p C 2 K i M h T p r l z W X f O Q J w M Q Y p 7 X 3 Q v n P Q M f g V O e p G r j 4 J p h P d k G c I R q m b F V A n h P V 6 4 g a 3 0 J T V V 5 d D V s B Y o a J A f V 2 5 K y z H S Y e o H C L O Z V d W i E r T r N / h C n a 4 q g N o I z x q x w j b o y 7 n R Q 8 t + k R e F v b l B V x I 8 N q z J 6 y q e v 6 C 9 G k v i L s B G / g F 7 + E X v I b b g i Q 7 l U Z J X e j x f S V x e 6 B W R I n g g J d B J O G r y l 3 2 H f j I o Z M w 2 B G r r f 3 o j F 6 a a / 4 u a U I M Z c n / d J j f 6 D B H p s U c f R J u 1 w N X v g + f f u H 4 Q + / 6 4 t X B L J z P w p + / n T x 7 2 n 4 E 3 v G + 8 r 7 2 9 r 3 Q e + I 9 8 5 5 7 J 9 6 Z F 3 t v v V + 9 P 7 w / f e b / 4 v / m / 9 6 Y 3 r 7 V c r 7 0 B o / / 1 7 9 X k / q u < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " p I n w 7 N O O D 3 7 2 o j 9 K Y Y h W 8 e t y 6 J w = " > A A A K 6 n i c n V Z b a 9 x G F F b S W 6 L e k v a x L 6 I b g w O b z c q 9 J C + G g E 1 J o K H u Y i c B a z E z 0 k g a d i 7 K z C j e 9 a A / 0 Z d S + t J C / 0 t / Q / 9 N z + i 2 K 9 k P p Y J d 5 n z n f G f O b U b C B a P a z O f / 3 L r 9 3 v s f f P j R n b v + x 5 9 8 + t n n 9 + 5 / 8 U r L U s X k L J Z M q j c Y a c K o I G e G G k b e F I o g j h l 5 j V d H T v / 6 H V G a S n F q N g V Z c p Q J m t I Y G Y C W U Z z T i 9 V + x H G w f n h x b z K f z e s n u L 4 I 2 8 X E a 5 + T i / t 3 / 4 4 S G Z e c C B M z p P V 5 O C / M 0 i J l a M x I 5 U e l J g W K V y g j 5

A U i B O 9 t H X U V b A H S B K k U s F P m K B G / S H F p E + X l o q i N E T E D S M t W W B k 4 H I J E q p I b N g G F i h W F D Y N 4 h w p F B v I e O D K I q d L n o Q k + X I 5 E q n I x R s n e I a q D c c V / e 2 E F h + H q Y K i c J L f n S 1 v 3 Q J B 5 6 w n w o x 9 S Q k d u U b A Q v H q H S y K F n I R V H j I D v E v 6 H f s j a I K X k 5 S g b D Y 3 J S T I E q Z Y J M m T o 3 W Y K F T m N h 9 n Y d d O v Q R x Y y h V H a q V F y T F R J D k 0 q i T T H p Y F E T
W 0 t D l 0 S i m S V n 4 Q 1 W t N T F k 0 Q w B T u 9 L T P T / Y f R x Y a w / P V Y a X d j 6 d z 7 6 B 3 / f V N B i Z u t L 9 R 9 N S s Z s t h 4 n l E H 5 G x e H s C R X T d 6 0 Q z k I q l j Y j k h O j N q N S i g T G P 6 V K G 1 C 4 c 6 j d C U w s s F x b m w p d U p N T Y c n b s j 5 4 l Y W x q B d + J M h l L D l H I r H R D 4 v q / G B p o 4 T q g q G N N h t G o h Q m 2 k Z T O w m r a F r Z y U E 1 Y q W q Y X E s 1 / b B D f Y P x g w I I B K y C Q 1 U i u w q F Q d t f T T 4 N Z 3 J S S q l E R J m 1 k Z w 1 j C N b Y 9 U c O j a I d I F O L A 2 p d l j B w + 3 h 4 E B d t 0 P 6 9 a j 6 D A r S a + v h Z E B 7 L H p D W p h Z J C z T t 2 0 O 5 z O p 7 P v 6 v g S k k Y Y + h Z h k u 1 2 p G p U J A E V A R 9 j B c 5 Q x 0 G M Z q I J u 6 G g h j L C c V Z 0 j A L K q e h 6 y y g a x g j H m e 4 Y u s R d B H p L 0 w 1 t r N x N X a o E e n o e L p s e x o j Z n 6 p 9 N w w P W z c J t V 1 / L Q T b o s d b 8 H g L d q b O T Q 0 3 + O k W P e 1 s O W 5 A n H a A b g A d q w 5 J G w Q m d N V B c e + q R W L W B x K z z n k B m R X u v d K b s c T 2 7 u 2 P f c T 5 L v y 8 h 6 8 A x U j Z q x 5 h 0 D S G R A b v q V p W I K t G b g y 0 s 9 j e n n V o U G + z j c 5 J M + c P r j R G R G b y q B S 0 X d q Q 8 5 C n a 4 q g N o I z x q x w j b o y 7 n R Q 8 t + k R e F v b l B V x I 8 N q z J 6 y q e v 6 C 9 G k v i L s B G / g F 7 + E X v I b b g i Q 7 l U Z J X e j x f S V x e 6 B W R I n g g J d B J O G r y l 3 2 H f j I o Z M w 2 B G r r f 3 o j F 6 a a / 4 u a U I M Z c n / d J j f 6 D B H p s U c f R J u 1 w N X v g + f f u H 4 Q + / 6 4 t X B L J z P w p + / n T x 7 2 n 4 E 3 v G + 8 r 7 2 9 r 3 Q e + I 9 8 5 5 7 J 9 6 Z F 3 t v v V + 9 P 7 w / f e b / 4 v / m / 9 6 Y 3 r 7 V c r 7 0 B o / / 1 7 9 X k / q u < / l a t e x i t > ∆ζ k (x) < l a t e x i t s h a 1 _ b a s e 6 4 = " H N Q Z k U r Q + w n a j L f Z n W 4 p n Z V I j S E = " > A A A K 8 3 i c n V Z b a 9 x G F F b S W 6 L e n L Z v f R H d G B z Y b F b u J X k x B G x K A g 1 1 j X M B a z E z 0 k g a d i 7 q z C j e 3 U G / p C + l 9 K W F / o 7 + h v 6 b n t F t V 7 I f S g W 7 z P n O O d + c 2 4 y E C 0 a 1 m c / / u X X 7 n X f f e / + D O 3 f 9 D z / 6 + J N P 9 + 5 9 9 k r L U s X k Z S y Z w A d w n / Q x 6 y M k g p e T X K R k N j c p I M Q a p l g g w Z s t t M o S K n 8 T A b u 2 r 6 N Y g D S 7 n k S C 2 1 K D k m i i R H R p V k 2 s O y I K K G F j a H T i l F 0 s o P o n q t i S m L Z g h g f J d 6 u u 8 H u 4 8 D a + 3 R h c r w w s 6 n 8 9 n X 8 P u u m g Y j U 1 e 6 / 2 h a K n a z 5 T C x H M L P q D i a P a Z i + r Y V w l l I x c J m R H h A z t s 6 g D a C I / b M c L 2 u M v 5 r I f O + k R e F P b F J V x I 8 N q z p 6 y q e v 8 z 0 q d 9 R t w N 2 M D P e Q 8 / 5 z X c F i T Z q T R K 6 k K P 7 y u J 2 w O 1 J E o E h 7 w M I g m f V + 6 y 7 8 C H D p 2 E w Y 5 Y b e 1 H Z / T K X O O 7 o g k x l C X / k z C / k T B H p s W c + y T c r g d U v g + f f u H 4 Q + / 6 4 t X h L J z P w p + + m T x 9 0 n 4 E 3 v G + 9 L 7 y D r z Q e + w 9 9 Z 5 5 p 9 5 L L / Y 2 3 q / e H 9 6 f f u n / 4 v / m / 9 6 Y 3 r 7 V + n z u D R 7 / r 3 8 B Y z b 9 6 w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " H N Q Z k U r Q + w n a j L f Z n W 4 p n Z V I j S E = " > A A A K 8 3 i c n V Z b a 9 x G F F b S W 6 L e n L Z v f R H d G B z Y b F b u J X k x B G x K A g 1 1 j X M B a z E z 0 k g a d i 7 q z C j e 3 U G / p C + l 9 K W F / o 7 + h v 6 b n t F t V 7 I f S g W 7 z P n O O d + c 2 4 y E C 0 a 1 m c / / u X X 7 n X f f e / + D O 3 f 9 D z / 6 + J N P 9 + 5 9 9 k r L U s X k Z S y Z w A d w n / Q x 6 y M k g p e T X K R k N j c p I M Q a p l g g w Z s t t M o S K n 8 T A b u 2 r 6 N Y g D S 7 n k S C 2 1 K D k m i i R H R p V k 2 s O y I K K G F j a H T i l F 0 s o P o n q t i S m L Z g h g f J d 6 u u 8 H u 4 8 D a + 3 R h c r w w s 6 n 8 9 n X 8 P u u m g Y j U 1 e 6 / 2 h a K n a z 5 T C x H M L P q D i a P a Z i + r Y V w l l I x c J m R H h A z t s 6 g D a C I / b M c L 2 u M v 5 r I f O + k R e F P b F J V x I 8 N q z p 6 y q e v 8 z 0 q d 9 R t w N 2 M D P e Q 8 / 5 z X c F i T Z q T R K 6 k K P 7 y u J 2 w O 1 J E o E h 7 w M I g m f V + 6 y 7 8 C H D p 2 E w Y 5 Y b e 1 H Z / T K X O O 7 o g k x l C X / k z C / k T B H p s W c + y T c r g d U v g + f f u H 4 Q + / 6 4 t X h L J z P w p + + m T x 9 0 n 4 E 3 v G + 9 L 7 y D r z Q e + w 9 9 Z 5 5 p 9 5 L L / Y 2 3 q / e H 9 6 f f u n / 4 v / m / 9 6 Y 3 r 7 V + n z u D R 7 / r 3 8 B Y z b 9 6 w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " H N Q Z k U r Q + w n a j L f Z n W 4 p n Z V I j S E = " > A A A K 8 3 i c n V Z b a 9 x G F F b S W 6 L e n L Z v f R H d G B z Y b F b u J X k x B G x K A g 1 1 j X M B a z E z 0 k g a d i 7 q z C j e 3 U G / p C + l 9 K W F / o 7 + h v 6 b n t F t V 7 I f S g W 7 z P n O O d + c 2 4 y E C 0 a 1 m c / / u X X 7 n X f f e / + D O 3 f 9 D z / 6 + J N P 9 + 5 9 9 k r L U s X k Z S y Z w A d w n / Q x 6 y M k g p e T X K R k N j c p I M Q a p l g g w Z s t t M o S K n 8 T A b u 2 r 6 N Y g D S 7 n k S C 2 1 K D k m i i R H R p V k 2 s O y I K K G F j a H T i l F 0 s o P o n q t i S m L Z g h g f J d 6 u u 8 H u 4 8 D a + 3 R h c r w w s 6 n 8 9 n X 8 P u u m g Y j U 1 e 6 / 2 h a K n a z 5 T C x H M L P q D i a P a Z i + r Y V w l l I x c J m R H h A z t s 6 g D a C I / b M c L 2 u M v 5 r I f O + k R e F P b F J V x I 8 N q z p 6 y q e v 8 z 0 q d 9 R t w N 2 M D P e Q 8 / 5 z X c F i T Z q T R K 6 k K P 7 y u J 2 w O 1 J E o E h 7 w M I g m f V + 6 y 7 8 C H D p 2 E w Y 5 Y b e 1 H Z / T K X O O 7 o g k x l C X / k z C / k T B H p s W c + y T c r g d U v g + f f u H 4 Q + / 6 4 t X h L J z P w p + + m T x 9 0 n 4 E 3 v G + 9 L 7 y D r z Q e + w 9 9 Z 5 5 p 9 5 L L / Y 2 3 q / e H 9 6 f f u n / 4 v / m / 9 6 Y 3 r 7 V + n z u D R 7 / r 3 8 B Y z b 9 6 w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " H N Q Z k U r Q + w n a j L f Z n W 4 p n Z V I j S E = " > A A A K 8 3 i c n V Z b a 9 x G F F b S W 6 L e n L Z v f R H d G B z Y b F b u J X k x B G x K A g 1 1 j X M B a z E z 0 k g a d i 7 q z C j e 3 U G / p C + l 9 K W F / o 7 + h v 6 b n t F t V 7 I f S g W 7 z P n O O d + c 2 4 y E C 0 a 1 m c / / u X X 7 n X f f e / + D O 3 f 9 D z / 6 + J N P 9 + 5 9 9 k r L U s X k Z S y Z w A d w n / Q x 6 y M k g p e T X K R k N j c p I M Q a p l g g w Z s t t M o S K n 8 T A b u 2 r 6 N Y g D S 7 n k S C 2 1 K D k m i i R H R p V k 2 s O y I K K G F j a H T i l F 0 s o P o n q t i S m L Z g h g f J d 6 u u 8 H u 4 8 D a + 3 R h c r w w s 6 n 8 9 n X 8 P u u m g Y j U 1 e 6 / 2 h a K n a z 5 T C x H M L P q D i a P a Z i + r Y V w l l I x c J m R H J i 1 H p U S p H A + K d U a Q M K d y C 1 O 4 q J B S / X 1 q Z C V 9 T k V F j y c 1 m f w M r C W N Q L P x L k K p a c I 5 H Y 6 P u z 6 u J w Y a O E 6 o K h t T Z r R q I U J t p G U z s J q 2 h a 2 c l h N f J K V e P F s V z Z + z f Y 3 k x l C X / k z C / k T B H p s W c + y T c r g d U v g + f f u H 4 Q + / 6 4 t X h L J z P w p + + m T x 9 0 n 4 E 3 v G + 9 L 7 y D r z Q e + w 9 9 Z 5 5 p 9 5 L L / Y 2 3 q / e H 9 6 f f u n / 4 v / m / 9 6 Y 3 r 7 V + n z u D R 7 / r 3 8 B Y z b 9 6 w = = < / l a t e x i t > χ(x) < l a t e x i t s h a 1 _ b a s e 6 4 = " H 1 x 5 b F / r y z l e g b M o s S n p 0 9 9 V v o E = " > A A A K 6 H i c n V Z b a 9 x G F F b S W 6 L e k v a x L 6 I b g w O b z c q 9 J C + G g E 1 J o K H u Y i c B a z E z 0 k i a 7 l z U m V G 8 6 0 H / o S + l 9 K W F / p n + h v 6 b n t F t V 7 I f S g W 7 z P n O + c 6 c 2 4 y E C 0 a 1 m c / / u X X 7 n X f f e / + D O 3 f 9 D z / 6 + J N P 7 9 3 / 7 J W W p Y r J W S y Z V G 8 w 0 o R R Q c 4 M N Y y 8 K R R B H D P y G q + O n P 7 1 W 6 I 0 l e L U b A q y 5 C g T N K U x M g C d R 3 F O 9 y O O g / X D i 3 u T + W x e P 8 H 1 R d g u J l 7 7 n F z c v / t 3 l M i 4 5 E S Y m C G t z 8 N 5 Y Z Y W K U N j R i o / K j U p U L x C G T m H p U C c 6 K W t Y 6 6 C P U C S I J U K f s I E N e o P K S Z 9 u r R U F K U h I m 4 Y a c k C I w O X S Z B Q R W L D N r B A s a K w a R D n S K H Y Q L 4 D V x Z x 7 X b R g 5 g s R y Z X O h 2 h Y O s U 1 0 C 9 4 b j y 9 8 Y O C s P X w 1 Q 5 S W j J l 7 b u h i b x 0 B P m Q z m m h o z c p m Q j e P E I l U Y O P Q u p O G I E f J f w P / R D 1 g Y p J S 9 H 2 W h o T E 6 S I U i 1 T J A h Q + 8 2 U 6 j I a T z M x q 6 b f g 3 i w F K u O F I r L U q O i S L J o V E l m f a w L I i o o a X N o V N K k b T y g 6 h e a 2 L K o h k C m N m V n u 7 5 w e 7 j w F p 7 e K 4 y v L T z 6 X z 2 F f y + r a b B y N S V 7 j + a l o r d b D l M L I f w M y o O Z 0 + o m L 5 t h X A W U r G 0 G Z G c G L U Z l V I k M P 4 p V d q A w p 1 C 7 c 5 f Y o H l 2 t p U 6 J K a n A p L f i 7 r Y 1 d Z G I t 6 4 U e C X M a S c y Q S G 3 2 3 q M 4 P l j Z K q C 4 Y 2 m i z Y S R K Y a J t N L W T s I q m l Z 0 c V C N W q h o W x 3 J t H 9 x g / 2 D M g A A i I Z v Q Q K X I r l J x 0 N Z H g 1 / T m Z y k U h o h Y W Z t B G c N 0 9 j 2 S A W H r h 0 i X Y A D a 1 O a P X b w c H s Y G G D X / b B u P Y o O s 5 L 0 + l o Y G c A e m 9 6 g F k Y G O e v U T b v D 6 X w 6 + 6 a O L y F p h K F v E S b Z b k e q R k U S U B H w M V b g D H U c x G g m m r A b C m o o I x x n R c c o o J y K r r e M o m G M c J z p j q F L 3 E W g t z T d 0 M b K 3 d S l S q C n 5 + G y 6 W G M m P 2 h 2 n f D 8 L B 1 k 1 D b 9 d d C s C 1 6 v A W P t 2 B n 6 t z U c I O f b t H T z p b j B s R p B + g G 0 L H q k L R B Y E J X H R T 3 r l o k Z n 0 g M e u c F 5 B Z 4 d 4 r v R l L b O / e f t 9 H n O / C z 3 v 4 C l C M l L 3 q E Q Z N Y 0 h k 8 J 6 q Z Q W y a u T G Q D u L 7 e 1 Z h w b 1 N t v o n D R z / u B K Y 0 R k J o 9 K Q d u l D T l v a U b 1 n G h q F B z K V n G 6 q z j d U R w T s 6 M B a a s y E u K k W d 5 c 9 p 0 j A B d j k N L e B + 0 7 B x 2 D X 5 G j b u T q k 2 A 6 0 Q 1 5 h m C U u l k B d U J Y r y d u c A t N W X 1 1 O W Q F j B U q C t T X l b v C c p x 0 i M o h 4 l x 2 Z Y W o N M 3 6 H a 5 g h 6 s 6 g D b C o 3 a M s D 3 q c l 7 0 0 K J P 5 G V h X 1 7 A h Q S v P X v C q q r n L 0 i f 9 o K 4 G 7 C B X / A e f s F r u C 1 I s l N p l N S F H t 9 X E r c H a k W U C A 5 4 G U Q S v q n c Z d + B j x w 6 C Y M d s d r a j 8 7 o p b n m 7 5 I m x F C W / E + H + Y 0 O c 2 R a z N E n 4 X Y 9 c O X 7 8 O k X j j / 0 r i 9 e H c z C + S z 8 8 e v J s 6 f t R + A d 7 w v v S 2 / f C 7 0 n 3 j P v u X f i n X m x J 7 1 f v T + 8 P / 2 f / F / 8 3 / z f G 9 P b t 1 r O 5 9 7 g 8 f / 6 F 6 3 l + d A = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " H 1 x 5 b F / r y z l e g b M o s S n p 0 9 9 V v o E = " > A A A K 6 H i c n V Z b a 9 x G F F b S W 6 L e k v a x L 6 I b g w O b z c q 9 J C + G g E 1 J o K H u Y i c B a z E z 0 k i a 7 l z U m V G 8 6 0 H / o S + l 9 K W F / p n + h v 6 b n t F t V 7 I f S g W 7 z P n O + c 6 c 2 4 y E C 0 a 1 m c / / u X X 7 n X f f e / + D O 3 f 9 D z / 6 + J N P 7 9 3 / 7 J W W p Y r J W S y Z V G 8 w 0 o R R Q c 4 M N Y y 8 K R R B H D P y G q + O n P 7 1 W 6 I 0 l e L U b A q y 5 C g T N K U x M g C d R 3 F O 9 y O O g / X D i 3 u T + W x e P 8 H 1 R d g u J l 7 7 n F z c v / t 3 l M i 4 5 E S Y m C G t z 8 N 5 Y Z Y W K U N j R i o / K j U p U L x C G T m H p U C c 6 K W t Y 6 6 C P U C S I J U K f s I E N e o P K S Z 9 u r R U F K U h I m 4 Y a c k C I w O X S Z B Q R W L D N r B A s a K w a R D n S K H Y Q L 4 D V x Z x 7 X b R g 5 g s R y Z X O h 2 h Y O s U 1 0 C 9 4 b j y 9 8 Y O C s P X w 1 Q 5 S W j J l 7 b u h i b x 0 B P m Q z m m h o z c p m Q j e P E I l U Y O P Q u p O G I E f J f w P / R D 1 g Y p J S 9 H 2 W h o T E 6 S I U i 1 T J A h Q + 8 2 U 6 j I a T z M x q 6 b f g 3 i w F K u O F I r L U q O i S L J o V E l m f a w L I i o o a X N o V N K k b T y g 6 h e a 2 L K o h k C m N m V n u 7 5 w e 7 j w F p 7 e K 4 y v L T z 6 X z 2 F f y + r a b B y N S V 7 j + a l o r d b D l M L I f w M y o O Z 0 + o m L 5 t h X A W U r G 0 G Z G c G L U Z l V I k M P 4 p V d q A w p 1 C 7 c 5 f Y o H l 2 t p U 6 J K a n A p L f i 7 r Y 1 d Z G I t 6 4 U e C X M a S c y Q S G 3 2 3 q M 4 P l j Z K q C 4 Y 2 m i z Y S R K Y a J t N L W T s I q m l Z 0 c V C N W q h o W x 3 J t H 9 x g / 2 D M g A A i I Z v Q Q K X I r l J x 0 N Z H g 1 / T m Z y k U h o h Y W Z t B G c N 0 9 j 2 S A W H r h 0 i X Y A D a 1 O a P X b w c H s Y G G D X / b B u P Y o O s 5 L 0 + l o Y G c A e m 9 6 g F k Y G O e v U T b v D 6 X w 6 + 6 a O L y F p h K F v E S b Z b k e q R k U S U B H w M V b g D H U c x G g m m r A b C m o o I x x n R c c o o J y K r r e M o m G M c J z p j q F L 3 E W g t z T d 0 M b K 3 d S l S q C n 5 + G y 6 W G M m P 2 h 2 n f D 8 L B 1 k 1 D b 9 d d C s C 1 6 v A W P t 2 B n 6 t z U c I O f b t H T z p b j B s R p B + g G 0 L H q k L R B Y E J X H R T 3 r l o k Z n 0 g M e u c F 5 B Z 4 d 4 r v R l L b O / e f t 9 H n O / C z 3 v 4 C l C M l L 3 q E Q Z N Y 0 h k 8 J 6 q Z Q W y a u T G Q D u L 7 e 1 Z h w b 1 N t v o n D R z / u B K Y 0 R k J o 9 K Q d u l D T l v a U b 1 n G h q F B z K V n G 6 q z j d U R w T s 6 M B a a s y E u K k W d 5 c 9 p 0 j A B d j k N L e B + 0 7 B x 2 D X 5 G j b u T q k 2 A 6 0 Q 1 5 h m C U u l k B d U J Y r y d u c A t N W X 1 1 O W Q F j B U q C t T X l b v C c p x 0 i M o h 4 l x 2 Z Y W o N M 3 6 H a 5 g h 6 s 6 g D b C o 3 a M s D 3 q c l 7 0 0 K J P 5 G V h X 1 7 A h Q S v P X v C q q r n L 0 i f 9 o K 4 G 7 C B X / A e f s F r u C 1 I s l N p l N S F H t 9 X E r c H a k W U C A 5 4 G U Q S v q n c Z d + B j x w 6 C Y M d s d r a j 8 7 o p b n m 7 5 I m x F C W / E + H + Y 0 O c 2 R a z N E n 4 X Y 9 c O X 7 8 O k X j j / 0 r i 9 e H c z C + S z 8 8 e v J s 6 f t R + A d 7 w v v S 2 / f C 7 0 n 3 j P v u X f i n X m x J 7 1 f v T + 8 P / 2 f / F / 8 3 / z f G 9 P b t 1 r O 5 9 7 g 8 f / 6 F 6 3 l + d A = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " H 1 x 5 b F / r y z l e g b M o s S n p 0 9 9 V v o E = " > A A A K 6 H i c n V Z b a 9 x G F F b S W 6 L e k v a x L 6 I b g w O b z c q 9 J C + G g E 1 J o K H u Y i c B a z E z 0 k i a 7 l z U m V G 8 6 0 H / o S + l 9 K W F / p n + h v 6 b n t F t V 7 I f S g W 7 z P n O + c 6 c 2 4 y E C 0 a 1 m c / / u X X 7 n X f f e / + D O 3 f 9 D z / 6 + J N P 7 9 3 / 7 J W W p Y r J W S y Z V G 8 w 0 o R R Q c 4 M N Y y 8 K R R B H D P y G q + O n P 7 1 W 6 I 0 l e L U b A q y 5 C g T N K U x M g C d R 3 F O 9 y O O g / X D i 3 u T + W x e P 8 H 1 R d g u J l 7 7 n F z c v / t 3 l M i 4 5 E S Y m C G t z 8 N 5 Y Z Y W K U N j R i o / K j U p U L x C G T m H p U C c 6 K W t Y 6 6 C P U C S I J U K f s I E N e o P K S Z 9 u r R U F K U h I m 4 Y a c k C I w O X S Z B Q R W L D N r B A s a K w a R D n S K H Y Q L 4 D V x Z x 7 X b R g 5 g s R y Z X O h 2 h Y O s U 1 0 C 9 4 b j y 9 8 Y O C s P X w 1 Q 5 S W j J l 7 b u h i b x 0 B P m Q z m m h o z c p m Q j e P E I l U Y O P Q u p O G I E f J f w P / R D 1 g Y p J S 9 H 2 W h o T E 6 S I U i 1 T J A h Q + 8 2 U 6 j I a T z M x q 6 b f g 3 i w F K u O F I r L U q O i S L J o V E l m f a w L I i o o a X N o V N K k b T y g 6 h e a 2 L K o h k C m N m V n u 7 5 w e 7 j w F p 7 e K 4 y v L T z 6 X z 2 F f y + r a b B y N S V 7 j + a l o r d b D l M L I f w M y o O Z 0 + o m L 5 t h X A W U r G 0 G Z G c G L U Z l V I k M P 4 p V d q A w p 1 C 7 c 5 f Y o H l 2 t p U 6 J K a n A p L f i 7 r Y 1 d Z G I t 6 4 U e C X M a S c y Q S G 3 2 3 q M 4 P l j Z K q C 4 Y 2 m i z Y S R K Y a J t N L W T s I q m l Z 0 c V C N W q h o W x 3 J t H 9 x g / 2 D M g A A i I Z v Q Q K X I r l J x 0 N Z H g 1 / T m Z y k U h o h Y W Z t B G c N 0 9 j 2 S A W H r h 0 i X Y A D a 1 O a P X b w c H s Y G G D X / b B u P Y o O s 5 L 0 + l o Y G c A e m 9 6 g F k Y G O e v U T b v D 6 X w 6 + 6 a O L y F p h K F v E S b Z b k e q R k U S U B H w M V b g D H U c x G g m m r A b C m o o I x x n R c c o o J y K r r e M o m G M c J z p j q F L 3 E W g t z T d 0 M b K 3 d S l S q C n 5 + G y 6 W G M m P 2 h 2 n f D 8 L B 1 k 1 D b 9 d d C s C 1 6 v A W P t 2 B n 6 t z U c I O f b t H T z p b j B s R p B + g G 0 L H q k L R B Y E J X H R T 3 r l o k Z n 0 g M e u c F 5 B Z 4 d 4 r v R l L b O / e f t 9 H n O / C z 3 v 4 C l C M l L 3 q E Q Z N Y 0 h k 8 J 6 q Z Q W y a u T G Q D u L 7 e 1 Z h w b 1 N t v o n D R z / u B K Y 0 R k J o 9 K Q d u l D T l v a U b 1 n G h q F B z K V n G 6 q z j d U R w T s 6 M B a a s y E u K k W d 5 c 9 p 0 j A B d j k N L e B + 0 7 B x 2 D X 5 G j b u T q k 2 A 6 0 Q 1 5 h m C U u l k B d U J Y r y d u c A t N W X 1 1 O W Q F j B U q C t T X l b v C c p x 0 i M o h 4 l x 2 Z Y W o N M 3 6 H a 5 g h 6 s 6 g D b C o 3 a M s D 3 q c l 7 0 0 K J P 5 G V h X 1 7 A h Q S v P X v C q q r n L 0 i f 9 o K 4 G 7 C B X / A e f s F r u C 1 I s l N p l N S F H t 9 X E r c H a k W U C A 5 4 G U Q S v q n c Z d + B j x w 6 C Y M d s d r a j 8 7 o p b n m 7 5 I m x F C W / E + H + Y 0 O c 2 R a z N E n 4 X Y 9 c O X 7 8 O k X j j / 0 r i 9 e H c z C + S z 8 8 e v J s 6 f t R + A d 7 w v v S 2 / f C 7 0 n 3 j P v u X f i n X m x J 7 1 f v T + 8 P / 2 f / F / 8 3 / z f G 9 P b t 1 r O 5 9 7 g 8 f / 6 F 6 3 l + d A = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " H 1 x 5 b F / r y z l e g b M o s S n p 0 9 9 V v o E = " > A A A K 6 H i c n V Z b a 9 x G F F b S W 6 L e k v a x L 6 I b g w O b z c q 9 J C + G g E 1 J o K H u Y i c B a z E z 0 k i a 7 l z U m V G 8 6 0 H / o S + l 9 K W F / p n + h v 6 b n t F t V 7 I f S g W 7 z P n O + c 6 c 2 4 y E C 0 a 1 m c / / u X X 7 n X f f e / + D O 3 f 9 D z / 6 + J N P 7 9 H a 5 g h 6 s 6 g D b C o 3 a M s D 3 q c l 7 0 0 K J P 5 G V h X 1 7 A h Q S v P X v C q q r n L 0 i f 9 o K 4 G 7 C B X / A e f s F r u C 1 I s l N p l N S F H t 9 X E r c H a k W U C A 5 4 G U Q S v q n c Z d + B j x w 6 C Y M d s d r a j 8 7 o p b n m 7 5 I m x F C W / E + H + Y 0 O c 2 R a z N E n 4 X Y 9 c O X 7 8 O k X j j / 0 r i 9 e H c z C + S z 8 8 e v J s 6 f t R + A d 7 w v v S 2 / f C 7 0 n 3 j P v u X f i n X m x J 7 1 f v T + 8 P / 2 f / F / 8 3 / z f G 9 P b t 1 r O 5 9 7 g 8 f / 6 F 6 3 l + d A = < / l a t e x i t > ∆ζ(x) < l a t e x i t s h a 1 _ b a s e 6 4 = " x < l a t e x i t s h a 1 _ b a s e 6 4 = " H A X U G S v U q T p C n 0 9 L 5 S t D t Q a 9 A y O B B l E C n 8 m q o v + x 6 8 X 6 O z M N g T q 5 3 9 5 I y e 2 0 v + z l k C l v H k f z r M r 3 S Y E 9 t h N X 0 W 7 t Y j V 7 6 P n 3 7 h 9 E P v 8 u L V w S J c L s K f v p k 9 f t R 9 B N 7 w v v C + 9 O 5 5 o f f Q e + w 9 8 1 5 6 J 1 7 s M e 9 X 7 w / v T z / x f / F / 8 3 9 v T a 9 f 6 z i f e 6 P H / + t f 2 i H 3 s w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " H A X U G S v U q T p C n 0 9 L 5 S t D t Q a 9 A y O B B l E C n 8 m q o v + x 6 8 X 6 O z M N g T q 5 3 9 5 I y e 2 0 v + z l k C l v H k f z r M r 3 S Y E 9 t h N X 0 W 7 t Y j V 7 6 P n 3 7 h 9 E P v 8 u L V w S J c L s K f v p k 9 f t R 9 B N 7 w v v C + 9 O 5 5 o f f Q e + w 9 8 1 5 6 J 1 7 s M e 9 X 7 w / v T z / x f / F / 8 3 9 v T a 9 f 6 z i f e 6 P H / + t f 2 i H 3 s w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = "   effects. Preheating happens at scales tens of e-folds smaller than the CMB scale, and thus has little impact on large-scale observables. Only a handful of indirect observable effects of preheating are known, including a slight shift of inflation observables (scalar tilt and tensor-to-scalar ratio), and a stochastic gravitational wave background at high frequencies.
A recent review could be found in [12]. It is thus desirable to look for more direct observable effects of preheating, especially the effects from its non-perturbative dynamics.
In this paper, we propose a scenario where the non-perturbative preheating dynamics could affect the large-scale fluctuation in a more direct manner. The key idea is to probe/perturb the preheating dynamics by an additional light scalar field χ, which we call the modulating field and which is ubiquitous in beyond SM physics. 2 During inflation, χ acquires a nearly scale-invariant Gaussian background χ 0 (x). After inflation, χ 0 (x) becomes the effective coupling that controls preheating. Later in this paper we will show with an example that preheating is triggered only when |χ 0 | is larger than a critical value χ c . Consequently the patches with |χ 0 | below or above χ c will experience very different expansion histories. In this way, the modulated preheating process will generate a characteristic curvature perturbation ∆ζ(x). We illustrate this process with a cartoon in figure 1.
In figure 1, we highlight the nonlinear nature of preheating, in contrast to modulated (perturbative) reheating. In the perturbative case, a small variation ∆χ in the modulating field (and thus the coupling strength) would induce a small variation in the decay rate 2 The SM Higgs could be a candidate for the modulating field, though it is subject to more constraints.
This includes the local non-Gaussianity generated by the Higgs self-interaction during the inflation, as well as the post-inflationary evolution of the Higgs. See [13] for more discussions.

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∆Γ of the inflaton and therefore a small curvature perturbation, ζ = ∆N , where N is the e-foldings of local expansion during the reheating era. In particular, ∆N depends smoothly on χ and a Taylor expansion in ∆χ is always feasible, resulting in a linear relation between the "incoming" χ wave and the "outgoing" ζ wave. However, in the modulated preheating case as we will see, the nonlinear nature of the particle production induces an abrupt dependence on χ, effectively breaking regions with different χ 0 into two phases. The resulting ζ will have an almost bivalued distribution. So the preheating dynamics has the effect of turning the original sinusoidal waves in χ into square waves in ζ, like a nonlinear lens.
One may wish to realize this scenario by requiring the χ field to modulate the nonperturbative inflaton decays. However, as we will show, such a scenario with all energy density ρ total going through preheating would result in a curvature perturbation ζ that is either nonlinear but too large, or of the right size but linear. To see the aforementioned nonlinear effect without invoking too large ζ, we find it better to have only a small fraction of energy going through preheating, with the rest of energy released smoothly through perturbative reheating. In section 3, we will describe a model realizing this partial preheating idea, in which we introduce a spectator field σ in addition to the inflaton φ. σ has a subdominant but non-negligible energy density ρ σ during inflation. After inflation, the σ sector will go through the modulated preheating while the inflaton still decays perturbatively. This model may appear a bit complicated with multiple fields but serves as a proof of principle. We will work out the corresponding curvature perturbation of this modulated partial preheating in section 4. As one can expect, the curvature perturbation generated from this process is always suppressed by ρ σ /ρ total and can be easily consistent with observation.
It remains to see how to probe such a "square wave" effect. We explore this issue in section 5. The main takeaway message is that the effect will manifest itself through the local non-Gaussianity of n-point correlators (n ≥ 3). Thanks to the modulating field and its nonlinear distortion, we could observe the nonlinear dynamics of preheating at super tiny scales directly at CMB scales. It is in this sense we call our scenario a cosmic microscope.
In general the "square wave" effect contributes only a small fraction of the curvature perturbation, while the major contribution is still from the inflaton fluctuation. Ideally, we may want to distill a bivalued distribution directly from the observed distribution of ζ. But this is very challenging, since the modulated preheating happens within a Hubble patch, which is much smaller than the resolution, or the size of pixels, of any practical observation. Consequently, the observed value of ζ in each pixel is necessarily the average over many Hubble patches during preheating, making the result no longer bivalued. Therefore it is important to look for effects that survive this averaging process. We will show that the averaged ζ has little scale dependence, so one could not look for it in the power spectrum. However, there could be potentially large local non-Gaussianity. In particular, since we have a fixed square-wave relation between the incoming Gaussian χ k 's and outgoing ζ k 's, we would expect that the sizes of n-point correlations, for all n ≥ 3, have a definite relation. At least in principle, if we could measure local non-Gaussianity in all n-point correlators,

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we should be able to identify this "square wave" component. But in practice, we will be able to access only a handful of them starting from n = 3. Thus in this work we only focus on the 3-point function as a case study.
We collect further discussions in section 6 and computational details in two appendices.
Comparison with previous works. Readers familiar with the literature may recognize that some ingredients of our scenario also appear in scenarios of modulated reheating [13][14][15][16][17] and modulated preheating [18]. In the former (latter) scenario, the inflaton perturbative decay (preheating) is controlled by the modulating field. In modulated reheating, the variation of the local e-fold number could be described by a linear function of χ, as mentioned above. In the modulated preheating that has been studied in the literature, due to the observed size of ζ ∼ 10 −5 , all the Hubble patches have to be either in an inefficient preheating phase with little energy transfer from inflaton to radiation [18] or in the efficient preheating phase [19,20]. It is not allowed that some Hubble patches have little particle production and negligible back-reaction while others have sufficient particle production and back-reaction triggered by it. Staying within the same region without phase transitions makes it possible to generalize the idea of modulated reheating directly to preheating, but misses the interesting opportunity of probing the nonlinear effects of preheating. In our scenario, the observational constraint is circumvented by requiring a spectator field with a sub-dominant energy fraction, instead of an inflaton, to undergo the nonlinear processes. To differentiate it from the modulated preheating in the literature, we will refer our mechanism as "modulated partial preheating". The modulating field can scan different preheating phases in our scenario and probe the full nonlinear dynamics. The traditional view that preheating happens at too small a scale to be observable is circumvented by this scenario, which enhances the nonlinear effects of preheating all the way to CMB scales. Thus, again, we call this "distorting the scale-invariant perturbation by the lens of preheating" mechanism a cosmic microscope.
Another class of preheating models that could generate large non-Gaussianities is the curvaton-type preheating scenario. The key difference is that in those scenarios, the additional scalar field with primordial fluctuations (the curvaton) directly participates in preheating either as a field created during preheating [18,[21][22][23][24], or the field responsible for the particle production [25]. All the couplings in the models are fixed. More details and references could be found in recent reviews of preheating [12,26].

Overview of modulated (p)reheating
In this section we review briefly (p)reheating and its modulated scenarios. This section is meant to provide a basic physical picture and some relevant formalism for non-experts. Readers who are familiar with the subject could safely skip this section.
There is rich physics between the end of inflation and the onset of the thermal universe. As mentioned briefly in the introduction, there could be both perturbative decays and nonperturbative particle production. These will be reviewed in section 2.1.
(P)reheating typically depends sensitively on various parameters. When such parameters vary over different spatial patches, (p)reheating, and therefore the expansion history JHEP01(2021)021 of local universes, could be spatially dependent, too. This could generate additional curvature perturbations at large scales, providing us a unique window to probe the (p)reheating physics. The required spatially dependent parameter can come from the background value of a light field χ. We will review this modulated scenario in section 2.2.

Reheating and preheating
In typical inflation scenarios, one needs a mechanism to stop inflation and then convert the inflaton energy to thermal radiation. The establishment of a thermal universe after inflation is called reheating. Typically, one can imagine that the inflaton falls into the bottom of its potential after inflation. Then it starts to oscillate and decay into other particles. Generically we expect the inflaton potential around the minimum to be quadratic, Then the equation governing the inflaton's evolution is The Hubble parameter H(t) decreases after the inflation, and the inflaton φ starts to oscillate when H(t) < m φ . A scalar oscillating in a quadratic potential behaves as cold matter when averaging over oscillation periods 1/m φ , and has an effective equation of state p = 0. So the universe expands as a ∝ t 2/3 , Here we assume that the perturbative decay rate Γ of the inflaton is a constant and much smaller than H(t) at the initial stage of inflaton oscillation, so that φ feels little friction from decays. Eventually, H decreases to Γ and the perturbative decays dominate the friction. The inflaton energy is completely transferred to radiation when H(t) Γ. At this point the universe is effectively dominated by radiation with an equation of state p = ρ/3 and thus expands as a ∝ t 1/2 .
The picture of perturbative decays described above is quite generic, but also often incomplete. There could be various sources of non-perturbative particle production, on top of the perturbative decays. Usually, such non-perturbative processes, called preheating, happen within a couple of e-folds after inflation while perturbative reheating usually happens significantly later. Another generic feature is that preheating could not complete the energy transfer from the mother scalar field to radiation so perturbative reheating is still needed to complete the transition after inflation.
Various non-perturbative mechanisms of particle production are known for preheating. See ref. [12] for a pedagogical introduction. The essential idea is that fast oscillations of the inflaton could introduce a time dependence in the effective mass of particles the inflaton couples to directly. In favorable parameter space, this time dependence can trigger proliferation of the associated particles through resonant production. Here we focus on one generic class of preheating mechanism known as tachyonic resonance [27]. In the tachyonic resonance scenario, there is an interaction term, − 1 2 gφσ 2 between the inflaton φ and a scalar field σ. In Fourier space, the equation of motion for σ(t, k) at the linearized level (ignoring the self-interaction of σ) is, σ(t, k) + 3H(t)σ(t, k) + m 2 σ + gφ 0 (t) + a −2 (t)k 2 σ(t, k) = 0. To understand how σ(t, k) evolves in time, it is helpful to consider the same problem in a non-expanding space (a = 1 and H = 0), in which case the inflaton background is simply φ 0 (t) = φ cos(m φ t). Then the equation above reduces to the Mathieu equation, , and the prime denotes d/dz. The most general solution of this equation is well known, and can be written as where f (±) are periodic functions, f (±) (z + π) = f (±) (z) for any z, and the exponent µ k = µ k (A k , q). Therefore, in regions of parameter space with Im µ k (A k , q) = 0, we will get an exponential enhancement of σ. The exponent µ k (A k , q) is conventionally displayed in the (A k , q) plane, known as the Floquet chart, as shown in figure 2. The picture becomes more complicated after including the cosmic expansion and the back-reaction to the background geometry. When the expansion is slower than the characteristic time scale of the system, namely H < m φ (which has to hold for the inflaton to oscillate), one could still use the adiabatic intuition. That is, one can think of q = 2gφ(t)/m 2 φ as a time-dependent parameter and slowly decreasing towards zero asφ(t) ∝ 1/t. (Note that physics is invariant under q → −q.) Inspecting the Floquet chart, we see that the cosmic expansion would eventually bring the system out of resonance bands for a generic initial value of q, since the small q region has no resonance. However, it is also possible that the expansion brings a previously non-resonant q into a resonant band en route. This complicated process due to expansion of the universe is called stochastic resonance [11]. In addition, once sufficiently many daughter particles are produced, they will back-react on the mother field and excite high-momentum modes. Both fields will eventually be fragmented spatially. Depending on the couplings in the model, preheating may end in the JHEP01(2021)021 first stage with little particle production or go through the back-reaction stage, which we will refer to as efficient preheating later. Note that efficient preheating in this paper does not imply full energy transfer from mother to daughter particles. It only means that an order one fraction of energy is transferred. We will discuss more about these interesting dynamics in our model in section 3.
For the purpose of this section, it suffices to point out that the equation of state of the universe can be altered quickly by preheating. Since the resonant production usually excites high k modes through back-reaction, the equation of state p = wρ can quickly evolve towards w = 1/3 in the case of efficient preheating.

Modulated (p)reheating
All the causal dynamics described above happens within a Hubble patch after inflation, whose size is extremely small compared with the CMB scale. In the simplest picture of single field inflation, different patches evolve similarly. In particular, preheating would happen either in all patches or in none.
There are alternatives when we go beyond the single field picture. Suppose that, in addition to the inflaton φ, there is a light scalar field χ with mass m χ H inf , the Hubble scale during inflation. During inflation, it develops a spatially varying background χ = χ(x) at cosmological scales. The fluctuation δχ is almost scale invariant since χ is light, and typically has the size δχ ∼ H inf . Then, after inflation, the χ background within each Hubble patch will be a constant over space (but not necessarily in time 3 ).
It is plausible that the variation in the background value χ 0 would affect the expansion history in each individual Hubble patch. Consider perturbative reheating first. In this case, the cubic coupling 1 2 gφσ 2 that has been considered in the previous section is actually from a dim-4 operator 1 2 g χφσ 2 with g = g χ 0 . Then the perturbative decay rate Γ(φ → σσ) ∝ χ 2 0 . As a result, the time of reheating t reh , determined by Γ = H(t reh ), depends on χ 0 through its dependence on Γ. Consequently, varying χ over large scales induces a change of expansion histories across different Hubble patches, and thus provides a new source generating curvature perturbations. This is known as the modulated reheating scenario, and we call χ the modulating field from now on.
The idea above can be neatly formulated by the so-called δN formalism [28][29][30]. In short, the curvature perturbation ζ(t 2 , x) at a later time t 2 , long after the reheating completes, receives contributions from two distinct sources. One is the "primordial perturbation" generated during inflation ζ (t 1 , x), where t 1 is chosen before inflation ends but after the modes of interest go outside of the horizon. The other one is the contribution from the "reheating" era, namely between t 1 and t 2 , in the form of perturbed e-folding number N ,

5)
3 Depending on its potential V (χ), the χ field would in general have nontrivial time dependence after inflation. For instance, if V (χ) = 1 2 m 2 χ χ 2 , then χ will start oscillating when H drops to mχ. In this section we will ignore the evolution of χ for simplicity, without affecting the main conclusion. Practically, one can achieve this by assuming mχ H(t) for all t of interest. However, the post-inflationary evolution could be crucial in some cases, see for example [13].

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where N is the unperturbed e-folding number. In the single field story, the curvature perturbation comes mainly from ζ(t 1 , x) and δN can be neglected. But in the modulated reheating, δN is non-negligible. In fact, assuming the equation of state is w = 0 before reheating and w = 1/3 afterwards, we can find, Here we have used H = 2/(3t) for matter domination and H = 1/(2t) for radiation domination. Now we perturb Γ, and thereby perturb t reh . From the relation H = 2/(3t) (t ≤ t reh ) and H(t reh ) = Γ, we get δt reh /t reh = −δΓ/Γ, and therefore, Since δΓ ∝ δχ, the curvature perturbation gets a contribution from δχ in this scenario. The formalism above can be directly adapted to include preheating. Instead of the perturbative decay rate Γ, the preheating efficiency is controlled by the µ k parameter introduced in eq. (2.4), which depends on the coupling g, and thus on the modulating field χ, in a highly nonlinear way, as illustrated in figure 2. The parameter µ k then determines the time preheating occurs, t pre , which roughly sets the boundary between effective matter domination (due to inflaton oscillation) and effective radiation domination. A variation in t pre would perturb the expansion history, similar to the perturbative case discussed above. The crucial difference, though, is that the parameter µ k (and thus t pre ) depends on χ in a highly nonlinear way. Consequently, a small linear change in χ could induce a large nonlinear perturbation in the expansion history. We would then expect a dramatic nonlinear contribution to the curvature perturbation through this mechanism. In fact, we will show in the following sections that the curvature perturbation induced by such a modulated preheating scenario is almost always too large to be consistent with the observed value ζ ∼ 10 −5 . This leads us to consider a preheating scenario with sufficient nonlinear effects in the curvature perturbation but a small overall amplitude of ζ. The model will be discussed in the following sections.

The model and phase diagram
In this section we will first present our main model in which a fraction of curvature perturbation is from modulated preheating. In this model, 1) the size of the curvature fluctuation can be easily made consistent with the observation while 2) it still carries the non-perturbative information of the preheating processes. We argue in last section that these two properties are difficult to be realized simultaneously in the inflaton preheating scenario. To circumvent this problem, we introduce a spectator field σ which goes through the non-perturbative decay, while the inflaton still decays perturbatively.
To be concrete, we will still use the trilinear coupling, analogous to what we have reviewed in the previous section, but between the spectator σ and an additional scalar ψ, and the associated tachyonic resonance mechanism. We will also present a phase diagram of the mechanism in the parameter space of the model. 1. Larger q 0 (or equivalently, larger trilinear coupling) leads to faster exponential growth.
2. The cosmic expansion slows down the growth so that the comoving occupation number n k grows with an exponent proportional to √ t instead of t as in the case of a non-expanding universe.
3. Increasing the wave number k would reduce the growth exponent. For fixed t, the largest wave number k max that can be excited can be found by asking the two terms in eq. (3.10) to cancel. This leads to In the linearized analysis, particle production is terminated by the cosmic expansion at t term , which satisfies [32] The analytical formula in eqs. (3.10) and (3.12) agree well with numerical results obtained by solving eq. (3.9) numerically, as shown in figure 3. Note that eq. (3.10) breaks down when q 0 5. For smaller q 0 , there is simply not much particle production as demonstrated in the first panel of figure 3.

Phase diagram
Particle production transfers energy from σ to ψ. In the linearized analysis, the fraction of energy density of ψ at any time before t term could be estimated to be (3.13) More details of the derivation for the equation above could be found in appendix A. Requiring δ(t term (k = 0)) = 0.1, we find that (3.14) When q 0 < q c (q 0 > q c ), the fraction of energy density in χ is below (above) 10% at t term . If at a time, t com , there is a significant fraction (e.g. 10%) of energy density in the ψ field comparable to what remains in the σ field, the back-reaction from ψ to the mother field σ through the trilinear coupling could not be ignored. The linearized analysis breaks down. Instead a full numerical simulation based on coupled equations of motion is needed as in ref. [33]. Once the back-reaction is effective, the system will quickly evolve into an energy equipartition state with similar amounts of energy in the ψ and σ fields. The effective equation of state, of the sub-system of σ and ψ, also quickly rises to a plateau ∼ 0.3, signaling a mixed radiation-matter state [33]. In the discussions below, we will approximate the asymptotic value of w sub to be that of radiation 1/3, for simplicity of the calculation without modifying the conclusions. One additional complication arises from the self-interaction of the ψ field. If the quartic coupling λ is large enough and sufficiently many ψ particles are produced, the self-interaction of ψ turns into an effective positive mass term λ ψ 2 ψ 2 and slows down the particle production.
Putting the considerations together, we have three possible "phases" (regions) in the (q 0 , λ) plane, as depicted in figure 4: • Region 1: q 0 < q c . In this region, t term < t com . Particle production is stopped by expansion of the universe before there are comparable energies in σ and ψ. Backreaction is negligible and the analysis based on the linearized equation of motion is valid. The fraction of energy transferred from σ to ψ is exponentially sensitive to q 0 , as shown in eq. (3.13), but it is always 1. Thus w sub is always approximately zero.
• Region 2: q 0 > q c and 0.1 1 2λ g mσ 2 ≤ 1. In this regime, t term > t com . At around t com , the linearized equation of motion breaks down and the system will quickly rises to a constant w sub close to 1/3. When q 0 is large enough, this is approximately a step function in time. Notice that there could be more details in the time evolution of w sub as shown in figure 2 in ref. [33]. For instance, before settling down to the asymptotic value, w sub could have an intermediate oscillation stage.

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Region 1 q c |q 0 | λ potential unbounded from below 0 sub Slow particle production Fast particle production and efficient preheating Fast particle production stopped by self-interaction of daughter particles t com Figure 4. Schematic phase diagram of the spectator tachyonic resonance mechanism in the parameter space (q 0 , λ). w sub = p sub /ρ sub defined for the (sub-)system of σ and ψ. The black curve corresponds to λ = , below which the potential is unbounded. The green dashed curve corresponds to a large enough λ, which leads to a large positive λ ψ 2 ψ 2 to stop particle production and separates region 2 and 3.
• Region 3: q 0 > q c and 1 2λ g mσ 2 0.1. In this region, back-reaction is non-negligible as in region 2 and the linearized analysis breaks down. In addition, the quartic interaction, λ ψ 2 ψ 2 becomes important and comparable to the trilinear term g σ ψ 2 . It acts as an effective positive mass term for ψ and slows down the tachyonic particle production process. This is probably the most tricky region with no simple description of the time evolution. The schematic picture of time evolution of w sub in figure 4 merely serves as an illustration.
Note that numerical simulations in ref. [33] supporting this phase diagram are implemented in the inflaton preheating scenario. We have carried out lattice simulations for the preheating of the spectator field σ with a fixed cosmic expansion and checked that this phase diagram still holds. More details of the simulation results could be found in appendix B.

Modulated partial preheating
At the end of last section, we presented the "phase diagram" for our preheating scenario, as in figure 4. Throughout our treatment in the previous section, we took the coupling g responsible for preheating to be a constant. It is indeed a constant in time within each JHEP01(2021)021 our scenario, we choose the two slices to be a slice shortly after inflation at t 0 and the inflaton reheating slice at t reh . Since the inflaton φ redshifts as matter, N between the end of inflation and inflaton reheating is given by where ρ φ (t 0 ) is the energy density of inflaton immediately after inflation and ρ φ (t reh ) is the energy density of inflaton when it reheats at t reh . Since the inflaton reheats when the Hubble scale drops to be its perturbative decay width Γ φ (which is taken to be fixed), we have where ρ σ (t reh ) is the energy density of the decay products from either preheating or reheating of σ at the inflaton reheating time.
For patches in region 2 with |q 0 | > q c , approximating σ and its daughters to redshift as radiation after t com as in eq. (4.5), 4 we have In the equation above, we take into account that after σ's preheating, the energy density of σ and its decay products dilutes faster than that of the inflaton and thus the fraction of energy density stored in the sub-system is reduced further by a factor of a(tcom) a(t reh ) . Given that to the leading order of γ, the whole system redshifts as matter, we have where we use that in a matter domination phase, a(t) ∝ t 2/3 , H = 2 3t and H reh = Γ φ . Combining all the equations above, we obtain that in patches with efficient spectator preheating (region 2), In patches without efficient preheating (region 1) with |q 0 | < q c , we have

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Note that the formula above is based on our assumption that σ reheats before inflaton reheats. If the order is switched, after the inflaton reheats, σ still redshifts as matter and its energy fraction increases. We will not discuss this possibility further. In summary, the e-folding number N as a function of q 0 (and thus the modulating field χ) is where ϑ is the Heaveside step function and where in the last step, we use the fact that preheating happens (much) earlier before reheating, t com 1/Γ σ < 1/Γ φ . Thus the value of A is independent of the modulating field χ. So the local e-folding number across different Hubble patches behaves as a square wave as the trilinear coupling varies, as shown in panel (a) (III) of figure 1. Note that A is proportional to γ, the initial energy fraction of σ, but suppressed further by (Γ φ /Γ σ ) 2/3 . In the limit that inflaton reheating happens much later than the reheating of σ, A → 0.

Non-Gaussianity as a probe of preheating history
In the last section we show that the modulated partial preheating is able to generate a contribution to the curvature perturbation in a highly nonlinear fashion but with a suppressed amplitude. The total curvature fluctuation ζ has two contributions, where ζ inf = −(H inf /φ 0 )δφ is from the usual inflaton fluctuation, and ζ mp is from the modulated preheating, where A γ/3 (Γ φ /Γ σ ) 2/3 , χ c = q c Λm 2 σ /(2σ 0 ), and χ 0 (x) is the background of the modulating field χ generated during inflation, which we take to be Gaussian and scale invariant. We assume that the mixing between the inflaton and χ fluctuations is small and thus ignore cross correlators.
In figure 1 we demonstrate with a simple cartoon that ζ mp (x) looks like a square wave for a single k-mode of χ. But the scale-invariant modulating field χ is a superposition of many k-modes and there is no simple square-wave like behavior after the superposition. Thus we need to look elsewhere for its observable consequences.
In this section we will show that one possible set of observables is local non-Gaussianities of ζ. The relation in eq. (5.2) distorts the original Gaussian spectrum of χ in a distinctive way, and it is in principle possible to reconstruct this "square-wave function" in eq. (5.2) by measuring all n-point correlations of ζ. Practically this is surely very challenging, and we have access to only a handful of n-point correlations that could possibly be extracted from data, starting from n = 3. So in this section we will use the 3-point function as an example and leave a more exhaustive study for future work. We will also JHEP01(2021)021 assume that the non-Gaussianity is dominated by the contributions from ζ mp , while the contribution from ζ inf is negligible.
It is well known that the local non-Gaussianity that is actually observable in a finite region does not necessarily agree with the one predicted by a model (as we will do below). See for instance [34]. This is caused by the biasing of super-horizon modes via the coupling of modes with different sizes. The finite-chart bias is particularly important when the fluctuation is highly non-Gaussian. In our model, the curvature perturbation is never highly non-Gaussian, since we assume partial modulated preheating, in which the dominant contribution to the curvature perturbation is still from the nearly Gaussian inflaton perturbations. But the small non-Gaussianity in our model can be non-perturbative, in the sense that the curvature perturbation from the modulated preheating sector does not allow a Taylor expansion in terms of Gaussian variables. Given the small non-Gaussianity, we expect the finite-chart bias discussed in [34] to be small in our model. But it is useful to keep in mind this small ambiguity when confronting our model parameters with observation constraints (like in figure 10). It is also interesting to study how the n-point function in our model changes with the size of observable universe. We leave this for future studies.
Since we work in a fully nonlinear regime where the usual perturbative expansion does not work, we will first present a formalism in section 5.1 that works for this special case. A similar treatment was developed to compute non-Gaussianity in curvaton-type preheating in ref. [35]. In section 5.2 we will apply this formalism to the study of modulated partial preheating.
We note in passing that the form of the curvature perturbation ζ mp in eq. (5.2) is reminiscent of the barrier-crossing model of Press-Schechter type for halo formation [36]. Several formalisms are known to calculate halo correlation functions, see for example [37]. For instance, one could expand the Heaviside step function in eq. (5.2) in terms of Hermite polynomials. This is particularly useful to calculate halo power spectrum in momentum space where the expansion converges rapidly. In our case, as we will show below, the correction to the primordial power spectrum is manifestly scale invariant (unlike the halo power spectrum with strong scale dependence), up to soft breaking by finite observation range, provided that the modulating field has a nearly scale invariant power spectrum. This has no independent observational consequence, so we have to go beyond the two-point level and study non-Gaussianities. In this case the method of Hermite polynomial expansion is not simpler than the one we will present below. But it still provides an interesting possibility for calculating the correlation functions in momentum space. It could also be useful to quantify the relative sizes of n-point correlators of ζ mp for different values of n. We leave a systematic study of this possibility for future work.
Since we will focus only on ζ mp , we will drop the subscript in the rest of this section unless confusion may arise.

General formalism
The nonlinear nature of the preheating process means that ζ depends on the background value of the modulating field χ in a nontrivial way, invalidating the Taylor expansion. So we must treat the function ζ = ζ(χ) nonlinearly. This will introduce characteristic local-type JHEP01(2021)021 non-Gaussianities of ζ at large scales. Since the functional dependence ζ(x) = ζ(χ(x)) is local in position space, it is more convenient to consider the fluctuations in position space instead of momentum space. Going from the usual formulation in momentum space to position space introduces some subtleties, which we will discuss first before presenting the general formalism.
The vacuum fluctuation of the light modulating field χ (m χ H inf ) is governed by a classical Gaussian distribution outside the horizon. In momentum space, it has a zero mean δχ k = 0. The 2-point function (power spectrum) can be parameterized as δχ k 1 δχ k 2 = (2π) 3 δ (3) (k 1 + k 2 )P χ (k 1 ). For a light field χ, the power spectrum will be nearly scale-invariant, We would like to understand how this looks like in position space. To this end one could try a straightforward Fourier transform as follows, which is divergent. The divergence is in the IR, meaning that we would see increasing correlations by going to larger scales. But observation-wise we are always limited by the total amount of data available to us -the size of the observable universe, which introduces an IR cutoff L, explicitly breaking the scale invariance. As a result, the position-space correlator becomes where sinc(x) = sin(x)/x and the cosine integral Ci(x) = ∞ x cos(t)/t dt. For |x 1 −x 2 | L, we have, where L ≡ Le 1−γ E with γ E the Euler gamma constant. The 2-point function also diverges in the coincident point limit. This divergence is cut off by the resolution of the sampling. Suppose the size of the pixel is , the above 2-point function tells us that χ(x) at a given point is a random Gaussian variable with a zero mean and a variance As a check of the position space correlator, we could Fourier transform it back to the momentum space and obtain

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where the sine integral Si(x) = x 0 sin(t)/t dt and in the second line, we keep leading terms in the kL 1 limit. Indeed we recover the original scale invariant power spectrum in momentum space, with corrections of the form sin(kL). This is a typical artifact of the sharp cutoff. Such fast oscillatory terms have characteristic scale L, which sets the resolution of measuring k and is averaged to zero. Now we consider the non-Gaussianities induced by the nonlinear preheating processes. By nonlinear we mean that the curvature perturbation ζ generated during preheating is a nonlinear function of the modulating field fluctuation, ζ(x) = ζ(δχ(x)). In the standard δN formalism, one often Taylor expands this function in terms of small fluctuation δχ as ζ(x) ⊃ δN = N − N = N a δχ(x) + N b (δχ(x)) 2 + · · · , which directly leads to a nonzero local non-Gaussiainity even when δχ itself is purely Gaussian. However, as we have shown, in modulated partial preheating, δN is not a smooth function and a Taylor expansion is impossible, which makes a non-perturbative treatment essential. The question is then to find the correlation function ζ(δχ) · · · ζ(δχ) given the function ζ = ζ(δχ) and the fact that δχ is scale-invariant and Gaussian. As we mentioned above, it is easier to compute in position space as the relation ζ = ζ(χ) is local in position.
In general, the n-point correlation function could be computed as a functional integral 9) where N is the overall normalization factor and D(x, y) is the functional inverse of the 2-point correlator G(x, y) ≡ χ(x)χ(y) , satisfying To calculate the correlator of curvature perturbations ζ, we can just take O i [χ(x i )] = ζ(χ(x i )) and carry out the functional integral above. However, there is a simpler way that turns the functional integral above into an ordinary integral [35]. The idea is to expand the function ζ(χ) in plane waves e iωχ rather than in powers χ n . The plane wave e iωχ can be treated as a generating function of moments χ n . To see how it works, let's define ζ(χ) = dω 2π e iωχ ζ ω , (5.11) which is point-wise in x. Then, the n-point correlator of the curvature fluctuation ζ can be found by ζ(x 1 ) · · · ζ(x n ) = dω 1 2π · · · dω n 2π ζ ω 1 · · · ζ ωn e iω 1 χ(x 1 ) · · · e iωnχ(xn) . (5.12) The correlator on the right side could be calculated explicitly,

Correlators in modulated partial preheating
Now we apply the formalism above to the modulated partial preheating scenario and calculate the two-and 3-point correlators. The starting point is the function ζ mp = ζ mp (χ) in eq. (5.2). Note again that we will drop the subscript for simplicity. The Fourier transformed curvature perturbation ζ ω , defined in eq. (5.11), is It is more appropriate to define the perturbation with the one-point functionζ = ζ(x) subtracted, δζ = ζ −ζ. We then find the 2-point function, using eq. (5.6), (5.7) and (5.14), where r ≡ |x − y|. Note that we have used the 2-point function of χ in eq. (5.6), which is only valid when r L. It is difficult to carry out the integral analytically and we have evaluated it numerically. By fitting to the numerical results, we find that where the coefficient c 2 > 0 is a function of log( /L) and ∆ χ /χ c . This is shown in figure 6 and 7. Note that the maximum value of c 2 is of order 10 −4 , which is purely a numerical fact and has no parametric dependences. We emphasize that c 2 , and thus P (mp) ζ (r), depend on χ c and ∆ χ only through the ratio ∆ χ /χ c . This is also true for n-point functions in general. Intuitively the perturbation δζ should go to zero when ∆ χ /χ c goes to either zero or infinity, since in these two limits the fluctuation cannot see the rectangular pulse. We demonstrate c 2 's dependence on ∆ χ /χ c in left panel of figure 7. We also show the 2D spatial slices of P (r) for different choices of ∆ χ /χ c in figure 8. In the right panel of figure 7, we show c 2 as a function of /L. When  we vary /L by several orders of magnitude, c 2 remains the same order of magnitude. This proves that c 2 only depends logarithmically on /L. In momentum space, the Fourier transform of P χ (r) gives rise to P (mp) ζ (k) = d 3 x e −ik·x P (mp) ζ (r) 4π 2 A 2 c 2 log(kL) k 3 + · · · (5.18) where we ignore unphysical oscillation terms due to the cutoff L and higher order terms of JHEP01(2021)021 It is useful to consider the squeezed limit of the 3-point function with r ≡ r 12 = r 13 r 23 ≡ r s . Numerically we find that in this limit, we have A 3 c 31 log r s L + c 32 log 2 r s L log 2 r L , (5.25) where c 31 and c 32 , analogous to c 2 , are functions of log( /L) and ∆ χ /χ c . The maximum values of c 31 and c 32 are also suppressed numerically and are of order 10 −6 . In momentum space, this corresponds to lim k 1 /k 3 →0 δζ(k 1 )δζ(k 2 )δζ(k 3 ) (2π) 7 δ 3 (k 1 + k 2 + k 3 )A 3 c 31 + 2c 32 log(k 3 L) log(k 1 L) (5.26) This is close to the form of the local non-Gaussianity [39]. Ignoring the mild scale dependence, we could estimate the local f NL to be

Conclusion and outlook
In this paper we have studied a scenario of modulated partial preheating, in which the space variation of a modulating field χ triggers the preheating of a spectator field σ after inflation. This scenario generates characteristic local non-Gaussianities at large scales with observably large f NL . This provides a unique chance to observe the nonlinear dynamics of preheating era which is in general difficult to probe directly. We have showed that the expansion histories of local patches during preheating have a two-phase structure controlled by the background value of the modulating field χ: the patches with |χ| larger than a critical value χ c have efficient particle production, while patches with |χ| < χ c do not. This induces "square wave" behavior in the curvature perturbation, which is a unique feature of the non-perturbative dynamics. We have also calculated the corresponding power spectrum as well as non-Gaussianity of this "square wave" component, and found it to be nearly scale invariant and with fairly sizable local non-Gaussianity. Another potential observable of this scenario is the isocurvature mode of the modulating field χ, if it remains today as a fraction of dark matter. But this is very model dependent. For instance, the isocurvature mode of χ could disappear completely if χ decays after reheating of the inflaton.
One may want to search for this effect by distilling the bivalued square wave distribution directly out of the observed fluctuations, as shown in figure 8. But this is likely impossible. The reason is that the bivalued distribution depends on the "UV cutoff" . Physically, this UV cutoff is provided by the horizon size of each Hubble patch during preheating, which is way below any resolution we can imagine for observing large-scale fluctuations. Consequently, what we actually observe is necessarily the average of the bivalued distribution over many Hubble patches at the time of preheating. The averaged distribution is no longer bivalued. So we have to look for effects that survives this average, among which the local non-Gaussianity is a good example. It survives the average because it has very weak scale dependence and is present practically at all scales. It is interesting to explore other possible observables of this effects that survive the horizon averaging.

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More open questions and directions remain to be explored: • Couplings in low-energy effective theories, instead of being constants, could originate from dynamical fields in their UV completions. The time evolution of such fields could lead to quite a variety of novel phenomenon, which has always been an interesting direction of model building in the particle physics community. What we have explored instead is to use the spatial variation of a dynamical field governing a coupling to probe the phase diagram of nonlinear dynamics in the early universe. Could our toy model be embedded in a full-fledged particle physics model? What will happen if the modulating field in our model is the SM Higgs?
• Recently the use of non-Gaussianity to probe particle physics, in particular heavy particles beyond the reach of ordinary colliders, has been developed rapidly in the context of cosmological collider physics [13,. What we have studied in this paper is, to a certain extent, a non-perturbative version of the cosmological collider. What other phase diagrams of non-perturbative processes in the early universe could be probed using our modulating mechanism? We have only studied the 3-point function in the squeezed limit. How about non-Gaussianity of general shapes and higher point correlations?
JHEP01(2021)021 Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.